Analysis and design of marine structures 9780203732854, 0203732855

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ANALYSIS AND DESIGN OF MARINE STRUCTURES

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PROCEEDINGS OF THE 4TH INTERNATIONAL CONFERENCE ON MARINE STRUCTURES (MARSTRUCT 2013), ESPOO, FINLAND, 25–27 MARCH 2013

Analysis and Design of Marine Structures

Editors

C. Guedes Soares Instituto Superior Técnico, Technical University of Lisbon, Portugal

J. Romanoff Aalto University, Finland

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2013 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20130620 International Standard Book Number-13: 978-0-203-73285-4 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Table of contents

Preface

ix

Committees

xi

Wave loads Long-term prediction of combined wave and whipping bending moments of containership M. Ćorak, J. Parunov & C. Guedes Soares Semi-empirical assessment of long-term high-frequency hull girder response of containerships—an update A. Kahl, H. Rathje, J. Rörup & T.E. Schellin

3

11

Numerical studies on deepwater dry tree semisubmersible R. Sundaravadivelu, N. Gogula, N. Srinivasan & V.P.K. Kaundinya

19

Numerical simulation of structural response under bow flare slamming load T. Yoshikawa & M. Maeda

25

Vibrations Vibrations of superyacht structures: Comfort rules and predictive calculations D. Boote, T. Pais & S. Dellepiane Noise on board RO-Pax vessels: Measured levels on existing ships and new pre-normative requirements T. Gaggero & E. Rizzuto An analytical method for cabin deck fundamental frequency A. Laakso, J. Romanoff, H. Remes & A. Niemelä

37

45 53

The use of seismic isolation with the method of finite element to reduce vibrations caused by the earthquake in the offshore platform H. Rostami & A.V. Oskouei

61

On the non-linear hydroelastic response in irregular head waves of a structural optimized container ship I. Rubanenco, I. Mirciu & L. Domnisoru

67

Simplified formulations of mass and geometric stiffness matrices in vibration and stability analyses of thin-walled structures I. Senjanović, N. Vladimir & D.S. Cho

79

Response to accidental loads Numerical analysis of warship’s structural panel subjected to airblast and underwater explosions M. Ali, S. Mondal, A. Samanta & G. Narendran Behavior of stiffened panels exposed to fire M.R. Manco, M.A. Vaz, J.C.R. Cyrino & A. Landesmann

v

91 101

Structures on ice Cost optimization for ice-loaded structures S. Ehlers & P. Kujala

111

Measured ice loads and design ice loads M. Suominen, P. Kujala, R. von Bock und Polach & J. Kiviranta

119

L-year maximum values of local ice loads on ship hulls A. Suyuthi, B.J. Leira & K. Riska

125

Impact and collision A new super-element for estimating the collision resistance of an inclined ship side L. Buldgen, H. Le Sourne & P. Rigo

137

Damage of ships for engine room section struck by bulbous bow C.F. Hung, Y.D. Hsieh, W.L. Chien, Y.T. Huang & Z.M. Zhou

147

Modeling aspects of strength capacity of intact and damaged ship girders D. Koukounas & M.S. Samuelides

157

Failure characteristics of strength-equivalent aluminium and steel plates in impact conditions B. Liu, R. Villavicencio & C. Guedes Soares

167

Shear and tensile failure of thin aluminium plates struck by cylindrical and spherical indenters B. Liu, R. Villavicencio & C. Guedes Soares

175

Material modeling for finite-element simulation of ship impacts J.N. Marinatos & M.S. Samuelides

187

Experimental and numerical investigations of an alternative stiffening system for ship side structures to increase collision safety M. Schöttelndreyer, I. Tautz, W. Fricke, B. Werner, C. Daske, H. Heyer & M. Sander Hydraulic modelling of submerged oil spill including tanker hydrostatic overpressure M. Sergejeva, J. Laanearu & K. Tabri Plastic mechanism analysis of structural performances for stiffeners on bottom floor plating during shoal grounding accident Z. Yu, Z. Hu & G. Wang

199 209

219

Fatigue strength Stress intensity factor analysis using digital image correlation: A post-processing approach J.H. den Besten, M.L. Kaminski & R.H.M. Huijsmans Realistic fatigue life prediction of weld toe and weld root failure in load-carrying cruciform joints by crack propagation analysis C. Fischer & W. Fricke Fatigue strength of laser-welded thin plate ship structures based on nominal and structural hot-spot stress approach W. Fricke, H. Remes, O. Feltz, I. Lillemäe, D. Tchuindjang, T. Reinert, A. Nevierov, W. Sichermann, M. Brinkmann, T. Kontkanen, B. Bohlmann & L. Molter

233

241

249

Influence of surface integrity on the fatigue strength of high strength steel in balcony openings of cruise ship structures E. Korhonen, H. Remes, J. Romanoff, A. Niemelä, P. Hiltunen & T. Kontkanen

255

Numerical study of stress concentration factors in damaged FPSO side panels under in-plane compression loads B.C. Pinheiro, I.P. Pasqualino & C.F.C. Ferreira

263

Implementation of fatigue properties of laser welds into classification rules H. von Selle, J. Peschmann & S. Eylmann

vi

273

A FE based numerical tool for crack assessment in ship structures employing the CSR loading scheme I.K. Zilakos, V.A. Karatzas, E.V. Chatzidouros, V.J. Papazoglou & N.G. Tsouvalis

281

Ultimate strength Design of Y stiffened panels in double hull tanker under axial compressive loads A.S. El-Hanafi, S.F. Badran & H.W. Leheta

291

Nonlinear buckling behavior of stiffened ship panels M. Ozdemir & A. Ergin

301

Finite element modelling of the ultimate strength of stiffened plates with residual stresses M. Tekgoz, Y. Garbatov & C. Guedes Soares

309

Elastic buckling and elasto-plastic collapse behaviors with torsion of a longitudinal stiffener under axial compression D. Yanagihara & M. Fujikubo Ultimate strength of river-sea container ships W. Zhang, X. Luo & W. Wu

319 329

Experimental analysis Experiments on three mild steel box girders of different spans under pure bending moment J.M. Gordo & C. Guedes Soares

337

Scale model tests for the post-ultimate strength collapse behavior of a ship’s hull girder under whipping loads K. Iijima, Y. Suzaki & M. Fujikubo

347

Solutions to improve accuracy in experimental measurement of the dynamic response of resilient mountings for marine diesel engines L. Moro, M. Biot, N. Mantini & C. Pestelli

355

Composite structures Numerical analysis of cracked marine structures repaired with composite patches E.I. Avgoulas, V.A. Karatzas, I.K. Zilakos & N.G. Tsouvalis

367

Hybrid composite and metallic hulls, the best of both worlds R.G.S. Barsoum

377

Buckling studies on ship hull imperfect composite plates E.F. Beznea & I. Chirica

383

Calibration of a finite element composite delamination model by experiments M. Gaiotti, C.M. Rizzo, K. Branner & P. Berring

389

Comparison of load-carrying behavior between web-core sandwich, stiffened and isotropic plate J. Jelovica & J. Romanoff

397

An experimental and numerical study of corroded steel plates repaired with composite patches V.A. Karatzas, E. Kotsidis & N.G. Tsouvalis

405

On the scope of using composites as major structural parts of large commercial ships K. Kunal & S. Surendran

413

Weld simulation Simulation of the weld overlay procedure for corrosion repair of pressure vessels L. Gannon

423

A study on computational fluid dynamics simulation of friction stir welding S.W. Kang & B.S. Jang

433

vii

Experimental investigation of welding deformations of hybrid structural joint T. Urbański & M. Taczała

441

Welding residual stress and its effect on fatigue crack propagation after overloading K. Yuan & Y. Sumi

447

Structural analysis Equivalent shell element for ship structural design E. Avi, A. Niemelä, I. Lillemäe & J. Romanoff

459

Thermal and strength analysis of LNG tanks and their supporting structures in the early design stage M. Biot, N. Mantini & L. Moro

469

Shear response of prismatic passenger ship hull-girders K. Melk, J. Romanoff, H. Remes, P. Varsta, H. Naar & A. Niemelä

477

Linear and nonlinear FE analyses of a container vessel in harsh sea state J.W. Ringsberg, Z. Li, A. Tesanovic & C. Knifsund

485

Structural design Revisit of a 1970s semi-submersible pipe layer B. Boon

497

A new generation of offshore structures F.P. Brennan & C.M. Rizzo

507

Multi-objective optimisation of ship hull structure by genetic algorithm with combined fitness function Z. Sekulski Optimization of structural design to minimize lifetime maintenance cost of a naval vessel D. Temple & M. Collette

515 525

Structural reliability and risk models Hull girder reliability assessment of FPSO N.-Z. Chen, G. Wang & J. Bond

535

A study on random field model for representation of corroded surface M.M. Htun, Y. Kawamura & M. Ajiki

545

Lifecycle structural optimization of mid-ship of double hull tanker based on holistic risk evaluation Y. Kawamura, Y. Ohba & Y. Kaede Probabilistic pit depth corrosion model of subsea gas pipeline M.H. Mohd, D.K. Kim, D.W. Kim & J.K. Paik

555 565

Renewable energy devices Structural design of a floating foundation for offshore wind turbines in red sea K.R. Hussein, A.W. Hussein, E.H. Hegazy & A.A. Amin

575

Performance of tethered floating breakwater supporting small scale wind turbine of 750 kw R. Sundaravadivelu, V.K. Gopikrishnan & C. Yuvraj

585

Author index

591

viii

Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Preface

The papers presented at the 4th International Conference on Marine Structures, MARSTRUCT 2013, held in Espoo, Finland between 25 and 27 March are collected in this book. This is the fourth edition of the MARSTRUCT Conference series which follows on from previous events held in Glasgow—Scotland, Lisbon—Portugal and Hamburg—Germany in 2007, 2009 and 2011 respectively. The principal objective of the MARSTRUCT Conferences is to create a specialised forum for academics, researchers and industrial participants whose areas of activity are directly related with structural analysis and design of marine structures. It was our intention that the MARSTRUCT Conferences be specifically dedicated to marine structures, which complements other available conferences on ships and offshore structures. This series of conferences is one of the main activities of the MARSTRUCT Virtual Institute, an association of research groups interested in cooperating in the field of marine structures which was created in 2010 after the end of the Network of Excellence on Marine Structures (MARSTRUCT), which was funded by the European Union. The MARSTRUCT Virtual Institute was founded with the same members as the EU project but it is expected that membership of other European groups will be possible and welcomed in the future. The conference reflects the work conducted in the analysis and design of marine structures in order to reflect the full range of methods, modelling procedures and experimental results for the structural assessment of marine structures. The aim is to make marine structures more efficient, environmentally friendly, reliable and safe using the latest methods and procedures for design and optimisation. This book also deals with the fabrication and new materials of marine structures. The papers, around 60, are categorized in the following themes and areas of research: • Methods and tools for establishing loads and load effects—Wave loads, Vibrations, Response to accident loads • Methods and tools for strength assessment—Structures on ice, Impact and collision, Fatigue strength, Ultimate strength • Experimental analysis of structures—Experimental analysis • Materials and fabrication of structures—Composite structures, Weld simulations • Methods and tools for structural design and optimisation—Structural analysis, Structural design • Structural reliability, safety and environmental protection—Structural reliability models • And, Renewable Energy Articles were accepted after a review process, based on the full text of the papers. Thanks are due to the Technical Programme Committee and to the Advisory Committee who had most of the responsibility for reviewing the papers and to the additional anonymous reviewers who helped the authors deliver better papers by providing them with constructive comments. We hope that this process contributed to a consistently good level of the papers included in the book. Carlos Guedes Soares & Jani Romanoff

ix

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Committees

CONFERENCE CHAIRMEN Prof. Jani Romanoff, Aalto University, Finland Prof. Carlos Guedes Soares, IST, Technical University of Lisbon, Portugal TECHNICAL PROGRAMME COMMITTEE Prof. Carlos Guedes Soares, IST, Technical University of Lisbon, Portugal (Chair) Prof. Jani Romanoff, Aalto University, Finland Prof. I. Chirica, University “Dunarea de Jos” at Galati, Romania Prof. L. Domnisoru, University “Dunarea de Jos” at Galati, Romania Prof. R.S. Dow, University of Newcastle-upon-Tyne, UK Prof. W. Fricke, Technical University Hamburg-Harburg, Germany Prof. Y. Garbatov, IST, Technical University of Lisbon, Portugal Prof. J. M. Gordo, IST, Technical University of Lisbon, Portugal Prof. A. Incecik, University of Strathclyde, UK Prof. M. Kaminski, Technical University of Delft, The Netherlands Prof. B.J. Leira, NTNU, Norway Dr. P. Noury, Det Norske Veritas, Norway Prof. T. Moan, NTNU, Norway Prof. P. Rigo, University of Liège, Belgium Prof. J. Ringsberg, Chalmers University of Technology, Sweden Prof. E. Rizzuto, University of Genova, Italy Prof. M. Samuelidis, NTUA, Greece Prof. A. Shenoi, University of Southampton, England, UK Prof. M. Taczala, West Pomeranian University of Technology, Poland Prof. P. Temarel, University of Southampton, UK ADVISORY COMMITTEE Prof. F. Brennan, Cranfield University, UK Dr. F. Cheng, Lloyd’s Rregister, UK Prof. Y.S. Choo, National University of Singapore, Singapore Prof. W.C. Cui, CSSRC, China Prof. C. Daley, Memorial University, Canada Prof. A. Ergin, ITU, Turkey Prof. M. Fujikubo, Osaka University, Japan Prof. T. Fukasawa, Osaka Prefecture University, Japan Prof. C-F. Hung, National Taiwan University, Taiwan, ROC Prof. H.W. Leheta, Alexandria University, Egypt Prof. N.R. Mandal, Indian Institute of Technology, Kharagpur, India Dr. O. Valle Molina, Mexican Institute of Petroleum, Mexico Dr. Y. Ogawa, National Maritime Research Institute, Japan Prof. J. K. Paik, Pusan National University, Korea Prof. J. Parunov, University of Zagreb, Croatia

xi

Dr. N.G. Pegg, DND, Canada Prof. M. Salas, University Austral of Chile, Chile Prof. Y. Sumi, Yokohama National University, Japan Prof. R. Sundaravadivelu, Indian Institute of Technology, Madras, India Dr. H. Thorkildsen, Det Norske Veritas, Norway Dr. M.A. Vaz, COPPE/UFRJ, Brazil Dr. P. Videiro, Petrobras, Brazil Dr. G. Wang, American Bureau of Shipping, USA Dr. X. Wang, American Bureau of Shipping, USA LOCAL ORGANIZING COMMITTEE Prof. J. Romanoff, Aalto University, Finland Prof. P. Kujala, Aalto University, Finland Prof. H. Remes, Aalto University, Finland Prof. J. Jelovica, Aalto University, Finland TECHNICAL PROGRAMME SECRETARIAT Maria de Fátima Pina, IST, Technical University of Lisbon, Portugal

xii

Wave loads

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Long-term prediction of combined wave and whipping bending moments of containership M. Ćorak & J. Parunov University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture, Zagreb, Croatia

C. Guedes Soares Centre for Marine Technology and Engineering (CENTEC), Instituto Superior Técnico, Technical University of Lisbon, Portugal

ABSTRACT: The purpose of the paper is to propose a practical computational procedure for long-term distribution of combined wave and whipping bending moments of containerships. The problem is formulated in the frequency domain using standard engineering tools for load computation: a seakeeping code for rigid-body response and a beam FE model for transient vibratory response. The simplified von Karman approach for bow flare slamming is employed. Correlation between wave and whipping bending moments is considered. Long-term distributions of combined bending moments are computed using the standard IACS scatter diagram for the North Atlantic environment and two additional scatter diagrams for custom containership shipping routes. The speed profile required for long-term load prediction is calculated by the seakeeping analysis respecting operability limiting criteria. The procedure is demonstrated on the example of a 9200 TEU containership. 1

INTRODUCTION

HydroSTAR 2011), while the transient vibrations of the ship hull are solved using Timoshenko beam theory. Bow flare slamming load is estimated according to ABS Guidance notes (2010). The procedure is demonstrated on the example of a 9200 TEU containership with the main particulars specified in Table 1. while hull form is presented in Figure 1. In the past several years there is a considerable development in time domain simulations of transient hydroelastic responses of containerships that include a coupling of structural and nonlinear hydrodynamic model (Derbanne et al. 2010, Tuitman 2010). 2D MLM and Wagner methods are used in those studies to simulate bow flare slamming loads. Beside the fully coupled analysis, approaches based on decoupled analysis have been developing too. Fonseca et al. (2006) in the

Vertical wave bending moments represent the most important global load component having a decisive impact on the structural design of merchant ships. However, for Ultra Large Containerships (ULCS) important transient vibrations occur due to bow flare slamming, which can considerably increase wave bending load in the vertical plane and endanger the safety of the structure. Therefore, these whipping moments need to be considered in the ship structural design. Since extreme values of wave and whipping loads are determined by different calculation procedures, and since extreme values do not occur at the same time instant and even at the same environmental conditions, it is necessary to combine them in a probabilistic way. The goal of this paper is to present a procedure for the calculation of extreme vertical wave and whipping bending moments and for their combination. Furthermore, the aim is to investigate the influence of operational and environmental uncertainty on these moments. The problem is approached in the manner that is convenient in the engineering practice: transfer functions of vertical wave bending moment, relative motion, and relative velocity are determined in the frequency domain by employing a 3D panel method (BV,

Table 1. Main particulars of the 9200 TEU containership. Lpp [m] B [m] T [m] D [m] Cb vn [kn]

3

334 42.8 13.15 27.3 0.63 25.2

by computation of their long-term distribution by standard procedure (IACS Recommendation No. 34, 2000). By assuming that the process is narrow banded, amplitudes of the VWBM (Mw) in the short-term sea state follow Rayleigh distribution:

(

)

F Mw HS ,TZ ,v, β ,C = 1 − e

⎛ Mw 2 ⎞ ⎜− 2⎟ ⎝ 2σ R ⎠

(1)

Figure 1. HydroSTAR model of the 9200 TEU containership.

where process variance is calculated as area below response spectrum:

first stage of the study used a non-linear time domain simulation for determination of rigid body motions, while structural responses and the impact loads due to bow flare and bottom slamming are calculated in a second stage. Similar approach is employed by Andersen and Jensen (2012). They used a non-linear time-domain strip theory for the hydrodynamic analysis of the vertical bending moment while hull flexibility is modelled as a nonprismatic Timoshenko beam. They considered bow flare slamming by employing von Karman added mass variation method. Limitations of the structural idealization namely, a 2D beam model and a 3D finite element model (3D plate) for the use in the hydroelasticity analyses are studied by Santos et al. (2009). They have shown that 2D beam model could be too simple for the hydroelasticity analysis of the high speed patrol boats. Recently, slamming load is trying to be more accurately predicted by the 3D methods based on computational fluid dynamics especially for the cases of severe slamming (Seng et al. 2012). Such methods could be further improved by numerical and experimental studies of hydrodynamic impact and elastic response (Luo et al. 2012, Wang & Guedes Soares 2012). However, implementations of sophisticated simulations is rather time consuming, and consequently they are not suitable for preliminary design and conceptual studies. The methodology proposed within this paper is considered to be appropriate for use in the preliminary design stage. Its efficiency is demonstrated by performing parametric study of the influence of shipping route and operational and environmental restrictions on extreme combined bending moments induced by waves and whipping.

σ R2

2

∞

∫ SR (ω e HS TZ v β )

(2)

ωe

0

The spectral density of response SR is obtained as product of square of the transfer function and the spectral wave density represented by the two-parameter Pierson–Moskowitz spectrum. Short-term ship response is then combined with the probabilities of occurrence of sea state, ship headings, speed, and loading condition in order to obtain long-term probability of exceeding of the VWBM. Long-term distribution is given as: nβ

n n Δβ ⎛ H T ⋅ ⎜ ∑ Fi i =1 2 ⋅ π ⎝ j k

FL (Mw ) = ∑

(

) (

j ,k

(M

⋅ r TZk , β i ⋅ p HS j TZk

w

))

HS j TZk , β i

) (3)

where p(HS,TZ) is the probability of occurrence of sea state, while the relative number of response cycles in each short-term sea state r (TZ , β ) is given as: r (TZ , β ) =

TZ TZk

(4)

Average zero upcrossing frequency υ of all sea states and zero upcrossing period TZ of each sea state read: nT

υ=∑ k

1 1 pkTZk = TZk TZ

TZ = 2π

WAVE BENDING MOMENTS

The most common way for determination of extreme Wave Bending Moments (VWBM) is

M0 M2

(5)

(6)

After establishing long-term distribution in discrete form by applying equation (3), the theoretical

4

two-parameter (2P) Weibull distribution may be fitted: FWeibulll (Mw ) = 1 − e

⎛M ⎞ −⎜ w ⎟ ⎝ ϑ ⎠

for the half cone neglecting Wagner correction the added mass reads: m

(7)

,

ϑ=e

−

⎛ b⎞ ⎝ a⎠

1 1 = N sl TC ⋅ υ

(8)

Mw*

tbf =

(9)

)

(12)

2H 3 rɺ

(13)

Finally, impact force Fbf (t ) due to bow flare slamming as a function of time can be calculated as: Fbf (t ) =

b

2 I impact

(10)

2 tbf

t

(14)

With such defined time variant load, hydroelastic response based on Timoshenko beam theory can be obtained. The program DYANA is used, where the forced vibration problem is solved by the mode superposition method, while the time integration of modal equation is solved by the harmonic acceleration method (Senjanović et al. 1989). If such procedure is performed for different impact velocities, it can be shown that whipping bending moment Md is proportional to the square of relative impact velocity:

In order to implement the described procedure, transfer functions of VWBM are calculated by a 3D panel method (BV, HydroSTAR 2011). 3

(

Time duration tbf of bow flare impact load can be obtained from the cone’s height H and relative impact velocity rɺ as:

Finally, the most probable VWBM for a given return period can then be calculated as: ln(- ln( q ) a e

4 ρ r − r rɺ 3

I impact

For the long-term return period (e.g. TC = 20 years), the probability of exceeding the Mw* ) is given as: most probable VWBM (M q=

(11)

Based on such added mass, impulse Iimpact of the force due to bow flare slamming can be calculated as:

where ϑ and α are the Weibull scale parameter and shape parameter respectively. The Weibull 2P distribution may be presented in the linear scale, and then the Weibull parameters are calculated by the least square method as:

α

4 3 ρr 3

α

WHIPPING BENDING MOMENTS

For the analysis of the transient vibration it is necessary to know the intensity and time variability of impact loads. Generally, three types of slamming loads exist, stern slamming, bottom slamming and bow flare slamming. Despite the fact that containerships can experience severe stern slamming, it is not considered within this paper. Regarding slamming in the forward part of the ship, for the ultra large container ships it is justified to consider only bow flare slamming since the probability of occurrence of bottom slamming is rather small. The simplified von Karman added mass variation method is employed in the paper (Jensen & Pedersen 2009, ABS Guidance notes 2010). Within that method the bow forward of fore perpendicular is considered as a simple half cone for which momentum impact is determined based on added mass variation of the disc with the same radius as the cone’s penetration of the still water plane (Jensen & Pedersen 2009). The added mass of the disc in the infinite fluid is 8/3r3ρ, so

Md

k rɺ2

(15)

Confirmation of the linearity assumption for the example used in the present paper can be seen from Figure 2. Based on the assumption of linearity it can easily be shown by transforming the Rayleigh distributed relative velocity that the bending moment due to transient vibrations follows an exponential distribution. Short-term probability density function of whipping bending moment thus reads:

(M d ) = λ e − λ M 5

d

(16)

peaks (Jensen & Pedersen 2009). In this paper, however, such phase difference is ignored, i.e. it is conservatively assumed that the maximum bending moment in hogging coincides with the maximum hogging wave bending moment. The second correlation takes into account simultaneous occurrence of the maximum whipping bending moment and the maximum wave bending moment in the short-term sea state. Without time simulations and model tests it is hard to estimate extent of the first two types of correlation, but two limiting cases may be identified. The first one is a fully independent load, where according to Jensen & Mansour`s (2002), the total combined load is slightly greater than the higher one of the considered loads. Upper limit of the combined load is fully correlated load, where the combined bending moment MCorr can be obtained as the sum of short-term bending moment quintiles:

Figure 2. Relationship between maximum whipping moment and square of relative velocity for the 9200 TEU containership.

where λ is dynamic coefficient which can be obtained by combining coefficient k and variance of relative impact velocityσ r2ɺ as: 1 λ= 2 kσ r2ɺ

MCorr

(17)

(18)

Short-term probability distribution of fully correlated bending moment Mcorr based on the sum of short-term bending moments for a given probability can be obtained as (Jensen & Mansour 2002):

Long-term probability of whipping bending moments may be obtained by the procedure presented in Section 2, using an exponential distribution for whipping bending moment instead of the Rayleigh distributed wave load. The described procedure is employed for the 9200 TEU containership, where a coefficient k for hogging condition reads 9381 kNs2/m (Figure 2). Transfer functions of relative motion and relative velocity are calculated using a 3D panel method (BV, HydroSTAR 2011). 4

MW + M d

P (MCorr ) = e

2 ⎡ 1 −σ MW + σ MW + 2σ Md ⋅MCorr − ⎢ 2⎢ σ Md ⎢⎣

⎤ ⎥ ⎥ ⎥⎦

2

(19)

where σMW and σMD represent standard deviation of the wave and whipping bending moment respectively. Due to the fact that in reality partial correlation of the considered loads exists, the total combined load will be placed somewhere between the two limiting cases. The third type of correlation is due to sea state influence and can only be taken into account in the long-term probability distribution. That type of correlation actually takes into account the fact that at the lower sea states whipping influence can be higher due to higher ship speed. The “true”value of combined load may be obtained through time simulations or the model tests from which load combination factors can be determined. Within this work full correlation is assumed for two firstly mentioned correlation types, while the third correlation is implicitly taken into account by the life-time weighted sea method (Section 2).

COMBINED LOAD

Wave load and whipping load can be considered as two correlated random processes of different frequencies, in which one process in a short period of time is stationary, while the other one is nonstationary. Since peak values of considered loads do not occur at the same time and at the same conditions, the probabilistic combination of these loads is necessary. Basically, three types of correlation between these loads exist. The first correlation is due to phase angle where an impact occurs just before maximum sagging wave bending moment. Since a slight variation in phase angles can cause large difference in peak values, this type of correlation is quite important. It can be handled by considering reduction in whipping amplitude because of the time difference between maximum whipping bending moment during slamming impact and wave bending moment

5 OPERATIONAL AND ENVIRONMENTAL UNCERTAINTIES As a consequence of different input parameters, comparative analysis of long-term distributions

6

of VWBM calculated by direct procedures shows large scatter (e.g. Nitta et al. 1992, Guedes Soares 1996, Parunov & Senjanović 2003, Parunov & Ćorak 2010). To avoid such discrepancies in wave load calculations, the International Association of Classification Societies (IACS) standardized direct hydrodynamic analysis and calculation of extreme wave loads which occur in long-term exploitation of ships (IACS Recommendation No. 34, 2000). For ships with unlimited navigation area, design sea environment of the North Atlantic is proposed. Some assumptions adopted within the Recommendation No. 34 are questionable for the assessment of UCLSs (Parunov et. al. 2011). There are several operational and environmental uncertainties that can influence long-term combined bending moments of ULCS. Within this paper the uncertainty of selection of the shipping route is considered, where three different routes are studied. Probability density functions of significant wave heights for different shipping routes can be seen on Figure 3 (Parunov and Ćorak 2010). The effect of choosing alternative routes was also studied by Guedes Soares and Moan, (1991). Heavy weather avoidance is considered by the truncated scatter diagrams as described by Parunov and Ćorak (2010). Furthermore, heavy weather manoeuvring, with main manoeuvre of voluntary ship speed reduction is also analysed. Firstly, general speed reduction curve proposed by ABS (2010) is employed. After that, a specific speed reduction profile which is constructed particularly for the analysed containership is considered (Fig. 4). This speed reduction curve is computed based on seakeeping operability analysis taking into account several criteria, such as number of slams, green water and vertical accelerations at fore perpendicular (Moan et al. 2006, Šperanda 2012). As can be seen from Figure 4, as a result of operability analysis, zero ship speed is obtained for wave heights Hs ≥ 11 m. In the ABS speed reduc-

Figure 4. Speed profile for the 9200 TEU containership calculated by the operability analysis and generic speed profile proposed by ABS (2010).

tion curve zero speed is not an option and the smallest considered speed is 25% of nominal speed for Hs ≥ 12 m. For lower sea states, a higher sustainable ship speed is assumed by ABS compared to the operability assessment, as may be seen in Figure 4. 6

LONG TERM BM PREDICTIONS OF THE 9200 TEU CONTAINERSHIP

Software implementation of life-time weighted sea method, described by equations (1) to (10) is done at the University of Zagreb (the FSB code in further text). The code is verified by comparison of rigid body long-term VWBMs obtained from developed code and from SESAM postprocessor program for general responses, POSTRESP (DNV, POSTRESP 1993), using the same transfer functions. As can be seen from Figure 5 and Table 2, agreement between the calculated long-term VWBMs with different software is fairly good. Difference between the calculated values is less than 1%, probably caused by some numerical inaccuracies. As expected, the assumption of short crested waves results in about 10% lower VWBMs compared to long-crested seas. All calculations are done for the 9200 TEU containership from Table 1 and also for the North Atlantic environment. For comparison, linear Rule VWBM reads 6826 MNm. The developed code is extended to include whipping effects. Firstly, whipping, wave, and combined

Figure 3. Probability density functions of significant wave heights for different shipping routes.

7

Figure 6. Long-term bending moments of the 9200 TEU containership (short crested sea, IACS sea environment, speed profile from operability analysis).

Figure 5. Comparison of developed code (FSB) with POSTRESP (long-term VWBMs, IACS route, v = 0). Table 2.

Comparison of long-term VWBMs for 10−8. VWBM 10−8 (MNm)

POSTRESP FSB code Δ (%)

Long-crested

Short-crested

9467 9533

8588 8640

0.7

0.6

bending moments are calculated for the short crested waves using the speed profile determined by operability analysis. Results are presented in Figure 6, where the linear IACS rule VWBM is included for reference. As can be seen from Figure 6, whipping bending moment reads about 26% of VWBM. VWBM with a return period of 20 years exceeds the linear IACS Rule value for about 20%. Combined bending moment exceeds the Rule value for about 33%. Combined bending moment is lower than the sum of VWBM and whipping bending moment due to the third type of correlation which is implicitly taken into account in the long-term computation. The influence of shipping route and the effect of heavy weather avoidance are studied next. The effect of the heavy weather avoidance is analysed by truncating probability distribution of significant wave heights at Hs = 10 m. The truncation procedure is described in detail by Parunov et al. (2011). As can be seen from Figure 7 and Table 3, the selection of the Far East-North Europe shipping route through the Suez Canal could reduce long-term total bending moment up to 24% with respect to the IACS North Atlantic sea environment. It appears from Figure 7 that heavy weather avoidance may cause additional reduction of extreme wave bending moments by about 8%. The North Pacific route (Yokohama–San Francisco) is more severe sea environment compared to the shipping route through the Suez

Figure 7. Influence of the Suez shipping route and heavy weather avoidance (long-term combined BMs, short crested sea speed profile from operability analysis).

Table 3. Long-term combined BMs for 10−8 (short crested sea, speed profile of ULCS with 9200 TEU). VWBM 10−8 (MNm) IACS North Pacific route North Pacific truncated Suez route Suez truncated

10211 9591 7787 7802 7201

Canal, so the selection of the North Pacific route will reduce long-term total bending moment only up to 6% compared to the North Atlantic sea environment. Heavy weather avoidance is more important within the North Pacific route so avoidance of extreme sea states can reduce long-term combined bending moment up to 19% as can be seen from Figure 8 and Table 3. The effect of the heavy weather manoeuvring is presented in Figure 9 for the short crested waves and the IACS sea environment. ABS speed reduction curve can reduce long-term total bending

8

which may be used in preliminary stage of ship structural design. Within this paper the developed procedure is used to calculate long-term distribution of bending moments and to investigate the influence of operational and environmental uncertainties on combined bending moments of a 9200 TEU containership. As expected, the IACS Rule VWBM is largely exceeded by combined bending moment calculated for the North Atlantic environment. The longterm combined bending moment could be reduced by 6% in the case of the North Pacific and 24% in the case of the Far East-Northern Europe shipping route through the Suez Canal. Heavy weather avoidance can further decrease combined bending moments, especially in the case of severe sea environment. Voluntary speed reduction has considerable influence on the total bending moment. By employing a realistic speed profile, the combined bending moment could be reduced by 35% with respect to the full ship speed.

Figure 8. Influence of the North Pacific shipping route and heavy weather avoidance (long-term combined BMs, short crested sea, speed profile of ULCS with 9200 TEU).

ACKNOWLEDGEMENTS The work of the first two authors is funded by EU FP7 Project “Tools for Ultra Large Container Ships (TULCS)”, which is gratefully acknowledged. Figure 9. Influence of heavy weather manoeuvring— voluntary speed reduction (long-term combined BMs, short crested sea, IACS sea environment).

REFERENCES ABS, December 2010, Guidance Notes on Whipping Assessment for Container Carriers, American Bureau of Shipping. Anderson I.M.V & Jensen J.J. 2012, On the effect of hull girder flexibility on the vertical wave bending moment for ultra large container vessels, OMAE, Rio de Janeiro, Brazi. BV, March 2011, HydroSTAR for experts user manual, Bureau Veritas, Paris. Derbanne Q., Malenica Š., Tuitman J.T., Bigot F. & Chen X.B. 2010, Validation of the Global Hydroelastic Model for Springing & Whipping of Ships, PRADS, Rio de Janeiro, Brazil. Det Norske Veritas 1993, POSTRESP—Interactive Postprocessor for General Response Analysis, SESAM User’s Manual, Høvik. Fonseca N., Antunes E., & Guedes Soares C. 2006, Whipping response of vessels with large amplitude motions, OMAE, Hamburg, Germany. Guedes Soares C. 1996, On the Definition of Rule Requirements for Wave Induced Vertical Bending Moments, Marine Structures, 9 (3–4): pp. 409–426. Guedes Soares, C. & Moan, T. 1991, Model Uncertainty in the Long-Term Distribution of Wave Induced Bending Moments for Fatigue Design of Ship Structures, Marine Structures, Vol. 4, pp. 295–315. IACS 2000, Recommendation No. 34. Standard Wave Data. Rev. 1.

moment for about 27% compared to the full ship speed. Speed reduction curve that is constructed by ship operability analysis, can reduce combined bending moment for about 35%. 7

CONCLUSIONS

The paper deals with the long-term prediction of a combined wave and whipping bending moments of ultra large containerships. The computational procedure for combined, fully correlated hogging bending moments is proposed, where the wave bending moment is calculated based on seakeeping analysis, while the whipping bending moment is obtained based on transient dynamic response of Timoshenko beam. This approach is justified by the fact that hydroelastic coupling between whipping response and impact load is not significant as for the springing response. The methodology for determination of extreme wave, whipping and combined load is formulated in efficient way,

9

Jensen, J.J. & Mansour, A.E. 2002, Estimation of Ship Long-Term Wave-Induced Bending Moment Using Closed-Form Expressions, The Royal Institution of Naval Architects, W291. Jensen, J.J. & Pedersen, T.P. 2009, Estimation of hull girder vertical bending moments including non-linear and flexibility effects using closed form expressions, Journal Engineering for the Marine Environment, Vol. 223, Part M, pp. 337–390. Luo H., Wang H. & Guedes Soares C. 2012, Numerical and experimental study of hydrodynamic impact and elastic response of one free-drop wedge with stiffened panels, Ocean Engineering, Vol. 40, pp. 1–14. Moan T., Shu Z., Drummen I., & Amlashi H. 2006, Comparative Reliability Analysis of Ships—Considering Different Ship Types and the Effect of Ship Operations on Loads, SNAME Transactions pp. 16–54. Nitta, A., Arai H. & Magaino A. 1992, Basis of IACS Unified Longitudinal Strength Standard, Marine Structures 5, pp. 1–21. Parunov, J. & Ćorak, M. November 2010, Influence of environmental and operational uncertainties on vertical wave bending moments of containerships. Proceedings of: The William Froude Conference— “Advances in Theoretical and Applied Hydrodynamics, Past and Future”, RINA, Portsmouth, UK. Parunov, J., Ćorak, M. & Senjanović, I. 2011, Environmental and operational uncertainties in long-term prediction of slamming loads of containerships, Analysis and Design of Marine Structures, Guedes Soares C. & Fricke W. (eds), Taylor & Francis group, UK, pp. 67–74.

Parunov J. & Senjanović I. 2003, Incorporating Model Uncertainty in Ship Reliability Analysis, SNAME Transactions, 111 (2003), pp. 377–408. Santos F.M., Temarel P. & Guedes Soares C. 2009, On the limitations of two- and three-dimensional linear hydroelasticity analyses applied to a fast patrol boat, Journal Engineering for the Marine Environment, Vol. 223, Part M, pp. 457–478. Seng S., Anderson I.M.V & Jensen J.J. 2012, On the influence of hull girder flexibility on the wave induced bending moments, Hydroelasticity in marine technology, Tokyo, Japan, 2012. Senjanović, I., Agustinović, T., Čorić, V. & Fan, Y. 1989, DYANA, Program for Ship Hull Vibration Analysis, User’s Manual, FSB, Zagreb. Šperanda Z. 2012, Influence of operability limiting criteria on wave—induced loads of containerships, Master’s thesis, University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture. Tuitman J. 2010, Hydro-elastic response of ship structures to slamming induced whipping, PhD thesis, Delft Technical University. Wang S. & Guedes Soares C. 2012, Analysis of the water impact of symmetric wedges with a multi-material Eulerian formulation, RINA Transactions.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Semi-empirical assessment of long-term high-frequency hull girder response of containerships—an update A. Kahl, H. Rathje, J. Rörup & T.E. Schellin Germanischer Lloyd SE, Hamburg, Germany

ABSTRACT: Based on measurement campaigns onboard a Panamax and a post-Panamax containership, high-frequency hull girder vibrations on fatigue damage were assessed by calculating cumulative damages. High-frequency hull girder response extracted from these measurements was superimposed on damage obtained form numerical seakeeping calculations of the rigid ships. Numerical analyses accounted for the duration of measured seaways, the ship’s heading relative to prevailing directions of encountered seaways, and the ship’s forward speed. Routes traveled by the post-Panamax ship during the Far East measurement campaign were characterized by relatively mild seaways, a situation that was reflected by relatively low damage ratios. To represent world-wide service routes, the measured high-frequency contributions were utilized to predict high-frequency response of the two ships operating also in severe seaways occurring in the North Atlantic and the North Pacific. 1

INTRODUCTION

Then, the high-frequency hull girder response extracted from measurements was superimposed. Unfortunately, routes travelled by the postPanamax ship during the Far East measurement campaign were characterized by relatively mild seaways, a situation that was reflected by relatively low damage ratios. To represent typical world-wide service routes, these measured high-frequency contributions were utilized to predict high-frequency response of the post-Panamax ship operating in severe seaways occurring in the North Atlantic and/or the North Pacific. Nonetheless, measurement campaigns of post-Panamax ships sailing in severe seaways characteristic of the North Atlantic and/or the North Pacific are urgently needed to validate the proposed procedure.

Two recently conducted measurement campaigns extending over three years onboard a Panamax containership and over one year and four months onboard a post-Panamax containership revealed that high-frequency contributions under conditions leading to springing and whipping significantly increased global hull girder loads (Kahl & Menzel, 2008; Kahl et al., 2009). Based on measured spectra, the contribution of high-frequency hull girder vibrations on fatigue damage was assessed by calculating cumulative damages, indicating their significant influence. The presented effects of high-frequency ship response have not yet been explicitly incorporated in strength related design rules, because various uncertainties must be clarified before present design rules can be changed (Rathje et al., 2012a). In addition, numerical methods still need to be validated before they can be used to obtain reliable predictions (Oberhagemann & el Moctar, 2011). However, the increased elasticity of large container ships calls for a practical technique to assess high-frequency response. The semi-empirical technique proposed here is an attempt to assess long-term high-frequency hull girder response based on numerical seakeeping computations and full-scale measurements. First, linear seakeeping calculations of the rigid ship were performed to obtain transfer functions of wave-induced ship motions, accelerations, and vertical hull girder bending moments needed for the longterm analysis of the ship’s response in natural seaways.

2

FULL-SCALE MEASUREMENTS

Full-scale measurement campaigns were recently conducted onboard a Panamax containership and a post-Panamax containership to gather long-term data of wave-induced hull girder strains. Table 1 lists main particulars of these ships. The campaigns were meant to consolidate current design rules and to contribute to future rule development as well as to enhance the theoretical basis of hydrodynamic and fatigue strength aspects. A significant aspect was to assess high-frequency response (whipping and springing) of typical, currently operating containerships. Operating world-wide, the trade route (PAX) of the Panamax ship extended from Europe to North

11

Table 1. Principal particulars of the two containerships. Containership

Panamax

Post-Panamax

Container capacity Length bet. perp. Breadth Design draft Service speed Block coefficient Displacement

4600 TEU 274.7 m 32.25 m 12.0 m 23 knots 0.672 77,000 t

14,000 TEU 356.0 m 51.2 m 13.5 m 26 knots 0.676 172,300 t

America, continued through the Panama Canal to East Asia, and returned on the same route to Europe. The trade route of the post-Panamax ship ranged from Europe to East Asia via the Suez Canal and the Indian Ocean and returned on the same route to Europe. During these measurement campaigns, only the Panamax ship acquired data under conditions representing severe seaways as its trade route spanned crossing the North Atlantic and the North Pacific. The post-Panamax ship sailed only under conditions representing relatively mild seaways as its trade route covered transiting the Indian Ocean. 2.1

Figure 1.

Sensor locations onboard the containerships.

Measurement arrangements

On both ships, decentralized data acquisition characterized the layout of the measurement system. Mounted at five locations, data acquisition units gathered measured data from close-by sensors. These units processed part of the gathered data, which was then transmitted to the main switchboard of the measurement system, located in the deckhouse or the engine room. Optical cables connected the data acquisition units to the main switchboard. To monitor global loads, strain gages were attached to primary structural members of the hull girder at three ship stations. On the Panamax ship, these stations were located at 0.35 LPP, 0.43 LPP and 0.75 LPP, where LPP stands for ship length between perpendiculars. On the post-Panamax ship, they were located at 0.48 LPP, 0.64 LPP and 0.77 LPP. The white dots in Fig. 1 show sensor locations. To distinguish between stress components caused by different kinds of global loads, strain gages were arranged to enable the decomposition of stresses caused by vertical bending, horizontal bending, and torsion, see Fig. 2. Strain gages were attached close to the free edges of structural members, allowing the use of unidirectional strain gages. The x-location of strain gages was chosen to minimize the influence of local loads. A gyro located in the deckhouse on the main deck and accelerometers installed in the fore and aft part of the ship monitored vertical and horizontal acceleration, roll and pitch angle, as well as yaw rate. To relate wave-induced loads and

Figure 2. Arrangement of strain gages at 0.35 LPP and 0.43 LPP onboard the Panamax ship.

ship motions to environmental conditions, a wave sensor, as part of the installed shipboard routing assistance systems, measured the seaway via an x-band radar scanner mounted on the foremast. These systems also traced the ship’s loading condition (cargo distribution and masses, tank levels, and metacentric heights) as well as navigational data (e.g., position, speed, and heading). All strain gage measurements were statistically evaluated, yielding maximum, minimum, and mean values as well as standard deviations for successive 30 minute intervals. The recorded time series were triggered by measured data of specific sensors exceeding predefined threshold values. The measured data identified high-frequency hull girder response. Rathje et al. (2012b) presented typical sample time series for a whipping and springing event. Stress spectra derived from measured data yielded information on stress ranges and their distribution. These spectra were obtained by applying the rain flow counting method on continuously measured time series from all sensors.

12

Hull girder stresses in the upper deck were obtained from strain gage measurements. Associated unfiltered as well as low-pass filtered stress spectra were stored to investigate high-frequency response on fatigue strength. The low-pass filter separated low-frequency, seaway-induced responses from high-frequency hull girder vibrations.

high-frequency response on long-term stresses in the upper deck structure at ship station 0.43 LPP of the Panamax ship and at ship station 0.48 LPP of the post-Panamax ship. In the vicinity of these stations, located close to amidships, maxima of high-frequency response occurred because the dominant two-node vibration mode of the hull girders was excited. The measurement-based stress spectra, plotted in Figs. 3 and 4, were extrapolated to the same probability level as rule values. The extrapolation was performed by increasing the spectra’s number of load cycles as follows:

2.2 Measured high-frequency response To indicate the influence of high-frequency response on long-term stresses and fatigue damage, spectra of low-pass filtered and unfiltered stress ranges were determined. They were based on strain measurements and stress computations and are plotted as normalized stress range versus number of load cycles in Fig. 3 for the Panamax ship and in Fig. 4 for the post-Panamax ship. (Stress ranges were normalized against the maximum stress range of the considered spectra.) They display the influence of

ni

ni meas (20

Tmeas )

(1)

where ni meas is the cumulative number of load cycles of a block i in the measurement-based spectrum, and Tmeas is the measured period in years. The magnitude and number of stress ranges were considerably increased by high-frequency loads. For both ships, stress ranges with low probability of occurrence were increased more than those occurring more frequently. This was because whipping was more pronounced in severe seaways. However, depending on the length of the measured period, load cycles that contributed most to fatigue damage were in the range of about 103 to 106. Furthermore, in the fatigue-relevant load cycle range, the increase of stress ranges caused by highfrequency loads was higher for the post-Panamax than for the Panamax ship. To quantify effects of high frequency loads on the life to failure of typical ship structural details, the fatigue damage was determined for the measured spectra of low-pass filtered and unfiltered stresses. The cumulative damage ratio, D, was calculated according to the Palmgren-Miner linear cumulative fatigue damage hypothesis as follows:

Figure 3. Spectra of measured stress in the upper deck at location 0.43 LPP for the Panamax ship.

I

D=∑ i =1

ni Ni

(2)

where I is the number of blocks comprising the stress range spectrum, ni is the total number of load cycles in block i, and Ni is the number of endured load cycles before failure in block i, based on the S-N curve, for stress range of block i. The ratio of cumulative damage for the highfrequency response, DHF, and that for the unfiltered response, Dtotal, reflected the contribution of high-frequency loads on total response. For the Panamax ship in PAX service, high frequency loads contributed 35%; for the post-Panamax in Europe-East Asia trade, 57%. For both ships, this high-frequency effect was obtained by considering all sea areas encountered during the entire measurement period.

Figure 4. Spectra of measured stress in the upper deck at location 0.48 LPP for the post-Panamax ship.

13

An evaluation for the Panamax ship in restricted sea areas indicated that fatigue damage from highfrequency loads was proportionally higher in mild seaways than in severe seaways. The measured high-frequency loads contributed 32% to the total fatigue damage on the route in the North Atlantic and 41% on the PAX route that excluded the North Atlantic and the North Pacific. From a physical standpoint, this was expected because in mild seaways other than wave-induced high-frequency components contributed a relatively greater part to fatigue damage than in severe seaways. To represent typical world-wide service routes, these measured high-frequency contributions of the Panamax ship were utilized to extrapolate the high-frequency response of the post-Panamax ship operating in severe seaways. For the PAX route, this resulted in a high-frequency contribution of 51%. Although this assumption was not been validated for the post-Panamax ship, the resulting damage ratios seemed to reflect realistic values for such ships in world-wide service. Therefore, measurement campaigns of post-Panamax ships sailing in severe seaways characteristic of the North Atlantic and/or the North Pacific are urgently needed to validate our proposed procedure. Measurements onboard the post-Panamax ship indicated the predominant influence of the two-node vertical hull girder bending vibration mode. In the vicinity of stations located close to amidships, high-frequency response maxima were excited by this mode. This was concluded by comparing stress range spectra from different strain gages attached to a hatch corner in the upper deck close to amidships. Figure 5 compares unfiltered and low-pass filtered normalized stress spectra from gages located at the inner and outer radius of this hatch corner. The outer strain gage measured considerably higher increases of stress ranges caused by high-frequency

loads than the inner strain gage. The inner gage measured strains caused mainly by bending of the transverse bulkhead. This bulkhead deflection resulted from the warping of the hull girder induced by torsional loads. At the outer gage, strains were caused mainly by vertical bending. Thus, it was concluded that the influence of torsional vibration was negligibly small compared to the influence of vertical hull girder bending vibration. 3

SEMI-EMPIRICAL ASSESSMENT

Assessing elastic hull girder response called for a practical procedure that evades the present shortcomings of sophisticated numerical prediction methods. The semi-empirical procedure presented here relied on recent findings based on full-scale measurements and on reliable seakeeping computations. This technique to assess long-term highfrequency response, schematically illustrated in Fig. 6, comprises the following steps: − Predict ship response for the rigid (inelastic) ship using standard seakeeping codes. − Predict long-term response for the rigid ship using spectral analysis. − Predict total long-term response by superposition of high-frequency hull girder response extracted from full-scale measurements. − Determine fatigue life for the total long-term response. This procedure was used to investigate the effect of high-frequency response on the predicted fatigue life of the Panamax ship and the post-Panamax ship on which onboard measurement campaigns were conducted. 4

RESPONSE OF THE RIGID SHIP

Seakeeping calculations were performed for the rigid ships, using the linear, frequency-domain boundary element code GL-PANEL (Papanikolaou and Schellin, 1992) to yield transfer functions of wave-induced ship motions and sectional forces and moments needed for the long-term analyses of ship response in natural seaways. For these numerical analyses, two aspects were found to significantly influence results. First, all seaways encountered during the voyage were individually considered and, second, the ships’ forward speed taken from shipboard recorded data was accounted for. This code incorporates the zero-speed Green function and relies on a forward speed correction based on the so-called encounter frequency approach (Rathje et al. 2000). The wetted hull was discretized into a finite number of small

Figure 5. Stress range spectra for a hatch corner on the post-Panamax ship.

14

Figure 6.

Prediction of long-term high-frequency hull girder response.

functions together with the ensemble of short-term sea states representing the long-term wave climate encountered during the measurement campaigns. Ship speeds less than one knot were considered as harbor time and amounted to 13.8% of the service life for the Panamax containership in PAX service. For the routes travelled by the Panamax ship and post-Panamax ship, composite wave climates were generated from the GWS annual all directions wave scatter tables and also from the directional annual wave scatter tables. The first scatter tables comprised the annual probabilities of seaways from all directions, whereas the latter scatter tables provided adapted data for eight seaway directions N, NE, E, SE, S, SW, W, NW. The composite wave climates generated from the directional wave scatter tables were accumulated for each visited GWS sea area and weighted with their probabilities of occurrence to obtain a best match between predicted fatigue life for the rigid ship and fatigue life based on low-pass filtered stress spectra.

Table 2. Load parameters for seakeeping computations. Containership

Panamax

Post-Panamax

Draft GM Speed Gyradii abt. CG Displacement

10.50 m 1.02 m 20.0 knots 13.5, 67.7, 67.8 m 64, 093 t

13.25 m 2.42 m 20.0 knots 22.2, 89.2, 88.9 m 166,133 t

triangular or rectangular surface patches (panels) in a way that represents the hull surface without creating ‘leakage’ gaps. A total of 7953 and 6280 surface panels idealized the hull of the Panamax ship and the post-Panamax ship, respectively. Of these, 4083 and 4006 panels represented the wetted part of these ships, respectively, beneath the calm waterline. Load data gathered by the onboard shipboard routing systems over the duration of the respective measurement campaigns comprised container weights and their locations, masses of tank contents, ship drafts, metacentric heights (GM), ship dis-placements, and trim. Ship positions, speeds, and courses were also monitored. Based on the recorded data, for each ship a representative loading condition used for the computations was decided on, see Table 2. (The three radii of gyration refer to radii about the longitudinal, transverse and vertical ship axes, respectively.) The ships’ headings relative to the principal direction of seaways were obtained on the basis of recorded ship positions, ship courses, and wave data from Global Wave Statistics (GWS), BMT (1986). To obtain long-term distributions of hull girder vertical wave-induced bending moments, a standard statistic/probabilistic procedure processed transfer

5

FATIGUE LIFE PREDICTION

Spectra of hull girder stresses in the upper deck were obtained by dividing vertical hull girder bending moments by the section modulus of the hull girders at the sensor locations. We assumed wave-induced vertical bending moments at the evaluated midship locations to be dominant and stresses caused by horizontal bending and torsion to largely cancel each other. To match measurement-based fatigue strength stress spectra with predictions, it was important to obtain a representative approximation for those

15

respectively. The corresponding life to failure based on our predictions for the rigid hull resulted in 24.9 years. From the difference of the spectra for the filtered and unfiltered measured stresses, the contribution of high-frequency loads was found to be 35% of total damage. Figure 8 shows the three stress spectra that correspond to these lifetimes, namely, those based on unfiltered measurements, on low-pass filtered measurements, and on predictions for the rigid hull. Although the shape of the spectra based on low-pass filtered measurements differs from that based on predictions for the rigid hull, they nearly coincide over those stress ranges that contributed most to fatigue damage. This kind of comparison was also made for the post-Panamax ship. Figure 9 shows the corresponding stress spectra based on measurements and on predictions for the rigid hull. The life to failure obtained from measurements based on unfiltered and filtered stress spectra resulted in 55.1 and 129.9 years, respectively. The life to failure predicted for the rigid hull was 108.9 years. Thus, also for this ship, the life to failure based on measurements compared favorably with the life to failure predicted for the rigid hull. Here, the contribution of high-frequency loads was found to be 57% of total damage.

Figure 7. Distribution of damage ratios for low-pass filtered stress ranges.

stress ranges that caused most fatigue damage. For a typical stress range spectrum based on measured strains onboard the Panamax ship, Fig. 7 plots stress ranges against the associated fatigue damage distribution, here expressed as damage ratios. As seen in this figure, load cycles of stress ranges between 0.1 and 0.4 times the maximum stress range caused the major part of fatigue damage, i.e., stress ranges outside this interval barely contributed to total fatigue damage. Thus, it sufficed to calculate total damage by neglecting the highest load cycles not accounted for by this extrapolation method. This was also justified in that their low probability of occurrence over the measurement period meant that they were unreliable. These calculations relied on the usual linear damage accumulation and the rainflow counting method to account for all high-frequency ranges. They were carried out according to rules of Germanischer Lloyd (2012). For the Panamax ship, we used an S-N curve with a detail category FAT 56. A typical example for this detail category is a gusset plate welded on deck. For the post-Panamax ship, we used an S-N curve with a detail category FAT 63. Here, a typical example is a doubling plate welded on a hatch coaming. Categories FAT 56 and FAT 63 correspond to 2⋅106 endured load cycles at a constant stress range of 56 and 63 MPa, respectively, for a confidence level of 97.5 percent. For the post-Panamax ship, the higher detail category FAT 63 was required to account for the increased global dynamic stress level compared to that of the Panamax ship. All structural details in way of the assessed locations needed to comply with the above detail categories to achieve the rule-based life to failure. For the Panamax ship, a life to failure obtained from measurements based on unfiltered and filtered stress spectra resulted in 15.9 and 24.4 years,

Figure 8. Spectra of measured and predicted stress in the upper deck at location 0.43 LPP for the Panamax ship.

Figure 9. Spectra of measured and predicted stress in the upper deck at location 0.48 LPP for the post-Panamax ship.

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Figure 10. Spectra of stress in the upper deck at location 0.48 LPP for the post-Panamax ship trading world-wide.

Figure 11. Envelope curves of predicted and rule-based vertical wave bending moment for the Panamax ship.

To predict the life to failure of the post-Panamax ship for world-wide trade, an additional computation was made for this ship, but now utilizing the same durations of encountered sea states in sea areas and the same relative wave headings the Panamax ship encountered during its measurement campaign. The resulting stress spectrum is shown in Fig. 10, and the calculated lifetime turned out to be 36.3 years. (Stress ranges were normalized against the maximum stress range of the predicted 20-year spectrum.) To account for high-frequency loads, the number of load cycles of the predicted spectrum was multiplied by a factor of 2.05. This factor was based on measurements on the post-Panamax ship during its East-Asia trade and corrected for more severe seaways and represented the ratio of life to failure excluding high-frequency response and life to failure including high-frequency response. Based on this modified stress spectrum, which now accounted also for high-frequency response, the life to failure resulted in 17.7 years. The associated stress spectrum is also shown in Fig. 10. For use in the structural design and analysis tool POSEIDON (Germanischer Lloyd, 2012) and for comparison to rule-based envelope curves of vertical wave-induced bending moment (VBM), predicted spectra of VBM needed to be transformed into straight-line spectra with a maximum number of load cycles equal to 5⋅107. First, the predicted VBM-spectra were approximated by a Weibulldistribution as follows: log

⎡ ⎛ VBM ⎞ llogg N mmax ⎢1 − ⎜ ⎟ ⎢⎣ ⎝ VBM mmax ⎠

h

⎤ ⎥ ⎥⎦

Figure 12. Envelope curves of predicted and rule vertical wave bending moment for the post-Panamax ship.

was determined from the maximum vertical bending moment VBMmax of the predicted Weibullapproximated spectrum using correction factors for h and Nmax. The correction factors were chosen to obtain straight-line spectra that yielded the same damage ratios as predicted by the Weibullapproximated spectra. The predicted spectra, which include highfrequency response, were transformed into straight-line spectra. Distributions of maximum values of VBM spectra in world-wide service over ship length are displayed in Fig. 11 for the Panmax ship and in Fig. 12 for the post-Panamax ship. For both ships the predicted values exceed the envelope curve representing the average rule-based value of the hogging and sagging wave-induced VBM (Germanischer Lloyd, 2012) applied for fatigue strength checks. For the Panamax ship the rule curve is exceeded by 30%; for the postPanamax ship, by 20%.

(3)

6

where N is the number of load cycles, h is the shape factor, Nmax is the maximum number of load cycles, and VBMmax is the maximum vertical bending moment. The maximum vertical bending moment of the straight-line spectrum

DISCUSSION AND CONCLUSION

The following three steps summarize our semiempirical method used to assess long-term high-frequency hull girder response of a

17

Panamax containership and a post-Panamax containership:

Lloyd cooperates with the Hamburg University of Technology (TUHH) to clarify this and other related issues (Fricke & Paetzold, 2012). In future, the widened Panama Canal will see a greater number of post-Panamax containerships in world-wide trade, where they are expected to also encounter severe seas. Therefore, measurements onboard post-Panamax ships sailing in severe seaways are urgently needed to validate the procedure presented here.

− Rigid body linear seakeeping computations yielded hull girder loads used to obtain longterm stress spectra of low-frequency hull girder response, representative for the 20-year lifetime of these ships. Based on these spectra, low-frequency fatigue damage was assessed by calculating cumulative damages. − By multiplying the number of load cycles by the factor equal to 20 years/Tmeas (see eq. 1), we matched measurement-based spectra of low-pass filtered stresses with predictions and obtained stress spectra for the same probability level as rule values. From the difference of stress spectra based on unfiltered and low-pass filtered strain measurements obtained from measurements onboard the Panamax and the post-Panamax ships, we calculated the contribution of high-frequency loads. − We used the damage increase caused by high-frequency response (factor 2.05 for the post-Panamax ship) to modify the predicted lowfrequency stress spectrum for the post-Panamax ship, thereby accounting for high-frequency response for this ship in world-wide trade. We assessed the life to failure that also accounts for high-frequency response by calculating cumulative damages.

REFERENCES BMT (1986). “Global Wave Statistics.” British Maritime Technology, Authors: Hogben, N., da Cunha, L.F., and Oliver, H.N., Unwin Brothers Ltd., London. Fricke, W. & Paetzold, H. (2012). “Experimental Investigation of the Effect of Whipping Stresses on the Fatigue Life of Ships.” Proc. 11th Int. Marine Design Conference, Glasgow, Scotland. Germanischer Lloyd (2012). “Rules for Classification and Construction, I – Ship Technology, Part I – SeaGoing Ships, Chapter I – Hull Structures.” Hamburg. Hong, S.Y. (2011). “Wave Induced Loads on Ships Joint Industry Project-II.” MOERI Technical Report No. BSPIS503 A-2207–2, Daejeon, Korea (confidential). Kahl, A., Brünner, E., & Menzel, W. (2009). “Full-Scale Measurements on a Panamax Containership.” Trans. German Society for Maritime Technology, Vol. 103, pp. 278–289. Kahl, A. & Menzel, W. (2008). “Full-Scale Measurements on a Panamax Containership,” Proc. Int. Symp. on Ship Repair Technology, SRT, Newcastle upon Tyne, pp. 59–66. el Moctar, O., Oberhagemann, J., & Schellin, T.E. (2011). “Free Surface RANS Method for Hull Girder Springing and Whipping.” Proc. SNAME Annual Meeting, Paper A56, Houston, 18 pages. Oberhagemann, J. & el Moctar, O. (2011). “Numerical and Experimental Investigation of Whipping and Springing of Ship Structures,” Proc. 21st Int. Offshore (Ocean) and Polar Engineering Conf. & Exhibition, Hawaii. Papanikolaou, A.D. & Schellin, T.E. (1992). “A ThreeDimensional Panel Method for Motions and Loads of Ships with Forward Speed. J. Ship Technology Research, Vol. 39, pp. 147–156. Rathje, H., Kahl, A., & Schellin, T.E. (2012a). “HighFrequency Ship Response Assessment of Large Containerships.” J. Offshore and Polar Engineering, Vol. 22 (1), pp. 1–8. Rathje, H., Kahl, A., & Schellin, T.E. (2012b). “SemiEmpirical Assessment of Long-Term High-Frequency Hull Girder Response of Containerships.” Proc. 22nd Int. Offshore (Ocean) and Polar Engineering Conf. & Exhibition, Rhodos, Greece. Rathje, H., Schellin, T.E., Otto, S., and Östergaard, C. (2000). “Predicting Nonlinear Wave-Induced Design Loads for Ships.” Proc. 19th Int. Offshore and Arctic Eng. Conf., ASME paper OMAE 00–6122, New Orleans.

Recently conducted hydroelastic hull structural analyses and comparable model test experiments (see, e.g., el Moctar et al., 2011) demonstrated that the wave-induced VBM seems to contribute most to high-frequency loadings by exciting the two-node vibration mode of the elastic hull girder. Several research projects worldwide have been initiated to investigate this issue, among them the ongoing joint industry project WILS-II, coordinated by Korea’s Maritime and Ocean Engineering Research Institute (Hong, 2011), of which Germanischer Lloyd is a partner. Our assessments of long-term high-frequency response for the Panamax ship (based on the stress spectrum derived form unfiltered measurements) and for the post-Panamax ship (based on the stress spectrum modified to account for high-frequency response) yielded lives to failure of 15.9 and 17.7 years, respectively. Screening our damage data base for Panamax containerships, however, indicated that no significant amount of damage occurred on similar ships, even for ships operating worldwide for more than 20 years. Our assessed service lives, therefore, seem too conservative as they are less than the design service life of these ships. The contribution of high-frequency loads in assessing the overall strength of large containerships does not seem to be fully resolved. Germanischer

18

Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Numerical studies on deepwater dry tree semisubmersible R. Sundaravadivelu, Nagendra Gogula, N. Srinivasan & V.P.K. Kaundinya Department of Ocean Engineering, Indian Institute of Technology, Chennai, India

ABSTRACT: The deep water offshore platforms (>300 m) is expected to grow from 3% in 2003 to10% in 2020. The semi submersible is generally used for wet tree operation and is not suitable for East Coast in the Bay of Bengal since the 100 year design waves are of 20 m significant height and 14 s period. Hence, a dry transportation with self-erection capability is looked with a semi-submersible concept with telescopic keel tank. The new vessel technology is suitable for dry-tree support production and this reduces the cost of the entire field development in the East Coast of India. The new platform produces acceptable heave, pitch and surge motions and highly stable and reliable to operate with moderate storms. The risers, moorings and umbilical hanging loads are significant in a deepwater and could be supported by the extra buoyancy in the keel tank and thus the deck pay load is not affected by the riser, mooring and umbilical pull down loads. 1

HYDRODYNAMIC ANALYSIS OF WAMIT

The theoretical hydrodynamic motion analyses are very important tasks for floating offshore platforms. As new deep-water oil and gas production systems are introduced, the need to investigate the hydrodynamics of the body using numerical tool helps understanding the motions beforehand. For the hydrodynamic analysis wamit is the most widely used and well-proven 3-d diffraction and radiation computer program in the frequency domain. In present study, for the global analysis of dry tree truss pontoon semisubmersible (DTSP) with keel tank in full telescopic condition (the draft of the main columns is 40.53 m and extended length of telescopic column below the main column is 36.575 m) the wamit program was used. Wamit is a panel program designed to solve the boundary-value problem for the interaction of water-waves with prescribed bodies in finite- and infinite-water depth. Viscous effects of the fluid are not considered throughout and thus the flow field is potential without circulation. 2

Figure 1.

3D View of the DTSP.

The purpose of the telescopic keel tanks is: 1. To provide required mass balance to bring the overall cg of the vessel down. 2. To provide considerable amount of added mass and separated flow damping and to bring down the heave natural frequency and thus the heave motion. 3. To support columns as a base to facilitate the fabrication process in the fabrication yard and to support the entire weight of the vessel with required deck load and free float with its minimum draft such that load on/out to the barge and self installations are feasible. 4. To support predominant load of the riser pretension and mooring loads.

DRY TREE TRUSS PONTOON SEMI SUBMERSIBLE

DTSP is a column stabilized floating unit and the columns are connected by simple truss structure all around. The conventional shell pontoon is replaced by the truss-pontoon. Figure 1 shows the 3D diagram of DTSP with description of different structural units.

19

Table 1.

3

Details of DTSP.

Particular

Details

Displacement (T) Draft (m) Free board (m) Column width (m) Column length (m) Column height (m) Keel-tank length (m) Keel-tank breadth (m) Keel-tank depth (m) Column center to center (Longitudinal) (m) Column center to center (Transverse) (m) Deck pay load

80605 79.9 15.47 16.764 19.815 53.35 85.4 85.4 2.8 64.081

NUMERICAL INVESTIGATION

The numerical investigation is carried out using WAMIT. The discretization of the structure is shown in Figure 2. The radiation damping obtained from will not be realistic due to the large

60.97 17150 T Figure 5.

Heave RAO.

Figure 6.

Roll RAO.

Figure 2.

Discretization.

Figure 3.

Surge RAO.

Figure 7.

Pitch RAO.

Figure 4.

Sway RAO.

Figure 8.

Yaw RAO.

20

viscous damping associated with keel plate. Hence a physical model study is carried out and the heave damping of 9.05% and pitch & roll damping of 10.66% obtained using the free oscillation study is used in the numerical study. The numerical study is carried out for head sea (0 deg), Beam Sea (90 deg) and Quarter Sea (45 deg) and the results are shown in Figure 3 to Figure 8. The results indicate that the heave RAO is less than 0.1 for wave period up to 20 s, and the maximum heave RAO of 1.6 is observed at the heave natural period of about 30 s. 4

on DTSP with and without mooring lines is studied. The drilling riser results are shown in Figure 9(b) & production riser results are shown in Figure 10(b) for DTSP with and without mooring. The effective tensions in production riser are 850 KN for DTSP without mooring and 410 KN for DTSP with mooring. The effective tension in drilling riser increases from 12400 KN for DTSP without mooring to 12590 KN for DTSP with mooring.

PARAMETRIC STUDY

The parametric study is carried out varying the water depth as 600, 900 and 1800 m and the details of drilling riser and the production risers are given in Table 2. The pretension is 1600 kN for 600 m water depth, 2400 kN for 900 m water depth and 3200 kN for 1800 m water depth in the production risers. The wave height and period used for different water depth is also given in Table. 4.1

Figure 9a. Drilling riser effective mean tension. Water depth = 600 m.

Water depth-600 m

The effect of pretension of 1600 KN in the production riser for the water depth of 600 m on DTSP with and without mooring lines is studied. The drilling riser results are shown in Figure 9(a) & production riser results are shown in Figure 10(a) for DTSP with and without mooring. The maximum effective tension in drilling riser increases from 10000 kN for DTSP without mooring to 11100 kN for DTSP with mooring. The effective tensions in production riser are 595 KN for DTSP without mooring and 593 KN for DTSP with mooring. 4.2

Water depth-900 m

The effect of pretension of 2400 KN in the production riser for the water depth of 900 m Table 2.

Figure 9b. Drilling riser effective mean tension. Water depth = 900 m.

Riser details.

Water depth (m)

600 m

900 m

1800 m

Drilling riser Outer diameter (mm) Thickness (mm) No. of production risers Diameter (OD) mm Thickness (mm) Wave height (m) Wave period (sec) Pretension (kN) in the production riser

1 533.4 mm 33.4 mm 4 279.4 25.4 2m 9 Sec 1600 kN

1 533.4 mm 33.4 mm 4 279.4 25.4 4m 9 Sec 2400 kN

1 533.4 mm 33.4 mm 4 279.4 25.4 6m 9 Sec 3200 kN

21

Figure 9c. Drilling riser effective mean tension. Water depth = 1800 m.

Figure 10c. Production riser effective mean tension. Water depth = 1800 m.

Figure 10a. Production riser effective mean tension. Water depth = 600 m.

Figure 11. 1800 m water depth for drilling riser for different wave period.

Figure 10b. Production riser effective mean tension. Water depth = 900 m.

4.3

Figure 12. 1800 m water depth for production riser for different wave period.

Water depth-1800 m

DTSP without mooring to 10060 kN for DTSP with mooring.

The effect of pretension of 3200 KN in the production riser for the water depth of 1800 m on DTSP with and without mooring lines is studied. The drilling riser results are shown in Figure 9(c) & production riser results are shown in Figure 10(c). The effective tensions in production risers are 1645 KN for DTSP without mooring and 802 KN for DTSP with mooring. The effective tension in drilling riser increases from KN 9460 kN for

4.4

Effect of wave period

The wave period is varied as 9, 12 and 15 sec and the wave height is kept constant as 6 m for the DTSP in 1800 m water depth and the results are given in Figures 11 and 12. The increase in wave period is found to be insignificant.

22

5

CONCLUSIONS

REFERENCES Chakrabarti, S.K. (2005) Technical Editor, Handbook of Offshore Engineering, Elsevier Publications, Oxford, UK, 464–501. Chris Mungal, Kevin Haverty, Shankar bhat, Davit Anderson, Indranil Sarkar & Jack Wu-KBR (2004) Semisubmersible based Dry-Tree platform with Complaint Vertical Access Risers, OTC-16199. Halkyard J., J. Chao, P. Abbott, CSO Aker; J. Dagleish, Ocean Energy; H. Banon, BP & K. Thiagarajan (2002) A Deep Draft Semisubmersible with a Retractable Heave Plate, OTC-14304. John Murray, Arcandra Tahar & Chan K. Yang (2007) Hydrodynamics of Dry Tree Semisubmersibles, ISOPE-2007, JSC-491. Leiv Wanvik, Clive Norman Kvaerner Oil & Gas USA Inc., & John Magne Johnsen, Kvaerner Oil & Gas Field Development Norway (2001) Deep Water Moored Semisubmersible with Dry Wellheads and Top Tensioned Well Risers, OTC-12986. Nagan Srinivasan, Subrata Chakrabarti & R. Radha (2006) Response analysis of a truss-pontoon semi-submersible with heaves plates, Journal of Offshore Mechanics and Arctic Engineering, 128, 100–107, OMAE2005-67522.

The heave motion and heave motion of DTSP is within the acceptable limit of +/−5 m in operating condition to accommodate the dry-tree facility in deck. Since, the DTSP structure has the high natural period than the storm wave period in deepwater region; it is suitable for production in the extreme sea condition. It is possible to use the DTSP structure like TLP and SPAR for drilling, and production using drytree facility on the deck. The effect of parameters such as water depth, wave height, period, pretension and direction has been studied. The riser tension increases with water depth and the pretension required to maintain positive tension especially for production riser is estimated to be 1600 kN, 2400 kN and 3200 kN respectively for 600 m, 900 m and 1800 m water depth respectively. The increase in wave period does not change the riser tension. The riser tension for DTSP with mooring is more than that for DTSP without mooring and this is attributable to lesser motions of DTSP with mooring.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Numerical simulation of structural response under bow flare slamming load Takao Yoshikawa & Masahiro Maeda Kyushu University, Fukuoka, Japan

ABSTRACT: In numerical simulation, the bow flare slamming phenomenon is often treated as the wedge impact to water surface. Up to now, in many numerical simulation, the impact pressure under bow flare slamming has been firstly calculated by CFD in which the structure have been assumed to be rigid, and then the structural response has been calculated utilizing FEA under the pressure obtained by CFD. But, the impact pressure is possible to be influenced by the structural rigidity. In this paper, in order to establish the assessment procedure of the structural response under bow flare slamming load, the general purpose FEM software; LS-DYNA, which can simulate the structure-fluid coupling behavior by ALE, is applied for the impact problem of elastic wedge. The parametric calculations are performed to obtain the pressure and the structural response by changing the velocity and the attacking angle of wedge to water surface. And, the effect of the structural rigidity on the impact load and the structural response are investigated by series of calculations. 1

INTRODUCTION

the structure-fluid coupling behavior by ALE, is applied for the impact problem of elastic wedge. The parametric calculations are performed to obtain the pressure and the structural response by changing the velocity and the attacking angle of wedge to water surface. And, the effect of structural rigidity on the impact load and the structural response are investigated by series of calculations.

Recently, the container ships have come to be possible to travel without avoiding heavy weather by virtue of improvement of machinery performance. And, the smaller bow flare angle is adopted for large size of container ships. Under such circumstances, the larger impact loads are apt to act on the bow flare part, and the design criteria of this part must be investigated again. In numerical simulation, the bow flare slamming phenomenon is often treated as the wedge impact to water surface. Up to now, in many numerical simulations, the impact pressure under bow flare slamming has been firstly calculated by CFD in which the structure have been assumed to be rigid, and then the structural response has been calculated utilizing FEA under the pressure obtained by CFD. But, the impact pressure is possible to be influenced by the structural rigidity. Therefore, in order to calculate the impact pressure and the structural response accurately, the coupling analysis of structure and fluid under the bow flare slamming load must be performed. In recent years, Tassin (2010) and many researchers have investigated on the water impact of elastic wedge, but the systematic calculation by changing the rigidity of structure have not been performed yet within the extend known to the authors. In this paper, in order to establish the assessment procedure of the structural response under bow flare slamming load, the general purpose FEM software; LS-DYNA, which can simulate

2 2.1

VALIDATION OF NUMERICAL SIMULATION Impact of rigid wedge on water surface

The pressure acted on the rigid wedge at water impact is calculated. The calculation model is shown in Figure 1. The wedge is assumed to have

Figure 1. wedge.

25

Calculation model for water entry of rigid

Table 1.

Property of water (Kimura, 1995).

Mass density: ρ0 [kg/m3] C0 [MPa] C1 [MPa] C2 [MPa] C3 [MPa] C4 C5 C6 E0 [MPa]

1000.0 0.0 2060 8432 8014 0.4394 1.3937 0.0 0.2623

The state equation of fluid is Figure 2.

P = C0 + C1µ + C2µ2 + C3µ3 + (C4 + C5µ + C6µ2)E0 µ = (ρ/ρ0) − 1 where P: pressure of fluid, ρ: current density, ρ0: initial density.

an infinite length and both of water and wedge are modeled with a unit length in wedge length direction. The water entry velocity of wedge is assumed to be 5 m/sec. The deadrise angle of wedge is assumed to be 1,3,4,5,10, and15 degree. In this calculation, the general purpose software LS-DYNA, in which the ALE procedure is equipped for structure-fluid coupling analysis, is used. The fluid region is modeled by Euler elements which have the material property shown in Table 1, and the wedge part is model as rigid body by Lagrange elements. The contact between water and wedge is judged by adopting the penalty coupling method. The infinite boundary condition is introduced at the boundary of water region. The aim of this paper is to investigate the coupling effect of structure and fluid on the impact pressure and the structural response considering of the structural deformation. Therefore the wedge must be treated as an elastic body. To unify the modeling method of wedge, the elastic plate are placed to the bottom surface and connected to the rigid wedge with about 300 mm pitch in this calculation. The contact is judged only between the elastic plate and the water surface. The plate thickness is assumed to be 30 mm. Because this thickness is thick enough considering the plate span, the plate can be assumed to be rigid. In the ALE method of LS-DYNA, not only the structure but also the fluid is calculated as Lagrange elements firstly. Next, the nodes of fluid region move to the original position and the physical quantity in fluid region, such as displacement, velocity, acceleration, and pressure, are mapped to the original elements. The calculated results of peak pressure on the plate are shown in Figure 2. The calculated pressure corresponds well to the theoretical value by

Figure 3. surface.

Impact pressure acting on wedge surface.

Time history of impact velocity at water

Wagner’s 2D model. In the case of deadrise angle lower than 3 degree, the impact peak pressure has to be lower than that of Wagner due to the cushion effect of entrapped air. But, the air is not modeled in this calculation, the above phenomenon cannot be captured. As the compressibility of fluid is treated in calculation, the impact pressure has to be lower than the limit pressure, ρcV, of compressibility. But the calculated pressure exceeds considerably this value. To examine this reason, the velocity of water surface in vertical direction is shown in Figure 3. The velocity of water surface is zero before impact, but increases rapidly just after the water entry in the vicinity of contact region between water and wedge. The rapid increase of velocity of water surface is due to the building up of water, and the velocity is getting large at farther position from the wedge center. It is confirmed that the impact pressure does not exceed the limit pressure of compressibility which is calculated by using the increasing velocity of water surface. 2.2 Effect of mesh size on the calculation accuracy In order to examine the effect of mesh division in fluid on the impact pressure, the numerical calculation for the solid water attacking to the plane structure was performed.

26

Figure 4. Calculation model for impact of rectangular solid water to rigid wall (Nakashima, 2011). Figure 5. Time history of pressure acting on plate (the crude calculation result, Fluid mesh size is 5 mm).

The calculation model is shown in Figure 4. The elastic plate with thickness of 30 mm is placed on the rigid bottom wall and supported at each 80 mm pitch. The region of water is 400 mm width and 500 mm height. The rigid walls are also placed to both sides of water so that the water may not flow out. The initial velocity of 5 m/sec was imposed to the region of water. The contact was judged only between the elastic plate and the water surface. The behavior of water attacking to the plate and rebounding from that was calculated. The duration of contact between water and rigid structure can be estimated by eq. (1). Tcontact = 2 H/V = 2 × 500/(1.5 × 106) = 670 µsec (1) where, H; height of water, V; velocity of sound in water. The periodic time of the first flexural eigen mode of elastic plate is 0.14 msec considering the span of 100 mm. As this time is shorter enough compared to the above contact time, the effect of deformation of elastic plate on the impact pressure is negligible, and the structure can be assume to be rigid in this case. The impact pressure can be estimated as following when the perfect elastic rebounding of water is assumed. And this pressure acts constantly on the plate during the duration of contact. P = 2 MVwater/Tcontact = 7.7 N/mm2

(2)

where, M; mass of water, Vwater; initial impact velocity of water. In Figure 5, the crude calculation results of impact pressure acting on the plate are shown. In Figure 6(a), the results of moving average with averaging time of 0.03 sec are shown. The results with averaging time of 0.06 sec are almost the same as this figure, but omitted for cutback of pages. The spinout pressure presented in Figure 5 is trivial for the response of structure because the duration of spinout peak pressure is short enough

Figure 6. Time history of pressure acting on plate (the result of moving average with averaging time of 0.03 sec) (Nakashima, 2011).

compared with both of the whole duration of pressure and the natural period of structure. The purpose of this paper is to investigate the structural response at water impact. The spinout peak pressure shown in Figure 5 does not affect on the response of structure. In examining the structural

27

response under water impact, it is supposed to be no problem that the calculated pressure is presented by averaging with the time smaller than 10% of the duration of pressure and the natural period of structure. The duration of pressure and the maximum pressure in Figure 6(a), in which the mesh size of fluid is 5 mm, coincide with the values of eq. (1) and (2). In Figure 6(b) with the fluid mesh size of 10 mm, the pressure rises up at early time and the gradient of increasing pressure is moderate when comparing with Figure 6(a), and the duration of pressure and the maximum pressure are a little bit different from the theoretical values. In Figure 6(c) with the fluid mesh size of 25 mm, this tendency is emphasized more. In the following, the reason of above will be discussed. The judgment of contact between water and structure in LS-DYNA will be explained using Figure 7. The occurrence of contact is judged when the fluid enters to the Euler mesh adjacent to structure. In the case of water location shown in Figure 7(a), the contact between water and structure is judged

not to occur in the both case of large and small mesh size. In Figure 7(b), the contact is judged to occur in large mesh case, but not to occur in small mesh case. In Figure 7(c), the contact is judged to occur in both cases. To be exact, the water has not yet contacted to structure in all cases of Figure 7. The starting time of contact is judged earlier and earlier when the mesh size is large. When the Euler mesh adjacent to the structure has been filled partially, the density of this mesh is smaller than that of water. Therefore, the contact pressure at that time will be reduced much from that of fully entering of water to the Euler mesh adjacent to structure. This is the reason that the pressure in Figure 6(b) and (c) rise up at earlier time and the gradient of increasing pressure are moderate when comparing with Figure 6(a). Even if large fluid mesh is used, the impulse acted on the structure will not change much. Therefore, the structural response will not be affected much by fluid mesh size when the duration of pressure is much smaller than the natural period of structure. But the historical shape of pressure will affect on the structural response when the duration of pressure approaches to the natural period of structure. Therefore, the fluid mesh size must be selected so that the rising up time of pressure, which is the passing time of water in the adjacent Euler element to structure, is smaller enough than the whole duration of pressure. In the calculation of sec. 2.1, the large fluid mesh size seems to be applicable because the water exist initially 100% for every Euler mesh. But, as the up of water surface will take place, the partially fluid filled mesh can exist after the wedge entry. At this time, the mesh size of Euler region must be determined so that the passing time of structure over the adjacent element to fluid is much smaller than the time of natural period of structure.

3

3.1

PRESSURE ACTING ON ELASTIC WEGED AND STRUCTURAL RESPONSE IN WATER IMPACT Calculating condition

In order to establish the assessment procedure of the structural response under bow flare slamming load, the general purpose FEM software; LSDYNA, which can simulate the structure-fluid coupling behavior by ALE, is applied for the water entry impact problem of elastic wedge. The parametric calculations are performed to obtain the pressure acted on the elastic wedge and the structural response by changing the velocity and the attacking angle of wedge to water surface.

Figure 7. Schematic diagram of contact judgment between fluid and structure.

28

The deadrise angle of wedge is varied in range from 10 to 20 degree. The impact velocity of wedge to water surface is varied from 5 to 20 m/sec. The wedge is modeled with the rigid body and the elastic plate is attached to bottom surface of wedge. The contact was judged only between the elastic plate and the water surface, this is same as previous section. The skin plate thickness at bow flare of container ship, in which the strength under bow flare slamming load is important, is 10 mm to 30 mm. In this paper, the plate thickness attached to wedge bottom is assumed to be 5, 10, 20, 30, 50, 100, and 200 mm in order to examine the effect of flexural rigidity of plate on the impact pressure and the structural response. The supported span of plate along to inclined plate is set to be 800 mm considering the spacing of longitudinal stiffener of bow section in actual ships. At the supporting location, the nodes of plate are connected to the nodes of rigid wedge. The plates in FEA represent the behavior of skin plates at middle span of longitudinal stiffener. As the range of deadrise angle of wedge is within 10 to 20 degree, the effect of air entrap at impact is not necessary to be considered. The dynamic response of several container ships were calculated by utilizing strip theory in frequency domain, and the time historical responses of these ships under 10−5 sea state of north Atlantic sea which is most severe to these ships were calculated utilizing the result of strip method. Under the head sea condition for two hours, the relative angle between the bow surface and the wave surface is 10 to 20 degree when the relative velocity is large. At that time, the relative velocity is from 9 m/sec to 15 m/sec. Based on the above results, the deadrise angle and the impact velocity in calculation are selected. In the case of oblique sea, the relative angle between the bow surface and the wave surface is possible to be smaller value. In this case, the effect of air entrap at impact must be considered. A sample of calculation model is shown in Figure 8. The right half of model is shown in this figure. The plate is divided with 8 elements between the adjacent supported points, that is the length of

mesh is 100 mm. And the mesh size of fluid region is 20 mm. This value is larger than the model of section 2.2, but the region of calculating model is large and the height difference of adjacent longitudinal stiffener is about 140 mm even in the case of deadrize angle of 10 degree. Therefore, the mesh size of fluid region is considered to be small enough.

Figure 8. Calculation model for water entry of rigid wedge with elastic panel.

Figure 9. Time history of pressure on the plate in the case of β = 10°, Vwedge = 15 m/sec.

3.2

Pressure acting on elastic wedge

The calculated results of the case that the deadrise angle of wedge is 10 degree and the water entry velocity is 15 m/sec are shown in Figure 9. Figure 9(a), (b), and (c) show the time history of pressure acted on the plate whose thickness is 10,20, and 30 mm, respectively. The duration in which the high pressure acts on can be representative by the time that the original water surface passes over the panel. This time can be obtained as follows. ΔTpressure = (yi + 1 − yi)/V

(3)

where, yi + 1 and yi are the y coordinates at both side of plate, V is the velocity of wedge.

29

In Figure 9, this time is about 14 msec. The higher peak pressure tends to act on the plate which is far from the apex of wedge. As shown in section 2.1, this is due to the buildup of water surface. By comparison of Figure 9(a),(b), and (c), it seems that the higher pressure tends to act on the thinner plate. There is a possibility that the deformation of thin plate reduces the pressure, but the countertrend is found. The reason of this should be investigated further. The relation of the peak pressure on the rigid plate and deadrise angle β is shown in Figure 10. The peak pressure is normalized by the result of deadrise angle β = 15 degree. The peak pressure is almost proportional to cotβ. The relation of the peak pressure on the rigid plate and the wedge impact velocity V is shown in Figure 11. The peak pressure is normalized by the result of wedge impact velocity V = 15 m/sec. The peak pressure is almost proportional to V2. 3.3

shown in Figure 12. Figure 12 (a) and (b) show result of plate thickness of 10 mm and 30 mm, respectively. The deflection of thin plate is larger than that of thick plate. The relation between the maximum stress at each plate and the duration of impact load are shown in Figure 13 and Figure 14. Figure 13 show the maximum stress of panel 2 in Figure 8 when the water entry velocity of wedge is 15 m/sec. In this figure, the result of β = 10, 15 and 20 degree are plotted. The vertical axis is normalized the maximum stress by the stress when the peak pressure on the rigid plate assumes to act statically on each plate. The horizontal axis is normalized the

Response of panel

The time histories of deflection of panels in the case of the β = 15 degree and V = 15 m/sec are

Figure 10. The relation between the peak pressure and deadrise angle of wedge.

Figure 12. Time history of deflection of plate in the case of β = 15°, V = 15 m/sec.

Figure 13. The relation between the peak bending stress of plate and the duration of impact load in the case of V = 15 m/sec.

Figure 11. The relation between the peak pressure and water entry velocity of wedge.

30

Figure 15. The structural response of 1 degree model under various load curves (Harris, 1976).

Figure 14. The relation between the peak bending stress of plate and the duration of impact load in the case of β = 15 degree.

From the calculation results above, the structural response of elastic wedge can be estimated as follows.

duration of impact load by the first natural period of each plate. The natural period of plate is calculated by considering the added mass wetted one side of plate In the case of thicker plate, the value of horizontal axis is larger because the first natural period of thicker plate is smaller. The duration of impact load is the passing time of water surface over the panel, as it explained in sec. 3.2. The structural response of thick plate over 50 mm is considered to be static because the natural period of plate is larger enough by comparison with the duration of impact load. Therefore the value of vertical axis saturates to 1.0 at large value of horizontal axis, that means the case of thick plate. In the case of thin plate, that is the small value of horizontal axis, the natural period of plate is smaller by comparison with the duration of impact load, and the normalized structural response will be reduced. In thin plate, the structural response is proportional to impulse not to peak pressure. In the case of middle thickness, that is near to 1.0 of horizontal axis, the structural response is larger than the static one because the natural period of plate is almost equal to the duration of impact load. Thickness of 15–30 mm corresponds to this, and the skin plate thickness of bow flare in actual container ship falls within this range. Figure 14 shows the results of panel 2 when the deadrise angleβis 15 degree. In this figure, the results of V = 10, 15 and 20 m/sec. are plotted. The relation between the structural response and the duration of impact pressure is similar to Figure 13. The structural response of one degree of freedom model under various load curves with unit impulse are shown in Figure 15. By comparison between Figures 13, 14 and 15, it is found that the response under the versine shock wave will give the safety side prediction as for the response of water entry of elastic wedge.

1. The peak pressure of water entry of wedge is calculated using Wagner’s method, 2. The magnification factor of structural response under impact load is estimated by the result of versine impact load shown in Figure 15. Where, the duration of impact load is the passing time of water surface over the panel and the natural frequency of panel will be calculated by considering of added mass of water. In the above procedure, the rate of increase in velocity, which is resulted by buildup of water surface in water entry of wedge, must be counted by some method. 4

THE EFFECT ON PANEL RESPOSE WITH ELASTIC SUPPORT BY LONGITUDINAL STIFFENER

The shell plates at bow part of ship are supported by the longitudinal stiffeners, and the longitudinal stiffeners are supported by transverse frames. When the impact pressure acts on the bow part of ship, the longitudinal stiffeners deform as well as plates. In this chapter, how the deformation of longitudinal stiffeners affects to the impact pressure acted on the plates will be investigated. The calculation model is shown in Figure 16. The longitudinal stiffeners are modeled as the springs and the concentrated masses to present those dynamic properties. The masses are located at the longitudinal stiffeners with considering modal mass, and the spring constants are decided so that the natural frequency of these spring and mass coincide with the first natural frequency of longitudinal stiffeners with both end simply supported by transverse members. The space of transverse frame is assumed to be 3200 mm, and the scantling of longitudinal stiffener is assumed to be hw × bf × tw × tf = 300 × 90 × 13 × 17 mm.

31

The time histories of pressure acted on the panel are shown in Figure 17. Figure 17(a) and (b) show the result of with rigid and flexible longitudinal stiffeners, respectively. The pressure acted on the panel with flexible longitudinal stiffeners is smaller than with rigid ones. The deformation of longitudinal stiffeners will reduce the mutual velocity between the fluid and structure, and it results in reduction of pressure. Figure 17(c) shows the result of pressure with both longitudinal stiffeners and girder. The location of girder is between panel 4 and panel 5. The rigidity of girder is considerable larger than longitudinal stiffener. Therefore, the spring at girder is eliminated and the plate is fixed to the rigid body. The pressure of panel 4, which is forehand of the girder increases much by comparison with Figure 17(b). This is due to the change of deadrise angle caused by the different deformation at the girder. From these calculation results, it is found that the deformation of longitudinal stiffeners and girders must be considered in order to calculate precisely the impact pressure of bow flare slamming.

Figure 16. Calculation model for water entry of rigid wedge with elastic panel supported by longitudinal stiffener.

5

CONCLUSIONS

In this paper, in order to establish the assessment procedure of the structural response under bow flare slamming load, the general purpose FEM software; LS-DYNA, which can simulate the structure-fluid coupling behavior by ALE, is applied for the impact problem of elastic wedge. The parametric calculations are performed to obtain the pressure and the structural response by changing the velocity and the attacking angle of wedge to water surface. And, the effect of structural rigidity on impact load and structural response are investigated by series of calculations. 1. The impact peak pressure act on rigid wedge can be calculated by Wagner’s 2D theory. The higher peak pressure tends to act on the plate which is far from the apex of wedge due to the buildup of water surface. 2. In the case of thick plate, the structural response is considered to be static because the natural period of plate is larger enough by comparison with the duration of impact load. 3. In thin plate, the structural response is proportional to impulse not to peak pressure. 4. In the case of middle thickness, the structural response is larger than the static one because the natural period of plate is almost equal to the duration of impact load. The skin plate thickness of bow flare in actual container ship falls within this range. The magnification fac-

Figure 17. Time history of pressure on the plate in the case of β = 15°, V = 15 m/sec.

32

tor of structural response under impact load is estimated by the result of versine impact load. In above, the duration of impact load can be representative by the time that the water surface passes over the panel. 5. The pressure acted on the panel supported by the flexible longitudinal stiffeners is smaller than the panel supported by rigid ones. The pressure acted on the panel located below of the girder is larger than the panel supported by longitudinal stiffener, because the rigidity of girder is considerable larger than longitudinal stiffener.

Kimura T., K. Asada, K. Inoue, K. Ida, 1995, Study on Dynamic Response of Structure Due to Underwater Pressure Wave, Mitsubishi Heavy Industries technical review, Vol. 32, No. 2. Lewis S.G., D.A. Hudson, S.R. Turnock, D.J. Taunton, 2010, Impact of free-falling wedge with water; synchronized visualization, Pressure and Acceleration Measurement, Fluid Dynamics Research, Vol. 42, pp. 1–30. Makoto Arai, 1998, Non-linear impact phenomena and analysis technique, Bulletin of The Japan Soiciety of Naval Architects of Japan, Vol. 8:29, pp. 22–30. Masakazu Nakashima, Masahiro Maeda, Takao Yoshikawa, 2011, A Study of water impact problem utilizing numerical simulation considering fluidstructure interaction, Proceedings of the 25th AsianPacific Technical Exchange and Advisory Meeting of Marine Structure, pp. 309–316. Sun H., O.M. Faltinsen, 2009, Water entry of a bow-flare ship section with roll angle, Journal of Marine Science and Technology, Vol. 14:1, pp. 69–79. Tassin A., N. Jacques, A.E.M. Alaoui, A. Neme, B. Leble, 2010, Assessments and comparison of several analytical models of water impact. International Journal of Multi-physics, Vol. 4:2, pp. 125–140. Temarel P., 2009, Comparison of theoretical slamming impact pressure and forces with model test results, MARSTRUCT European Network of Excellence, Reoprt MAR-D1-3-UoS-04. Wagner, 1932, Uber Stoss und Gleitvorgange an der Oberflache von Flussigkeiten, Zeitschrift fur Angewante Mathematik und Mechanik,12. Yoshikawa T., T. Kawasaki, H. Nishikawa, A. Murakami, H. Shimizu, 2001, Water impact response of container ship bow flare by numerical method, Journal of The Kansai Society of Naval Architects of Japan, No. 236, pp. 211–219.

ACKNOWLEDGEMENTS This study is financially supported by grant in aid for scientific research No. 23656551. REFERENCES Brizzolara S., N. Coutry, O. Hermundstad, A. Ioan, T. Kukkanen, M. Viviani, P. Termarel, 2008, Comparison of experimental and numerical loads on an impacting bow section. Ship and Offshore Structures, Vol. 3:4, pp. 305–324. Fairlie-Clarke A.c., T. Tveitnes, 2008, Momentum and gravity effects during the constant velocity water entry of wedge shaped sections, Ocean Engineering, Vol. 34:7, pp. 709–719. Harris C.M., C.E. Crede, 1976, Shock and Vibration Handbook, McGraw-Hill.

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Vibrations

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Vibrations of superyacht structures: Comfort rules and predictive calculations D. Boote & T. Pais University of Genova, Italy

S. Dellepiane Cantieri Navali Benetti, Livorno, Italy

ABSTRACT: The comfort level on board modern superyachts has been recently the object of specific attention by most important Classification Societies which issued new rules and regulations for the evaluation of noise and vibration maximum levels. Such rules, usually named as “Comfort Class Rules”, contain both the general criteria for noise and vibration measurement in various yacht areas, and maximum limit values which such measurements should fall into. Comfort Class Rules follows the ISO Standard 1984 formulation, recently removed by the ISO 2000 “Mechanical vibration—Guidelines for the measurement, reporting and evaluation of vibration with regard to habitability on passenger and merchant ships”. In this work, performed in cooperation between Azimut|Benetti Shipyards and the Department of Naval Architecture of the University of Genova, a complete review of the existing rules is presented, together with a case study developed on a 60 meters superyacht, on which a detailed FEM investigation has been carried out in order to investigate the dynamic behaviour of hull and superstructures. The results of modal and transient analyses are compared with a first series of experimental data gathered during the vessel construction. 1

INTRODUCTION

the natural frequency of the hull and of local structures, such as decks and bulkheads, and then their response to exciting loads induced by propellers, engines and waves. Vibration problems, in particular, are certainly more critical in the case of steel and light alloy yachts, even if it’s well known that also FRP vessels are not free from this kind of problem.

Vibrations and noise are crucial aspects for cruise ships and large yachts. For what the second category is concerned the current global crisis and the consequent demand contraction exasperated competition between shipyards, which are continuously looking for new solutions to reduce construction costs and to improve the quality and innovation of their products. When performances are not a primary objective for the achievement of new customers, the strongest effort of technical offices is focused mostly on other aspects related more to the aesthetic impact of the project (internal and external) and to the comfort on board. From this point of view, vibrations and noise represent most difficult issues to deal with for designers, both in the initial phase of the project, when it is necessary to have preliminary information about the response of the structure not yet defined, and during construction, in case some critical behaviors arise in any part of the structure. Given the objective difficulty in making any change to the dynamic behaviour of hull structure after construction, it is extremely important to perform FEM predictive analyses to identify

2

THE COMFORT ON BOARD IN ACCORDANCE WITH THE RULES

The problem of vibration and noise on board large yachts falls within comfort characteristics required by owner specifications and suggested by Classification Society Rules. This subject was initially regulated by ISO 6954, 1984 edition and then by the new, more restrictive ISO 2000 standards. Recently, most important Classification Societies assumed new Comfort Class Rules based on maximum levels of vibration and noise. Baker and McSweeney (2009), as an example, present a complete analysis of present ABS Rules concerning vibrations and noise published in the ‘Guide for the Class Notation Comfort—Yacht’ (2008). Two notational options are considered:

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COMF(Y), which establishes a level of comfort based on ambient noise and vibration alone and COMF+(Y) which adds slightly more demanding criteria for noise and vibration, and provides additional criteria for the assessment of motion sickness. ABS Yacht Comfort guide, however, have been recently revised in some aspects. In Table 1, a synthesis of the new version is reported for yacht below and over 45 meters in length when sailing and at rest in harbour; in this table the following units have been assumed to measure vibration intensity:

it is indispensable to carry out detailed analyses at the design stage. Although the simplified approach based on variable section beams with concentrated masses remain a valuable tool for the calculation of the first natural frequencies of the hull, the presence of large openings such as aft and side garage doors, considerably complicates the dynamic behaviour of the hull. In this case only a FEM modelling of the entire structure allows to obtain reliable results. As a matter of fact hull natural frequencies are very low and far from the frequencies of usual exciting forces. For what local vibrations are concerned, the most critical areas are represented by decks, bulkheads and superstructures. The danger of resonance exists with reference to four main sources of vibrations represented by main engines, electric generators, main propellers and bow thruster. Engines and generators are usually mounted on resilients and this significantly reduces their contribution to hull global and local vibrations. On the contrary main propellers induce high dynamic forces at harmonics corresponding to their Blade Passing Frequencies (BPF); this depends on the number of blades, shaft rate and operating conditions. As stated by Roy (2008), for large motor yachts (between 50 and 100 m in length) blade passing frequencies could be between 15–20 Hz at cruise speed condition, up to 30–40 Hz at maximum speed. Aft and central parts of the hull are the most exposed areas of the vessel to the propeller exciting forces and their dynamic response mainly depends on the distance from the propellers. The bow thruster propeller should be considered as well; his blade passing frequency BT, much higher than BPF (over 50 Hz) but with minor intensity, can affect mainly the fore part of the vessel (see Fig. 1). Several practical approaches are assumed by designers but, as a common general rule, in order to assure that decks do not resonate at any point in the speed range, the design philosophy should be to ensure that the first mode frequency of every deck panel is in excess of BPF. This goal is not so simple to be achieved in case of larger structural components, such as decks and bulkheads, the only possible action in this respect being the increase of the structure stiffness. While a simple plate thickness increase is not advisable because of excessive consequences on hull weight, an accurate selection of deck secondary and primary structure can provide better results. In case of superstructures the problem is relatively mitigated by the higher distance from propellers and, consequently, by lower exciting forces; on the other side large spans of unsupported decks lower their natural frequencies making necessary the addition of cumbersome and unaesthetic pillars. Even so, often it become really impossible to

aw = multi axis acceleration value calculated from the root-sums-of-squares of the weighted Root Mean Square (RMS) acceleration values in each axis (axw, ayw, azw) at the measurement point; v = spectral peak of structural velocity in mm/s. Other Classification Societies produced similar rules for the assessment of comfort levels on board yachts and superyachts. In the following a list of the most important ones are reported: • Bureau Veritas (2011), Part E, Sect 5, ‘Additional Requirements for Yachts’; • Det Norske Veritas (2011), Part 6, Chapt. 12, ‘Noise and Vibration’; • Germanischer Lloyd (2003b), Part 1, Chapter 16, ‘Harmony Class’; • Lloyd’s Register (2011), Chapt. 6, ‘Passenger and Crew Accommodation Comfort’; • RINA (2011a), Part E, Chapt. 5, ‘Comfort on board’. Some examples of maximum allowable levels for vibration are shown in Table 2 for Bureau Veritas, Lloyd’s Register and RINA. From a structural point of view vibrations can be assessed at global and local level. As well known in the first case it’s quite impossible to apply any corrective action after the construction and thus Table 1. Maximum whole-body vibration according to ABS COMF(Y) class for yachts below and over 45 metres in length. Yacht length [m]

Notation

Freq. (Hz) Measure

Max. levels

L ≤ 45 m L ≤ 45 m L ≤ 45 m L ≤ 45 m

COMF(Y) COMF(Y) COMF+(Y) COMF+(Y)

1–80 1–80 1–80 1–80

aw [mm/s2] v [mm/s] aw [mm/s2] v [mm/s]

89.4 53.50 2.5 1.50 53.5 45.00 1.5 1.25

L > 45 m L > 45 m L > 45 m L > 45 m

COMF(Y) COMF(Y) COMF+(Y) COMF+(Y)

1–80 1–80 1–80 1–80

aw [mm/s2] v [mm/s] aw [mm/s2] v [mm/s]

71.5 45.00 2.0 1.25 53.5 35.75 1.5 1.00

Sail

Harb

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Table 2. Maximum whole-body vibration according to Bureau Veritas, Lloyd’s Register and RINA Comfort Rules for yachts. Bureau Veritas

Lloyd’s Reg

RINA

Location

Hz

v

Hz

v

Hz

v

Cabins and lounge Public spaces Open decks

1–80 1–80 1–80

1.0–3.0 1.0–3.0 1.0–3.0

1–80 1–80 1–80

1.8–2.5 1.8–2.5 1.8–2.5

0–100 0–100 0–100

1.0–3.0 1.0–3.0 1.0–3.0

3

FEM MODELING

For this study the project of a typical three-decks superyacht, 60 meters long and 9.5 meters wide has been kindly made available by AZIMUT|BENETTI Shipyard, within a joint research program with the Department of Naval Architecture of the University of Genova. The ship has a steel hull and aluminum superstructures; the whole structure is longitudinally framed with web frame interval equal to 1200 mm. In order to perform accurate and reliable FEM calculations a detailed numerical model of the stern part of yacht structures has been realized by the multipurpose code ANSYS version 13.0 (Swanson Analysis System Inc., 2011). The hull geometry and structure lay out has been imported from a 3D model previously created by a rendering software. Since the beginning of this investigation the numerical model of the hull structure has been created considering the large calculation time required by transient dynamic analyses. For this reason several tests with different mesh refinements have been carried out and a good compromise between structure definition and mesh “weight” appeared to be that corresponding to an average panel diagonal around 300 millimeters. The mesh was created using “SHELL63” elements of the ANSYS library for plating and main reinforcements such as keelsons, floors and girders. For secondary stiffeners simple “BEAM44” elements have been preferred to keep the model dimension as low as possible. The mass and loads of the main deck and the two superstructure decks have been modeled by “SURF154” elements, which are particularly suitable for dynamic analyses. The geometry of the complete model is represented in Figure 2, while in Figure 3 a view of the inside structure is shown by a longitudinal section of the mesh. In Figures 4 and 5 some details of the bottom and of the upper deck structure is shown. In total the numerical model consists of 59,000 nodes and 61,500 elements. On each deck a distributed load has been applied corresponding to the finishing and outfit

Figure 1. Subdivision of the vessel into areas for local vibrations assessment.

keep natural frequencies higher than BPF and the only possibility is to remain below the cruise speed BPF. Even in this case a detailed FEM analysis is the only way to identify critical areas and to look for possible solutions in advance. Nevertheless it is plain that every action addressed to the structure vibration reduction lead to an increase of the hull weight; by way of example it has been estimated that, for a superyacht in the range between 90 and 100 meters, the weight increase to reduce vibrations is more than 100 tons. The purpose of this work is to verify the dynamic behaviour of the stern area of superstructure decks of a superyacht. In this area the “upper deck” and the “sun deck” are characterized by a considerable overhang that, in conjunction with the lower aluminium stiffness, may cause annoying vibrations for passengers close to the frequencies of the exciting forces induced by the propellers. The study has been carried out by a general purpose FEM code in two phases: in the first phase, after a detailed modeling of the stern part of the yacht, the natural frequencies of the superstructure have been determined. In the second phase a series of transient analyses has been performed in order to investigate the structure response by varying the intensity of the exiting forces and the structure damping. The calculation results, in terms of vibration velocity of decks, have been compared with the limit values imposed by Classification Society Rules.

39

Figure 2. yacht.

Model geometry of the stern part of the

Figure 3.

Longitudinal section of the numerical model.

Figure 4.

View of the mesh in the bottom area.

Figure 5.

Detail of the upper deck structures.

Figure 6. Distributed loads applied on decks by SURF154 elements.

Figure 7. Tender garage with distributed and concentrated loads (yellow masses).

weight (about 300 N/m2) as shown in Figure 6; on the main deck the concentrated load of the tender crane in the garage has been considered as well. The tender garage is located in the stern area below the main deck (as shown in Fig. 7). The corresponding load was applied to the reinforced beams of the main deck simulating the presence of a crane hanging under the deck; the crane weight is about 25,000 N and it was applied by placing 10 concentrated mass of 2500 N each. The yacht mass has been increased in way of 80% of displacement to take into consideration the water added mass. The numerical model has been constrained in correspondence of section n.17 where the hull

has been “cut”. All nodes located on this section have been completely clamped, thus forcing the structure to behave as a cantilever. Constraints are located far enough from the area of interest; as a consequence the constraint effect on the results of the analysis can be assumed irrelevant. 4

MODAL ANALYSIS

As already mentioned in the previous chapters the analysis has been carried out in two phases.

40

In the first one the natural frequencies of the whole structure have been investigated by modal analysis; the Lanczos mode extraction method (LANB approach in ANSYS code) has been employed. This solver is particularly suitable for large models consisting of shells or a combination of shells and solids. As expected the most significant vibration modes have been individuated on the main deck and on the superstructure decks. The numerical values have been compared with those measured on the same vessel during different construction phases: on the separate decks and after their assemblage without outfitting (only steel). In this way a first interesting evaluation can be made between the single structure response and when connected with the rest of the hull. In Table 3 the most significant natural frequencies, calculated and measured (assembled), are reported. Another aspect which the authors wanted to investigate on is the influence of the outfitting on local structure natural frequencies. As a matter of fact, when this paper was in preparation, the yacht outfitting was in course and it has not been possible to perform any measurement. This will be the subject of next activities regarding this research.

Figure 9. plot.

5

Calculated (Hz)

Measured (Hz)

Main deck—1st mode Upper deck—1st mode Sun deck—1st mode Sun deck—2nd mode

12.9 11.2 11.0 22.0

13.5 11.5 11.3 –

TRANSIENT ANALYSIS

After the natural frequencies of decks have been identified a direct dynamic analysis in the time domain has been performed in order to investigate the structural response to the propeller exciting forces. The propeller action has been simulated by a pulsating sinusoidal pressure applied on a portion of bottom plating on the stern counter in correspondence of the propeller disk. Two values of the propeller maximum pressure have been assumed on the base of gathered data made available by the shipyard: 1 KPa for the half loaded propeller and 2 KPa for the propeller 100% loaded. The pressure has been applied with a sinusoidal law and with two different frequencies; the first one has been chosen very close to the first natural frequency of the sun deck (10 Hz). The second frequency has been assumed equal to the BPF of the propellers this yacht is equipped with (33 Hz). Another parameter which has been investigated in the transient analysis is represented by the damping coefficient β of the structure. Many sources exist in literature providing damping values for steel structures; they are usually related to the structural typology and to the stress intensity. In case of steel welded structures subjected to low stresses, such as those considered in this analysis, the value of the damping coefficient can be assumed equal to β equal to 0.005. Nevertheless, being unlikely the stress distribution could be uniform on a complex structure like a large yacht is, it is not so simple to identify a single damping value for the complete vessel. In order to quantify the influence of this choice on final results, a series of calculations has been carried out on the same structure with five different β values: 0.001, 0.0025, 0.005, 0.010 and 0.025.

Table 3. Calculated and measured natural frequencies of main deck and superstructure decks. Item

Sun deck first natural frequency displacement

Figure 8. Main deck first natural frequency: displacement vector plot.

41

The propeller pressure has been applied to two rectangular areas shown in Figure 10. The structural response has been monitored in correspondence of five points shown in Figure 11. While only one point has been monitored on main deck, on the sun deck and on the upper deck displacements were considered in two different points: on the extreme stern part (points n.1) and at the deck centre (points n.2) where, from the experimental surveys, the largest vertical displacement took place. From the ANSYS data base all the result have been extracted in the five monitored points. So the displacement, velocity and acceleration time histories have been plotted and examined making reference to the limit values of the CS Rules. Obviously the performed calculations are relative to the yacht in “sailing” conditions. As an example the results for “upper deck point n.1” are reported in Figures 12, 13 and 14. Displacements, velocities and accelerations time histories for a short time of 0.2 sec are plotted relatively to a propeller passing frequency equal to 33 Hz,

Figure 12. Time history of vertical displacement on the monitored point n.1 of upper deck.

Figure 13. Time history of vertical velocity on the monitored point n.1 of upper deck.

Figure 10. applied.

Areas where propeller induced pressure are

Figure 14. Time history of vertical acceleration on the monitored point n.1 of upper deck.

propeller maximum pressure equal to 2 KPa and all the considered damping coefficients. Being the structural velocity the most common way used by the rules to characterize vibrations, all results have been synthesized in Tables 4 and 5 in terms of peak velocity of the five monitored points, as a function of the damping coefficient β and of two different loading conditions of propellers. The corresponding plots of velocity time histories in the monitored points have been included as well in Figures 15, 16 and 17 for BPF equal to 33 Hz, full loaded propeller and all the considered damping coefficients.

Figure 11. Monitored points for the analysis of structure dynamic response.

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Table 4.

Peak velocity [mm/s] in the monitored points on decks. Half loaded propeller

Item

β = 0.001

β = 0.0025

β = 0.005

β = 0.01

β = 0.025

Main deck Up. deck 1 Up. deck 2 Sun deck 1 Sun deck 2

1.87 1.98 2.12 1.46 0.59

0.99 1.01 1.08 0.89 0.41

0.62 0.60 0.68 0.55 0.29

0.36 0.38 0.44 0.34 0.21

0.30 0.24 0.00 0.22 0.15

Table 5.

Peak velocity [mm/s] in the monitored points on decks. Full loaded propeller

Item

β = 0.001

β = 0.0025

β = 0.005

β = 0.01

β = 0.025

Main deck Up. deck 1 Up. deck 2 Sun deck 1 Sun deck 2

3.69 3.90 4.19 2.87 1.17

1.98 2.01 2.17 1.73 0.80

1.24 1.18 1.36 1.05 0.56

0.71 0.75 0.86 0.68 0.41

0.58 0.47 0.01 0.43 0.30

Figure 15. Time history of vertical velocity on the monitored point of bottom.

Figure 17. Time history of vertical velocity on the monitored point n.1 of sun deck.

exceeds the lower limit of 1.0 mm/s suggested by BV and RINA Comfort Class Rules. − For more realistic damping value equal to 0.005 the lower limit is exceeded only in the bottom area, the closest one to the propeller exciting action. The situation on the main and superstructure decks is fully satisfactory. − For what acceleration is concerned, the plot of Figure 14 shows that, for β equal 0.005, the limit values of ABS COMF(Y) equal to 71.5 mm/s2 is slightly overcome. However this approach should be carefully considered and further verified with more accurate experimental measurements.

Figure 16. Time history of vertical velocity on the monitored point on main deck.

At this point the time histories have been compared with the limit values suggested by Comfort Class Rules and the following considerations can be done.

At the end of this first part of the research it can be said that the numerical approach herein developed is fully suitable to analyze and verify the main aspects of superyacht vibrations and to perform

− For what the structural velocities are concerned, for β lower than 0.005, the velocity always

43

a first, reliable comparison with the CS Comfort Class Rules resumed in Tables 1 and 2. 6

below the limit values, confirming the comfort qualities of the vessel. At present the research continues on other vessels with similar dimensions but different structural and outfitting solutions in order to better define the dependence of the structural response on every detail, even if apparently unimportant. In this perspective new measurement tools are under evaluation to achieve a wider data base on the pressure values induced by propellers under different operating conditions and a more powerful instrumentation for vibration and noise measurement and real time analysis.

CONCLUSIONS

The study herein presented is the first part of a research program carried out in cooperation between Azimut|Benetti Shipyards and the University of Genova aimed at thoroughly analyzing the structural problems of modern motor yachts. In this phase attention has been devoted to those aspects having a direct impact on the comfort characteristics of the vessel. At first a complete overview of recent rules issued by ISO and Classification Societies has been carried out in order to highlight the most important parameters to be considered. Then attention has been focused on hull vibrations, at local level. The evaluation procedures and limit values suggested by the rules has been applied to a case study, consisting of a 60 meters steel yacht at present close to completion, where the dynamic behaviour has been constantly monitored during the various construction phases. Taking advantage of the data availability a detailed numerical model of the stern part of the hull and superstructures has been set up with the aim of identifying the natural frequencies of local structures such as bulkheads, main deck and aluminum superstructures decks. The results of this first analysis perfectly matches the experimental measurements carried out on board. In a second phase the structure response to the exciting forces induced by the propellers has been investigated to verify possible local resonances. In order to achieve a wide result spectrum calculations have been performed for different BPF, different maximum propeller pressure and several values of the damping coefficient. The obtained results, in terms of vibration peak velocity, have been compared with the limit values suggested by the regulations and they proved to be reasonably

REFERENCES American Bureau of Ships. 2008. Guide for the Class Notation Comfort Yacht (COMF(Y)) and Comfort Plus Yacht (COMF+(Y)). New York, USA. Baker, C. and Mc Sweeney, K. 2009. Setting a Standard for Luxury and Comfort, Design, Construction and Operation of Super and Mega Yachts Conference, Genova, Italy. Bureau Veritas. 2012. Rules for the Classification and Certification of Yachts. Paris, France. Det Norske Veritas. 2011. High Speed, Light Craft and Naval Surface Craft, Hovik. Norway. Germanischer Lloyds. 2003. Part 1—Seagoing Ships, Chapter 16, Harmony Class—Rules on Rating Noise and Vibration for Comfort, Cruise Ships. Hamburg, Germany. Lloyd’s Register of Shipping. 2011. Rules and Regulations for the Classification of Special Service Craft, London, UK. Registro Italiano Navale. 2011. Rules for the Classification of Yachts Designed for Commercial Use, RINA S.p.A., Genova, Italy. Roy, J. et al. 2008. Longitudinal versus Transversely Framed Structures for Large Displacement Motor Yachts, 20th International HISWA Symposium on Yacht Design and Yacht Construction, Amsterdam, The Netherlands. Swanson Analysis System Inc. 2011, ANSYS— Engineering Analysis System, Version 13.0, Houston, Pennsylvania (U.S.A).

44

Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Noise on board RO-Pax vessels: Measured levels on existing ships and new pre-normative requirements T. Gaggero & E. Rizzuto University of Genoa, Genoa, Italy

ABSTRACT: The problem of noise annoyance on board ships has been addressed since a long time. The first document setting limits on onboard noise was the “Code on noise levels on board ships”, issued by the International Maritime Organisation in 1981. More recently all the major Classification Societies introduced in their requirements the so called “Comfort Classes”. They generally provide different ratings of the acoustic comfort of ships. The growing needs of passengers and workers for acoustical comfort on board pushed both Classification Societies and IMO to consider an update of limits and criteria for noise annoyance assessment. The normative framework in the field is therefore presently in an evolution phase. In the present work, a series of existing measurements of noise carried out on a set of Ro-Pax vessels is analysed and compared with the existing requirements and with the new proposals of the SILENV collaborative project (Ships oriented Innovative soLutions to rEduce Noise & Vibrations), funded within the 7th Framework Programme of the E.U. and running in the period 2009–2012. The comparison provides an indication of the possibility for shipbuilding to cope with the new requests in terms of noise performances, provided that such requests are inserted in the design process. Further efforts are needed similar to those already implemented to fulfill the present formulations of compulsory requirements and reflected in the measured values. 1

INTRODUCTION

of noise impact has been therefore evolving from including mainly the most evident physical effects to accounting for more subtle psychological effects related to wellness or well-being perception. In parallel, the technological development made available means to achieve better acoustical performances on board, thus allowing to follow the surge in requirements. In the present work, existing measurements carried out on a series of Ro-Pax vessels have been analysed. The aim is to assess the present state of the art in the field of internal noise for a segment of the European fleet. Measured levels are compared with the present requirements and with recently issued pre-normative limits for the noise internal to the ships. The latter were proposed within the SILENV project and represent a possible formulation of the more severe requirements that are likely to be introduced in the next future.

The noise impact on crew members of ships can range from health impairment to a decrease in working performances or to a loss of comfort. The first issue can be represented by a reduction in the capabilities of the auditory system. The second aspect can be related to a loss of concentration or an accumulation of fatigue causing potentially dangerous situations. The loss of comfort can affect the crew particularly during off-duty hours, inducing a lack of recovery and again an overstress on work. The most dramatic effects are related to cases of exposure to high noise levels, but most of noise effects are associated to prolonged exposures, due to long periods spent on board by the crew members. Passengers are usually exposed for shorter periods (which, however, can be of the order of several days in cruise ships) and to lower levels; expectations for comfort, however, are likely to be higher, particularly considering recreational motivations for the stay on board (cruise). As in many other engineering fields, the control of noise level has focused first on safeguarding the workers’ health and on ensuring acceptable working conditions, and later on enforcing a comfortable living on board. The concept and the definition

2 2.1

EXISTING NORMS: REVIEW IMO code on noise levels on board ships

The first available document dealing with the impact of noise on board ships is the “Code on Noise levels on board ships” issued by the International Maritime Organisation in 1981 (IMO, 1981).

45

Main aim of the Code is to guarantee suitable working conditions for the crew. To this aim, the Code contains limits on:

instrumentation that is to be used in surveys and the environmental conditions during the trials. In spite of its age, the Code is still in force unmodified and is considered a major reference document for noise rating in the shipbuilding field.

• The noise levels in different spaces • The daily noise dose for workers • The insulation properties of the vertical subdivisions of spaces present on board.

2.2

Limits on the maximum allowable noise levels in dB(A) are fixed in a set of spaces covering five main categories: work spaces, navigation spaces, accommodation spaces, service spaces, normally unoccupied spaces (see Table 1). Limits are set depending on the needs dictated by the use of the space and on the existing technical limitations in terms of results achievable by noise control. Fixing lower levels in accommodation spaces has the aim of allowing the crew to recover during the off-duty period. In navigation spaces and in work spaces the aim is to ensure safety by proper communication and audibility of emergency acoustic signals and to prevent loss of attention and performances because of stress and fatigue. To account for the total exposure to noise of the seafarers with different noise level during the 24 hours, as mentioned above, a limit noise dose is introduced in the code. A limit to the total acoustic energy received in 24 hours is fixed. It corresponds to an equivalent continuous exposure of 24 hours to a level of 80 dB(A). In the Code are also specified: the operating conditions in which the limits are to be verified; the

Table 1.

International Labour Organisation framework (ILO)

The main aim of the ILO is the protection of workers’ health through the enforcement of acceptable working conditions. A few documents issued by ILO deal with working conditions on board ships making specific reference to noise: • the Maritime Labour Convention, with the Guidelines B3.1.12—Prevention of noise and vibration, B4.3.2—Exposure to noise and B4.3.3—Exposure to vibration; • the ILO Convention No.188—Concerning Work in the Fishing Sector; • ILO Recommendation No.199—Recommendation Concerning Work in the Fishing Sector. These documents set general motivations and suggest means to improve the conditions of workers on board. They give practical guidelines on how to arrange the layout of ships in view of a mitigation of the noise impact and indications on the main countermeasures that can be adopted. However, they do not provide precise or quantitative information on limit levels.

Limits on sound pressure levels imposed by IMO Res. A.468(XII). Limits [dB(A)]

Space type

Spaces

Work spaces

Machinery spaces (continuously manned) Machinery spaces (not continuously manned) Machinery control rooms Workshops Non-specified work spaces Navigating bridge and chartrooms Listening post, including navigating bridge wings and windows Radio rooms (with radio equipment operating but not producing audio signals) Radar rooms Cabins and hospital Mess rooms Recreation rooms Open recreation areas Offices Galleys, without food processing equipment operating Serveries and pantries Spaces not specified

Navigation spaces

Accommodation spaces

Service spaces

46

90 110 75 85 90 65 70 60 65 60 65 65 75 65 75 75 90

2.3

3

Comfort Classes (CC)

In the '90s all the major Classification Societies (CS) started issuing additional voluntary class notations, commonly named Comfort Classes (CC), dealing with noise and vibrations onboard. These rules are aimed at assessing the level of comfort on board for both passengers and crew, setting standards on: noise, vibrations, sound insulation and impact noise levels. Limits are set in dependence on various parameters reported in Table 2. In general three grades of comfort are defined, corresponding to three different limits for the categories above listed. The general structure of CC requirements is pretty similar to the one in IMO (1981), with a set of limits in different spaces. As regards the crew spaces, reference is directly made to IMO (1981) for noise limits but vibration limits (not contained in IMO) are provided in addition. In passenger spaces limits are in general more restrictive than in crew spaces. Requirements (in terms of lower limits) on the isolation performances of vertical and horizontal subdivisions of spaces are given, too. The aim is to ensure privacy and quietness in the private spaces on board.

3.1

Type of ship

Length (m)

Speed (knots)

Other charact.

ABS

Pax

=

=

Cargo

=

=

With berth. cabins WO berth. cabins

Pax Cargo All ships Pax HSLC

LBP ≥ 65 LBP ≥ 65 LBP < 65 = LBP > 50 LBP < 50

= = = = =

BV

DNV

GL

Cargo Cargo Pax

LR RINA

High speed craft Pax Cargo Pax & Cargo All ships

LBP > 80 LBP < 80 = =

• discussing the criterion of application of the Code, making it mandatory for new ships (see also IMO (2009)) • upgrading the standards for measurements • updating the limit levels for the spaces on board. In general the updates under discussion move in the direction of more severe limits. A lowering of 5 dB(A) for most of the spaces on board and a similar increase [+ 5 dB(A)] in the minimum insulation indexes have been proposed, see IMO (2011). This is in line, on one hand, with the modern needs for increased comfort and better working conditions for seafarers and, on the other hand, with the improved technologies available for noise control on board. 3.2

= = =

LBP < 65

=

SILENV

One of the aim of the SILENV projects was to define a set of new pre-normative requirements able to enforce a significant reduction of the noise impact of shipping activities in the various fields affected. The control of noise impact was sought with an holistic approach, working on the various aspect of the noise radiation: inside the ship, outside in air, outside in water. In particular the new requirements for internal noise were set on the basis of: • an analysis of existing requirements • a collection of existing and new data of noise measurements on different types of vessels, giving a picture of the European fleet. • a correlation of noise measurement results with the human perception, acquired through a large number of questionnaires, distributed on board ships of different typology to both passengers and crew. They contain objective questions on status (age, gender, etc.) and location on board during the trial and subjective questions about the feeling on noise and vibrations.

= = = V ≤ 25 V > 25 VMax > 7.16*Δ1/6

= = LBP ≥ 65

IMO

Recently, within IMO, an updating process of the IMO Code dated 1981 has been undergoing. A correspondence group [last issued document IMO (2011)] has been established on this matter aiming at:

Table 2. Ships’ categorization used by Comfort Classes (HSLC: High Speed Light Crafts). CS

FUTURE TRENDS IN NORMS AND REQUIREMENTS

The final formulation of limits was based on a balance between comfort and work impairment on board and technical feasibility of the target of control. SILENV limits are reported in Table 3: for more details see SILENV (2012).

47

Table 3.

Group # Group name

Location example

1

Cabins

Passenger cabin Crew cabins Hospital

2 3

Offices Public space A

4

Public space B

5

Public space C

6

Outdoor areas

7

Wheelhouse

8

Workspace A

9

Workspace B

10

Workspace C

11

Workspace D

4

2007 in the shipbuilding industry for this type of units. In Table 4 the complete list of the ships analysed is reported together with some macro parameter of the vessels. As it can be seen in Table 4, two families of “sister” ships are present in the data analysed (RP-14 to RP-21 and RP22 to RP25). Some of the comparison that are carried out below are performed within the same family, while other ones are carried out considering all the ships together.

SILENV noise limits.

Libraries Calm public spaces Restaurant, lounge Mess room Shops Disco Ballroom Corridor Staircase Open recreational area Bridge wings/open deck working areas Wheelhouse Radio room Engine control room Galleys Pantries Stores Laundries Workshops Garage Continuously manned machinery spaces Not continuously manned machinery spaces

dB(A) 50 53 55 60 65

4.1

Sister ships comparison

A first interesting aspect is to compare the acoustical performances of sister ships that in principle are identical. In Table 5 and Table 6 the percentage of locations surveyed on board where the measured levels are not complaint with SILENV requirements is reported for each ship belonging to the same family. Wide differences are present among the sister ships, thus suggesting that, despite the same design, the ships are not actually completely similar. The above mentioned Tables refer to all spaces on board considered together; Table 7 and Table 8 are focused specifically on two categories of spaces for which a large number of measurements are

70

60 65 75

90

Table 4.

Analysed ships. Main dimensions

105 RP-13 RP-14 RP-15 RP-18 RP-19 RP-21 RP-22 RP-23 RP-25 RP-26 RP-27

DATA ANALYSIS

A statistical analysis of a number of measurements in different spaces on board is carried out with the aim of assessing how far is the existing fleet from the present formulation of requirements (compulsory: IMO and optional: CC) and the newly pre-normative version developed within SILENV. It is interesting to note that the surveys used in the present study were not devoted to verify specifications, nor to assess the acoustical performances of the ships in view of assigning a Comfort Class notation, but measurements were part of a standard process of quality assurance internal to the shipyards. They do not reflect, therefore, an effort in achieving a control of noise (at least for passenger cabins), but a picture of the state of the art of what was achieved in the decade between 1997 and

L [m]

B [m]

D [m]

# of decks

Year

169.5 169.5 169.5 169.5 169.5 180 180 180 203.7 166.4 107.4

25.6 25.6 25.6 25.6 25.6 25.9 25.9 25.9 31.5 25.5 20

27.2 27.2 27.2 27.2 27.2 34 34 34 33.5 T = 5.3 T=4

7 7 7 7 7 8 8 8 10 5 8

2002 2001 2004 1997 1997 2000 1999 1999 2001 2004 2007

Table 5. Overall performances for sister ships (family 1).

RP-13 RP-14 RP-15 RP-18 RP-19

48

Not complaint

Total measured

%

21 27 30 23 27

81 94 61 82 55

26% 29% 49% 28% 49%

compliance for the 3 ships of the series without cabins on the 5th deck. On the contrary, the other two ships (those with cabins on deck 5) feature a very different behavior: in one case, all the cabins on the upper deck are below the limit, whereas, in the other one, a percentage of above 40% of non compliance is found. On the other hand, all the measured cabins on deck 5 of the 2 ships having this feature did not comply with the limit. It seems possible to conclude that cabins adjacent to the garage have more noise problems than those at an upper deck, but the presence of cabins on a lower deck (instead of larger public spaces) has not a definite influence on the performances on the upper deck. Looking at the spaces that are distributed on two or more decks (see Table 9) the trend is of an increasing percentage of not fulfillment for the lower decks, but the garage, for which deck 3 has the worst performances. This is probably due to the fact that deck 3 is the car deck closest to machinery spaces. Furthermore, it covers a wide range of the ship length and for this kind of spaces noise control is quite difficult.

Table 6. Overall performances for sister ships (family 2).

RP-21 RP-22 RP-23

Not complaint

Total measured

%

31 10 33

78 27 55

39% 37% 60%

Table 7. Mean dB(A) levels for Crew and passenger cabins for ships belonging to the same family (1). Cabins Mean dB(A) level

Crew

Pax

RP-13 RP-14 RP-15 RP-18 RP-19

48.5 47.8 48 48.9 46

47.2 45.8 49 47.3 52.2

Table 8. Mean dB(A) levels for Crew and passenger cabins for ships belonging to the same family (2). RP-21 RP-22 RP-23

54.1 50* 54.2

4.3

54.2 52* 54.4

In this section the performances of the single space types will be analysed, taking into account the measurements of all the ships together. It is to be underlined that SILENV limits do not depend on the characteristics of the ship (dimensions, speed, type). Looking at Table 10 it can be noticed that most of the spaces with a percentage of not compliance greater than 50% belong to crew spaces and specifically to navigation and work spaces. A significant value regards the space type ‘Engine Control Room’ with a percentage of 88.9% of non-compliance. This highlights the need for a careful design of the acoustical characteristics of this type of space. Crew cabins, located in the accommodation spaces, feature a percentage of non-compliance of 41%. This value means that, in the investigated cabins, a large part of the seafarers cannot recover completely during the off-duty shift with a consequent potential decreasing of the working performances. Only 16% of the typical locations occupied by the passengers in RO-pax vessels (cabins and restaurants) do not fulfil the limits. It is to be remembered that what above relates to the application of SILENV limits (that were not even formulated when the ships were built).

* Only one datum available.

available. Comparisons of the mean dB(A) level are presented, again for the two groups of sisters ships. As regards family 1, significant differences are present between the passenger cabins and the crew cabins and the differences are not always in the same direction. The performances of family 2 are in general worse, but almost no differences in the mean values are present both between the ships and the two classes of cabins. 4.2

Performances of the various space types

Spaces distribution

For the family 1 of ships (RP 13-19) the distribution of spaces along the seven decks of the vessel has been analysed, too. Aim of the analysis was to check if a correlation between the noise level and the vertical location on board is present. For this specific ship type, however, many categories of spaces are concentrated on a single deck. For example, crew cabins are all located on deck 7 for the 5 sister ships of family 1. Passengers cabins are located mainly on deck 6: for two ships of this series, however, a few passenger cabins were placed in addition on deck 5. It is interesting to notice that the cabins on deck 6 have quite a stable percentage (about 10%) of non

4.4

Histograms

For the categories of spaces for which a larger number of measurements was available, a more

49

Table 9. Measured spaces distribution on board [% of measured spaces located in the specific deck (% of measured not fulfilling)]. Deck 7 Crew cabins Day cabins Hospital Offices Mess room Engine control room Workshops Not cont manned machinery Listening posts, bridge wings Garage Passenger standard cabin Outside installation Restaurant, lounge Corridors, Staircase Mean % of not compliance

Deck 6

100% (16%) 100% (0%) 100% (44%) 14% (0%) 100% (0%)

Deck 5

Deck 4

Deck 3

Deck 2

Deck 1

100% (56%) 100% (71%) 4% (0%)

48% (17%)

46% (23%)

33% (11%)

33% (13%)

11% (2%)

11% (4%)

11%

35%

9.5%

13.5%

86% (25%)

100% (0%) 89% (4%) 17% (13%)

65% (6%) 10% (0%)

17% (10%) 90% (11%) 83% (9%)

3.3%

10.3%

17% (0%) 9.1%

11% (2%) 11% (5%)

Table 10. Spaces performance for all the ships (decreasing order). Space type Open deck recreation Radio room Chart rooms Engine control room Workshops Cargo control room Hospital Garage Galleys Outside installation Corridors, staircase Offices Crew cabins Not cont. manned machinery spaces Open deck working areas Stores Fan rooms Passenger standard cabin Restaurant, lounge Mess room Wheelhouse Closed public spaces Recreation Laundries Listening posts, bridge wings

Not comp.

Tot. meas.

accurate analysis was possible. Histograms of the distribution of noise levels were derived and presented in Figures 1 to 3 together with the SILENV limits as well as other types of requirements. Percentages of overcoming the limits are reported in Table 11. In the machinery spaces, the SILENV limit is slightly lower than the modal value of the histogram (Figure 1). A little increase in the acoustic planning of such spaces, i.e. a little decrease in the levels present in that location, can bring to significant improvements in the compliance to the limit. The histogram presents a tail towards lower levels and only 13% of points overcome the compulsory IMO limit. This kind of shape suggests that efforts have been made to fulfill this compulsory limit. It is to be noticed that the SILENV proposed limit corresponds to the requirement contained in the proposal of the new IMO code (IMO, 2011). In restaurants the limit is not fulfilled only by the 16% of the measurements (Figure 2). For passengers cabins, the histogram presents a tail towards very high levels, while the mean value is lower than for crew cabins. The shape of the histogram in Figure 3 can be related to the fact that no compulsory limits were taken into account in the design of the cabins. Nevertheless, efforts have been made to keep the levels as low as possible in order to ensure passengers’ comfort. This results in a histogram with a lower mean but with a considerable upper tail (no truncation).

%

2 1 1 8 6 3 6 30 5 17 12 5 39 27

2 1 1 9 7 4 9 47 9 31 22 12 94 66

100.0% 100.0% 100.0% 88.9% 85.7% 75.0% 66.7% 63.8% 55.6% 54.8% 54.5% 41.7% 41.0% 40.9%

3 3 2 47 9 1 1 0 0 0 0

8 8 7 295 57 11 11 2 4 2 9

37.5% 37.5% 28.6% 15.9% 15.8% 9.1% 9.1% 0.0% 0.0% 0.0% 0.0%

50

Table 11. limits.

Probability of exceeding different noise Probability of exceeding

Machinery Restaurant Crew cabins Pax cabins

Figure 1.

5

Machinery spaces noise levels distribution.

Figure 2.

Restaurants noise levels distribution.

Figure 3.

Cabins (crew&pax) noise levels distribution.

Higher CC

SILENV

IMO

Lower CC

– 3% – 2%

40.1% 15.8 41% 15.9%

13% – 0% –

– 84% – 60%

COMMENTS AND CONCLUSIONS

As mentioned in the introduction, there is an increasing raise in requirements for working and living conditions on board. In the present paper, this issue has been discussed with reference to the specific aspect of noise levels in the various locations on board, but other aspects may be cited, too: limits on vibrations, on noise and vibration doses, on sound insulation and on impact noise are already covered by requirements and presently in the process of being made more severe in a general trend towards more restrictive limits. The above reported data show, for a small but significant segment of ships, that the European fleet is generally compliant with the present normative framework, and needs some further efforts to comply with the new requirements that are likely to be issued in the next future. SILENV Green Label requirements are taken here as an example of possible pre-normative implementation. A long term possible development indicated in the SILENV project (SILENV, 2012b) is a re-formulation of the indicators that are used to quantify (and limit) the annoyance due to noise. This has not yet been implemented in the shipbuilding context, but has already been attempted in other industrial fields (see Badino et al., in press). Noise indicators are at the moment represented mainly by dB(A) levels, which, as known, represent a simplified quantification of the total acoustic power perceived by the human ear in the whole audible range. A step forward would be represented by the definition and the calibration of more complex indicators, involving e.g. the distribution in frequency and in time of the acoustic energy (giving in other words a vector representation of noise, as opposed to the scalar dB(A) units). This would provide a more detailed quantification of the actual perception of noise and related annoyance, and could pave the way towards a new definition, on the long term, of acoustic wellness on board and a new generation of requirements in the field.

In SILENV requirements the same limit was set for passenger and crew cabins, and this limit is lower than the original IMO limit (set for crew only). This results in a higher probability of exceeding for crew cabins than for passengers, due to the different shape of the histogram.

51

ACKNOWLEDGEMENTS

ILO 2007a. ILO Convention No. 188 concerning work in the fishing sector. ILO 2007b. ILO Recommendation No. 199 concerning work in the fishing sector. IMO 1981. Resolution A.468(XII): Code on Noise Levels on Board Ships. IMO 2009. Document DE 53/10, Proposals for the development of amendments to SOLAS regulation II-1/36 and a revision of the Code on noise levels on board ships. IMO 2011. Document DE 56/11/1, Protection against noise on board ships. SILENV 2012a. Deliverable 5.2, Noise and Vibration Label Proposal (Green Label), www.silenv.eu. SILENV 2012b. Deliverable 1.1, Review of the Existing Requirements for Noise & Vibration Control, www. silenv.eu.

This work was developed in the frame of the collaborative project SILENV—Ships oriented Innovative soLutions to rEduce Noise & Vibrations, funded by the E.U. within the Call FP7-SST-2008RTD-1 Grant Agreement SCP8-GA-2009-234182. REFERENCES Badino, A., Borelli, D., Gaggero, T., Schenone, C., Rizzuto, E. 2012. Normative framework for ship noise: Present Situation and Future Trends. Noise Control Engineering Journal, 60 (6), in press. ILO 2006. ILO MLC Maritime Labour Convention.

52

Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

An analytical method for cabin deck fundamental frequency A. Laakso, J. Romanoff & H. Remes Department of Applied Mechanics, Aalto University, Espoo, Finland

A. Niemelä STX Finland, Turku, Finland

ABSTRACT: Present paper introduces an analytical method for calculating the fundamental frequency of a cabin deck. Cabin deck structure consists of T-beam grillage and a stiffened deck plate. The fundamental mode is assumed to be either transversal or longitudinal global mode, or local deck plate mode. Shapes of the global modes are approximated by applying Newton’s laws of motion, and static beam and plate theories. The shape approximations include local deformations of the deck plate and its stiffeners. Rayleigh’s method is used to calculate corresponding eigenfrequencies of the approximated mode shapes. The lowest of the calculated frequencies provides the fundamental frequency. Present method is validated by finite elements method in case analyses for a variety of deck geometries that form boundaries of a typical design space. Sufficient accuracy is obtained for structural analyses in conceptual design. Furthermore, the paper shows that the effect of local deformations is significant in certain cases. 1

INTRODUCTION

2

Vibration analyses are highly important in passenger ship design due to increased awareness of comfort issues. It would be beneficial to begin these analyses already in the earliest design phase— the conceptual design. However, limited time and resources in the conceptual design prevents usage of detailed finite elements analyses. Method suitable for conceptual design should produce reliable and useful information in basis of limited knowledge about the structure (the main frame). Cabin decks form significant part of passenger ship’s structure, while being essential for comfort. In order to avoid harmful vibration, it is important to prevent resonances between the structure and the excitation frequencies. In case of cabin decks the resonances are usually prevented by designing the structure to be supercritical, i.e. such that its eigenfrequencies are above the excitation frequencies. Lowest eigenfrequency, the fundamental frequency, is needed for designing supercritical structures. Reliable method suitable for conceptual design doesn’t exist. Existing analytical methods are applicable only for analysis of single structural members. However, the structure vibrates as a whole including interaction between the structural members. This paper presents a method that takes the combined effect into account.

CABIN DECK STRUCTURE, NOTATIONS

Cabin decks are horizontal structures in a passenger ship, typically in its superstructure. This study considers a deck that is symmetric relative to the centerline of the ship, and supported by pillars and side structures as illustrated in Figure 1. Direction parallel to the centerline is referred as the longitudinal direction, and direction normal to that as transversal. The deck is assumed to be

Figure 1.

53

Cabin deck structure in a cruise ship.

infinitely long in the longitudinal direction so that periodic unit of the structure is sufficient to define its modal characteristics (Mead 1996). The structure consists of a continuous deck plate of thickness t and its stiffening system. In this study, the stiffening system includes high number of small longitudinal stiffeners with spacing S, and less numerous T-beams in longitudinal and transversal directions. Transversal T-beams are referred as web frames, and longitudinal T-beams as girders. 3 3.1

energy is considered for one dimensional member in harmonic motion W(x,t) and uniform mass per length m. The peak kinetic energy T is defined as: 2 ⎡m ⎤ ∂ T = max ⎢ ∫ ⎛ W ( x, t )⎞ dx ⎥ ⎝ ⎠ ⎣ 2 ∂t ⎦

(2)

The harmonic motion can be presented as a product of the maximum deflection wmax(x), and a harmonic function for the time dependency, here sin(ωt).

METHOD DEFINITION

⎡m = max ⎢ ∫W ⎣2

Frequency by Rayleigh’s method

Vibration of the structure is assumed to be free and undamped. The amplitude of vibration is assumed to be small, so that the effects of large deformations and material nonlinearity are negligible. Mass of the structure is assumed constant and uniformly distributed within each structural member. Motion is assumed to be harmonic so that the magnitude of deformation varies as harmonic function of time. All structural members involved in the vibration are assumed to vibrate in same phase. Rayleigh’s method is feasible for calculating the frequency of the assumed vibration. Total energy of the structure consists of strain energy of structural deformation, and kinetic energy of the motion. The kinetic energy vanishes at the time of maximum deformation, and strain energy at the time of zero deformation. The peak values of these energies are thus equal as the total energy of the system remains constant. The equality for a structure consisting of j members is presented in Equation 1. Peak value of the strain energy of i:th member is Ui, and kinetic energy Ti.

⇒T =

⎤ ( x )w cos(wtt )2ddx ⎥ ⎦

mω 2 wmax ( x )2 dx 2 ∫

(3)

(4)

The peak kinetic energies in rotation and for two-dimensional structures have similar form than Equation 4. Square of the vibration angular frequency ω2, is common factor for all kinetic energies of the structural members. Thus the kinetic energy of a member can be written as a product of the square of angular frequency, and the remaining kinetic energy factor as follows. Ti

w 2Ti*

(5)

The frequency of the vibration f can now be solved from the Equation 1.

ω 1 f = = 2π 2π

∑ ∑

j

Ui

i =1 j

Ti

∗

(6)

i =1

j

j

i =1

i =1

∑Ui ∑Ti

(1)

Energies of the whole structure are obtained by summing the energies of all individual members. Deck structure contains large numbers of stiffeners and deck plate segments between them. The summing of their energies is simplified by deriving their energies only for unit length of web frame having unit deflection, and then integrating the result over the web frame deflection curve. The frequency of vibration having certain maximum deformation can be solved from Equation 6. Amplitude of the undamped vibration does not affect the frequency (Weaver et al. 1990). Therefore the vibration frequency is only affected by the shape of the deformation –the mode shape. Mode shapes in certain cases of this study are functions of the vibration frequency. In such cases the final frequency needs to be solved iteratively.

Strain energies of common structural models can be found in literature. This study considers a deck structure that is combination of several structural members. Stiffeners are small in relation to their span length so that Euler-Bernoulli beam theory is sufficient. Timoshenko beam theory is used for web frames and girders. Kirchoff plate theory is used for the deck plate. The peak deformations and stiffness properties are required for the strain energy of all these models. Total kinetic energy consists of translational and rotational kinetic energies of the structural members. Velocity and mass and/or equivalent rotational properties are needed for calculation of the kinetic energy of a member. Translational kinetic

54

3.2

Mode shape estimation

3.2.1 General Freely vibrating structure is bent only by inertia. In harmonic motion, this inertia is directly proportional to the mode shape of a structural member with uniform mass distribution. Ideally, mode shapes of such structures could be solved from static load case where the shape of the load distribution equals the resulting deformation shape. In this study the load is estimated by functions that roughly estimate the possibly fundamental mode shapes. Some assumptions about the modal characteristics of the structure are needed to define such functions. Flexural waves may appear in transversal or longitudinal direction. In the fundamental mode, single half wave appears between adjacent pinned supports (Ellington and McCallion 1959). These supports are the pillars and sides in global behavior of the deck. In addition, deck plate may form local deformation waves so that the beams act as supports. It is assumed that the fundamental mode is either transversal or longitudinal global mode, or the local deck plate mode. These modes for typical deck are illustrated in Figure 2. Beam bending is the primary type of deformation in the global vibration modes. Estimates of these mode shapes are created by solving static bending differential equations for beams. 3.2.2 Transversal mode The transversal global vibration mode is first considered. Girder deformation is assumed to be small so that all web frames bend similarly. This assumption can be justified by the fact that aim of this study is to find the lowest frequency only. The transversal mode is not likely to be the lowest mode in a case with low stiffness girders relative to the web frames. The transversal mode can thus be modeled as single web frame with web frame spacing of attached stiffened deck plate. The web frame is supported in side and pillar locations by supports that are assumed to be pinned. Centerline of the ship is modeled as symmetry if it is not a pillar line or anti-symmetry if there are pillars in centerline. Lowest vibration mode of such continuous beam involves single half wave in each span of the web frame. Half sine waves are used as functions to estimate the inertia load shape. Relative heights of these waves are presented by constant factors. The mode shape functions for each span can now be solved from group of Timoshenko beam bending differential equations for all spans. The supports provide required boundary conditions. Pinned supports between spans request that the deflection is zero, and the angle due bending, and

Figure 2. The three mode shapes assumed to include the fundamental mode: A) Transversal B) Longitudinal C) Local deck plate mode.

the bending moment are continuous. The constants factors used for relative load amplitudes of spans can be solved by requesting relations between span midpoint deflections to be similar in mode shape, and in the applied load. In addition to the web frame deformation also the stiffeners and deck plate deform in the vibration mode. Web frames provide vibrating symmetry supports for the stiffeners. The inertia load bending the stiffeners has thus uniform part from the translation. In addition full 1-cos shaped wave is used to approximate the stiffener deformation. Stiffener deformation as function of the support point amplitude can be solved from bending differential equation. Stiffener deformation amplitude in relation to web frame amplitude at support point can be solved for certain vibration frequency by requesting it to be equal to the value used in the load shape in middle of the stiffener. Deformation of a deck plate segment in the mode is assumed to form full waves in both

55

directions. The magnitude of this deformation is estimated by uniform load that is sum of inertia forces by the web frame deformation and portions of stiffener and deck plate midpoint deformations. The portions of stiffener and plate deformations are chosen so that the deformation caused by the uniform load is close to deformation in the actual vibration. 3.2.3 Longitudinal mode The longitudinal global mode is created using similar principles as the transversal mode. However, more structural members are involved and the support conditions differ. In longitudinal mode web frames with pillar support are in torsion and without pillar support in flexure. The supported web frames act as anti-symmetry boundaries for stiffeners and girders, and the unsupported web frames as symmetry boundaries. The web frames without pillar support are supported by stiffeners and girders. Support reaction by stiffeners is modeled as distributed spring force. Magnitude of the stiffener support is twice the shear force of a stiffener at their web frame end divided by the stiffener spacing to get value per unit length. Girder supports are modeled as conditions that request deflections of girders and the web frame to be same at the crossings. Parameter values are given to these deflections at crossing points. One of these parameters can be chosen, but the other(s) need to be chosen iteratively so that the vibration frequency gets its smallest value. The mode shape of the unsupported web frame is created by solving static bending equation. Applied inertia load is combination of constant (translation), linear (rotation) and sine half wave (deformation) terms. In addition the stiffener support is modeled as load that is product of spring constant per unit length, and the inertia load shape (instead of final shape for simplicity). The stiffeners vibrate so that their end at the pillar supported web frame is pinned and the other end moves with symmetry constraint at the unsupported web frame. The used inertia load shape is sum of linear term of web frame motion and half wave of sine of stiffener deformation. The relative amplitude of the motion is solved by requesting equality between the mode shape and load shape in the middle. The load shape and resulting deformation shape are illustrated in Figure 3. Girder shape is created similar way as the stiffener with the exception of using Timoshenko’s beam theory. Torsional mode shape of the supported web frame is assumed to be same as the angle of stiffeners at the web frame. For small vibrations this is approximately the derivative of the stiffener mode shape. Deck plate deformation is created similarly as in the transversal mode.

Figure 3. Load shape (dashed) and the resulting mode shape estimate of a stiffener in longitudinal vibration mode.

3.2.4 Local deck plate mode The local deck plate mode is calculated based on the assumption that the number of stiffeners is high, and thus the boundary effects from girders and sides are negligible. Single stiffener spacing is considered. The mode shape is simply assumed to be half wave between stiffeners and full wave between web frames. In addition, torsion of stiffeners is included by assuming torsional angle to be same as the plate partial derivative (angle) at the stiffener location. 3.3

Stiffness properties

Stiffness properties of the structure are needed. Effective breadth by (Paik 2008) is used for defining the breadth of deck plate included in the calculation of the section properties of beams. Increased value for the elastic modulus is used for taking into account the stress state in the deck plate. In stiffener bending the transverse axial strain is assumed constrained because of the continuity of the plate. Therefore according to Hooke’s law the effective elastic modulus in stiffener direction increases. Es* =

E 1 − v2

(7)

In web frame bending, the axial strain in stiffener direction is assumed to be constrained. The effective elastic modulus of the deck plate in the web frame direction thus increases. The cross section area of the stiffener profiles AS and the stiffener spacing S affect this increase. The effective elastic modulus in web frame direction is derived from the Hooke’s law. Ew* =

56

E A⎞ ⎛ 1 − v 2 ⎜1 + s ⎟ St ⎠ ⎝

(8)

Shear correction factor by (Senjanović and Fan 1990) for wide flange I-beams is used for T-beam cross sections. Shear correction factors of validation cases vary between 0.0944 and 0.169. 4

in this case analysis as suggested by (Asmussen et al. 2001). Finite elements models are created for validation of the selected cases. Femap 10.3 software is used for creating and post processing the models, and Abaqus 6.10 for solving the models. Linear 4-noded shell elements are used for all structures except the bulb-parts of the stiffeners (beam elements). Aspect ratios of the shell elements are under 2:1. Six or more elements are used in the deck plate for stiffener spacing. The deck plate is modeled using higher density material so that the deck mass is included in the mass of the deck plate. Sample of the used mesh is presented in Figure 4. Similar mesh quality is used in all of the models.

CASE STUDY

4.1 Cases Validation cases are selected to represent a typical cruise ship deck structure, and a structure consisting of a single 7.2 m span of the deck. Web frame spacing 2.8 m is used in both cases. The deck structure has pillar lines at 4.05 m and 10.35 m from centerline. Total breadth of the deck is 35.1 m. These dimensions are typically fixed at the point when structural design is started. Variables to be chosen during the structural design are varied to form boundaries of typical design space. The design variables are beam sections, stiffener spacing, and the plate thickness. Web frames and girders have the same profiles. Two values are used for each variable; representing lower and upper limits of the design space. Totally 16 different combinations of the design variables can be made. These cases are presented in Table 1 Material used in the case study is linear elastic common structural steel with density 7880 kg/m3, elastic modulus 206 GPa, and Poisson’s ratio 0.3. In addition to structural mass, some deck mass is used to represent the mass of outfitting materials. Uniform 40 kg/m2 is used

Figure 4. Sample of FE-mesh of case 6 validation model.

Table 1. Validation cases, and comparison of the deck structure fundamental frequencies and modes by present analytical method and FEM. Dimensions in mm, and frequencies in Hz. L = Longitudinal mode, P = Local plate mode & T = Transversal mode. Case

T-Profile

Stiffener profile

Plate thickness

Stiffener spacing

FEM

Analytical

Δ [%]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

300 × 7 + 150 × 10 300 × 7 + 150 × 10 300 × 7 + 150 × 10 300 × 7 + 150 × 10 300 × 7 + 150 × 10 300 × 7 + 150 × 10 300 × 7 + 150 × 10 300 × 7 + 150 × 10 500 × 10 + 200 × 15 500 × 10 + 200 × 15 500 × 10 + 200 × 15 500 × 10 + 200 × 15 500 × 10 + 200 × 15 500 × 10 + 200 × 15 500 × 10 + 200 × 15 500 × 10 + 200 × 15

FB 80 × 6 FB 80 × 6 FB 80 × 6 FB 80 × 6 HP 160 × 8 HP 160 × 8 HP 160 × 8 HP 160 × 8 FB 80 × 6 FB 80 × 6 FB 80 × 6 FB 80 × 6 HP 160 × 8 HP 160 × 8 HP 160 × 8 HP 160 × 8

3 3 8 8 3 3 8 8 3 3 8 8 3 3 8 8

900 300 900 300 900 300 900 300 900 300 900 300 900 300 900 300

7.30 P 8.89 L 7.87 L 8.10 L 8.79 T 8.88 T 9.18 T 8.43 T 6.94 P 15.00 L 11.62 L 14.13 L 9.60 P 17.44 T 16.50 L 17.50 T

6.76 P 8.67 L 7.52 L 8.12 L 8.89 T 8.85 T 9.25 T 8.47 T 6.76 P 13.90 L 11.54 L 13.80 L 9.87 P 17.05 L 16.67 L 17.59 T

−7.99 −2.59 −4.65 0.20 1.20 −0.31 0.68 0.49 −2.70 −7.91 −0.74 −2.45 2.74 −2.27 1.02 0.49

57

4.2

structure in Table 3. Table 2 shows that the case 14 frequencies of transversal and longitudinal modes are close. It can be seen that the differences in the modal frequencies between analytical and FEM are relatively small even though the fundamental modes are different. Same can be seen in case 3 of the single span structure. Agreement between the analytical method and FEM is best in the transversal mode frequencies of the deck structure. They differ less than 2%

Results

Fundamental frequencies and modes of the deck structure cases by present analytical method and FEM are presented in Table 1. The fundamental modes are same by both methods except the case 14. The differences in fundamental frequencies of the deck structure vary between −8 and 3%. Frequencies of the transversal, longitudinal and local deck plate modes are presented for the deck structure in Table 2, and for single span Table 2.

Comparison of the modal frequencies of the deck structure by the analytical method and FEM. Transversal mode

Longitudinal mode

Local deck plate mode

Case

FEM

Analytical

Δ [%]

FEM

Analytical

Δ [%]

FEM

Analytical

Δ [%]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

– 10.01 9.05 9.11 8.79 8.88 9.18 8.43 – 18.26 – 17.40 – 17.44 18.21 17.50

8.78 9.92 9.00 9.19 8.89 8.85 9.25 8.47 11.63 17.92 14.94 17.61 12.08 17.10 18.39 17.59

– −0.89 −0.53 0.81 1.20 −0.31 0.68 0.49 – −1.91 – 1.19 – −1.99 0.96 0.49

– 8.89 7.87 8.10 – 13.47 10.98 13.54 – 15.00 11.62 14.13 – 17.71 16.50 17.82

8.28 8.67 7.52 8.12 10.74 13.85 11.29 13.89 11.00 13.90 11.54 13.80 12.38 17.05 16.67 17.81

– −2.59 −4.65 0.20 – 2.79 2.77 2.54 – −7.91 −0.74 −2.45 – −3.83 1.02 −0.07

7.30 – – – 9.18 – – – 6.94 – – – 9.60 – – –

6.76 49.84 21.08 166.51 9.87 40.00 21.53 122.10 6.76 49.84 21.08 166.51 9.87 40.00 21.53 122.10

−7.99 – – – 7.05 – – – −2.70 – – – 2.74 – – –

Table 3.

Comparison of the modal frequencies of the single span structure by the analytical method and FEM. Transversal mode

Longitudinal mode

Local deck plate mode

Case

FEM

Analytical

Δ [%]

FEM

Analytical

Δ [%]

FEM

Analytical

Δ [%]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

8.63 10.07 9.07 9.12 8.82 8.88 9.19 8.40 11.26 18.48 15.16 17.56 12.38 17.53 18.45 17.54

8.75 9.88 8.93 9.11 8.86 8.81 9.16 8.40 11.64 17.98 14.93 17.57 12.09 17.19 18.35 17.55

1.37 −2.02 −1.53 −0.20 0.51 −0.80 −0.25 0.02 3.29 −2.79 −1.53 0.09 −2.36 −2.03 −0.55 0.03

– 11.26 9.51 9.89 – 15.77 12.71 15.18 – 18.00 12.26 16.58 – 22.75 20.37 22.06

8.73 10.49 8.76 9.62 10.81 14.96 12.41 14.80 11.22 17.09 12.45 16.32 12.43 20.53 19.66 20.98

– −7.33 −8.55 −2.80 – −5.43 −2.41 −2.56 – −5.33 1.49 −1.56 – −10.81 −3.60 −5.13

7.23 – – – 9.19 – 21.94 – 6.84 – – – 9.61 – 22.60 –

6.76 49.84 21.08 166.51 9.87 40.00 21.53 122.10 6.76 49.84 21.08 166.51 9.87 40.00 21.53 122.10

−6.92 – – – 6.94 – −1.94 – −1.24 – – – 2.63 – −4.97 –

58

bending of the deck plate. The local plate deformation is also visible in the Figure 6 that presents the mode shape by analytical method and FEM. Figures 6 and 7 present mode shape comparisons between the analytical method and FEM.

between the methods. The difference is larger in the longitudinal and local modes, remaining under 8%. The differences are slightly larger in the single span structure cases as can be seen in Table 3. Transversal mode frequencies differ up to 3.3%, and longitudinal mode frequencies up to 11%. Figure 5 illustrates the effect of stiffener and deck plate deformations. It shows transversal mode frequencies of the single span structure with different local deformations enabled. Case 8 geometry is such that the web frame is small and the stiffeners and deck plate relatively stiff. In that case the local deformations are insignificant. In case 9 the web frame is stiff in comparison with the other structures. In that case the local deformations of both, stiffeners and deck plate are significant. In case 11 the stiffeners are small compared to the other members. Therefore stiffener deformation in that case is significant. The case 13 has low stiffness deck plate compared to the other structures; in that case the effect of the deck plate deformation is remarkable. In case 5 the transversal mode is the fundamental mode, and the local deformations are significant. The local deformation in the case is mostly caused by

Figure 6. Transversal mode shapes of case 5 single span structure by A) the analytical method and B) FEM.

Figure 5. Transversal mode frequencies of single span structure with different deformations enabled.

Figure 7. Longitudinal mode shapes of case 10 single span structure by A) the analytical method and B) FEM.

59

Figure 6 presents transversal mode of case 5, and Figure 7 longitudinal mode of case 10. The figures show that the analytical method can predict the mode shapes including local deformations of the plate and stiffeners. 5

Financial support by FIMECC SBI (Innovation and Networks within the scope of the Finnish Metals and Engineering Competence Centre) is acknowledged. Computational resources provided by CSC (CSC-IT Center for Science Ltd.) are also acknowledged.

CONCLUSION

REFERENCES

Method for calculating the fundamental frequency of a cabin deck structure is presented. The method considers the structure as a combination of several structural members. Estimates of assumed mode shapes are created by using static bending equations. These mode shape estimates include local deformations of stiffeners and the deck plate. Corresponding frequencies are then calculated by Rayleigh’s method. The effect of the local deformations is found to be significant in certain cases. The presented method is validated against fine mesh Finite Elements Method for variety of cases representing typical design space boundaries. The obtained accuracy in all validation cases can be considered sufficient for conceptual design purposes.

Asmussen, I., Menzel, W. & Mumm, H. 2001. GL Technology Ship Vibration. Hamburg: Germanisher Lloyd Aktiengesellschaft. Ellington, J.P. & McCallion, H. 1959. The Free Vibrations of Grillages. Journal of Applied Mechanics 26: 603–607. Mead, D. 1996. Wave propagation in continuous periodic structures: research contributions from Southampton 1964–1995. Journal of Sound and Vibration 190: 495–524. Paik, Jeom Kee. 2008. Some recent advances in the concepts of plate-effectiveness evaluation. Thin-Walled Structures 46: 1035–1046. Senjanović, I & Fan, Y. 1990. The bending and shear coefficients of thin-walled girders. Thin-Walled Structures 10: 31–57. Weaver, W., Timoshenko, S. & Young, D. 1990. Vibration Problems in Engineering. John Wiley & Sons, Inc.

ACKNOWLEDGEMENTS The present work was made in Department of Applied Mechanics in Aalto University.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

The use of seismic isolation with the method of finite element to reduce vibrations caused by the earthquake in the offshore platform H. Rostami Islamic Azad University, Science and Research Sirjan Branch, Iran

A.V. Oskouei Shahid Rajaee Teacher Training University, Lavizan, Iran

ABSTRACT: In this paper, with using Lead Rubber (LR) isolation system, the performance of the offshore platform is evaluated under earthquake excitation. LRIs are known isolators in structural control systems which dissipate the imposed energy at isolation level. For this aim, time history analysis method has been used for seismic assessment of the offshore platform in Persian Gulf location. Nonlinear finite element method is utilized for simulation of the structure in ANSYS software. Because of consideration the randomness nature of ground motions, four earthquakes (ElCentro (1940), Kocaeli (1999), Manjil (1990) and Kobe (1995)) have been studied as the imposed excitation. Engineering Demand Parameters (EDPs) such as displacement and acceleration of deck are compared in both unisolated and isolated state. The results of this study show the application of isolation systems is a practical solution for reducing the seismic and dynamic loads on the jacket. 1

INTRODUCTION

system was proposed (Yang & Ou, (2006)) which is composed of isolators (rubber bearings) and viscous dampers. The vibration isolation system is placed between the upper level of structure and lower level of deck on the offshore platform and the location is called isolation level. Between different approaches in retrofitting technics, vibration isolation is selected for this study. In this method, with using Lead Rubber (LR) isolations in isolation level, the target improvement results in the response of the platform. Various parameters of LR separators are studied for seismic control of the platform under the earthquake loads. Also, the effect of dynamic forces which are exerted by LR bearing is considered for evaluation of Engineering Demand Parameters (EDPs). In 2009, newly designed platform, SPD9 was deployed in Persian Gulf location. One of the goals of seismic isolated structures is to divert the main frequency of the offshore structure from the dominant frequency earthquake. Another purpose of this separator system is to provide additional means for damping imposed energy and thus reduced momentum transferred to deck. The main point in this method is to separate the structure from the ground (usually horizontal) in order to reduce the transmission of earthquake excitations. Also, for reduction of possible damages of offshore platforms in harsh marine environments, further studies must be carried out.

In recent decades, special attention for achievement more production from offshore oil and gas resources carried out specially in the Persian Gulf and the Caspian Sea locations. Due to seismic activity of faults, especially in Persian Gulf region, seismic assessment and retrofitting of offshore structure is a valuable subject. There are various vibration mitigating techniques which have been used successfully in offshore engineering to reduce dynamic response which imposed by wind and earthquake (Housner et al. 1997) or wind, wave and current Lee (1997), Vandiver & Mitome (1979), Vincenzo & Roger (1999). Making use of the jacket structure cases in the Bohai Sea, lots of research attempts have been made on the ice induced vibration mitigations. Terro et al. investigated the effciency of some active and passive control mechanisms to moderate the dynamic response of a steel jacket platform due to wave induced forces (Terro, et al. 1999). The results showed that due to the limitations on the placement positions of the dampers and relatively low vibrating displacement of the platforms, the damping ratios added by the dampers were relatively small, and the effect of dampers to reduce ice induced vibration is not remarkable. In order to solve these problems, the vibration isolation

61

The contents of this article mainly include the consideration of the influence of LR isolation system on vibration control of the offshore platform under earthquake excitations. Some significant conclusions are mentioned at the end of the paper. 2

THEORETICAL BASIS OF VIBRATION ISOLATION

Vibratory systems are composed of a means for storing potential energy (spring) and a means for storing kinetic energy (mas or inertia) and a means by which the energy is lost (damper). The vibration of a simple vibratory system involves the alternating transfer of energy between potential and kinetic forms. In a damped system, some energy is dissipated at each cycle of vibration and must be replaced from an external source if a steady state vibration is to be maintained. An ordinary method of exhibition the elements of a simple vibratory system is to study that the system is composed of a spring and a damper which are connected to a rigid infinite body and a mass which is suspended by that spring and damper (Ding (2001)), as shown in Figure 1. Vibration isolation system can be categorized as two cases: base isolated system and mid isolated system which shown in Figure 2. By simulation the offshore structure and deck with simplified two degree of freedom system (fig. 3), the motion equation of vibration isolated structure can be written as M s xɺɺs + Cs xs + K s xs = Fg (t ) + FI (t )

(1)

Fg (t ) = − M s xɺɺg

(2)

FI (t ) = CI xd + K I xd

(3)

Figure 2. Element of a vibratory system simplified isolation model.

Figure 3.

where Ms, Cs and Ks are equivalent mass, damping coefficient and stiffness of the simulated structure, respectively. xs, xɺ s and xɺɺs denote the relative displacement, velocity and acceleration regard to

Element of a vibratory system.

earth motion, respectively. xɺɺg is defined as exerted earthquake acceleration. Motion equation of deck can be define as follows: M d ( xs

xd ) CI xɺ d

K I xd

M d xɺɺg

(4)

where Md is mass of the deck and xd, xɺ d and xɺɺd denote the relative displacement, velocity and acceleration regard to earth motion, respectively. 3

Figure 1.

STEEL RUBBER BEARING OF JACKET OFFSHORE PLATFORM

Recently, the use of steel rubber bearing for the vibration isolation of offshore platform has

Element of a vibratory system.

62

Table 1.

Figure 4. Geometric layouts of steel rubber bearing used for vibration isolation of offshore jacket platform (Long, et al. 2006).

Figure 5.

Geometric properties of SPD9.

Parameters

Value/information

Topside mass Jacket structure mass Water depth Facilities Gas production

2545 ton 750 ton 64 (m) Drilling SubSea pipeline Production Utilities 1000 MMscfd 4

Number of columns

Detail of the isolation system.

become a common and well recognized method of providing protection against ice induced, wave force and seismic damage. The geometric layouts of steel rubberbearing used for vibration isolation of offshore jacket platform are shown in Figure 3 (Ou, et al. 2002). Isolation systems have been successfully used for reducing the dynamic response of structures subjected to lateral loads such as wind and seismic forces. The main purpose of using seismic isolator and energy dissipation devices is relatively low cost. Rubber bearing is regarded as one of the most popular isolators used in structural control. Isolation system consists of three isolators as shown in Figure 5. Stiffness behavior in nonlinear state and its value are important parameters for designing an LR isolation system. These characters can be determined according to the vibration control objectives.

Figure 6.

SPD9 platform.

(3) Main deck, (4) weather deck. It consists of two main components: substructure and superstructure. The elevation view of SPD9 platform in the longitudinal (X) direction is shown in Figure 7. The superstructure is located on the deck which is rigidly connected to the main structure. The Superstructure is composed of the living module and the utility module (Figure 7). The substructure (main structure) is a steel tubular jacket structure. Under sea level, jacket has four level units at EL. −62 m, EL. −45 m, EL. −28 m, EL. −11 m, respectively. There are diagonal brace members in both vertical and horizontal planes in the units to enhance the structural stiffness. Above sea level, the jacket also has one level at EL. +6 m.

4 OVERVIEW OF THE JACKET PLATFORM The SPD9 platform is located in the Persian Gulf which is the world’s third largest Gulf. The peak acceleration of the design earthquake excitation for the offshore structure is 0.27 g. Table 1 provides some geometric properties data for SPD9 platform. The SPD9 platform is a four leg jacket structure, as illustrated in Figure 6. Each deck has four floors: (1) Caller Deck, (2) Cellar mezzanine deck,

63

4.1

FE models and input parameters

The Geometrical, FE and material properties of SPD9 platform for the model is given in Table 2. FE model is constructed in ANSYS program. The material nonlinearity of tubular members is assumed elastic-perfectly plastic. The earthquake forces exerted as base acceleration excitation. The FE model of the platform is presented in Figure 8.

Figure 7.

SPD9 platform.

Table 2. Geometrical, FE and material properties of SPD9 platform. Figure 9.

Geometrical properties Water depth Jacket height Top level dimensions Bottom level dimensions Total no. of jacket legs Pile dimensions

64.0 m 76.0 22 × 12 28 × 27 4 1650 × 20

FE properties Elements Nodes Analysis Solution method

895 3784 Nonlinear Full newton Raphson

Material properties Young modulus (MN/m2) Poisson ratio Density (kg/m3) Yield stress (MN/m2)

2 × 105 0.3 7850 240

Figure 8.

FE model of SPD9 platform.

64

Push over analysis.

Figure 10.

Rigid deck on isolation system.

Figure 11.

Rigid deck on isolation system.

4.2

As mentioned above, the yield force of LR isolators can be defined as follow:

Determination of LR isolation parameters based on offshore structural characteristics

Large interstory drift of the isolation level is not allowed for the jacket platform to satisfy the drilling and production requirements. The yield force and stiffness of LR isolators are major effective parameters to determine the interstory drift of the isolation level. In this study, these parameters are defined such that the ratio of total stiffness and yield of the structure to LR isolator is 0.1. Total stiffness of the structure is determined by the period of the first mode in the modal analysis. By using this method for SPD9 platform, the period of the first mode is determined as 1.026 second. Total stiffness can be considered as follows: 2

⎛ 2π ⎞ Ks = ⎜ × (M s ⎝ Tdom ⎟⎠ KI =

M d ) = 42.0 ( MN m )

(

)

y s

n

= 0.

(MN )

(7)

The members of a fix offshore platform, depends on changing of nonlinear shapes, are divided in two categories: braced and frames. The behavior of braced members is affected by compressive load, low resistance and stiffness. Although, the behavior of frame members in nonlinear domain is carried with yielding. The deck is modeled as a rigid beam and it weight is applied on legs. Also, we exclude the rotary inertia effects. Based on Table 1, we consider scalar mass as well (Fig. 10). Ground motion is the movement of the earth’s surface from earthquakes or explosions. Ground motion is produced by waves that are generated by sudden slip on a fault or sudden pressure at the explosive source and travel through the earth and along its surface. Among different methods for simulation of ground motion effects, time history method has valuable advantage in assessment of nonlinear and complicated models. Time history analysis of the seismic responses of the platform is performed using the ElCentro (1940), Manjil (1990), Kobe (1995), Kocaeli (Izmit 1999) earthquake ground motion records (Fig. 11).

(6)

In which, Tdom is the fundamental period of the offshore platform, α is ratio of structure to LR isolator, n is the number of LR isolator in isolation level. In the current study, α and n are assumed equal 0.1 and 3, respectively. Total yield force of the structure can be defined by performing pushover study on unisolated state. The result of pushover analysis for deck displacement is presented in Figure 9. Table 3.

α ×(

4.3 Simplified model of the deck in FE modeling of the offshore platform

(5)

α × Ks = 1.40 ( MmN ) n

) =

y I

Summary of the computational results under the four earthquake excitations.

Earthquake

PGA (cm/s2)

State

Max deck displacement (m)

Max deck acceleration (m/s2)

ElCentro

412

Manjil

541

Kobe

598

Kocaeli

824

Unisolated Isolated Unisolated Isolated Unisolated Isolated Unisolated Isolated

21.2 8.82 15.36 8.3 37.1 11.2 44.1 15.3

2133.2 858.6 2983.7 784.2 2515.6 1232.2 4183.6 1726.6

Figure 12. Computational time history of the deck acceleration under Manjil earthquake excitation.

Figure 13. Computational time history of the deck displacement under Manjil earthquake excitation.

65

The numerical simulations are carried out for the unisolated and isolated structure under the mentioned excitations. 5

REFERENCES Abdel Rohman, M. Structural control of a steel jacket platform, Structural Engineering and Mechanics 4 (2) (1996) 125–138. Ding, J.H., 2001. Theoretical and experimental study on structural vibration repressed system using viscous fluid dampers. Ph.D. dissertation. Harbin Institute of Technology [in Chinese]. Housner, G.W., Bergman, L.A., Caughey, T.K., Chassiakos, A.G., Claus, R.O., Masri, S.F., 1997. Structural control: past, present, and future. Journal of Engineering Mechanics, ASCE Vol. 123, 897–908. Jinping Ou, Xu Long, Q.S. Li, Y.Q. Xiao., 2006. Vibration control of steel jacket offshore platform structures with damping isolation systems. Engineering Structures. Lee, H.H., 1997. Stochastic analysis for offshore structures with added mechanical dampers. Ocean Engineering Vol. 24 (9), 817–834. Meirovitch L. Elements of vibration analysis. McGrawHill, Inc.; 1975. Ou, J.P., Long, X., Xiao, Y.Q., Wu, B., 2002. Damping isolation system and its vibration suppressed effectiveness analysis for offshore platform jacket structures. Earthquake engineering and engineering vibration Vol. 22 (3), 115–122. Terro, M. J., Mahmoud, M. S. and Abdel Rohman, M., “MultiLoop Feedback Control of Offshore Steel Jacket Platforms”, Computers and Structures, Vol. 70, No. 2, (1999), 185202. Vandiver, J.K., Mitome, S., 1979. Effect of liquid storage tanks on the dynamics response of offshore platform. Applied Ocean Research Vol. 1, 67–74. Vincenzo, G., Roger, G., 1999. Adaptive control of flowinduced oscillation including vortex effects. International Journal of NonLinear Mechanics Vol. 34, 853–868. Yang, Y., Ou, J.P., 2006. Experimental research on isolation structure model of jacket offshore platform with MR damper. Journal of vibration and shock Vol. 25 (5), 1–6.

RESULTS AND DISCUSSIONS

The computational results are presented in Table 3. Figures 12 and 13 show the comparison of the time history of the deck acceleration and the time history of the jacket cap relative displacement among the results for the cases of the unisolated and isolated models under the Manjil earthquake excitation. 6

CONCLUSION

A finite element model was developed for simulation of LR isolation system. The response of offshore jacket platforms installed with passive control devices (i.e. LR isolators) under earthquake loading was studied. For this aim, an offshore platform (SPD9) which is located in seismic region (Persian Gulf) was selected. A nonlinear finite element model was applied for seismic assessment of isolated and unisolated offshore platform. Also, parameters of LR isolator were defined based on total behavior of unisolated state. The computational results for SPD9 platform demonstrated that the isolation strategy was a very effective way to reduce the seismic responses of the jacket offshore platform. For the ElCentro, Manjil, Kobe and Kocaeli earthquake excitations, both deck acceleration and displacement were considered. The results showed about 73% and 20% in average for Maximum displacement and acceleration, respectively.

66

Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

On the non-linear hydroelastic response in irregular head waves of a structural optimized container ship I. Rubanenco, I. Mirciu & L. Domnisoru University “Dunarea de Jos” of Galati, Naval Architecture Faculty, Galati, Romania

ABSTRACT: This paper focuses on the linear and non-linear hydroelastic structural response analysis of a container ship induced by irregular head waves, model Longuet-Higgins. The numerical analysis are carried out with own program codes package DYN, based on hydroelasticity theory, improved in the frame of the Marstruct Project. The motion equations system solution and the structural wave induced loads are based on linear frequency domain procedures and also non-linear time domain implicit integration procedures. The dynamic structural response includes: the linear and non-linear oscillations, taking into account the bottom and side slamming phenomena, and the global vibrations on the first and higher natural modes, taking into account the springing and whipping phenomena. The numerical analysis is developed on an 1100 TEU container ship, with overall length of 173.42 m, under full loading condition. The ship structure in the cargo hold domains is considered on two cases: with initial design structure and the optimized structure, with minimum weight objective function and strengths multi-criteria restrictions. The numerical results on short-term prediction outlined the extreme hydroelastic wave loads induced in the elastic container ship hull, also making possible to compare the initial and the optimized ship structure hydroelastic response. 1

INTRODUCTION

Price & Temarel 2003, Perunovic & Jensen 2005, Park & Temarel 2007).

The standard seakeeping theory (Bhattacharyya 1978, Bertram 2000) includes only the ship rigid hull oscillations, wave induced dynamic response. For advanced design of an elastic ship hull, the dynamic loads and the motions of the ship in waves have to be obtained based on the hydroelasticity theory, including the low frequency (oscillations) and high frequency (vibrations) components (Bishop & Price 1979, Guedes Soares 1999, Ozsoysal 2004, Hirdaridis & Chunhua 2005, Domnisoru 2006). The ship hull design process includes several steps of structural optimization, having the minimum weight objective function with strengths multi-criteria restrictions (Hughes 1988, GL 2011), and even reference exploitation life changes. The differences between the initial and the optimized ship structure hydroelastic responses have to be analyzed at the global ship strengths assessment. This study is focused on the wave induced dynamic response analysis for an 1100 TEU container ship (GL 2011), at full cargo loading condition, with two amidships versions: the initial design structure (case 1) and the optimized ship structure (case 2). The dynamic analysis is based on linear and non-linear hydroelasticity theory (Bishop & Price 1979, Domnisoru 1998, Hirdarides,

2

THEORETICAL BACKGROUND

The hypotheses of the linear and non-linear hydroelastic approach are the following: – The ship hull model is based on the finite elements method, with 1D-FEM elastic beam finite elements, Timoshenko model (Bishop & Price 1979); – The hydrodynamic forces are modelled according to the strip theory (2D) and hydroelasticity theory, using non-linear terms and the slamming components (Hirdaris, Price & Temarel 2003, Fonseca & Guedes Soares 2005, Park & Temarel 2007); – The hydrodynamic coefficients are calculated for the instantaneous ship-water free surface position, at the non-linear ship motion equations, based on a potential fluid flow method (Bishop & Price 1979); – Based on modal analysis technique, the ship dynamic response is obtained with low and high frequency components (Bishop & Price 1979); – The ship speed is considered constant; – The first order wave spectrum density is ITTC type (Price & Bishop, 1974) and the wave model Longuet-Higgins (L-H) is used to obtain

67

where: mz, Nz are the hydrodynamic mass and damping coefficients; FFK is the Froude-Krilov force; Fimp is the impact slamming force; us is the ship speed. The relative ship-wave displacement is as following:

the second order interference components (Perunovic & Jensen 2005, Domnisoru 1998), at head wave condition. Based on the above hypotheses, the linear and non-linear ship hydroelastic theoretical model is presented in synthesis (Domnisoru, 1998, 2006). Based on the modal analysis technique, the ship girder dynamic response is decomposed on heave (r = 0) & pitch (r = 1) oscillation modes and the first natural global vertical vibration modes (r = 2,n). w ( x,tt )

n

∑ w ( x ) p (t ) r

r

r=0

x ∈[ , L ]

zr ( x,t x t) zr ( x,t x t)

(1)

(2)

zro ro ( x t ) + wnl ( x t ) zro ( x, t ) = wo ( x,t x t ) w ( x, t )

FFK ( x,tt )

L

Fh s (t ) = ∫ Fh ( x t ) ws ( x ) dx d ; s = 0, n ⇒ {Fh (t )} o

∑ ⎡⎣η (x,ω )

(ω e )

c w

e

Fimmp ( x t )

ω et ηws ( x ω e ) i ω et ⎤⎦

ggbo ( x ) wnl ( x,t ) (6)

K imp

zr

⎡ Dzr ( x,t ) ⎤ ⎥ ⎣ Dt ⎦

2

( x, t ) ⎢

(7)

where Kimp|zr(x,t) is the impact slamming pressure factor, taking into account the bottom ship geometry. The hydrodynamic coefficients, at the instantaneous ship-water position, have the following expressions:

(3) where: ωe is the encountering ship-wave equivalent circular frequency of a wave component; ηwc,s(x,ωe) are the ωe wave frequency domain components. Based on the generalized strip theory of Gerristma & Beukelman (Bishop & Price 1979), taking into account the non-linear components, the hydrodynamic force has the following expression: Dzr ( x,t ) ⎤ D⎡ Fh ( x t ) = − ⎢ mz ( x,t ) ⎥ Dt ⎣ D Dt ⎦ Dz ( x,t ) − N z ( x, t ) r + FFK ( x,,t ) Dt + Fimmp ( x,t ) ; D Dt = ∂ ∂ t us ∂ ∂ x

ρ gb gbo ( x ) zrroo ( x,t ) + ρ ggA Anl zr ( x,t )

where: ρ ,g are the water density and the gravity acceleration; bo(x) is the still water plane breadth; Anl|zr(x,t) is the non-linear correction for the instantaneous wet transversal section surface. The impact slamming force, according to Ochi & Kawakami (Bishop & Price 1979), is as following:

where: [a ], [ b ], [c ] are the generalized structural inertial, damping and rigidity y matrix; { p (t )} is the principal coordinate’s vector; {Fh (t )} is the generalized hydrodynamic force vector. The Longuet-Higgins (L-H) time domain wave elevation, with random component phases (Perunovic & Jensen 2005), has the following expression:

ηw ( t )

(5)

where: wo(x,t) is the linear component, steady state, at the vertical side hypothesis, without slamming occurrence, at Longuet-Higgins wave (Equation 3); wnl(x,t) is the non-linear component that includes: the nonlinear correction of the steady state dynamic response and the transient whipping dynamic response, in the case that bottom and/or side slamming occurs. The hydrostatic component Froude-Krilov, with geometric nonlinearities, has the following expression:

Where: pr(t), r = 0,n are the principal modal coordinates; wr(x), r = 0,n are the modal form functions of the displacement field; x is the space coordinate in the ship system; t is the time coordinate; L is the ship length. The motion equations system in the ship vertical plane, based on the principal modal coordinate’s formulation (Equation 1), has the following expression:

[a ]{ p (t )} [b]{ p (t )} + [c ]{ p (t )} {Fh (t )}

w ( x t ) − ηw ( x,tt ) ; w ( x,t ) = w0 ( x,tt ) wnl ( x,t )

mz ( x,t x t)

m0 ( x ) + mnnll

= N0 ( x ) + N nl

( x, t ) ; N z ( x, t ) z ( x, t )

zr

r

(8)

where mo(x), No(x) are calculated for the still water position. From Equations 4–8 the hydrodynamic force becomes: Fh ( x t )

(4)

68

Fh 0 ( x,t ) + Fh 0 ( x t ) Fh1 ( x,t )

(9)

where for each wave component ωe the motion equations system (Equation 12) is solved in the frequency domain, based on the following equivalent system:

D ⎡ Dz ⎤ Dz m0 ( x ) ro ⎥ − N 0 ( x ) ro − ρ ggb0 ( x ) zro Dt ⎢⎣ D Dt ⎦ Dt D D ⎡ Dwnl ⎤ Dwnl Fh 01 ( x,, t ) = − m0 ( x ) − N0 (x ) − ρ ggbb0 ( x ) wnl Dt ⎢⎣ Dt ⎥⎦ D Dt D ⎡ Dz ⎤ Dz Fh1 ( x, t ) = − mnl zr ( ) r ⎥ − N nl zr ( ) r ⎢ Dt ⎣ Dt ⎦ Dt 2 ⎡ Dzr ⎤ + ρ gAnl zr ( ) + K im x, t ) ⎢ imp zr ( ⎣ Dt ⎥⎦ Fh 0 ( x, t ) = −

{

The same as Equation 5, the principal modal coordinate’s vector is decomposed as following:

{ p (t)} = { po (t)} + { pnl (t)}

(10)

where: Fho s (t ) = − H s (t ) + Fws (t ) ; s

H (x t)

D 2wo + ⎡ No ( Dt 2 ⎣ + ρ ggbbo ( ) wo mo ( x )

Fh

L

w

{

s

0

) − us mo′ ( )⎤⎦

t

nl

nl

}

= Fh (t { pnl } { pnl } , { pnl })

Dwo Dt

nl

[A] [a ] + [ Ah ] |ωω [B ] = [b] [ Bh ] |ωω [C ] [c ] + [Ch ] ωω 2

et

e

Bhsr pnlr (t ) + Chsr pnlr (t )}

The non-linear component equations system becomes:

0, n

{

− Fws (

e

)} i

{ p ( )} s o

e

e

t

; (16)

where [Ah], [Bh], [Ch] are calculated for rigid modes at ship heave circular frequency ωosc and for elastic modes at ship first order natural vibration circular frequency ω2. Because {Fh1 (t )} is function to the non-linear component response { pnl (t )}, it is necessary to apply a step by step procedure for the non-linear system (Equation 16). At each step, the ß-Newmark procedure for direct time domain integration is applied (Domnisoru 2006), with simulation time t [ Ts ] Ts = 80 s and time step δ t = 0.01 s corresponding to 100 Hz triggering frequency.

⎤

et ⎦

]

o osc 2

o osc 2

(12)

e

ω e ⎡⎣ B (ω )⎤⎦

s = 0, n

where: [Ah], [Bh], [Ch] are the generalized inertial, damping and elasticity hydrodynamic matrix, for each ωe wave component; {Fwc,s(ωe)} are the ωe wave excitation generalized force frequency domain components. The linear dynamic response in principal modal coordinates has the following expression: ( e)

(t ) = − ∑ {Ahsr pnlr (t )

(11)

[A] [a ] + [ Ah ] [B ] = [b] [ Bh ]; [C ] [c ] + [Ch ]

o

s

r=0

[A]{ po } + [B ]{ po } + [C ]{ po } = {Fw (t )}

{ p ( t )} = ∑ [ { p ( ) }

ω e2 ⎣⎡ A (ω )⎤⎦ + ⎡⎣C (ω )⎤⎦ ;δ 2

(15)

The linear component equations system becomes:

c o

δ1

n

D 2 ηw Dηw + ⎣⎡ No ( x ) − us mo′ ( x ) ⎦⎤ Dt 2 Dt + ρ ggbo ( x ) ηw

( e)

⎡δ 1 ⎢δ ⎣ 2

[a ]{ pnl } + [b]{ pnl } + [c ]{ pnl } = {Fh (t )} + {Fh (t )}

Fw ( x,t ) = mo ( x )

{Fw (t)} = ∑ ⎡⎣{Fwc ( e )}

⎧{ pc }⎫ ⎧{F c }⎫ −δ 2 ⎤ ; {X } = ⎨ os ⎬ ; {F } = ⎨ ws ⎬ (14) ⎥ δ1 ⎦ ⎩{ po }⎭ ⎩{Fw }⎭

[D ]

∫ F ( x t ) w ( x ) dx [A]{ p (t )} [B ]{ p (t )}ɺɺɺ+ [C ]{ p (t )}

L

H s (t ) = ∫ H ( x t ) ws ( x ) dx d Fws (t ) 0

}

From Equations 2, 9 & 10 the equations system for non-linear component has the following expression:

where: {po(t)} is the linear component, steady state dynamic response; {pnl(t)} is the non-linear component, including the nonlinear correction of the steady state dynamic response and the transient dynamic response. From Equations 2,3,9 & 10, the motion equations system for the linear component has the following expression:

[a ]{ po (t )} [b]{ po (t )} + [c ]{ po (t )} {Fho (t )}

} {

⎡⎣ D (ω e )⎤⎦ X (ω e ) = F (ω e )

Step 0:

{ p (t)}( ) = 0 ⇒ {F nl

h

(t { p } nl

{ pnl } { pnl }

)}

()

Step 1: With ß-Newmark procedure it is solved:

[A]{ pnl }( ) + [B ]{ pnl }( ) + [C ]{ pnl }( ) () () = {Fh (t )} ⇒ { pnl (t )}

(13)

69

(k )

Step k: { pnl (t )}

Table 1. The main dimensions of the 1100 TEU container ship.

(17)

Step k+1: With ß-Newmark procedure it is solved: ( k+ )

[A]{ pnl (t )}

{ (

k 1) ( k+

[B ]{ pnl (t )} k

k ) ( k+

+ [C ]{ pnl (t )}

= Fh1 t,, { pnl } , { pnl } , { pnl } k

LOA [m] LBP [m] BMLD [m] DMLD [m]

k

)}

( k+ )

173.42 164.00 27.30 14.60

dMLD [m] cB TEU Δ [t]

8.50 0.758 1100 29673.2

( k+ )

⇒ { pnl (t )}

Until the convergence test is accomplished: max r ,t pnl(k r+ ) (t ) − pnl(k )r (t ) max r ,t pnl(k )r (t ) t

[

≤ ε = 0.001; r = 0, n;

,Ts ]

The total time domain dynamic response, on each mode (oscillations and vibrations), has the expression: pr (t ) = po r (t ) + pnl r (t ) r

0, n t

[

Ts ]

(18) Figure 1. The 1100 TEU container ship offset section lines.

and based on the modal analysis technique the total time domain displacements and deformations (Equation 1), shear forces, bending moments and stresses dynamic hydroelastic responses are obtained. The theoretical model is numerically implemented with the own program code DYN (Domnisoru 1998), experimental validated (Domnisoru 2000), with pre and post-processing facilities, having the linear solver STABY and the non-linear solver TRANZY. Based on the linear solver, the steady state ship hull dynamic response is obtained, including the linear oscillations and springing phenomenon components. Based on the non-linear solver, the non-linear and transitory ship dynamic response is obtained, including nonlinear oscillations and springing, bottom and side (flare) slamming, whipping components. In order to obtain the short-term prediction statistical parameters, the spectral analysis is applied to the total ship dynamic response, based on the FFT Fast Fourier Transformation Method (Domnisoru 2006). 3

Table 2.

The container amidships mass [t] changes.

Items

Structural case 1

Structural case 2

Changes [%]

Plates Stiffeners Frames

121.403 40.644 35.256

105.529 40.644 36.761

−13.07 0 +4.27

Total steel

197.303

182.934

−8.80

R = 25 years and also a supplementary added corrosion thickness, according to the initial shipping company requirements. The structural case 2 is obtained based on a optimization procedure (LBR 2007) of the structural case 1, including first a reduction of the reference exploitation life to R = 20 years and also only a standard added corrosion thickness (GL 2011). The optimization procedure is applied on the amidships sections, having the minimum mass objective function and as design variables: stiffeners spacing, plate thicknesses and stiffeners dimensions. Each design variable has a lower and upper limit and a set of corresponding constraints, which can be local (yield stress limit, buckling criteria), global strengths criteria (bending moments sagging and hogging), ultimate bending moment criteria, geometric (predefined profiles or thickness values). Table 2 presents the values for the mass of longitudinal plates, web frames and stiffener profiles, for structural cases 1 and 2, resulting that the

THE 1100 TEU CONTAINER SHIP MODEL

This numerical analysis is focused on an 1100 TEU container ship, with the main dimensions and model characteristics from Table 1. The ship offset section lines are presented in Figure 1. The structural case 1 represents the preliminary hull structure design of the container ship (GL 2011), taking as reference the exploitation life of

70

Table 3. The 1100 TEU container amidships section characteristics. Structural case 1 Iy [m4] A [m2] Afz [m2] Kτn-n [m−2] WDRL[m3] WB [m3] WD[m3] WDB[m3]

Table 5.

Full cargo case Δ [t]

Structural case 2 100.188 2.893 1.515 0.582 10.530 15.447 12.347 20.095

Iy [m4] A [m2] Afz [m2] Kτn-n [m−2] WDRL[m3] WB [m3] WD [m3] WDB [m3]

The 1100 TEU container ship analyzed cases.

91.928 2.616 1.391 0.574 9.883 13.723 11.634 17.683

dm [m] daft [m] dfore [m] us [Knots]

Structural 29673.20 8.502 8.502 case 1 Structural 29473.15 8.454 8.454 case 2

8.502

18

8.454

18

Table 4. The material isotropic steel characteristics (GL 2011). E [N/m2] G [N/m2]

2.1 1011 8.1 1010

ρm[t/m3] Structural damping

ReH-A [N/mm2] σadm-A [N/mm2] τadm-A [N/mm2]

235 175 110

ReH-AH32 [N/mm2] σadm-AH32 [N/mm2] τadm-AH32 [N/mm2]

Table 6.

7.7 0.001 315 224 141

Figure 2. The 1100 TEU container ship mass diagram (case 2).

The 1100 TEU container ship natural modes frequencies.

Vertical modes f [Hz]

Oscillations

Full cargo case

0

1

2

3

4

– 0.1256

– 0.1379

1.771 1.253

3.743 2.732

5.924 4.457

– 0.1259

– 0.1381

1.703 1.205

3.602 2.627

5.694 4.282

Structural case 1 Dry Wet Structural case 2 Dry Wet

Vibrations

Table 5 presents the two analyzed cases, where Δ is the displacement; dm,daft,dfore are the ship still water equilibrium parameters (draughts medium, aft peak and fore peak); us is the ship design speed. Figure 2 presents the 1100 TEU container ship mass diagram, full cargo load, structural case 2. The structural characteristics are considered with a trapeze distribution over the ship length, based on the design data for the amidships cargo holds part. Table 6 presents the natural hull oscillation and vibration mode frequencies, dry hull and with hydrodynamic masses (wet hull), based on the 1D-FEM model of the 1100 TEU container ship.

amidships steel structure mass has been reduced by 8.80%. Table 3 presents the amidships structure characteristics: the moment of inertia Iy [m4], the total transversal section area A[m2], the equivalent shearing area Afz [m2], the coefficient of shearing tangential stress in the neutral axis Kτn-n [m−2], main deck (D), hatchway girder upper flange (DRL), bottom (B) and Double Bottom (DB) bending modules WD, WDRL, WB, WDB [m3]. WZ =

I yy eZ

;

Kτ

n n

=

0.5 ⋅ Syn − n I yy tn

; tn

n

tS + tDS (19)

n

where: Syn-n is the transversal section static moment at the neutral axis; tS, tDS, are the thickness of the side and double side;; eZ is the distances of the neutral axis to the reference horizontal shells. Based on the Germanischer Lloyd’s Rules (GL 2011), Table 4 presents the material isotropic steel characteristics and the admissible stresses.

4

THE 1100 TEU CONTAINER SHIP HYDROELASTIC RESPONSE ANALYSIS

The numerical analyses are focused on the dynamic loads, induced into the ship girder from external waves, under the hydroelasticity hypotheses

71

(Section 2), for the structural cases of the 1100 TEU container ship (Table 5). This study represents a short-term prediction of the ship dynamic response, at extreme head waves, having as output the significant response statistical values. The dynamic analyses are carried on for the head wave first order spectra ITTC (Price & Bishop, 1974) with the significant wave height h1/3 = 0–12 m, step δh1/3 = 0.5 m, according to the Beaufort scale Blevel = 0–11, using the own programs package DYN (Domnisoru 1998, 2006). The 1100 TEU container ship 1D-FEM model is presented in Section 3, for the two structural cases. The numerical results for the hydroelastic response analysis are synthesized in Tables 7–9 and Figures 3–9, for both structural cases. In Figures 3–9 are included the following numerical results by hydroelastic analysis:

moment, head wave h1/3 = 9.326 m, at amidships section x/L = 0.5, structural case 2 (optimized); – Figures 6a,b,c & 7a,b,c present the maximum significant normal stresses for the upper and lower ship girder panels: hatchway girder upper flange and bottom, the neutral axis tangential significant stress, for both structural cases; – Figures 8a,b & 9a,b present the maximum significant non-linear vertical displacements (oscillations and vibrations) over the ship length, for both structural cases. In Tables 7–9 are included the following numerical results by hydroelastic analysis: – the ratio for the significant deformation on the fundamental natural vibration mode and the significant vertical displacement of the ship rigid hull oscillations %w1/3vib/w1/3osc (Tables 7a,b), for both structural cases; – the maximum ratios for the significant bending moments and shearing forces, on fundamental natural vibration mode and the ship rigid hull oscillations, max(%M1/3vib/M1/3osc,%T1/3vib/T1/3osc) (Tables 8a,b), for both structural cases; – the maximum significant normal main deck, hatchway girder upper flange, double bottom and bottom shells, tangential neutral axis stresses (according Equation 20), added to still water (σsw,τsw) stresses (Tables 8a,b), for both structural cases.

– Figures 3a,b present the time record and the amplitude spectrum for the Longuet-Higgins (L-H) wave, with first order wave spectrum ITTC head wave h1/3 = 9.326 m, at amidships section x/L = 0.5; – Figures 4a,b present the time record and the amplitude spectrum FFT, for the linear bending moment, head wave h1/3 = 9.326 m, at amidships section x/L = 0.5, structural case 2 (optimized); – Figures 5a,b present the time record and the amplitude spectrum FFT, for the non-linear bending

Table 7a. The ratios between the significant displacements (oscillations) and deformations (vibrations) %w1/3vib/w1/3osc, structural case 1 (initial design), for the reference wave significant height h1/3 = 9.326 m. Section

x/L

%vib/osc linear

%vib/osc non-linear

Bottom slamming

Side slamming

Aft L/4 L/2 3L/4 Fore

0.05 0.25 0.50 0.75 0.95

3.50 3.59 2.68 2.29 2.37

3.59 3.60 2.79 2.36 2.52

>2.5 – – – >9.5

Yes – – – Yes

Average

2.89

2.97

Slamming occurrence

Table 7b. The ratios between the significant displacements and deformations %w1/3vib/ w1/3osc, case 2 optimized (h1/3 = 9.326 m). Section

x/L

%vib/osc linear

%vib/osc non-linear

Bottom slamming

Side slamming

Aft L/4 L/2 3L/4 Fore

0.05 0.25 0.50 0.75 0.95

3.70 3.30 3.07 3.37 3.59

3.84 3.35 3.17 3.33 3.65

>2.5 – – – >9.5

Yes – – – Yes

Average

3.41

3.47

Slamming occurrence

72

Table 8a. The maximum ratios for the significant bending moments & shearing forces max (%M1/3vib/M1/3osc,%T1/3vib/T1/3osc), case 1 initial design (reference h1/3 = 9.326 m). Section

x/L

%vib/osc linear

%vib/osc non-linear

Aft L/4 L/2 3L/4 Fore

0.05 0.25 0.50 0.75 0.95

4.79 5.07 4.92 3.98 3.11

31.66 42.23 46.91 40.33 29.24

Average

3.39

38.07

Springing

Whipping

Linear: Very reduced

High

Non-linear: Small

Table 8b. The maximum ratios for the significant bending moments & shearing forces max (%M1/3vib/M1/3osc,%T1/3vib/T1/3osc), case 2 optimized (reference h1/3 = 9.326 m). Section

x/L

%vib/osc linear

%vib/osc non-linear

Aft L/4 L/2 3L/4 Fore

0.05 0.25 0.50 0.75 0.95

5.87 5.94 5.37 3.94 2.84

35.65 46.66 51.16 43.73 31.72

Average

4.79

41.78

Springing

Whipping

linear: Very reduced

High

Non-linear: Small

Table 9a. The maximum significant stresses (dynamic response), added to still water values, case 1 initial design (bottom, side and double bottom σadm-A = 175 N/mm2 & τadm-A = 110 N/mm2; deck and hatchway girder (RL) σadm-AH32 = 224 N/mm2). Maximum stress [N/mm2] (taking as reference h1/3 = 12 m) σmax_LIN + |sw| deck σmax_LIN + |sw| deck RL σmax_LIN + |sw| bottom σmax_LIN + |sw| DB bottom σmax_NL + |sw| deck σmax_NL + |sw| deck RL σmax_NL + |sw| bottom σmax_NL + |sw| DB bottom τmax_LIN + |sw| n-n τmax_NL + |sw| n-n

88.83 104.16 71.00 54.58 120.17 140.91 96.05 73.84 58.62 72.89

Strength criterion

h1/3 [m] limit

Beaufort limit

0.396 < 1 0.464 < 1 0.406 < 1 0.312 < 1 0.536 < 1 0.628 < 1 0.549 < 1 0.422 < 1 0.533 < 1 0.663 < 1

12 12 12 12 12 12 12 12 12 12

11 11 11 11 11 11 11 11 11 11

Table 9b. The maximum significant stresses (dynamic response), added to still water values, case 2 optimized (bottom, side and double bottom σadm-A = 175 N/mm2 & τadm-A = 110 N/mm2; deck and hatchway girder (RL) σadm-AH32 = 224 N/mm2). Maximum stress [N/mm2] (taking as reference h1/3 = 12 m) σmax_LIN + |sw| deck σmax_LIN + |sw| deck RL σmax_LIN + |sw| bottom σmax_LIN + |sw| DB bottom σmax_NL + |sw| deck σmax_NL + |sw| deck RL σmax_NL + |sw| bottom σmax_NL + |sw| DB bottom τmax_LIN + |sw| n-n τmax_NL + |sw| n-n

99.56 117.20 84.40 65.50 134.29 158.08 113.84 88.35 61.90 77.37

73

Strength criterion

h1/3 [m] limit

Beaufort limit

0.444 < 1 0.522 < 1 0.482 < 1 0.374 < 1 0.599 < 1 0.705 < 1 0.651 < 1 0.505 < 1 0.563 < 1 0.703 < 1

12 12 12 12 12 12 12 12 12 12

11 11 11 11 11 11 11 11 11 11

Figure 3a. The Longuet-Higgins (L-H) wave elevation time record [m], h1/3 = 9.326 m, amidships section (x/L = 0.5).

Figure 5a. Bending moment time record [kNm], nonlinear analysis, h1/3 = 9.326 m, us = 18 Knots, x/L = 0.5, structural case 2 (opt.).

Figure 3b. The Longuet-Higgins (L-H) wave amplitude spectrum FFT [m], h1/3 = 9.326 m, amidships section (x/L = 0.5).

Figure 5b. Bending moment amplitude spectrum FFT [kNm], non-linear analysis, h1/3 = 9.326 m, us = 18 Knots, x/L = 0.5, case 2 (opt.).

Figure 4a. Bending moment time record [kNm], linear analysis, wave h1/3 = 9.326 m, us = 18 Knots, x/L = 0.5, structural case 2 (opt.).

Figure 6a. Maximum significant normal deck hatchway stress, non-linear analysis+still water, h1/3 = 0–12 m, us = 18 Knots, case 1 (initial).

Figure 4b. Bending moment amplitude spectrum FFT [kNm], linear analysis, h1/3 = 9.326 m, us = 18 Knots, x/L = 0.5, structural case 2 (opt.).

Figure 6b. Maximum significant normal bottom stress [N/mm2], non-linear analysis+still water, h1/3 = 0–12 m, us = 18 Knots, case 1 (initial).

74

Figure 6c. Maximum significant side neutral axis tangential stress [N/mm2], non-linear analysis+still water, h1/3 = 0–12 m, us = 18 Knots,case 1.

Figure 8a. Maximum significant vertical displacement [m], non-linear analysis + still water, h1/3 = 0–12 m, us = 18 Knots, case 1 (initial).

Figure 7a. Maximum significant normal deck hatchway stress, non-linear analysis + still water, h1/3 = 0–12 m, us = 18 Knots, case 2 (optimized).

Figure 8b. Maximum significant vertical displacement [m], non-linear analysis + still water, h1/3 = 0–12 m, us = 18 Knots, structural case 1 (initial design).

Figure 7b. Maximum significant normal bottom stress [N/mm2], non-linear analysis + still water, h1/3 = 0–12 m, us = 18 Knots, case 2 (optimized).

Figure 9a. Maximum significant vertical displacement [m], non-linear analysis + still water, h1/3 = 0–12 m, us = 18 Knots, structural case 2 (optimized structure).

Figure 7c. Maximum significant side neutral axis tangential stress [N/mm2], non-linear analysis + still water, h1/3 = 0–12 m, us = 18 Knots,case 2.

Figure 9b. Maximum significant vertical displacement [m], non-linear analysis + still water, h1/3 = 0–12 m, us = 18 Knots, structural case 2 (optimized structure).

75

criterion is satisfied on both structural cases, according to Germanischer Lloyd’s Rules (GL 2011). 6. The numerical results obtained from this study (Tables 9a,b) are pointing out that in order to put in evidence the extreme wave loads in the ship girder, induced by the slamming and whipping phenomena, it is necessary to use a nonlinear hydroelastic theory approach (Figs. 5a,b compared to Figs. 4a,b). 7. The optimized structure is more flexible as the initial design case (Table 3), with smaller natural vertical vibration frequencies (Table 6) and higher significant vibration response (Tables 7a,b and Tables 8a,b). 8. Based on the numerical results from Tables 9a,b and Figures 6,7a,b,c, it results that the stress values into the ship girder panels (deck, hatchway girder upper flange, bottom, double bottom and side shells) are higher on the optimized structural case, around 12%, in compare to the initial design structural case, so that higher damage cumulative factors on a long-term fatigue analysis are also expected (Domnisoru 2008).

σ max_ Z M1// max WZ τ max_ n n = T1// max Kτ n n (20) σ max_ Z + σ sw _ Z τ max_ a n n + τ sw _ n n ≤1 ≤1 σ adm τ adm where Z 5

{D DLR B DB}.

CONCLUSIONS

Based on the numerical results from Section 4 with the theoretical model from Section 2, for the 1100 TEU container ship with the structural model presented in Section 3, it results the following conclusions: 1. The ratios between the significant displacements and deformations %w1/3vib/w1/3osc, at initial design structure and optimized structure cases, with the reference wave h1/3 = 9.326 m, speed us = 18 knots, are presented in Tables 7a,b. The ship dynamic response analysis, based on the hydroelasticity theory indicates that the elastic ship girder deformations are small comparing to the ship oscillation displacements, 2.89÷3.41% in the case of linear analysis (Tables 7a,b) and 2.97÷3.47% in the case of non-linear analysis (Tables 7a,b), so that the ship motion parameters can be evaluated with standard seakeeping analysis (only oscillations). 2. Bottom and side slamming have high probability to occur at ship extremities, for both structural cases (Tables 7a,b, Figs. 8 & 9), due to the ship high oscillation amplitudes (with frequency around 0.1256–0.1381 Hz ≈ 0.13 Hz, Table 6), resulting in a whipping phenomenon with high intensity (Tables 8a,b, Figs. 5a,b). 3. The linear springing has small intensity (Tables 8a,b, Figs. 4a,b), because the wave components around the first ship natural vibration frequencies 1.205–1.253 Hz ≈ 1.23 Hz (Table 6) have reduced energy (Figs. 3a,b). The non-linear springing phenomenon has medium intensity (Table 8a,b), induced by the hydrodynamic nonlinearity sources (Tables 7.a,b), due to the ship high motion amplitudes and geometric fine ship lines at bow (Fig.1, Figs. 8 & 9). 4. From the hydroelastic ship response, taking as reference the oscillations significant values for bending moments and shearing forces, at statistical short-term prediction, as average, the first order vibration component represent 3.39÷4.79% at linear analysis, 38.07% initial design structural case and 41.78% optimized structural case at non-linear analysis (Tables 8a,b). 5. From the numerical results (Tables 9a,b, Figs. 6,7a,b,c) the global ship structure strength

ACKNOWLEDGEMENTS This study has been accomplished in the frame of the national projects TOP ACADEMIC POSDRU 107/1.5/S ID-76822 2010–2013 and EFICIENT POSDRU 88/1.5/S ID-61445 2009–2012, granted to Galati “Dunarea de Jos” University. REFERENCES Bertram, V. 2000. Practical ship hydrodynamics. Oxford: Butterworth Heinemann. Bhattacharyya, R. 1978. Dynamics of marine vehicles. New York: John Wiley & Sons Publication. Bishop, R.E.D. & Price, W.G. 1979. Hydroelasticity of ships. Cambridge: University Press Cambridge. Domnisoru, L. & Domnisoru, D. 1998. The unified analysis of springing and whipping phenomena. Transactions of the Royal Institution of Naval Architects London 140(A): 19–36. Domnisoru, L. & Domnisoru, D., 2000. Experimental Analysis of Springing and Whipping Phenomena, International Shipbuilding Progress Delft 47(450): 129–140. Domnisoru, L. 2006. Structural analysis and hydroelasticity of ships. Galati: University “Lower Danube” Press. Domnisoru, L., Dumitru, D. & Ioan, A. 2008. Numerical methods for hull structure strengths analysis and ships service life evaluation, for a LPG carrier. OMAE 15–20 June 2008, Estoril:509–518.

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Fonseca, N., Guedes Soares, C., 2005. Comparison between experimental and numerical results of the non-linear vertical ship motions and loads on a containership in regular waves. International Shipbuilding Progress Delft 52(1), 57–89. GL. 2011. Germanischer Lloyd’s Rules & Design GL/ Poseidon Program. Hamburg. Guedes Soares, C. 1999. Special issue on loads on marine structures. Marine Structures 12(3): 129–209. Hirdaris, S.E., Price, W.G & Temarel, P. 2003. Two and three-dimensional hydroelastic modelling of a bulk carrier in regular waves. Marine Structures 16: 627–658. Hirdaris, S.E. & Chunhua, G. 2005. Review and introduction to hydroelasticity of ships. Report 8. London: Lloyd’s Register. Hughes, O.F. 1988. Ship structural design. A rationallybased, computer-aided optimization approach. New Jersey: The Society of Naval Architects and Marine Engineering.

LBR. 2007. LBR-5 Software User Guide, ANAST, University of Liege, Faculty of Applied Sciences. Ozsoysal, R. 2004. A Review of Recent Ship Vibration Papers, The Shock and Vibration. Digest 5(36), 207–214. Park, J.H. & Temarel, P. 2007. The influence of nonlinearities on wave-induced motions and loads predicted by two-dimensional hydroelasticity analysis. ABSPRADS 1–5 Oct. 2007, Houston (1):27–34. Perunovic, J.V. & Jensen, J.J. 2005. Non-linear springing excitation due to a bidirectional wave field. Marine Structures 18: 332–358. Price, W.G. & Bishop, R.E.D. 1974. Probabilistic theory of ship dynamics. London: Chapman and Hall.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Simplified formulations of mass and geometric stiffness matrices in vibration and stability analyses of thin-walled structures I. Senjanović & N. Vladimir University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture, Zagreb, Croatia

D.S. Cho Pusan National University, Department of Naval Architecture and Ocean Engineering, Busan, Korea

ABSTRACT: Ship hydroelastic analysis is a complex task of determining the interaction between coupled structure motion and vibrations with water. In the governing equation of motion the unified restoring and geometric stiffness plays an important role. This paper deals with simplified geometric stiffness formulation which has some advantages in hydroelastic analysis comparing to the consistent one used in stability analysis. From a mathematical point of view, buckling and natural vibrations are similar eigenvalue problems. However, due to the dependency of geometric stiffness on imposed load, buckling is more complicated. Vibration analysis of a thin-walled structure can be performed with a consistent mass matrix determined by the shape functions of all degrees of freedom (d.o.f.) used for construction of conventional stiffness matrix, or with a lumped mass matrix related to deflection d.o.f. In similar way stability of a structure can be analysed with consistent geometric stiffness matrix or geometric stiffness matrix with lumped buckling load, related only to the rotational d.o.f. In this paper, first, the simplified mass matrix for beam element is constructed employing shape functions of in-plane displacements for deflection, and then the same approach is used for construction of simplified geometric stiffness matrix for beam, and triangular and rectangular plate elements. Application of newly developed matrices is illustrated by analyzing natural vibrations of simply supported beam, as well as buckling of simply supported beam, and simply supported plate with different mesh densities. The results of direct calculations are compared with the analytical solution. Also, a comparison with commercial software results is provided. Finally, combinations of simplified and lumped matrices, called hybrid matrices, are analysed in order to increase accuracy of vibrations and stability analyses, respectively. The performed analyses show that the usage of simplified mass matrix in vibration analysis, as well as usage of simplified geometric stiffness matrix in buckling analysis leads to quite good results. In that sense, the application of developed geometric stiffness matrix in ship hydroelastic analysis is reasonable choice. 1

INTRODUCTION

intuitively condensed to the neighbouring nodes. Natural frequencies determined by the consistent and lumped mass matrices usually converge from the opposite sides, and so called coupled mass matrix as their average is used to increase the accuracy (Kilroy, 1997). In this paper, first, a simplified mass matrix for beam element is shown. It is derived by employing the in-plane (membrane) displacement shape functions for deflection and ignoring rotations (Senjanović et al., 2010). In a similar way the displacement interpolation functions and numerical integration technique used in the formulation of lumped mass matrix can also be adopted to the formulation of geometric stiffness matrix with lumped buckling load, (Shah, 2001). Moreover, the in-plane displacement shape functions can be used for deflection in simplified geometric stiffness matrix

The finite element method is very efficient tool for structural analysis of thin-walled structures (Zienkiewicz, 1971, Bathe, 1996). Free vibration and stability analyses are similar eigenvalue problems from mathematical point of view. In the former case it is necessary to determine the conventional stiffness matrix and mass matrix of finite elements, while in the latter determination of conventional stiffness and geometric stiffness matrices is required. For this purpose shape function of all d.o.f., i.e. deflection and rotations are used. It is well known that instead of consistent mass matrix also lumped mass matrix can be used without losing accuracy. Nodal-quadrature rule or the Gauss rule is usually employed for derivation of the lumped mass matrix. However, in case of complex structures like ships, some equipment masses are

79

(Senjanović et al. 2012). As a result the rotational d.o.f. are ignored. In hydroleastic analysis of ships and offshore structures geometric stiffness occurs as a constitutive part of the restoring stiffness (Huang & Riggs, 2000, Senjanović et al., 2011). Actually, it consists of three terms, i.e. ordinary geometric stiffness with deflection and rotation d.o.f., well-known in buckling analysis, and two additional terms with in-plane displacements, which complete expression for ship stability analysis, and are a novelty in structural analysis. In that sense, the same approach used for derivation of mass matrices is applied for for construction of simplified geometric stiffness matrix for beam, and triangular and rectangular plate elements. Numerical examples include natural vibration analysis of simply supported beam, as well as buckling analysis of simply supported beam and plate with different mesh densities. The results of direct calculations are compared with the analytical solution. Also, a comparison with commercial software results is provided. The performed analyses show that the usage of simplified mass matrix in vibration analysis, as well as usage of simplified geometric stiffness matrix in buckling analysis leads to quite good results. Based on that simplified geometric stiffness may be further used in ship hydroelastic analysis without loosing computational accuracy. 2

⎡ 156 22l 54 −13l ⎤ ⎢ ⎥ 4l 2 13l 3l 2 ⎥ ml ⎢ m1 = , 156 −22l ⎥ 420 ⎢ ⎢ ⎥ ⎢⎣Sym. 4l 2 ⎥⎦

where l is the element length and m is mass per unit length. The consistent mass matrix can be determined in a simpler way by the first order polynomials shape functions related only to beam deflection. In that case one obtains ⎡2 ⎢ ml ⎢0 m2 = 6 ⎢1 ⎢ ⎣0

⎡1 ⎢ ml ⎢0 m3 = 2 ⎢0 ⎢ ⎣0

)

ωn =

(1)

(

)

−3l 6

3l ⎤ ⎥ l2 ⎥ , −3l ⎥ ⎥ 2l 2 ⎥⎦

(5)

0 0 1 0

0⎤ 0 ⎥⎥ . 0⎥ ⎥ 0⎦

(6)

(L )

2

EI , m

(7)

β2L / 2

2.365,

β 4 L / 2 5.497, (8)

and for the skew (antisymmetric) modes

β3L / 2 3.925,

β1

(2)

−6

0 0 0 0

( β n L )2

β0

β5 L / 2

7.068. (9)

The finite element stiffness and mass matrices, derived with the shape functions in the form of the third order (Hermitian) polynomials, yield (Senjanović, 2002): 3l ⎡ 6 ⎢ 2l 2 2 EI k = 3 ⎢⎢ l ⎢ ⎢⎣Sym.

0⎤ 0 ⎥⎥ . 0⎥ ⎥ 0⎦

where the roots of the frequency equations for the symmetric modes are the following:

Where K is global stiffness matrix, M is global mass matrix, δ represents displacement vector and ω is natural frequency. Its solution, i.e. eigenvalues and eigenvectors which represent natural frequencies and natural modes, respectively, is obtained from the condition Det K − ω 2M = 0

1 0 2 0

Natural frequencies of simply supported beam can be also determined analytically according to the formula

Governing equation of beam flexural natural vibrations yields:

ω 2M δ = 0

0 0 0 0

The third possibility is to directly use lumped masses, i.e.

MASS MODELLING IN VIBRATION ANALYSIS OF A BEAM

(K

(4)

3

LUMPED GEOMETRIC STIFFNESS

3.1 Beam element Governing equation of beam buckling yields:

(K

(3)

K

) δ = 0,

(10)

where KG is global geometric stiffness matrix. Geometric stiffness matrix of finite element

80

depends on external compression load (axial force) N, Figure 1. A procedure for deriving the lumped geometric stiffness matrix is shown in (Shah, 2001). It is constructed by employing the trapezoidal numerical integration scheme. The integration points are located at the element nodes resulting in a diagonal matrix for in-plane forces. Here, a lumped geometric stiffness matrix for beam and plate elements are presented for the need of correlation analysis. The consistent geometric stiffness matrix for beam finite element (Cook et al., 1989) reads 3l ⎡ 36 ⎢ 4l 2 N ⎢ kG1 = 30l ⎢ ⎢ ⎢⎣Sym.

3 3l 36

l ⎤ ⎥ −l 2 ⎥ , −3l ⎥ ⎥ l 2 ⎥⎦

Figure 1.

⎡0 ⎤ ⎢ 1 ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎢ ⎥ 0 ⎢ ⎥ ⎢ ⎥ 1 ⎢ ⎥ 0 ⎢ ⎥ + N yab ⎢ ⎥, 0 ⎢ ⎥ ⎢ ⎥ 1 ⎢ ⎥ 0 ⎢ ⎥ ⎢ ⎥ 0 ⎢ ⎥ 1 ⎥ ⎢ ⎢ 0 ⎥⎦ ⎣

(11)

while the lumped one (Shah, 2001) is

kG 2

⎡0 ⎢ Nl ⎢0 = 2 ⎢0 ⎢ ⎣0

0 1 0 0

0 0 0 0

0⎤ 0 ⎥⎥ , 0⎥ ⎥ 1⎦

(12)

(13) where Nx and Ny are membrane normal forces per unit length while a and b is one half of the plate length and width, respectively. The corresponding nodal displacement vector δT = w1 ϕ1,ψ 1 w2 ,ϕ 2 ψ 2 , w3 ϕ 3 ,ψ 3 w4 ,ϕ 4 ψ 4 is where w is displacement and ϕ and ψ rotation around y and x axis, respectively. Coefficients different from zero in Eq. (13) are related to the rotational d.o.f. A disadvantage of the lumped geometric stiffness formulation is that the problem of shear buckling due to Nxy is not captured. Also, how to derive lumped geometric stiffness matrix for a triangular element remains an open question.

where N is constant axial force and l is the element length. The above matrices are related to the nodal displacement vector δT = w1 ϕ1, w2 ϕ 2 , where w and ϕ is beam displacement and rotation of crosssection respectively. 3.2 Plate element The consistent geometric stiffness matrix for an ordinary 4-noded rectangular plate element has a quite complex form (Szilard, 2004). On the contrary, the lumped geometric stiffness matrix is quite simple (Shah, 2001)

G2

Beam buckling.

⎡0 ⎤ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ 1 ⎢ ⎥ 0 ⎢ ⎥ ⎢ ⎥ 0 ⎢ ⎥ 1 ⎢ ⎥ = N x ab ⎢ ⎥ 0 ⎢ ⎥ ⎢ ⎥ 0 ⎢ ⎥ 1 ⎢ ⎥ ⎢ ⎥ 0 ⎢ ⎥ 0 ⎥ ⎢ ⎢ 1⎥⎦ ⎣

4 4.1

SIMPLIFIED GEOMETRIC STIFFNESS Beam element

The general formula for geometric stiffness matrix for beam element (Przemieniecki, 1968) reads kG

l ⎧ ∂φ ⎫ ∂φ j N∫ ⎨ i ⎬ dx, ∂x ⎭ ∂x 0⎩

(14)

where φi are deflection shape functions. Bracket {⋅} and 〈⋅〉 denote column and row vector, respectively. For simplified geometric stiffness shape functions of bar tension are used for beam deflection, while those for rotations are zero

81

x l

φ1

φ2 = 0,

φ3 1

x l

φ4 = 0.

(15)

This leads to the following geometric stiffness matrix

kG 3

N = l

⎡1 ⎢0 ⎢ ⎢ −1 ⎢ ⎣0

0 −1 0 0 0 1 0 0

0⎤ 0 ⎥⎥ 0⎥ ⎥ 0⎦

(16) Figure 2.

for the nodal displacement vector δ = 〈w1,ϕ1,w2,ϕ2〉, where wi and ϕi are displacements and rotations respectively.

where

α1 4.2

⎡ ∂φ ∂φ ⎤ ⎡ N x kG = ∫∫ ⎢ i i ⎥ ⎢ ∂x ∂y ⎦ ⎢⎣ N yx A ⎣

⎡ ∂φ j ⎤ N xxy ⎤ ⎢ ∂x ⎥ ⎥ dxdy, ⎥⎢ N y ⎥⎦ ⎢ ∂φ j ⎥ ⎢ ⎥ ⎣ ∂y ⎦

(17)

kG Gx x + kGxy

kGy ,

xy

⎡ ∂φ ∂φ j ∂y

∫∫ ⎢⎣ ∂xi

∂φi ∂φ j ⎤ ⎥ dxdy, ∂y ∂x ⎦

⎡ ∂φ ∂φ j ⎤ N y ∫∫ ⎢ i ⎥ dxdy. ∂y ∂y ⎦ A ⎣

xij

xi − x j

A

1 (x y 2

φ1 φ2 φ3

x y

yj ,

i j = 1, 2, 3,

).

(24) (25)

∂φ1 1 = x32 ∂y 2 A ∂φ 2 1 = x13 1 ∂y 2 A ∂φ3 1 = x21 ∂y 2 A

(26)

Using Eq. (26), the stiffness matrices (19), (20) and (21) can be derived as follows:

(19)

kGx

(20)

(21)

kGxy x

Shape functions for in-plane displacements of the triangular element (Zienkiewicz, 1971) as shown in Figure 2 are used as 1 (α1 + 23 + 32 ) 2A 1 (α 2 + 31 + 13 ) 2A 1 (α 3 + y12 x + x21y) 2A

yij = yi

∂φ1 1 = y23 , ∂x 2 A ∂φ 2 1 = y31, ∂x 2 A ∂φ3 1 = y12 , ∂x 2 A

(18)

⎡ ∂φ ∂φ j ⎤ N x ∫∫ ⎢ i ⎥ dxdy, ∂x ∂x ⎦ A ⎣

A

kGy

α 2 = x3 y1 − x1y3 , α 3 = x1y2 − x2 y1,

The derivatives of the shape functions are constants as follows:

where

kGxy x

3 y2

(23)

where Nxy = Nyx are the shear membrane forces and A is the element area. The matrix kG can be split into three matrices

kGx

2 y3

Triangular plate element

General formula for plate geometric stiffness matrix (Przemieniecki, 1968) reads

kG

Triangular finite element.

2 ⎡ y23 ⎢ N = x⎢ 4A ⎢ ⎢⎣Sym

y23 y31 2 y31

y23 y12 ⎤ ⎥ y31y12 ⎥ , 2 ⎥ y12 ⎥⎦

(27)

⎡⎡22 y x y x + x32 y31 y23x21 + x32 y122⎤ Nxxy ⎢ 23 32 23 13 ⎥ = 2 y31x13 y31x21 + x13 1 y112⎥ , 4A ⎢ ⎢⎣ Sym 2 y112 x221 ⎥⎦ (28)

kGy (22)

2 ⎡ x32 x32 x13 Ny ⎢ 2 = x13 ⎢ 4A ⎢ ⎢⎣Sym.

x32 x21 ⎤ ⎥ x13x21 ⎥ . 2 ⎥ x21 ⎥⎦

(29)

The above matrices are derived for displacements d.o.f. δ = 〈w1,w2,w3〉 due to reason of

82

simplicity. Therefore, it is necessary to spread them to the complete 9 × 9 displacement field, which includes displacement as well as rotations, i.e. δT = 〈w1,ϕ1,ψ1,w2,ϕ2,ψ2,w3,ϕ3,ψ3〉. Hence, the matrix elements related to the rotations as well as those related to their coupling with deflection d.o.f. are zero. 4.3

Rectangular plate element

A rectangular element in the Cartesian coordinates is shown in Figure 3. The shape functions for inplane displacements expressed by non-dimensional coordinates ξ = x/a and η = y/b read (Holand and Bell, 1970)

φi

1 ( 4

ξi ξ ) (

ηiη )

(30)

Figure 4. Rectangular finite element in non-dimensional coordinates.

where ξi and ηi are the nodal coordinates as shown in Figure 4. These shape functions are used for displacement and one finds for their derivatives ∂φi 1 = ξi ( + ∂x 4 a ∂φi 1 = (+ ∂y b

)

ij kGx xy =

(31)

ij kGy =

)ηi

⎛ ηiη j ⎞ Nx b ξi ξ j 1 + , 4a 3 ⎟⎠ ⎝

4

N ya 4b

(

ηi

i

j

⎛ j

⎝

+

1+

j i

),

ξi ξ j ⎞ . 3 ⎟⎠

(33)

(34)

Furthermore, by taking into account the value of the nodal coordinates as shown in Figure 4, Eqs. (32), (33) and (34) lead to the geometric stiffness matrices

By substituting (31) into (19), (20) and (21) and performing integration over the element area, one finds the matrix elements ij kGx =

N xy

(32) kGx

−2 −1 1 ⎤ ⎡ 2 ⎢ 2 1 −1⎥⎥ N b = x ⎢ , 2 −2 ⎥ 6a ⎢ ⎢ ⎥ 2⎦ ⎣Sym.

(35)

0 −1 0 ⎤ ⎡ 1 N xxy ⎢ − 1 0 1 ⎥⎥ ⎢ = , 1 0⎥ 2 ⎢ ⎢ ⎥ −1⎦ ⎣Sym.

(36)

kGxy x

kGy

1 −1 −2 ⎤ ⎡ 2 ⎢ N ya 2 −2 −1⎥⎥ ⎢ = . 2 1⎥ 6b ⎢ ⎢ ⎥ 2⎦ ⎣Sym.

(37)

The above matrices are derived for displacement d.o.f. δ = 〈w1,w2,w3,w4〉, and therefore they have to be spread to the complete displacement field of 12 × 12 terms, which includes also rotational d.o.f.

Figure 3. Rectangular finite element in Cartesian coordinate system.

83

δ = w1 ϕ1,ψ 1 w2 ,ϕ 2 ψ 2 , w3 ϕ 3 ,ψ 3 w4 ,ϕ 4 ψ 4 , in a similar way as explained in the triangular element. 5

Discrepancies of the results for different mass modeling with respect to the analytical solution are also included in Table 1. The accuracy of results obtained by m2 specification is decreasing at the higher modes comparing to that of m1, but for the first mode, which is very important in ship hydroelasticity, it is acceptable. So it seems rational to investigate the influence of ignored rotational d.o.f. in geometric stiffness matrix on buckling analysis results. Since m2 and m3 mass formulation overestimates and underestimates the results, respectively, one can use the hybrid mass matrix m23 = (m2 + m3)/2. Thus, the discrepancies are considerably reduced, Table 1. Natural vibration analysis of the considered beam has also been performed using commercial packages SESAM (2007) and NASTRAN (MSC, 2005), by taking into account consistent and lumped mass distribution as well as the coupled mass matrix, m13 ( m1 + m 3 ) / 2 , into account. The superior behavior of mass matrices computed from averaged consistent and lumped mass matrix is shown in (Hughes, 2000). The beam is discretisized in the same way as in the previous case. The obtained results are listed in Table 2. Application of the coupled mass only slightly increases accuracy with respect to the lumped mass, and it is not so effective as in the case of longitudinal vibrations (Kilroy, 1997). Much better results are obtained with hybrid matrix m23, Table 1. In order to demonstrate convergence of the numerically determined results to the analytical solution, the beam vibrations calculation is repeated by taking 16 finite elements into account, Table 3. By comparing values of discrepancies in Table 3 with those in Table 2, it is obvious that δ3 and δ13 are considerably reduced. Values of δ1 are slightly increased but are still very small.

ANALYTICAL SOLUTIONS FOR BUCKLING OF SIMPLY SUPPORTED BEAM AND PLATE

The analytical formula defining the critical beam buckling force, Figure 1, reads (Timoshenko and Gere, 1961) Ncr =

π 2 EI . L2

(38)

According to Timoshenko and Gere (1961), the analytical solution of critical axial load of a rectangular plate reads N xcr

k

π 2D , b2

D=

Et 3

(

12 1 − ν 2

)

.

(39)

where t is plate thickness and k is factor which depends on the aspect ratio a/b, i.e. plate length and width a and b respectively. For square plate k = 1. The critical buckling force due to uniformly distributed shear loading can be also determined analytically by the use of the Fourier series (Timoshenko and Gere, 1961), and can be presented in the form N xycr

k

π 2D , b2

(40)

where k = 9.4. 6

6.2

NUMERICAL EXAMPLES

Beam buckling

Buckling of the same beam as in the case of natural vibrations, Section 6.1, is considered. Critical buckling force calculated according to Eq. (38) yields Ncr = 215.7 MN. The critical buckling force is also determined by the finite element method for consistent, simplified, lumped and hybrid geometric stiffness matrix

6.1 Natural vibrations of simply supported beam Application of different mass modeling and resulting accuracy is illustrated for case of free beam of the following properties: Length L = 40 m Breadth B=2m Height H=1m Cross-section area A = 2 m2 Moment of inertia of cross-section I = 0.1667 m4 Mass M = 6.28 ⋅ 105 kg Young’s modulus E = 2.1 ⋅ 1011 N/m2

kG

1 ( kG 2 + kG 3 ) . 2

(41)

The beam is divided into 8 and 16 finite elements. The obtained results are listed in Table 4. They converge with increasing the number of finite elements. The consistent geometric stiffness introduces quite small error. The simplified and lumped stiffness overestimate and underestimate the results

The beam is divided into 8 finite elements. Table 1 contains the obtained natural frequencies as well as the analytically determined values according to Eq. (7).

84

Table 1.

Natural frequencies of beam flexural vibrations ωi [Hz], 8 finite elements.

Mode no.

Consistent mass, m1

Simplified mass, m2

Lumped mass, m3

Hybrid mass, m23

Analytical

δ1 (%)

δ2 (%)

δ3 (%)

δ23 (%)

1 2 3 4

3.323 9.165 17.994 29.841

3.374 9.687 20.149 35.746

3.171 8.481 16.180 26.079

3.268 9.025 17.834 29.749

3.323 9.151 17.951 29.678

0 0.15 0.24 0.55

1.51 5.53 10.90 16.98

−4.79 −7.90 −10.95 −13.80

−1.70 −1.40 −0.66 0.24

Table 2.

Natural frequencies of beam flexural vibrations ωi [Hz], SESAM and NASTRAN, 8 finite elements.

Discrepancy

SESAM

NASTRAN

Mode no.

Consistent mass, m1

Lumped mass, m3

Discrepancy δ1 (%)

Discrepancy δ3 (%)

Coupled mass, m13

Discrepancy δ13 (%)

1 2 3 4

3.319 9.125 17.832 29.394

3.123 8.239 15.338 24.224

−0.12 −0.28 −0.67 −0.97

−6.40 −11.07 −17.04 −22.51

3.168 8.451 16.052 25.692

−4.89 −8.28 −11.83 −15.51

Table 3.

Natural frequencies of beam flexural vibrations ωi [Hz], SESAM and NASTRAN, 16 finite elements. SESAM

NASTRAN

Mode no.

Consistent mass, m1

Lumped mass, m3

Discrepancy δ1 (%)

Discrepancy δ3 (%)

Coupled mass, m13

Discrepancy δ13 (%)

1 2 3 4

3.319 9.118 17.786 29.211

3.231 8.696 16.416 26.240

−0.12 −0.36 −0.93 −1.60

−2.84 −5.23 −9.35 −13.10

3.280 8.944 17.324 28.248

−1.31 −2.31 −3.62 −5.06

Table 4. Number of finite elements 8 16

Table 5.

Beam buckling force Nxcr [MN]. Discrepancies, % Analytical

Consistent kG1

Simplified kG3

Lumped kG2

Hybrid kG4

δ1

δ3

δ2

δ4

215.7 215.7

215.91 215.89

218.69 216.59

213.09 215.21

216.58 216.07

0.090 0.088

1.370 0.412

−1.226 −0.230

0.408 0.171

Buckling factor k for simply supported square plate, uniform axial compression Nx, analytical value k = 1. Discrepancies, %

Mesh

Consistent kG1

Simplified kG3

Lumped kG2

Hybrid kG4

NASTRAN kG5

δ1

δ3

δ2

δ4

δ5

4×4 8×8

0.9449 0.9888

1.0905 1.0213

0.7655 0.9339

0.9452 0.9882

1.0280 1.0003

−5.83 −1.13

8.30 2.09

−30.63 −7.08

−5.79 −1.20

2.72 0.03

85

respectively. Hence, the hybrid stiffness reduces the discrepancy considerably. 6.3

buckling force is illustrated in the case of a simply supported plate. A numerical example is performed by the use of finite element method for a square plate of the following dimensions a b = 2 m, and t = 0.01 m. The plate is divided into 4 × 4 = 16 and 8 × 8 = 64 square finite elements, respectively. The conventional

Plate buckling

The application and evaluation of the simplified geometric stiffness matrix for defining the critical Table 6.

Buckling factor k for simply supported square plate, uniform shear load Nxy, analytical value k = 9.4. Discrepancies, %

Mesh

Consistent kG1

Simplified kG3

NASTRAN kG5

δ1

δ3

δ5

4×4 8×8

8.3481 8.9280

15.5617 10.5616

14.7962 10.1736

−12.60 −5.29

39.60 11.00

36.47 7.60

Figure 5.

Buckling modes of simply supported plate for axial load and different mesh density.

Figure 6.

Buckling modes of simply supported plate for shear load and different mesh density.

86

It should be also mentioned that derived simplified geometric stiffness eliminates drawback of lumped geometric stiffness, i.e. it can take shear buckling effect into account.

stiffness matrix is determined by employing the non-conforming rectangular finite element, (Szilard, 2004). Consistent, simplified, lumped and hybrid geometric stiffness matrix are taken into account. The obtained results are compared with the analytical solution in Table 5. The consistent geometric stiffness matrix gives quite good results, which converge from below. They are very close to those shown in (Holand & Bell, 1970). The results obtained by the simplified and lumped matrices converge from the opposite sides and the solution obtained by the hybrid matrix is very good and close to the one of the consistent matrix. Commercial software NASTRAN is also used to define the buckling factor k, due to illustration, Table 5. The results obtained by NASTRAN seem to be the best ones among the numerical solutions, (MSC, 2005). Furthermore, the buckling of the same square plate due to uniformly distributed shear forces is analyzed here. The finite element results are shown in Table 6. The best results are obtained by the consistent geometric stiffness matrix, which is better than the one obtained by NASTRAN. The simplified matrix produces results of the same order of discrepancies as NASTRAN. A mesh density 4 × 4 elements gives very large discrepancies for the both formulations. Buckling modes obtained by NASTRAN in the case of axial and shear load, and for the mesh density 4 × 4 and 8 × 8 elements are shown in Figures 4 and 5, respectively. The higher mesh density smoothes the deflection field. 7

ACKNOWLEDGEMENTS This work was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean Government (MEST) through GCRCSOP (Grant No. 2011-0030669).

REFERENCES Bathe, K.J. 1996. Finite Element Procedures. Prentice Hall. Cook, R.D., Malkus, D.S., Plesha, M.D.. 1989. Concepts and Applications of Finite Element Analysis. 3rd ed. John Wiley & Sons. Holand, I., Bell, K. 1970. Finite Element Methods in Stress Analysis. Tapir Forlag, Trondheim. Huang, L.L. and Riggs, H.R. 2000. The hydrostatic stiffness of flexible floating structure for linear hydroelasticity. Marine Structures, 13(2):91–106. Hughes, T.J.R. 2000. The Finite Element Method, Dover Publications, Inc., New York, 2000. Kilroy, K. 1997. MSC/NASTRAN, Quick Reference Guide, The MacNeal-Schwendler Corporation. MSC. 2005. MSC.NASTRAN2005: Installation and Operations Guide, MSC Software. Przemieniecki, J.S. 1968. Theory of Matrix Structural Analysis. McGraw-Hill Book Company. Senjanović, I. 2002. Finite element method in analysis of ship structures, University of Zagreb, Zagreb. (textbook, in Croatian). Senjanović, I., Hadžić, N., Tomić, M. 2011. Investigation of restoring stiffness in the hydroelastic analysis of slender marine structures. ASME Journal of Offshore Mechanics and Arctic Engineering, 133(3). Senjanović, I., Vladimir, N., Cho, D.S.: 2012. A simplified geometric stiffness in stability analysis of thin-walled structures by the finite element method. International Journal of Naval Architecture and Ocean Engineering, 4(3):313–321. Senjanović, I., Vladimir, N., Hadžić, N. 2010. Some aspects of geometric stiffness modelling in the hydroelastic analysis of ship structures. Transactions of FAMENA. 34(4):1–10. Szilard, R., 2004. Theories and Applications of Plate Analysis. John Wiley & Sons. Shah, S.J. 2001. Finite-element geometric stiffness matrix lumping by numerical integration for stability analysis. Transactions, SMiRT 16. Washington DC, Paper 2057. SESAM. 2007. User’s manual, Det norske Veritas, Høvik. Timoshenko, S.P. and Gere, J.M. 1961. Theory of Elastic Stability. McGraw Hill Book Company. Zienkiewicz, O.C. 1971. The Finite Element Method in Engineering Science. London: McGraw-Hill.

CONCLUSIONS

The finite element formulation of the unified restoring and geometric stiffness in ship and offshore hydroelastic analysis is rather complex task. By ignoring rotational terms in the geometric stiffness matrix, the problem is simplified, without loss of accuracy. In this paper, natural vibrations of a simply supported beam with different formulations of mass matrix, are analyzed. It is shown that simplified mass matrix gives quite accurate results. Then, the same procedure as in case of simplified mass matrix has been used to derivation of simplified geometric stiffness for buckling problems. Vibration analysis is considered prior buckling analysis, since both problems are the same from the mathematical point of view. Physical background of particular terms in geometric stiffness matrix is much more complex to explain than of those in mass matrix. The numerical examples of simply supported beam and plate, modelled with rectangular elements, show that the simplified geometric stiffness matrix can be successfully used in restoring stiffness for the hydroelastic analysis of ship and offshore structures.

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Response to accidental loads

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Numerical analysis of warship’s structural panel subjected to airblast and underwater explosions Masud Ali, Susmita Mondal, Asokendu Samanta & Girish Narendran Research and Rule Development Division, Indian Register of Shipping, Mumbai, India

ABSTRACT: A numerical methodology is developed to analyze structural panel of ship structure subjected to blast loading using a nonlinear explicit solver Dytran along with Patran as pre-processor and post-processor. The present study applied the General Lagrangian–Eulerian (GLE) coupling technique, a fluid–structure interaction approach, to simulate the interaction between structure and surrounding fluid (air or water). Strain rate sensitivity effects are also considered by Cowper-Symonds relation. To validate the present methodology standard problems are taken from the literature. For validation of airblast problem square steel panel with specific dimensions and two different thicknesses are analyzed. Similarly for underwater explosion problem rectangular steel panel with specific dimension is analyzed for different shock factors. Relevant specifications of explosive mass, air and water medium are taken as specified in the literature. The central permanent displacements obtained from the present analysis are compared with the displacements published in the literature. The permanent deformation of plate obtained from numerical analyses is in good agreement with the permanent deformation determined experimentally. After validation for bare steel panels, the present methodology is applied in Underwater Explosion (UNDEX) to analyze stiffened panel which is the basic structural component of ship structure and damage inflicted by blast load is estimated. 1

INTRODUCTION

this primary shock wave, mainly due to bubble formation. However, the bubble effect is beyond the scope of this paper as the objective of this paper is to determine the local damage inflicted by underwater explosion shock wave which is an early time phenomena. The bubble simulation will be subsequently included in the future investigations. This paper addressed the study of stiffened panel response to underwater shock loading through the code Dytran. For reliability of the results of stiffened panel, validation is done on bare plate with the experimental results of Houlston et al. (1993) on airblast and experimental and numerical results of Ramajeyathilagam et al. (2000) on underwater explosion.

A warship can be subjected to airblast by missile attack above water surface or under water explosion by the attack of torpedo or mine below water surface, which can damage the structure in combat operations. In blast loading the ship structure is subjected to high pressure wave. So, the survivability of naval vessel to blast loading is a very important factor for proper functioning of ship structure. As stiffened panels are the basic structural components of ship structures, it is important to estimate possible damage of the stiffened panels incurred by blast loading. Airblast and underwater explosions are complex phenomenon occurring in air and water medium respectively. The general sequence of a blast begins with the chemical reaction of the explosive material known as detonation. The product of the reaction is a hot gas with very high pressure. This immense pressure generates blast wave which propagates through the surrounding air medium and shock wave which propagates through water medium. The wave propagates in a spherically symmetric fashion unless they encounter any objects setting up complex interaction and reflection patterns. In underwater explosion, the ship structure may be affected by some late time phenomenon behind

2

LITERATURE REVIEW

Research on airblast and underwater explosion has been subjected to analytical/empirical, numerical and experimental study since the 1950s. The analytical/empirical methods include some empirical equations and are applicable to simple cases. Experimental methods provide a good understanding of the event but are costly and have potentially negative impact on environment. So, the ability to predict the structural damage subjected to airblast

91

or underwater explosion loading by numerical method is of great interest. The structural response of ship-panels subjected to air blast was investigated by Houlston and Slater (1985). Experiment on steel plates and stiffened panels were conducted and results were compared with numerical finite element analysis. Houlston and Desrochers (1987) studied the nonlinear structural response of a square steel plate subjected to air blast loading by finite element analysis and correlated the displacement-time histories with experimental results. Research was carried out by Houlston and Slater (1991) to assess the solution accuracy and sensitivity with respect to the loading and structural details subjected to air blast loads. It was found that central displacement-time response could be improved with a progressively refined finite element mesh towards the centre of the plate. Yanchao Shi et al. (2008) studied the mesh size effect in numerical simulation of blast wave propagation and interaction with structures. It was found that the positive incident/reflected pressure is more sensitive to the mesh size, compared with the wave front arrival time and the positive incident/reflected impulse. A formula was derived for calculating the plastic deformation of protective bulkhead subjected to blast loading using energy method, and the ultimate capability of the protective bulkhead calculated by Xing-ning et al. (2009). The calculation was compared with external experiment. Experimental and numerical investigations were carried out by Ramajeyathilagam et al. (2000) on clamped rectangular plates subjected to underwater explosion loading. The test plate was considered to be air-backed by a box model set up and submerged in a water tank. For various shock factors the numerical results were compared with experimental results. Ramajeyathilagam and Vendan (2004) investigated the failure modes and extent of failure of flat, concave and convex hull panel and concluded that the concave panels suffer lesser deformation than other panels subjected to same intensity explosion loading but the convex panels can withstand larger shock loads before reaching the tensile tearing failure mode. Jen and Tai (2009) studied the deformation behavior of a stiffened panel subjected to underwater shock loading using the non-linear transient finite element method. The shock factor is adopted to describe the shock severity. The success of the numerical method for solving structures subjected to airblast or underwater explosion loading requires the coupling between fluid and structure and the strain rate sensitivity effect. So, in this present study a methodology is established to solve different components of a warship numerically.

3

BLAST LOADING

In general, sudden release of huge amount of energy from an explosive material leads to the formation of highly pressurized, superheated gas. The fluid domain surrounding the explosive is pushed back and as a result pressure pulse is generated. As the blast wave or shock wave travels outward from the explosion centre, pressure of the wave front decreases. 3.1

Airblast

A generic blast wave profile at a point in the medium is shown in Figure 1. Blast wave arrives at that point ta times after the explosion occurs and the pressure suddenly jumps to the peak value followed by an exponential decay. The first phase of shock wave is positive phase and the phase immediately following it is negative phase. The impact of the positive phase is found to be much more than that of the negative phase. So, the duration of the blast wave is considered as positive duration. The instantaneous pressure P(t) of the positive phase at time t after the arrival of shock wave is obtained by the Friedlander equation as given by Kinney and Graham (1985) is P (t ) = P0

b t ⎡ t⎤ − Pi ⎢1 − ⎥ e td ⎣ td ⎦

(1)

where, P0 is the ambient pressure, td is the positive duration of the pressure pulse and b is waveform parameter, can be obtained by the equation given as Z 2 − 3.7Z + 4.2

(2)

where, Z is the scaled distance. Thus, any distance R from an explosive of equivalent trinitrotoluene (TNT) charge weight W can be transformed into

Figure 1.

92

Blast wave pressure-time history.

3.2 Underwater explosion

a characteristic scaled distance Z given by Kinney and Graham (1985) is Z=

R

The major characteristics associated with the shock wave are peak pressure and pressure time history. The peak pressure of shock wave front is given by Cole (1948) and Swisdak (1978) as

(3)

1

W3 The use of Z allows a compact and efficient representation of blast wave data for a wide range of situations. The arrival time for the shock wave at a standoff distance R, is given by Kinney and Graham (1985) as ⎡ 1 1 R 6 Pi ta = ∫ ⎢ a0 rc ⎢1 + 7 P0 ⎣

Pmax

(4)

Pi

P0 1+

⎛ Z ⎞ ⎝ 0.048 ⎠

2

1+

⎛ Z ⎞ ⎝ 0.32 ⎠

2

1+

⎛ Z ⎞ ⎝ 1.35 ⎠

2

⎛ 1⎞ W 3⎟ θ = W ⋅ k2 ⎜ ⎜ R ⎟ ⎝ ⎠

A2

(9)

With the peak pressure from equation (8) and decay constant from equation (9), the pressure time history is given by equation (10) P (t ) = Pmax e

(5)

−tt t1 θ

+ PD

(10)

where, PD is the hydrostatic pressure at the depth of the point from the water surface and t1 is the time, the shock wave takes to reach the point of interest.

Kinney & Graham (1985) has given the following equation for calculating positive phase duration. ⎡ ⎛ Z ⎞ ⎤ 980 ⎢1 + ⎥ td ⎢⎣ ⎝ 0.54 ⎠ ⎥⎦ = 1 3 6 2 W 3 ⎡1 + ⎛ Z ⎞ ⎤ ⎡1 + ⎛ Z ⎞ ⎤ 1 + ⎛ Z ⎞ ⎢ ⎥⎢ ⎥ ⎝ 6.9 ⎠ ⎢⎣ ⎝ 0.02 ⎠ ⎥⎦ ⎢⎣ ⎝ 0.74 ⎠ ⎥⎦

(8)

1

where, a0 represents the speed of sound in air and rc is the radius of TNT charge. Many solutions exist for the peak overpressure. Here, the expression for the peak overpressure given by Kinney & Graham (1985) is presented. ⎡ ⎛ Z ⎞2⎤ 808 ⎢1 + ⎥ ⎢⎣ ⎝ 4.5 ⎠ ⎥⎦

A1

where, R is the distance to the point of interest from the center of explosive and W is the charge weight. The peak pressure is assumed to have exponential decay. The decay constant θ, represented by Swisdak (1978) as

1

⎤2 ⎥ dr ⎥ ⎦

⎛ 13 ⎞ W ⎟ k1 ⎜ ⎜ R ⎟ ⎝ ⎠

t1 =

R C

(11)

10

where, C is the speed of sound in the fluid. k1, k2, A1 and A2 are constants which depend on explosive charge type and given by Table 1 for TNT explosive. The shock wave pressure field is modified due to reflection from the structure and the effect of fluid structure interaction. The final pressure considering all these effects is called total pressure. The magnitude of peak value of total pressure i.e. total peak pressure is twice the pressure given by equation (8).

(6)

When the blast wave encounters a large wall on the way, the wave gets reflected and magnified. The peak reflected pressure Pr can be obtained from Rankine-Hugoniot relationships for an ideal gas as Pr

2 Pi

(7 P0 4 Pi ) (7 P0 Pi )

Table 1. Shock wave parameters for TNT explosive charges.

(7)

Thus in the frontal face of the plate the pressure that is acting is reflected pressure. Replacing Pi by Pr in equation (1), results the reflected pressure– time history that acts on the surface which is normal to the shock front.

93

Constants

Value for TNT explosive

k1 A1 k2 A2

52.16 1.13 92.50 −0.22

The response of a structure depends on the pressure time history of the shock wave. So, the severity of shock wave is generally represented by the energy released from the pressure loading at the structure [Keil (1961)]. The energy density can be approximately expressed as E C1 W2 and R the Shock Factor (SF) can be represented as in equation (12). SF = 0.45

W R

Figure 2.

(12)

Table 2.

So, theoretically the energy released from the pressure loading is equal for any combination of charge weight and standoff distance keeping the Shock Factor (SF) constant. 4 4.1

NUMERICAL METHODOLOGY

Airblast problem description.

Weight of TNT charge (kg)

Stand-off distance (m)

Peak reflected pressure (MPa)

14.50

2.44

5.56

Computational code

Dynamic response of structure subjected to airblast and underwater explosion loading can be simulated by various methods. Computational mechanics tools are available that simulate the response of both solid and fluid material under highly dynamic conditions (e.g. detonation), where shock wave propagation is a dominant feature. MSC Dytran, LS Dyna etc. are the most commonly used commercial software to solve blast problem. In this paper MSC Dytran is used to obtain the structural response subjected to blast loading. Dytran is a three-dimensional analysis code for analyzing the dynamic, nonlinear behavior of solid structures, and fluids. It uses explicit time integration and incorporates features that simulate a wide range of material and geometric nonlinearity. Meshes within Lagrangian and Eulerian solver can be coupled together to analyze fluid-structure interaction problem. This is based on the creation of coupling surfaces on Lagrangian structures. The coupling surface acts as a boundary to the flow of material in the Eulerian mesh. At the same time, the pressures in the Eulerian elements cause forces to act on the coupling surface, distorting the Lagrangian elements. 4.2

Schematic diagram of the problem.

Plate properties Thickness: 0.0034 m Yield stress: 260 MPa Thickness: 0.0015 m Yield stress: 285 MPa

Table 2 summarizes the problem with peak reflected pressure acting at the centre of the plate. 4.3

UNDEX problem definition

For validation of the methodology for underwater explosion problem, test results given by Ramajeyathilagam et al. (2000) are used. The high strength steel plate used for the UNDEX problem had effective dimension 0.25 m × 0.3 m and thickness of 0.004 m. Plate was kept at a stand-off distance of 0.15 m and different charge weights were considered to simulate different shock factors. Table 3 summarizes the detail of the problem with total peak pressure. 4.4

Numerical model

Patran is used for modeling purpose. To discretize the square plate, Lagrangian shell element is used. Additional surfaces known as dummy surfaces are generated to obtain a close structure for the purpose of general coupling. These additional surfaces are meshed with dummy shell elements. In Dytran through Von Mises yield model entry properties of isotropic, elastic perfectly plastic behavior of material with failure is defined. To consider the high strain rate effect Cowper-Symonds relation is used to compute the dynamic yield stress (σdy), i.e.

Airblast problem definition

For validation of the numerical analysis methodology of structure subjected to airblast load test data given in Houlston et al. (1993) is used. The plate considered for numerical analysis had effective dimension of 0.508 m × 0.508 m and thickness of 0.0034 m and 0.0015 m. A TNT charge of weight 14.5 kg is kept at a stand-off distance of 2.44 m from the centre of the plate. The schematic diagram of the problem is shown in Figure 2.

σ dy

94

⎛ εɺ = σ y ⎜1 + D ⎜⎝

1 n

⎞ ⎟ ⎟⎠

(13)

Table 3.

in combination with a cavitation model where the pressure P is defined as, for no cavitation (ρ > ρC)

UNDEX problem description.

Weight of TNT charge (kg) 0.005 0.010 0.020 0.050 0.070

Stand-off distance (m) 0.15 0.15 0.15 0.15 0.15

Shock factor (kg1/2/m) 0.212 0.300 0.424 0.671 0.794

Total peak pressure (MPa)

P

127.1 165.0 202.0 302.6 343.5

and for cavitation (ρ ≤ ρC) P

Water density Constant A0 Constant A1 Constant Gamma (γ)

= 210 GPa = 0.3 = 7800 kg/m3 = 0.23 = Elasto-Plastic

) ρe

P

⎛ ωη ⎞ − R1 A ⎜1 − e R1 ⎟⎠ ⎝

⎛ ωη η ⎞ − Rη2 B 1− e + ωηρ η 0 e (16) ⎝ − R2 ⎟⎠

Properties assigned for explosive using JonesWilkins-Lee (JWL) equation of state are

(14)

Properties assigned to Euler domain for Ideal gas equation of state are Air density Ratio of specific heat Gas constant Constitutive model

= 1025 kg/m3 = 1 × 105 = 3.31 × 108 = 7.15

Hydrostatic pressure is not considered in the present study as Tait cavitation model does not support hydrostatic card in Dytran. However, with polynomial equation of state it is observed that the effect of hydrostatic parameter on shock wave is negligible as magnitude of hydrostatic pressure is very less compared to shock wave pressure. However, hydrostatic pressure cannot be ignored if the bubble phenomenon is considered into the analysis. Explosive can be modeled as a sphere, cylinder or explosive property can be assigned to some fluid elements itself. In present problem explosive is modeled as a sphere. The Jones-Wilkins-Lee (JWL) equation of state is used to model the TNT explosive charge. In JWL EOS pressure is defined as

The fluid model is a box shaped solid geometry which surrounds the plate and the explosive completely. It is well known that accuracy of numerical results is strongly dependent on the mesh size used for the analysis. On the other side the mesh size is also limited by the dimension of the model and the hardware capacity. Numerical study is conducted with several Eulerian mesh sizes and the mesh size is finalized to match the total peak pressure accurately. To model air, water and explosive Equation of States (EOS) are used in Dytran. The EOS correlates pressure with Specific Internal Energy (SIE), density and volume of specified material. EOS is used to initialize the internal characteristics of elements assigned with air and explosive properties. The Ideal gas EOS is used in this study to model the air. In Ideal gas EOS pressure is defined as given by Dytran user’s guide (2008)

(

(15)

Pc

where, ρ is overall material density, ρ0 is reference density and ρc is the critical density which produces cavitation pressure Pc. Properties assigned to the Euler domain for the Tait cavitation model are

where, σy is the static yield stress and D and n are other material parameters. In the present calculations, D = 40/s and n = 5, being commonly accepted values for high strength steel, have been used. Other parameters assigned to plate material properties are Young’s modulus Poisson’s ratio Density Maximum plastic strain Constitutive model

⎡⎛ ρ ⎞ γ ⎤ A0 + A1 ⎢⎜ ⎟ − 1⎥ ⎢⎣⎝ ρ0 ⎠ ⎥⎦

Density Constant A Constant B Constant R1 Constant R2 Constant ω Constitutive model

= 1.225 kg/m3 = 1.4 = 287 J/kg ⋅ K = Ideal gas

The water is modeled as a Tait cavitation model. The Tait equation of state is based on Tait model

= 1654 kg/m3 = 3.71213 × 1011 = 3.231 × 109 = 4.15 = 0.95 = 0.3 = JWL Explosive

Boundary conditions are applied to the FEA model. In this current study fixed boundary

95

Figure 3. Numerical model of plate-explosive-fluid system.

condition is assigned along the plate boundary and non-reflecting boundary condition is assigned along the exterior surface of the fluid domain. The non-reflecting boundary condition applied to the exterior surface of air or water domain prevents shock wave reflection from the exterior surface of the fluid domain and allows the shock wave to continue through the end of the fluid. Non-reflecting boundary condition simulates the infinite boundary. To introduce interaction between Lagrangian and Eulerian domain general coupling is used in Dytran. Figure 3 shows the FE model including plate enclosed by dummy surfaces, spherical explosive and fluid domain.

Figure 4. problem.

5

Table 5. Charge weight and stand-off sensitivity study.

5.1

Table 4.

Comparisons of airblast results. Central displacement (m)

Stand-off Houlston Present Thickness distance et al. numerical Variation (m) (m) (1993) (Dytran) (%) 0.0034 0.0015

NUMERICAL SIMULATION Airblast problem

The analysis is carried out with end time of 6 millisecond and time step of 10−4 millisecond. The central maximum displacement for 0.0034 m and 0.0015 m thick plates are 0.0399 m and 0.0801 m respectively. The central displacement vs. time plot for the above case is shown in Figure 4. The central permanent deformation obtained from the plot for 0.0015 m and 0.0034 m thick plate are 0.0362 m and 0.0783 m respectively. The central permanent deformations of the square steel panel obtained in the present analysis are compared with the Houlston et al. (1993) results, shown in Table 4. The central displacements obtained numerically using Dytran code are in good agreement with the Houlston et al. (1993) results. 5.2

Central displacement time history for airblast

2.44 2.44

0.0340 0.0730

0.0362 0.0783

6.47 7.26

distance

Shock factor (kg1/2/m)

Charge weight (kg)

Stand-off distance (m)

Central displacement of plate (m)

0.832

0.0770 0.2140 0.0055 0.0100

0.15 0.25 0.15 0.20

0.0165 0.0162 0.0060 0.0063

0.225

UNDEX problem

A sensitivity study of structural response to charge weight and stand-off distance is carried out keeping the shock factor constant and the results are presented in Table 5 which shows that for a constant shock factor the central displacements of plates are not varying.

Figure 5. Central displacement time history for UNDEX problem.

96

The analysis of problem defined in section 4.3 is carried out with end time of 1 millisecond and time step of 10−4 millisecond. Different charge weights and corresponding shock factors are considered. The central permanent deformation of the plate for the shock factors of 0.212, 0.300, 0.424, 0.671 and 0.794 are reported. The central

Table 6.

displacement vs. time plot for the above case is shown in Figure 5. The central permanent deformations of the rectangular steel panel obtained in the present analysis are compared with the Ramajeyathilagam et al. (2000) results. Table 6 shows the comparison of present numerical central displacements of the

Comparisons of UNDEX results. Central displacement (m) Ramajeyathilagam et al. (2000)

Shock factor (kg1/2/m) 0.212 0.300 0.424 0.671 0.794

Table 7.

Experimental

Numerical

Present numerical (Dytran)

Variation with experimental (%)

0.012 0.023 0.032 0.059 0.072

0.013 0.024 0.032 0.054 0.063

0.014 0.024 0.035 0.051 0.061

16.6 4.3 9.4 13.6 15.3

Dimensions of plate and stiffeners.

Area of plate Thickness of plate Size of ‘T’ stiffeners

1.5 m × 1.2 m 0.01 m 0.12 m × 0.06 m × 0.006 m

Figure 7. Central displacement time history for stiffened panel subjected to UNDEX loading.

Figure 6.

Plate with stiffener arrangement.

Table 8. panel.

Underwater explosion results for stiffened

Charge weight (kg)

Standoff distance (m)

Shock factor (kg1/2/m)

Permanent central displacement (m)

0.30 0.60 1.25 2.00 3.00 5.00

1.5 1.5 1.5 1.5 1.5 1.5

0.16 0.23 0.33 0.42 0.52 0.67

0.071 0.102 0.139 0.166 0.194 –

Figure 8. Failure pattern of stiffened plate at shock factor 0.67 (time 8 ms).

97

APPENDIX

plate with experimental and numerical central displacements of Ramajeyathilagam et al. (2000) for various cases. The numerical simulation results show good correlation with experimental results. However, the maximum variation of the deflection obtained numerically with respect to experimentally obtained deflection is around 16%. Such variations occurred may be due to considerable uncertainty involved in underwater explosion.

Notation: a0

Speed of sound

t

Time

b

Waveform parameter Constant

ta

W

P

Cowper-Symond’s parameter Energy Specific internal energy Cowper-Symond’s parameter Pressure

P0

Ambient pressure

η

P(t)

Instantaneous pressure Incident peak overpressure Peak reflected pressure Stand-off distance Radius of TNT charge

σy

Shock wave arrival time Positive duration of pressure Explosive charge weight Scaled distance Overall material density Reference density of explosive Ratio of specific heat Ratio of overall and reference density Static yield stress

σdy

Dynamic yield stress

εɺ

Strain rate

θ

Decay Constant

C1 D

5.3

E e

Stiffened plate subjected to UNDEX

Rectangular stiffened panel with two parallel stiffeners placed symmetrically with a spacing of 0.4 m is analyzed for underwater explosion by the methodology established. The TNT charge is placed at a stand-off distance 1.5 m. The dimensions of steel panel and stiffener are given in Table 7 and the stiffener arrangement is shown in Figure 6. Six different shock factors are considered and the permanent central displacements of the stiffened plates are reported in Table 8. Figure 7 shows the central deformation vs. time plot for different shock factors. The central displacement-time history of the plate for shock factor 0.67 is not stabilized as failure occurs along the centre line of the plate and along the supports when the plastic strain of the loaded plate exceeds the maximum plastic strain defined in the material model. Figure 8 shows the failure pattern of the plate. 6

n

Pi Pr R rc

td

Z ρ ρ0 γ

REFERENCES Cole, R.H. 1948. Underwater Explosion. Princeton, New Jersey: Princeton University Press. Dytran user’s guide, 2008 r1. Houlston, R., Slater, J.E., Pegg, N., Desroachers, C.G. 1985. On analysis of structural response of ship panels subjected to air-blast loading. Comput. & Structures 21: 273–289. Houlston, R., DesRochers, C.G. 1987. Nonlinear structural response of ship panels subjected to air-blast loading. Comput. & Structures 26: l–15. Houlston, R., Slater, J.E. 1991. Global and local modeling of naval panels subjected to shock loads. Comput. & Structures 40:353–364. Houlston, R., Slater, J.E., Ritzed, D.V. 1993. Damage assessment of naval steel panels subjected to free-field and enhanced air-blast loading. Advances in Marine Structures-2, Elsevier Science Publishers Ltd. Jen, C.Y., Tai, Y.S. 2009. Deformation behavior of a stiffened panel subjected to underwater shock loading using the non-linear finite element method. Journal of Materials and Design, doi:10.1016. Keil, A.H. 1961. The response of ships to underwater explosions. Annual Meeting of The SNAME, New York. Kinney, G.F., Graham, K.J., 1985. Explosive shocks in air. 2nd ed. New York: Springer.

CONCLUSIONS

Under blast loading the stiffened panels of ship structures are subjected to lateral pressure. The paper deals with the numerical simulation of blast phenomena through the code Dytran. For validation, the simulation results have been compared with the published experimental results. On the basis of the comparison it is concluded that the present methodology has the ability to predict the structural response due to airblast and underwater explosion loading. The present work addresses the response of the stiffened panel to underwater shock loading. Shock severity is described by shock factors and the displacement-time histories for different shock factors are presented. The stiffened plate for the present case shows failure under a shock factor of 0.67. The results are useful in designing hull panel of naval ship and the present methodology can be extended to analyze other structural components also.

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Ramajeyathilagam, K., Vendhan, C.P., Bhujanga Rao, V. 2000. Non-linear transient dynamic response of rectangular plates under shock loading. International Journal of Impact Engineering 24:999–1015. Ramajeyathilagam, K., Vendhan, C.P. 2004. Deformation and rupture of thin rectangular plates subjected to underwater shock. International Journal of Impact Engineering 30:699–719. Swisdak, M. 1978. Explosion Effects and Properties: Part II—Explosion Effects in Water. NSWC/WOR TR 76-116.

Xing-ning, P., Wu, N., Bo,Y. 2009. Capacity of surface warship’s protective bulkhead subjected to blast loading. J. Marine. Sci. Appl. 8: 13–17. Yanchao, S., Zhongxian, L., Hong, H. 2008. Mesh size effect in numerical simulation of blast wave propagation and interaction with structures. Trans. Tianjin Univ & springer-verlag 14:396–402.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Behavior of stiffened panels exposed to fire M.R. Manco, M.A. Vaz, J.C.R. Cyrino & A. Landesmann COPPE/UFRJ, Rio de Janeiro, Brazil

ABSTRACT: This paper presents a numerical investigation of the behavior of stiffened steel panels, accounting for different configurations of the Passive Fire Protection (PFP) layer, under fire conditions. In the simulations, nominal temperature-time curves, as well as variations in mechanical and thermal properties of steel due to temperature, determined by EUROCODE1 and 3 [EC1 e EC3] (2004) were used, and a shell finite element model was developed using ABAQUS® commercial software. The thermal and mechanical analyses were decoupled; therefore, the transient temperature field due to fire was first assessed and, then, this thermal load was applied to the structure for the evaluation of its mechanical behavior. The initial geometry of the model took into account the initial imperfections recommended by ISSC (2012). The fire conditions were assumed as a result of a typical process involving combustion of hydrocarbons. Once the fire scenario was set, it was possible to assess the development of the temperature field with respect to time. Induced thermal loads were taken into account in the analysis of the structural model in combination with the pre-existing operating loads, thereby allowing the evaluation of the panel behavior. The purpose of this paper is to present the methodology employed to assess a steel ship structure in case of fire, which can be generalized to represent diverse conditions. The results show that the use of protective materials delays fire heating, improving its behavior during the fire. The choice of the best PFP solution depends on the load case and panel configuration. Finally, it is concluded that the methodology employed in this study can be used in the optimization process of the PFP layer. 1 1.1

INTRODUCTION Current scenario

The design of steel structures for the offshore exploration and production of oil and gas provides for the consideration of different scenarios related to severe accidents, including: large waves, extreme winds, earthquakes, collision between vessels and fires. For all these conditions, the integrity of the facility must be ensured and, in some cases, damages must be limited, ensuring the maintenance of safe operating characteristics for a given period of time. Due to the presence of large volumes of flammable materials (liquid and gas), equipment operating at high temperatures, active flames (e.g., flares) and human lives, living and interacting in confined spaces, the manufacturers and operators of these facilities are required to follow strict fire protection criteria (deemed the most severe among all industrial facilities). The occurrence of a fire in this type of structure is considered one of the most unfavorable conditions, for the combustion of hydrocarbons presents a very high rate of temperature increase in the initial stage of the fire, causing a very rapid loss in mechanical properties of steel, thus generating the possibility of deaths and economic and environmental damages.

Tragic examples of this kind of accidents are those which took place in the Piper Alpha platform in the UK, in July 1989, killing 167 of the 229 occupants in less than 22 minutes and, most recently, at the Deepwater Horizon platform, in the Gulf of Mexico in April 2010, where 11 people disappeared and a large environmental impact was generated. As a result of the analysis of the Piper Alpha platform disaster, made by Cullen (1990), the adoption of security measures based on principles of structural performance (performance-based analysis) was suggested in substitution of merely prescriptive recommendations, deemed too generic. Cullen (1990) noted that the sole suggestion of fire protective coatings for structural elements (even in large quantities) does not necessarily guarantee safety; in other words, each situation must be evaluated in order to rationally dimension the PFP layer required. Additionally, it should be noted that the presence of passive fire protective results in significant facility cost implications, either as far as their application or even their preventive maintenance are concerned. Other operational issues must also be considered, for example: additional weight, reduced internal spaces, difficulties to inspect structural components, among others. These issues significantly increase the facility costs.

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In the late 90s, PETROBRAS initiated a development program in order to rationalize the use of passive protection systems that would allow addressing the consequences of a fire in a global manner. According to Mendes (1996), the methodology was applied to different platforms, and resulted in a significant reduction of costs associated with passive protection systems. 1.2

Scope and objective

In this context, this work presents only the application of a methodology of computer numerical analysis to evaluate the behavior of a stiffened panel exposed to fire with different configurations of the PFP layer. The numerical analyses are performed using the commercial code ABAQUS®, according to the Finite Element Method (FEM), taking into account the structural and thermal effects resulting from the proposed fire. Variation in thermal and mechanical properties of materials in case of high temperature conditions are taken into account in the analysis, in accordance with the applicable standard recommendations, such as is the case of part 1.2 of EC3 (2004). The fundamentals of the applied analysis model are described in item 2 below. A case study is proposed and briefly described in item 3 hereto. A panel is submitted to a fire scenario, caused by the burning of hydrocarbons, Part 1.2 of EC1 (2004), allowing to evaluate the thermo-structural behavior for different instances of the fire. The main results obtained with the numerical model are presented and critically assessed in item 4 of this work, taking into account the fulfillment of security requirements. The main conclusions drawn from the analyses developed are referenced in item 5, indicating that this methodology not only allows meeting the safety requirements, but also shows great potential for application in the reduction of the use of passive protection structure. 2

behavior as a function of the elapsed time of fire, in other words, depending on the thermal conditions of fire exposure and applied external loads (mechanical). Computational characteristics adopted in this final stage of the numerical simulations are briefly described in item 2.2 hereto. 2.1

Thermal analysis

The numerical model used FEM to solve the twodimensional transient heat conduction problem, as shown in Cook (2002), Skallerud and Amdahl (2002), Lewis et al. (2004) and Landesmann et al. (2010). The DS4 element was used to model panels and the DC1D2 element was used to represent the thermal protection layer, respectively made of 4 and 2 nodes. The partial differential equation which expresses the temperatures (in degrees Celsius) T(x,y,z,t) is shown in Equation (1), subject to a temperature field defined in its contour Ts, which is represented in this analysis by fire-temperature curves (Part 1.2 of EC1 (2004)). ∂ ⎛ ∂T⎞ ∂ ⎛ ∂T⎞⎞ ∂ ⎛ ∂T⎞ ∂T k⋅ + ⎜k ⋅ ⎟ = ρ ⋅ c ⋅ (1) ⎜k ⋅ ⎟ + ∂x ⎝ ∂x⎠ ∂y ⎜⎝ ∂y⎟⎠ ∂z ⎝ ∂z ⎠ ∂t where ρ is the specific mass of steel (assumed temperature independent), ρ = 7850 kg/m3, c is the specific heat and k is the thermal conductivity. In this paper, the thermal properties of steel, as a function of temperature, are provided by part 1.2 of EC3 (2004) and shown in Figure 1. When prescribed temperatures are different from temperatures on the surface, a heat flux qn with two portions is generated: (i) one due to convection and another (ii) resulting from radiation, which can be written in a single equation through the linearization of the radiation portion, as below: qn = qc + qr

(

c

r

)

(Ts − Tg )

(2)

ANALYSIS METHODOLOGY

The analysis is carried out using FEM, the model includes a direct and rigorous consideration of nonlinear physical and geometric effects on the numerical formulation, allowing the estimate of the possible structural collapse modes. The proposed analysis procedure begins with the review of the panel layout with the selection of the fire scenario. Then, the thermal analysis is performed, which purpose is to determine the variation of the temperature in the elements exposed to fire. The main numerical formulation aspects of this stage are addressed in item 2.1. The final stage of the procedure aims at determining the structural

Figure 1. Specific heat and thermal conductivity of carbon steel as a function of the temperature.

102

(

)(

)

+ , εr is the where: α r ε r .σ r . Ts 2 − Tg 2 ⋅ resulting emissivity, defined as 0.8 (for steel); σr is the Stefan-Boltzmann constant (5.67.10−8W/m2 K4); and σc is the convective heat coefficient adopted as 50 W/m2K (part 1.2 EC1 (2004)). Denoting C as capacitance matrix, Kl and Kc are conductivity matrices (Kl + Kc = Kt), fb as vector of nodal flux due to convection, Equation (1) can be rewritten: ⋅

∂T (t ) + Kt T (tt ∂t

fb (t )

(3)

Solution of Equation (3) is based on FEM, being possible to determine the temperature at time n + 1 based on data at time n: Tn +1 =

{⎡C ⎣⎡C

(

)

tKt⎤⎦ Tn + t ⎡f tK ⎣ fbn + C K ( t t)

(fbn +

}

fbn)⎤⎦

(4)

of 4 nodes and 6 degrees of freedom per node (translations and rotations around global axes X, Y and Z), with capacity for developing nonlinear physical and geometrical analyses. The complete Newton-Raphson solution process is adopted to update the matrices and the linear solution of equations. The Von Mises criterion is adopted for determining the element plastification criterion. Apart from the thermal deformation imposed on the structural model, variations in the mechanical properties of steel as a result of temperature, as shown in Figure 2, are also taken into account, including reduction: of yield strength (fy,θ), modulus of longitudinal elasticity (Eθ) and yield point (fp,θ) obtained based on recommendations of part 1.2 of EC3 (2004). Defining fy,20 as the characteristic yield stress and E20 as the modulus of elasticity, at environment temperature, the reduction factors (ratio between the values of the property considered at a

where Δt is the time interval, γ is the temporal integration factor (taken as 0.9) and the initial temperature throughout the solid is assumed to be equal to 20°C (To). Analyses presented here uses the nominal fire curve corresponding to the burning of hydrocarbons (Part 1.2 of EC1 (2004)), as given by Equation (5):

(

Tg (t ) = To 1080 1 0.325 e−0.167.t − 0.675 ⋅ e

2.5.t

)

(5) where: t is the elapsed time of fire (in minutes), Tg is the temperature in the middle (in °C) and To is the initial temperature (equal to 20°C). 2.2

Figure 2. Stress-strain relationship for carbon steel at elevated temperatures.

Structural analysis

Since the variation of the temperature field was established in the previous analysis stage, the finite element mesh used, i.e., the nodal coordinates, the elements connectivity and the results for heat fluxes are used in the simulation of structural behavior under the postulated fire conditions. The procedure is initialized by the application of external loads, including the own structural weight, fluid action and other operational loads. At this stage, deformations and their respective stresses, corresponding to normal operating conditions of the panel, can be seen. The variation of the temperature field determined in the thermal analysis is imposed to the structural model along with other external loads applied. In building the mesh of finite elements for the structural analysis, the S4R element is used for the panel simulation. This element is composed

Figure 3. Reduction factor for the stress-strain relationship and thermal expansion coefficient for carbon steel at elevated temperatures.

103

temperature θ and initial temperature of 20ºC, e.g. kE,θ = Ea,θ/Ea,20), as well as the coefficient of linear thermal expansion of steel, as a function of temperature (αa,θ), are shown in Figure 3. The thermal protection material was not considered in the structural analysis due to the fact that it has many mechanical properties lower than steel, which would cause the addition of very small terms in the stiffness matrix. 3

CASE STUDY

A stiffened panel is considered, as part of a platform deck, under fire resulting from the burning of hydrocarbons. In this panel, shown in Figure 4, points were selected to present the results of analyses. The temperature of the hot gases (on the side of stiffeners) is described by Equation (5) while the outdoor temperature was considered constant and equal to 20°C. For simplicity, we considered only the weight of the structure itself and a lateral pressure of 0.10 MPa. As boundary conditions, we considered the four panel edges clamped. The geometry and initial geometric imperfections patterns of the plate and stiffener of the panel were considered equal to the benchmark study of ISSC (2012), which is presented in Table 1 and Equations 6 and 7, respectively. w( x , y ) =

bp 200

sin

mπ x π y Lu πx πy sii sin + sin sin i Lu bp 1000 Lu NS ⋅ bp (6)

Figure 4.

Figure 5. Settings of the thermal protection layer on the reinforced panel.

where m = Lu/bp + 1 and NS is the number of stiffeners v( x , z ) =

4.1 Table 1.

Geometry of stiffened panel.

Length of stiffened panel Width of stiffened panel Plate thickness Stiffener web height Stiffener web thickness Stiffener flange width Stiffener flange thickness Yield strength Slope of linear elastic range

Lu bp tp hw tw bf tf fy,20 E20

2550 mm 850 mm 16 mm 235 mm 8 mm 90 mm 10 mm 324 MPa 210 GPa

(7)

Once the panel is subject to fire, we seek to ensure the maintenance of its functionality over a period of two hours, considered sufficient for the arrival of rescue brigades. In order to guarantee such time, three different configurations of the PFP layer, shown in Figure 5, were considered. This plaster layer is 5 mm thick and has the following properties: specific heat of 1700 J/kg ⋅ K, thermal conductivity of 0.2 W/mK and density of 800 kg/m3, deemed independent from temperature, Vila Real (2003). 4

Stiffened panel model.

hw L z mπ x πz si sin si sin i + u 200 Lu hw 1000 hw

RESULTS Thermal analysis

Figures 6–8 show the distribution of temperatures on the plate, web and flange of stiffener, respectively (taken at points shown in Figure 4), for different time instants after the start of the fire. A 400°C temperature is taken as reference, since above this, the mechanical properties show a very pronounced decline (see Figures 2 and 3). On the plate (Figure 6), we saw that, between points A and B, in cases where there is an equivalent thermal protection, the temperature distribution is the same, while between points B and C, the

104

Figure 6. in time.

Temperature at points A, B and C of plate,

Figure 7. Temperature at points C, D and E in the web of the stiffener, in time.

more, it should be pointed out that the temperature on the plate does not reach 1100°C, which is the maximum temperature in the considered fire curve (Hydrocarbon Temperature-Time Curve, HT-HT), since one of its sides is in contact with the environment, where temperature is 20°C constantly. The stiffener web (Figure 7) has the highest rate of warming, due to the large ratio of the area exposed to fire and little volume of the element. Thus, for cases 1 and 3, the temperature of 400°C, at point D, is reached within 4 minutes, while in cases 2 and 4, it is reached in 55 min. The difference between the temperature distribution at points C and E, in the analyzed cases, is due to heat flux from the web to other elements. Finally, on the flange of the stiffener (Figure 8), we have the same temperature distribution for cases 1 and 2, however, in cases 3 and 4, the same trend is found, but with different magnitudes. In cases 1 and 2, the 400°C temperature is reached in less than 4 min. On the other hand, in case 3, this temperature is reached only at points E and F. In case 4, the temperature does not exceed the threshold value over the time interval examined. Considering the severity of the fire in case 1, where there is no thermal protection, temperatures in the different panel components quickly reach temperatures above 400°C. This indicates that the properties of the structural elements will be significantly affected by the fire, reaching their capacity limit, in other words, there will be a structural failure, besides the possibility of large areas under plastic regime. The continuity of the fire, with a progressive rise in temperature, could cause the local perforation of a portion of the panel, enabling the propagation of smoke, heat and flames to other areas of the platform. It should be noted that the protection layer considered in cases 2, 3 and 4 was not able to reduce the temperature and keep it below 400°C, over the whole panel, thus its resizing shall help increase the safety of the platform. This protection layer should be optimized, reducing cost and weight in these facilities, notwithstanding security. 4.2

Figure 8. Temperature at points E, F and G in the flange of the stiffener, in time.

distribution changes according to the heat flux in the stiffener. The reference temperature (400°C) in case 1 is reached, across the plate, in about 4 minutes, while, in other cases, times change significantly between points A, B and C, which should be taken into consideration upon the mechanical analysis. Further-

Structural analysis

Based on information on the variation of the temperature field in the panel, we have checked the state of stresses and deformations in different postulated fire instants. Here, in addition to actions resulting from thermal deformations (fire), external actions (mechanical) applied on the structure were also taken into account. It should be mentioned that the load condition considered significantly affects behavior of the stiffened panel, altering the time of resistance to fire. In this paper we will con-

105

sider only the action of gravity and uniform lateral pressure of 0.10 MPa. Figure 9 shows the stress fields after application of considered loads. Case 1 evaluates the behavior of the panel without any thermal protection, and is therefore the most critical case among the four ones considered, which justifies its separated analysis. In this case, the structural analysis was stopped after 78 minutes. Figure 10 shows the distribution of Von Mises stress in the panel at different time instants. We can see the fast growth of vertical displacements, indicating the rapid loss of strength due to the high rate of heating in the panel. The stiffener web is the most critical element, since in about 15 minutes, its stiffness decreases, starting to twist and losing, therefore, its load capacity and behaving as a membrane. Figure 11 illustrates the variation of vertical (U2) and transverse (U3) displacements, respectively, at points A, D, E, H and I over time. This figure shows the beginning of relative displacements (vertical and horizontal) between points D-E and

Figure 11. Transverse and vertical displacements in points A, D, E, H and I for different instants of time for Case 1. Figure 9.

Figure 10. Case 1.

Stress field after application of loads.

Stress field for different instants of time for

H-I, indicating the beginning of the stiffener torsion. We can see the different final configurations between the right and left stiffener, due to the influence of boundary conditions. It is important to note, also, that the vertical displacement of point A reaches 0.6 m, approximately. We can see that stresses at the ends of the panel are higher due to the temperature distribution in those areas, where temperature is lower, which is reflected in a lower decrease of mechanical properties, as shown in Figures 2 and 3. Figure 12 presents the stress states for cases 2, 3 and 4. It can be seen that, in case 2, stiffener behavior is similar to case 1, stiffener begins to twist off after 60 minutes of analysis as the plate, up until then, had load capacity. In case 3, the largest deformations are located on the plates between stiffeners, since, in this case, the plate is the most weakened element. In case 4, the mechanical properties of both the plate and the stiffener are weakened, but the panel behaves well due to the fact that the stiffener flange has a good resistance preventing the rotation of the stiffener. One should also mention that this behavior is valid only for the

106

Figure 13. Vertical displacement of point A for different instants of time for all cases.

ical properties, improving the structural behavior of the panel. In Figure 13 we can see the vertical displacement of point A in all analyzed cases, as well as the final configuration of the panel in half section. As expected, case 4 shows the best performance, plus it has also the highest weight of the protective layer. 5

Figure 12. Stress field for different instants of time for Cases 2, 3 and 4.

assumed loading condition. In a real situation, where the structure undergoes the action of waves, a distinct behavior can be seen. Results show that the thermal protection layer delays, in all cases, the heating of the protected element by helping to maintain over time the mechan-

CONCLUSIONS

The numerical-computational methodology for analyzing the behavior of steel structures under fire conditions presented in this paper was applied to evaluate the behavior of a stiffened panel subject to a fire scenario. In spite of idealized loading conditions, we can see the thermomechanical behavior for different instants of the postulated fire. Based on the results of case 1, the need to apply PFP elements is evident. In cases 2 through 4, we find the final configurations of the panel, showing the difference between behaviors for each case. It should be noted that considerably severe fire conditions were assumed with the assignment of a standard curve for the burning of hydrocarbons. Finer studies can be applied in order to make more realistic simulations of fire scenarios, for example, the use of models under CFD (Computational Fluid Dynamics) and, therefore, the mechanical properties of the structure could be estimated in a more reliable manner. Conclusions reached based on numerical simulations discussed in this work indicate that the methodology presented hereto can be applied to assess the structural performance of offshore structures, with different loading conditions as well as different thermal protection materials, with potential application in the reduction of the use of PFP, notwithstanding the preservation global security levels. Finally, despite considering ideal-

107

ized thermal and mechanical loads, for more complex and real situations, the analysis procedure is the same. ACKNOWLEDGEMENTS The authors wish to express their gratitude to the National Petroleum Agency of Brazil (ANP) and COPPE-UFRJ for their support for the development of this work. REFERENCES Cook, R.D., Malkus, D.S., Plesha, M.E. and Witt, R.J. (2002), Concepts and Applications of Finite Element Analysis, 4th Ed., John Wiley and Sons, New York. Cullen, L. (1990), “The Public Inquiry into the Piper Alpha Disaster”, HM Stationery Office. European Committee for Standardization (2004), EUROCODE No. 1: Actions on Structures, Part 1–2: Actions on Structures exposed to Fire, ENV 1991-1-2, British Standards Institution, London, UK. European Committee for Standardization (2004), Eurocode No. 3: Design of steel structures, Part 1.2: Structural fire design, ENV 1993-1-2, British Standards Institution, London, UK.

ISSC (2012). Report of Specialist Committee III.1 Ultimate Strength, Proceedings of the 18th International Ship and Offshore Structures Congress (ISSC 2012), Edited by Wolfgang Fricke and Robert Bronsart, Rostock, Germany, Vol. 1, pp. 285–363. Landesmann, A., Mendes, J.R., Ellwanger, G. (2010), “Numerical Model for the Analysis of Offshore Structural Elements under Fire Conditions” Proceedings of XXXIV Jornadas Sudamericanas de Ingeniería Estructural, San Juan, Argentina. Lewis R.W., Nithiarasu, P. and Seetharamu, K.N. (2004), Fundamentals of the Finite Element Method for Heat and Fluid Flow, John Wiley and Sons, England. Mendes, M.F., “A Methodology for Fire Computational analysis in Offshore Instalations” (Portuguese), D.Sc. Thesis, Civil Engineering Program, COPPE/UFRJ, 1996. Skallerud, B. and Amdahl J. (2002), Nonlinear Analysis of Offshore Structures, Research Studies Press Ltd., Baldock, Herforshire, England. Vila Real, P.M. (2003), Fire on Steel Structures— Structural Calculation (Portuguese), 1st Ed., ORION Editions, Portugal.

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Structures on ice

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Cost optimization for ice-loaded structures S. Ehlers Norwegian University of Science and Technology, Department of Marine Technology, Trondheim, Norway

P. Kujala Aalto University, Department of Applied Mechanics, Marine Technology, Finland

ABSTRACT: Economic ship transport in Arctic and Baltic waters requires the vessel to be competitive both in the ice-infested and ice-free season. Thus, the aim of this paper is to present an optimization procedure to identify a production cost and weight optimum hull structure, i.e. stiffened panel scantlings. Furthermore, the current uniform design ice load is utilized as well as a non-uniform load. The latter is identified based on typical measured damages occurring over the lifetime of a vessel. Consequently, the scantlings for the plating and frames will be identified under both types of loading in view of production and repair cost. Additionally, the identified sensitivity of the scantlings to the loading type will contribute to an increase in safety for ships operating in ice-infested waters. 1

INTRODUCTION

The industry standard for the design of ice going vessels concerning hull strengthening and propulsion in ice is set by the Finnish-Swedish Ice Class Rules (FSICR, 2011) for the lower polar, i.e. Baltic sea, ice classes. They originate from the FinnishSwedish winter navigation system, which has been developed, because during an average winter every Finnish port and every Swedish port north of Stockholm is ice bound. Furthermore, every port in Estonia is ice bound during an average winter. Thus, economical feasibility of the shipbased import and export depends on safe winter navigation through the ice-covered waters for these countries in the Baltic Sea. However, it is also this economical feasibility which conflicts with the ship design criteria as the current rules develop along a thin line of acceptable ice induced damages, see e.g. Hänninen (2005), and acceptable CA-, VO— and OPEX additions for the vessels ice capabilities. Nevertheless, these rules have been adopted by all major classification societies. Under these rules, vessels are engineered on the basis of vast experience with single year ice, which appears to be applicable to a large extent for the Barents Sea, also given the fact that the Baltic Sea ice is considerably harder due to its low salinity. Concerning the higher polar classes, the IACS polar code (2011) seeks to harmonise a variety of different approaches into a single framework. Currently, both FSICR and IACS utilize a simplified rectangular pressure patch as means to introduce the ice load into the side plating. The influence of this simplification compared

to a non-uniform load resulting from actual icestructure interaction will be investigated in this paper. The reason for this investigation arises from the fact that ships are typically damaged due to ice (Hänninen, 2005) and that the FSICR and IACS accept some plastic deformations, even though the actual design limit in case of the FSICR is yielding and the exact amount of plasticity is not clearly defined in IACS (Kaldasaun and Kujala, 2011). Furthermore, Kujala (1991) presented detailed non-uniform measurements of permanent plate deflections as a result of ice-structure interaction. Consequently, the aim of this paper is to investigate the influence of the design ice load distribution over the simplified pressure patch. Therefore, a non-uniform load will be created as a result of the permanent deflection measurements from Kujala (1991). Furthermore, the sensitivity to the choice of stiffener spacing, plate thickness and stiffener scantlings to the weight and production cost will be assessed by means of an optimization algorithm. Therein, direct calculations using the nonlinear Finite Element Method (FEM) will be carried out to assess the stresses as function of the load carrying mechanism following the stiffened plating supported by webframes and stringers. The scantlings will be considered as variables to further identify convergence of the design, respectively optimum compliance to the objectives. However, in order to reach compliance to the objectives, here the minimization of cost and mass, conflicts are faced. For example, being incapable to create a lightweight and inexpensive design might lead to conflicting objectives. Therefore, this paper will

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combine these objectives in a straightforward fashion allowing for their combined minimization at equal treatment. Recently the use of Genetic Algorithms in multiobjective optimisation is quite popular (e.g. Okada and Neki 1992, Klanac et al. 2009). Furthermore, Ehlers (2010) utilized a Particle Swarm Optimisation (PSO) algorithm to identify optimized crashworthy Marine Structures consisting of stiffened and supported plates. Therefore, a PSO algorithm coupled to nonlinear FE-simulations and a production cost assessment will be utilized in this paper to optimize the design alternatives in compliance with the uniform and non-uniform ice loading. As a result, we will obtain the sensitivity of the design alternatives to the production cost and mass as well as the utilization of the individual structural components by means of stress state following the chosen design load. Furthermore, we will identify if both types of loading result in the same scantlings and if the concept of a uniform pressure patch suffices. The latter will be presented for a case study vessel operating in the Baltic Sea. 2 2.1

Figure 1. Long-term measure load on a frame at bow, midship and aftship of MS Kemira (Muhonen, 1992).

PROBLEM DEFINITION Configuration of typical ice induced loads and hull damages

The ice-induced loads are known to have strong stochastic nature due to the stochastic nature of ice strength properties and ship-ice interaction process. As ice is formed in nature, numerous variables affect the mechanical and physical properties of ice. In addition ship operations in ice can have various forms: independent navigation or navigation with icebreaker assistance e.g. in level ice, ice floes and ridged ice with various amounts of first year and multiyear ice features. Further, the possibility of the ship getting stuck in moving ice has to be included. The ice-breaking process has been successfully simulated lately in level ice with independent navigation (Su et al. 2011), but still the physical process of ice breaking is not captured in all operative scenarios and ice conditions. Therefore, the long-term full-scale measurements give the most reliable basis to evaluate the load level as a function of occurrence frequency (return period). Figure 1 shows an example the long-term loads measured on-board MS Kemira during a 7-year period from 1985–1991 (Muhonen, 1992). Typical ice induced damage can be seen in Figure 2 where the plating between frames has experienced a few centimetres of permanent deflections whereas the transverse frames have somewhat minor deformations. This type of damage has to be prepared after the winter season.

Figure 2.

Typical ice induced damage (Kujala, 1991).

Figure 3. A sketch of design point definition (Riska, 2007, Riska and Kämäräinen, 2011).

For the design purposes, the knowledge given in Figure 1 and 2 should be combined. The main question for risk-based design purpose is how often such damage case can take place or if any permanent deformations shall be permitted. This process can also be seen in Figure 3, which presents the current approach used in the FSICR rules. Therein, local yielding of the plating and

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frames is accepted annually and the plastic limit state, i.e. no noticeable permanent deformations, can occur once during the ship’s lifetime (Riska and Kamarainen, 2011, Kaldasaun and Kujala, 2011). Consequently, the typical permanent damage presented in Figure 2 is likely to occur and most importantly it is non-uniform, i.e. the load acting on the plating due to ice contact is different from a constant pressure applied over a given area. However, such simplified ice loading with the pressure, p, acting on a load patch of height, hc, and length, L, given as p × hc × L is defined in the FSICR. This load is always to be placed so that the height is at the mid-span for the frames and plating. The longitudinal location is symmetric with respect to the structural element investigated. Hence, in this paper we will at first derive a nonuniform load based on the typical deflection measurements presented in Figure 2. The absolute load level, i.e. average over the application area, will be set equal to load obtained from the FSICR. This becomes necessary to allow for a straightforward comparison of both loading types; hence, for the latter also the basic application area will be kept equal. Furthermore, both loads will be utilized in an optimization procedure to assess their influence of the optimum scantlings in terms of weight and production cost. 2.2

Example ship: MS Kemira

The example ship used in this paper to obtain the ice loading is shown in Figure 4. Her main particulars are given in Table 1. The webframe spacing of the panel to be analysed with the Finite Element Method is 2.1 m with an aspect ration of 1; see Figure 5.

Table 1.

Length between perpendiculars Beadth Draft v (reference) DWT MCR

105 m 17.5 m 8.0 m 14 kn 8145 t 3400 kW

Figure 5. Panel overview and model extent for the finite element analysis.

3

THE PARTICLE SWARM ALGORITHMBASED OPTIMIZATION

An economically viable design is a lightweight design that is cheap to manufacture and fulfils the operational requirements. In other words, it is the design with the lowest Cost per Mass (C/M) ratio, which reduces both cost and mass. The choice of this single objective comprising the two characteristic and conflicting measures arises from the fact that this tends to describe the physical target behaviour of the structure best. Further, it does not identify alternatives at the extreme ends of the objective space, i.e. lightweight structure at high cost, due to the relative representation. As a result, the following optimisation formulation can be stated

max f ( x ∈X

Figure 4. MV Kemira and the used instrumentation to obtain the data given in Figure 1 (Kujala, 1989).

Main particulars of MV Kemira.

⎡ ⎢ )⎢ ⎢ ⎣

∑

gi ( x ) ≥ 0 i =1,..., 4

R ( Rgi

⎤ ⎥ )⎥ ⎥ ⎦

(1)

where we search for a vector of variables x=( ) that minimizes the design objective f ( x ) = ( , as well as satisfying )/( the design constraints gi( ). R is the penalty factor, which helps to generate feasible solutions

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from marginally infeasible solutions and thereby allows the algorithm to progress also from the infeasible side. The length of x arises from the 7 strake variables defined in Figure 6 and their range shown in Table 2. In the latter, the change in stiffener spacing, s, is realized by altering the amount of stiffeners placed on the strake. The webframe spacing is kept constant at 2.1 m according to MV Kemira’s scantlings. Alternatives satisfying the constraints are called feasible alternatives. Feasibility of the design constraints is assessed by evaluating the design constraints with the nonlinear finite element method based on the exceedance of local yielding for each structural element loaded in compliance with the equivalent ice loading for 1 AS and for the non-uniform patch load following the typical ice induced damage.

The design limit state given for 1 AS according to FSICR is yield, which is also adopted for this comparison. The non-uniform load follows the typical damage presented in Figure 2 and results in a displacement field as shown in Figure 7. The non-uniform load itself consists of a uniform basic load applied over the entire patch (2.1 m × 0.35 m), followed by one patch applied at one half of the webframe spacing with a width of 0.2 m and a peak load with a length of one frame spacing and a width of 0.1 m, see Figure 8. The peak load is applied right next to the webframe with a three times larger weight factor compared to the other two loads while maintaining the same average load for the entire patch area as applied in the standard FSICR case. The resulting out-of-plane displacement shown in Figure 7 approximates the deflection measurements; see Figure 2, with reasonable accuracy. 3.1 The optimization procedure A Particle Swarm Optimisation (PSO) algorithm is chosen to identify the conceptual design

Figure 6. Sketch of a strake section with adjacent structure and optimization variables. Table 2.

Discrete optimization variables.

Thickness [mm] Stiffeners Stiffener type

10, 11 … 30 1, 2 … 16 HP100 × 6, HP120 × 8, HP140 × 8, HP160 × 8, HP180 × 10, HP200 × 10, HP220 × 10, HP240 × 10, HP260 × 12, HP280 × 12, HP300 × 12, HP320 × 13, HP340 × 14, HP370 × 13, HP400 × 16, HP430 × 15

Figure 7. Relative out-of-plane displacement shown for the non-uniform patch load and the uniform patch load.

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Table 3.

Bilinear material behaviour.

Density Young’s modulus σyield Tangent modulus

7850 kg/m3 210 GPa 235 MPa 1.14 GPa

simulations; see Hallquist (2007). A bilinear material behaviour is implemented using material 3 of LS-DYNA (Hallquist, 2007); see Table 3. 3.2 Production cost Figure 8.

Uniform versus non-uniform load.

alternatives. PSO is a population-based optimisation technique developed by Kennedy & Eberhart (1995). Particle Swarm Optimization (PSO) algorithms utilized in this paper is based on Jalkanen (2006) and described in detail in Ehlers (2012) for ship structures. In this study, the PSO algorithm is used as a tool to rationally increase the objective function value, even though a global optimum might not be reached at termination. The latter also arises from the fact that the employed PSO algorithm is a stochastic process, which typically requires multiple runs to verify the overall convergence of the problem. The explicit nonlinear Finite Element Method (FEM) solver LS-DYNA version 971 is used for the ice patch load simulations. The ANSYS parametric design language is used to build the finite element model for the nine-field strake model with variable structural dimensions; see Table 2 and Figure 5, and a full webframe and stringer height of 0.9 m, which remains constant. For details on the parametric modelling see Ehlers et al. (2008). The three dimensional parametric model is clamped at the transverse edge and simply supported at the longitudinal edges. As a result, web frames and stringers surround the centre strake, thus resembling realistic boundary conditions for this panel of interest. Consequently, both ice loads are applied to the structural elements of this centre panel as a uniform and non-uniform pressure patch. The structure is modelled using four noded, quadrilateral Belytschko-Lin-Tsay shell elements with 5 integration points through their thickness. The finite element length is variably defined to result in three elements per stiffener spacing and height. This element size is well justified for this primarily elastic analysis, yet it sufficiently allows for non-linear structural deformations, which may occur in the vicinity of the design point. Standard LS-DYNA hourglass control is used for the

The steel structure production cost of each alternative is calculated with a cost module according to Rigo (2003). The cost is based on a simplified calculation of labour and material costs. The calculated cost is calibrated referring to the cost of a straight stiffened panel using unitary production costs of the yard. The production cost is calculated as a sum of three components: Cost total =

∑ (costt material + costconsumables

panel

+ cost labour )

(2)

Material cost includes raw material cost for the plate and stiffeners. Cost of consumables consists of the costs from welding (energy, gas, electrodes and provision for equipment depreciation). Labour cost is based on the workload for surface preparation and welding. 4

RESULTS OF THE OPTIMIZATION PROCEDURE

The progression of the normalized objective best particles in terms of mass and cost are shown in Figure 9. Normalization of the cost and mass is done for convenience of plotting in the form of the ratio to the maximum value found for all objectives. Furthermore, it can be seen that 30 iterations resulted in a reasonable convergence of these characteristic values, thus indicating the sensitivity of the different design variables. Additionally, Figure 10 shows the best particles of each optimisation run for all iterations and clearly visualizes the range in mass and cost found for the best particles, which totals to as much as 11%/11% and 27%/31%, for the uniform and non-uniform load respectively. The progression of the shell plate thickness can be seen in Figure 11, the progression of the stiffener spacing in Figure 12, the stiffener type following the order given in Table 2 in Figure 13 and the volume of the eventual bracket

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Figure 12. Progression of the stiffener spacing for both loads.

Figure 9.

Progression of normalized mass and cost.

Figure 10.

Best particles of each optimisation run.

Figure 11. loads.

Progression of the plate thickness for both

Figure 13. Progression of the stiffener type for both loads.

in Figure 14. Furthermore, the progression of the utilization in terms of stress of these structural components is shown in Figure 15. Consequently, it can be seen in all figures that the non-uniform load results in a significantly higher demand of

Figure 14. loads.

Progression of the bracket volume for both

the structure and thus results in an increase in weight and production cost. Furthermore, the optimization procedure suggests that a bracket is not required for the uniform load, while it is suggested for the non-uniform load. The latter will

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indicates that the simply application of a uniform patch load may not suffices. The latter is clearly confirmed by the typical permanent deflection measurements presented. Hence, the utilization of non-uniform ice loads for the scantling design of ice going vessels must be explored further to contribute to the minimization of ice induced damages and thereby to an increase in safety in ice-infested waters. REFERENCES

Figure 15. Progression of the stress levels in each component for both loads. Table 4.

Resulting best design alternative.

Load

Uniform

Non-uniform

Difference

Cost Mass t [mm] s [mm] Type Bracket σplate σstiffener σbracket

0.59 0.64 10 150 10 No 0.94 0.99 –

0.69 0.73 25 190 11 Yes 0.82 0.99 0.58

+0.1 +0.09 +15 +40 +1 +bracket −0.12 0 +0.58

contribute to the utilization of smaller stiffeners and thereby compensate the increased production cost for the bracket installation. The resulting best particle after 30 generations is presented in Table 4. 5

SUMMARY AND CONCLUSION

This paper presented the a brief background to ice loading and the utilization in terms of simplified pressure patches for the scantling design according to current rules and regulations. Furthermore, a non-uniform patch load was introduced, which was obtained according to a typical measured permanent deflection of the outer shell. Both loads are further utilized to optimize a stiffened panel from a case study vessel to minimize its cost and mass. As a result, it can be said that the large range in mass and cost found for the best particles using both loads clearly indicates the potential gain from optimized structures under ice loading in general. Furthermore, the increase demands in scantlings, and consequently mass and cost, arising from the utilization of a non-uniform load

Ehlers, S. 2012. A particle swarm algorithm-based optimization for high-strength steel structures. J of Ship Production and Design 28(1): 1–9. Ehlers, S., Klanac, A., Kõrgesaar, M. 2008. A design procedure for structures against impact loading. Jahrbuch Schiffbautechnische Gesellschaft, Springer. FSICR, 2010: Ice Class Regulations 2010: Finnish-Swedish Ice Class Rules 2010 Finnish Transport Safety Agency, 23.11.2010 TRAFI/31298/03.04.01/2010, p. 48. Hallquist, J.O. 2007. LS-DYNA. Keyword User’s Manual, Version 971, Livermore Software Technology Corporation. Hänninen, S. 2005. Incidents and Accidents in Winter Navigation in the Baltic Sea, Winter 2002–2003. Winter Navigation Research Board, Research Report No. 54, Helsinki, p. 39. IACS, 2011. Requirements concerning polar class. Intl. Association of classification societies. Jalkanen, J. 2006. Particle swarm optimization of load carrying structures, J. Structural Mechanics 2, 23–35. Kaldasaun, J., Kujala, P., 2011. Risk-Based Approach for Structural Design of Ice-Strengthened Vessels Navigating in the Baltic Sea. Proceedings of the 21st International Conference on Port and Ocean Engineering under Arctic Conditions, Montréal, Canada. Kennedy, J. & Eberhart, R. 1995. Particle swarm optimization. Proceedings, EEE Int. Conf. Neural Networks, Piscataway, 1942–1948. Klanac, A., Ehlers, S., Jelovica, J. 2009. Optimization of crashworthy marine structures. Marine Structures. 22(4); 670–690. Kujala, P. 1991. Damage statistics of ice-strengthened ships in the Baltic Sea 1984–1987. Winter Navigation Research Board. Report. No. 50. 61 p. + app. 5 p. Kujala, P. 1989. Long term ice load measurements onboard chemical tanker Kemira during winters 1985 to 1988. Helsinki. Winter Navigation Research Board. Report No. 47. 55 p. + app. 139 p. Muhonen, A., 1992. Ice Load Measurements on Board the MS Kemira, Winter 1991. Otaniemi, 1992. Helsinki University of Technology, Laboratory of Naval Architecture and Marine Engineering, Report M-121. ISBN 951-22-1173-4. Okada, T., Neki, I. 1992. Utilization of Genetic Algorithm for Optimizing the Design of Ship Hull Structure, J.S.N.A., Japan, 171; 71–83. Rigo, P. 2003. An Integrated Software for Scantling Optimization and Least Production Cost. Ship Technology Research 50: 126–141.

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Riska, K., Kämäräinen, J. 2011. A review of ice loading and the evolution of the Finnish-Swedish Ice Class Rules. Proceedings of the SNAME Annual Meeting and Expo. November 16–18, Houston (TX), USA.

Su, B., Riska, K., Moan, T., 2011. Numerical simulation of local ice loads in uniform and randomly varying ice conditions. Cold Regions Science and Technology, 65:145–59.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Measured ice loads and design ice loads M. Suominen, P. Kujala, R. von Bock und Polach & J. Kiviranta Aalto University, School of Engineering, Department of Applied Mechanics, Marine Technology, Espoo, Finland

ABSTRACT: The number of marine operations in ice infested waters has been increasing significantly in the past two decades. However, the level of knowledge on how ice affects the performance and safety of ships has still not been sufficiently explored. To increase the knowledge in this field more full scale ice measurements are required. In this context the Finnish Funding Agency for Technology and Innovation (Tekes) launched the PSRV full scale ice trial project in September 2011. The project aims to create a scientific basis for the design of ice-going ships. Within the project, a full scale ice trial in the Baltic Sea was conducted with PSRV S.A. Agulhas II during March 2012. During the ice trial the consortium conducted measurements on ice loads on the ship hull and propulsion system, ice induced vibration and noise, ship motions in ice, global ice loads, underwater noise, and mechanical and physical sea ice properties. For the ice load measurements three areas of the ship hull were instrumented: the bow, bow shoulder, and stern shoulder. This paper focuses on the ice loading measurements on the ship hull and a comparison to existing load prediction approaches. The load prediction methods presented here are based on the FinnishSwedish Ice Class Rules (FSICR) and the International Association of Classification Societies’ (IACS) Unified Requirements for Polar Ships. The ice loads measured during the ice trial are put into context with the design ice loads in design ice conditions. The comparison of the measurements and calculations shows a good compliance of the design line load based on FSICR ice class 1A Super and IACS PC 6 on the bow area with the line load. However, at the stern shoulder the measured line load is significantly greater than the design load proposed by FSICR and IACS Polar Class 6. 1

INTRODUCTION

Marine operations are extending more and more towards polar waters, which increase the significance of design ice loads. The design ice loads are important to withstand ice impacts during operations. The knowledge of the load characteristics is limited and classification societies use simple mechanical models to approach the problem. A deeper understanding of the ice loads is required to make seaborne traffic in ice safer and to improve the current rules. Full scale load measurements are expensive and hence not often conducted, but they are the most promising approach for increasing the knowledge. In the rules incorporated methods must be broken down to easily applicable formulations and therefore a deeper insight into the nature of the ice load is required. The comparison of load levels between various ice rules is a sensitive task as one has to account for the limit states used in the rules when comparing the load levels (Kaldasaun and Kujala, 2010, Riska and Kämäräinen, 2012). This paper contributes to highlighting the current differences between the Finnish Swedish Ice Class Rules (FSICR, 2010) and the guidelines of the International Association of Classification Societies (IACS, 2011) for ice loads.

A comparison of the ice loads proposed by IACS and ice loads with a non-constant distribution over the load patch (von Bock und Polach, 2006) indicated that a plastic design approach might cause plastic deformation of the ship hull. Quinton (2012) stated that the calculation methods used for the unified requirements (IACS, 2011) for static glancing impacts differ from dynamic loads. However, the research is based on laboratory experiments and would require additional validation against full scale measurements for better proof. Riska et al. (2002) stated that earlier findings on the nature of ice loads from full scale measurements have already been incorporated into the Finnish-Swedish ice class rules. In addition, the measured ice loads have been compared with loads derived from existing rules and a modification of the rules was suggested based on those measurements (Kujala et al., 2007). It is stated by Riska that the shortcomings of the rule formulations are known, but there is a need for new full scale data for additional development (Kujala et al., 2007, WP7.4). In order to gain more knowledge on the effect of ice conditions on ships and comfort on board, a new project funded by the Finnish Funding Agency for Technology and Innovation (Tekes)

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was launched in September 2011. The project aims to create a scientific basis for the design of ice-going ships. The main focus of the project is on relating ice loading on the hull and propulsion system of a ship to the prevailing ice conditions. In addition, the main causes of the noise and vibration on board the ship in various types of ice conditions are another main focus. In this paper the IACS and FSICR loads are compared with full scale measurements in the Baltic Sea in 2012 to highlight the load level observed during the ice trial of the PSRV S.A. Agulhas II. 2

DESCRIPTION OF THE SHIP AND INSTRUMENTATION

Prior to the launching of the Polar Supply and Research Vessel (PSRV) S.A. Agulhas II, several areas were equipped for measurements during the ice trial. The main dimensions of the ship are presented in Table 1. During the ice trial the consortium conducted measurements on ice loads on the ship hull and propulsion system, ice induced vibration and noise, ship motions in ice, global ice loads, underwater noise, and mechanical and physical sea ice properties. In addition to these, the ice conditions, such as thickness, concentration etc., were observed visually. The ice thickness was measured with an electromagnetic device and with a stereo camera system. Furthermore, the mechanical properties of ice were measured at two locations. In addition, the navigation and machinery data were recorded. As mentioned above, this paper focuses on the ice load measurements on the ship hull. Therefore only the measurements of the ice load on the hull are discussed in this paper. Three areas of the hull were instrumented for the ice load measurements: the bow, bow shoulder (on the bow intermediate ice belt), and stern shoulder (on the ice belt), see Figure 1. In total nine frames were instrumented to measure the ice induced loading on the ship hull; two at the bow, three at the bow shoulder, and four at the stern shoulder. In addition, the ice induced stress on the hull plating was measured with two strain sensors at the bow, two sensors at the bow Table 1.

shoulder, and six sensors at the stern shoulder. The instrumentation of the frames and hull plating is presented in Figure 2. The ice load acting on the frame is determined by measuring the shear strain on the upper and lower parts of the frame between the brackets (see Fig. 2). The shear force acting between the sensors is calculated from the difference between the measured shear strains on the upper and lower parts of the frame. The stress on the hull plating is determined by measuring the normal strain on the plating. The setup of the measurement instrumentation was calibrated through calibration measurement before the ice trial. In the calibration measurement, the force is applied to the structure by pulling on each frame separately with a winch while measuring the strains that occur and the added load. The added load is measured with a conventional load sensor between the frame and the winch. The calibration coefficients are defined from the known added loading and the strains resulting from it. A similar study was conducted with FE models of the instrumented areas and calibration coefficients were determined. As the bow instrumentation (frames Nos. 134 and 134+400) was in

Figure 1. Instrumented areas of hull indicated with black boxes.

Main dimensions of the ship.

Length, bpp. Breath mould. Draught, design Deadweight at design draught Speed, service Ice class

121.8 m 21.7 m 7.65 m 5000 t 14.0 kn +1A1, ice class IACS PC5

Figure 2. Instrumentation of the frames and hull plating at the bow (on the right), bow shoulder (in the middle), and stern shoulder (on the left).

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a ballast water tank which was filled before the measurements, it was not possible to conduct the calibration measurements in this area because of the tight schedule. As the FE model gave similar results for the stern and bow shoulder areas than the calibration measurements, the calibration coefficients for the bow area were determined with the FE model. 3

ICE TRIAL MEASUREMENTS

The ice trial took place between the 19th and 24th of March 2012. Because of the mild winter, ice infested waters could only be found in the northern part of the Bay of Bothnia. The ice conditions on March 21st are presented in Figure 3. The ship was operating in ice during the daytime of the 21st and 22nd of March. All the above-mentioned measurements were ongoing during this time and a total of 24 hours of measurement data were recorded. In order to gain more knowledge on the effect of the manoeuvring of the ship on the measured ice loads, different manoeuvring operations were conducted during the ice trials. These operations included turning tests in ice, breaking out of the channel, straight forward in level ice, straight forward in a channel, and ice ridge penetration tests. The maximum ice loads have the highest significance for the design. As this study compares the measured ice loads to the design ice loads, only the measured maxima are discussed. Figure 4 presents the time history of the measured ice loads

Figure 4. The time history of the measured ice loads on the bow when the maximum ice load occurred.

on the bow frame when the highest measured maximum ice load during the ice trial occurred. As can be seen from Figure 4, the ice loading has a clear stochastic nature as the ship was operating straight forward in visual constant ice conditions during the time period presented in Figure 4. The stochastic nature makes the prediction of the ice loading difficult and emphasises the importance of the full scale measurements, as discussed above. The measured maximum ice loads on the bow and bow shoulder were 404 kN and 575 kN, respectively. Both of these maxima occurred when the ship was sailing forward at speeds of 5.7 m/s and 0.72 m/s at the moment when the maximum ice load occurred. The maximum ice load measured on the stern shoulder was 458 kN, which occurred during a turning test. Following the rules, the measured ice loads on the frames are converted to line loads by dividing the measured loads with the frame spacing. Using this method, the measured maximum line loads were 1010 kN/m, 1437.5 kN/m, and 1145 kN/m for the bow, bow shoulder, and stern shoulder, respectively. 4

DESIGN ICE LOAD CALCULATION

In this chapter, the calculation methods for design ice loads based on the Finnish-Swedish Ice Class Rules and International Association of Classification Societies (IACS) are presented. The design ice loads are calculated for ice classes 1A Super (FSICR) and PC 6 (AICS). 1A Super and PC 6 are chosen for the calculations as those are corresponding ice classes in FSICR and IACS (IMO, 2002). 4.1 Figure 3. Ice conditions in the northern part of the Bay of Bothnia and the route of the ship on March 21st.

Finnish-Swedish ice class rules

FSICR is defined for ships operating in first-year ice only. 1A Super is the highest ice class in FSICR.

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In FSICR, the design ice pressure is calculated with the equation p

cd c pca p0 [ MPa M ]

(1)

where cd, cp, and ca are factors referring to the ship size and the engine output, the probability that the design ice pressure will occur in a certain region of the hull, and the probability that the full lengths of the area under consideration will be under pressure, respectively. cd equals 1.0 as a maximum and is calculated with the equation cd =

a*k + b 1000

(2)

where k is k=

Δ*P 1000

(3)

a and b are determined from the table presented in FSICR (2010, p. 12), based on the value of k. For S.A. Agulhas II the values of a and b obtain values of 30 and 230 for the bow and 8 and 214 for the stern, respectively. Δ is the displacement of the ship in tons at the maximum ice class draught and P is the actual continuous engine output in kilowatts. The values for cp are taken from the table presented in FSICR (2010, p. 12). For 1A Super the values are 1.0 and 0.75 for the bow and stern respectively. The maximum and minimum for ca are 1.0 and 0.35. ca is calculated with the equation ca =

l0 la

(4)

where l0 = 0.6 and la is determined from the table presented in FSICR (2010, p. 13). For a transversely framed ship la is taken as the frame spacing, which in this case is 0.4. Equation (4) gives the value of 1.225 for S.A. Agulhas II and ca obtains the value 1.0 as it is the maximum. p0 is the nominal pressure, which is taken as 5.6 MPa according to the rules. The design pressure (using Equation (1)) for the bow is now 3.15 MPa and for the stern 1.27 MPa. The design ice load height of the area actually under ice pressure is taken as 0.35 metres for 1A Super. The design line load can now be calculated by multiplying the design ice pressure by the design load height. The design line load for the bow is 1.10 MN/m and for the stern it is 0.45 MN/m. It should be noted that the FSICR rules are based on the elastic strength, while the IACS polar rules

are based on the plastic stength. The requirements stated in the rules can be compared when the elastic section modulus determined on the basis of FSICR is multiplied by 1.25 to obtain the plastic section modulus. It should be noted that this is not entirely correct since the correlation factor between the elastic and plastic strength depends on the profile and the shape of the frame (Kaldasaun and Kujala, 2010). Since the load required to exceed the yield strength of the frames can be shown to depend linearly on the section modulus, the design line loads for plastic design can be determined by multiplying the line loads in elastic design by the factor 1.25. The design line loads based on FSICR that are determined for plastic deformation are 1.38 MN/m for the bow and 0.56 MN/m for the stern. 4.2

International association of classification societies

The specification of Polar class PC 6 states that these ships are designed for summer/autumn operation in medium first-year ice, which may include old ice conditions. In the bow area, the force, F, line load, Q, pressure, p, and load patch ratio, AR, are functions of the hull angles measured at the upper ice waterline. The influence of the hull angles is accounted for by the bow shape coefficient fa. The waterline length of the bow is generally divided into 4 sub-regions of equal length and the force, line load, pressure, and load patch ratio are calculated for each region with respect to the mid-length position of each region. As the ice loads were measured only on two adjacent frames at the bow during the ice trial, the design ice loads are calculated only for this location in this paper in order to comply with the measurements. The shape coefficient, fa, is determined as follows: fa = min( fa1; fa f 2 ; fa3 )

(5)

fa1, fa2, and fa3 are determined as follows: (6)

(7) fa3 = 0.60

(8)

where L is the ship length as defined in UR S2.1 (but measured on the upper ice waterline), x is

122

the distance from the forward perpendicular to the load location of interest, α is the waterline angle, β′ is the normal frame angle, D is the ship displacement in kilotons, CFC is the crushing failure class factor, and CFF is the flexural failure class (see Table 1 in IACS 2011, pp. I2-3). For S.A. Agulhas II, Equations (6)–(8) give values of 0.268, 1.117, and 0.6 for the factors fa1, fa2, and fa3. According to Equation (5), fa is the smallest of these and the value of fa1 is chosen for further analysis. The force is defined with the equation f CFC D 0.64 [ fa

F

(9)

]

The load aspect ration is given as (10) Furthermore, the line load is calculated with the equation Q=

F 0 61 CF CFD [ 0.35 AR

(11)

/ ]

where CFD is the load patch dimensions class factor (see Table 1 in IACS 2011, pp. I2-3). The forces occurring in the other areas than the bow are determined with the equation FNonBow

0 36 CF CFC DF [

(12)

]

where DF equals D0.64 as in this case D ≤ CFDIS. CFDIS is a displacement class factor determined from the table in IACS (2011, pp. I2-3). Additionally, the line load for the areas other than the hull is calculated with the equation QNonBow 0 639 FNonBow 0.61 ⋅CF FD [

/ ]

(13)

The magnitude of the load expected in different areas is calculated by using area factors, AF, which are presented in Table 3 in IACS (2011, pp. I2-7). As mentioned earlier, the instrumented areas are at the bow, bow shoulder (bow intermediate ice belt by the rules), and stern ice belts. For class PC 6, these area factors are 1.00, 1.00, and 0.40, respectively. The design line load on the bow frames is calculated with Equations (5)–(11) and the hull area factor for the bow to be 1.479 MN/m. Multipying Equation (13) by the hull area factors gives design line loads on the bow intermediate ice belt and on the stern ice belt of 1.896 MN/m and 0.758 MN/m, respectively.

5

COMPARISON OF MEASURED AND CALCULATED ICE LOADS

The maximum line loads derived from the measurements are 1010 kN/m, 1437.5 kN/m, and 1145 kN/m for the bow, bow shoulder, and stern shoulder. The measured maximum load on the stern is of the same order of magnitude as the measured maximum loads on the bow area of the ship. In sailing forward operations, large loads mainly occur in the bow area, whereas loads at the stern area are small compared to the ones at the bow. When the ship turns greater maximum loads are measured on the stern as it starts to break the ice. Comparing the design loads for the plastic design of the structure, the line loads obtained following FSICR are 1.38 MN/m for the bow and 0.56 MN/m for the stern and 1.479 MN/m for the bow, 1.896 MN/m for the bow shoulder, and 0.758 MN/m for the stern based on IACS. Although a difference between the rules in the design line loads can be noted with the methods used, it should be kept in mind that the extrapolation of the elastic design of FSICR to plastic design is done in a simplified manner, as shown in Chapter 4.1. If a different factor were used to scale the elastic section modulus into the plastic section modulus, different design line loads would be obtained with FSICR. The comparison is highly sensitive to the chosen value and the value depends strongly on the profile and the shape of the frame. 6

CONCLUSION

The comparison between the line loads derived from the measurements and the rules (FSICR and IACS) shows that the measured and design line loads on the bow areas are of the same order of magnitude. In addition, the design line loads based on IACS are clearly higher in the bow shoulder area than the measured ones. This is reasonable as the ships might encounter multiyear ice in polar regions and hence the severity of the ice loading is increased. It was found out that the measured maximum ice loads on the stern shoulders are significantly higher than the loads proposed by the rules. This would indicate that the rules underestimate the ice loads that occur. It must be kept in mind that the ship was operating during an ice trial with an operational profile comparable to that of icebreakers. The highest maximum loads measured on the stern shoulder occurred when the ship was turning. Normal merchant vessels do not perform fast manoeuvres when they operate independently in ice conditions and they try to avoid ice conditions. Manoeuvres in ice conditions increase the occurrence and the

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magnitude of the measured maximum ice loads on the stern shoulder significantly as the stern is in contact with the ice and breaks it. In conclusion, it can be stated that the design line loads of FSICR and IACS are of the same magnitude for the measurement-related maximum line loads in the bow and bow shoulder areas, but underestimate the loading at the stern. ACKNOWLEDGEMENTS The authors would like to thank Tekes for the funding of the project. Furthermore, all the project partners, University of Oulu, University of Stellenbosch, Aker Arctic, Rolls-Royce, STX Finland, Wärtsilä, DNV, and the Department of Environmental Affairs of South Africa, are gratefully acknowledged. REFERENCES FSICR 2010. Finnish-Swedish Ice Class Rules 2010. Transport Safety Agency, Helsinki, Finland. TRAFI/ 31298/03.04.01.00/2010. IACS 2011. Requirements concerning POLAR CLASS. International Association of Classification Societies. Structural Requirements for Polar Class Ships IACS Req. 2006/Rev.2, 2010.

IMO 2002. Guidelines for Ships Operating in Arctic Ice-Covered Waters. MSC/Circ.1056. Kaldasaun, J. & Kujala, P. 2011. Risk-Based Approach for Structural Design of Ice-Strengthened Vessels Navigating in the Baltic Sea. Proceedings of the 21st International Conference on Port and Ocean Engineering under Arctic Conditions, July 10–14, 2011. Montréal, Canada. Kujala, P., Suominen, M., Jalonen, R. (ed.) 2007. Increasing the safety of icebound shipping—Final scientific report: volume 2. Helsinki University of Technology, Laboratory of Naval Architecture and Marine Engineering, Espoo, Finland. Report M-302. WP7.4. Quinton, B., Daley, C., Gagnon, R. 2012. Realistic Moving Ice Loads and Ship Structural Response. Proceedings of 22nd International Offshore and Polar Engineering Conference, June 17–22, 2012, Rhodes, Greece. Riska, K., Uto, S., Tuhkuri, J. 2002. Pressure distribution and response of multiplate panels. Cold Regions Science and Technology, Volume 34, pp. 209–225. Riska, K. & Kämäräinen, J. 2012. A Review of Ice Loading and the Evolution of the Finnish-Swedish Ice Class Rules. SNAME Transactions. SNAME Annual Meeting & Expo and Ship Production Symposium, Nov 16–18, 2011, Hyatt Regency, Houston, TX, USA. von Bock und Polach, R. 2006. Nonlinear Finite Element Analysis of Ship Frames under Ice Load. Master’s Thesis, Technical University of Berlin, Berlin, Germany.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

L-year maximum values of local ice loads on ship hulls A. Suyuthi & B.J. Leira Norwegian University of Science and Technology, Trondheim, Norway

K. Riska Center of Ships and Offshore Structures, Trondheim, Norway TOTAL ASA, Paris, France

ABSTRACT: For the purpose of design of an ice-going vessel, the L-year maximum value of the iceinduced load acting on the ship hull is important. Such a maximum value can be obtained by means of classical statistical analysis for which it required to establish a long term distribution. In the present paper, a methodology is proposed which is referred to as the m-nautical mile maximum approach. The approach relies on the conditional distribution of ice load for given stationary ice conditions and/or vessel’s speed. Two statistical models were applied and compared, i.e. the Weibull and the three-parameter exponential models. Full scale measurement data, which was obtained on board KV Svalbard during the winter of 2007 in the Northwestern part of the Barents Sea, was employed in order to demonstrate the application of the procedure. It was found that for this particular data set application of the Weibull model as the initial distribution leads to less conservative extreme values as compared to the measured maximum values. Predictions which were on the safe side of the observed maximum value were produced by instead applying the three-parameter exponential model as the initial distribution. The key benefit of the present proposed method is that it requires a quite limited amount of ice load data as long as the data is accompanied by a simultaneous ice thickness record. 1

INTRODUCTION

The magnitude of extreme ice loads acting on ship hulls during the entire service life is required in order to achieve an adequate design. Here, such an estimated extreme value is referred to as the L-year maximum of the ice-induced load. This extreme value should only be exceeded with a sufficiently small probability with respect to the applicable long term extreme distribution of the ice load. Previous works dealing with long term statistic of ice load were mostly based on the asymptotic approach; see e.g. Vuorio et al. (1979), Kujala and Vuorio (1985, 1986), etc. The approach utilized 12-hours or daily maxima obtained from fullscale measurements. The Gumbel distribution was applied in order to model these maxima. In addition, Kujala and Vuorio (1986) observed that the peak amplitudes follow the exponential distribution. A systematic statistical approach was established by Jordaan et al. (1993). They started the analysis by exploiting the fact that the upper tail of the ice load process follows the exponential distribution. The extreme value distribution formally can be obtained by raising the corresponding distribution to the number of events corresponding to a specific return period. They proposed

several statistical models for the extreme processes which differ with respect to the assumption of the number of events, i.e. whether it is a fixed integer, or binomially distributed, or Poisson distributed. Their method was then followed and applied by e.g. Frederking (2005) and Taylor et al. (2010), and was also compared with another method, e.g. Li et al. (2010). The two methods described above do not relate their estimated values to a specific ice condition or specific vessel operation characteristics. The data which was utilized was assumed to represent the long term variation. Therefore, both methods predict the extreme value directly. Accordingly, these methods do not aim at taking into account any future variation of ice conditions, even when such variations can be well predicted. Furthermore, these methods require a significant amount of data collected through several years in order to produce reliable predictions. Kujala (1996) proposed a semi-empirical evaluation of long term ice loads on a ship hull. The work seems to be the first of its kind to include variation of ice conditions throughout the lifetime of the vessel as part of the statistical analysis. The ice condition was represented by a winter maximum equivalent ice thickness, heq, which varies from year to year. However, the method requires long term

125

data collection (several years) of full scale measurement since the basic reference period is one year. The equivalent ice thickness is an equivalent average level ice thickness which gives the same load level as encountered by a ship when it navigates in ice conditions with varying amounts of level and ridged ice (Kujala, 1996). The data basis for this evaluation consisted of 12-hour maximum ice load records obtained from four years of measurement on board the chemical tanker MS Kemira in the Baltic Sea. The annual distribution of the ice load was modeled by the Gumbel distribution, for which the parameters were estimated based on the mean and variance of the 12-hour maxima. Since the parameters were expressed as functions of heq, the annual distribution was considered to be a conditional distribution. The long term distribution of the ice load for each zone is then determined by the integral of the product of the annual ice load distribution and the distribution of heq. The final expression for the long term distribution is obtained by summing up (with a certain weight) the annual ice load distribution for different zones. The present paper proposes a different approach in order to establish the long term distribution of the ice-induced load on a ship hull. The available data of the present analysis correspond to records of peak amplitudes of ice loading obtained from o measurement on board KV Svalbard during one week in the winter of 2007. There were 8 sensors installed on the bow part, i.e. four sensors on each side. Apart from the vessel’s speed and propulsion power records, there was also an EM (electromagnetic) ice thickness measurement device installed on board of the vessel, such that a quite detailed resolution of the ice thickness time series was available. Based on the available data, the authors selected 18 time records for which the ice condition with respect to the average ice thickness and the voyage characteristics with respect to the vessel’s speed and propulsion power are stationary. During each stationary condition, a short term statistical analysis was performed, see e.g. Suyuthi et al. (2012b, c), and the resulting statistical model can be considered as being conditional on the associated average ice thickness, h. This conditional model can be regarded as representing a short term m-nautical mile extreme distribution. It corresponds to the distribution the largest ice load peak amplitude during the m-nautical mile travel distance. Therefore the method is referred to as the m-nautical mile maximum approach. The long term distribution is then obtained by integrating the product of the conditional ice load distribution and the distribution of the average ice thickness h. The distribution of the average ice

thickness should be considered as a long term distribution, which takes into account all variations of ice thickness throughout the travelling route of the vessel during the entire life of the vessel. The key benefit of the proposed method is that it can be applied to a quite limited (a couple of days) database of ice-induced loads on a ship hull as long as it is accompanied by a corresponding ice thickness record. It is also important to state that the vessel must travel through various ice thicknesses with constant propulsion power and/or speed in order to achieve a stationary condition. 2

FORMULATION OF LONG TERM ICE-INDUCED LOADS

Let us assume that the process of peak amplitudes of ice-induced load within a stationary ice condition and/or vessel speed is denoted by X. The stationary ice condition can be characterized by the average ice thickness, see e.g. Suyuthi et al. (2013). Due to the necessity of applying a lower threshold, γ, we need to introduce another random variable Y, such that: Y

X−γ

(1)

Sometimes an ice load measurement system needs to employ a lower threshold in order to distinguish the ice load from other loads. However, in the absence of such a threshold, γ simply could be set equal to zero. Based on the deducted peak amplitude data Y, a statistical model can be established. Here, we should consider the statistical model as a conditional distribution for a given average ice thickness, FY|Hi(y|h), or alternatively as a conditional distribution for a given average ice thickness and vessel’s speed, FY|HiVs(y|h,v). The conditional distribution can be well represented by the Weibull and the three-parameter exponential models, see e.g. Suyuthi et al. (2012a, b), which can be formulated as follows. The Weibull model is expressed as: ⎧⎪ ⎛ y ⎞ FY |Hi ( y | h ) = 1 − exp⎨ − ⎜ ⎟ ⎪⎩ ⎝ θ ( h ) ⎠

k(h) ⎫

⎪ ⎬ ⎪⎭

(2)

The three-parameter exponential model is expressed as: FY |Hi ( y | h )

a( h )

{

+ ( − a( hh)))

(

{−

y

}

( h) y) (

y

}

( h y)

(3)

The m-nautical mile maximum approach considers the largest peak amplitude of the ice-induced load during a stationary condition

126

for an m-nautical mile travelling distance. If the m-nautical mile largest peak amplitude is denoted by Ym, the conditional distribution for a given average ice thickness can be obtained as FYm Hi ( y | h)

⎡ FY |H ( y h) ⎤ i ⎣ ⎦

k ( m , h)

(4)

where k(m,h) = m.νd is the number of peak amplitudes during the m-nautical mile travelling distance for a given average ice thickness h. It is noted that νd is the frequency of peak amplitudes per unit distance, which will be described in Section 3.3. The marginal (long term) distribution (FY) is then obtained by integrating the conditional distribution and the relative frequency of various average ice thickness values, such that: FY ,m ( y ) =

hL

∫0

FYm Hi ( y | h fHi ( h )dh

(5)

No additional weighting is needed because the frequency of the maximum load for each m-nautical mile stationary condition is the same, i.e. 1 per stationary condition. If the vessel speed Vs is also considered in addition to the average ice thickness Hi in order to characterize the stationary condition, the marginal (long term) distribution (FY,m) is obtained as a double integral of the conditional distribution and the relative frequency of various ice conditions and vessel speeds, such that: FY ,m ( y ) =

hL

vL

∫0 ∫0

FYm HiVs ( y | h,v ) fHiVs h,v )dvdh (6)

The maximum value of m-nautical mile traveling distance corresponding to a return period of L-years can be determined by solving the following equation:

(

)

1 − FY ,m y( L ) =

1 r( L )

(7)

where r(L) is the number of m-nautical mile events in L-years, i.e. r(L) = L ⋅ S/m and S is the average annual travelling distance. 3

CONSIDERATION OF PRACTICAL APPLICATION

A systematic procedure for assessment of the maximum ice-induced load on a ship hull was outlined in the previous section. With respect to practical application of this formulation, there are several

expressions which need some clarification, i.e. the conditional distribution of ice-induced load for a given stationary condition (FYm|Hi(y|h)), the distribution of ice thickness (fHi(h)), and the frequency of peak amplitudes (νd) per unit distance. The following discussion focuses on how these entities can be obtained in practice. 3.1 Conditional distribution of ice-induced load for given stationary condition (FY|Hi(y|h) or FY|HiVs(y|h,v)) In principle, FY|Hi(y|h) or FY|HiVs(y|h,v) can be determined by statistical analysis of the ice load peak amplitude data available for a particular condition h or (h,v). Such data can be obtained by means of a numerical model, ice model testing, or full scale measurements. A numerical model focusing on local ice-induced loads on ship hulls, e.g. Su et al. (2011), can be a good candidate in order to produce the required data. The ice conditions can be easily varied and a number of numerical simulations can be run with limited computational efforts. It is also possible to perform ice model tests to generate such ice load data if the hull of the model is equipped with adequate strain sensors. However, many of the ice model tests at the moment are focused on the ice resistance for the purpose of evaluating the ship’s performance in ice. Tests for the purpose of producing ice load data which are relevant for extreme load analysis will typically require significant costs in terms of resources and time due to the necessity of repeating the tests for a range of different ice conditions. The third method that can be applied to acquire the ice load data is utilization of full scale measurements, which are not commonly available. If such measurements are at hand, they frequently tend to be incomplete in the sense that they do not cover all the ice conditions in question. Evaluating these three options, the numerical model developed by Su et al. (2011) seems to be quite promising. However, more validation is needed before it can be fully implemented. 3.2

Distribution of ice thickness (fHi(h))

The distribution of average ice thickness (fHi(h)) in Equation (5) is considered as a long term distribution. We can expect that the average ice thickness during a voyage varies for different zones. We can also expect that the distribution of the average ice thickness for different voyages varies even when the route is the same. Due to long term weather variations, the distribution of the average ice thickness also varies for different years. All of these variations should be ideally represented by fHi(h).

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In practice, collection of ice charts throughout the year, which have been published by various meteorological institutes, can be used as the main data to develop the ice thickness distribution along the route. In the absence of such charts, temperature records of the sea surface throughout the year can be used as an initial estimate of the duration of periods where the average daily temperature is below the freezing point and corresponding sea-ice formation will occur. The thickness of the sea ice can be determined according to Lebedev’s formulation, see e.g. Serreze and Barry (2005), as follows. hice

.

F FDD 0.58

(8)

where FDD is freezing degree days in (°C) and hice is ice thickness in cm. 3.3

Frequency of peak amplitude (νd) per unit distance

Figure 1. The impact frequency per unit distance: theoretical vs. data (Bridges et al., 2006). nm = nautical mile. It is noted that the line is not based on data fitting, instead it was developed based on Equation (9).

The frequency of peak amplitudes per unit distance νd for a particular travelling condition varies from one realization to another even when the travelling conditions are stationary. Theoretically the amplitude frequency is governed by the floe size (dfloe) which is broken in bending. This floe size depends on the speed, such that the higher the speed the smaller the floe size becomes, see Varsta (1983). On the other hand, the ship’s speed depends on the ice thickness. Therefore, Bridges et al. (2006) proposed the following formulation for frequency of impact per traveling distance: 1852

νd = 10.4

3/ 4 heq

.

⎛ heeq ⎞ vOW ⎜ − 1⎟ ⎝ hL ⎠

(9)

where νd is number of events per nautical mile (1/nm), heq is the equivalent ice thickness in meter, vOW is open water speed in m/s, and hL is the ice thickness limit in meters. Bridges et al. (2006) plotted the theoretical frequency of impact in Equation (9) together with the available data as shown in Figure 1. We can see that there is a significant scatter. Bridges et al. (2006) explained that the scatter was due to variation in the ice conditions which the ship encountered. Figure 1 bears clear evidence of limited data being available, which was concentrated at 0.4 m < heq < 0.7 m and around heq = 1.0 m. The frequency of impact based on full scale measurements can also be obtained based on the results provided by Suyuthi et al. (2012c), see Figure 2. In this figure, more data are avail-

Figure 2. The impact frequency per unit distance (1/nm), generated from data given in Suyuthi et al. (2013). nm = nautical mile.

able as compared to Figure 1 and they are spread out between 0.2 m < h < 1.2 m. Similar values of the frequency of impact per unit distance can be observed in this figure as compared to Figure 1. The dashed line in the figure is the upper theoretical line of the frequency of impact per unit distance according to Bridges et al. (2006). The solid line is the upper theoretical line for the frequency of impact per unit distance based on the inverse of the characteristic length of the ice plate. The characteristic length of the ice plate, l, is the distance (radius) between the point of load to

128

the location where the crack deformation occurs at the ice plate, which can be formulated according to the thin plate theory as follows, see e.g. Kerr (1976) and Fransson (2009). ⎛ ⎞ E h3 l= ⎜ 2 ⎟ 12 1 − υ ) ⎠ ⎝ ρw g12(

1/ 4

(10)

Assuming a 100% concentration of level ice and no effect from relative speed on the broken ice floe length, the frequency of impact per unit distance (1/nm) can be taken as the inverse of the characteristic length of the ice plate in Equation (10). The solid line in Figure 2 was developed based on this principle and the constants involved were the Young’s modulus of ice E = 3.5 GPa, the density of sea water ρw = 1025 kg/m3, g = 9.81 m/s2, and the Poisson’s ratio of ice υ = 0.3, such that: 1852 ν d ( h) = 13.3617 h3 / 4

(11)

The theoretical line in Figure 2 as well as its formulation in Equation (11) should be understood to be valid up to the ice thickness limit, hL, when the ship cannot maintain its continuous progressive movement. For merchant vessels, icebreaker assistance is needed at this stage. Equation (11) does not account for the effect of vessel’s speed. This means that the predicted broken floe length is longer and consequently the frequency of impact per unit distance, which is calculated by Equation (11), is lower than the one which is calculated by Equation (10). Most of the data points which are presented in Figure 2 are located below the two theoretical lines. The main reason for this is that there is a certain probability that a particular part of the ship hull missed an ice floe because it has already been broken by other parts of the ship hull. Another reason for observing lower values of the frequency of impact per unit distance is the implementation of a lower threshold. Such a threshold, which was intended to remove the noise, can also unintentionally remove a number of ice-induced load/stress peaks. Consequently the frequency of impact is lower than it should be. Both Bridges et al. (2006) and Suyuthi et al. (2013) applied a lower threshold. However, the lower threshold which was applied by Suyuthi et al. (2013) was higher than the former one. This can be the main reason for why the frequency of impact given by Suyuthi et al. (2013) is a bit lower than the other one. The quite low values of the frequency of impact for thickness ≤ 0.3 m in Figure 2 as compared to

the theoretical lines can be explained as follows: During the voyage with data collection, level ice was rarely encountered. Most of the ice field was some sort of broken ice with significant variation in thickness. In fact, some stretches of open water were encountered as reflected by the ice concentration being as low as 58%. This is especially true for cases with quite low average thickness, i.e. ≤0.3 m. Encountering open water implies a lower frequency of impact. Moreover, it seems that the load sensors have poor performance with respect to detection of low values of the loads. Noise interfering with the low load signals implies that all these loads were removed due to implementation of the lower threshold. 4

CASE EXAMPLE

In this case example, the maximum long term ice induced load acting on a ship hull is evaluated by application of the approach described in Sect. 2, i.e. the m-nautical mile maximum approach. It is assumed that the ice-induced load is affected by variation of the ice condition only and no variation of the vessel’s speed is considered. In this case, the average ice thickness is assumed to be sufficient in order to represent the variation of the ice conditions. The vessel is assumed to travel in ice covered waters at a distance of 1500 nautical miles annually. The available data consists of time series of ice-induced loads for various ice conditions. The following steps were applied: 1. define the conditional distribution of ice-load for each stationary condition, FYm|Hi(y|h). 2. define the distribution of the ice thickness, fHi(h). 3. calculate the long term distribution. 4. estimate the L-year maximum value. 4.1 Conditional distribution of ice-load for given stationary condition FYm|Hi(y|h) The available data consisted of 144 sets of peaks of the ice-induced load (in kN/m). Each data set was collected during a stationary condition with respect to ice thickness and vessel’s propulsion power. The m-nautical mile maximum approach as described in Section 2 was implemented. The m-nautical mile maximum distribution can be established by raising the conditional distribution FY|Hi(y|h) in Equations (2) and (3) to the power of k(m,h) as defined in Equation (4). k(m,h) is the number of peak amplitudes during the m-nautical mile travelling distance for a given average ice thickness h. Utilizing the frequency of

129

peak amplitudes per unit travelling distance, νd, we know that: k ( m, h )

m

d ( h)

(12)

It is noted that νd(h) can be estimated based on Equation (11). Based on the 144 data sets, the m-nautical mile maximum conditional distribution (of which m = 1) FYm|Hi(y|h) is established. The Gumbel distribution is a proper choice in order to model this short term extreme distribution FYm|Hi(y|h), see e.g. Suyuthi et al. (2012b). It is defined as follows. ⎡ ⎧ (yy h )) ⎫ ⎤ FYm Hi ( y | h ) = exp ⎢− exp ⎨− ⎬⎥ β ( h) ⎭ ⎦ ⎩ ⎣

Figure 3.

Figure 4 presents results for the case that the initial distribution FY|Hi(y|h) is represented by the Weibull model, i.e. the estimated parameters of the short term extreme α and β for FYm|Hi(y|h) Figure 5 presents corresponding results for the case that the initial distribution FY|Hi(y|h) is modeled by the three-exponential model, i.e. the estimated parameters of the short term extreme α and β for FYm|Hi(y|h) 4.2

Distribution of ice thickness (fHi(h))

During her expedition in the winter of 2007, KV Svalbard was equipped with an electromagnetic ice thickness measurement device, such that the ice thickness along the route of the voyage was recorded. More details about the system are given in Pfaffling (2007). The ice thickness data presented in Figure 3 was then evaluated and fitted by the two parameter gamma distribution, which is formulated in Equation (14). The estimated parameters are also presented in Figure 3.

fH i h ) =

1 ⎛−h ⎞ ha −1exp ⎜ ⎟ ⎝b ⎠ b (a ) a

The average ice thickness distribution, fHi(h).

(13)

(14)

This average ice thickness distribution was developed based on ice thickness records for a single voyage. Therefore it is a sort of short term distribution. As has been indicated in Section 3.2, the long term distribution of average ice thickness, which takes into account the yearly variation, is needed to estimate the long term maximum ice load. Due to data limitation, let us assume that

the long term distribution of average ice thickness, fHi(h), can be represented by the one fitted in Figure 3. 4.3

Long term distribution (FY(y))

Based on the conditional distribution of the m-nautical mile maximum, i.e. FYm|Hi(y|h) (for which the associated Gumbel parameters as function of average ice thickness h are indicated in Figure 4 based on an initial Weibull distribution and in Figure 5 based on an initial threeparameter exponential distribution), the long term distribution FY(y) based on the m-nautical mile maximum approach as defined in Equation (5) can be established. The value applied here is m = 1 nm. These two FY(y) which were produced from two different initial distributions are presented in Figure 6. 4.4

The L-years maximum values

In Figure 6 some points are also indicated along the long term distribution curves which represent the L-year maximum value and the corresponding return periods. For clarity, these values are summarized in Table 1. A challenging question to be answered is which of the long term distributions in Figure 6 (and the corresponding extreme values in Table 1) that is most relevant. In order to deal with this issue, a plot of the maximum ice load for each data set is presented in Figure 7. Eventually there are 144 data points in the figure. Among those points, there is one point, i.e. Ymax = 458.9 kN/m for an average ice thickness of h = 0.78 m, which could possibly be assumed to be an outlier. Unfortunately, the authors cannot see any specific physical/mechanical reason for such a peculiar load, and therefore we cannot reject it. This is mainly because the ice load

130

Figure 4. The estimated parameters of the Gumbel distribution (based on the Weibull model as the initial distribution). This is employed as the conditional distribution of the m-nautical mile maximum distribution for a given average ice thickness. Here, m = 1 nautical mile. The dashed-lines correspond to the 95% confidence interval of the fitted parameters.

Figure 5. The estimated parameters of the Gumbel distribution (based on application of the three-parameter exponential as the initial distribution). This is employed as the conditional distribution of the m-nautical mile maximum distribution for a given average ice thickness. The dashed-lines correspond to the 95% confidence interval of the fitted parameters.

131

Figure 6. The long term distribution, FY(y), based on the m-nautical mile maximum approach. The thin line indicates results based on the Weibull distribution. The thick line indicates results based on the three-parameter exponential distribution.

Table 1. The maximum ice-induced load with respect to various long term return periods of L-years. y(L) (kN/m)

Return period [L (years)]

Long term dist. [FY(y(L)) = 1-1/r(L)*]

Weibull

3-exp

1 5 10 20

0.9993333 0.9998667 0.9999333 0.9999667

350 407 436 454

582 687 735 785

*Note: r(L) = L ⋅ S/m; S = 1500 nm. r(L) is number of m-nautical mile maximum during L-years.

magnitude is frequently surprising, which means that a very high peak load as shown in Figure 7 or even higher could occur rather suddenly. This assessment is in accordance with Gumbel (1960) who stated in the commentary to the work of Anscombe (1960) that “The rejection of outliers on a purely statistical basis is and remains a dangerous procedure. Its very existence may be a proof that the underlying population is, in reality, not what it was assumed to be”.

Comparing Table 1 (column 3) and Figure 7 with the Weibull distribution as the conditional model, we can observe that three data points exceed the predicted 1-year maximum value and one data point exceeds the predicted 20-years maximum value. It is noted that all data sets utilized in the present paper were collected during a short period of 5 days only. Accordingly, this evaluation shows us that that application of the Weibull distribution leads to an un-conservative extreme value prediction. This is in accordance with e.g. Suyuthi et al. (2012c) who proposed the three-parameter exponential distribution as an alternative. Comparing Table 1 (column 4) and Figure 7 with the three-parameter exponential distribution as the conditional model, we can observe that the predicted extreme values are more sensible, although there seems to be some over prediction. As compared to the measured maximum value, the predicted maximum value for L = 1 year and L = 20 years are approximately 20% and 40% higher, respectively. These values of the predicted extreme values can be considered to be both acceptable and on the safe side.

132

Figure 7.

5

The maximum ice-induced load (kN/m) for each data set.

CONCLUSIONS

A long term statistical approach is presented which aims at determining the L-year maximum value of the ice-induced load on a ship hull has been presented. Two different statistical models of the ice load peak amplitudes, i.e. the Weibull and the three-parameter exponential models, were applied. Based on the observations from the present example calculations, the Weibull distribution tends to give underestimated maximum values, which is undesirable. On the other hand, the maximum values predicted by application of the three-parameter exponential model are slightly overestimated which is more acceptable. The present example only considers the variation of ice conditions and does not consider the variation associated with the vessel’s speed. When both sources of variation i.e. (ice properties and vessel speed) are taken into account, such an evaluation requires that a bivariate statistical analysis is performed in order to take care of the conditional ice load, FY|HiVs(y|h,v) and FYm|HiVs(y|h,v). In principle, the calculations can then be based on Equation (6) as long as the required bivariate data, i.e. the ice load peak amplitude for a given ice thickness and vessel’s speed, is available. This issue

should be considered as part of future work. The same applies to the case where there is variation of operational modes (independent operation or navigation with ice breaker assistance), and also to the case where there is a long term variation of the ice conditions due to long term variation of the global weather conditions. The key benefit of the present proposed method is that it requires a quite limited database of ice-induced loads on ship hulls as long as it is accompanied by a simultaneous ice thickness record. REFERENCES Anscombe, F., 1960. Rejection of outliers. Technometrics 2 (2), 123–147. Bridges, R., Riska, K., Zhang, S., 2006. Fatigue assessment for ship hull structures navigating in ice regions. In: Proceedings of the Icetech 2006. Banff, Canada. Frannson, L., 2009. Ice Handbook for Engineers. Luleå University of Technology. Frederking, R., 2005. Local ice pressures on the Oden 1991 polar voyage. In: Proceedings of the 18th International Conference on Port and Ocean Engineering under Arctic Conditions. Potsdam, USA.

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Gumbel, E., 1960. Discussion of the papers of Messrs. Anscombe and Daniel. Technometrics 2 (2), 157–166. Jordaan, I.J., Maes, M., Brown, P.W., Hermans, I.P., 1993. Probability analysis of local ice pressures. Journal Offshore Mechanics and Arctic Engineering 115 (1), 83–89. Kerr, A.D., 1976. The bearing capacity of floating ice plates subjected to static or quasi-static loads. Journal of Glaciology 17 (76), 229–268. Kujala, P., 1996. Semi-empirical evaluation of long term ice load on a ship hull. Marine Structures 9 (1), 849–871. Kujala, P., Vuorio, J., 1985. On the statistical nature of the ice-induced pressures measured on board I.B. Sisu. In: Proceedings of the 8th International Conference on Port and Ocean Engineering under Arctic Conditions. Greenland. Kujala, P., Vuorio, J., 1986. Results and statistical analysis of ice load measurements on board icebreaker Sisu in winters 1979 to 1985. Tech. rep., Technical Research Centre of Finland, Helsinki. Li, C., Jordaan, I.J., Taylor, R.S., 2010. Estimation of local ice pressure using up-crossing rate. Journal of Offshore Mechanics and Arctic Engineering 132. Pfaffling, A., 2007. Ship-borne sea ice thickness electromagnetic measurements. Tech. rep., Pfaffling Geophysics. Serreze, M.C., Barry, R.G., 2005. The Arctic Climate System. Cambridge University Press.

Su, B., Riska, K., Moan, T., 2011. Numerical simulation of local ice loads in uniform and randomly varying ice conditions. Cold Regions Science and Technology 65, 145–159. Suyuthi, A., Leira, B.J., Riska, K., 2012a. A generalized probabilistic model of ice load peaks on ship hulls in broken-ice fields. Cold Regions Science and Technology, (submitted for publication). Suyuthi, A., Leira, B.J., Riska, K., 2012b. Short term extreme statistics of local ice loads on ship hulls. Cold Regions Science and Technology 82, 130–143. Suyuthi, A., Leira, B.J., Riska, K., 2012c. Statistics of local ice load peaks on ship hulls. Structural Safety 40, 1–10. Suyuthi, A., Leira, B.J., Riska, K., 2013. Fatigue damage of ship hulls due to local ice-induced stresses. Applied Ocean Research, (submitted for publication). Taylor, R.S., Jordaan, I.J., Li, C., Sudom, D., 2010. Local design pressures for structures in ice: Analysis of fullscale data. Journal of Offshore Mechanics and Arctic Engineering 132. Varsta, P., 1983. On the mechanics of ice load on ships in level ice in the Baltic Sea. Ph.D. thesis, Technical Research Centre of Finland, Espoo, Finland. Vuorio, J., Riska, K., Varsta, P., 1979. Long term measurement of ice pressure and ice-induced stresses on the Icebreaker Sisu in Winter 1978. Tech. rep., Winter Navigation Research Board, Helsingfors and Stockholm.

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Impact and collision

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

A new super-element for estimating the collision resistance of an inclined ship side L. Buldgen University of Liège, Liège, Belgium

H. Le Sourne Institut Catholique d’Art et Métiers, Nantes, France

P. Rigo University of Liège, Liège, Belgium

ABSTRACT: This article is directly related to the well-known super-elements technique, used for rapidly estimating the collision resistance of a ship struck by another one. This paper introduces a new type of super-element for considering properly the case of inclined plating. The first part of the article is devoted to a short presentation of the mathematical developments leading to the crushing resistance. The proposed formula is then validated by numerical simulations on individual components. In the second part of the study, we consider the example of a collided ship having inclined plating elements. A comparison is made between the results provided by numerical simulations and those given by a simplified analytical tool involving the above-mentioned new super-element. 1

INTRODUCTION

Amongst all the loads that may act on a ship, it is important to consider the case of a collision between two vessels, especially for ships transporting polluting substances or for military vessels. For checking the resistance to collision, it is of common practice to use finite elements software. As this approach may be time-expensive, some simplified tools were developed for rapidly assessing the ability of a vessel to withstand a collision. Such methods are very useful for optimization in the pre-design phase of the impacted ship. The basic idea of these simplified procedures is to divide the vessel into structural macro-components, called “super-elements”. Each of them is characterized by a relation giving the crushing resistance with respect to the penetration of the striking ship. As the impacting vessel is moving forward into the struck structure, these super-elements are successively activated and their contribution to the total collision force is evaluated. As a consequence, one of the most important steps is to coherently derive the individual crushing resisting force for the various components. Some results are already available in the literature, especially for horizontal or vertical members. However, the side shell of a vessel may also presents some inclined parts. Some additional research is therefore

needed, and it is the aim this paper to present a new super-element for considering properly the case of inclined plating. 2 2.1

ANALYTICAL DEVELOPMENTS Displacements field

The super-element under consideration is an inclined plate impacted by a ship bow (Fig. 1).

Figure 1. Three dimensional view of the impact scenario.

137

This one is characterized by the parameters ϕ and ψ corresponding respectively to the stem and side angles. The local coordinate frame associated to the vessel is (xs, ys, zs). In this frame, the curve Γ describing the shape of the uppermost deck has the following mathematical equation: Γ ≡ zs = RZ − RZ

xs2 RX2

(1)

where RX and RZ are the two radii of the parabola (Fig. 1). The angle between the plate and the uppermost deck is denoted by α. The plate is assumed to be simply supported on its four edges. The initial contact point P defines four regions characterized by the dimensions a1, a2, b1 and b2. Local axes (x, y, z) are also attached to the inclined plane and are obtained by rotating the absolute frame (X, Y, Z). As the ship is moving forwards, the plate is submitted to strong tensile forces implying internal dissipation and it is our purpose to estimate the corresponding energy rate. To do so, it is necessary to postulate the displacements field occurring on the structure during the impact. For a given penetration δ parallel to the global Z axis, we assume that the plate is only submitted to a displacement w(x,y) acting perpendicularly to its plane, i.e. along the local z axis. Let us first consider the plane of the uppermost deck. For a given indentation δ, the plate is forced to follow the shape of the impacting bow, which leads to the deformation depicted on Figure 2. In order to avoid any redundancy, we will only describe the displacements field for the region 0 ≤ x ≤ a1, but developments are similar for a1 ≤ x ≤ a1 + a2. The displacement field W(x) in the plane of the uppermost deck is composed of two parts. The first part W1(x) is limited to the zone 0 ≤ x ≤ x0 and is defined so as to be tangent to the curve Γ for x = x0. We also assume a horizontal

Figure 2. Displacement W(x) in the plane of the uppermost deck.

tangency for x = 0. Considering equation (1) leads to: W1 ( x ) = δ

x2 a1x0

(2)

The second part W2(x) has to perfectly fit the curve Γ, because the plate is supposed to take the same shape as the impacting bow. Consequently, W2(x) may be written as: W2 ( x ) = δ − RZ

(x

a RX2

)2

(3)

The two previous components W1(x) and W2(x) are separated by the abscissa x = x0. Initially, for δ = 0, we have x0 = a1 but for further indentation, x0 tends to 0. If we account for the tangency conditions explained here over, it can be shown that x0 is given by: x0 (δ )

a1 −

RX2 δ RZ a1

(4)

The displacement W(x) is now entirely characterized by equations (2) to (4). It is worth noting that W(x) is strictly horizontal, i.e. parallel to the global Z axis. However, we postulate that each point (x,y) is only submitted to a movement perpendicular to the plate. To illustrate this assumption, the displacements profile in a plane perpendicular to the X axis is depicted on Figure 3. It can be seen that the shape function W(x) is simply linearly interpolated so as to have w(x,y) = 0 for y = 0 and y = b1 + b2. If we consider the plane perpendicular to the X axis and passing through point P (left

Figure 3. Displacements in a plane parallel to (Y,Z) plane.

138

part of Fig. 3), we see that P is submitted to the horizontal indentation W(a1) = δ, but the points of the plate only suffer a displacement δ sin α decreasing linearly with y. Similarly, in any plane perpendicular to the X axis, the maximal displacement is simply given by w(x,y) = W(x) sin α and is linearly interpolated along the y axis. This can be written as:

carefully chosen. Under the hypothesis of a plane stress state, Simonsen (1997) shows that the plastic flow theory leads to the following expression of the internal energy rate:

– If 0 ≤ y ≤ b2 − W(x) cos α, then:

where the dot is used for denoting the derivative ∂/∂δ, A is the area of the plate, tp is the thickness and σ0 is the flow stress. In this last expression, ∂εxx, ∂εyy and ∂εxy are the components of the strain rate tensor. They may be found by taking the incremental form of the Green-Lagrange relations:

w ( x, y ) =

b2

W ( x ) sin α y W ( x ) cos α

(5)

– If b2 − W(x) cos α ≤ y ≤ b1 + b2, then: w ( x, y ) =

W ( x ) sin α (b b1 W ( x ) cos α

b − y)

(6)

where W(x) is given by equations (2) or (3) according to the position along the x axis. This assumption of having w(x,y) perpendicular to the plate means that the structure is submitted to a plastic flow, for which each material point is simply “flowing” over the bow. Of course, this implies that no friction is likely to take place between the stem and the structure, which seems rather conservative as the effective penetration δ tends thus to be overestimated. 2.2

Internal energy rate

Equations (5) and (6) describe the displacements function w(x,y) for every point (x,y) belonging to the surface of the plate. This function is said to be kinematically admissible, as it is compatible and respects all the boundary conditions. For this reason, w(x,y) may be used for applying the upperbound principle. According to Jones (1997), this theorem states that if the work rate of a system of applied loads during any kinematically admissible collapse of a structure is equated to the corresponding internal energy dissipation rate, then that system of loads will cause (incipient) collapse of the structure. If we denote by ∂Eint the internal energy rate produced by the velocity field ∂w(x,y), the resistance P opposed by the super-element during the collision may be found by the virtual work principle: ∂

int

∂Eint = P ⋅ ∂δ ⇔ P = ∂δ

(7)

According to the above-mentioned upper-bound theorem, it is clear that equation (7) may give an excessive approximation of the real plate resistance if the collapse displacements field w(x,y) is not

Eɺ int =

ε xx

2σ 0t p 3

2 2 2 εɺxxx + εɺ yy + εɺxy + εɺxxx εɺ yyy dxdy

∫∫

(8)

A

1 ⎛ ∂w ⎞ 2 ⎝ ∂x ⎠

2

ε yy =

1 ⎛ ∂w ⎞ 2 ⎜⎝ ∂y ⎟⎠

2

ε xy =

1 ∂w ∂w 2 ∂x ∂y (9)

So the internal energy rate ∂Eint may be found by introducing equations (5) and (6) into (9), and then deriving the obtained expressions with respect to δ. This leads to the strain rates associated with the field w(x,y). ∂εxx, ∂εyy and ∂εxy are then introduced in equation (8) in order to obtain the corresponding energy rate. However, the mathematical formulations of w(x,y) are too complex for allowing an analytical integration of equation (8). As a numerical approach is not directly sought, we will rather make another approximation to get a closed-form solution of ∂Eint. In fact, the plate is supposed to be made of independent fibers oriented along the x and y axes. During the collision, these fibers are only submitted to membrane deformations along x or y, without suffering any shearing force. Consequently, εxy = 0 and the strain energy associated to εxx and εyy may be evaluated separately: Eɺ int

σ 0t p ∫∫ ɺxxx dxdy d d + σ 0t p ∫∫ ɺ yyydxd dy A

(10)

A

Introducing (9) into the previous equation and taking account of (7) leads to the following expression of the resistance opposed by the plate: P

⎛ ∂w ∂ 2 w ∂w ∂ 2 w ⎞ σ 0t p ∫∫ ⎜ + ddxdy ∂x ∂x∂δ ∂y ∂y∂δ ⎟⎠ A⎝

(11)

where w(x,y) has been defined previously by (5) or (6). Equation (11) may be analytically solved in order to derive a closed-form expression of P. However, the developments are quite fastidious

139

and will not be detailed in this paper. So far, it is important to bear in mind that the total resisting force P is the sum of two contributions Px and Py given by: Px

t p ∫∫ A

Py

t p ∫∫ A

larger than a given threshold value εc. This may be written as: 1 ⎛ ∂w ⎞ 2 ⎝ ∂x ⎠

2

ε xx

1 ⎛ ∂w ⎞ 2 ⎝ ∂y ⎠

2

ε yy

2

∂w ∂ w d dxdy ∂x ∂x ∂δ

(12)

2

∂w ∂ w d dxdy ∂y ∂y∂δ

(13)

where Px and Py are due to the membrane elongation of the fibers oriented along x or y respectively (the detailed mathematical expressions of Px and Py are provided as an appendix). The assumption of a non-shearing displacements field is a conservative approach because it neglects the energy that is also dissipated through the component εxy of the strain tensor. 2.3 Failure All the previous developments are valid provided that there is no failure in the material. However, as the indentation is growing, the plate is submitted to increased deformations that finally result in a material rupture. The membrane energy is then released, which in turn causes an abrupt decrease of the resistance. However, after this sudden transition, the resistance tends to stabilize at a non-zero level (Fig. 4). This is due to the fact that tearing is first located in a confined area of the plate, before extending to a larger region. Of course, as tearing develops, the resistance is becoming smaller. In our simplified approach, we will simply admit that the resistance is set to zero immediately after the initiation of tearing (Fig. 4). In order to account for this phenomenon, we assume that the fibers oriented along the x (resp. y) axis are no more able to support tensile forces if their deformations εxx (resp. εyy) becomes

> ε c → Px = 0

(14)

> ε c → Py = 0

(15)

where εc is the threshold value of the deformation for which rupture is assumed to take place on the super-element. According to Zhang (1999) or Lützen (2001), typical values for εc are between 0.06 and 0.12. Equations (14) and (15) together with equation (11) completely define the resistance P offered by the plate for a given penetration δ of the striking ship. The law P(δ) is precisely the relation sought for characterizing the present new super-element. Because of all the previous hypotheses, the method for deriving P may appear to be over-conservative, but one has to bear in mind that the application of the upper-bound method always provides an excessive approximation for the real resistance of a collapsing structure. 3

INDIVIDUAL VALIDATION

In order to validate the expression of P(δ) for the new inclined super-element, some comparison are made with numerical solutions given by the finite elements code LS-DYNA. To do so, the bow of the striking ship is modeled with rigid shellelements. Its geometrical properties (as defined on Fig. 1) are listed in Table 1. This ship bow is used for impacting different plates having various lengths (a1 + a2) and (b1 + b2). For each of them, the inclination varies from α = 45° to α = 90° with a step of 15°. So intermediate simulations are also performed for 60° and 90°. The mesh in the contact area is similar for the plate and for the ship, with an element size of 5 × 5 cm. The plate is simply supported on its four edges and is collided by the vessel travelling at the initial speed of 2 m/s. In this paper, we only present results obtained for the following characteristics:

Table 1. Geometrical dimensions of the striking bow as defined on Figure 1.

Figure 4. Resistance before/after rupture and present simplified model.

140

p (m)

q (m)

ϕ (deg)

ψ (deg)

6

8

84

84

a1 = 2.5 m, a2 = 2.5 m, b1 = 2 m, b2 = 2 m and tp = 0.02 m. The material law used for the plate is depicted on Figure 6. The stress-strain curve is made of two parts. The first one corresponds to the elastic phase, where a linear behavior is assumed. The stresses are directly proportional to the strain through the Young’s modulus E, until the elastic capacity σ0 is reached. The second phase stands for a plastic behavior, with a linear hardening characterized by the tangent modulus ET. If an element is deformed beyond the ultimate strain εu, then rupture occurs and the element is simply removed from the numerical model. For our simulations with LS-DYNA, the parameters listed in Table 2 are used. They are assumed to fit reasonably with classical steel. Figures 7 to 10 show comparisons between results obtained numerically and the present inclined super-element for the dimensions listed here over. As it can be seen, the resistance is increasing with the angle α, which means that vertical panels (α = 90°) oppose the largest force to the indentation of the striking vessel.

Figure 7. Comparison between analytical and FEM, α = 45°.

Figure 8. Comparison between analytical and FEM, α = 60°.

Figure 5. Finite elements models of the striking ship and the struck plate.

Figure 6.

Material law used for numerical simulations.

Table 2. Parameters defining the stress-strain curve for classical steel, as depicted on Figure 6. E (MPa)

ET (MPa)

σ0 (MPa)

εu

210000

1015

240

0.2

Figure 9. Comparison between analytical and FEM, α = 75°.

These figures show that the agreement between the curves is quite good. In some cases however, the analytical failure appears for greater penetrations than in the numerical model. This is mainly due to the difficulty of choosing an appropriate value for the critical strain εc. In the present paper, we choose εc = 8%, as recommended by Zhang (1999) and

141

Figure 10. Comparison between analytical and FEM, α = 90°.

Lützen (2001) or Buldgen (2012) who mentioned that typical values of εc are between 6 and 12%. In fact, we practically determined εc by making comparisons with the rupture levels predicted by finite elements analyses, so that the rupture strain predicted by the super-elements method corresponds to the one given by LS-DYNA. For the present inclined super-element, we observed that the most adequate value was εc = 8% because this choice was fitting with most of the numerical results. However, it remains difficult to choose a proper value for εc that satisfactorily fit with all numerical simulations. In the numerical model, rupture is supposed to appear when deformations overcome the value of εu = 0.2, but they are highly dependent on the contact conditions between the stem and the plate. For example, if the striking bow is modeled with too sharp edges, this may results in bad contact conditions involving rupture since the very beginning of the impact. Moreover, as shown by Le Sourne (2001), the numerical failure is also related to the mesh size. For these reasons, we decided not to accord a too large confidence to the precise value of δ associated with rupture. In addition to that, the discrepancy between numerical and analytical failures remains acceptable. As mentioned earlier, after failure, the resistance is not immediately set to zero. The plate still withstands further indentations because of the progressive extension of tearing. Since this phenomenon is neglected in the present super-element, our approach remains conservative as it tends to underestimate the total energy dissipated by the structure. This is also an additional argument for not caring about rupture occurring too late. 4

scenario will be treated by using both the finite elements code LS-DYNA and the super-elements method. The latter has already been successfully applied by (amongst others) Pedersen & Zhang (1998), Zhang (1999), Lützen, Simonsen & Pedersen (2000), Lützen (2001) and Le Sourne (2007) for treating ship-ship collisions. Currently, the applications are limited to vessels having an internal structure involving only horizontal and vertical components. However, for some vessels like frigates or aircraft carriers, the shell plating may exhibit a non negligible inclination. So far, this was not tackled by using the super-element approach and the presented new developments are therefore a first step for extending the method. For the numerical simulations with LS-DYNA, the ship is only represented by one of its sections limited by two transverse bulkheads. The total length is of 11 m and the first impact point is exactly located at mid-section. Moreover, we suppose that the ship is at rest against a quay, so that no sway motion may occur during the collision. The material characterizing the ship is still the same than the one depicted on Figure 6, with the values listed in Table 2. In order to point out the importance of accounting for the inclination of the shell plating, we consider two different ships. 4.1

Ship with slight side inclination

For this first collision scenario, the shell plating of the ship is not too deeply inclined. A three dimensional view of the internal structure of the vessel is proposed on Figure 11. In order to model the presence of the quay, we apply boundary conditions so as to prohibit any point of the plane X = 0 from moving in the Z direction. For this first model, the inclination of the shell plating α = 80° is quite

IMPACT ON A FULL SCALE SHIP

In this section, the case of a ship collided by the rigid bow depicted on Figure 5 is considered. This

Figure 11. Three dimensional view of ship model 1.

142

modest. The first contact point between the rigid striking bow and the struck vessel is denoted by P and depicted on Figure 12. The new inclined super-element developed in this paper has been integrated in a calculation code that allows the ship to be treated as a set of superelements. By so doing, it is possible to derive analytically a rough estimation of the curve showing the evolution of the force P opposed by the ship to the penetration of the rigid bow. On Figure 13, we first propose a comparison between the curve P(δ) given by LS-DYNA (“numerical” curve), and the one obtained by the super-elements method including the new developments presented in this paper (“analytical” curve). The agreement is seen to be reasonably good, even if the resistance tends to be overestimated at the beginning of the impact. This is mainly due to a coupling occurring between the uppermost deck (Fig. 11) and the inclined plating. In fact, as the ship is moving forward, this deck is bent around the X axis (Fig. 11) because of the tensile forces transmitted at the boundary with the plating. However, the inclined super-element has been

Figure 12. Section of ship model 1 and corresponding “staircase” model.

Figure 13. Comparison between numerical and analytical results for ship model 1.

developed under the assumption of a plate simply supported on its four edges. Since the deck exhibits flexural vertical displacements, this hypothesis is not completely realistic and we think that further investigations are still needed in order to account for this phenomenon. In order to illustrate the new contribution of our developments, we also decomposed the ship only with horizontal and vertical components, as it could be achieved with the super-elements developed so far. By so doing, we obtain the so-called “staircase” model depicted on Figure 12. If we apply the super-elements method to this simple case, we obtain the curve P(δ) denoted by “staircase” model on Figure 13. As it can be stated, this curve shows an important discrepancy with results given by LS-DYNA. 4.2

Ship with strong side inclination

For this second model of ship, we consider a much more inclined shell plating, with α = 45° (Fig. 14). We choose this particular configuration in order to show the importance of considering the side inclination when developing simplified analytical tools. Except this new value of α, the ship is identical to the one presented in the previous section. In particular, the scantling is very similar to the one depicted on Figure 11. Comparisons between the numerical curve given by LS-DYNA and the analytical ones are shown on Figure 15. It is now clear that the “staircase” model is really inappropriate for modeling ships with strongly inclined plating. On the contrary, the “analytical” curve obtained with help of our new super-element appears to be much more suited. However, there is still a little discrepancy between the “analytical” and the “numerical” curves. As explained in section 4.1, this is still due to the coupling between the decks and the plating. In the present case, the divergence remains

Figure 14. Section of ship model 2 and corresponding “staircase” model.

143

Figure 15. Section of ship model 2 and corresponding “staircase” model.

moderate because rupture occurs before having too large flexural displacements of the decks. But if we remove the failure condition in our models, we observe that the two curves exhibit a discrepancy increasing with the penetration δ. For this reason, it seems to be important to perform additional research in order to account for this phenomenon. 5

CONCLUSION

In this paper, we present a new super-element developed to treat collisions on ships having an inclined shell plating. For such components, a relation between the global resisting force P and the penetration of the striking vessel δ is derived. The analytical developments are then verified by comparing the calculated crushing force with numerical results obtained by simulating collisions between a rigid bow and an oblique plate. We found a quite good agreement with the closedform expression of P(δ) derived for the new superelement. In order to check the coherence of our developments, collisions on full scale ship models are also considered. On one hand, we performed numerical simulations with LS-DYNA for a ship impacted by a rigid bow. On the other hand, we integrate our present developments in a global super-elements treatment of the ship. Comparisons between the results of the two approaches show a reasonable agreement. However, simulations on entire models point out a lack in our calculations. When developing the law P(δ) for an inclined plate, we assumed that it is simply supported on its four edges, where displacements are constrained. On a real ship, this assumption has to be relaxed because the oblique plate is supported by bulkheads and decks. As these structures are not infinitely rigid, they may exhibit some bending deformation, especially when large tensile forces occur in the plating. As a consequence,

the boundaries of the plate are likely to move, which implies a reduced internal dissipation in the super-element. In this sense, our calculations may not be conservative. It is quite difficult to handle with this problem of coupling, because the super-elements method precisely assumed that each structural component is decoupled from the others. As a consequence, some additional research is still needed for a better integration of this phenomenon. Nevertheless, the developments exposed in this paper already allows for a first treatment of collisions on ships having inclined plating. In particular, results are much more confident than those obtained by modeling the vessel only with vertical and horizontal superelements (“staircase” models). This however shows the real need of establishing new laws, as it has been done throughout this work. 6

APPENDIX

In this appendix, we provide the mathematical expressions required to analytically evaluate the resistance of an impacted inclined plate. In this section, we will in fact detail equations (12) and (13). The resisting force provided by the fibers parallel to x axis is given by: 12

t p ∑ Px ,i

Px

(16)

i =1

where the corresponding expressions for 1 ≤ i ≤ 6 are as follow: ⎡ ⎛ 1 ⎞⎤ ⎢1 − f1 arctanh ⎜ ⎟⎥ ⎝ f1 ⎠⎥⎦ ⎢⎣ −1 4b δ ssin2 α ⎛ 1⎞ = 2 1 − ⎜⎝ f ⎟⎠ 3a1 1 ⎤ ⎛ a x0 ⎞ 4RZ b22 sin2 α ⎡ = ⎢ f 2 arctan ⎜ 1 ⎟ − f3⎥ 2 3RX b2 − δ cos α ⎢⎣ ⎥⎦ ⎝ f2 ⎠ 2 2 ⎛ 1 ⎞⎤ 16b1 sin α ⎡ = ⎢1 − f4 arctan ⎜ ⎟⎥ 3x0 cos α ⎢⎣ ⎝ f4 ⎠⎥⎦ −1 4b δ sin2 α ⎛ 1⎞ = 1 1 − ⎜⎝ f ⎟⎠ 3a1 4 2 2 ⎤ ⎛ a x0 ⎞ 4RZ b1 sin α ⎡ = ⎢ f5 arctanh ⎜ 1 ⎟ − f6 ⎥ 2 3RX b1 + δ cos α ⎢⎣ ⎥⎦ ⎝ f2 ⎠

Px ,1 = Px , 2 Px ,3 Px ,4 Px ,5 Px ,6

8b22 sin2 α 3x0 cos α

For 7 ≤ i ≤ 12, the formulae of Px,i are simply obtained by substituting a1 by a2 in the previous expressions of Px,i and fi.

144

The resisting force provided by the fibers parallel to y axis is given by:

f3 =

a1

x0

1 + (a1 − x0 )

2

f 2−1

f6 =

a1

x0

1 + (a1 − x0 ) f5−1 2

16

t p ∑ Py i

Py

(17)

f7 =

i =1

where the corresponding expressions for 1 ≤ i ≤ 6 are as follow: ⎡ ⎛ ⎢1 − f1 arctanh ⎜ ⎝ ⎢⎣ −1 2 x sin i α⎛ 1⎞ = 0 1− ⎟ 2 cos α ⎜⎝ f1 ⎠

Py,1 = Py, 2

Py,3 = Py,4 =

Py,6

b2 (b2 −

) sin2 α

cos

2 (b2 − cos

)2 cos α

b22 sin2 α

2 (b2 − δ cos α )

2

Py,7 =

⎡ a1 x0 ⎤ ⎢ 2 ⎥ cos α ⎣ 1 + f7 ⎦

Py,8 =

3a1b1 sin2 α 2δ cos 2 α

b1 (b1 +

cos

2 (b1 + cos

)

2 (b1 + δ cos α )

) sin2 α

2

− b12 sin2 α

2

( f7 )

f2

⎡ ⎛ ⎢1 − f4 arctan ⎜ ⎝ ⎢⎣ −1 2 x sin α ⎛ 1⎞ = 0 1+ ⎟ ⎜ 2 cos α ⎝ f1 ⎠

Py,5 =

1 ⎞⎤ ⎥ f1 ⎟⎠ ⎥⎦

3a1b2 sin2 α 2δ cos 2 α

cos α

1 ⎞⎤ ⎥ f4 ⎟⎠ ⎥⎦

f5

( f8 )

⎡ a1 x0 ⎤ ⎢ 2 ⎥ cos α ⎣ 1 − f8 ⎦

with: f1 =

a1b2 x0δ α

f2

RX2

f4 =

b2 − δ α RZ cos α

a1b1 x0δ α f5

RX2

a1

x0 f2

f8 =

a1

x0 f5

For 9 ≤ i ≤ 16, the formulae of Py,i are simply obtained by substituting a1 by a2 in the previous expressions of Py,i and fi. REFERENCES Buldgen, L., Le Sourne, H., Besnard, N. & Rigo, P. 2012, Extension of the super-elements method to the analysis of oblique collision between two ships, Marine Structures 29, 22–57. Jones, N. 1997, Structural Impact, Cambridge University Press. Le Sourne, H. 2001, Surface ship collisions: bibliography of fracture models used for fast dynamic finite elements simulations. Assessment of the threshold strain values to be used in a ship collision simulation, IRCN Report 01/RT070. Le Sourne, H. 2007, A ship collision analysis program based on super-element method coupled with large rotational ship movement analysis tool, Int. Conf. on Collision and Grounding of Ships ICCGS-2007, 131–138. Lützen, M. 2001, Ship Collision Damage, PhD. diss., Technical University of Denmark. Lützen, M., Simonsen, B.C. & Pedersen, P.T. 2000, Rapid prediction of damage to struck and striking vessels in a collision event, Int. conf. on ship structure for the new millennium: supporting quality in shipbuilding. Pedersen, P.T. & Zhang S.M. 1998, On the mechanics in ship collisions, Marine Structures 11, 429–449. Simonsen, B.C. 1997, Ship grounding on rock—I. Theory, Marine Structures 10, 519–562. Zhang, S.M. 1999, The mechanics of ship collisions, PhD. diss., Technical University of Denmark.

b1 + δ α RZ cos α

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Damage of ships for engine room section struck by bulbous bow C.F. Hung & Y.D. Hsieh Department of Engineering Science and Ocean Engineering, National Taiwan University, Taiwan

W.L. Chien, Y.T. Huang & Z.M. Zhou Department of Design, CSBC Corporation, Taiwan

ABSTRACT: The engine room is the most important part of ship, which holds the power resource and the control center of whole ship. Damage of engine room after collision may induce the most serious loss of all collision conditions. In this paper, the LS-DYNA nonlinear finite element analysis program was used to simulate the ship collision, in which the engine room section of an 8,236TEU container was struck by a similar size of conventional bulbous bow and a high deformable bow structure. The damage conditions and crashworthiness of striking bow and struck hull structures have been compared. The impact level for evaluation of injuries of crews and damage of main engine or facilities will be investigated. The different collision cases were defined under combination of five parameters, which are the initial collision speeds, the attack angles, the local structural parts of struck hull, and the material properties of steel and structural type of striking bow. The relative motion, damage conditions, crash force and energy dissipation of struck and striking vessels for different collision cases were compared. The root mean square and maximum acceleration of check points specified on struck hull structures and center of main engine have been gathered to evaluate the impact level. 1

INTRODUCTION

Although the velocity of ship motion is not fast, the ship has huge mass, and carries enormous momentum and kinetic energy. During the collision the velocity of striking ship will be reduced very quickly and very high impact loading between two colliding ships is induced, and serious damage may be taken place. The engine room, which holds the power resource and the system control center of whole ship, is the most important part of ship. Damages of this part caused by collision may induce the most serious loss of all collision conditions. Under the consideration of the consequent catastrophe in coast area or ocean environment after damage of struck hull, since 23 years, the double hull and mid-deck design concepts were requested to avoid the oil or chemical material escaping from tank in order to prevent the calamity induced by ship collision and grounding. Numerous researches about the crashworthiness were carried out in different approach, e.g. theoretical, experimental, and numerical approaches (Minorsky 1959; Jones & Jouri 1987; Pedersen et al. 1993; Pedersen & Zhang 2000; Wang 2000; Kitamura 2002; Lehmann & Peschmann 2002; Ehlers et al. 2008; Yamada & Endo 2008; Yagi et al. 2009; Karlsson 2009).

Because of the complexity of ship structures, most of the structural components damages are multi-damage modes. Performing a small scale experiments may have significant uncertainties, whereas the large scale experiments is too expensive. And hence the experiment was performed only in some special projects used as benchmarks for other researches (Rodd 1996; Wevers & Vredevelt 1999, Ehler et al. 2008; Karlsson 2009). Recently, the analysis of structural damage in ship grounding and collision by nonlinear Finite Element (FE) method becomes a principle approach in past years. Nevertheless, it needs large computation resource and the establishment of analysis model and evaluation of analysis results for ship collision need tedious time. Therefore, in the research of ship collision, many researchers use simplified model to analyze. They turn the structural damages to basic damage modes, which can be classified into stretching mode, tearing mode and penetration mode. Then, by analyzing the statistics, the whole reaction force and internal energy were approximated. (Pedersen & Zhang 2000, Wang 2000, Hung & Chen 2007, Yamada & Pedersen 2008, Paik & Seo 2007). This research investigated the impact condition of a bulbous bow striking the hull structures in engine room section of a container. The shock levels were compared with different collision cases,

147

the results will be used as reference for selection of anti-shock measures for engine-room section in ship collision. 2

MODELS OF COLLISION ANALYSIS

2.1

Motion of ships in collision

In case of a ship with speed V2 striking a ship with zero speed in perpendicular direction, the momentum before and after collision is V2M 2 = (M1

M 2 )V

(1)

Figure 1.

Analysis model of striking bow structure.

where M1 = mass of struck vessel and its added mass; M2 = mass of striking vessel and its added mass; V2 = initial velocity of striking ship; V = combined velocity of both vessels after collision. The combined velocity of both ships after collision becomes: V2M 2 / (M + M

V

)

(2)

The lost kinetic energy after collision is: 1 M 2V22 2

ΔE =

(M

Figure 2. section.

+ M )V 2

1 M1 M 2V22 2 M1 M 2

(3)

Most of the lost kinetic energy will be transferred to the internal energy of damaged structural members. 2.2

Analysis model of struck engine-room

FE-analysis model

In this research, the bulbous bow structure of an 8,236TEU container striking the side hull in engine room section of the same size of container was simulated. In the FE analysis model, the resistance of water around the ship was represented by added mass of ships. During collision, the striking ship is moving longitudinally, its water resistance is smaller, and the struck ship is moving toward lateral, its water resistance is much higher than striking ship, so we took the added mass to be 10% of displacement for striking ship and 50% for struck ship. The engine room is located at after part of ship; and the struck ship will be rotating during collision. Nevertheless, the motion direction of ships during collision will be not changed significantly, the added mass was assumed to be unchanged during collision. The FE-model of striking bow structure and the struck hull in engine room section are shown in Figure 1 and Figure 2, respectively. The bow

Figure 3. Analysis model of engine-room section with two rigid extended parts of an 8,236 TEU container.

model contains its own mass the added mass, the material density was adjusted, so the model mass is equal to bow mass and its added mass. The mass of other parts of ship including added mass are applied on the afterward bulkhead of bow model, uniformly. For struck ship, the steel density of struck hull model was adjusted so that the mass in engine room section contains the structure mass, fluid mass in tank and the added mass. The main engine was established as rigid parts; its model contains the engine mass, and its c.g. is kept in design position. The other structure parts before and after the engine room section is established as rigid hull without inner structural parts. The mass includes ship mass and added mass except the engine room section, the mass was distributed following the load curve of hull girder, and the distribution of mass (ship mass and added mass) was adjusted, so that c.g. is located on the design position, shown in Figure 3. The material constants of regular steel of analysis model are listed in Table 1. The rupture strain for regular steel is 15% to 22%, this work selects 0.2. The Nippon Steel (2010) has developed

148

Table 1.

Material constants of steel analysis model.

Item

Regular steel

High ductile steel

Elastic modulus (E) Mass density (ρ) Static yield stress (σy) Tangent modulus (Et) Rupture strain (εrup) Ultimate stress (σult) Poisson’s ratio (ν) Friction coefficient (µ)

2.1 × 105 MPa 7860 kg/m3 300 MPa 400 MPa 0.2 380 MPa 0.3 0.2

2.1 × 105 MPa 7860 kg/m3 300 MPa 200 MPa 0.4 380 MPa 0.3 0.2

Table 2.

NS-Ship-Safety235 in 2009, the rupture strain of high ductile steel is 1.5 to 2.0 times of regular steel, and in this study 0.4 is used for high ductile steel, temporarily. In the FE analysis model, the shell element and beam element were used. The element size was set 200 mm.

Case V2

θ

Bow type

Hull type Hull mat.

1 2 3 4 5 6 7

90° 45° 90° 90° 90° 90° 90°

Normal Normal Normal Normal Normal High defom. High defom.

Original Original Original M1 M2 Original Original

5 m/s 5 m/s 10 m/s 5 m/s 5 m/s 5 m/s 5 m/s

ANALYSIS CASES

Different collision conditions may have different effects on the damaged conditions. The different collision cases will be defined under combination of five parameters, which are the initial collision speeds, attack angles, type of local structural parts of struck hull, materials properties of steel and structural type of striking bow. In this paper two cases of angle of collision (θ) are considered.θ = 90°, the moving direction of striking ship is perpendicular to struck ship; θ = 45°, the moving direction of striking ship has 45° to struck ship. Two cases of initial velocity of striking ship (V2) are considered, V2 = 5 m/s and V2 = 10 m/s. Two bow structure types are considered, the regular bow design, and the modified bow design with smaller stiffness and higher ductile material, which is called high deformable bow. The comparison case studies are summarized below and Table 2. 1. Different angle of collision Case 1: θ = 90°, V2 = 5 m/s Case 2: θ = 45°, V2 = 5 m/s. 2. Different initial velocity of striking ship Case 1: θ = 45°, V2 = 5 m/s Case 3: θ = 45°, V2 = 10 m/s. 3. Different type of local structure Case 1: θ = 90°, V2 = 5 m/s, original hull design Case 4: θ = 90°, V2 = 5 m/s, modified local hull structure, type M1 Case 5: θ = 90°, V2 = 5 m/s, modified local hull structure, type M2.

Regular Regular Regular Regular Regular Regular High ductile

4. Different type of material and bow type Case 1: initial bow type, regular hull steel Case 6: high deformable bow, regular hull steel Case 7: high deformable bow, high ductile hull steel. 4

ANALYSIS RESULTS

4.1 3

Parameters of all 7 analysis cases.

Different angle of collision

Figure 4 shows damage condition of hull structure at 6th second after collision in two different cases of collision angle. The 90° collision case has a more serious ruptured structural member, and the 45° collision has wider permanent deformed structural parts. Figure 5 shows the four check points for investigation of impact acceleration during collision process, which are: Point a: FR34 web-frame connected with bottom deck and longitudinal floor and the bottom support of main engine. Point b: Connection point of inner hull and bottom deck and web frame at FR37. Point c: Inner hull of impact position at FR37 Point d: FR41∼FR45 where bottom web-frame connected with bottom floor and the support of main engine. The root mean square (RMS) and the maximum acceleration at 4 check points are listed In Table 3. In both case the RMS and maximum acceleration occur not at inner hull of impact position. In Case 1 (θ = 90°), it occurred at check point b (connection point of inner hull, bottom deck and web frame at FR37), the acceleration is reduced remarkably at pint a. In Case 2 (θ = 45°), it occurred at check point d; And the acceleration at point b is relative low. Figure 6 shows the time history of acceleration at the center-of-gravity of main engine. The time history and magnitude of acceleration of Case 1 and Case 2 are very different. Figure 7 shows the time history of internal energy. The struck point (the engine room) is located at after part of ship; the struck ship

149

Figure 4. angles.

Damaged conditions of two cases of collision

Figure 7. Time history of internal energy of struck hull in Case 1 and Case 2.

Figure 5. 4 check points for observation of impact acceleration on hull structure. Table 3. RMS and the maximum acceleration (m/s2) at four check point for Case 1 and Case 2. Check point

a

b

c

Figure 8. The extruded bow profile in the struck hull in Case 1 and Case 2 after collision.

d

RMS acceleration 31.6 θ = 90° 17.2 θ = 45°

81.7 1.4

50.2 21.3

31.3 40.5

Max acceleration 265.8 θ = 90° 256.3 θ = 45°

2433.5 22.6

651.0 415.7

401.6 1194.7

collision angle. Figure 8 shows the extruded bow profile in the struck hull. From the analysis results some points can be summarized:

Figure 6. Time history of acceleration at c.g. of main engine.

will be rotating during collision, and the motion resistance of struck ship is smaller than the collision case when the striking point is located at c.g. of struck ship. There is no significant difference in internal energy between the two cases with different

− The hull structure has larger RMS and maximum acceleration at check point b and point c for 90° collision case; and at check point d and point c for case of 45° collision. The acceleration at point b is relative small in 45° collision case. − In the 45° collision case, the impact force on struck ship is much smaller than 90° collision case. − The maximum acceleration of 90° collision case is 30% higher than 45° collision case, and average acceleration is also higher. − The internal energy of 45° collision case is little lower than 90° collision case in first 1.0 second after collision, and from 1.0 second to 4.3 second is little higher, after 4.5 second the internal energy becomes little lower. In generally, there is no significant difference in internal energy between the two cases with different collision angle. − The 90° collision case has deeper indentation, and the 45° collision has wider indentation area.

150

− The RMS and maximum acceleration at local structure may b3 very high, the maximum acceleration at main engine is lower than 4 m/s2 for initial velocity 5 m/s. 4.2

Different initial velocity of striking ship

Figure 9 shows 4 check points on struck ship (A, B, C, D) and point E on striking ship for observation of ship motion and deformation during collision. The time history of velocity of check points of C and E in Case 1 is shown in Figure 10. From 4 seconds after collision, the velocity of struck ships higher than striking ship, it means after collision the two ships are separated. Figure 11 shows the time history of displacement at all 4 check points. The results show the translation and rotation of

Figure 9.

Check points for observation of ship motion.

struck ship, the relative displacement between points C and E indicate the indentation depth of struck hull. The maximum depth is 8.4 meters. The difference of displacement between check point A (bow) and point D (stern) shows the rotation of ship, the difference between the displacement the mid-ship and mean displacement of point A and point D is the horizontal deflection of struck hull at mid-ship, and represents the horizontal bending of struck hull. Figure 12 shows the time history of velocity of check points of C and E in Case 3. Both ships are linked together from 5 seconds after collision. Figure 13 shows the time history of displacement at all 4 check points. The maximum indentation depth is 13.21 meters. The rotations of Case 1 and Case 3 at 6 seconds after collision are almost the same. The horizontal deflection at mid-ship of Case 3 is about 4.2 time of Case 1. Theoretically the lost kinetic energy for case of two collision ships being separated after collision is smaller than for the case of two ships being linked together. The lost kinetic energy after collision approximated by eq. (3) is a critical case; nevertheless the energy loss due to friction is not taken into consideration in eq. (3). Table 4 lists the lost kinetic energy after collision approximated by eq. (3) and the FEM calculated lost kinetic energy of Case 1 to Case 3. Although eq. (3) is used for approximation of kinetic energy for 90°

Figure 10. Time history of velocity of two ships in Case 1.

Figure 12.

Figure 11. Time history of displacement of 4 check points in Case 1.

Figure 13. Time histories of displacement of 4 check points in Case 3.

151

Time history of velocity of ships in Case 3.

Table 4.

The lost kinetic energy (MJ) in different case.

ΔE from eq. (3) FEM calculated ΔE

Case 1

Case 2

Case 3

− The lost kinetic energy approximated by eq. (3) is only about 10% lower than Case 1, Case 2 and Case 3 including contact-friction loss.

390 431

453 482

1864 2033

4.3

Figure 14. Damaged condition of struck hull in different initial velocity cases.

Table 5. Root mean square and the maximum acceleration (m/s2) at check points on struck hull for Case 3 and Case 4. Check point

a

RMS acceleration 31.59 V2 = 5 m/s 43.72 V2 = 10 m/s

b 81.65 124.66

c

To investigate the effect local structural type on the crashworthiness of struck hull in engine room section, three types of local structures are examined, shown in Figure 15, Figure 16 and Figure 17, respectively, which are the Case 1, Case 4 and Case 5. Case 1 is the initial design. In Case 4 a horizontal deck in side tank is added. In Case 5 a web ring is added to original design. The added structural parts in Case 4 and Case 5 are small change only, the purpose of comparison study of three cases is to show the design sensitivity of transverse member and longitudinal member. Figure 18 shows the time history of internal energy of struck hull, and Figure 19 shows the

d 50.17 89.27

Max acceleration 265.8 V2 = 5 m/s

2433.5

650.9

554.5

3191.9

995.4

V2 = 10 m/s

Different type of local structures of struck hull

31.26 96.51 401.55 3284.6

collision, and striking point at c.g. struck of ship, it is not suitable for this analysis. The results calculated by FEM (including contact-friction loss) have only about 10% higher. Figure 14 shows the damage conditions of struck hull after collision. Table 5 shows the RMS and the maximum acceleration of both cases at 4 check points shown in Figure 5. Some point can be summarized from the calculated results: − The faster speed the striking ship is the more momentum and energy it has. And the damage will become more serious. − The rotations of Case 1 and Case 3 at 6 seconds after collision are almost the same, the deflection at mid-ship for Case 3 (V2 = 10 m/s) is about 4.2 times of Case 1 (V2 = 5 m/s). − The relationship between damage conditions and the initial velocity of striking ship is nonlinear.

Figure 15. (Case 1).

The initial design of local side tank structure

Figure 16. Type M1 (a horizontal deck is added to initial design) (Case 4).

152

Figure 20. Time history of acceleration at c.g. of main engine in Case 1, Case 4 and Case 5.

Figure 17. Type M2 (a web-ring is added to initial design) (Case 4).

Table 6. RMS and the maximum acceleration at c.g. of main engine in Case 1, Case 4 and Case 5. Case

Case 1

Case 4

Case 5

RMS acceleration Max acceleration

1.44 3.79

1.55 3.93

1.63 4.62

From the results a brief summary can be made:

Figure 18. Time history if internal energy struck hull for Case 1, Case 4 and Case 5.

− Compared with tree cases, Case 5 has higher internal energy and lower average reaction forces; and Case 4 has a higher reaction force. − The maximum acceleration at c.g. of main engine in Case 5 has highest peak value and also RMS of acceleration. − Both Case 4 and Case 5 have higher RMS and maximum acceleration of main engine than Case 1. − The Case 5 is superior in crashworthiness, but inferior in acceleration effects on main engine. 4.4

Figure 19. Time history of reaction force in Case 1, Case 4 and Case 5.

time history of reaction force of three collision cases. Figure 20 shows the time history of acceleration at c.g. of main engine. The RMS and maximum acceleration at center gravity of main engine are listed in Table 6.

Different type of striking bow structures

To investigate the effect bow structures on the crashworthiness, a soft bow structure modified from initial design of Case 1, shown in Figure 21 was examined. The soft bow structure has some lower stiffness than initial design. To keep the same rupture strength of soft bow structure as the initial design, the high ductile material was used. It means the soft bow will have higher deformation than initial design, and both types will absorbed almost the same energy, when the bow structure is ruptured during collision. The soft bow is also called high deformable bow. The material constants are listed in Table 1. In another alternative the high ductile material is used for stuck hull. In this study, the Case 1 has initial bow type; in Case 6 the high

153

Figure 21.

Different type of bow structure.

Figure 25. Time history of acceleration at c.g. of main engine in Case 1, Case 6 and Case 7.

Table 7. RMS and the maximum acceleration (m/s2) at c.g. of main engine in Case 1, Case 6 and Case 7.

RMS acceleration MAX acceleration

Case 1

Case 6

Case 7

1.44 3.79

1.47 3.73

1.77 4.18

Figure 23. Internal energy of damaged structural parts in Case 1, Case 6 and Case 7.

deformable bow type is used; in the Case 7 the high deformable bow and stuck hull with high ductile are considered. Figure 22 shows the extruded bow profile in struck hull of Case 1, Case 6 and Case 7. Figure 23 shows the time history of internal energy of struck hull, and Figure 24 shows the total reaction force of three cases. Figure 25 shows the time history of acceleration at c.g. of main engine. The RMS and the maximum acceleration at c.g. of main engine in Case 1, Case 6 and Case 7 are listed in Table 7. The indent depth of struck hull, energy dissipation of all damaged parts in collision, and the time when the outer and inner hull of struck ship was ruptured in all 7 cases are summarized and listed in Table 8. From the calculated results the brief summation are made:

Figure 24.

− Case 6 and Case 7 have significant higher internal energy of damaged structural parts and total reaction force than Case 1. − From the impact acceleration at the c.g. of main engine, Case 6 has better buffer effect for main engine. − After collision the damaged contact area between two colliding ships in high deformable bow is wider than Case 1. − When the out hull damaged, the case with high deformable bow has higher indented depth of struck hull, higher energy dissipation and higher reaction force than the normal bow case.

Figure 22. Extruded bow profile in struck hull in Case 1, Case 6 and Case 7.

Reaction force of Case 1, Case 6, Case 7.

154

Table 8. The indent depth, energy dissipation, and the time when the outer and inner hull structures of struck ship were ruptured in different case. When outer hull ruptured

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7

When inner hull ruptured

Indented depth (m)

Energy (MJ)

Time (sec)

Indented depth (m)

1.793 2.581 4.075 1.778 1.739 1.812 2.517

37.7 377.2 45.9 32.1 36.7 39.3 376.0

0.38 1.76 0.40 0.31 0.35 0.39 1.55

3.133 110.3 2.593 401.5 5.068 65.6 3.354 110.9 3.095 94.5 2.457 157.3 Inner hull no ruptured

− In case of high deformable bow and struck hull with high ductile material, the indented depth of struck hull and energy dissipation are significant higher than Case 1 when the out hull is ruptured. After collision the inner hull is not ruptured.

5

CONCLUSION

This paper investigated the damage and impact conditions of struck hull in engine room section for an 8,236 TEU container in ship collision. Based on the previous analysis results, the following conclusions are summarized: 1. Effect of different collision angle and collision speed. a. In 90° collision case the indented depth of stuck hull is higher, and more structural parts were ruptured. In 45° collision case, the struck hull has wider permanent deformed area, and has less ruptured parts. b. The struck point in engine room section is located after part of ship; the struck ship will be rotating during collision, and the motion resistance of struck ship is smaller than the collision case when the striking point is located at c.g. of struck ship. The difference in internal energy between 45° and 90° collision cases is not significant. c. The relationship between damage conditions and the initial velocity of striking ship is nonlinear. d. For very high collision speed (in case of initial speed 10 m/s) the indented depth of struck hull are much higher than the case of low collision speed. Nevertheless the energy dissipation is very low when outer and inner hull of struck ship are ruptured.

Energy (MJ)

Time (sec) 0.53 1.95 0.51 0.52 0.65 0.89

e. Although eq. (3) is used for approximation of kinetic energy for 90° collision, and striking point at c.g. struck of ship, it may not suitable to use for the analysis cases in this paper. The results calculated by FEM (including contactfriction loss) have only about 10% higher. The lost kinetic energy approximated by eq. (3) can be used as a first approximation of energy dissipation of structures after collision. 2. Effect of different local structural type of struck hull. a. Compared with different local structures, Case 5 with added vertical web-ring has higher internal energy and lower average reaction forces during collision. b. Case 4 with an added horizontal deck in side tank has higher reaction forces. c. The Case 5 with added vertical web-ring is superior in crashworthiness, but inferior in acceleration effects on main engine. d. Both Case 4 and Case 5 have higher RMS and maximum acceleration of main engine than Case 1. e. The local structure of struck hull has small effects on indented depth and energy dissipation of struck hull. 3. Effect of bow type and material properties of steel. a. Compared with the initial bow case, in case of high deformable bow and struck hull with regular steel, the internal energy of damaged structural parts and total reaction force are significant higher. The RMS and peak acceleration at c.g. of main engine have only small difference from the case of normal bow type. b. Besides high deformable bow the high ductile steel is used for struck hull, the internal energy of damaged structural parts and total reaction force will be increased remarkably;

155

the inner hull will be not ruptured after collision. I.e. the crashworthiness is raised obviously. Nevertheless the RMS and peak acceleration at c.g. of main engine is raised. ACKNOWLEDGEMENTS The authors would like to acknowledge that this work was supported by the National Science Council of ROC under grant NSC99-2221-E-002228-MY3, and also partially supported by CSBC Corporation, Taiwan.

REFERENCES Ehlers, S., Broekhuijsen, J., Alsos, H.S., Biehl, F., Tabri, K., 2008, Simulating the collision response of ship side structures: A failure criteria benchmark study, International Shipbuilding Progress, vil. 55, pp. 127–144. Hung, C.F., Chen, C.P., 2007, The approximate method to predicate the crashworthiness of ship double hull structures, J. Taiwan SNAME, vol. 26, no. 3, pp. 139–150. Jones, N., Jouri, W.S., 1987, A Study of Plate Tearing for ship Collision and Grounding Damage, Journal of Ship Research, vol. 31, no. 4, pp. 253–268. Karlsson, U.B., 2009, Improved collision safety of ships by an intrusion—tolerant inner side shell, Marine Technology, vol. 46, no. 3, pp. 165–173. Kitamura, O., 2002, FEM approach to the simulation of collision and grounding damage, Marine Structures, vol. 15, pp. 403–428. Lehmann, E., Peschmann, J., 2002, Energy absorption by the steel structure of ship in the event of collisions, Marine Structures, vol. 15, pp. 429–441.

Minorsky, V., 1595, An analysis of ship collisions with reference to protection of nuclear power plants, Journal of Ship Research, pp. 1–4, 1959. Nippon Steel Corporation, 2010, Steeling for Global Competition, Annual Report 2010, March 31, 2010 Paik, J.K., Seo, J.K., 2007, A method for progressive structural crashworthiness analysis under collisions and grounding, Thin-Walled Structures, vol. 45, no. 1, pp. 15–23. Pedersen, P.T., Valsgaad, S., Olsen, D., Spangenberg, S., 1993, Ship impact: bow collision. International J. Impact Engineering, vol. 13, no. 2, pp. 163–187. Pedersen, P.T., Zhang, S., 2000, Absorbed Energy in Ship Collisions and Grounding-Revising Minorsky’s Empirical Method, Journal of ship Research, vol. 44, no. 2, pp. 140–154. Rodd, J.L., 1996, Observation on Conventional and Advanced Double Hull Grounding Experiments, Int. Conf. On Design and Methodologies for Collision and Grounding Protection of Ship, San Francisco, USA, pp. 13.1–13.13. Wang, G., 2000, Behavior of a double hull in a variety of stranding or collision scenarios, Marine Structures, 13, pp. 147–187. Wevers, L.J., Vredevelt, 1999, Full Scale Ship Collision Experiment, TNO Report, Delft. Yagi, S., Kumamoto, H., Muragishi, O., Takaoka, Y., Shimoda, T., 2009, A study on collision buffer characteristic of sharp entrance angle bow structure, Marine Structures, vol. 22, no. 1, pp. 12–23. Yamada, Y., Endo, H., 2008, Experimental and Numerical Study on the Collapse Strength of the Bulbous Bow Structure in Oblique Collision, Marine Technology, vol. 45, no. 1, pp. 42–53. Yamada, Y., Endo, H., Pedersen, P.T., 2008, Effects of buffer bow structure in ship-ship collision, International Journal of Offshore and Polar Engineering, vol. 18, no. 2, pp. 133–141.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Modeling aspects of strength capacity of intact and damaged ship girders Dimitris Koukounas University of Patras, Greece

Manolis S. Samuelides National Technical University of Athens, Greece

ABSTRACT: Goal based standards require that classification rules include requirements related to the ultimate strength capacity of ship structures, whereas the recent release of the Unified Common Structural Rules for Tankers and Bulk Carriers contain a section dedicated to this aspect. Researchers have published numerous finite element simulation of bending of ship girders that aim to determine the ultimate strength of the hull in intact condition. However, most of the published work addresses symmetric problems, i.e. symmetric hulls and loading, whereby the neutral axis of the cross section moves but does not rotate as the loading increases. In the case of Finite Element simulation of the bending of girders with non-symmetric cross-sections, such as the girder of damaged ships, particular attention should be given to the boundary conditions applied to the ends of the modeled length of the girder in order that the forces and moments that are induced in the end sections of the model are realistic and represent the actual behavior of the hull girder. The paper addresses modeling aspects for the strength capacity of intact and damaged ship girders in light of the related requirements of Unified Common Structural Rules. 1

INTRODUCTION

In July 2012 IACS published a draft version of the Harmonized Common Structural Rules (IACS, 2012). In comparison to the version of CSR currently in force, the Harmonized-CSR contain provisions related to the residual strength of tankers and bulk carriers, i.e. the ultimate strength in damaged conditions. In accordance to the draft, the residual strength is assessed for specific collision and grounding damages against a linear combination of the design bending moment in calm water and the design wave induced bending moment, both in hogging and sagging conditions. For the determination of the residual strength the draft suggests a formula, including a factor to account for the rotation of the neutral axis in the case of collision damage, which results in a non-symmetric damaged cross-section. Various methods have been proposed for the calculation of the residual strength. Recently Paik et al. (2012) and Villavicencio et al. (2011) have published relevant work. Paik et al. (2012) introduced a Grounding Damage Index (GDI) representing the loss of material of the inner and outer bottom plating with stiffening. Once the damage has been

defined the residual strength is determined using one of the following procedures: a) the modified Paik—Mansour formula (Paik et al. 2011), b) the method described in CSR, which is a “Smith-type” method, c) ANSYS finite element explicit code and d) the ALPS/HULL (2011) super-size finite element method. The paper does not make any reference to the rotation of the neutral axis, which is expected when analyzing the non-linear bending response of thin walled beams. It is noted that the Paik-Mansour formula (Paik et al. 2011) is derived for cross sections that are symmetric about an axis normal to the Neutral Axis, the CSR method is an iterative method that allows for the translation of the Neutral Axis and the FE element methods do not require the a priori definition of the Neutral Axis. Having compared the residual strength that has been obtained using the four different methods, the authors plotted the residual strength obtained by the Paik—Mansour formula versus the damage index for a number of grounding scenarios defined using a sampling technique. The method suggested by Villavicencio et al. (2011) is based on the following steps: a) Generate a strain-stress relationship for a double bottom considering both the rotation of the end sections and the compression of the

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length under investigation. The rotation and the compression are dependent, because they follow the rotation of the cross section of the ship’s hull around its neutral axis. The location of the neutral axis is determined under the assumption that the hull is intact. b) Use the stress-strain relationship generated in previous step to calculate the moment—curvature relationship of the damaged hull. In this step the neutral axis is allowed to move from its initial position because of the buckling of double bottom elements. c) Repeat step a to generate an updated strainstress relationship of the double bottom, taking into account that the neutral axis of the ship’s hull was shifted in accordance with the results of step b. The applicability of the specific procedure that the paper suggests is limited only to symmetrical damages. The work reported herein addresses the determination of the ultimate and residual strength of bulk carrier using Abaqus CAE software. However, the resulting curvature-moment curve and consequently the ultimate strength Mu depend on the modeling parameters, in particular the solution algorithm, the mesh size, the boundary conditions, the extent of the model and the model of the material. The goal of the work is to investigate the sensitivity of the results on the modeling parameters. The paper is divided into the following sections: Target Vessel and FE Model: The characteristics of all the models used in all cases, are presented in this section, together with the material models. Numerical Simulations and Results: Brief description of the analysis procedures and algorithms, and results presentation. Comparison of Simulation and Modeling procedures: Comparison of the results obtained using different algorithms or modeling procedures. Ultimate and Residual strength of a Bulk Carrier: Comparison of the Ultimate and Residual strength for the target vessel. 2

The FE models were discretized using S4 reduced integration elements (see Fig. 1). The mesh size was set to approximately 100 mm and in most surfaces of the model is structured. The size of the elements was selected on the basis of the work of Pollalis et al. (2012). The thickness of the structural elements was taken net, with or without the half of the corrosion margin, depending on the case. Six different materials are used: steel grade A, AH32, AH36, DH32, DH36, and EH36. The Young Modulus is 210 GPa. The yielding stress for grade A is equal to 235 MPa, for AH32 and DH32 is 315 MPa, and for AH36, DH36, EH36 is 355 MPa. When the material was assumed to exhibit hardening, the hardening properties were taken as in Table 1 and in Figure 2, in terms of true stress—true strain. The density for all the cases is considered equal to 7800 Kg/m3. After creating and meshing the model in Abaqus, all the nodes of the fore and aft cross sections, are coupled to define a rigid body for each section. Translation and rotation of the nodes that are attached on each rigid body, are controlled by the control point that has been selected in the centre of area of the related section. For all the analysis cases, rotation control is used. This means that rotation is applied to each control point, imposing curvature on the model and the associated bending moment capacity is the reaction moment that is developed to the control points to achieve the applied curvature. Two different solution algorithms have been used: static implicit and dynamic explicit, i.e. “Static General” and “Dynamic Explicit” in Abaqus terminology. Five cases of different boundary conditions are investigated. These are presented in Table 2, which indicated the constrained degrees of freedom, the degrees of freedom where the rotations are applied and the solver, static or explicit dynamic that has been used in each case.

TARGET VESSEL AND FE MODEL

The target vessel is an 180,000 tonne DWT bulk carrier with length equal to 283 m, breadth 45 m, depth of 24.7 m and design draught 16.5 m. Two models have been tested: most of the simulations were performed using a strip of the hull structure between two successive transverse frames i.e. floors, that have a distance of 3.7 m—1 block model-, and a second model that has a length of three floor space strips, i.e. a length of 11.1 m—3 block model.

Figure 1. 1 block model meshed with s4R elements.

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Table 1.

Hardening parameters for steel in terms of true stress—true strain.

Grade A

AH32, DH32

AH36

DH36, EH36

Stress

Plastic strain

Stress

Plastic strain

Stress

Plastic strain

Stress

Plastic strain

237 388

0 0.0179

315 439

0 0.0177

355 449

0 0.0177

355 490

0 0.0175

Figure 2. Stress-strain hardening-see Table 1.

curves

showing

material

Table 2. Boundary condition cases—the red colour means constrained degree of freedom. The symbol * means that rotations are applied in the d.o.f. or indicates the algorithm used (T: translation, R: rotation, Longitudinal axis: x, Transverse vertical and horizontal axes: y and z respectively). Boundary condition case d.o.f

1

NODE 1 Tx Ty Tz Rx Ry Rz NODE 2 Tx Ty Tz Rx Ry Rz Implicit Explicit

3 3.1

* *

2

3

*

*

* * *

* *

4

5

* *

* *

NUMERICAL SIMULATIONS AND RESULTS Static analysis

As the stiffness of the model is suddenly reduced sharply there were convergence problems when

using static “Static General” solution algorithm, which caused the analysis to stop, before reaching the ultimate strength. One measure to overcome this problem and to achieve a solution of static equilibrium is to use the automatic stabilization mode, offered by “Static General”. Automatic stabilization introduces artificial damping forces in the model, that are proportional to an artificial velocity vector Vi = ΔUi/Δt, where ΔUi = Ui(t+Δt)-Ui(t), Ui(t) being the nodal displacement vector for all the degrees of freedom i, and Δt is the current time increment. The damping forces are Fi = c ⋅ M ⋅ Vi where c is the damping factor and M is an artificial mass matrix computed by the software. It is very important to control the amount of damping used in the analysis because too much damping leads to an overestimation of the Ultimate Strength. Damping is controlled through the Dissipated Energy Fraction (DEF), which is the ratio of the energy dissipated by the artificial damping, to the total internal energy of the model. This parameter must be checked after the analysis, to ensure that the percentages of DEF ratio are small enough. There are three ways to control the amount of damping used in the model: The first is by changing the damping factor c, which remains constant during the analysis. The second method is adaptive automatic stabilization, in which, a damping factor c for the first time increment and a maximum DEF are defined by the user. The algorithm, automatically adjusts the damping factor of the other increments, in a way not to exceed the defined maximum value of DEF. The third method requires the definition of DEF in the initial time increment. For the present study, the third method has been selected, because it results to lower amount of damping applied to the model. Boundary condition cases 1 to 4 described in Table 2 were simulated for DEF values ranging from 5 ⋅ 10−7 to 4 ⋅ 10−4. The net scantlings and materials without hardening are used. The investigation of the ultimate strength has been performed for both the sagging and hogging conditions. Sagging: Figures 3–6 show the moment curvature curves for the four boundary condition cases, for each DEF. Table 3, presents the values of ultimate

159

Figure 3.

Figure 6.

Sagging BC case 1.

Sagging BC case 4.

Table 3. Comparison of the ultimate strength in GNm resulting from different BC cases. The symbol—indicates that the analysis did not reach the point of ultimate strength.

Figure 4.

DEF

BC case 1

BC case 2

BC case 3

BC case 4

4E-04 1E-04 2E-05 5E-06 5E-07

14.89 14.78 14.70 14.70 14.71

15.61 15.08 14.78 14.77 14.77

15.75 15.11 14.78 14.77 -

14.97 -

Sagging BC case 2.

Figure 7. Convergence of Mu for Sagging and Hoggingsee Table 3. Figure 5.

Sagging BC case 3.

strength for each case. The convergence of the ultimate strength with DEF for each BC case is presented in Figure 7. As it may be observed, the ultimate strength converges with the reduction of the DEF. The ultimate strength is converged to 14.7 GNm for BC case 1 and to 14.77 GNm for BC cases 2 and 3. The convergence is achieved for DEF values lower than 5E-5. The difference between BC case 1 on one hand and BC cases 2 and 3 on the other is probably because the boundary conditions of case 1 introduce vertical shear forces. BC case 4 does not exhibit as good convergence as the other cases. The lowest DEF value, for which the analysis is completed, is DEF = 0.0006 and the resulting Mu is equal to

14.98 GNm. For DEF = 0.0002, the last point of the analysis, that is very close to the ultimate strength point, gives Mu = 14.84 GNm. This indicates a tendency of the Mu values to converge for the BC case 4 too, as it can also be seen from Figure 7. It is further noted that the moment-curvature curves obtained from BC cases 2 and 3 are very close, and they differ from the curve obtained from BC case 1. This difference is greater for higher DEF values. Also, it can be seen from Figure 7, that the convergence is faster for BC cases 2 and 3. The following parameters that affect the convergence and consequently the reliability of the numerical results are checked: a. The dissipated stabilization energy over the internal energy ratio (ALLSD/ALLIE as named in Abaqus) at the point of the ultimate strength, as

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shown in Figure 8. It can be observed, that as the DEF parameter is reduced, the amount of damping at the ultimate point is converged to zero. b. The total reaction force in y—direction ΔRFy is studied as a function of time for each analysis. This force, normally must be zero. A nonzero total reaction force is introduced by the damping and converges to zero as damping decreases. In Figure 9, the total reaction force is presented for BC case 2. From what has been discussed above, it can be seen that the more suitable BC case, to be used together with static general, is the case 2, for the following reasons:

BC case, are shown in Figures 10–12. Figure 7, that shows the convergence also for the hogging simulations, illustrates that Mu generally exhibits convergence with the reduction of DEF. For the BC case 1, the Mu of convergence is 16.31 GNm for DEF lower than 0.0001. For the BC case 2, the Mu of 16.48 GNm is obtained for DEF = 0.0001. This is not a convergence value, but is the value for the lowest DEF, for which, the analysis does not stop before the ultimate strength is reached. For the BC case 3, the only value that can be obtained is Mu = 16.82 GNm for DEF = 0.0004. For the BC case 4, the analysis does not reach the ultimate point for all the values of DEF studied.

a. The BC case 2 does not introduce shear forces to the model. b. The analysis passes the ultimate point even if the amount of damping is small. c. For DEF = 5E-5, a value that introduces a small amount of damping, the analysis is completed, giving an ultimate strength value of 14.93 GNm, which is very close to the ultimate strength obtained with DEF = 5E-7. Hogging: The curvature—moment curves for sagging condition for all the DEF values and for each Figure 10.

Hogging BC case 1.

Figure 11.

Hogging BC case 2.

Figure 12.

Hogging BC cases 3 and 4.

Figure 8. Dissipated to internal energy ratio at the ultimate strength point for Sagging and Hogging.

Figure 9. Reaction force in y direction for Sagging BC case 2.

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As in the case of sagging the parameters that affect the reliability of the results are checked: a. The dissipated stabilization energy to internal energy ratio (ALLSD/ALLIE as named in Abaqus) at the point of the ultimate strength is presented in Figure 8. As arises from the results, the dissipated energy to internal energy ratio decreases as the DEF parameter is reduced. b. As it can be observed, the total reaction force in y direction converges to zero as the DEF is reduced. In Figure 13, is shown the reaction force’s convergence to zero, for BC case 1. 3.2

Explicit time-integration

As mentioned in the previous section the case 2 of the boundary conditions was found to be appropriate for the analysis. The same scenario is simulated employing the dynamic solver in combination with a time integration scheme. A further boundary condition case, BC case 5, is also simulated using the same solver, to study possible effect that can caused to the solution by allowing an end section to move freely as a rigid end. The time period, the time increment size and the amplitude of the applied rotation are of critical importance when using the explicit time integration scheme. The time period is set equal to 2 second, which is large enough that leads to very small ratio of the kinetic energy over the total internal energy, for all the increments during the analysis. This ratio must be checked after each analysis, to ensure that the kinetic energy doesn’t plays an important role to the results. The size of the time increment is set equal to 1E-6 sec that is a small enough value to provide error transfer during the elements of the model. The amplitude of the applied rotation, which is the function of the applied rotation vs. time, is set linear.

The following modeling cases are studied with Abaqus Explicit solver: a. One block model with the net scantlings for BC case 2 and case 5. The material models do not include strain hardening. Rotations are applied to simulate both hogging and sagging conditions. b. One block model with the net scantlings plus half of the corrosion margin for BC case 5. The material models exhibit strain hardening. Rotations are applied to simulate sagging condition. c. Three block model with the net scantlings plus half of the corrosion margin for BC case 5. The material models do not include strain hardening. Rotations are applied to simulate sagging condition (see Fig. 14). d. Three block model with the net scantlings plus half of the corrosion margin for BC case 5. The material models exhibit strain hardening. Rotations are applied to simulate sagging condition. e. Three block model in damaged condition with the net scantlings plus half of the corrosion margin for BC case 5. The material models exhibit strain hardening. Rotations are applied to simulate sagging condition. The damage is assumed to result from a collision and has the following characteristics: height: 75% of the depth or 18.53 m from the strength deck, breadth: B/16 or 2.81 m length one block, i.e. 3.7 m (see Fig. 15).

Figure 14.

Figure 13. Reaction force in y direction for Hogging BC case 1.

3 Block model meshed with s4R elements.

Figure 15. 3 Block model damaged by a collision, meshed with s4R elements.

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One of the above mentioned cases, i.e. case (d) is also simulated under dynamic loading. This is achieved by reducing the period of the analysis step, while the maximum applied rotation remains constant. The applied rotation vs. time function is linear. By this way, higher rates of curvature application are introduced. Two speeds of applied curvature are imposed: Case 1: the time period is set to 0.5 second when the maximum applied rotation on the free node is 0.0009 rad. This leads to a rate of 0.000162 1/m*s and case 2: time period of 0.1 second with a maximum applied rotation of 0.0009 rad, leads to a rate of 0.00081 1/m*s. It is noted that in quasi static condition a rate of 0.0000405 1/m*s is applied. The results in terms of Mu for the scenarios a mentioned above are presented in Table 4. The curvature vs. moment curves for sagging and hogging are shown in Figure 16. It is important to be noted that the resulting curvature—moment curves for the two BC cases, are the same, except small differences in the post ultimate region of sagging. The ultimate strength values for sagging condition from modeling cases b-e, are presented in the following table and the M-k curves, in Figure 17. As it can be seen from the Table 5, material strain hardening causes only a slight increase of 1.9% on the ultimate strength. From the results obtained from the simulations with the explicit solver, it can be observed that the amount of kinetic energy introduced to the model is very small compared to the external work for all the quasi static cases. For the cases of dynamic simulation the amount of kinetic energy introduced to the model is large, comparatively to quasi static

Figure 17. Table 5.

Explicit, modeling cases b-e, see Table 5.

Results from explicit time-integration.

3 block model Intact without strain hardening Intact with strain hardening Intact dynamic case 1 Intact dynamic case 2 Damaged

15.20 GNm 15.49 GNm 15.86 GNm 16.94 GNm 13.17 GNm

1 block model Intact with strain hardening

15.66 GNm

Table 4. Ultimate strength in GNm from explicit time integration no hardening—1 block model.

BC case 2 BC case 5

Sagging

Hogging

14.00 14.00

15.79 15.79

Figure 18.

Explicit—reaction force in y direction.

problems. In addition, as arises from the Figure 18, the reaction forces in y direction, that applied to the model, increase as the kinetic energy increases.

4 4.1

Figure 16.

Explicit, modeling case a, see Table 4.

COMPARISON OF SIMULATION AND MODELING PROCEDURES Static vs. explicit time-integration

The resulting curvature-bending moment curves (M-k curves) from static and explicit time integration analyses are compared for BC case 2, for

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sagging and hogging condition, see Figures 4 and 11. This study indicates how much the amount of damping—even if it is small—affects the solution obtained from the static solver. An increase in ultimate strength that caused by the damping can be seen. The difference is 0.77 GNm or 5.5% of the Mu derived from explicit method, for sagging and 0.69 GNm or 4.3% of the Mu derived from explicit, for hogging. In addition, as arises from the Figures 4 and 11, the slope of the M-k curves obtained by static analysis are slightly higher than those obtained by explicit time-integration. The M-k curves obtained by Explicit exhibit nonlinearity sooner. 4.2 Extent of model The one block model, models a strip of the hull between two strong transverse frames, while the three block model consists of 3 blocks, with strong transverse stiffening structure between them. The models are analyzed using explicit time integration and the resulting curvature-moment curves for quasi static conditions, are compared. The purpose of this comparison is to indicate possible differences between the ultimate strength of the models, that can caused by different supporting of the plate and stiffeners, between the one block model and the middle block of the three block model. As it can be observed from the Table 5 and from the Figure 17, a small reduction of 1% on ultimate strength appears when using 3 block modeling instead of 1 block modeling. 4.3 Dynamic vs. quasi-static analysis The higher rate of curvature application leads to higher inertial forces development inside the material and for that reason the model exhibits higher ultimate strength. In this study, higher rates of curvature application are introduced to the model, by reducing the period of the analysis step. In the first case, the curvature is applied in a rate four times higher than the one used for quasi static condition. This results to an increase of 2.4% on the ultimate strength. In the second case, a ratio 20 times the one, used in quasi static analysis, cause a 9.4% increase on the ultimate strength, as it can be seen from the Table 5 above. The M-k curves from dynamic and quasi static analysis are shown together in Figure 17. 5

ULTIMATE AND RESIDUAL STRENGTH OF A BULK CARRIER

5.1 Ultimate strength The hull girder Ultimate Strength (Mu) in quasi static condition, is determined for sagging, using

material models with strain hardening, instead of elastic completely plastic models. The scantlings used in this analysis are the net scantlings plus half of the corrosion margin as defined by the Common Structural Rules for Bulk Carriers and Tankers. The problem is solved in Abaqus Explicit using the 3 block model for sagging condition. The resulting ultimate strength is Mu = 15.49 GNm as arises from the Table 5. The M-k curve, can be seen in Figure 17. 5.2

Residual strength

In this study, damage is introduced to the 3 block model. This is a modelling of a damage can caused by a collision. Materials with strain hardening are used, and the net scantlings plus half the corrosion margin. The problem in both cases, intact and damaged, is solved in quasi static condition using Abaqus Explicit. The comparison of the results indicates a 15% reduction on Ultimate Strength caused by the damage as it can be observed from the Table 5. In addition, as arises from the Figure 17, where the intact and damaged M-k curves are compared, there is a considerable reduction on the slope of the damaged M-k curve.

6

CONCLUSIONS

The problem of the determination of the hull girder ultimate and residual strength using nonlinear Finite Element Analysis, is affected by various parameters the most important of which, are the mesh size, the material modeling, the boundary conditions, and the algorithm that is used for the solution. From the results discussed above, it can be supported that a static solution can be obtained for the ultimate strength problem using Abaqus Static General step, using dissipated energy fraction—DEF—equal to 5E-5 and imposing restriction in the translation of one section along the transverse axes and in the rotation of the section about the transverse vertical and longitudinal axes. The loading is imposed by symmetric rotations about the transverse horizontal axes in both end sections. The solution may be also obtained using explicit time integration with appropriate values of period, increment size and loading amplitude to achieve a quasi static behavior. No significant difference was observed between the results that were obtained from the one block and three block models. The procedure has been used for the determination of the intact and damaged hull of a bulk carrier.

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REFERENCES ALPS/HULL (2011). A computer program for progressive collapse analysis of ship hulls, Advanced Technology Center, DRS C3 Systems, Inc., MD, USA (www.proteusengineering.com, www.maestromarine. com). IACS (2012). Harmonized common structural rules for bulk carriers and double hull oil tankers (draft version). International Association of Classification Societies, London, UK, July 2012. Paik, J.K., Kim, D.K., Park, D.H., Kim, H.B., Mansour, A.E. & Caldwell, J.B. (2011). Modified Paik-Mansour formula for ultimate strength calculations of ship hulls. Advances in Marine Structures, C. Guedes Soares and W. Fricke, (Eds.), Taylor & Francis Group, London, UK.

Paik, J.K., Kim, D.K. Park, D.H, Kim, H.B. & Kim, M.S. (2012). A new method for assessing the safety of ships damaged by grounding. Trans RINA, Vol. 154, Part A1, Intl J Maritime Eng, Jan-Mar 2012. Pollalis, C. & Samuelides, M. 2012. Ultimate strength of damaged hulls. Submitted to the 6th International Conference of Collision and Grounding of Ships, Trondheim, Norway. Villavicencio, R., Liu, Z., Amdahl, J. & Soares, C.G. (2011). Influence of the neutral axis displacement on the residual strength of a damaged tanker double bottom structure. Advances in Marine Structures, C. Guedes Soares and W. Fricke, (Eds.), Taylor & Francis Group, London, UK.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Failure characteristics of strength-equivalent aluminium and steel plates in impact conditions B. Liu, R. Villavicencio & C. Guedes Soares Centre for Marine Technology and Engineering (CENTEC), Instituto Superior Técnico, Technical University of Lisbon, Portugal

ABSTRACT: Experimental drop weight impact tests have been performed to examine the failure characteristics of small-scale clamped rectangular plates stuck laterally by a mass with a spherical indenter. The experiments are conducted on aluminium and steel plates of the same bending stiffness. Thus, the impact tests describe the behaviour of strength-equivalent ship building materials subjected to rapidly varying loads. The experimental results are presented in terms of the force-displacement responses and the failure modes of the specimen plates, which are very sensitive to the diameter of the indenters. Also, the experimental results show that the critical deflection is similar for both the aluminium and steel plates while some differences are observed in the critical force and energy. In addition, the magnitudes of the critical deflection, force and energy are compared a static theoretical prediction to allow the selection of equivalent impact strength plates. 1

INTRODUCTION

Table 1.

The aluminium can replace the steel in the construction of the hull structures in order to reduce the total weight of the ship, particularly for high-speed ships. Design criteria can convert the steel to the strength-equivalent aluminium structure by the same deflection and stiffness (Lamb and Beavers 2010; Kasten 2010). Nevertheless, ships suffer rapidly varying loads, such as slamming, ice-impact and collision and grounding. In the bottom and bow flare slamming, the shell plating and its supporting structure are subjected to high impact pressure forces, which accelerate and deflect all of this structure and set up vibrations, particularly in the plating. During ship collisions and groundings, the ship structure should be capable to absorb the impact energy without undergoing certain types of failure. Since the aluminium and the steel behave as brittle and ductile material, respectively, their impact strength should be compared to guarantee the safety of the design. Some advantages and disadvantages of aluminium and steel plates with the same stiffness are listed in Table 1. The coefficients used to replace the steel plates by aluminium are also indicated in Table 1. Usually, the steel experiences larger yield stress, ultimate tensile stress and fracture strain. Also, the material strain rate sensitivity is more evident for the steel, improving its structural crashworthiness. The thickness of the replaced aluminium structure should be larger to

Comparison of aluminium and steel plates.

Aluminium: Lightweight; expensive; no corrosion; low yield stress; low ultimate tensile stress; low fracture strain; may melt in fire. Steel: Heavy; cheap; corrosion; simple to fabricate. Thickness (t) Young’s modulus (E) Bending stiffness (Et3) Deflection (w) Density (ρ) Weight (m)

tal = 1.42 tst Eal = 0.35 Est Eal tal3 = Est tst3 wal = wst ρal = 0.34 ρst mal = 0.48 mst

*Subscript al and st represents aluminium and steel, respectively.

achieve the same steel stiffness due to the smaller Young’s modulus. Experimental tests of laterally loaded rectangular plates have been conducted to derive analytical expressions of the plastic response and failure of aluminium and steel plates. For example, Jones (1971) proposed an energy approach, based on the rigid-plastic theory, to estimate the permanent deflection of rectangular plates under uniformly distributed loads. Zhu et al. (1994) used the energy approach of Jones to study the dynamic behaviour of plates subjected to low velocity impacts. The initiation of rupture is an important aspect in the analysis of laterally impacted plates.

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Therefore, Shen et al. (2002a, b) proposed a theoretical method to predict the onset of failure for thin circular plates stuck by a conical indenter, and Simonsen and Lauridsen (2000) and Lee et al. (2004) developed analytical formulae to estimate the deformation and failure of thin clamped plates punched by a spherical indenter. Moreover, it has been demonstrated that the plastic deformation until failure is strongly dependent on the shape of the indenter as summarised by Jones et al. (2008, 2012) when proposed a theoretical method to predict the dimensionless perforation energy for plates struck by indenter with different shapes. The present paper investigates the plastic response and failure characteristics of strengthequivalent aluminium and steel plates. This is done through drop weight impact tests. Different diameters of hemi-spherically ended indenters are used in order to study their influence on the impact response of the specimens. The impact tests provide the actual force-displacement response and failure mode of the plate specimens. The magnitudes of the critical deflection, force and energy are compared with the theoretical predictions proposed by Simonsen and Lauridsen (2000). The actual experiments and the theoretical predictions allow the definition of equivalent impact strength aluminium plates that can absorb similar incident energy than that of steel plates. In addition, the specific energy absorption is used to evaluate the weight effectiveness of the steel and aluminium plates subjected to lateral impact. The presented results are of considerable practical importance to assess the safety of aluminium and steel structural elements subjected to dynamic loads, since these calculations require accurate estimates of the large deformations provoked by the impact. 2

EXPERIMENTAL DETAILS

is used to determine the engineering stress-strain behaviour of the material. The mechanical properties of the aluminium and steel materials are summarized Table 2 and the engineering stress-strain curves are shown in Figure 2. The steel has better elastic properties since the yield stress is larger than that of the aluminium (∼82%). Also, the steel withstands larger stresses before necking because of its larger ultimate tensile strength (∼42%). The aluminium fractures at the maximum load while the steel necks decreasing the level of stresses until fracture occurs. The plates are fully clamped by four M8 bolts between two thick rectangular steel plates with an internal cut-out of 127 × 76.2 mm (Fig. 3). The bolts compress the specimen and restrict its longitudinal displacement between the supports. The support plates are fixed to a strong structural base to prevent any movement. The impact tests are performed using a fully instrumented Rosand IFW5 falling weight machine

Figure 1.

Table 2.

The experimental program evaluates the plastic response until failure of rectangular plates stuck laterally by a mass with a spherical indenter. Aluminium and steel plates of the same bending stiffness are selected. The plates are aluminium alloy 5083/H111 (2.0 mm thickness) and mild steel ST12 (1.4 mm thickness). The mechanical properties are obtained by quasi-static tensile tests carried out on material cut from the same plates from which the impact specimens are taken. The dimensions of the machined tensile test pieces are shown in Figure 1. Three tensile tests are conducted for each material at a rate of 1.0 mm/min until fracture occurs. The displacement-controlled tension tests are carried out with a tensile test machine INSTRON 3369. It records the force-elongation curve, which

Dimensions of the machined test pieces.

Mechanical properties of material.

Property

Units

Aluminium 2.0 mm

Steel 1.4 mm

Density Young’s modulus Poisson’s ratio Yield stress Ultimate tensile strength Fracture stress Fracture strain (100 mm) Strength coefficient C0 Strain hardening exponent n

kg/m3 GPa MPa MPa

2650 72 0.33 125 257

7850 206 0.3 228 364

MPa -

257 0.15

272 0.23

MPa

405

585

168

-

0.160

0.172

Figure 2. Engineering stress-strain curves of aluminium and steel material.

Figure 4. Fully instrumented Rosand IFW5 falling weight machine.

absorbed energy and the velocity are calculated from the measured force-time data by successive numerical integrations. Four hemispherically ended strikers of diameters 30, 20, 16 and 10 mm are evaluated. The incident kinetic energy is 250 J for the aluminium plates and 400 J for the steel plates. The impact response of the aluminium and steel plate is compared at the critical incident energy, i.e. when failure occurs. Therefore, the incident energy could be overestimated for some specimens. However, the tendencies of the force-displacement curves are not affected with this overestimation. The last statement is valid as far as the material strain rate sensitivity is negligible.

Figure 3.

3

Experimental set up.

(Fig. 4). A small, light, hemispherically ended indenter is dropped from a known, variable height between guide rails onto horizontally supported specimens. A much larger mass is attached to the indenter and a load cell between the two gives the variation of the impact force with time. An optical gate provides the incident velocity of the impact head. The time histories of the displacement, the

EXPERIMENTAL RESULTS

In all cases, the specimens suffer large plastic deformation and fail at the impact point. The magnitudes of the critical forces, deflections and absorbed energies at failure are summarized in Table 3. These magnitudes are measured at the peak force, which occurs at the initial rupture of the plates. The critical energy is smaller than the incident kinetic energy. The diameter of the indenter strongly influences the experimental results, particularly the specimens impacted with larger indenters absorb

169

Table 3.

Summary of experimental results.

Specimen*

Input energy (J)

AL-D30 AL-D20 AL-D16 AL-D10 ST-D30 ST-D20 ST-D16 ST-D10

250 250 250 250 400 400 400 400

Values at failure Force (kN)

Defln (mm)

Energy (J)

23.8 17.9 15.4 9.7 27.5 21.8 18.9 11.8

18.7 18.1 16.9 13.1 18.7 17.8 16.0 12.9

199.2 149.5 125.4 61.9 257.1 209.6 149.4 76.9

*Specimen Notation. AL and ST: aluminium and steel, respectively; D30, D20, D16 and D10: diameter of indenter 30, 20, 16 and 10 mm, respectively.

more energy and, consequently, larger forces and deflections develop. The aluminium plates absorb less energy than the steel plates (22%). Also, the critical force of the aluminium plates is about 17% smaller than for the steel plates. However, both aluminium and steel plates suffer similar deflection at failure (2.2%). Since the strength-equivalent aluminium plates present unfavourable energy absorbing capabilities, it is not appropriate to replace steel plates in the design of structures subjected to lateral impact. Therefore, the thickness of the aluminium plates should increase to improve the impact strength characteristics. This is possible since the weight of the aluminium is smaller than the one of the steel (Table 1). The shape of the deformation of the aluminium and steel plates is shown in Figure 5. The plates struck by the indenter with diameter 10 mm are not included in Figure 5, since the final perforation is affected by the geometry of this striker. Large plastic deformation develops in the specimens in order to absorb the external dynamic energy. The deformation of the specimens is divided into two parts, global deflection and local indentation (Shen et al. 2002a; Liu et al. 2012), as indicated in Figure 6. The local indentation has the shape of the striker head. The large external dynamic energy produces structural in-plane tearing failure due to the excessive in-plane tensile strain (Shen et al. 2002a). The spherical indenter penetrates the plate with ductile enlargement to elongate the material in a plastic flow field below the indenter, forming a circular edge named necking circle, as shown in Figure 6. In general, the diameter of the necking circle is smaller than the one of the striker. As the supports of the laterally impacted plate are restrained axially, the centre line of the plate is longer in the deformed

Figure 5. Shape of the deformation. (a): AL-D30; (b): ST-D30; (c): AL-D20; (d): ST-D20; (e): AL-D16; (f): ST-D16.

Figure 6. Deformation profile of the rectangular plates.

configuration. This stretching rises into an axial membrane strain and associated membrane force. Thus, the plate deforms plastically due to the membrane forces at the necking circle. When the specimens suffer considerable plastic strain, the crack is initiated at the necking circle in the direction of the short span (Fig. 6) since the membrane force is much larger in this direction. In this paper the characteristic failure mode is named ‘tensile failure’ since the necking circle experiences tensile stretching, after severe indentation, and thinning in the thickness direction. Finally, the fracture is propagated through the necking circle while the indenter pushes the plate. The failure mode of the specimen plates is similar to the mode observed in plate quasi-static punching tests (Lee et al. 2004, Ehlers 2010, Atkins et al. 1998) and dynamic impact tests (Shen et al. 2002a).

170

Figure 7.

Force-displacement responses. Different diameters of indenters. (a): aluminium plate; (b): steel plate.

It must be mentioned that the energy absorbing mechanisms are similar between dynamic and static tests of ship structures with the same structural arrangement. However, the energy absorbed in dynamic tests could be larger than that absorbed in the corresponding static tests, a circumstance attributed to the phenomenon of material strainrate sensitivity (Jones and Wierzbicki 1983). The aluminium and steel plates show similar global deflection and local indentation since they suffer close deflection at failure. The equivalent strain distribution near the impact point can influence the deformation mode of the specimens. In particular, this strain distribution is strongly influenced by the coefficient of friction between the interfaces of the indenter and the specimen (Lee et al. 2004). If this coefficient is zero the plate rupture at the centre, while if this coefficient grows the diameter of the necking circle may increase, as numerical simulations demonstrated. Since the static friction between aluminium-steel and steel-steel are similar (Villavicencio et al. 2012, Liu et al. 2012, Lee et al. 2004), the necking circle for both types of specimens follow the same pattern at the local indentation. In the actual experiments, the final shape of the deformation of the aluminium plates, compared with the steel plates, exhibits some small cracks which initiate in the necking circle and extend towards the indentation. This occurs because the aluminium has less plastic deformation capabilities as observed in the engineering stress-strain curve (Fig. 2). The impact behaviour of the specimens during the impact event is better described by the recorded force-displacement data (Fig. 7). The critical absorbed energy is the integration of the force-displacement curve until the peak force. The diameter of the indenter strongly influences the impact response of the plates since it changes the dimension of the local indentation. The slope of the force-displacement curves decreases for specimens impacted with smaller diameter of indenter. This slope is the instantaneous stiffness of the specimen, indicating its ability

to resist the impact load. It is also a measure of the resistance of the specimen to plastic deformation until failure. The reduction of the slope is due to more localised contact surface between the striker and the specimen. This smaller contact effect minimizes the instantaneous reaction force for a particular deflection. Furthermore, the absorbed energy decreases because the specimens impacted by smaller indenters experience smaller local indentation and, consequently, form a smaller necking circle. This provokes easier rupture of the specimens. It should be mentioned that the slope of the steel plates is larger than the one of the aluminium plates since the steel plates have larger impact resistance before failure. 4

ANALYTICAL APPROXIMATION

Analytical approaches have been proposed to predict the deformation and failure of plates under a quasi-static spherical punch loading. In the present paper, the theoretical approach of Simonsen and Lauridsen (2000) is selected to analyse the failure characteristics of the current aluminium and steel plates. The comparison with quasi-static formulae is valid since the incident impact velocity at failure is relatively low. Simonsen and Lauridsen (2000) investigated the onset of failure for static lateral punch indentation of a rigid sphere into a thin clamped circular plate. The analytical approach predicts the plate failure and the energy absorption up to this point. It was demonstrated that the response of the thin plates depends on the strength coefficient C0 and the strain hardening exponent n of the material. The strength coefficient and the strain hardening exponent define the true stress (σt) and true strain (εt) material curve based on the simple power law relation. This curve gives an approximate true indication of the deformation characteristics of a metal (Dieter 1986):

σt

171

C εt n

(1)

The parameters of the power law relation are determined from the engineering stress-strain curve. The exponent n is the linear slope of a loglog plot of the logarithmic true stress and true strain up to the maximum load expressed in terms of the engineering stress and engineering strain. The coefficient C0 is the true stress at εt = 1.0. The tensile tests of aluminium plates show that the stress at maximum load is almost coincident with the fracture stress and that very little necking occurred in the width direction. However, necking is observed in the thickness direction. For that reason, the true material relation for the aluminium plates is also defined beyond the maximum load (Børvik et al. 2009; Beese et al. 2010). Here, the parameters C0 and n are obtained using the set of equations proposed by Zhang et al. (2004). The coefficients of the power law relation for both aluminium and steel materials are summarised in Table 2. Simonsen and Lauridsen (2000) estimated the normalized penetration until failure for circular plates by:

δf R0

= 1.41n0.33ξ00.52

(2)

where δf is the critical punch penetration to failure, R0 is the radius of the clamped plate, n is the strain hardening exponent and ξ0 = Rb/R0 is the dimensionless radius of the applied punch where Rb is the radius of the spherical punch. This equation is valid as far as 0.1 < ξ0 < 0.5 and 0.1 < n < 0.3. The deflection of these thin plates is independent of the plate thickness and the yield stress since the bending resistance can be neglected. Thus, the plate response is governed by the membrane force. This phenomenon is manifested in tensile test results of specimens manufactured with the same material but different thickness since they should result in the same engineering elongation. Also, materials with different yield stress but similar strain hardening exponent could result in similar elongation until fracture. This is due to the fact that the strain hardening exponent determines the elongation at fracture of the material. The penetration until failure for circular plates is similar to the penetration for square plates where the initial half-width corresponds to the initial radius R0 (Lee et al. 2004). Therefore, the theoretical model of Simonsen and Lauridsen (2000) is used to evaluate the parameters that influence the deformation process of the actual plates. Equation (2) shows that the strain hardening exponent determines the deflection at failure of geometrically similar specimens manufactured with different materials. In the actual aluminium

and steel material the hardening exponent is very similar, 0.160 and 0.172, respectively. Therefore, Equation (2) estimates similar deflections for the aluminium and steel plates when impacted by the same type of indenter (difference 2.4%). This is demonstrated in the experimental results (Table 3). In Equation (2), the relationship between the deflection and the diameter of indenter is δf ∼ Rb0.52. This implies that the deflection at failure decrease accelerated with the diameter of indenter, as also observed in the current impact tests. The theoretical approach of Simonsen and Lauridsen (2000) was followed by Lee et al. (2004) to evaluate the influence of the material ductility. Lee et al. (2004) demonstrated that the strain hardening exponent can be replaced by the critical damage value for ductile fracture:

δf R0

=

2 Dc ξ00.38

(3)

where Dc is the critical damage value for ductile fracture. This equation is valid for 0.1 < ξ0 < 0.75. The critical damage value is a function of the critical failure strain obtained from the tensile test simulations. Since the accuracy of these simulations is relatively low beyond localisation, the failure strain is only approximated, and thus this new approach is less accurate. In addition, Equation (3) is not appropriate to estimate the deflection of the plate for larger failure strains, because the deflection at failure is determined by the failure at the necking circle (Fig. 6). For example, the deflection at failure is similar for a range of plastic strains between 0.7 and 1.2, which are obtained from numerical simulation of tensile and impact tests (Liu et al. 2012). However, Equation (3) is very sensitive to the critical damage coefficient, thus it is appropriate for low failure strains, as the ones obtained in the actual aluminium plates. The reaction force until failure for the circular plates of Simonsen and Lauridsen (2000) can be expressed as: 2π C0t0 Rb [ − l

ψ c ]n cos cosψ c i 2 ψ c

(4)

where C0 is the strength coefficient, t0 is the plate thickness and ψc is the critical wrapping angle to failure: c

≈ 0.957 + 0.399 n

(5)

The absorbed energy until failure is: E

172

π C0t0R0Rb { ξ00.607 − 0.387ξ0 + + 0.067( n − 0.2 )}

ξ02

(6)

Equations (4) to (6) are valid within the range 0.1 < ξ0 < 0.5 and 0.1 < n < 0.3. It should be noticed that the reaction force (Equation 4) is independent of the radius of the plates. Simonsen and Lauridsen (2000) demonstrated the reaction force is similar for plates with circular, square and rectangular geometries, i.e. the influence of the specimen geometry is negligible to estimate this force. This is possible since the specimens suffer the local indentation far away from their boundaries (Simonsen and Lauridsen 2000). Thereby, Equation (4) is adopted to analyse the current rectangular plates. Equation (4) predicts similar reaction forces for the aluminium and the steel plates; the difference is ∼ 1.1%. However, the experimental results demonstrate that the peak force for the steel plates is ∼17% larger than that for the aluminium plates (Table 3). Since the theoretical predictions are similar to the experimental results of the aluminium plates, the difference in the experimental reaction forces is attributed to small strain rate effects of the mild steel plates when subjected to dynamic impact loads. It must be mentioned that the quasi-static approach should be similar to the dynamic experimental response of the aluminium plates because the aluminium alloy material is essentially strain rate insensitive (Jones 1989). The absorbed energy until failure is the numerical integration of force-displacement relationship. Therefore, the theoretical energy (Equation 6) includes the parameters used to estimate both the deflection and reaction force. As the analytical deflections and reaction forces until failure are very similar for the aluminium and steel plates, the theoretical critical energies predict also close magnitudes. This similitude differs from the experimental results as shown in Table 3. The difference between the experimental forces is responsible for this difference in the energy absorption. 5

DISCUSSION

The present paper investigates the impact characteristics of strength-equivalent aluminium and steel plates for application in ship structural design. For converting a parent steel ship, that satisfies rule requirements in terms of structural strength, to an equivalent aluminium ship, three design parameters should be considered: stress, deflection and weight. The equivalent ship manufactured in aluminium should have an equal or better structural integrity than the parent steel ship. The coefficients used to replace the steel plates by aluminium plates (Table 1) are valid within the elastic limit, so they use a safety factor smaller than the yield stress. However, the design of structures subjected to dynamic impact requires the inclusion of the nonlinear behaviour of the material.

The critical energy, for a given plate geometry (width × length) and material, is proportional to the plate thickness (Equation 6). Thus, the absorbing capability of this particular plate can be improved by increasing its thickness. In the actual experiments, the aluminium plates require to absorb 1.3 times more energy to make equal the energy absorbed by the steel plates. This implies that the thickness of the aluminium plates should be increased to 2.6 mm. It can be concluded that the impact tests can estimate the thickness necessary to absorb the same energy absorbed by the parent plate. Therefore, the relation tal = 1.42tst for strength-equivalent aluminium and steel plates should increase to tal = 1.85tst in order to satisfy the same impact strength. This linear relation is valid for thin plates and assumes that the critical energy varies proportional to the plate thickness. The latter is based on quasi-static punching tests and could be used to estimate the impact strength of aluminium plates since the aluminium is essentially strain rate insensitive. The absorbed energy is often divided by the weight of the structure to give the specific energy absorption (Se) which is widely used to judge the relative effectiveness of various energy absorbing devices and structures (Jones and Wierzbicki 1983). The materials with high Se offer the potential for significant weight reduction in energy absorbing components, usually used in the design of cars, airplanes, ships, etc. In the present work, the same principle is used to evaluate the absorbing capability of the steel and aluminium plates. The aluminium has higher Se due to its smaller density, offering potential for significant weight reduction in energy absorbing components. Moreover, the larger thickness for the aluminium plates (tal = 1.85tst) reduces the deflection compared with the steel plates, improving the designed structural yield and buckling strength. In the actual experiments, the Se of the aluminium is larger than the one of the steel (about 63%). Thus, the steel plate weighs 63% more than the aluminium plate if both have the same energy absorption until failure. This implies that the aluminium material is advantageous respect to the weight effectiveness to resist lateral impact. The Se estimates the reduced weight with the same energy absorption until failure for the preliminary design. Thus, the carrying capacity, stability and cruising speed based on the steel parent ship can be evaluated. 6

CONCLUSIONS

The failure of the laterally impacted aluminium and steel plates is analysed through drop weight impact tests and theoretical analyses. This provides

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good understanding of the impact strength of equivalent aluminium and steel structures and help to evaluate the ship structural design using aluminium plates. The strength-equivalent aluminium and steel plates experience similar deflections until failure although the forces and the energy are larger in the steel plates. In order to obtain the same equivalent impact strength of the steel plates, the thickness of the aluminium plates can be increased since the aluminium presents higher specific energy absorption. The experimental deformation process until failure is well described by the theoretical method presented in this paper. The local indentation plays an important role in the total deformation of the plates. The rupture is initiated at one point of the necking circle due to the development of membrane force, so the failure is named tensile failure. The analysis of the impact tests with the analytical approximations describes the parameters that influence the critical deflection, force and energy of laterally impacted metal plates. It is shown that the absorbed energy until failure is improved by increasing both material power law coefficients and not simply the yield stress, as in the comparison of the strength-equivalent plates. In this study, the experimental results in terms of the force-displacement responses and the failure modes provide relevant information to compare the impact strength of aluminium and steel plates. ACKNOWLEDGEMENTS The work of the first author has been financed by a PhD scholarship from ABS, the American Bureau of Shipping. The authors are grateful to Dr George Wang for his initiative to promote this scholarship. The second author has been financed by the Portuguese Foundation for Science and Technology, under contract SFRH/BD/46369/2008. REFERENCES Atkins AG, Afzal khan M, Liu JH. 1998. Necking and radial cracking around perforations in thin sheets at normal incidence. International Journal of Impact Engineering; Vol. 21, No. 7, pp. 521–539. Beese AM, Luo M, Li Y, Bai Y, Wierzbicki T. 2010. Partially coupled anisotropic fracture model for aluminum sheets. Engineering Fracture Mechanics; 77: 1128–1152. Borvik T, Forrestal MJ, Hopperstad OS, Warren TL, Langseth M. 2009. Perforation of AA5083-H116 aluminium plates with conical-nose steel projectiles—Calculations. International Journal of Impact Engineering; 36: 426–437.

Dieter GE. 1986. Mechanical behavior under tensile and compressive loads. ASM Handbook; 8: 99–10. Ehlers S. 2010. Strain and stress relation until fracture for finite element simulations of a thin circular plate. Thin-Walled Structures; 48: 1–8. Jones N. 1971. A theoretical study of the dynamic plastic behaviour of beams and plates with finitedeflections. International Journal of Solids Structures; 7: 1007–1029. Jones N. 1989. Structural Impact. Cambridge University Press. Jones N, Wierzbicki T. 1983. Structural Crashworthiness. Butterworth & Co (Publishers) Ltd. Jones N, Birch RS, Duan R. 2008. Low-velocity perforation of mild steel rectangular plates with projectiles having different shaped impact faces. Journal of Pressure Vessel Technology, transactions of the ASME; Vol. 130, No. 3. Jones N, Paik JK. 2012. Impact perforation of aluminium alloy plates. International Journal of Impact Engineering; 48: 46–53. Kasten M. 2010. Strength of aluminum vs strength of steel. http://www.kastenmarine.com/alumVSsteel.htm. Lamb T, Beavers N. 2010. The all aluminum naval ship— the way to affordable naval ships. In Proc. 10th International Naval Engineering Conference and Exhibition: HM Naval Base, Portsmouth, United Kingdom. Lee YW, Woertz JC, Wierzbicki T. 2004. Fracture prediction of thin plates under hemi-spherical punch with calibration and experimental verification. International Journal of Mechanical Sciences; 46: 751–781. Liu B, Villavicencio R, Guedes Soares C. 2012. Experimental and numerical plastic response and failure of laterally impacted rectangular plates. In. Proc. 31st International Conference on Ocean, Offshore and Arctic Engineering (OMAE 2012): Rio de Janeiro, Brazil; Paper: OMAE2012-84015. Shen WQ, Rieve NO, Baharun B. 2002a. A study on the failure of circular plates struck by masses. Part 1: experimental results. International Journal of Impact Engineering; 27: 399–412. Shen WQ. 2002b. A study on the failure of circular plates struck by masses. Part 2: theoretical analysis for the onset of failure. International Journal of Impact Engineering; 27: 413–432. Simonsen BC, Lauridsen LP. 2000. Energy absorption and ductile failure in metal sheets under lateral indentation by a sphere. International Journal of Impact Engineering; 24: 1017–1039. Villavicencio R, Sutherland LS, Guedes Soares C. 2012. Numerical simulation of transversely impacted, clamped circular aluminium plates. Ships and Offshore Structures; 7 (1): 31–45. Zhang L, Egge ED, Bruhns H. 2004. Approval procedure concept for alternative arrangements. Proceedings of the 3rd International Conference on Collision and Grounding of Ships, Izu, Japan; pp. 87–96. Zhu L, Faulkner D, Atkins AG. 1994. The impact of rectangular plates made from strain rate sensitive materials. International Journal of Impact Engineering; 15 (3): 245–255.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Shear and tensile failure of thin aluminium plates struck by cylindrical and spherical indenters B. Liu, R. Villavicencio & C. Guedes Soares Centre for Marine Technology and Engineering (CENTEC), Instituto Superior Técnico, Technical University of Lisbon, Portugal

ABSTRACT: Experimental drop weight impact tests have been performed to examine the failure characteristics of small-scale clamped rectangular aluminium plates struck by a mass with cylindrical and spherical indenters. The laboratory results are compared with numerical simulations. Six plate specimens are impacted at the same kinetic energy in order to study the influence of the shape and the diameter of the indenter. The experiments are conducted using a fully instrumented impact testing machine. The obtained force-displacement responses show a good agreement with the simulations performed by the LSDYNA finite element solver. The strain hardening of the material is defined using experimental data of quasi-static tensile tests and the critical failure strain is evaluated measuring the thickness and the width at fracture of the tensile test pieces. The results show that the impact response and the failure mode of the specimens are highly sensitive to the geometry of the indenters. The failure modes are described by the matrix of the infinitesimal strain tensors and the shape of the deformation of the first failing element. In addition, the process of initiation and propagation of the material fracture is presented. 1

INTRODUCTION

The plastic response of small-scale ship structural components subjected to lateral impact has been well predicted by finite element simulations that use accurate true material relationships and define precise boundary conditions (Ehlers et al. 2008; Grytten et al. 2009; Villavicencio and Guedes Soares 2011; Villavicencio and Guedes Soares 2012a; Liu et al. 2012). However, the failure due to material rupture is still not well resolved numerically, because the fracture length is much smaller than the side length of the elements in the finite element model. Thus, it is difficult to establish a procedure suitable for prediction of failure in the engineering practice. The onset of failure is an important aspect for the analysis of laterally impacted plates. Therefore, experimental tests using different types of indenter have been conducted to derive analytical expressions. Shen et al. (2002a, b) proposed a theoretical method to predict the failure of thin circular plates stuck by a conical indenter. Chen et al. (2005) presented an analytical model to study the material failure due to shear banding in circular plates struck by a blunt projectile. They analyzed the global bending of the plate and the localized shear at the impact point. Børvik et al. (2002a) and Gupta et al. (2007) studied the influence of the projectile shape on the absorbed energy until failure for circular plates

struck by blunt, hemispherical and conical nosed projectiles. It was demonstrated that the impact response depends on the shape of the indenter and that the blunt nosed projectile absorbs the lowest energy. Jones et al. (2008) and Jones and Paik (2012) proposed a theoretical method to predict the dimensionless perforation energy for plates struck by various shapes of indenter. The finite element analysis is a useful tool to predict the extent of damage in the marine structural components. However, the results of the nonlinear dynamic analysis should be validated with experimental tests before being implemented in the structural design. Unfortunately, experiments using full-scale prototypes are extremely expensive and thus rarely performed. In this respect, prior to performing analyses of largescale structures, it is necessary to verify the experimental-numerical models of the dynamic large deformation in small-scale structural elements. This provides the basis for the design of complex marine structures subjected to dynamic impact loads. An important numerical definition that must agree with the characteristics of each particular impact test and specimen is the material nonlinearity. It is known that plastic strain hardening, plastic strain rate and true fracture strain are required for prediction of the extent of damage in structural components under impact loads.

175

Numerical simulations of small-scale plates impacted with different strikers have used the Johnson-Cook model (Johnson and Cook 1983) to define the plastic flow, isotropic strain hardening, strain rate effect and softening due to adiabatic heating (Børvik et al. 2002b; Gupta et al. 2006; Grytten et al. 2009; Arias et al. 2008; Børvik et al. 2010). These references used the JohnsonCook failure criterion (Johnson and Cook 1985) to evaluate the damage, including the effect of stress triaxiality, strain rate and temperature on the equivalent failure strain. Other authors have been performed simulations of tensile test experiments to predict the critical failure strain of circular and rectangular plates (Simonsen and Lauridsen 2000; Tabri et al. 2007; Liu et al. 2012). In addition, the failure strain for simple structural elements has been evaluated by measuring the thickness and the width at fracture of tension test specimens (Beese et al. 2010). A more accurate method was presented by Ehlers (2010). He selected an optical system to obtain the true stress-strain curve until failure from tensile pieces and used this information to predict the critical failure strain of circular steel plates punched by a spherical indenter. The present paper investigates the plastic response until failure of small-scale rectangular aluminium plates subjected to lateral impact. This is done through drop weight impact tests and finite element simulations. The aluminium has been widely used to analyze the numerical impact response of plate elements since it is essentially strain rate insensitive and, thereby, the scale of the yield stress can be omitted in the material definition (Grytten et al. 2009, Børvik et al. 2009, Børvik et al. 2010, Villavicencio et al. 2012). Thus, the critical failure strain can be evaluated with the results of quasi-static tension tests. The plate specimens are impacted by spherical or cylindrical indenters of different diameters so as to evaluate their influence in the impact response. The experimental and numerical results are discussed in terms of their force-displacement response and the failure mode provoked by each type of indenter. The failure modes are described by the matrix of the infinitesimal strain tensors and the shape of the deformation of the first failing element. In addition, the process of initiation and propagation of the material fracture is presented. The definitions adopted in the present work are of considerable practical importance in naval architecture when it is necessary to assess the safety of the structural elements subjected to dynamic loads, since the calculations require accurate estimates of the large deformations provoked by the impact.

2

EXPERIMENTAL DETAILS

The experimental program evaluates the impact response until failure of rectangular aluminium plates stuck laterally by a mass with different indenters. The thickness of the specimen plates is 2.0 mm. The material is aluminium alloy 5083/ H111, which mechanical properties are obtained by in-house quasi-static tensile tests using standard procedures (ASTM, E8). The dimensions of the machined test pieces are shown in Figure 1. Five tensile tests are conducted at a rate of 1.0 mm/min until fracture occurs. The displacement-controlled tension tests are carried out with a tensile test machine INSTRON 3369. It records the forceelongation curve, which is used to determine the engineering stress-strain behaviour of the material. The mechanical properties of the aluminium material are summarised in Table 1. The specimens are fully clamped by four M8 bolts between two thick rectangular steel plates (upper and lower support plates) with an internal cut-out of 127 × 76.2 mm, as shown in Figure 2. The bolts compress the specimen and restrict its longitudinal displacement between the supports. The upper and lower support plates are fixed to a strong structural base to prevent any movement. The impact tests are performed using a fully instrumented Rosand IFW5 falling weight machine (Fig. 3). A small, light indenter is dropped from a known, variable height between guide rails onto horizontally supported specimens. A much larger

Figure 1. Table 1.

Dimensions of the machined test pieces. Mechanical properties of material.

Property

Units

Aluminium 2.0 mm

Density Young’s modulus Poisson’s ratio Yield stress Ultimate tensile strength Fracture stress Fracture strain (100 mm)

kg/m3 GPa – MPa MPa MPa –

2650 72 0.33 125 257 257 0.15

176

Figure 2.

mass (24.3 kg) is attached to the indenter and a load cell between the two gives the variation of the impact force with time. An optical gate provides the incident velocity of the impact head. The time histories of the displacement, the absorbed energy and the velocity are calculated from the measured force-time data by successive numerical integrations. In all cases the impact velocity is 4.5 m/s, thus the incident kinetic energy is estimated at 250 J. Two shapes of indenter are used to impact the specimens: spherical and cylindrical. Moreover, each shape is evaluated with diameters of 30, 20 and 10 mm. The illustration of all indenters is provided in Figure 4 where S and C indicate spherical and cylindrical indenter, respectively, and the number indicates the diameter of the indenter. The shape of the deformation for the specimens struck by spherical indenters differs to that experienced for the specimens impacted by cylindrical indenters. In the present paper, the former specimens are named ‘spherical specimens’ while the latter are named ‘cylindrical specimens’. The deformation modes of both spherical and cylindrical specimens are shown in Figure 5. In the spherical specimens (Fig. 5a), the shape of the deformation is divided into two parts: global deflection and local indentation (Shen et al. 2002a, Liu et al. 2012). The local indentation has the shape of the striker head. The plate specimen elongates below the indenter forming a circular edge named ‘necking circle’. In general, the diameter of the necking circle is smaller than the one of the striker. As the supports are restrained axially, the centre line of the plate is longer in the deformed configuration. This stretching rises into an axial membrane strain and associated membrane force at the necking circle. The crack is initiated at this

Experimental set up.

Figure 3. Fully instrumented Rosand IFW5 falling weight machine.

Figure 4.

177

Shapes of indenters.

Figure 5.

Deformation profile of the rectangular plates. (a): Specimen S20; (b): Specimen C20.

circle in the direction of the short span since the membrane force is much larger in this direction. Finally, the fracture is propagated through the necking circle while the indenter pushes the plate. In the cylindrical specimens (Fig. 5b), the indenter provokes a local shear ring at its perimeter which is named shear circle or shear banding (Børvik et al. 2002a; Chen et al. 2005). These specimens suffer small local indentation and plastic deformation outside the shear circle. The area below the cylindrical indenter does not suffer plastic deformation although it should be subjected to high level of stresses during the initial contact. The material fracture initiates at one point of the shear circle and extends along this circle. Finally, the circular plug is ejected from the specimen plate. The inspection of the spherical specimens determines that the circle necking suffers tensile stretching and thinning in the thickness direction, thus the characteristic failure mode is named ‘tensile failure’. On the other hand, the fracture of the cylindrical specimens is mainly due to the cutting force given by the indenter, which provokes a shear effect at the shear circle, so the failure mode is named ‘shear failure’. The values of the forces, deflections and energies at failure are summarised in Table 2 and the forcedisplacement responses are shown in Figure 6. The magnitudes presented in Table 2 are measured at the peak force, which occurs at the initial rupture of the plates. In all cases, the critical absorbed energy is smaller than the incident kinetic energy (250 J). The shape of the indenter strongly influences the impact responses particularly the cylindrical specimens experience smaller deflection and energy than those suffered by the spherical speci-

Table 2.

Summary of experimental results. Values at failure

Specimen

Force (kN)

Defln (mm)

Energy (J)

S30 S20 S10 C30 C20 C10

23.8 17.9 9.7 22.3 15.2 8.9

18.7 18.1 13.1 11.1 10.9 9.7

199.2 149.5 61.9 110.3 77.4 41.8

Figure 6.

Force-displacement response.

mens (40%). However, the critical force is less influenced by the shape of the indenter (∼10%). The large difference in terms of deflection and absorbed energy is due to the different contact area between the indenter and the specimen. The larger contact area in the cylindrical specimens increases the forces quickly. Moreover, the stresses concentrated at the shear circle provoke easier fracture

178

of the specimen. This results in smaller deflection and, consequently, smaller absorbed energy. The slope of the force-displacement curves decreases for specimens impacted with smaller diameters of indenter. This slope is the instantaneous stiffness of the specimen, indicating its ability to resist the impact load. It is also a measure of the resistance of the specimen to plastic deformation until fracture. The reduction of the slope is due to more localised contact surface between the striker and the specimen. This smaller contact effect minimizes the instantaneous reaction force for a particular deflection. Furthermore, the absorbed energy decreases because the specimens impacted by smaller indenters experience smaller local indentation and, consequently, form a smaller necking or shear circle, provoking easier fracture of the specimens.

the failure mode. In the remaining surface of the plate, the mesh size is 1.0 mm. The supported part is modelled with a coarser mesh of 1.0 × 3.0 mm.

The specimen plate is modelled by four-node shell elements with five-integration points throughout the thickness, defining the Belytschko-Lin-Tsay shell element formulation (Hallquist 2010). The mesh is finer at the impact point (0.5 mm) which is evaluated according to its ability to predict the experimental force-displacement response and

3.1.1 Material The mechanical properties of the material (Table 1) used in the finite element model are obtained from in-house quasi-static tensile tests carried out on material cut from the same plates from which the specimens are taken. Since the engineering stressstrain curve does not give a true indication of the deformation characteristics of a metal, it is necessary to use the true stress-strain curve that represents the basic plastic-flow characteristics of the material (Dieter 1986). The engineering and true stress-strain curves are shown in Figure 8. The flow stress-strain is characterised by the ‘combined material relation’ (Villavicencio 2012). In this material model, the true stress-strain curve is divided in two parts with respect to the onset of necking. Thus, the logarithmic true stress-strain up to the maximum load (Dieter 1986) defines the process before necking and a simple power expression (Zhang et al. 2004) defines the process after necking. The material relation is named combined material since it combines two approximations for the flow curve. This true material was proposed by Villavicencio and Guedes Soares (2012b) to predict the numerical plastic response and the critical failure strain of quasi-static tensile test specimens and dynamic models of transversely impacted prenotched beams. Moreover, the combined material has been selected to predict the plastic behaviour until fracture of laterally impacted small-scale rectangular plates (Liu et al. 2012) and tanker vessel side panels (Villavicencio 2012). The tensile tests of the actual aluminium plates show that the stress at maximum load is almost coincident with the fracture stress and that very little necking occurred in the width direction. However, necking is observed in the thickness direction. For that reason, the true material relation should be

Figure 7. indenter.

Figure 8.

3

NUMERICAL MODEL

The computations are carried out using the finite element package LS-DYNA Version 971 (Hallquist 2010) which is appropriate for non-linear explicit dynamic simulations with large deformations. The numerical model is designed with the following components (Fig. 7): specimen, striker and supports. The definitions adopted in the numerical model are in agreement with previous experimental-numerical studies (Villavicencio et al. 2012; Liu et al. 2012), thus they are only briefly explained in the text. On the other hand, the emphasis is put on the material characterisation. 3.1

Specimen

Details of finite element model. Spherical

179

Engineering and true material curves.

defined beyond the maximum load (Børvik et al. 2009; Beese et al. 2010). The critical failure strain of the impact specimens should be estimated for one finite element of dimensions t × b × l = 0.5 × 0.5 × 0.5 mm. Here, the three principal tensor strains of this small element are assumed to be similar to the tensor strains under uniaxial tension. Numerical simulations of aluminium tensile test experiments have demonstrated that the initial fracture occurs at the neutral axis of the tensile piece and that the deformation of this first failing element is homogeneous. Moreover, as very little necking occurs in the width direction, the strain in the width of the failing element is equal to the average strain in the width direction of tensile specimen. Therefore, the equivalent strain to failure (critical failure strain) can be determined from the tensile piece by measuring the thickness and width at fracture, as illustrated in Figure 9 (Beese et al. 2010). This implies that the strain in the width direction (ε2) and the strain in the thickness direction (ε3) can be estimated as:

ε 2 = ln( /b0 )

(1)

ε 3 = ln( / t0 )

(2)

where b0 and be are the initial and reduced width; t0 and te are the initial and reduced thickness. The component in the longitudinal direction of the principal tensor strain (ε1) can be found from the incompressibility condition:

ε1 = ln( e / l0 ) = ln( ln(b0t0 /

e e)

(3)

where l0 and le are the initial and elongated length. Finally, these assumptions allow the definition of the equivalent strain to failure by:

ε eq =

2 2 (ε1 + ε 22 + ε 32 ) 3

(4)

The ‘average’ equivalent strain to fracture for the five tensile specimens is estimated at 0.619

Figure 9. section.

Schematic of dimensions of fractured cross-

(see Table 3). The thickness and the width at fracture and the principal strains are also summarised in Table 3. Published experimental results for aluminium alloy beams (Liu and Jones 1987; Jones 1989) showed that they are essentially strain-rate insensitive, and for the plates considered here, the inclusion of nominal strain-rate coefficients in the numerical simulations resulted in larger level of the forces than seen in the experimental results. Hence, strain-rate sensitivity is not included in the numerical model. The material selected from the library of LSDYNA is ‘Mat.024-Piecewice linear plasticity’, which allows the definition of a true stress-strain curve as an offset table. Also, failure based on a plastic strain can be defined. 3.2 Striker The striking mass and the indenter assembly, shown in Figure 3, are modelled as a simple solid sphere or cylinder. The solid elements avoid initial penetrations of the upper surface of the plate specimen. As no information is obtained from the striker elements, zero integration points are defined. A rigid material is defined to ensure no deformation, assigning steel mechanical properties. However, an artificially large density is used to give the same mass as the one used in the experiments. The mechanical properties of this rigid material are used to determine the sliding interface parameters in the contact definition when the rigid body interacts. It is necessary to give realistic values of these properties, since any unrealistic value may cause numerical problems in the contact definition (Hallquist 2010). The material ‘Mat.020-Rigid’ is selected from the library of LS-DYNA. The contact between the striker and the specimen is defined as automatic surface to surface (Hallquist 2010). In this LS-DYNA’s automatic contact, the nodes on the slave side are first checked for penetration through the master surface and then the master nodes are checked for penetration through the slave surface. The static coefficient of friction between the striker and the specimen is assumed at 0.3 (Lee et al. 2004, Rusinek et al. 2009, Villavicencio et al. 2012). This contact also allows the definition of the dynamic coefficient of friction. However, the preliminary simulations demonstrated that the response of the specimens is insensitive to this coefficient and, thereby, it is omitted in the contact card. The initial impact velocity (4.5 m/s) is assigned in the free vertical translation of the rigid body. The solid-rigid striker used to represent the actual experimental mass-indenter assembly is in agreement

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Table 3.

Reduced-area measurement of fracture strain in dog-bone specimens.

Specimen no.

Fracture thickness

Fracture width

ε1

ε2

1 2 3 4 5

1.23 1.24 1.20 1.27 1.19

11.45 11.52 11.51 11.49 11.55

0.574 0.560 0.593 0.538 0.598

0.618 −0.088 −0.486 0.605 −0.082 −0.478 0.643 −0.083 −0.511 0.579 −0.084 −0.454 0.650 −0.079 −0.519 Average failure strain: 0.619

with previous simulations of similar small-scale structural components (Villavicencio 2012). 3.3

Supports

The load carrying capacity of a structure is strongly dependent on the restraints at the supports. Therefore, when developing numerical models that are to be compared with experimental results, it is necessary to represent the real boundary conditions, instead of the ideal theoretical ones. This must be done, since the assumptions of simply supported or fully clamped boundary conditions can produce errors in the load carrying capacity of the analyzed structure (Villavicencio and Guedes Soares 2011). The boundary conditions include the representation of the whole experimental set up, that is, the upper and lower support plates interacting with the specimen. This numerical representation of the supports is initially proposed by Villavicencio et al. (2012) and later improved by Liu et al. (2012) to simulate the experimental restraint of similar plate specimens subjected to lateral impact. The support plates compress the specimen as occurred in the experiments (see Figs. 2 and 7). No gap between the supports and the specimen plate is considered. The lower support plate is constrained in all degrees of freedom. The upper support plate is free in the vertical translation since a prescribed vertical motion is imposed to compress the specimen. The compression of the upper support plate is due to the torque applied on the bolts. This compression only fixes the specimen between the support plates and does not cause any plastic deformation of the specimen. As the bolts prevent the longitudinal displacements of the supported part of the specimen, nodal axial restrictions are imposed where the bolts pass through the plate. The support plates are modelled with shell elements and use a rigid material to ensure no deformation. The “Mat.020-Rigid” is selected from the material library of LS-DYNA, assigning mild steel mechanical properties and mass density. The LS-DYNA’s automatic surface to surface contact between the surfaces of the support plates

ε3

εeq

and the supported part of the specimen is defined. The contact includes a static coefficient of friction equal to 0.2, as proposed by Villavicencio and Guedes Soares (2011). They estimate this value for a similar support-specimen contact in sliding surfaces when the specimen is struck transversely by a mass. The dynamic coefficient of friction is also omitted here. 4

NUMERICAL RESULTS

The failure modes are captured accurately by the numerical simulations (Fig. 10). The global deflection of the cylindrical specimens is slightly smaller than that of the spherical specimens (∼7.0%). Therefore, the larger critical energy obtained for the spherical specimens (see Table 2) could be attributed to their pronounced local indentation. The simulations reveal that the maximum level of stresses is concentrated near the local indentation while they are smaller in the remaining surface of the plate and at the support. The experimental and numerical force-displacement responses are plotted in Figure 11. In general, the impact behaviour is well predicted by the simulations. However, few discrepancies are noticed at the critical force and deflection, particularly for specimens S30 and S10. This can be attributed to the fact that the neutral axis of the first failing element (Fig. 12) could be displaced from the necking circle and thus, the lateral displacement at the impact point is not well predicted. Since this difference is acceptable for dynamic analysis, the mesh size 0.5 mm is good enough to predict the ‘very local’ phenomenon of failure. In order to suppress this discrepancy, a new mesh design can be adopted, e.g. defining a radial orientation of the elements at the impact point and making coincident the circle necking with the first failing element. Since this solution lacks general application, it is not adopted here. The good correspondence between the numerical simulations and the experimental results indicates that the material relation describes the stress

181

Figure 10.

Shape of the deformation. Effective stress. (a): Specimen S20; (b): Specimen C20.

Figure 11. Force-displacement response. (a): Specimens S30 and C30; (b): Specimens S20 and C20; (c): Specimens S10 and C10.

and strain state of the impact specimens with sufficient accuracy. This is due to the fact that the true stress and strain relationship used as input in the material definition of the numerical model is based on the tensile test experiments. Moreover, the critical failure strain is well predicted since it is

obtained from actual measurements of the tensile test pieces. The histories of the current material yield stress versus the effective plastic strain on the surface of the first failing element for a spherical (S20) and a cylindrical (C20) specimen are

182

Figure 12.

Position of first failure element. (a): Specimen S20; (b): Specimen C20.

Figure 13. Current material yield stress versus effective plastic strain. The histories are measured on the surface of the first failure element (Fig. 12). (1): Specimen S20. (2): Specimen C20. (3): True stress-strain relationship of the plate material (Fig. 8).

shown in Figure 13. The cross plot of these histories could not follow precisely the true stressstrain material relation since the post-processed equivalent plastic strain includes all the combined loads, such as tension, compression and bending, developed during the impact event, while the true strain of the base material depends purely on the uniaxial load. In both specimens the current material stress deviates at some point of the curve becoming smaller than that defined by the true stress-strain relation. Since specimen S20 deviates at a larger strain (strain ∼0.4), it is assumed that the deformation of the first failing element is influenced by the effect of uniaxial forces. On the other hand, specimen C20 should have a strong influence of shear forces, because the curve deviates earlier (strain ∼0.1). The strain state of the first failing element can be illustrated with the infinitesimal strain tensors obtained from the numerical post-processing. For better understanding the resulting strains in a three-dimensional body are shown in Figure 14.

Figure 14.

Strains in a three-dimensional body.

The matrices of the infinitesimal strain tensors measured at the middle surface of the first failing elements are: ⎡ε xx ⎢ ⎢ε yx ⎢ ⎢⎣ ε zx

ε xy ε xz ⎤ ⎥ ε yy ε yz ⎥ = ⎥ ε zy ε zz ⎥⎦ ⎡0.230 (Specimen S20) ⎢⎢0.022 ⎢⎣0.470 ⎡ 0.239 (Specimen p C20) ⎢⎢0.022 ⎢⎣ 0.501

0.022 0.116 −0.074 0.022 0.033 −0.035

0.470 ⎤ −0.074 ⎥⎥ −0.276 ⎥⎦ 0.501 ⎤ −0.035 ⎥⎥ ⎥⎦ −0.163 .

These matrices show that the first failing element of both specimens suffers membrane and shear forces. Specimen S20 suffers higher biaxial stretching in the plane surface (εxx, εyy) and thinning in the thickness (εzz) than specimen C20 due to the compression provoked by the spherical striker, as illustrated in Figure 12a. Specimen C20 shows slightly higher shear strain in the vertical plane (εxz, εzx) than specimen S20, since the cylindrical striker transmits shear forces, as appreciated in

183

Figure 12b. It should be mentioned that the shear strain (εxz, εzx) in specimen S20 is due to the compression of the indenter. The strain state of the first failing elements described above can be illustrated graphically in Figure 15. It is convenient to recall that an element subjected to pure tension forces elongates along the axial direction and decreases its cross-sectional area (Fig. 15a), while an element under pure shear forces deforms as an isochoric plane and does not change the length and the orientation during its deformation (Fig. 15b). The first failing element of the spherical specimens is subjected to the combined tensile and compressive forces (Fig. 15c), thus it shows stretching in the biaxial directions of the plane surface and thinning in the thickness direction. On the other hand, the first failing element of the cylindrical specimens is subjected to the combined tensile and shear forces (Fig. 15d), thus it shows stretching in the uniaxial direction, thinning in the thickness direction and keeps the original length in the side of element ejected by the striker. The matrix of the strain tensors indicates that both spherical and cylindrical specimens suffer shear effects (εxz, εzx). However, the source of the shear effect is different as Figure 15 and visual inspection of the tested specimens demonstrated. Since the shear strain for the cylindrical specimens is due to the external load given by the indenter, it is assumed that the elements fail by shear. However, the shear strain for the spherical specimens should be due to the necking of the failing elements (thinning), thus it is assumed that these elements fail by tension. The post-processed magnitudes of the principal strains (εx, εy, εz) for the first failing element are shown in Table 4. The magnitudes of the principal strains in all directions are very similar to the average principal strains obtained from the tensile

specimens. In the impact specimens the strains in the normal (εx), tangential (εy) and thickness (εz) directions should be compared with the strains in the longitudinal (ε1), width (ε2) and thickness (ε3) directions for the tensile specimens. The magnitude of the strains in tensile (εx, ε1) and thickness (εz, ε3) direction is similar and much larger than the one of the strains in transverse direction (εy, ε2). Since the three principal strains at failure are similar, the critical failure strain obtained from the tensile test pieces (Equations 1 to 4) is adequate for the definition of the critical failure strain for the actual impact models. The results of the stress distribution and deformation process of the specimens are not recorded in the experimental program. Thus, once the forcedisplacement response and the failure modes are validated, some extra information from the numerical models is given. The steps of the deformation process for specimens S20 and C20 are shown in Figures 16 and 17, respectively. It is observed that when the stress exceeds the yield strength, the plates undergo large plastic deformation (Figs. 16a and 17a). For specimen S20 the maximum stress is concentrated in the local indentation below the head of the indenter, but for specimen C20 the maximum stress concentrates in the shear circle. Once the elements exceed the critical failure strain, the crack initiates on the necking circle for specimen S20 (Fig. 16b) and on the shear circle for specimen C20 (Fig. 17b). As the rectangular plate is more rigid along the short span, the material fracture is initiated in this direction (Figs. 16b and 17b). When the absorbed energy is larger than the critical impact energy, the crack initiates and extends in the two directions along the necking circle for specimen S20 (Fig. 16c) and the shear circle for specimen C20 (Fig. 17c). This propagation is much faster for specimen C20. Finally,

Figure 15. Type of failure element. (a): Pure tension; (b): Pure shear; (c): Combined tension and compression. (d): Combined tension and shear.

184

Table 4. Comparison of principal strains in the tensile and impact specimens.

Specimen

Strain in X direction (εx, ε1)

Strain in Y direction (εy, ε2)

Strain in Z direction (εz, ε3)

S20 C20 Tensile

0.511 0.578 0.573

0.123 0.036 −0.083

−0.565 −0.505 −0.490

Figure 16. Deformation process of specimen S20. Effective stress.

The impact response of the specimen plates is strongly affected by the shape and the diameter of the indenters. The global deflection of the specimens struck by the spherical and cylindrical indenters is similar. However, the local deflection of the spherical specimens is much larger and, consequently, they absorb more energy until failure. The critical absorbed energy also increases with the diameter of the indenter since it provokes larger plastic deformation. On the other hand, smaller indenters provoke easier fracture and smaller stiffness slope in the force-displacement response due to the more localised surface of contact. The strain state of the first failing element illustrates that this element suffers combined tension and compression when impacted by the spherical indenter and combined tension and shear when impacted by the cylindrical indenter. Since the three principal strains at failure are similar, the critical failure strain obtained from the tensile test pieces is adequate for the definition of the critical failure strain for the actual impact models. The true material relation until failure used in this study is valid in small-scale structures subjected to impact where such a small mesh size can be defined.

ACKNOWLEDGEMENTS

Figure 17. Deformation process of specimen C20. Effective stress.

specimen S20 exhibits some small cracks which initiate in the necking circle and extend towards the indentation (Fig. 16d). For specimen C20 the shear circle is completely cut by the indenter (Fig. 17d). 5

CONCLUSIONS

Detailed information of the impact response of rectangular plates, struck laterally by spherical and cylindrical indenters is obtained through drop weight impact tests and nonlinear explicit dynamic simulations. The numerical models reproduce the plastic response and the onset of failure with sufficient accuracy, indicating that the defined true stress-strain curve and the critical failure strain describe the stress-strain state of the impacted specimens. The good agreement with the experiments is also achieved by the definition of a very small mesh size.

The work of the first author has been financed by a PhD scholarship from ABS, the American Bureau of Shipping. The authors are grateful to Dr George Wang for his initiative to promote this scholarship. The second author has been financed by the Portuguese Foundation for Science and Technology, under contract SFRH/BD/46369/2008.

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Material modeling for finite-element simulation of ship impacts J.N. Marinatos & M.S. Samuelides National Technical University of Athens, Athens, Greece

ABSTRACT: The Finite Element simulation of strongly non-linear response of structures, which results in large strains, heavily depends on the material model that is chosen by the user to represent the actual material behavior. Finite element codes require as input the true Mises stress-strain relation of the material in the plastic domain, which commonly the user determines from the measurements of the force and the elongation acquired during standard tensile tests. However, there is more than one procedure to determine the required relationship that the user inputs to the code, from the measured parameters. The paper reviews the alternatives regarding the modeling of the material, i.e. material curve and rupture criterion, that have been suggested by various researchers to be used in the simulation of ship impacts and investigates their effect on the simulation results. On the basis of the present investigation, the authors provide guidance concerning the modeling techniques. 1

INTRODUCTION

Simulation of ship impacts with the use of finite elements results in strain patterns including strains that exceed the strain corresponding to the maximum load of the load-displacement curves obtained from tensile tests. These curves incorporate the localized damage that occurs during tension and capture the tri-axial stress field generated in the neck, which depends on the thickness of the specimen. Thus, it is important for the user to supply the finite-element code with a true Mises stress-strain curve (true stress-strain curve) that corresponds to the actual behavior of the material in large strains. An implicit question to this observation is, if the true stress-strain curve until rupture that is obtained from tensile tests represents in a realistic manner the material behavior over the whole range of strains that appear in modes of response, other than those observed in tensile tests, for example bending, tearing, crushing. In particular it is reasonable to question, if the necking and the subsequent softening of the response of the tensile specimen, is a phenomenon that occurs also in structural elements of a ship’s hull that is subjected to impact collision and grounding actions, and if it is appropriate to use a true stress-strain curve that captures and incorporates the effects of necking in the simulation of such impacts. The above considerations are also related to the selection of the rupture criterion that is used to simulate the rupture of the material during the impact and the removal of the element that meets the criterion from the finite-element model. If the criterion triggers the removal of the element before

the “necking strain”, or if as a result of the criterion the strain patterns on the structure do not include considerable areas of strains much larger than the “necking strain” then the representation of the true stress-strain behavior beyond necking might not influence the simulation results. However, if the strains are allowed to become larger than the “necking strain” before the rupture criterion is triggered, the importance of the true stress-strain curve over its whole range becomes important. Both, the determination of the true stress-strain curve beyond necking as well as of the rupture criterion from tensile tests depend on the gauge length, which is the initial length of the strain free specimen. Further, the results of a simulation, which refers to the behavior of a ship’s structure under impact loading, depend strongly on the size and type of the elements that have been selected by the user. Ideally the user should select the material model that is appropriate for the specific mesh size and the numerical results should be independent from it. Thus, in order to perform a finite element simulation of a ship impact, the user needs to select i. a mathematical representation of the true stressstrain relationship (true stress-strain curve). The model is used by the code to compute at the end of each load increment, all the components of the true stress, on the basis of the stresses from the previous increments and the incremental strains. ii. a criterion to identify when the material ruptures that triggers the removal of an element from the FE model, and

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iii. the discretization of the structure, in terms of element type and size. A number of researchers have suggested alternative methodologies for the calculation of the true stress-strain curve from the engineering stressstrain curves determined from uniaxial tensile tests, in particular in the range beyond the ultimate applied force, and performed benchmark studies to investigate how the structural response of a loaded structure depends on it. Further, numerous papers suggesting rupture criteria to be incorporated in Finite element codes have been published. The paper reviews the various procedures that have been suggested to determine the true stressstrain curves as well as the rupture criteria that are appropriate to use for the FE simulations of ship impacts. Further, the paper presents the results and conclusions from the application of the various true stress-strain curves and rupture criteria to simulate tests that have been performed by two different research groups in order to suggest, if possible, generally applicable guidelines for the FE modeling and simulation of structures under extreme loading conditions.

2

TRUE STRESS-STRAIN CURVE

The curve that represents the relation between true stresses and strains in a multi-axial stress field is calculated on the basis of the applied force and displacement measurements obtained during uniaxial tensile tests, i.e. the engineering stress-strain curve. However, such a procedure implies that the axial stress field in the cross section is uniaxial, which is not the case in the neck region of a specimen. The multi-axial stress field in the neck region may be observed in the results from finite element simulation of tensile tests (see for example Ehlers et al. 2009). Further, the engineering stress-strain curve and consequently the true stress-strain curve depend on the gauge length. The effect of this dependency on the result of the simulation of the response of a structure is not significant, when the response under consideration is limited to small plastic strains, i.e. for mild steel the strains in the region of the yield plateau, but it becomes important, when the simulation involves the regions of the material curve beyond necking. The dependency of the stress-strain material curves from the gauge length has been investigated by Ehlers at al. (2009) and Hogström et al. (2009). Ehlers et al. (2009) investigated if both the constitutive equation and the failure criterion—both elements compose the material model employed in a finite element simulation—need to take into

account the mesh size. The authors performed benchmark tests using two material models: one incorporates a material power law and a mesh independent failure strain according to ASM Handbook; the other is introduced by the authors and uses a mesh dependent true stress-strain relationship. The true strain is measured on a reference length, which varies from 0.88 mm to 4.4 mm and the true stress is calculated using the average of the actual cross sectional area of the specimen along the reference length. The rupture criterion was defined as the maximum effective plastic strain being assumed to be equal with the axial maximum strain, which is measured during the tensile tests using, as described above, different reference lengths and consequently being mesh dependent. The curves were used to simulate tensile tests and the authors concluded that the mesh dependent true equivalent stress-strain curve predict more accurately the engineering stress-strain curve for 7 different mesh sizes between 0.88 mm to 4.4 mm. Rupture, i.e. the last point of the engineering stressstrain curve, was also more accurately predicted with the mesh dependent curve and in particular with mesh sizes less or equal to 2 mm. In Ehlers (2010a) the author concluded that when simulating punching of 4.12 mm thick plates, the fracture and the plate thinning is captured accurately when the true stress-strain relationship used, is determined on the basis of a gauge length corresponding to the element size. However, the forcedisplacement curves obtained from the simulation of the punching tests show relatively low sensitivity to the various true stress-strain curves. In a further publication Ehlers et al. (2010) compared the effect of the above mentioned material models, when used to simulate the structural response of a hull struck by a bow: the author found that the energy absorption capacity was much more sensitive to the mesh size when using the material model with the power law constitutive equation, rather than when using the material model presented in Ehlers et al. (2009). However, it has not been investigated if the relatively better performance is to be attributed to the mesh dependency of the constitutive equation or the rupture criterion—the issue is addressed by Ehlers (2010c)—and it remains open to investigation how to define the appropriate true stress-strain curve in case of relatively large mesh size, i.e. 50 mm ≤, commonly used in ship impact simulations. Hogström et al. (2009) defined true stress-strain curves for three materials-NVA mild steel, Domex 355 high strength steel and NV5083 aluminum— from measurements obtained from tensile tests on 4 mm thick specimens. From the figures of the publication it can be observed that the steel curves show considerable dependency on the gauge length,

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which varies from 6 mm to 78 mm, whereas the aluminum curve shows a relatively low dependency on the value of ultimate strain for gauge lengths varying from 6 mm to 42 mm. The authors applied a piecewise curve of mild steel for an FE simulation of the tensile tests on models with mesh size varying from 2 mm to 8 mm. The results showed no dependency from the mesh size up to a strain of 22%, which has been defined by the authors as the onset of damage. ISSC Committee V.III (2003) defined a true stress-strain curve of mild steel with and without including the necking effect, although it is not described how the curve, which includes necking is defined. It is understood that the curve that does not account for necking is taken to follow the power law with the constants being calculated on the basis of the measurements up to the point of maximum load. The curve that includes necking shows a considerable decrease of the true stresses after the point of maximum load. The resulted curves were used in a finite element simulation of tensile tests and it was found that the curve including necking effects gave considerable better results than the curve that did not include them. However, when the material models were used for the simulation of response of more complicated specimens, the results were influenced to a lesser extent from these models. Following Okazawa et al. (2004), there are two sources that soften the behavior of a structure during tension: one is attributed to the material, i.e. to development of micro-voids, temperature rise and shear band formation and is incorporated in the constitutive equation. The other is related to the global tangent stiffness matrix. Okazawa et al. (2004) performed FE simulations of tensile test employing LS-DYNA and using BelytschkoLin-Tsay plate elements which may have constant thickness or incorporate changes in thickness. The relationship between true stress and true plastic strain was determined on the basis of the direct measurements of the axial and one transverse displacements during uniaxial tensile tests. The other transverse displacement was taken equal to the one measured since the cross-section of the specimen was square. The axial displacement measurements were made by an optical system, which monitors the displacement of points spaced 1 mm apart. The side of the square cross-section was 10 mm. From the measurements the authors defined the material parameters C and n of the relationship: = σ y ( + Ceepl )n

(1)

where σ is the true stress, epl the true plastic strain, and σy the yield stress. For the Japanese Industrial Standard SS400 steel—Young’s Modulus

E = 194 GPa, Poisson’s ratio v = 0.3 it was found that σy = 262 MPa, C = 74.3 and n = 0.274. The paper does not clarify how the constants were calculated and if the measurements were taken until breaking of the specimen or up to maximum load. From the results of the simulations it can be concluded that the selection of the elements with the option to change the thickness makes it possible to capture necking and the subsequent decrease of the applied load. The drop of the stress after necking shows a dependency on the element size, which size varied from 6 mm to less than 1 mm. Zhang et al. (2004) suggested a power type curve for modeling the true stress-strain relationship:

σ = Ce pl n

(2)

where n = ln(1 + eu) and C σ u (e/n )n σu being the ultimate engineering stress, and eu the maximum engineering uniform strain related to the ultimate tensile stress σu and e the natural logarithm. For shipbuilding steel with σu less than 355 MPa, the author suggested that, if εu is not known the following relationship may be used:

εu =

0.24

1 0.01395σ u

(3)

Using the above the true stress-strain curve for mild steel with σu = 355 MPa is-the stress in MPa:

σ = 575e pl 0.176

(4)

Villavicencio et al. (2011) defined a true stress relationship using the suggestion of Zhang et al. (2004) for the region beyond ultimate load and a power law curve obtained from the measurements of the engineering values for the pre-necking region. Both the curve of Zhang and the combined curve were used to simulate a tensile test of a 4 mm thick specimen with 5 mm and 2 mm shell elements. The results show that the combined curve simulated more accurately the response of the specimen in terms of the engineering stress-strain curves, although the differences are not considered significant. Beyond necking an influence of the mesh on the numerically obtained engineering stress-strain curve has been observed. Ling (1996) suggested to use a true stress-strain curve obtained from the linear interpolation of a power law curve that is defined on the basis of the values of ultimate stress and the corresponding strain and a curve that is obtained by a linear combination of the power law curve up to the point of ultimate stress and a linear relationship thereafter that is tangential to the power law. Thus, the

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author suggests using a true stress-strain curve having the form:

σ

⎡ σ u ⎢w ( ⎢ ⎣

e pll

εu ) (

ε ⎛ e pl ⎞ u ⎤ w)⎜ ⎟ ⎥ ⎝ εu ⎠ ⎥ ⎦

(5)

where σu and εu are the true values of stress and plastic strain at the point the neck starts and W is an interpolation factor varying from 0 to 1. When w = 0, the true curve (5) is denoted as ‘powerlaw type’ curve and when w = 1 as ‘tangent type’ curve, (Marinatos et al. 2012). For intermediate values of σw, true stress-strain curves between the powerlaw and tangent type are derived. Thus, powerlaw type curve represents the lower bound and tangent type the upper bound of the curves obtained by relationship (5), (Ling 1996). If at maximum load the stress keeps the value σu, regardless of the increase in strains, then there is no hardening and the material is considered to behave as perfect plastic beyond that point. This type of true curve is denoted as ‘experimental type’ (Marinatos et al. 2012). The curve in (5) has been used to simulate tensile tests of copper alloys and mild steel. The best results for the mild steel were obtained when w = 0.45, (Ling 1996).

3

MODELING OF RUPTURE

The procedure that is used to deal with material rupture in finite element analysis is to remove the element, which models the part of the structure where rupture occurs. This technique although it is not the only possible method to deal with rupture—an alternative could be to simulate loss of material integrity by removing the attachment between nodes of adjacent elements—is preferred by almost all researchers performing simulations of collisions and grounding with finite elements. The simpler rupture criteria that have been used in FE codes are based on a damage parameter, which is an integral along the strain path of the element, see for example (Lehmann et al. 2002; Simonsen et al. 2004). The most common of such a criterion is based on a maximum allowable plastic strain, either the plastic component of the Mises stress (Lehmann et al. 2002) or the thru thickness plastic strain (Zhang et al. 2004). However, the reliability of the criterion depends on the mode of deformation of the structure under consideration, i.e. the criterion has inconsistent performance. This is understandable, because the criteria have been derived on the basis of measurements obtained from particular tests, usually uniaxial tensile tests. This issue is addressed by Bao et al.

(2004), who investigated the performance of nine damage criteria when applied to predict the rupture that occurred during two sets of uniaxial compression tests and three tests of uniaxial compression tests. The comparison of the test results versus the prediction that are obtained by the various criteria revealed that the general Rice-Tracey (1969) and the hydrostatic stress criteria were consistent in predicting rupture in the case of the tensile tests and the Cockcroft-Latham (1968) criterion in the case of compression. Thus, it seems that a unique criterion, at least to our present understanding, is not adequate to predict rupture during all modes of structural response that occur in ship-ship collisions and it is more appropriate to use a set of criteria. This has been done by Törnqvist (2003), who combined the Rice-Tracey and the CockcroftLatham criteria to establish the so-called RTCL criterion. Servis et al. (2006) have incorporated the T-criterion in ABAQUS explicit. The T-criterion correlates the reversible elastic energy density storage process with both brittle and ductile failure and is currently used to predict the initiation of preexistent macroscopic cracks, as well as in the calculation of forming limit diagrams—FLDs—for metal forming processes (Andrianopoulos et al. 1966). This is achieved by defining a dual limit to the strain energy that it is stored by the material: the one is determined by the capacity of the material to store elastic energy density due to volume change, the dilatational strain energy, and the other by the capacity of the material to store elastic energy due to change in shape, the distortional strain energy. The critical values of these energies are independent of the shape of the volume of material and the loading conditions, and depend on the strain rate and temperature. More recently, Alsos et al. (2008) combined the Hill (1952) criterion for zero and negative values of biaxial straining and the Bressan-Williams (1983) criterion for respective positive values. The criterion is based on a critical membrane stress level, which is reached when the material becomes instable and loses its capacity to carry loads. Criteria that distinguish the initiation of rupture on the basis of trixialities, use as transition values of triaxialities from one behavior to another the value of 1/3 that corresponds to uniaxial tension field and −1/3 that corresponds to uniaxial compression field. A further distinction that has been used in the treatment of rupture concerns the initiation of rupture in the weld lines. This failure mode, which has been observed in collision tests has been simulated by inserting particular elements to simulate the welds and by incorporating a rupture criterion that is based on a critical stress value—see for

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example (Lehmann et al. 2002; Huatao et al. 2001; Ehlers et al. 2010). All criteria are sensitive to mesh size. In particular criteria that are based on plastic strains are relatively more sensitive, because plastic strains increase relatively faster as a collision progresses and the distribution of plastic stains largely depends on the size of the element. Criteria that use as control parameters stress fields (Alsos et al. 2008) or the elastic strain energy (Servis et al. 2006), although they are relative less sensitive to mesh size, they too exhibit a certain sensitivity. A widely applied remedy to this fundamental problem is to define the critical values of control parameters as a function of the size of the element—see for example (Ehlers et al. 2009; Hogström et al. 2009; Peschmann 2001; Kitamura 1997). A further challenge related to the use of the above criteria is the definition of the parameters that are associated with them, i.e. the specific values which when they are reached, rupture allegedly occurs and the element is removed. Various combinations of experiments and numerical simulations have been proposed for their determination. As far as the removal of element is concerned, there are various options that the user may select. In the case of a shell element with more than one integration points through thickness, the element may be removed when the criterion is met in all points or in the mid point—for example material 123 of LS DYNA allows the user to select the number of through thickness points, which should meet the criterion to remove the element. When the user selects to consider only the mid-point, bending strains are ignored in the initiation of rupture, i.e. it is assumed that rupture occurs as a result solely of the axial forces that are developed in the cross section. In the case of solid elements, the element is removed when the criterion is met in the single integration point and bending strain field is taken into account via the number of elements through the thickness of the plate. Yuen et al. (2005) have used eight-node brick (C3D8R) and six-node prism (C3D6) elements of LS-DYNA to simulate the response of small plates under transverse blast load. 4

INVESTIGATION OF UNCERTAINTIES

Ehlers et al. (2008) presented a study on the effect of the mesh size and modeling of material rupture on the results of simulations of large scale collision tests, using LS-DYNA. The authors used for the simulations the four node Belytschko-Lin-Tsay element with 5 integration points. To the author’s knowledge, this is one of the few systematic convergence studies, as many publications, which present

FE simulations of structural response of large size structures under extreme loading conditions, lack adequate evidence to support the selection of mesh. In general, results obtained with models made with 25 mm elements were closer to the test results rather that with models made with 50 mm and 100 mm elements. However, the load penetration curves do not show a similar consistent trend in all three cases that were analyzed, i.e. the proximity of these curves to the experimental curve varies with the penetration and in some cases it is not obvious, which criteria should be used for the determination of the best correlation. For one of the tests, where the side structure modeled a conventional double hull, the authors presented a comparison of the experimental force-penetration curve, and i) the respective curves that they obtained from their simulations with three models, all having a mesh of 25 mm and three different rupture models, ii) the curve obtained from Peschmann, who included the effect of the supports of the side structure to the hull of the struck vessel and iii) a force-penetration curve that was obtained using the model of the structure from Peschmann with the model of rupture based on the RTCL criterion. The comparison highlights the difficulties associated with the definition of an appropriate modeling technique. Further the authors showed that the propagation of fracture depends highly on the material failure criterion, even in the case of the most dense model that employed the 25 mm elements. The test of the double hull model that is mentioned in the previous paragraph, was also analyzed by Alsos et al. (2007) using two models with mesh size of 25 mm and 12.5 mm and the BWH criterion to simulate material rupture. The forcepenetration curves correlate better with the experimental results rather than the respective curves, which were obtained with models having mesh size of 100 mm, 50 mm and 25 mm and by simulating rupture on the basis of the RTCL or maximum plastic strain criteria (Ehlers et al. 2008). Hogström et al. (2012) studied the effect of the uncertainty in the material model and the parameters and concluded that they affect significantly the assessment of the survivability of a ship involved in a collision. The authors simulated the response of a model of a double hull structure loaded transversely by a rigid intender. Eighteen simulations were performed, using six material rupture criteria and three sets of material properties. The best performance was obtained with a rupture criterion based on Mises strain and with the values of material parameters being equal to the average values obtained from tensile tests minus two standard deviations. When the same eighteen material models were used to investigate a collision, the results showed a large scatter and the damaged area of

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the side structure varied from 5.3 m2 to 69.1 m2. However it is noted that the material model that gave the best results when simulating the test, gave the largest damaged area, i.e. 69.1 m2, in the collision investigation. This observation illustrates the need for further investigation of the uncertainties that are involved in the investigation of contact accidents. 5

NUMERICAL ANALYSIS

In Marinatos et al. (2012), the authors simulated five tests performed by two independent research groups (Alsos et al. 2009; Paik et al. 1999) using ABAQUS 6.10-2 (2010) explicit. Combinations of three alternative material true stress-strain curves beyond necking, three alternative rupture criteria and three element sizes were used to investigate how the modeling parameters affect the comparison between the experimental and numerical results. The material properties are summarized in Table 1. The tests conducted by Alsos et al. (2009) refer to the transverse loading of three plates, one being unstiffened, one with a single stiffener and a third one with two parallel flat bar stiffeners, defined as US-Plate, 1-FB and 2-FB, respectively. The tests conducted by Paik et al. (1999) refer to the transverse loading of a double bottom model where the impact location of the cone shape indenter hit between webs or on webs.

The models were defined as ST-3-BW and ST-3-OW, respectively. Concerning the material curves, it is noted that the true stress-strain curve, σtrue-εtrue, was derived up to the point where necking starts—this is assumed to coincide with the occurrence of maximum load in the stress-strain engineering curve—by simply transforming the engineering stress-strain curve to true, adopting the volume incompressibility concept. Beyond the point of necking three different types of true stress-strain curves were assumed and implemented in the simulations, i.e. the experimental, the powerlaw and the tangent type (Marinatos et al. 2012). In order to simulate rupture, three rupture criteria were incorporated into the explicit finite element code ABAQUS 6.10-2 (2010). The critical true plastic strain criterion or SHEAR criterion, the BWH instability criterion and the RTCL damage criterion. It is noted that two forms of the RTCL and SHEAR damage criteria were implemented in the simulations, the unscaled (RTCL, SHEAR) and the scaled (RTCLS, SHEARS) form. In the first case the critical true plastic strain was set equal to εn, namely fracture was determined regardless of the element length and set to the value predicted for t = le through uniaxial simulations, while in the second case the critical true plastic strain became dependent of the element length according to the adopted fracture scaling laws, as described in Marinatos et al. (2012), in order to capture fracture after onset of local necking.

Table 1. Powerlaw material parameters for the various structural components. It is noted that σue and εfr are the engineering values of ultimate stress and fracture strain and K, n the strength coefficient and the strainhardening index, respectively. Component

E (GPa)

σy (MPa)

σue (MPa)

εfr

K (MPa)

n

US Plate FB Stiff. ST-3 models

210.0 210.0 197.7

285.0 340.0 245.3

416.0 442.0 337.8

0.35 0.35 0.46

740 760 590

0.240 0.225 0.221

Table 2. Best reproduction of the experimental force-penetration curve for the US-Plate, 1-FB, 2-FB and ST-3 models. Marinatos et al. (2012)

Alsos et al. (2009)

AbuBakar et al. (2010)

Model

le (mm)

Criterion

Curve

le (mm)

Criterion

Curve

le (mm)

Criterion

Curve

US Plate 1-FB 2-FB ST-3-BW ST-3-OW

18 18 5 25 25

RTCL SHEARS RTCLS SHEARS SHEARS

Tangent Tangent Tangent Tangent Tangent

18 10 18 – –

RTCL BWH RTCLS – –

Powerlaw Powerlaw Powerlaw – –

15 15 15 – –

FLD FLD FLD – –

Powerlaw Powerlaw Powerlaw – –

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Various element sizes were selected to investigate the effect of the mesh on the results. The FE models of the specimens tested by Alsos et al. (2009) were modeled with 5, 10 and 18 mm, while the specimens tested by Paik et al. (1999) with 25, 40 and 50 mm. Furthermore in the cases of the ST-3 models, and in the use of the SHEARS rupture criterion and a tangent type material true curve the FE models were modeled with three more mesh sizes, namely 18, 12.5 and 5.6 mm. S4R shell elements with five through thickness section points were used for the simulations. In the cases of the RTCL and SHEAR rupture criteria, in scaled or unscaled form, elements were removed from the mesh when the specific criteria were satisfied in all of through thickness points of the element, while in the case of BWH rupture criterion, when only at the mid-point of the element. The comparison between the experimental results and the numerical results is not a straight forward task, because in the highly non linear region the response cannot be described by one or more numbers, but by the force-penetration curve as well as by the mode of response. In the present case the comparison was based on the comparison of the trend and limit values of the force-penetration curves as well as on the visual comparison of the deformation modes. From Marinatos et al. (2012) it has been concluded that best reproduction of the various experimental force-penetration curves is not achieved in all of the examined models by the same rupture criterion, material true curve and element length as seen in Figure 3. The parameters that gave the best comparison are presented in Table 2. This non consistent behavior of all modeling alternatives, which could be also observed in the results presented by Ehlers et al. (2008), renders difficult to suggest general applicable modeling guidelines for the simulation of the structural behavior of large structures under extreme loading conditions. However in the present work, the authors attempt to define on the basis of their previous work, the unique modeling parameters, which produce results that correlate well with all tests, even if in particular cases there are modeling parameters that are considered to yield results closer to the experimental measurements and subsequently to investigate if there is a solid basis to suggest the unique modeling parameters for other simulations. From the previous analysis (Marinatos et al. 2012) the authors could observe that the unique modeling parameters, is the combination of the true stress-strain curve with a tangent region beyond the point that corresponds to the maximum engineering stress, and the SHEARS criterion, with the critical rupture strain being depended on the size of the element. The investigation of whether these

Figure 1. US-Plate, 1-FB & 2-FB models. Forcepenetration diagrams. Comparison between best results from three different numerical analyses.

parameters are generally appropriate, is based on the comparison of the results obtained using the unique modeling parameters and the parameters which simulate better each particular case, as well as on observations and conclusions from work of other researchers (Alsos et al. 2009; AbuBakar et al. 2010). These have been summarized in

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Figure 2. ST-3-BW & ST-3-OW models. Force-penetration diagrams. SHEARS criterion in combination with a tangent type material true curve and six element lengths.

Figure 3. US-Plate, 1-FB, 2-FB, ST-3-BW & ST-3-OW models. Force-penetration diagrams. Comparison between best results from three different numerical analyses, rupture criteria and SHEARS criterion in combination with a tangent type material true curve.

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Table 2 that presents the modeling parameters that produce the best results from Alsos et al. (2009) and AbuBakar et al. (2010), together with the respective modeling parameters from Marinatos et al. (2012). The numerical results of these three different research groups (RG), i.e. Alsos, Amdahl et al. (2009)-(RG1), AbuBakar et al. (2010)-(RG2) and Marinatos et al. (2012)-(RG3) are presented in the diagrams in Figures 1 and 3 concerning the US, 1-FB and 2-FB models for comparison. RG1 and RG2 use a powerlaw fit to the experimental true stress-strain curve to describe the true stress-strain relation prior and beyond necking, while RG3 uses

the transformed engineering stress-strain curve to true up to the onset of neck and beyond necking applies three different types of true material curves, namely the experimental, the powerlaw and the tangent stress-strain curve, (Marinatos et al. 2012). 6

DISCUSSION

According to Figure 1, that illustrates the best results, as they were obtained by the three different research groups for each examined model, in the cases of the

US PLATE

1-FB

2-FB

Figure 4. US-Plate, 1-FB & 2-FB models. Representation of damage according to SHEARS failure criterion for each model in combination with a tangent type material true curve and three element lengths.

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US-Plate and 1-FB models the best reproduction of the experimental force-penetration curve is achieved by RG3, in the use of a tangent type material true curve and RTCL, SHEARS failure criteria, respectively. In the case of the 2-FB model both best results produced by RG1 and RG3 show satisfactory reproduction of the force-penetration experimental curve. Specifically, RG1 results capture more accurately the part of the curve which concerns the residual strength of the examined model, while RG3 results, the part of the curve up to the point the force is maximized. It is noted that the results of RG1 were obtained using the RTCLS failure criterion in combination with the power law and the results of RG3 using the RTCLS failure criterion in combination with a tangent type material true curve.

In Figure 3 SHEARS failure criterion and a tangent type material true curve is compared to the best results obtained by all research groups for various failure criteria. One may observe that in all cases SHEARS failure criterion responds fairly well. This is also in agreement with the results obtained by Hogström et al. (2012), as previously referred to in section 4, where it was found that the best performance, through simulations to investigate the response of a model of a double hull structure loaded transversely by a rigid intender, was obtained with a rupture criterion based on Mises strain in combination with damage evolution model. Furthermore in the cases of the ST-3 models, results with the SHEARS failure criterion and

Figure 5. ST-3-BW & ST-3-OW models. Representation of damage according to SHEARS failure criterion for each model in combination with a tangent type material true curve and six element lengths.

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a tangent type material true curve for six mesh sizes have been checked for convergence. As seen in Figure 2 in both ST-3 models numerical results tend to converge with reducing mesh size. In addition Figures 4 and 5 illustrate the deformation modes of the examined models for various mesh sizes and SHEARS rupture criterion in combination with a tangent type material true curve. It is observed that a good representation of the deformation mode of the each experiment is achieved with element lengths equal to 10, 18 and 10 mm in the cases of the US-Plate, 1-FB and 2-FB models, respectively. In both cases of the ST-3 models an element length equal to 25 mm captures more accurately the deformation mode of the each experiment. Interesting is the fact that also in Ehlers et al. (2008), it has been concluded that, models made with 25 mm elements were closer to the test results, in simulations of large scale collision tests. Numerical results with element lengths 10, 18 and 10 mm in the cases of the US-Plate, 1-FB and 2-FB models and 25 mm in both cases of the ST-3 models in combination with SHEARS rupture criterion and a tangent type material true curve, also exhibit the best correlation with the various experimental force-penetration curves. In Figure 5 in the case of the ST-3-OW model it is obvious that the finer the mesh the less stiff the behavior of the structure. This is depicted through the buckling of the side webs, which is not the case in the experiment, when a ratio element length over thickness approaches the value of 2. 7

CONCLUSIONS

In the present work, a detailed and systematic procedure was followed for the simulation of five indentation tests performed by two different research groups, referring to the transverse loading on plates and double bottom models with plate thickness varying from 2.8 to 5 mm. The influence of three modeling parameters to the numerical results was taken into account, namely the true stress-strain curve, the rupture criterion and the mesh size. Through this investigation it was found that the SHEARS failure criterion in combination with the ‘tangent type’ true stress-strain curve corresponds fairly well in all cases of the examined models. Considering also its simple form, this combination appears appropriate for the simulation of the response of structural elements under extreme loading conditions. Furthermore, it is observed that mesh sizes with ratio of element length over thickness, which varies between 2 to 3.6, seem to be the best choice concerning FE simulations

of simple structures as in the case of Alsos et al. (2009) tests, while for more complicated structures, which involve in-plane loading of structural elements, as in the case of Paik et al. (1999) tests, a coarser mesh yields better correlation with the experimental measurements. ACKNOWLEDGEMENTS This research has been co-financed by the European Union (European Social Fund—ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)—Research Funding Program: Heracleitus II. Investing in knowledge society through the European Social Fund. REFERENCES ABAQUS 6.10-2. 2010. Documentation. AbuBakar, A. & Dow, R. 2010. Simulation of Grounding Damage using the Finite Element Method. 2010. 5th International Conference on Collision and Grounding of Ships, Espoo, Finland (pp. 208–216). Alsos, H.S., Hopperstad, O.S. & Amdahl, J. 2007. Prediction of rupture in collision and grounding of ships using the BWH failure criterion. 4th International Conference on Collision and Grounding of Ships, Hamburg, Germany (pp. 155–162). Alsos, H.S., Hopperstad, O.S., Tornqvist, R. & Amdahl, J. 2008. Analytical and numerical analysis of local necking using a stress based instability criterion. International Journal of Solids and Structures 45(7–8): 2042–2055. Alsos, H.S. & Amdahl, J. 2009. On the resistance to penetration of stiffened plates, part I: experiments. International Journal of Impact Engineering 36(6): 799–807. Alsos, H.S., Amdahl, J. & Hopperstad, O.S. 2009. On the re-sistance to penetration of stiffened plates, part II: numerical analysis. International Journal of Impact Engineering 36(7): 875–887. Andrianopoulos, N.P., Dodd, B., Kourouklis, S.K. & Limin, L. 1966. Forming-limit diagrams for Al 2124 and Al-Li 8090 through fracture mechanics and perturbation analysis. Journal of Materials Processing Technology 275–282. Bao, Y. & Wierzbicki, T. 2004. A Comparative Study in Various Ductile Crack Formation Criteria. Transactions of the ASME 126: 314–324. Bressan, J.D. & Williams, J.A. 1983. The use of a shear instability criterion to predict local necking in sheet metal deformation. International Journal of Mechanical Science 25: 155–168. Cockcroft, M.G. & Latham, D.J. 1968. Ductility and the workability of metals, J. Inst. Metals 96: 33–39. Ehlers, S., Broekhuijsen, J., Alsos, H.S., Biehl, F. & Tabri, K. 2008. Simulating collision response of ship structures: a failure criteria benchmark study. International Shipbuilding Progress 55: 127–144.

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Ehlers, S. & Vastra, P. 2009. Strain and stress relation for non-linear finite element simulations. Thin-Walled Structures 47: 1203–1207. Ehlers, S., Tabri, K., Romanoff, J. & Varsta, P. 2010. Numerical and Experimental Investigation on the Collision Resistance of the X-core Structure. Journal of Ships and Offshore Structures, Special Issue on Collision and Grounding: 1–9. Ehlers, S. 2010a. Strain and stress relation until fracture for finite element simulations of a thin circular plate. Thin-Walled Structures 48: 1–8. Ehlers, S. 2010b. A material relation for numerical ship collision analysis. In Proceedings of 5th International Conference on Collision and Grounding of Ships, Helsinki, Finland. Ehlers, S. 2010c. The influence of the material relation on the accuracy of collision simulations. Marine Structures 23: 462–474. Hill, R. 1952. On discontinuous plastic states with special reference to localized necking in thin sheets. Journal of the Mechanics and Physics of Solids 1: 19–30. Hillerborg, A., Modeer, M. & Petersson, P.E. Analysis of Crack Formation and Crack Growth in Concrete by Means of Fracture Mechanics and Finite Elements. Cement and Concrete Research, vol. 6, pp. 773–782, 1976. Hogström, P., Ringsberg, J.W. & Johnson, E. 2009. An experimental and numerical study of the effects of length scale, and strain state on the necking and fracture behaviours in sheet metal. International Journal of Impact Engineering 36(10–11): 1194–1203. Hogström, P., Ringsberg, J.W. & Johnson, E. 2010. Analysis of a struck ship with damage opening—influence from model and material properties uncertainties. An experimental and numerical study of the effects of length scale, and strain state on the necking and fracture behaviours in sheet metal. In Proceedings of 29th Conference on Ocean, Offshore and Arctic Engineering (OMAE2010), Shanghai, China, June 6–11, 2010. Hogström, P. & Ringsberg, J. 2012. An extensive study of a ship’s survivability after collision—A parameter study of material characteristics, non-linear FEA and damage stability analyses. Marine Structures 27: 1–28. Hogström, P. 2012. RoPax Ship Collision-a Methodology for Survivability Analysis. Chalmers University of Technology. Ph.D. thesis. Huatao, J. & Röhr, U. 2004. Investigation of the Rupture Failure of Welding Lines. 3rd International Conference of Collision and Grounding of Ships. Izu, Japan. ISSC Committee V.III (2003). Collision and Grounding. In Proceedings of the 15th International Ship and Offshore Structures Congress, San Diego, USA, edited by A.E. Mansour & R.C. Ertekin.

Kitamura, O. 1997. Comparative Study on Collision Resistance of Side Structure. Marine Technology 34: 293–308. Lehmann, E. & Peschmann, J. 2002. Energy Absorption by the Steel Structure of Ships in the Event of Collisions. Marine Structures 15: 429–441. Ling, Z. 1996. Uniaxial true stress-strain after necking. AMP Journal of Technology 5: 37–48. Marinatos, J.N. & Samuelides, M. 2012. Material characterization and implementation of the RTCL, BWH and SHEAR failure criteria to finite element codes for the simulation of impacts on ship structures. Submitted to the 6th International Conference of Collision and Grounding of Ships, Trondheim, Norway. Okazawa, S., Fujikubo, M. & Hiroi, S. 2004. Static and dynamic necking analysis of steel plates in tension In Proceedings of 3rd International Conference on Collision and Grounding of Ships, Izu, Japan, 25–27. Paik, J.K., Chung, J.Y., Choe, I.H., Thayamballi, A.K., Peder-sen, P.T. & Wang, G. 1999. On rational design of double hull tanker structures against collision. Society of Naval Architects and Marine Engineers 107: 323–363. Peschmann, J. 2001. Berechnung der Energieadsorption der Stahlstruktur von Schiffen bei Kollisionen und Grundberuehrungen. Dissertation, TU HamburgHarburg. Rice, J. & Tracey, D. 1969. On the ductile enlargement of voids in triaxial stress fields, Journal of the Mechanics and Physics of Solids 17: 201–217. Servis, D. & Samuelides, M. 2006. Implementation of the T-failure criterion in finite element methodologies. Computers and Structures 84: 196–214. Simonsen, B.C. & Törnqvist, R. 2004. Experimental and numerical modelling of ductile crack propagation in large-scale shell structures. Marine Structures 17: 1–27. Törnqvist, R. 2003. Design of crashworthy ship structures. Ph.D. thesis, DTU, Lyngby, Denmark. Villavicencio, R. & Soares, C.G. 2011. Numerical prediction of impact loads in rectangular plates. Proceedings of Advances in Marine Structures MARSTRUCT 2001, Hamburg, Germany edited by Guedes Soares and Fricke. Yuen, S.C. & Nurick, G.N. 2005. Experimental and numerical studies on the response of quadrangular stiffened plates. Part I: subjected to uniform blast load. International Journal of Impact Engineering 31: 55083. Zhang, L., Egge, E.D. & Brunhs, H. 2004. Approval Procedure Concept for Alternative Arrangements. 3rd International Conference of Collision and Grounding of Ships. Izu, Japan.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Experimental and numerical investigations of an alternative stiffening system for ship side structures to increase collision safety M. Schöttelndreyer, I. Tautz & W. Fricke Hamburg University of Technology (TUHH), Hamburg, Germany

B. Werner, C. Daske, H. Heyer & M. Sander University of Rostock, Rostock, Germany

ABSTRACT: This paper reports about the current state of a joint research project called SideColl. In the case of a ship-ship collision, the struck vessel as a normal rule reacts as the weak collision partner. In the past, several alternative side structures and double bottom designs with improved collision resistance already were presented. Most of them are attended with a general redesign of the ship structure, hence it is difficult to implement them only in critical parts of a ship. With this background the Institute of Structural Mechanics of the University of Rostock proposed an alternative stiffening system for double hull side structures called PVPS in a previous research project. PVPS is a German acronym for: PLATES STRENGTHENED STIFFENERS. The basic idea of this design is to connect the bulbs or face plates of two neighbouring profiles with a curved shell without changing the conventional structure. This alternative stiffening system was shown to significantly increase the collision resistance, but requires further investigations with respect to manufacturing as well as experimental and numerical investigations which demonstrate the influence of the PVPS on collision safety. In this context the follow-up project SideColl includes—among other investigations—three quasi-static collision experiments, which were carried out on the existing test facility of the Institute for Ship Structural Design and Analysis of TUHH. The present paper describes the two previously performed collision experiments where one side structure is equipped with PVPS and one is a conventional side structure. Furthermore the experiments are accompanied by simulations. The results of these simulations will be shown and compared with the measured results. One requirement of simulating and validating the collision experiments is to investigate the applied steel by uniaxial tensile tests. With the measured results of the uniaxial tensile test, true stress-strain curves are evaluated. The method for generating these true stress-strain curves will also be presented. 1 1.1

INTRODUCTION General

The strengthening of ship side structures and ship double bottoms for improving collision safety were part of several investigations in the past. In the 1950s and 1960s investigations of large-scaled collision experiments were in the spotlight of research caused by the development of nuclear merchant ships. The German company Gesellschaft für Kernenergie in Schiffbau und Schiffahrt mbH GKSS developed an innovative cellular side structure with quadratic cells. To prove the strengthening of the cellular side structure, several collision tests were carried out on the German shipyard Howaldswerke-Deutsche Werft AG in Hamburg, see Woisin (1976). In the 1990s the Dutch shipyard Royal Schelde designed a double hull structure with Y-core. This side structure was tested in the TNO test series

in the end of the nineties, see for e.g. Tabri et al. (2004). Present numerical methods offer engineers the possibility to develop new structural designs regarding collision safety. To this day new structural designs become part of current numerical investigations. The new concepts were presented e.g. in form of hat profiles, steel sandwich panel, see for e.g. Naar et al. (2001) as well as different kinds of corrugated-cores and X-cores, see e.g. Ehlers et al. (2007). The results of these numerical investigations show that an increase of absorbed energy between 30–50% could be achieved with innovative ship structures. Most of these structures require a general redesign of the ship structure. Also renewals of class certificates are difficult caused by the inaccessibility of the cells and void spaces. This paper presents an alternative stiffening system which improves the strengthening of the side structure without changing the conventional design.

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It is an adaptive system offering the possibility of inspection at any time. 1.2

University Rostock with their innovators U. Röhr and H. Heyer who are still engaged in the actual project SideColl.

Background

This paper reports on current research work carried out in collaborative joint research project called SideColl. The project started 2010 and will end in 2013. SideColl stands for: “Alternative stiffener systems considering ship collision and ice load”. One major research objective is to carry out large-scaled collision tests and to analyse these. The academic research is conducted by the TUHH with the Institute of Ship Structural Design and Analysis, the University of Rostock with the Institute of Structural Mechanics and Fraunhofer Institute for Manufacturing Engineering and Automation with their Application Center Large Structures in Production Engineering. The industrial partners consist of the classification society Germanischer Lloyd, the technical office for design and naval architecture Neptun Ship Design GmbH as well as the German shipyards P+S WERFTEN GmbH. 1.3

Previous research project

In a previous project the University Rostock proposed the alternative stiffening system for double hull side structures of tank ships, see Röhr (2008). The basic idea of this alternative stiffening system is to connect two neighboured profiles with a covered shell, see Figure 1. The profiles together with the covered shell were entitled as PVPS. This name is a German acronym for: PLATES STRENGTHENED STIFFENERS. The PVPS were investigated by several numerical simulations. The results showed that the average of absorbed energy is 34% higher with a maximum value of 78%, see Röhr (2008). This alternative stiffening system is protected by a patent with the number: 10 2007 058 060. Holder of this patent is the

Figure 1.

Ship side structure equipped with PVPS.

2 2.1

EXPERIMENTS Collision test-plant

The collision tests are carried out on the existing test-plant of the Institute of Ship Structural Design and Analysis of TUHH. This test plant was developed for the project ELKOS which is described in Tautz et al. (2010). Actual details will be published by Tautz et al. (2013) and Schöttelndreyer et al. (2013). The Collision forces are applied by four hydraulic cylinders which support a cross-beam. The maximum load capacity of 4,000 kN can be applied with these hydraulic cylinders with a maximum displacement of 400 mm. Larger displacements will be implemented by using appropriate interim pieces between the bulbous bow and the cross-beam. The hydraulic cylinders and the two supports are installed on two longitudinal girders. The bulbous bow structure is mounted underneath the cross-beam and is driven against the side structures with a collision angle of 90°. All collision experiments are quasi-static with a speed of 0.2 mm/sec. Figure 2 shows the schematical laboratory test configuration. 2.2

Test model of the bulbous bow

The shape of the rigid test model was developed within the scope of the project ELKOS, see Tautz et al. (2010). It is a cylindrical construction with a diameter of 813 mm and a length over all of 1,700 mm shown in Figure 3.

Figure 2.

200

Collision test plant of TUHH.

Figure 3. bow.

Geometry of rigid test model of the bulbous Figure 4. Conventional side structure without shell plate.

The wall thickness amounts to 30 mm in the cylindrical part and up to 150 mm in the forepart of the bow. supportconstruction

2.3 Test models of the two side structures stringer x y

z bulb profile inner shell web frame

cover shell

Figure 5. Side structure equipped with PVPS without shell plate (1/4 model). 1260 10°

knuckle knuckle

175

R175

470 65

The test model of the conventional side structure is deduced from the Anchor Handling Tug/Supply Vessel AHTS 480 which was designed and built by the German shipyards P+S Werften. For this type of ship an increasing resistance against collision is of high importance because the probability of a collision with offshore structures cannot be suspended. The test model represents a part of the parallel middle body of the AHTS 480, built with transverse framing. The design of the test side structure is restricted by the dimension of the existing test plant including the mounted rigid bulbous bow and the required space for manufacturing the structure. Hence the distance between inner and outer shell was scaled down with the factor 0.71. The distance between the two stringers, between the two web frames and the height of the profile were reduced by the scale factor 0.41. For getting a typical height, the bulb profile HP 120 × 6 was chosen which is half of the original profile. The designed test model has a length of 5.88 m, a breadth of 3.50 m and a height of 0.94 m over all. The investigated area within the support-constructions measured a length of 3,62 m, a breadth of 2.31 m. The distance between the two stringers amounts to 1.32 m and between the two web frames to 0.79 m. The frames of the side structure consist of six bulb profiles. The spacing between them is 270 mm and the bulbs are facing each other, see Figure 6. A wall thickness of 5 mm was chosen for the two web frames as well as for the two stringers. The shell plates have a thickness of 4 mm. The arrangement of the mentioned components is shown in Figure 4. For the investigation of the alternative stiffener system the geometry of the conventional side

30°

770

Figure 6. Details of PVPS, left: expanded cover shell without knuckle; right: cross section.

structure was not changed. It was modified with the cover shells which are located on the bulbs of the profiles of the outer shell, see Figure 5. The geometry of the cover shell is different from the geometry which was suggested in the previous research project because of the accessibility for the welding and cost concerns in manufacturing the PVPS. Figure 6 shows the expanded cover shell on the left side and the cross section of the PVPS on the right side. All cover shells are not connected to the stringers. 3 3.1

EXPERIMENTAL RESULTS General

Up to 131 measured values were recorded by a multi-channel data acquisition unit with a sample

201

rate of 2 Hz. The results for the comparison are plotted without breaks for visual analysis, i.e. the progress of deformation as well as the unloading and reloading operations in order to insert the interim pieces for realising higher displacements. 3.2

Effect of the PVPS

In Figures 7 and 8 the measured results of both experiments are compared with each other to demonstrate the influence of the PVPS. Furthermore the force-displacement curves build the basis for validating the numerical calculations further on. The measured results of the conventional test side structure are represented by the grey curve and the results of test structure equipped with PVPS by the black graph. The meaningful characteristics of both curves are marked by the numbers 1 to 4 in Figure 7. The first crack in the outer shell is observed at number 1 and is followed by the collapse of the stiffeners located at the area of impact marked by number 2. In the experiment with the conventional side structure, two bulb profiles failed one after another at a displacement of 349 mm to 435 mm. The failures occurred at different profile ends less than 320 mm in front of the stringers. In contrast to the conventional side structure, the bulb profiles of the modified side

Figure 7.

Comparison of measured reaction forces.

structure failed at once and on one end just behind the cover shell at a displacement of 458 mm. It is obvious that the cover shell postponed the collapse of the profiles. The result is a high increase of reaction force. After that failure the PVPS was still connected at the two bulb profiles on the other side which failed one after another at a displacement of 872 mm and 968 mm. It is easy to see that influence of the PVPS ends with the final separation. The cracks in the inner shell were visible at number 3. The collapse of the longitudinal stiffeners of the inner shell can be monitored at number 4. In total a significant increase of the reaction force of 77% (first peak) was achieved by the side structure equipped with PVPS. A final statement about the benefit by using the PVPS is possible with the energy-displacement curve which is shown in Figure 8. The collapse of the inner hull at number 3 is chosen for the comparison. The use of PVPS enables an improvement of absorbed energy of 56%. 4

DETERMINATION OF TRUE STRESS- STRAIN RELATIONS

For nonlinear analysis of structural impact reliable True Stress Strain Relations (TSSR) are required. Therefore the determination of TSSR is a fundamental base for the analysis of structural impact scenarios. By performing tensile tests force-displacement curves of a material were determined. Hence the Engineering Stress-Strain Relation (ESSR) for this material is specified. With onset of necking a polyaxial state of stress is generated. During the tensile test the load change and the cross section change have to be identified. Because of the non-uniform distribution of strain it is only possible to determine the averaged tensile stresses all over the cross section. As an alternative in this paper the TSSR is represented as a function of the equivalent stress in terms of von Mises depending on the accumulated equivalent plastic strain. In the uniform stress area of a tensile test the distribution of strain is uniform and the true strain ε can be approximated with the engineering strain εt:

ε

l

(

+ε

)

(1)

Supposing the incompressibility of the material the true stress σ can be determined with the engineering stress σt:

σ = σt ( + ε Figure 8.

Comparison of the measured energies.

)

(2)

Just as the specimen is necking, a multiaxial state of stress occurs. As a consequence the TSSR

202

cannot be determined in an analytic way by using the Equation 1 and 2. As an alternative method to investigate the TSSR Peschmann (2001) used the numerical simulation of the tensile test with the finite element method. Using this method the TSSR is iteratively modulated as long as the ESSR of the numerical simulation of the tensile test is congruent with the ESSR of the experiment. The numerical simulations are carried out with the finite element software MARC MENTAT. Solid elements were used. Furthermore, material properties with isotropic hardening and the von Mises flow rule were applied. The TSSR presented in Figure 9 are based on averaged ESSR, which were determined by testing three batches of ship steel grade A. The legend of Figure 9 indicates the quotient of the applied average element length and the extracted square root of the specimen cross section. As illustrated in Figure 9 the numerical determination of TSSR depends on the applied finite element mesh. By exceeding a strain limit εL the TSSR only illustrates a parameter curve that is adapted to the current finite element model. This curve does not represent a reliable TSSR that is independent from the respective finite element model. The stain limit εL of a numerical determined TSSR was specified by comparing the numerical determined TSSR curve with another numerical TSSR curve that was determined by using a finer mesh. At the point where these two TSSR curves begin to diverge, the strain limit εL is reached. Practically the allowed discrepancy between the curves has to be specified to define the diverge. In this paper the allowed discrepancy was set to 1%. To specify reliable TSSR curves in a codomain beyond the strain limit εL a half-analytical method from Ling (1996) was applied. In the range of uniform strains the TSSR is determined by using the Equations 1 and 2. In the

range of necking strain Ling refers to the frequently used connectedness

σ

(3)

where K and n are empirical constants. He established an upper

σ

σu ⋅ ( + ε −

)

(4)

and a lower

σ

⎛ε⎞ σu ⎜ ⎟ ⎝ n⎠

n

(5)

bound for the TSSR, where σu represents the true stress at the beginning of tensile necking computed with Equation 1. The parameter n equals the true strain εu according to the true stress σu and is determined by n = εu = l

(

+

g

)

(6)

where εg represents the engineering uniform strain. Ling weighted the equations 4 and 5 with the weight parameter w in the codomain 0 ≤ w ≤ 1 to the Equation

σ

⎡ σ u ⋅ ⎢w ⋅ ( + ε − ⎢⎣

)+(

−

) ⋅ ⎛⎜⎝

n ε⎞ ⎤ ⎟⎠ ⎥ n ⎥⎦

(7)

and determined w by inserting a known point of the TSSR. As known point one approved point of the numerical determined TSSR was used. More specifically that point was used, which was calculated with the finest mesh and which is located at the strain limit εL of the next coarser mesh. Subsequently the weight parameter w was determined with

w=

Figure 9. TSSR determined by method of Ling (1996) with equations 1, 2 und 7 vs. numerical determined TSSR.

K εn

σ L ⎡ εL ⎤ − σ u ⎢⎣ n ⎥⎦

n

n ⎡⎛ ⎡ε ⎤ ⎞ ⎤ ⎢⎜ 1 + ε L − n − ⎢ L ⎥ ⎟ ⎥ ⎣ n ⎦ ⎠ ⎥⎦ ⎢⎣⎝

(8)

where σL is the true stress of the finer mesh according to εL. As a result it was investigated a TSSR which is independent of meshing and which can be calculated by using the Equation 1, 2 and 7. The TSSR of the steel batches and the filler metal which were implemented in the finite element model of the tested side structures are investigated by the demonstrated procedure and are

203

1200

σ [N/mm²]

1000

800

600

filler metal grade A shells

400

grade A stiffener grade A stringer 0

0.2

0.4

0.6

0.8

1

Figure 10. TSSR for the steel batches implemented in the finite element model determined with equations 1, 2 and 7.

Table 1. True stress-strain relations of the finite element model with the power law coefficient and exponent as well as the belonging components of the test structure. TSSR

K (MPa)

n

Component

Filler metal Grade A shell stiff. Grade A shells Grade A stringer

841.9 749.5 742.7 720.6

0.08 0.178 0.162 0.155

Welds Stiffener Shell plates/PVPS Stringer/web frame

represented in Figure 10. Table 1 gives an overview of the TSSR with the determined power law coefficient and exponent. In addition the components of the tested side structures with the related TSSR are shown. The material of the stiffeners is not investigated yet. For now a TSSR based on three averaged batches of ship steel with grade A for the stiffeners is used. All sorts of ship steel grade A were tested with several flat tension specimens. The filler metal is tested as a round tensile specimen shaped with a lathe from a melted filler rod. To the test structure and the tensile specimens comparable welding parameters and identical specification are applied. A total number of six specimens was tested. 5 5.1

NUMERICAL ANALYSIS OF THE COLLISION EXPERIMENT WITH PVPS Finite element model

Simulations of the collision experiment with PVPS are carried out with the explicit finite element software LS-DYNA. The finite element model includes only the test structure as shown in Figures 4 and 5. The connection between test structure and test plant is taken into account through boundary conditions. The structure is meshed with an

element length of approximately 33, 16 and 8 mm to prove their influence on the failure mechanism. In addition the mesh with 8 mm element length is used to implement fillet welds between shell and stiffener (see Section 5.3). Shell elements with the formulation of Belytschko-Tsay with five integration points through their thickness are used. For the self-contact of the structure as well as for the contact between bulbous bow and structure a friction coefficient of 0.3 is applied. To predict failure in the structure a fracture criterion, developed by Scharrer et al. (2000), is applied. This criterion is actually developed for large finite element sizes. It defines the fracture strain εf as

ε f = ε g + εe

t . le

(9)

The uniform strain εg and necking strain εe are determined at fragments of steel plates belonging to several ships involved in collision and grounding accident. Both strain values are determined to εg = 0.056 and εe = 0.54. The mesh dependency of fracture strain is considered through the ratio of plate thickness t and element length le. From Equation 9 in conjunction with the constancy of volume it is possible to calculate a limiting value of thickness reduction

εd = −

εf 1+ ε f

(10)

of the element. In this context the fracture criterion is applied as a limiting value of thickness reduction. 5.2

Comparison of experimental and numerical results

In Figure 11 the force-displacement curves of the experiment and three finite element simulations with varying mesh size of the large-scale test structure with PVPS are compared. The force-displacement curve of the experiment shows two maxima. The first one occurs at the penetration of the outer shell and the failure of the stiffeners and the second one at the penetration of the inner shell (see also Figure 7). All three simulations predict the failure of the outer shell, at a displacement of around 100 mm and a reaction force of around 750 kN, in a very good agreement with the experiment. In finite element analysis crack initiation and propagation is realised through element deletion. This approach implicates the disappearance of material and causes a drop down of the reaction force after crack initiation in the outer shell.

204

At a displacement of around 400 mm the reaction force of all three simulations decline significantly due to the failing stiffeners and PVPS. In the simulation with 33 mm element length the stiffeners and the PVPS tear in the middle (see Figure 12a). In the simulation with 8 and 16 mm element lengths the stiffeners tear at the cut-outs of the stringer. First they tear on the left-hand end and later on the right-hand side (see Figure 12b). A fragment of outer shell, PVPS and stiffeners is perforated and is pushed into the structure by the bulbous bow. Both simulations are reproducing the failure mechanism of the experiment very well. However, the maxima of the reaction force of all three simulations with 1100 kN are clearly too low

Reaction force [kN]

2000 FEGL 33 mm FEGL 16 mm FEGL 8 mm Experiment

1500

1000

500

0 0

500 1000 Displacement [mm]

1500

Figure 11. Force-displacement curve of the collision experiment with PVPS compared with numerical results of three different mesh sizes.

since the experiment shows a maximum reaction force of 1540 kN. The differing failure mechanism of outer shell and PVPS in the simulation with 33 mm element length on one side and the simulations with 8 and 16 mm element length on the other side (see Figure 12) are influencing the failure of the inner shell. In the case of the coarse mesh it leads to a poor correlation of the force-displacement curve between simulation and experiment whereas the simulations with 8 and 16 mm element length show an acceptable agreement of the force-displacement curves. 5.3

Implementing fillet welds

In the test structures the fillet welds have leg length between 5 and 7 mm. One can assume that they have an influence on the failure mechanism of the structure. Alsos et al. (2009) implemented welds at the stiffener–shell intersection by increasing the thickness of the elements (see Figure 13). For elements of the shell a thickness gain of 2 mm and for the stiffener elements a thickness gain of 4 mm is applied. Since the large-scale structure of Alsos et al. (2009) had similar plate thicknesses and weld dimensions the values for the thickness increase are adopted and implemented in the finite element model with 8 mm element length. In contrast to them also the material behaviour of the filler metal are assigned to the weld elements. The described criterion by Scharrer was developed from penetrated plates of damaged ships with ship steel grade A. For this reason it is not recommended for the use of the filler metal and therefore failure of the weld elements is considered through the RTCL fracture criterion. A damage indicator D is determined for every element by the function D

1 p f (T )RTCL d ε eeq . ε0 ∫

(11)

D is a normalized value of the uniaxial fracture strain ε0. If D reaches unity in an element, it gets deleted and a crack is initiated or propagated.

Figure 12. Cross section of the finite element model with PVPS at a penetration depth of 500 mm a) 33 mm element length, b) 16 mm element length.

Figure 13. Cross section of the stiffener-shell intersection with weld elements.

205

The function of stress triaxiality f(T)RTCL and further detailed information of the RTCL criterion can be found in Törnqvist (2003). The value ε0 of the filler metal is calibrated with a finite element simulation of the uniaxial tensile test. The uniaxial fracture strain depends on the mesh size and the element formulation. Since the tension specimen is a smooth round bar axisymmetric elements instead of shell elements are used and an element length of 8 mm in load direction is chosen. The uniaxial fracture strain is determined to ε0 = 0.24. Figure 14 shows the force-displacement curve of the experiment with PVPS and the finite element simulation with 8 mm element length and implemented weld elements. It is obvious to note that the simulation agrees very well with the experiment. Both maxima of the simulation have a similar reaction force like the experiment, however, they also show a small offset of the displacement. 5.4

Future work

Including fillet welds in the finite element model, in the described manner, leads to a major influence of the reaction force. Further investigations have to clarify which detail of the presented method leads to a higher reaction force. The weld elements differ from the elements of the shells by a different fracture criterion, an overmatching material behaviour and a thickness increase. To prove the influence of the fracture criterion the results of the simulation with and without weld elements should be reproduced by using different fracture criteria for the welds as well as for the plates. The RTCL criterion could be used for the plates, too. In a second step the influence of the overmatching filler metal (see Table 1) on the reaction force should

be quantified. This could be done by comparing simulation results between weld elements with filler metal and weld elements with ship steel grade A. In a third step the sensitivity of the thickness increase on the reaction force has to be investigated. If the influence of welds by using small element sizes in the context of large scale collision test is confirmed, it is to investigate if they have a crucial influence in finite element simulations of ships. Due to a different ration between plate thickness and leg length of the welds they might be negligible. 6

CONCLUSIONS

This paper describes two collision experiments which were performed on the test facility of the Institute for Ship Structural Design and Analysis of TUHH. One conventional side structure is compared with a side structure equipped with an alternative stiffener system PVPS. These two side structures are encountered with a rigid bulbous bow. The experimental investigations show that the PVPS are able to absorb 56% more energy. Finite element simulations of one large-scale collision experiment with three different mesh sizes by using the fracture criterion developed by Scharrer are presented. In addition weld elements are implemented in the finite element model with the fine mesh where the RTCL criterion is predicting failure of these elements. This investigation shows that considering fillet welds through a thickness gain of shell elements and applying the material behavior of the filler metal is leading to a very good correlation between simulation and experiment. Finally this paper implicates the importance of validating high non-linear finite element simulations of large-scale collisions experiments. ACKNOWLEDGEMENTS

Figure 14. Force-displacement curve of the collision experiment with PVPS compared to the simulation with 8 mm element length and weld elements.

The work was performed within the research Projects ELKOS and SideColl, funded by the German Federal Ministry of Economics and Technology (BMWi) under project no. 03SX284B and 03SX309B. The authors are responsible for the content of this paper and wish to thank those, who supported this project. The authors’ gratitude is particularly addressed to U. Röhr and H. Heyer for providing their invention; the German shipyard Flensburger Schiffbau-Gesellschaft for delivering the crossbeam, the two supports for the test-plant and the test model of the rigid bulbous bow and the German shipyards P+S Werften for delivering the two side structure test models.

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REFERENCES Alsos, H.S., Amdahl, J. & Hopperstad, O.S. 2009. On the resistance to penetration of stiffened plates. Part II: Numerical analysis. International Journal of Impact Engineering 36 (7): 875–887. Ehlers, S., Klanac, A. & Tabri, K. 2007. Increased safety of a tanker and ropax vessel by implementing a novel sandwich structure. Proc. of 4th Int. Conference on Collision and Grounding of Ships (ICCGS). Hamburg: Hamburg University of Technology. Ling, Y. 1996. Uniaxial True Stress-Strain after Necking. AMP Journal of Technology 5. Naar, H., Kujala, P., Simonsen, B.C. & Ludolphy, H. 2001. Comparison of the crashworthiness of various bottom and side structures. Proc. of 2nd Int. Conference on Collision and Grounding of Ships (ICCGS). Lyngby: Technical University of Denmark. Peschmann, J. 2001. Berechnung der Energieabsorbtion der Stahlstruktur von Schiffen bei Kollision und Grundberührung. Dissertation. Technische Universität Hamburg-Harburg. Röhr, U. 2008. Forschungsvorhaben Fertigungs- und sicherheitstechnisch alternative DoppelhüllenKonstruktionen im Tankschiffschiffbau, Abschlussbericht zum Vorhaben A 234/S 24/10107/05, Universität Rostock. Scharrer, M., Zhang, L. & Egge, E.D. 2000. Verbundvorhaben Life Cycle Design, Abschulßbericht zum Vorhaben MTK 0588 4, Versagensverhalten von Schiffen bei Kollision. Germanischer Lloyd Report 2000.017. Hamburg.

Schöttelndreyer, M., Tautz, I., Fricke, W. & Lehmann, E. 2013. Side structure filled with multicellular glass hollow spheres in a quasi-static collision test. submitted for publication at Proc. of 6th Int. Conference on Collision and Grounding of Ships (ICCGS). Trondheim: Norwegian University of Science and Technology. Tautz, I., Schöttelndreyer, M., Fricke, W. & Lehmann, E. 2010. Experimental investigation on collision behaviour of Bow Structures. Proc. of 5th Int. Conference on Collision and Grounding of Ships (ICCGS). Espoo: Aalto University. Tautz, I., Schöttelndreyer, M., Fricke, W. & Lehmann, E. 2013. Collision tests with rigid and deformable bulbous bows driven against double hull side structures. submitted for publication at Proc. of 6th Int. Conference on Collision and Grounding of Ships (ICCGS). Trondheim: Norwegian University of Science and Technology. Törnqvist, R. 2003. Design of Crashworthy Ship Structures. PhD thesis. Technical University of Denmark. Tabri, K., Broekhuijsen, J. Matusiak, J. & Varsta, P. 2004. Analytical modelling of ship collision based on fulle scale experiments. Proc. of 3rd Int. Conference on Collision and Grounding of Ships (ICCGS). Izu: The Society of Naval Architects of Japan. Woisin, G. 1976. Die Kollisionsversuche der GKSS, Jahrbuch der Schiffbautechnischen Gesellschaft 70: pp.465–491, Berlin, Heidelberg, New York, SpringerVerlag.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Hydraulic modelling of submerged oil spill including tanker hydrostatic overpressure M. Sergejeva, J. Laanearu & K. Tabri Department of Mechanics, Tallinn University of Technology, Tallinn, Estonia

ABSTRACT: Internal hydraulic theory is employed to deal with submerged oil leak from damaged tank of single- or double-hull tanker. Variable locations and sizes of damage opening are considered to calculate the oil-spill volumes and durations. The exchange-flow solutions are presented for the side-hull damage opening in the case of balanced and un-balanced hydrostatic-pressure situations. The hydrostatic overpressure is associated with the initial uni-directional oil outflow that can fill the ballast tank in the case of double-hull tanker. In the case of balanced internal and external hydrostatic pressures, the internal flow through the side hole is bi-directional, with the upper-layer less-dense liquid (oil) flowing against the direction of the lower-layer denser liquid (sea water). The time-dependent oil outflow through the bottom damage results from the reduction of internal hydrostatic pressure in the tank as compared to the external hydrostatic pressure due to the sea level. 1

INTRODUCTION

Ship collision and groundings are still one of the major types of accidents in maritime transportation yielding to significant consequences. Looking at the IMO data shows that the share of collision accidents alone is about 20% of all serious and very serious accidents (Ehlers 2009). For example, in the Gulf of Finland, there are on average 20–30 collision or grounding accidents per year making it one of the riskiest regions in maritime traffic (Kujala et al. 2009). A ship participating in collision or grounding accident is often damaged to extent where its inner hull or tank bottom is breached. Therefore, ship collision and grounding accidents are the main reasons for large scale oil spills. A comprehensive analysis of possible collision and grounding accidents for a certain ship or for a certain sea region, should not only assess the structural damage, but should also include the duration and extent of a possible oil spill. This would allow to understand the nature of accidents and to develop risk control options accordingly. This paper proposes a simulation model that allows evaluating the extent and duration of oil spill in collision and grounding accidents once the ship dimensions and the description of damage are obtained. When two liquids of different densities are connected at damage opening, the exchange flow can take place. Under balanced internal and external hydrostatic pressure, the internal flow through a tank side-hole is bi-directional, with the upper

less-dense liquid (oil) flowing against the direction of the lower denser liquid (sea water). During this process, the oil in the tank is replaced by the inflowing water until the oil interface is ascending above the upper lip of the opening. In addition to the oil amount also the duration of the oil spill is an important parameter, which depends also on the oil and sea-water interactions. In the case of internal overpressure for the side damaged tank the unidirectional oil outflow is expected at the beginning of oil spill process. Hydrostatic balance between the liquids of different density deletes the net baratropic component from the oil outflow layer. The oil outflow through the side damage is essentially bi-directional under hydrostatic balance situation. However, the unidirectional internal flow is mainly possible for the bottom hole, with the less-dense liquid (oil) that is flowing out from the tank under comparatively high internal hydrostatic overpressure. The internal hydrostatic under-pressure situation would result in the sea-water intake. Loading procedure of oil tankers requires that the internal pressure in tank is larger compared to the external pressure (see Tavakoli et al. 2011). The aim of the present study is to demonstrate how the internal hydraulics theory can be applied for the modelling of stratified bi-directional flow of connected liquids of different densities and hydrostatic pressures. Hydraulic control is the generic principle that can be used to determine internal-flow dynamics through an opening due to internal pressure gradient. The functional

209

approach of this theory has been used in some engineering applications (Dalziel & Lane-Serff 1991). The Bernoulli type analytical solutions of internal flow are used in Tavakoli et al. (2010). For this purpose the hydraulics model for exchange flow, proposed by Laanearu & Davies (2007) for the quadrtatic constrictions, is used for the submerged oil leak from damaged tank. Regarding the geometric shape of the flow, the internal flow has one horizontal (length) maximum (l0) and one vertical (depth) maximum (d0) (both not necessary located at the centre lines of the hole). The areal shapes of flow are then classified quantitatively by the value of shape factor ξ (≥1), representing the ratio of equivalent area (l0 × d0) to the actual area A of the damage opening. While in the case of single-layer flow the hydraulic modelling is usually simplified for the maximum transport and its application requires the calibrating procedure of discharge coefficient. In the case of two-layer flow several internal-flow regimes represent maximum transport (Armi 1986) and the total two-layer sheared flow resistance is due to the flow separation from the hole edges and also due to the dynamical interaction (mixing) between the coupled liquids. The stratified bi-directional flow is parameterized by the densimetric Froude-numbers (F12) and (F22) corresponding to the upper- and lower-layer hydraulic regime, respectively. The internal-flow solutions are parameterized by the combined Froude number (G2 = F12 + F22). A simulation model is developed in present study for the oil-leak calculations in uni- and bidirectional flow cases. The model can be applied quite generally to predict the oil spill for different tank filling levels. The practical outcome of the model is the estimation of the volume of spilled oil and duration for the spilling of different tanker configurations and cargo oils. In the first chapter the general system of a tanker is presented. In the next section general principles of internal-flow hydraulics are briefly explained and some numerical examples are presented for the exchange flow with non-linear filling of tank. In the third chapter the hydraulic formula are applied to the system under unbalanced and balanced external and internal hydrostatic-pressure situations in the side damage case. The hydraulic formulae are applied to the system with the overpressure and bottom damage in the fourth chapter. The proposed model herein is used to simulate the model-scale experiment results by Tavakoli et al. (2011) to estimate the spilled and retained oil for the cases of bottom and side holes in the single and double hull situations. The validation of the method and overall results established are concluded and discussed in the last chapter.

2

SYSTEM DESCRIPTION

The most tankers are loaded such that the internal pressure due to oil level in the tank is larger than the external pressure due to the sea level. Thus, uni- or bi-directional oil spills are possible from the damaged tank. The unidirectional oil outflow results from the high internal hydrostatic pressurehead (ΔI) as compared to the external hydrostatic pressure-head (ρ2/ρ1 ΔO). In the case of the balanced internal and external hydrostatic pressure situation the oil flow is due to density difference between the cargo oil density ρ1 and the sea-water density ρ2 (ρ2 > ρ1). If the tanker carries substantially less cargo oil such that hydrostatic balance is established at- or several meters above the tank bottom, water enters the ship through the hole in the hull as long as the highest point of damage is below the hydrostatic balance level (National Research Council 1991). Four cases of the single- and double-hull tank damages are studied in this paper as sketched in Figure 1. Side damages in Figure 1 i) & ii) correspond to the submerged oil-leak from the single- and double-hull holes, respectively. Bottom damages in Figure 1 iii) & iv) correspond respectively to the submerged oil-leak from the single- and double-hull holes. The central case under investigation herein is the balanced internal and external hydrostatic pressure situation i.e. (ΔI + za) = ρ2/ρ1 (ΔO + za) with ΔI and ΔO representing the inside and outside height of liquid above the hole upper lip, respectively, and the parameter za is a quantity that depends from the hole orientation. The uni-directional oil-flow from the side-damaged

Figure 1. Submerged damages of the single- and doublehull tankers: i) side hole of single-hull tank, ii) side hole of double-hull tank, iii) bottom hole of single-hull tank and iv) bottom hole of double-hull tank.

210

tank occurs if the unbalanced internal and external hydrostatic pressure situation exists. In the case of the hydrostatic overpressure condition i.e. (ΔI + za) > ρ2/ρ1 (ΔO + za), only the uni-directional oil-spill from the damaged tanker can occur. Thus the difference between the sea level outside and the oil level inside of the tank, and the difference between the liquid densities, determine the oil-leak dynamics after accident. Typical tanker contains several cargo tanks (see Fig. 2), which can be damaged in accident. Denoting the dimensions of a single tank as length (LT), width (WT) and depth (DT), which define the tank volume VT = LT × DT × WT. The oil volume in the tank can differ from the tank volume because of the partial filling to satisfy the safety requirements. Here we assume that the oil-filled tank is of rectangular shape and the oil surface area is determined by ST = LT × WT (see Fig. 2). Double-side and bottom tank ballast volumes are defined herein as Vds and Vdb, respectively. The area of the damage opening in the tank structure is denoted with A. Equivalent dimensions of side damage (see Fig. 2 inset i)) are characterized by the depth d0 and the length l0. The area of the side opening is thus A = d0 × l0. Hole equivalent dimensions for the bottom damage (see Fig. 2 inset ii)) are characterized by width w0 and length l0 and the area is A = w0 × l0. However, the area of the damage opening in outer and inner hull in the case of double-hull can differ. Henceforth, the sub-indexes O and I are used for the system parameters defined of outer- and inner-skin, respectively. The vessels collision in the offshore region results with the side damage having hole with different opening area, vertical position and shape. The bottom damages have fixed vertical position in the system. The set of model parameters such as oil outflow volume VT, volumetric flux QT and oil outflow time

TT are made non-dimensional by the following volume, flux and time scales. VSCALE

LT ⋅ H 0 WT ,

(1a)

QSCALE

A ⋅ 2 g H0 ,

(1b)

TSCALE =

VSCALE . QSCALE

(1c)

The ship draft is denoted by H0. (Example tanker scales are: HO = 17.8 m, LT = 50 m, WT = 26.5 m). 3

INTERNAL-FLOW HYDRAULICS

An aim of the internal-hydraulics model calculations is to determine the oil outflow volumes and durations for the different side-hole areas A and depths Δ. It is considered that the oil-leak takes place through a hull equivalent-size hole, which connects the less-dense liquid (oil) of density ρ1 in the tank and the denser liquid (water) of density ρ2 in the sea. The oil surface is considered to be under atmospheric pressure (p0) and its vertical position can differ from that of outside sea water. The balanced internal and external hydrostatic pressure situation is represented by (ΔI + d0/2) = ρ2/ρ1 (ΔO + d0/2). In this paper the bi-directional flow is considered only for the side damage opening. In this central case the internal flow through a side-hole is bi-directional, with the upper-layer less-dense liquid (oil) flowing against the direction of the lowerlayer denser liquid (sea water) (see Fig. 3). The oil outflow through the bottom damage opening is considered to be unidirectional. The Pascal’s Law of connected vessels of different density liquids apply in both cases.

2 – Sea-water side (Outer)

O/I

1 – Oil-tank side (Inner) Balanced oil surface ρ1

Sea surface

ΔI ΔO

HO

d0O

hO

Q1 Q2 Oil-water interface

d0I

HI

hI I

ρ2

Figure 2. Schematic representation of the system: i) tank side opening and ii) tank bottom opening.

O

Figure 3. Sketch of the bi-directional flow through the single-hull side damage hole and notations.

211

An essential consideration in the theoretical analyses of the two-layer flow is existence of critical-flow sections. The maximal exchange-flow approximation requires that two controls exist simultaneously (Armi 1986). For instance in the case of the double-hull tanker it is distinct to locate the control section positions in the inner and outer holes of tanker hulls, yielding maximal transport estimate for the exchange flow. The flow through the single-hull tank side damage can only be sub-maximal i.e. is controlled only by single flow section. However, the sub-maximal solutions for exchange flow are mainly exploited in the present study, and this is justified by the damaged character where tank inner hull is always less breached. In the internal hydraulics theory the internal-head parameter Ki = (B2 − B1)/g′ is represented by the difference of the Bernoulli heads in lower (B2) dense and upper (B1) less-dense layer. The reduced gravity g′ = g (1 − r) is fixed by the density ratio r = ρ1/ρ2. The internal flow regime is parameterized by the combined Froude number G2. According to Laanearu & Davies (2007) the composite Froudenumber equation and the internal-head equation for the exchange flow, respectively, are given as G2 = ξ

u12 d2ξ − 1 u22 + ξ , g′ D g ′ d2 d2ξ

⎛ ξ D (ξ Ki = ⎜ l0 ⎝ + d2

2 ⎡⎛ ⎤ 1 ⎞ q2 ⎢ ⎥ ⎟ K ii ⎢⎜ ξ ⎟ − ξ ξ 2⎥ ( D d2 ) ⎠ ⎝ d2 ⎠ ⎣ ⎦

− )⎞2

h.

(2)

(3)

The volumetric-flux parameter (determined here by the flow rate in the lower dense layer) is Kii = Q22/(2g′). The ratio of upper layer (Q1) and lower layer (Q2) discharges is expressed conveniently in terms of the parameter q2 = Q12/Q22. The velocity of upper (1) less-dense and lower (2) dense liquids are u1 and u2, respectively. The net velocity is defined by un = u1 − u2. The lower water-layer depth is d2 and the upper oil-layer depth can be determined by the relationship d2 = d0 − d1, where the depth maximum of the hole is d0 (see Fig. 2 inset i)). The maximum length of the hole is l0. It should be mentioned that the actual hole-area A can be related to the equivalent depth and width (d0, l0) only by fixing the hole shape i.e. the shape factor ξ. The rectangular-shape hole represents the limiting case with the shape factor ξ = 1, and it will be most exploited in the present study. For instance, the triangular-shape hole corresponds to the “quadratic-shape” hole with the shape factor ξ = 2 and the circular-shape hole has the shape factor ξ = 4/π due to similarity principle ξ = l0 × d0/A.

(For instance in the case of the environmental engineering study on river channel flow, the maximum width is at surface, maximum depth is at Thalweg and the shape parameter ξ was 1.8 in Laanearu et al. (2011).) In the case of single-hull tank the equations (2) and (3) can be evaluated directly to estimate the sub-maximal exchange through a hole, and in the case of double-hull tank the equations (2) and (3) should be evaluated separately for inside- and outside-hull holes, and the combined solution of the four equations can be employed to work out numerically the maximal exchange-flow flux. It should be mentioned that Equations (2) and (3) can be applied quite generally to predict the layer depths of the out- and inflowing fluids for sub-maximal exchange with the inner-hull control only. 3.1

Exchange flow

The oil spill from the single-hull tank side hole is schematically represented in Figure 3. The exchange flow through side hole is determined by the model parameters, such as the hole areal size A = d0 × l0 (ξ = 1), the stratification that is given by the reduced gravity g′, and the system parameters, such as oil thickness (HI = hI + d0I + ΔI) inside of the tank and the sea-water relative thickness (HO = hO + d0O + ΔO) outside of the tank. In the case of balanced internal and external hydrostatic pressure situation ((ΔI + d0/2) = ρ2/ρ1 (ΔO + d0/2)) the oil thickness HI is a dynamical parameter, which depends only on the parameter hI during the bidirectional-flow process. The sea-water thickness HO (and hO) is assumed to be constant and fixed by the ship draft. The hull thickness in the case of single-hull tank is considered negligible. During tanker emptying process the oil with the depth hI(t) below the lower lip of hole is regulated by the sea-water amount inside of the tank, and the sea-water depth hO below the lower lip of hole is dependent on the sea-water intake depth outside (what in the present model calculations is fixed by the tank bottom). 3.2

Application example

The oil-spill dynamics for the dimensionless cases of hole size (A/(L × HO)): with A = (5 × 5) m2, (4 × 4) m2 and (3 × 3) m2, and with the hole lower lip dimensionless depth (ΔO + d0)/HO: with (ΔO + d0) = (5 + 5) m, (10 + 4) m and (15 + 3) m from the oil level inside of the tanker, respectively, is shown by full curves in Figure 4. The non-linear emptying of the initially oil-filled tank is associated with the changing oil outflow flux, having peak value in the case of equal liquids

212

flux is determined by the standard Torricelli’s formula Q1 Cd A 2 g

((

d0

)

ρ2

1

(

)

d0 2 ) , (4)

where Cd is the discharge coefficient. The timedependent oil outflow through the side hole results from the reduction of internal pressure due to descending of oil surface in the tank as compared to the sea level of fixed position i.e. HO = const. In the case of single-hull side hole the oil outflow duration and volume with changing internal pressure can be calculated by the analytical formulae: Figure 4. Bi-directional flow with the oil outflow volumetric flux equal to the sea-water inflow volumetric flux (q2 = 1) for different dimensionless hole sizes (A/ (LT × HO)) and the dimensionless depths. Case 1: Balanced system, 3 × 3 m2, ΔI = 15 m; Case 2: Overpressure, 3 × 3 m2, ΔI = 14 m; Case 3: Balanced system, 4 × 4 m2, ΔI = 10 m; Case 4: Overpressure, 4 × 4 m2, ΔI = 9 m; Case 5: Balanced system, 5 × 5 m2, ΔI = 5 m; Case 6: Overpressure, 5 × 5 m2, ΔI = 4 m.

depths at hole i.e. d1 = d2. It can be recognized from the results in Figure 4 that comparatively large holes correspond to high peak volumetric flux, and comparatively deeply submerged holes correspond to larger outflow time. Also solution curves for the oil leak in the cases of un-balanced hydrostatic pressure situations are included in Figure 4 with dashed curves. The peak fluxes are “right-ward” shifted due to the uni-directional oil outflow in the case of hydrostatic overpressure condition at beginning. The exchange flow analytical calculations by Tavakoli et al. (2010) were limited to the maximum oil outflow flux, which is employed essentially in the estimation of shortest oil outflow duration below.

4 4.1

Toil =

⎛⎛ d ⎞ d ⎞⎞ ⎛ Voil = Δ I + ⎟ − ΔO + 0⎟⎟ S. ⎝ ⎠ ⎝ 2 ρ1 2 ⎠⎠ ⎝

Toil =

The uni-directional leak from the side-damaged tank occurs if the unbalanced internal and external hydrostatic pressure situation exists. Herein the example calculations are given for the oil outflow through the rectangular-shape hole (ξ = 1). In the model calculations the oil thickness HI is a dynamical parameter that depends on the parameter ΔI(t) during the uni-directional flow process (Fig. 4). The sea-water thickness HO (and ΔO) is assumed to be constant and fixed by the ship draft. In the hydraulic model calculation the uni-directional oil outflow volumetric

(5a)

(5b)

The total oil outflow volume for uni-directional flow is fixed due to the difference in the hydrostatic pressure of the oil and water columns relative to the hole axis. The total oil outflow duration for uni-directional flow is dependent on the discharge coefficient, and the shortest duration is associated with the inviscid case i.e. Cd = 1. The oil spill uni-directional flow calculations for different side dimensionless holes A/(HO × LT) (with the actual areal sizes of A = 1 × 1 m2, 2 × 2 m2, 3 × 3 m2, 4 × 4 m2) of the single-hull tank and inviscid case are concluded in Table 1. In the case of double-hull tank, however, the Torricelli’s Formula should be integrated with the side ballast volume VDS. Several limitations of the double-hull volume may apply due to the tanker construction (see Fig. 5). The time-dependent oil outflow duration and volume can be calculated by modified analytical formulae:

SIDE DAMAGE: ANALYTICAL MODEL Uni-directional flow

2S 1 Voil /S , A 2g Cd

2S 1 Voi*l / S , A 2 g Cd

⎛⎛ d ⎞ d ⎞⎞ ⎛ Voi*l = Δ*I + ⎟ − ΔO + 0⎟⎟ S , 2 ⎠ ρ1 ⎝ 2 ⎠⎠ ⎝⎝

(6a)

(6b)

where ΔI* is fixed by the relationship ΔI* = ΔI − VDS/S. Here the ballast volume VDS that is available for the oil, is dependent on the trapped air pillow i.e. non-ventilation condition apply. It should be mentioned here that in the case of submerged and low-positioned damage no oil outflow may occur from the tanker, because the oil outflow volume

213

Table 1.

Side damage oil spill analytical calculations. Single hull unidirectional

Single hull bi-directional VSPILL/VSCALE

A/ASCALE

ΔI/H0

VSPILL/VSCALE

0.0011 0.0045 0.0101 0.0180

1.40 1.12 0.84 0.56

0.31 1.11 0.35 1.19 0.40 1.26 0.44 1.33 Double-hull unidirectional

0.07 4.39 0.35 15.37 0.63 22.57 0.91 28.22 Double-hull bi-directional

0.0011 0.0045 0.0101 0.0180

1.40 1.12 0.84 0.56

0.23 0.29 0.36 0.42

0.07 0.35 0.63 0.91

TSPILL/TSCALE

0.95 1.08 1.19 1.29

Figure 5. Sketch of the uni-directional flow through the double-hull side damage hole and notations. Nonventilated side ballast volume VDS is available for the initial oil volume loss.

available due to the high internal overpressure condition may be smaller as compared to the side ballast volume VDS. In such case the oil outflow from the tanker is determined only by the bidirectional flow. The oil spill uni-directional flow calculations for different side holes of dimensionless areas A/(HO × LT) (with the actual areal sizes of A = 1 × 1 m2, 2 × 2 m2, 3 × 3 m2, 4 × 4 m2) of the double-hull tank and viscid case (Cd = 1.0) are also concluded in Table 1. 4.2

Bi-directional flow

The bi-directional flow through the double-hull tank side hole is sketched in Figure 6. An essential consideration in the theoretical analyse of the two-layer internal hydraulics is existence of critical-flow sections. Several submaximal flow regimes are possible by maximising

TSPILL/TSCALE

4.39 15.37 22.57 28.22

Figure 6. Sketch of the bi-directional flow through the double-hull side damage hole and notations. Nonventilated side ballast volume VDS is filled during the unidirectional flow.

the exchange flow through the side hole of damaged tank. The critical-flow solutions corresponding to the sub-maximal flow can be determined by using the implicit-function differentiation theorem as applied to Equation (3) with respect to the lower layer depth variable d2. The result corresponds to the composite Froude number formula G2 = F12 + F22 = 1. The sea-water volumetric flux maximum of bi-directional flow is given by the formula

Q22 =

g ′w02d03 3

ξ

⎡ 2ξ ⎢⎛ d 0 ⎞ ⎢⎜ ⎟ ⎢⎝ d 2 ⎠ ⎣

1

−

(

−1

⎤ ⎥ . 3⎥ ⎥ ⎦

−1)

( d2 d0 ) (1 − ( d2 d0 )ξ ) q

2

(7)

The oil-leak volumetric flux of bi-directional flow is determined by Q12 = q2Q22. It should be noted that the sea-water volumetric flux by Equation (7)

214

has a single maximum associated with the fully controlled flow at the oil-interface position d1 = d2 in hole. In the case of balanced internal and external hydrostatic pressure situation the upper-layer less-dense liquid (oil) is flowing against the direction of the lower-layer denser liquid (sea water) without net flux i.e. Qn = Q1 − Q2 = 0. The oil spill bi-directional flow calculations for different side holes dimensional areas A/(HO × LT) (with the actual areal sizes of A = 1 × 1 m2, 2 × 2 m2, 3 × 3 m2, 4 × 4 m2, 5 × 5 m2) of the single- and double-hull tank are also concluded in Table 1. 5 5.1

BOTTOM DAMAGE: ANALYTICAL MODEL Uni-directional flow

Q1 Cd A 2 g (

−

2

1

),

(8)

where Cd is the discharge coefficient. The timedependent oil outflow through the bottom hole results from the reduction of internal pressure due to descending of the oil surface in the tank as compared to the sea level of fixed position i.e. HO = const. In the case of single-hull bottom hole the timedependent oil outflow duration and volume can be calculated by the straightforward analytical formulae: Toil =

The uni-directional leak from the bottom-damaged tank occurs if the un-balanced internal and external hydrostatic pressure exists. In the case of hydrostatic overpressure i.e. ΔI > ρ2/ρ1ΔO (with ρ2 > ρ1), only the uni-directional oil-spill from the damaged tanker can take place. Here the example calculations are given for the oil outflow through the rectangular-shape hole (ξ = 1). In the case of double-hull tanker the inside hull rectangularshape hole area AI = w0I × b0I is set equal to the outside hull area AO = w0O × b0O i.e. A = AI = AO. It should be mentioned that in the model calculations similar to the uni-directional flow through the side hole the oil thickness HI is a dynamical parameter, which depends directly from the parameter ΔI(t) during the uni-directional-flow process (see Fig. 7). The sea-water thickness HO is assumed to be

Figure 7. Sketch of the uni-directional flow through the double-hull bottom damage hole and notations. The initial oil volume loss is due to the bottom ballast of volume VDB.

constant and fixed by the ship draft. In the model calculation the uni-directional oil outflow flux is determined by the standard Torricelli’s Formula:

Voil

2S 1 Voil S , A 2g C d ⎛ ⎝

I

ρ2 ρ1

⎞ O

⎠

S.

(9a)

(9b)

The total oil outflow volume for uni-directional flow is fixed due to the difference in the hydrostatic pressure of the oil and water columns relative to the tank bottom. The total oil outflow duration for uni-directional flow is dependent on the discharge coefficient, and the shortest duration is associated with the inviscid case i.e. Cd = 1. The oil spill unidirectional flow calculations for different bottom holes of dimensionless areal sizes A/(HO × WT) (with the actual areal sizes of A = 1 × 1 m2, 2 × 2 m2, 3 × 3 m2, 4 × 4 m2, 5 × 5 m2) of the single-hull tank and inviscid case are concluded in Table 2. In the case of double-hull tank, however, the Torricelli’s Formula should be integrated with the bottom ballast volume VDB, which is available for the oil volume loss. Several limitations of the double-hull volume may apply due to the tanker construction (see Fig. 7). In this case the oil outflow duration and volume can be calculated by modified analytical formulae: Toil =

2S 1 Voi*l S , A 2 g Cd

(10a)

Voi*l

⎛ * I ⎝

(10b)

ρ2 ρ1

⎞ O

⎠

S.

where ΔI* is fixed by the relationship ΔI* = ΔI − VDB/S. It should be mentioned here that no oil outflow occurs from the tanker if the bottom ballast volume VDB is large enough as compared to the oil outflow volume available due to the internal overpressure.

215

Table 2.

Bottom damage oil spill calculations. Single hull unidirectional

Double hull unidirectional

A/ASCALE

ΔI/H0

VSPILL/VSCALE

TSPILL/TSCALE

VSPILL/VSCALE

TSPILL/TSCALE

0.0021 0.0085 0.0191 0.0339 0.0530

1.40 1.12 0.84 0.56 0.28

0.30 0.30 0.30 0.30 0.30

1.10 1.10 1.10 1.10 1.10

0.064 0.064 0.064 0.064 0.064

0.51 0.51 0.51 0.51 0.51

The oil spill uni-directional flow calculations for different bottom holes of dimensionless areas A/(HO × WT) (with the actual areal sizes of A = 1 × 1 m2, 2 × 2 m2, 3 × 3 m2, 4 × 4 m2, 5 × 5 m2) of the double-hull tank and inviscid case are concluded in Table 2. 6

Table 3. Modelled spilled and retained oil percentages. Tavakoli et al. (2011) experimental results in parentheses.

DISCUSSION

Here the developed method is compared due the hydraulic characteristics with the model-scale experiments by Tavakoli et al. (2011). Due to flow separation effects at hole, the real discharge is lower than the maximal discharge. For instance, the discharge coefficient that is estimated for the unidirectional flow through single bottom puncture (C1, diameter = 2.2 cm) was found to be 0.67 in Tavakoli et al. (2011). The model-scale experiment discharge coefficient was found to be 0.68 in the case of single side puncture (S1, diameter = 2.2 cm). In the simulation model proposed, the case of the side hole below the waterline 0.4 meter and overpressure-head with 0.27 m the discharge coefficient of circular hole was estimated to be Cd = 0.77. This small discrepancy found is apparently due to vena contracta of the streamlines at the puncture, reported in Tavakoli et al. (2011). With the equal hydrostatic pressures on both sides of the hole, the bi-directional flow occurred in the modelscale experiment of Tavakoli et al. (2011). In the two-layer phase, the oil surface was constant and the oil-water interface ascended in the cargo tank. The two-layer flow in the experiment continued for approximately 5 h and finally stopped when the hole was covered by water on both sides. In the two-layer model proposed herein the circular hole is approximated by the shape factor ξ = 4/π. The model calculation for the bi-directional inviscid flow under same initial and boundary conditions proved that the oil outflow from the cargo tank lasted 1,53 h. This proves well that the total twolayer sheared flow resistance through the circular hole (S1) of area 3.8 cm2 was due to the flow separation from the hole edges and also due to the

Design

Hole

Spilled oil (%)

Retained oil (%)

Single bottom Single side hole Double bottom Double side

C1 S1 C1 S1

36.3 (34) 45.7 (57) 28.3 (26) 34.9 (40)

0 (0) 0 (0) 11.7 (35) 10.8 (19)

dynamical interaction (mixing) between oil and water. The total oil outflow volume was also found to correspond with a model-scale experiment by Tavakoli et al. (2011). The oil outflow of the simulation model can be: i) uni-irectional, ii) bi-directional and iii) retained in the ballast. Calculated spilled oil and retained oil volumes are presented and compared with the model-scale experimental results by Tavakoli et al. (2011) in Table 3. The model-scale experimental tank was built with a horizontal section of 100 × 50 cm2 and height of 100 cm. The water level was 47 cm and the oil level varied between 70–85 cm in the modelscale tank experiments. The largest puncture at bottom (C1) and side (S1) had diameter of 2.2 cm (A = 3.8 cm2). 7

CONCLUSIONS

The aim of present study was to demonstrate how the internal hydraulics theory can be applied for the modelling of submerged oil spill. The practical outcome of the model was estimation of the volume and duration of spilled oil from damaged tanker. In the case of un-balanced internal and external hydrostatic pressure situation the initial flow through the holes was uni-directional and after establishment of the balanced hydrostatic pressure situation at the side hole axis the bi-directional flow followed. However, distinction between the single- and double-hull tankers was done due to the ballast volume, associated with the oil loss.

216

A numerical model was used to calculate the stratified flow through the side hole with the nonlinear water filling of initially oil-filled tank. This numerical solution was restricted to the zero netflow (un = u1 − u2 = 0). This case represented realistic situation with the bi-directional flow starting and ending with the zero oil outflow volumetric flux. This condition corresponds to the uni-directional flow situation that was ended after oil surface inside of the tank was descended to the balanced hydrostatic pressure level and before the oil interface was ascended above the hole upper lip. The oil outflow flux maximum (Q1 = Q2 → max.) was associated with the oil interface position at hole of the equal liquid depths i.e. d1 = d2. The oil uni-directional outflow velocity was time dependent due to gradually reducing internal pressure (due to descending of the oil surface inside of the tank), and was associated with the non-linear emptying of the oil-filled tank until the balanced internal and external hydrostatic pressure situation was established. The linear emptying of the oil-filled tank was associated with bi-directional flow through the side hole, which was fixed to the maximum volumetric flux without net flow. In all simulation model solutions the oil-water interface was ascending from tank bottom to the upper-lip of hole. To conclude the study it should be mentioned that the developed simulation model can be applied also in the cases of not submerged single- and double hull side holes. The uni-directional flow phase in this case corresponds to the oil surface descending until the sea level, and occurs without the oil loss into ballast volume for the double-hull tanker case. ACKNOWLEDGEMENTS This research work has been financially supported by Estonian Science Foundation (grant agreement ETF8718) and by Central Baltic Interreg IV program through MIMIC project (“Minimizing

risks of maritime oil transport by holistic safety strategies”). This help is here kindly appreciated. REFERENCES Armi, L. 1986. The hydraulics of two flowing layers of different densities. Journal of Fluid Mechanics: 163: 27–58. Cuthbertson, A.J., Laanearu, J. & Davies, P.A. 2006. Buoyancy-driven two-layer exchange flows across a slowly submerging barrier. Environ. Fluid Mech: 6 No 2(19): 133–151. Dalziel, S.B. & Lane-Serff, G.F. 1991. The hydraulics of doorway exchange flows. Building and Environment: 26: 121–135. Ehlers, S. 2009. Material Relation to Assess the Crashworthiness of Ship Structures, Doctoral Dissertation, Helsinki University of Technology, available at: http:// lib.tkk.fi/Diss/2009/isbn9789522481443/. Ehlers, S. Tabri, K. 2012. A combined numerical and semianalytical collision damage assessment procedure, Marine Structures: 28(1): 101–119. Kujala, P., Hanninen, M., Arola, T., Ylitalo, J. 2009. Analysis of the marine traffic safety in the Gulf of Finland. Reliability Engineering & System Safety: 94(8): 1349–1357. Laanearu, J. & Davies, P.A. 2007. Hydraulic control of two-layer flow in quadratic type channels. Journal of Hydraulic Research: 45(1): 3–12. Laanearu, J., Vassiljev, A. & Davies, P.A. 2011. Hydraulic modelling of stratified bi-directional flow in a river mouth. Engineering and Computational Mechanics I: 164(EM4): 207–216. Tavakoli, M.T., Amdahl, J. & Leira, B.J. 2010. Analytical and numerical modelling of oil spill from a side damaged tank. 5th International Conference on Collision and Grounding of Ships, Helsinki, 10–14 June 2010. Tavakoli, M.T., Amdahl, J. & Leira, B.J. 2011. Experimental investigation of oil leacage from damage ships due to collision and grounding. Ocean Engineering: 38(17–18): 1894–1907. Zhu, Z.Z., Fouli, H. & Okyere, A.Y. 2001. Exchange flow through opening. Journal of Hydraulic Research: 40(3): 341–350. Zhu, D.Z. 2003. Hydraulic control of exchange flows. Journal of Hydraulic Research: 41(5): 503–511.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Plastic mechanism analysis of structural performances for stiffeners on bottom floor plating during shoal grounding accident Zhaolong Yu State Key Lab of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, China

Zhiqiang Hu State Key Lab of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, China State Key Lab of Structure Analysis for Industrial Equipment, Dalian, China

Ge Wang ABS Greater China Division, Shanghai, China

ABSTRACT: A theoretical model is introduced in this paper for structural performances of the stiffeners on the double bottom floor plating in a shoal grounding accidental scenario. It is based on a study of the progressive deformation process of numerical simulation results and the plastic analytical methods. Special emphasis is laid on establishing the characteristic deformation mechanism and identifying the major energy dissipation pattern. When calculating the structural performances of ship bottom during ship grounding accident, a prevailing method is to smear the stiffeners into the plating thickness for simplicity. Thus, the corresponding structural responses of the stiffeners are involved in the smeared plating. However, this method may not provide an accurate prediction of the structural performance characteristics of the stiffeners, and the deformation process of the stiffeners cannot be mastered. Therefore, a theoretical model is proposed in this paper to provide a deep insight into the deformation patterns with reasonable predictive accuracy. Using the plastic analytical method, the expressions of distortion energy composed of the crushing, bending, and membrane stretching effect of stiffeners on the floor plating are formulated, and the formulae of subsequent grounding resistances are obtained. The numerical simulation software LS_DYNA is employed to verify the proposed method. It covers a wide range of slope angles and indentations of shoal grounding scenarios. The results of the proposed simplified analytical method compare favorably with those of the numerical simulations and good predictive accuracy is acquired. Furthermore, the theoretical model proposed can conveniently lend itself for quick assessments of the performances of ship double bottom structures during shoal sliding grounding scenario, and it will benefit the application of accidental limit state design concept in the ship design stage. 1

INTRODUCTION

This paper deals with the collapse model of the stiffeners on the bottom floor plating during shoal grounding scenario. This type of collapse model is mostly observed when a ship slides over seabed obstacles with large contact surfaces. By understanding the deformation modes and energy dissipation pattern, the proposed simplified analytical method helps to perform quick assessment of ship performance during grounding accidents. Ship grounding over seabed obstacles has been identified as one of major hazards to ship sailing safety and crude oil transportation. Potential consequences may vary from minor structural damages to the major damages where ship structural integrity is threatened, great economic loss, severe

oil pollution, and which in its ultimate state, may sink the vessel and deprive numerous people’s lives. The grounding accident of Exxon Valdez in Alaska 1989 is considered one of the most devastating man-made environmental disasters ever to occur at sea. The accident resulted in the pouring of approximately 40,000 tons of oil into a pristine wilderness area, and it is now still suffering from adverse effects of the pollution (see Fig. 1) (Lin & Amdahl, 2008). These disasters caused great public sensations and more rational safety regulations for ships were demanded to enhance the sailing safety and protect the environment. A large number of tools and analysis procedures have been developed into understanding the response of ships subjected to collision and grounding. Generally, the prevailing approaches

219

Figure 1. Examples of oil spills with disastrous consequences (left: Torrey Canyon; right: Exxon Valdez), (Alan 2010).

Figure 2. Seabed topology with reference to bottom sizes (a) rock; (b) reef; (c) shoal (Alsos and Amdahl, 2007).

can be divided into 4 categories, i.e. experimental methods, empirical methods, plastic analytical methods, and nonlinear numerical simulation methods. It is advantageous to apply plastic analytical methods of analysis in developing simplified methods, since mechanism analysis can provide significant insight into the governing physical processes. Many simplified analytical formulae have been developed based on the construction of realistic deformation mechanisms identified during either actual ship accidents or model tests. Wang (Wang G. et al. 1997) predicted the strength of a ship hull in the event of grounding by considering four primary failure modes: the stretching failure of transverse structures, denting, tearing and concertina tearing failure of bottom plates. Vaughan (Vaughan, 1980), Ohtsubo and Wang (1995), Simonsen and Wierzbicki (1997), Zhang (2002) have all contributed substantially to identification and development work of fundamental theoretical models for ship structures subjected to accidental loads. During ship grounding accident, the deformation of ship structures is much related to the seabed topology. Different shape and size will generally lead to different deformation mechanisms. Alsos and Amdahl (2007) defined three types of seabed indenter, namely “rock”, “reef ” and “shoal” as shown in Figure 2. Most analytical methods are concerned with sharp seabed obstacles, although grounding with large contact

surface are more likely to occur. Unlike grounding with sharp obstacles, shoal grounding (grounding on the large contact surfaces) is more likely to degrade the global hull-bending capacity significantly and eventually trigger collapse of the hull girder by bending or shear, causing hazardous consequences (Lin & Amdahl, 2010). Therefore, the shoal grounding accident calls for a more profound understanding of the governing structural collapse mechanisms. The stiffeners are indispensable structural components of vessels which support plating against global loads and lateral pressures, and they provide also considerable strength to resist grounding actions. A prevailing method is to smear the stiffeners onto their attached plating which is known as the smeared thickness method or equivalent thickness method. This approach greatly simplifies the calculating process. However, the “effectiveness-ratio” factor in Paik’s smeared method (Paik, 1996) very much depends on the type of stiffeners and the grounding scenario, which greatly affects the predictive accuracy of consequences. The factor is often taken as 1.0 for simplicity. But Hu & Amdahl (2011) found that it might underestimate the role of stiffeners during the shoal grounding accident. Additionally, the collapse pattern of the stiffeners cannot be traced during the deformation process by using the smeared thickness method. The present work is undertaken on a need for accurate predictions of the performance of stiffeners during ship shoal grounding scenario. The proposed theoretical approach is based on a careful study of the progressive deformation process of numerical simulation results and the plastic analytical methods. It provides a comprehensive description of the deformation mode, energy dissipation pattern and structural resistance of stiffeners on the bottom floor plating when ship slides longitudinally during shoal grounding. A comprehensive numerical analysis using LS_ DYNA code is carried out to make verification, and it covers a wide range of slope angles of the indenter and indentations. The results are encouraging, and this analytical model will contribute greatly to the establishment of efficient tools for quick and reliable assessment of the performances of ship bottom structures during shoal grounding, for example in conjunction with emergency decision support systems or in preliminary design stages. 2

PROBLEM DESCRIPTION

Lin & Amdahl (2010) discussed the structural response of a double bottom tanker subjected

220

to the shoal grounding accident, neglecting the effect of stiffeners. However, for a typical tanker, stiffeners are indispensable, since they provide substantial strength and rigidity in resisting accidental actions and supporting the bottom structure. Generally, a double bottom structure mainly consists of three stiffened components: longitudinal girders, transverse floors and the bottom plate, and all of them are equipped with stiffeners (see Fig. 3). In this study, special emphasis is laid on the performance of the stiffeners on the bottom floor plating when a typical double tanker slides over the “shoal” type seabed obstacle. The numerical code LS-DYNA is employed to simulate the grounding event. Through the numerical simulation results, the progressive deformation process is clearly exhibited. Figure 4 shows the grounding scenario for the stiffened bottom floor plating and the attached stiffeners, and the other components are all hidden for ease of observation. As the rigid indenter slides along the double bottom in a longitudinal direction with a constant indentation, the stiffener on the floor plating produces a complicated deformation. The deformation pattern includes three different modes (see Fig. 5), among which the effects of bending, stretching and crushing are identified. A detailed deformation patterns are presented in Figure 6. It is observed that Mode 1 occurs only for stiffeners on the initial floor plating, which means it is not a steady state and they are not taken into consideration in this paper. Mode 3 is a typical stable deformation mode and occupies most of the deformed floor stiffeners while Mode 2 occurs occasionally during the deformation process. Therefore, Mode 3 is analyzed analytically as a typical deformation mode for stiffeners on the floor plating. A deep insight of the deformation mode of Mode 3 is undertaken. Through observation, it is indicated that the upper and lower 1/4 length of the stiffener plating seems to keep undeformed, and the distortion mainly concentrates in the middle part, where bending about the plastic hinges, bending and stretching due to the formation of curvature are identified. Figure 7 presents the process on how the stiffener plating deforms with a growth of the grounding indentation in Mode 3. It is indicated that an obvious arch-like deformation form occurs at the middle portion of the plating. As indentation increases, it will fold more intensively, and the arch-like deformation portion is going to bend about the plastic hinges.

Unlike the situation under simply axial crushing loads, the lower portion of the stiffener is found to move horizontally a distance in a shoal grounding accident, which has been observed by Lin & Amdahl (2010). The displacement tends to grow larger as indentation increases.

Figure 3. stiffeners.

A typical double bottom structure with

Figure 4. plating.

The grounding scenario of stiffened floor

Figure 5. The deformation of floor plating stiffener after grounding.

Figure 6. Detailed deformation patterns of floor stiffeners after grounding.

221

Figure 7. The development of Deformation Mode 3 with the growth of indentation.

3

3.1

Figure 8. Symmetric collapse model of a cruciform joint (Amdahl, 1983).

SIMPLIFIED ANALYTICAL METHOD FOR STIFFENERS ON THE BOTTOM FLOOR PLATING Review of the deformation of bottom floor plating

This paper is mainly concerned with the grounding performances of stiffeners on the bottom floor plating. The stiffeners provide substantial stiffness for the appended plating in maintaining stability and resisting accidental actions, such as collision and grounding. Also, the deformation patterns of the stiffeners and their appended plating are very much related. Based on observation, it is found that the joint line, which is also the bottom line of the stiffener, share the same deformation pattern with the attached plate. Therefore, considering the common similarities of stiffeners and the appended plate, a review of the deformation of floor plating is presented necessarily, before the analysis of the stiffener deformation. For a typical double bottom tanker, the floor plating and the longitudinal girder form a cruciform structure together, which processes an intense interaction with each other. Thus, the two structural components are generally considered as a whole. Wierzbicki and Abramowicz (1982) discussed the performance of a tube with a rectangular cross section under axial compression. Amdahl (1983) described the collapse of a four flange cruciform which deforms in a symmetric mode and gave its analytical expressions of a complete single fold of the length 2H (see Fig. 8). While considering the failure mode of transverse members by a concentrated load, Wang et al. (1997) applied a beam model dominated by membrane force. Simonsen & Wierzbicki (1997) modified the denting model of transverse members for application to the bottom raking process. When the shoal grounding accident is considered, the longitudinal flange of the cruciform are no more subjected to the axial load (see Fig. 4). Lin & Amdahl (2008) studied the collapse of the longitudinal girders during shoal grounding and found a repetitive wave-like deformation pattern.

Figure 9. Across section of the central part of the transverse floor before and after horizontal crushing (Lin & Amdahl, 2010).

A horizontal movement of the intersection between girder and flange plating is observed. For the collapse of the transverse floor, Lin did not give new mechanisms. Instead, Amdahl’s symmetric model for a four flange cruciform under axial compression loads is applied considering the horizontal displacement u (see Fig. 9). This approach may partly reflect the energy dissipation pattern. However, since the external loads are no more axial, the original method may neglect many significant clues. Thus, a new mechanism for the central part of the transverse floor is needed when grounding over a shoal type seabed topology. 3.2

Theoretical modeling for bottom floor plating stiffeners

The deformation pattern of the stiffeners on the bottom floor plating largely depends on the deformation of the attached floor plating, which is reviewed in the previous section. Based on a careful observation of the numerical simulation results and a review of the transverse floor plating research, assumptions are made as follows: • The stiffeners are firmly welded on the floor plating so that the deformation of the floor stiffener base line conforms to that of its attached floor plating.

222

• In the simulations, the bottom part of the stiffener is observed to move horizontally a distance u. Due to the mutually interaction with the longitudinal girders, u can be determined by the theoretical model of the longitudinal girder (Lin & Amdahl, 2010): u

2 H tan ς

(1)

where u is the horizontal displacement and: 2

1.0836 D + 0.0652

(2)

2ς

0 94ψ − 0.0048ψ 2

(3)

ψ is the slope angle of the indenter and D is the indentation. The central part of the transverse floor contacted with the front surface of the indenter is loaded uniformly along the long edge, and its side length ratio is considerable. Therefore, it is assumed that the stiffener conforms to the theory of cylindrical bending and it can be treated as a strip beam (Chen & Chen). Through numerical simulation, it is also observed that the strip beam is perfectly rigidly supported at both ends. Therefore, a strip beam model is established in Figure 10. According to assumption 1, the stiffener will share a similar deformation situation as that of floor plating. The stiffener on floor plating will deform as shown in Figure 11. According to Amdahl’s symmetric theory (see Fig. 8), it is subjected to axial crushing loads. The Point C stands for a stationary hinge line where discontinuities occur. This satisfies Klanac’s analysis of discontinuity (Klanac, 2010): ⎡ du ⎤ ⎡ ∂u ⎤ ⎢ dt ⎥ = ⎢ ∂t ⎥ + v ⋅ [∇u ] = 0 ⎣ ⎦ ⎣ ⎦

(4)

Figure 11. The floor stiffener deformation model during axial crushing.

where, u denotes the displacement field for the surface. Amdahl assumed that the displacement u and the velocities ∂ ∂∂t are continuous in space at any time and at any point of the shell. v is the velocity of the yield line. If ∂ ∂∂t is continuous, ∇u can be discontinuous only across stationary hinge lines for which v = 0, just as the case in Figure 11. However, for the scenario of shoal grounding accident, there occurs a horizontal displacement u. Then, the hinge line C is no longer stationary and will move (i.e. v ≠ 0) during the deformation process. Then, according to Eq. 4, ∇u must be 0 which means ∇u must be continuous across the hinge line C. Therefore, BC and CD can no longer be straight lines. And also based on an observation of the numerical simulations, it is assumed hereby line BC and CD are two arcs tangent to Point C. Therefore, a new model is established in Figure 12(a) for small indentations (D < Ls/2, Ls is the length of the stiffener). When D is larger or equal to Ls/2 but less than 3Ls/4, the theoretical model is presented in Figure 12(b). This analytical model well conform to the numerical simulations observed. When D exceeds 3Ls/4, the additional energy is assumed to dissipate through the plastic rolling about the hinge line B. These two models are established hereby to predict the responses of bottom floor stiffeners when grounding over the shoal type obstructions. Applicability may, more promisingly, be valid for the central part of the transverse floor plate. 3.3 Energy dissipation Based on the established theoretical model mentioned above, the energy is mainly dissipated through three major modes:

Figure 10. The strip beam model for the central part of the transverse floor plating.

1. Bending about the plastic hinges at point B and D.

223

the ultimate tensile strength σu. In the following discussion, σ0 is taken as the average value of the two parameters, i.e. σ0 = (σy+σu)/2. Based on the assumptions made above, Eq. 5 can be separated into two parts representing the rate of bending and membrane energy dissipation, respectively: Eɺ int

Eɺ b + Eɺ m n

∑∫

Eɺ b

(6)

M 0 βɺi dli

(7)

i = 1 li

∫

Eɺ m

(a)

σ 0εɺeqq dV dV

ɺ N0udt

(8)

V

M0 =

σ 0 h2 4

N0

(9) (10)

0A

In the expressions above, σ 0 is the flow stress. εɺeq is the equivalent strain rate. uɺ represents p the components of the principal strain. βɺi and lɺi are the rate of curvature change over i th plastic yield line and the length of ith plastic yield hinge line, respectively.

(b) Figure 12. Theoretical models for floor stiffener during shoal grounding accident (a) D < Ls/2, (b) Ls/2 < = D < 3Ls/4.

3.3.1 When D < Ls/2 The scenario is presented in Figure 12(a), it can be derived: Geometry relations: R=

2. Bending due to the formation of the arc deformation of BC and CD. 3. Membrane stretching due to the formation of the arc deformation of BC and CD. According to the plastic upper-bound theorem, the work rate of the external rate of the external loads on the structure can be equaled to the internal plastic energy dissipation rate. F δɺ

Eɺ int

r=

(11)

x2 2 sin(2α − θ )

(12)

set straight line BC as x1, CD as x2, then the governing geometrical equations are: x1 cos α x2 cos α = u ( x1 x2 ) cos α Ls / 2 D  + CD  = L /2 ⇒ BC s x1θ x2 (2α θ ) + = Ls / 2 sinθ sin(2α θ )

(5)

where F is the external load, δɺ is the velocity along the load action direction and Eɺ int denotes the rate of internal energy dissipation. The bottom material is assumed to be perfectly rigid plastic. The conventional metals used for construction exhibit some work hardening, which is taken into account in the present approach by choosing a flow stress σ0. It is larger than the initial yield stress σy but obviously smaller than

x1 2sinθ

(13)

Energy dissipation equations: For bending of the arc, energy dissipated is derived as:

224

2θ

Eb1

∫

0

M 0tφ R Rddφ =

M 0ttx x1θ 2 sinθ

(14)

: In a similar way for arc CD 2 ( 2α − θ )

∫

Eb 2

M 0tφ rdφ =

0

M 0ttx x2 (2α − θ )2 sin(2α − θ )

(15)

Energy dissipation expressions:  , energy dissipated is For bending of the arc BC derived as: 2θ

For bending energy about the plastic hinges at Point B and D: 4M 0tα

Ehinges

∫

Eb1

0

M 0ttx x1θ 2 sinθ

(23)

: In a similar way for arc CD

(16)

 , energy dissipated For stretching of the arc BC is derived as:

M 0tφ R Rddφ =

π 2 ( 2α −θ + ) 2

∫

Eb 2

0

π M 0ttx x2 ⎛ 2α − θ + ⎞ ⎝ 2⎠ M 0tφ rdφ = cos(2α − θ )

(24)

h

Es1

∫

k ⋅ σ 0tdk d N0θ h

(17)

0

h

∫

(

) k ⋅ σ 0t ddk k

For bending energy about the plastic hinges at Point B and D:

N0 (2 ( 2 − )h

(18)

(25)

 , energy dissiFor the stretching of the arc BC pated is derived as:

0

h

Es1

Then: ET

M 0t( 4 + )

Ehinges

: In a similar way for arc CD Es 2

2

∫

k ⋅ σ 0tdk d N0θ h

(26)

0

Eb1 + Eb Es1 + Es Ehinges M ttx x θ 2 M tx (2α − θ )2 = 0 1 + 0 2 + 2 N0α h + 4M 0tα sinθ sin(2α − θ )

: In a similar way for arc CD h

Es 2

(19)

∫ 0

⎛ ⎝

π⎞ π⎞ ⎛ k ⋅ σ 0tdk d N0 2α − θ + h ⎝ 2⎠ 2⎠ (27)

Then:

3.3.2 When Ls/2 < = D < = 3Ls/4 The situation is observed in Figure 12(b), it can be postulated:

ET

r=

x1 2sinθ x2 x2 = π ⎞ 2 cos(2α − θ ) ⎛ 2 sin 2α − θ + ⎝ 2⎠

Ehinges

2

2

(20)

(21)

Set straight line BC as x1, CD as x2, then the governing geometrical equations are: x1 cos α x2 sin i α=u x2 cos α x1 sin i α = Ls / 2 D   BC + CD = Ls / 2 ⇒ π ) x1θ x2 ( 2 = L /2 + s sinθ cos(2α θ )

Es1 + Es

π M 0ttx x2 ⎛ 2α − θ + ⎞ ⎝ M 0ttx x1θ 2⎠ = + sinθ c 2α θ ) cos( π ⎞ ⎛ + N0 2α + h + M 0t( ) ⎝ 2 ⎠

Geometry relations: R=

Eb1 + Eb

(28)

3.3.3 When D > 3Ls/4 When D exceeds 3Ls/4, the additional energy is assumed to be dissipated through the plastic rolling about the hinge line B. The rolling angle is assumed . The basic indentation is usually set 3Ls/4, and Ebasis is the energy dissipation at this indentation condition. Eadditional

(22)

(29)

Then, the total energy consumed is ET

225

M 0 t( D − Dbasis ) ⋅ π

Ebasis + Eadditional

(30)

3.3.4 Enegy reducing effects The above analytical formulae tend to overestimate the dissipated energy for two main reasons. One is that during the formation of the arc BC and CD, the bending moment and the axial forces function together on the cross section of the stiffener along the arc BD. The yielding condition for each section according to Reference 0 is: m + n2 = 1

(31)

where m and n are normalized bending moment and membrane force. The corresponding curve is shown in Figure 13. The generalized forces on each stiffener cross section along the arc BD conform to a certain point, which represents the bending moment and the axial force in the ultimate state. This makes it difficult to calculate analytically. In this paper, the envelope line of the original curve is adopted for simplicity. However, this will overestimate the dissipated energy to a certain extent. The second reason is that the model has been formulated based on an idealized assumption: the stiffener will keep in the in-plane during bending and stretching and not buckle to either side. However, in reality, stiffeners have the uniqueness compared with a common plate: given an identical cross section area, a stiffener cross section tends to process far larger inertia moment than that of a corresponding plate, which is much related to the magnitude of bending moment. (see Fig. 14). During the in-plane bending and stretching, the stiffeners can hardly maintain in-plane deformation, and are more likely buckle to either side of the original plane (with small inertia moment) so as to minimize the dissipated energy. This effect reduces the total energy dissipated to a

Figure 14. Cross sections of (a) stiffener and (b) plate with an identical area.

certain extent compared with the proposed model and affects the predictive accuracy. The energy reducing effect is considered by multiplying a reducing factor λ with empirical methods as follows:

λ=

{

0.833, Model .1 0.714, Model .2

(32)

3.3.5 Solution of equations by Newton’s method The governing equation group (13) is somewhat complicated, which has 3 governing equations with 4 unknown parameters α ,θ , x1,x , x2 . In line with the upper bound theorem, θ should be chosen so as to give the minimum energy dissipation, i.e. the minimization of Eq. 19. A similar way goes for equation group (22) and Eq. 28. Due to complexity, it is difficult to directly obtain the analytical expressions for minimum value via derivation. Therefore, a numerical method is adopted using Newton’s method. The equation group (13) and (22) can be written in the following form: f1 x1, x2 ,α ,θ ) = 0 f2 x1, x2 ,α ,θ ) = 0 f3 x1, x2 ,α ,θ ) = 0

(33)

Discretize the variable θ from 0 to π . For a specific value θ 0 , the Taylor y expansion is used about an estimated value fɶi xɶ1, xɶ 2 ,αɶ ,θ 0 ),(i , 2, ) with first-order terms reserved and higher order terms neglected:

Figure 13. The ultimate condition under the combined effects of the bending moment and the axial forces.

226

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

∂fɶ 1k ∂fɶ 1k ∂x1 ∂x2 ∂fɶ 2k ∂fɶ 2k ∂x1 ∂x2 ∂fɶ 3k ∂fɶ 3k ∂x1 ∂x2

∂fɶ 1k ∂α ∂fɶ k

⎤ ⎥ ⎥ 2 ⎥ ⎥ ∂α ⎥ ∂fɶ 3k ⎥ ⎥ ∂α ⎦

⎡ Δx1k ⎤ ⎡ − fɶ1k ⎤ ⎢ ⎢ ɶk ⎥ k ⎥ ⎢ Δx2k ⎥ = ⎢ − f2k ⎥ ⎢⎣ Δα ⎥⎦ ⎢⎣ − fɶ3 ⎥⎦

(34)

Assume

T

[x [f

x

] T , then: T f ]

4

f f By y solving Eq. q 34, we can get ΔT , and Then, T k = T k + ΔT k This process is repeatedly executed until final convergence is obtained. The value of f ∞ is set as a standard to judge convergence. When f k < tolerance, T k is viewed as a satisfactory answer∞ with good precision. Until now, the right T has been obtained for a specific θ 0 . Then, for different values of θ , the right T can also be derived with the previously introduced method. Finally, the expected T andθ value is the one which minimizes the objective function, and the minimum objective value is obtained. 3.4

Grounding resistance

The total energy consumed is obtained based on the deformation of bottom structures and the plastic mechanism. It is mainly dissipated by the horizontal and vertical resistance. The horizontal and vertical displacement is u and D, respectively. The internal force FH plasticity t and FV plasticity t are directly related. Therefore, the energy dissipation process can be expressed as: Etotal FV

FH , plasticity l i i ⋅u

plasticity plasticity t

FV

plasticity t ⋅

FH , plasticity / tanψ

D

4.1

g(

) ⋅ FH , plasticity

Numerical simulations

The simplified analytical method for the stiffeners on the bottom floor plating is proposed on the basis of the numerical simulation when the ship slides over seabed obstacles with large contact surfaces. A double bottom shuttle tanker is chosen as object with scantlings as listed in Table 1. A side view of the double bottom tanker is presented in Figure 16 and the finite-element model using numerical code LS_DYNA is shown in Figure 17. The seabed topology “shoal” is represented by a flat indenter, which is defined as rigid and slides along the double-bottom structure with a velocity of 5 m/s. An elastic-plastic material with a yielding stress of 235 Mpa is assumed. The model is restrained with all six degrees of freedom fixed. Thus, ship motions are not considered. The Table 1. Scantling of the double bottom of the tanker Units (m).

(35) (36)

Based on balance on forces in the horizontal direction according to Figure 15, FH is derived as: FH

COMPARISON BETWEEN THE PROPOSED SIMPLIFIED METHOD AND THE NUMERICAL SIMULATION

Item

Value

Total length Scantling breadth Scantling height Scanting drought Length of one hold

265 42.5 22.0 16.5 32.0

(37) (38)

The g( µ ,α ) in Eq. 37 has been proposed by Ohtsubo and Wang (1995) in the plate tearing model for bottom raking in dealing with the friction effect. Then, FV is obtained as: FV

g ( ψ ) ⋅ FH , plasticity / tanψ

Figure 16. A side view of the double bottom tanker (Hu & Amdahl 2011).

(39)

Figure 15. Relative motion of bottom plate and indenter for friction factor calculation (Lin & Amdahl 2010).

Figure 17. The nonlinear finite-element model of a double bottom tanker.

227

automatic single-surface contact is used to treat the contacts among structural components occurring during the simulation, with a static friction coefficient of 0.3.The height of the indenter Hi is 2.68 m. The stiffener model is built with the width t = 12 mm the height h = 150 mm, and the length Ls = 2.971 m. 21 analysis cases are defined and presented in Table 2. The slope angle of the indenter is assigned as 20 deg, 30 deg, 45 deg, 60 deg, respectively. For each slope angle, indentation magnitude ranges from 10%–90% of the double bottom height, and at most 9 groups of indentation magnitudes are adopted. 4.2

Figure 18. Comparison of the deformation pattern with results by the simplified method when (a) D = 0.2 H; (b) when D = 0.5 H.

Resuls comparison

Table 3. The comparison consequences of distortion energy.

The grounding process has been analyzed numerically. The results of numerical simulations are compared to the prediction using simplified analytical method proposed in this study. Figure 18 presents a comparison of the deformation patterns with that of the proposed simplified method. The comparisons of distortion energy for all simulation cases are summarized in Table 3, when side stiffeners are neglected for small indentations while set as an entire entity for large indentations. Relative error is applied as Error (Rsisimplified implified lifi d − Rnumerical ) /Rnumerical ⋅

% (40)

Table 2. Case definition for the grounding performance of stiffeners. Model

θ (deg)

D (m)

D/Hi

M 21 M 23 M 24 M 29 M 31 M 32 M 33 M 34 M 35 M 36 M 37 M 38 M 39 M 41 M 43 M 44 M 49 M 61 M 63 M 64 M 69

20 20 20 20 30 30 30 30 30 30 30 30 30 45 45 45 45 60 60 60 60

0.268 0.804 1.072 2.412 0.268 0.536 0.804 1.072 1.34 1.608 1.876 2.144 2.412 0.268 0.804 1.072 2.412 0.268 0.804 1.072 2.412

10% 30% 40% 90% 10% 20% 30% 40% 50% 60% 70% 80% 90% 10% 30% 40% 90% 10% 30% 40% 90%

Model

Numerical method

Simplified method

error

M 21 M 23 M 24 M 29 M 31 M 32 M 33 M 34 M 35 M 36 M 37 M 38 M 39 M 41 M 43 M 44 M 49 M 61 M 63 M 64 M 69

4.3E+06 8.4E+06 1.1E+07 2.2E+07 4.6E+06 6.4E+06 8.3E+06 1.0E+07 1.2E+07 1.6E+07 1.8E+07 2.0E+07 2.0E+07 5.0E+06 7.8E+06 9.8E+06 2.0E+07 4.7E+06 7.4E+06 9.2E+06 1.9E+07

4.4E+06 8.3E+06 1.0E+07 2.1E+07 4.5E+06 6.4E+06 8.1E+06 1.0E+07 1.2E+07 1.7E+07 1.9E+07 1.9E+07 2.0E+07 4.5E+06 8.1E+06 9.9E+06 1.8E+07 4.3E+06 8.0E+06 1.0E+07 1.6E+07

1.2% −1.7% −7.5% −2.2% −3.7% 0.1% −1.8% 0.0% 1.8% 5.5% 2.0% −3.6% 0.3% −10.1% 3.0% 0.5% −11.3% −7.2% 8.0% 9.7% −17.9%

It is indicated that the proposed analytical model has basically grasped the major deformation mode and is capable of representing the true deformation pattern. The horizontal displacement observed is properly considered according to Lin & Amdahl (2008), where the minor influence due to the varying slope angle is well explained. For the distortion energy, energy dissipated in Model 1 and Model 2 is replaced by that in the typical Model 3 for simplicity and fairly good coherence is achieved considering the energy reducing effect, and good predictive accuracy is acquired.

228

5

CONCLUSIONS

In the present study, a theoretical analytical model for stiffeners on the bottom floor plating during the shoal grounding accident is proposed. It is based on a careful study of the progressive deformation patterns of the numerical simulations and a review of the grounding responses of the transverse floor. The sliding process of the stiffeners is composed of three deformation modes: bending about the plastic hinges, bending and stretching due to the formation of the arc shape deformation. A horizontal displacement is properly considered during modeling. By using the plastic analytical method, the expressions of distortion energy and grounding resistance are formulated. A combination of the bending moment and the axial forces act on the cross section of the arc shape deformation and each cross section has a different group of generalized force values in the ultimate state. This makes it difficult to calculate. The envelope line of the original curve is adopted for simplicity. However, this will overestimate the dissipated energy to a certain extent. Besides, the bending and stretching of the arc shape deformation is idealized as in-plane behavior. However, due to the considerable height of the stiffener plate, it can hardly maintain in-plane deformation, and is more likely buckle to either side of the original plane. These effect leads to a sharp decrease of the distortion energy of the proposed model. An energy reducing factor λ is employed with empirical values. The proposed theoretical approach compares favorably with the results from nonlinear FE analysis, which means it is capable of predicting the grounding performance of stiffeners on the floor plating during sliding grounding with good accuracy. The corresponding deformation pattern is proved to basically comprehend the major energy dissipation approaches. This method is established hereby to predict the responses of bottom floor stiffeners when grounding over the shoal type obstructions. More promisingly, it is also able to be applied for the central part of the transverse floors. This new mechanism will contribute substantially to the establishment of efficient tools for fast and reliable assessment of the outcome of the collision and grounding events, and may also be used to design the stiffener arrangement for crashworthiness in the preliminary design stage and incorporated into the decision support systems for crisis handling in emergency situations. ACKNOWLEDGEMENTS

Division, and Foundation of State Key Lab of Ocean Engineering, Shanghai Jiao Tong University, China (Grant No. GKZD010056-12), all of these supports are gratefully acknowledged by the authors. REFERENCES Alsos HS, Amdahl, J. 2007 on the resistance of tanker bottom structures during stranding. Marine Structure. 20 (4):218–237. Amdahl J. 1983. Energy absorption in ship-platform impacts [PhD thesis]. [Trondheim (Norway)]: Department of Marine Technology, Norwegian University of Science and Technology. Chen T., Chen B. Ship Structural Mechanics; Shanghai Jiao Tong University Press (In Chinese). Hu, Z., Amdahl, J. (2011). A study on The Effect of Plate Stiffeners of Double Bottom During Ship Grounding with Large Contact Surface, OMAE2011-49056 June. 2011, Rotterdam, The Netherlands. Klanac, A. et al. (2010) Environmental risk of collision for enclosed seas: Gulf of Finland, Adriatic, and implications for tanker design. The 5th International Conference on Collision and Grounding of Ships. Lin, H., Amdahl J. 2008. Plastic mechanism analysis of the resistance of ship longitudinal girders during grounding and collision. Ship and Offshore Structures. 3(3):159–171. Lin, H., Amdahl J. Rapid assessment of ship grounding over large contact surfaces. In: Proceeding of 5th International Conference on collision and grounding of ships; 2010. June 14th–16th, Espoo, Finland, 2010. Ohtsubo H, Wang G. An upper-bound solution to the problem of plate tearing. Journal of Mar Science and Technology 1995; 1:46–51. Paik J.K., On Qusai-Static Crushing of a Stiffened Squared Tube, Journal of Ship Research, 1996, Vol. 40, No. 3, pp. 258–267. Simonsen, B.C., Ocakli, H. Experiment and theory on deck and girder crushing. Thin-Walled Structures 34(1999) 195–216. Simonsen, B.C., Wierzbicki T. 1997. Plasticity, fracture and friction in steady state plate cutting. Int J Impact Eng. 19(8):667–691. Vaughan H. The tearing strength of mild steel plate. Journal of Ship Research 1980; 24(2):96–100. Wang G, Ohtsubo H, Liu D. 1997. A simple method for predicting the grounding strength of ships. J Ship Res. 41(3):241–247. Wierzbicki, T., Abramowicz, W.: “On the Crushing Mechanics of Thin-Walled Structures”, Mass. Inst. of Tech., Dept. of Ocean Engineering, Rep. No. 82-4, 1982. Xu B., Liu X. 1985. “Ultimate state analysis of the structural plasticity”, China Building Industry Press. (In Chinese). Zhang S. Plate tearing and bottom damage in ship grounding. Marine Structures 2002; 15:101–17.

This work was financially supported by China Offshore Technology Center, ABS Great China

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Fatigue strength

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Stress intensity factor analysis using digital image correlation: A post-processing approach J.H. den Besten, M.L. Kaminski & R.H.M. Huijsmans Delft University of Technology (Section Ship Hydromechanics and Structures), Delft, The Netherlands

ABSTRACT: Fatigue testing series have been developed to identify crack growth in aluminium arcwelded joints adopting a typical fillet weld T-joint geometry. Digital Image Correlation (DIC) is used to obtain the required far field and notch region parameters; results presented here are for 1 set images of 1 welded small scale specimen only. Displacement fields are estimated on a general kinematic basis using commercial DIC software (Istra4D, Dantec Dynamics). A posteriori, as a—mechanical—filtering process, the displacement fields are decomposed onto a selected kinematic basis, i.e. an Airy stress function and Williams’ asymptotic solution. The displacement amplitudes, least squares solutions, present in a one-to-one correspondence the crack growth governing parameters: linear far field stress distribution, SIF and crack tip location. 1

INTRODUCTION

Fatigue, a governing damage mechanism in aluminium high-speed ship structures, thin plate/ shell structures—joints are often critical in that respect—is concerned with crack initiation, crack growth (in plate thickness direction) and crack propagation (in plate length or width direction). For arc-welded joints, notch geometries, it is assumed that crack growth dominates since it is inevitable that flaws already exist and throughthickness cracks are considered to be both a design and repair criterion. To identify the crack growth behaviour, the Stress Intensity Factor (SIF) as governing parameter, crack length (crack tip position), far field stress distribution, plastic zone size and T-stress have to be determined, i.e. measured. 1.1

prevent for related uncertainties because the SIF is both geometry and loading determined. For the first series, (non-welded) fillet-weld T-joint geometries—Figure 1, a notch geometry, no standard crack growth specimen—are milling machined from aluminium 5083H321, 10 [mm] sheet material. In order to obtain the same geometry for the second series, the welded SSS, coarse contours are obtained first using a laser cutter machine. Fillet-welds are applied using MIG welding equipment and, subsequently, the T-joints are milling machined. Finally, for both series, artificial cracks are applied at the weld toe (one side only), typical size is ∼0.1 [mm], using a laser engraving technique. Notes: – To prevent for weld root failure in any case, the coarse contour approach for the welded SSS is

Small scale fatigue testing

Fatigue testing series are developed in a quasi 2D setup to validate a crack growth model and joint SN-curve formulation (den Besten, in prep.). Aim is to investigate initial crack size effects, influence of mode I far field loading combinations (membrane and bending) and notch plasticity consequences. It is recognised that crack growth behaviour in arcwelded joints is a complex combination of properties in the alternating material zones: weld material, Heat Affected Zone (HAZ) material and base material, and in a two-series approach non-welded and welded Small Scale Specimens (SSS) are used. The specimen geometry (including initial crack size) as well as the far field loading is controlled to

Figure 1. Non-welded SSS (left), welded SSS in 3 steps (right).

233

preferred rather than full penetration welding of base and cross-plates. The latter may introduce large deformations and aim is to keep the weld material removal to a minimum. – To obtain an initial crack, an artificial laser engraved crack is preferred rather than (time consuming) pre-cracking. Size and shape control is defined as decisive. For pre-cracking—in contrast to a laser engraved crack—it is uncertain at which notch the crack will appear first and it is likely to obtain a non-constant size and shape over specimen width. Besides, for a laser engraved initial crack, the size can be verified after testing using a stereo microscope as shown in Figure 2. These arguments are considered more important than a possible influence of laser heat input, which eventually may simulate the welding process induced heat input up to some extent.

notch region far field region

image: 2452 x 2056 pixels

F max

R = F min/F max R = 0.1 [-]

2 [s]

1 [s]

snapshot snapshot

F min

1.2

Experimental setup

zero load condition

To identify the crack growth behaviour at a weld toe of T-joints, the SSS are cyclically loaded (Fig. 3); load ratio R = 0.1 [−] and frequency f = 10 [Hz]. To capture the SIF as governing crack growth parameter, its notch and far field information—respectively crack length (crack tip position) and (linear) stress distribution—is required and can be obtained by measuring displacements. A direct optical observation technique, Digital Image Correlation, is adopted and images are taken every 500 cycles at

N = 500

N

initial crack

Figure 3. Specimen notch and far field region (top), cyclic loading scheme (middle), exp. setup (bottom).

Figure 2. Laser engraved initial crack, stereo microscope image after specimen failure.

maximum and minimum load condition Fmax and Fmin using a stereo camera setup. It is recognised that the SIF notch and far field information is available at different scales. However, aim is to capture all information from the same image, meaning that the notch region requirements are governing. For a good spatial resolution, 5 MP cameras are used; the physical pixel size corresponds to 6 [µm].

234

1.3

Goal ux (x,y)

To obtain the required notch and far field region information accurately, the displacement field kinematic basis is a key issue. As a first step in the analysis procedure, displacement fields are estimated on a general kinematic basis using commercial DIC software (Istra4D, Dantec Dynamics) for 1 set images of 1 welded SSS. A posteriori, the displacement fields are decomposed onto a selected kinematic basis, providing the far field stress, crack tip location and SIF. A note in advance: it is known that an integrated approach estimating the displacement field directly using the selected kinematic basis, increases the accuracy (Roux & Hild, 2006), in particular close to the specimen edges and around the crack(tip). However, it is considered to be the next step.

P(xp ,yp ) y Q(xq ,yq ) x

uy (x,y)

P* (x p*,yp* ) y* Q* (xq*,yq*) x* y

x

2

GENERAL DIC PRINCIPLES

Processing 2 digital images, 1 captured before and 1 after object deformation using a single or stereo camera setup, 2D or 3D displacements u(x) and displacement derivatives are estimated by matching (correlating) the image textures, i.e. light intensity (grey level) distributions, respectively f (x) and g(x). It is called: Digital Image Correlation (DIC), a non-destructive, non-contact, optical surface measurement method. Optical flow conservation requires: g ( x ) = f x u( x ))

(1)

Areas with pixels (equivalent: subset, facet, Zone Of Interest (ZOI), correlation window) rather than single pixels are used to perform the correlation process because single pixel has no unique signature. Alternatively, a subspace of suited functions can be adopted, e.g. Finite Element (FE) based formulations (Hild & Roux, 2006). In particular cases, Airy stress functions or Williams’ (generalised) asymptotic solution can be selected as well and operate as a mechanical filter. In order to ensure a unique grey level distribution for each subset, the texture should be random. If the observed texture does not meet this criterion, an artificial random pattern must be applied. Selecting a single subset from a 2D image before deformation (the reference image), the image after deformation is searched for the same grey level distribution (Fig. 4). Assumption: the grey level distribution does not change during deformation, meaning a one-to-one correspondence exists. The displacements ux(x,y) and uy(x,y)

g (x* ,y* )

y'

x'

Figure 4. Subset before and after deformation (top), sub-pixel interpolation principle (bottom).

force the subset centre point P(xp,yp) to shift to P*(xp*,yp*): ⎧⎪ x p* = x p + ux ( x, y ) ⎫⎪ ⎨ * ⎬ ⎪⎩ y p = y p + uy ( x, y ) ⎪⎭

(2)

For a linear deformation approximation—the subset size should be small enough, the displacement gradients are constant in the subset and the location Q(xq,yq) in the same subset becomes Q*(xq*,yq*): ∂ux ∂ux ⎫ ⎧ * ⎪⎪ xq = xq + ux + ∂x dx + ∂y dy ⎪⎪ ⎨ ⎬ ∂u ∂u ⎪ yq* = yq + uy + y dx + y dy ⎪ ∂x ∂y ⎪⎩ ⎪⎭

(3)

The 2 displacement components and 4 displacement gradients (providing strain information) have to be solved for. Note: the subset deforms as a parallelogram. The adopted subset deformation approximation is basically a smoothness

235

assumption, meaning each subset behaves as a low-pass filter. Higher order displacement gradients can be included to allow for more complicated subset deformations. Generally speaking, P* and Q* are located in between pixel positions. Grey levels are not available at these points and interpolation is required. Adopting a bi-linear interpolation scheme (alternatives: bi-cubic or spline interpolation schemes), the grey level g(x*,y*) between 4 surrounding pixels (Fig. 4) is obtained as: g ( x* , y* ) = a0 + a1 ⋅ x ′ + a2 ⋅ y′ + a3 x ′ ⋅ y′

(4)

Note that g(x*,y*) is a continuous grey scale distribution. The constants an are calculated using the locations and grey level values of the surrounding pixels. To represent the correlation for each subset, a least squares or relative correlation coefficient Φ is defined:

or Φ=

∑ ⎡⎣ f

pixel

∑

x, y ) − g ( x* , y* )⎤⎦

2

f x, y ) ⋅ g ( x* , y* )

(5)

TEXTURE QUALITY

To estimate a priori the predominantly texture quality determined DIC displacement measurement performance, global and local quality indicators are introduced to analyse the notch and far field area using the reference image. The texture, shown in Figure 5, is obtained by painting the specimen respectively satin white (using a spray can) and black (using an airbrush system). Figure 6a represents a global texture quality indicator, i.e. the far field area (8-bit depth) grey level distribution histogram: as wide as possible to employ the full grey level range to maximise the grey level characteristic, and without any (white colour) saturation to prevent for reflection. Considering the texture random character, each subset grey level distribution is unique. To evaluate whether a subset contains enough information for a proper analysis, the subset grey level standard deviation, a local quality and sensitivity indicator, is introduced (Hild & Roux, 2008). Adopting the normal distribution, a practical criterion has been found to be 1 [%] of the subset grey level range: for an 8-bit depth grey level distribution the subset grey

pixel

∑

pixel

f x , y )2

∑

far field and notch region

( x* , y* )2

12.0

pixel

10.0

Respectively minimising or maximising the correlation coefficient, an optimisation problem, the design variables, displacements and displacement gradients, can be obtained. It is a key issue that the subset size can strongly influence measurement accuracy (Huang, 2012). For regions with small (uniform) displacement or strain distributions, like the far field region, the random or statistical error (noise and illumination fluctuations) associated with subset quality dominates—generally speaking—the systematic error (e.g. DIC algorithm induced). It means that an increased subset size is preferred to reduce the random error as it contains more information in terms of grey level distribution. However, the accuracy close to the specimen edge decreases because the change of increased mismatch between subset boundary and specimen edge. On the other hand, for the notch region, containing relative large strains because of the (weld) geometry related strain concentration and a crack induced displacement field discontinuity, a smaller subset size is recommended because of the increased systematic error. Consequently, DIC is a trade-off between displacement (and strain) accuracy and spatial resolution. Increasing the subset size increases accuracy, correlation, but decreases spatial resolution. The same applies vice versa.

8.0 6.0 y [mm]

Φ=

3

far field region

4.0 2.0 notch region

0.0 -2.0 -4.0 -10.0

-8.0

-6.0

-4.0

-2.0

0.0 x [mm]

2.0

4.0

6.0

8.0

10.0

Figure 5. Reference image, notch and far field region (top), selected area at centre of far field region (bottom).

236

far field region (8 bit) greyscale histogram 0.28 0.26 0.24 0.22

relative frequency

0.20 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00

0

15

30 45

60 75

90 105 120 135 150 165 180 195 210 225 240 255 greyscale

mean subset grey level standard deviation criterion relative mean subset grey level standard deviation [-]

0.20

subset standard deviation: mean subset standard deviation: minimum

0.16

0.12

0.08

0.04 1 [%] criterion 0.00

0

5

10

15 subset size [pixel]

20

25

30

Figure 6. Grey level histogram (a), subset grey level standard deviation criterion (b).

level range should approximately be in between 11, i.e. N(µ ± 2σ), and 15, i.e. N(µ ± 3σ). Figure 6b shows that already for 5-pixel subsets this criterion is satisfied: σ = 15.9 [pixels]. It is recognised that this criterion is developed for the Q4-DIC procedure, i.e. a bi-linear FE based subset displacement formulation. However, the linear kinematic basis approximation holds for many DIC procedures, regardless the different correlation procedures. 4

Figure 7. Far field displacement ux at Fmax (top), far field displacement uy at Fmax (bottom).

Savitzky-Golay smoothing filter or Airy stress function, has the important advantage that strain, spatial displacement derivatives, quite easily can be obtained. An appropriate Airy stress function ϕ is adopted:

ϕ = C13 ⋅ x 2 y + C02 ⋅ x 2

(6)

By definition, the stress components σij become:

FAR FIELD REGION ANALYSIS

Following some a-priori quantative displacement field analyses results as well, a subset size of 21 [pixel] has been selected and the displacement fields ux and uy are obtained using Istra4D (Fig. 7). Some noise can be identified. Note: the displacement range is in the order of a few microns. Subsets close to the specimen edge are ignored because of the subset boundary and specimen edge mismatch induced inaccuracy. The SIF related far field information, the linear far field stress distribution, requires strain. The expected strain amplitudes are very small and accurate and smooth strain fields can only be obtained using (mechanically) filtered displacement fields—disregarding an increased subset size for the moment. A polynomial filter, e.g. a

237

∂ 2ϕ = C13 y + C02 ∂x 2 2 ∂ ϕ = =0 ∂y 2 ∂ 2ϕ =− = −2 ⋅ C13 ⋅ x ∂x∂y

σ xx = σ yy σ xy

(7)

Introducing the stress-strain relations

ε xx = ε yy = ε xy =

(κ + )

σ xx − ( − κ ) σ yy

(κ + )

8⋅ G σ yy − ( − κ ) σ xx

σ xy 2⋅ G

8⋅ G

(8)

with

κ=

( (

−ν) +ν)

assuming a plane stress condition—DIC is a surface measurement technique—and adopting the small strain-displacement relations

ε xx

∂u y ∂ux 1 ⎛ ∂uy ∂ux ⎞ , ε yy = , ε xy = ⋅ ⎜ + ∂x ∂y 2 ⎝ ∂x ∂y ⎟⎠

(9)

the displacement field in complex coordinates is obtained: u (z) = ux

i ⋅ uy

u (z) = Cx i ⋅C C y (i ⋅ x y) ⋅ Cr + 2 ( + ) x − 2 (3 − κ ) i ⋅ y ⎛C02 ⎞ ⋅⎜ + ⎝ G ⎟⎠ 8 − { + (κ +

)} i

Figure 8. Airy stress function far field displ. ux (top), Airy stress function far field displ. uy (bottom).

x 2 + 2 (κ + ) xy − ( − κ ) i ⋅ y2 ⎛C13 ⎞ ⋅⎜ ⎟ ⎝G⎠ 8 (10)

The constants {Cx, Cy, Cr} represent the rigid body terms. Now the Istra4D displacement fields are decomposed on u(z), Equation 10, and the displacement amplitudes {Cx, Cy, Cr, C02/G, C13/G}, least squares solutions, are obtained assuming the Poisson ratio ν = 0.33 [−]. Figures 8 and 9 show respectively the Airy stress function based displacement fields and errors. The displacement error is 1 order of magnitude smaller compared to the displacement (range) itself; the relative error is smaller than 0.5 [%]. The strain field εxx (Fig. 10) can be obtained using Equation 8. It is confirmed that the strain amplitudes are very small. A small bending component can be identified. At the same time each image is taken, the force Fx is measured. For the considered image, Fx = 5462 [N]. Taking the actual specimen width w = 9.65 [mm] and base plate thickness tb = 10.0 [mm] into account, the membrane stress component can be calculated: σm = 57 [MPa]. Using Equation 7, the shear modulus G can be obtained:

σ m = 2 ⋅ C02 G

(11)

However, G = 25.2 [GPa], quite small compared to an expected value of 26.3 [GPa]; approximately 4 [%] difference. The other way around, σm = 59 [MPa] using the expected shear modulus.

Figure 9. Error far field displacement ux (top), error far field displacement uy (bottom).

For comparison, the Finite Element (FE) method has been used and εxx ≈ 0.084 [%] is obtained in plane stress condition, assuming G = 25.2 [GPa]. Anyway, it seems to demonstrate that the measured force is in agreement with the obtained strain field adopting a plane stress assumption. It turns out that the bending stress component σb, i.e. the linear εxx variation, is smaller than 1 [MPa] as expected: the far field loading was intended to be a pure membrane case. Note that the linear stress distribution is independent of plane stress or plane strain assumptions.

238

Figure 10.

5

Airy stress function strain field εxx.

NOTCH REGION ANALYSIS

The Istra4D displacement fields ux and uy in the notch region (Fig. 11) are obtained using the same subset size as for the far field region: 21 [pixel]. To take the displacement discontinuity around the crack into account, the notch region requires a subset match with the crack tip and edges. However, no crack(tip) information is involved meaning a mismatch and consequently less accurate displacement field predictions. Aim of the notch region analysis is to obtain a reliable SIF estimate and a subspace of suited functions, Williams’ asymptotic displacement field solution in complex polar coordinates (Mathieu, Hild & Roux, 2011), is introduced:

υ ⎧ω ⎫ u ( z ) = ∑ ⎨ n ⋅ Ωn ( z ) + n ⋅ Ψn ( z ) ⎬ G G ⎭ n ⎩

(12)

with Ωn (z)

1− n 2

(− )

n

r2

2 ⋅ 2 ⋅π i( ⎡ i n⋅θ ⋅ ⎢κ e 2 − ⎛ n⎞ e ⎜⎝ ⎟⎠ 2 ⎢ ⎣

n)⋅θ 2

n⎞ − ⎛ n + (−1) + ⎟ e ⎝ 2⎠

i ⋅n⋅θ ⎤ 2 ⎥

⎥ ⎦

Figure 11. Notch displacement ux at Fmax (top), notch displacement uy at Fmax (bottom).

ω2 and υ2 coincide with the rigid body rotation and T-stress component. The amplitudes ω1 and υ1, related to Ω1 and Ψ1, the only singular terms with respect to the stress components, are respectively the mode I and mode II SIF. Note that the SIF is directly obtained; no crack opening displacement or J-integral needs to be calculated (numerically). Assuming a crack tip location, the first super-singular term can be used to calculate the crack tip offset d (along the crack) with respect to the exact location (Roux, Réthoré & Hild, 2009): d = 2⋅

and Ψn (z)

i⋅

1− n 2

(− )

2 ⋅ 2 ⋅π

n ⋅r2

⎡ i n⋅θ ⎛ n⎞ i ( 2 + ⎜⎝ ⎟⎠ e ⋅ ⎢κ e 2 ⎢ ⎣

n)⋅θ 2

n⎞ − ⎛ n + (−1) − ⎟ e ⎝ 2⎠

i ⋅n⋅θ ⎤ 2 ⎥

⎥ ⎦

The amplitudes ω0 and υ0, associated with Ω0 and Ψ0, correspond to the rigid body translations;

ω−1 ω1

(13)

On the other hand, sub-singular terms account for large-scale effects like boundary conditions. For the notch region analysis, nmin = −1 and nmax = 5 have been used. The Istra4D displacement fields are decomposed onto u(z), Equation 12, and the displacement amplitudes {ωi/G,υi/G}, least squares solutions, have been obtained. The decomposed displacement fields ux and uy are shown in Figure 12; corresponding errors in Figure 13. The relative

239

crack tip offset estimate along y-axis using 1st super-singular term 0.10 0.08 0.06 estimated offset [mm]

0.04 0.02 0.00 -0.02 -0.04 -0.06 -0.08 -0.10 -0.10 -0.08 -0.06

-0.04 -0.02 0.00 0.02 0.04 assumed crack tip position [mm]

0.06

0.08

0.10

Figure 14. Crack top offset estimates using equation 13.

6

Figure 12. Williams’ asymptotic solution notch displacement ux (top), Williams’ asymptotic solution notch displacement uy (bottom).

CONCLUSIONS

Provided a sufficient texture quality, DIC seems a reliable technique to measure at the same time far field and notch displacements for a fillet weld T-joint geometry with a rather small crack at the weld toe, the geometry edge, and obtain crack growth parameters with sufficient accuracy. The displacements estimated on a general kinematic basis (Istra4D, Dantec Dynamics), a posteriori decomposed onto respectively an Airy stress function or Williams’ asymptotic solution based displacement fields, mechanical filters, show errors within 0.5 [%]. The displacement amplitudes, least squares solutions, present in a one-to-one correspondence the crack growth parameters, i.e. the linear far field stress components and notch related SIF, showing accuracies within 5 [%]. The crack tip location can accurately be predicted using the first super-singular term from Williams’ asymptotic solution. REFERENCES

Figure 13. Relative error notch displacement ux (top), relative error notch displacement uy (bottom).

displacement error for uy is somewhat larger compared to ux. Crack tip position convergence has been verified (Fig. 14). The SIF KI, including the normal approximate 95 [%] confidence bounds, yields: KI

82.7

4.3 [MPa mm ]

(14)

Besten, J.H. den. 2013 in prep. Fatigue in high-speed ships: a total stress approach and joint SN-curve formulation. PhD thesis. Delft University of Technology. Hild, F. & Roux, S. 2006. Digital Image Correlation: from Displacement Measurement to Identification of Elastic Properties—a Review. Strain 42 (2): 69–80. Huang, J. et al. 2012. Digital Image Correlation with Self-Adaptive Gaussian Windows. Experimental Mechanics 52. Roux, S. & Hild, F. 2006. Stress Intensity Factor Measurements from Digital Image Correlation: Post-Processing and Integrated Approaches. Int. J. of Fracture 140: 141–157. Roux, S., Réthoré, J. & Hild, F. 2009. Digital Image Correlation and Fracture: an Advanced Technique for Estimating Stress Intensity Factors of 2D and 3D Cracks. J. of Physics D: Applied Physics 42 (21): 214004. Mathieu, F., Hild, F. & Roux, S. 2012. Identification of a Crack Propagation Law by Digital Image Correlation. Int. J. of Fatigue 36 (2): 146–154.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Realistic fatigue life prediction of weld toe and weld root failure in load-carrying cruciform joints by crack propagation analysis C. Fischer & W. Fricke Hamburg University of Technology (TUHH), Hamburg, Germany

ABSTRACT: At load-carrying cruciform joints weld fracture due to cyclic loading occurs either at weld toe or at weld root, mainly depending on the ratio between plate and weld throat thickness. For a weld shape optimization a reliable prediction of the critical fatigue crack is of interest and can be assured by limit curves derived from experiments or, alternatively, by different approaches of fatigue assessment. In this paper three common approaches are applied to a selected geometry aiming to benchmark their quality with respect to the fatigue assessment and the prediction of the failure critical notch. Here, disagreement with tests occurs when a two-dimensional crack with a constant depth along the weld seam is assumed. Therefore, different possibilities reaching a more realistic fatigue life assessment are analyzed and it is shown how a semi-elliptical crack front at the weld toe can be included in the crack propagation analysis. Moreover, several solutions are presented in order to estimate a proper aspect ratio together with their influence on the stress intensity factors and life cycles. For the improved crack propagation analysis a solid model is generated by finite elements and different crack sizes at the weld toe are investigated. It turned out that creating the crack front and its direct connection to the weld seam as well as the estimation of stress intensity at the plate surface are the challenging tasks. The three-dimensional crack propagation analyses prove that the total remaining lifetime is more realistic if a semi-elliptical crack front is taken into account. This, finally, leads to identify the critical location of crack initiation correctly and confirms the results of experimental evaluation and stress-based approaches. 1

INTRODUCTION

Many structures need to fulfill design criteria with respect to both various static load cases as well as fatigue strength. In the latter case, dynamic loads affect crack initiation which results in fatigue fracture and local collapse under certain conditions. These conditions are an inappropriate stress ratio and/or a local stress concentration due to a notch. The latter especially occurs at welded joints and cracks can start at the weld toe or at weld root due to an incomplete weld penetration. Figure 1 shows these

Figure 1. Potential failure modes at a load-carrying cruciform joint.

two failure modes for the case of a load-carrying cruciform joint where the crack propagates in the main plate or, respectively, through the fillet weld. Different approaches exist in order to assess the fatigue strength of welded joints (Hobbacher, 2009), also during design progress. They differ with respect to their parameters of assessment and their complexities of application. Moreover, an optimization of the weld shape is often of interest to produce efficiently so that the prediction of the critical fatigue crack is also relevant. In this respect, a possible failure starting from the weld root is to be excluded since crack growth can be only detected by optical methods when the crack penetrates the weld surface. Fricke and Feltz (2010) performed fatigue tests on fillet welds at cover plates and lap joints where fracture starting from both the weld root and the weld toe occurred. In additional, they applied the common fatigue approaches and compared them with the experimental fatigue strengths whereby some disagreements regarding the failure critical notch were observed. This comparative investigation was later extended in terms of the crack propagation approach (Fischer et al., 2011). Here, the two-dimensional, numerical analysis determines

241

the kind of failure for all geometries correctly, but the number of cycles and the fatigue strength assessed were considerably conservative due to the disregard of semi-elliptical crack shapes probably. In case of load-carrying cruciform joints Maddox (1974) derived limit curves for different plate thicknesses which evaluate the occurrence of cracks at weld toe and weld root with equal probability (Fig. 2). Here, the stress increase inside the weld throat is the driving parameter which is described by means of the plate thickness 2b, the weld leg length c and the gap length 2a due to partial weld penetration. The limit curves are based on S-N curves, which reflect a confidence level of 95%, from different experimental tests in literature. Maddox (1974) re-analyzed the data statistically to gain a reference value being valid uniformly for both failure modes. For this purpose, an integral correction factor is included in the S-N curve for weld root fracture. Similar conclusions can be drawn from parametric studies by Nykänen (1999). He applied the crack propagation analysis numerically to various geometric variations of this welded detail and simulated crack growth from both the weld toe and the root. His results are in agreement with the fatigue classes according to Hobbacher (2009) and allow assessing the failure critical notch. This paper deals with the application of three fatigue approaches to a selected geometry of a cruciform joint in order to benchmark the quality of the assessment. In particular, different parameters influencing the crack propagation analysis, i.e. the crack shape and the initial crack size, are investigated. The crack shape is implemented on the one hand by means of approximate solutions which are documented in the common guidelines of the International Institute of Welding IIW (Hobbacher, 2009) or in the British Standard (BS 7910, 2005). On the other hand, Finite Element (FE) calculations using different solid models are

Figure 2. Limit curves for fracture at weld root or weld toe in cruciform joints after Maddox (1974).

performed. It is described in detail how a semielliptical surface crack directly located at the weld toe is modeled and the Stress Intensity Factors (SIF) are determined appropriately. 2

GEOMETRY AND STRESS-BASED APPROACHES

For the considered load-carrying cruciform joint the geometric parameters are selected according to the labeling in Figure 2 as follows: Plate thickness: Weld leg length: Gap length:

2b c 2a

= = =

25.0 mm 17.7 mm 12.5 mm

This geometry is slightly below the limit curve for a plate thickness of 25 mm shown in Figure 2. Hence, a failure originating from the weld root is expected. The fatigue approaches should indicate this by a higher stress value at the notch located there. The horizontal plates are loaded with a nominal stress range of Δσ = 100 MPa and a stress ratio of R = 0 in the calculation. The nominal stress approach assigns classified structural details to different fatigue classes which vary with respect to the allowable stress range. Correction factors consider additional influences, such as yield strength. A cruciform joint which has fillet welds or incomplete root penetration and fails due to cracks at weld toe is frequently associated with fatigue class FAT 63 (No. 413 in Hobbacher, 2009). Weld root failure belongs to fatigue class FAT 36 referring, however, to the nominal stress in the weld throat (No. 414). If the smallest cross section, for instance, is defined by the distance between the end of the gap and the weld surface, i.e. 16.9 mm for this geometry, the ratio between the stresses in the weld throat and in the plate is 0.739. Consequently, the fatigue class rises to FAT 49 when relating this also to the nominal stress in the plate. By comparing the allowable stress ranges for both failure modes, the value for weld root cracks is smaller than for the weld toe and, hence, root fatigue is more critical. This is in agreement with Figure 2. The effective notch stress approach is an alternative which represents the real notch geometry and the local material behavior by means of a fictitious notch radius rref = 1 mm (Hobbacher, 2009). Here, the required maximum linear-elastic stress is based often on the 1st principal stress which can be determined by FE analysis. This investigation uses a two-dimensional quarter model with elements that have a quadratic shape function (PLANE183 in ANSYS®) and assumes plane strain conditions (Fig. 3). Boundary conditions at the left and bottom side represent the symmetry conditions. The end of the gap is built as a keyhole notch without

242

Figure 4. Different crack shapes at the weld toe of a butt joint.

Figure 3. Quarter FE model for the effective notch stress approach and 1st principal stress determined at weld toe and weld root.

reducing the cross section inside the weld throat. Along the notch radii the element size follows the common recommendations of Fricke (2008). When comparing the obtained stress, it turns out that the value at the weld root (σe,r = 443 MPa) is higher than at the weld toe (σe,t = 375 MPa). Hence, the effective notch stress approach leads also to a complying assessment regarding the critical failure mode. This holds true also if small misalignment is considered which would increase primarily the effective notch stress at the weld toe. 3

CRACK PROPAGATION APPROACH

The third fatigue approach applied here is the crack propagation analysis. This method, on contrary, determines the remaining load cycles for a specific initial crack and defined nominal stress range. For welded joints the phase of crack initiation and micro-crack propagation is neglected as a simplification since possibly existing defects as well as a gap due to the partially penetrated weld can be regarded as initial cracks (Radaj et al., 2006). When assessing these defects or the gap, the stage of the stable crack propagation is described properly using the SIF. This value can be estimated by approximate formulae provided in handbook, analytically by the weight function approach and even directly by FE analysis. Finally, the number of cycles is calculated via integration of the Paris power law (Paris & Erdogan, 1963) from an initial crack size ai up to the predefined final length af. The Paris law describes the relation between the crack propagation rate da/dN and the cyclic SIF ΔKI: da = C ( K I )m dN

(1)

In this investigation the material parameters are selected as C = 3 ⋅ 10−13 and m = 3 (Units: N and mm) so that they correspond to Maddox’s (1974) analysis. However, equation (1) requires a discrete integration because the SIF varies non-linearly due to the crack depth and the crack shape. In order to avoid an under- or overestimation of load cycles the mean value of the cyclic SIF ΔK K I is utilized, when the crack grows from size 1 to 2. This value averages the SIFs determined for the both considered crack depth as follows: log(Δ

I)

.

⎡ log( g( ⎣ l (ΔK I ,1 ) log(

⎤

, 2 )⎦

(2)

In the following, the numerical simulation stops when the weld toe crack reaches half the plate thickness or half the leg length of the weld for weld root cracks. The load cycles remaining until unstable fracture can be neglected as they are rather short. The choice of the initial crack, indeed, affects highly the lifetime (Radaj et al., 2006). Besides the crack depth, also the shape has a significant influence so that calibration with test results can be necessary. By taking a butt joint as an example, Figure 4 pictures two different crack shapes at the weld toe where a semi-elliptical shape with the depth a and the width 2c is often observed. After crack coalescence the elliptical shape is relatively shallow due to an escalating width. Here, the occurring crack can be simplified to be two-dimensional like a long surface crack which has a constant depth. 3.1 Crack propagation from weld root The gap caused by the partial penetration is regarded as an initial crack with a length of 2ai = 12.5 mm which propagates symmetrically into the lower and upper weld throat. This assures a conservative assessment and conforms to observations in experiments. Due to its dimension a crack shape with a continuous depth is assumed so that a two-dimensional quarter model is used under plane strain conditions for the FE analysis (Fig. 5). Symmetry conditions as well as elements with

243

Figure 5. crack.

Two-dimensional FE model with weld root

element size of 0.1 mm near the crack path, but it includes only the vertical symmetry condition so that the plate is split by just crack. If no defects are detected, Hobbacher (2009) recommends to assume an initial crack depth of ai = 0.15 mm. The integration of the estimated SIF yields about 250,000 cycles when the crack tip reaches half of the plate thickness. This value is obviously smaller than that for the weld root crack so that this failure is dominant. However, this conclusion does not agree with both the stress-based concepts and the limit curve of Figure 2. The reason is that the propagation rate of the weld toe crack increases rapidly. By choosing a smaller initial crack size of ai = 0.05 mm, the final number of load cycles rises somewhat (N ≈ 276,000) whereas the prediction of the failure mode remain unchanged. The smaller initial crack size is below the validity range of the Paris law, however it is considered here to present its effect on fatigue life. The SIF KI includes, among other things, the effect of the crack shape which is represented by means of the correction function Y in the analytical description: KI

Figure 6.

SIF and load cycles for root failure.

quadratic shape function are applied. The mesh is relatively fine (edge length ca. 0.1 mm) in the region of the expected crack path. The initial mesh is generated in ANSYS®, whereas the crack propagation with a constant growth of Δa = 0.1 mm is simulated automatically by Franc2D (2010). The integration of equation (1) leads to approximately 460,000 cycles at the stop criterion. Moreover, the numerical SIF values and, consequently, the increase of load cycles are satisfactorily in agreement with formulae that are published in the guidelines of the IIW (Hobbacher, 2009) or in the BS 7910 (2006), see Figure 6. The progressive rise of values determined by the FE analysis after a relative crack depth of 0.5 is caused by the quicker reduction of the cross section than in the formulae due to the curved crack path. The influence on lifetime, however, is negligible. 3.2

Here, σ is the nominal stress, a the crack depth and Mk the magnification factor due to a nonlinear peak stress caused by the welded joint. The literature offers approximate formulae of Y for a multitude of different crack configurations, i.e. in case of a one-sided surface crack with a long surface flaw (Tada et al., 1985) or with semi-elliptical shape (Newman & Raju, 1983), both for a plate in tension. The numerically determined SIF can be dispersed in the four parameters of equation (3) so that a different crack shape is caught by exchanging the correction function Y. However, the questions arise which initial shape should be chosen and how would the aspect ratio of the ellipsis a/c develop during crack propagation. Hobbacher (2009) advises a constant aspect ratio of a/c = 0.1 for a conservative assessment. When starting with an initial crack shape of ai = 0.15 mm and ci = 1.5 mm, the analysis achieves finally about 340,000 cycles. The number increases further if the aspect ratio varies. Engesvik & Moan (1983) evaluated a multitude of fracture surfaces and found out a linear relationship for the aspect ratio (Unit: mm):

Crack propagation from weld toe using simplified analyses

c = 6.34a 34a − 0.27

At first, the crack propagation from the weld toe is also performed under plane strain conditions in Franc2D (2010) assuming a one-sided, long surface crack with constant depth. Here, the model employs

(3)

a ⋅Y Y Mk

a / 2c = 0

fo 0.06 mm < a ≤3 mm

(4)

for a>3mm

The deepest semi-elliptical surface crack they observed was approximately 3 mm. Afterwards

244

Figure 8. Figure 7. Load cycles for weld toe cracks based on simplified analyses with pre-scribed aspect ratios of the crack shape.

crack coalescence leads to a jump in width so that the crack face is assumed to run from edge to edge for the sake of simplicity (Engesvik & Moan, 1983). By applying this relation together with an initial semi-circular crack shape of ai = ci = 0.15 mm, the calculation yields about 410,000 load cycles. Figure 7 shows the load cycles against the relative crack depth for the different considered kinds of surface cracks and for the weld root crack. The crack depth is normalized by either the plate thickness 2b (weld toe crack) or the maximum cross section inside the weld seam c + b (weld root crack). A smaller initial flaw affects the lifetime less than a semi-elliptical crack shape. This holds especially true, when an altering aspect ratio is taken into account since the propagation rate decreases obviously at relative small cracks. In the end, however, all additional modifications of the correction function Y do not lead to changing the dominant failure mode. 3.3

Individually modeled semi-elliptical cracks and determination of SIF

The direct numerical calculation of the SIF along a semi-elliptical crack face offers another possibility to determine the crack growth. The Paris power law, here, is integrated for a defined number of cycles at the surface and at the deepest point of the crack separately and results in an individual increase. For that, a solid model assuming a plate width of B = 50 mm is used in ANSYS® (Fig. 8). It represents the geometry by elements with quadratic shape function (SOLID186) in parts of the main plate and the fillet welds in order to keep the computation effort in check. Furthermore, the symmetry in z-direction is taken into account and nodes located on the gap are unsupported. Due to

Solid FE model for the crack at the weld toe.

the long weld leg length, the applied fixing does not affect the local stress field at the weld toe. The SIFs are determined by means of the J-integral which uses six contour integrations at each node of the crack face. Thereby, the contour integral is almost independent of the applied FE mesh when considering linear-elastic material behavior (Anderson, 1991). Furthermore, the local stress field at the crack tip fulfills plane strain conditions (Gross & Seelig, 2007). The relation between the SIF and the J-integral is described under pure mode I loading by the Young’s modulus E and the Poisson’s ratio ν: KI

J ⋅ E /(1 v 2 )

(5)

The process how the crack is built up numerically is divided into three stages: extruding singular elements along the crack front, filling remained crack surface and generating the weld toe by rotation of plate surface elements. These singular elements are characterized by midsize nodes which are shifted ¼ lengths towards the crack face and are sized by 10% of the crack depth. Figure 9 shows the modeled crack in detail. Here, the front part of the crack surface is removed to reveal the crack face and the connection with the weld toe. However, when the notch radius at the weld toe is vanished, an additional singularity arises there. Figure 10 points out the steep increase for a crack with dimensions of a = 0.236 mm and c = 0.537 mm selected arbitrarily. Here, the obtained SIFs are plotted against the expanded normalized crack front length where the deepest face point is arranged at l/lmax = 0 and the surface point is at l/lmax = 1. Additionally, the abscissa is elongated to damp down the slope optically. For this reason, the SIF is estimated at the surface location by means of a quadratic extrapolation which is based on the values of the three previous nodes and an equal spacing.

245

Table 1. Results for crack aspect ratio determined iteratively. SIF, KI [Nmm−3/2]

Figure 9.

#

Cycles [dN]

a [mm]

c [mm]

a/c

Crack root

Surface

0 1 2 3 4 5 6 7 8

9,000 22,000 30,000 44,000 130,000 130,000 69,000 74,000

0.150 0.157 0.185 0.246 0.365 0.918 1.852 2.557 3.557

0.150 0.199 0.334 0.542 0.886 2.368 4.792 6.603 9.151

1.00 0.79 0.56 0.45 0.41 0.39 0.39 0.39 0.39

131 149 178 199 220 266 312 338 375

262 268 282 287 309 365 429 463 511

Σ

508,000

Modeled semi-elliptical crack in detail.

+ 55,500 cycles up to af = 12.5 mm (a/c = 0)

Table 2. Results for crack aspect ratio according to eq. (4). SIF, KI [Nmm−3/2]

Figure 10. Evolution of SIF along the crack front length for an arbitrary selected crack.

3.4

#

Cycles [dN]

a [mm]

c [mm]

a/c

Crack root

Surface

0 1 2 3 4 5 6 7 8

3,726 13,490 24,746 38,701 117,594 116,336 60,201 63,304

0.150 0.157 0.185 0.246 0.365 0.918 1.852 2.557 3.557

0.150 0.363 0.452 0.644 1.021 2.775 5.736 7.970 11.141

1.00 0.433 0.41 0.38 0.36 0.33 0.32 0.32 0.32

131 186 195 208 227 276 325 355 396

262 267 275 288 311 365 425 462 514

Σ

438,000

Crack propagation from weld toe using three-dimensional FE models

The three-dimensional analysis assumes a semi-circular initial crack of ai = ci = 0.15 mm and is divided into eight steps each with a defined number of cycles. The crack growth in both direction—depth and width—is determined by integrating the Paris power law iteratively. Here, up to six loops at each step yield a sufficient accuracy so that, in total, 38 individual calculations of different crack shapes were performed. The load cycles used at each step are listed in Table 1, together with the finally achieved crack dimensions and the corresponding SIFs. The weld toe crack accumulates about 508,000 load cycles when it reaches a depth of a = 3.56 mm. In accordance with Engesvik & Moan’s (1983) observation, the following crack shape runs from edge to edge for which the beginning the two-dimensional analysis makes 55,500 cycles additionally.

+ 55,500 cycles up to af = 12.5 mm (a/c = 0)

Moreover, another calculation which applies the altering aspect ratio according to equation (4) is also performed with the three-dimensional model. The same cracks depths as achieved in Table 1 are used for a final comparison. Advantageously, the high number of iterative calculations is unessential but different SIF values occur at the deepest face point due to the modified crack shape. The integration of the Paris power law which refers to the deepest point results finally in less cycles (ca. 493,000). Table 2 lists each value in detail. Both methods to consider a varying aspect ratio—the iteration and equation (4)—achieve longer lifetimes for the weld toe crack than for the root notch. At last, the failure critical location is identified in agreement with the two stress-based

246

Table 3. Results of different crack propagation analyses. Cycles Weld root crack Symmetric crack propagation; 2D crack (a/c = 0)

460,000

Weld toe crack 2D crack; (a/c = 0); ai = 0.15 mm 2D crack; (a/c = 0); ai = 0.05 mm 2D crack; manipulating Y; a/c acc. to eq. (4); ai = 0.15 mm 3D crack; a/c iterative; ai = 0.15 mm 3D crack; a/c acc. to eq. (4); ai = 0.15 mm

250,000 276,000 410,000 563,500 493,500

Figure 11. Growth of semi-elliptical shapes for differently determined aspect ratios. Table 4. Characteristic fatigue strength Δσc based on nominal stress in plate for the different approaches.

approaches and the limit curve of Figure 2. Moreover, in both cases the initial semi-circular flaws evolve into semi-elliptical shapes (Fig. 11). Here, the application of equation (4) leads to wider shapes with smaller aspect ratios what makes two consequences. On the one hand, the aspect ratio changes a lot in the first step, but affects the total cycles slightly since the increment of crack growth is small. On the other hand, the SIF at the deepest point of the crack front increases whereas the surface value remains unaffected. For this reason, the final lifetime is assessed conservatively. 4

COMPARISON OF RESULTS

The fatigue assessment and the identification of the critical notch with respect to failure (here at the weld root) had been performed by three approaches: the nominal stress, the effective notch stress and the crack propagation approach. The latter determines the remaining cycles for a crack which starts either at the weld root or at the weld toe (Table 3). First, a crack having a constant depth (a/c = 0) is assumed for both positions but this underestimates the cycles for the weld toe flaw and, consequently, the failure mode. By considering a semi-elliptical crack front the lifetime increases significantly, whereby the direct numerical calculations result in an agreement of all approaches and the limit curve as to Figure 2. Furthermore, the aspect ratio according to equation (4) has been applied to both the twodimensional and solid model but deviations as to SIF and cycles are obtained. Here, the Franc2D calculation offers slightly higher values since the mesh is relatively coarse in front of the weld toe. In order to compare the quality of the applied approaches directly, the characteristic fatigue strength Δσc is determined (Table 4). This refers to the permissible nominal stress range in the main

Δσc [MPa]

Reference by tests (Maddox, 1974) Nominal stress approach Effective notch stress approach Crack propagation approach*

Weld toe

Weld root

68

63

63 60

49 51

63–66

61

semi-elliptical crack shape at weld toe.

*)

plate at 2⋅106 cycles and a survival probability of 97.7%. It turns out that all approaches provide a conservative fatigue assessment. Hereby, the notch being critical to fracture is characterized by a lower value Δσc. The stress-based methods are relatively similar at both notches whereas the crack propagation analysis nearly reaches the reference values. It should be noted that the cruciform joint is assumed to be well-aligned in the effective notch stress and crack propagation analysis so that their characteristic fatigue strengths would reduce if misalignments were considered. In the nominal stress approach the characteristic fatigue strength corresponds to the fatigue class suggested by Hobbacher (2009) whereas in the effective notch stress approach the S-N curve (FAT 225) is divided by the obtained stress concentration factor. At last, the lifetime estimated for the weld root and the weld toe by crack propagation are converted into stresses. The reference values are based on test results which are statistically evaluated by Maddox (1974). The S-N curve for weld toe failures is described by: (

247

)3 ⋅ N = 6.3 ⋅ 1011

(6)

In the case of the other fracture mode Maddox (1974) uses a correction factor I which results from the integration of the Paris power law. In this manner, the S-N curve refers also to the nominal stress range Δσ in the main plate: (

)3 ⋅

a +w I

⋅ N = 3.3 ⋅ 1012

(7) REFERENCES

here, ai is the initial crack length (half gap depth) and w the weld leg length. For the selected geometry a correction factor of I = 0.85 is estimated. 5

– By considering semi-elliptical crack shapes at the weld toe, the crack propagation approach also leads to an agreement with respect to the failure critical notch; but the effort for modeling and computations are obviously larger than the one of stress-based approaches

CONCLUSIONS

The failure mode of load-carrying cruciform joints depends mainly on the ratio of plate and throat thickness and can be assessed using limit curves derived by Maddox (1974). In this paper a selected geometry for which failure starting from weld root is expected is investigated by three fatigue assessment approaches. While the nominal stress and effective notch stress approach identify the failure critical notch correctly, the result of the crack propagation analysis is essentially influenced by the chosen initial crack shape. The analyses draw following conclusion with respect to the application of the crack propagation approach using FE method: – For cracks initiated at the weld root a continuously deep crack face and a symmetrical crack growth can be assumed; this matches with common approximate solutions – When the same crack shape is applied at the weld toe, this results in a significant underestimation of cycles until failure so that, consequently, the dominant initial crack is assessed wrongly – The effect of a smaller initial crack size is low compared to varying the crack shape – A semi-elliptical crack front can be included by a manipulation of the geometric function whereby an alterable aspect ratio is advantageous – Alternatively, the influence of a semi-elliptical crack shape can be directly included by using a solid FE model where the crack propagation analysis yields a satisfying result – The direct simulation can evaluate the aspect ratio either iteratively, or using relations given in literature (e.g. Engesvik & Moan, 1983) are used – The iterative solution requires significantly more effort whereas it ends up in a more precise assessment

Anderson, T.L. 1991. Fracture Mechanics: Fundamentals and Applications. 1st Edition. Boca Raton, Florida: CRC Press. BS 7910. 2005. Guide to methods for assessing the acceptability of flaws in metallic structures. British Standard Institution. Engesvik, K.M. & Moan, T. 1983. Probabilistic analysis of the uncertainties of the fatigue capacity of welded joints. Eng. Fract. Mech., Vol. 18, No. 4, 743–762. Fischer, C., Feltz, O., Fricke, W. & Lazzarin, P. 2011. Application of the notch stress intensity and crack propagation approaches to weld toe and root fatigue. Welding in the World, Vol. 55, No. 7/8, R30–R39. Franc2D. 2010.: http://www.cfg.cornell.edu/software /franc2dl.htm. Fricke, W. 2008. Guideline for the fatigue assessment by notch stress analysis for welded structures. IIW-Doc. XIII-2240r1-08/XV-1289r1-08. Int. Institute of Welding. Fricke, W. & Feltz, O. 2010. Fatigue tests and numerical analyses of partial-load and full load carrying fillet welds at cover plates and lap joints. Welding in the World, Vol. 54, No. 7/8, R225–R233. Gross, D., & Seelig, T. 2007. Bruchmechanik: mit einer Einführung in die Mikromechanik. 4th Edition. Berlin: Springer. Hobbacher, A. 2099. Recommendations for fatigue design of welded joints and components. IIW-Doc. 1823-07. New York: Welding Research Council Bulletin 520. Maddox, S.J. 1974. Assessing the significance of flaws in welds subject to fatigue. Welding Journal Research Supplement, 401s–409s. Newman, J.C. & Raju, I.S. 1983. Stress-intensity factor equations for cracks in three-dimensional finite bodies. ASTM STP 791, I-238–I-265. Nykänen, T. 1999. Geometric dependency of fatigue strength in a transverse load-carrying cruciform joint with partially-penetrating K-welds. Welding in the World, Vo. 43, No. 5, 47–53. Paris, P.C. & Erdogan, F. 1963. A critical analysis of crack propagation law. Trans ASME, J. Basic Eng., 528–539. Radaj, D., Sonsino, C.M. & Fricke, W. 2006. Fatigue assessment of welded joints by local approaches. 2nd Edition. Cambridge: Woodhead Publishing Ltd. Tada, H., Paris, P.C. & Irwin, G.R. 1985. The stress analysis handbook. St. Louis: Paris Prod. Inc.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Fatigue strength of laser-welded thin plate ship structures based on nominal and structural hot-spot stress approach W. Fricke Hamburg University of Technology, Hamburg, Germany

H. Remes Aalto University, Espoo, Finland

O. Feltz Hamburg University of Technology, Hamburg, Germany

I. Lillemäe Aalto University, Espoo, Finland

D. Tchuindjang Hamburg University of Technology, Hamburg, Germany

T. Reinert Meyer Werft GmbH, Papenburg, Germany

A. Nevierov Fincantieri—Cantieri Navali Italiani SPA, Trieste, Italy

W. Sichermann ThyssenKrupp Marine Systems, Hamburg, Germany

M. Brinkmann Flensburger Schiffbau-Gesellschaft mbH & Co. KG, Flensburg, Germany

T. Kontkanen STX Europe, Turku, Finland

B. Bohlmann University of Applied Sciences, Kiel, Germany

L. Molter Center of Maritime Technologies e.V., Hamburg, Germany

ABSTRACT: To improve the energy efficiency, the demand for new light weight solutions has been increased significantly in the last decades. The weight reduction of the current ship structures is possible using thinner plates, i.e. plate thickness between 3 and 4 mm. However, at present this is not possible due to the 5 mm minimum plate thickness requirements given by classification societies. The present paper investigates the fatigue strength of thin plated ship structures. In the European research project BESST—“Breakthrough in European Ship and Ship Building Technologies”, the extensive fatigue test programme was carried out for butt and fillet welded specimens, which were manufactured by the arc, laser and laser-hybrid welding methods. The test programme covered also the different production quality, and thus, a large variation of misalignments was included. Fatigue test results were analysed using the nominal as well as the structural stress approach, where the actual geometry of the specimens was taken into account. The results show that the present design S-N curve with slope value of three is

249

applicable to thin plates, but it is slightly non-conservative. The fatigue test results for thin plates show better agreement with the slope value of five. For thin plates and slender ship structures, the secondary bending stress due to angular misalignment plays an important part and changes as the function of the applied tension load. Therefore, it is important to consider the plate straightening effect in structural stress analysis. 1

INTRODUCTION

2

In order to increase the energy efficiency, the demand for new light weight solutions has been increased significantly in the last decades. The weight reduction of the current ship structures is possible using thinner plates i.e. down to plate thicknesses of 4 or even 3 mm. However, at present the use of thin plates is not allowed due to the 5 mm minimum plate thickness requirements given by classification societies. In thin plates, one of the main challenges is distortions caused by fabrication process. Especially, axial and angular misalignments are commonly significantly larger for thin than thick plates (Eggert et al., 2012; Lillemäe et al., 2012). However, if low heat input welding such as laser welding is used instead of conventional arc welding, the amount of the misalignments can be reduced (Roland et al., 2004). The misalignments are harmful since they cause secondary bending stress on the weld notch reducing the fatigue strength of the joint. The fatigue strength of thin welded plates has been studied in several papers (Radaj et al., 2009, Eibl et al., 2003; Sonsino et al., 2010). These studies focused mainly on the automotive industry application, the production of which differs from ship production. In the production of large ship structures, the control of the distortion is significantly more difficult. Therefore, further investigation is needed to develop a solid design basis for the fatigue strength assessment of thin ship structures. This paper investigates the fatigue strength of thin ship structures welded by the arc, laser and laser-hybrid welding methods. The special emphasis of the study is paid on the influence of the misalignments on the fatigue strength. Therefore, the different production quality is simulated having a large variation of misalignments. The extensive fatigue test programme was carried out for butt and fillet welded specimens within the European research project BESST. Fatigue test results were analysed using the nominal as well as the structural hot-spot stress approach, where the actual geometry of the specimens was taken into account. This investigation focuses on the fatigue strength of the welded joint, and thus, the effects of surrounding structures are neglected.

2.1

FATIGUE EXPERIMENTS Fatigue test programme

The summary of the fatigue test programme is presented in Table 1. The investigations were focused on the plate thickness t between 3 mm and 5 mm. Different laser welding arrangements and edge preparation suitable for the shipyard production were covered. The test programme included also some conventional arc welded, i.e. SAW, FCAW, MAG welded joints and 6–8 mm thick joints for comparison. In total, 14 different butt joint and 14 different T-joint series were tested. For each test series ten specimens were fabricated out of welded plates produced at four shipyards. The base plates had the yield strength ReH between 293 MPa and 466 MPa. The tensile strength Rm of the base plates was between 439 MPa and 572 MPa, and the failure strain A was 31% on average. The produced test specimens were tested under transversal or longitudinal loading. The load ratio of R = 0 was applied. In some tests, where test arrangements (4-point-bending) did not allow R = 0, other R-values were chosen. The test programme included also some series, which were tested with the load ratio of R = 0.25 and R = 0.5 for comparison. 2.2

Geometry measurements

Geometry measurements were carried out with the optical measuring technique using high resolution cameras or laser scanner. The geometry points on the top of the specimen surfaces were recorded and numerical 3D model of the specimen were created. From the 3D model the section perpendicular to the weld seam was generated for 2D structural stress analysis (Lillemäe et al., 2012). From the geometry measurements data the axial and angular misalignment of the joints are defined as shown in Figure 1. 2.3

Fatigue tests

Constant amplitude fatigue tests were carried out using hydraulic and resonance test machines. Special clamping jaws were applied to avoid additional bending stresses during clamping. The load frequency was 10 Hz for the hydraulic test machines

250

Table 1.

Fatigue test programme.

Series

Joint description

Loading

Properties of base plate/stiffener

Symbol

Welding method

Type

t [mm]* ReH [MPa]

Rm [MPa] A [%]

B.3.LA.1 B.3.HY.1 B.3.CV.1

Laser welded butt joint Laser-hybrid welded butt joint Conventionally arc welded butt joint, block joint Conventionally arc welded butt joint Conventionally arc welded butt joint, faired after welding Laser welded butt joint, thickness step Laser-hybrid welded butt joint, thickness step Conventionally arc welded butt joint, thickness step Laser welded butt joint Laser welded butt joint Laser-hybrid welded butt joint Conventionally arc welded butt joint Conventionally arc welded butt joint, faired after welding Laser-hybrid welded butt joint, plasma cut edges Laser welded T-joint Laser-hybrid welded T-joint Laser-hybrid welded T-joint, one side weld Conventionally arc welded T-joint Conventionally arc welded T-joint, one side weld Conventionally arc welded T-joint Laser welded T-joint Laser-hybrid welded T-joint Laser-hybrid welded T-joint Laser-hybrid welded T-joint Conventionally arc welded T-joint Conventionally arc welded T-joint Laser welded T-joint Laser-hybrid welded T-joint

Transversal, R = 0 Transversal, R = 0 Transversal, R = 0

3 3 3

414 399 466

567 531 564

24,7 26 31

Transversal, R = 0 Transversal, R = 0.5

3 3

466 408

564 490

31 41

Transversal, R = 0

3/5

466/432

564/521

31/30

Transversal, R = 0

3/5

466/432

564/521

31/30

Transversal, R = 0

3/5

466/432

564/521

31/30

Transversal, R = 0 Transversal, R = 0.5 Transversal, R = 0 Transversal, R = 0 Transversal, R = 0

4 4 4 4 4

443 443 414 356 356

547 547 537 474 474

31 31 24 34 34

Transversal, R = 0

6

343

472

34

Transversal, R = 0 Transversal, R = 0 Transversal, R = 0.25

3/5 3/5 3/5

399/400 348/358 348/358

531/533 439/492 439/492

26/33 39/29 39/29

Transversal, R = 0 Transversal, R = 0.25

3/5 3/5

466/432 348/358

564/521 439/492

31/30 39/29

Longitudinal, R = 0.1 Transversal, R = 0 Transversal, R = 0 Transversal, R = 0.25 Transversal, R = 0.5 Transversal, R = 0 Longitudinal, R = 0.15 Transversal, R = 0 Transversal, R = 0

3/5 8/5 8/5 8/5 8/5 8/5 8/5 8/7 8/7

399/400 384/400 293/377 293/377 293/377 333 384/400 384/448 293/336

531/533 546/533 445/514 445/514 445/514 450 546/533 546/572 445/470

26/33 28/33 30/23 30/23 30/23 40 28/33 28/29 30/32

B.3.CV.2 B.3.CV.3 B.35.LA B.35.HY B.35.CV B.4.LA.1 B.4.LA.2 B.4.HY.1 B.4.CV.1 B.4.CV.2 B.6.HY.1 F.35.LA.1 F.35.HY.1 F.35.HY.2 F.35.CV.1 F.35.CV.2 F.35.CV.3 F.85.LA.1 F.85.HY.1 F.85.HY.2 F.85.HY.3 F.85.CV.1 F.85.CV.2 F.87.LA.2 F.87.HY.1

*) Two values characterize either a thickness step or give the thicknesses of the base plate and stiffener of T-joints.

force divided by the cross sectional area in the middle of the specimen. 3 Figure 1.

Axial e and angular α misalignments.

and 30 Hz for the resonance test machines. During the tests applied force and number of load cycles to failure were recorded. The number of cycles to failure was deined at the occurrence of the inal fracture. The nominal stress was deined as the

STRUCTURAL STRESS ANALYSES

The structural hot-spot stresses for the measured joint geometries were calculated using the finite element method. The analysis was performed according to IIW guidelines (Hobbacher, 2009). A 2D plane stress element model was created for each of the test specimens. ANSYS 13.0 and Abaqus 6.11–2 software with geometrical non-linearity was used, where

251

the load is applied step-by-step and the geometry is intermediately updated according to the straightening of the specimen. A linear-elastic material behaviour with Young’s modulus of 206, 800 MPa and Poisson’s coeficient 0.3 was assumed. The FE simulations were validated by the measured strains at the specimen surface (Lillemäe et al., 2012). 4 4.1

RESULTS Joint geometry

The measured axial and angular misalignments are summarised in Figures 2–4. The minimum, average and maximum values are presented for each test series. The results are compared with the limit values of the quality classes B, C and D according to ISO 5817. The limit values of angular misalignment refer to the standard version until 2006. The axial misalignment of the butt welded joints, shown in Figure 2, varies between different joints. For all conventionally arc welded joints, the maximum values are above of e/t = 0.15 limit. For the hybrid and laser welded joints almost all of the axial misalignments are inside of e/t = 0.1 limit. In this comparison, the thickness step induced axial misalignment was neglected. The angular misalignment given in Figure 3 and 4 reveals similar behaviour. For conventionally welded joints, the angular misalignments were significantly larger than those of hybrid and laser welded joints. For the conventional welded joints, the maximum absolute value is 6.7 degrees. For hybrid and laser welded joints the absolute maximum values are 2.2 and 1.8 degrees, respectively. In general, the average value of the angular misalignments is higher for T-joints than for butt welded joints. 4.2

Fatigue strength

The fatigue test results can only be shown in a summary here. As expected for relatively thin-plated

Figure 2.

Axial misalignments of butt welded joints.

Figure 3.

Angular misalignments of butt welded joints.

Figure 4. T-joints.

Angular misalignments of fillet welded

joints, the axial and angular misalignments affect the fatigue lives considerably. Therefore, the test specimens were associated to three classes according to ISO 5817 mentioned above: • Misalignment low: • Misalignment medium: • Misalignment high:

if e/t ≤ 0.1 and α ≤ 1°. if 0.1 < e/t ≤ 0.15 and/ or 1° < α ≤ 2°. if e/t > 0.15 or α > 2°.

The results based on the nominal stress acting in the test specimens are shown for the butt joints in Figures 5–7 separately for those welded conventionally (CV), with laser-only (LA) and with laserhybrid process (HY). It can be seen at a first glance that the majority of CV belongs to the class with highest misalignment, whereas the majority of LA to the lowest and HY to the intermediate. An influence of the misalignment class is particularly large for CV, but also visible for LA. In the S-N diagrams the specimens with thickness step which induce an additional misalignment and which usually belongs to a lower fatigue class according to the nominal stress approach are indicated. Also the design S-N curve FAT80 (80 MPa at

252

Figure 5. Fatigue test results for conventionally welded butt-joints based on nominal stress.

Figure 8. Fatigue test results for conventionally welded T-joints based on nominal stress.

Figure 6. Fatigue test results for laser-only welded buttjoints based on nominal stress.

Figure 9. Fatigue test results for laser-only welded T-joints based on nominal stress.

Figure 7. Fatigue test results for laser-hybrid welded butt-joints based on nominal stress.

Figure 10. Fatigue test results for laser-hybrid welded T-joints based on nominal stress.

2 ⋅ 106 cycles) which is usually applied to butt joints not meeting higher requirements regarding e.g. weld reinforcements is included. All specimens with small misalignments and without thickness step are above FAT80, whereas those with higher misalignment are partly far below the FAT80 line. The fatigue test results for the T-joints based on nominal stress are plotted in Figures 8–10, again separately for the three welding processes. Here, only angular misalignment is possible. An effect of the welding process can be seen in all cases. Almost all specimens with low angular misalignment meet the design S-N curve FAT80 commonly used for transverse stiffeners, whereas those with higher misalignment are below this curve, particularly for CV

and HY. Another observation is that the slope of the S-N results is obviously flatter than that of the design S-N curve (m = 3), which was previously observed particularly for thin-walled components resulting in a proposal of m = 5 (Sonsino et al., 2010). In addition to the nominal stress approach, also the structural hot-spot stress approach was applied which considers the effect of secondary bending stress due to misalignment in the structural stress. The results based on the structural hot-spot stress approach are plotted in Figure 11 for all butt-joints and in Figure 12 for all T-joints. Almost all test results are above the design S-N curve FAT100 proposed by Hobbacher (2009) for the structural hot-spot stress approach. Pronounced

253

joints resulting in fatigue strength below the fatigue class FAT80. • Based on the structural hot-spot stress, which includes the effects of misalignment, almost all results are above the relevant fatigue class FAT100. Here, the differences between the different welding processes are small.

Figure 11. Fatigue test results for all butt-joints based on structural hot-spot stress.

A more refined evaluation of the fatigue test results and further influence factors on fatigue strength is beyond the scope of this paper. Such investigations have been performed being published in separate papers, such as (Lillemäe et al., 2012). Conclusions regarding the implementation of the results into classification rules are drawn by von Selle et al. (2013). ACKNOWLEDGEMENTS The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007–2013) under grant agreement n° 233980. All the inancial support is gratefully appreciated.

Figure 12. Fatigue test results for all T-joints based on structural hot-spot stress.

differences between the different welding processes and misalignment classes are not visible except for T-joints, where laser welding seems to show superior results. 5

DISCUSSION AND CONCLUSIONS

Extensive fatigue tests were performed for buttjoints and T-joints welded conventionally, with laser-only and laser-hybrid process. The welding was performed by different European shipyards using thicknesses between 3 mm and 8 mm for the base plates. Axial and angular misalignment could not be avoided with these plate thicknesses and these were recorded for each specimen. From the fatigue test results, which were evaluated according to the nominal and the structural hot-spot stress approaches, the following conclusions can be drawn: • The misalignments considerably affect the fatigue strength if based on the nominal stress approach. • The fatigue strength based on nominal stress is above the fatigue class FAT80 used in codes as long as the misalignments are relatively small, which is the case for almost all laser welded joints. • Relatively large misalignments have been observed particularly for conventionally welded

REFERENCES Eggert, L., Fricke, W. & Paetzhold, H. Fatigue strength of thin-plated block joints with typical shipbuilding imperfections. Weld World 2012; 56(11–12). Eibl, M., Sonsino, C,M,, Kaufmann, H. & Zhang, G. Fatigue assessment of laser welded thin sheet aluminium. Int J Fatigue 2003; 25: 719–31. Fricke, W., Lilienfeld-Toal, A. & Paetzold, H. 2012. Fatigue strength investigations of welded details of stiffened plate structures in steel ships, International Journal of Fatigue, Vol. 34: 17–26. Hobbacher, A. 2009: Recommendations for Fatigue Design of Welded Joints and Components, Iiw doc.1823-07, Welding Research Council Bulletin 520, New York. Lillemäe, I., Lammi, H., Molter, L. & Remes, H. 2012. Fatigue strength of welded butt joints in thin and slender specimens, International Journal of Fatigue, Vol. 44:98–106. Radaj, D., Sonsino, C.M. & Fricke, W. Recent developments in local concepts of fatigue assessment of welded joints. Int J Fatigue 2009; 31:2–11. Roland, F., Manzon, L., Kujala, P., Brede, M. & Weitzenböck, J. 2004. Advanced joining techniques in European shipbuilding. Journal of Ship Production. Vol. 20:3. p. 200–210. Sonsino, C.M., Bruder, T. & Baumgartner, J. 2010, S-N lines for welded thin joints—suggested slopes and FAT values for applying the notch stress concept with various reference radii. Welding in the World 54 (2010), No. 11/12, R375–R392. von Selle, H., Peschmann, J. & Eylmann, S. 2013: Implementation of fatigue properties of laser welds into classification rules. Submitted for publication in Proc. of 4th Intl. Conf. on Marine Structures MARSTRUCT’2013, Espoo.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Influence of surface integrity on the fatigue strength of high strength steel in balcony openings of cruise ship structures E. Korhonen, H. Remes & J. Romanoff Department of Applied Mechanics, Aalto University, Espoo, Finland

A. Niemelä, P. Hiltunen & T. Kontkanen STX Finland, Turku, Finland

ABSTRACT: This paper investigates experimentally the influence of surface integrity to the fatigue strength of high strength steel used in balcony openings of cruise ship structures. The fatigue test specimens, having a dog-bone shape and yield strength of 355 MPa, 460 MPa or 690 MPa, were cut by plasma. After the cutting, the specimens were treated by grinding or by grinding followed by sandblasting, i.e. using post-cutting treatments suitable for shipyard conditions. The resulting surface roughness and hardness profile were measured. The fatigue tests with load ratio R = 0.1 were carried out until failure of the specimens. The investigation shows that post-cutting treatments suitable for shipyard conditions can considerably increase the fatigue strength of high strength steel used in balcony openings. 1

INTRODUCTION

During recent years the general arrangement of cruise ships has undergone dramatic changes. One of these changes is the size of openings in load carrying bulkheads. The balcony openings have become larger making the corners of these openings strength critical; see Figure 1. While static strength can be increased using High Strength Steel (HSS), fatigue strength in shipbuilding applications does not increase with the same rate as the yield or ultimate strength of the material. This is due to the fact that the cutting of steel plates introduces surface defects. The surface defects can be considered as initial cracks for fatigue crack growth. If the defects have certain size the crack growth dominates the fatigue process over crack initiation, which leads to the loss of benefits of HSS; see Figure 2. This leads to two problems; added weight and vertical center of gravity increases, both caused by the increased material thickness. The influence of surface treatments and different cutting methods on steels fatigue life has been discussed for example by Sasahara (2005), Sperle (2007) and Gao et al. (2007). As shown by Murakami and Endo (1994) the surface profile and hardness need to be considered when investigating fatigue of specimens with different treatments. The aim of this study is to examine the influence of surface integrity on the fatigue strength of high strength steel when production processes suitable for the shipyard are considered. Sandblasting,

Figure 1.

255

Balcony opening of a cruise ship.

Figure 2. Kittagawa-diagram showing the relation between surface defect size and fatigue strength at 2 × 106 cycles (Sperle, 2007).

required before painting, and grinding are selected as post processing methods after plasma cutting. Steels having yield strength of 460 MPa and 690 MPa are considered and compared with the reference material with yield strength of 355 MPa. The reference material represent the high strength steel commonly used in the shipbuilding. Specimens are fatigue tested until complete fracture.

Figure 3. Dog-bone-specimen simulating the 1stprincipal stress at stress concentration of balcony opening. Left: opening under pure shear; right opening under pure tension; bottom: the shape of the specimen. Table 1. Test program and specimens. All specimens are plasma cut first. Surface treatment

2 2.1

TESTS Test specimens

The balcony openings of passenger ships fail typically under the maximum principal stress (Bäckström, 2009). This stress can be due to global hull girder shear force or bending moment. As shown in Figure 3 the fatigue critical location changes in the window opening depending on the ratio of shear force and bending moment-induced stresses. In any case the failure occurs in the direction of the principal stress. Therefore, in testing the problem has been simplified and dog-bone specimens are considered. The radius of the specimens has been selected to be equal with that of typical balcony opening (R = 500 mm). In the case of the axial loaded test specimen, this radius means the stress concentration factor of Kt = 1.02. Thus, the influence of stress-concentration was neglected and the results analyses are based on the nominal stress, σ = F/A, of the test cross-section. The nominal width of the specimen at narrowest cross-section was 30 mm while the thickness was 15 mm or 17 mm. The steel materials were produced by Ruukki, S460ML (EN10025-4:2004) and S690QL (EN10025-6:2009) with nominal yield strength of 460 MPa and 690 MPa respectively. In all specimens the corner between cut edges and plate surfaces were grinded to omit the stress concentration both due to cutting and sharp corner. In first set of specimens this treatment was all that

Yield strength (MPa)

None (#)

Grinding (#)

Grinding & Sandblast (#)

355 460 690

0 5 10

5 5 10

0 5 10

was done, while in the second set also the cut edges were grinded to smooth the surface. In the third set also sandblasting was performed to decrease the sharpness of the manufacturing induced flaws. As a reference a set of steel with 355 MPa yield strength was used (NVA-36; DNV 2005). This led to seven sets of specimens and 50 specimens in total; see Table 1. 2.2

Surface geometry measurements

The size of surface defects was measured using surface roughness measurements according to SFS-EN ISO 4288 (1996). The measurements were carried out for both specimen edges along three different lines, i.e. at the edges and center of the cut edge; see Figure 4A. The tests were carried out using replica technique and Surface Roughness Tester TR200 and Taylor Hobson Surtronic 3+ -devices. Both the arithmetic average, Ra, and average of the five largest peak to peak average, Rz, were defined. Due to the fact that fatigue crack

256

for the edge of the specimen cross-section which was affected by the various treatments and at middle which was unaffected by these. From the Vickers hardness the yield strength of the material was estimated by using relation σy = HV1*k, where k is a hardness conversion factor varying between 2 and 3 (Pavlina and Van Tyne, 2008). In the present study, the mean value 2.5 was applied. 2.4

Figure 4. Surface geometry measurements. A) surface roughness Ra and Rz, and B) definition of the notch radius ρ.

Fatigue tests

The fatigue tests were carried out using cyclic tensile tests with load ratio R = 0.1; see Figure 5. The specimens were fixed to hinged clamping jaws which allowed rotation of the specimen within its’ own plane. The frequency of the test system was 1.8–3.5 Hz depending on the load level defined for the test. Run-out-point was defined at two million cycles. The specimens were instrumented with strain gauges both sides of the specimens to verify that secondary bending was not present during the experiment. The test was load-controlled and stopped at complete failure of the specimen. In addition to fatigue tests, static tensile tests were carried out for one specimen from each series. 3

RESULTS

3.1 Surface roughness and hardness Figure 6 presents the measured typical surface profiles for the various treatments. Figure 7 presents the extracted surface roughness, pictures of the sample surfaces and determined defect radius ρ of the specimens. Figure 8 presents examples of the macro-sections used in hardness measurements, while Figures 9–11 present the hardness

Figure 5.

Fatigue test set up.

nucleates at the defect that has the maximum depth and fatigue process is affected by the radius of the notch, also notch radius was measured by fitting circle to the measured surface height profile; see Figure 4B. 2.3

Hardness measurements

The Vickers hardness measurements were carried out for each test series according to SFS-EN ISO-6507 (2005). The specimens were manufactured by sawing from the dog-bone specimens. The hardness measurements were carried out for both

Figure 6. Examples of surface profile and defect radius, ρ, for different treatments of the specimens.

257

Figure 7. Surface roughness and defect radius, ρ, for different treatments of the specimens.

Figure 10. Hardness of plasma cut and grinded specimens having yield strength 355 MPa, 460 MPa or 690 MPa. Solid lines mark the average of the base material.

Figure 8. Definition of hardness measurement distance from macro sections of specimens having yield strength 460 MPa or 690 MPa.

Figure 11. Hardness of plasma cut, grinded and sandblasted specimens having yield strength 460 MPa or 690 MPa. Solid lines mark the average of the base material.

Figure 9. Hardness of plasma cut specimens having yield strength 460 MPa or 690 MPa. Solid lines mark the average of the base material.

distributions of the base material and the processed materials. Table 2 presents the measured and hardness-based estimations of the yield strength for the base materials. Figures 8–11 show that the plasma cutting has affected the material close to the edge of the specimen. The Heat-Affected-Zone (HAZ) is 0.5 mm–1.0 mm and is higher in specimens, which

have low yield strength. Grinding and sandblasting do not affect this distance considerably. The hardness of the plasma-cut edge of the specimen is considerably higher than that further inside the specimen. Regardless of the base material hardness the maximum value is between 460 HV1 and 510 HV1 being highest for the specimens, which were plasma cut, grinded and sandblasted; the effect of sandblasting is about 50 HV1. When compared to the yield strength of the base material given in Table 2, this means that the surface of the specimens has considerably higher yield strength than that inside the specimen. 3.2

Fatigue strength

The fatigue test results have presented in Figure 12 for the grinded specimens with different yield strengths; 50% probability of survival has been considered due to the fact that some of the test

258

Table 2. Yield strength comparison of the specimens. Measured values taken from steel producer certificates. Surface treatment

Nominal σy [MPa]

Hardness HV1

Estimated σy [MPa]

Plasma Grinded Sandblasted Plasma Grinded Sandblasted

460 460 460 460 690 690

199 210 205 299 300 272

498 526 513 747 750 680

Figure 12. Fatigue strength for specimens with different yield strength. The specimens have been plasma cut and grinded.

series had only few specimens tested. In Figure 13, the influence of different treatments on fatigue strength has been compared for series having yield strength 690 MPa. There the 97.7% probability of survival is considered. In both figures the comparison to DNV (2008) design curve is also presented; corrosion protection is to be used, N < 107, m = 4, FAT160. The applied stress range

508

744–843

and fatigue life for each test specimens are given Table 3. Figure 12 shows that the fatigue strength and the slope of SN-curve increase considerably with the increasing yield strength in grinded specimens. The design curve of DNV (2008) is well exceeded when 50% probability of survival is considered. For specimens with yield strength 690 MPa also the 97.7% curve has been derived. It can be seen that even in this curve, on the range of 104 to 105 cycles, the fatigue strength is close to the measured monotonic yield strength of 744 MPa–843 MPa; see Table 2. Figure 13 shows the change of slope caused by the surface treatment. When the specimens have been grinded or grinded and sandblasted the slope increases considerably when compared to the specimens, which have been only plasma cut. It can be also seen that all trendlines meet on the range of 104 to 105 cycles. All results in this case are well above the design curve of DNV (2008). Grinding the specimens increases the fatigue strength, but this followed by the sandblasting slightly decreases it. 4

Figure 13. Fatigue strength of the specimens having various treatments and yield strength 690 MPa. In trendlines the 97.7% probability of survival is considered.

Measured σy [MPa]

DISCUSSION

The investigation shows that the fatigue strength of the balcony openings can be greatly increased by surface treatment of the plasma cut surfaces. While best results in terms of fatigue strength are obtained by grinding the cut surfaces after plasma cutting, the positive effects are somewhat lost due to sandblasting required before painting the steel plates. This can be partly due to increased surface roughness of the sandblasted specimens when compared to the initial state after plasmacutting or grinding; see Figures 6 and 7. However, sandblasting has positive effect on the shape of the defects and hardness of the specimen surface, i.e. it makes the notches radius somewhat larger and increases the hardness. This makes the sandblasted specimens better in terms of fatigue strength than only plasma cut specimens.

259

Table 3. Yield strength

Summary of fatigue test results. Treatment

Specimen name

Plate thickness

Width, w

Load ratio

Max load

Min load

Mean load

Stress range

Nf

355

Plasma cut & grinded

P2_355_1 P2_355_2 P2_355_3 P2_355_4

17.2 17.1 17.1 17.5

30.4 30.2 30.1 30.2

0.10 0.10 0.10 0.10

203 245 248 241

21 25 25 24

112 135 136 132

349 427 434 412

Run out 107,852 99,839 141,924

460

Plasma cut

P1_460_1 P1_460_2 P1_460_3 P1_460_4 P1_460_5 P2_460_1 P2_460_2 P2_460_3 P3_460_1 P3_460_3 P3_460_4 P3_460_5

15.3 15.2 15.1 15.3 15.1 15.0 14.9 14.8 15.1 15.0 15.1 15.0

30.4 30.5 30.4 30.1 30.5 30.1 30.3 30.1 30.2 30.0 30.3 30.4

0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

200 210 230 221 221 240 250 249 230 220 225 225

21 22 24 22 22 24 25 25 23 22 22 22

111 116 127 121 122 132 137 137 126 121 124 124

386 409 450 432 431 479 499 504 455 440 444 445

Run out Run out 128,052 345,089 406,532 340,454 191,725 180,729 132,519 Run out 133,496 1,683,779

P1_690_1 P1_690_2 P1_690_3 P1_690_4 P1_690_5 P1_690_6 P1_690_7 P1_690_8 P1_690_9 P1_690_10 P2_690_1 P2_690_2 P2_690_3 P2_690_4 P2_690_5 P2_690_6 P2_690_7 P2_690_9 P3_690_4 P3_690_5 P3_690_6 P3_690_7 P3_690_8 P3_690_9 P3_690_10

15.1 15.3 15.1 15.4 15.1 15.3 15.1 15.4 15.2 15.1 14.9 15.0 14.9 15.0 15.0 14.9 15.0 15.1 15.0 15.1 14.9 15.0 15.1 15.1 15.0

30.4 30.5 30.4 30.5 30.4 30.6 30.1 30.6 31.0 31.0 30.2 30.5 30.7 30.1 30.6 30.1 30.4 30.1 30.0 30.5 30.2 30.4 30.3 30.5 30.1

0,08 0,09 0.10 0.10 0.10 0.10 0.10 0.10 0,11 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

255 210 183 259 210 281 240 340 310 340 335 340 320 350 349 340 330 350 290 330 320 310 315 315 320

21 19 18 26 21 28 25 33 35 35 34 34 32 35 35 34 34 35 29 33 32 31 31 32 32

138 115 101 142 116 154 133 187 173 188 184 187 176 192 192 187 182 193 160 182 176 171 173 173 176

511 409 360 497 411 540 475 650 584 654 670 669 630 700 685 680 649 696 580 647 638 613 624 618 639

200,710 359,047 Run out 322,686 Run out 338,862 555,330 118,407 190,356 122,227 132,574 204,120 1,784,428 140,676 159,009 468,579 207,989 142,475 Run out 119,358 90,681 752,909 377,856 495,229 479,482

Plasma cut & grinded Plasma cut, grinded & sandblasted 690

Plasma cut

Plasma cut & grinded

Plasma cut, grinded & sandblasted

The finding is supported by investigation by Sasahara (2005) who concluded that in addition to increased surface hardness and better shape of the crack-like flaws compressive residual stresses increase the fatigue strength. Residual stresses were not measured in present investigation, so the influence of those is difficult to analyze here. In addition, the fracture surfaces identifying the failure locations need to be carefully analyzed. This is left for future work.

The utilization of the results depends on the statistical treatment of the experimental results. According to DNV (2008) the recommended slope value is m = 4.0 for the base material in noncorrosive environment when number of cycles is less than 107. Present study shows that plasma-cut specimens are in agreement with this value considering safety factor of fs = 1.65 between the design curve and tested specimens (97.7% probability level). However, for grinded and sandblasted speci-

260

mens the slopes observed in tests are considerable higher than that of plasma cut specimens and proposed by DNV (2008). This finding is in line with that from Gao et al. (2007) who showed that with different surface treatments the slope is changed considerably in ultra high strength steel. This results also in fatigue strength at two million cycles; in present investigation this was above 590 MPa for both series. Meanwhile the results on grinded specimens indicate that the slope value is also affected by the yield strength of the material when specimens have high surface quality. In this case, fitting the results to fixed slope m = 4 lead to situation where the scatter of the tests results form the average curve becomes very high and the fatigue strength at range of 104 to 105 cycles becomes higher than the measured yield strength of the material. In order to utilize the potential created by the surface treatment many aspects need to be taken into account. The quality of grinding and sandblasting needs to be ensured in production process. In addition, the post-treatment protection must be established and finally the effect of wave-induced loading needs to be considered in the fatigue assessment. 5

CONCLUSIONS

This paper investigated experimentally the influence of surface integrity to the fatigue strength of high strength steel plates used in balcony openings of cruise ship structures. Different surface treatments were considered that are due to manufacturing process of the shipyard. The fatigue test specimens, having a dog-bone shape and yield strength of 460 MPa or 690 MPa, were cut by plasma. After the cutting, the specimens were treated by grinding or by grinding followed by sandblasting. The resulting surface roughness and hardness profile were measured. The fatigue tests with load ratio R = 0.1 were carried out until failure of the specimens. The investigation shows that post-cutting treatments suitable for shipyard conditions can considerably increase the fatigue strength of high strength steel used in balcony openings.

ACKNOWLEDGEMENTS Finnish research project funded by the Tekes, Technology Agency of Finland. The work is related the Light and Innovations and Network projects within the scope of the Finnish Metals and Engineering Competence Centre. All the inancial support is gratefully appreciated. REFERENCES Bäcktröm, M. and Kivimaa, S., 2009, Estimation of crack propagation in a passenger ship’s door corner, Ships and Offshore Structures, Vol. 4, No. 3, pp. 241–251. Det Norske Veritas, 2008, Classification Notes, No. 30.7, Fatigue Assessment of Ship Structures, Det Norske Veritas, Hovik, Norway. Gao, Y., Li, X., Yang, Q. and Yao, M., 2007, Influence of Surface Integrity on Fatigue Strength of 40CrNi2Si2MoVA Steel, Material Letters, Vol. 61, pp. 466–469. Murakami, Y. and Endo, M., 1994, Effect of defects, inclusions and inhomogenities on fatigue strength, International Journal of Fatigue, Vol. 16, No. 3, pp. 163–182. Pavlina, E.J. and Van Tyne, C.J., 2008, Correlation of Yield Strength and Tensile Strength with Hardness for Steels, Journal of Materials Engineering and Performance, Vol. 17, No. 6, pp 888–893. SFS-EN ISO 4288, 1996 Geometrical Product Specifications (GPS)—Surface texture: Profile method—Rules and procedures for the assessment of surface texture. SFS-EN ISO-6507, 2005 Metallic materials. Vickers hardness test. Part 1: Test method. Sasahara, H., 2005, The effect on fatigue life of residual stress and surface hardness resulting from different cutting conditions of 0.45% C steel, International Journal of Machine Tools and Manufacture, Vol. 25, pp. 131–136. Sperle, J.-O., 2007, Influence of Parent Metal Strength on the Fatigue Strength of Material with Machined and Thermally Cut Edges, International Institue of Welding, Swedish Delegation, IIW Document No. XIII-2174-07.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Numerical study of stress concentration factors in damaged FPSO side panels under in-plane compression loads B.C. Pinheiro & I.P. Pasqualino Subsea Technology Laboratory, COPPE/UFRJ, Rio de Janeiro, Brazil

C.F.C. Ferreira Ocean Engineering Department, UFRJ, Rio de Janeiro, Brazil

ABSTRACT: Side panels of FPSO (floating production, storage and offloading) units can be subjected to collisions by supply vessels and the resulting damage may lead to a fatigue failure, considering the long in-service period and the action of dynamic loads. The aim of this work is to evaluate Stress Concentration Factors (SCFs) in damaged panels and their effect on the fatigue life through a theoretical study. A finite element model is developed to reproduce a supply vessel collision and evaluate resulting SCFs under in-plane compression load. A parametric study is carried out considering different damage magnitudes and the results obtained are used to develop an analytical expression to provide SCFs as a function of dimensions of damage and panel. These SCFs could be used in a theoretical fatigue life study, which can help to forewarn a fatigue failure under the event of accidental collision. 1

INTRODUCTION

FPSO (floating production, storage and offloading) units can be subjected to mechanical damage in their side panels as a result of accidental collisions by visiting or passing vessels. The most frequent type of collision involves supply vessels that are designated to reach offshore installations and can accidentally collide with them (HSE 2003). Even if the static strength (ultimate tensile strength) of the panel may not be significantly affected by small magnitude damage, the stress concentration in collided regions can lead to the initiation of cracks considering the long in-service period and dynamic loads undergone by the offshore installation. Then, the fatigue life of damaged FPSO side panels should be assessed under the event of accidental damage. A metallic component or structure under dynamic loads can fail by fatigue at a stress well below its ultimate tensile strength, or, usually, even below its yield strength. Fatigue failures usually initiate at the surface of metals or at high local stress concentration sites, such as interfaces precipitate-matrix or inclusion-matrix and grain boundaries, in microscopic scale, and geometrical discontinuities or defects caused by mechanical damage, in macroscopic scale. In the last decades, the phenomenon of fatigue has been subject of several studies due to the importance of fatigue life prediction of large-scale metallic structures,

as side panels of FPSO units, schematically shown in Figure 1. A number of works regarding SCFs in side panels available in the literature are devoted to the study of fatigue behaviour of welded joints (Chen 2011, Da-wei 2012, Fricke 2002). However, studies about SCFs in damaged FPSO side panels were not found in the literature until now. The aim of this work is to study the effect of stress concentration in a FPSO side panel resulting from accidental collisions by a supply vessel (Fig. 2) on the fatigue life of the damaged structure. A numerical model is developed according to the Finite Element Method (FEM) with the aid of the commercial code

Figure 1.

263

Example of a FPSO unit.

initial geometric imperfections, is reinforced with stiffeners of T-profile cross-section. 2.1

Figure 2. Supply vessel model (de Andrade et al. 2010).

ABAQUS (2009) to reproduce the collision by a supply vessel on a FPSO side panel in high strength AH36 steel and estimate the stress concentration generated by the resulting mechanical damage. The numerical model is used in a parametric study considering different penetration depths. Stress Concentration Factors (SCFs) are then estimated under the application of an in-plane compressive load, representing the loading acting on the vessel beam. From the results obtained in the parametric study, an analytical expression is developed, with the aid of the Buckingham Π theorem, to provide stress concentration factors as a function of the dimensions of damage and side panel. This expression is of practical use in the field and the SCFs provided could be employed in a theoretical fatigue analysis. The duration of residual fatigue life of FPSO side panels, under the event of an accidental collision by a supply vessel, could then be estimated as a function of the damage magnitude in order to forewarn a fatigue failure. 2

Geometric parameters

The dimensions of a typical FPSO side panel considered in the present work are schematically shown in Figure 3. The geometry of the collision bulb was adopted based on the structural arrangement of a supply vessel presented in de Andrade et al. (2010). Figure 4 presents the structural arrangement of the supply vessel model responsible for the collision on the side panel and the introduction of mechanical damage. Only the upper region of the panel was modelled, between transversal frames, from the second to the eighth stiffeners (Fig. 3), totalizing six stiffeners. Table 1 presents the dimensions of the panel and stiffeners adopted in the numerical model. In Figure 5 the side panel modelled with three-dimensional thin shell elements is shown. Figure 6 shows the bulb of the collision supply vessel modelled with three-dimensional rigid elements.

Figure 3. Dimensions (mm) of a typical FPSO side panel.

NUMERICAL MODEL

A three-dimensional numerical model was developed according to the finite element method, with the aid of the commercial code ABAQUS (2009), to reproduce the collision by the bulb of a supply vessel on a FPSO side panel and estimate the stress concentration generated in the damaged region. The side panel is modelled by thin shell elements, while the supply vessel bulb is modelled by rigid elements. The side panel, represented without

Figure 4. Structural arrangement of a supply vessel (de Andrade et al. 2010).

264

Table 1. Geometric parameters of the side panel and stiffeners. Value (mm)

Parameter Plate length, a Plate width (between stiffeners), b Panel thickness, t Web height, h Web thickness, tw Flange width, w Flange thickness, tf Number of stiffeners, n

Figure 5.

Value (∈Z)

5240 860 19 495 11 125 22.5 ≥2

Finite element mesh of the FPSO side panel.

2.2

Material properties

The material adopted for the side panel is the high strength AH36 steel (ASTM 2011), usually used for naval ship buildings. The mechanical properties of the AH36 steel were adopted according to Estefen (2009), in which the material characterization was carried out by uniaxial tension tests. The average mechanical properties of the AH36 steel are presented in Table 2, where E is the Young modulus and ν is the Poisson coefficient. The material was modelled assuming linear elasticity and elastic-plastic behaviour. A plastic constitutive behaviour was adopted within the potential low rule, assuming the von Mises yield function under combined isotropic and kinematic hardening. The kinematic hardening reproduces the material behaviour under reverse loads taking into account the Bauschinger effect (Mendelson 1968). Since the cyclic uniaxial behaviour of the material was not evaluated experimentally, an approximation routine was adopted, incorporated in the kinematic hardening model, to estimate this effect using half cycle data from the stress-strain curve (ABAQUS 2009). The plastic material behaviour is defined in terms of the true stress versus logarithmic plastic strain curve (Estefen 2009). This curve is obtained from the engineering stress-strain curve, which is defined in terms of nominal stress and strain (σnom and εnom), as follows

σt

σ nom (

εt

l (

nom ) n

(1)

ε nom )

(2)

where σt and εt denote the true stress and the true (or logarithmic) strain, respectively. Figure 7 shows the true stress versus logarithmic plastic strain for the AH36 steel used as input material data in the numerical model. 2.3

Finite element mesh

The side panel was modelled with three-dimensional thin shell elements of the type S8R5 (ABAQUS 2009). S8R5 are second-order quadrilateral elements with ive degrees of freedom per

Table 2. steel.

Figure 6. Finite element mesh of the supply vessel bulb.

Average mechanical properties of the AH36

E (GPa)

ν

Yield strength* (MPa)

Ultimate tensile strength (MPa)

208

0.3

365

500

*0.20% offset.

265

Figure 7. True stress versus logarithmic plastic strain for the AH36 steel used as input data in the numerical model.

Table 3. model.

Summary of the FE mesh refinement of the Number of elements

Direction

Parameter

Value

Plate length, a Plate width, b Web height, h Flange width, w

Na Nb Nh Nw

84 14 8 4

Figure 8. FE model composed by the FPSO side panel and supply vessel bulb.

2.4

node (three translations and two in-surface rotations) and reduced integration. The formulation of this element type assumes arbitrary large rotations and small strains and satisfies the Kirchhoff constraint, according to which the change in thickness with deformation is ignored. In this element type, the Kirchhoff constraint is satisied numerically. The Finite Element (FE) mesh of the plate was generated with 84 elements in the longitudinal direction (x axis) and 14 elements between stiffeners in the transverse direction (y axis). The FE mesh refinement comprises 8 elements in the web height and 4 elements in the flange width. The mesh refinement in the longitudinal direction of both web and flange follows that of the plate. The FE mesh refinement is summarized in Table 3. The FE mesh of the collision bulb (Fig. 6) was generated with three-dimensional rigid elements of the type R3D3 and R3D4 with three and four nodes, respectively. The whole FE mesh satisfies an aspect ratio of 1:1 and comprises a total of 12,335 elements, corresponding to 11,928 elements of the type S8R5, 48 elements of the type R3D3 and 359 elements of the type R3D4. The FE mesh formed by the side panel and supply vessel bulb is presented in Figure 8.

Boundary conditions and loadings

Three load steps were prescribed, corresponding to indentation of the bulb on the panel (mechanical damage generation), removal of the bulb, and application of in-plane compressive load on the panel, reproducing the loading undergone by the vessel beam. In the load steps of indentation and bulb removal, the boundary conditions in all edges of the plate were imposed constraining the translations in the directions y and z and the three rotational degrees of freedom. In the compressive load step, the boundary conditions of the previous steps were maintained and, additionally, a displacement of 0.0025a = 6.55 mm in the longitudinal direction (x axis) was prescribed at the transverse edges. In the axis of longitudinal symmetry the displacement in the x direction was constrained in all load steps. The displacement of the bulb in the z direction, normal to the panel surface (Fig. 8), can be prescribed to introduce different damage depths. 2.5

Contact properties

For the modelling of the contact between the panel and bulb, the bulb surface was defined as master, with the panel surface set as slave. The interaction between the contact surfaces was modelled assuming small sliding. Elements of the type S8R5 in the slave surface are automatically converted to S9R5, which possess one additional

266

central node. This conversion is adopted to minimize convergence difficulties. The contact pressure between the surfaces was assumed to follow an exponential law for the pressure-overclosure relation between them. This law is defined considering the contact pressure at zero clearance (p0 = 365 MPa) and the clearance necessary to the beginning of the contact pressure transmission by the surfaces (c0 = 9.5 mm). 3

NUMERICAL RESULTS

Figures 9 to 12 show numerical results in terms of von Mises stress distribution obtained for penetration depths d of 10.75 mm and 107.50 mm. The higher stress values indicated are observed at remote points in the panel edges, being out of concern for this work.

Figure 9. FE results of von Mises stress distribution (MPa) in the panel internal surface for d = 10.75 mm.

Figure 11. FE results of von Mises stress distribution (MPa) in the panel internal surface for d = 107.50 mm.

Figure 12. FE results of von Mises stress distribution (MPa) in the panel external surface for d = 107.50 mm.

4 4.1

Figure 10. FE results of von Mises stress distribution (MPa) in the panel external surface for d = 10.75 mm.

PARAMETRIC STUDY FE analyses

The FE model was used in a parametric study considering different penetration depths for the supply vessel collision on the side panel. The stress concentration in the damaged panel is estimated under the application of an in-plane compressive load, reproducing the loading undergone by the vessel beam. With the aim of carrying out a wide parametric study, an algorithm was developed in the programming language FORTRAN to optimize the generation of the FE mesh for different geometries (varying dimensions and number of stiffeners). The generated FE mesh was imported to the commercial program ABAQUS (2009) and the model

267

was completed. The FE analyses were then carried out with the aid of ABAQUS (2009). Table 4 presents the penetration depths of the collision bulb prescribed in the numerical analyses of the parametric study. The stress concentration factors Kt were calculated as (Pilkey 1997) Kt =

σm max σ nom

(3)

where σ max is the maximum von Mises stress in the damaged panel and σ nom is the nominal von Mises stress in the intact panel, i.e. without mechanical damage (d = 0). Stress concentration factors were considered in the centre of the panel, preventing edge effects. Higher stress concentration factors were obtained in the internal surface of the panel. The obtained FE results are summarized in Table 5, where d′ is the damage depth after bulb Table 4. Penetration depths of the collision bulb on the panel prescribed in the parametric study.

Table 5.

d/b (%)

d (mm)

0 1.25 2.5 3.75 5 6.25 7.5 8.75 10 11.25 12.5

0 10.75 21.50 32.25 43.00 53.75 64.50 75.25 86.00 96.75 107.50

removal (elastic return), and Kt are referred to the internal surface. Figure 13 shows the FE results of stress concentration factors (Kt) in the internal panel surface as a function of the damage depth d′/b (%). These results were fitted to a second order polynomial curve. 4.2 Analytical formulation Although sophisticated FE models can precisely evaluate stress concentration factors, a direct approach through a simple equation is obviously more practical, provided the precision of the results is assured. For this reason, an analytical methodology was proposed using the available FE results to develop expressions to estimate Kt. The maximum stress on the collided panel can be assumed to depend on the primary involved variables (Buckingham 1914), that is

FE results in the panel internal surface.

σ max

d (mm)

d/b (%)

d′/b (%)

σ max (MPa)

Kt

0 10.75 21.50 32.25 43.00 53.75 64.50 75.25 86.00 96.75 107.50

0 1.25 2.5 3.75 5 6.25 7.5 8.75 10 11.25 12.5

0 0.06 0.48 1.15 1.88 2.70 3.77 4.88 5.98 7.10 8.22

296* 352 418 444 440 473 474 475 506 573 569

1 1.20 1.42 1.51 1.49 1.61 1.60 1.60 1.72 1.95 1.94

*σ nom .

Figure 13. Stress concentration factors Kt (panel internal surface) obtained in the parametric study as a function of d′/b.

f (σ nom , d , b, t )

(4)

It follows from the Buckingham’s Π theorem that the Equation 4 can be reduced to a relationship between non-dimensional variables (Buckingham 1914), such as

σ max ⎛ b d′⎞ =F⎜ , ⎟ ⎝t b⎠ σ nom

(5)

or, from Equation (3) Kt

268

⎛ b d′⎞ F⎜ , ⎟ ⎝t b⎠

(6)

Equation (6) can be expressed as the following series (Buckingham 1914)

Kt

⎡⎛ b ⎞

∞

∑ An ⎢⎜⎝ t ⎟⎠

α1

⎢⎣

n=0

⎛d ⎞ ⎜⎝ ⎟⎠ b

α2 ⎤n

⎥ ⎥⎦

(7)

Since Kt ≥ 1, it can be assumed that for n = 0, A0 = 1. Then, neglecting terms with powers n > 1, the following first order expression was proposed to fit the FE results Kt

1 A1B

(8)

where A1 is the angular coefficient of the linear equation and the non-dimensional geometric parameter B is given by ⎛ b⎞ B=⎜ ⎟ ⎝t⎠

α1

⎛ d′⎞ ⎜⎝ ⎟⎠ b

α2

(9)

FATIGUE LIFE ASSESSMENT

Stress concentration factors obtained in the parametric study were used in a theoretical fatigue life study to estimate the duration of residual fatigue life of a side panel collided by a supply vessel as a function of the damage magnitude. Fatigue failures usually start at stress concentration sites, where cracks are initiated. In the present work, the mechanical damage in the FPSO side panel resulting from the bulb collision represents a region of high stress concentration and the fatigue life of the damaged panel should be assessed. Fatigue failures occurring at number of cycles between 1 and 103 are classified as low cycle fatigue, while failures taking place after 103cycles concerns the high cycle fatigue domain. The high cycle fatigue is usually evaluated by the S-N curve, also known as Basquin equation, relating the number of cycles to failure (N) to the alternating stress (Sn) Sn

The parameters A1, α1, and α2 were determined in order to achieve an accurate correlation between Equation (8) and the available FE results. Finally, the analytical methodology led to the expression ⎛ b⎞ K t = 1 5.70 ⎜ ⎟ ⎝t⎠

5

−0.31 31

⎛ d′⎞ ⎜⎝ ⎟⎠ b

0.29

(10)

In Figure 14, this expression is compared to the available FE results of stress concentration factors. The mean deviation of the linear fitting is about 9.77%. Further FE analyses are being carried out in order to define a more precise linear fit.

CN b C

(11)

where parameters C and b depend on the material properties and testing conditions (Shigley 2001). In logarithmic scale, the S-N curve renders itself as a straight line. Steels, like other metals, will not fail by fatigue if the alternating stress is lower than a threshold, called the endurance limit (Se′ ) and usually related to N ≥ 106. For standard testing conditions, the endurance limit for steels can be inferred from the ultimate tensile strength, Su (Shigley 2001) S′e = S

i

{

}

(12)

Some empirical factors convert the endurance limit obtained in standard conditions to the one expected in a real structure. Considering the fatigue stress concentration factor (Kf) and the surface finish factor (ka), the endurance limit of component or structure (Se) can be calculated as Se

Figure 14. Linear fit of FE results of stress concentration factors Kt (internal surface) using the Buckingham Π theorem.

Se′

ka Kf

(13)

where ka and Kf are the surface finish factor and the fatigue stress concentration factor, respectively. To completely deine the high cycle fatigue domain, in addition to the endurance limit point, conflicting approaches to deine another point in the S-N curve can be found in the literature. In the absence of experimental data, two S-N curves were determined analytically (Cunha et al. 2009, Pinheiro et al. 2009): (1) the first S-N curve was defined as a straight line in logarithmic scale connecting (0.9Su) at N = 103 cycles and Se at

269

N = 106 cycles, and (2) the second S-N curve was defined according to Graham (1968), which suggests that the high cycle portion of an S-N curve would cross the one cycle axis at the trues tress of failure in a tensile test (σf). The surface finish factor was estimated as (Shigley 2001) ka

aSu b

(14)

where parameters a and b were adopted as 57.7 (Su in MPa) and −0.718, considering hot-rolled surface finish. Fatigue stress concentration factors were estimated from theoretical stress concentration factors, disregarding the notch sensitivity factor, i.e. Kf = Kt. In Figure 15 the two S-N curves determined analytically for Kt = 1 are compared to fatigue testing results obtained by Wang et al. (2010) and Crupi et al. (2009) for the AH36 steel. The analytical S-N curve (1) correlates better to the experimental results obtained in the literature. In Figure 16

Figure 15. Comparison between analytical S-N curves (Kt = 1) and experimental results found in the literature.

this curve is corrected by the maximum stress concentration factor obtained in the parametric study (Ktmax = 1.95) to evaluate the effect in the fatigue life of the panel of the most severe collision damage reproduced. 6

CONCLUSIONS

In this work, a Finite Element (FE) model is developed to evaluate stress concentration factors in a FPSO side panel damaged due to a supply vessel collision. The FE model is used in a parametric study considering different penetration depths of the supply vessel on the side panel, resulting in different damage magnitudes. The Stress Concentration Factors (SCFs) are estimated under the application of an in-plane compressive load in the panel, reproducing the loading undergone by the vessel beam. The results obtained in the parametric study are used to develop an analytical expression, with the aid of the Buckingham Π theorem, that linearly fits the numerical data and gives SCFs as a function of dimensions of damage and panel (d′, b and t). This expression allows a practical evaluation in the field of SCFs in side panels subjected to a collision by a supply vessel. A mean deviation of 9.77% is found for the linear fitting of numerical results of SCFs by the analytical expression developed. Further FE analyses are being carried out in order to define a more precise linear fit. Then, once the in-plane loading frequency undergone by the side panel is known, SCFs obtained by the analytical expression could be used in a fatigue life study as a function of the damage magnitude, which can help to forewarn a fatigue failure under the event of an accidental collision by a supply vessel. ACKNOWLEDGEMENTS The authors would like to thank the Brazilian National Agency of Petroleum, Natural Gas and Biofuels (ANP) for the financial support to this research work.

REFERENCES

Figure 16. Analytical S-N curve (1) for stress concentration factors Kt = 1 and 1.95.

ABAQUS 2009. User’s and Theory Manuals, Release 6.9, Boston: Hibbitt, Karlsson & Sorensen, Inc. ASTM 2011. ASTM Standards, West Conshohocken: American Society for Testing and Materials. Buckingham E. 1914. On physically similar systems, illustration and the use of dimensional equations. Physical Review 4(4): 345–376.

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Chen, N., Wang, G., Guedes Soares, C. 2011. Palmgren–Miner’s rule and fracture mechanics-based inspection planning. Engineering Fracture Mechanics 78: 3166–3182. Cunha, S.B., Pasqualino, I.P., Pinheiro, B.C. 2009. Stress-life fatigue assessment of pipelines with plain dents. Fatigue Fract. Eng. Mater. Struct. 32: 961–974. Crupi, V. et al. 2009. Fatigue analysis of butt welded AH36 steel joints: Thermographic Method and design S-N curve. Marine Structures 22: 373–386. de Andrade, R.H.R. & Nassar, R.M. 2010. Report 2, Course Naval Design III. Rio de Janeiro: Naval and Ocean Engineering Department, Federal University of Rio de Janeiro. Da-wei, G., Gui-jie, S., De-yu, W. 2012. Residual ultimate strength of hull structures with crack and corrosion damage. Engineering Failure Analysis 25: 316–328. Estefen, T.P. 2009. Influence of the Fabrication Distortions on the Structural Behaviour of Stiffened Panels of Semi-Submersible Platforms under Axial Compression. Master Dissertation, Rio de Janeiro: COPPE/UFRJ.

Fricke, W., Cui, W. et al. 2012. Comparative fatigue strength assessment of a structural detail in a containership using various approaches of classiication societies. Marine Structures 15: 1–13. Graham J.A. 1968. SAE Fatigue design handbook, New York: Society of Automotive Engineers. HSE 2003. Ship/platform collision incident database, Research Report 053, Suffolk: HSE (Health and Safety Executive) Books. Mendelson A. 1968. Plasticity: Theory and Application. New York: Macmillan. Pilkey, W.D. 1997. Peterson’s stress concentration factors, New York: John Wiley & Sons. Pinheiro, B. & Pasqualino, I. 2009. Fatigue analysis of damaged steel pipelines under cyclic internal pressure. International Journal Fatigue 31: 962–973. Shigley, J.E. & Mischke, C.R. 2001. Mechanical engineering design, New York: McGraw-Hill. Wang, X.G. et al. 2010. Quantitative Thermographic Methodology for fatigue assessment and stress measurement. International Journal of Fatigue 32: 1970–1976.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Implementation of fatigue properties of laser welds into classification rules H. von Selle, J. Peschmann & S. Eylmann Germanischer Lloyd SE, Germany

ABSTRACT: Within the European funded research project ‘Breakthrough in European Ship and Shipbuilding Technologies’ (BESST) several measures were investigated to strengthen the European shipyards industry. This paper describes the implementation of laser welding into classification Rules as one project activity. The potential of laser welding opens the door to lightweight ship structures as the heat input is much smaller in comparison to conventional welding resulting in smaller manufacturing imperfections. Two main topics will be covered by this paper: First of all several fatigue tests were performed. The test evaluation was done with respect to Rule development. Here special focus was on thin plates in the range of 3–5 mm. Besides fatigue testing finite element calculations and the application of several assessment methods will be presented. The influence on fatigue will be described for parameters based on geometric properties as well as for those based on the laser welding process itself. Secondly the compliance of the application cases planned by the shipyards with existing rule requirements was investigated. Deviations from the rules were detected and a Failure Modes, Effects and Criticality Analysis (FMECA) was carried out to investigate the risks associated to the rule deviations and identify possible Risk Control Options (RCO). 1

INTRODUCTION

This paper describes the outcome of one work package of the European funded project “BESST—Breakthrough in European Ship and Shipbuilding Technologies”. BESST has been formed by Europe’s leading shipbuilders, including STX Finland, STX France, Fincantieri, Meyer Werft, ThyssenKrupp Marine Systems, Flensburger Schiffbau-Gesellschaft FSG and Damen Group. In addition, Germanischer Lloyd along with twenty research institutes and universities, four classification societies and 31 industrial companies are part of the research network. The primary goal is to increase the competitiveness of European built ships through decreased life cycle cost, drastically reduced environmental impact and continually improved safety. This paper deals with the potentials of laser welding in shipbuilding industries. Due to the low heat input by laser welding this technology is well suitable for the fabrication of thin walled structures as welding distortions are expected to be small in comparison to conventional welding. The fatigue properties of laser welds are one of the main objectives. A large testing program has been set up and extensive finite element calculations have been performed in parallel. The way of result transfer into Rules will be described first.

In part two the compliance of selected laser application cases planned by the shipyards with existing rule requirements will be described. The risks related to the usage of laser-hybrid welding for certain selected application cases under consideration of decreased plate thicknesses have been identified and ranked. Deviations from the rules were detected and a Failure Modes, Effects and Criticality Analysis (FMECA) was carried out to investigate the risks associated to the rule deviations and possible Risk Control Options (RCO) were identified. 2

TEST RESULTS

The comprehensive test program discussed in this paper was established mainly due to two reasons. Firstly, there is only insufficient knowledge of the fatigue strength of welded joints in ship structures made of plate thicknesses below 5–6 mm. The reason is the minimum plate thickness required by today’s classification rules. Secondly, the laser welding technology was expected to be advantageous compared to conventional arc welding methods especially when thin walled structures are welded. The test program consisted of 29 series of small scale specimens. Typical welded connections like

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butt welds and T-joints were fabricated by the participating shipyards applying different geometries and welding procedures. In order to be able to identify the differences in fatigue strengths between the laser- and laser-hybrid welding procedures on the one hand and the conventional methods on the other hand, nearly all series were manufactured applying all three kinds of welding procedures. The test work was done by the AALTO University in Espoo, Finland and the Hamburg University of Technology (TUHH), Germany. An extract of the test program containing series dealt with in this paper is presented in Table 1. The series codes are explained within the column ‘Naming scheme’ at their first occurrence. Each series consisted of at least eight specimens. More detailed information on the test program and the evaluation work can be found in Fricke et al. (2013). The test series shown in Table 1 may be grouped into butt welds connecting plates with equal thickness, butt welds joining plates with different thicknesses and fillet welds, either welded from both sides or from one side only. The results for each group of series are presented in the following sub sections. 2.1 Butt welds In Table 2 the fatigue test results for the butt weld series are summarised. Firstly each series was statistically evaluated considering a fixed slope of the SN-curve of m = 3. The mean values represent the fatigue strength at 2 million cycles for a

Table 1.

Extract of the test program.

Series

Naming scheme

Institute

B.3.LA

Butt (B), t = 3 mm, Laser (LA) Hybrid (HY) Conventional (CV)

AALTO

B.3.HY B.3.CV B.4.LA B.4.HY B.4.CV B.3/5.LA B.3/5.HY B.3/5.CV B.3.CV.BJ F.8/5.CV.T F.8/5.LA.T F.8/5.HY.T F.3/5.CV.T F.3/5.LA.T F.3/5.HY.T B.3.CV.F B.4.CV.F

t = 3 mm/5 mm thickness step Block Joint (BJ) Fillet (leading ‘F’)

Faired (ending ‘F’)

AALTO AALTO TUHH TUHH TUHH AALTO AALTO AALTO AALTO TUHH TUHH TUHH AALTO AALTO AALTO TUHH TUHH

Table 2.

Butt weld series test results. Nom. stress appr.

Struct. stress appr.

Series

Mean FAT TN (MPa) (MPa) 1:

Mean FAT TN (MPa) (MPa) 1:

B.3.LA B.3.HY B.3.CV B.3.CV.F B.3.CV.BJ B.4.LA B.4.HY B.4.CV B.4.CV.F

178 81 34 71 112 90 119 57 67

183 156 150 137 174 141 188 140 150

123 48 21 48 68 72 85 44 41

4.4 8.0 6.4 4.8 7.1 2.4 3.7 2.6 7.0

137 117 113 92 131 109 127 100 116

2.5 5.9 1.7 4.9 8.0 2.8 5.0 3.7 2.7

probability of survival of 50%, while the values shown in column ‘FAT’ are based on a probability of survival of 97.5% (mean minus two standard deviations). This represents the rule value used in common standards. The results were analysed both based on nominal as well as structural stress. The results based on the nominal stress approach generally show the tendency that conventional welds achieve worse fatigue lives than laser welds. But, considering FAT80 as the lower limit for butt welds in the classification rules in force, the results for the conventional welds do not even reach today’s rule requirements. In this context it must be kept in mind that the FAT-classes provided by the rules cover only a limited amount of stress increase due to axial and angular misalignments, in this case 30%. Higher stress increases have to be considered separately. This is done by application of the structural stress approach taking all stress increases caused by axial and angular misalignments into account. Now all series but ‘B.3.CV.F’ reach the required FAT100. On the other hand the laser welds are not as much better as expected, e.g. for the 3 mm series the laser-hybrid welds do not reach considerably better fatigue lives than the conventionally welded specimens do. This fact proves that misalignments are the most influencing parameters (Fricke et al., 2013.), the expected technological advantage of the laser and laser-hybrid welding procedures could not be confirmed. 2.2

Butt welds with thickness step

The test results of the butt weld with thickness step series are presented in Table 3. These results are quite similar to those of the butt welds without thickness steps: For the nominal stress concept the conventionally welded series

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Table 3.

Butt weld with thickness step series results. Nom. stress appr.

Struct. stress appr.

Series

Mean (MPa)

FAT (MPa)

TN 1:

Mean (MPa)

FAT (MPa)

TN 1:

B.3/5.LA B.3/5.HY B.3/5.CV

104 88 44

90 66 32

1.8 3.0 3.6

160 139 160

120 104 120

2.7 4.6 5.0

Table 4.

T-Joint series results. Nom. stress appr.

Struct. stress appr.

Series

Mean FAT TN (MPa) (MPa) 1:

Mean FAT TN (MPa) (MPa) 1:

F.8/5.LA.T F.8/5.HY.T F.8/5.CV.T F.3/5.LA.T F.3/5.HY.T F.3/5.CV.T

123 101 69 104 91 121

151 140 122 – – –

91 90 63 74 44 57

3.2 1.6 1.4 3.9 18.0 18.5

111 125 107 – – –

3.2 1.6 1.8 – – –

shows unsatisfactory fatigue life. Taking again the axial and angular misalignments into account by the structural stress approach the reached FAT classes are above the current requirements, but the laser welding procedures do not develop significant advantages compared to the conventional welding method. 2.3

T-joints

Conventionally welded T-Joint specimens consist of fillet welds, the laser welded seams are fully penetrated and show a sharper notched weld profile compared to the conventional ones. The test results are assembled in Table 4. The conclusions for the ‘F.8/5.*’-series are in line with the experiences gained from the butt weld series. The ‘F.3/5.*’-series exhibit very high scatters for the la-ser-hybrid and conventional welded specimens. Un-fortunately the structural stress evaluation was not available when this paper was edited. Therefore, the ‘F.3/5.*’-series will not be further considered within this paper. 3

HOW TO CONSIDER PROJECT RESULTS IN RULES AND GUIDELINES

In this section a possible assessment concept for laser welds is introduced. The development of a new fatigue assessment procedure especially for

laser and laser-hybrid welds is not desirable. For easy application the aim is to expand the procedure which is already implemented in current classification rules. It is assumed that today’s rules for fatigue assessment are generally applicable to conventionally welded thin plates, at least in a conservative way. For welded connections a slope of m = 3 is to be applied according to current rules. Other investigations indicate that for thin walled welded structures different slopes may be more realistic, e.g. m = 5. But in the assessment concept the slope m = 3 shall be kept as it is in line with well accepted Rules and standards. Nevertheless the slope will be topic of further discussions. According to the recommendations of the International Institute of Welding IIW (Hobbacher, 2009) FAT classes determined from tests carried out at a stress ratio of R = 0 shall be reduced by 20%. Up to now this recommendation has not been followed as it is deemed to be extremely conservative. The FAT class to be applied to butt welds was chosen to be FAT80 (Type A3 as provided by Ger-manischer Lloyd’s classification rules (GL 2011)) for reasons of partly huge weld reinforcements and abrupt weld seam transitions. The laser welded specimens are welded from one side only, which allows for FAT71. This means that laser welds would be disadvantaged according to current rules. The achieved fatigue lives are similar to those of the both side welded conventional welds. Therefore, one sided laser welds may be assigned FAT80 as well. Greater benefits cannot be justified, as the results for series ‘B.4.LA’ (109 MPa, see Table 2) is worse than the results of the conventionally welded series ‘B.3.CV’ and ‘B.4.CV.F’. However, a greater benefit for laser welds would not be helpful, as today block joints are welded conventionally and transverse members need to be arranged. For those details also FAT80 is to be applied. Both side welded butt welds with thickness step may be assessed using FAT71, but one side welding is not covered by today’s rules. For this connection the following is proposed: FAT80 minus one FAT class for single side welding minus one FAT class for thickness step. This approach results in FAT63. The achieved results in Table 3 show equality of both laser and conventionally welded details for the structural stress approach. Thus, the application of FAT71 for the laser welded specimens can be justified. The T-Joint series may be provided with FAT80 according to detail type C7 of current rules. The results of the ‘F.8/5’-series are in accordance with the experiences gained from the butt weld series. Therefore the applicability of FAT80 is verified. The results of the test program showed that axial and angular misalignments are the most influencing parameters with respect to fatigue. Taking them into

275

account, the fatigue lifes of the laser and conventional welds are in the same order of magnitude. The advantage of the laser welds can mainly be found in the lower fairing effort in order to reduce the misalignments to acceptable dimensions. It would be possible to include laser and laser hybrid welds into the existing assessment procedure of Germanischer Lloyd. 4

COMPLIANCE OF SELECTED APPLICATION CASES WITH EXISTING RULES

Based on expected higher fatigue performance of welds fabricated by laser assisted welding techniques the possibility of further decrease of material thickness in various parts of the hull structure was investigated. A number of possible areas were suggested by shipyards, mainly areas where the design of the steel structure is often based on minimum requirements of class rules (Table 5). Finally accommodation decks for cruise liners and watertight bulkheads in engine room area of Mega Yachts were selected for further investigations. Both structures are designed as stiffened steel panels. Usually the decks are longitudinally and the bulkheads vertically stiffened by steel bars (Bulb profile (HP)- or L-sections). Both structures are located within closed, air conditioned spaces. The decks of the reference design are usually of a plate thickness of 5 or 6 mm stiffened by bulb profiles with a spacing of about 700 mm. The distance of the transverse supporting girders is of about 2.8 m. Usually these decks do not represent a separation between main fire zones. Subdivision classes A-0 or A-15 are required in terms of fire protection. Therefore an additional insulation has to be applied only in special cases. The main difference of the alternative (new) design compared with the reference design is the

Table 5.

thickness of the plating, which will be reduced down to 3 mm. Stiffener dimensions and arrangement will be changed accordingly considering required strength and buckling resistance. The bulkhead reference design is of a plate thickness of 6 mm and stiffeners of HP 140 × 7 spaced 600 mm. The required subdivision class with respect to fire safety in dependency of the number of passengers that can be carried on board (in this case not more than 36) is A-0 class (within engine room). The bulkhead has to be watertight and can be used as tank boundary within the engine room area. The difference of the alternative (new) design in this case is also a reduced plate thickness (3.5 mm) and accordingly changed stiffener arrangement. The material used is in both cases (reference and alternative design) higher tensile steel D36. Existing rule requirements are related to aspects on strength, fire safety and welding. Fire safety is in principle regulated by the SOLAS convention of IMO. In case of strength assessment and welding a number of class requirements and international standards have to be observed (see e.g. GL 2000). The requirements that are obviously challenged by the alternative (new) designs are the minimum thickness requirements set by the applied class rules (in this case the rules of Germanischer Lloyd GL 2011). The minimum thickness for engine rooms, accommodation decks and decks of short deckhouses is 5 mm (GL 2011) while the minimum thickness for the bulkhead plating of watertight bulkheads is defined as: tmin

f

where f is defined as 235/ReH. This results in a minimum thickness of 6 mm for mild steel and about 5 mm for higher tensile steel with ReH of 355 MPa. The new designs obviously do not fulfill this minimum requirement.

Possible application cases for thin plates.

Accommodation decks (near neutral axis) Transversal/longitudinal bulkheads on intermediate accommodation decks Deck superstructures and covers Watertight bulkheads (engine room) Balcony platforms Stair case landings

PAX

Ferries

Naval ships

Mega yachts

River cruiser

√

√

√

√

√

√

√

√

√

√

√

√ √

√ √

√

276

5

DESCRIPTION OF FMECA AND IDENTIFICATION OF RCO’S

Scope of the investigation described in the following was to show the possibility and identify requirements that have to be fulfilled with respect to get an “Approval in Principle” for the alternative (new) design solutions. The “Approval in Principle” is an intermediate step within the approval process for alternative designs, showing that the design is in principle acceptable (refer to MSC.86/5/3 and GL 2009). Guidelines regarding the approval of alternative (risk-based) designs are available from classification societies (e.g. GL 2009) or from IMO (refer to MSC86/5/3 and MSC.1/Circ 1212). The present investigation was based on the guideline proposal submitted to IMO by Denmark (MSC86/5/3). The first part of the approval process until the “Approval in principle” (Approval of the generic design) is shown in Figure 1. This general procedure will be followed in different levels of detail dependent from the novelty and complexity of the alternative design under consideration. For the definition of the level of detail to be applied an approval matrix is included in the guideline (MSC86/5/3), where the novelty of the design is subdivided in 4 categories from category 1 which is characterised by a proven technology in a known application area to category 4 which is characterised by an unproven technology in a new field of application. The present case, the use of laser assisted welding technology in fabrication of decks and bulkheads however can be categorised as a proven technology which is applied in an area with limited field history where no specific rules and standards are currently in force. For such a case category 2 of level of detail can be assigned which requires a basic risk assessment independently of the severity of rule challenges (in this case the under-usage of minimum thickness requirements of class rules).

Figure 1.

First part of approval process (MSC86/5/3).

Figure 2.

Definition of the Risk Index (RI).

The basic risk assessment in the present case was carried out by assessing the risks of the alternative generic design in comparison with a reference design. Therefore the safety equivalence of the generic design is assessed. The analysis procedure used was a modified Failure Modes, Effects and Criticality Analysis (FMECA). The Risk Index (RI) for this analysis is defined as shown in Figure 2. The frequency index here is not given in absolute, but in relative categories. It indicates, if the occurrence of a failure of the alternative generic design will be more or less probable than for the reference design. Furthermore it was assumed, that the consequences will be in general the same for both, the alternative as well as the reference design, because the general structural arrangement as well as the function and the boundary conditions of the system are the same for both designs. Therefore, with respect to the need of establishing Risk Control Options (RCO) the following could be derived. If the RI identified by the FMECA is equal or lower than 3, no RCO will be needed because the risk is equal or lower than the already accepted risk for the reference design. For failures with an associated RI higher than 4, RCOs have to be established. It has to be noted, that the way for risk ranking and assessment shown here can not be used in general. In principle the RI assessment of the subsystems of the generic design in comparison with the reference design gives no clear picture of the safety level of the whole system. However the present investigation was intended to identify differences between both designs with respect to risk to identify possible RCOs and derive requirements that have to be fulfilled by the alternative design to get an “Approval in Principle”. Because of the low level of detail (category 2) as described above and the very high similarity of the alternative compared to the reference design, the described way of analyses was estimated as practicable and acceptable. The identification of Hazards was carried out based on a generic model for a cabin deck of a passenger ship and a watertight bulkhead of a Mega Yacht as described in detail in Peschmann et al., (2011). The generic models are characterised by

277

a segmentation of the considered structure in structural components like plate fields, stiffeners and girders as well as the welding connections between these components. Possible failure causes could therefore assigned to major failure modes and to the predefined components of the structure as shown in Table 6. 24 failures with an associated RI of 4 and higher were identified. Table 6.

15 out of these 24 entries are directly related to the welding connections. Therefore the development of RCOs capable to improve and ensure a sufficient quality of welded joints with regard to the fatigue and strength performance of the connection is of importance. In Table 6 it has to be noted that a number of failures are related to the conventional welded

Summary of failures for which RCO have to be established (Peschmann et al., 2011).

No.

RI

Failure mode

Primary failure cause

0.1

4

Fracture

Accidental load

1.6

4

0.2

5–6

3.1

4–5

3.3 3.4 3.5 3.9

3–4 3–4 3–4 4–5

6.7

4–5

3.10

3–4

3.11 3.12 3.13

3–4 4–5 4–5

0.4

4

1.9 0.5

4 4

1.10

4

0.6

4

1.11 1.12

4 4

3.14

4

6.8

4

3.15

4

6.9

4

Accidental load (local impact) Flooding Fracture/ fatigue

Fatigue

Linear misalignment Notches (undercut) Root concavity Weld porosity Local angular distortion

Block joint conventional

Appropriate welding procedures, sequences and shrinkage evaluation, fairing

Residual stresses after welding

Handling (lifting, transport)

Global buckling

Deck (overall system) Plate field Check of hull girder capacity acc. to rules Possible improvement through transverse stiffening in block joint fixed backing for butt welds

Initial deflections

Local buckling

Risk control option

Deck (overall system) Block joint conventional

Excess weld reinforcement Excess penetration Weld flange angle Fairing Buckling

Component

Pre-fabrication (paint workshop) Welding deformation (local deflections) Welding deformation (global deflections)

Plate/girder weld conventional Block joint conventional

Deck (overall system) Plate field Deck (overall system) Plate field Deck (overall system) Plate field Block joint conventional Plate/girder weld conventional Block joint conventional Plate/girder weld conventional

278

Induction heated possible improvement through transverse stiffening in block joint fixed backing for butt welds Appropriate welding procedures, sequences and shrinkage evaluation Check by parametric investigations, determination of new limits (smaller plate fields due to smaller stiffener distances) Pre-check of transport procedures by buckling evaluation (additional load cases for design)

Appropriate welding procedures, sequences and shrinkage evaluation, fairing Appropriate welding procedures, sequences and shrinkage evaluation, fairing

block joint (Nos. 3.3, 3.4, 3.5, 3.10, 3.11) and the related quality of the weld. In this case and for these failure causes the requirements given in the existing standard (ISO 5817) are seen as sufficient and no additional RCOs are necessary. Only 3 of the 24 entries in Table 6 are related to accidental failure modes. One of these three is assessed as a catastrophic failure, the flooding case followed by a possible collapse of the upper decks, which will become under pressure loads in a sagging condition of the ship. However a proof of the ultimate hull girder capacity considering flooding cases used for the proof of the damage stability of the ship is required since a rule change of GLRules in 2010 (GL, 2011). Regarding the failure 3.1 and 3.13 an appropriate risk control option could be a change in design. To limit the initial deflection within the block joint and to decrease the linear misalignment and the related heat input due to fairing, a strengthening of the joint due to a fixed permanent backing or a transverse stiffener could be reached. However this solution will limit the fatigue performance due to the necessary transverse fillet joints. Another option is the insert of thicker plating in the area of block joints. To limit the risks related to the failure mode buckling different RCOs with respect to the primary failure cause are possible as also given in the last row of table 6. In summary the following RCOs (without any ranking) were identified: • Definition of limit values for weld imperfections with respect to fatigue and establishing of quality monitoring of the weld process • Optimisation of clamping during the weld process. Definition of limit values for initial and weld deflections with respect to buckling and systematic check of deflections after welding/block assembling and fairing if necessary • Require an accurate shrinkage evaluation during production planning • Buckling check for transportation and prefabrication procedures/optimisation of transport technique • Check of ultimate hull girder capacity during design phase considering flooded conditions according to the damage calculations (refer GL 2011) • Design measures to limit initial deformations and/or heat input by fairing works (e.g. stiffening of block joints by thicker plating or stiffeners). As indicated above one of the most important problems related to the applicability of laser assisted welding in connection with thin plated structures within ship hull structures is to ensure a sufficient weld quality with respect to the strength

and fatigue performance of the welded joints. This includes the quality requirements regarding the shape of the joint, inclusions and other imperfections as well as misalignments and residual stresses. The development of methods and tools to estimate the influence of weld quality aspects on the strength and fatigue behaviour of the considered structure is therefore a matter of great interest. 6

CONCLUSIONS

Laser as well as laser hybrid welded joints exhibit smaller angular misalignments due to lower heat input compared to conventionally welded joints. As angular misalignments are one of the most influencing parameters on fatigue life, especially when thin plates are considered, it was expected that laser- and laser-hybrid welds reach higher fatigue lives than conventional ones. The forecasted lower angular misalignments of the laser welded joints could be found in the specimen geometries. However, their fatigue lives did not significantly exceed those of the conventional welds when misalignments are fully taken into account by application of the structural stress approach. Most of the initial test specimens did not fulfill well accepted production standards as ISO5817 (2003), VSM (2003) and IACS (2010). Nevertheless the fatigue behaviour could be assessed by today’s methods quite well. But it has to be emphasized that thin walled structures are very sensitive to geometric imperfections. If the additional bending stress due to angular distortion and axial misalignment is higher than that what is already covered by Rules, the FAT class has to be lowered accordingly. The test results revealed the following possible amendments to the existing classification rules: • Application of FAT80 for one sided laser/laserhybrid welded butt welds instead of FAT71 for similar conventionally welded joints • Application of FAT71 for one sided laser welded butt joints with thickness step. Up to now only butt welds with thickness step welded from both sides are included in the catalogue of details. The main advantage of the laser welds can be found in the smaller misalignments resulting in less fairing effort. Some questions still remained open: • Which are realistic magnitudes of misa-lignment in full scale structures? • How to measure misalignments? Dealing with small plate thicknesses local and global deformations should be distinguished.

279

Furthermore the investigations related to the identification of possible RCOs showed clearly, that the strength and fatigue behaviour of the weld connections are a key factor with respect to a further decrease of material thicknesses in special areas of the ship steel structure. ACKNOWLEDGEMENTS The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007–2013) under grant agreement n° 233980. The work has been performed within work package 05 and the partners were Fincantieri, Meyer Werft, STX Finland, Blohm & Voss, Flensburger Schiffbau-Gesellschaft FSG, Aalto University, Hamburg University of Technology (TUHH), Centre of Maritime Technology CMT and Germanischer Lloyd GL. We like to thank all involved project partners for the good cooperation and especially the TUHH and AALTO universities for their comprehensive test and evaluation work. REFERENCES DIN EN ISO 5817:2003–12 Welding—Fusion-welded joints in steel, nickel, titanium and their alloys (beam welding excluded)—Quality levels for imperfections (ISO 5817:2003 + Corr 1:2006) English version of DIN EN ISO 5817.2006-10. Eggert, L. Fricke, W.,. Paetzold, H. (2011) Fatigue strength of butt welded assembly joints considering shipyard-typical misalignments (in German). Jahrbuch der Schiffbautechn. Ges., Vol. 105, pp. 27–36.

Fricke, W. et al. (2013) Fatigue strength of laser-welded thin plate ship structures based on nominal and structural hot-spot stress approach, MARSTRUCT Conference 2013. German Shipbuilding and Ocean Industries Association (VSM). Production standard of the German shipbuilding industry,7. Edition/2006. GL 2000. Rules for Classification and Construction, IIMaterials and Welding, Part 3 Welding, Chapter 1–3. Edition 2000. Germanischer Lloyd SE. Hamburg. GL 2011. Rules for Classification and Construction, I-Ship Technology, Part 1 Seagoing Ships, Chapter 1 Hull Structures. Edition 2011. Germanischer Lloyd SE. Hamburg. GL 2009. Rules for Classification and Construction, V-Analysis Techniques, Part 2 Risk Analysis, Chapter 1 Guidelines for the Analysis of Alternative Design and Arrangements. Edition 2009. Germanischer Lloyd SE. Hamburg. Hobbacher, A. (ed.) 2009. Recommendations for fatigue design of welded joints and components. IIW doc.1823-07, Welding Research Council Bulletin 520, New York. IACS Recommendation No. 47: Shipbuilding and Repair Quality Standard, Rev. 5, Oct 2010. MSC86/5/3 2009. Guidelines on approval of risk-based ship design. International Maritime Organization (IMO). February 2009. submitted by Denmark. MSC.1/Circ.1212, 2006. Guidelines on Alternative Design and Arrangements for SOLAS Chapters II-1 and III. International Maritime Organization, London. Peschmann, Jörg (ed.) 2011. Report on Risk Identification and Control Options. Deliverable D5–02. EU funded project BESST—n.233980 of FP7/2007–2013. unpublished. SOLAS. International Convention for the Safety of Life at Sea 1974 and its protocol of 1988 with all amendments until 1st January 2011. International Maritime Organization (IMO).

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

A FE based numerical tool for crack assessment in ship structures employing the CSR loading scheme I.K. Zilakos, V.A. Karatzas, E.V. Chatzidouros, V.J. Papazoglou & N.G. Tsouvalis Shipbuilding Technology Laboratory, School of Naval Architecture and Marine Engineering, National Technical University of Athens, Greece

ABSTRACT: Among the most common defects found in ship structures, which potentially could lead to critical loss of structural integrity, are cracks. A methodology is proposed that combines the Common Structural Rules (CSR) fatigue loadings with the Finite Element Method (FEM) for the study of cracks on actual marine structure subjected to high-cycle low-stress fatigue. The aim is to establish a common base for evaluating different crack arrest technologies. To this end a three-compartment model of an Aframax tanker was created using the FEM and following the guidelines proposed by the CSR. The structure was subjected to fatigue loads and different crack scenarios were investigated for two loading conditions. The stress intensity factors were calculated and by using the well-known Paris law formula an estimation of the crack growth rate was derived. The proposed method is simple and based on rules and regulations widely accepted in the ship industry. 1

INTRODUCTION

The sustainability of the global maritime industry depends on enhancing ship safety and mitigating the consequences of ship incidents and accidents. To a very large extent, a comprehensive system of international rules covers every conceivable aspect of the construction and use of ships, including their safe navigation and the welfare of their crew members. Nevertheless there are reports (Bloor et al. 2006, Papanikolaou et al. 2006, Papanikolaou et al. 2007, Knudsen and Hassler 2011, Eliopoulou et al. 2012) showing that vessels continue to sail with grave irregularities and defects even after recent inspections (Knudsen and Hassler 2011). For example, in the case of Double Hull Tankers, non-accidental structural failures appear frequently (Eliopoulou et al. 2012), in ships 0–5 years old. Although critical damages are the ones that require immediate rectification actions, non-critical damages such as certain types of cracks might be underestimated and soon develop into critical damages. The aforementioned issue should be emphasized in the light of studies (IACS 2010, Hamann et al. 2011) that show significant underreporting of relevant non-serious accidents. Among the most common non-accidental structural failures found in ship structures, which potentially could lead to critical loss of structural integrity, are cracks. Cracks are formed predominantly due to low stress-high cycle fatigue loading of the structure during its operational life as well as

by various other causes, such as harbor operations during cargo loading/unloading, poor initial design of structural details, overloading due to extreme sea conditions, dormant defects caused by inefficient workmanship during repairs, to name but a few. Hitherto the detection of cracks in a ship leads inevitably to its immediate repair which in most cases means the cease of operation for the ship. Therefore there is an increasing interest in developing the means and technology to address both the creation of cracks in the ship’s structure and finding alternative ways to arrest already existing cracks (e.g. the Co-Patch and MOSAIC European Funded Projects). The suitability of different crack arrest methods for marine applications, have been studied in laboratory conditions (Tsouvalis et al. 2009) and in a few cases on actual operating structures, proving their trust-worthiness (Grabovac et al. 2009, McGeorge et al. 2009). A step that would facilitate the wide acceptance of the aforementioned trend is the use of existing rules and regulations for the preliminary study of cracks on a case by case basis, aimed at assessing the most suitable course of action. Within this framework, a platform is developed and is described in this paper for the implementation of proper fatigue loads. A methodology is proposed that can be used as a common base for evaluating different crack arrest technologies. In this work the proposed methodology was employed for the comparative study of two cracks in an Aframax tanker following a simple approach. The stress intensity

281

factors were derived at the crack tips under different loading conditions and compared for each loading case. This well-known method, though no more state-of-the-art, is simple, straightforward and can serve as a first step for the assessment of the criticality of cracks. 2

Aframax tanker depicting the modeled

PLATFORM DESCRIPTION

An Aframax Tanker was used in order to study the development of defects (cracks) on the ship structure. This ship type was selected as a test bed for the developed procedure, because of reports which state that double bottom tankers have a relatively high frequency of occurrence of non-accidental structural failures (Eliopoulou et al. 2012), (Larsen et al. 2010) relative to their average age. Most frequent cause of developing cracks is high cycle fatigue loads which are characteristic of normal seagoing operation. 2.1

Figure 1. section.

FE model

In marine applications the loading that acts on the structure is a combination of different loads (cargo, wave induced loads etc.) that is impossible to reproduce in controlled conditions. Therefore the use of the FEM represents the most appealing alternative for structural assessment. Another characteristic of marine vessels is that their structural response due to the loads exerted from the marine environment, varies greatly, depending on ship type and its operating conditions. To this end the classification societies have developed the CSRs for certain shiptypes, which are the most widely accepted rules for the structural assessment of ships. This work is the result of the further development of a tool that integrates the prescribed CSR into the FE analyses of the ship’s structure. Originally this tool was developed to study the strength assessment of the ship’s structure (Zilakos et al. 2011); recently it was enriched so as to take into account fatigue loads. The methodology used for the FE modeling and the loading scheme is in full accordance with the CSR. The ship used in this study is an Aframax tanker; the principal dimensions of the ship are presented in Table 1. The modeled section of the ship is shown in Figure 1. This three-compartment model consists of the Table 1. Basic ship data. LBP BMLD DMLD Td = TSC

239 m 44 m 21 m 14.6 m

Figure 2. Bulkhead stiffeners modeled using shell elements (red) and Euler-Bernoulli beams (cyan).

Figure 3.

Stiffener mesh size.

cargo tank amidships along with one tank aft and one fore. The cargo holds No. 3, 4 and 5 have been modeled using fully integrated shell elements. In order to reduce the complexity of the geometry and meshing, it has been decided to model the stiffeners of the bulkhead and stringers with Euler-Bernoulli beam elements (Fig. 2). A global element length of approximately 200 mm has been chosen for the mesh generation. This value ensures that the web and flange of the T-profile stiffeners will be meshed with at least two shell elements (Fig. 3), given the fact that the height of the web of the stiffeners of the ordinary section range from 300 mm to 450 mm, according to the available drawings (Fig. 4). The thicknesses of the shell elements are in accordance with the actual thicknesses of the

282

Figure 5. The FLD (top) and WBD (bottom) conditions. Figure 4.

Aframax tanker’s web frame. Table 2.

structure. The material used in the analysis is high strength steel AH36 according to the ABS classification rules. The mechanical properties of the AH36 steel are: Young’s Modulus E = 210 GPa; Poisson’s ratio ν = 0.3 and yield stress = 355 MPa. 2.2

Loading and boundary conditions

The types of loads that were imposed on the threecompartment model for the fatigue analysis are both static and dynamic (IACS 2010). The loads taken into consideration are: • • • • •

weight of the structure and the cargo/ballast static sea pressure dynamic wave pressure dynamic tank pressure hull girder vertical/horizontal bending moments

Analyses were carried out for two representative loading conditions, according to the intended ship’s operation as prescribed by the CSR, namely, for oil tankers, for the Full Load Departure (FLD) condition and for the Water Ballast Departure condition (WBD). The implementation of the aforementioned loads was achieved using a subroutine that was written in FORTRAN and was called by ABAQUS during the FE analysis of the threecompartment model. The boundary conditions imposed on the model were in full accordance with the requirements of CSR. In all studied cases linear analyses were conducted. At this point it must be underlined that the proposed methodology could also take into consideration non-linearities. In total four indicative analyses have been conducted to demonstrate the potential of the aforementioned methodology. These cases consist of the ship operating in FLD and WBD (Fig. 5) in hogging and sagging (Table 2). The fatigue loading of these cases consists of cyclic alternation between hogging and sagging. However the aforementioned

Studied loading cases.

FLD-hogging FLD-sagging

WBD-hogging WBD-sagging

methodology provides the ability to study a vast combination of loading scenarios for several loading patterns. All four analyses were performed to the intact ship structure. Subsequently, and aimed at minimizing the computational cost, a sub-model of the structural member where the cracks were to be introduced was subtracted from the threecompartment model. Another reason that favors the proposed sub-modeling technique, in addition to the flexibility provided considering the mesh refinement and element type, is the modeling of details that might have been omitted in the threecompartment model. 2.3

Creation of the sub-model

In this study the web floor (Fig. 6) located amidships and starboard, was selected for the introduction of cracks. The selection of this particular structural member was based on existing surveys’ data for oil tankers. These reports indicate that openings or/and manholes in that region are prone to the development of cracks (Larsen et al. 2010). To this end two crack scenarios were envisaged, regarding the position of the cracks. Both cracks had a total length equal to 300 mm, originating from the flange stiffener of the manhole and extending to the web plate. To this end, the 200 mm flange was interrupted in both crack locations and a 100 mm crack length was also modeled on the web plate. Both cracks were inclined at 45o with respect to the vertical axis of the structure. Their location and nomenclature are depicted in Figure 7.

283

Figure 6. The starboard side web floor of the three compartment model.

Figure 8. Derivation of the Uy displacement analytical field for the hopper floor boundary.

Figure 7.

Regions where the cracks were introduced.

The boundary conditions that were imposed on the sub-model were, for each scenario, derived from the three-compartment model. The displacements and rotations were extracted from the web floors boundary regions. Since the mesh refinement level between the three-compartment model and the sub-model will in most cases differ, it is essential to express the displacements and rotations of the boundary analytically. In order to acquire these analytical expressions, polynomial fitting was performed to the initial values extracted from the three-compartment model (Fig. 8). As depicted in Figure 9, the values of the rotational degrees of freedom at the web floor boundary were negligible and therefore omitted in the present analyses.

Figure 9. Distribution of the nodal rotations for the hopper floor boundary.

3

RESULTS AND DISCUSSION

The evaluated von Mises stresses were significantly lower than the yield stress of the AH36 steel (355 MPa). The region in the vicinity of the crack tip that had surpassed the yield stress can be characterized as highly local (grey area in Fig. 10), compared to the size of the crack. Therefore the assumption of linear elastic analysis is valid and subsequently the use of Linear Elastic Fracture Mechanics (LEFM) can be applied.

284

Figure 11. Distribution of stresses normal to the Crack-2 edges for hogging (top) and sagging (bottom). Figure 10. Von Mises stress distribution near Crack-1 for hogging (top) and sagging (bottom).

The value of the J integral was derived only for Crack-1 and for all loading cases, and the Stress Intensity Factor (SIF) was calculated with the use of the following well-known LEFM formula: J = G =

K2 E

(1)

where G is the strain energy release rate, K is the SIF and E is the Young’s Modulus of Elasticity. The J integral value of Crack-2 was not taken into account as this crack did not exhibit mode-I opening. This is justified by the distribution of stresses that are normal to the crack edges (S11). As depicted in Figure 11, the stress distribution is virtually uniform for both sides of the crack edges and remains constant for hogging and sagging. Additionally, the stresses that are parallel to the crack edges (S22) exhibit a mode II behavior for Crack-2 (Fig. 12). Once again the distribution of S22 stresses in the Crack-2 area is practically identical for both hogging and sagging. Furthermore, the values of these stresses in the crack edges are significantly low. Reviewing the aforementioned results, it can be concluded that although the hopper tank floor

Figure 12. Distribution of stresses parallel to the Crack-2 edges for hogging (top) and sagging (bottom).

285

is subjected to fatigue loading, the area around Crack-2 is in effect experiencing static loads. This is also indicated by the almost identical von Mises stress distribution around the crack area for both hogging and sagging (Fig. 13). Hence the Paris law could not be applied for the calculation of the crack growth rate of Crack-2. As earlier mentioned, the crack locations were selected according to reports that cite the most commonly encountered crack locations in oiltankers (Larsen et al. 2010). The fact that Crack 2 is in reality not subjected to fatigue for the specific loading patterns, indicate that this crack could have been formed due to different loading patterns of the ship. Additionally, since a linear analysis cannot employ any contact algorithms, it is possible that element penetration at the crack edges may occur. This was indeed observed in the FLD condition and in these cases the J integral was not calculated (Fig. 14). On the contrary this was not observed in the WBD Crack-1 case. In Table 3 the SIF value is listed for the Crack-1 WBD case where the J integral could be evaluated.

Figure 14. Indicative figure showing the crack edge penetration.

Table 3.

Value of K and J for the WBD.

Crack case

Loading case

J Integral (J/m2)

K (MPa ⋅ m1/2)

Crack-1

WBD hogging WBD sagging

10611.40 1428.82

46.41 17.03

Table 4.

Calculated ΔΚ and crack growth rate.

Crack case

Loading case (WBD)

ΔK (MPa ⋅ m1/2)

dα/dN (mm/cycle)

Crack-1

Hogging–Sagging

29.38

0.00184

Once the K value was derived, an estimation of the Crack-1 growth rate was accomplished, based on the Paris law: da m =C( K) dN

Figure 13. Distribution of von Mises stresses around Crack-2 area for hogging (top) and sagging (bottom).

(2)

where ΔΚ is the difference of the SIF in hogging and sagging and C = 2.3E-12 and m = 3 are material constants in the fatigue crack growth expression (BS 7910:2005). The values of ΔK and crack growth rate of Crack-1 for the WBD scenario are listed in Table 4. In order to assess the criticality of the crack, the evaluated (ΔK, da/dN) point was plotted on a logarithmic chart along with the crack propagation curve of the AH-36 steel (Fig. 15). In Figure 15 the crack propagation curve of AH36 steel was assumed to be equal to the one described in BS 7910 for marine steels. Furthermore, the ΔΚth value was chosen equal

286

Figure 15. Crack propagation curve of AH36 steel along with the evaluated points (ΔK, da/dN) for Crack-1.

to 2 MPa ⋅ m1/2 following BS7910 and the value of KIC of AH36 was selected equal to 170 MPa ⋅ m1/2 (Sielski 1992). The ΔΚmax is: ΔK max

K max − K min =

( K max R

)

REFERENCES

= 107.73 MPa m

where Kmax = KIC and R = 2.73 for Crack-1. As observed in Figure 15, the (ΔΚ, da/dN) point of Crack-1 is further away from the midpoint of the AH36 steel’s Paris law curve, thus indicating that Crack-1 will propagate rapidly and rectification measures have to take place. Once again it must be noted that these cases were selected as a demonstrator of the proposed method; the J integral values could have been derived for the FLD condition as well through the implementation of contact algorithms but this was outside the scope of the present study. The difference in the exhibited response of the cracks for WBD condition is significant, especially when considering the fact that the studied cracks have the same characteristics, are located in the same structural member and in close proximity between each other. From the findings of the results, the need for a case by case approach of cracks in marine structures is once again being pointed out, due to the complexity of the structure and the constantly varying acting loads.

4

developed and a demonstration of its potential was presented through the introduction of two cracks with the same characteristics, in neighboring positions in the floor of a web frame of an Aframax oil tanker. The response of the two considered cracks under hogging and sagging loading was evaluated for the FLD and WBD conditions. It was observed that for different loading patterns (FLD, WBD), the response of the modeled cracks was different. Furthermore under the same loading pattern (WBD) the two cracks exhibited contradicting results, as Crack-1 would propagate while Crack-2 remained stationary. This remark underlines once again the necessity of a case by case study of cracks and the importance of creating a common base concerning the fatigue loading of structures operating in the marine environment.

CONCLUSIONS

A methodology based on the implementation of the fatigue loadings prescribed by CSR in the FEA has been presented. This methodology is proposed as a common base for the implementation of proper fatigue loads in the study of the effectiveness of different crack arrest technologies on real applications in marine structures. A platform was

ABAQUS 2010. Theory Manual, Version 6.10, Dassault Systèmes Simulia Corp. Providence: RI, USA. BS 7910:2005. Guide to methods for assessing the acceptability of flaws in metallic structures. Bloor, M., Datta, R., Gilinskiy, Y. and Horlick-Jones, T. 2006. Unicorn among the cedars: On the possibility of effective ‘smart regulation’ of the globalized shipping industry. Social and Legal Studies 15:534–551. Eliopoulou, E., Papanikolaou, A., Diamantis, P. and Hamann, R. 2012. Analysis of tanker casualties after the Oil Pollution Act (USA, 1990). Proceedings of the Institution of Mechanical Engineers, Part M: Journal of Engineering for the Maritime Environment 226:301–312. Grabovac I. and Whittaker D. 2009. Application of bonded composites in the repair of the ships structures—A 15-year service experience. Composites: Part A: Applied Science and Manufacturing, 40, pp. 1381–1398. Hamann, R., Eliopoulou, E. and Konovessis, D. 2011. Standard risk models for collision and grounding events of passenger vessels. Document Deliverable 5.1, GOAL Damage Stability, DG Research—FP7 2nd call. IACS. 2010. International Association of Classification Societies. General cargo ship safety FSA study— step 2 (Risk Analysis). Document MSC 87/INF.4. Maritime Safety Comittee, International Maritime Organization, London, UK, 2010. IACS. 2010. Common Structural Rules for Double Hull Tankers, International Association of Classification Societies. Knudsen, O.F. and Hassler, B. 2011. IMO legislation and its implementation: Accident risk, vessel deficiencies and national administrative practices. Marine Policy 35:201–207. Larsen, A.T. and Wilson, E.D. 2010. “Definition of Application Cases (Marine)”, Co-Patch Project Report, Composite Patch Repair for Marine and Civil Engineering Infrastructure Applications.

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McGeorge, D., Echtermeyer A.T., Leong K.H., Melve B. Robinson M. and Fischer K.P. 2009. Repair of floating offshore units using bonded fibre composite materials. Composites Part A: Applied science and manufacturing, 40(9): p. 1364–1380. Papanikolaou, A., Eliopoulou, E., Alissafaki, A., Mikelis, N., Aksu, S. and Delautre, S. 2007. Casualty analysis of Aframax tankers. Proceedings of the Institution of Mechanical Engineers, Part M: Journal of Engineering for the Maritime Environment 221:47–60. Papanikolaou, A., Eliopoulou, E. and Mikelis, N. 2006. Impact of Hull Design on Tanker Pollution. 9th International Marine Design Conference, Ann Arbot—Michigan.

Sielski, R.A. 1992, Fracture mechanics of ship structures, Naval Engineers Journal, 104 36–45. Tsouvalis, N.G., Mirisiotis, L.S. and Dimou, D.N. 2009. Experimental and numerical study of the fatigue behaviour of composite patch reinforced cracked steel plates. International Journal of Fatigue 31:1613–1627. Zilakos I.K., Karatzas V.A., Chatzidouros E.V. and Papazoglou V.J. 2011. Simulation of External Application of SuSy devices on an Aframax Tanker that has been Structurally Compromised, RINA, Royal Institution of Naval Architects—International Conference on Design and Operation of Tankers, Proceedings, pp. 121–130.

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Ultimate strength

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Design of Y stiffened panels in double hull tanker under axial compressive loads Ahmed Shawki El-Hanafi Department of Naval Architecture and Marine Engineering, Alexandria University, Alexandria, Egypt

Sherif Farouk Badran Department of Marine Engineering Technology, Arab Academy for Science, Technology and Maritime Transport, Alexandria, Egypt

Heba Wael Leheta Department of Naval Architecture and Marine Engineering, Alexandria University, Alexandria, Egypt

ABSTRACT: In typical tanker ship structures, deck and bottom panels are reinforced by longitudinals (stiffeners) in the longitudinal direction and transversely supported by widely spaced transverse structures (such as transverse bulkheads, deck beams and bottom floors). The conventional longitudinals are usually Tee, angle, bulb or flat bar profiles, while the transverses are typically T-beam sections. The objective of the present work is to replace the longitudinal conventional stiffener profiles with new longitudinal Y (Hat + Tee/Angle) stiffener profiles in order to obtain more strength/safety margin to weight ratio based on the advantage of using Hat section (closed section) to give more effective plate allowing an increase of the stiffener spacing, hence a reduction of the number of stiffeners. The weight of the stiffened deck and bottom panels and the unstiffened panel width using the new Y stiffener profiles are less than those panels with the as-built conventional stiffener profiles. The section modulus of the new Y stiffener profiles with the effective plate and the safety margin (ultimate strength minus applied compression stress) of deck and bottom panels with the new Y stiffener profiles are larger than for panels with the as-built conventional stiffener profiles. 1

INTRODUCTION

Stiffened panels are very important components in ship and offshore structures, and they form the backbone of most of the ship’s structure, and they are by far the most commonly used element in a ship. They can be found in bottom structures, decks, side shell, and superstructures. During the loaded condition, deck panels are subject to uniaxial compression stress in the longitudinal direction from the effect of the membrane compressive stress induced by hull girder vertical sagging bending moment, while during the ballast condition, inner and outer bottom panels are subject to uniaxial compression stress in the longitudinal direction from the effect of the membrane compressive stress induced by hull girder vertical hogging bending moment and the outer bottom panels are subject to additional lateral pressure from both seawater and double bottom ballast water. 2

ULTIMATE STRENGTH AND FAILURE MODES

The buckling collapse strength of stiffened panels, particularly in the deck and bottom structures, is

an important design requirement for the safety level of the hull girder, since its ultimate capacity under vertical bending is mainly governed by the buckling collapse strength of these structural components (Gaspar et al. 2011). Ultimate strength failure modes of deck and bottom panels include local failure of unstiffened panels, collapse of the longitudinally stiffened panels, and overall collapse of the gross panels. Gross panel (or grillage) buckling occurs when the transverse frames are not stiff enough to provide undeflecting support to the longitudinal stiffeners and they buckle together with the longitudinal stiffeners. If the transverse frames are rigid and provide adequate support to the longitudinals, failure will occur in the longitudinally stiffened panel between the transverse frames. In most cases, the transverse frames have substantially deeper webs and are more rigid than the longitudinal stiffeners, eliminating the possibility of grillage buckling. With the grillage buckling eliminated, the longitudinal stiffened panels between transverse frames must be analyzed (Chen 2003). The behaviour and design of stiffened panels under predominantly compressive loads has been studied by several authors based on analytical,

291

numerical and experimental approaches. A complete review on this topic can be found, for example, in Paik & Thayamballi (2003), Moatsos & Das (2006) and ISSC (2000, 2006, and 2009). It is generally accepted that the buckling collapse failure modes of a stiffened panel under predominantly compressive loads can be classified by collapse pattern in the following categories (Paik & Kim 2002): overall collapse after overall buckling of the plating and stiffeners as a unit, plate-induced failure by yielding at the corners of plating between stiffeners, plate-induced failure by yielding of stiffeners with attached plating at mid-span, stiffener induced failure by local buckling of stiffener web and stiffener-induced failure by lateral-torsional buckling or tripping of stiffeners. Some of them may in some cases interact and occur simultaneously. However, for design purposes, they are typically treated separately, i.e., the buckling collapse strength of a stiffened panel is usually assumed to be equal to the minimum strength value, obtained with each buckling collapse failure mode separately (Gaspar et al. 2011). The overall failure of the longitudinally stiffened panel is highly undesirable since it reduces the hull girder capacity to resist the applied bending moments. A well-designed structure does not collapse when the local plate fails as long as the stiffeners can resist the extra load due to the plate failure. However, if the lateral rigidity of the stiffeners is not sufficiently high, they may buckle as columns due to the increased compressive load. This failure mode is called beam-column buckling. Another failure mode of stiffeners is torsional (tripping) buckling due to inadequate torsional rigidity. As the stiffeners usually have thin-walled open cross-sections, such as T or angle crosssections. Due to the low torsional rigidity of the thin-walled open cross-section, tripping will occur prior to beam-column type flexural buckling when the stiffeners are subjected to axial compression. It has been recognized that tripping should be fully taken into consideration in estimating the ultimate longitudinal strength of ship hulls. Many classification societies have also required in their design rules that tripping strength of stiffeners should be checked. If stiffeners fail, the plate panels will lose almost all the lateral support to sustain inplane compressive load and consequently induce the overall failure of the stiffened panel. Hence, the overall stability of longitudinally stiffened panels under longitudinal in-plane compressive loads is governed by the lateral and torsional rigidity of stiffeners (Guo 2010). As the present study aims at applying the IACS-CSR design rules for double hull oil tankers in the analysis and design of stiffened panels against buckling failure modes. The following

buckling failure modes implemented in IACS-CSR section 10 buckling and ultimate strength (IACS 2010) for longitudinal stiffened panels are explicitly considered in the design formulation for conventional stiffener profiles shown in Figure 1. 1. Uniaxial buckling of the plating between the conventional stiffeners.

σu

p

⎧ηallowσ y − p ⎪ ⎛ 2.14 0.89 ⎞ =⎨ ⎪ηallow ⎜⎝ β − β ⎟⎠ σ y ⎩

for β ≤1.58 p

β >1.558 (1)

where:

β=

s

σ y− p

t p− net

E

(2)

The allowable buckling utilisation factor ηallow to be considered for design purposes is defined as a function of the vertical position of the structural element in the hull girder cross section. Typically, for structural elements above 0.5D, with D the ship moulded depth, the allowable buckling utilisation factor is defined as ηallow = 1.0. For structural elements below 0.5D, this factor is reduced to ηallow = 0.9. This reduction in ηallow for structural elements in the lower region of the hull girder cross sections is an implicit but simplified way of accounting for the effects neglected in the buckling strength formulation and which may reduce the buckling strength, e.g., the effect of the lateral pressure P and the effects of the compressive stress σy and shear stress τ. These effects are typically more significant in the lower region of the hull girder cross sections due to the higher lateral pressures applied and induced local bending of secondary structures [IACS 2010]. Hence, ηallow = 1.0 is applied here for deck panels and ηallow = 0.9 is applied here for both inner and outer bottom panels.

Figure 1. Geometrical characteristics conventional T stiffener profiles.

292

of

the

2. Beam-Column buckling of the conventional stiffeners with attached effective plating. σ cr − bc =

2

H − 4ZeQ (ηallow allowσ y

H

sZeC f

σ ET =

− M1C f )

2ZeQ

ε = 1+

(3) where: H

ZeQ

allow

π 2 stt p

Q=

net net

l2

Cf = CP =

wQF FE − M1Q

ZeC f

y s

⎛ As ⎞ ⎜ 1+ ⎟ stt p net ⎠ ⎝

π 2 FE (1 Cp ) l2

(4) (5)

(6)

1

(7)

⎞ 0.91 ⎛ 12 I e 1+ − 1⎟ ⎜ Ca ⎝ stt 3p net ⎠ 2

⎧ ⎪ ⎪ Ca = ⎨ ⎪ ⎪ ⎩

⎛ 1 2s ⎞ + ⎟ ⎜⎝ 2s l ⎠ 2 2 ⎡ ⎛ l ⎞ ⎤ ⎢ 1+ ⎜ ⎥ ⎝ 2 s ⎟⎠ ⎥ ⎢⎣ ⎦

for l ≥ 2 s (8)

⎞ π 2Is E ⎛ επ + 0.385IT ⎟ ⎜ 2 IP ⎝ l ⎠

(17)

l4 4( 3 4 ⎛ s π Is ⎜ 3 + 4 ⎝ t p net

f

0.5t f − net ) ⎞ ⎟ 3 3tw net ⎠

(18)

The polar moment of inertia of the stiffener IP, the St. Venant’s moment of inertia of the stiffener IT, and the sectorial moment of inertia of stiffener Is are calculated at point C (see Figure 1). The local buckling of the stiffener web and flange is controlled by slenderness limits. The effects of the welding-induced initial imperfections on the buckling collapse strength are not explicitly considered in this design formulation, expect for the beam-column buckling mode, where the amplitude of the stiffener initial distortion is explicitly included as a design variable in IACS-CSR. For new Y stiffener profiles shown in Figure 2 the following buckling failure modes are considered: 1. Uniaxial buckling of the plating between new stiffeners, using the same procedure for uniaxial buckling of the plating between conventional

for l < 2 s (9)

w = wo + w1 s ⎛ l ⎞ wo = min ⎜ , ,10 ⎝ 250 250 ⎟⎠

(10)

w1 =

P sl 4 384 EII e

(11)

M1 =

P sl2 24

(12)

3. Lateral-torsional buckling or tripping of the conventional stiffeners.

σ cr

ηallow CT σ y

t

(13)

s

where: ⎧1.0 ⎪ 1 CT = ⎨ ⎪φ + φ 2 − λ 2 T ⎩

for λT > 0.2

λT − 0. 0 2 ) + λT2 )

φ λT =

for λT ≤ 0.2

σy

s

σ ET

(14)

(15) (16)

Figure 2. Geometrical characteristics of the new Y stiffener profiles.

293

Figure 3.

2.

3.

4.

5.

New Y stiffener profiles spacing.

stiffeners failure mode (1) and replacing s with the maximum of s1, s2 shown in Figure 3. Beam-column buckling of the conventional (Tee/ Angle) stiffeners with the attached plating of the hat section flange, using the same procedure for conventional stiffener profile failure mode (2) and replacing s, tp-net with bhf/2 and thf-net, respectively. Beam-column buckling of Y stiffeners with attached plating, using the same procedure for conventional stiffener profile failure mode (2). The attached effective plating is calculated based on Eurocode (Johansson et al. 2007) and (IACS 2010). Lateral-torsional buckling or tripping of the conventional (Tee/Angle) stiffeners attached to the hat section flange, using the same procedure for conventional stiffener profile failure mode (3) and replacing s, tp-net with bhf/2 and thf-net, respectively. This failure mode is based on the assumption that closed Hat sections are rare to buckle torsionally because of their large torsional rigidity. Local buckling of the hat section web, using the same procedure for uniaxial buckling of the plating between conventional stiffeners failure mode (1) and replacing s, tp-net with bhw and thw-net, respectively.

Equations for ultimate strength and safety margin were programed and solved using EES (Engineering Equation Solver) software. EES is a general equation-solving program that can numerically solve thousands of coupled non-linear algebraic and differential equations. 3

VERY LARGE CRUDE CARRIER CHARACTERISTICS

A VLCC is used as an example. The midship section and principal particulars are shown in Figure 4 and summarized in Table 1, respectively. 3.1

Characteristics of as-built conventional panels

3.1.1 Geometrical characteristics of the as-built conventional panels Geometrical characteristics of bottom and deck panels with as-built conventional stiffener profiles are summarized in Table 2.

Figure 4. As-built characteristics.

Table 1.

VLCC

midship

section

Principal particulars of the VLCC.

Ship type Hull type LOA LBP L B D Td Tsc Displacement at Tsc (Ton) Deadweight, DWT (Tonnage) Design SWBM (Sagging) Design SWBM (Hogging)

VLCC Double 332 320 315.82 58 31 20.8 22 341,000 300,000 6160680 7553700

3.1.2

Ultimate strength characteristics of the as-built conventional panels Ultimate strength characteristics for bottom and deck panels with as-built conventional stiffener profiles are summarized in Table 3. Table 3 shows that the stifferer torsional buckling (tripping) is the most critical failure mode for deck and bottom panels with as-built conventional stiffener profiles. 3.2

Generation of equivalent new Y stiffener profiles

Fabrication of Y stiffener profiles are based on welding of Tee/Angle stiffener profiles on the

294

Table 2. Geometrical characteristics of the as-built panels. Parameter

Outer bottom

Inner bottom

Deck

Bp tpo Stiffener type s l dw two bf tfo Steel type σy-p, σy-s N Ast AP Zgross Ze

41860 20.684 T-Profile 910 5120 580 12 200 20 AH32 315 42 460320 1326152.24 3.44*106 2.804*106

41860 19.5 T-Profile 910 5120 630 12 200 20 AH32 315 42 485520 1301790 3.8*106 3.057*106

50960 21 T-Profile 910 5120 400 12 150 18 AH32 315 53 397500 1467660 1.66*106 1.225*106

Table 4. Bottom and deck panels’ modification cases with Y stiffener profiles. Case

N = N1 + N2

Figure

Outer and inner bottom panels 1 910 910 12 10 2 840 840 12 12 3 780 780 14 12 4 680 680 16 14 5 610 910 14 12 6 390 910 16 14

22 24 26 30 26 30

5

Case

s1

s2

s1

Deck panels 1 800 2 700 3 490 4 340 5 810 6 710 7 640

N1

N2

s2

N1

N2

N = (N1 + N2)+ 1 Figure

910 910 910 910 810 710 640

12 12 14 16 12 14 16

16 18 20 22 18 20 22

29 31 35 39 31 35 39

6

Table 3. Compressive stress and ultimate strength characteristics of the as-built conventional panels. Parameter

Outer bottom

Inner bottom

Deck

Unstiffened panel σx Psw PBallast σu–p σu–p/AP S.M.un S.M.un/AP

180.709 0.2212 0.0302 239.20 180.360 58.476 44.094

141.03 – – 216.39 166.225 75.36 57.889

219.9 – – 258.1 175.86 38.2 26.03

Stiffened panel σcr–bc σcr–t σu(St.Panel) σu(St.Panel)/AP S.M.st S.M.st/AP

253.1 216.8 216.8 163.48 36.091 27.215

275.4 207.7 207.7 159.55 66.67 51.214

292.9 245.9 245.9 167.55 26.0 17.715

Overall panel σu σu/AP S.M. S.M./AP

216.8 163.48 36.091 27.215

207.7 159.55 66.67 51.214

245.9 167.55 26.00 17.715

Figure 5. Outer and inner bottom panel modification cases.

Figure 6.

Deck panels modification cases.

Table 5. Geometrical characteristics of outer bottom panels with Y stiffener profiles in comparison with those panels with as-built conventional stiffener profiles.

top of hat section flange. T and angle stiffener profiles used in the fabrication of Y profiles are selected from different shipbuilding industries (rolled sections) and from existing ships. Additional fabricated T profiles are generated based on the restrictions on the stiffener profile dimensions stated in the IACS-CSR. Results of generation of

Parameter

Hat + T

Hat + L

Ast Ast% Ap Ap% Ze Z e% Zgross Zgross%

371536 19.29 1237368.24 6.69 4.24E+06 51.35 5.22E+06 51.71

380875 17.26 1246707.24 5.99 3.77E+06 34.59 4.87E+06 41.66

295

Table 6. Geometrical characteristics of inner bottom panels with Y stiffener profiles in comparison with those panels with as-built conventional stiffener profiles.

Table 9. Ultimate strength characteristics of inner bottom panels with Y stiffener profiles in comparison with those panels with as-built conventional stiffener profiles.

Parameter

Hat + T

Hat + L

Parameter

Hat + T

Hat + L

Ast Ast% Ap Ap% Ze Ze% Zgross Zgross%

385088 20.69 1201358 7.71 4.66E+06 52.31 5.83E+06 53.42

394834 18.68 1211104 6.97 3.96E+06 29.52 5.18E+06 36.32

σu–p σu–p% σu–p/Ap σu–p /Ap% S.M.Un S.M.Un% S.M.Un/Ap S.M.Un/Ap% σu(N.St.Panel) σu(N.St.Panel)% σu(N.St.Panel)/AP σu(N.St.Panel)/AP% S.M.St S.M.St% S.M.St/AP S.M.St/AP% σu σ u% σu/AP σu/AP% S.M. S.M.% S.M./Ap S.M./Ap%

267.90 23.80 207.21 24.65 126.30 67.60 97.53 68.48 258.10 24.27 201.96 26.58 116.80 75.19 89.92 75.57 243.10 17.04 189.39 18.70 101.60 52.39 78.84 53.94

243.10 12.34 191.13 14.98 101.60 34.82 79.25 36.90 247.70 19.26 190.73 19.54 106.30 59.44 81.85 59.82 230.10 10.78 181.95 14.04 88.60 32.89 69.35 35.41

Table 7. Geometrical characteristics of deck panels with Y stiffener profiles in comparison with those panels with as-built conventional stiffener profiles. Parameter

Hat + T

Hat + L

Ast Ast% Ap Ap% Ze Ze% Zgross Zgross%

367710 7.49 1437870 2.03 1.53E+06 24.64 2.10E+06 26.20

384160 3.36 1454320 0.91 1.45E+06 18.34 1.99E+06 19.72

Table 8. Ultimate strength characteristics of outer bottom panels with Y stiffener profiles in comparison with those panels with as-built conventional stiffener profiles. Parameter

Hat + T

Hat + L

σu–p σu–p% σu–p/Ap σu–p/Ap% S.M.Un S.M.Un% S.M.Un/Ap S.M.Un/Ap% σu(N.St.Panel) σu(N.St.Panel)% σu(N.St.Panel)/AP σu(N.St.Panel)/AP% S.M.St S.M.St% S.M.St/AP S.M.St/AP% σu σu% σu/AP σu/AP% S.M. S.M.% S.M./Ap S.M./Ap%

266.90 11.58 208.39 15.53 86.10 47.24 65.58 48.74 249.20 14.94 196.79 20.38 67.70 87.58 51.47 89.11 248.10 14.44 190.66 16.63 66.80 85.09 50.50 85.56

283.50 18.52 215.40 19.42 101.70 73.92 77.27 75.24 242.30 11.76 185.35 13.37 61.30 69.85 46.26 69.99 239.20 10.33 185.19 13.28 58.20 61.26 44.04 61.81

the hat sections are shown in Table 4 with different cases for deck and bottom panels. For all cases shown in Table 4 the following additional restrictions are used in order to generate the Y stiffener profiles: • bhf is assumed to be equal to (bhw, 1.5 bhw and 2 bhw). • θh = 30 °, 45 °, 60 ° and 90°. • Total Y stiffeners area (weight) ranging from 65% to 100% of the total as-built stiffeners area (weight). • The unstiffened panel width using the new Y stiffener profiles (maximum of s1 and s2) is less than those panels with the as-built conventional stiffener profiles. • The section modulus of the new stiffener profiles with the effective plate (Ze and Zgross) and the safety margin (ultimate strength minus applied compression stress) of deck and bottom panels with the Y stiffener profiles are larger than for panels with the as-built conventional stiffener profiles. 3.2.1 Comparison between Y stiffener profiles and as-built conventional stiffener profiles Geometrical charateristics of Y stiffener profiles in comparison with as-built conventional stiffener profiles are shown in Tables 5, 6 and 7.

296

Tables 5, 6 and 7 show that using Y stiffener profiles for deck and bottom panels can save weight represented by (Ast%, Ap%), give more effective net and gross stiffener section modulus represented by (Ze%, Zgross%) than using as-built conventional T stiffener profiles. Ultimate strength characteristics of bottom and deck panels with Y stiffener profiles in comparison with those panels with as-built conventional stiffener profiles are shown in Tables 8, 9 and 10. Table 8 and Table 9 show that using Y stiffener profiles can give the following advantages for unstiffened, stiffened and overall outer/inner bottom panels: more ultimate strength, more ultimate strength to panel area (weight) ratio, more safety margin, and more safety margin to panel area (weight) ratio than using as-built conventional T stiffener profiles. Table 10 shows that using Y stiffener profiles can give the following advantages for stiffened and overall deck panels: more ultimate strength, more ultimate strength to panel area (weight) ratio, more safety margin, and more safety margin to panel area (weight) ratio than using asbuilt conventional T stiffener profiles. While deck unstiffened panel ultimate strength remains unchanged because the cases that satisfy the restrictions stated in section 3.2 are (1, 2, 3, and 4) have s2 value equal to the as-built conventional T stiffener spacing.

Table 10. Ultimate strength characteristics of deck panels with Y stiffener profiles in comparison with those panels with as-built conventional stiffener profiles. Parameter

Hat + T

Hat + L

σu–p σu–p% σu–p/Ap σu–p/Ap% S.M.Un S.M.Un% S.M.Un/Ap S.M.Un/Ap% σu(N.St.Panel) σu(N.St.Panel)% σu(N.St.Panel)/AP σu(N.St.Panel)/AP% S.M.St S.M.St% S.M.St/AP S.M.St/AP% σu σu% σu/AP σu/AP% S.M. S.M.% S.M./Ap S.M./Ap%

258.10 0.00 179.50 2.07 37.80 −1.05 25.76 −1.02 275.20 11.92 187.94 12.17 54.70 110.38 37.35 110.86 258.10 4.96 176.26 5.20 37.80 45.38 25.76 45.42

258.10 0.00 177.47 0.92 37.00 −3.14 25.22 −3.11 258.90 5.29 176.77 5.51 37.70 45.00 25.74 45.30 258.10 4.96 176.37 5.27 36.90 41.92 25.19 42.22

Table 11.

Dimensions of Y (Hat + T) stiffener profile for outer bottom panels.

Stiffener gives maximum of

bhf

bhw

tho

θh

dw

two

bf

tfo

Case

S.M.% S.M.st% S.M.un% S.M./Ap% S.M.st/Ap% S.M.un/Ap% Ap%

468 305 285 468 305 285 422

312 305 285 312 305 285 282

9.5 9.5 9 9.5 9.5 8 8

60 60 30 60 60 30 30

300 350 400 300 350 460 450

12 12 15 12 12 12 12

200 220 200 200 220 200 200

18 20 20 18 20 18 18

3 5 3 3 5 3 1

Table 12.

Dimensions of Y (Hat + L) stiffener profile for outer bottom panels.

Stiffener gives maximum of

bhf

bhw

tho

θh

dw

two

bf

tfo

Case

S.M.% S.M.st% S.M.un% S.M./Ap% S.M.st/Ap% S.M.un/Ap% Ap%

455 455 282 455 455 282 533

455 455 282 455 455 282 267

12 12 8 12 12 8 8.5

60 60 45 60 60 45 45

285 285 482 234 285 482 482

10.5 10.5 11.5 11.5 10.5 11.5 11.5

100 100 150 90 100 150 150

15 15 18 16 15 18 18

1 1 4 1 1 4 1

297

Table 13.

Dimensions of Y (Hat + T) stiffener profile for inner bottom panels.

Stiffener gives maximum of

bhf

bhw

tho

θh

dw

two

bf

tfo

Case

S.M.% S.M.st% S.M.un% S.M./Ap% S.M.st/Ap% S.M.un/Ap% Ap%

323 305 282 468 305 282 607

323 305 282 312 305 282 303

9.5 11 8.5 9.5 11 8 8

45 60 45 60 60 45 60

350 300 482 300 300 460 350

15 15 12 12 15 12 12

200 200 175 200 200 200 200

20 20 18 20 20 18 18

3 5 4 3 5 4 1

Table 14.

Dimensions of Y (Hat + L) stiffener profile for inner bottom panels.

Stiffener gives maximum of

bhf

bhw

tho

θh

dw

two

bf

tfo

Case

S.M.% S.M.st% S.M.un% S.M./Ap% S.M.st/Ap% S.M.un/Ap% Ap%

504 455 468 348 455 468 607

336 455 312 348 455 312 303

11 12.5 9.5 10 12.5 8.5 8

60 60 60 45 60 60 60

377 284 482 482 284 482 482

11.5 11.5 11.5 11.5 11.5 11.5 11.5

120 100 150 150 100 150 150

23 16 18 18 16 18 18

2 1 3 2 1 3 1

Table 15.

Dimensions of Y (Hat + T) stiffener profile for deck panels.

Stiffener gives maximum of

bhf

bhw

tho

θh

dw

two

bf

tfo

Case

S.M.% S.M.St% S.M./Ap% S.M.St/Ap% Ap%

179 141 179 141 252

179 141 179 141 168

8 9 8 9 8

30 45 30 45 45

250 250 250 250 282

15 12 15 12 11

150 150 150 150 150

22 22 22 22 18

3 4 3 4 3

Table 16.

Dimensions of Y (Hat + L) stiffener profile for deck panels.

Stiffener gives maximum of

bhf

bhw

tho

θh

dw

two

bf

tfo

Case

S.M.% S.M.St% S.M./Ap% S.M.St/Ap% Ap%

252 252 252 252 252

168 168 168 168 168

9 9 9 9 8.5

45 45 45 45 45

332 332 332 332 332

11.5 11.5 11.5 11.5 11.5

120 120 120 120 120

18 18 18 18 18

3 3 3 3 3

Tables 11–16 show the dimensions of Y stiffener profiles to give maximum safety margin, maximum safety margin to panel area ratio and maximum panel area reduction percentages for different bottom and deck panels.

4

CONCLUSION

From the above results we can conclude that using Y stiffener profiles in shipbuilding instead of using conventional stiffener profiles in deck and bottom

298

panels subjected to axial compressive loads can lead to greater ultimate strength, safety margin, section modulus and weight saving. 5

NOMENCLATURE

Ap Ap% As Ast Ast% bf bhf bhw B Bp Cf CT dw D ef

E FE Ie IP Is IT l L LBP LOA M1 N N1

Total Panel Cross Section Area (mm2) Reduction Percentage in Ap Stiffener Cross Section Area (mm2) Total Stiffeners Cross Section Area (mm2) Reduction Percentage in Ast Breadth of the Stiffener Flange (mm) Breadth of the Hat Section Flange (mm) Height of the Hat Section Web (mm) Breadth of the Ship (m) Panel Breadth (mm) The Elastic Support Provided by the Stiffener (N/mm2) Torsional Buckling Coefficient Stiffener Web Depth (mm) Depth of the Ship (m) The Vertical Distance from the Center of the Stiffener Flange to the Stiffener Intersection Point with the Attached Plate (mm) Young’s Modulus of Elasticity (N/ mm2) Ideal Elastic Buckling Force of the Stiffener (N) Net Moment of Inertia of the Stiffener with Effective Width of Attached Plating (mm4) Stiffener Polar Moment of Inertia (mm4) Stiffener Sectorial Moment of Inertia (mm6) Stiffener St. Venant’s Moment of Inertia (mm4) Stiffener Span (Frame Spacing) (mm) Ship Rule length (m) Ship Length between perpendiculars (m) Ship Length overall (m) Bending Moment Due to Lateral Load (N.mm) Total Number of Stiffener in the Panel Number of Y stiffeners in central tank

N2 P PBallast Psw s s1 s2 S.M. S.M.% S.M./Ap S.M./Ap% S.M.St. S.M.St.% S.M.St./Ap S.M.St./Ap% S.M.Un. S.M.Un.% S.M.Un./Ap S.M.Un/Ap% SWBM tfo tf-net tho thf-net thw-net tPo tp-net two tw-net Td Tst w wo w1 Ze Ze%

299

Number of Y stiffeners in wing tanks Lateral Load (Pressure) (N/mm2) Ballast Tank Pressure (N/mm2) Still water Pressure (N/mm2) Stiffener Spacing (mm) Inner Spacing between Hat Section Inclined Webs (mm) Outer Spacing between Hat Section Inclined Webs (mm) Panel Safety Margin (N/mm2) Increasing Percentage in S.M. Panel Safety Margin to Total Panel Area Ratio (MN/m4) Increasing Percentage in S.M./Ap Stiffened Panel Safety Margin (N/ mm2) Increasing Percentage in S.M.St. Stiffened Panel Safety Margin to Total Panel Area Ratio (MN/m4) Increasing Percentage in S.M.St./Ap Unstiffened Panel Safety Margin (N/mm2) Increasing Percentage in S.M.Un. Unstiffened Panel Safety Margin to Total Panel Area Ratio (MN/m4) Increasing Percentage in S.M.Un./Ap Still-Water Bending Moment (KN ⋅ m) As-Built (Gross) Stiffener Flange Thickness (mm) Stiffener Flange Net Thickness (mm) Hat Section As-Built (Gross) Thickness (mm) Hat Section Flange Net Thickness (mm) Hat Section Web Net Thickness (mm) As-Built (Gross) Plate Thickness (mm) Plate Net Thickness (mm) As-Built (Gross) Stiffener Web Thickness (mm) Stiffener Web Net Thickness (mm) Design Draft (m) Scantling Draft (m) Later Deformation of the Stiffener (mm) Assumed Initial Imperfection (mm) Deformation of the Stiffener Due to Lateral Load (mm) Net Section Modulus of Stiffener Including Effective Plating (mm3) Increasing Percentage in Ze

Gross Section Modulus of Stiffener Including Effective Plating (mm3) Zgross% Increasing Percentage in Zgross Critical Beam-Column Buckling σcr-bc Stress (N/mm2) Critical Torsional Buckling Stress σcr-t (N/mm2) Reference Torsional Buckling Stress σET (N/mm2) Ultimate Strength (N/mm2) σu Increasing Percentage of σu σu% Ultimate Strength to Total Panel σu/Ap Area Ratio (MN/m4) σu/Ap% Increasing Percentage in σu/Ap Ultimate Strength of the Convenσu(St.Panel) tional Stiffened Panel (N/mm2) Increasing Percentage in σu(St.Panel) σu(St.Panel)% σu(St.Panel)/Ap Ultimate Strength of the Conventional Stiffened Panel to Total Panel Area Ratio (MN/m4) σu(St.Panel)/Ap% Increasing Percentage in σu(St.Panel)/Ap Ultimate Strength of the New Stiffσu(N.St.Panel) ened Panel (N/mm2) σu(N.St.Panel)% Increasing Percentage in σu(N.St.Panel) σu(N.St.Panel)/Ap Ultimate Strength of the New Stiffened Panel to Total Panel Area Ratio (MN/m4) Increasing Percentage in σu(N.St.Panel)/ σu(N.St.Panel)/ Ap% Ap Axial Compressive Stress (N/mm2) σx Yield Stress (N/mm2) σy Yield Stress of the Unstiffened Plate σy-p (N/mm2) Yield Stress of the Stiffener (N/ σy-s mm2) Ultimate Strength of the Unstiffσu-p ened Plate (N/mm2) Increasing Percentage in σu-p σu-p% Ultimate Strength of the Unstiffσu-p/Ap ened Plate to Total Panel Area Ratio (MN/m4) Increasing Percentage in σu-p/Ap σu-p/Ap% Allowable Utilization Factor ηallow Reference Degree of Slenderness for λT Torsional Buckling Angle of Hat Section Inclined Web Θh (Deg.) Stiffener Degree of Fixation ε Zgross

β

Slenderness Ratio of the Unstiffened Plate

REFERENCES Chen Y. (2003). “Ultimate Strength Analysis of Stiffened Panels Using a Beam-Column Method”. Ph.D Thesis, Faculty of the Virginia Polytechnic Institute and State University. Gaspar B., Teixeira A.P., Guedes Soares C., Wang G. (2011). “Assessment of IACS-CSR Implicit Safety Levels for Buckling Strength of Stiffened Panels for Double Hull Tankers”, Marine Structures, 24, 478–502. Guo J. (2010). “Reliability-Based Inspection Planning with Application to Deck Structure Thickness Measurement of Corroded Aging Tankers”. Ph.D Thesis, Department of Naval Architecture and Marine Engineering, University of Michigan. IACS (2010). “Buckling and Ultimate Strength”, Common structural rules for double hull oil tankers. Section 10. International Association of Classification Societies, London, UK. ISSC (2000). “Ultimate Strength”, International Ship and Offshore Structures Congress, Committee III.1, (prepared by M.L. Kaminski et al.), 2–6 October 2000, Nagasaki, Japan. ISSC (2006). “Ultimate Strength”, International Ship and Offshore Structures Congress, Committee III.1, (prepared by Yao T et al.), 20–25 August 2006, Southampton, UK. ISSC (2009). “Ultimate Strength”, International Ship and Offshore Structures Congress, Committee III.1, (prepared by J.K. Paik et al.), 16–21 August 2009, SEOUL, KOREA. Johansson B., Maquoi R., Sedlacek G., Müller C. and Beg D. (2007). “Commentary and Worked Examples to EN 1993-1-5 Plated Structural Elements”, Luxembourg: Office for Official Publications of the European Communities, October. Moatsos I. & Das P.K. (2006). “Structural Reliability Framework for Floating Production, Storage and Offloading Vessels/Floating Surface Units”, Universities of Glasgow and Strathclyde for the Health and Safety Executive 2006. Paik J.K. & Thayamballi A.K. (2003). “Ultimate Limit State Design of Steel-Plated Structures”. John Wiley & Sons, Ltd. Paik J.K. & Kim B.J. (2002). “Ultimate Strength Formulations for Stiffened Panels under Combined Axial Load, In-Plane Bending and Lateral Pressure: A Benchmark Study”. Thin-Walled Structures; vol. 40:45–83.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Nonlinear buckling behavior of stiffened ship panels M. Ozdemir & A. Ergin Istanbul Technical University, Istanbul, Turkey

ABSTRACT: Correct buckling strength calculation of stiffened panels leads to effective ultimate strength analysis. Under this point of view, the nonlinear buckling behavior of different stiffened panels is examined. Firstly, a series of finite element eigenvalue analysis has been carried out to determine the bucking mode shapes of stiffened panels. Then, a nonlinear finite element analysis was carried out by considering initial imperfections in order to investigate the effects of stiffener type and several geometric parameters on the buckling behavior of the stiffened ship panels. Beside this, the effects of lateral pressure on the nonlinear buckling behavior of the stiffened panels were also investigated. It is observed that the stiffener type has a great effect on the nonlinear buckling behavior of the stiffened ship panels. Tee bar stiffeners increase the stiffness of panels more than angle bar and flat bar stiffeners. 1

INTRODUCTION

Stiffened panels are widely used as structural components of ships, bridges, offshore structures etc. A reliable estimation of buckling strength for the stiffened panels has great importance. Caldwell (1965) calculated the ultimate strength of a ship hull. Caldwell’s method had some deficiencies, thus Smith (1977) proposed a new method called simplified beam column method. The simplified beam-column method is used for design purpose because of that the method is simple and gives acceptable results. This method gives more accurate results when stiffener characteristics are dominant on the panel behavior. The simplified beam-column methods consider elastoplastic behavior of panels and local buckling. Ueda & Rashed (1984) proposed a new method for ultimate strength analysis of stiffened panels and ship hull. This method was proposed to reduce computational time of ultimate strength analyses. ISUM aims to decrease degree of freedom and nodes on the system. ISUM is based on the finite element method. This method models the structure as a large structural unit. The behavior of the structural unit is represented by special purpose shape functions. The ISUM provides effective solution process for large structures. Paik et al. (2001) proposed a large deflection orthotropic plate approach for the ultimate strength estimation of stiffened panels under biaxial compression/thrust and lateral pressure. The investigated panel has a number of one-sided small stiffeners in either one or both orthogonal directions.

Large deflection orthotropic plate approach gives accurate results for overall collapse. Results may be inconsistent for local buckling between stiffeners and stiffener tripping. Chen (2003) investigated the ultimate strength of several panels by using a simplified method and compared results with those found in literature. Hughes et al. (2004) investigated the buckling mode of stiffened panels depending on the stiffener properties. Fujikubo et al. (2005a) carried out a series of elastic/elastoplastic large deflection finite element analysis to investigate ultimate strength of continuous plating under combined transverse thrust and lateral pressure. Ultimate strength formulations are derived by using finite element results. Fujikubo et al. (2005b) investigated the effect of stiffener type, plate-stiffener interaction and continuity of rotation on the ultimate strength of stiffened panels. In that study, local collapse of stiffened panels was investigated considering hungry horse mode initial imperfections. Gordo & Soares (2008) investigated the effect of stiffener type on the buckling and collapse behavior of stiffened panels. In that study, flat bar, L and U shaped stiffeners are investigated. Yao (2009) investigated the elastic local buckling behavior of stiffened panels considering platestiffener interaction using principle of minimum potential energy. Stiffened panels undergo several collapse modes depending on the stiffener properties. If the stiffeners are rigid enough, local plate buckling occurs between the stiffeners. In case of lower stiffener bending rigidity, overall buckling may be observed.

301

The buckling strength of a simply supported plate is evaluated by FEM eigenvalue analysis. The results are compared with the analytical solution obtained by the equilibrium method. 2.1

Formulation of elastic plate buckling

Differential equation for elastic plate buckling problem can be given as Eq. (1). Figure 1.

Overall collapse of stiffened panel.

∂4w ∂4w ∂4w + 2 + ∂x 4 ∂x 2 ∂y 2 ∂y 4 ⎛ 1 ∂2 w ∂2 w ∂2 w ⎞ = ⎜ N x 2 + 2N 2 N xy + Ny 2 ⎟ D⎝ ∂x∂y ∂x ∂y ⎠

(1)

where Nx, Ny and Nxy denote the internal forces. For a simply supported plate under uniaxial thrust Nx = -qx, Ny = 0 and Nxy = 0 terms are substituted into the differential equation. Eq. (1) leads to following form:

Figure 2.

D∇ 2∇ 2w + N x

Tripping of stiffener.

Beside this, if the stiffeners are very slender, tripping of stiffeners may be occurred. In this study, the effects of stiffener type and stiffener dimensions on the local buckling strength were investigated by carrying out both the linear and nonlinear finite element analyses. Firstly a linear eigenvalue analysis is carried out to determine the buckling strength and buckling mode shapes of the stiffened panels. Furthermore, a nonlinear finite element analysis is carried out by considering initial imperfections as overall buckling mode shape. In literature, effect of stiffener dimensions, plate thickness and lateral pressure on the ultimate strength of stiffened panels investigated considering local collapse of plate between stiffeners or local collapse of stiffeners. In this study, overall collapse of stiffened panels is considered, effects of stiffener type, stiffener web height, plate thickness and lateral pressure on ultimate strength of stiffened panels are investigated. Initial imperfections are imposed as overall buckling mode of stiffened panels. 2

∂ 2w =0 ∂x 2

(2)

Buckling mode shape of simply supported plate can be assumed as Eq. (3). w ( x, y )

∞

∞

⎛ mπ x ⎞ ⎛ nπ y ⎞ i ⎜ ⎟ . sin ⎝ b ⎟⎠ a ⎠

∑ ∑ wmn sin ⎜⎝

m =1 n =1

(3)

Eq. (3) satisfies simply supported boundary conditions. m and n denote buckling half wave numbers in longitudinal and transverse directions, respectively. By substituting Eq. (3) into Eq. (2), Eq. (4) is obtained. 2 ⎡ 2 n2 ⎞ m2 ⎤ 4⎛m ⎢ D π + ∑ ∑ ⎢ ⎜⎝ a2 b2 ⎟⎠ qxπ 2 a2 ⎥⎥ m =1 n =1 ⎣ ⎦ ⎛ mπ x ⎞ ⎛ nπ y ⎞ wmn sin ⎜ . sin ⎜ =0 ⎝ a ⎟⎠ ⎝ b ⎟⎠ ∞

∞

(4)

Trivial solution for Eq. (4) is wmn = 0. Minimum longitudinal buckling strength is obtained by substituting n = 1 into Eq. (4). Longitudinal buckling

LINEAR BUCKLING ANALYSIS

A finite element eigenvalue analysis was carried out to determine the effect of stiffener type and stiffener properties on buckling strength of stiffened panels.

Figure 3. Simply supported plate under uniaxial thrust.

302

strength for simply supported plate is obtained as in Eq. (5) per unit length.

π 2 D ⎛ mb a ⎞ Nx = 2 + ⎟ mb ⎠ b ⎝ a

2

(5)

2.2 FEM eigenvalue analysis Elastic local buckling strength of a continuous stiffened panel subjected to uniaxial thrust may be calculated by FEM eigenvalue analysis. Fine mesh is applied to the model (see Fig. 4). Shell181 finite strain shell element is used for modeling the plate, web and flange. Stiffened panel material’s Young modulus is 205.8 GPa and Poisson’s ratio 0.3. Transverse frame is not modeled, instead of that; displacements in the z-direction along the transverse frame are constrained. When the buckling half wave number in the longitudinal direction, m, is an odd number, the local buckling behavior can be simulated considering region between ab and cd in Figure 5 and imposing the symmetry conditions along the boundaries (Fujikubo & Yao, 1999). 2.3

Table 1. Buckling strength of simply supported plate (MPa).

Results

The elastic buckling strength of the simply supported plate is evaluated by Eq. (5) and FEM eigenvalue analysis. The FEM eigenvalue analyses are carried out for 27 different stiffened panels. The results are compared with the analytical solutions given by Yao (2009).

a (mm) b (mm)

tp (mm)

σnFEM

σnAnalytical

2400

10

108.25

116.25

800

Table 2. Comparison of FEM and analytical results given by Yao (2009). Stiffener

hw

tw

bf

tf

FEM (σ1/σ0)

Yao (σ1/σ0)

Flat Flat Flat Flat Flat Flat Flat Flat Flat Flat Flat Angle Angle Angle Angle Angle Angle Angle Tee Tee Tee Tee Tee Tee Tee Tee Tee

100 100 100 195 195 195 310 310 310 400 400 150 150 250 250 250 400 400 150 150 250 250 400 400 400 400 400

10 15 20 10 15 20 10 15 20 10 20 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

0 0 0 0 0 0 0 0 0 0 0 90 150 90 120 150 90 150 90 150 90 15 90 120 90 150 150

0 0 0 0 0 0 0 0 0 0 0 12 12 16 16 16 16 16 12 12 16 16 12 12 16 16 12

1.033 1.14 1.269 1.044 1.197 1.544 1.013 1.209 1.557 0.772 1.552 1.48 1.487 1.451 1.453 1.454 1.423 1.425 1.475 1.486 1.449 1.454 1.421 1.422 1.423 1.425 1.423

1.069 1.136 1.441 1.077 1.303 1.563 0.989 1.296 1.618 0.836 1.62 1.607 1.66 1.555 1.577 1.593 1.477 1.517 1.567 1.617 1.507 1.578 1.425 1.449 1.486 1.51 1.462

σn: buckling strength of simply supported plate. σ1: buckling strength of stiffened panel. Panel: axbxt: 2400 × 800 × 10 mm. Hw, tw, bf, tf: in mm. Figure 4.

Finite element model of stiffened panel.

Figure 5.

Continuous stiffened plating.

Figure 6. Buckling mode shape of stiffened panel under longitudinal thrust (flat: 195 × 10 mm).

303

Figure 7. Buckling mode shape of stiffened panel under longitudinal thrust (tee: 250 × 90 × 12/16 mm). Figure 8.

The calculated results show that the stiffener type and stiffener geometric properties have considerable effect on the buckling strength. It is observed that the finite element analyses underestimate the buckling strength of the stiffened panels. 3

NONLINEAR BUCKLING ANALYSIS

Ultimate strength analysis of stiffened panels can be carried out by several methods, such as Nonlinear Finite Element Analysis, ISUM (Idealized Structural Unit Method) and Simplified Beam-Column Methods. These methods are the most popular techniques used for ultimate strength analysis. Nonlinear FEA is one of the strongest methods for the ultimate strength calculations. This method generally requires lots of modeling effort and computational time. Furthermore, the analyst should have enough experience to assess the results. Nowadays, the nonlinear finite element analysis is used as control mechanism for new studies. A series of nonlinear finite element analyses are carried out to estimate the ultimate strength of the stiffened panels. The calculated results are compared with those in literature. A finite element eigenvalue analysis is carried out to determine the mode shapes. The calculated mode shapes are imposed to finite element model as initial imperfections. Finite element model must capture all the mechanisms that could lead to collapse of the structure. For inelastic analysis 1-bay model cannot capture the collapse accurately because of that inter-frame bay deflect in upward or downward half wave, while the next bay would deflect in opposite sense. The boundary condition at the frame is intermediate between simply supported and clamped, and cannot be accurately modeled as a loaded edge (see, for instance, Chen (2003)). Because of these reasons, the finite element model is represented as a symmetric 1 1 / 2 bay model as seen in Figure 8. In the finite element analysis, Shell181 finite strain shell element is used. A fine mesh is used to allow for free development of the buckling modes.

Finite element model of stiffened panel.

Table 3.

Properties of stiffened panels.

Panel

a

tp

ns

hw

tw

bf

tf

P1 P2 P3 P4

1800 3600 3600 3600

21 16 21 10

3 3 5 5

42 100 80 80

12 9 12 6

100 100 100 80

15 14 15 10

ns: number of stiffeners.

3.1

Boundary conditions

• The long edges are considered as simply supported. The displacements in the z-direction and rotations about the y and z-axes, as in Figure 9, are constrained to impose simply supported boundary conditions. • The transverse frame is not modeled, but displacements in the z-direction along the transverse frame are constrained. • The short edge, which is the mid-length of the mid-bay of the full 3-bay model, has symmetric boundary condition. The symmetric boundary condition is satisfied by constraining displacements in the x-direction and rotation about the y-axis. • The displacements in the y-direction are constrained at the mid-width node in each of two short edges to prevent rigid body motions. 3.2 Imperfections In the nonlinear buckling analysis we are not interested in bifurcation buckling. The problem has to be considered as a continuous response problem (see, for instance, Chen (2003)). Because of that, the initial imperfections are introduced for the stiffeners and plating. The imperfect geometry is assumed as overall buckling mode shape obtained from the eigenvalue analysis. The selected mode shape has an upward deflection in full bay and a downward deflection in half bay as seen in Figure 9.

304

Figure 9. panel.

Overall buckling mode shape of stiffened

Table 4. Comparison of nonlinear FEM results with results obtained by Chen (2003). Panel

Nonlinear FEM (σu/σY)

Chen (σu/σY)

P1 P2 P3 P4

0.402 0.476 0.517 0.412

0.418 0.507 0.549 0.448

Figure 10.

Effect of stiffener type for P1.

Figure 11.

Effect of stiffener web height for P1.

Figure 12.

Effect of plate thickness for P1.

Figure 13.

Effect of lateral pressure for P1.

Figure 14.

Effect of stiffener type for P2.

305

Figure 15.

Effect of stiffener web height for P2.

Figure 16.

Effect of plate thickness for P2.

Figure 17.

Effect of lateral pressure for P2.

Figure 18.

Effect of stiffener type for P3.

Figure 19.

Effect stiffener web height for P3.

Figure 20.

Effect of plate thickness for P3.

306

Figure 21.

Effect of lateral pressure for P3.

Figure 22.

Effect of stiffener type for P4.

Figure 24.

Effect of plate thickness for P4.

Figure 25.

Effect of lateral pressure for P4.

The scaling factor for initial imperfection of the stiffened panel is w0 0.0025a, where a is the length of one bay. The initial imperfections are not introduced to the model of the finite element analysis which is carried out to investigate effect of lateral pressure. 3.3 Results

Figure 23.

Effect of stiffener web height for P4.

A series nonlinear finite element analyses are carried out to estimate the ultimate strength of stiffened panels. The obtained results are compared with those in literature. The effects of stiffener type and stiffener web height on the ultimate strength are investigated. The load–displacement curves are evaluated for the stiffened panels.

307

4

CONCLUSIONS

A series linear and nonlinear finite element analyses was carried out to determine the buckling strength and ultimate strength of several stiffened ship panels. Following findings are obtained from the finite element analyses: • The results of the linear analyses show that the effect of stiffener type and stiffener properties has great influence on the local buckling strength of the stiffened panels. • The FEM eigenvalue results are compared with the analytical results obtained by Yao (2009). It is observed that the finite element method underestimates the buckling strength of the stiffened panels. • The results obtained from the nonlinear finite element analyses are compared with the results of Chen (2003). The finite element analyses give lower ultimate strength values and it can be accepted as safer. • The stiffener web height has great effect on the ultimate strength of the panels. In this study only overall buckling mode shapes are investigated. In case of large increments in stiffener web height, the local buckling of plating between stiffeners or tripping of stiffeners may be observed. • The plate thickness has no considerable effect on the ultimate strength of the panels because of the overall collapse. However, the effect of the plate thickness on the post-collapse region must be taken into consideration. • The difference between the tee bar stiffened panels and angle bar stiffened panels is small for all the panels, but in case of flat bar, the ultimate strength reduces considerably. • The lateral pressure generally reduces ultimate strength of the panels. But, when the lateral pressure is increased, the ultimate strength may also increase.

Chen, Y. 2003. Ultimate strength analysis of stiffened panels using a beam-column method. Doctoral dissertation, Virginia Polytechnic Institute and State University, Virginia. Fujikubo, M., Harada, M., Yao, T., Khedmati, M.R., Yanagihara, D. 2005. Estimation of Ultimate Strength of Continuous Stiffened Panel Under Combined Transverse Thrust and Lateral Pressure Part 2: Continuous Stiffened Panel, Marine Structures, V18: pages 411–427. Fujikubo, M., Yao, T. 1999. Elastic Local Buckling Strength of Stiffened Plate Considering Plate/Stiffener Interaction and Welding Residual Stress, Marine Structures, V12: pages 543–564. Fujikubo, M., Yao, T., Khedmati, M.R., Harada, M., Yanagihara, D. 2005. Estimation of Ultimate Strength of Continuous Stiffened Panel Under Combined Transverse Thrust and Lateral Pressure Part 1: Continuous Plate, Marine Structures, V18: pages 383–410. Gordo, J.M., Soares, C.G. 2008. Compressive Tests on Long Continuous Stiffened Panels, 27th International Conference on Offshore Mechanics and Arctic Engineering, Estoril, Portugal. Hughes, O.F., Ghosh, B., Chen, Y. 2004. Improved Prediction of Simultaneous Local and Overall Buckling of Stiffened Panels, Thin Walled Structures, V42: pages 827–856. Paik, J.K., Thayamballi, A.K., Kim, B.J. 2001. Large deflection orthotropic plate approach to develop ultimate strength formulations for stiffened panels under combined biaxial compression/tension and lateral pressure, Thin-Walled Structures, V39: pages 215–246. Smith, C.S. 1977. Influence of Local Compressive Failure on Ultimate Longitudinal Strength of A ship’s Hull, International Symposium on Practical Design in Shipbuilding, Tokyo, Japan. Ueda, Y., Rashed. M.H. 1984. The idealized structural unit method and its application to deep girder structure, Computers and Structure, V18 ISSN 0045-7949: pages 277–293. Ventsel, E., Karuthammer T. 2001. Thin Plates and Shells, Theory, Analysis, and Applications, New York: CRC Press. Yao, T. 2009. Buckling/Plastic Collapse Behavior and Strength of Ship Structures lecture notes.

REFERENCES ANSYS Release 13 User’s manual. Caldwell, J.B. 1965. Ultimate Longitudinal Strength, Transactions of the Royal Institution of Naval Architects, 107, 411–430.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Finite element modelling of the ultimate strength of stiffened plates with residual stresses M. Tekgoz, Y. Garbatov & C. Guedes Soares Centre for Marine Technology and Engineering (CENTEC), Instituto Superior Técnico, Technical University of Lisbon, Portugal

ABSTRACT: The objective of this work is to analyze the effect of residual stress on the ultimate strength assessment of a stiffened panel. The analysis is performed by a nonlinear finite element method. The ultimate strength is evaluated for 3 finite element models accounting for different levels of residual stresses and boundary conditions in order to develop modified stress strain curves that can be used directly for nonlinear finite element analyses of ultimate strength of stiffened panels. 1

INTRODUCTION

Residual stresses may be induced in structural components during manufacture such as welding, forging, casting, age hardening, machining and other processing applications. Due to their negative effect on structural strength, a wide range of studies have been performed in order to evaluate the reasons that are leading to their development. Residual stresses, that arise from non-uniform induced heat and plastic strain development, differ from the service loading stresses that the structure is subjected during its service life. Structural response is in equilibrium with the external loading. In contrast, the residual compressive stresses are only in equilibrium with the tensile residual stresses. The ultimate strength of structural components and systems is a real measure of its structural capacity in the sense that the ultimate strength is the maximum load carrying capacity that they can withstand. Achieving the ultimate strength implies that the remaining capacity deteriorates due to plasticity, which appears since the axial structural rigidity is set to zero. Ultimate strength assessment involves a large amount of uncertainties and many factors may affect it (Guedes Soares, 1988b, 1997). Garbatov et al. (2011) implemented a Monte Carlo simulation in order to find the most influential parameters on the ultimate strength. It has been found that the slenderness ratio and plate thickness have the most significant effect on the ultimate strength of the stiffened panels. In addition to that, residual stresses and different loading conditions, imperfection magnitude and material properties have an important effect on ultimate strength. Guedes Soares (1988a) has analyzed the effect of residual stresses and initial imperfections on the

ultimate strength of plates concluding that residual stresses are important for stocky plates while initial imperfections are more important for slender plates, which is also reflected in stiffened panels (Guedes Soares and Soreide, 1983). Ueda and Yao (1991) showed that, both welding residual stresses and initial geometrical imperfections reduce the compressive buckling and ultimate strength of plates, and this reduction achieves its maximum when the plate slenderness is about 1.8. Welding residual stresses reduce the compressive buckling and ultimate strength of stiffened panels when local buckling takes place, but increase them when overall buckling occurs. Ueda et al. (1997) concluded that, welding residual stresses reduce the buckling strength remarkably, but have a little effect on the ultimate strength when the plate is thin. On the other hand, when the plate is thick, welding residual stresses reduce the ultimate strength remarkably if there is an initial geometrical imperfection accompanied by local bending stresses. Grondin et al. (1999) applied three levels of residual stresses and showed that with plate slenderness β larger than 1.7 the residual stresses in the plate decrease the strength roughly in direct proportion to the magnitude of the compressive residual stresses in the plate. However, when yielding occurs before buckling, the effect of residual stresses is diminished. When failure of stiffened plates is due to overall Euler buckling, the effect of the residual stresses in the plate is less pronounced. The effect of different finite element models on the ultimate strength assessment of plates and stiffened panels has been discussed by Tekgoz et al. (2012) and the objective of this work is to analyze the effect of residual stress on the ultimate strength assessment of stiffened panels. The analysis is

309

performed by finite element method. The ultimate strength is evaluated for 3 finite element models accounting for different material stress-strain relationships, levels of residual stresses and boundary conditions. A modified stress strain relationship is developed to be used directly for the nonlinear finite element analysis of ultimate strength accounting for residual stresses and structural configuration.

This equation accounts implicitly for average levels of initial deflection and it can be complemented with others that dealt explicitly with the effect of residual stresses. Guedes Soares (1988b) has extended that formulation by deriving a strength assessment expression for the compressive strength of plate elements under uni-axial load, which deals explicitly with initial defects as:

2

φu (ϕ p B p ) (Rr , Br )(Rδ Bδ )

RESIDUAL STRESS MODELLING

The residual stress reduces the strength of the structural components and moreover it reduces the structural stiffness. For a better approximation in simplified design equations some authors (e.g. Guedes Soares (1992)) decided to divide the ultimate strength in various terms, accounting for initial geometrical imperfection and residual stresses as:

φ pRδ Rr

φu

(1)

where φp is the perfect ultimate strength, is the reduction factor due the initial geometry imperfection and Rr is the reduction factor due to the residual stresses. Dwight and Moxham (1969) related the level of compressive residual stress to the width of the tensile block ηt:

φ=

σr 2ηt = σ y b ηt

(2)

where t and b are, respectively, the thickness and breadth of the plate, σr if the residual stress and σy is the yield stress, and the reduction factor due to residual stresses may be calculated as: Rr = (1 − φr )

(3)

where φp is given by Eqn (4), Bn, Br and are model uncertainty factors and Brδ and Rδ are strength reduction factors which are due to the presence of weld induced residual stresses and initial distortions respectively. These expressions are: B p = 1.08 Rr = 1 −

a1 a2 − , β β2

β ≥ 1.0

β=

b σy t E

where E is the Young modulus of the material.

(5)

Δφ 1.08φ p

(8) (9)

Rδ = 1 − (0.626 − 0.121β )

δ0 t

(10)

δ0 + 0.1β (11) t where δ0 is the initial imperfection and Rr is the strength reduction due to the residual stress is in which:

Brδ = 0.76 + 0.01η + 0.24

Δφ p φr

Et E

(12)

where the tangent modulus of elasticity, Et/E ratio accounts for the development of plasticity. This modulus can be approximated by the expression was used by Guedes Soares and Faulkner (1987):

(4)

where the constants a1 and a2 are given as a1 2 0 and a2 = 1.0 for simple supports and a1 2 5 and a2 = 1.56 for clamped supports and the plate slenderness is defined as:

(7)

Br = 1.07

Faulkner (1975) developed an expression for ultimate strength fitting it to data of the ultimate plate strength leading to:

φp =

(6)

Et β − 1 = , for 1 ≤ β ≤ 2.. E 1.5

(13)

Et = 1, f E

(14)

β ≥ 2.5

The approaches presented here can be directly used for design and they estimate only the ultimate strength of steel structure subjected to compressive load. The second group of approaches, dealing with the residual stress modelling for the ultimate strength assessment, are those by using a modified stress strain elasto-plastic curve or by using direct prescribed pre-stresses, simulating the residual stresses. These methods employ the finite element method and estimate not only the ultimate

310

strength but also the pre and post collapse regime behaviour. Faulkner (1977) suggested a model to account for residual stresses by implementing a modification in the elastic perfectly plastic stress-strain curve. Through this type of stress strain curve, the structure is subjected to premature yielding and in turn reduces both stiffness and strength. The structural yielding stress point is reduced allowing the structure to absorb more strain and reduces the strength per strain. This implies that the linear part of the stress strain curve is reduced. The pre and post collapse regime assessment, based on modified stress-strain curve, is influenced by the structural configurations, boundary conditions of edges and the level of the residual stresses. The first step in the modification of the stress-strain curve is to define the stress level of proportional limit in order to capture the first yielding point. Once the first yielding limit has been defined, the next point in the stress-strain curve has to be identified by changing the slope angle or by adding a new point. The iterations are ended when the two structural responses from both methods have been found almost identical. The residual stress may also be modelled as direct pre-stresses induced to the finite element model before the start of non-linear ultimate strength analysis. The results achieved for the ultimate strength, when direct prescribed pre-stresses are used to model the residual stresses, are twice smaller if the residual stresses are modelled by equivalent temperature (Paik et al., 2009). In the present study pre-stresses induced to the finite element model and modified elastic perfectly plastic stress-strain curve are used for residual stress modelling in the finite element analysis. To find out the shape of the modified stressstrain curve an iterative procedure is created here in which the material stress-strain curve is modified according to the best fitted results of pre and post collapse estimated by directly prescribed residual stresses. The entire procedure is divided into two stages. In the first one, for three levels of the residual stresses defined as slight, average and severe prestresses are induced into the finite element model. The defined residual stresses are applied over the structure according to predefined residual stress distribution. Then the structural normalized strength against strain is estimated. In the second stage, the iterative procedure is implemented to identify the best modification to the material stress-strain curve used for finite element calculations of ultimate strength in order to find out the best fit of the structural response to that of the first stage, which has been based on the direct prestress modelling of residual stresses.

3

FINITE ELEMENT MODELLING

Three structural models are analyzed here. These models have been defined on a box girder geometry that was tested and used in several studies performed by Saad-Eldeen et al. (2011). Models 1 to 3 are shown in Figure 1, where the Model 1 represents a stiffened plate between two neighbouring transversal frames; Model 2 is also a stiffened panel with a transversal frame in the middle. Model 3 is composed of ½ + 1 + ½ plates and two transversal frames. The ultimate strength of the panels is analyzed based on finite element method using commercial software ANSYS (2009). The software enables modelling of elastic plastic material properties and large deformations. Four-node quadrilateral shell elements have been used to model the plates and stiffeners. Three distinct stiffened models are studied here as can be see them in Figure 1. The length of the stiffened panels of the Model 1 is l = 400 mm and the breadth is b = 150 mm respectively. The web thickness is 3.6 mm and the height is 25 mm and the plate thickness is 2.7 mm. Plate slenderness is β = 1.87. The aspect ratio, l/b of the plate is 2.66. The stiffener has a cross section of a standard type flat bar. The area of the stiffener, Ast is approximately 22 per cent of the area of the plate, Apl = bl. The non-dimensional column slenderness of the stiffened panel is λ = 0.69, calculated with the full plate width. The thickness, length, breadth of plates and the height of stiffeners are equal for all finite element models. The initial geometric imperfection of the plates and stiffeners are generated by a pre deformed surface. Faulkner (1975), Smith et al. (1988) have reported that the maximum imperfections in a plate can be assumed to be proportional to β 2. They suggested that the maximum initial deformation for an average imperfection can be calculated as wmax = 0.1tβ2

Figure 1.

311

Model structural configurations.

(15)

The maximum camber tolerance, for a standard shape is usually assumed to be 0.2% of the length. The initial geometry surface imperfection is modelled as proposed by Smith et al. (1988): w ( x, y )

⎛ mx ⎞ ⎛ ny ⎞ wmax sin π sin π ⎝ l ⎠ ⎝ b⎠

(16)

where l is the length of the panel and b is the breath of the panel, x and y are the Cartesian coordinates of any location on the plate and m and n are number of half waves. The transverse frame for Model 2 and Model 3 is not modelled by finite elements and its effect is accounted for by the respective boundary conditions applied to the nodes associated with the connection between plate and stiffener. Several studies have been performed in order to find the appropriate finite element type. Four-node quadrilateral shell 181 element has been chosen to be used in order to model the plate and stiffener for all models. The kinematic assumption of the finite element analysis calculation is large displacement and rotation, but small strain. The material stress strain model, when the residual stresses are not accounted for, is assumed to be bilinear elastic-perfectly-plastic without hardening with the yield stress of σy = 235 MPa, the elastic modulus is E = 2.06E + 11 Pa and the Poison coefficient is v = 0.3. Geometric and material non-linearity has been taken into account. Stress stiffening effect, in order to avoid sharp post-collapse regime, has been accounted for. The applied load is uni-axial compression. 4

and the symmetry boundary conditions are applied to the longitudinal edges, since the maximum displacement is considered to occur there. Coupling conditions are employed on the longitudinal edges in order to keep the structural net section plane. Three levels of the compressive part of the applied residual stress distribution are considered as proposed by Smith et al. (1988): slight, average and severe as defined by Eqn (2) and only the longitudinal residual stresses are investigated.

σ r ⎧⎪−0.05, slight = ⎨−0.15, average σ y ⎪−0.3, severe ⎩

(17)

The compressive and tensile parts of the residual stresses are equilibrated, leading to a specific residual stress distribution, as can be seen in Figure 3. The residual stresses reduce the first yielding point of the structural response and due to the plasticity it results in large strain absorption, but less strength (see Fig. 4). However, the strain absorption after the first yielding point may be affected by the boundary conditions and may lead to a less strain absorption. The strength reduction, resulted from the initial geometry imperfection and residual stresses, tends to be a linear function of b/t ratio as can be seen in Figures 5 and 6.

ULTIMATE STRENGTH ASSESSMENT

4.1 Model 1 4.1.1 Directly prescribed residual stresses Finite element Model 1, presenting a stiffened panel, is located between two neighboring transverse frames as can be seen in Figures 1 and 2. The applied boundary conditions are shown in Figure 2, where U-Displacement, R-Rotation, C-Coupling

Figure 2.

Model 1.

Figure 3.

Model 1, Residual stress distribution.

Figure 4. Stress strain relationship, directly prescribed residual stress, Model 1 (RS-residual stresses).

312

thickness, which gives more strength and therefore more plastic propagation may be achieved. As can be seen in Figures 7 and 8, modelling the residual stresses by the use of a modified stress strain curve, the stress strain response matches very well the stress strain response calculated by the use of directly prescribed residual stresses, except for the post collapse regime. By increasing the severity of the residual stress, the first yielding stress is reduced as can be seen in Figure 9.

Figure 5. Ultimate strength, Model 1—one half wave geometry imperfection (RS-residual stresses).

Figure 7. Stress-strain response, Model 1 (SSM— modified stress-strain, RS-residual stresses, WRS— welding residual stresses).

Figure 6. Ultimate strength, Model 1—three half wave geometry imperfection (RS-residual stresses).

For the larger plate slenderness values, the residual stress becomes less influential on the ultimate strength. In addition to that, as the structural capacity is reduced, through increasing the number of the longitudinally induced half-waves of the initial geometry imperfection, the residual stresses become almost insignificant as can be seen in Figure 6. 4.1.2 Modified stress-strain curves accounting for residual stress When structures are subjected to a tensile loading, their structural stress-strain responses follow the material stress-strain response. This means that the transition from elastic to plastic behavior is smooth and there is no disturbance. When the structure is subjected to compressive loading the transition from elastic to plastic behavior is with disturbance due to the presence of buckling and depends upon the structural configurations and boundary conditions, which lead to a non-uniform plastic propagation. It is strongly dependent on the plate

Figure 8. Stress-strain response, Model 1 (SSM— modified stress-strain, RS-residual stresses, WRS-welding residual stresses).

Figure 9.

313

Material stress strain relationship, Model 1.

Once the first yielding point is achieved, the next point has been found according to the structural response which is highly affected by the boundary conditions of the edges. As the first yielding stress point is reduced, the structure absorbs more strains. However, if the structure is not well constrained, then this may lead to less strain absorption. As for Model 1, the first cut of σ − ε curve leads to more strain absorption. The second cut of σ − ε curve makes the structure to absorb less strain. Therefore, the modified stress-strain curve used for modelling residual stresses has to be defined for any particular structural configuration and applied boundary conditions at the edges. The modified material stress-strain data points have been shown in Table 1 where εy is the material yielding strain. 4.2

Model 2

4.2.1 Directly prescribed residual stresses Model 2 is defined as a stiffened panel between two half bays, ½ + ½. The boundary condition may be seen in Figure 10. The longitudinal and transverse edges are subjected to the symmetry boundary conditions, since the initial maximum imperfection has been generated in the longitudinal and transverse edges and also the coupling conditions are employed in order to keep the section plane during loading.

Table 1.

As for Model 2, longitudinal residual stresses have been taken into account and three levels of residual stress have been applied, where the magnitude of compressive stresses are defined as Eqn (2) where for severe residual stresses are considered as −0.2. Three different initial geometry imperfection shapes have been analyzed in order to find the residual stress effect on the ultimate strength. The shapes of initial geometry imperfects are shown in Figure 11. The welding residual stresses reduce the structural first yielding point and strength (see Figs. 12 and 13). The strain absorption is also affected by the geometry non-linearity. Due to the premature yielding, more strain is absorbed and geometrical non-linearity increases and reduces the linear part of the structural behavior. As can be seen in Figure 14, the increased structural complexity, which depends on the shape of the initial geometric imperfection, leads to the ultimate strength reduction. This can also be interpreted in the way that as the geometry non-linearly increases the residual stress reduction on ultimate strength becomes more significant.

Model 1, Modified stress—strain descriptors.

Slight level

Severe level

Average level

Stress Stress Pts (MPa) Strain (MPa) Strain

Stress (MPa) Strain

0 1 2 3 4

0.00 160 235 235

0.00 200 210 235 235

Figure 10.

0.00εy 0.85εy 0.94εy 4.00εy 5εy

Model 2.

0.00 145 220 235 235

0.00εy 0.617εy 2.1εy 3.00εy 5εy

0.00εy 0.680εy 2.500εy 5εy

Figure 11.

Geometry shape imperfections, Model 2.

Figure 12. Stress strain response, directly prescribed residual stress, Model 2, Shape 1 (left) and Shape 2 (right).

314

Table 2.

Model 2, Modified stress—strain descriptors. Severe level

Average level

Pts

Stress (MPa)

Strain

0 1 2 3 4

0 85 95 235 235

0.00εy 0.361εy 0.5εy 3.40εy 5εy

Stress (MPa)

Strain

0 85 235 235

0.00εy 0.361εy 2.70εy 5εy

Figure 13. Stress strain response, directly prescribed residual stress, Model 2, Shape 3.

Figure 15. Stress strain response, Model 2 (left) and material stress strain relationship (right).

Figure 14.

Ultimate strength, Model 2.

4.2.2

Modified stress-strain curves accounting for residual stress Shape 3 has been selected as a reference to identify the modified stress strain descriptors. As opposed to Model 1, after the first yielding point, the structure may have less strain by increasing the material strain since the boundary edges are imperfect which leads to less plasticity propagation. The modified stress-strain curve is influenced by the boundary conditions at the edges, which in turn increases the edge rotations and reduces the stiffness and structural strain. Therefore, for the modified severe stress-strain, the second cut has been defined closer to the first cutting point. The modified stress-strain descriptors have been shown in Table 2, where εy is the material yielding strain. As for Model 2, the average and severe levels of residual stresses have been considered in identifying the descriptors of the modified stress-strain curve. The post collapse regime is highly affected by the plasticity propagation, which in turn is affected by the boundary conditions of edges. The post-collapse regime of structural response

Figure 16.

Model 3.

accounting for the presence of residual stresses is less smother than the one developed based on the modified stress-strain curve (Fig. 15). 4.3

Model 3

Model 3 represents a stiffened panel between two halves and one bays, ½ + 1 + ½. The applied boundary conditions are presented in Figure 16. The longitudinal and transverse edges are subjected to the symmetry boundary conditions since the maximum imperfection has been considered in the longitudinal and transverse edges. The coupling effect is employed in order to keep the section plane during loading.

315

Modified stress-strain curves have been found as a good and fast method to introduce the effect of residual stress. It has been found that the modified stress-strain curve descriptors are influenced by the structural configurations and boundary conditions at the edges.

ACKNOWLEDGEMENTS

Figure 17. Stress strain response, directly prescribed residual stress, Model 3.

The work reported here is a contribution to the activities of the MARSTRUCT VIRTUAL INSTITUTE, (www.marstruct-v.com) in particular its Technical Subcommittee 2.3 on Ultimate Collapse Strength.

REFERENCES

Figure 18. Ultimate strength reduction comparison, Model 2 and 3, 1:slight level, 2:Average level, 3:Severe level.

The effect of the residual stresses on the ultimate strength reduction for the Model 3, which can be seen in Figure 17 is less significant in a comparison to Model 2 since the structural capacity of the Model 3 is larger than the one of Model 2 (see Fig. 18). There is no significant behaviour change between Model 2 and 3. Model 2 is better representative of the stiffened panel since the Model 3 always present more optimistic loading carrying capacity results. 5

CONCLUSIONS

The residual stress effect on the ultimate strength has been investigated using different structural finite element models. It has been found that the residual stresses decrease the first yielding point of the structure response. As the structural capacity increases, the effect of residual stresses on the ultimate strength decreases. Initial geometric imperfections increase the strength reduction along with the residual stresses.

ANSYS, 2009. Online Manuals, Release 11. Dwight, J.B., Moxham, K.E., 1969. Welded steel plates in compression. Structural Engineering 47, pp. 49–66. Faulkner, D., 1975. A Review of Effective Plating for use in the Analysis of Stiffened Plating in Bending and Compression. Journal of Ship Research 19, pp. 1–17. Faulkner, D., 1977. Compression tests on welded eccentrically stiffened plate panels, in: Dowling, P.J., Harding, J.E., Frieze, P.A. (Eds.), Steel Plated Structures. Crosby Lockwood Staples, London, pp. 130–139. Garbatov, Y., Tekgoz, M., Guedes Soares, C., 2011. Uncertainty Assessment of the Ultimate Strength of a Stiffened Panel, in: Guedes Soares, C., Fricke, W. (Eds.), Advances in Marine Structures. Taylor & Francis Group, London, UK, pp. 659–668. Grondin, G.Y., Elwi, A.E., Cheng, J.J.R., 1999. Buckling of stiffened steel plates-a parametric study. Journal of Constructional Steel Research 50, pp. 151–175. Guedes Soares, C., 1988a. Design Equation for the Compressive Strength of Unstiffened Plate Elements with Initial Imperfections. Journal of Constructional Steel Research 9, pp. 287–310. Guedes Soares, C., 1988b. Uncertainty Modelling in Plate Buckling. Structural Safety 5, pp. 17–34. Guedes Soares, C., 1992. Design Equation for Ship Plate Elements under Uniaxial Compression. Journal of Constructional Steel Research 22, pp. 99–114. Guedes Soares, C., 1997. Probabilistic Modelling of the Strength of Flat Compression Members, in: Guedes Soares, C. (Ed.), Probabilistic Methods for Structural Design. Kluwer Academic Publishers, p. 402. Guedes Soares, C., Faulkner, D., 1987. Probabilistic Modelling of the Effect of Initial Imperfections on the Compressive Strength of Rectangular Plates, Proceedings of the International Symposium on Practical Design of Ships and Mobile Units, (PRADS’87), pp. 783–795. Guedes Soares, C., Soreide, T.H., 1983. Behaviour and Design of Stiffened Plates Under Predominantly Compressive Loads. International Shipbuilding Progress 30, pp. 13–27.

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Paik, J., Branner, K., Choo, J., Czujko, J., Fujikubo, M., Gordo, J.M., Parmentier, G., Iaccarino, R., O’Neil, S., Pasqualino, I., Wang, D., Wang, X., Zhang, S., 2009. Committee III.1 Ultimate Strength, in: Jang, C., Hong, S. (Eds.), 17th International Ship and Offshore Structures Congress (ISSC2009). University of Seoul, Seoul, South Korea, pp. 375–475. Saad-Eldeen, S., Garbatov, Y., Guedes Soares, C., 2011. Experimental assessment of the ultimate strength of a box girder subjected to severe corrosion. Marine Structures 24, pp. 338–357. Smith, C.S., Davidson, P.C., Chapman, J.C., Dowling, J.P., 1988. Strength and Stiffness of Ships’ Plating under In-plane Compression and Tension. Transactions RINA 130, pp. 277–296.

Tekgoz, M., Garbatov, Y., Guedes Soares, C., 2012. Ultimate Strength Assessment Accounting for the Effect of Finite Element Modelling, in: Guedes Soares, C., Garbatov, Y., Sutulo, S., Santos, T. (Eds.), Maritime Technology and Engineering. Taylor & Francis Group, London, UK, pp. 353–362. Ueda, Y., Yao, T., 1991. Fundamental Behavior of Plates and Stiffened Plates with Welding Imperfections. Transactions of JWRI 20, pp. 141–155. Ueda, Y., Yasukawa, W., Yao, T., Ikegami, H., Ohminami, R., 1997. Effect of Welding Residual Stresses and Initial Deflection on Rigidity and Strength of Square Plates Subjected to Compression (Report II). Transactions of JWRI 6, pp. 33–38.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Elastic buckling and elasto-plastic collapse behaviors with torsion of a longitudinal stiffener under axial compression D. Yanagihara Ehime University, Matsuyama, Ehime, Japan

M. Fujikubo Osaka University, Suita, Osaka, Japan

ABSTRACT: In this study, FE analyses are performed for stiffened plates with a flat-bar or deep tee-bar under axial compression, the characteristics of the torsional deformation of the stiffener is clarified. The analytical solutions based on elastic large deflection analysis are derived to simulate the elastic behavior of an isolated stiffener under axial compression. It is confirmed that the elastic behavior has a sufficient accuracy. The method for the stiffened plate is also proposed. The elastic behavior in the case with the flatbar is simulated with a sufficient accuracy, while the accuracy in the case with the deep tee-bar is not sufficient. However, it is acceptable to estimate the ultimate strength. The ultimate strength of the stiffened plate with the deep tee-bar is estimated using the analytical solutions. The estimated strength has a good correlation with that obtained by FEA, and it is found that the proposed method is beneficial. 1

INTRODUCTION

Stiffened plates which compose deck and bottom structures are subjected to in-plane compression in the stiffener direction due to a longitudinal bending moment of a ship hull girder. Since the buckling/plastic collapse of the stiffened plate might produce the overall collapse of the ship, it is very important to estimate the ultimate strength of the stiffened plates with the high accuracy. Several methods to estimate the ultimate strength of the stiffened plate have been proposed. In most of them, the stress developed in the stiffened plate is calculated by some way, then the applied load at the point when the stress reaches the yield stress is defined as the ultimate strength. The method based on a beam-theory is most simplified to predict the stress, and the stiffened plate is modeled by a beam-column. As typical examples, Carlsen (1980) proposed the simplified method to estimate the ultimate strength of stiffened plates. The common structural rule for bulk carriers adopts the estimation method assumed the beam-column (IACS 2006). In addition, authors also proposed the estimation method of the similar type (Fujikubo et al. 1999a, b). By the way, higher stiffeners tend to be attached to plates of the ship structure. When the higher stiffener with an angle or tee shape is subjected to axial compression, the torsional buckling easily takes place with the mode of one half-wave. On the other

hand, a flat-bar attached to the stiffened plate is torsionally deformed with the same short-waved mode as the plate buckling. When such a torsional deformation of the stiffener becomes large, it is difficult to predict the stress in the stiffened plate using the beam-column model and it is necessary to consider the stiffener as a plate structure. As one of the methods which enable such a treatment for the stiffener, there is the method using the analytical solutions based on the elastic large deflection analysis of the plate element. PULS (Plate Ultimate Limit State) software proposed by DNV (2005) is based on the elastic large deflection analysis, and enables to estimate the ultimate strength of the stiffener plate under various load condition. Because PULS assumes many modes of the deflection on the stiffened plate, the estimation has not only the high accuracy but also the problem to need relatively much calculation time. Therefore, the simplified method is needed from the viewpoint of structural design. In this study, firstly, Finite Element Analyses (FEA) are performed to examine the buckling/ plastic collapse behavior of the stiffened plate subjected to axial compression. As the stiffener shape, flat-bar and deep tee-bar, which are easily deformed with torsion, are selected. Focused on the torsional deformation of the stiffener, its characteristics are discussed. Next, The analytical solutions based on the elastic large deflection analysis are derived to simulate the elastic behavior of an isolated plate

319

and stiffener under axial compression, and their accuracy is validated through the comparison with FEA. The procedure to estimate the elastic behavior as the stiffened plate is also proposed and validated. Finally, the ultimate strength of the stiffened plate with the deep tee-bar, in which the torsion of the stiffener is most predominant, is estimated using the analytical solutions based on the elastic large deflection analysis. The usability of the proposed estimation method is discussed through the comparison with FEA and PULS.

2

2.1

BUCKLING/PLASTIC COLLAPSE BEHAVIOR OF STIFFENED PLATE WITH TORSION

Initial deflections are given to the stiffened plate. In the plate part, the initial deflection of the buckling mode is considered as the following equation: w0

v0

Figure 1. Continuous stiffened plate subjected axial compression.

mπ x πy sin i a b

(1)

The coefficient, W01, is set to be 1% of the plate thickness, tp. Based on the result by a buckling eigenvalue analysis, the number of half-waves, m, is decided. This is generally larger than that of the isolated plate due to the resistance of the torsional rigidity of the stiffener. In the stiffener, the initial deflection of a torsional mode is considered as the following equation:

FEA procedure

In order to examine fundamental the collapse behavior of a stiffened plate associated with torsional buckling of the stiffener, Finite Element Analyses (FEA) are performed. A continuous stiffened plate subjected to an in-plane compressive load in a stiffener direction is considered, and the shaded region shown in Figure 1 is selected as the analytical region. The region ABCD is called a double-span/double-bay model (Yao et al. 1998), and the symmetric boundary condition is applied to all the plate edges. The region ABEF is called a triple-span/triple-bay model, and while the symmetric boundary condition is applied to the longitudinal edges, the periodical symmetric boundary condition is applied to the transverse edges. The double-span/double-bay model is used when a buckling mode of the plate and stiffener has odd half-waves. Meanwhile, the triple-span/double-bay model enables to analyze both the cases of even and odd half-waves. Because this study focuses the torsional buckling deformation of the stiffener, the z-displacement is fixed on the joint line between the stiffener and plate to prevent the occurrence of Eulerian buckling deformation of the stiffener.

W01 sin

z πx V01 sin h a

(2)

The coefficient, V01, is set to be 1/1000 of the span length, a. In this study, a flat-bar stiffener and deep teebar stiffener are considered since these relatively easily produce the torsional buckling deformation. Although a deep angle-bar also produces the torsional deformation, this is ignored since the coupled flexible-bending and torsional buckling takes place with the very complicated behavior. Young’s modulus and yielding stress of the material are set to be 205.8 GPa and 313.6 MPa, respectively, and 1/65 of Young’s modulus is considered as strain hardening after yielding. In-house FEA code “ULSAS” developed by one of authors is used. 2.2

FEA results

The deformation at the ultimate strength of the stiffened plate with the flat-bar obtained by FEA is illustrated in Figure 2. The color in elements means the spread of the yielding in the thickness direction, and the red element represents fully yielded, the light blue fully elastic. Figure 2(a) shows the deformation in the case of the relatively thin plate and stiffener (a × b × tp = 2400 × 800 × 13, h × tw = 250 × 19 (mm)). The stiffener is torsionally deformed with the same number of half-waves as the local buckling of the plate although the overall mode (one half-wave mode) is given as the torsional initial deflection. On the other hand, Figure 2(b) shows the deformation in the case of the thick plate and thinner stiffener (a × b × tp = 2400 × 800 × 20, h × tw = 350 × 19 (mm)). Although this case is likely to produce the overall torsional deformation, according to the figure, the stiffener is hardly deformed with torsion. These results shows that the torsional

320

Figure 2.

Deflection mode and spread of yielded zone at ultimate strength of stiffened plate with flat-bar.

Figure 3.

Deflection mode and spread of yielded zone at ultimate strength of stiffened plate with deep tee-bar.

deformation of a flat-bar is local mode with the same number of half-waves as the plate buckling, and the overall mode does not occurs as long as the stiffener has such dimensions as is used in a ship structure. Figure 3(a) shows the deformation in the case of the thin plate and deep tee-bar stiffener (a × b × tp = 2400 × 800 × 10, h × tw + bf × tw = 433 × 12 + 150 × 17 (mm)). In this case, the plate buckling firstly takes place, and this buckling deforms the stiffener with the slight magnitude and the same number of half-waves as the plate buckling. However, this deformation in the stiffener does not grow even after the applied compressive load is increased, and instead of this, the overall torsional deformation of the stiffener grows as shown in Figure 3(a). In the case of the deeper

stiffener, the stiffener web buckling, which is that the stiffener web is deformed like a rectangle plate simply-supported at all the edges, might be caused. However, in the case shown in Figure 3(b) (a × b × tp = 4200 × 840 × 16, h × tw + bf × tw = 620 × 11 + 200 × 30 (mm)), such web buckling does not take place, and the overall torsional deformation is predominant in the stiffener. These results clarify that a stiffened plate with a deep tee-bar stiffener collapses with the overall torsional deformation of the stiffener even when the local buckling deformation is produced in the web by the plate buckling at early loading stage. By the way, according to Figure 3, the yielded zone spreads in the half breadth of the stiffener flange. That is, once the half breadth of the flange is yielded at the center of the span, the applied load reaches the ultimate strength.

321

3 3.1

THEORY OF ELASTIC LARGE DEFLECTION ANALYSIS

3.2

Deflection function

In this study, to accurately simulate the elastic behavior of the stiffened plate with the large torsional deformation of the stiffener, the analytical solutions based on Elastic Large Deflection Analysis (ELDA) of the plate and stiffener are derived. One of authors has developed the method to estimate the elastic buckling strength of a stiffened plate (Fujikubo and Yao 1999). In this method, the vertical deflection on the plate, w, and the horizontal deflection on the stiffener, v, are assumed as follows: ⎧ πy 1 ⎛ 2π y ⎞ ⎫ mπ x w ⎨W1 i + W2 ⎜1− cos ⎟ ⎬ sin ⎝ b 2 b ⎠⎭ a ⎩

(3)

⎧ z π z⎞ π z ⎫ mπ x ⎛ v ⎨V1 +V2 ⎜1− cos ⎟ +V3 siin ⎬ sin ⎝ 2h ⎠ 2h ⎭ a ⎩ h

(4)

(5) v0

z mπ x V01 sin i h a

αp

(

αp =

mπ x πy mπ x πy sin i , w0 W01 sin sin i a b a b

z mπ x v V1 sin h a

In order to derive the analytical solution of the panel part based on elastic large deflection analysis, the components of in-plane stress and strain are firstly derived from the compatibility equation using Airy’s stress function and the assumption of in-plane stress. Then, the components of bending strain are derived by directly differentiating the deflection function, and the components of the bending stress are also derived from the strain components and the assumption of in-plane stress. Applying the virtual work principle, the relationship between the deflection coefficient, W1, and the average stress acted on the plate part, σap, is derived as follows: 2

2

)

β p (W1 −W01 ) σ ap W1 = 0

(7)

where

Each deflection component is illustrated in Figure 4. Considering the interaction between deflections of the plate and stiffener, V1–V3 are given as a function of W1 and W2. Although the analytical solutions with high accuracy can be derived using the deflections shown in Equation (3) and (4), the solutions become very complicated and are not derived easily. Therefore, simplified equations for the deflections are used. That is, the terms of W2, V2 and V3 are ignored, and then, the interaction between W1 and V1 is not considered explicitly. The following equations are used as the deflections, w, v, and the initial deflection, w0, v0: w W1 sin

Analytical solutions of plate and stiffener

(6)

βp =

π 2E a2 ⎛ m4 1 ⎞ + ⎟, 16 m 2 ⎝ a 4 b 4 ⎠ 2

π 2E

(

12 1 − ν 2

)

⎛ tp ⎞ ⎛ m b a ⎞ ⎜⎝ b ⎟⎠ ⎜⎝ a + m b ⎟⎠

2

βp is equivalent to the elastic buckling stress of the plate part. Deriving the analytical solutions of the stiffener if the same procedure as the plate part is adopted, solutions, such that the vertical displacement (z-displacement) is restrained at the top of the stiffener and the excessive stress is produced in z-direction, is given. Therefore, the analytical solutions are derived by the following procedure. According to the FEA results, the in-plane stress component in x-direction (loading direction) is predominant, the value of the other components, which are in z-direction (depth direction) and shear, is as small as 1/5 of the x-component. Therefore, the x-component is only considered as the inplane stress components produced in the stiffener web and the other components are ignored.

σ xpw

E ε xpw , σ zp zpw w

0 τ xzpw = 0

(8)

The in-plane strain in x-direction, εxpw, is given as the following equation:

ε xpw =

Figure 4.

Assumed components of buckling mode.

∂u + ∂x

2

1 ⎧⎪ ⎛ ∂v ⎞ ⎛ ∂v0 ⎞ ⎨⎜ ⎟ − ⎜⎝ ⎟ 2 ⎩⎪ ⎝ ∂x ⎠ ∂x ⎠

2⎫

⎪ ⎬ ⎭⎪

(9)

Then, the following equilibrium condition is considered.

322

∂σ xpw

∂τ xzpw

+

∂x

∂z

= 0,

∂σ zpw zpw

+

∂z

∂τ xzpw ∂x

=0

(10)

From Equations (8)–(10), the following equation is derived. 2 2 ∂ 2u ⎛ ∂v ⎞ ⎛ ∂ v ⎞ ⎛ ∂v0 ⎞ ⎛ ∂ v0 ⎞ = − + ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎝ ∂x ⎠ ⎝ ∂x 2 ⎠ ⎝ ∂x ⎠ ⎝ ∂x 2 ⎟⎠ ∂x 2

(11)

Substituting Equation (6) into Equation (11), and substituting the equation obtained by the integration of Equation (11) into Equation (9), the in-plane strain, εxpw, is given. The in-plane strain components calculated by the above procedure do not satisfy its compatibility condition. The in-plane stress in x-direction, σxpw, is calculated as follows:

π 2 m 2 E ⎛ 3 z 2 3Af Aw ⎞ 2 2 − ⎜ ⎟ V1 V01 − σ as Af Aw ⎠ 12 a 2 ⎝ h2

(

σ xpw =

)

(12) where σas is the average stress acted on the stiffener. On the other hand, the bending strain and stress are derived by the same procedure as the panel part. The bending stress components are:

π 2m2E

σ xbw =

2

(

a h 1− ν

σ zbw

2

)

y z (V

V

) sin

mπ x a

(14)

ν σ xbw

τ xybw = −

(13)

π mE y(V − V a h( + ν )

) cos

mπ x a

(15)

The stress components in the stiffener flange are obtained by substituting z = h into Equations (12)–(15). Applying the virtual work principle, the relationship between the deflection coefficient, V1, and the average stress, σas, is derived as follows:

αs

(

2

)

2

β s ( V1 − V01 ) σ as V1 = 0

π 4 m 4 E Aw 90 a

3

(

(

Af + Aw

)

)

Af + Aw ξ1

π 4 m 4 E htw3

π 2 m 2 G tw3 βs = + 6 a h ξ1 72 a3 1 − ν 2 ξ1 4

+

(

4

)

π m EIf 3

2 a ξ1

+

π 2 m2 G J f 6 a h2 ξ1

bf t f 2

ξ1 = π m

2

Aw = htw I f

(A

f

)

+ Aw / ( a )

b3f t f /

, J f = b f t 3f / ,

βs is also equivalent to the elastic buckling stress of the stiffener. 3.3

Analytical solutions for stiffened plate

Although Equations (7) and (16) have been derived assuming that the deformations of the plate and stiffener are independent of each other, there actually exists the interaction between both the deformations. So, in this study, the increases in the plate buckling stress due to the stiffener torsional rigidity and the stiffener buckling stress due to the plate bending stiffness are considered as the interaction. As mentioned above, one of authors has developed the equations to estimate the buckling stress considering all the deflection components shown in Equations (3) and (4). The increases in the buckling stresses can be considered by applying the buckling stresses σcrp and σcrs, which are calculated by the developed equations, to Equations (7) and (16) instead of βp and βs. As the plate buckling stress, σcrp, the minimum value is selected changing the buckling half-waves, m. In accordance with the FEA results mentioned in 2.2, m regarding the stiffener buckling stress, σcrs, is fixed to be 1 in the tee-bar and the same number as the plate in the flat-bar. Furthermore, when σcrs is calculated, it is assumed that the plate part is not subjected to compression load, that is, the plate part has only the role to resist the stiffener torsional deformation. Equations (7) and (16) are solved assuming that each of the plate and stiffener has the same average strain, εa. Both the equations can be solved independently as long as εa is known. The average stress on the total cross-section of the stiffened plate, σa, is calculated as follows:

σa =

(16) 4

where

αs =

Af

4.1

Ap σ ap As σ as Ap As

, Ap

bt b t p , As = Af

Aw (17)

RESULTS OF ELASTIC LARGE DEFLECTION ANALYSIS Isolated stiffener

Firstly, the elastic buckling behavior of an isolated stiffener is focused. Figure 5 indicates the relationships between the average stress and horizontal deflection at the top of the stiffener and center of the span. The lines in the figure represent the results simulated by the Elastic Large Deflection Analysis (ELDA) and the symbols by FEA.

323

Figure 5.

Relationships between average compressive stress and horizontal deflection of isolated stiffener.

In the flat-bar, the results by ELDA have a good agreement with those by FEA. In the teebar, the small difference between both the results is observed in the deflection range that exceeds 60 mm. However, it is no problem from a practical viewpoint because the yielded zone must widely spread in this deflection range of the actual stiffener. The narrow dashed line in Figure 5 shows the results considering the in-plane stress component in z-direction. This case excessively underestimates the deflection, it is confirmed that the proposed method ignoring the in-plane stresses except x-component is reasonable.

at the top of the stiffener and the center of the span is plotted in Figure 7. The ELDA results are compared with the FEA results in Figure 7(a), ELDA excessively underestimates the deflection. This is because that the stiffener is inclined by the buckling deflection of the plate. In this study, the effect of the stiffener inclined by the plate deflection is considered as the increase in the initial deflection of the stiffener. That is, the initial deflection, V01, in Equation (16) is modified to be a function of the plate deflection, W1, as the following equation: V01

4.2 Stiffened plate with flat-bar Since the torsional deformation of a flat-bar attached on stiffened plates is local mode, the results in the case of the stiffened plate with the length of one buckling half-wave are shown here. Figure 6 represents the relationships between the average stress acted on the plate part and the maximum vertical deflection of the plate. Comparing the results estimated by ELDA with that obtained by FEA, although there is the difference between both the results in the range of large deflection, there is totally a good correlation between both. This indicates that the elastic behavior after buckling can be simulated using the simplified buckling mode shown in Equation (5). The relationships between the average stress acted on the stiffener and the horizontal deflection

η ( W1 − W01 ) V0o1

(18)

where Vo01 is the original initial deflection. η is the parameter to indicate the degree of the influence of the plate deflection, and is assumed as the following equation: 0 ⎧ ⎪⎪ ⎞ η = ⎨⎛ 2.9 h ⎞ ⎛ σ crs ⎪⎜⎝ b − 0.1⎟⎠ ⎜ σ − 1⎟ crp r ⎝ ⎠ ⎩⎪

σ crs / σ crp ≤ 1 W

σ crs / σ crp r >1 (19)

RW is the ratio between the horizontal deflection at the top of the stiffener and the maximum deflection of the plate, which are found when the buckling stress, σcrp, is calculated. That is, using

324

the coefficient of the buckling mode, W1, W2 and V1–V3, RW is given as follows: RW

Figure 6. Relationships between average compressive stress and maximum deflection of plate part in stiffened plate with flat-bar.

(V + V

V ) /(W

W

)

(20)

The relationships between the average stress and deflection of the stiffener estimated by the modified ELDA are compared with those obtained by FEA in Figure 7(b). Both the relationships relatively correspond with each other. Also in the average stress-average strain relationships as the stiffened plate shown in Figure 8, the results by modified ELDA have a good agreement with those by FEA. From these results, it is confirmed that the proposed ELDA has a sufficient accuracy to estimate the elastic buckling behavior of the stiffened plate with the flat-bar. The modification mentioned in this section is only applied to the flat-bar, not to the tee-bar. 4.3

Stiffened plate with tee-bar

The elastic buckling behavior of the stiffened plate with a deep tee-bar stiffener is focused here. The long plate whose size is a = 4350 mm and b = 800 mm and the deep stiffeners of h = 450– 650 mm, are selected. The number of buckling half-wave of the plate is set to be m = 7 in ELDA as well as initial deflection in FEA, and that of the stiffener m = 1. The relationships between the average stress of stiffener and the deflection at the top of stiffener are plotted in Figure 9. In the case of the stiffener with 650 mm height which is easily deformed with the torsion, the results by ELDA are in a good agreement with those by FEA. However, in the other cases that the plate buckling takes place before the

Figure 7. Relationships between average compressive stress and horizontal deflection of stiffener in stiffened plate with flat-bar.

Figure 8. Relationships between average stress and average strain of stiffened plate with flat-bar.

325

FEA is not good. These results mean that it is difficult to ensure the accuracy of the estimation using only simplified deflection functions such as Equation (5) and (6). However, in an actual ship structure, it is predicted that the yielding spreads in the wide region of the stiffener in the stress range in which the large difference between the ELDA and FEA is observed, and the difference may be acceptable from a practical point of view. So, the estimation of the ultimate strength of stiffened plates will be attempted using the unchanged ELDA in the next section. 5

5.1 Figure 9. Relationships between average compressive stress and horizontal deflection of stiffener in stiffened plate with deep tee-bar.

torsional buckling of the stiffener, although ELDA can be simulate the elastic behavior with a sufficient accuracy in the low stress and small deflection range, excessively overestimates the deflection in the range beyond this. This is thought to be because the local deformation in the stiffener web produced by the plate buckling prevents the growth of the overall torsional deformation. Also in the relationships between the average stress and average strain as the stiffened plate shown in Figure 10, the correlation between the results by ELDA and

Method to estimate ultimate strength

Finally, the ultimate strength of the stiffened plate with a deep tee-bar is estimated using ELDA. In this study, the applied load at the point when the stress component in x-direction exceeds the yield stress at a checking point located in the stiffener flange, is defined as the ultimate strength. The stress component in the flange is given by substituting z = h in Equations (12) and (13). Two yieldchecking points are considered. One is the end of the flange (CASE 1), that is, y = bf/2 is substituted in Equation (13). The other is the center of the half breadth of the flange (CASE 2), that is, y = bf/4. The latter corresponds to the collapse behavior observed in FEA, such that the full half breadth of the flange is yielded at the ultimate strength. Both the points are located at the center of the span, i.e. x = a/2. The dimensions of the stiffened plates are indicated in Table 1. The same settings of m as shown in 4.3 are applied. 5.2

Figure 10. Relationships between average stress and average strain of stiffened plate with deep tee-bar.

ESTIMATION OF ULTIMATE STRENGTH OF STIFFENED PLATE WITH DEEP TEE-BAR

Estimated results

Table 1 indicates the strength obtained by FEA and estimated by ELDA. In addition, the estimated ultimate strengths by PULS, which is the software to estimate the ultimate strength and developed by DNV, are also indicated in the table. σifea in the table represents the applied stress at the point when the yielding initially occurs in the flange. Focused on the estimated ultimate strength by the ELDA under CASE 1, σueld1 is smaller than σufea obtained by FEA. Meanwhile, σueld1 has a good correlation with σifea. Considering the fact that the flange is initially yielded at its end, this good correlation means that the elastic behavior before the initial yielding can be simulated by the proposed ELDA. Namely, ELDA is sufficiently available from a practical point of view although ELDA

326

Table 1.

Ultimate strength and yield strength of stiffened plate with tee-bar obtained by FEA, ELDA and PULS. Dimensions*

FEA

ELDA

PULS

Model

tp

h

σufea/σY

σifea/σY

σueld1/σY

σueld1/σufea

σueld2/σY

σueld2/σufea

σupls/σY

σupls/σufea

h45t10 h45t13 h45t18 h55t10 h55t13 h55t18 h65t10 h65t13 h65t18

10 13 18 10 13 18 10 13 18

425 425 425 525 525 525 625 625 625

0.7975 0.8479 0.9462 0.7381 0.8107 0.9207 0.6635 0.7415 0.8517

0.7241 0.7930 0.8821 0.6582 0.7254 0.8129 0.5683 0.6323 0.7113

0.7049 0.8187 0.9041 0.6171 0.7256 0.8245 0.5376 0.6295 0.7116

0.8839 0.9655 0.9554 0.8360 0.8951 0.8955 0.8103 0.8489 0.8355

0.7615 0.8661 0.9503 0.6783 0.7922 0.8937 0.6061 0.7060 0.8003

0.9549 1.0215 1.0043 0.9189 0.9772 0.9706 0.9135 0.9521 0.9397

0.8259 0.8418 0.8960 0.8291 0.8578 0.8992 0.8195 0.8737 0.9152

1.0335 0.9815 0.9168 1.1210 1.0519 0.9500 1.2364 1.1697 1.0494

*Other dimensions: a = 4350, b = 800, tw = 13.5, bf = 150, tf = 25, which are fixed; Length in (mm); σufea: Ultimate strength obtained by FEA; σifea: Strength of initial flange yielding obtained by FEA; σueld1: Estimated ultimate strength by ELDA (CASE 1); σueld2: Estimated ultimate strength by ELDA (CASE 2); σupls: Estimated ultimate strength by PULS; σY: Yield stress (=313.6 MPa).

Figure 11. Collapse mode obtained by PULS (tp = 10 and h = 650).

cannot simulate the large torsional deformation of the stiffener. The reason why σueld1 is smaller than σufea is that the applied load does not reach the ultimate strength until the yielding spreads in the half breadth of the flange. Since σueld2 is estimated considering the yield-checking point located at the center of the half breadth, σueld2 has a good agreement with σufea although a little overestimation in the case of h = 450 and underestimation in the case of h = 650 are observed. In many models, σupls estimated by PULS is larger than σufea, the difference is particularly large in the case of the deeper stiffener. Figure 11 illustrates the collapse mode obtained by PULS in the case of tp = 10 and h = 650. In PULS, it is assumed that the primary buckling mode grows without a mode change as shown in Figure 11. In fact, the overall torsional buckling mode of the stiffener also grows as shown in Figure 3, and this mode becomes predominant. Because of this, the estimation by PULS which does not consider the mode change has the tendency to overestimate. The correlations between the estimated ultimate strengths and the FEA results are plotted in Figure 12. The estimations by ELDA under CASE 2 have a very good correlation with the FEA results, it is confirmed that the proposed method is beneficial to estimate the ultimate strength.

6

Figure 12. Correlation between estimated ultimate strength and FEA result.

CONCLUSIONS

In this study, the buckling/plastic collapse behavior of the stiffened plate with the torsional deformation of the stiffener was examined through the FE analyses. It has been clarified that the torsional

327

deformation of the flat-bar stiffener is local mode with the same number of the buckling half-waves as the plate. Meanwhile in the tee-bar, the overall mode with one buckling half-wave is finally predominant even when the local buckling occurs in the stiffener web. The analytical solutions based on the elastic large deflection analysis were derived to simulate the elastic behavior of the plate and stiffener under axial compression. By ignoring the in-plane stress component except axial component, the elastic behavior of the isolated stiffener can be simulated with high accuracy. The method to simulate the elastic behavior of the stiffened plate was also proposed. This method considers the increase in the buckling stress due to the interaction between plate and stiffener. In the flat-bar, added to this, the influence of the inclining of the stiffener due to the plate buckling deflection is considered as the increase in the initial deflection. The elastic behavior of the stiffened plate with the flat-bar can be simulated with a sufficient accuracy, while the accuracy in the case with the deep tee-bar is not sufficient. However, this is acceptable to estimate the ultimate strength even in the case of the teebar. The improvement of the method for the teebar will be performed as a future work. Finally, the ultimate strength of the stiffened plate with the deep tee-bar was estimated using

the analytical solutions based on the elastic large deflection analysis. The ultimate strength estimated by proposed method has a good correlation with that obtained by FEA, and it has been found that the method is beneficial. REFERENCES Carlsen, CA. 1980. A parametric study of collapse of stiffened plates in compression, The Structural Engineer 58B(2):33–40. DNV. 2005. Nauticus hull user manual, PULS. Fujikubo, M. & Yao, T. 1999. Elastic local buckling strength of stiffened plate considering plate/stiffener interaction and welding residual stress. Marine Structures 12:543–564. Fujikubo, M., Yanagihara, D. & Yao, T. 1999a. Estimation of ultimate strength of continuous stiffened plates under thrust. Journal of the Society of Naval Architects of Japan 185:203–212. [in Japanese]. Fujikubo, M., Yanagihara, D. & Yao, T. 1999b. Estimation of ultimate strength of continuous stiffened plates under thrust (2nd report). Journal of the Society of Naval Architects of Japan 186:631–638. [in Japanese]. IACS. 2006. Common structural rules for bulk carriers. Yao, T., Fujikubo, M. & Yanagihara, D. 1998. On loading and boundary conditions for buckling/plastic collapse analysis of continuous stiffened plate by FEM. Proceedings of the twelfth Asian technical exchange and advisory meeting on marine structures 305–314.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Ultimate strength of river-sea container ships Wenhua Zhang & Xiaofeng Luo China Classification Society, China

Weiguo Wu Wuhan University of Technology, Wuhan, China

ABSTRACT: In this paper the authors investigated the ultimate strength of some river-sea container ships using Smith’s Method, analyzed the effects of the floor spaces, the scantlings of the deck longitudinal, the thicknesses of the deck plating and the sheer strakes on the ultimate loading capacity, and evaluated the ultimate strength of ships. The results indicated that for the river-sea container ships: a) the ultimate hogging bending moment was greater than the ultimate sagging bending moment, and the failure mode of hull girder was mainly the buckling of deck plating under the sagging condition; b) the floor space had slight influence on the ultimate bending moment Mu; c) the depth of ship D had a significant effect on Mu, and the ultimate loading capacity was generally proportional to D2; d) increasing the scantlings of the deck longitudinal, the deck plating and the sheer strakes increased Mu by approximately 10%; e) B/D had better not exceed 3.0. 1

INTRODUCTION

The frequently used approaches to analyze the ultimate strength of hull girder are the progressive collapse analysis, the non-linear finite element method and the Idealized Structural Unit Method (ISUM[1]. Among the methods for the progressive collapse analysis on the transverse section of a hull girder subjected to longitudinal bending, Smith’s method was adopted by the member societies of International Association of Classification Societies in their rules. In this paper, the authors carried out the analysis of the ultimate hull girder strength of a series of river-sea ships with critical dimension ratios. The ultimate hull girder strength is related to the transverse section configuration of the hull. The typical transverse section considered herein is a double hull structure with large deck openings, as shown in Figure 1. The distance from the plat-

form in the side tank to the baseline ranged from 0.57D to 0.62D, and differed in the arrangement of the shell longitudinals which were to be spaced as evenly as possible. The design depths of river-sea ships are always constrained by the loading/unloading condition of river terminals and the bridges along the river. Thus the longitudinal strength is a great concern at sea. This paper investigated the ultimate hull girder strength of the ships with critical dimension ratios and with invariable length L and breadth B but different depth D. The main parameters were L = 60 m, 80 m, 100 m, 110 m and 130 m respectively, L/B = 4.5, B/D = 3.0 and 3.5, draught T = 0.75D and Cb = 0.862. The ultimate bending moments were first calculated using the scantlings which meet the minimum requirements by the rule, then the increased scantlings that met the bending and torsional strength, and the comparison was made as well. 2

Figure 1.

Typical transverse section.

SMITH’S METHOD

In 1977 Smith[2] developed a method to incorporate the load shortening curves of the plate elements in the calculation of the hull girder collapse. In this method, a cross-section is divided into small elements composed of stiffener(s) and attached plating. The behavior of each plate was calculated by finite element method and their contribution to the overall behavior of the hull girder was accounted

329

as a function of the plate location in the cross section. The ultimate bending moments of a hull girder under hogging and sagging conditions, are defined as the peak values of the curve of bending moment M versus the curvature χ of the transverse section considered, as shown in Figure 2. M-χ curve is obtained through an incremental-iterative procedure[3]. Each step of the incremental procedure is represented by the calculation of the bending moment Mi which acts on the hull transverse section as the effect of an imposed curvature χi. The increment of curvature results in an increment of the rotation angle of the transverse section around its horizontal neutral axis, and induces axial strains ε in each hull structural element, whose value depends on the position of the element. The stress σ induced in each structural element by the strain ε is to be obtained from the loadend shortening curve σ-ε of the element, which takes into account the behavior of the element in the non-linear elasto-plastic domain, see Table 1. Plating panels and ordinary stiffeners such as the longitudinals may collapse in one of the failure modes specified in Table 1. Hard corners, such as bilge, sheer strake-deck stringer elements, girderdeck connections and face plate-web connections on large girders, collapse mainly according to an elasto-plastic mode of failure.

Figure 2. M-χ curve. Table 1. Failure mode σ–ε curve

Failure modes. Elasto-plastic Beam column Torsional collapse buckling buckling

The distribution of the stresses induced in all the elements determines a variation of the neutral axis position because of the non-linear σ-ε relationship. The new position of the neutral axis is to be obtained by means of an iterative process, imposing the equilibrium among the stresses acting in all the hull elements. Once the position of the neutral axis is known and the relevant stress distribution in the section structural elements is obtained, the bending moment of the section Mi around the new position of the neutral axis, which corresponds to the curvature χi imposed in the step considered, is to be obtained by summing the contribution given by each element stress.

3 3.1

ANALYSIS OF INFLUENCING FACTORS Floor space

Floor spaces were taken as 4 frame spaces, i.e. S ranged from 2000 mm to 2800 mm, with the increment of 200 mm. The ultimate bending moments decreased as S increased, but the decreasing amplitude was small. Set L = 130 m as an example, as shown in Table 2. Ultimate hogging bending moment Mu (+) decreased progressively by approximately average 1.96%, and ultimate sagging bending moment Mu (−) by about average 1.613%. This means that S had slight effect on the ultimate bending moments, and had weaker effect on Mu (−) than on Mu (+). The average decreasing amplitudes of ultimate bending moments of the ships, as shown in Table 2, decreased gradually as S increased, and decreased gradually as L increased, as shown in Table 3. The larger the ships are, the less the effects of the change of the floor spaces on the ultimate bending moments. The ratio of Mu (+) to Mu (−) for a ship decreased as S increased, but the decreasing amplitude is small, as shown in Table 2. The average ratio of Mu (+) to Mu (−) ranged from 1.15 to 1.18, see Table 4. 3.2

Scantlings of longitudinal members

The check of global longitudinal strength indicated that the minimum scantlings given by the formula in the rule cannot guarantee to meet the requirements of the global longitudinal strength and the buckling strength due to the large deck openings, so the scantlings of some longitudinal members had to be increased. 3.2.1 Increase the scantlings of deck longitudinal No doubt that increasing the scantlings of deck longitudinal will increase the ultimate bending

330

Table 2.

Mu when L = 130 m (×105 kN-m).

S (mm)

2000

2200

2400

2600

2800

Average

Mu (+) Mu (−) Hog (%) Sag (%) Mu (+)/Mu (−)

9.374 −7.970 – – −1.176

9.185 −7.831 −2.010 −1.744 −1.173

9.002 −7.709 −1.998 −1.552 −1.168

8.824 −7.583 −1.975 −1.629 −1.164

8.660 −7.468 −1.856 −1.527 −1.160

– – −1.960 ± 0.071 −1.613 ± 0.098 −1.168 ± 0.006

Table 3.

Average decreasing amplitudes of Mu.

L (m)

60

80

100

110

130

Average

Hog (%) Sag (%)

−1.947 −2.508

−2.416 −2.732

−2.259 −1.795

−2.299 −1.698

−1.960 −1.613

−2.176 ± 0.211 −2.069 ± 0.513

Table 4.

The ratio of Mu (+) to Mu (−).

Mu (+)/Mu (−)

60

80

100

110

130

Average

−1.150

−1.180

−1.164

−1.169

−1.168

−1.166 ± 0.011

Table 5. Comparison after and before adjusting the scantlings of deck longitudinal. L (m)

60

80

100

110

130

Average

Mu (+) Mu (−)

1.030 1.040

1.087 1.122

1.067 1.088

1.095 1.119

1.081 1.099

1.072 ± 0.06 1.094 ± 0.033

Table 6.

Comparison after and before adjusting the thicknesses.

L (m)

60

80

100

110

130

Average

Mu (+) Mu (−)

1.019 1.019

1.011 1.016

1.121 1.124

1.140 1.148

1.252 1.261

1.109 ± 0.099 1.114 ± 0.102

moment of hull girder. The average increase of Mu (−) was 9.35%, and Mu (+) 7.22%, as shown in Table 5. The increasing amplitude of Mu (−) was slightly greater than that of Mu (+). 3.2.2

Increase the thicknesses of deck plating and sheer strake Two kinds of plating are always synchronously increased because they are in high stress area and their thicknesses are interrelated in the rule. Another reason is that the large thickness difference probably results in the local stress concentration. Increasing thickness was more efficient to increase the ultimate bending moment than increasing deck longitudinal, as shown in Table 6.

3.3

Depth D

Let B be invariable, and B/D increased from 3.0 to 3.5, as a result, the ultimate bending moments decreased significantly. The decreasing amplitudes were about 30% (floor space S = 2800), as shown in Table 7. Ultimate loading capacity of hull girder depends on its sectional modulus. i.e. it is proportional to B and D2. Ultimate bending moments became (D1/D2)2 times the original values when B/D was increased from 3.0 to 3.5 and B was kept invariable. The result of (D1/D2)2 = (3/3.5)2 = 0.735 was very close to the Average value in Table 7. They can be regarded as the same if the accumulative errors in the calculation were taken into account. The results

331

Table 7.

Mu at different D (×1011 N-mm).

L (m)

60

80

100

110

130

Average

Mu (+) M’u (+) Ratio Mu (−) M’u (−) Ratio

1.330 0.894 0.672 −1.137 −0.767 0.675

2.216 1.714 0.773 −1.865 −1.497 0.802

4.455 3.120 0.700 −3.862 −2.725 0.706

5.622 3.969 0.706 −4.862 −3.507 0.721

8.660 6.265 0.723 −7.48 −5.531 0.741

– – 0.715 ± 0.037 – – 0.729 ± 0.048

Note: M’u (+) and M’u (−) correspond to B/D = 3.5; M’u (−) at B/D = 3.5 was approximately 73% of Mu (−) at B/D = 3.0, while M’u (+)at B/D = 3.5 was about 71.5% of Mu (+) at B/D = 3.0.

confirmed that t the moulded depth D has a significant effect on the ultimate bending moment. 4

ULTIMATE HULL GIRDER STRENGTH

The ultimate strength of hull girder is required to meet the following formula: Mu Ms

Mw

≥β

(1)

where: Mu—the ultimate bending moment of hull girder; Ms—the still water bending moment of hull girder; Mw—the wave-induced bending moment of hull girder; β—safety factor ≥ 1.15; α—non-linear correction factor; take α = 1.2 when the additional bending moment due to bow slamming is considered. If α = 1.0, no non-linear loads are considered, and the denominator in formula (1) stands for the resultant loads used for checking the global longitudinal strength. Ms was calculated in accordance with the loading conditions, and the maximum of the still water bending moment was used for the calculation in formula (1). Mw was calculated by the following formula[3]: Mw ( ) Mw ( )

0. MC Crs L2 B (Cb + 0. ) 0 19M MC Crs L2 BCb

(2) (3)

where: Crs—hydrodynamic coefficient; Crs = 0.03L+2.91; L, B—ship length and breadth respectively, in m; Cb—block coefficient; M—bending moment distribution factor from aft to fore perpendicular, as shown in Figure 3.

Figure 3.

4.1

Bending moment distribution factor.

B/D = 3.5

Before the scantlings were increased, hull girders might collapse after the ship exceeds certain length. Deck plating was buckled due to the sagging of hull girder of ships with length L ≥ 100 m, even non-linear correction due to slamming was not considered. The failure of hull girder occurred when L ≥ 80 m and non-linear correction was considered, and Mu (+) was greater than Mu (−). The failure mode was the buckling of deck plating, and this regularity is consistent with the longitudinal bending of hull girder and the buckling of deck panel. When L = 80 m, Mu (+) met the requirement, but Mu (−) met it only when S = 2000 mm. When L > 80 m, both Mu (+) and Mu (−) met the requirements. After the scantlings were increased, hull girders might collapse depending on whether non-linear correction of bending moment was considered. If no, the safety factors were greater than 1.0. If yes, the outcome will depend on the ship length and floor space. When L ≥ 130 m, neither Mu (+) nor Mu (−) met the requirement. When S = 2000 mm, the safety factors were greater than 1.0; When S > 2000 mm, if L ≥ 100 m, Mu (−) could not meet the requirement. 4.2

B/D = 3.0

Before the scantlings were increased, if non-linear correction of bending moment was not considered,

332

Table 8.

Mu when B/D = 3.0 and S = 2800 mm (×106 N-mm).

No.

L (m)

60

80

100

110

130

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16)

Mu (+) Mu (−) Mc (+) Mc (−) (1)/(3) (2)/(4) Mc1 (+) Mc1 (−) (1)/(7) (2)/(8) Muh (+) Mus (−) (11)/(3) (12)/(4) (11)/(7) (12)/(8)

1.330 −1.137 0.426 −0.533 3.118 2.134 0.501 −0.611 2.655 1.862 1.402 −1.209 3.289 2.269 2.800 1.979

2.216 −1.685 1.299 −1.464 1.707 1.152 1.497 −1.672 1.481 1.008 2.437 −2.116 1.877 1.446 1.628 1.266

4.455 −3.862 3.233 −3.255 1.378 1.186 3.663 −3.706 1.216 1.186 5.120 −4.500 1.583 1.382 1.398 1.214

5.622 −4.862 4.435 −4.469 1.268 1.088 5.035 −5.097 1.117 0.954 7.005 −6.199 1.580 1.387 1.391 1.216

8.660 −7.468 8.090 −7.263 1.071 1.028 9.176 −8.402 0.944 0.853 1.169 −1.031 1.445 1.420 1.274 1.228

Note: Mc = Ms+Mw, represents the resultant wave bending moments; Mc1 = Ms+1.2Mw, refers to the resultant moments after considering non-linear correction; Mu (+) & Mu (−)—Ultimate bending moments before adjusting the scantlings; Muh (+) & Mus (−)—Ultimate bending moments after adjusting the scantlings.

all ships met the requirement of formula (1). If the non-linear correction was considered, when L = 130 m, Mu (−) could not meet the requirement, and Mu (+) could not either if S ≥ 2400 mm. When L = 110 m, if S ≥ 2400 mm, Mu (−) could not meet the requirement; When L ≤ 110 m, and S ranged from 2000 mm to 2800 mm, both Mu (−) and Mu (+) met the requirements. After the scantlings were increased, whether the non-linear correction was considered or not, the safety factors were greater than 1.2. The ultimate loading capacity of hull girder was significantly improved. The check of the ultimate strength of hull girder can be exempted, provided that the ship meets the requirements for the global longitudinal strength and the buckling strength of the hull girder. The ultimate strength is relatively easier to meet when B/D = 3.0 than B/D = 3.5, therefore B/D = 3.0 is a better option than B/D = 3.5. As an example, ultimate loading capacity was analyzed when S = 2800 mm and B/D = 3.0, and the ultimate strength before and after adjusting the scantlings was evaluated, as shown in Table 8. After the scantlings were increased, whether the non-linear correction was considered or not, the safety factors were greater than 1.2. When S = 2000 mm, the safety factors exceeded 1.3. So the river-sea ships can have larger frame spaces and floor spaces like sea-going ships other than river ships. The checking of ultimate strength can facilitate the designer to choose the rational plate thickness and longitudinal stiffeners.

5

CONCLUSION

In this paper the ultimate loading capacities of hull girders of river-sea container ships with B/D = 3.0 and B/D = 3.5 were calculated respectively using Smith’s Method. The effects of the floor spaces, the moulded depth, the changing of the scantlings on the ultimate loading capacities were analyzed by comparing the changing pattern at the different floor spaces. The results demonstrated that: – For the river-sea container ships, the ultimate hogging bending moment was greater than the ultimate sagging bending moment, and the failure mode of hull girder was mainly the buckling of deck plating under the sagging condition; – the floor space had slight influence on the ultimate bending moment Mu; – the moulded depth of ship D had a significant effect on Mu, and the ultimate loading capacity was approximately proportional to D2; – increasing the scantlings of the deck longitudinal, the deck plating and the sheer strakes increased Mu by approximately 10%; – B/D had better not exceed 3.0. When B/D = 3.0, after the scantlings are increased enough to enable the ship to meet the global longitudinal strength and the buckling strength, the safety factors are greater than 1.2 whether the non-linear correction is considered or not. The ultimate loading capacity has a significant improvement.

333

– Smith’s Method is a very effective approach to calculate the ultimate bending moment of hull girder of the ship. The assessment of the ultimate hull girder strength is a very important way to guide the structure design. The plate thicknesses, the scantlings of longitudinal, and the floor space, even the ship depth, can be rationally chosen by this procedure. In addition, the limitation of the ration B to D in the Society’s rule can be reasonably defined in this way.

IACS. Common Structure Rules for Bulk Carrier. 1 April, 2006. Smith CS. Influence of local compressive failure on ultimate longitudinal strength of a ship’s hull. Proceedings of the International Symposium on PRADS, Tokyo, Japan, 1977. p. 73–9. Yao T. Hull Girder Strength. Marine Structures 16(2003) pp 1–13.

REFERENCES China Classification Society. Guidelines on River-Sea ships, 2008.

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Experimental analysis

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Experiments on three mild steel box girders of different spans under pure bending moment J.M. Gordo & C. Guedes Soares Centre for Marine Technology and Engineering (CENTEC), Instituto Superior Técnico, Technical University of Lisbon, Lisboa, Portugal

ABSTRACT: An experimental study is presented of three box girders made of mild steel subjected to pure bending moment with different spacing between frames. The moment curvature curves are presented, allowing for the analysis of elastic-plastic behaviour until collapse and the evaluation of the ultimate bending moment and post collapse behaviour for each experiment. The residual stress relief during loading and unloading path is also analysed. The effect of the span between transverse frames on the ultimate bending moment of the box girder is study, thus its dependence on the column slenderness of the panel under compression can be established. The results are compared with tests on similar box girders made of very high tensile steel. The effects of residual stresses on the behaviour of the box girder are analysed using a progressive collapse method for structures under longitudinal bending moment. 1

INTRODUCTION

The evaluation of the ultimate capacity of ships under bending moment is a very important issue for the structural design. It is associated with a global failure of the hull and the final result is normally the loss of the ship, its cargo and human lives. In the last years several works have been done on the subject, most of them on the evaluation of the ultimate bending moment of ships made of normal mild steel. The existing calculation methods may be divided into two groups: finite elements methods, and simplified methods. There has been a great activity and comparison between the different methods is available in the literature (Yao et al. 2000). The development of the design of structures under bending has been made on the assumption that the structure can be divided into several simple stiffened plate elements that act independently. The authors have been working on a method based on these assumptions (Gordo et al. 1996), which has been validated against data from a full scale accident (Rutherford 1990) where the loading conditions could be well established and compared against some small scale experiments of models representing simplified typical sections of ships (Dow et al. 1981, Faulkner et al. 1984, Gordo & Guedes Soares 1996, Nishihara 1984). The results of these comparisons showed that the method can be used confidently on typical hull configurations and for normal steel.

Changing the span between frames will affect the non-dimensional slenderness of those plate elements leading to different collapse strength despite using the same geometry. The change of frame spacing will induce collapse at different levels of column’s slenderness and this call for new experimental results, covering the appropriate range of the governing parameters of the plating. In this study the behaviour of three box girders made of the same material with the same configuration but different spans is compared. 2

HULL STRENGTH EVALUATION

There are several methods available to evaluate the ultimate moment in sagging or hogging that a hull may sustain. The authors have been working on a method (Gordo et al. 1996) that is able to predict the overall behaviour of the hull under bending moment. This method predicts not only the ultimate bending moment but also the pre and post collapse behaviour. It considers all the modes of collapse of the structure and it also includes an algorithm to deal with residual stresses and corrosion. This method and the software that has been developed to implement it, proved to give good prediction for normal steel ships when compared to the tests and hazards examples available in the literature. In order to provide data for those comparisons a plan of experiments was developed for box

337

girders subjected to pure bending moments. These box girders may reproduce in a simple manner the behaviour of the ship’s structure under bending, allowing the identification of the differences of using mild steel or high tensile steel, widening the range of validity of the method and covering the behaviour of panels of high column slenderness. The typical element of the box girders is a plate with a bar stiffener which has been proved to be representative of the actual type of structure of ship’s hull (Gordo and Guedes Soares, 1993). In order to obtain information about the carrying capacity of different panel arrangements, like plates reinforced by complex stiffeners, another series of experiments has to be planned due to the geometric limitations for the reproduction of such scantlings at the present scale and limitations on the total loading that one may use in these box girders experiments. 2.1

Main parameters of the structural design

The main parameters affecting the structural design of ship hulls subjected to bending moment are the plate and column slenderness, because they affect directly the effectiveness of the panels under compression. These parameters are defined as follows: – Plate slenderness, β – Column slenderness, λ

σ o /E σ o /E

and they depend directly from the geometry of the structural elements and from the material properties. The geometric characteristics of interest are the width (b) and the span (a) of the structural elements, as well as their thickness (t) and the radii of gyration (r) of the cross section of the stiffener with an appropriate associated plate. Other geometric characteristics may affect the behaviour of the stiffener in special cases. This may occur when the stiffener is very weak or it has low torsional rigidity, promoting a different mode of collapse known in the literature as tripping. The material properties of interest are the yield stress (σo) and the modulus of elasticity (E). The shear modulus of elasticity (G) has some influence on the tripping stress of the stiffener. Also the nature of the stress-strain curve of different steels may affect the elasto-plastic behaviour of the structural elements under compression, especially concerning on having or not having a constant yielding stress. For the same transversal geometry of the structure but changing the span between frames, the column slenderness changes and this will lead to a weaker structure with lower buckling and ultimate stresses for longer spans.

2.2

Assessment of the hull girder strength

The ability of the hull girder to sustain applied bending moment may be understood as the summation of individual contributions of each stiffened plate element that one may subdivide the entire cross section between two frames. This can be expressed as: M

∫ ( z − zn ) σ ( z ) ⋅ dA = ∑ ( zi i

A

zn ) ⋅ σ i ( zi ) Ai (1)

where the average stress σ on the stiffened plate element measured on its centroid is a function of the average strain ε and the latter is dependent of the location zi of the element and of location of the neutral axis zn:

σ ( zi )

f (ε i ) and ε i = g ( zi zn )

(2)

The relation between stress and strain depends on many parameters including residual stresses due to welding, geometric imperfections, transverse support due to frames rigidity, etc. The stress-strain curves may be obtained from a data base of pre-calculated load-shortening curves (Smith 1977) or by approximate methods (Gordo & Guedes Soares 1993, Yao & Nikolov 1991) based on the empirical formulas for the ultimate strength of panels under axial loading. 3

TEST SETUP AND BOX GIRDERS’ CHARACTERISTICS

3.1 General information The boxes are made of mild steel of 270 MPa yield stress and the Young’s modulus is considered to be of 200 GPa. The specimen N200 has five frames corresponding to four frame spacing of 200 mm each and a total length of 1400 mm, because there is 100 mm in each side of the top frames to allow the redistribution of stresses. The model has a nominal width of 800 mm and a nominal depth of 600 mm. The longitudinal stiffeners are five in total on the top panel 150 mm apart from each other. The other two girders, N300 and N400, are similar to N200 except that they have only four transverse frames and the spacing between adjacent frames is 300 and 400 mm, respectively for N300 and N400. The experiment setup is presented in Figure 1 for box N300, showing the bold connections to the supporting side structures and the loading device. 3.2 Type of experiment The tests consist on a four point bending of a beam like box girder. The beam is divided into three

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Figure 1.

Test setup for box N300.

parts: two symmetric supporting parts and, in the middle, one has the box girder structure. Each supporting part is 2 m long. The box girder is subjected to pure bending moment, inducing tension on the bottom and compression on the top of the box. So it is expected to have some plasticity in tension on the bottom and a marked buckling collapse of the top panel at the maximum load. Since the former is not very much affected by the span between frames, the maximum bending moment will be controlled by the buckling average stress of the top panel which is sensitive to the distance between frames. The average curvature of the box girder is measured by computing the angle of rotation, a, of the extreme of the box from the relative vertical displacement, d, of the opposite tops of the box apart of a distance L due to the curvature 1/R, as shown in Figure 2. It has mounted to two displacement gauges, one at each side of the box girder, allowing detecting any transversal rotation that may occur. 3.3

Figure 2. Relation between the radii of curvature and the angle of rotation of the extreme of the box girder.

Figure 3.

Cross section of the box girders.

Figure 4.

Geometry of the box girder N200.

Geometric properties of the models

The boxes are made of 4 mm thick plate. The spacing between stiffeners is 150 mm, Figure 3, which leads to width to thickness ratio of 37.5. The span between frames is 200, 300 and 400 mm, Figure 4. The nominal column slenderness covered is from 0.97 to 1.94. The plate slenderness β is constant at the top panel and equal to 1.38 with a b/t of 37.5, which a very common value in ship structures. The stiffeners are bars of 4 mm by 20 mm, leading to a cross sectional area of 680 mm2, for each relevant stiffened plate element. The plating area of the individual stiffened plate on the top panel is 600 mm2 and the stiffener area is 80 mm2.

4

EXPERIMENT OF BOX-GIRDER N200

The box girder, which was denoted as N200, was tested applying four cycles of loading followed by discharge as shown in Figure 5. The first cycle reached the total vertical load of 250.5 kN with a

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Figure 5. Vertical load versus vertical displacement on N200 test.

corresponding vertical displacement at the loading point of 10.16 mm. The following cycles achieved 501.1 kN at 23.87 mm, 619.6 kN at 37.68 mm and the maximum load was 643.0 kN when the vertical displacement at the loading point reached 45.3 mm. The different cycles of loading allow identifying and quantifying the shakedown of residual stresses due to plastic deformations on the initially high stressed parts of the box girder due to manufacturing. As known from the typical residual stress pattern, regions close to the welding are in tension with stresses close to the yield stress of the material. Thus, when these regions are loaded with external tensile loads they just yield at the squash stress without supporting any further load but retaining some permanent elongation. When the load is removed, the stress in those points reduces according to the Hooke’s law. The final result for the next cycle is to have a higher effective structural modulus in the initial stages of load until the load reaches the maximum level of load of the previous cycle. After that point the same process repeats resulting in an increase of the shake down of residual stresses until they disappear completely. However, note that this process only occurs on the panels under tension due to the bending of the structure. If the structure has asymmetric welding, which is the case for these box girders, the load may become unbalanced leading to the rotation of the structure, even if the structure is symmetric. That seems to be the reason for the differences on the measurements of the displacement transducers that read the rotation of the box girders during the first cycle of loading, as represented in Figure 6, by Rot_L and Rot_R transducers. These two transducers are used to evaluate the curvature of the structure at each loading step and they are located on opposite sides of the box, as shown in Figure 7. If one has no transverse rotation of the

Figure 6. Measurements on the displacement transducers on first cycle.

Figure 7. Setup of one of displacement transducers for measuring the rotation over top panel (N400).

box then the readings should be the same in both transducers. As may be seen from the figure, the left transducer, Rot_L, remains almost unchanged, while the right transducer displaces until 2 mm during the uploading and keeps some permanent of 0.3 mm after the downloading. Since there is a welding on the right side most probably this behaviour comes from the yielding of the welding in the early stage of the loading due to residual stresses. The global effect for all cycles of load is a constant difference of 25% between the readings of the two gauges Gordo and Guedes Soares (2008). Disp_L and Disp_R represent the vertical displacement at the opposite tops of the box and Disp_1/2 gives the vertical displacement at middle length of the box. As expected the top transducers give the same readings due to symmetry and the middle transducer gives higher values than the top ones due to curvature of the box. Again there are residual values after discharge meaning that plasticity and stress relief have occurred.

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4.1

Moment curvature relationship

Having the load and rotations, it is possible to generate the curve that relates the applied bending moment with the curvature. That relationship is plotted in Figure 8 where the curvature is the average curvature of whole box girder. It can be seen from the figure that if no discharge was done then the resulting moment curvature relationship would be the upper envelope of the four cycles of loading. In that case the behaviour would be elasto-plastic in the whole range of the curvatures due to permanent plasticity on the welding regions of the panel under tension and on the panel under compression in the late stage of loading. But the intermediate discharge of loading between cycles cancels the direct effects of the residual stresses during the following loading path and that allows identifying the elastic behaviour of the structure free of residual stresses. The linear nature of the relation between the bending moment and the curvature is perfectly identified in the third and forth cycles of loading; on the second cycle, the transversal rotation of the box already mentioned before introduces non-linear effects when the average curvature calculated from the two rotations is used. This affects directly the performance of the effective structural modulus, which is the slope of the bending moment curvature curve. 4.2

Effective structural modulus

The nominal structural modulus is EI, as known from the linear elastic beam theory, where E is the Young’s modulus of the material and I is the moment of inertia of the cross section of the box girder. The nominal structural modulus is the maximum value that one may expect for the effective structural modulus defined as the derivative of the bending moment in respect to the curvature. Initial imperfections and residual stresses cause

Figure 9. Structural modulus during loading path for different cycles.

a decrease in the rigidity in the load shortening curves of the panels that constitutes the structure, thus the effective structural modulus is always less than EI, which is 153 MNm2 for these boxes. On the first cycle the effective structural modulus has initially very high values with large scatter but reduce rapidly due to the plasticity developed on the welding of the panel under tension and rearrangement of the initial imperfections of the box. One has to note that the loaded box has panels under tension where the initial imperfections tend to reduce with increasing load and panels under compression where they tend to increase. These panels are connected and the performance of one affects the behaviour of the adjacent one. Figure 9 compares the evolution of the structural modulus during the loading path for the different cycles. There are two main tendencies to be observed: the constant value of the structural modulus in the range of loading already reached in previous cycles, between 140 and 160 MNm2; and a lower evolving curve corresponding to an experiment with only one cycle of loading. This lower curve is always decreasing smoothly and vanishes at the ultimate bending moment where one has a null structural tangent modulus. What is rather interesting is that the curve is almost straight from the maximum load of the first cycle until the collapse. This straight line intercepts the y axis at a value similar to those found for the effective structural modulus in the elastic range after shakedown of residual stresses. 5

Figure 8. Moment curvature relationship of N200 specimen.

EXPERIMENT OF BOX-GIRDER N300

The box girder N300 is similar to the previous one but the space between frames is 300 mm. Also the number of frames is 4 instead of 5. The total length of box is 1100 mm. The plate slenderness on the top panel subject to compression is 1.38 and

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column slender of the stiffened panel composed by stiffener and associated plate is 1.45. 5.1

Load vertical displacement relationship

This test was performed in 3 stages: the first cycle until a load of 200 kN which approximately one third of the maximum load, the second cycle until 400 kN, and the last cycle until collapse and beyond, as shown in Figure 10. The maximum vertical displacement achieved was 50 mm but the measurement of the vertical displacement is not very relevant because it depends on the length of the lateral supporting structure. Nevertheless it may be used to evaluate the absorbed energy in every cycle of load. This information can be used to estimate indirectly the residual stresses of the structure (Gordo & Guedes Soares, 2004). Most important aspects of the curve are the evolving curve of the 3 cycles that represents the behavior of the box under bending without residual stresses relief, a strong reducing in the slope above 400 kN and several small discharges around 500 kN that lead to the collapse of the entire upper panel in compression and consequently to the failure of the box under pure bending moment. On Figure 11 it is presented the measurements of displacements used for the evaluation of the rotation and curvature of the box during loading and in both sides. They agree very well until the collapse which means that there is not any substantial lateral rotation of the box due to asymmetric welding or initial imperfections. In fact, at small discharge close to 500 kN, one has some differences on the readings indicating that the buckling of the top panel in compression was initiated in the left side propagating quickly to the entire panel. After collapse the box acquired some lateral rotation, which is expected due to development of very large deformations in the entire structure after the collapse.

Figure 11. Measurements on the displacement transducers for N300.

Figure 12. Moment curvature relationship of N300 specimen.

5.2

5.3 Figure 10. Vertical load versus vertical displacement on N300 test.

Moment curvature relationship

Based on the data from Figure 11, one may generate the moment curvature relationship which is presented in Figure 12. The most important aspects on this curve are the same as those of the load versus vertical displacement presented in Figure 10. The maximum bending moment is 512 kNm and it located in platoon where the bending moment kept almost constant value in broad range of the curvature, i.e., from a curvature of 0.012/m to 0.014/m with a variation of less than 1% in the bending moment, which represents a very smooth collapse of the structure developing very large deformations at constant bending moment. The discharged of the buckled structure is almost vertical meaning that the permanently deformed box does not present much elastic recovery as expected. Effective structural modulus

The relation between the effective structural modulus and the bending moment presents similar properties than those observed for the box N200, as

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Figure 13. Structural modulus during loading path for different cycles (N300).

Figure 14. Vertical load versus vertical displacement on N400 test.

shown in Figure 13. However the calculated modulus is approximately 10% below the one of the stockier box girder, N200. Also the moment presents slightly lower values at lower loads than those at high loads after the relief of residual stresses in the bottom panel during the second cycle, as can be observed on the curve of the third cycle. 6

EXPERIMENT OF BOX-GIRDER N400

The box girder N400 has the same space between frames is 300 mm. Also the number of frames is 4. The total length of box is 1400 mm. The plate slenderness on the top panel subject to compression is 1.38 and column slender of the stiffened panel composed by stiffener and associated plate is 1.94. 6.1

Load vertical displacement relationship

This test was performed in 4 stages: the first cycle until a load of 200 kN, the second cycle goes to 300 kN, the third until 400 kN and the last cycle until collapse and beyond, as shown in Figure 14. The maximum vertical displacement achieved was 40 mm. Qualitatively one has obtained a similar curve to the ones of experiments N200 and N300. The maximum vertical load is the lowest of the 3 experiments which is in accordance to the theory of stability of structures that indicates a decrease of ultimate strength of compressed panels with the increase on column slenderness. The reduction on the ultimate strength of the top panel leads naturally to reduction on the ultimate moment supported by the structure. Figure 15 presents the displacements associated with rotations of the structure. In the first cycle a residual difference of rotation is generated, which is kept constant along the others intermediate cycles. This means that the structure have no transversal rotation in cycles 2 and 3.

Figure 15. Measurements on the displacement transducers for N400.

On the last cycle the structure presents some lateral rotation in the collapse phase due to the asymmetric nature of residual stresses on the whole structure and asymmetric large permanent deformations after collapse. 6.2

Moment curvature relationship

The moment curvature relationship is presented in Figure 16. The maximum bending moment is 475 kNm at a curvature of 0.0086/m. The platoon where the bending moment kept almost constant value is very narrow compared to the one found in experiment N300. One particular aspect is of worth interest: in cycles 2 and 3 the discharge of load is, initially, associated with an increase in the curvature, demonstrating that plasticity needs time to spread during the loading of the structure. 6.3 Effective structural modulus Figure 17 shows the variation of the structural modulus with the bending moment.

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Figure 16. Moment curvature relationship of N400 specimen.

Figure 18.

N400 before test.

Figure 17. Structural modulus during loading path for different cycles (N400).

Figure 19.

N400 loaded at 300 kN on 2nd cycle.

Figure 20.

N400 loaded at 400 kN on 3rd cycle.

The average value in the ‘elastic’ range is 143 MNm2 and it is located in a narrow band between 140 and 145 MNm2 on the fourth cycle for bending moments above 80 kNm. The others characteristics of the curves confirm the comments made for box N200. 6.4 Deformations induced by loading In this section it is presented the evolution of top panel deformations during the test. Figure 18 shows the initial deformations before test where one may detect some upwards deformations in the middle stiffeners of the first bay and a slight downward deformations in the stiffeners of the middle bay. The third bay panel is virtually straight. During the initial cycles of loading these initial deformations amplifies according to the level of loading, as shown in Figure 19 for 300 kN at the second cycle and in Figure 20 for 400 kN at the third cycle. After the unloading of the 400 kN cycle there are some remaining permanent deformations of small magnitude as shown in Figure 21. Figure 22 presents the deformation of the box girder at the moment of collapse with well-developed

deformations on the first and second bay, and much smaller deformations on the third bay. So the shape of collapse is not symmetric for a symmetry specimen and loading which demonstrates the importance of the shape and amplitude of the initial imperfections on the final result of this type of experiments. One should note that the deformations are alternately up and downwards. The residual deformations after collapse and total discharge of load are presented in Figure 23.

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They retain the shape of collapse indicating that one has two simultaneous different type of panel collapse. The first bay top panel failed by plate induced collapse or yielding of the stiffener in tension due to local bending and the middle bay’s panel by stiffener collapse in compression. There is no evidence of tripping in the middle bay’ stiffeners. 7

Figure 21.

N400 unloaded at the end of 3rd cycle.

Figure 22.

N400 at collapse load, final cycle.

EFFECT OF FRAME SPACING ON THE ULTIMATE BENDING MOMENT

The frame spacing on ship structures increases the column slenderness of the stiffener panel and, as a consequence, the critical stress of the panel reduces according to the elastic theory and the ultimate stress under axial compression. In these experiments having equal cross section, the top panel is under compression, thus the maximum strength of this panel decreases with increasing the space between frames. According to equation (1) the maximum bending moment that may be achieved reduces with a reduction on the compressive strength of the constitutive structural component. Figure 24 presents the compilation of test’ results for the ultimate bending moment and the structural modulus at last cycle of loading in the ‘elastic’ range. It confirms the decrease of ultimate bending moment with the increase on the frame’s spacing. As expected, the effective structural modulus is independent of this variable and the very slight decrease may be attributed to panel’s initial imperfections and experimental uncertainties. Figure 25 presents the moment curvature curve of all experiments together. One may identify several aspects of the effect of frame spacing on the behaviour of box girders under pure bending moment. The curvature at collapse and also the ultimate bending moment tend to decrease with the increase in the frame spacing. More slender boxes tend to be more flexible when residual stresses and imperfections are present.

Figure 23. Residual deformations of N400 after collapse.

Figure 24. Ultimate bending moment and structural modulus versus column’ slenderness of top panel.

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ACKNOWLEDGEMENTS This paper has been prepared within the project “MARSTRUCT—Network of Excellence on Marine Structures”, (www.mar.ist.utl.pt/ marstruct/), which has been funded by the European Union through the Growth program under contract TNE3-CT-2003-506141. REFERENCES Figure 25.

Moment curvature curve of experiments.

From comparison of the three curves at collapse range one may conclude that the presence of high levels of initial imperfections and residual stress on box N300 promotes a reduction in the tangent structural at low levels of loading and leads to a premature step by step collapse of top panel stiffeners increase the plateau at the collapse region. It may also be observed that the permanent deformed box N300 after collapse has lower level of elastic energy than the other two, as it is shown by the slope of discharge of load in the final cycle of each test. 8

CONCLUSIONS

Three tests on the bending of box girders were performed varying the span between transverse frames. It was found that the transverse frame spacing is an important parameter on the strength of thin walled box subjected to bending moment. The increase of the span reduces the ultimate bending moment of the structure due to a decrease of the buckling stress of the panels under compression. Residual stresses are very important in this type of experiment and the moment curvature curves depend very much on their level according to the manufacturing process. However it is possible to have a good understanding of the behaviour of the structure without residual stress by performing a series of loading cycles prior to the collapse of the structure. With those cycles one removes the residual stresses on the panels in tension allowing for the observation of the elastic behaviour of the structure. The column slenderness controls the type of collapse of the structure: high column slenderness leads to more sudden collapse, followed by large discharge of load during the failure of the structure. That was found during the experiments and it is represented by the shedding pattern of experimental moment curvature curves. The structural modulus in the elastic range of the structure free of residual stresses does not change with the frame spacing.

Dow, R., Hugill, R., Clark, J., & Smith, C. 1981. Evaluation of ultimate ship hull strength, Extreme Loads Response Symposium; 133–147. Faulkner, J.A., Clarke, J.D., Smith, C.S. & Faulkner, D. 1984. The loss of HMS Cobra—A reassessment. Transactions of RINA. 127:125–151. Gordo, J.M. & Guedes Soares, C. 1993. “Approximate load shortening curves for stiffened plates under uniaxial compression”. Integrity of Offshore Structures—5, D. Faulkner, M.J. Cowling A. Incecik and P.K. Das.; (Eds) Glasgow. Warley, U.K.: EMAS; 189–211. Gordo, J.M., Guedes Soares, C. & Faulkner, D. 1996. Approximate assessment of the ultimate longitudinal strength of the hull girder. Journal of Ship Research. 40(1):60–69. Gordo, J.M. & Guedes Soares, 1996; C Approximate method to evaluate the hull girder collapse strength. Marine Structures. 9(1):449–470. Guedes Soares, C. & Gordo, J.M. 1997. Design Methods for Stiffened Plates Under Predominantly Uniaxial Compression. Marine Structures. 10(6):465–497. Gordo, J.M. & Guedes Soares C. 2004. Experimental Evaluation of the Ultimate Bending Moment of a Box Girder. Marine Systems and Ocean Technology. 1 (1): 33–46. Gordo, J.M. & Guedes Soares, C. 2008. Experimental evaluation of the behavior of a mild steel box girder under bending moment. Ship and Offshore Structures. 3 (4): 347–358. Gordo, J.M. & Guedes Soares C. 2009, Tests on ultimate strength of hull girders made of high tensile steel. Marine Structures, 22 (4): 770–790. Nishihara, S. 1984. Ultimate longitudinal strength of mid-ship cross section. Naval Arch. & Ocean Engng.; 22:200–214. Rutherford, S.E. & Caldwell, J.B. 1990; Ultimate longitudinal strength of ships: a case study. Trans. SNAME: 98: 441–471. Smith, C.S. 1977, Influence of local compressive failure on ultimate longitudinal strength of a ship’s hull, Proc. 3th Int. Symposium on Practical Design in Shipbuilding (PRADS), Tokyo, 73–79. Yao T, et al. 2000. Ultimate hull girder strength, Proceedings of the 14th International Ship and Offshore Structures Congress (ISSC), Nagasaki, Japan, 321–91. Yao T, & Nikolov, P.I. 1991, Progressive collapse analysis of a ship’s hull under longitudinal bending. J. Soc. Naval Arch. of Japan. 170: 449–461.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Scale model tests for the post-ultimate strength collapse behavior of a ship’s hull girder under whipping loads Kazuhiro Iijima, Yuko Suzaki & Masahiko Fujikubo Department of Naval Architecture and Ocean Engineering, Osaka University, Osaka, Japan

ABSTRACT: In this paper, experimental investigations into the post-ultimate strength collapse behavior of a ship’s hull girder under whipping loads are presented following the author’s previous works based on numerical simulations. One of the important conclusions of the previous works is that given the same magnitude of the loads, the collapse extent is much smaller for the loads with the shorter duration. For the validation, a scale model with a scale ratio of 1/100 which follows a law of similitude in the ultimate bending strength as well as geometry is employed in tank tests. The whipping loads are produced by dropping a mass object and the time history of the whipping loads are pre-adjusted by tuning the object mass, cushion material and dropping height. The hull girder bending moment with the same magnitude and with a time duration ranging 0.5 s−1.5 s in real scale is applied to the hull girder. It was observed that the collapse extent was smaller for the loads with the shorter duration. 1

INTRODUCTION

Wave-induced vibrations superposed with the normal wave loads increase the structural loads. Such a problem known as whipping has long been a matter of concern to naval architects. Recent transition of ships to larger size, higher speed and larger usage amount of high tensile strength steel for lighter structure makes us focus on the problem again. In Goal Based Standard (GBS) discussed at International Maritime Organization (IMO), it is requested to submit information and document for verification of rules at TIER III level, which includes how the rules consider deformation or vibration levels that may damage or impair the ship structure. When it is considered that the hull girder strength is within the scope, a consideration of the influence of the whipping on the hull girder ultimate strength may be necessary. There are various researches into the increasing effects. To name a few, Jensen JJ et al. (2002) developed a formula to evaluate the maximum hull girder bending moment accounting for the effect of the wave-induced vibrations. Wu and Moan (2006) investigated probability distributions of the hull girder loads including the normal wave and vibration components based on time-domain simulations. They found that the wave-induced vibrations significantly increase the hull girder loads while pointing out the importance of the large statistical uncertainties. In these works, peaks of the bending

moment were counted without distinguishing the peaks by the normal-wave loads from those by the combined loads. From the viewpoint of risk-based design, however, its consequence should be considered because the collapse behavior under normal wave loads and that under whipping loads may differ. i.e. if the structural loads exceed the ultimate strength of the structure, their impact on the hull girder ultimate strength need to be evaluated. Although structural engineers have long been questioning about the importance of the whipping from the aspect of consequence, no clear answers have been given to it. Iijima et al. (2011) showed that the postultimate strength behavior can be predicted by a hydro-elastoplasticity approach, accounting for the interaction between structural deformation and hydrodynamic/hydrostatic loads. Xu et al. (2012) also showed that given the same amplitude of bending moment, loads with a shorter time duration result in the smaller collapse extent. This means that the consequence of collapse under whipping load is not very significant, and therefore, the importance of the whipping loads may be less in evaluating the ultimate strength. The collapse behavior of a hull girder under impact loads is investigated experimentally in the present paper. A series of dynamic collapse tests are carried out using a scaled model which has been developed for the collapse tests.

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2 2.1

THEORETICAL PREDICTION Analytical model

A simple analysis model to describe the dynamic collapse behavior of ship’s hull girder is proposed by Xu et al. (2012). It is outlined herein. The whole ship is modeled as a two-rigid-bodies system connected each other by a nonlinear rotational spring as shown in Figure 1. In the figure, F1w and F2w represent the external forces, F1R and F2R the restoring forces, M1w and M2w the external moments measured around the center of gravity, M1R and M2R the restoring moments due to hydrostatic force with respect to the center of gravity. fint and Mint represent the reaction force and moment due to the spring, respectively. θ1 and θ2 show the rotational displacement of the respective bodies. The rotational spring represents the nonlinear relationship between the bending moment and the rotational angle, including the elastic range, ultimate strength, reduction of load carrying capacity due to buckling and yielding, and recovery of the bending rigidity when it is unloaded. It may be measured or calculated in advance either by experiment or FE analysis. The actual relationship between the internal moment Mint and the rotational angle θ may be described by using the paths shown by the chain dotted lines OA’BFG in Figure 2. It follows the elastic range at the beginning path. Then, before the path reaches the ultimate strength at Point A’, the slope changes due to buckling and yielding.

Figure 1. Free body diagram of the two rigid bodies system connected by nonlinear rotational spring.

Figure 2. Theoretical relation between bending moment and rotational angle.

After the path reaches the ultimate strength, the load carrying capacity rapidly decreases with the development of inelastic buckling deformation and localization of yielding (path A’B), and then the capacity drop becomes relatively moderate (path BF) with further spread of yielding. When unloaded (path FG), the recovered slope is smaller than the original (path OA) due to the presence of plastic deformation in structural members. Thus, the internal bending moment Mint, or capacity of the hull is a function of the relative rotational angle θ between the two bodies. M int

M int (

)

(1)

It is further assumed that two bodies of the hull have the same masses, lengths, moments of inertia and longitudinal metacentric heights, denoting as m, l, I and GM, respectively. Then, the equilibrium equations of motion can be obtained by reference to the gravity center of the respective bodies and can be further reduced to the following one dof equation. Iθɺɺ 2M int

ρ ggV ⋅ GMθ = 2M ext

(2)

where θ denotes θ1 − θ2 and 2M ext M1w 1 − M 2 w as the external load. A solution to the above equation may be given to any time history of load by using a numerical time integration scheme. 2.2

Sample result

The curve showing the relationship between the bending moment and rotational angle may be approximated to the piecewise linear curves as shown by the solid line OABCD in Figure 2. Mu is the ultimate strength, or collapsing moment at Point A. Before Point A, the curve is approximated to a straight line assuming neither buckling nor yielding occurs until Point A. The slope is given by k1 = EI/h where EI is the bending rigidity of the hull girder and h is the distance between transverse members herein. Point A is followed by path AB with slope k2, then the path reaches B. Along path BC, the capacity is assumed to be constant with slope k3. The slope of unloading path CD is assumed to be the same as the original stiffness along path OA. For illustration, a dynamic collapse behavior of the hull girder is analyzed for a ship whose particulars are given as shown in Table 1. A load time history as shown in Figure 3 is applied. The amplitude of the external moment M0 is 1.167 × 1010 Nm, which is almost 130% of the ultimate strength MU = 0.917 × 1010 Nm. The circular frequency of the load is taken ω0 = 0.27 rad/s, meaning that the duration of the load is 2π/0.27/2 = 11.6 s. The stillwater load is not given.

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Table 1.

Principal specifications.

Item

Symbol

Value

Length of each body Breadth Depth Draught Slope of path OA Slope of path AB Ultimate strength Constant capacity

l (m) B (m) D (m) d (m) k1 (Nm/rad) k2 (Nm/rad) MU (Nm) MBC (Nm)

150 40 30 10 5.395 × 1012 −5.395 × 1011 9.172 × 109 7.338 × 109

Figure 3. moment.

Figure 4.

Sample time history of vertical bending

frequencies of the external moment is performed. The magnitude of the internal moment amidship should be kept constant to detect the dependences of the collapse extent on the frequency. To this end, preliminary calculations in an elastic range are firstly performed to calibrate the amplitude of the external moment M0 inducing the internal moment in an elastic hull with magnitude of 130% of the ultimate strength, or 1.3 × MU. Figure 5 shows the relationship between the amplitude of the external moment M0 calibrated and the circular frequency of the moment. The amplitude of the external moment reaches 1.3 × MU or 1.19 × 1010 Nm when the circular frequency approaches to zero. For the larger circular frequency, the magnitude of the external moment inducing the internal moment with the same magnitude decreases due to the dynamic effect such as whipping. Collapse behavior is evaluated by solving Eq (2) to the calibrated loads. Figure 6 shows the relationship between the collapse extent and the circular frequency of the external moment with the constant magnitude. In this figure, the wave circular frequency of the whipping loads may fall into a circular frequency range over 2.0 rad/s. Note that the

Figure 5. Relationship between the amplitude of the external moment inducing a constant bending moment and the circular frequency.

A sample time history of rotational angle.

Figure 4 shows the time history of the rotational angle. It is observed the rotational angle starts to grow and develop when the internal load exceeds the ultimate strength. Then, it is followed by vibratory response around a new equilibrium position after the extinction of the external load. The plastic rotational deformation at the end of the calculation or the collapse extent is about 0.115 rad. 2.3 Response to impulsive loads By employing the same model explained in the previous section, a series of calculations of collapse behaviors under loads with various circular

Figure 6. Relationship between the collapse extent and the circular frequency of the vertical bending moment.

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frequency 2.0 rad/s corresponds to 0.16 s of time duration. It is observed that the collapse extent diminishes almost monotonically with the increase of the circular frequency of the external moment, indicating that the collapse may not occur to a large extent under impact loads.

3 3.1

EXPERIMENT Experimental model

A scaled model which follows the law of similarity in strength has been developed by Wada, R., et al. (2010). A box-shape ship of 1/100 scale ratio with dimensions of 3 m in length, 0.4 m in breadth, 0.3 m in depth and 0.1 m in draft is employed as in Figure 7. The mechanical properties of the model such as the ultimate strength are scaled according to the law of similitude. The model consists of two rigid bodies, a hinge and a sacrificial specimen amidship. The two bodies are connected each other by the hinge at deck level with the sacrificial specimen fixed to the bottom of the ship’s fore body and connected to the ship’s aft body via a rigid boom. The specimen is bent as a cantilever subjected to a horizontal force at the lower end when two bodies rotate around the hinge to the bending moment. The collapse tests can be conducted repeatedly by changing the sacrificial specimens. The nonlinear relationship between the bending moments and rotational angle is reproduced by the collapse mechanism of the specimen. By adopting the specimen appropriately designed, various collapse behaviors may be realized. In the present research, trough shape specimens with two slits near the fixed end were

adopted (Fig. 8). The specimens collapse due to buckling and yielding near the root of the specimens, as the bending moment is the largest at the fixed end of the specimens (see Figure 7 bottom). It is expected that the buckling occurs on the bottom plate of the U-shape specimen located on the compression side of bending. The bottom plate is not under pure compression but eccentric loading. This makes the collapse behavior of specimens less susceptible to initial imperfections caused during machining process. The scantlings of the specimens are determined based on the nonlinear FE-analysis results taking the strain hardening effects into account. Three specimens are finally selected for fabrication, and are denoted TS1, TS2 and TS3, respectively. The main particulars of the specimens are presented in Table 2. The rotational moment can be measured by a potentiometer equipped at the hinge while the vertical bending moment can be measured at the cross section adjacent to the collapse section by using three load cells. Static tests are carried out to detect the load carrying capacity of the model for each specimen. The tests are conducted keeping the model afloat on the water. The rotational displacement is given by lifting the model at two points near the both ends of model via chain ropes and pulleys.

Figure 8. The schematic diagram and the section of the trough type specimen.

Table 2.

Main particulars of specimens.

Parameter (units)

Figure 7.

Experimental model.

H (mm) h (mm) b (mm) t1 (mm) t2 (mm) h1 (mm) b1 (mm) Moment of inertia (m4)

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TS1

TS2

TS3

40 10 12 2 2 7 6 3.72 × 10−10

40 11 12 2 4 6 6 5.36 × 10−10

40 11 12 2 3 6 6 5.20 × 10−10

The measured results are presented in Figure 9. The ultimate strength is about 130 Nm, 180 Nm, and 155 Nm, for TS1, TS2 and TS3, respectively. TS1 has the lowest ultimate strength among the three, which may be understood by comparing the scantlings of the three specimens. In TS1 case, the capacity decreases mildly after reaching the ultimate strength and then approaches to a constant value. The curve for TS2 does not reach a constant moment within the measured range since the bottom plate of the specimen has the largest thickness among the three. The behavior of TS3 is somewhat between TS1 and TS2. The natural frequency of the hull is 5–6 Hz. 3.2 Experimental setup A series of collapse tests under impact loads was conducted in a tank with dimensions of 4 m in length, 2 m in breadth and 0.4 m in water depth. Impact loads were generated by dropping an object from a prescribed height as shown in Figure 10. The object was dropped at the midship of the hull instead of the fore or stern part since a collapse behavior in sagging was expected. A cushion material was spread on the upper surface of the midship. Selection of the cushion material was critical since the time duration of the load could be varied mostly by changing the cushion materials. Dropping height is a measure showing the distance

from the surface of cushion material to the bottom of dropping object while clamped. 3.3

Load calibration

Calibration of the loads in terms of magnitude and time duration was firstly carried out by changing parameters including the mass of the object, dropping height and cushion material. Two types of dynamic impact loads which have the same amplitude, however, with different time duration were targeted. Table 3 summarizes the parameters for producing the two types of load. Figure 11 shows the time histories of the vertical bending moment measured amidship. The time duration of the two types of loads was 0.08 s and 0.16 s, respectively. Here, the time duration is defined as the half period of the first wave as shown in Figure 3. It is understood that the time durations correspond to the realistic whipping period when scaled according to Froude’s law. The repeatability of the loads under the same condition was also confirmed by repeating dropping attempts. The average values of the dynamic bending moments in three attempts were 103 Nm and 107 Nm respectively for short and long duration cases. The variation was within 1.5% to the total bending moment including the static and dynamic moments. Table 3.

Parameters adjusting the load. Drop height (mm)

Cushion material

Short duration 3.2

250

Long duration 4.9

300

NR sponge rubber Urethane foam

Parameter

Mass (kg)

Figure 9. Relationship between vertical bending moment and rotational angle for the scale model.

Figure 10.

Experimental setup.

Figure 11. Two time histories of vertical bending moments acting on model ship (The still-water bending moment is 66 Nm).

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As for the still water bending moment, ballasts amounting to 60 kg in total were distributed along the bottom of the hull to attain target still water bending moment as well as the draught of 0.1 m. Sufficiently large still water bending moment was necessary to induce collapse under the impact loads explained above. The values of the still water bending moment were set respectively to 66 Nm, 131 Nm, 98 Nm for TS1, TS2 and TS3. Then, the still water bending moment combined with the dynamic bending moment would induce the total bending moment with magnitude 30% larger than the ultimate strength. 4

RESULTS AND DISCUSSIONS

4.1

Experimental results

All the experimental cases and the results are summarized in Table 4. The dynamic collapse tests were conducted 15 times in total. Each test id shows the specimen type and trial number, respectively. For example, the test id TS1-2 shows the second trial of the collapse tests using TS1 specimen. The maximum moment in the table shows the total bending moment estimated by summing up the maximum dynamic and static moments. Figure 12 shows time histories of the rotational angle and bending moment for TS1-1. Immediately after the object was dropped and impacted against the hull, the deformation started developing. At the same time, the bending moment grew, overshot the ultimate strength (130 Nm) measured by the static

Table 4.

Experimental results.

Test id

Time duration (s)

Maximum moment (Nm)

Collapse (rad)

TS1-1 TS1-2 TS1-3 TS1-4 TS1-5 TS2-1 TS2-2 TS2-3 TS2-4 TS3-1 TS3-2 TS3-3 TS3-4 TS3-5 TS3-6

0.08 0.08 0.16 0.16 0.16 0.08 0.08 0.16 0.16 0.08 0.08 0.16 0.16 0.08 0.08

170 (1.3) 170 (1.3) 173 (1.3) 173 (1.3) 173 (1.3) 235 (1.3) 235 (1.3) 238 (1.3) 238 (1.3) 202 (1.3) 202 (1.3) 206 (1.3) 206 (1.3) 235 (1.5) 235 (1.5)

0.0024 0.0021 0.0064 0.0065 0.0076 0.0040 0.0045 0.0089 0.0096 0.0033 0.0033 0.0071 0.0070 0.0045 0.0055

Figure 12. Time histories of rotational angle and bending moment for TS1-1.

test to a small extent, and started unloading after reaching the maximum value (142 Nm). The fact that it exceeds the ultimate strength might be attributed partly to the effect of strain rate. Then, the rotational displacement shook down after several vibratory fluctuations around a new equilibrium position. It is observed from Figure 12 that the height of the second moment peak and peaks afterwards was all smaller than the ultimate strength. Thus, the bending moment reached the bending capacity only once and collapse developed during the first peak. In Figures 13–15, time histories of rotational angle for the respective series using the same specimens are presented. They are shown with time shifts of 1 s for clarification. It is observed that variability among the collapse behaviors is small for the same type of loads and specimens. In these cases, impact loads inducing the total bending moment which would have reached 130% of the ultimate strength of the respective specimens were given to the hull. It is also noted that the collapse extent is larger for the loads with the longer duration. Still water bending moments larger than those explained in the previous section were also applied to check the variation of the collapse extent to the change of the total bending moments. Figure 16 compares the time histories of the rotational angle for TS3 series with different magnitudes of bending moment. In these cases, the magnitude of the total bending moment was set 202 Nm and 235 Nm, respectively. The time duration was all set 0.08 s. It is observed that the collapse extent is larger for the loads with the larger magnitude when the time duration of the loads is kept constant. 4.2

Maximum moment shows a sum of maximum dynamic moment and still water bending moment. The value in the parenthesis shows the ratio to the ultimate strength.

Discussions

The collapse extent under the vertical bending moment induced by impact load which exceeds the ultimate strength by 30% has been shown to be within the range of 0.006 rad, 0.01 rad and

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Figure 13. Time histories of rotational angle for TS1-1– TS1-5. Comparison between loads with short and long time durations.

Figure 14. Time histories of rotational angle for TS2-1– TS2-4. Comparison between loads with short and long time durations.

Figure 15. Time histories of rotational angle for TS3-1 – TS3-4. Comparison between loads with short and long time durations.

0.007 rad respectively for TS1, TS2 and TS3. These values seem small compared to the relationship in Figure 9. It is observed that the collapse extent did not reach the rotational angle where the internal moment reached the ultimate strength. It is interpreted that unloading started before the internal moment reached the ultimate strength (maximum of the bending capacity) due to the short time duration and that the larger external moment or

Figure 16. Time histories of rotational angle for TS31,2 and TS3-5,6. Comparison between loads with different magnitudes of bending moment.

the larger time duration was necessary to exceed the ultimate strength. Once the ultimate strength is reached, it is expected that the collapse would develop to a more extent since the path in the curve enters an unstable path with a negative slope and smaller capacity. In order to compare the collapse behavior under wave loads, a numerical code based on the hydroelastoplasticity approach proposed by the authors is adopted for the same box ship model. The interaction between the elasto-plastic deformation and the loads are accounted for in the hydro-elastoplasticity analysis (Iijima et al). Nonlinear hydrodynamic and hydrostatic loads are evaluated by employing a nonlinear strip theory. The collapse behavior in a wave train may be simulated. A focused wave in the reference is selected for the simulation. Firstly, calibration of the wave train is performed. The focused wave is generated from a spectrum with mean period of 10 s. The magnitude of the focused wave is so adjusted that the maximum vertical bending moment reaching 130% of the ultimate strength would be induced. The curve for TS1 in Figure 9 is adopted in the simulation, however, the calibration is performed within an elastic range assuming no buckling or yielding of the structure. Then, collapse behavior analysis of the hull girder in the focused wave is performed. The collapse extent is found out to be 0.095 rad. The time duration of the vertical bending moment then is 6.7 s, or 0.67 s in the model scale. It is apparent in this case that the collapse behavior follows the unstable path after the ultimate strength in Figure 8. This is the reason why the collapse extent develops more largely under normal wave loads whereas the deformation develops to only a small extent under the whipping loads, given the same magnitude of the loads. The ratio of the collapse extent is found to be about 40 to that for TS1-1 case in Table 4. In Figure 17, the collapse extent for TS1 case is plotted against the circular frequency of the load.

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6. The collapse extent under whipping loads is limited as the time duration for development of the plastic deformation is short. 7. The characteristics of the capacity curve after the ultimate strength are among strong influential factors to the collapse extent.

Figure 17. Relationship between the collapse extent and the circular frequency of the vertical bending moment.

The circular frequency of the wave load is converted to 0.47 rad/s (=2 /6.7 s/2). In the figure, TS1-1 and TSS1-3 cases are also presented in real ship scale. It is found that the curve in Figure 5 resembles the results in Figure 17. Thus, the predictions by the simple analytical model given in Section 2 well explain the frequency dependences of the collapse behavior although the load time history in the simulation/experiment is not identical to that assumed in the simple analysis. 5

CONCLUSION

In this research, a series of scale model tests are performed to investigate the post-ultimate strength behavior of a ship’s hull girder under whipping loads. The test results have good agreement with the theoretical predictions. The following conclusions are derived. 1. Given the same amplitude of the whipping bending moment, the bending moment with the shorter time duration results in the smaller extent of collapse. 2. Given the same amplitude of the vertical bending moment, the collapse extent under normal wave loads is much larger than that under impact loads. 3. The collapse extent increases with the increase of the difference between the total bending moment consisting of dynamic and static components and the ultimate strength of the hull girder. 4. For the test model employed in this study, the collapse behavior under whipping loads does not enter the unstable path after the ultimate strength even when the magnitude of the loads exceeds the ultimate strength by 30%. 5. On the other hand, the collapse behavior under normal wave loads exceeding the ultimate strength by 30% follows the unstable path after the ultimate strength.

In the present simulations, the bending momentdisplacement relationship curves taken from the scale model tests have been adopted. The ultimate strength and bending rigidity similar to those of real ship structures might be reproduced in the scale model, however, the characteristics of the capacity curve are not necessarily similar in terms of the capacity reduction after the ultimate strength. As discussed, it is clear that the characteristics after the ultimate strength influence the results. To clarify the collapse behavior of real ships, it is necessary to employ the relationship curves calculated for the real ship structures. The stress and strain distributions at the collapse should also be carefully looked into by using the nonlinear FE analysis to evaluate the consequence of hull girder collapse. The research works are underway.

ACKNOWLEDGEMENTS This research was partly supported by the Ministry of Education, Science, Sports and Culture, Grantin-Aid for Scientific Research (A), (23246150), 2012.

REFERENCES Iijima, K., Kimura, K., Xu, W. and Fujikubo, M. Hydroelastoplasticity approach to predicting the post-ultimate strength behavior of a ship’s hull girder in waves, J Mar Sci and Technol, 2011; 16(4):379–389. Jensen, J.J. and Mansour A.E. Estimation of ship longterm wave-induced bending moment using closedform expressions, Royal Institution of Naval Architects. Transactions. Part A. International Journal of Maritime Engineering (ISSN: 1479-8751), pp. 41–55, 2002. MSC87/WP.4 Annex 3, page 2, ANNEX Guidelines for Verification of Conformity with the International Goal-Based Ship Construction Standards for Bulk Carriers and Oil Tankers. Wada, R. Iijima, K. Kimura, K. Xu, W. & Fujikubo, M. 2010. Development of a design methodology for a scaled model for hydroplasticity of a hull girder in waves, Proceedings of PAAMES 2010, pp. 248–253. Wu MK and Moan T. Statistical analysis of wave-induced extreme nonlinear load effects using time-domain simulations. J Appl Ocean Res, 2006; 28:386–297. Xu W, Iijima K, Fujikubo M. Parametric dependencies of the post-ultimate strength behavior of ship’s hull girder in waves, J Mar Sci and Technol, 2012; 17(1): 1–13.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Solutions to improve accuracy in experimental measurement of the dynamic response of resilient mountings for marine diesel engines L. Moro, M. Biot & N. Mantini Department of Engineering and Architecture, University of Trieste, Trieste, Italy

C. Pestelli Research & Development, Wärtsilä Corporation, Italy

ABSTRACT: In the paper, after an introduction to the standards by which tests are performed and a presentation of the test rig used at the Ship Noise and Vibration Laboratory of the University of Trieste, outcomes of a series of experiments is discussed, carried out on a resilient mounting specifically designed for medium speed marine diesel engines. The aim of the experimental campaign is to design laboratory tests leading to a measurement of the dynamic transfer stiffness comprehensive of some interaction effects due to the coupling between engine foot and resilient element. The research is explained by the need of better understanding how the boundary conditions at the top casting of the resilient mounting influence the dynamic transmissibility to the foundation. Results are widely discussed, giving also suggestions for further improvement of the laboratory tests. 1

INTRODUCTION

On board ships, to avoid high level of structure borne noise due to main engines, dynamic characterization of resilient mountings is necessary (Hynnä, 2002, Ran Lin, et al. 2009, Tao, et al. 2000). As no analytical methods are today available to predict non-linear behaviour of resilient mountings, special laboratory tests are carried out (Biot & Moro, 2012, Thompson, et al. 1998). A complete dynamic characterization of the marine resilient mountings as structural noise barriers is performed, by using specific test rigs, at frequencies ranging from about 100 Hz up to no less of 1 kHz. The specifically designed test rigs should be able to dynamically excite resilient mounting under testing while a static load is applied. Practice in experimental testing has been acquired in the last years by a number of research laboratories working in the marine sector, where test apparatus are operated and, basically, perform measurement of linear motion dynamic stiffness (Fulford & Gibbs, 1997). A common standard for the test procedure and the design of the laboratory facilities is the ISO standard 10846 “Acoustics and vibration— Laboratory measurement of vibro-acoustic transfer properties of resilient elements” which relevant parts are Part 1 (Principles and guidelines), Part 2 (on the direct method) and Part 3 (on the indirect method). Methods to maintain accuracy

at satisfactory levels in experimental testing are contemplated in the standard. These guidelines are not generally sufficient to guarantee the reliability of the tests, because it is leave to ability of researchers to make the experimental condition of resilient mounting to be true to real working condition. So, the subject has to be examined in depth of the simulation in laboratory tests of the real coupling between resilient mounting and engine frame on one side and engine foundation on the other. That is the case of the test of special resilient mountings. In some configurations, resilient mountings are equipped with fixtures for the adjustment of resilient mounting in height, by means of a ring nut, for a better engine alignment and an easier on board installation below engine. While the resilient mounting base is rather large and tightly joined to the bed plate of the foundation, the resilient mounting top with its adjustment flange may be weak to transverse flexural deformations. Being resilient mounting top and fixing rail of the engine foot firmly fastened each other, gives rise to a dynamic coupling of the two parts and so to coupled modes of vibration. With the aim of setting a proper test procedure for this kind of resilient mountings, a research activity has been developed in the Ship Noise and Vibration Laboratory of the University of Trieste. A comparative experimental research has been carried out in order to evaluate the influence of

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different boundary conditions at the top of the resilient element on its dynamic response. Laboratory tests have been carried out on a resilient mounting for medium speed marine diesel engines, and three configurations have been tested: with and without the fixture on the original top casting, and with a cylindrical shaped top casting, which provides a larger contact surface between the resilient element and the fixing rail of the engine foot. 2 2.1

THE EXPERIMENTAL APPROACH The single-point-connected systems

A simplified approach to measure the dynamic stiffness is based on the assumption of treating the problem as a one-dimensional problem in which the structure-borne noise prediction is carried out by considering independent from each other all the transmission paths through the single resilient mounting (Fig. 1). By means of this approach, resilient mountings interactions are neglected. A further assumption is made of considering any resilient mounting to develop only the vertical translational velocity V and the vertical force F, so neglecting the presence of moments and angular velocities. Under these assumptions, the system composed by engine, isolators and foundation can be treated as a series of 2-DOF lumped parameter systems. Figure 2 shows a 2-DOF lumped model system, where parameters characterizing the system are given: the mass ms and mr of source and receiver, the static stiffness k0 of the spring-like isolator, the input force F0 acting on the source, and forces and velocities mutually exerted by the bodies. The velocity level on the foundation is given by (Hynnä, 2002): LV,

( f ) = L ,s ( f ) + LZ, i ( f ) + LM,r ( f ) V

(1)

Figure 2. Model for the 2-DOF lumped parameter system.

where the foundation velocity level LV,r (r for receiver) is a function of the source velocity level LV,s (s for source), of the resilient mounting mechanical impedance level LZ,i and of the foundation mobility level LM,r. So, to be able to predict the foundation velocity level, the three terms on the right hand of the equation have to be known. With respect to the determination of the resilient mounting impedance LZ,i, it has to be emphasized that it cannot be simply determined by resorting to traditional FE methods. The resilient mounting vibro-acoustic transfer properties can be reliably determined only by means of measurements performed on laboratory test rigs. The resilient mounting transfer function can be expressed as a dynamic inertia, and defined as the ratio between the force F2b(f) measured on the blocked side (side 2)—that is the blocking force at the isolator output side that implies a null displacement at the same side, and the complex acceleration a1(f) measured on the driven side (side 1). Once the resilient mounting dynamic inertia curve has been achieved by direct measurements, the isolator impedance level is then given by: LZ, i ( f ) =

F2 ( f ) F (f) =2 f 2 ≈ 2 π f T2b,1 ( f ) V1 ( f ) a1 ( f )

(2)

where the dynamic inertia is usually referred to as a dynamic transmissibility, and so denoted as T2b,1(f), in consideration that it is an output to input force ratio where the input force is referred to a unit mass. 2.2 Figure 1.

Resilient mounting in marine application.

The experimental damper characterization

The dynamic parameters of a resilient mounting system are evaluated by referring to the diesel

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engine as a suspended vibrating source. They are defined by making reference to a three block system: the vibration source (i.e., the diesel engine), the N isolators and the receiving structure (i.e., the engine foundation). Application of the procedure is subject to the main assumption, that only passive linear-response resilient elements may be studied. It also supposes that the N contacts between the three blocks may be treated as a series of independent single-point-connected systems (as in Fig. 2). According to ISO 10846-1:2008, to describe the vibro-acoustic characteristics of a resilient mounting the most significant parameter is the dynamic transfer stiffness k2,1(f). The dynamic transfer stiffness is defined as the ratio between the dynamic force on the blocked output side of an isolator F2b(f) and the complex displacement on the driven side u1(f). To prove the statement about the significance of the dynamic transfer stiffness, the simple case of a unidirectional transmission of vibrations through an isolator is considered in the following. If k1,1(f) and k2,2(f) are the driven-point stiffnesses when the resilient element is blocked at the opposite side (that is, u2 and u1 are null, respectively), the equilibrium equations of the isolator can be written for each frequency by: {F} = [K] {u}

(3)

where the stiffness matrix [K] is a 2 × 2 symmetric square matrix. In the three block system, isolator excites the receiving structure with the force F2 which may be derived by the definition of the driven-point stiffness of the receiving structure, that is kr = −F2/u2. So, F2 may be expressed as: k2,1 F2 = u1 k 1 + 2,2 kr

(4)

From a practical point of view, if |k2,2| < 0.1 |kr| it comes out that F2 ≈ F2b = k2,1 u1, which is a very significant relationship. 2.3

The indirect method

As above shown, complete characterization of a resilient mounting may be achieved by knowledge of the dynamic transfer stiffness k2,1 which may be correlated to the transmissibility by relationship T2b,1(f) = k2,1(f)/(2πf)2 and to the dynamic impedance LZ,i(f) by Equation 2. Even if, by a theoretical point of view, dynamic transfer stiffness is the reference parameter, usually dynamic behaviour of resilient mountings is discussed in terms of transmissibility.

In the audio-frequency range, the dynamic transfer stiffness of resilient elements is determined using the so-called indirect method. Unlike the direct method and the driving point method, in the indirect method the blocking force is not directly measured and is derived from acceleration measurements performed on the blocking mass which is dynamically decoupled from the test rig chassis. The resilient mounting is not directly coupled with the vibration source, as a compact mass, called excitation mass, is interposed. The excitation mass function is to provide the condition of contact point at the input side of the resilient mounting and, as the blocking mass, it is dynamically decoupled from the test rig structure using auxiliary isolators. Under the resilient mounting, the so-called blocking mass is placed. It provides a high-stiffness contact point at the isolator output side, so that the forces between the isolator output side and the receiving mass are approximately equal to the blocking forces. The blocking mass must have a high inertia, both translational and rotational, whereas its decoupling isolators should have a suitable low stiffness so as to keep low the resonance frequencies of the 6 rigid body motions of the mass. An actuator is used for applying a static preload, so that the resilient mounting is tested as in working condition. In the indirect method test, the acceleration (or displacement or velocity) of the excitation mass and the acceleration (or displacement or velocity) of the blocking mass are measured and the dynamic transfer stiffness is derived as: k2,1 ≈

F2 u 2 = − (2 f ) m2 2 u1 u1

a

(2 f )2 m2 a2

(5)

1

where the third part of expression is the one used in the indirect method. 2.4 The prescriptions for the test accuracy The approach based on the indirect method is still valid if tests are performed in a frequency range which is not affected, at the low frequencies, by the vertical motion at resonance frequency f0 of the blocking mass constrained between a soft isolator bed and the resilient element, and, at the high frequencies, by the limit of the rigid-body like behaviour of the blocking mass at f3 frequency. An approximate formulation for the frequency of the lowest internal resonance of the resilient mounting is the following:

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k fe = 0.5 m0 el

(6)

where k0 is the low-frequency dynamic stiffness of the resilient mounting and mel is the mass of its elastic part. According to the ISO 10846 standards, measurements are valid for frequencies higher than fe/3, but just if the resonant frequency f0 is lower than fe/10. As for the upper limit, the ISO 10846-3 standard provides the frequency values f3 for steel blocking masses of cylindrical or cubic shape. Upper frequency limit may be also determined by means of experimental tests and referring to the concept of effective mass. When the effective mass m2,eff (f) has been determined, the upper frequency limit f3 has to be chosen as the lower frequency value where the effective mass does not vary more than 12% from the mass value m2. Moreover, both background noise level and unwanted vibration levels related to the degrees of freedom other than the investigated one, are to taken low. These noises are controlled by setting, according to the weight and static stiffness of the tested resilient mounting, a series of physical parameters of the moving system. This is made through a comprehensive prediction of the moving system behaviour. As for the background noise, as required in the ISO 10846-3 standard, acceleration levels measured in vertical direction at the blocking mass when the vibration source is turned on, must be at least 15 dB [ref. 10–6 mm/s2] higher than the corresponding levels measured while the vibration source is turned off. The second check has to be carried out to control the transverse acceleration levels (i.e. horizontal accelerations). The acceleration levels measured at the excitation mass in orthogonal direction to the driving direction must be 15 dB [ref. 10–6 mm/s2] lower than those measured in the driving direction. If that is not verified, the unwanted accelerations of the excitation mass could cause, on the blocking mass, accelerations which are a linear combination of the excitation mass transverse and vertical accelerations. Unwanted horizontal accelerations have to be measured on the excitation mass according to procedure given in the ISO standard (Fig. 3). 2.5

Figure 3. Plan of acceleration transducers on the moving system of the test rig.

The test facility at the NVL Lab

At the Ship Noise and Vibration Laboratory (NVL) of the University of Trieste is operating a test rig for the measurement of the dynamic behaviour of the elastomeric resilient mounting systems. The layout of the equipment is given in Figure 4. Characteristics of the test rig fulfil the guidelines given in the ISO-10846 standards for laboratory measurement of the resilient element transmissibility. Specifically, experimental investigations may be carried out by applying the indirect method for

Figure 4. Test rig layout for the laboratory experiments (according to the indirect method).

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tests in the audio-frequency range, and the direct method may be also implemented in case of low frequency tests. The core of the test rig are the vibrating bodies (Fig. 3), whose assembling is the so called moving system. The moving system, being tightened by a hydraulic piston between an upper and a lower elastic bed of auxiliary isolators, may vertically move under the action of an electrodynamic shaker. The moving system is made by the resilient element supported by a blocking mass and holding on the top the excitation mass, by which the whole system is connected to the elastically suspended shaker. Connection between moving system and shaker is a very significant part and is made by a “stinger” rod. Through the stinger, vibrational energy is transmitted to the excitation mass. Mechanical connection among such elements has been designed to avoid local resonance frequencies in order to preserve measurements from such noise sources. Length and diameter of the stinger rod, inertia of the excitation and blocking mass and stiffness of the two soft isolator beds (i.e., the auxiliary isolators) are all set according to the test type and the resilient element characteristics. Laboratory tests can be developed on small to big resilient elements as test controlling parameters may be adjusted in a wide range of values. The most distinctive feature of the test rig lies in its capacity of performing tests on the biggest resilient mountings today used in the suspension systems for marine engines, with a static load capacity up to 150 kN and a maximum dynamic load capacity of 4 kN up to a frequency of 2 kHz, which range largely covers the typical frequency range in the investigations on medium-speed marine diesel engines. Dimensions of test rig are remarkable too, being supporting frame for moving masses very large, so allowing tests to be carried out on resilient mountings 0,5 metres wide and weighting about 1 kN placed on a foundation of up to 8 kN weight. 3

by which a dynamic coupling appears between top casting and the exciting mass simulating the engine frame. 3.1

The resilient element

A set of tests has been carried out on a conicalshaped resilient mounting (Fig. 5), designed by Vulkan Italia Srl for a medium-speed marine diesel engine of a rated power of about 16000 kW. The core of the resilient mounting is the rubber part, which is a conical-shaped toroid body coupled by a conical-shaped seat with the resilient mounting base. The resilient mounting top is adapted to house a flange which allows the adjustment of resilient mounting in height, by means of a ring nut. 3.2

The setting up of the laboratory tests

The dynamic stiffness has been evaluated with reference the linear motion along the vertical axis (driving direction) of the tested resilient element. Dynamic transfer properties have been derived by measuring the accelerations on the exiting mass and on the blocking mass. In all tests, shaker was controlled in order to produce a flat acceleration autospectrum on the top of the excitation mass. Experimental tests have been performed in a frequency range from about 100 Hz to 1000 Hz. The resilient mounting has been tested at the expected working load of 75 kN. In a previous experimental campaign a series of tests have been carried out to verify linearity in the dynamic response of resilient element under different static loads. Linearity in dynamic response has been also verified by varying the levels of the applied dynamic loads. Background noise levels and unwanted horizontal accelerations have been then compared with the

THE LABORATORY TESTS

The aim of the experimental campaign was the identification of the mechanism by which the dynamic coupling takes place between the resilient mounting top and the fixing rail of the engine foot when the resilient element is equipped with fixtures on the top. Two different configurations have been studied of the resilient mounting, by applying or removing the fixture on the top casting. Removal of the flange on the top of the resilient mounting has been considered to try and evaluate the mechanism

Figure 5. The resilient mounting (without top flange) under testing at the NVL Lab.

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vertical acceleration. Measured values comply with the standard limit values. Finally, the stinger rod length and diameter have been selected according to the dynamic characteristics of the moving system. Environmental conditions were kept the same during all the test campaign. All test outcomes have been collected in terms of accelerations acquired by piezoelectric accelerometers controlled by a dedicated data acquisition system, while the shaker control has been made by means of separate measuring chains.

The acceleration levels measured on the base of the exciting mass are shown in Figure 6 for the resilient element fitted with the flange. Relevant axis of accelerations are schematically shown in Figure 3. The flat curve of az refers to the vertical acceleration level induced, by the application of the vibration generator, on the bottom of the exciting mass. The straight horizontal line for az proves the rigid body behaviour of the exciting mass, being the shaker connected and controlled on the top of the mass. Curves of ax and ay show the acceleration levels arising in the two coordinate horizontal directions, the ones related to the linear motions which should not be excited during the test. Test configuration was designed in order to be effective in maintaining low the level of these unwanted vibrations up to 1 kHz. The prediction of the useful frequency range for the test was made on the base of procedure discussed in Section 2.4 and, in addition, by evaluating the value of the first natural frequency of the exciting mass. The method was proved to be very reliable in a number of tests previously carried out in similar experimental conditions on similar resilient elements.

In effect, the ax and ay curves show to maintain a difference of more than 15 dB [ref. 10–6 mm/s2] lower than that measured in the driving direction. Unfortunately, in the specific case, an unexpected disturbance has been detected, being the output relevant to the horizontal linear motion in both coordinate directions unsatisfactory due to the presence of a peak at about 681 Hz. Clearly, the low difference in the levels of the vertical and horizontal accelerations is not due to the resonance of the exciting mass itself. So, with the aim to investigate the reason of such unexpected behaviour of the exciting mass, a second test has been carried out changing the contact mode between exciting mass and resilient mounting. This has been made by removing the movable fixture from the top casting of the resilient mounting. The acceleration levels measured on the base of the exciting mass are shown in Figure 7 for the resilient element without the flange. Test has been carried out by imposing an input vertical acceleration slightly larger than that generated in the previous case, and this is considered not to be a detriment in the comparison of the two experiments. The ax and ay curves now obtained have the same trend of the previous ones and they increase as the frequency increases. Their shapes diverge significantly from the previous ones since they do not show any clear peak in the useful frequency range, except for the smooth peak at about 950 Hz. Comparison of the results of the two tested configurations of the same resilient element points out that the two tested configurations have a different dynamic behaviour and differently interact, through the top casting, with the upper body which they are in contact with. So, new experiments were planned

Figure 6. Accelerations measured on the exciting mass (resilient element fitted with the top flange).

Figure 7. Accelerations measured on the exciting mass (resilient element without the top flange).

3.3

The experimental results

360

to understand reasons of discrepancy in the accelerations relevant to the horizontal motions. 4

INTERPRETATION OF THE EXPERIMENTAL RESULTS

With the aim of finding the reason of the transverse vibration of the exciting mass when test was carried out on the resilient element fitted with the top flange, a series of further investigations have been conducted. 4.1

The ODS based analysis

First of all, special attention has been paid to the correct functioning of each element of the moving system of the test rig. A comprehensive measurement of movement and deflection has been carried out on all the masses, and in particular on the excitation mass and on its connection to the shaker by the stinger rod. Such an analysis has been performed by the Operational Deflection Shape (ODS) technique. This method allows to obtain, basing on the displacement cross spectra derived by a series of acceleration measurements, the identification of each mode shape and its animate for each single body of the moving system. This technique has been applied by acquiring the cross spectra of accelerations at a series of significant points of the moving mass, while the shaker was producing, through the stinger rod connected to the top of the exciting mass, a flat acceleration autospectrum. A total of 8 and 4 points located at the same distance each other on the outer edge of both top and base surface of the exciting mass and blocking mass respectively was detected; as for the resilient mounting, just 4 points have been used to characterise motion and deformation of its top casting. After processing the measured values, it has been ascertained that in general all the parts move perfectly in the direction of the driving force and that they behave like rigid bodies within the investigation frequency range. At the same time, a transverse movement has been detected on both exciting mass and top casting of the resilient mounting at the frequencies close to 681 Hz, as shown in Figure 8. At this frequency range a pronounced transverse deflection of the top casting of the resilient element has been observe, so identifying an in-phase coupled mode of oscillation between top casting and exciting mass. Results of the analysis based on the ODS technique put a suggestion on the reason why an unwanted oscillation is triggered on the exciting mass, but further investigations are needed to

Figure 8. Mode shape of the bodies of the moving system at the resonance frequency of about 681 Hz (resilient element fitted with the top flange).

prove the hypothesis about the dynamic coupling of the two parts. 4.2

The FEM simulation of the moving system

A further analysis have been performed by a numerical simulation of the whole moving system of the test rig, performed within the linear elastic approach. The FE model has been specially designed to acquire information on the dynamic coupling between the top casting of the resilient element and the exciting mass. This means that for some parts a fine mesh model has been generated, with a special attention to their vibrational response, while others have been just rough modelled. The exciting mass, the blocking mass, the top casting and the base of the resilient mounting have been explicitly modelled by brick elements (hexahedral and tetrahedral elements) so giving an accurate modelling of their geometry and mass distribution. The rubber part of the resilient mounting has been simulated by a number of linear springs, so overcoming the difficulty of the simulation of the contact between rubber and inner and outer castings. The interactions between the other bodies in contact have been solved by two different ways: by the merging of all the nodes shared by the two bodies in contact, when the bodies are expected to remain perfectly in contact at any modal vibration

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frequency—that is the case of the contact between the base of the resilient element and the blocking mass; or by merging only a selected part of the nodes in common between the bodies, when the bodies are expected to undergo relative perpendicular movements at a given modal vibration frequency—that is the case of the contact between the top casting of the resilient element and the exciting mass. In the latter case, the number of nodes to be merged, has been chosen by an iterative process aimed to minimize the gap between the measured and the predicted lowest coupled resonance frequencies. The strategy of allowing a free relative movement to the nodes belonging to the two bodies in contact, has been proven to be the key for the interpretation of the dynamic coupling between the two bodies. The FE model, which is not accurate enough to give a proper prediction of the frequency response of any parts of the moving system, has been proven to be adequate to reproduce the mechanism by which a high transverse vibration arises at a given frequency on the exciting mass. The main results of the numerical modal analysis, are the mode shapes of the whole moving system at the lowest coupled resonance frequencies. As for the interaction between the top casting of the resilient element and the exciting mass, a dynamic coupling between the two bodies has been identified at a frequency of 775 Hz (+10% with reference to the peak identified by the ODS technique), very close to the frequency of the peak in the cross-spectrum of the acceleration levels ax and ay measured in the laboratory test on the base of the exciting mass. The corresponding mode shape is that shown in Figure 8, an angular oscillation around an horizontal axis of the two bodies moving with opposite phase. The deformation calculated on the two bodies is in perfect agreement with that directly derived, by the ODS technique, from the measured accelerations. The clear agreement between the outcomes of the mode shape analysis carried out through the ODS technique and the FE simulation, points out that the coupled mode of vibration identified by the FE simulation is the explanation of the transverse movement of the exciting mass observed in the laboratory test. The same modal analysis has been performed on the modified resilient mounting, where the flange has been removed. In this case, the lowest coupled resonance frequency between the exciting mass and the top casting is calculated at a frequency of about 980 Hz, a +3% with respect to the measured value. The mode shape is shown in Figure 9. Also in this case, the FE analysis gives a real contribution to the interpretation of the experimental results, showing that the smooth peak at about 950 Hz in the acceleration levels ax and ay (see Fig. 7) is due

Figure 9. Coupled mode shape of the exciting mass and the top casting (without flange) at the frequency of about 780 Hz.

Figure 10. Accelerations measured on the exciting mass (resilient element with cylindrical top casting).

to the repositioning to higher frequencies of the same disturbance measured on the resilient element without top flange, caused even now by the dynamic coupling of the two parts. 4.3

Further investigations

Once ascertained that the cause of the transverse oscillation of the exciting mass was due to the dynamic coupling of this mass with the top casting of the resilient mounting, and once established that the main responsible of that mechanism was the top flange, a further study has been performed to eliminate such unwanted transverse vibrations. The study has been carried out by laboratory testing on the same resilient mounting where the top casting has been modified in order to have a large contact surface with the exciting mass. The cylindrical top casting is made by aluminium alloy and is internally shaped to fit the conical-shaped rubber part. The accelerations measured to this configuration of the resilient mounting are shown in Figure 10.

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The moving of the lowest coupled resonance frequency of the exciting mass and the top casting of the resilient to a higher value, does confirm the hypothesis made about the cause of such transverse vibration. 5

CONCLUSIONS

With the aim of setting a proper test procedure for the resilient mountings, a research activity has been developed at the NVL Lab of the University of Trieste. A comparative experimental research has been carried out in order to evaluate the influence of different boundary conditions at the top of the resilient element on its dynamic response. The mechanism by which a dynamic coupling takes place between the resilient mounting and the upper mass has been deeply studied. The main practical implication of the results is the necessity of define new standards to properly simulate by laboratory tests the real coupling between resilient mounting and engine frame.

Fulford, R.A. & Gibbs B.M. (1997). Structure-Borne Sound Power and Source Characterization in Multi-Point-Connected Systems, Part 1: Case Studies for Assumed force Distributions. Journal of Sound and Vibration, 204(4) (1997) 659–677. Hynnä, P. (2002). Vibrational Power Method in Control of Sound and Vibration. Research Report No. BVAL37-021229, Technical Research Centre of Finland, Espoo, Finland, 2002. Ran Lin, T., Pan, J., O’Shea, P.J. and Mechefske, C.K. (2009). A Study of Vibration and Vibration Control of Ship Structures. Marine Structures, 22 (2009) 730–743. Tao,. J.S, Liu G.R. and Lam, K.Y. (2000). Design Optimization of Marine Engine-Mount System. Journal of Sound and Vibration, 235(3) (2000) 477–494. Thompson, D.J., Van Vliet, W.J. and Verheij, W. (1998). Developments of the Indirect Method for Measuring the High Frequency Dynamic Stiffness of Resilient Elements. Journal of Sound and Vibration, 213(1) (1998) 169–188.

REFERENCES Biot, M. & Moro, L. (2012). Experimental study of a resilient mounting for marine diesel engines. Proceedings of the IMDC 2012 Conference, 2012, Glasgow, UK.

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Composite structures

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Numerical analysis of cracked marine structures repaired with composite patches E.I. Avgoulas Department of Engineering, Division C: Micromechanics, University of Cambridge, Cambridge, UK

V.A. Karatzas, I.K. Zilakos & N.G. Tsouvalis Shipbuilding Technology Laboratory, School of Naval Architecture and Marine Engineering, National Technical University of Athens, Athens, Greece

ABSTRACT: The vast majority of ships and marine structures are inevitably facing typical defects such as corrosion and cracks. A new and quite promising repair method is using composite material patches. This work consists of a numerical simulation of such repairs in a cracked structural component of an Aframax tanker. Following IACS’s Common Structural Rules, cracks of different lengths and in two locations around the manhole of a hopper tank transverse web frame were modeled. Different composite patch configurations were applied and studied for each crack case. The stress field in the area around the crack was calculated, along with the stress intensity factors, prior and after composite patch repairing. Results showed that stresses are significantly reduced and that the reduction of the stress intensity factor after the repair ranges between 51% and 93% compared to the unpatched case, thus significantly slowing down, or in some cases even practically arresting, the crack growth. 1

INTRODUCTION

The majority of ship and marine structures will invariably develop many defects throughout their operational life, some of which can be classified as critical and will need to be repaired immediately after they are detected. Among the most common defects are corrosion and cracks. These defects usually appear in decks, superstructures, frames, plates, holds and double-bottom of the vessels and occur due to the intense corrosive environment inside which the ship is operating, the cyclic nature of the applied loading and the unavoidable welding defects. Conventional repair methods are primarily based on the replacement of the defected parts, which involve cutting and welding processes. Inevitably these repair methods involve hot work and in many occasions lead to the interruption of the ship service and require procedures such as dry docking and gas free in case of explosive environment. This in turn results in severe logistic problems and loss of revenue due to the cease of operation of the vessel. On the other hand, composite patch repairs can deal with a lot of the limitations faced in traditional repair methods. In composite patch repairing, part of the applied load is transferred to the composite patch through an adhesive layer. Apart from the absence of hot work, they are also attractive due to their adaptability in complex substrate

geometry, light weight, corrosion resistance, low maintenance cost, high durability in the marine environment, and their high fatigue resistance. The latter is of high importance due to the cyclic loading that a marine structure is subjected throughout its operational life. Moreover, composite patches can be manufactured to meet the specific needs of each case by choosing the suitable materials, layers stacking sequence, as well as the adhesive that will be used for the application of the patch. Only a few cases are reported in marine industry starting with Grabovac et al (1993, 2003, 2009) who used composite patches to repair the aluminum deckhouse of a Royal Navy Frigate that repeatedly exhibited fatigue cracking (Fig. 1). After 15 years of active service no cracking underneath the patch or in adjacent areas was initiated. Additionally, a time-cost estimation is given for the repairs. Another more recent work dealing with the application of composite patch repairs in marine structures is that of McGeorge et al (2009) where two composite patch repairs were carried out on an FPSO. The first repair was carried out in order to arrest a fatigue crack that had developed from the corner of a door opening, while the second one was carried out to restore material loss on a heavily pitted deck floor. The two demonstrators have shown the viability of bonded composite repairs to the two most widely encountered damage scenarios

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Figure 2.

Structural detail selected to be examined.

Figure 1. Composite patch placed at the superstructure of a FFG-7 class frigate (Grabovac & Whittaker 2009).

in floating offshore units, i.e. fatigue cracking and plate thinning due to corrosion. The aforementioned advantages of composite patch repairs, as well as the good output from the limited existing repairs in marine structures have triggered the interest towards the further investigation of this new technology so as to be widely accepted as a repair method. The work presented in this paper consists of numerical simulations of such repairs in a cracked structural component of an Aframax tanker. To this end, a three compartments model has been created using the Finite Element Method, following IACS’s Common Structural Rules (CSR). Different crack positions and lengths were considered as well as different composite patch configurations for the rehabilitation of these cracks. The Stress Intensity Factors (SIF) between the patched and the unpatched crack cases were calculated so as to evaluate the effectiveness of the repair.

2 2.1

NUMERICAL MODELING PROCEDURE Construction of the three compartment model

There are numerous structural members in marine structures and vessels that are prone to crack development depending on the ship type (Larsen & Wilson 2010). From the cases presented in the work of Larsen & Wilson regarding the existence of cracks in oil tankers’ structural members, one was selected for analysis (Fig. 2). This case involves the existence of cracks in the transverse web frame

Figure 3. Three compartments model (Zilakos et al. 2011).

plating inside the hopper tank of oil tankers. To this end, an Aframax tanker (112000 tons DWT) was selected. The tanker was made of common marine steel, and therefore typical marine steel properties were taken into consideration during modeling. According to CSR (IACS 2010), in order to conduct a strength analysis of a ship using the finite element method, a Three Compartments Model (TCM) of the ship is required (Fig. 3). The TCM consists of the cargo tank amidships along with one tank aft and one fore the middle one. The finite element software ABAQUS 6.10 was used to model and analyze the TCM. Three and four-node shell elements (referred to as S3 and S4 in ABAQUS) were used to model both the plating and the stiffeners, with the exception of the vertical stiffeners of the bulkheads and those attached to the horizontal stringers of the bulkheads, which were modeled using Euler-Bernoulli beam

368

Figure 4. 2010).

Cargo loading considered in the TCM (IACS

elements (referred to as B33 in ABAQUS) so as to minimize the computational cost and to facilitate the modeling procedure. The average element edge size was equal to 200 mm. According to CSR for oil tankers, the structural strength assessment is to be carried out for various loading patterns with different loading cases that take into consideration both static and dynamic loading (IACS 2010). The loads taken into consideration are: • • • • • •

Figure 5.

Mesh of hopper tank web frame sub-model.

weight of the structure and the cargo/ballast static sea pressure applied dynamic wave pressure green sea loads dynamic tank pressure hull girder vertical/horizontal bending moment and shear forces

One of the proposed loading patterns was chosen to be applied in the present work (Fig. 4). The implementation of the aforementioned loads was achieved using a special subroutine that was written in FORTRAN and was called by ABAQUS during the analysis of the TCM. The boundary conditions imposed in the model were in full accordance with the requirements of CSR (Zilakos et al. 2011). The analyses conducted were linear for all cases. Construction of the web frame sub-model

Figure 6. Displacement distribution in (a) x-axis, (b) y-axis and (c) z-axis.

The review of the results from the TCM analyses indicated that the hopper tank transverse web frame that is located amidships and in the starboard side was the one more stressed for the selected loading pattern. For this reason the crack scenarios were introduced in this specific web frame. Given the fact that all analyses were linear and in order to minimize the computational cost and increase the mesh refinement in the area of interest, a submodel of the specific web frame was developed. All crack scenarios and composite patch configurations were subsequently studied in this sub-model. The displacements and rotations of the boundary region of the web frame were extracted from the TCM and subsequently imposed to the sub-model as boundary conditions. It is well documented in the bibliography that the SIF in cracks that

have been repaired with composite patches varies through thickness when the patch is applied only on one side of the plate (Tsouvalis et al. 2009). In order to calculate the through thickness variation of the SIF with ABAQUS, 20-nodes solid elements (referred as C3D20R in ABAQUS) were used. The number of elements through thickness of the web plating was equal to 8 and the global mesh size was approximately 50 mm (Fig. 5). The boundary conditions imposed to the submodel, as mentioned earlier, were derived from the TCM. The calculated rotations were negligible and, therefore, only the displacements fields in x, y and z axes were introduced in the sub-model’s analyses (Fig. 6).

2.2

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2.3

Introduction of cracks

Table 2.

Three different crack scenarios were considered with respect to the crack length and location. The cracks were introduced in regions 1 and 2 as shown in Figure 7, originating from the openings’ circumferential flange stiffener. The cracks were inclined by 45o with respect to the vertical axis Z. The crack locations were selected based on existing surveys’ data for oil tankers. These reports indicate that openings or/and manholes in that region are prone to the development of cracks (Larsen et al. 2010). In addition, the work presented here was focused on calculating the SIF under mode I loading and, to this end, different crack locations around the opening were examined and the ones that would exhibit a dominant mode I behavior were selected to proceed with the analyses. The crack lengths in the web floor plate were considered equal to 200 mm and 400 mm. The crack cases considered are listed in Table 1. Regarding the mesh size, the edge length of the elements around the crack tip was equal to 2 mm. It must be noted that cracks were considered extending both in the flange around the opening and in the web frame plate. 2.4

Composite patch repair scenarios

Two different composite patch configurations were selected, namely Carbon/Epoxy using the Hand-Lay-Up method (HLU-CE) and Carbon/ Vinylester using the Vacuum Infusion method (VI-CV). The mechanical properties of the materials have been measured from a preceding series of characterization tests (Karatzas & Tsouvalis 2011).

Figure 7. Crack regions on the hopper tank web frame. Table 1.

Crack cases.

Crack case

Region

Length (mm)

Crack-1/200 Crack-2/200 Crack-2/400

1 2 2

200 200 400

Mechanical properties of steel and adhesives.

Material

E (MPa)

ν

σ0 (MPa)

Steel Vinylester Epoxy

207000 3050 3450

0.33 0.35 0.35

235 – –

Table 3. Mechanical properties of composite patches.

E1 (MPa) E2 (MPa) E3 (MPa) ν12 ν23 ν13 G12 (MPa) G23 (MPa) G13 (MPa)

VI-CV

HLU-CE

102600 7600 7600 0.488 0.919 0.488 4500 1981 4500

38700 6500 6500 0.355 0.518 0.355 1700 2142 1700

The composite patches were modeled using four conventional, continuum solid elements through thickness. The patches were modeled as homogeneous orthotropic, with a linear elastic response. The adhesive layer was considered to be in all cases the resin of the composite system. The adhesive was modeled as an isotropic linear elastic material using one continuum solid element through thickness. No special contact or cohesive elements were used at the patch/resin and the resin/steel interfaces. The mechanical properties of the resins were taken from the manufacturers’ data sheets. The material properties of the steel and the adhesive layers are listed in Table 2, where E = Young’s modulus; ν = Poisson ratio; and σ0 = steel yield stress. The mechanical properties of the composite patches are listed in Table 3. Axes 1, 2 and 3 refer to the principle directions of the composite material, whereas E1, E2 and E3 are the Young’s moduli of the composite patch, ν12, ν23, ν13 the corresponding Poisson ratios and G12, G23, G13 the shear moduli. 2.5

Parametric analysis

A numerical parametric analysis was conducted, with parameters like the region of the crack, its length, the geometry of the patch, the composite material of the patch (HLU-CE or VI-CV), its thickness (corresponding to a specific stiffness ratio, SR) and whether the patch is applied on both or on one side of the hopper tank web frame steel plate. Table 4 presents all cases of this parametric analysis.

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Table 4.

Parametric analysis cases.

Region

Patch Geom.

Crack case

Composite system

SR

Application side

1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2

A A A A B B B B C C C C C C C C C C C C

Crack-1/200 Crack-1/200 Crack-1/200 Crack-1/200 Crack-1/200 Crack-1/200 Crack-1/200 Crack-1/200 Crack-2/200 Crack-2/200 Crack-2/200 Crack-2/200 Crack-2/400 Crack-2/400 Crack-2/400 Crack-2/400 Crack-2/200 Crack-2/200 Crack-2/200 Crack-2/200

VI-CV VI-CV HLU-CE HLU-CE VI-CV VI-CV HLU-CE HLU-CE VI-CV VI-CV HLU-CE HLU-CE VI-CV VI-CV HLU-CE HLU-CE VI-CV VI-CV HLU-CE HLU-CE

0.4 0.6 0.2 0.4 0.4 0.6 0.2 0.4 0.4 0.6 0.2 0.4 0.4 0.6 0.2 0.4 0.4 0.6 0.2 0.4

Aft Aft Aft Aft Aft Aft Aft Aft Aft Aft Aft Aft Aft Aft Aft Aft Aft & Fore Aft & Fore Aft & Fore Aft & Fore

The patch geometries are analyzed in the following section, whereas two SR were considered for each composite material, namely SR = 0.4 and 0.6 for the VI-CV case and SR = 0.2 and 0.4 for the HLU-CE case. These values were selected on the basis of finally achieving a realistic patch thickness. This thickness is determined by Equation 1 below: tp =

Es t s ⋅ SR E1

(1)

where Es is the Young’s modulus of the steel plate (assumed to be 207 GPa), ts is the steel plate thickness with a value of 14 mm according to the actual ship drawings, E1 is the Young’s modulus of the composite patch in the fibre direction (defined in Table 3) and tp is the composite patch thickness. The patch thickness values derived from Equation 1 are equal to 11.3 and 17 mm for the VI-CV patch (corresponding to SR of 0.4 and 0.6, respectively) and equal to 15 and 30 mm for the HLU-CE patch (corresponding to SR of 0.2 and 0.4, respectively). 2.6

Crack region 1

A composite patch was modeled on top of crack1/200 and in the aft side of the web frame. Two different patch geometries were used, namely A and B. Patch geometry A is illustrated in Figure 8. The patch was designed with edges AB and CD being

Figure 8.

Patch geometry A at region 1.

as close as possible to the transverse stiffeners and parallel to them. Moreover, the distance of the crack tip (point F) from the edge BC was approximately equal to three times the crack length for patch geometry A and equal to two times the crack length for patch geometry B. Therefore, the difference between patch geometries A and B is that, in geometry B, the AB and CD dimensions were shortened by 200 mm (one crack length) with respect to those of geometry A. Finally, dimension AG is approximately equal to dimension EG. The dimensions of both patch geometries A and B are presented in Table 5. Figure 9 illustrates the von Mises stress distribution of the steel plate at the area of crack-1/200 before the application of the composite patch, whereas Figure 10 illustrates the

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Table 5. Patch dimensions at region 1 (in m). Dimension

A

B

AB BC CD DE EA

0.561 0.587 0.898 0.187 0.472

0.361 0.587 0.698 0.187 0.472

Figure 11. Through thickness SIF variation at the crack tip of crack-1/200 (100% thickness refers to the patched side).

Figure 9. Von Mises stress distribution in the steel plate in Crack-1/200 before the application of the patch.

Figure 10. Von Mises stress distribution in the steel plate in Crack-1/200 after the application of the patch.

same stress distribution after the application of a patch with geometry A, VI-CV composite system and SR = 0.4. Similar stress distributions were observed for all other composite systems and for all different values of SR. The upper limit in the contour legends of these two figures was set equal to the yield stress of the steel (235 MPa), thus areas with stresses exceeding this value are depicted in gray color. Figures 9 and 10 indicate that the application of the patch leads to a significant reduction of the von Mises stress values in the vicinity of the crack tip. As a consequence, a significant reduction of the Stress Intensity Factor (SIF) values at the crack tip is also observed.

Figure 11 presents the through thickness variation of the SIF, at the crack tip of crack-1/200, for the unpatched case and for the four different cases of geometry A patch configuration. The zero thickness value corresponds to the unpatched side of the steel plate, whereas the 100 thickness value corresponds to the patched side. The SIF variation of the unpatched plate is slightly tilted due to some bending loading that develops on the steel plate, originating from the global loading of the ship. Figure 11 however shows also a quite inclined SIF variation for all patched cases, which owes to the inherent bending of the plate, to the additional bending that the one-side patch implies and to the smaller stresses that develop in the steel plate towards the patch side in comparison to those developing in the steel near its unpatched side. It is evident from this figure that the application of a patch results in a very large reduction of the SIF values. Comparing the maximum values of SIF for the unpatched and the patched cases, a reduction of 57%, 64%, 63% and 68% was observed in the patched cases for the composite systems HLU-CE (SR = 0.2), HLU-CE (SR = 0.4), VI-CV (SR = 0.4) and VI-CV (SR = 0.6), respectively. The higher difference in the reduction of SIF (11%) was observed between the HLU-CE (SR = 0.2) and the VI-CV (SR = 0.6) cases. Moreover, negligible difference in the SIF reduction was observed between the two different composite systems with the same SR (0.4). However, for the same SR, the thickness of the composite patch was smaller in the VI-CV case compared to the HLU-CE one, due to the higher mechanical properties of the former (see Table 3). Nevertheless, the VI-CV is a more costly manufacturing method compared with the HLU-CE one. Regarding geometry B patch, no differences were obtained in the SIF variations from those of

372

the geometry A patch. Thus, it can be stated that, in this specific case, the use of the smaller geometry B patch is equally effective. 2.7

Crack region 2

2.7.1 Crack-2/200 A rectangular composite patch was modeled on top of the crack in region 2 (patch geometry C) and in the aft side of the web frame (Fig. 12). As previously, the patch edge CD was parallel and as close as possible to the existing stiffener. The length of the edges AB and CD were equal to three times the length of the crack. Finally, the dimension CE is equal to BE. The actual dimensions of patch geometry C are AB = CD = 0.6 m and BC = DA = 0.46 m. Figures 13 and 14 illustrate the von Mises stress distribution of the steel plate before and after the application of the composite patch, respectively. The results of Figure 14 have been calculated for a patch geometry C, using VI-CV as a composite system and SR = 0.4. Similar distributions were obtained for all other composite systems and for all different SR values. As before, the upper limit in the contour legend was set equal to the yield stress of the steel (235 MPa). Areas with stresses that exceed this value are depicted in gray color. By

Figure 12.

Figure 14. Von Mises stress distribution in the steel plate in Crack-2/200 after the application of the patch.

Figure 15. Von Mises stress distribution in the steel plate in Crack-2/400 before the application of the patch.

Patch geometry C at region 2. Figure 16. Von Mises stress distribution in the steel plate in Crack-2/400 after the application of the patch.

comparing Figures 13 and 14, it can be observed that the patch reduces significantly the von Mises stress concentrations at the area of the crack and especially at the crack tip.

Figure 13. Von Mises stress distribution in the steel plate in Crack-2/200 before the application of the patch.

2.7.2 Crack-2/400 The patch geometry was kept exactly the same for the Crack-2/400 case (length 400 mm) and all different composite systems and SR values were analyzed (see Table 3). Figures 15 and 16 illustrate the von Mises stress distribution before and after the application of the composite patch on top of Crack-2/400, respectively. Figure 16 illustrates

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the von Mises stress distribution at the crack area after the application of patch geometry C by using VI-CV as a composite system and SR = 0.4. In all other cases with patches from other composite materials and different SR values, similar results were obtained. Once more, the achieved stresses reduction when applying the patch is significant. 2.7.3

Comparison between crack-2/200 and crack-2/400 Figure 17 shows a comparison between the through thickness SIF variations at the crack of region 2, calculated at the crack tip before and after the application of the patch (geometry C) and for all different composite systems and SR values investigated. The results of Figure 17 refer to both crack-2/200 (dash lines) and to crack 2/400 (solid lines). The differences of the results between the two crack lengths are in general small, indicating that the geometry C patch is equally efficient in decreasing the stress state at the tip of either crack2/200 or crack-2/400. Comparing the maximum values of SIF of Figure 17 for the unpatched and the patched cases of Crack-2/200, a reduction of 51%, 60%, 57% and 62% is observed in the patched cases for the composite systems HLU-CE (SR = 0.2), HLU-CE (SR = 0.4), VI-CV (SR = 0.4) and VI-CV (SR = 0.6), respectively. For the Crack-2/400 case the corresponding reductions are 58%, 69%, 64% and 71%, slightly higher than in the previous case. In both crack cases, the higher difference in the reduction of SIF between the composite systems of the same crack case was 11% for Crack-2/200 and 13% for Crack-2/400. This difference was observed between HLU-CE (SR = 0.2) and VI-CV (SR = 0.6) composite systems. Finally, small differences in the SIF reduction (3% and 5%) are observed between the two different composite systems with the same SR (0.4).

Figure 17. Through thickness SIF variation at the crack tip of crack-2/200 (dash lines) and crack-2/400 (solid lines) (100% thickness refers to the patched side).

2.7.4 Bilateral composite patch A bilateral patch (applied on both sides of the steel plate) was modeled for the crack-2/200 case. Both patches were modeled using patch geometry C and taking into account all the different composite systems (see Table 3). As Figure 18 indicates, the application of a bilateral patch results in the development of very low stresses, both near the crack tip and throughout the crack seam. These stresses are significantly lower than the corresponding stresses when a one-side patch is applied (see Fig. 14). Figure 19 provides a comparison between the through thickness SIF variations when a unilateral (solid lines) and a bilateral (dash lines) patch is applied. Results show that in the case of the bilateral patch configuration, the SIF variation is approximately constant through the thickness of the steel plate, a fact owing to the lack of bending, since the cross section of the reinforcement is now symmetric. In the case of the bilateral patch, a SIF reduction of 86%, 89%, 91% and 93% is observed in the patched cases with respect to the unpatched one, for the composite systems HLU-CE (SR = 0.2),

Figure 18. Von Mises stress distribution in the steel plate in Crack-2/200 after the application of a bilateral patch.

Figure 19. Through thickness SIF variation at the crack tip of crack-2/200, using either a unilateral (solid lines) or a bilateral (dash lines) patch (100% thickness refers to the patched side).

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HLU-CE (SR = 0.4), VI-CV (SR = 0.4) and VI-CV (SR = 0.6), respectively. 3

CONCLUSIONS

The aim of the present paper was to demonstrate the efficiency of the composite patch repair technology in cracked marine structures. According to the results, the follow conclusions can be drawn: 1. The SIF values are lower at the patched side of the steel plate in the case of unilateral patches. On the other hand, the SIF values are constant throughout the thickness of the steel plate in the case of bilateral patches. 2. In all different crack cases and for all different composite systems, SR values and patch geometries, the use of a unilateral composite patch led to significant reductions of the SIF values at the crack tip, ranging from 51% to 71% with respect to the unpatched case. 3. The use of a bilateral composite patch reduced the SIF values by more than 86% with respect to the unpatched case. 4. If the loading considered was cyclic, it can be concluded that the above stated SIF reductions would result in even higher reductions of the crack growth rate. 5. The four composite material configurations considered did not result in major differences, i.e. the effectiveness of the four types of patches considered was comparable. 6. The patches which had a length equal to two times the crack length proved to be equally effective with the larger patches, which had a length equal to three times the crack length. 7. Provided that the adhesive bonding between the patch and the steel plate is of adequate strength, composite patching is a very effective means for reinforcing defected steel plates and delaying the crack propagation.

REFERENCES Grabovac, I., Bartolomeusz R.A., Baker A. 1993. Carbon fibre composite reinforcement of a ship structure— project overview. Composites 24(6): 501–509. Grabovac I. 2003. Bonded composite solution to ship reinforcement. Composites: Part A: Applied Science and Manufacturing, 34: 847–854. Grabovac I., Whittaker D. 2009. Application of bonded composites in the repair of the ships structures—A 15-year service experience. Composites: Part A: Applied Science and Manufacturing, 40: 1381–1398. IACS, 2010. Common Structural Rules for Double Hull Tankers, International Association of Classification Societies. Karatzas V.A. and Tsouvalis N.G. 2011. Composite Materials Characterization Tests, Co-Patch Project Report, Composite patch repair for marine and civil engineering infrastructures applications; Project No: SPC8-GA-2009-233969; Report NTUA-TR-WP3-3-v1. Larsen, A.T., & Wilson, E.D. 2010. Definition of Application Cases (Marine). Co-Patch Project Report, Composite Patch Repair for Marine and Civil Engineering Infrastructure Applications. McGeorge, D., Echtermeyer A.T., Leong K.H., Melve B. Robinson M., Fischer K.P. 2009. Repair of floating offshore units using bonded fibre composite materials. Composites Part A: Applied science and manufacturing, 40(9): 1364–1380. Tsouvalis, N.G., Mirisiotis, L.S. and Dimou, D.N. 2009. Experimental and Numerical Study of the Fatigue Behaviour of Composite Patch Reinforced Cracked Steel Plates, International Journal of Fatigue, 31: 1613–1627. Turton, T.J., Dalzel-Job, J., Livingstone, F. 2005. Oil Platforms, Destroyers and Frigates—Case Studies of QinetiQ’s Marine Composite Patch Repairs, Composites: Part A, Applied science and manufacturing 36: 1066–1072. Zilakos I.K., Karatzas V.A., Chatzidouros E.V., Papazoglou V.J. 2011. Simulation of External Application of SuSy devices on an Aframax Tanker that has been Structurally Compromised. RINA, Royal Institution of Naval Architects—International Conference on Design and Operation of Tankers: 121–130.

ACKNOWLEDGEMENTS The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007–2013) under grant agreement n° 233969 (www.co-patch.com).

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Hybrid composite and metallic hulls, the best of both worlds R.G.S. Barsoum Office of Naval Research, Ships and Engineering Systems Division, Arlington, VA, USA

ABSTRACT: Hybrid composite and metallic structures may provide cost-effective solutions for future ship hulls. We will discuss the advantages of the hybrid hull from the “Total Ship Engineering” approach. They offer the potential to reduce the manufacturing costs associated with complex shaped bow and stern sections while simultaneously providing a lower weight and non-magnetic structural design. Challenges to achieving these benefits include the unknown and uncertain structural performance of the compositesteel joint connection. This interface is a complex combination of in-plane and out-of-plane strength, bearing and friction, which represent several challenges for computations and design. The paper will discuss the research and development effort on lightweight, low-cost, improved survivability hybrid hull concepts and composite-to-steel joining technology for future hybrid hull structures. It will also discuss the use of the hybrid concept for hull modification to increase payload and improve fuel efficiency. It will address analytical assessment and the influence of design parameters on joint performance, tests of several hybrid joint components involving the in-plane attachment of a composite bow/stern component to a metallic mid-body ship hull, investigations of material and geometric design parameters for hybrid composite-to-steel joints for bonded, fastened, and bonded/fastened configurations and their performance under static, dynamic loading. 1 1.1

HYBRID HULL CONCEPT Types of hybrid hull

Figure 1, shows different hull forms (including flared bow and wave-piercing bow) of the proposed three basic types (or concepts) of hybrid hulls: A, B and C; while hybrid hull D is a variant of C, as will be discussed below. Hybrid hull Type A has composite bow and stern; and Type B has a composite bow and stern and the mid section is made of steel framing and composite panels, (Barsoum 2008, 2009). Type C has composite bow and stern, while its mid section is a conventional steel hull with composite side expansions (or blisters). For sake of weight reduction, these side expansions are constructed of steel framing with composite panels.

Hybrid Ship Hull Types A, B and C are the subject of US Patents: 6,386,131, and 6,941,888. Hybrid hull D, is similar to C, except that the entire hull is steel, including bow and stern, while the side expansions (or blisters) are hybrid steel framing with composite panels. 1.2

It is proposed that GRP composites (Glass Reinforced Plastic) are used, because the GRP/ Steel joints have very small thermal stresses and GRP is much cheaper and more resistant to fire than carbon fiber. We will discuss here that, although carbon fiber sandwich is a lightweight and efficient construction, it requires costly extensive testing. Full-scale testing of naval size structures have shown that its buckling load is unpredictable (see section 3.3 for deficiencies in FE Codes in the analysis of CFRP sandwich), while the buckling load of GRP reinforced construction is highly predicable, and thus the cost of large scale testing can be considerably reduced, or completely eliminated. This issue will be discussed in detail in the design section below. 1.3

Figure 1.

Hybrid hulls types.

Use of GRP in the hybrid hull concepts

Advantages of hybrid hull

The hybrid hull concept offers the best of both worlds: steel for strength and stiffness; and

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composites for its stealth properties, lightweight and low maintenance. In addition, the hybrid concept offers many other advantages required for large Navy combatants as discussed through the paper. Specifically, the hybrid concept takes advantage of “each material”, where it performs best from a “Total Ship Engineering” approach: • Steel has both high stiffness and strength. • Steel is better for supporting localized loading such as heavy machinery, especially when subject to shock loading. • Composites are lightweight, and their electromagnetic properties can be easily modified, and can be made in complex shapes with exact tolerance, but they are supposed to best carry distributed loads (sea loads). The hull girder loads are carried by the steel. • Bonding for lightning strikes: Composite structures are vulnerable and require additional hardware, however, hybrid hulls are made of steel and composite and hence require little or no additional hardware for protection from lightning (i.e. the steel portion of the hull will provide the necessary bonding to the ground). • Low Signature: If stainless steel is used instead of carbon steel in the hybrid concepts (in addition to the complex bow and stern for hydrodynamic advantage) it will provide a nonmagnetic hull. • Highly Survivable: Type A hybrid hull provides a highly survivable mid-ship, because whipping moments are low (analysis show that the whipping moments of the hybrid hull is about 30% less than corresponding steel hull). Whipping moment is even lower (less than 40%) for Type B hybrid hull. These reductions are a consequence of the stiff mid-ship section and lighter ends (bow and stern) of the hull. • For the case of internal explosion, Type B hybrid hull can be designed with “Blow-out Panels”. Fatigue tests showed that this hybrid hull had no composite failure at design sea loads and was able to carry additional 25% higher loads. • Light weight is achieved not only due to the use of composites, but also due to the reduced whipping moments. • Ship Construction: Shipyard practice of “Block Outfitting” is supported by all the proposed Hybrid concepts. • Low Maintenance: GRP composites are not subject to corrosion or galvanic degradation. • Reduced repair cost (removable Panels in the case of Type B hybrid). • The hybrid concept allows for use of low cost GRP composites and low cost fabrication as opposed to an all-carbon-fiber-composite-hull.

• The hybrid concept eliminates the danger of an entire hull delamination upon impact (or the need for delamination stoppers for an all composite hull). • Modularity: the center section of the hull can be modular with the bow and stern remaining the same. In addition Type B-hybrid allow removal of decks (or panels) for substitution of payload. • Embedded sensors and special purpose metamaterials can be integrated into the composite. In addition to their above advantages over an all carbon fiber composite hulls, the hybrid hulls minimize the impact of manufacturing problems associated with an all composite, such as voids, local porosity and uneven quality of the composite; and availability of large supply of high performance of carbon composites. The design of hybrid hulls is based on using low cost GRP high quality composites. 2

HYBRID HULL FOR SHIP MODIFICATION

2.1 Hydrodynamic optimization of hybrid ship hull The use of composites in hybrid ship hull allows for complex shapes that can be easily manufactured with high precision based on optimized complex shapes to achieve improved hydrodynamic performance in addition to other desired improvements. Assume, for instance, a ship hull which has to be modified in order to carry larger payloads or to improve on its fuel efficiency, with the least disruption to the ship a hybrid concept is the best approach. The same existing hull, with added composite appendages to the steel hull, such as in the case of the mid section of (Type C in Fig. 1), could also lead to the desired ship performance. Figure 2, (Stern 2011), shows a comparison of the hydrodynamic performance of baseline (original hull) and modified hull with an optimized hybrid composite with side expansions (blisters), Type D. The optimization of the blister design configuration was aimed at disrupting the diverging Kelvin wave and recovery of the wave energy. This hull is capable of a 20% increase in payload and in the mean time reduces the resistance of the hull by 8%, which leads to higher fuel efficiency, (Transport Factor increased by 25%) which is significant! For modifications requiring large increase in volume and payload, a modification of hybrid composite side expansions can go all the way to the deck and thus provide the additional hull volume. For smaller payload increases, the hybrid side expansions can be reduced to composite blisters below the waterline.

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Figure 2. (Stern 2011) Mechanisms and action of curved side expansions (blisters). Same hull w. optimized blister shape (on left) to reduce resistance. Notice Kelvin wave moved aft.

3 3.1

DESIGN ISSUES

3.4 Slamming loads

Fatigue and ultimate load of hybrid construction

The design of hybrid concept is accomplished by finding the sea load moments and shears based on ABS and DNV rules. Both fatigue and ultimate hogging and sagging loads were studied in several hybrid composite/steel scaled models (1/20th and 1/6th) by (Gerenstead, et.al. 2004, 2006, 2009). Test results show that the hybrid design is as good or better than conventional steel hull construction. 3.2

survivability under hull sagging conditions. Therefore, in addition to small scale testing of joint strength, full-scale deck plating structures (with joints) were tested for buckling. Recent buckling tests, at the unique grillage test facility at NSWC Carderock Div., of full-scale hybrid GRP composite-to-steel grillage, showed that predicted loads agreed very well with test results. For Carbon fiber sandwich construction, the predicted buckling loads were not in agreement with test results and the FE computations with publicly available commercial codes were on the non-conservative side. To address this issue ONR is sponsoring both theoretical and testing program, (Bazant 2012). Bazant developed a theory that shows that the stress rate (Jaumann Rate) used in all publically available FE codes, is not compatible with strain rate tensor. This deficiency is only apparent (significantly) in the case of buckling of sandwich plates with high modulus skins (like CFRP) and soft core (like Divinycell). Unfortunately, this error results in predictions of higher buckling loads, and thus it is on the non-conservative side, as shown in the top curve of Figure 3.

Scaling of composite-to-steel joints

Joints are the weakest part of any structural component and their failure is the most complex part of any analysis, for this reason ONR funded several theoretical and experimental programs, to address the scaling issue. Because the failure is by fracture at the interfaces (complex singularity), scaling laws were nonlinear, and close to parabolic, (Bazant 2009). Several Joining techniques (including the standard bolted –bonded joints) have been addressed and tested, e.g. the modified Comeld® (Shkolnikov, et.al. 2009), results in much stronger joint with low stress concentration, and is also cheaper than standard joints. 3.3 Deck buckling In the Design of Hybrid hull Type A, in Figure 1, buckling of the joint is fundamental to ship

In order to design the composite bow or the composite panels of the proposed hybrid hulls shown in Figure 1, slamming loads have to be accurately evaluated. Slamming or hydrodynamic impact is a phenomenon that all ships and boats encounter, and it is the cause of the highest loads on ships. Slamming pressures are of high intensity, are very dynamic, and interact with the elasticity of the hull. Because of the lack of experience of composites in ship hulls, ONR sponsored the development of Lehigh University’s. Slamming Load Test Facility (Grenestedt 2012) is a unique test bed for experimentally studying slamming under real sea conditions on composites. There is no similar facility worldwide at this time, it measures slamming based on horizontal ship speed when slamming into waves. This facility is built of hybrid construction similar to hybrid hull Type B in Figure 1, (Barsoum 2009), i.e. steel framing with composite panels as shown on the right in Figure 4. It has been designed according to DNV rules for high speed boats. The Slamming Load Facility is capable of reaching speed of 50–60 knots, and thus results can be easily scaled to Navy ships. The unique data acquisition and measurement equipments on the facility are able to characterize the slamming events and their dependence on ship motion and wave height and relate pressure sensor data, bottom panel strains, displacements and accelerations, angle of

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Figure 3. Cylindrical buckling behavior of thick CFRP sandwich composite panel (boundary conditions and loading, shown on the top, a) FE computations using different rate definitions, (Bazant 2012), b) panel deflection curve of first buckling mode.

slamming loaded panel relative to the water, panel stiffness and panel supports, to wave height and shape, horizontal and vertical speeds, ship motion (yaw, pitch, roll, heave, surge, sway).

4

Figure 4. Slamming load facility at Lehigh University (Grenestedt 2012).

CONCLUSIONS

Hybrid hull offers high potential for “Total Ship Engineering”, addressing reduced signature, reduced maintenance-costs of ownership, low weight, reduced construction costs, improved survivability, damage tolerant hull and offer modularity and greater architectural flexibility. It could be argued that the hybrid concept was shown to be applicable to both large combatant and lightweight fast craft. The proposed hybrid curved

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mid-body, made possible by the hybrid composite side expansion (or blister) was shown (using hydrodynamic optimization) to result in reduction in hull resistance (fuel saving) and increase in Transport Factor. REFERENCES Barsoum, R. “Hybrid Composite and Metallic Hulls— Stealth, Strength and Durability”, ASNE Enging. the Total Ship, Tyson Corner, VA, Sep 2008. Barsoum, R. “Application of Hybrid Hull Concept to High Speed High Performance Ships and Craft” ASNE High Performance Marine Vehicles Linthicum, MD, Nov. 2009. Bazant, Z. ONR final Report N000140710313, 2009. Bazant, Z. “Critical Loads for Buckling of Elastic Soft-Core Sandwich Plates: Choice of Finite Strain Measure and Errors of Commercial Codes”, to be published, ONR Report 2012.

Cao, J, Grenestedt J. “Design and testing of joints for composite sandwich/steel hybrid ship hulls” Composites Part A”, Applied Science and Manufacturing 2004; 35:1091–1105.5. Cao, J, Grenestedt, J, Maroun, WJ. “Testing and Analysis of a Six Meter Steel Truss/Composite Skin Hybrid Ship Hull Model”; Marine Structures, 2006. Grenestedt, J., Sause, R. Steel/composite hybrid girder, large scale fatigue testing, composite sandwich panels, welded steel truss, corrugated skin sandwich, adhesive bonding, 2009 ONR Final report N000140610872. www.dtic.mil/dtic/tr/fulltext/u2/a498352.pdf. See also: Reany, J. PhD thesis, Lehigh Univ. Grenestedt, J. ONR program review, May 2012. Shkolnikov, V, Khodorkovsky, Y. “Advantageous Joining Technology for Hybrid Ship Hulls” ONR report, N00014-09-C-0331 (2011). Stern, F. ONR program review, May 2012.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Buckling studies on ship hull imperfect composite plates E.F. Beznea & I. Chirica University “Dunarea de Jos” of Galati, Romania

ABSTRACT: This paper focus on the buckling behavior analysis of a clamped, composite laminated quadratic plate under a uniaxial and shear in-plane loading. The applied methods have been developed in the European Project MARSTRUCT and improved in a further work. The imperfection is considered as the initial deformation due to the manufacturing operations: cosine shape in both of the longitudinal and transverse direction. Usually, the initial deformation mode appears in the form of the fundamental mode of the buckling or vibration. Maximum initial deformation magnitude is considered as a rate from the thickness. The boundary conditions are considered as the usual condition of the structural composite ship panels. On the plate sides the loading is considered as an uniform pressure. The parametric numerical calculations were done for various values of loading rate. Variation of the maximum transverse displacement versus the inplane load (displacement controlled after nonlinear buckling analysis) for three cases, obtained in numerical analysis are performed. 1

INTRODUCTION

The lightweight structural parts of the ship hull are stiffened thin-walled plates or shells. So, an important key factor in the design analysis of these types of structures is overall buckling behavior. The outstanding performance of laminated composite plates, in terms of high strength properties and low specific weight have found an increasing use in many engineering areas, especially in marine and shipbuilding, where the corrosion is a dominant key factor. The imperfections reduce the elastic buckling load of the laminated composite structures and leads to global structural failure at loads below the design level. The presence of in-plane loading may cause buckling of stiffened panels and an accurate knowledge of critical buckling load and mode shapes are essential for reliable and lightweight structural design. Many studies were developed in last decade for postbuckling analysis of laminated imperfect plates. In Mittelstedt et al (2011), postbuckling behavior of rectangular orthotropic laminated composite plates with initial imperfection under in-plane shear load was investigated in a closedform analytical manner. Beznea & Chirica (2010) present the results obtained after axial buckling analysis of ship hull plates made of composite materials taking into account the transversal imperfection (spatial cosine form) due to fabrication. The buckling load is determined when the first failure occurs in an

element, based on the Tsai-Wu failure criterion, who provides the mathematical relation for the strength under combined stresses. Yang (2009) presents an analysis of especially orthotropic laminates based on the First-order Shear Deformation Theory. Here analytical solutions are developed for simply supported and clamped plates with uniaxial compression. The analysis method covers the cases with in-plane biaxial compression, in-plane shear loading and combined loadings related to simply supported plates. Further, the investigation is concerned about plates with an initial geometric imperfection. Shen (1990) presents a method based on the perturbation analysis for the buckling and postbuckling of imperfect antisymmetrically angleply laminated composite plates under uniaxial compression. The shape of initial geometric imperfection is assumed as initial buckling mode of rectangular plates. Meanwhile the effects of in-plane boundary conditions, angles, total number of layers on the postbuckling behavior of laminated plates are also discussed. Zou & Qiao (2002), developed a higher-order finite strip method based on the higher-order shear deformation plate theory for postbuckling analysis of laminated composite plates with initial geometric imperfection subjected to progressive end shortening. Geier & Rohwer (1989), perform analysis of the buckling behaviour of laminated composite plates and shells, by means of different shear flexible theories. They turns out that there is a rather limited range of plates where the transverse shear

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is of considerable influence. Results obtained with Mindlin- or Reissner-type theories prove almost as adequate as those obtained with Reddy’s theory. For the buckling analysis of shells the Kirchhoff–Love theory is precise enough. Snapthrough buckling and imperfection sensitivity appear in composite shells, too, especially if they are optimized with respect to high bifurcation buckling loads. Shen & Williams (1995), have performed a postbuckling analysis for a uniaxial in-plane loaded, simply supported, composite laminated rectangular plate resting on a “softening” non-linear elastic foundation. The analysis uses a perturbation technique to determine the buckling loads and postbuckling equilibrium paths, taking initial geometrical imperfection into account. Thurley & Marshall (1995) provides a comprehensive insight into buckling and postbuckling. Basic theory, methods of buckling analysis and their application, the effect of external variables such as temperature and humidity on the buckling response and buckling tests are all covered. Misirlis et al (2009), present the validation of finite element models against a series of plate tests that were performed within this framework and parametric studies that were carried out to identify the effects of geometric imperfections on the ultimate compressive strength of composite plates with three alternative lay-up configurations. Singh & Chakrabarti (2012) have developed a C0 FE model, based onhigher order zigzag theory, for buckling analysis of laminated composite plates. The C0 continuity of the FE model is compensated in the stiffness matrix calculations by using penalty parameter approach. Numerical results and comparison with other existing solutions show that the present model is very efficient in predicting the buckling responses of laminated composites. The methodology to evaluate the influence of the imperfections on the buckling and postbuckling behavior of the composite plates under shear and compression loading, used in ship hull structure, is presented in Chirica & Beznea (2012). The parametric numerical calculations was done for various positions of delaminations and seven values of loading ratios. 2

It is of practical importance to consider load ranges beyond bifurcation buckling and to develop analysis methods that allow for a postbuckling analysis and design to be used in day-to-day engineering practice, in order to fully exploit the lightweight potential of such thin-walled structures. The behaviour of ship deck plating normally depends on a variety of influential factors, such as geometric/material properties, loading characteristics, initial imperfections, boundary conditions and deterioration arising from interlaminar fatigue cracking. The analysis is presented for a uniaxial and bi-axial in-plane loading, clamped, composite laminated quadratic plate. The imperfection is considered as the initial deformation due to the manufacturing operations: cosine shape in both of the longitudinal and transverse direction. Usually, the initial deformation mode appears in the form like the fundamental mode of the buckling or vibration. The boundary conditions are considered as the usual condition of the structural composite ship panels. On the plate sides the loading is considered as an uniform pressure. Variation of the maximum transversal displacement versus the inplane load (displacement controlled after nonlinear buckling analysis) for three cases, obtained in numerical analysis are performed. For maximum initial deformation magnitude three values are considered: 1.06 mm, 3.2 mm and 9.6 mm. A methodology to evaluate buckling and postbuckling behavior of the ship deck composite plates with imperfections under biaxial compression loading is presented. The parametric numerical calculations was done for various values of loading rate k = q/p (see Fig. 1). The numerical tests

ORTHOTROPIC PLATES BUCKLING

Postbuckling analysis is essential to predict the capacity of composite plates carrying considerable additional load before the ultimate load is reached, and manufacturing-induced geometric imperfections often reduce the load-carrying capacity of composite structures.

Figure 1.

384

In-plane loading of imperfect plate.

are analyzed with respect to combined boundary biaxial compression buckling on the imperfect composite plates. The results are be presented, in terms of deflection, evolution of buckling load versus transversal displacement in the middle point of the panel, for various loading ratios k and transversal imperfection magnitude. The equilibrium of a plate deformed by forces acting in the plane of the middle surface is unique. The equilibrium is stable if the forces are sufficiently small. If, while maintaining the distribution of forces constant at the edge of the plate, the forces are increased in magnitude, there may arise a time when the basic form of equilibrium ceases to be unique and stable and other forms become possible, which are characterized by the curvatures of the middle surface. Linear buckling of beams, membranes and plates has since been studied extensively. A linearized stability analysis is convenient from a mathematical viewpoint but quite restrictive in practical applications. What is needed is a capability for determining the nonlinear load-deflection behaviour of a structure. Considerable effort has also been expended on this problem and two approaches have evolved: class-I methods, which are incremental in nature and do not necessarily satisfy equilibrium; and class-II methods, which are self-correcting and tend to stay on the true equilibrium path (Thurley & Marshall 1995). Historically, class-I was the first finite element approach to solving geometrically non-linear problems (Ambarcumyan 1991). In this method the load is applied as a sequence of sufficiently small increments so that the structure can be assumed to respond linearly during each increment. To solving of geometrically and material nonlinear problems, the load is applied as a sequence of sufficiently small increments so that the structure can be assumed to respond linearly during each increment (Beznea & Chirica 2011). For each increment of load, increments of displacements and corresponding increments of stress and strain are computed. These incremental quantities are used to compute various corrective stiffness matrices (variously termed geometric, initial stress, and initial strain matrices) which serve to take into account the deformed geometry of the structure. A subsequent increment of load is applied and the process is continued until the desired number of load increments has been applied. The net effect is to solve a sequence of linear problems wherein the stiffness properties are recomputed based on the current geometry prior to each load increment. The plate material is damaged according to a specific criterion.

For various materials classes three dimensional failure criteria are developed. These include both isotropic and anisotropic material symmetries, and are applicable for macroscopic homogeneity. In the isotropic materials form, the properly calibrated failure criteria can distinguish ductile from brittle failure for specific stress states. Although most of the results are relevant to quasi-static failure, some are for time dependent failure conditions as well as for fatigue conditions. The buckling load determination may use the Tsai-Wu failure criterion in the case if the general buckling does not occurred till the first-ply failure occurring. In this case, the buckling load is considered as the in-plane load corresponding to the first-ply failure occurring. The Tsai-Wu failure criterion provides the mathematical relation for strength under combined stresses. Unlike the conventional isotropic materials where one constant will suffice for failure stress level and location, laminated composite materials require more elaborate methods to establish failure stresses. The strength of the laminated composite can be based on the strength of individual plies within the laminate. In addition, the failure of plies can be successive as the applied load increases. There may be a first ply failure followed by other ply failures until the last ply fails, denoting the ultimate failure of the laminate. Progressive failure description is therefore quite complex for laminated composite structures. A simpler approach for establishing failure consists of determining the structural integrity which depends on the definition of an allowable stress field. This stress field is usually characterized by a set of allowable stresses in the material principal directions. The failure criterion is used to calculate a Failure Index (F.I.) from the computed stresses and usersupplied material strengths. A failure index of 1 denotes the onset of failure, and a value less than 1 denotes no failure. The failure indices are computed for all layers in each element of your model. During postprocessing, it is possible to plot failure indices of the mesh for any layer. The Tsai-Wu failure criterion (also known as the Tsai-Wu tensor polynomial theory) is commonly used for orthotropic materials with unequal tensile and compressive strengths. The failure index according to this theory is computed using the following equation F.I. = F1 1 F2 ⋅ σ 2 + F11 σ 12 F22 ⋅ σ 22 + F66 σ 62 + 2F F12 ⋅ σ 1 ⋅ σ 2

(1)

The coefficient F12, which represents the parameter of interaction between σ1 and σ2, is to be obtained by a mechanical biaxial test.

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The other coefficients are obtained from equations

Table 2.

w0 [mm]

1 1 1 F1 = T − C ; F11 = T C ; R1 R1 R1 R1 1 1 1 F2 = T − C ; F22 = T C ; R2 R2 R2 R2 F66 =

Buckling load [MPa].

(2)

1 . 2 R12 1

In the equations (2), the strength parameters Ri are the compressive strength (C) and tensile strength (T) in the material in longitudinal direction (i = 1) and transversal direction (i = 2). The parameter R12 is in-plane shear strength in the material 1–2 plane. According to the Tsai-Wu failure criterion, the failure of a lamina occurs if

k = q/p

0

1.06

3.2

9.6

0 0.2 0.4 0.6 0.8 1.0 ∞

110 233 241 243 244 285 348

106 220 233 239 240 270 298

98 212 220 246 230 260 289

85 134 199 210 228 250 270

F.I. > 1. The failure index in calculated in each ply of each element. In the ply where failure index is greater than 1, the first-ply failure occurs, according to the Tsai-Wu criterion. In the next steps, the tensile and compressive properties of this element are reduced by the failure index. If the buckling did not appeared until the moment of the first-ply failure occurring, the in-plane load corresponding to this moment is considered as the buckling load. 3

AXIAL AND SHEAR BUCKLING OF LAMINATED COMPOSITE PLATES

In this paper, the numerical analysis carried out using COSMOS/M finite element package software is presented. The square plates (320 × 320 mm), are made of E-glass/polyester concerning 16 biaxial layers having the orthotropic directions and thickness (4.96 mm) according to the topological code of the plate: [02/45/902/45/02]s. Table 1.

Material characteristics.

Mechanical characteristic Young’s modulus Ex Young’s modulus Ey Shear modulus Gxy Poisson’s rate µxy Tension strength in x direction Tension strength in y direction Compression strength in x direction Compression strength in y direction Shear strength

38.6 GPa 8.27 GPa 4.14 GPa 0.3 1.06 GPa 0.031 GPa 1.58 GPa 0.118 GPa 0.72 GPa

Figure 2. Variation of axial loading p versus transversal displacement, for pure compression (q = 0).

The material characteristics, determined in experimental tests are presented in the Table 1. The direction of the axial loading is considered along the symmetry geometrical axis of the plate.

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The loading was considered to be a combination between axial compression (p) and shear (q) acting on the plate sides as it is presented in the Figure 1. The combination is determined by the loading rate k = q/p. For the loading rate, 6 values were considered: 0 (pure axial compression); 0.2; 0.4; 0.6; 0.8; 1.0 and ∞ (pure shearing). In the Table 1, the values of ultimate strength (buckling load) for the all analyzed loading ratios, for the three cases of initial transversal imperfection magnitude are given. For each case in the parametric numerical calculations, according to the changing in the slope of the curves, the value of the buckling load (determined by graphical method, by drawing a tangent line in the point of suddenly changing the curve slope) is presented in the Table 2. In the Figures 2–4 the variation of the axial loading p (q for pure shearing) versus the magnitude of the transversal displacement are presented for pure compression, loading rate equal to 1 and

Figure 4. Variation of axial loading p (shear q) versus transversal displacement, for pure shearing (p = 0).

Table 3. Buckling load [MPa], according to Tsai-Wu criterion.

p/q 0 0.2 0.4 0.6 0.8 1 Figure 3. Variation of axial loading p (shear q) versus transversal displacement, for loading rate q/p = 1.

∞

387

FAIL type 1 2 1 2 1 2 1 2 1 2 1 2 1

w0 [mm] 0

1.06

3.2

9.6

256 – 249 – 250 – 187 – 156 – 125 – 200

189 193 77 141 76 131 50 91 43 85 31 71 150

177 181 60.5 131 50 121 50 87 40 71 30 56 150

69 138 403 121 40 99 30 80 30 70 30 56 150

pure shearing for the three cases of initial transversal imperfection magnitude are presented. In the Table 3, the values of ultimate strength (buckling load) determined by Tsai-Wu criterion for all cases are presented. As it is seen, the first failure for all cases occurs for the tension case (FAIL 1). As it is seen the buckling load is decreasing as the magnitude of the transversal deformation is increasing. But for the magnitude of 1.06 mm the plate behavior looks to be as a corrugated plate, having a good endurance to the buckling. 4

CONCLUSIONS

This paper presented a detailed numerical investigation on the post-buckling behaviour of composite panels subjected to shear and axial loads. No experimental validation has been carried out. From the comparative analysis it can be concluded that the numerical model provides a good approximation to the actual behaviour of imperfect plates. Thus, it can be used as an useful analysis tool in order to develop and establish new design rules for the composite ship structures. Moreover, the phenomenon study requires a stress analysis in order to improve the evaluation of the structural response of the composite plates with transversal imperfection. As it is seen in the Table 2, the buckling load is decreasing as the magnitude of the transversal deformation is increasing. But for the magnitude of 1.06 mm the plate behavior looks to be as a corrugated plate, having a good endurance to the buckling. For nonlinear numerical calculus, the analysis can become numerically no stable for larger values of the transversal displacement. Therefore the calculus have to be stopped because the interested buckling value can be determined according to the changing in the slope of the curves. The proposed methodology accounts for failure, material non-linearity/degradation, geometric imperfections and geometric non-linearity effects. The numerical results indicated that the post-buckling behaviour of the panels prior to collapse was not significantly affected by the geometric imperfections and their magnitudes. The numerical results obtained from the tests and allow to reach certain conclusions related to the behaviour of imperfect composite plate under combined axial and shear loads.

ACKNOWLEDGEMENTS The authors acknowledge the financial support received for this work from the Romanian UEFISCDI Project 168 EU (2012–2014).

REFERENCES Ambarcumyan S.A. 1991. Theory of Anisotropic Plates: Strength, Stability, and Vibrations, Hemispher Publishing, Washington. Beznea E.F. & Chirica I. 2011. Buckling and Post-buckling Analysis of Composite Plates, Advances in Composite Materials—Ecodesign and Analysis, Dr. Brahim Attaf (Ed.), ISBN: 978-953-307-150-3, InTech, 383–408. Chirica I. & Beznea E.F. 2012. Buckling behavior of the multiple delaminated composite plates under shear and axial compression, Computational Materials Science, 64: 173–178. Geier B. & Rohwer K. 1989. On the analysis of the buckling behaviour of laminated composite plates and shells, International Journal for Numerical Methods in Engineering, v.27, issue 2, 403–427. Misirlis K., Downes J., Dow R.S., Delarche A., Lundsgaard-Larsen C., Berggreen C., Hayman B., Tsouvalis N., Yang N. & Das P.K. 2009. Investigations on the ultimate compressive strength of composite plates with geometrical imperfections, ICCM 17, Edinburgh, UK: 147–160. Mittelstedt C., Erdmann H. & Schroder K.-U. 2011. Postbuckling of imperfect rectangular composite plates under inplane shear closed-form approximate solutions, Arch Appl Mech 81: 1409–1426. Shen H.S. 1990. Buckling and postbuckling behavior of antisymmetrically angle-ply laminated composite plates, Applied Mathematics and Mechanics, Shanghai, v. 11, No. 12, 1155–1165. Shen H.S. & Williams F.W. 1995. Postbuckling analysis of imperfectcomposite laminated plates on non-linear elastic foundations, International Journal of Non-Linear Mechanics, Volume 30, Issue 5: 651–659. Singh S.K. & Chakrabarti A. 2012. Buckling analysis of laminated composite plates using an efficient C0 FE model, Latin American Journal of Solids and Structures 9(2012): 353–365. Thurley G.J. & Marshall I.H. 1995. Buckling and Postbuckling of Composite Plates, Ed. Chapman & Hall, London. Yang Q.J. 2009. Simplified Approaches to Buckling of Composite Plates, Thesis for the degree of Master of science, University of Oslo. Zou G. & Qiao P. 2002. Higher-Order Finite Strip Method for Postbuckling Analysis of Imperfect Composite Plates, Journal of Engineering Mechanics, 128(9): 1008–1015.

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Calibration of a finite element composite delamination model by experiments M. Gaiotti & C.M. Rizzo DITEN, Faculty of Engineering, University of Genova, Italy

K. Branner & P. Berring Department of Wind Energy, Technical University of Denmark, Denmark

ABSTRACT: This paper deals with the mechanical behavior under in plane compressive loading of thick and mostly unidirectional glass fiber composite plates made with an initial embedded delamination. The delamination is rectangular in shape, causing the separation of the central part of the plate into two distinct sub-laminates. The work focuses on experimental validation of a finite element model built using the 9-noded MITC9 shell elements, which prevent locking effects and aiming to capture the highly non linear buckling features involved in the problem. The geometry has been numerically defined by a previously established modeling strategy (Branner et al., 2011; Gaiotti & Rizzo, 2011), using a pure shell model where the delamination is accounted for by properly offsetting its surfaces and connecting them to the intact plate via rigid link constraining algorithms. The numerical model developed by the University of Genova is compared with the experimental results provided by an extensive experimental campaign conducted by the Department of Wind Energy at the Technical University of Denmark (Branner & Berring, 2011). Along with the experimental/numerical comparison, an attempt to identify the fracture modes related to the production methods is presented in this paper. A microscopic analysis of the fracture surfaces was carried out in order to better understand the failure mechanisms. 1 1.1

INTRODUCTION Overview

The marine industry has traditionally been applying reinforced plastics for decades, particularly in the field of small recreational boats. Larger marine vessels, such as anti mines, and wind turbine blades are nowadays finding a widespread use of composites to provide light slender structures able to reduce the total weight and being cost effective at the same time. However, previous works established that marine environmental loads and ships’ structural behavior highlight the limited inter laminar shear strength of these materials (Baley et al., 2003), giving rise to delamination induced failures, often originating from impact loads. Moreover delamination is usually the most critical type of damage that composite and sandwich structure experience under compressive loads (Abrate 1991, Pavier & Clarke, 1995). 1.2

selected geometry presenting a shallow initial delamination have been analyzed in detail, namely to their compressive behavior able to induce out of plane effects due to elastic instability. It is worth noting that two different manufacturing methods were chosen to fabricate the specimens: resin infusion in vacuum bag and overlapping of epoxy prepreg. Beside the experimental campaign, carried out by the Department of Wind Energy at the Technical University of Denmark, the authors focus on the validation of a numerical model based on a 9-noded MITC9 shell elements, able to account for embedded delaminations (Gaiotti & Rizzo, 2011) using a two dimensional modeling strategy. A similar approach was also followed by Huimin & Yongbo (2011). 2 2.1

Aim

The present work aims at analyzing the collapse modes of fiberglass reinforced plastic plates with an embedded delamination. Four specimens of a

MATERIAL PROPERTIES Elastic moduli

Both the prepreg and the infusion made plates are constituted by a similar stacking sequence, mainly involving unidirectional layers broken by light scatter-oriented plies with the purpose of absorbing

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the exceeding inter laminar resin. Several ±45° thin layers are also used on the surface. Mixture rule is a widely applied method to evaluate the mechanical properties of a composite. In order to properly account for the transversal behavior, the well-established Halpin-Tsai equations are taken into account (Halpin & Nicolais, 1971). Due to the glass content sensitivity, the authors decided to start off an experimental campaign based on burn out tests conducted on specimens extracted from the tested Fiberglass Reinforced Plastic (FGRP) plates. Four separate burn out tests were carried out, that allowed to measure the total fiber weight fraction, along with the partial contribution of the different layers involved in the stacking sequence. By chance, for both the prepreg and the infusion made plates, an overall weight glass content Wf = 0.72 was found, with a 9.5% in weight of ±45° plies. The corresponding volume fraction is then identified as Vf = 0.54. With those data available, the Halpin-Tsai formulations return the elastic properties reported in Table 1. 2.2

Ultimate strength

The stresses at failure were instead taken from a previous experimental campaign, already presented by the authors in a previous work (Gaiotti & Rizzo, 2009), and relevant to unidirectional prepreg laminates having a similar glass content fraction. Ultimate strength values are reported in Table 2. Since these properties, reported from an experimental campaign based on standard ASME tension and bending tests on small specimens, refer to a slightly different material in terms of glass content and manufacturing method (Vf = 0.52, Table 1. Mechanical properties of UD fiberglass considered. E1 [MPa]

E2 [MPa]

G12 [MPa]

ν12 [-]

40642

15172

7374

0.257

Table 2.

Considered failure stresses of UD fiberglass.

Tensile stresses Compressive stresses

Shear stresses

Axis*

1

2

3

MPa MPa

590 505

90 137

90 137

Plane

12

13

23

MPa

64

48

8

*1 is the fibers’ direction, while 2 is the one in-plane and perpendicular to the fibers; 3 is the out of plane direction.

prepreg only), a sensitivity numerical analysis on the longitudinal compressive stress limit, identified as the governing parameter of the limit state, has been carried out. The above mentioned analyses, based on the limit state curves reported by Naik & Kumar (1999), investigate the numerical failure flag up to a compressive stress limit of 565 MPa. The influence on the failure stress parameter is by the way very limited, as the collapse is suddenly induced by bending effects introduced after elastic instability occurs. As a matter of fact, a sudden longitudinal stress growth takes place in agreement with the behavior noticed by Peck & Springer (1991).

3

EXPERIMENTAL TESTS

3.1 Geometry and test At DTU Wind Energy compressive loading tests on delaminated specimens were carried out (Branner & Berring, 2011). Four specimens from this test campaign, approximately 400 × 380 mm, are examined further in this paper. Two different panel types were tested. The former made with prepregs with an initial embedded delamination of 202 × 160 mm at 6.0 mm depth and a total thickness of 20.0 mm and the latter using the vacuum infusion technique with an initial embedded delamination of 156 × 128 mm at 4.0 mm depth and a thickness of 19.0 mm. Despite the different size of the delaminations, comparison of manufacturing methods is still possible taking into account that in both cases the delamination is close to the panel surface, thus leading to local buckling in the early stage of the loading history. Previous numerical works (Gaiotti et al., 2011; Sørensen et al., 2010) show that for shallow delaminations the model sensitivity to delamination size is relatively limited. Even though the specimens were made similar to the load carrying laminates in a typical wind turbine blade, the material properties are rather similar but not fully representative to those in actual pleasure boat and wind turbine blade industry, as the specimens were produced in a lab under controlled conditions and obviously contain much less fabrication imperfections. A specially designed test rig (Sørensen et al., 2009) was used in a 5 MN Instron testing machine as shown in Figure 1. The rig is designed to limit rotation and out-of-plane deflection of the edges of the panels. The panels are supported by steel blocks along all four edges and on both sides, so the panel area free to experience out-of-plane deflections is approximately 320–325 mm in both the vertical and horizontal direction. The panels were loaded in

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3.2

Figure 1. Experimental setup. Deflections were measured by a DIC measurement system on one side of the panel and by conventional displacement transducers on the other side.

The experimental data available show a similar stiffness, in terms of compressive modulus, for both the infusion and the prepreg made panels. On the other hand the collapse behavior appears different, and a lower compressive strength for the prepregs is observed. In order to depict this observed behavior, a microscopic analysis of the failed specimens was carried out. A visual analysis of a cut out slice of the original panel is formerly considered. When looking at the cross section, several new delaminations can be identified, all located in the thicker sub-laminate, while no or only little growth of the originally embedded one is detected. See Figure 3. New delaminations extend in length l/L ≈ 0.7 for prepreg and l/L ≈ 0.4 for infusion, where l is the length of the delamination and L is the length of the specimen. Moreover, the delaminations observed in the prepregs are purely inter-ply, located in the ±45° inner layers, while only a few intra-ply delaminations are found in the infusion made panels. 3.3

Figure 2. Experimental results presentation, non dimensional in-plane displacements vs. in-plane load.

compression to ultimate failure and a Digital Image Correlation (DIC) measurement system was used on one side of the panel to monitor full field displacements and conventional displacement transducers were used on the other side to monitor the opening of the delamination under the entire load history. Owing this configuration, the experimental load vs. in plane displacement curve can be plotted, showing the maximum in plane strength of each panel along with the compressive stiffness (Figure 2). The slightly non linear behavior observed in the early stage is likely to be attributed to an initial setting of the test rig and panel slip over the edges until full contact with the clamping devices is achieved.

Discussion

Microscopic analysis

The newly opened surfaces originated by the delamination process were scanned using an optical microscope aiming at justifying the reduced strength of the prepregs. The idea is to identify, if any, a specific fracture mechanism related to the production method. The prepreg panel presents a translucent surface with no fiber trays emerging from the resin, thus clearly indicating the delamination grew in the inter-laminar resin between two adjacent layers with limited interaction with the fibers. On the surface the transversal stitches constituting the weft of the UD fiberglass layer are easily identified. Even if they look emerging from the fractured edge, those fibers are still covered by the resin, hence indicating the fracture mechanism did not affect the fiber/ matrix interface, as shown in Figure 4. Failure mode is consequently due to interlaminar de-bonding as the most of the unidirectional fibers is still intact.

Figure 3.

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Specimen section taken from panel PP7.

4 4.1

Figure 4. Particular of the delaminated surface ±45° fibers (prepreg).

Figure 5. surface.

FINITE ELEMENT MODEL Modeling strategy

The FE modeling strategy presented in this paper refers to a previously developed approach (Gaiotti & Rizzo, 2011), which is nowadays gaining ground in literature as shown by Huimin & Yongbo (2011). It consists on a shell model defined by a single surface for the intact area of the panel and by two surfaces for the delaminated area, each surface representing one side of the delamination, as shown in Figure 6. The mid-plane of the model is defined at the half thickness of the actual plate and on this plane shell elements are meshed to describe the intact part of the model. The two sides of the delamination are modeled onto two surfaces, having a proper offset above and below the mid-plane in order to simulate the two sub-laminates resulting from the delamination. The nodes along the edge of the delamination are coupled to the nodes on the main surface by a rigid link constraint, where the master node is the node lying on the edge of the intact mid-plane and the nodes on both sub laminates edges are its slaves. Frictionless contact conditions were defined on the abutting surfaces of delamination.

Fiber trays particular on the delaminated

In the infusion made panels, instead, the opening of the delamination occurs inside the reinforcement layers, involving the fibers in the fracture mechanism. The following Figure 5 shows the fiber trays emerging from the surface. In this case the delaminations tend to grow in the intra laminar resin, involving the fiber/matrix interface in the fracture mechanism and developing through the transversally oriented layers. Such layers were aimed at reducing the inter-laminar resin thickness. The fibers in way of the delaminated edge appear free from any signs of resin, therefore suggesting that fibers are sliding in the matrix during the failure process. Such interactions often lead to fiber bridging, which causes the nominal delamination resistance to increase as the crack extends, as described by Spearing & Evans (1992). The interpretation of the different observed fracture mechanism may justify the different collapse behavior depending on the manufacturing method, whereas the fiber bridging process typically delays the initiation and propagation process of newborn delamination, thus avoiding early collapse.

Figure 6. Shell model with offset translation of the delaminated layers.

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The model is able to account for widely non-linear geometrical effects due to large displacements. Contact conditions also introduce non-linearities in the model. 4.2 Element choice A 9-nodes shell elements mesh is generated on the surfaces. The element type is the MITC9 as suggested by Buncalem & Bathe (1993), and Bathe et al. (2000), to prevent element locking problems for thin laminates. Element locking is, as widely discussed in relevant literature, the phenomenon of an element being much too stiff compared with reality. In essence, the phenomenon arises because the interpolation functions used for an element are not able to represent zero (or very small) shearing or membrane strains. If the element cannot represent zero shearing strains, but the physical situation corresponds to zero (or very small) shearing strains, then the element becomes very stiff as its thickness over length ratio decreases (see e.g. ADINA, 2008). The use of this element in the present case is justified by the shallow embedded delamination induced in the tested panels, originating a very thin sub-laminate. 4.3 Fixities and loads Being the experimental results in terms of displacements instead of stresses/strains as usual, the proper definition of the boundary condition is important to obtain an accurate experimental/ numerical comparison. The clamped edges of the tested specimens were modeled by fixing all degrees of freedom of the MITC9 shell elements except in-plane displacements in the loading direction of the loaded edge. This was felt necessary in order to maintain the actual effective plate length preventing the plate elastic constant from being inappropriately accounted for. The panel area bounded in the clamping devices is then transversally constrained giving rise to a reduced actual free span for out of plane displacements, not corresponding to the real length of the panel. The supports on the lateral edges, fixing the out of plane translation, are also placed accounting for the actual geometry of the experimental set. The load is applied considering an assigned inplane displacement on one of the clamped edges. The corresponding in-plane reaction on the opposite edge identifies the total force (Figure 7). As widely discussed in literature (Sørensen et al., 2010), a preliminary linearized buckling analysis is performed to identify the mode shapes, that will be exported as initial imperfection in the following non-linear analysis. The maximum amplitude

Figure 7. Boundary conditions and applied load. The dark area represents the delaminated sub-laminate.

of the initial imperfection corresponds to the 2.6% of the plate total thickness. This value could sound relatively high, but it is necessary to solve contact convergence problem and it is justified by the fact that it refers to a very local mode. So the initial imperfection basically involves the thinner sublaminate only. In the non-linear analysis, to provide smooth in-plane displacements vs. in-plane load curve, the load increment is divided into 200 identical steps. Two FE models were developed accounting for the different delamination sizes of the tested specimens. The considerations of the following sections are valid for model with both delamination sizes. 4.4

Buckling modes

The linearized analysis provides an early local buckling mode, involving the thinner sub-laminate only, occurring at the beginning of the loading history. Being the delamination very close to the surface, the non linearity introduced is too weak to be clearly visible, as it affects a very limited part of the panel. This buckling mode is also quite difficult to be described. The non-linear numerical prediction shows that it tends to shift from mode 1 (one half wave) to mode 2 (complete wave), which is however forbidden by contact conditions. This produces a migration of the wave peak from the center towards the delamination edges. This fact, observed in the numerical model, could explain some experimental mismatching in terms of out of plane displacements measured at mid-span. Actually, some specimens experienced a mode change while others a pure mode 1. When increasing the in plane displacements, an expected global buckling mode takes place giving rise to a slope change in the in-plane displacement vs. in-plane load curve. The boundary conditions on the four edges limit the softening effect although the non-linearity is still visible.

393

On the other hand, the bending-induced strains introduced by the global buckling condition have a strong influence over the global strains because they sum up to the in-plane compressive effects. 4.5

Failure

The failure criterion has been set according to the well established Tsai-Hill criterion, considering the compressive limit state curve. As already mentioned in Section 2.2, due to uncertainties related to the failure stresses, several configurations were tested, with the compressive limit stress ranging from 505 MPa to 565 MPa. The bending stresses induced by elastic instability, whose occurrence only depends on elastic moduli, leads to an immediate stress growth causing sudden failure in any of the tested configuration. A post first ply failure analysis in order to establish the ultimate stress of a multi layered composite is usually hard to achieve. However, in this case, failure spreads throughout the elements in one single step, basically involving the whole unidirectional layers, from top to bottom as shown in Figure 9. The buckling load is then calculated by means of the widely used ‘robust’ method described by Sørensen et al. (2010), which detects the change of linear behavior off-setting the initial linear part of the load shortening curve and extends it until the intersection point of the curves identifying eventually the buckling load (Figure 10). For further explanations on this method see Gaiotti & Rizzo (2011). It is worth noting that the buckling load perfectly matches with the failure load shown in Figure 9. The critical condition is thus clearly identified, for normalized in-plane displacement d = 0.77, where global buckling leads to sudden failure.

Figure 9. Failure flag showing the sudden collapse which basically involves the whole unidirectional layers in one single (small) load step increment.

Numerical curve linearly extended 30% over the actual loading history

Numerical in-plane displacement vs. inplane load

In plane displacement (normalized)

Figure 10. Buckling load identified by extending the initial linear part of the in-plane displacement vs. inplane load curve.

5 5.1

Figure 8. Experimental normalized in-plane displacement vs. out of plane displacement measured at midspan. Mode shift is observed for PP7.

EXPERIMENTAL/NUMERICAL COMPARISON Discussion

The sudden failure identified in the numerical analyses seems to be confirmed by the experimental results, where, except for panel PP7-03, the maximum load is fairly linearly reached, thereafter an immediate drop of the sustained load occurs. When attempting to compare the in-plane displacements vs. in-plane load curves, in order to achieve the validation of the numerical FE model, the main task to overcome is the non-linear behavior observed in the very early stage of the experimental loading history. Since the compressive modulus shows a good matching with the

394

experimental results, the hypothesis that this non linearity might be due to a stabilization of the boundary condition takes place. A constant offset to be added to the numerical in-plane displacements is then considered. This does have no influence on the predicted collapse load, only acting on the in-plane displacements for a better comparison in terms of stiffness, which is represented by the steepness of the curve, also not affected. 5.2

Plots

An interesting experimental/numerical comparison, both in terms of stiffness and failure load, is obtained by reporting the FEM results on the plot relevant to experimental testing already reported in Figure 2. See Figure 11. The compressive stiffness of the tested panels looks quite accurately predicted by the numerical model. It is worth noting that the glass content has been derived from out tests, including the glass weight fraction of each oriented layer. Once these data are available, the ´mixture rule´ properly corrected using the Halpin-Tsai equation provides rather reliable results. The prepreg panels suffered early delamination due to a pure inter-laminar fracture mechanism, not involving fiber bridging as described in Section 3. This critical aspect led to an unexpected early failure, whose prediction needs the capability of simulating the mechanism giving rise to the opening of new delaminations. Unfortunately, if no information about the distribution and entity of the initial defects is available, predicting such a complex behavior is impossible.

6

CONCLUSIONS

In the present paper the work focused on the experimental analysis of four fiberglass plates built up by two different manufacturing methods, followed by the validation of a previously developed numerical modeling strategy through the comparison with experimental results. The experimental measurements led to the identification of two distinct failure modes, related to the fabrication method. In particular early delaminations leading to sudden failure are reported for the prepreg panels, due to an inter-laminar fracture mode not involving fibers bridging. The proposed numerical model matches very well the experimental data especially for infusion made panels. It is able not only to represent the correct compressive stiffness of the panels, but also to predict with noticeable accuracy the critical load leading to elastic instability and followed up by sudden collapse for the infusion made panels not affected by early failure. The FE model simulating the prepreg panel behaves very similar to the infusion one. This confirms, as expected, that the delamination size has rather limited influence onto global buckling collapse. Global buckling is believed to be influenced mainly by the thicker sub-laminates which, in the considered cases, were similar. The slightly higher stiffness of the prepregs models is clearly due the higher thickness of the plate. In this latter case the experimental/numerical mismatching is therefore attributed to the early delaminations suffered by the prepregs, which obviously cannot be reproduced by the proposed finite element modeling strategy. The 9-noded MITC9 shell model is thus a reliable modeling strategy to reduce computational efforts required by traditional 3 DoF solid element models, allowing the definition of a delaminated model by appropriately offsetting the contact surfaces of the delamination.

REFERENCES

Figure 11. Experimental/numerical comparison. Normalized In-plane displacements vs. In-plane load.

Abrate, S. 1991. Impact on laminated composite materials. Applied Mechanical Review 44: 155–190. ADINA 2008. Theory and modeling guide, v. 8.5.3, Watertown (MA), ADINA R&D Inc. Bailey, C. Davies, P. Grohens, Y. Dolto, G. 2003. Application of interlaminar tests to marine composite, a literature review. Applied Composite Materials 11(2): 99–126. Branner, K. Berring, P. Gaiotti, M. Rizzo, C.M. 2011. Comparison of two finite element methods with experiments of delaminated composite panels. Proc. of the 18th International Conference on Composite Materials, ICCM18, 21–26 August 2011. pp. 133–139, Jeju Island, Korea.

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Branner, K. & Berring, P. 2011. Compressive Strength of Thick Composite Panels, in: Proc. of 32nd Risø International Symposium on Materials Science, 5–9 September 2011, Roskilde, Denmark. Gaiotti, M. & Rizzo, C.M. 2009. An experimental/ numerical study on bending behaviour of composite sandwich panels. Proc. of the 16th International Conference of Ship and Shipping Research NAV 2009, 26–27 November 2009. Messina, Italy. Gaiotti, M. & Rizzo, C.M. 2011. Buckling behavior of FRP sandwich panels made by hand layup and vaacum bag infusion procedure. In Rizzuto & Guedes Soares (eds), Sustainable Maritime Transportation and Exploitation of Sea Resources, pp. 385–392, Proc. of the Congress of the International Maritime Association of the Mediterranean, 13–16 September 2011, Genova, Italy London Taylor & Francis Group. Halpin, J.C. & Nicolais, L. 1971. Ingegnere Chimico Italiano. 7, 173. Huimin, F. & Yongbo, Z. 2011. On the distribution of delamination in composite structures and compressive strength prediction for laminates with embedded delaminations. Applied Composite Materials 18(3): 253–269. Naik, N.K. & Kumar, S. 1999. Compressive strength of unidirectional composites: evaluation and comparison of prediction models, Composites Structures 46: 299–308.

Pavier, M.J. & Clarke, M.P. 1995. Experimental Techniques for the Investigation of the Effects of Impact Damage on Carbon-Fibre Composites. Composites Science & Technology 55: 157–169. Peck, S.O. & Springer, G.S. 1991. The behaviour of delaminations in composite plates—analytical and experimental results. J. Composite Materials 25: 907–929. Sørensen, B.F. Toftegaard, H. Goutanos, S. Branner, K. Berring, P. Lund, E. Wedel-Heinen, J. Garm, J.H. 2010. Improved Design of Large Wind Turbine Blade of Fibre Composites (Phase 4). Summary Report, Risø-R-1734(EN), Risø National Laboratory for Sustainable Energy, Denmark. Sørensen, B.F. Branner, K. Lund, E. Wedel-Heinen, J. Garm, J.H. 2009. Improved Design of Large Wind Turbine Blade of Fibre Composites (Phase 3)—Summary Report, Risø-R-1699(EN), Risø National Laboratory for Sustainable Energy, Denmark. Spearing, S.M. & Evans, A.G. 1992. The role of fiber bridging in the delamination resistance of fiberreinforced composites. Acta Metallurgica et Materialia. 40(9): 2191–2199.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Comparison of load-carrying behavior between web-core sandwich, stiffened and isotropic plate J. Jelovica & J. Romanoff Department of Applied Mechanics/Marine Technology, Aalto University School of Engineering, Aalto, Finland

ABSTRACT: This paper investigates theoretically the differences in load-carrying behaviour between web-core sandwich plate, stiffened plate and isotropic plate. Buckling and post-buckling is studied. The study is carried out using two approaches, both solved with the finite element method. The first is a threedimensional model of the plates. The second approach is the equivalent single-layer theory approach. First-order shear deformation theory is used. The second approach allows plates to the viewed through ABD- and DQ stiffness coefficients. Plates are axially loaded in the web plate/stiffener direction. Simply supported boundary condition is considered with loaded edges kept straight and unloaded edges free to move in-plane. The results show that the buckling load of sandwich plate is 42% to 65% higher than the stiffened plates. The reason is that sandwich plate is a symmetrical structure where coupling between in-plane and out-of-plane displacements does not exist (B-matrix is equal to zero). Furthermore, breadthto-thickness ratio (representing local plate slenderness) is about two times lower in sandwich plate than in stiffened plate which prevents local buckling. On the other hand, buckling load of sandwich plate can be improved by increasing the transverse shear stiffness, e.g. by filling the voids in the core. For the same structural weight, post-buckling stiffness of stiffened plate is somewhat lower than in sandwich, also owing to the B-matrix. Isotropic plate of the same bending stiffness as sandwich plate has higher postbuckling stiffness due to larger in-plane stiffness (A-matrix) or structural weight. 1

INTRODUCTION

Increased payload of a ship can be achieved by reducing her structural weight. Optimized stiffened plates have traditionally been used for this purpose. On the other hand, sandwich plates have long been recognized to have high bending stiffness for their weight. Their usage in large ships is far less than stiffened plates. To explore their benefits as loadcarrying members, this paper studies web-core steel sandwich plates; see Figure 1a. These plates have been shown in Klanac et al. (2005) and Ehlers et al. (2012) to save space and increase crashworthiness. Global bending of a ship causes compression of the decks. The behaviour of stiffened plate in compression has been extensively studied; see e.g. Paik and Thayamballi (2003), Guedes Soares and Gordo (1997), Gordo and Guedes Soares (2011). Byklum and Amdahl (2002) and Byklum et al. (2004) have developed a two stage approach for the buckling and post-buckling assessment of stiffened plates. Local buckling is calculated first and the non-linear ABD-matrix is derived for global analysis. However, the investigations do not consider the influence of out-of-plane shear deformations. These have shown to have high influence on the response of web-core

Figure 1. The plates under investigation: a) web-core sandwich plate, b) stiffened plate, c) isotropic plate.

397

sandwich plate; see Romanoff and Varsta (2007), Nordstrand (2004) and Jelovica et al. (2012). In difference to stiffened plate, web-core sandwich plate under in-plane compression has only been investigated in a few studies. Kolsters and Zenkert (2006a), Kolsters and Zenkert (2006b), and Kolsters (2004) studied the local buckling and post-buckling behaviour of face plates. Taczala and Banasiak (2004) presented the difference in buckling mode and critical stress between sandwich and stiffened plate. Kozak (2006) studied the ultimate strength of sandwich columns. Jelovica et al. (2012) studied the influence of laser-weld stiffness on global bifurcation buckling strength of sandwich plates. Laser-weld stiffness is ratio of the moment to the rotation angle at the face-plate-web-plate intersection. Jelovica and Romanoff (2012) studied geometrically non-linear load-carrying behaviour of web-core sandwich plates. The plate response was compared to isotropic plate of the same bending stiffness. In this paper we continue the former study by comparing the load-carrying behaviour of sandwich plate, stiffened plate and isotropic plate. The reasons for differences in their response are outlined. Further, stresses along the load-deflection curve are presented. The investigation is carried out with two theoretical approaches that have different kinematic assumptions, both of which are solved with the Finite Element Method (FEM). The first is a 3-D shell model of the structure. Global buckling and postbuckling is simulated using large deflection Equivalent Table 1.

Single-Layer (ESL) theory approach, described for example in Reddy (2000). There, first-order shear deformation theory is used. Plates are loaded parallel to the web plates or stiffeners. Geometric imperfections in the form of lowest buckling mode are scaled and imposed on the plates prior to load-carrying analysis. Simply supported boundary condition is considered with loaded edges kept straight and unloaded edges free to move in-plane. The investigation is limited to linear-elastic material behaviour. 2

DESCRIPTION OF THE PLATES

Three types of plates are studied: a web-core sandwich plate (Figure 1a), a stiffened plate (Figure 1b) and an isotropic plate (Figure 1c). The studied sandwich plate is a standard web-core sandwich plate for marine and civil applications. The thickness of the face plates is 2.5 mm and the web plate 4 mm. Height of the core is 40 mm and spacing of the web plates is 120 mm. The area weight of the sandwich plate is 50 kg/m2. Dimensions of the stiffened plate are selected such that the structure has the same extensional stiffness as the sandwich plate; see Table 1. This results also in the same area weight. Through different combination of plate thickness, size and spacing of the bulb profiles, three stiffened plates are created. Their buckling behaviour ranges from global buckling or local plate/stiffener buckling, as is presented in continuation.

Geometric and stiffness properties of the plates. Sandwich plate tf/tw × hc/s [mm]

Stiffened plate 1 tp/HP/s [mm]

Stiffened plate 2 tp/HP/s [mm]

Stiffened plate 3 tp/HP/s [mm]

Isotropic plate h [mm]

2.5/4 × 40/120

5/80 × 5/400

5/100 × 6/600

4.5/120 × 7/600

30.7

β local

48 ⋅ σ y E

89 ⋅ σ y E

120 ⋅ σ y E

120 ⋅ σ y E

117 ⋅ σ y E

A11 [MN] A22 [MN] A12 [MN] A33 [MN]

1 406 1 132 339 396

1 410 1 132 339 396

1 397 1 132 339 396

1 377 1 019 306 356

6 960 6 960 2 090 2 440

D11 [kNm] D22 [kNm] D12 [kNm] D33 [kNm]

548 511 153 179

772 119 36 42

1 100 162 48 56

1 921 367 110 128

548 548 164 191

B11 [kN/rad] B22 [kN/rad] B12 [kN/rad] B33 [kN/rad]

0 0 0 0

0 11 516 3 454 4 031

0 13 444 4 033 4 705

0 19 304 5 791 6 756

0 0 0 0

DQx [kNm] DQy [kNm]

68 ⋅ 103 419

178 ⋅ 103 330 ⋅ 103

177 ⋅ 103 330 ⋅ 103

265 ⋅ 103 297 ⋅ 103

2 033 ⋅ 103 2 033 ⋅ 103

398

The isotropic plate features the same bending stiffness as the sandwich plate and, as a result, much higher in-plane stiffness. The area weight is 240 kg/m2, almost five times higher. The calculation of extensional, bending, coupling and shear stiffness coefficients is presented in Appendix A for the sandwich plate and in Appendix B for the stiffened plate. The values are tabulated in Table 1. It can be seen that sandwich plate has much lower shear stiffness than other plates, especially in the transverse direction. The bending stiffness of stiffened plates is 2 to 3 times higher than in isotropic and sandwich plate. In-plane stiffness is, however, five times higher in isotropic plate than in the other plates. Sandwich and isotropic plate are symmetrical and thus their coupling coefficients are zero, while that is not the case with stiffened plate. Further, breadth to thickness ratio, b/t, (representing the plate slenderness β) is about two times lower in sandwich plate than in other structures. The size of plates is 3600 × 3600 mm. This is close to typical web frame spacing in a ship and is a result of multiplying considered stiffener spacing with integer number. The material behaviour is linear elastic, characterised by a Young’s modulus E = 206 GPa and Poisson’s ratio ν = 0.3. 3

ANALYSIS

Analysis is carried out in two steps: the bifurcation buckling shape is obtained in the first step and scaled to the required magnitude for the second step, in which an increasing load is applied. The analyses are carried out using the Abaqus software, version 6.9. A modified Riks procedure is used in the second step; see Abaqus (2012). The shape of the initial imperfection is taken as the shape of the first eigenmode. Shape and magnitude of different production and exploitation imperfections has been measured for stiffened plates in numerous studies (see ISSC 2009 for summary on imperfections) and their influence on behaviour in compression presented (Paik 2007, Masaoka and Mansour 2008). However, such rigorous studies have not been shown for web-core sandwich plate. Thus in the current analysis the imperfection shape for all structures is given minimal magnitude to allow successful running of the solver. In case of global bifurcation buckling the shape is multiplied with a/10 000 or with t/10 in case of local plate buckling.

nodes in the neutral axis. Six elements per web plate height are used. The face plates have six elements between the webs. Mesh density is considered sufficient since the difference in bifurcation buckling load to analytical solution (Reddy 2000, Robinson 1955) is less than 2%. Simply supported boundary conditions are considered with loaded edges kept straight and unloaded edges free to move in-plane. The deflection restraint is set only on the nodes at the geometric mid-plane. This allows the rotation of the plate around the midplane edge. Furthermore, all the nodes at a certain web plate have the same displacement in the y-direction, v; see Figure 2. Additionally, the nodes at the geometric mid-plane at x = 0 are required to have the same displacement in x-direction, u. The same is required at x = a. Stiffened plate is modeled with shell elements (S4) except the tip of the stiffener which is modeled with beam elements (B31). Concentrated nodal forces act on the nodes in the neutral axis. Ten elements between the stiffeners are used. The height of the stiffener is divided in five elements. Mesh density is considered sufficient since the increase by 50% resulted in difference in bifurcation buckling load by less than 2%. Deflection constraint is imposed on the plate edges and stiffeners are not able to rotate; see Figure 3. Further, plate edges at x = 0 and x = a are required to stay straight.

Figure 2. FE mesh boundary condition for the 3-D model of the sandwich plate.

3.1 3-D model Face and web plates of the sandwich plate are modelled with shell elements to form an actual topology. Shell elements with four nodes (S4) are used. Concentrated nodal forces act on the

Figure 3. FE mesh boundary condition for the 3-D model of the stiffened plate. In the analysis the flange of the stiffener has an offset to resemble actual shape of HP profile.

399

10 9 8

Load, N [MN]

7 6 5 4 3 2 1 0

Figure 4. Simply supported 2-D model with unloaded edges free to move in-plane. Dashed lines show unloaded plate. Solid lines show (exaggerated) post-buckling shape.

3.2

4.1

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

Deflecon, w [m]

Figure 5.

Load-deflection curves with 3-D models.

2-D model

The structure is considered as orthotropic plate, described through a single layer in its neutral axis, where the loads and boundary conditions are also introduced. Equivalent stiffness properties for extension, coupling, bending and shear are described through ABD- and DQ-matrices, respectively; see Table 1. The plate is divided into 100 elements in the length and width directions. The correspondence in bifurcation buckling load to analytical solution for sandwich plate (Reddy 2000, Robinson 1955) or FEM solution for stiffened plate is within 2%. Deflection constraint is imposed on the edges and no rotation is allowed around axis perpendicular to the edge on loaded sides. Loaded edges are required to stay straight. The typical deformed shape of the plate is shown in Figure 4.

4

0.00

RESULTS Load-carrying behaviour

Load-deflection and load-end shortening curves for studied structures are presented in Figure 5 and Figure 6. The curves are obtained with 3-D models. The deflection of the sandwich plate is measured at the centre of the face plate (a/2, b/2) on the convex side. The deflection of the stiffened plate is measured at the stiffener-plate intersection either in the centre of the plate (a/2, b/2) or on the nearest stiffener if the stiffener is not at b/2 (a/2, b/2-s/2). The presented deflection includes initial imperfection. The sandwich plate buckles at 3.40 MN. Buckling load of the stiffened plates is in the range between 1.20 MN and 1.95 MN, stiffened plate 3 being the lowest and stiffened plate 1 being the highest. This is from 65% to 42% lower than the sandwich plate.

Figure 6. Load-end shortening curves with 3-D models.

The sandwich plate and stiffened plate 1 buckle globally (as well as isotropic plate). Local buckling in stiffened plate 2 is present in plates between the stiffeners and stiffeners as well. Only plates buckle locally in stiffened plate 3. Both stiffened plates continue to global buckling as the load increases. Isotropic plate, being almost five times heavier than the other structures, buckles at 6 MN. The response of the isotropic plate is in excellent agreement with experiments (Rhodes et al. 1975) and textbook results (Jones 2006). Further load increase is possible as a result of the high stiffness of the straight edges. However, Figure 5 and Figure 6 show that analyzed structures have different post-buckling stiffness. Isotropic plate has significantly higher post-buckling stiffness than the rest. Sandwich plate is somewhat stiffer than the stiffened plates. 4.2

Influence of stiffness coefficients

The influence of stiffness coefficients on loadcarrying behavior is conveniently studied here with 2-D analysis model. Its accuracy is presented in Figure 7 in comparison to 3-D model on the results from previous chapter. From stiffened plates, only

400

Figure 7. Comparison of 3-D and 2-D analysis models for sandwich plate and stiffened plate 1.

no. 1 is investigated since it buckles globally. Others are neglected since their local buckling modes cannot be captured with 2-D model (Jelovica and Romanoff 2012). The figure shows that the accuracy is very good in case of sandwich plate and somewhat lower in case of sandwich plate. Figure 8 shows the effect of stiffness coefficients on load-deflection behaviour of sandwich plate. Figure 9 shows the same effect on load-end shortening behaviour. The buckling load is almost doubled by increasing the transverse shear stiffness, DQy, to the value of longitudinal shear stiffness. Small further increase in buckling load is achieved by increasing the shear and the bending stiffnsses, other than D11, to that of the isotropic plate. The load-end shortening is not affected. However, these alterations do not affect the post-buckling stiffness. The post-buckling stiffness is increased by increasing extensional stiffness coefficients or A-matrix, mostly due to A11 which is the loading direction. Figure 10 shows the effect of stiffness coefficients on load-deflection behaviour of stiffened plate 1. Increasing the shear stiffness to the one from isotropic plate doesn’t have any influence on response. The buckling load is first increased by 50% when neglecting the coupling stiffness matrix (B-matrix) and then doubled by equating the bending stiffness matrix to isotropic plate. Disregarding the B-matrix has slight positive influence on postbuckling stiffness. Post-buckling stiffness equal to isotropic plate is obtained by equating the extensional stiffness coefficients (A-matrix). 4.3

Figure 8. Influence of stiffness coefficients on loaddeflection curve starting from sandwich plate.

Figure 9. Influence of stiffness coefficients on load-end shortening curve starting from sandwich plate.

Figure 10. Influence of stiffness coefficients on loaddeflection curve starting from stiffened plate.

Stress levels

Figure 11 shows the maximum value of von Mises stress in the plates along the load-deflection curve. The stress rapidly increases as a result of out-ofplane deformations in all structures. The 200 MPa stress in the sandwich plate appears in the plate mid-span, at the face plate–web plate intersection,

Figure 11. Maximum von Mises stress in the structure along the load-deflection curve.

401

on the concave side. Further loading continues to increase deflections and the location of the highest stress moves next to unloaded plate edges. This is where 350 MPa and 500 MPa stress appears, also at the face plate–web plate intersection, both on concave and convex side. The equivalent stress in the plate mid-span remains or slowly increases from 200 MPa. The 200 MPa in the stiffened plate 1 (global buckling) happens in the stiffener flange which is under tension. At the same location the 350 MPa is found, as well as next to unloaded edges. The 500 MPa occurs is the last location as a result of local buckling. Stiffened plate 2 and 3 buckle locally between stiffeners and the global deformation follows afterwards. The considered equivalent stress levels are found in locally buckled plates, firstly in the midspan and later next to unloaded edges. 5

DISCUSSION AND CONCLUSION

This study presents the load-carrying behaviour of web-core sandwich plates in compression. The behaviour is compared to the stiffened plates of the same in-plane stiffness, or A-matrix, and isotropic plate of the same bending stiffness in the loading direction, D11. The sandwich plate and stiffened plate 1 buckle globally (as well as isotropic plate). Local buckling in stiffened plate 2 is present in plates between the stiffeners and stiffeners as well. Only plates buckle locally in stiffened plate 3 due to increased size of stiffeners. Both stiffened plates continue to global buckling as the load increases. The buckling load of sandwich plate is 42% to 65% higher than the stiffened plates. The reason is that sandwich plate is a symmetrical structure where coupling between in-plane and out-of-plane displacements does not exist (B-matrix is equal to zero). Better performance of sandwich over stiffened plate was shown also in Taczala and Banasiak (2004). Furthermore, b/t (representing local plate slenderness) is about two times lower in sandwich plate than in stiffened plate which prevents local buckling. On the other hand, buckling load of sandwich plate is almost doubled by increasing the transverse shear stiffness to be equal to the longitudinal. Thus significant improvements in global buckling can be expected by filling the voids in the core. Kolsters and Zenkert (2006a) came to same conclusion for local buckling. The sensitivity of buckling load on variations in DQy was further shown in Jelovica et al. (2012). For the same structural weight, post-buckling stiffness of stiffened plate is somewhat lower than in sandwich, also owing to the B-matrix. Both structures have significantly lower post-buckling stiffness than isotropic

plate. It is shown that this is due to lower in-plane stiffness (A-matrix) of the sandwich and stiffened plate, coming from their low cross-sectional area. Jelovica and Romanoff (2012) showed that sandwich plate is less efficient in carrying the membrane forces than isotropic plate. The effective width was 10% lower at 35% in access to buckling. The stress rapidly increases as a result of out-of-plane deformations in all structures. The 200 MPa stress in the sandwich plate appears in the plate mid-span, at the face plate–web plate intersection, on the concave side where compressive stresses are highest due to bending. Further loading continues to increase deflections and the stress in the plate mid-span continues to change from in-plane governed to out-of-plane. In other words, membrane stress reduces while the bending stress increases. This results in almost constant equivalent stress on the concave side during the load increase. The stress decreases on the convex side. Thus the location on the highest stress moves to unloaded edges where the influence of bending is small. This is where 350 MPa and 500 MPa stress appears, also at the face plate–web plate intersection, both on concave and convex side. The mechanism is similar in stiffened plate 1 (global buckling), however, it reaches 200 MPa through tension in the stiffener flange and higher stresses next to unloaded edges as a result of local plate buckling. High stresses in stiffened plate 2 and 3 occur in the local buckles. Post-buckling strength of an isotropic plate is allowed to be used in the design of stiffened plates, provided that the stiffener support system is strong enough to prevent the overall panel field buckling; see Hughes and Paik (2010) and DNV (2005). There, the plates are allowed to buckle elastically and it is assumed that the supporting girders have adequate strength to retain safety of the structure. This design philosophy allows for more stable collapse behaviour, observed in stiffened plate 3 in this study. To see the collapse behaviour of webcore sandwich plate, material nonlinearity needs to be included in the further studies, at least until the the point of yielding as is done in Byklum et al. (2004) for a stiffened plate. Then conclusions on desirability of different modes of failure can be drawn.

6

APPENDIX A—STIFFNESS PROPERTIES OF SANDWICH PLATE

The extensional, extensional-bending, and bending stiffness matrices respectively are (Reddy 2000):

([A],[B ],[D]) ∫ [E ] (

402

h/2

−h / 2

i

di di z ) ddz, i = t,c,b

(1)

where the distance from the mid-plane of the plate is:

7

h tt − , 2 2 d c z, h t db = − + b . 2 2

The extensional, extensional-bending, and bending stiffness matrices follow Eq. (1), however, the integration over the height of the sandwich plate is replaced by the height of the flange of the stiffener, height of the web of the stiffener and the plate thickness of the stiffened plate (Aavi 2012). The elasticity matrix of the plate and the web of the stiffener are given in Eq. (3) and Eq. (4), respectively. The elasticity matrix of the flange of the stiffener is:

dt =

(2)

The elasticity matrix [E] of the face plates is:

[E ] i

⎡ Ei 1 ⎢ = ν i Ei 1 − νi2 ⎢ ⎢ 0 ⎣

Ei 0 Ei 0 0 Gi ( − i

i

⎤ ⎥ ⎥ , i = t,b, (3) )⎥⎦

[E ] f

while the core has the elasticity matrix:

[E ] c

⎡1 0 0 ⎤ E wtw ⎢ = 0 0 0 ⎥⎥ . s ⎢ ⎢⎣0 0 0 ⎥⎦

(4)

12 Dw s

⎛ 2

⎛D d⎞ D d⎞ kQ ⎜ w + 6 ⎟ + 12 w − 2 ⎟ s⎠ kθ s s⎠ ⎝ ⎝ Dt

(5)

kQ =

Dt ⎛ 1 1⎞ D d − +6 t s ⎝ kθt kθb ⎟⎠ Dw s D d D 1 + 12 t + t Dw s Db

t ⎛ ⎞ k 2 Gttt + Gbtb + w Gw hc ⎟ , ⎝ ⎠ s

(6)

Gw Gf

(7)

)

(

)

⎛ ⎞ ⎛ τ ⎞ A ⎜ ∑ ∫ ⎜ i ⎟ ti dsi ⎟ ⎜⎝ i ⎝ QQx s ⎠ ⎟⎠ 2

, i = t,c,b.

(11)

tw , s bf Gp . s

Gp

(8)

(12)

Shear correction factor in longitudinal direction kxz is: kxz =

( xz )avg , ( xz )max

(13)

where average shear stress is approximated with:

w

1

(10)

kxz G pt p + G h + G f h f ,

(τ xz )avg ≈ A

where k11 =

(

kyz G t p ,

where Gp is the shear stiffness of the plate and Gw and Gf are shear stiffness of the web and flange of the stiffener:

and laser weld rotation stiffness, kθ, is equal to infinity in this study; see Romanoff et al. (2007) and Jelovica et al. (2012) for the effects of finite weld stiffness in bending and buckling, respectively. The shear stiffness in the longitudinal direction is: DQx

(9)

where shear correction factor kyz is 5/6 and tp is the plate thickness. The shear stiffness in longitudinal direction is (Aavi 2012): DQ

where kQ is defined as: 1 + 12

⎡1 0 0 ⎤ E bf ⎢ = 0 0 0 ⎥⎥ , s ⎢ ⎢⎣0 0 0 ⎥⎦

where bf is the width of the stiffener flange. The shear stiffness in transverse direction is: DQy

The shear stiffness in transverse direction is (Romanoff et al. 2007): DQy =

APPENDIX B—STIFFNESS PROPERTIES OF STIFFENED PLATE

Qz , twt p tw h f

(14)

and the maximum shear stress is calculated using:

(τ xz )max 403

(

Qz ⎡ Ap 2 zna ⎣ ≈

)

(

)

2 t p + tw zna − t p ⎤⎥ ⎦. 2 I ztw

(15)

Aw is the area of the web and Ap is the area of the plate between two stiffeners. tw is the thickness of the web of the stiffener and hf is the height of the flange of the stiffener. zna is the distance from the tip of the stiffener to the neutral axis and Iz is the second moment of area of stiffener and associated plate. ACKNOWLEDGEMENTS The authors gratefully acknowledge the support of Aalto University School of Engineering. REFERENCES Aavi, E. 2012. Equivalent shell element for ship structural design. M.Sc. thesis, Aalto University. ABAQUS. 2012. User’s manual, version 6.9. Byklum, E., Amdahl, J. 2002. A simplified method for elastic large deflection analysis of plates and stiffened panels due to local buckling, Thin-Walled Structures 40: 925–953. Byklum, E., Steen, E., Amdahl, J. 2004. A semi-analytical model for global buckling and postbuckling analysis of stiffened panels, Thin-Walled Structures 42: 701–717. Det Norske Veritas. 2005. Rules for the classification of steel ships. Høvik. Ehlers, S., Tabri, K., Romanoff, J., Varsta, P. 2012. Numerical and experimental investigation on the collision resistance of the X-core structure, Ships and Offshore Structures 7: 21–29. Guedes Soares, C., Gordo, J.M. 1997. Design methods for stiffened plates under predominantly uniaxial compression, Marine Structures 10: 465–497. Gordo, J.M., Guedes Soares, C. 2011. Compressive tests on stiffened panels of intermediate slenderness, ThinWalled Structures 49: 782–794. Hughes, O.F., Paik, J.K. 2010. Ship structural analysis and design. Jersey City: The Society of Naval Architects and Marine Engineers. ISSC. 2009. Ultimate Strength Committee. S. Korea. Jelovica, J., Romanoff, J., Ehlers, S., Varsta, P. 2012. Influence of weld stiffness on buckling strength of laser-welded web-core sandwich plates, Journal of Constructional Steel Research 77: 12–18. Jelovica, J., Romanoff, J. 2012. Load-carrying behaviour of web-core sandwich plates in compression. Submitted to Thin-Walled Structures. Jones, R.M. 2006. Buckling of bars, plates, and shells. Blacksburg: Bull Ridge Publishing.

Klanac, A., Ehlers, S., Tabri, K., Rudan, S., Broekhuijsen. J. 2005. Qualitative design assessment of crashworthy structures. In: Proceedings of the International Maritime Association of Mediterranean, Portugal: 461–469. Kolsters, H., Zenkert, D. 2006a. Buckling of laser-welded sandwich panels: Part 1: elastic buckling parallel to the webs, Journal of Engineering for the Maritime Environment 220: 67–79. Kolsters, H., Zenkert, D. 2006b. Buckling of laser-welded sandwich panels: Part 2: elastic buckling normal to the webs, Journal of Engineering for the Maritime Environment 220: 81–94. Kolsters, H. 2004. Structural Design of Laser-Welded Sandwich Panels for Marine Applications, Doctoral Dissertation, Royal Institute of Technology, Department of Aeronautical and Vehicle Engineering, Stockholm. Paper D: Buckling of laser-welded sandwich panels: Part 3: Ultimate strength and experiments. Kozak, J. 2006. Problems of strength modeling of steel sandwich panels under in-plate loads, Polish Maritime Research 1: 9–12. Masaoka, K., Mansour, A. 2008. Compressive strength of stiffened plates with imperfections: simple design equations, Journal of Ship Research 52:3 227–237. Nordstrand, T. 2004. On buckling loads for edge-loaded orthotropic plates including transverse shear, Composite structures 65: 1–6. Paik, J.K. 2007. Empirical formulations for predicting the ultimate compressive strength of welded aluminium stiffened panels, Thin-Walled Structures 45: 171–184. Paik, J.K., Thayamballi, A.K. 2003. Ultimate limit state design of steel-plated structures. Chichester: John Wiley & Sons. Reddy, J.N. 2000. Mechanics of laminated composite plates and shells—Theory and analysis, second ed. Boca Raton: CRC Press. Rhodes, J., Harvey, J.M., Fok, W.C. 1975. The loadcarrying capacity of initially imperfect eccentrically loaded plates, International Journal of Mechanical Sciences. 17: 161–175. Robinson, J.R. 1955. The Buckling and Bending of Orthotropic Sandwich Panel With All Edges SimplySupported, The Aeronautical Quarterly p.125–148. Romanoff, J., Remes, H., Socha, G., Jutila, M., Varsta, P. 2007. The stiffness of laser stake welded T-joints in web-core sandwich structures, Thin-Walled Structures 45: 453–462. Romanoff, J., Varsta, P. 2007. Bending response of web-core sandwich plates, Composite Structures 81: 292–302. Taczala, M., Banasiak, W. 2004. Buckling of I-core sandwich panels. Journal of Theoretical and Applied Mechanics 42(2): 335–348.

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An experimental and numerical study of corroded steel plates repaired with composite patches V.A. Karatzas, E. Kotsidis & N.G. Tsouvalis Shipbuilding Technology Laboratory, School of Naval Architecture and Marine Engineering, National Technical University of Athens, Athens, Greece

ABSTRACT: Most of the structures operating at sea will suffer corrosion in areas that have been left unprotected or where the anticorrosion measures are inadequate. The current common repair method consists primarily of replacing damaged steel. An alternative repair method is using patches made of composite materials. This method counters the majority of the problems faced by conventional renewal repairs. The work presented in this paper consists of an experimental study of artificially corroded steel plates repaired with composite patches. In total eight specimens were tested in tension. The effect of aging was taken into consideration by three different aging scenarios. Part of the experimental results was subsequently validated with the use of numerical simulations that encompassed cohesive elements so as to simulate the debonding procedure. Results showed that composite patch repairing was able to rehabilitate the defected steel plates and improve their load bearing capacity. The numerical results are in good agreement with the experimental ones. 1

INTRODUCTION

The vast majority of ships and marine structures that are currently in service are manufactured using steel and will inevitably face typical defects such as corrosion and cracks throughout their operating life. The current methods of repair of the aforementioned defects found onboard these structures consist mainly of the renewal of the defected steel parts. These methods often prove to be time consuming and in some cases quite complicated, which in turn leads to severe logistic problems and loss of revenue for the ship-owners. Composite material patching is a very promising method for repairing and/or reinforcing steel structures. In a composite patch repair, part of the applied load is transferred from the base plate through an adhesive layer to the composite patch, thus reducing the stress levels in the steel substrate. Composite patch repairs and/or reinforcements overcome many, if not all the disadvantages of the traditional repair methods (Allan et al. 1988, Turton et al. 2005). They do not involve hot works in any way and, therefore, existing deadweight loading or proximity to explosive environments has no particular consequences. Patches can be applied directly on corroded steel members by performing a simple surface preparation. Composite patching has proven its effectiveness and cost benefits by its application in the aerospace industry for several years now (Roach & Graf 1996, Baker et al. 1999). However, there are

several fundamental differences between the aerospace applications and marine/offshore steel applications, which dictate a separate approach and investigation of the problem. These differences include the different stiffness of the base metal (stiffer steel versus the more similar to composites flexible aluminium), the completely different geometries involved (significantly thicker plating and larger beams in steel structures), the different loading cases and the different operating and environmental conditions. In the marine industry only a few cases are reported starting with Grabovac et al. 1993, 2003, 2009 who used composite patches to repair the aluminum deckhouse of a Royal Navy Frigate that repeatedly exhibited fatigue cracking. After 15 years of active service no cracking underneath the patch or in adjacent areas was initiated. Additionally a time-cost estimation is given for the repairs. Another more recent work dealing with the application of composite patch repairs in marine structures is that of McGeorge et al. 2009 where two composite patch repairing were carried out on an FPSO. The first repair was carried out in order to arrest a fatigue crack that had developed from the corner of a door while the second one was carried out to restore material loss on a heavily pitted deck floor. The two demonstrators have shown the viability of bonded composite repairs to the two most widely encountered damage scenarios in floating offshore units—fatigue cracking and plate thinning due to corrosion. The present work

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has been performed within the context of the EU FP7 Co-Patch research program and presents an experimental and numerical study on the effectiveness of composite patch repair method as an alternative way for repairing corroded plates in marine structures. In order to account for the harsh environmental conditions during the operating life of the structure and to study its effect on the repair, three different aging scenarios were considered. 2

SPECIMENS GEOMETRY

The geometry and main dimensions of the plates tested are presented in Figure 1. The nominal thickness of the steel plates, ts, is 5 mm which is a typical plate thickness for secondary structural members in marine structures. The central orthogonal reference part of the steel specimen has length Ls equal to 400 mm and width Ws equal to 100 mm. In the one side of the plates, an area in the center having length 100 mm and width 80 mm has been subjected to artificial corrosion. On top of the corroded area side a composite patch was laminated. The composite patch was tapered. The full patch thickness is denoted as tp. The effective length was equal to Lp = 200 mm while the total length

Figure 1.

Geometry of specimens.

Table 1. Measured nomenclature.

dimensions

and

specimen

Specimen

Ws (mm)

Ls (mm)

ts (mm)

tp (mm)

P2D1-A P2D1-B P2D2-A P2D2-B P2D2-C P2D2-D P2D3-A P2D3-B

100 100 100 100 100 100 100 100

400 400 400 400 400 400 400 400

5.30 5.27 5.30 5.25 5.24 5.32 5.30 5.40

3.34 3.36 3.33 3.74 3.48 3.33 3.24 3.31

(with the tapered ends) was equal to 300 mm and the width was equal to Wp = 100 mm. The nomenclature and the actual measured geometry of the specimens are listed in Table 1. The depth of the artificial corrosion was measured prior to lamination and had an average value of 0.26 mm. 3

MATERIALS AND MANUFACTURING PROCEDURE

Once the prescribed central area of the plates had been corroded, they were removed from the environmental chamber and grit blasted so as to clear the plates from corrosion products and more importantly to achieve a good bonding of the laminate to the steel surface. Lamination was carried out directly after the surface preparation (Klanac 2012), so as to avoid contamination of the surface. The composite patches used for the rehabilitation of the plates were carbon/epoxy using the vacuum infusion method, and in all cases comprised of unidirectional plies that had the fiber direction parallel to the length of the specimens. In particular, CST 200 carbon unidirectional fibers with area density equal to 200 g/m2 provided by SGL GROUP were used. The number of layers for each patch was 16. The resin used was LH 288 Epoxy resin with H283 hardener by HAVEL. Additionally, a 280 g/m2 E-glass woven roving, twill weave by AEROGLASS was laminated first as a separator to prevent galvanic corrosion between the carbon fibres and the steel substrate. The infusion of the specimens took place in two stages infusing half of the plies and subsequently infusing the 8 remaining plies. The vacuum pressure was equal to 0.75 bar. The environmental conditions during the manufacturing of the specimens were 35°C and 29% humidity for the first step and 35°C and 34% humidity for the second. The carbon fibers were cut with varying length in groups of four so as to achieve the desired tapering at the edges of the patch. The steel used was typical marine grade steel. The mechanical properties of the materials have been measured from a preceding series of characterization tests (Karatzas & Tsouvalis, 2011) and are listed in Table 2. In the above table, E1T and E2T are the tensile moduli of the patch in principal directions 1 and 2, respectively, ν12 the Poisson ratio and G12 the shear modulus; E, ν and Sy are the modulus, Poisson ratio and yield stress of steel, respectively. From the set of eight specimens, six were subjected to accelerated cyclic corrosion conditions into a salt spray chamber (see Fig. 2) for 100 days (300 cycles) according to ISO 14993:2001 with 5% NaCl solution. From these six specimens, two have

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Table 2.

Material properties.

Properties

Patch

Properties

Steel

E1T [GPa] E2T [GPa] ν12 G12 [GPa]

112.3 ± 11.93 5.5 ± 0.4 0.38 2.0 ± 0.2

E [GPa] ν Sy [MPa]

203 0.3 314

Figure 3. Strain gage positions for the P2D1 specimens.

Figure 2. Six P2D (2 painted and 4 unpainted) specimens after 75 days in the salt spray chamber.

Figure 4. P2D3.

been painted with a common marine paint before being placed in the salt spray chamber. The areas that would subsequently enter the grips of the testing machine were protected with vinyl tape.

uneven surface created from the corrosion of the steel. Therefore, a different scheme was used and all three strain gages were positioned on the patch surface as shown in Figure 4. Debonding initiated from the edges of the composite patch and gradually progressed until the patch did not contribute to the strength of the structure. The first local maximum in the forcecrosshead displacement curves was considered as the patch failure load, as it is in this point that substantial debonding has occurred which affects the behaviour of the specimens. In reality the patch was still bonded to the plate to a certain extent and did not lose completely its load bearing capacity. The patch failure loads of all specimens are listed in Table 3. The non-aged specimens presented the higher failure load as expected, followed closely by the painted specimens that had undergone aging. The unpainted specimens presented the lower failure loads. The first comment that can be made from the study of the results in Table 3 is that there is very good repetability of the failure loads of the nominally identical specimens. The existing variation among the failure loads of the aged, unpainted specimens is attributed to two different factors. The first one is that the corrosion thickness reduction of specimens P2D2-A and P2D2-B is less compared to that of specimens P2D2-C and P2D2-D. In particular, the average thickness

4

TEST PARAMETERS AND RESULTS

All specimens were tested in tension. The specimens were mounted in a special fixture in order to achieve uniform loading and then positioned in a 250 kN capacity MTS hydraulic testing machine. A force control pretension of 25 kN was performed in all specimens so as to make sure there are no tolerances in the fixture-assembly-specimen system. After the pretension, the specimens were unloaded and tested up to failure. The tensile tests were performed using displacement control with a rate of 0.5 mm/min. During testing, the force, crosshead displacement and the measurements from the strain gages were recorded. Strains were recorded using 5 mm long strain gages in three locations, as shown in Figure 3. Strain gages 1 and 2 (SG-1, SG-2) were positioned on the back side of the specimens, 100 mm from the centre of the corroded area for the P2D1-A and P2D1-B plates. Strain gage 3 (SG-3) was positioned on the patch and at its centre. In the aged cases it was not possible to position strain gages in positions SG-1 and SG-2 due to the

407

Strain gage positions for specimens P2D2,

Table 3.

Patch failure load and aging scenario.

Specimen

Patch failure load (kN)

Aging scenario

P2D1-A P2D1-B P2D2-A P2D2-B P2D2-C P2D2-D P2D3-A P2D3-B

170.30 168.72 145.61 152.02 128.83 129.25 166.30 168.17

N/A N/A 300 cycles 300 cycles 300 cycles 300 cycles 300 cycles painted 300 cycles painted

reduction of specimens P2D2-A & B was measured equal to 0.66 mm, while in the P2D2-C & D specimens the thickness reduction was equal to 1.2 mm. The other factor that has an impact on the patch failure load is the ingression of moisture that took place along the length of the unpainted specimens, near their edges (Fig. 5). The area underneath the patch that was affected from this moisture ingression was 20% of the bond area for the P2D2-A & B specimens, whereas in the P2D2-C &D specimens this percentage was equal to 23%. These differences are the result of the specimens’ position inside the environmental chamber. Moreover, comparing the average failure loads for the non-aged specimens and the aged, unpainted ones a 18% reduction is noted on average. However, the comparison between the non-aged and the aged, painted specimens shows that there was no notable deterioration in the patch failure load of the latter. Thus nornal marine paint seems to adequately protect the patch and the bondline from the environmental effects. The area beneath the patch was inspected after testing. In all cases there was no trace of fibers on the steel surface and no visible damage to the patch. In the painted specimens there was no sign of corrosion underneath the patch area or on the steel. In Figure 6, the force—crosshead displasement curves are concentrated for all tested specimens. As the subsequent finite element analysis will show, the change of slope in the force—displacement curves is attributed to the yield of steel, which precedes patch debonding in all cases. The variation in the yielding load between the non-aged and the aged specimens is due to the different corrosion thickness reduction of the steel. The thickness reduction of the base metal during aging was not uniform and varied for each specimen, therefore the yield load varies too, since it depends on the cross sectional area of the metal. The deviation in the curves of the P2D1-A, P2D2-C and P2D2-D specimens is due to local yielding in the holes, in

Figure 5.

Specimen P2D2-C after testing.

Figure 6.

Force-displacement curves for all specimens.

the area inside the fixture that was used for fixing the specimens in the testing machine. Concerning the measurement from the strain gages, excellent repeatability of the strains was noted for the non-aged cases (see Fig. 7). Strains measured on the steel (i.e. SG-1 and SG-2) increased linearly until yielding. The increase of the strains measured in the SG-3 position ensured that there was a proper load transfer from the metal

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Figure 7. Representative force-strain graph of nonaged specimen. Figure 9.

The bilinear traction separation law.

Table 4. Material properties of epoxy resin for the cohesive law.

Mode I Mode II

Figure 8. Representative force-strain graph of specimen with all strain gages positioned on the patch.

substrate to the patch, but since it was in the center of the patch it provided no information concerning the initiation and propagation of debonding which started from the edges. In the aged cases there was a good repeatability of the measured strains. Additionaly the inflexion point in the strains in positions SG-1 and SG-2 (Fig. 8) is a sign of debonding propagation in the patch-steel interface under these sensors. The slightly increased inflection point of SG-3 curve indicates the difference between the time instance when debonding propagated under the edges of the full thickness patch and the time instance when debonding reached the centre of the patch. 5

NUMERICAL MODELING

A 2D finite element simulation of the specimens that had not undergone aging was developed using ANSYS 13.0 software. 2D high order 8-node elements were used to model the steel substrate and the patch, these elements referred to as PLANE183 by ANSYS. The length of the elements edges was

Maximum stress (MPa)

Critical fracture energy (N/mm)

7.3 53.0

0.14 0.28

1 mm. Only half of the specimen was modeled due to symmetry so as to reduce computational time. ANSYS includes also contact elements that can be used to simulate interface fracture using the cohesive zone model. These interface finite elements include a cohesive mixed-mode damage model based on the indirect use of Linear Fracture Mechanics. The damage model combines aspects of strength-based analysis and fracture mechanics and includes a linear softening process, which allows for a smoothly decrease of stresses as the relative displacements grow. The area under the softening curve is equal to the critical energy release rate. These finite contact elements allow the simulation of damage initiation and growth in the adhesive (Alfano & Crisfield 2001) and were used in the present model throughout the interface of the steel-CFRP. There exist several different traction separation (or cohesive) laws so as to simulate different material behaviour. The currently incorporated traction separation law is bilinear and the material behaviour is characterized by a linear elastic loading, followed by a linear softening (Fig. 9). The bilinear law is well suited for the modeling of resins and non-ductile adhesives (Tvergaard and Hutchinson, 1993). The material properties for the patch were those stated in Table 4. In the present model, both material and geometric nonlinearities were taken into account. It is of critical importance for the correct

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modeling to implement the proper corresponding material properties by performing experiments specifically designed for this purpose. The material properties required for the cohesive law are: the maximum normal stress, the maximum tangential stress (tractions), the critical fracture energy for normal separation (mode I) and the critical fracture energy for tangential slip (mode II). These properties were taken from the literature (Lee et al. 2010) and were implemented in the present model (Table 4). The developed model was able to realistically represent the initiation and propagation of damage (Fig. 10). Debonding started from the tapered edges of the patch and gradually progressed to the center of the patch. After a certain point, due to the excessive increase of the debonded area, the patch contributes neither in stiffness nor in the load bearing capacity of the specimen. The FE results showed that yield of the steel (grey area in Fig. 11) initiated in the areas outside the patch and

Figure 10.

Figure 11. debonding.

Progression of debonding.

preceded the initiation of debonding. The area of the steel that is bonded to the patch started yielding only after the patch has debonded. In Figure 12 the force—crosshead displacement of the non-aged specimens along with the results obtained by the numerical models are presented. The deviation in the slopes between the experimental and the numerical modeling is attributed to the fact that the displacement measured is actually a combination of the specimen’s and the testing machine’s stiffness. The patch failure load obtained from the FE analysis is 162.39 kN. The average experimental patch failure load for the two non-aged specimens P2D1-Α and P2D1-Β is equal to 169.51 kN. The deviation between the calculated failure load and the experimental load is only 4%. The strains calculated from the numerical model are also in perfect agreement with the experimental results (Fig. 13). Apart from the patched FE model, two additional models were developed, one for the nondefected (non-corroded) plate and one for the corroded but unpatched plate. This was done in

Figure 12. Comparison of experimental and numerical force—displacement curves.

Von Mises Stress distribution before

Figure 13. Comparison of experimental and numerical force—strain curves.

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order to evaluate the effect of the patch on the strength of the corroded plate, as well as to compare the strength of the rehabilitated plate to that of the initial, non-defected plate condition. In Figure 14 the corresponding force—displacement graphs are shown. The patched model exhibits a strength increase compared to the unpatched case, whereas it also reaches the strength of the initial non-defected plate. Concerning the yield loads

Figure 14. Force—displacement graphs for the nondefected, corroded unpatched and corroded patched cases.

Figure 15. Numerical model with 1 mm thickness reduction in the corroded area.

of the three models, as it was expected the lower yield load is presented in the unpatched, corroded model due to reduction of the cross sectional area and is equal to 150.0 kN. The yield load is the same for the initial plate and the patched case and is equal to 158.3 kN. This is due to the fact that, in the patched case, yielding occurs in the steel area that is beyond the patch and since the cross section in this area is the same as the initial non-defected plate, the yield load is the same. Having validated the numerical models, another case was considered for study. In this case the corrosion thickness was taken equal to 1 mm. This thickness reduction is about 20% of the initial steel thickness and is the maximum allowable thickness diminution according to IACS rules. Once again, the patched and the unpatched cases were considered. The material properties and geometry are the same as in the previous cases, apart from the thickness reduction in the corroded area. The corresponding FE model is depicted in Figure 15. The force—displacement curves for the patched and the unpatched cases for thickness reductions equal to 0.26 and 1 mm are presented in Figure 16. As it was expected, the unpatched case with 1 mm thickness reduction yielded in the lower load (125.0 kN). Once again the yield load of the patched cases and the non-defected plate was the same (158.3 kN), as yielding once again occurred in the steel areas beyond the patch for the patched cases. Another significant remark is that the corrosion thickness reduction of the patched plates has almost no effect, as the patch repaired cases exhibited the same stiffness and almost the same patch failure load, independently of the thickness reduction in the corroded area. Therefore, the effect of the repair is more significant in the case where thickness reduction is 1 mm. 6

Figure 16. Force displacement graphs for the nondefected steel plate, defected unpatched and composite patched cases for 0.26 and 1 mm thickness reduction.

CONCLUSIONS

The effectiveness of composite patch repairing of corroded steel members was studied both experimentally and numerically. To this end, corroded steel specimens that were repaired with a CFRP patch were tested in tension. Three different aging scenarios in cyclic corrosion conditions were taken into consideration, i.e. non-aged, aged unpainted and aged painted specimens. Results showed that the unpainted specimens yielded the lower failure loads due to the degradation of the mechanical properties of the patch and the partial ingression of moisture under the patch from the edges. The painted specimens showed the same failure load as the non-aged ones, a fact which simplifies the installation procedure of such repairs, as it appears that a typical marine paint is sufficient to protect

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the repair from the aggressive marine environment. Yielding of the steel was observed prior to patch debonding, a fact which was later on validated with the FE simulations. The numerical simulations encompassing interface elements with a bilinear mixed-mode cohesive law were proved capable of accurately predicting the initiation and propagation of debonding in the patch-steel interface, as well as the corresponding patch failure load. The numerical simulation results are in very good agreement with the experimental ones. The repaired case was also numerically compared to the non-defected and the corroded unrepaired steel case. Additionally, a second scenario with a corrosion thickness reduction of the steel plate equal to 20% was investigated numerically. Results showed that composite patch repairing was able to rehabilitate the defected steel plates and reinstate the original, non-defected condition in all cases, with the repair being more effective for the case where the thickness reduction of the steel was equal to 20%. ACKNOWLEDGEMENTS The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007–2013) under grant agreement n° 233969 (www.co-patch.com). The authors gratefully acknowledge the contribution of project partners AS2CON (A. Klanac and D. Bolf) and CARDAMA (J. Sanchez). REFERENCES Alfano, G. and M.A. Crisfield, 2001 Finite element interface models for the delamination analysis of laminated composites: Mechanical and computational issues. International Journal for Numerical Methods in Engineering, 50(7): p. 1701–1736. Allan, R.C., J. Bird, and J.D. Clarke., 1988 Use of adhesives in repair of cracks in ship structures. Materials Science and Technology, 4(10): p. 853–859.

Baker, A., 1999 Bonded composite repair of fatiguecracked primary aircraft structure. Composite Structures,. 47(1–4): p. 431–443. Grabovac, I., Bartolomeusz R.A., Baker A., 1993 Carbon fibre composite reinforcement of a ship structure— project overview. Composites. 24(6): p. 501–509. Grabovac I., 2003 Bonded composite solution to ship reinforcement. Composites: Part A: Applied Science and Manufacturing, 34, pp. 847–854. Grabovac I., Whittaker D., 2009 Application of bonded composites in the repair of the ships structures—A 15-year service experience. Composites: Part A: Applied Science and Manufacturing, 40, pp. 1381–1398. ISO 14993:2001 Corrosion of metals and alloys— Accelerated testing involving cyclic exposure to salt mist, “dry” and “wet” conditions. Karatzas V.A. and Tsouvalis N.G., February 2011 Composite Materials Characterization Tests, Composite patch repair for marine and civil engineering infrastructures applications; Project No:SPC8-GA2009-233969; Report NTUA-TR-WP3-3-v1. Klanac A., 2012, Private communication, Co-Patch meeting presentation, Genova. Lee M.J., Cho T.M., Kim W.S., Lee B.C., Lee J.J., 2010 Determination of cohesive parameters for a mixed mode cohesive zone model. International Journal of Adhesion & Adhesives 30322-328. McGeorge, D., Echtermeyer A.T., Leong K.H., Melve B. Robinson M., Fischer K.P., 2009 Repair of floating offshore units using bonded fibre composite materials. Composites Part A: applied science and manufacturing, 40(9): p. 1364–1380. Roach D.P., Graf D., 1996 Validation of Bonded Composite Doubler Technology Through Application Oriented Structural Testing, Proceedings of 11th DoD/NASA/ FAA Conference on Fibrous Composites in Structural Design. Turton T.J., Dalzel-Job J., Livingstone F., 2005 Oil Platforms, Destroyers and Frigates—Case Studies of QinetiQ’s Marine Composite Patch Repairs, Composites Part A, 36, 1066–1072. Tvergaard, V. and J.W. Hutchinson, 1993, The influence of plasticity on mixed mode interface toughness. Journal of the Mechanics and Physics of Solids, 41(6): p. 1119–1135.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

On the scope of using composites as major structural parts of large commercial ships K. Kunal & S. Surendran Indian Institute of Technology Madras, Department of Ocean Engineering, India

ABSTRACT: Use of Steel and Aluminium is very well accepted in marine industry. For metallic structures coatings are to be given to prevent corrosion due to salty atmosphere existing in tropical countries. Ship building industry has been concentrating on the application of composite materials as part of ship hulls to build high speed crafts and naval ships. Most of such vessels are made of composite laminates strengthened by stiffeners of various sectional and mechanical properties. The structural parts vulnerable to higher loads in sea operations are paid attention in this paper. In marine field uses of composite materials are increased drastically due to their superior properties such as ratio of strength to weight, improved corrosion, environmental resistance and life cycle cost. It is susceptible to damages under impact load. These damages cause considerable reduction in strength and stiffness. Therefore, the impact response of fiber reinforced laminated composites are to be analyzed. Usually ships are subjected to loads from wave breaking, bottom slamming, bow slamming and stern slamming. In the past there were accidents due to collapse of hatch cover because of wave breaking. In impact loads there are two types namely low velocity impact and high velocity impact. In marine context low velocity impact is more relevant. The deterioration of steel in saltwater is well known since decades. There are advantages if composites are used for ships as structural parts. The designer has to develop alternate composition of composite panel using chemicals addition. He/she should measure the impact strength of the above mentioned panel prepared using chemical additives. It is highly desirable to have a record of the composition of composite panel for most favourable operating conditions.The methodology covers, experimental technique to find impact strength, finite element analysis using the FEM packages ANSYS for static analysis and ABAQUS for impact load analysis. 1

INTRODUCTION

The structural damages of hatch cover may result in flooding of some compartments and the loss of an entire ship. Ersdal et al. (2000) have shown structural damage as a result of shipping of green water. Composites can be considered for superstructures, masts, hatch cover etc. of commercial ships during the design stage. To keep the vertical centre of gravity in favourable height it is advisable to use strengthened composite panels while building the ship. Composite can be designed with reinforcement of fibers in a mixture of resin and suitable filler. Strength to weight ratio for composite material is higher than conventional material steel. Composites are high corrosive resistant materials compared to steel and aluminium. Mouritz et al. (2001) reviewed advanced composite structure for naval ships. He also discussed the reasons for more uses of composite materials in marine industry. The advantages of use of fiber

reinforced composite panels subjected to low velocity impact is well established and proven in submarine technology. Chalmers (1994) discussed structural engineering application for marine hulls. The fiber reinforced marine composite plates subjected to low velocity impact can cause significant damage, such as matrix cracking, delamination and fiber breakage. These damages imply significant reduction in the strength and stiffness of materials. Theoretically, many works have been developed with an aim of studying the behaviour of composite plate under low velocity impact. Karakuzu et al. (2008) analysed numerically and experimentally the low velocity impact effects for few parameters and calculated contact force history. A few researchers used fillers to improve the impact strength of resin and composite panels. Fillers are the ingredients mixed with resin to enhance specific properties of composite materials. Magnitude of changed properties depends on the

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particle content, size distribution, bonding with the polymer matrix, and extent of dispersion. Effect of CaCO3 on wood based composite was investigated by Kord (2011). William et al. (2002) had studied the effect of calcium carbonate on the impact performance on rigid PVC (polyvinyl chloride) compounds. The author analyzed the effect of filler level and particle size of filler on impact strength of PVC compound. In the present analysis, effect of Calcium Carbonate (CaCO3), Silicon Carbide (SiC) and Alumina (Al2O3) fillers on impact strength of E-glass epoxy composite are employed. E-glass fiber is cost effective among all which can be used in marine environment to get required strength. Hatch cover of capsize bulk carrier is designed with cost effective composite materials. Hybrid composite is also used to design a composite hatch cover. Response of composite panels against low velocity impact is calculated. Parametric studies are done considering different cases such as plate dimensions, impactor density, impactor energy, impactor shape, impactor surface area, stiffener effect and effect of loading eccentricity for dynamic characterisation. 2 2.1

Experimental set up for impact test.

Figure 2.

Specimens after impact tests.

METHODOLOGIES Experiments for impact strength

In present work attempts are made to increase the impact strength of E-glass epoxy composite with suitable fillers, which can be used in marine field. Three different fillers Calcium Carbonate, Silicon Carbide and Alumina are considered with E-glass epoxy composite for experimental study. Un-notched specimens are prepared and Izod impact strength tests are done according to ASTM D256 standards. The dimensions of unidirectional E-glass epoxy specimens are 65 × 13 × 3 mm. Impact tests are done by pendulum type testing machine of Karl Frank Gmbh 53568 on unnotched specimens. The un-notched specimens are kept in cantilever position and pendulum having weight 453 g swings to break the specimens. Pendulum striking velocity is 3.5 m/s and fall angle is 124.34 degrees. Experimental set up and tested specimens are shown in Figures 1 and 2 respectively. 2.2

Figure 1.

Finite element analysis of hatch covers

Hatch covers are analyzed numerically with the help of well known FEM package ANSYS. CSR steel cover is modeled with elements Shell 181 and Beam 188. Plates are modeled with Shell 181, which are 3 dimensional 4 noded linear elements. Steel is

considered as an isotropic material with 210 GPa Young’s modulus and 0.3 Poisson’s ratio for present analysis. Dimensions of CSR cover are shown in Figure 3 with 12 mm thickness. Dimensions of stiffeners and girders are shown in Figures 4–6. Finite element model of steel hatch cover is shown in Figure 7. Structured meshing is done since there is global load acting on structure. The longitudinal stress was seen converged for a number of elements 60,000. Simply supported boundary conditions are applied for all hatch covers analysis and numerically it is shown in Equations 1 and 2. Static load of 2.58 tonnes/m2 is applied on the faces of hatch covers.

414

Figure 3.

Figure 4.

Plan view of CSR hatch cover.

Stiffener.

Uz = Ux = 0, θy = θz = 0 on X = 0, X = 15 m.

(1)

Uz = Uy = 0, θx = θz = 0 on X = 0, X = 10 m.

(2)

Effect of stiffeners is explained by Paik et al. (2001) on flexural rigidity and relation is also shown. The analytical method used by Paik et al. is used by authors to optimize the dimensions of composite stiffeners and girders. Composite is an orthotropic material and plate is modelled with lay-up process of lamina in ANSYS. Optimization of composite shell thickness and fiber angle is done in ANSYS. Hat type stiffeners and girders are used in composite hatch cover having highest first ply failure load and was analysed by Thinh et al. (2010). Girders position and dimensions are shown in Figure 8. Stiffeners dimensions and spacing are shown in Figure 9. Finite element model used to analyze for given load in ANSYS is shown in Figure 10. Fiber angle and shell thickness are the design variables. Deflection is considered as a state variable. Total weight of hatch cover is the objective in the present optimization process. The result of optimization process is shown in Table 1. Finally feasible fiber angles are arrived at for design process. Design of the composite structures is based on the DnV rules for High Speed, Light

Figure 5.

Central girder.

Figure 6.

Side girder.

Figure 7. cover.

Finite Element Model (FEM) of CSR hatch

Craft and Naval Surface Craft. According to DnV HSLC&NSC Pt. 3 Ch. 4 Sec. 6 A202, the maximum allowable deflection of laminate is W ≤ 2t, where t is the thickness and σd ≤ 0.3 σmax. Mechanical properties used for analysis is in Table 8. 2.3

Low velocity impact on composite panel

Low velocity impact load or dynamic phenomenon is solved with an explicit algorithm available

415

Table 1. cover.

Figure 8.

Figure 9.

Figure 10.

Side view of composite hatch cover.

Front view of composite hatch cover.

FEM of suggested composite hatch cover.

in ABAQUS. Contact force is the response force of a structure against impactor and proportional to stresses. It gives the time at which the stress will be the highest. Jackson et al. (1992) have described that impact force (contact force) can be used as a scalar parameter in the context of the impact response of composite laminates.

Fiber angle and thickness for composite hatch

Layer number

Actual fiber angle

Feasible fiber angle

Laminate thickness

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

316.5 309.8 343.43 53.6 64.32 6.1 289 278 278 289 6.1 64.32 53.6 343.4 309.8 316.5

−45 −45 0 45 60 0 60 90 90 60 0 60 45 0 −45 −45

1.136 0.57 0.6 0.729 0.635 0.783 4.5 3.5 3.5 4.5 0.783 0.635 0.729 0.6 0.57 1.136

Conventional shell element is considered for composite structure. For impactors of all types, solid element is considered. Four noded shell elements with six degree of freedom are considered for composite shell. Unstructured mesh pattern is used for finite element analysis for both master and target. Optimization of element size is also done for getting results. Impactor can be modeled as a deformable or rigid body. For our analysis it is considered as rigid body, since it affects or damages the composite structure more compared to deformable one. Prismatic impactor is modeled for numerical analysis. Irregular shape of water load or accidental load is to be modeled in prismatic shape, though strictly speaking it may not be same. Material properties of impactor except mass density and Young’s modulus are not assigned, since the main aim is to analyze the composite structure. In present analysis 100 × 100 mm E-glass epoxy with 8 mm thickness is considered and 10 kg mass of prismatic steel impactor is impacted with 4 m/s velocity. Modeled impactor and target are shown in Figure 11. Hard contact law is used to solve low velocity impact problem. When clearance between impactor and lamina becomes zero then contact constraint is applied. Contact pressure can be transferred between surfaces. When the surfaces will separate contact pressure will become zero, then the constraint is removed. The impactor is called master surface and plate nodes are referred as slave nodes. The contact force is applied to the slave nodes to oppose the penetration. Low velocity impact load or dynamic phenomenon is solved

416

Figure 12. Effect of filler types, particle size and chemical treatment on impact strength. Figure 11. Model with prismatic impactor and applied velocity in ABAQUS.

with an explicit algorithm available in ABAQUS. Mechanical properties of composite laminate are put in Table 8. 3

RESULTS AND DISCUSSION

3.1

Experimental study

Filler changes the flaw distribution of materials and hence strength. Filler also changes the brittleness of materials and the mechanical properties are also changed.The effect of fillers type, loading level, particle size and chemical treatment on impact strength of E-glass epoxy composite is shown in Figure 12. It can be observed from Figure 12 that maximum impact strength was observed at approximately 10% weight content for Al2O3 and SiC having impact energy values 17.1 and 15 joules respectively. Impact energy for 0.8 microns CaCO3 was maximum 22 Joules at 6% of weight loading. 3.2 3.2.1

Hatch cover analysis

Hatch cover as per Common Structural Rule (CSR) CSR suggested hatch cover is analysed for a DnV design load. Green water load is converted into equivalent uniform transverse load. According to DnV magnitude is 2.58 tonnes/m2. Von Mises stress contour is shown in Figure 13 and central node deflection is obtained as 9.63 mm. From responses it is clear that actual stresses are well below the permissible stresses. This is seen to be not so critical conditions.

Figure 13.

Contour for von Mises stress.

3.2.2 Suggested composite hatch cover In the present work the authors have suggested to design of composite hatch cover of capsize bulk carrier. Composite is made with E-glass fibers, epoxy resin, hardener and Calcium Carbonate fillers. It satisfies the DnV strength criteria under design load. Longitudinal stress contour is shown in Figure 14 and other responses are shown in Table 2. Weight comparison is done for CSR and composite hatch cover and is shown in Table 3. 3.2.3 Suggested hybrid composite hatch cover Hybrid composite is widely accepted in aerospace industry. It is the combination of two or more composites. It improves the strength to weight ratio further. In the present work composite hatch cover is also tried to replace with hybrid composite. E-glass epoxy and Aramid epoxy materials are considered while designing

417

hatch covers. Finally weight and cost comparisons are done between composite and hybrid hatch cover designed for capsize bulk carrier. In hybrid case, plate is designed with E-glass epoxy and stiffeners are made with Aramid epoxy. Cost comparison of materials is based on the price of SP Systems Product (Newport, England). E-glass epoxy 300 gsm rates are 2.191 EUR/m2 and Aramid epoxy rate is 9.438 EUR/m2. Thickness of shell is 26 mm and 23 mm in the case of glass epoxy and hybrid composite respectively. Dimensions of stiffeners and girders are optimized in MATLAB and results are tabulated in Tables 4. Stresses developed and deflections of hybrid composite hatch as per DnV design load are shown in Table 5. Weight comparison between composite and hybrid composite is done in Table 6. Table 7 shows the ratio of cost of materials only used for designing hatch cover in this paper. 3.3

Low velocity impact analysis

3.3.1

Effect of impactor density for same volume of impactor In this case, steel and aluminium impactors of prismatic shape with same dimensions are impacted with same velocity and height on same composite plate. Figure 15 indicates that for more mass density impactor causes higher contact force even if the volume is same.

3.3.2 Effect of laminate inplane dimensions A prismatic impactor with 108.2 mm dimensions and different plate dimensions are considered for analysing the effect of in plane dimensions on contact force. The effect of in-plane dimensions on contact force is analysed on 8mm thick plate with 0 degree fiber angle. Figure 16 depicts that maximum contact force decreases with an increase in the in-plane laminate dimensions but, central deflection increases with increase in in-plane dimensions of plate. Contact time is not affected significantly due to change in in-plane dimensions of composite plate. 3.3.3 Effect of impactor surface area Various approach on modeling green water mass for dynamic analysis is become a challenge to analysist. If it is of prismatic shape then the question is how much should be the surface area to be considered. Here surface area effects on contact force and central deflection are shown in Figure 17. If surface area of impactor is increased then contact force increases and deflection also increases. 3.3.4 Effect of stiffeners in plate As per Figure 18, it can be concluded that the contact force increses due to the attached stiffener in glass epoxy composite plate at the center. Central deflection will decrease in stiffened plate due to increase in stiffeness matrix. 3.3.5 Effect of loading eccentricity A prismatic impactor with low velocity 4 m/s is dropped at different locations on plate. From Figure 19, it can be obserbed that contact force is maximum when prismatic load is impacted at center of plate. There is no more change in contact duration due to loading eccentricity.

Table 3. Weight comparison of steel and composite hatch cover.

Figure 14. Longitudinal stress contour of composite cover under DnV design load.

Table 2.

Wt. of 1 hatch cover (tonnes)

CSR steel hatch

Composite hatch

58.484

19.9

Structural response of composite hatch cover.

Water pressure (t/m2)

δ

Compressive σx (MPa)

Tensile σx (MPa)

Compressive σy (MPa)

Τ (MPa)

2.58

55.3

42.5

128.2

11

7.2

418

Table 4.

Stiffeners (mm) designed with Aramid epoxy. Web*

Part of hatch cover Stiffener Central girder Side girder

Crown

Flange

Length*

Thickness

Length

Thickness

Length

Thickness

635 920 560

5 5 5

95 100 100

9 45 50

50 50 50

5 7 5

Table 5. Suggested hybrid composite hatch cover analysis under water pressure. Water pressure δ (t/m2)

Compressive Tensile Compressive σx σx σy τ

2.58

94

45

136.41

12.5

8.97

Table 6. Weight comparison between glass epoxy composites and hybrid composites.

Weight (tonnes)

E-glass epoxy hatch cover

Hybrid hatch cover

19.9

16.5

Table 7. Cost comparison between glass epoxy and hybrid composites hatch covers.

Cost ratio Table 8.

E-glass epoxy hatch cover

Hybrid hatch cover

1

2.84

Figure 16. Contact force history for various plate inplane dimensions.

Elastic property of composite materials.

Elastic properties

Magnitude

Longitudinal Young’s modulus Transverse Young’s modulus Shear modulus Poison’s ratio

38 GPa 9.8 GPa 3.6 GPa 0.3

Figure 17. Contact force history for different contact area of cubical impactor.

*Deflection and dimension of stiffeners are in mm and stresses are in MPa for all Tables.

Figure 15. Contact force history for different material impactor.

Figure 18.

419

Contact force history for stiffened laminate.

REFERENCES

Figure 19. Contact force history for different impact locations.

4

CONCLUSIONS

Feasibility of fabricating large sized structural component using composites is shown in this study. A steel hatch cover can be replaced with E-glass epoxy composite with CaCO3 filler at 8% loading level and 65.97% weight can be reduced. Hybrid composite material with E-glass epoxy and aramid epoxy can be also used for a hatch cover and 71.39% can be saved as compared to steel. Using E-glass epoxy composite cost can be reduced by 1.84 times the cost required using hybrid composite. There is an increment in contact force if surface area of prismatic impactor increases. Keeping the total energy of impactor constant, increase in mass causes decrease in contact force. If in-plane dimensions of laminate will be increased then contact force will decrease. Loading eccentricity also changes the contact force. Stiffeners increase the contact force and decrease deflection due to increment in stiffness matrix. ACKNOWLEDGEMENTS

Chalmers, D.W., 1994, The Potential for the Use of Composite Materials in Marine Structure. Marine structures 7: 441–456. Ersdal, G. and Kvitrud, A., 2000, Green Water on Norwegian Production Ships. International Offshore and Polar Engineering Conference. Faulkner, D., 2001, An Analytical Assessment of the Sinking of the M.V. Derbyshire, The Royal Institute of Naval Architects. International Association of Classification Society, 1997, Evaluation of Scantlings of Hatch Covers of Bulk Carrier Cargo Holds. IACS Requirement 1997, Vol. 1. Jackson, W.C. and Poe, C.C., 1992, The Use of Impact Force as a Scale Parameter for the Impact Response of Composite Laminates. NASA TM 104189:981-998. Karakuzu, R., Erbil E. and Aktas M., 2008, Damage Prediction in Composite Plates Subjected to Low Velocity Impact. Composite Structures 83: 73–82. Kord, B., 2011, Effect of Calcium Carbonate as a Mineral Filler on the Physical and Mechanical Properties of Wood Based Composites. World applied Sciences Journal 13: 129–132. Mouritz, A.P., Gellert, E., Burchill, P. and Challis K., 2001, Review of Advanced Composite Structures for Naval Ships and Submarines. Composite Structures 53: 21–41. Paik, J.K., Thayamballi, A.K. and Kim, B.J, 2001, Large Deflection Orthotropic Plate Approach to Develop Ultimate Strength Formulations for Stiffened Panels Under Combined Biaxial Compression/Tension and Lateral Pressure. Thin-walled Structures 39: 215–246. Thinh, T.I. and Quoc, T.H., 2010, Finite Element Modelling and Experimental Study on Bending and Vibration of Laminated Stiffened Glass Fiber/Polyester Composite Plates. Computational Materials Science 49: 83–89. William, S., Brayent and Wiebking, H., 2002, The Effect of Calcium Carbonate Size and Loading Level on the Impact Performance on Rigid PVC Compounds Containing Varying Amounts of Acrylic Impact Modifier. ANTEC Conference Proceedings, Vol-3.

The authors are thankful of Prof. J.S. Mani, Head of the Department of Ocean Engineering, IIT Madras, for encouragement.

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Weld simulation

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Simulation of the weld overlay procedure for corrosion repair of pressure vessels L. Gannon Defence R&D Canada, Atlantic, Canada

ABSTRACT: The use of pressure vessels in a salt water environment may result in damage due to corrosion, which in severe cases can cause a reduction in the collapse pressure and limit operational capabilities. The weld overlay procedure may be used to restore material damaged by corrosion, where the damaged material is removed by grinding and replaced by multiple passes of weld metal. As a first step in understanding the potential benefits of weld overlay repair of pressure vessels, finite element analysis is used to simulate the weld overlay procedure and the resulting residual stresses and distortions are compared with experimental measurements. Because welding simulations are complex and computationally expensive, different modeling approaches are investigated in order to simplify the welding simulation. The study demonstrates that a general purpose finite element software can be used to accurately predict weldinginduced residual stress and distortion using relatively modest computational resources. Results indicate that the mesh density in the vicinity of the weld and modeling of weld metal protruding above the surface of the plate have the greatest influence on weld overlay simulation accuracy. 1 1.1

INTRODUCTION Background

Pressure vessels operated in a marine environment typically consist of cylindrical and conical compartments made from high strength steel and may be internally or externally stiffened by ring stiffeners. Over time, the exterior face of the shell plating may suffer damage due to corrosion that occurs during operation in a salt water environment. Although corrosion damage can be mitigated by protective coatings or cathodic protection systems, these may not be fully effective and can be damaged while in service. The loss of shell plating material due to corrosion of the external surface can reduce the capacity of a pressure vessel to resist hydrostatic pressure and thus the maximum depth at which it can operate. The load-bearing capacity of the structure is reduced not only because there is less material available to resist hydrostatic pressure during submerged operation, but also because the loss of material on one side of the shell plating produces an eccentricity in the internal resisting forces, inducing undesirable bending stresses in some cases. An experimental study of the effects of corrosion on the strength of internally stiffened ring stiffened cylinders by MacKay (2010) found that for approximately 25% thinning of the cylinder plating at the outer surface, a serviceability limit state defined by local instability in the corroded

region reduced the strength by 27% whereas the ultimate limit state (interframe collapse) pressure was reduced by approximately 9% compared to the undamaged cylinder. For those test specimens that failed by overall collapse, the reduction in collapse pressure due to corrosion of the shell plating was on average, equal to the percent depth of plate thinning. In order to mitigate the potential reduction in collapse pressure due to corrosion, remedial measures may be taken, including: − Removal and replacement of damaged plating which may be expensive and time-consuming. Also, the residual stresses and distortions introduced by cutting and welding operations could be more harmful to the strength of the structure than the damage due to corrosion. − Removal of damaged material by grinding with no replacement of lost material. − Removal of damaged material by grinding followed by replacement of the material with multiple layers of weld metal. This procedure is often referred to as weld overlay or weld buttering. Again, the lost material is reclaimed, but the residual stresses and distortions induced by welding may lead to a greater loss in structural capacity than the original damage due to corrosion, depending on the amount of overlay. In relation to the last point above, preliminary experimental results have shown that weld

423

overlay repair allowed a pressure vessel to regain 80% of the strength lost due to corrosion damage. Although this is an important result, further study is required in order to determine the efficacy of the weld overlay process for a range of welding parameters and damage extents. This would be prohibitively expensive and time consuming. An attractive alternative to experimental methods is numerical modeling. In particular, Finite Element Analysis (FEA) may be used to simulate the weld overlay process and to predict the residual stresses and distortion that result. After verification by comparison with experimental results, welding simulations can be used to evaluate different welding procedures with the goal of minimizing adverse effects that welding-induced imperfections might have on the performance of pressure vessels repaired by weld overlay. Furthermore, these simulations can be used to provide guidance of the efficacy of the repair under various conditions. 1.2

Welding simulation

Welding simulations vary in their complexity from relatively simple 2D models (Ueda & Yamakawa, 1971), typically based on plane strain conditions to full three-dimensional models using all solid elements (Gannon et al., 2012) or a combination of solid and shell elements. Physics-based models of the welding process might consider an assortment of physical processes including the temperaturedependent microstructural properties, phase changes of the materials, convection within the weld pool, and the dependence of mechanical properties on microstructure. Although an ideal model would consider all of the potential physical processes and interactions between the fields of mechanics, microstructure, heat flow and fluid dynamics, each additional level of complexity increases the time required to run a simulation. Fortunately, the couplings between physical processes are not all strong enough to necessitate rigorous modelling and so they can either be neglected, or simplifying assumptions may be made to incorporate their effects in a simulation with a lesser degree of coupling between fields. The focus of welding simulation in this study is determination of the transient temperature field and the associated thermal strains. With this in mind, the following simplifying assumptions are made: − Microstructural changes are accounted for by specifying experimentally measured temperaturedependent material properties. − The effect of stress state on phase change is not included, as it only needs to be considered in a detailed analysis of weld pool dynamics.

− The heat generated by deformation is negligible in comparison to the heat generated by the welding torch and so may be omitted. Only a brief description of the aspects of welding simulation that are salient to the study of weld overlay are discussed herein. A comprehensive review of the development of finite element welding simulation is provided by Lindgren (2001). 1.3 Objectives The primary objective of this study is to simulate the weld overlay process by FEA. The simulation results are compared with experimental measurements (Bayley and Goldak, 2012) to verify the accuracy of the welding simulation method. Although Bayley and Goldak (2012) also performed a numerical analysis of the weld overlay procedure, this was done using specialized welding software. The current work is unique in that it demonstrates a method for simulation of weld overlay using a general purpose FEA package and should therefore be of interest to a wider audience that does not have access to specialized welding simulation tools. Also, guidance is given regarding the fidelity required of the welding simulation in order to obtain accurate results in a reasonable amount of time using moderate computational resources. Thus, the second objective of the project is to simplify the welding simulation as much as possible in order to reduce the analysis time while still achieving reasonably accurate results. This will be valuable in reducing the time required for further study of the various parameters that affect welding-induced imperfections. It will also help to reduce solution time in models where the structure of the surrounding pressure vessel is included so as to be able to study the effects of weld overlay on the strength of pressure vessels. 2

WELD OVERLAY EXPERIMENT

2.1 Geometry In the experiment described in detail by Bayley and Goldak (2012), the weld overlay procedure was applied to a high strength steel plate specimen with nominal yield strength of 550 MPa. The plate specimen was cut to 427 mm × 427 mm square. A 100 mm × 100 mm square groove, 3 mm deep was machined out of the centre of the plate and it was then welded to a frame made of hollow structural sections made from 300 W steel as shown in Figure 1. The HSS frame was intended to represent approximately the restraint that would be provided by the surrounding pressure vessel shell plating.

424

3

Figure 1. Plate with simulated corrosion (Bayley and Goldak, 2012).

The weld overlay simulation consists of sequentially coupled transient thermal and nonlinear structural finite element analyses. The complete analysis is performed using ANSYS® finite element analysis software. The first stage is a thermal analysis where the heat from the welding torch is represented by a moving heat source and the transient temperature distribution in the welded plate is calculated. The temperatures from the thermal analysis are then applied to the same finite element mesh, but with thermal element types changed to structural elements for a nonlinear structural analysis wherein the residual stress and distortion due to welding are determined. For most simulations, weld elements are activated progressively, simulating the deposition of weld metal as each weld pass is laid. 3.1

Figure 2.

2.2

Welding sequence.

Welding

Before welding, the plate was instrumented with a number of strain gauges and thermocouples that were used to record strain and temperature measurements during the overlay process. The plate was preheated to 120 °C by wrapping it in insulating blankets and induction heating cables. After the minimum preheat temperature was achieved, welding began along the middle axis and progressed towards the sides of the groove, alternating about the first weld pass. A total of 27 weld passes were required to fill the groove in two layers using a flux-cored arc welding method with an average welding speed of 3.3 mm/s. After each weld pass was complete, the plate, still wrapped in the insulating blankets, was allowed to cool until the temperature of the surface of the next weld pass reached the maximum interpass temperature of 150 °C. The average cooling time between weld passes was 306 s. For all weld passes, the current was set at 150 Amps and the voltage ranged from 22.5 V to 23.5 V. The sequence in which the weld passes were deposited is illustrated in Figure 2. Following the weld overlay, distortions were measured using a coordinate measurement machine and residual stresses were measured by neutron diffraction. These measurements are compared with the results of finite element welding simulations in the following sections.

WELDING SIMULATION

Geometry and mesh

Since a primary objective of this study is to develop a method of simulating weld overlay that takes the least amount of time while still achieving reasonably accurate results, several finite element meshes were used. These are associated with three models. The first (model A) is a high fidelity model where the weld overlay geometry was meshed in greatest detail as shown in Figure 3. Model B is less detailed than model A and does not include weld elements protruding above the top surface of the surrounding plate. The weld metal in model B is modelled more simply, using two layers of hexahedral elements as shown in cross-section in Figure 4. In order to determine whether modeling the protruding weld elements would affect the simulation results, a third model (model C) was created that is the

Figure 3.

425

High fidelity finite element model (model A).

Figure 4.

Table 1.

Summary of meshes and weld patterns.

Model

A (b)

B (b)

Weld pattern*

Ambient temp. (°C)

A B1 B2 B3 B4 C1 C2

4 (1) 4 (1) 4 (1) 6 (1) 8 (1.5) 8 (1.5)

12 (6) 12 (6) 12 (8) 12 (8) 12 (8) 12 (8)

Incremental Incremental Incremental Incremental Incremental Incremental Pass

130 20 130 130 130 130 130

(b): element size bias through thickness, smaller elements at top face of plate; *incremental: weld passes activated in increments of 8.3 mm per load step; pass: complete weld pass activated at once, 100 mm per load step.

Typical model B cross-section.

3.3

Figure 5. Model C mesh showing protruding weld elements.

The first stage of the weld overlay simulation is a transient thermal analysis where the transient temperature distribution resulting from a moving heat source representing the welding torch is calculated. The majority of the elements used in the models were linearly interpolated, isoparametric hexahedrons with one degree of freedom at each node. A convection and radiation boundary condition was applied to all exterior surfaces of the models by specifying a convection film coefficient value that accounts for heat loss by both mechanisms. The film coefficient is given by Goldak et al. (1984) as: H

same as model B except that the weld elements were allowed to protrude 1.5 mm above the top surface of the plate as shown in Figure 5. For all of the models, the plate is meshed with hexahedral elements with limited triangular prisms where necessary. The HSS frame surrounding the plate was meshed with shell elements. For models B and C, the mesh density was varied to evaluate what influence these changes have on both solution accuracy and run time. With reference to Figure 5, Table 1 summarizes the mesh configurations used in each model. 3.2

Material properties

Welding raises the temperature of the base metal enough to cause significant changes in both the thermal and mechanical properties of the material. Measured temperature-dependent properties of a high strength steel similar in composition to HY-80, which is commonly used for pressure vessel construction were used for all models. The nominal yield strength and elastic modulus are 550 MPa and 200 GPa, respectively.

Thermal analysis

24.1 10 4 εT 1.61

(1)

where ε is the emissivity which is taken as 0.9 for hot-rolled steel (Goldak et al., 1984), and T is the surface temperature in Kelvin. For models B and C, the heat input was divided into two parts: one representing the heat flux incident on the surface of the plate from the hot plasma surrounding the welding torch; and the other representing the heat transferred to the plate from the molten weld metal. The heat flux on the surface of the plate accounts for 40% of the total heat input, while a volumetric heat generation within the weld elements accounts for the other 60% (Allum and Quintino, 1985). For the high fidelity model, only the volumetric heat generation was specified. The total heat input for each weld pass is given by: Q

VI V

(2)

where η is the welding process efficiency, which is assumed to be 0.8 (Deng et al., 2007); V, is the voltage and I, is the current supplied by the welding machine. These parameters were recorded during the weld overlay experiment giving an average

426

heat input of 2786 W accounting for the process efficiency. The process efficiency and division of the heat input into a surface heat flux and volumetric heat generation rate are based on a gas metal arc welding process whereas the experiment simulated here used a flux-cored arc welding process. Therefore, it is possible that more suitable values for these parameters could have been used if they had been measured during the experiment. The weld elements were initially de-activated using the ANSYS element birth and death feature. User-defined sub-routines were created using the ANSYS® parametric design language to model the moving heat source and to control the activation of weld elements as the heat source progressed. The volumetric heat generation was applied evenly to the weld elements directly below the position of the surface heat flux as the heat source incrementally progressed along the weld path. The surface heat flux was assumed to have a Gaussian power distribution acting over a circular area as shown in Figure 6. The surface heat flux was discretized and applied to the model as a nodal heat flux on the surface weld elements according to the element shape functions. The distribution of power Q, over the surface is defined relative to a coordinate system moving with the heat source of radius c, (Fig. 6) and is given by: Q(ξ ,γ ) =

3Q −3ξ 2/ c2 e e π c2

γ 2/c 2

(3)

Models C1 and C2 have the same geometry and mesh, however the welding simulation was altered. For C2, all of the elements in each of the weld passes were activated at once so that the surface heat flux and volumetric heat generation were applied to an entire weld pass for 2.43 s. This corresponds to the time for which each element along the length of each pass would be loaded if the

Figure 6.

Gaussian distribution of surface heat flux.

deposition of each pass had been modeled in full (incrementally), as is the case for model A, model C1 and all series-B models. 3.4

Structural analysis

For all three models, the same mesh was used in the structural analysis as in the thermal analysis. Only the element types and material properties were changed. Uniform reduced integration was used for both the solid and shell elements in the structural analysis in order to reduce analysis time and to avoid the possibility of overly stiff bending behaviour that can occur in fully integrated, linearly interpolated elements. Displacement boundary conditions were applied on three of the four corners of the HSS frame (Fig. 7), allowing thermal strain to occur freely along all three coordinate axes while preventing rigid body motion. The temperature time history from the thermal analysis was used to define a series of load steps in the structural analysis, where each increment in the position of the heat source made up one load step. All of the weld elements were deactivated in the first load step by multiplying their stiffness by a severe reduction factor. This was implemented using the ANSYS® element birth and death feature. As the weld progressed, a user-defined subroutine was used to apply loads from the thermal analysis at the corresponding time and to calculate the average temperatures of the weld elements based on the nodal temperatures. Those elements whose average temperature had fallen below the solidification temperature (1500 °C) were activated. The average nodal temperature of each element was applied to that element as a uniform body force in order to resolve an inconsistency between the linearly varying strains caused by direct application of the temperature field and the constant strains allowed for by the linear shape functions of the structural elements. The application of constant

Figure 7.

427

Structural boundary conditions.

temperatures to each element also prevented excessive element distortion in areas with a high thermal gradient, enhancing solution convergence. 4 4.1

RESULTS

not fall as quickly as the measured temperature. The ambient temperature that gave the best results was 130 °C and was used for all further weld overlay simulations. 4.2 Structural analysis

Thermal analysis

Figure 8 shows a comparison of results from the thermal analysis stage of the welding simulation against temperatures measured by a thermocouple located on the bottom (non-welded) surface of the plate at the mid-length of the weld beads and 20 mm off-centre in the direction transverse to the weld path. When the ambient temperature used in calculating heat loss due to convection and radiation was taken as 20 °C, the temperature at the thermocouple location in the finite element model was consistently lower than the experimentally measured value (Fig. 8). This difference is attributed to the use of insulating blankets around the plate in the experiment that retained an unknown amount of heat that would otherwise have been lost due to convection and radiation. In order to correct the model boundary conditions for the unknown insulating characteristics of the blankets, it was assumed that the insulating blankets trapped hot air around the plate, reducing the potential for convective heat loss and increasing the effective ambient temperature around the plate. The ambient temperature in the FE model was adjusted by trial and error until the temperature time history in the FE model at the location of thermocouple matched the measured temperatures with an acceptable degree of accuracy. There was still some inaccuracy in the temperature time history from the model because in the model, the ambient temperature was constant, whereas in the experiment, the ambient temperature would have fallen during the cooling period between weld passes. This difference is discernible in Figure 8, where the model temperature after the peaks does

4.2.1 Distortion Figure 9 shows the measured out-of-plane distortion at the bottom surface of the plate (0 mm at the corners and 1 mm at the centre). The effect of specimen tilt was removed from both the measurements and the finite element model results using a planar regression to find the best fit plane to the measured coordinates (displaced nodal coordinates in the finite element models) at the bottom face of the plate. The plane was then translated along the z-axis (Fig. 7) so that the minimum displacement value is zero. Table 2 lists the maximum distortion (near the bottom centre of the plate) calculated by the various welding simulations. Although the distortion predicted by model A was closest to the measured

Figure 9. Measured out-of-plane displacement (mm, Bayley and Goldak, 2012). Table 2.

Figure 8. Measured and calculated temperature time histories.

Distortion predicted by welding simulation.

Model

Plate distortion at bottom centre (mm)

Run time*

A B1 B2 B3 B4 C1 C2

0.94 0.78 0.72 0.79 0.80 0.91 0.75

7.0 weeks 14.6 hrs 14.0 hrs 25.5 hrs 23.1 hrs 25.7 hrs 21.8 hrs

*Using 2 cores on a desktop PC with a 3.07 GHz Intel i7 processor and 12 Gb of RAM.

428

value (1 mm), the level of detail in that model led to an excessive run time of seven weeks. Distortions predicted by models B1 though B4 were between 20% and 30% less than the measured value. The difference in distortion between models B1 and B2 (−8%) shows that the higher ambient temperature associated with the use of insulating blankets in the experiment decreased distortion. Increasing the element size bias through the thickness of the plate from model B2 to B3 had a significant influence on results. The distortion in model B3 was 9% higher than in model B2, illustrating the importance of a dense mesh near the weld where there is a high thermal gradient. Increasing the mesh density in the far-field on the other hand, had little effect on the predicted distortion as evidenced by the small difference between the distortions of model B3 and B4. Other than an increase in mesh density in the far-field region of the plate, which does not have a significant effect on distortion, the mesh density in model C1 is the same as in models B3 and B4. The primary difference between the B and C-series models is that the C-series of models have a layer of weld metal protruding above the top surface of the plate by 1.5 mm. This protruding layer of elements increases the distance between the force resulting from weld metal shrinkage and the neutral axis of the model. As indicated by the 14% increase in distortion from model B4 to C1, including the protruding layer of weld elements in the model results in a significant improvement in the accuracy of distortion calculated by the welding simulation. Aside from model A, model C1 gave the most accurate maximum distortion, within 9% of the measured value. This is consistent with numerical simulations performed by Bayley and Goldak (2012) who predicted peak displacements within 15% of measured values. The change in welding simulation method from model C1 to C2 caused a marked decrease in solution accuracy. Loading and activating each complete weld pass sequentially instead of simulating their incremental deposition element by element, reduced the calculated distortion by around 18% while the solution time was reduced by 15%. It should be noted however, that the time for solution of models C2 and C1 could be improved with little loss of accuracy if the mesh density in the far field is reduced to be the same as model B3. 4.2.2 Residual stress Residual stress through the thickness of the plate, calculated by FEA is compared against measurements at the two locations shown in Figure 10, where point A is located at the center of the weld overlay area. Model A has the highest level of detail and is likely to give the most accurate results,

Figure 10. Residual stress data sampling locations (mm).

Figure 11.

Model A residual stress.

therefore residual stresses predicted by the welding simulation are shown for model A first in Figure 11. Stress directions are consistent with the coordinate axes shown in Figure. Residual stresses calculated using model C1 are compared with measurements next (Fig. 12), as this model gave the most accurate maximum distortion for a more reasonable solution time than model A. When comparing the welding residual stress calculated by model A against measurements, there was some discrepancy between the values in all three directions. This is attributed in part to

429

The residual stresses calculated using model C1 also agreed reasonably well with measured values once hot-rolling stresses were included as shown in Figure 11. This slightly less accurate solution is attributed to the lower mesh density used in model C1 and also to the fact that uniform reduced integration reduced the integration point density to just one per element. This could be improved by using a different element formulation that does not reduce integration point density and is left for future consideration. 5

Figure 12.

Model C1 residual stress.

residual stress-inducing fabrication steps that were not included in the welding simulation including hot-rolling and welding of the HSS frame to the plate specimen. It is also possible that some of the simplifying assumptions and values assumed for the welding process such as efficiency and emissivity may have differed from the test conditions. Although it is beyond the scope of the present project to model the welding of the HSS sections, the hot rolling stresses in the plate were calculated by finite element modeling in order to determine if the superposition of these stresses along with welding residual stresses would improve solution accuracy. To this end, a finite element model of the plate alone was assigned an initial temperature of 1500 °C (melting temperature) and allowed to cool to an ambient temperature of 20 °C, producing residual stresses due to hot-rolling consistent with measured values in hot-rolled plates (Alpsten, 1968). These were added to the welding residual stress from Model A as shown in Figure 11. Including the hotrolling residual stress improved the solution accuracy at both points A and B although point B did not agree as well with measured values as point A. This is attributed to the unmodelled residual stress due to welding of the HSS sections to the plate that would be more pronounced at point B than point A because it is closer to the plate edge.

CONCLUSIONS

The weld overlay procedure was simulated by FEA, giving the welding-induced residual stress and distortion. Results of the welding simulations were compared against measured values and good agreement was observed between model A and model C1. Model C1 is preferred because the geometry of the model is simpler and a solution can be arrived at in approximately one day compared to seven weeks for model A. The calculated maximum distortion with model C was 91% of the measured value. By using various modeling techniques and mesh configurations, it was found that the mesh density in the vicinity of the weld and inclusion of weld material protruding above the plate surface have the greatest influence on solution accuracy. Residual stresses predicted by the welding simulation using Model C1 were in reasonable agreement with measured values. The agreement between computed and measured values improved when residual stresses due to hot-rolling were superimposed on the welding residual stress. This shows that although residual stress due to welding is predominant, those induced by preceding fabrication processes can influence the final state of residual stress by a notable amount. REFERENCES Allum, C. & Quintino, L. 1985. Control of fusion characteristics in pulsed current MIG welding, Part 2—simple model of fusion characteristics. Metal Construction, 17(5): 314r–317r. Alpsten, G.A. 1968. Residual stresses in thick welded plates: Thermal residual stresses in hot-rolled steel members. Fritz Engineering Laboratory Report No. 337.3. Bayley, C. & Goldak, J. 2012. Welding induced distortions and strains of a built-up panel, experiment and numerical validation. ASME Journal of Pressure Vessel Technology, 134. Deng, D., Liang, W. & Murakawa, H. 2007. Determination of welding deformation in fillet-welded joint by

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means of numerical simulation and comparison with experimental results. Journal of Materials processing Technology, 183: 219–225. Gannon, L., Liu, Y., Pegg, N. & Smith, M.J. 2012. Effect of welding-induced residual stress and distortion on ship hull girder ultimate strength. Marine Structures, 28(1): 25–49. Goldak, J., Chakravarti, A., & Bibby, M. 1984. A new finite element model for welding heat sources. Metallurgical Transactions B, 15B: 229–305. Lindgren, L.E. 2001. Finite element modeling and simulation of welding part 1: Increased complexity. Journal of Thermal Stresses, 24(2): 141–192. Lindgren, L.E. 2001. Finite element modeling and simulation of welding part 2: Improved material modeling. Journal of Thermal Stresses, 24(33): 195–231.

Lindgren, L.E. 2001. Finite element modeling and simulation of welding part 3: Efficiency and integration. Journal of Thermal Stresses, 24(44): 305–335. MacKay, J.R., Smith, M.J., van Keulen, F., Bosman, T.N. & Pegg, N.G. 2010. Experimental investigation of the strength and stability of submarine pressure hulls with and without artificial corrosion damage. Marine Structures, 23: 339–359. Ueda, Y. & Yamakawa, T. 1971. Analysis of thermal elastic-plastic stress and strain during welding by finite element method. Japan Welding Society Transactions, 2(2): 90–100.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

A study on computational fluid dynamics simulation of friction stir welding Sung Wook Kang & Beom Seon Jang RIMSE, Department of Naval Architecture and Ocean Engineering, Seoul National University, Seoul, Korea

ABSTRACT: Friction stir welding is a new technology which makes it possible to join two materials at a solid—state. The welding is performed by a non-consumption tool which rotates at a high speed to generate the frictional heat. The tool consists of a shoulder and a pin, and it is inserted into the work pieces and moves at a constant speed along the welding line. By this method, it is able to weld some hardto-weld materials, such as aluminum alloy, magnesium alloy and dissimilar materials. Nowadays friction stir welding is mainly used for vehicles like vessel, aircraft, railcar and automobile. In this paper, a three dimensional numerical model is constructed for a simulation of the friction stir welding process using a commercial computational fluid dynamics code “Fluent”. Through this simulation, asymmetric temperature distribution of advancing side and retreating side is calculated and compared with experimental results. A case study is performed to investigate an effective diameter of rotation affected cell zone in the CFD simulation. 1

INTRODUCTION

The Friction Stir Welding (FSW), which is solid state welding is mainly used for welding of lightweight metals including aluminum alloy. The friction stir welding is applied to various structures due to many advantages like high quality, no need for protective gas and little welding deformation and residual stress. FSW was invented at The Welding Institute UK in December 1991. Many researchers have been made on the heat transfer modeling of the friction stir welding over 10 years. Chao et al. (1998) presented a 3-dimensional heat source model of parent metal based on an assumption that a constant heat flux occurs from tool shoulder, and the suggested model was adjusted by a comparison with a measured temperature distribution. Frigaard et al. (2001) developed process model on the parent metal based on finite difference method which has moving heat source. Bendzsak et al. (2000) and Smith et al. (2000) conducted modeling of heat transfer through finite difference method, assuming the parent metal welded by friction stir welding as non-Newtonian fluid materials. Gould et al. (1998) used method of Rosenthal developing analysis model on heat transfer of the parent metal in friction stir welding. Khandkar et al. (2001) modeled heat transfer of welding model on lap welding using friction stir welding, which has moving heat source occurring from tool shoulder. Song et al. (2004) presented more precise 3-dimensional heat

transfer model of tool and work piece based on finite difference method for a simulation of shortterm temperature. Here, the heat input from tool pin was modeled as moving heat source. Colegrove et al. (2000) proposed an improved analysis method on heat generation toward tool with threaded tool pin. A theory is describing how the material flows around the pin. Schmidt et al. (2004) established an analytical model for heat generation by friction stir welding, based on different assumptions of the contact condition between the rotating tool surface and the weld piece. Also researchers have been made on a simulation using computational fluid dynamics. This simulation can provide asymmetric temperature distributions between advancing side and retrieving side. Ulysse (2002) calculates temperature distributions considering tool speed using a three-dimensional visco-plastic model. Fourment et al. (2008) performed a heat transfer analysis using FORGE3 FE software. Comparison of the temperature distribution with threaded tool pin and unthreaded tool pin. Aljoaba et al. (2009) performed a heat transfer analysis using STAR CCM+. Some case studies for different welding speeds and rotational speeds are carried out for temperature, strain rate, flow stress and material velocity. Kim et al. (2010) performed a heat transfer analysis using Fluent. Temperature dependent materials properties are considered by user define function. It investigated the effects of tilted tool and backing plate on the backing plate.

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For a correct assessment of welding deformation and welding residual stress using thermal elastoplastic analysis, it is necessary to predict heat distribution on the cross section as exactly as possible. The heat distribution should be obtained from a heat transfer analysis based on a proper modeling of heat source to be imposed on the surface of welding joint. As a first step toward the development of heat source model, this paper provides a study on 3D CFD simulation which can offer similar heat distribution to that from an experiment. Especially, the simulation can realize asymmetric temperature distribution similarly to experiment results. A parametric study on diameter of rotation affected zone is carried out to find out the best value. This study is expected to be utilized for a development a new heat source model to generate asymmetric temperature distribution. 2

3

In this study, two friction stir welding experiments are referred (Bang et al. 2008 and Kim et al. 2010). Two aluminum materials are used in the experiments: AL6061-T6 and AA5058-H18. The heat distributions on the surface measured in the experiments are used to verify the validity of the proposed CFD simulation. The chemical composition of AL6061-T6 and AA5058-H18 are summarized in Table 1 and Table 2, respectively. Plate size, tool dimensions and welding parameters of the FSW experiment for AL6061-T6 and AA5058-H18 are listed in Table 3 and Table 4, respectively. 4

FRICTION STIR WELDING

The principle of friction stir welding is to probe into jointing part of the plate by using pin on the tip of rotating tool, which is referred to as stirring or mixing process. The rotating tool generates friction heat by rotating at constant speed when pressing plate with constant load. The friction stir welding is made by the heat generated in the stirring process, and the heat sources are divided into two heat sources: one from the friction between tool shoulder and the base plate, and the other from plastic deformation of solid state material around the rotating pin. In this study for CFD simulation, a full sticking condition at the interface between the tool and the material is adopted so that heat was generated only by plastic deformation without heat generation by friction. Figure 1 is schematic of the friction stir welding process.

FRICTION STIR WELDING EXPERIMENT DETAILS

4.1

THERMO-MECHANICAL ALALYSIS Heat source theory

FSW 2D CFD simulation was conducted by Seidel et al. (2001) and FSW 3D CFD simulation by KIM et al. (2010) and Khandkar (2005). FSW 3D simulation can take into account tool shape and a tilt of the tool. In addition, more detailed boundary conditions can be reflected into the simulation. Table 1. The chemical composition of AL6061-T6. Component

Percentage (%)

Aluminum, Al Chromium, Cr Copper, Cu Iron, Fe Magnesium, Mg Silicon, Si Zinc, Zn

Remainder 0.04–0.35 0.15–0.40 Max 0.7 0.8–1.2 Max 0.4–0.8 Max 0.25

Table 2. The chemical composition of AA5083-H13.

Figure 1. Schematic of the friction stir welding process.

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Component

Percentage (%)

Aluminum, Al Chromium, Cr Copper, Cu Iron, Fe Magnesium, Mg Manganese, Mn Silicon, Si Titanium, Ti Zinc, Zn

Remainder 0.05–0.25 ⱕ0.1 ⱕ0.4 4.0–4.9 0.4–1.0 ⱕ0.4 ⱕ0.15 ⱕ0.25

Table 3.

Plate, tool dimension and welding parameters of FSW for AL6061-T6.

Al6061-T6

W (mm)

L (mm)

t (mm)

Plate size

140

100

12

Tool size

Welding parameters

Shoulder diameter (mm)

Pin diameter (mm)

Pin length (mm)

Tool tilt (°)

Rotation speed (RPM)

Welding speed (mm/min)

32

10

9

0

600

50

Table 4.

Plate, tool dimension and welding parameters of FSW for AA5083-H13.

AA5083-H18

W (mm)

L (mm)

t (mm)

Plate size

200

600

1.64

Tool size

Welding parameters

Shoulder diameter (mm)

Pin diameter (mm)

Pin length (mm)

Tool tilt (°)

Rotation speed (RPM)

Welding speed (mm/min)

10

4

1.34

3

1000

100

In this CFD simulation, aluminum material is assumed incompressible non-Newtonian fluid. The conservation of mass (continuity), conservation of momentum (Navier-Stokes) and conservation of energy (first law of thermodynamic) are governing equations. The temperature field is calculated by solving the equations considering temperature dependent material properties and conduction/ convection heat transfer boundary conditions. Since sticking condition is applied between the plate and tool, heat generation is induced only by plastic deformation, not by friction between plate and tool. Constitutive model based on normality rule is adopted. (Kim et al. 2010) S

2 µD

σ µ(

3 S S 2

Qɺ

S:D

(5)

where α = conversion factor (= 1). The convective heat boundary condition is ∇

=

h (T − TB ) k

(6)

where h = convective heat transfer coefficient; and TB = surrounding temperature.

(1) 2 d εɺ = D⋅D 3

σ( ) εɺ ) = ɺ 3ε

4.2 Temperature dependent material properties (2) (3)

where µ = viscosity; S = deviatoric Cauchy stress; and D = rate of deformation; σ = yield stress; εɺ = strain rate; and T = temperature. The energy conservation equation is (Spencer 2004)

ρC p

where ρ(T) = density; Cp(T) = specific heat; k(T) = thermal conductivity; and Qɺ = heat generation rate by plastic dissipation. Heat generation rate Qɺ becomes

DT = ∇ ⋅ ( k∇T ) + Qɺ Dt

(4)

To calculate the temperature field, material nonlinear CFD analysis is performed using temperature dependent properties such as density, specific heat and thermal conductivity. These material properties are plotted in Figures 2–4, respectively (Chao et al. 1998 and Kim et al. 2010). 4.3

Modeling

CFD 3D modeling is shown in Figure 5. It consists of base plate cell zone, rotation affected cell zone and tool zone. The boundary condition and welding parameters are shown in Figure 6. Steady state simulations

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Figure 2. Temperature dependent AL6061-T6 and AA5083-H18.

density

of Figure 5. (a) 3 dimension modeling; (b) Base plate cell zone; (c) Rotation affected cell zone; (d) Tool cell zone.

Figure 3. Temperature dependent specific heat of AL6061-T6 and AA5083-H18. Figure 6. Boundary conditions and welding parameter.

Figure 4. Temperature dependent thermal conductivity of AL6061-T6 and AA5083-H18.

are performed for two welding conditions as the laminar incompressible flow. The material at room temperature flows in at the inlet boundary and flows out at the outlet boundary at the welding speed. The top and side surface of the plate are in contact with ambient air and the convective heat

transfer coefficient of 30 W/m2°C is used (Chao et al. 2003). The bottom surface of the workpiece supported by a backing plate. The heat transfer coefficient is determined by the tool’s downward force and the thermal material properties of the backing plate. In this study, conduction coefficient of the bottom surface of 200 W/m2°C is used (Kim et al. 2010). Case studies for different rotation affected cell zone diameters (Fig. 5c) are carried out. The range of rotation affected cell zone diameters are listed in Table 5 and Table 6 along with tool size. From the experiment performed by Murr et al. (1998), the size of weld nugget zone is slightly larger than the pin diameter, and this study assumes that the diameter of the rotation affected cell zone would be similar with that of weld nugget zone. Table 5 and Table 6 are shown case studies range of rotation affected cell zone diameter of AL6061-T6 and AA5083-H18, respectively. The range of rotation affected cell zone diameter is considered shoulder and pin diameter. The tilt of

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Table 5.

Range of rotation affected cell zone diameter of FSW for AL6061-T6. Rotation affected cell zone diameter

Tool size Shoulder diameter (mm)

Pin diameter (mm)

Pin length (mm)

32

10

9

Table 6.

Range of rotation affected cell zone diameter of FSW for AA5083-H18. Rotation affected cell zone diameter

Tool size Shoulder diameter (mm)

Pin diameter (mm)

Pin length (mm)

10

4

1.34

4.4

Figure 7. diameter.

12 mm, 16 mm, 20 mm, 24 mm 28 mm, 32 mm, 34 mm, 38 mm 42 mm, 46 mm

Definition of rotation affected cell zone

5 mm, 6 mm, 8 mm, 10 mm 12 mm, 15mm, 20 mm, 25 mm

Cross section microstructure

Welded joint can be divided into four zones: base material, heat affected zone (HAZ), thermo-mechanically affected zone (TMAZ). Figure 8 shows the cross section microstructure of friction stir welded plate (Mishra et al. 2007). When the rotation effect cell zone diameter is 34mm, Figure 9 shows temperature distribution across the cross section of AL6061-T6 obtained from CFD analysis result during FSW. HAZ of AL6061-T6 can be generally defined as a zone experiencing temperature beyond 380°C. (Chao et al. 1998 and Jang 2010). HAZ is represented between two dotted lines The HAZ shape of AL6061-T6 is similar to that of experiment as shown in Figure 8(a). On the other hand, plate of AA5083-H18 is quite thin and HAZ is quite broad as depicted in Figure 8(b). 4.5 Temperature distribution results

Figure 8.

Cross section microstructure of FSW.

the tool can have effects on the welding process. It can affect how easily the tool can move across the joint line because less pressure is put in the direction of the joint line and prevent defect formation such as tunnel defects and lack of penetration (Haver et al. 2010). A general range for tool tilt is between 2 and 4 degrees.

3D CFD simulation mentioned in Section 4.3 is performed for different rotation affected cell zone diameters listed in Table 5 and 6. The resultant temperature distributions for AL6061-T6 and AA5083-H18 are shown in Figures 10 and 11, respectively. On the whole, as the diameter increases, the maximum temperature also increases. It is because larger tangential speed and larger rotating zone leads to high temperature. As identified in Figure 11, when the tool is tilted, CFD simulation and experiments give more asymmetric temperature distribution than without being tilted.

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Figure 12. The most well matching rotation effected cell zone diameter with AL6061-T6 experiment results. Figure 9. FSW CFD temperature distribution of cross section. (a) AL6061-T6 (t: 12 mm); (b) AA5083-H18 (t: 1.64 mm).

Figure 13. The most well matching rotation effected cell zone diameter with A5083-H18 experiment results.

Figure 10. Temperature distributions for different rotation affected cell zone diameters of AL6061-T6.

AL6061-T6 and AA5083-H18 temperature distributions of experiment results show the best agreement those when the rotation affected cell zone diameters are 34 mm and 12 mm, respectively. In Figures 12 and 13, only those results are compared with experiment results. In the experiment, temperature of advancing side is measured higher than retreating side. In the two CFD simulations, the same tendency is observed especially for A5083-H18. In two experiments, when the rotation affected cell zone diameter is 2 mm larger than tool shoulder diameter, it gives the best agreement with experiment result. This results matches well with the initial assumption that the diameter would be similar with the weld nugget which is slightly larger than the tool shoulder diameter. 5

Figure 11. Temperature distributions for different rotation affected cell zone diameter of AA5083-H18.

CONCLUSION

In this study, material nonlinear CFD analysis is performed using temperature dependent material

438

properties. Some case studies for different rotation affected cell zone diameter are carried out. When rotation affected cell zone diameter is 2 mm larger than tool shoulder diameter, temperature distribution shows a good agreement with experiment. ACKNOWLEDGEMENTS This work was supported by Research Settlement Fund for the new faculty of SNU. REFERENCES Aljoaba, S.Z., Jawahir, I.S., Dillon Jr. O.W., Ali, M.H., and Khraisheh, M.K. 2009. Modeling of friction stir processing using 3D CFD analysis. International Journal of Material Forming Vol. 2: 315–318. Bang, H.S., and Bang., H.S. 2008. Transient Thermal Analysis of Friction Stir Welding using 3D-Analytical Model of Stir Zone. Journal of KWJS Vol. 26: 580–585. Bendzsak, G.B., North, T.B., and Smith, C.B. 2000. An experimentally validated 3D model for friction stir welding. Proceedings of the Second International Symposium on Friction Stir Welding, Sweden. Chao, Y.J., and Qi, X. 1998. Thermal and ThermoMechanical Modeling of Friction Stir Welding of Aluminum Alloy 6061-T6. Journal of Materials Processing & Manufacturing Science Vol. 7: 215–233. Chao, Y.J., Qi, X., and Tang, W. 2003. Heat Transfer in Friction Stir Welding-Experimental and Numerical Studies. Journal of Manufacturing Science and Engineering Vol. 125: 138–145. Colegrove, P. 2000. Three-dimensional flow and thermal modeling of the friction stir welding process. Proceedings of the Second International Symposium on Friction Stir Welding, Sweden. Colegrove, P., and Shercliff, H.R. 2005. 3-Dimensional CFD modeling of flow round a threaded friction stir welding tool profile. Journal of Materials Processing Technology Vol. 168: 320–327. Fourmet, L., and Guerdoux, S. 2008. 3D numerical simulation of the three stages of Friction Stir Welding based on friction parameters calibration. International Journal of Material Forming Vol. 1: 1287–1290. Frigaard, Ø., Grong, Ø., and Midling, O.T. 2001. A Process Model for Friction Stir Welding of Age Hardening Aluminum Alloys. Metallurgical and Materials Transactions A, Vol. 32: 1189–1200.

Haver, W.V., Meester, B.D., Geurten, A., and Jacques, D. 2010. Innovative joining of critical aluminum structures with the friction stir welding technique. Jang, Y.J. 2010. Welding Deformation Analysis of Friction Stir Welded Aluminum Alloy Structures using Equivalent Load Method based on Inherent Strain. Master dissertation, Department of Naval Architecture and Ocean Engineering, Seoul National University. Khandkar, H., Zahedul, M. and Khan, J. 2001. Thermal modeling of overlap friction stir welding for Al-alloys. Journal of Materials Processing and Manufacturing Science Vol.10: 91–105. Khandkar, H. 2005. Thermo-mechanical modeling of friction stir welding. Ph.D. dissertation, Department of Mechanical Engineering, University of South Carolina. Kim. D.G., Badarinarayan, H., Kim, J.H., Kim, C.M., and Okamoto, K. 2010. Numerical simulation of friction stir butt welding process for AA5083-H18 sheets. European Journal of Mechanics A/Solids Vol.29: 204–215. Mishra, R.S., Mahoney, M.W. 2007. Friction Stir Welding and Processing. ASM International. Murr, L.E., and McClure, J.C. 1998. A TEM study of precipitation and related microstructures in frictionstir-welded 6061 aluminum. Schmidt, H., Hattel, J., and Wert, J. 2004. An analytical model for the heat generation in friction stir welding. Modeling and Simulation in Materials Science and Engineering Vol. 12: 143–157. Seidel, T.U., and Reynolds, A.P. 2003. Two-dimensional friction stir welding process model based on fluid mechanics. Science and Technology of Welding & Joining Vol. 8: 175–183. Smith, C.B., Bendzsak, G.B., North, T.H., Hinrichs, J.F., Noruk, J.S., and Heideman, R.J. 2000. Heat and material flow modeling of the friction stir welding process. Proceedings of the 9th International Conference on Computer Technology in Welding, Detroit, Michigan. Song, M., and Kovacevic, R. 2003. Numerical and experimental study of the heat transfer process in friction stir welding. Proceedings of the Institution of Mechanical Engineers Vol. 217: 73–85. Song, M., and Kovacevic, R. 2004. Heat Transfer modeling for both workpiece and tool in the friction stir welding process: A coupled model. Journal of Engineering Manufacture Vol. 218: 17–33. Ulysse, P. 2002. Three-dimensional modeling of the friction stir-welding process. International Journal of Machine Tools & Manufacture Vol 42: 1549–1557.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Experimental investigation of welding deformations of hybrid structural joint T. Urbański & M. Taczała West Pomeranian University of Technology, Szczecin, Poland

ABSTRACT: The results of the experimental investigation of the hybrid joint are presented in the paper. The hybrid joint being a part of the large-size steel structure is formed as a result of joining I-core panel with conventional steel plate by the process of welding. Welding deformations are among the most severe problems in prefabrication of the large-size blocks composed of sections. The welding deformations due to complexity of their arising should be analyzed experimentally including the influence of technological and structural parameters on the mode and size of deformations. 1

INTRODUCTION

Ship hull is a large-size spatial object composed of many structural elements, typically built in steel. Modern sea-going ships are assembled of several and often more than a hundred sections of various types. Despite geometrical diversity among the structural modules, planar unidirectionally stiffened sections (panels) are essential components of each large-size spatial section (block). The planar section is typically stiffened with flat bulbs, Figure 1.

Despite structural simplicity, conventional flat sections are characterized by large labour consumption due to the fabrication processes. A large part of the expenditures (both in terms of labour and material consumption) are related to the amendment works, mainly thermal straightening, improving bad section quality which is unaccepted according to the quality standards of prefabrication of ship structures, (Quality Standard T100-1, 2001, Production Standard of the German Shipbuilding Industry, 1977, Shipbuilding and Repair Quality Standard IACS 1996). The production

Figure 1. Main elements composing ship hull structure, A—hull, B—spatial section, C—semi-spatial section (orthogonally stiffened), D—planar section (unidirectionally stiffened).

441

Figure 2.

Hybrid joint (on the basis of Urbański 2009).

quality of the flat sections severely influences the assembly usability in the latter phases of building ships (Metschkow 2001). The welding deformations are the most disadvantageous phenomenon having large impact on the ship section quality. This is the reason why elimination of the stiffeners becomes nowadays a key to improvement of efficiency of prefabrication and assembly operations of the ship sections. Application of a new innovative structural element—I-core panel— which has appeared on the market enables complete elimination of the stiffeners (Fig. 1), what is the greatest advantage of application of the innovative steel sandwich panels for the ship structures related to the ship production technology. Application of the innovative panels causes many problems appearing during design, building and operation of structures. During prefabrication of the structural elements joining of details and subassemblies is the essential fabrication operation. In the ship structure it is the junction I-core panel—conventional plate which is the greatest problem (Figure 2 ) (Urbański 2009). 2

IDENTIFICATION OF INVESTIGATION OBJECT

A specific fragment of the large-size steel structure where two various, with respect to structure and technology, elements are connected is referred to as a hybrid joint. The hybrid joint is an integral

part of the hybrid (mixed) structure and is composed of: I-core panel, conventional plate as well as connecting element—Figure 2 (Urbański 2009). Possibility of using the hybrid application in ship structures is large as it can be applied everywhere together with the I-core panels. Potential application of the I-core panels are given in: Kozak 2002, Pyszko 2006, Urbański, Graczyk 2005, Urbański 2009. Among many shapes of the connecting elements which can be used in the hybrid joint, the element was selected for the analysis presented in Figure 2. Other elements and the method of selection the best for the specific purpose regarding the assembly usability are given in: Urbański 2009. The important aspect of the problem is determination of the width of the hybrid joint. The width should be considered as the sum of the width of the connecting element and the widths of the plastic deformation zones covering the conventional plate and innovative structural element in the largest extent. The width of plastic deformation zone can be evaluated using mathematical equations (Jakubiec, Lesiński, Czajkowski, 1980, Jastrzębski, Taczała, 1995, Metschkow 2001, Myśliwiec 1972, Ranatowski 1999). 3

EXPERIMENTAL INVESTIGATION

The problem was investigated experimentally according to the principles of theory of experiment planning.

442

Figure 3.

Black box of planned experiment (on the basis of Urbański 2009).

Experiment was planned using cybernetic approach; the hybrid joint was considered as a black box. Parameters responsible for deformations, referred to as independent variables formed the input to the box while the response—dependent variables—were the output. Two another groups of parameters are also related to the black box—disturbing and constant factors. However, due to the fact that they are neither controllable nor measurable they were considered irrelevant. The black box of the planned experiment together with its components, is presented in Figure 3. The following quantities were taken as the independent variables, depending on the type of the hybrid joint weld: linear energy of welding, thickness of the connecting element, welding gap, width of the lower edge of the connecting element and width of the fragment of the I-core panel upper plate. Several modes of the welding deformations were identified as the dependant variables including transversal and longitudinal deformations of the I-core panel plating and connecting element as well as the shrinkage close to the butt weld. In the case of selection of three independent variables which can be set either to minimum or maximum values—Table 1, the problem was investigated using two-value fraction plan (Montgomery 2001, Polański 1984). Typical plan of experiment for detection of the transversal shrinkage close to the butt weld is presented in Table 1. Number of experiments in one required block is 8. The minimum and maximum

Table 1.

Plan of experiment—normalized values.

No of measurent

Linear energy (ql)

Thickness of connecting element (g2)

Welding gap (a)

1 2 3 4 5 6 7 8

−1 +1 −1 +1 −1 +1 −1 +1

−1 −1 +1 +1 −1 −1 +1 +1

−1 −1 −1 −1 +1 +1 +1 +1

Independent variables corresponding to normalized values take following values: ql ∈ [1,0; 2,6], kJ/mm, g2 ∈ [6; 10], mm, a ∈ [4; 8], mm.

values of the independent variables correspond to the dispersion of selected technological and structural parameters. Experiments were done in the conditions representing real production conditions with the use of equipment and methods employed in the shipbuilding industry. The samples were produced by a welder having several year experience. 4

RESULTS OF INVESTIGATION

Having obtained the results theoretical models can be developed describing relationships

443

between independent and dependent variables. Typical approach employed to define the relationships is analysis of regression. The theoretical models can be used for predicting welding deformations. Polynomials representing relationship between the modes of welding deformation of the hybrid joint and the technological and structural parameters are given by Eqs: 1, 2 and 3: yDPP yDWP W ySP 3

b + b1ql + b3c + b12ql g 2 + b23 g2c b + b ql + b2 g2 + b12ql g2 b0 + b3a − b12 ql g

b ql a

(1) (2) (3)

where: yDPP1, yDWP2, ySP3—approximated values for transversal deformation of the lower plating of the I-core panel, longitudinal deformation of the upper plating of the I-core panel, transversal

shrinkage of the butt weld, b0, … ,b23—regression coefficients, a, c, e, g2, ql—independent variables. Estimating the values obtained for the analyzed welding deformations of the hybrid joint—Eqs 1–3, it can be found that for these welding deformations modes the welding linear energy is the most important variable. The thickness of the connecting element is also an important factor. The presented approximation polynomials are mathematical models suitable for prediction of the amplitude of the specific welding deformation for arbitrary combination of technological and structural parameters. The only condition is that these parameters must belong to the definition space of the actual experiment. Accuracy of the prediction can be verified comparing the values calculated using Eqs 1–3 against those found experimentally. The comparison is presented in Figure 4. Good agreement of predictions and actual deformations can be observed in Figure 5a–c. Adjustment of the regression surface to the experimental

Figure 4. Estimation of accuracy of predicted deformations, A—diagram for transversal deformation of the lower plating of the I-core panel, B—diagram for longitudinal deformation of the upper plating of the I-core panel, C—diagram for transversal shrinkage of the butt weld, D—diagram for normality of residuals for transversal shrinkage of the butt weld, DPP1, DWP2, SP3—values obtained experimentally, yDPP1, yDWP2, ySP3—approximated values.

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Figure 5.

Deformations of: A—lower, B—upper plating of I-core panel.

Figure 6.

Deformations of connecting element.

data was: 95% for DPP1, 90% for DWP2 and 88% for SP3. The adjustment factors were obtained calculating corrected square of the multiple correlation coefficient (Urbański 2009). The agreement is also confirmed by the diagrams of the normality of residuals for the analyzed models (the diagram for transversal shrinkage of the butt weld is presented in Figure 4D). From the locations of the points on this diagram we can see that the distribution of the residuals in the adopted model is close to the normal distribution what unequivocally confirms good adjustment of the linear model to the experimental data.

The results of the experimental investigation for the sample with variables equal to [+1, −1, +1] are presented in Figures 5 and 6 where the analyzed deformations are shown in the form of the 3D produced using the data from the measurement points. To obtain better visibility of results the values for the Z axis (deformations) were rescaled using factor of 10. Values for the other axes of the coordinate system are left unchanged (points created by intersection of grid lines correspond to the measurement points on the sample surface). The effect of mirror reflection of the measurement device readout was not accounted for.

445

From Figures 5 and 6 it can be seen that the longitudinal deformations of the I-core panel (DWP1 and DWP2) appear throughout the width of the innovative structural element, transversal deformations (DPP1 and DPP2) do not appear beyond the first internal stiffener of the panel, from he weld towards the end of the panel. All deformation modes of the I-core panel are the modes for which straightening is relatively difficult. Apart from the deformations associated with elongation of elements, also the angular deformation of the connecting element is important for the assembly usability—Figure 6. It is a decisive factor for the possibility of joining the element with the conventional plate which is critical to fit elements for the butt joint. 5

CONCLUSIONS

The hybrid joint is a new challenge for welding of ship structures. This type of joining elements in the large-size steel structures produces many technological problems. Estimation of the results of the planned experiment allows for developing mathematical models for prediction of modes and size of welding deformations. Using the approximation equations it is possible to prove which of the parameters, found relevant, in fact influence the deformation mode. The theoretical models can be applied for seeking the technological procedures allowing for maintaining the joint geometry in the assumed tolerances of dimensions and, consequently, controlling the producibility of the innovative structure or its fragment. The results of the experimental investigations of the welding deformations of the hybrid joints can also be used for verification of results obtained using numerical methods such as the finite element method as well as artificial intelligence or calibration of the results obtained using these methods. REFERENCES Jakubiec M., Lesiński K., Czajkowski H., 1980. Production technology of welded structures (in Polish), Wydawnictwo Naukowo-Techniczne, Warszawa. Jastrzębski T., Taczała M., 1995. Influence of fabrication thermal processes on deformations and stresses in ship structures (in Polish), report developed in framework of Project “New production technologies for reducing time of building cycles and increasing quality standards”, Szczecin University of Technology.

Kozak J., 2002. All-steel sandwich panels—innovative elements of ship hull (in Polish). Proc. of 20th Session of Shipbuilders. Metschkow B., 2001. Influence of welding on quality of prefabricated structures (in Polish), Proc. of Laboratoria Technologiczne—Aspekty Utrzymania Wysokiej Jakości Wyrobu, XI Międzynarodowa Konferencja, Zintegrowane Systemy Zarządzania w Przemyśle. Metschkow B., 2001. Estimation of size of welding deformation using practical methods (in Polish). Proc. of 20th Session of Shipbuilders. Montgomery D.C., 2001. Design and analysis of experiments. John Wiley & Sons, Inc. fifth editio. Myśliwiec M., 1972. Thermo-mechanical fundamentals of welding (in Polish). Wydawnictwo NaukowoTechniczne, Warszawa. Polański Z., 1984. Planning experiments in technology (in Polish), Państwowe Wydawnictwo Naukowe, Warszawa. Production Standard of the German Shipbuilding Industry, 1977 (revised edition with the first edition— November 1974 and second edition—August 1977). Pyszko R., 2006. Application of all-steel sandwich panels to ship and offshore structures (in Polish), PhD dissertation, Katedra Technologii Okrętu, Systemów Jakości i Materiałoznawstwa, Wydział Oceanotechniki i Okrętownictwa Politechniki Gdańskiej. Quality standard: T100-1, 2001. Steel hull of ships. Accuracy of hull structure (in Polish), Szczecin Shipyard Inc. Ranatowski E., 1999. Elements of physics of joing metals (in Polish). Wydawnictwa Uczelniane Akademii Techniczno-Rolniczej. Shipbuilding and Repair Quality Standard IACS, London 1996: Part A. Shipbuilding and Repair Quality Standard for New Construction; Part B. Repair Quality Standard for Existing Ships. Urbański T., Graczyk T., 2005. Application of innovative materials in waterborne transport means—identification of technological problems, International Conference on Innovative Materials and Technologies for Surface Transport (INMAT 2005), Gdańsk, Poland, 7–8 November. Urbański T., 2009. Method for prediction of welding deformations of hybrid joint using experimental approach (in Polish). PhD dissertation, Zakład Konstrukcji, Mechaniki i Technologii Okrętów, Wydział Techniki Morskiej, Zachodniopomorski Uniwersytet Technologiczny w Szczecinie. Urbański T., 2012. Hybrid joint—prediction of welding deformations of I-core panel using two-value planned experiment (in Polish), Przegląd Spawalnictwa, 3/2012: 16–22.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Welding residual stress and its effect on fatigue crack propagation after overloading K. Yuan Graduate School of Engineering, Yokohama National University, Yokohama, Japan

Y. Sumi Faculty of Engineering, Yokohama National University, Yokohama, Japan

ABSTRACT: Numerical simulation is used to investigate the effect of welding residual stress on fatigue crack growth lives of fillet weld with and without preloads. T-joint fillet weld and boxing fillet weld are chosen as analysis models. The as-welded residual stress distribution and that after shakedown are both predicted by three-dimensional uncoupled thermo-mechanical finite element analysis. Considering the predicted residual stress, a crack propagation life of toe crack located at fillet welded joint is predicted based on a crack opening and closure simulation method. From the simulation results, it is found that when tensile residual stress nearby the weld toe is severe, the crack growth lives do not exhibit considerable change except for the extremely high tensile preloads. 1

INTRODUCTION

Residual stresses induced in the welding process of marine structures such as ships and offshore structures may significantly affect their fatigue strength. Many research works have been carried out to investigate the welding residual stress distributions experimentally like neutron diffraction or sectioning methods (Fricke 2005, Kim et al. 2001) and numerically (Ma et al. 1995, Teng et al. 2001, Chang & Lee 2009). Generally, the tensile residual stress up to yield stress of the material is assumed in fatigue design recommendations. This assumption seems conservative because the welding residual stress is likely to be relaxed due to overloading, and uncertainties still exist about the effect of actual residual stress on fatigue failure mechanism in welded structures. For fatigue crack life evaluation, a crack opening and closure model based on strip yield model was originally proposed by Newman (1981) and Toyosada et al. (2004a, b). Considering the contact of plastic wake along the crack surfaces, the effective tensile plastic stress intensity range, ΔKRP, is calculated (Toyosada et al. 2004a). These methods have been successfully incorporated into crack propagation simulation code CP-System (Okawa & Sumi 2008). In this paper, the authors make efforts to investigate the effect of welding residual stress on fatigue crack growth lives with and without preloads. T-joint fillet weld and boxing fillet weld, which are widely used in ship structures, are selected as analysis models. The residual stress distributions are

predicted by three-dimensional uncoupled thermomechanical finite element analysis using commercial code SYSWELD. The phenomenon of residual stress shakedown can also be simulated by this code. Considering the predicted residual stresses, a crack propagation life of toe crack located at fillet welded joint is predicted based on the crack opening and closure simulation method by CP-System. 2

WELDING RESIDUAL STRESS ANALYSIS

The solution procedure consists of two steps. First, the temperature distribution and its history in the welding model are computed by thermal analysis. Then, the temperature history is employed as a thermal load in mechanical analysis. 2.1

Thermal analysis

In SYSWELD, a volumetric heat source in the shape of double ellipsoid, which was first proposed by Goldak (1984), is employed for the simulation of welding processes like MIG, TIG, etc, as illustrated in Figure 1. The power density distribution inside the front heat source is expressed as: q f (x , y , z ) =

447

6 3 ffQ

⎛ x′2 ⎞ exp −3 2 ⎟ ⎝ a ⎠ abc1π π ⎛ z′2 ⎞ ⎛ y′ 2 ⎞ exp 3 2 exp ⎜−3 2 ⎟ ⎝ b ⎠ ⎝ c1 ⎠

(1)

Figure 2. Figure 1.

Goldak’s double ellipsoid heat source model.

Similarly, for the rear heat source the power density inside the ellipsoid becomes: qr ( x , y , z ) =

6 3 frQ ⎛ x′2 ⎞ exp −3 2 ⎟ ⎝ a ⎠ abc2π π ⎛ z′2 ⎞ ⎛ y′ 2 ⎞ exp 3 2 exp⎜−3 2 ⎟ ⎝ b ⎠ ⎝ c2 ⎠

(2)

where x′, y′ and z′ are local coordinate fixed to the moving heat source, the parameters a, b, c1 and c2 are related to the shape characteristics of the welding arc, and Q is the power input from the welding arc. Parameters ff and fr define the fractions of the heat deposited in front and rear parts, respectively. Note that ff + fr = 2.0. To consider the heat losses, both the thermal radiation and convection on the weld surface are assumed. The total temperature-dependent heat transfer coefficient is applied as: h=

εσ σ [(T

273)4 (T0 + 273)4 ] + hc T T0

(3)

where T0 = 20°C is the ambient temperature, the convection coefficient hc = 25W/(m2K), σ = 5.67 × 10−8 J/(m2K4s) is defined as the Stefan-Boltzmann constant for radiation, and the emissivity is defined to be ε = 0.8, respectively. 2.2

Schematic illustration of shakedown process.

The components on the right-hand side of Equation (4) represent strain rate due to elastic, plastic, thermal load, volumetric change and transformation plasticity, respectively. The material is assumed to follow the Von Mises yield criterion, temperature dependent mechanical properties, and isotropic hardening model. 2.3

3 3.1

Mechanical analysis

During the welding process, besides the elastic, plastic and thermal strain, the phase transformation gives rise to two additional strains. One is strain due to volumetric change, the other is transformation plasticity strain. Therefore, the total strain rate can be written as the summation of the individual components: εɺ = εɺ E + εɺ P + εɺT + εɺ ΔV + εɺTrrp

(4)

Shakedown analysis

Residual stress is likely to be relieved due to high static loads. This phenomenon is called as “shakedown” and its mechanism is shown as Figure 2. Point A represents the initial residual stress, and external load σex is applied to weldment. If the summation of initial residual stress and external load is not greater than yield stress (point B), residual stress returns to point A after unloading. This is no-shakedown cycle “A-B-A”. On the other hand, if their summation exceeds the yield stress, a part of elastic strain related to residual stress will be converted to the plastic deformation. When external load is removed, residual stress moves, through C, to point D. This is shakedown cycle “A-B-C-D”.

SIMULATION MEHTOD FOR FATIGUE CRACK PROPAGATION Crack opening and closure simulation

The change process of the plastic deformation and working stress distribution ahead of the crack tip is shown in Figure 3, where c is the crack length, a the plastic zone length, σY the yield stress and λ the plastic constraint factor due to stress triaxiality. When the crack is subjected to the maximum load Pmax, there is a tensile plastic zone generated at the crack tip. As the load is decreased down to minimum load Pmin, contact may occur between the crack surfaces. Toyosada et al. (2004a) have

448

Figure 4.

The crack opening and closure model.

Figure 5.

Simulation flow of CP-system.

Figure 3. Plastic deformation and working stress distribution during fatigue crack propagation.

pointed out that the crack does not propagate in the load range from crack opening load Pop to load PRP, where tensile yielding starts again, because no plastic work is proceeded. The crack starts to propagate when the tensile plastic zone just initiates to develop ahead of crack tip at load PRP. Thus, the effective tensile plastic stress intensity range, ΔKRP, is defined as ΔK K RP

K max − K RP

(5)

The crack propagation rate based on ΔKRP is determined as da/dN d

C ( K RP )m

(6)

where C and m are the material constants. The effective stress intensity range ΔKRP can be calculated based on the crack opening and closure simulation method developed by Okawa & Sumi (2008). As shown in Figure 4, the bar elements of elastic-perfectly plastic body are connected on the crack surfaces based on the crack opening displacement calculated by strip yield model, where n is the total number of bar elements and k is the number of bar elements in the plastic zone. The ith bar element is located at [bi, bi+1], and xi is the midpoint of the bar element. For instance, the crack opening displacement at the maximum load, vmax (xi), is given by

where max is the stress intensity factor corresponding to maximum load at hypothetical crack tip, E is the Young’s modulus and ν the Poisson ratio. As far as the stress distribution is within elastic range, the superposition properly account for re-distribution of the residual stress due to crack growth. From Equation (7)

(9) where res is the stress intensity factor at hypothetical crack tip caused by residual stress. 3.2

(7)

E′ =

{

E E

2

for f plane stress ffor plane strain

(8)

Simulation based on CP-system

Figure 5 shows the main simulation flow of crack propagation simulation code, CP-System, which has been developed to deal with fatigue crack propagation in three-dimensional plate structures. In order to simulate the practical crack propagation problem in large-scale structures, the super-element technique is utilized. The crack propagation behavior is simulated by step-by-step

449

finite element analysis. Based on above mentioned crack opening and closure model, the crack growth life is predicted, considering the welding residual stress. 4 4.1

FATIGUE LIFE PREDICTION OF FILLET WELDED JOINTS Analysis model

In this work, two kinds of fillet welded joints, non-load-carrying T-joint fillet weld and boxing fillet weld, are selected as analysis specimens. The geometries of specimens are illustrated in Figure 6. The 6 mm fillet leg length and existence of 0.1 mm root gap between attachment and main plate are considered in both specimens. For the three dimensional FE models, one quarter of the specimens are shown in Figure 7, considering the symmetry features. The models mainly employ 8-node hexahedron elements except for few tetrahedron and wedge elements in the weld bead region. The minimum element size is approximately 1 mm in the direction perpendicular to the welding line near the weld toe. The material used in specimens is selected as SM490, the widely used structural carbon Figure 7. (a) FE model of T-joint fillet weld. (b) FE model of boxing fillet weld.

Figure 6. (a) Geometry of T-joint fillet weld. (b) Geometry of boxing fillet weld.

steel. Its thermal and mechanical properties are dependent on temperature, as shown in Figure 8. The stress-strain curves which vary with phases and temperature are given by discrete points in SYSWELD material database. The welding conditions chosen for this study are as follows: welding method, single pass metal inert gas arc welding; heat input Q = 1300 J/mm; arc efficiency Eff = 0.85 and welding velocity v = 6 mm/s. The function of Goldak’s heat source has been predefined in simulation code, in which the inclined motion of heat source for fillet weld bead is considered, and the used values of heat source parameters are summarized in Table 1. In order to simulate residual stress shakedown phenomenon, a group of static preload cases are designed, tabulated in Table 2. User-defined function file is created to introduce the load patterns of preloads onto as-welded models. The experimental and simulated results (Sumi et al. 2010, Feltz et al. 2010) imply the fatigue cracks often initiate from the weld toes in nonload-carrying joints. For the fatigue life prediction, it is difficult to directly analyze the crack closure behavior of surface crack in welded joints. In order

450

to simplify the problem, we simulate the fatigue crack growth of a two-dimensional semi-infinite crack subjected to the same stress intensity ranges as those at the deepest point of surface cracks. The crack propagation lives are calculated by the crack opening and closure model presented in this paper. The initial crack depth is assumed as 0.05 mm, the order of grain size, and the parameters we use in these calculations are summarized in Table 3. Since the repeated plastic stress-strain cycle ahead of the crack is taken into account by ΔKRP, the threshold phenomenon is not observed for fatigue crack propagation (Toyosada et al. 2004a). 4.2 Simulated results of T-joint model

Figure 8. (a) Temperature dependent thermal properties. (b) Temperature dependent mechanical properties. (c) Phase specific yield stresses. Table 1.

Parameters of heat source.

Parameters

a (mm)

b (mm)

c1 (mm)

c2 (mm)

ff

fr

Value

5.0

8.0

4.5

7.5

1.33

0.67

Table 2.

The as-welded longitudinal residual stress σz and transverse one σx distributed along x-axis at middle cross-section are shown in Figure 9. Ignoring the region on the left hand-side of broken line (location of the weld toe), it can be seen that the longitudinal welding residual stress may be very close to yield stress at the weld toe. With some distance away from the weld toe, the tensile longitudinal stress is changed to compression for selfequilibrium. As for transverse residual stress σx, it is almost tensile and gradually decreases to zero at the plate edge. In addition, the residual stress in the thickness-direction, σy, is considerably small compared with the other components. It can be observed that the transverse stress σx at the weld toe is locally in compression, but it changes to tension at slight distance away from the weld toe. The occurrence of compressive residual stress is caused by the effect of volume change due to martensite transformation (Deng et al. 2003), which has been considered by SYSWELD. During rapid cooling, the austenite with face centered cubic structure changes into martensite with body centered tetragonal structure, and then volume increases. The predicted phase proportions nearby the weld bead are shown in Figure 10. The microstructure of point D at weld toe is mainly a mixture of martensite and bainite, and point E still keeps 100% ferrite state.

Static preload conditions.

Normalized

As-welded

T195

T312

T380

C195

C312

C380

Preload* Load pattern

0

0.5σY 0.5σY

0.8σY 0.8σY

0.97σY 0.97σY

−0.5σY

−0.8σY

−0.97σY

−0.5σY

−0.8σY

*Normalized with respect to yield stress of 390 MPa.

451

−0.97σY

Table 3. Calculation parameters for fatigue life prediction. C Parameters (MPa m ) Value

Figure 9. x-axis.

m

E ν (Gpa) (MPa) σY

3.514 × 10−11 2.692 206

0.3

λ

390 1.04

As-welded residual stress distributions along

Figure 10. Phase proportion distributions nearby weld bead.

Figure 11. Temperature and transverse stress history of element at weld toe.

The temperature and corresponding transverse stress histories of the element at weld toe during first 500 second are illustrated in Figure 11. During heating process, compressive stress is produced because of the expansion constrained by surrounding material. Upon cooling, a tensile stress arises due to shrinkage of weld. When martensite transformation occurs, compressive stress is generated again because of large volume change and recovery of elastic modulus at lower temperature. It is also easy to understand the point E slightly away from the weld toe just experienced “compressive to tensile” stress change, because no phase transformation occurs. If surface crack occurs at the weld toe of T-joint fillet weld, the transverse residual stress σx, which works as mean stress, will affect the stress ratio. The preloads are applied onto sides of main plate along x-axis to bring out the change of σx. The stress histories of σx at weld toe in T312 and C312 are compared as example shown in Figure 12. It can be seen that in the case of compressive preload, the material at the both ends of the weld deforms in fully plastic condition during loading process. After complete unloading, the considerable shakedown takes place at these portions of the weld. For the tensile preload, almost no shakedown can be seen, because the summation of as-welded residual stress and preload is lower than yield stress. The through-thickness transverse stress distributions right underneath weld toe at middle section after different preloads are compared in Figure 13. The values of transverse stress on the surface of main plate are summarized in Table 4. In compressive preload, the degree of shakedown becomes more significant with the increase of preload. On the other hand, we cannot expect any considerable shakedown effect after tensile preload. Therefore, in the following we just discuss the effect of shakendown residual stress by compressive preloads on crack growth life. We assume that the T-joint fillet weld is subjected to constant amplitude cyclic load with stress

Figure 12. Stress history in T312 and C312. *Notation: AW: as-welded; Max: at the maximum value of preload; SH: after shakedown.

452

Figure 13. Through-thickness transverse stress after preloads. Table 4.

Figure 14. Comparison of crack growth lives considering different residual stress distributions in T-joint fillet weld. *Notation: RF: residual stress free.

Transverse residual stress after preloads. Middle section

At ends*

Preload unit: MPa

σx

Δσx

σx

Δσx

As-welded T195 T312 T380 C195 C312 C380

−125.2 −124.8 −113.4 −138.8 −90.8 −58.3 −32.5

–

−599.8 −599.4 −604.4 −550.5 −400.0 −269.4 −198.3

–

0.4 11.8 −13.4 34.4 66.9 92.7

0.4 −4.6 49.3 199.8 330.4 410.5

*Averaged value of two weld ends.

Figure 15.

range ΔS = 100 MPa and stress ratio R = 0.0. Considering the as-welded and shakendown residual stresses, the crack growth lives are compared in Figure 14. It can be found that the compressive preloads will reduce the beneficial contributions of compressive residual stress. Usually, the transverse welding residual stress is in tension at the weld toe. The reason why localized compressive residual stress occurs is that we have chosen a relatively low martensite-transformation temperature (420ºC). Certainly, some newly developed LTT (low transformation temperature) welding wires have introduced the compressive residual stress to the weld bead and weld toe, resulting in the improvement of fatigue strength (Ohta et al. 1999, Nose & Okawa 2011). It should be noted that this kind of favorable effect is reduced due to the compressive overloads.

σz along x-axis at wrap-end. We can find that the tensile residual stress nearby wrap-end is severe, as high as yield stress, which may accelerate the crack propagation. To exemplify the change of longitudinal residual stress due to shakedown, the preload conditions in Table 2 are applied onto sides of main plate along z-axis. Same as former section, the stress histories of σz in T312 and C312 are compared in Figure 16. The shakendown residual stress σz distributed through thickness right underneath weld toe are compared in Figure 17, and values on the surface of main plate are summarized in Table 5. From these figures and a table, it can be observed that shakedown occurs at the wrap-end by both tensile and compressive preloads, which becomes more evident with higher loads. Because the tensile residual stress nearby wrap-end has reached the tensile yield stress level, the shakedown by tensile preloads occurs easily. In Figure 18, the simulated crack growth curves of toe crack located at the wrap-end are illustrated. The loading condition is constant amplitude load with stress range ΔS = 100MPa and stress ratio R = 0.0. It is found that with and without considering welding residual stress, the predicted crack

4.3 Simulated results of boxing fillet model In general, the fatigue crack is more prone to occur at the weld toe of wrap-end than that of fillet weld, so that it is necessary to pay attention to residual stress distribution around the wrap-end. Figure 15 shows the as-welded longitudinal residual stress

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As-welded residual stress σz along x-axis.

growth lives appear different. In each case, the crack initiation life (upon to 1mm depth) accounts for above half of total crack growth life. Although the through-thickness residual stress relieved by preloads (see Fig. 17), they are still in relatively high tension. Consequently, preloads do not significantly affect the crack growth except for the cases of extremely high tensile preloads such as “T312” and “T380”.

Figure 16.

Figure 17. preloads.

5

Stress history in T312 and C312.

Through-thickness longitudinal stress after

Table 5. Longitudinal residual stress after preloads. At weld toe of wrap-end Preload Unit: MPa

σz

Δσz

As-welded T195 T312 T380 C195 C312 C380

460.4 283.3 111.2 10.8 433.4 303.1 227.6

– 177.1 394.2 449.6 27.0 157.3 232.8

CONCLUSIONS

In the present work, the effect of welding residual stress, with and without preloads, on fatigue crack propagation in fillet weld has been numerically investigated. The welding residual stress and shakedown phenomenon can be both simulated by finite element analysis code SYSWELD. The degree of residual stress shakedown is mainly dominated by level of the summation of as-welded residual stress and preload stress exceeding the yield stress. The crack propagation life of toe crack located at fillet welded joint is calculated based on a crack opening and closure simulation method by CPSystem, in which the predicted residual stress can be considered. With and without considering the welding residual stress, the predicted crack growth lives appear different. Sometimes the beneficial compressive residual stress is relieved by preloads, the crack growth life may decrease. On the other hand, when the tensile residual stress nearby the weld toe is very high, the crack growth lives do not exhibit considerable change except for the cases of exceedingly high tensile preloading. REFERENCES

Figure 18. Comparison of crack growth lives considering different residual stress distributions in boxing fillet weld.

Chang, K.H. & Lee, C.H. 2009. Finite element analysis of the residual stresses in T-joint fillet welds made of similar and dissimilar steels. International Journal of Advanced Manufacturing Technology 41:523–538. Deng, D., Luo, Y., Serizawa, H. & Murakawa, H. 2003. Numerical simulation of residual stress and deformation considering phase transformation effect. Trans. JWRI 32-2:325–333. Feltz, O., Fisher, C. & Fricke, W. 2010. Fatigue assessment of weld toe and root cracks with the notch stress intensity factor and crack propagation approach. Proceedings of the 11th International Symposium on Practical Design of Ships and Other Floating Structures: 1357–1365. Fricke, W. 2005. Effects of residual stresses on the fatigue behavior of welded steel structures. Mat.wiss.u.Werkstofftech 36–11:642–649. Goldak, J., Chakravarti, A., Bibby, M. 1984. A new finite element model for welding heat sources. Metallurgical Transactions B 15B:299–305.

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Kim, W.S., Kim, D.H. & Lee, Y.K. 2001. Fatigue strength of load-carrying box fillet weldment in ship structures. Proceedings of the 8th International Symposium on Practical Design of Ships and Other Floating Structures: 1161–1167. Ma, N.X., Ueda, T., Murakawa, H. & Maeda, H. 1995. FEM analysis of 3D welding residual stresses and angular distortion in T-type fillet welds. Trans. JWRI 24–2:115–122. Newman Jr, J.C. 1981. A crack-closure model for predicting fatigue crack growth under aircraft spectrum loading. ASTM STP 748: 53–84. Nose, T. & Okawa, T. 2011. Approaches for fundamental principles 2: total solution for fatigue of steel. NSC Tech Rev 391: 156–161 (in Japanese). Ohta, A., Suzuki, N., Maeda, Y., Hiraoka, K. & Nakamura, T. 1999. Superior fatigue crack growth properties in newly developed weld metal. International Journal of Fatigue 21: 113–118. Okawa, T. & Sumi, Y. 2008. A computational approach for fatigue crack propagation in ship structures under random sequence of clustered loading. Journal of Marine Science and Technology 13–4:416–427.

Sumi, Y., Nakamura, M. & Mohri, M. 2010. Crack paths in weld details under combined normal and shear loading. Engineering Fracture Mechanics 77–11: 2115–2125. SYSWELD v2011. ESI Group. Teng, T.L., Fung, C.P., Chang, P.H. & Yang, W.C. 2001. Analysis of residual stresses and distortions in T-joint fillet welds. International Journal of Pressure Vessels and Piping 78:523–538. Toyosada, M., Gotoh, K. & Niwa, T. 2004. Fatigue crack propagation for a through thickness crack: a crack propagation law considering cyclic plasticity near the crack tip. International Journal of Fatigue 26: 983–992. Toyosada, M., Gotoh, K. & Niwa, T. 2004. Fatigue life assessment for welded structures without initial defects: an algorithm for predicting fatigue crack growth from a sound site. International Journal of Fatigue 26: 993–1002.

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Structural analysis

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Equivalent shell element for ship structural design E. Avi & A. Niemelä STX Europe, Turku, Finland

I. Lillemäe & J. Romanoff Aalto University, Espoo, Finland

ABSTRACT: This paper presents an equivalent shell element for assessing the ship global and local static and vibration response in early design phases. The element provides a computationally economic tool for global analysis and the same mesh can be used in primary, secondary and in some cases tertiary level. The stiffened panel is considered as a three layer laminate element, where the first layer represents the plate, the second layer the stiffener web and the third the stiffener flange. The layers are described as 2D iso—and orthotropic materials, where the elasticity matrices are found by applying the Rule of Mixtures. The element includes the in-plane, membrane-bending coupling, bending and additionally also shear stiffness, which follows the Reissner-Mindlin plate theory for anisotropic homogenous shells. The local plate bending response between the stiffeners is considered as well. The developed shell formulation has been implemented in commercial FE software FEMAP with NX Nastran and demonstrated through a case study. Results are validated against 3D fine mesh analysis and good agreement is observed. 1

INTRODUCTION

In the design of modern passenger ships new and challenging concepts are utilized, which means that advanced numerical methods are needed to accurately evaluate global, i.e. primary, and local, i.e. secondary and tertiary responses, Figure 1. Generally two types of global response analysis are performed: equivalent quasi-static and vibration. In the static analysis the stresses and deformations of the hull girder due to hogging, sagging, racking, torsion and docking load conditions are obtained (DNV 1991). If large lateral loads occur, the exam-

Figure 1. The definition of primary, secondary and tertiary response (Edward 1988).

ination of the local structures becomes essential as well. Vibration response analysis is especially important for cruise ships, where vibration limits are commonly determined from the passenger comfort and machinery lifetime requirements. The standard procedure for preventing undesired vibrations is to obtain the natural frequencies of the ship structures correctly and restrain them above the allowable resonant limit (Okumoto et al. 2009). A complete vibration analysis of a ship does not only include global vibration modes but also an examination of local structures, such as cabin areas, restaurants, engine rooms, etc. Nowadays, fine mesh Finite Element Analysis (FEA) is regarded as the most reliable method for evaluating the cruise ship structural behavior (Moan & Berge 1997). The 3D finite elements analysis of the ship has been performed e.g. by Naar (2007), Kurki (2010) and Jang et al. (2008). However, this method is computationally very expensive. Creation of the global cruise ship model can take few months and calculation some additional weeks. Hence, fine mesh FEA is unsuitable for cruise ships in initial design phases, where the analysis has to be performed relatively quickly and several large changes must be possible as the ship general arrangement changes. Coarse mesh global FE analysis is the most convenient way to obtain the response of the complete vessel. The ship global FE model, Figure 2, is created so that the secondary stiffeners, Figure 3, are

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Figure 2.

Global FE model of a modern cruise ship.

Figure 3. Primary and secondary stiffeners (Mobasher et al. 2009).

incorporated into the plate or shell element formulation in a way that it results in equivalent stiffness, bending and vibration response. The primary stiffeners such as girders and web frames, Figure 3, are explicitly modeled due to their large participation in overall stiffness of the structure. They are usually discretized with offset beam elements according to Timoshenko theory and their length is identical to the size of the neighbouring equivalent plate element. Most common way to represent the equivalent plate element is by lumping the stiffeners (Hughes 1983). In the lumping process all secondary stiffeners are put inside of the topological beam elements, which are located at the edges of the plate element. Each stiffener causes an increase in the cross sectional properties of the lumped beam elements. Lumping is one of the simplest approaches but it is also less accurate. For more precise analysis, Hughes (1983) suggested the orthotropic plate technique, where the stiffeners are blended with the plating so that the plate has different stress-strain properties in two directions. Since the stiffened panel is represented as one orthotropic plate, this approach does not allow analysing plate and stiffener separately and the obtained normal stress is average of the plate stress and the stiffener stress. For more advanced analysis Hughes (1983) pro-

posed a separate “stiffener element”, where the stiffeners are described as a special membrane element, which extends over the same area as the plating and has the same corner nodes and Degrees of Freedom (DOF). The stiffness matrix of the complete stiffened panel is obtained by summing the stiffness matrices of the plate and the “stiffener element”. All the equivalent element techniques proposed by Hughes can only represent the in-plane stiffness of stiffened panel and therefore the ship global FE model is not sufficient for secondary response analysis and local areas need to be separately analysed with 3D mesh. Satish Kumar & Mukhopadhyay (2000) added the plate and stiffener bending into the formulation of equivalent plate element by combining the Allman’s plane stress element (Allman 1984) with the discrete Kirchoff-Mindlin triangular plate bending element (Katili 1993) and applied it in the ship static 3D FE analysis. However, the element has several weaknesses. It neglects the effect of bending caused by the membrane. Also, the shear correction factor of 5/6 is used, which does not express the stiffened panel shear stress distribution correctly. More advanced equivalent elements have been used for web-core sandwich panel response analysis. Lok & Cheng (2000 and 2001) derived the expressions for the elastic constants of the symmetric truss-core sandwich panel. The constants represent bending, twisting and transverse shear stiffness and enable the transformation of a 3D structure into an equivalent homogeneous orthotropic thick plate continuum. Romanoff & Varsta (2007) developed a theory for the bending response of asymmetric web-core sandwich structure. The web-core sandwich structure is described as a laminate plate, which follows the thick-face-plate kinematics. The stiffness properties of the sandwich panel were obtained from laminated shell theory (Reddy & Ochoa 1992). Also, the effect of local bending of the face plates was taken into account by adding the local plate response to the overall laminate solution. The objective of this paper is to present an equivalent shell element, where the secondary stiffeners are incorporated into the shell element formulation, to assess ship global and local static and vibration response in limited time frame. Unlike the existing equivalent element formulations for stiffened panels, the element includes the membrane-bending coupling, bending and additionally also shear stiffness, which follows the ReissnerMindlin plate theory for anisotropic homogenous shells. The relationships between the homogenized internal forces, strains and curvatures are derived according to the laminated shell theory. The local plate bending between the stiffeners is considered

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as well, by adding the local plate bending response to the laminate element solution. The validation of proposed element is achieved with 3D FEM by considering typical case study of the ship cabin area. 2

THEORY

The actual stiffened panel structure is described mathematically using equivalent shell elements, where only homogenized stiffness properties are considered. The in-plane, membrane-bending coupling and bending stiffness of the structure are expressed by ABD-matrix and shear stiffness by DQ-matrix. The stiffness properties are derived with respect to the geometrical mid-plane of the stiffened panel, Figure 4a. 2.1

where subscript 0 denotes the mid-plane displacements and the slope θ with subscripts x and y describes the rotations around the x and y axis, respectively. The abbreviations p, w and f denote the plate, web and flange, respectively. The corresponding strain vector {ε} is the gradient of the displacements and it can be written as: ⎧ ε ⎫ ⎧ ε x0 ⎫ ⎧κ ⎫ ⎪ x ⎪ ⎪⎪ 0 ⎪⎪ ⎪ x⎪ ⎨ ε y ⎬ = ⎨ ε y ⎬ + zi ⎨ κ y ⎬ , i = p,w,f, ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎩γ xy ⎭i ⎪⎩γ xy ⎪⎭i ⎩κ xy ⎭

where {ε}0 is the mid-plane strain vector and {κ} is the vector of curvature, i.e. the second derivatives of the displacement: ⎧ ε0 ⎫ ⎫ ⎪⎪ x ⎪⎪ ⎧⎪ ∂u0 ∂x ⎪ 0 ⎨ ε y ⎬ = ⎨ ∂v0 ∂y ⎬ ⎪ 0 ⎪ ⎪ ∂u ∂y + ∂v / ∂x ⎪ 0 ⎭ ⎪⎩ γ xy ⎪⎭ ⎩ 0 2 2 ⎧ −∂ wg / ∂x ⎫ ⎧κ ⎫ ⎪⎪ 2 ⎪⎪ ⎪ x⎪ 2 ⎨ κ y ⎬ = ⎨ −∂ wg ∂y ⎬. ⎪ ⎪ ⎪ ⎪ 2 ⎪⎩ − ∂ wg / ∂ ∂y ⎪⎭ ⎩ κ xy ⎭

Notations

The stiffened panel is considered as a three layer laminate element, where the first layer represents the plate, the second layer the stiffener web and the third the stiffener flange. Plate layer has the thickness of the plate tp, web layer has the thickness of the web height hw and the flange layer has the thickness of the flange height hf. The cross section of the flange is idealized as a rectangle, which dimensions result in equal mass and second moment of area as the original HP-profile, Figure 4b. Strain-displacement relationship

According to the Kirchoff plate theory (Roylance 2000), the displacements u, v, w corresponding to the directions x, y and z, respectively, can be expressed as a linear function of the depth z: ui u (x x y ) − zi x ( x, y ), vi v (x x y ) − zi y ( x, y ), wi w(x x y ), i = p,w,ff

(1)

(3)

The out-of-plane shear strain can be written as (Whitney & Pagano 1970): ∂wi , i = p,w,f, ∂x ∂w p θy + . ∂y

i γ xz

2.2

(2)

θx +

p γ yz

2.3

(4)

In-plane elasticity matrix

The plate layer is described as a 2D isotropic shell element, where ε3 = γ13 = γ23 = 0. According to the Hooke’s law, the elasticity matrix [E] of the plate layer is: [ ]p =

1 (

0 ⎡ E νE ⎢ν E E 0 2 ⎢ ) ⎢⎣ 0 0 G(

⎤ ⎥. ⎥ ) ⎥⎦

(5)

The web and flange layers are described as 2D orthotropic shell elements, where the components of the elasticity matrix ([E]w and [E]f) are found by applying the Rule of Mixtures: Figure 4. Stiffener panel division into three layer laminate element and the notations used in this paper (a) and the idealisation of the HP-profile flange (b).

⎡E tw ⎢ [ ]w = ⎢ 0 s ⎢⎣ 0

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0 0⎤ 0 0 ⎥⎥ , 0 0 ⎥⎦

(6)

[ E] f

⎡E bf ⎢ = 0 s ⎢ ⎢⎣ 0

0 0⎤ 0 0 ⎥⎥ . 0 0 ⎥⎦

(7)

The normal stress vector {σ} = {σx σy τxy}T for layer i is obtained by multiplying the strains {ε} = {εx εy γxy}T with the elasticity matrixes [E]:

{σ }i [ ]i {ε }i , 2.4

i = p,w,f.

Relationship between the internal forces and strains

1 [E ] zi di dz , 3 − h∫/ 2 i

∫ [E ]i { }i dz

−hh /

[ A]]

−hh /

[ ]

(16) h/

hf

∫ [E ] f z

hp

z+ −hh /

∫ [E ]w zdz hf

h/2

∫ [E ]p zdz ,

h/

(10) −hh /

{ N } [ A] { } + [ B ] { } .

hf

hp

−h / 2

−h / 2

Taking into account Eq. (2), the normal force becomes:

−hh /

hp

∫ [E ]wdz

∫ [E ]pdz ,

+

+ i = p,ff ,w w.

(15)

h/2

h/2

∫ [E ]i z { }i dz

z

−h / 2

(9)

and the bending moment vector {M} = {Mx My Mxy}T is:

h/

hf

∫ [E ] f h/

i = p,f ,w

i = p,w,f .

Inserting elasticity matrices, Eq.-s (5), (6) and (7), into the ABD-matrix expressions, Eq-s (13), (14) and (15), the [A], [B], and [D] stiffness matrices for stiffened panel are obtained:

h/2

−h / 2

{M }

h/2

]

(8)

Internal forces and moments in the laminated plate are related to the stresses in the layer and to the deformations in the laminate and they are obtained by integrating Eq. (8) over the thickness of the stiffened panel. The normal force vector {N} = {Nx Ny Nxy}T is then:

{N }

where di is the distance between the geometrical mid-planes of the whole stiffened panel and the separate layer. The Dij components relate the bending moment with the curvatures in the laminate in case of pure bending and its components are defined as:

[ ]

(17)

hp

h/

hf

∫ [E ] f z

−h / 2

z+ −hh /

hp

2 ∫ [E ]w z dz hf

h/2

+

(11)

2 ∫ [E ]p z dz .

(18)

h / 2 − hp

Similar expression can be written for the moment resultants:

{M } [B ] { } + [D ] { }.

(12)

The Aij components relate the force flows with the strains in the laminate in the absence of curvatures and its components are defined as:

[ A]

h/2

∫ [E ]i dz

i = p,w,f.

(13)

−h / 2

The Bij components express the membranebending coupling, which is the effect of bending caused by the membrane loads and its components are defined as:

[B ]

Because the plate and stiffener layers have the same material orientation angle, which is 0 or 90 degree, the A-matrix components A16 and A26 in Eq. (16) become zero. Thus, the applied axial forces cause no shear strains and the applied shear forces cause no axial strains. Also B-matrix components B16 and B26 in Eq. (17), which relate the in-plane direct stresses to the laminate twisting and the applied torque to the in-plane direct strains, respectively, are zero. The same assumption is made for D-matrix components D16 and D26, Eq. (18), which relate the bending moments to the plate twisting and the torque to the plate curvature, respectively (Marsden & Irving 2002). 2.5

h/2

∫ [E ]i di dz

−h / 2

i = p,w,f, f

(14)

Relationship between out-of-plane shear forces and shear strains

An additional deflection will be produced by the shearing force, Qz, due to the mutual sliding of

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adjacent cross sections along each other. This phenomenon can be taken into account by applying Reissner-Mindlin plate theory. According to that the relationship between the shear force {QQ} and average shear strain {γ} is written as (Whitney & Pagano 1970): ⎧ QQ x ⎫ ⎡ DQx ⎨ ⎬=⎢ ⎩ QQ y ⎭ ⎣ 0

0 ⎤ ⎧ γ xxz ⎫ ⎨ ⎬, DQy ⎥⎦ ⎩ γ yyz ⎭

(19)

where DQx is laminate shear stiffness in stiffener direction and DQy transverse to stiffener direction. 2.5.1 Shear stiffness in the stiffener direction The shear stiffness DQx can be written as:

(

)

kxz G pt p + Gw hw + G f h f ,

DQx

(20)

where Gp is the shear modulus for the plate layer and Gw and Gf are the shear modules for web and flange layers, respectively. These can be obtained from the following relations: tw , s bf Gs . s

Gw

Gs

Gf

(21)

kxz is the shear correction factor in the xz-plane. Shear correction factor k relates the maximum shearing stress (τxz)max, i.e. the shearing stress at the centroid of the cross section, to the average stress (τxz)avg: k=

(

xz )avg

(

xz )max

The maximum shear stress (τxz)max in stiffened panel is defined as: (

h

xz )avg =

∫0 τ xz ( h

)dz

DQy

(23)

where h is the total length of the shear flow area in xz-plane, which for stiffened panel structure is equal with h = tp+hw+hf. The average shear stress can also be obtained by using the well-known approach by dividing the shear force by the effective shear area: (

xz )avg

≈

Aw

Qz . twt p tw h f

na

p ) + tw ( na

p)

2⎤

2 I ztw

⎦,

(25)

(

)

kyz G pt p ,

(26)

where the shear correction factor kyz 5/6 is derived from the plate shear energy and is included in Reissner-Mindlin plate theory (Whitney & Pagano 1970). 2.6

Relationship between homogenized internal forces and strains and curvatures

If [A], [B], [D] and shear stiffness matrix [D]Q are obtained, the Eq.-s (11), (12) and (19) can be summarized as a single matrix form: ⎧ N⎫ ⎡A B 0 ⎤ ⎥ ⎪ ⎪ ⎢ ⎨ M⎬ = ⎢ B D 0 ⎥ ⎪ Q⎪ ⎢ 0 0 D ⎥ ⎩ ⎭ ⎣ Q⎦ 2.7

,

Qz ⎡⎣Ap (

2.5.2 Shear stiffness transverse to stiffener direction The laminate shear stiffness transverse to the stiffener direction DQy can be written as:

For stiffened panel structure, the average shear stress can be calculated from:

(

=

where zna is the distance from the neutral axis and Iz is the second moment of area. Shear correction factor in stiffener direction depends on different parameters such as plate thickness, effective flange breadth and stiffener type. It is found here that it varies between 0.7 to 0.8, being smaller when the stiffener spacing and plate thickness is smaller and being larger in the structures, where the neutral axis is more close to the plate surface.

(22)

.

xz )max

⎧ ε⎫ ⎪ ⎪ ⎨κ ⎬. ⎪γ ⎪ ⎩ ⎭

(27)

Local bending of the deck plate

Due to lateral loads an additional plate bending between the stiffeners occurs, Figure 5, which cannot be predicted by the homogenized model, where

(24) Figure 5.

463

Local bending of the deck plate.

it is assumed that the stiffener spacing divided by the panel breadth equals zero (s/B = 0). In global ship analysis such tertiary effects are usually neglected, because the main focus is on the primary and secondary response. However, if the plate top or bottom stress is of interest, then the previously described theory is limited and the additional bending response should be included. The plate between the stiffeners can be considered as a clamped plate under pure bending, where the normal stress vector {σ} = {σx σy τxy}T is obtained by multiplying the strain vector{ε}l with elasticity matrix [E]p, which is defined according to Eq. (5):

{ σ }l [ ] p { ε }l .

(28)

Under pure bending behaviour the strain vector {ε}l is defined as: ⎧ε ⎫ ⎧κ ⎫ ⎪ x⎪ ⎪ x⎪ ⎨ ε y ⎬ = − z ⎨ κ y ⎬, ⎪ ⎪ ⎪ ⎪ ⎩ε xy ⎭l ⎩κ xy ⎭

(29)

where {κ} is the curvature, see Eq. (3). Taking into account Eq. (10), the bending moment vector becomes:

[ D] l { },

{M}

3 3.1

(34)

CASE STUDY General

The developed laminate element formulation is applied for the bending and vibration analysis of the typical deck structure in passenger ship, Figure 6a. The thickness of the deck plate is 6 mm and it is stiffened with HP100 × 6 profiles with the spacing of 680 mm. The web frame (T-440 × 7+FB150 × 10) spacing is 2400 mm and a pillar (Ø150 × 15) is located at every second frame. All structural parts are made of steel with Young's modulus of 206 GPa, Poisson ratio of 0.3 and mass density of 7850 kg/m3. All edges of the model have clamped boundary conditions. The local coordinate system and its origin is shown in Figure 6a. The FE analysis is carried out using NX Nastran 7.1 software. The pre—and post-processing has been done with FEMAP 10.2. 3.2

tp / 2

1 [E ]p z 2dz. 3 −t ∫ / 2

wtot = wg + wl .

(30)

where: [ ]l

where subscripts t and b denote top and bottom surface, respectively. Superscript g and l denote the global and local deflection, respectively. The total deflection in the plate thickness direction, wtot, is obtained by the same principle:

(31)

Laminate element model

The stiffened panel is modeled using laminate element, where each layer is described as CQUAD4 (quadrilateral) shell element, which has four

p

Carrying out the integration of Eq. (31) gives: 0 ⎤ ⎧κ x ⎫ ⎡ E νE ⎪ ⎪ 1 ⎢ νE E 0 ⎥⎥ ⎨κ y ⎬ . {M} = − ⋅ 2 ⎢ 12 (1 − ) ⎢⎣ 0 0 G (1 − )⎥⎦ ⎪κ xy ⎪ ⎩ ⎭ t p3

(32) 2.8

Total response

The total stress of the plate layer is found by adding the local plate bending stresses to the laminate solution, i.e.: ⎧ σ xtot ⎫ ⎧ σ xg ⎫ ⎧ σ xl ⎫ ⎪ tot ⎪ ⎪ g ⎪ ⎪ l ⎪ ⎨ σ y ⎬ = ⎨ σ y ⎬ + ⎨ σ y ⎬ , i = t,b, ⎪ tot ⎪ ⎪ g ⎪ ⎪ l ⎪ ⎩ τ xy ⎭ i ⎩ τ xy ⎭ i ⎩ τ xy ⎭ i

(33)

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Figure 6. (a) The dimensions of the deck structure (mm) and local coordinate system; (b) mesh density in fine mesh model.

Table 1.

Material properties of the laminate element. E [Pa]

G [Pa]

Layer

x

y

xy

xz

yz

ρ [kg/m3]

Plate Web Flange

2.06 ⋅ 1011 6.59 ⋅ 109 1.81 ⋅ 1010

2.06 ⋅ 1011 – –

7.92 ⋅ 1010 – –

7.92 ⋅ 1010 5.74 ⋅ 108 5.74 ⋅ 108

7.92 ⋅ 1010 – –

7850.0 60.6 232.3

Figure 7. Stiffened panel modeled as laminate element and T-girder as offset beam.

nodes and 24 Degrees of Freedom (DOF). The mesh size is 85 mm. According to the presented theory, the plate layer is described by isotropic material. Orthotropic material is applied for the web and flange layers, where the material properties are calculated according to the Rule of Mixture, see Eq. (6) and (7). Calculated material properties are presented in Table 1. Defining the correct location of the reference plane in the laminate element is very important. By default NX Nastran sets the plane at the geometrical mid-plane of the laminate, i.e. −0.5 times the element thickness. However, in order to properly represent the structural mass distribution, which is essential in the vibration analysis, the reference plane should coincide with the centre of gravity of the stiffened panel as shown in Figure 7. Primary stiffeners of the structure are modeled using offset beam elements (CBEAM), which follow Timoshenko theory. 3.3

3D fine mesh model

The plating and webs of the stiffeners and girders are modeled using four-node shell elements (QUAD4). The flanges of the girders and stiffeners are modeled with the offset beam elements, see Figure 6b. The mesh size is 85 mm, which is eight elements per stiffener spacing. The girder webs have 5 elements and stiffener webs one element in height (z) direction. 3.4

Bending response

The deck plating, is loaded with the uniform pressure of q = 10 kPa. This presents a typical

Figure 8. The deflection of the deck structure under uniform pressure loading of q = 10 kPa in (a) x-direction and (b) y-direction.

secondary and tertiary response problem in ship design, where the rigidity of the deck structure is tested under the lateral loads. The analysis results indicate that the maximum deflection as well as the deflection distributions can be predicted with good accuracy using the laminate mesh, Figure 8. The maximum deflection in x-direction, measured at the longitudinal girder (y = 0 m), is evaluated with less than 3% and at the stiffener line (y = 2.72 m) with less than 1% difference compared to the fine mesh model, Figure 8a. The deflections in y-direction show good agreement as well and the maximum deflection at the transversal girder (x = 7.2 m) is evaluated with less than 3% difference. The local plate bending between the stiffeners is also successfully taken into account and the maximum deflection at x = 8.4 m are predicted with less than 1% difference, Figure 8b.

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Figure 9 presents the normal stresses in x-direction at the mid, top and bottom surfaces of the deck plate at y = 2.72 m. Both the shape and the magnitude of the stresses are in good agreement with the fine mesh results. The results reveal that the maximum membrane stress of the deck plate is significantly lower than the local bending stress. Therefore, it is very important to consider the local plate bending, especially when the plate thickness is small and stiffener spacing is large. Figure 10 shows the average normal stresses in stiffener flange at y = 2.72 m. Again, both the shape and the maximum stress are predicted with good accuracy. The average difference from 3D FE model is about 8%. Figure 11 compares the shear stresses, τxy, in deck plate and the maximum bottom stress is evaluated with less than 2% difference. 3.5

Natural frequency analysis

Due to low structural stiffness of deck structures, the prediction of correct natural frequencies is

Figure 9. Comparison of deck plating normal stress distribution, σy, at y = 2.72 m.

Figure 10. Average normal stresses, σx, in stiffener flange at y = 2.72 m.

Figure 11. Comparison of deck plating shear stress distribution, τxy, at y = 0.

Table 2. Natural frequencies (Hz) for the deck structure. Fine mesh

Laminate h = 0.085

Laminate m = 0.680

Laminate m = 1.200

Mode nr

[Hz]

[Hz]

Dif. [%]

[Hz]

Dif. [%]

[Hz]

Dif. [%]

1 2 3 4

32.5 34.2 38.4 39.0

33.0 34.5 41.4 42.4

1.5 1.1 7.6 8.6

32.5 33.9 40.3 41.2

−0.2 −0.7 4.8 5.6

31.9 33.5 39.8 40.6

−2.0 −2.0 3.6 4.1

very crucial from the viewpoint of passenger comfort. Four most important frequencies are analysed using the fine mesh model and the laminate models with three different mesh densities: eight (h = 0.085 m) and one element (h = 0.680 m) per stiffener spacing and two elements (h = 1.2) per web frame spacing. The mass matrix is formulated using the lumped scheme, where the distributed mass of the element is simply divided between the nodes. According to that, a diagonal mass matrix is built, which provides higher computational economy. The results are listed in Table 2 and the mode shapes are presented graphically in Figure 12. The natural frequencies calculated using the laminate element models agree well with the fine mesh ones, even if only 4 laminate elements are used to describe the vibration shape (two elements per web frame). In case of coarser mesh the laminate model gives better results. This is because in the lumped mass distribution the element mass is divided between nodes, which makes the structure less stiff and therefore to some extent balances out the error of having larger elements.

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calculated more accurately than in Satish Kumar & Mukhopadhyay (2000). There they use the factor of 5/6, which follows the parabolic shear stress distribution. However, the shear distribution in stiffened panel is more complex and it is found here that the correction factor generally varies between 0.7 to 0.8 depending on the stiffener type, spacing and plate. From the orthotropic material properties it is possible to derive stiffener parameters such as spacing and the thickness of web and flange. Therefore, the utilization of the laminate element in the optimization process of the ship structures seems possible and attractive. Future work may also be done for including fluid effects in vibration analysis. ACKNOWLEDGEMENTS The research work carried out in this paper was funded by STX Turku Shipyard and the financial support is gratefully appreciated.

REFERENCES

Figure 12. Analyzed vibration modes with (a) laminate element model (h = 0.085 m) and (b) fine mesh model.

4

DISCUSSION

The equivalent shell element presented in this paper is able to assess ship global and local static and vibration response accurately and relatively fast. This makes it very suitable in early design phases. The mesh sensitivity analysis showed that two 4-noded elements per web frame are an optimal size to evaluate the lower natural frequencies of deck structure with good accuracy. Therefore, this mesh density is recommended for global FE model if secondary responses are of interest. Compared to existing equivalent elements the developed element includes membrane-bending coupling, which stiffness properties have been calculated using laminated shell theory. Due to layer-formulation, it is possible to extract the normal and shear stresses from plate, stiffener web and flange separately. Secondly, local plate bending between the stiffeners is added to the overall laminate solution. This enables to analyse the tertiary response, i.e. deck plate top and bottom stresses, straight from the global model and loses the need for the tedious sub-modeling. Thirdly, the shear correction factor in stiffener direction is

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Thermal and strength analysis of LNG tanks and their supporting structures in the early design stage M. Biot, N. Mantini & L. Moro Department of Engineering and Architecture, University of Trieste, Trieste, Italy

ABSTRACT: In the paper, some calculations are presented aimed to the definition of a procedure for the structural design of a type C tank and its supporting structures. More specifically, calculations have been performed on a bi-lobe type C tank for a LNG carrier. Different solutions are proposed for the numerical simulation of interaction between thanks and ship structures, giving suggestions to designers about the modelling of the contact of tank with its supports and on how to partially model the ship structures below tank and their boundary conditions. Moreover, a simulation procedure for heat transfer performed by FE analyses is presented, useful for an early stage evaluation of temperature distribution on ship hull structures. 1

INTRODUCTION

The increase in the worldwide demand for natural gas leads to ever safer containment systems for gas transportation on the ocean (Arai, et al. 2012). Moreover, the pressing requirement of reducing CO2 emissions forces owners to equip new ships with dual-fuel diesel engines which require storage tanks for natural gas. In general, for both storage and transportation, type C pressure vessels give some advantages compared to other tank alternatives when a small to medium storage capacity is required. At the moment, rules and guidelines are well established for the structural design of pressure tanks, while standards are under development to guarantee higher degrees of safety for LNG fuel systems. Structural design of a type C tank has to be performed according to the basic requirements given by the IGC Code (IMO, 1993) and also by referring to the standard practice given by ISO (ISO, 2007). Additional requirements are given by the classification societies for the assessment of the interaction between tank and hull structure. Direct structural calculations need in general to be performed to check agreement with rules and guidelines. Setting of FE models is not so straightforward being the tank simply supported by hull structures, and such interaction implies a nonlinear approach to be solved. Moreover, to properly choose the grade of the steel for both the supporting structures and the surrounding ship structures, a thermal analysis needs to be performed (Han, et al. 2011, Zhang, et al. 2008). Such analyses are also useful for the dimensioning of the thermal isolators

of the tanks. Both structural and thermal analysis may be carried out by applying the FE method. 2

DESIGN STANDARDS FOR TYPE C TANKS

Pressure vessels are usually designed according to the ISO or ASME standards: the ISO 16528-1:2007 is considered as a sound reference in pressure vessel design and the EN 13445-1:2009 is its European equivalent standard. The latter establishes in the Annex C a design method based on different stress categories. The proposed approach leads to the calculation of an equivalent stress that takes into account both bending and membrane stresses. According to the ISO standard, the equivalent stress is calculated considering all the loads acting on the tank structures in the standard operating condition. Furthermore, such equivalent stress is required to satisfy the following relation: σ m+b ≤ 1.5 f

(1)

where σm+b is the equivalent stress and f is the nominal design stress defined as: f

i

⎛ Rp 0.2 / T Rm / 20 ⎞ f1, f2 ) = min ⎜ , 2.4 ⎟⎠ ⎝ 1.5

(2)

where Rp0.2/T is the yield strength at 0.2% offset at the analysis reference temperature T, and Rm/20 is the tensile strength at 20 °C. The pressure vessel design performed according to the EN 13445-1:2009

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European Standard is considered as a minimum scantling requirement by the IGC Code. The IGC Code outlines a different structural analysis according to the type of tanks that are installed on-board a ship, divided in three different categories. As for the type C independent tanks, the IGC Code sets that the equivalent membrane stress σmem resulting from a FE analysis is to be compared with an allowable stress σall as follows: ⎛R R ⎞ σ mem ≤ σ all = ⎜ m , e ⎟ ⎝ A B⎠

(3)

where Rm is the minimum tensile strength and Re is the minimum yield strength, both evaluated at room temperature; A and B are coefficients depending on the tank material. Ship classification societies, as the Italian RINA, set down a series of guidelines for the estimation of the design loads and structural analysis, according to the IGC Code specifications. As for the type C tanks, the RINA rules are focused on the structural scantling and analysis of the internal stiffening rings in way of the tank supports. According to these rules, the equivalent stress σm+b resulting from FE analysis should consist of both membrane and bending stresses, and it must comply with the allowable stress σall, obtained as follows: m+b

≤ σ all = min (0.57

0.85 85 eeH m, 0

)

(4)

where ReH and Rm are the minimum yield strength and the minimum tensile strength respectively. 3

IDENTIFICATION OF THE CRITICAL LOADS

As mentioned before, the cargo containment systems for LNG ships are regulated by the International Maritime Organization through the International Code for the Construction and Equipment of Ship Carrying Liquefied Gas in Bulk (the so called IGC Code). According to such international code, type C tanks are designed by considering both the vapour pressure acting inside the tank, which is set to a minimum design value, and the inertial loads, which are to be evaluated considering the long-term distribution of ship motions on irregular sea. In the structural analysis of tank supporting structures and tank shell plating, design load is given by applying both the own weight of the tank structures and the total pressure exerted by the cargo. The latter is accounted by the equivalent internal pressure Peq resulting from the design

vapour pressure P0, defined as the maximum gauge pressure at the top of the tank, and the internal liquid pressure Pgd, that is the dynamic contribution, due to ship motions, to the equivalent internal pressure, as follows: Peq P0 +(Pgd )max

(5)

The internal liquid pressure Pgd is generated by the acceleration of the centre of gravity of the cargo and is obtained by the combination of the effects of gravity and those of ship motions as follows: Pgd =

aβ Zβ ρ 1.02 ⋅ 10 4

[bar]

(6)

where aβ is the acceleration, dimensionless respective to the acceleration of gravity, resulting from inertial loads in an arbitrary direction β, Zβ is the largest liquid height above the point where the pressure is determined, measured from the tank shell plating in the arbitrary direction β, and ρ is the cargo density at the design temperature. As shown in Equation 6, the internal liquid pressure Pgd is function of the dimensionless acceleration aβ and the largest liquid height, which are in turn both function of the direction β. In order to obtain a straightforward structural analysis of the tank, once the internal liquid pressure Pgd has been assessed by varying the direction β in its own range, the equivalent internal pressure Peq is calculated according to Equation 5, considering the maximum value of internal liquid pressure obtained by Equation 6 acting on each structural element. Accelerations due to ship’s motions should be evaluated according to the worst condition experienced by the ship in its operating life. If the ship is designed to operate world-wide, the IGC Code suggests to calculate the inertial accelerations due to ship’s motions by making reference to a probability level of 10−8 in the North Atlantic. In this regard, simple formulae are provided in order to calculate inertial acceleration components—the longitudinal acceleration ax, the transverse acceleration ay and the vertical acceleration az. The acceleration components so obtained are used for evaluating the dimensionless accelerations aβ. In the procedure set by the IGC Code, those three linear accelerations are considered as the semi-principal axes of the so called “acceleration ellipsoid”, whose centre lies below the centre of gravity of the tank at a unit-magnitude distance. So, the dimensionless inertial acceleration aβ for an arbitrary direction is equal to the magnitude of the vector, forming an angle β with the vertical axis, that joints the tank’s centre of gravity with the point laying in the (lower) curve of the ellipse.

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The procedure suggested by the IGC Code for the determination of loading conditions on tanks, is considered as a sound reference by the main classification societies. Also in the RINA rules the design loads are to be calculated according to IGC Code prescriptions. In particular, RINA rules state that the inertial internal liquid pressure is to be calculated in ship conditions that are mutually exclusive: the upright ship conditions and the inclined ship conditions. In the upright ship conditions, ship’s motions are all in the X-Z plane and they are produced by bow waves, whereas in the inclined ship conditions the ship’s motions are all in the Y-Z plane. So, for determining the dimensionless acceleration aβ, two different “acceleration ellipses” have to be defined. The procedure for determining the design loads suggested by the IGC Code leads to define the equivalent internal pressure for each calculation point. This is a straightforward procedure when an analytical approach is implemented for the strength analysis. Otherwise, in a direct calculation approach carried out by means of FE analyses, the outlined procedure should be modified in order to obtain a series of Load Cases, each of them corresponding to a proper angle β. In that way, the entire tank structure is analysed for each Load Case and the worst results are used for the structural design. 4

TECHNIQUES FOR THE FEM MODELLING

4.1 The simulation of the tank-support contact In the FE analysis of type C tank structures and their supporting structures, particular attention should be paid to the modelling of constraint and boundary conditions, and several considerations are necessary. Tanks are placed inside holds and are supported by cradles. In order to thermally insulate tanks, thermal insulating materials cover the tanks shell plating while a special type of wood is placed between tank and cradles and it is glued to the tanks shell plating. The wooden seats insulate tanks and so protect the steel of the hull against very low temperatures. As a matter of fact, low temperatures could lead to the use of high grade steels, and so to high construction costs, to prevente the risk of brittle fracture. Tanks are simply supported by the cradles which, along with the anti-flooding installed on deck, prevent the vertical rigid body motions of tanks. So, the cradle is the interface of a unilateral contact between two deformable bodies, the thank and the ship structure. Such a constraint type introduces a non-linearity in the model simulating the interaction between the two bodies.

A simplified approach to the contact simulation consists in replacing the non-linear unilateral constraint with an equivalent linear bilateral constraint. That is made by means of rod elements connecting tank shell plating and cradle upper plate. According to this approach, the wooden supports are not explicitly modelled and the rod elements are characterized by a stiffness equal to that of the replaced wooden section. So, FE analysis becomes a linear static analysis. As the rod elements could undergo a tensile action which is inconsistent with the actual contact mechanism, an iterative process needs to be carried out, where at each step all elements in tension are removed from the FE model. The iterative process ends when all the rod elements simulating the wooden supports are only subject to compressive actions. An improved way to assess strength of tanks and cradles structures is carrying out a non-linear FE analysis. The non-linear FE analysis can be performed by setting a series of parameters for optimizing an iterative algorithm that simulates the unilateral contact, so avoiding the interpenetration between the bodies but allowing their mutual detachment. At each step of the iterative procedure, loads are gradually increased, contact forces are calculated and, once the balance of such forces is verified, a new iteration starts. All parameters of the iterative procedure, such as trial load step, minimum percentage of load at each step and maximum allowable error in the calculation of the balance of contact forces, should be properly set (MSC, 2009). This analysis allows the identification of the contact points which belong to both the deformable bodies, and the evaluation of the contact forces. The tangential forces may be calculated if a friction coefficient is introduced. Moreover, the presence of rigid body motions can also be checked. 4.2

The simulation of the tank support

Tanks are supported by the cradles structures and to strengthen the shell plating of the tanks in view of the cradles, stiffening rings are placed inside the tanks. Deformability of the two elements in contact needs to be properly modelled in order to come to an accurate evaluation of the contact forces. The simplest way to perform the calculation is to model the cradle upper plate as a rigid body, which can clearly leads to overestimate the reaction forces on the cradle, and in turn to overestimate the supporting structures strength. On the other hand, explicitly modelling of the entire ship hull structure is a very time consuming simulation, whose benefits in refinement of values of the reaction forces need to be proven. This is why

471

a compromise should be reach on the extension of the explicitly modelled support structure. In order to do that, FE simulations should be carried out, which differ from each other for the extension of the hull structure that is taken into account. 4.3

The simulation of the thermal effects

Liquefied natural gas is carried in pressure vessels at very low temperature. Since the early design phases, it is useful to predict the temperatures inside the ship’s hold in order to set the most proper grade of the steel for the hull structures. Temperature inside the ship’s hold is due to three different heat transfer mechanisms: conduction, convection and radiation. The thermal conduction is responsible of heat transfer to structures that are in direct contact with the tank, such as the cradles. The thermal convection is the prevalent mechanism in heat transfer within the tank surrounding cavities, that are the volume between tank and inner hull plating, and the cavities in the double bottom. The air inside such cavities is moved by buoyancy forces that occur for the density variations due to temperature variations, which is a typical case of free convection. Also the thermal radiation needs to be considered in evaluation of the thermal energy transfer. As the thermal loads are considered not varying over time, a steady-state thermal analysis can be executed for determining the temperatures of each part of hull structures. In such an analysis, an iterative solving procedure is used to calibrate the heat transfer coefficients, as they depend on the average temperatures inside the cavities and on the wall temperatures. Steady-state heat transfer analyses may be performed by applying the FE method, by using specific tools generally implemented in the FEA software for the structural analysis. The analysis should be carried out considering the hardest ambient condition. The IGC Code sets the design ambient temperatures and specifies that for a cargo temperature below −10 °C, the ambient temperatures to be considered are 5 °C for the air and 0 °C for the seawater. These temperatures are valid for analysis of world-wide service ships, whereas lower temperature values may be fixed for ships engaged on particular routes. Once the temperatures have been calculated, the stresses due to thermal loads can also be evaluated. 5

have been performed on an innovative LNG carrier whose main dimensions are reported in Table 1. The overall load capacity of LNG is around 40000 m3. Cargo is loaded in four bi-lobe type C tanks. Each large-size tank is made by two cylindrical shaped bodies divided from each other by a longitudinal bulkhead, as shown in Figure 1. Tanks are about 30 meters in length, have a diameter of cylindrical body of about 15 meters and the cylindrical body ends with two torispherical end caps. Tanks are designed to sustain a vapour pressure P0 of 6 bar(g) and the design temperature is equal to −164 °C. Tanks shell plating is supported by two internal stiffening rings, as shown in Figure 1. The inner structure of each bi-lobe tank is not symmetric, because the inner bulkhead is stiffened on one side and also the webs of stiffening rings on the longitudinal bulkhead have different height. As the independent tanks do not form part of the hull structures, they are not effective to the hull strength and are not stressed by the ship hull girder loads. Thereby, cradles structures and tank structures deform independently from each other. Tank rigid body motions are restrained by stop devices. Three different significant load cases have been considered for the tank structure scantling. Each Load Case specifies the loads to be applied to the tank and cradle structures for a given inclining ship angle, which has been determined according to the IGC Code. Such load cases correspond to the following ship conditions: upright ship Table 1.

Main characteristics of the LNG carrier.

Length between perpendiculars Breadth Depth at freeboard deck Draught for scantling Service speed

180.00 m 30.00 m 16.00 m 9.00 m 16.00 kn

THE CASE STUDY

The above-mentioned operative procedure for calculating the tank and cradle structures, has been implemented in a specific case study. The FE structural and thermal analyses of a tank and its support structures

Figure 1. FE model of the bi-lobe type C tank: view of tank shell plating and inner structures.

472

condition (βt = βl = 0°, Load Case 0°), trimmed ship condition, where inertial loads refer to a maximum pitch angle of 8° (βl = 8°, Load Case 8°) and inclined ship condition, where the maximum roll angle is of 20° (βt = 20°, Load Case 20°). For each load case, both static and dynamic loads have been taken into account in the simulations. 6

ANALYSIS OF THE CONTACT SIMULATION

A comparative study is here discussed, concerning the analysis of the contact between a tank and its support structure. Two different methods have been applied to solve the contact simulation problem: a simplified one, based on the linear bilateral equivalent constraint approach, and an improved one, based on the explicit non-linear simulation. In both approaches, the FE model includes tank structure, and supporting structures have been limited to the cradle upper plates, whose nodes have been rigidly restrained to linear motions. This assumption does not affect the robustness of the comparative study. In the FE model for the non-linear analysis, the wooden supports have been explicitly modelled by means of brick elements, whereas in the simplified linear simulation they have been replaced by equivalent linear-elastic rod elements. All calculations have been performed by using the MSC Patran/ Nastran software. Results will be discussed making reference to the aft inner stiffening ring and its associated tank shell plating, being that the most critical part of the tank structure. In Figure 2 the deformation shape is shown of the tank in way of the cradle (partial view) evaluated according to the non-linear FE simulation. In Figure 3, the FEA outcomes on the aft inner stiffening ring are shown in terms of displacements and stresses respectively, relevant to the linear-elastic analysis and inclined ship condition (Load Case 20°). In order to compare the results achieved by the two approaches, the deformations of the aft inner stiffening ring have been analyzed by measuring

Figure 2. Example of deformation shape (values in mm) of the tank in way of the cradle (partial view) evaluated according to the explicit non-linear FE approach.

Figure 3. Results of the linear bilateral equivalent constraint approach for the Load Case 20°: displacement field on the aft inner stiffening ring (in mm) and relevant von Mises stresses (in MPa). Table 2. Out of roundness (in mm) δUY calculated on the aft inner stiffening ring of the tank by the two FEA approaches. Load case

Explicit non-linear approach

Linear equivalent approach

βt = βl = 0° βl = 8° βt = 20°

−0.71 3.55 −0.82

−0.97 3.21 −0.99

the out of roundness of such structural element. Relative displacements of two pairs of nodes have been measured on the horizontal plane (points P2 and P4) and on the vertical plane (points P1 and P3) and displacement of each node has been considered positive when the node moves away from the lobe centre. So, the out of roundness on the horizontal Y-direction and on the vertical Z-direction comes out to be respectively: δU Y δU Z

uP 2 + uP 4 uP1 + uP 3

(7)

In Table 2 and in Table 3 the δUY and δUZ values are shown for the non-linear and linear analysis and for each Load Case. Agreement between the two approaches is clear, as the deformation shape is the same for each different load case. On the other hand, values differ from each other up to 50%. The forces mutually exerted between the parts on tank to cradle contact are shown in Figure 4 and Figure 5. The forces coming from the non-linear FE simulation have a smoother distribution than the forces given by the simplified approach. Distributions of contact forces obtained by the two methods are in general not so diverging from each other, even if maximum

473

Table 3. Out of roundness (in mm) δUZ calculated on the aft inner stiffening ring of the tank by the two FEA approaches.

Table 4. Contact forces FC, total values (in MN), calculated on the aft tank-cradle contact by the two FEA approaches.

Load case

Explicit non-linear approach

Linear equivalent approach

Load case

Explicit non-linear approach

Linear equivalent approach

βt = βl = 0° βl = 8° βt = 20°

1.99 1.34 2.58

1.13 0.72 1.87

βt = βl = 0° βl = 8° βt = 20°

53.8 51.9 46.4

62.7 62.8 57.0

Table 5. Equivalent von Mises stresses (in MPa) calculated on the tank structure by the two FEA approaches. Load case

Explicit non-linear approach

Linear equivalent approach

βt = βl = 0° (*) βl = 8° (*) βt = 20° (**)

319.0 307.0 339.0

289.0 275.0 348.0

(*): in way of the Y-joint. (**): on the tank shell plating at the aft inner stiffening ring. Figure 4. Contact force distribution (from starboard side to portside of ship) calculated according to the linear equivalent approach.

Figure 5. Contact force distribution (from starboard side to portside of ship) calculated according to the explicit non-linear approach.

peak values at the cradle ends (at midship) differ from each other in the range from about 0% to 21%, and the higher values are relevant to the non-linear approach (divergences are higher when the cradle ends at ship sides are considered). As for the total value of the contact forces (see Table 4), maximum difference between the two approaches is in the range from about 16% to 23% and the lower values are relevant to the non-linear approach. Analysis of stresses on the tank structures shows (see Table 5) that the maximum stress in terms of von Mises equivalent stress arises on two different parts: the so called Y-joint, that is the welded joint connecting the inner longitudinal bulkhead and the two end cap shell plating, and the tank shell plating in the

area of the contact between tank and support. Stress values differ from each other up to 12%, with the larger values relevant to the non-linear approach. As for the strength assessment of the tank supporting structures, basing on the comparison of results in terms of both contact force distribution and total value of the contact forces, it can be pointed out that the simplified approach leads to an approximation on the safe side, except for the central part of the support at midship, where cradle suddenly ceases to offer support to tank—here, peak values are higher using the non-linear approach. On the other hand, results in terms of stresses give a clear indication on the convenience to use the non-linear approach for the assessment of the tank structures. In conclusion, the simplified approach, even if more time consuming than the non-linear one, being founded on the basics of the structural analysis is of ease application for solving the tank-cradle contact. So the simplified approach is recommended by a practical point of view, only in the early design stage of tanks and their supporting structures. 7

ANALYSIS OF THE SUPPORT EFFECT

The tank supporting structures are special beams protruding upwards from the double bottom and are tightly connected to the main structures of the hull. Deformation of the cradles, when subject to the loads transmitted by the tank, is strictly dependent on the deformability of the main structures of the hull, that are the main transverse frames and the longitudinal elements of the hull girder. So, in the strength assessment of tanks and their supporting

474

structures it becomes of great importance to accurately simulate the behaviour of hull structures close to the tank supporting structures. Moreover, in case of FEM models carried out on a partial model of the ship, also boundary conditions on longitudinal elements need to be accurately defined. In the case here presented, a FEM model has been generated extending for the length of a ship’s hold at the mid-length of the ship. The effects of hull girder deformation have been neglected and the hypothesis has been made of all tanks fully loaded. Three different configurations of tank supporting structures have been taken into account, which are shown in Figure 6. In the first configuration, hereinafter called “Support A”, just the cradle upper plates have been modelled, each of them rigidly constrained at their lower surfaces. In the second configuration, called “Support B”, the whole tank supporting structure has been modeled for a longitudinal extension, at each tank support, of two main frame intervals. In the Support B model, the hull structures are rigidly constrained at their ends. In the third configuration, called “Support C”, the whole hull is modelled for the longitudinal extension of the hold containing the tank. In the Support C model, the hull structures are rigidly constrained at the ends of the model, that is at the nodes located at the transverse bulkheads. All FE analyses have been carried out by considering the upright ship condition (Load Case 0°).

For the third configuration also the effect of the sea pressure on the hull plating has been studied, by considering three different ship conditions: hull girder on the wave crest, on the wave hollow and on still water. Here, being the sea pressure effects on the interaction between tank and hull structures almost secondary, just the still water condition has been taken into consideration. The support effect calculated by the different FE models is discussed on the basis of deformations, contact forces and stresses. In Table 6, the δUY and δUZ values are shown for the three support types. Even if the deformation shapes are the same for the different cases, differences in values are in the range from −13% to 24% when related to the Support C case. Distribution of the forces mutually exerted between aft cradle and tank are shown in Figure 7. The differences in the forces distribution may be explained by the different deformability of the considered support structures. The small deviations in the total contact force values shown in Table 7 is the prove of the higher level of accuracy of the non-linear contact simulation. As for the stresses arising on the tank structures (see Table 8), the calculated values give a clear indication on the convenience to use the FE model of the entire hold—in any case, the simplified models give indications on the safe side. In conclusion, differences in the relative displacements and in the stresses lead to consider not negligible the role of the deformability of the support especially in the strength optimization process applied to the inner supporting structures of the tank. Table 6. Out of roundness (in mm) δUY and δUZ calculated on the aft inner stiffening ring of the tank by the three FE models.

Figure 6. Types of supports considered for the study of the support effect.

Out of roundness

Support A Support B

Support C

δUY δUZ

1.98 −1.15

1.64 −0.93

1.29 −0.81

Figure 7. Contact force distribution (from starboardside to portside) calculated according to the non-linear contact simulation approach and for different support types.

475

Table 7. Contact forces FC, total values (in MN), calculated on the aft tank-cradle contact by the three FE models. Contact force

Support A

Support B

Support C

FC

53.8

53.3

53.1

Table 8. Equivalent von Mises stresses (in MPa) calculated on the tank structure by the three FE models. Location

Support A

Support B

Support C

Aft inner ring Shell plating (*) End caps

272.0 482.0 448.0

263.0 449.0 447.0

227.0 440.0 446.0

9

(*): at the aft inner stiffening ring.

8

Figure 8. Calculated temperatures (in °C) on the ship’s structures in the area of the contact with the tank.

ANALYSIS OF THE HEAT TRANSFER

A thermal analysis for determining the temperature distribution on the hull structures has been carried out. The initial thermal loads have been set according to IGC Code (ambient temperatures set to 5 °C for the air and to 0 °C for the seawater) and the tank temperature has been set to −164 °C, which is the design cargo temperature. The radiation of sun has been neglected as well as the thermal convection due to the seawater flow around the hull while the ship is moving, so considering the worst situation of ship at rest in the night time. A series of simulations has been performed by the FE method, which differ from each other for the method used in modeling the mechanism of heat transfer in the thermal convection. This has been made by changing the degree of thermal coupling between walls of the same cavity and by applying different criteria to identify the coupled surfaces in complex cavities, such as the hold volume between tank and inner hull plating. A comparison of the temperature distributions predicted by the different simulations has been made, in order to improve the method for heat transfer calculation. Figure 8 shows the calculated temperatures. By this study it has been shown that the FE method may be successfully applied for approaching the evaluation of the temperature distribution on the structures of the ship in the zone affected by the presence of the LNG tank. Clearly, further investigations are necessary to confirm that the criteria applied in the heat transfer simulation are adequate to model the heat transfer phenomenon.

CONCLUSIONS

With the aim of setting a procedure for the strength analysis of a type C tank and its supports, a case study has been performed. The main practical outcomes are valuable suggestions for designer on the implementation of the FEA simulation for both the structural and thermal analysis of type C tanks. Specifically, comparison of outcomes of FE models generated according to different criteria leads to the following conclusions: the contact should be simulated by an explicit non-linear simulation, while the linear bilateral equivalent constraint approach could be advisable only in an early design stage; the tank supporting structures need to be modelled for the entire hold length, especially in a strength optimization process of the same; the temperature distribution and the relevant stresses may be evaluated by a relatively simple FEA simulation. REFERENCES Arai, M., Bogaert, H., Graczyk, M., Ha, M.-K., Kim, W.-S., Lindgren, M., Martin, E., Noble, P., Tao, L., Valle, O., Xiong, Y. (2012). Natural Gas Storage and Transportation. Proceedings of the 18th International Ship and Offshore Structures Congress, ISSC2012; Rostock, 10–13 September 2012. Volume 2, Pages 67–112. Han, S., Bae, J., Jhon, K., Suh, Y., Eom, J.-K. (2011). Assessing structural of inner hull structure under cryogenic temperature, Proceedings of the 30th International Conference on Offshore Mechanics and Artic Engineering, OMAE2011; Rotterdam, 19–24 June 2011.Volume 2, Pages 961–967. International Maritime Organization (1993). International Code for the Construction and Equipment of Ships Carrying Liquefied Gases in Bulk (IGC Code). IMO, London. International Organization for Standardization (2007). ISO 16528-1:2007—Boilers and pressure vessels— Part 1: Performance requirements. ISO, Geneva. MSC.Software Corporation (2009). MSC Nastran, Contact Analysis, NAS133—Course Notes, Introduction to Contact Analysis. Zhang, W.-X., Mo, J.-H., Jin, L.-M., Cai, Z.-Y., Liu, C.-T. (2008). Analysis of thermal transfer and thermal stresses for 138000 m3 membrane type LNG carrier, Journal of Ship Mechanics, Volume 12, Pages 770–777.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Shear response of prismatic passenger ship hull-girders K. Melk, J. Romanoff, H. Remes & P. Varsta Department of Applied Mechanics, Aalto University, Espoo, Finland

H. Naar Tallinn Technical University, Tallinn, Estonia

A. Niemelä STX Finland, Turku, Finland

ABSTRACT: Present paper investigates the shear-induced secondary normal stresses in the balcony openings of modern passenger vessel with narrow superstructure. The investigation is carried out using Finite Element Method. Two loading schemes are considered, i.e. cosine shape loading simulating realistic loading caused by waves and four-point bending load enabling deeper analysis on the shear-induced responses. The investigation shows that the shear-induced normal stresses have considerable effect to the overall stress state around balcony openings. It also shows that the passenger ship hull girder shear response is extremely complicated phenomenon and therefore entire ship length needs to be modeled when assessing the bending response of the ship. 1

INTRODUCTION

During recent decades, there have been significant changes in the size and structural complexity of passenger ships. For example, in the early 1970s operated passenger ships with a gross tonnage (GT) of 20000, while in 2010 the largest cruise ship in active service had the corresponding figure around 225000. In these ships, the increasing demand for large and open spaces, restaurants, atriums and theatres has caused complex structural behavior, which does not follow the basic beam theory as it does for many other ship types such as tankers and bulkers; see for example Bleich (1952), Fransman (1988), Heder and Ulfvarson (1991), Naar et al. (2004) and Andric and Zanic (2010). Therefore, direct calculations based Finite Element Method (FEM) are required; see Paulling, J.R. and Payer (1968); McVee (1980); Payer (1980); ISSC (1997); Gudmunsen (2000); DNV (2007); Zanic et al. (2007, 2010). The deficiency of direct FE-calculations is that the analytical understanding of the hull girder response is lost. Numerous balcony, door and window openings in the superstructure form structural discontinuities that reduce the shear stiffness of the superstructure (Muckle, 1962, 1966; Fransmann, 1988; Naar et al., 2004; Andriz and Zanic, 2010). As the large balcony openings are exposed to shear, the plate strips between the openings are exposed to secondary bending (Fransman, 1988; Gudmunsen 2000;

Bäckström, 2009; Andric and Zanic, 2010). This is taken into account in global strength analysis using various FE-modeling approaches as summarized by Andric and Zanic (2010). Then the aim of the strength analysis is modeling of global hull girder stiffness correctly. The deformations of the

Figure 1.

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Balcony opening of a cruise ship.

global model are then transferred as displacement boundary conditions to local model in which the focus is on stress response evaluation of a single opening. Romanoff and Varsta (2006) and Romanoff et al. (2007) have investigated the secondary

Figure 2.

bending stresses on web-core sandwich beams with discrete core. There is analogy to passenger ships with balcony openings. The vertical plate strips between balcony openings are similar to webs of sandwich panels. Both of these bend under horizontal shear. However, it was shown that in

Main frame of the investigated ship.

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addition to bending of webs the face plates bend and are therefore exposed to high secondary normal stresses. Following the analogy, horizontal side shell plate strips between balcony openings bend as well. When coarse mesh modeling is used, these stresses cannot be captured accurately since too few elements are often used to model the opening. The purpose of this paper is to investigate numerically the shear-induced secondary response of passenger ship hull girders. In order to simplify the analysis and to create the analogy to beams, a prismatic structure is considered. The investigation is carried out with 3D-Finite Element Method where fine meshing of the balcony openings throughout the entire ship is used to capture the secondary bending response accurately. The hull girder is first exposed to four-point-bending which enables separation of the global shear force and bending moment induced effects on the response. Then the ship is exposed to more realistic cosine shape pressure load on bottom plating of the ship. Both the displacement and stresses are investigated. The ship considered is simplified version of a new concept developed by STX Finland called xP-tray (STX, 2012), which was optimized by Bergström for minimum weight (2010); see Figure 2. 2 2.1

CASE DESCRIPTION Case ship

The considered passenger ship with narrow superstructure has a length of 246 m, a height of 62,4 m and a width of 38,6 m. A wide open sundeck (8th deck) is positioned on top of the hull, which extends the overall width of the ship up to 49 m. Ship has in total 22 decks, a double bottom, lifeboats areas and an extremely narrow superstructure; see Figure 2 for the main frame and scantlings. The web frame and pillar spacing is 2565 mm and 5130 mm respectively. Lower decks in the hull are supported by the longitudinal bulkheads and by the pillar lines. Upper decks in the superstructure are supported only by the pillars and by the side shell. Side shell at the superstructure has large balcony openings with dimensions of 1955 × 1767 mm starting from the 9th deck until to the deck 20th. The main frame of the ship is presented in Figure 2. 2.2

shell with the opening, it was modeled using four parabolic elements between two consecutive openings; see Figure 3. The pillar, web-frames and girders are modeled with parabolic beam elements using offset. 2.3

Load and boundary conditions

The load and boundary conditions were selected in the way that the bending moment and shearinduced stresses can be clearly separated. The basis was the standard relation from beam theory between external distributed loading, q, shear force, Q, and bending moment M: d 2M dQ = = −q dx dx 2

(1)

This allows the beam to be completely free from external shear force over large distance in length direction, i.e. x-direction, in case of four-pointbending; this region is between x = L/3 … 2L/3; see Figure 4. Similarly at region x = 0 … L/3 the effect of shear is present meanwhile the bending moment increases linearly. This allows in the neighborhood of x = 0 investigations on pure shear-induced effects (M = 0). In case of cosine shape loading, these two modes are not as clearly separated as in four point bending; however still the shear and bending moment-induced reactions can be extracted; see Figure 4. The cosine loading, M(x) = M0/2(1 − cos(2πx/L)), was modelled as self-equilibriating pressure on

Finite element mesh

The finite element mesh was created with the aim to obtain accurately the shear-induced normal stresses at balcony openings. This meant that instead of using the common practice, i.e. orthotropic equivalent shell element (e.g. Fransmann 1988; Heder & Ulfvarson 1991) to model the side

Figure 3. Finite Element Mesh of the investigated ship.

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Figure 4. The cosine shape loading and four-point bending.

bottom plating of the quarter FE-model. The symmetry conditions were applied in the xz- and yzplanes by fixing the displacement and rotations, i.e. uy = Rx = Rz = 0 and ux = Ry = Rz = 0 respectively. The four point bending loads were modelled as nodal forces at x = L/3 on bottom of trasverse bulkhead. In addition the model was constrained in vertical direction at transverse bulkhead at x = 0. 3 3.1

RESULTS Displacements

Figure 5 presents the displacements of the hull girder and the balcony opening for the cosine shape pressure load and the four point bending. It is clear that the global shear force induces warping of the cross-section, which can be seen at global scale at the ends of the superstructure and at the local scale at the balcony openings. It is also seen that the shear deformation occurs both in vertical and horizontal direction. At the region of zero global shear force, the warping is not present. It is important to notice that the warping of balcony openings induces secondary bending to the side shell of the ship superstructure as well as in the decks. 3.2

Normal and shear stresses at the hull

Figure 6 presents the normal stress distribution of the hull for different decks for ship where the

superstructure is considered while Figure 7 presents the case when superstructure has been removed. As Figure 6 shows the normal stress at decks follows to some extent the shape of the bending moment distribution. However, the shape does not match the shape perfectly; that is in case of four-point bending it seems that the hull carries less stress and in different way than the external bending moment distribution would indicate. As Figure 7 shows when superstructure is removed, this effect is lost, i.e. both bending moment induced normal stresses and shear force induced shear stresses follow the shape of the external loading. The normal stress caused by the bending moment follows exactly the shape of external bending moment distribution. In case of shear stresses the same correspondence is seen with shear force. At the location of transverse bulkhead, the shear lag is present that causes decay on the distribution of shear stresses. Thus, hull-superstructure interaction is complex phenomena which makes the stress distributions to deviate from those predicted directly from the bending moment and shear diagrams. 3.3 Normal stresses at the side shell Figure 8 presents the normal stresses at the side shell of the ship. The elements selected are those just below the balcony opening. It is seen that the normal stress oscillates along the length of the ship with period equal to balcony opening length. The amplitude of oscillation is highest at the area of the high global shear force. It can be also seen that the oscillations cause higher normal stresses than the membrane stress (induced by the bending moment) at decks alone does. Oscillations are lost at the location of zero global shear force as four-point bending indicates. As can be seen the oscillations are not however zero everywhere at the region of zero shear force—shear lag can be seen at this region, causing a smooth decay of oscillations to zero; however the sum of shear forces in different decks is found to be zero. The cosine shape loading shows the same effect, i.e. maximum oscillations are not at the location of maximum global shear force at all decks. The smooth decay is most pronounced at the lowest deck of the superstructure where also the deviation from the global shear force distribution is most significant. At top decks the oscillations follow more accurately the global shear force distribution, while the magnitude of the oscillations are much smaller. This indicates that the way external loads are balanced by internal stress resultants is much more complex than the basic beam theory would indicate.

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Figure 5.

Deformations of the hull girder and the balcony opening.

Figure 7. Normal and shear stress distibutions of the main hull without superstructure. Figure 6. Normal stress distributions of the main hull at different decks. Top: four-point-bending, bottom cosine shape pressure loading.

4

Thus, the equivalent shear angle of the opening consists of two parts:

γ = γ H + γV

DISCUSSION

The investigation shows that the global shear force induces considerable warping deformations to the balcony openings of the superstructure. These deformations are both horizontal and vertical.

(2)

where subscripts H and V denote the horizontal and vertical shear angles respectively. Naar et al. (2004) derived analytical stiffness equations accounting the horizontal flexibility of the vertical plate strips between the balcony openings. In addition, they

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Figure 8.

Comparison of normal (longitudinal) stresses on the superstructure side openings (Melk, 2011).

considered horizontal plate strips as Timoshenko beams. Considering the discrete behavior due to balcony openings could be used to model analytically the balcony opening deformations. These analytical expressions could be also used to estimate the normal stresses induced by the shear. The fact that the shear angle consist of two parts causes also challenges to the FE-modeling of the openings with single element (see Fig. 1) as the typical shell element contains only the total shear angle and not separately the vertical and horizontal contributions. If the total shear angle could be obtained with the vertical and horizontal contributions, in principle the oscillating stress could be calculated using submodels and the contributions of these two angles. As shown in Romanoff et al. (2007) for sandwich panels, averaging odd functions over unit cell length, lead to vanishing shear-induced normal stress within the unit cells. This means that in present case the warping induced secondary normal stresses are lost unless the warping is reconsidered in the postprocessing of the global FE-results based on single orthotropic element modeling the opening. It is

also seen that the primary, hull girder, responses are coupled with secondary responses occurring between the web frames. In order to go around these problems, fine mesh is required over the entire bulkhead containing the openings. In order to reduce computational efforts, the analysis can be simplified by making the fine mesh analysis only at the neighborhood of the global shear force. This is inline with the modeling approaches proposed by Fransman (1988), Gudmunsen (2000) and Andric and Zanic (2010). The extension of the model in the length direction of the hull girder is however much more difficult issue to define. As seen from Figure 8 the stresses can get maximum away from the maximum of the global shear force when coupled with the effect of the bending moment. 5

CONCLUSIONS AND FUTURE WORK

Present paper investigated the shear induced normal stresses on the balcony openings of the

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passenger ships. It is clear that these stresses are significant and should be considered on global strength analysis of modern passenger ships. It was also shown that the interaction between the hull and superstructure is complex and requires modeling the entire length of the ship. The study was limited to fine mesh analysis of prismatic ships. In future more efficient methods to assess shearinduced effects with coarse mesh models should be developed. In addition, the influence of nonprismatic hull girder together with the end effects should be investigated. ACKNOWLEDGEMENTS The research project was funded by the Tekes, Technology Agency of Finland. The work is related the Innovations and Network project within the scope of the Finnish Metals and Engineering Competence Centre. All the inancial support is gratefully appreciated. REFERENCES Andric, J. and Zanic, V., 2010, The Global Structural Response Model for Multi-Deck Ships in Concept Design Stage, Ocean Engineering, Vol. 37, pp. 688–704. Bergström, M., 2010, Longitudinal Strength Analysis of a Cruise Ship with a Narrow Superstructure, Aalto University, School of Science and Technology. Bleich, H.H., 1952, Nonlinear Distribution of Bending Stresses due to Distortion of the Cross Section, Journal of Applied Mechanics, Vol. 29, pp. 95–104. Bäcktröm, M. and Kivimaa, S., 2009, Estimation of Crack Propagation in a Passenger Ship’s Door Corner, Ships and Offshore Structures, Vol. 4, No. 3, pp. 241–251. Det Norske Veritas, 2007, Direct Strength Analysis of Hull Structures in Passenger Ships. Fransman J., 1988, The Influence of Passenger Ship Superstructure on the Response of the Hull Girder, Transactions of RINA, pp. 1–12. Gudmunsen, M.J., 2000, The Structural Design of Large Passenger Ships, Lloyd’s Register of Shipping, pp. 1–16.

Heder M. and Ulfvarson A., 1991, Hull Beam Behaviour of Passenger Ships, Marine Structures, Vol 4(1). pp. 17–34. ISSC, 1997, Committee II.1—Quasi-Static Response, 13th International Ship and Offshore Structures Congress, 18–22 August 1997, Trondheim, Norway, pp. 158–165. McVee, J.D., 1980, A Finite Element Study of Hull-Deckhouse Interaction, Computers and Structures, Vol. 12, pp. 371–393. Melk, K., 2011, Shear-Induced Secondary Bending Response of Balcony Openings at Passenger Ships, Master’s Thesis, Aalto University, School of Engineering. Muckle, W., 1962, The Influence of Large Side Openings on the Efficiency of Superstructures, Transactions of RINA, pp. 301–308. Muckle, W., 1966, Superstructures with Large Side Openings: a Comparison between Theory and Experiment, Transactions of RINA, pp. 177–187. Naar, H., Varsta, P. and Kujala, P., 2004, A Theory of Coupled Beams for Strength Assessment of Passenger Ships, Marine Structures Vol. 17, No. 8, pp. 590–611. Paulling, J.R. and Payer, H.G., 1968, Hull-Deckhouse Interaction by Finite Element Calculations, Proceedings of the SNAME Annual Meeting, November 13–16, New York, pp. 281–307. Payer, H.G., 1980, Hull Strength—The Effectiveness of Long Superstructures and Deckhouses under Static and Dynamic Loading, Germanischer Lloyd, Annual Report, pp. 8–10. Romanoff, J. and Varsta, P., 2006, Bending Response of Web-core Sandwich Beams, Composite Structures, Vol. 73, No. 4, pp. 478–487. Romanoff, J., Varsta, P. and Klanac, A., 2007, Stress Analysis of Homogenized Web-Core Sandwich Beams, Composite Structures, Vol. 79, No. 3, pp. 411–422. Zanic, V., Andric, J. and Prebeg, P., 2007, Decision Support Methodology for Concept Design of Multi-deck Ship Structures, 10th International Symposium of Practical design of Ships and Other Floating Structures, Houston, Texas, pp. 468–476. Zanic, V., Andric, J., Prebeg, P., Stipcevic, M. and Piric, K., 2010, RoPax Structural Design-Multi Level Decision Support Methodology, 11th International Symposium of Practical design of Ships and Other Floating Structures, Rio de Janeiro, Brazil, 2010, pp. 490–501. http://www.stxeurope.com/sites/Finland/ Products/Technology%20and%20Engineering/Pages/ XP-Tray.aspx, downloaded 04.09.2012.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Linear and nonlinear FE analyses of a container vessel in harsh sea state J.W. Ringsberg, Z. Li, A. Tesanovic & C. Knifsund Chalmers University of Technology, Department of Shipping and Marine Technology, Division of Marine Design, Gothenburg, Sweden

ABSTRACT: Container vessels have a challenging structural design with respect to fatigue; they are long and slender and have large openings in the deck. Their structural design in combination with high wave loads, makes these ships sensitive to fatigue damage. Classification societies mainly base their assumptions in the fatigue rules on elastic material response and the fatigue analysis is carried out using the stress-based approach. However, for harsh sea states the assumption of elastic material response is not fulfilled in local details where the stress concentration is high; cyclic plasticity may occur, which requires a strain-based fatigue assessment. The objective of this study is to present a methodology which combines the softwares SESAM and ABAQUS to enable realistic hydrodynamic, structure and fatigue analyses of a container vessel in a harsh sea state. Both of the softwares are needed in order to be able to choose a solver in the structure analysis to be either linear, or nonlinear, depending on the cyclic elastic or plastic material response. Analyses of a case study of a 4,400 TEU container vessel on the North Atlantic route are presented. A long-term fatigue analysis is performed in order to search for the fatigue-critical locations in the ship. A part of the side-shell structure was chosen for detailed study comparing results from a linear and nonlinear finite element analysis for two significant wave heights: 6.0 and 7.5 m. The results show that for both of these wave heights cyclic plasticity occurs locally, but at a very low accumulation rate and continued cyclic loading will most likely result in elastic shakedown of the material response. Hence, the overall conclusion is that, in the studied location, a strain-based approach to fatigue is not necessary. 1

INTRODUCTION

Container vessels are long and slender and the ship structure, or the deck beam, is partly open crosssection geometry. The structural design is demanding with respect to fatigue resistance and stress ranges could locally become significant. This could eventually result in fatigue cracks after only a few years in service, i.e. earlier than expected according to design rules. Fatigue analysis based on the stress-based approach is used when the cyclic stresses are assumed to be less than the yield stress of the material, and Hooke’s law can be used to describe the linear relationship between stress and strain. The stress-based approach emphasizes nominal stresses rather than local stresses and strains, and uses Stress-Concentration Factors (SCF) instead of calculating the stresses and strains in local regions; see Dowling (2007). In highly stressed regions where the stresses are beyond the yield stress limit and the material deforms plastically (i.e. there is no elastic shakedown of the material response after some cyclic loading: see Li et al. 2007), the fatigue life should

be based on a strain-based approach; see Dowling (2007). It emphasizes the cyclic plastic deformation that occurs in the localized regions where fatigue cracks could begin. A numerical simulation of a ship’s structural response to wave loading conditions is often divided into hydrodynamic, structure and fatigue analyses. The structure analysis is further separated into a full-ship (global) analysis followed by a detailed structure (local) analysis. There are several commercial codes available that can be used for this purpose. However, most of them have limitations in the solver used for structure analysis, which restricts the Finite Element (FE) analysis to be linear; see SESAM (DNV 2010a), ShipRight (Lloyd’s Register 2012) and SafeHull (ABS 2012). 1.1

Objective of current study

General structure analysis based on arbitrary load and sea state conditions should not be limited to linear FE analysis. Hence, the main objective of this study is to present the development of a methodology with an interface between the SESAM (DNV 2010a) and ABAQUS (ABAQUS 2007)

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softwares which enables either linear, or nonlinear, structural analysis on a global and local structure level; the latter software enables e.g. nonlinear structure analysis. The methodology is used to simulate numerically and assess the structure response of a case study container vessel in a harsh sea state on the North Atlantic route during winter conditions (December to March). Section 2 gives a brief presentation of the methodology and the case study vessel. The numerical simulation procedures are described in more detail in Section 3. In Section 4, long-term fatigue analyses using a full-ship FE model are presented; the purpose is to identify the fatigue-critical areas in the vessel. Linear and nonlinear FE analyses of a part of a local side-shell structure are carried out and the results are presented and discussed in Sections 5 and 6, respectively. Finally, the conclusions from the study are presented in Section 7. 2

METHODOLOGY AND CASE STUDY VESSEL

In previous work by the authors (Li 2011, Li et al. 2012), direct analysis in the time domain and spectral analysis in the frequency domain of container vessels have been carried out using the commercial software SESAM developed by Det Norske Veritas (DNV). The structure analysis solver in SESAM cannot be used for nonlinear structure analysis. For example, in order to assess more thoroughly local responses for arbitrary and harsh sea states, and to be able to include residual stresses from welding, etc, a nonlinear structure analysis has to be carried out. This study presents a methodology which comprises the following three types of numerical analyses and the possibility to choose the solver of the structure analysis depending on if the material response is linear or nonlinear: • Hydrodynamic simulations. • Structure response analyses; linear and nonlinear. • Fatigue evaluation. Each of these parts will be described briefly with respect to the methods and software that are used. Figure 1 represents the flowchart of the methodology for this study and the comparisons that are presented in the following sections. The DNV software package SESAM and the commercial software ABAQUS version 6.7 (ABAQUS 2007) are used for the analyses, see Figure 1. The DNV software package SESAM consists of different modules which depend on the simulation that is supposed to be carried out. The following SESAM-related softwares are used; WASIM version 5.1-01 (DNV 2011a), SESTRA

Figure 1. Flowchart of the methodology for this study: name of the softwares used within parenthesis.

version 8.4-01 (DNV 2010a), STOFAT version 3.4-01 (DNV 2011b), PATRAN PRE version 2010.2 (MSC Software 2010), and SUBMOD version 3.2-01 (DNV 2004). All the software that is mentioned in the study subsequently always refers to the versions and manuals mentioned here. The hydrodynamic simulation is carried out in WASIM. The hydrodynamic simulations in the current study are performed in the frequency domain and in the time domain for a ship operating between Europe and North America during the winter season (December to March). In the structural analysis, two levels of FE models are used in the simulations; a full-ship FE model with a relatively coarse mesh that represents the global model of the container vessel. A detailed structure analysis cannot be carried out using this model. Instead, a local sub-model (the detailed local model) with a more detailed and fine mesh is used to achieve a more accurate and sufficient resolution of, for example, stresses and strains in local structure details. The FE simulations of local details are carried out in SESTRA for a linear structural FE analysis, and in ABAQUS for a nonlinear structural FE analysis. Since SESTRA can only handle a linear structural FE analysis, a change of solver to ABAQUS is necessary in order to perform a nonlinear structural FE analysis. The features given from SESTRA are, however, still used as an input for the nonlinear structural FE analysis in ABAQUS; see Section 3. The fatigue evaluation for obtaining fatigue-critical locations in the full ship, i.e. on a global level, is carried out in STOFAT. In the detailed local model, STOFAT can also be used. There is also an option, indicated by the dashed box in Figure 1, to use in-house codes which provide the possibility of using other fatigue criteria that may be more suitable for general stress—or strain-based multiaxial fatigue assessment. A 4,400 TEU container vessel that operates between Europe and North America is used in a case study; see Table 1 for its main particulars.

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Table 1.

Main particulars of the case study vessel.

Length over all, LOA [m] Length between perpendiculars, LPP [m] Breadth, B [m] Draft, D [m] Service speed at design draft, U [knots] Deadweight at design, DWT [metric tonnes] Max twenty-foot equivalent unit, TEU

294.0 281.0 32.26 10.78 23.0 47,754 4,400

This vessel has been studied thoroughly in previous work by Balatsos (2010), Li (2011) and Tesanovic & Knifsund (2012). A detailed description of the hydrodynamic model and simulation, and the linear FE models and simulations, are presented in those investigations. Two sea states, a moderate sea state with a significant wave height of Hs = 6.0 m and a high sea state with Hs = 7.5 m, representative during winter conditions for the North Atlantic, are chosen for study. The speed of the ship is a constant 23 knots. These wave heights are reported from the period during December to March. Measurements show that the occurrence probability is very low for waves with a significant wave height higher than 7.5 m. Therefore, waves higher than 7.5 m have not been studied in the current work. Waves that come from a bow sea, with a 135-degree heading angle, are studied using the local sub-model, since it is very common for a vessel to operate in oblique bow sea on the studied route; see Mao (2010). Waves that come from this direction will, according to Li (2011), give a large accumulation of fatigue damage in the ship (for example in the hatch corners and in the side-shell structure near the bilge area) due to a combination and superposition of horizontal and vertical bending and torsion loading conditions. 3

NUMERICAL SIMULATION PROCEDURES

This section aims to give a brief description of the numerical procedures divided into hydrodynamic, structural and fatigue analyses. For more details of each of these parts, see Balatsos (2010), Li (2011) and Tesanovic & Knifsund (2012). 3.1

Hydrodynamic simulations

WASIM simulations are carried out in the frequency domain and in the time domain, respectively. In the former, the wave loads are described by a PiersonMoskowitz spectrum with 19 frequencies and 24 wave directions—heading angles—are chosen, with a significant wave height of Hs = 1.0 m. The results

are used in a linear elastic FE analysis using SESTRA followed by a full-ship fatigue analysis using STOFAT. See DNV (2010a & 2011b) for the full description of this procedure. The time domain analyses are performed with two significant wave heights, Hs = 6.0 m and Hs = 7.5 m. Significant wave heights that are higher than 7.5 m have not been studied, since the occurrence probability is very low for Hs larger than 7.5 m. The results from the time domain analyses are used (i) in a linear elastic FE analysis using SESTRA and ABAQUS (for a comparison when changing the solver, see Section 3), and (ii) in a nonlinear FE analysis using ABAQUS’s solver to study the local nonlinear structural response and evaluate if a stress—or strain-based fatigue approach should be used in the fatigue analysis. 3.2

Structure response analyses

The linear structural FE analysis for the full-ship model is performed in SESTRA. Finite element analyses of detailed local models using sub-models are carried out using both SESTRA and ABAQUS for a comparison of results and verification that a script created for optional use of either SESTRA or ABAQUS solvers works; more details are presented later in this section. The nonlinear structural FE analyses are performed in ABAQUS in order to find out the amount of accumulated plastic deformation for the simulated harsh sea states. The meshes for the global model and the local sub-models were created in PATRAN PRE. The global model consists of 16980 4-node quadrilateral and 4858 3-node triangular thin shell elements. The local sub-models used were created by Balatsos (2010). The most detailed local model consists of 10850 8-node quadrilateral and 30 6-node triangular shell elements. The mesh size for the global model is approx. 3.0 m and for the detailed local model it is t × t (t = plate thickness in mm) close to stress-raisers, and 50 × 50 mm in other locations. The hydrodynamic simulation in WASIM gives the pressure variations from the wave loads that act on the ship. These pressure variations are directly transferred from WASIM to SESTRA. In the current study, both surface pressure loads from internal ballast and external seawater pressure is considered. The load transfer between different levels of models (full-ship model to detailed local models) was carried out using the sub-modelling facility SUBMOD. Figure 2 presents the local sub-model in contrast to the full-ship model (global model). The displacements from the full-ship analysis in SESTRA were transferred to the intermediate local model and detailed local model as boundary node displacements in SUBMOD. By using the sub-modelling technique it is possible to obtain the

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Table 2. Material properties (DNV 2010b) for the cyclic stress-strain curve depicted in Figure 3. Yield stress, σy [MPa] Young’s modulus, E [GPa] Poisson’s ratio, υ Material coefficient, H’ [MPa] Material coefficient, n’

Figure 2. Sub-modelling from full-ship model to intermediate model to detailed local model.

277 210 0.3 602.8 0.117

A half-cycle stress-strain curve for mild steel.

Figure 4. Definition of the wave direction (heading angle) in this investigation.

nodal displacements in six degrees of freedom for every node in the boundary region of the local submodel. Local loads from the surface pressure loads from internal ballast and external seawater pressure are also accounted for in the local sub-models. The modules SESTRA and SUBMOD were complemented with a script to enable the change of FE solver from SESTRA to ABAQUS for all levels of FE models. The script that has been created for the nonlinear structural FE analysis in ABAQUS consists of four parts:

presents the true stress-strain curve for the mild steel grade used. The curve has been obtained from the Ramberg-Osgood relationship (Dowling 2007) using the material properties for this steel grade in DNV (2010b), which is presented in Table 2. The asterisks in Figure 3 indicate the stress-strain values which are used in ABAQUS pre-processor CAE to define the material. The current “combined hardening” is activated by selecting this option in material properties in the CAE module; see ABAQUS (2007) for details.

Figure 3.

• Transfer of the mesh of the local sub-model given from the software PATRAN PRE to ABAQUS. • Substitution of the elastic constitutive material model in SESTRA to a nonlinear constitutive material model in ABAQUS. • Mapping of surface pressure loads from internal ballast and external seawater pressure. • Connection and representation of boundary displacements from the global model to the local sub-model given from SUBMOD. The nonlinear constitutive material model in ABAQUS is, in the current study, represented by the constitutive material model for “combined hardening”. It is a nonlinear hardening model which associates the properties from both isotropic hardening and nonlinear kinematic hardening; see ABAQUS (2007) for details. Figure 3

3.3

Fatigue evaluation

A long-term fatigue evaluation is carried out. It is restricted to the identification of fatiguecritical areas in the full-ship FE model by means of a linear spectral fatigue analysis. The fatigue damage calculation was carried out according to DNV (2010b) for welded plates and shells using STOFAT. The accumulated partial damage was calculated and weighted over sea states and 24 wave directions. The wave directions that are considered are 0, 15 and up to 360 degrees. The definition of the wave direction, or heading angle, is presented in Figure 4. The case study container vessel is designed to operate for 20 years in service and is assumed to be in full load condition in 85 per cent of

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its operational time span. The wave climate is represented by a scatter diagram that represents the North Atlantic. Hence, a Pierson-Moskowitz wave spectrum is used and cos2 is the wave-spreading function. 4

FATIGUE ANALYSIS—FULL-SHIP MODEL

The wave loads calculated in the frequency domain are transferred from WASIM to SESTRA, which obtains the structure response due to wave loading. STOFAT uses the principal stresses from SESTRA and calculates the accumulated partial damage— the usage factor. The DNVC-I for welded joints is used as a Stress-Life (SN) curve; see DNV (2010b). The parameters for this curve are presented in Table 3, where N is the number of cycles to failure for stress range, log a is the intercept of the N-axis by the SN-curve, and m is the negative inverse slope of the SN-curve. A critical location in the full-ship model is defined as an area or location where the partial fatigue damage, i.e. usage factor, is high. The usage factor is defined as the design life—life in service— divided by the calculated fatigue life. For example, if the usage factor is 1.0, it will result in fatigue failure after 20 years, or if the usage factor is 0.5, it will result in fatigue failure after 40 years. The structure analysis in SESTRA using the full-ship FE model gives nominal stress values. To obtain the local stress response, which is required also in the fatigue evaluation, Stress Concentration Factors (SCF) are used. A SCF is the ratio between the local stress, which is the increased stress caused by, for example, a notch or weld, and the nominal stress. The stress concentration factors that are used in the fatigue analysis in STOFAT are presented in Table 4; see DNV (2010b & 2011b) for details.

4.1

Results from fatigue analysis

In the STOFAT fatigue analysis, an equal probability of 0.04167 for all wave directions is assumed; the wave directions 0, 15 and up to 180 degrees are studied. The fatigue-critical locations are presented in Figure 5: the midships hatch corners, the engine room bulkhead and in the bilge region. A detailed analysis of the contribution to partial fatigue damage for each heading direction was carried out. The emphasis was on the 135-degree wave direction, since it is the most common wave direction for the case study container vessel on its route from the EU to North America (Mao 2010). This analysis confirms that the areas indicated in Figure 5 are critical for this heading direction. In addition to these locations, which are also presented in Li (2011), the side-shell structure midships near the bilge region is a location of interest due to the combination and superposition of stresses caused by horizontal and vertical bending, as well as torsion (warping-induced stresses due to Vlasov torsion). 4.2

Discussion

The usage factor in the most critical region, shown in Figure 5, is relatively high, with a usage factor = 3.8, which corresponds to a calculated fatigue life of 5 years. The calculated fatigue life is too low to be realistic and can obviously not be trusted. One reason for the high value of the usage factor is due to the mesh size of the global model that is coarse with element lengths up to 3 m, and that the SCFs used in the fatigue analysis may be uncertain. Note that values from a global model fatigue assessment should not be considered as

Table 3. SN-curve parameters of welded joints, DNV (2010b). N

log a

m

≤107 >107

12.164 15.606

3.0 5.0

Table 4. Stress concentration factors (SCF) used in STOFAT. Geometric stress concentration Weld stress concentration Eccentricity stress concentration

1.0 1.0 1.0

Figure 5. Critical regions in the ship with maximum usage factors located in the midships hatch corners (3.8), engine room bulkhead (3.7) and in the bilge region (0.5).

489

exact. They should be interpreted as relative values for comparison, i.e. the objective with this model is to identify critical locations which require intermediate or detailed local models. Note that in the global model, no specific details have been modelled, such as cut outs and longitudinals. Hence, in every critical location, a detailed local model with a finer mesh and more detailed structure has to be analysed in order to get more accurate and reliable results from the structure and fatigue analyses. Note that a nonlinear structural FE analysis on a global model with a fine mesh is not possible, since the calculation time would be too long. It is far more computation-efficient to utilize sub-modelling of local details for nonlinear FE analysis. It is shown in Figure 5 that the maximum usage factors are in the hatch corners of the midships deck region, in the corners of the forward engine room bulkhead and in the midships bilge region. In contrast to similar investigations in the literature on the subject, see Li (2011), the bilge region should be chosen for further study. The structural response to the wave loading on this part of the vessel is, according to Li (2011), more complicated than in the deck. This is due to the significant contribution from wave-induced sea pressure loads, warping stress, and the stresses caused by vertical and horizontal bending. Hence, a detailed local model close to the region of interest, developed in previous work (Balatsos 2010), is used. The local sub-model is located in the side-shell midships region on the port side of the vessel. 5

LINEAR FINITE ELEMENT ANALYSES OF LOCAL DETAIL

The location of the detailed local model, in contrast to the global model, is presented in Figure 2. An intermediate sub-model exists in between these two levels of models, in order to decrease the difference in mesh size between the global model and detailed local model. Figure 6 represents the detailed local model that is studied in the upcoming analyses. The dimensions and location of the detailed local model are presented in Table 5. In highly stressed regions, such as, for example, in the cut out, the mesh size is the same as the thickness. In other regions that are less stressed, such as, for example, in the web frame, the mesh size can be up to 50 × 50 mm. 5.1

Linear finite element analyses in SESTRA

In this section, the results from a linear structural FE analysis in SESTRA are presented. The hydrodynamic loads are obtained from 20-minute time

Figure 6. Table 5.

The detailed local model. Dimensions of detailed local model.

Sub-model length, Δx [m] Sub-model width, Δy [m] Sub-model height, Δz [m] Position and extension below waterline [m] Element size (depends on the location in the sub-model) [mm] Range of element thickness, t [mm]

3.40 2.10 0.90 3.00–3.90 t × t, or 50 × 50 11–17

domain simulations in WASIM. As is described in previous sections, the vessel is operating in the North Atlantic, thus, a Pierson-Moskowitz wave spectrum is used in WASIM and a ship constant service speed of 23 knots, for two significant wave heights, Hs = 6.0 m and Hs = 7.5 m. The waves are described by 19 frequencies and a 135-degree wave direction, see Figure 4. The linear structural FE analysis in SESTRA is performed during a period of 2,400 time steps, where 1 time step = 0.5 second. Thus, the boundary displacements are time-dependent. The time instant that gives the highest structural response (von Mises effective stress) in the stress-history is studied. The contour plot from the linear FE analysis, for Hs = 6.0 m, is presented in Figure 7, at the time step when the maximum von Mises stress occurs during the stress-history. The highest stressed region is obtained in the cut out of the outer side-shell surface. Hence, Figure 8 illustrates the contour plot around the cut out. It can be seen that element 11,630 suffers the most, and high stresses are locally active and the yield stress (σy = 277 MPa) is exceeded. Figures 9 and 10 present the stress-histories for element 11,630 in terms of maximum principal stress from FE analyses with Hs = 6.0 m and Hs = 7.5 m. The highest stress response is indicated by the circle markers. Note: a mesh refinement here did not change the stress values. The stress-histories show that the yield stress

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Figure 7. The von Mises effective stress [unit: Pa] for the detailed local model from a linear structural FE analysis in SESTRA, with Hs = 6.0 m.

Figure 10. Maximum principal stress [unit: Pa] in element 11,630 from a linear structural FE analysis in SESTRA with Hs = 7.5 m, where 1 time step = 0.5 sec. The circle indicates the peak maximum principal stress.

of the material is exceeded repeatedly. Hence, a nonlinear FE analysis should be carried out for these sea states in order to study how much plastic deformation that is accumulated. 5.2

Figure 8. The von Mises effective stress [unit: Pa] in the cut out for the detailed local model from a linear structural FE analysis in SESTRA, with Hs = 6.0.

The FE analyses presented in Section 5.1 were repeated using ABAQUS’s solver for comparison. Hence, the methodology presented in Section 2 using the ABAQUS-script presented in Section 3.2 was used. Note that in this comparison between the SESTRA and ABAQUS solvers, the constitutive material model in both softwares was a linearelastic material model. A comparison in results from a linear structural FE analysis of the structure in Figure 6 using SESTRA and ABAQUS, respectively, showed excellent agreement since there was only some 2–5% deviation of local stresses. Hence, the approach presented in Figure 1 and the ABAQUS-script was deemed reliable for a nonlinear structure FE analysis using ABAQUS’s solver. 6

Figure 9. Maximum principal stress [unit: Pa] in element 11,630 from a linear structural FE analysis in SESTRA with Hs = 6.0 m, where 1 time step = 0.5 sec. The circle indicates the peak maximum principal stress.

Comparison of results using different linear solvers: SESTRA and ABAQUS

NONLINEAR FINITE ELEMENT ANALYSES OF LOCAL DETAIL

The same FE analysis cases, i.e. time-histories with Hs = 6.0 and Hs = 7.5 m, presented in Section 5, were repeated with ABAQUS’s solver and with a nonlinear constitutive material model; see Section 3.2 for details of this model. The purpose was to study the magnitude of the accumulated equivalent plastic strain according to von Mises for the time-histories, respectively.

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Figures 11 and 12 show the accumulated equivalent plastic strain according to von Mises, after a 20 -minute sea state simulation in WASIM, from a nonlinear structural FE analysis with Hs = 6.0 m and Hs = 7.5 m. The results show that only a few elements suffer from a small quantity of plastic behaviour, which is somewhat higher locally for Hs = 7.5 m compared with Hs = 6.0 m. There are only a few elements around the cut out where plastic strains are accumulated. The accumulation rate of the equivalent plastic strain in all the elements was studied and it showed a clear decay with continued cyclic loading. Despite the harsh sea conditions, the material response reached an almost elastic shakedown steady-state due to strain hardening of the material; the plastic

Figure 11. Accumulated equivalent plastic strain [unit: -] in the cut out from a nonlinear structural FE analysis in ABAQUS, with Hs = 6.0 m.

Figure 12. Accumulated equivalent plastic strain [unit: -] in the cut out from a nonlinear structural FE analysis in ABAQUS, with Hs = 7.5 m.

strain range in the stress-strain cycles was very low. Consequently, the position of the detailed local model, in combination with the significant wave height of Hs = 6.0 m and Hs = 7.5 m, should not be considered as a critical region requiring strainbased fatigue evaluation. To conclude, a fatigue analysis of this structure detail can be carried out using a linear FE analysis and the stress-based fatigue approach. 7

CONCLUSIONS

The structural response of a container vessel was analysed in this study. First, a global model—a full-ship model—was studied followed by a detailed local model. The fatigue analysis and the linear structural FE analysis were performed for the global model. Linear and a nonlinear structural FE analyses were performed for the detailed local model. A linear structural FE analysis and spectral fatigue analysis were performed for the global model in order to localize the fatigue-critical locations. The analyses showed that the critical regions are located in the midships hatch corner, the engine room bulkhead and in the bilge region. An in-depth analysis of the partial fatigue damage accumulation in the vessel for a 135-degree wave direction—a course direction which is common for the studied container vessel—showed that a local sub-model near the bilge area was of interest for the study of nonlinear structural response. The lower part of side-shell region on the port side of the container vessel was studied more in detail with the help of a detailed local model. Analyses with two different significant wave heights, Hs = 6.0 m and Hs = 7.5 m, were studied. Results from linear structural FE analysis in ABAQUS were identical to the results from linear structural FE analysis that was performed in SESTRA. Nonlinear structural FE analysis was performed in ABAQUS. The nonlinear structural FE analysis confirmed that the highest accumulation of equivalent plastic strain is located in the cut out in the outer shell of the model. However, the plastic strain around this region is fairly low, and the rate of accumulation during cyclic loading decreased for every cycle of loading. The conclusion is that the current location of the local sub-model in the container vessel does not require a nonlinear FE analysis and strainbased fatigue assessment. Hence, a stress-based approach can be used for the fatigue assessment for the studied local sub-model.

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REFERENCES ABS. 2012. SafeHull software. [http://www.eagle.org/ eagleExternalPortalWEB; accessed 2012-12-07.] Balatsos, J. 2010. Stochastic fatigue assessment of a container vessel. MSc thesis, Department of Shipping and Marine Technology, Chalmers University of Technology, Gothenburg, Sweden. Dassault Systèmes [ABAQUS]. 2007. Abaqus version 6.7 documentation. Analysis user’s manual. Vélizy Villacoublay, France: Dassault Systèms. DNV [Det Norske Veritas]. 2004. SESAM user manual SUBMOD: transfer displacement from global model to sub-model. Version 3.0. Hövik: Det Norske Veritas. DNV [Det Norske Veritas]. 2010a. SESAM user manual SESTRA: superelement structure analysis. Version 8.4. Hövik: Det Norske Veritas. DNV [Det Norske Veritas]. 2010b. Classification Note No. 30.7: fatigue assessment of ship structures. June 2010. Hövik: Det Norske Veritas. DNV [Det Norske Veritas]. 2011a. SESAM user manual WASIM: wave loads on vessels with forward speed. Version 5.1. Hövik: Det Norske Veritas. DNV [Det Norske Veritas]. 2011b. SESAM user manual STOFAT: fatigue damage calculation of welded plates and shells. Version 3.4. Hövik: Det Norske Veritas. Dowling, N.E. 2007. Mechanical behavior of materials. Third edition. London: Prentice-Hall International. Li, L., Zhang, B. & Moan, T. 2007. Residual stress shakedown in typical weld joints and its effect on fatigue of FPSOs (OMAE2007-29285). In: Proceedings of ASME 26th International Conference on Offshore Mechanics and Arctic Engineering (OMAE2007); San Diego, CA, 10–15 June 2007. New York, NY: American Society of Mechanical Engineers (ASME).

Li, Z. 2011. Direct calculation of wave-induced loads and fatigue damage of container vessels. Licentiate thesis, Department of Shipping and Marine Technology, Chalmers University of Technology, Gothenburg, Sweden. Li, Z., Mao, W. & Ringsberg, J.W. 2012. A comparison of direct calculation approaches applied on the fatigue strength assessment of a Panamax container ship (OMAE2012-83332). In: Proceedings of the ASME Thirty-first International Conference on Ocean, Offshore and Arctic Engineering (OMAE 2012); Rio de Janeiro, 1–6 July 2012. New York, NY: American Society of Mechanical Engineers (ASME). Lloyd’s Register. 2012. ShipRight software. [http://www. lr.org/sectors/marine/products/index.aspx; accessed 2012-12-07.] Mao, W. 2010. Fatigue assessment and extreme response prediction of ship structures. PhD thesis, Department of Mathematical Sciences, Chalmers University of Technology, Gothenburg, Sweden. MSC Software. 2010. PATRAN PRE, version 2010.2. [http://www.mscsoftware.com; accessed 2012-12-07.] Tesanovic, A. & Knifsund, C. 2012. Global and detailed local fatigue assessment of a container vessel—a comparison between linear and nonlinear FE-analyses. MSc thesis, Department of Shipping and Marine Technology, Chalmers University of Technology, Gothenburg, Sweden.

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Structural design

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Revisit of a 1970s semi-submersible pipe layer B. Boon Bart Boon Research and Consultancy, Aerdenhout, The Netherlands

ABSTRACT: In 1972–1975 yard Gusto designed and built the semi-submersible pipe-lay barge Viking Piper. The vessel after nearly 40 years is still successfully in operation, now called Castoro Sette owned by Saipem, Italy.When built the design was very novel in several respects. No rules or regulations existed. Gusto had themselves to set design methods and criteria in consultation with the classification society. As the vessel by far outsized the fabrication capacities of the yard innovative new construction methods were implemented with all the impact this had on the structural design. The paper reviews the original structural design approach. The 1970s design is compared with today’s standards and practice. 1

INTRODUCTION

The start of offshore drilling took place in 1947. Since then many novel floating structures were conceptualised, designed, built, put in operation and finally scrapped again. When these structures are called ships, as does the author, this meant an unprecedented rapid and all-encompassing development in naval architecture. The impact of this development was not restricted to mobile offshore units only, but in fact had an indelible influence on the design of all types of ships. All of this took place in a time frame spanning just one or two human generations. Documenting this history still can be based upon living memories and does not need to be restricted to paper information sources. In the early 1970s Gusto Shipyard of Schiedam, Holland, designed and built the then new generation semi-submersible pipelay barge Viking Piper (Fig. 1). The design basis had to be set by the yard

Figure 1. Viking Piper at delivery (Gusto MSC).

itself as hardly any rules or accepted standards did exist. Today, some 40 years later, the vessel still is in operation, now named Castoro 7, and cherished by her present owners SAIPEM as one of the world’s most efficient semi-submersible pipelay barges. This proofs the value of the original design principles and the quality of the fabrication work then performed. Concentrating on structure and strength this paper describes the original design approach (Groeneveld 1973) and compares it with the way in which this would be performed today. 2

BACKGROUND

In the 1960s gas and later oil was found in the North Sea. Constructing production platforms, laying pipes and other offshore activities were originally performed using equipment from the Gulf of Mexico (GoM) where such work already followed an established pattern. Soon it became clear that the environmental conditions in the North Sea were so much more harsh that many days were lost in down-time. Simple crane and pipelay barges were not suitable in this new environment. In 1968 Santa Fe from the USA designed a semi-submersible crane/pipelay barge which was built in Holland at the van der Giessen/de Noord yard (Fig. 2). Although new in its concept this vessel in several respects still possessed characteristics of the GoM lay barges (combined crane and pipelay capabilities, pipelay located at vessel side, hinged stinger, single joint pipelay). It was not the success its designers had hoped for, because at semisubmersible draft its stability was insufficient for heavy lifting operations. In 1972 a group of offshore entrepreneurs and investors, mainly from The Netherlands, supposed

497

Figure 2. Choctaw II (www.kombuispraat.com).

that a new generation of pipelayer would be needed for North Sea conditions. IHC Holland recently had acquired RJBA (RJ Brown and Associates), a specialist company in pipelay engineering and studies. Its founder, Bob Brown, came up with the idea of the 3GLB (3rd generation lay barge) semi-submersible. Together with the recently formed R&D department of Gusto Shipyard, another IHC Holland subsidiary, a feasibility study for such a large vessel was performed. The positive outcome led to the decision to build the vessel (the later-named Viking Piper, VP) and the order thereto was given to Gusto in November 1972. 3

STATE-OF-THE-ART IN SEMI DESIGN

Semi-submersibles for drilling were built in the USA since the early 1960s. Late 1960s in Holland RDM built the SEDCO 135D and the Sedneth I and, as already mentioned, GNK the Choctaw II. All were of American design. Notwithstanding the vicinity to those building yards the background to such designs was not known to Gusto other than through a small number of papers presented at the OTC, Offshore Technology Conference in Houston. Even the total loss, only a few years before, of the semi-submersible Ocean Prince in the Humber Wash off the English coast was completely unknown in those days without internet and all todays offshore related magazines (Fig. 3). The only knowledge available in the yard was the experience in designing other (offshore) vessels such as jack-ups and drillships. And of course what was taught about ships and engineering mechanics at university. Gusto used opportunities to train their young engineers. Thus the author by coincidence was send to a one-week introductory course in offshore engineering given at the University of

Figure 3. Ocean Prince Ocean%20Prince1.htm).

(www.veritas-assoc.com/

Berkeley. Although given before the VP-contract, it showed similarities and differences between ship strength courses and structural strength of offshore units. Splitting loads and racking were not normally considered for ships, but governing in semi-submersibles. This fresh knowledge even in its condensed form, assisted tremendously in the further design of VP. The classification society involved, Bureau Veritas, as well had very limited experience with semi-submersibles and certainly no rules. Actually the number of published design rules in existence was very small, mainly a small booklet published by ABS. Both yard and class, who performed the early global structural analysis of the vessel with a finite plate element analysis, had no other choice than to use an approach from first principles. Only where applicable existing rules could be used, such as for instance in determination of the thickness of plates subjected to lateral water pressure. When a plate would perform a global and a local strength role, those were analysed separately and not in combination (as often is done today). 4

ASSEMBLY METHOD

VP dimensions were much larger than the Gusto slipways. This alone already meant that the vessel had to be assembled afloat. In the feasibility

498

and conceptual design stage the assembly method had not explicitly been taken into account. Setting large blocks using floating sheerlegs was assumed, but this was not translated into design consequences. Once the construction contract was signed that assembly method was one of the first things to be tackled. The 2000 tons maximum block weight anticipated required too many floating sheerlegs to make that system practical. Instead it was decided to set the blocks with a dedicated jack-up. After lifting the sections the partly assembled vessel was floated underneath and the blocks put down (Fig. 4). When opting for this construction method it was assumed that later it would be possible to cut the jack-up in two, add legs and sell them as two individual jack-ups for civil construction. Such platforms were much in demand and the commercial optimism in this sense fortunately proved to be true. This assembly method was found to be quite efficient. But of course, practical problems were experienced. One of earliest tasks was positioning the two floats relative to each other and connecting them with the first set of braces (see later). Notwithstanding all the measurements taken (such as hose water levels) it was (later) found that the floats had

Figure 4.

a slight trim relative to each other. This translated in the deck blocks being somewhat rotated in the horizontal plane. The underdeck girders for the deck crane rails consequently could not be aligned properly. Doublers on the girder webs had to be installed in order to properly support the crane rails. Today a vessel like this probably would be built in a large drydock making assembly much easier. And even if assembled afloat, much more capable equipment exists today than in those days. Still, the elevating platform idea was good given the circumstances then. 5

THE BRACING SYSTEM

Semi-submersibles consist of a number of elements such as floats, columns and work areas tied together by a space frame generally consisting of tubulars. The Ocean Prince is a typical example thereof (Fig. 3). The concept of VP was based upon the same idea: large blocks supported by a space frame (Fig. 4). Not visible in this sketch a longitudinal horizontal member was foreseen at point A between two cross-sections as shown at the ends of the columns (not between the brace systems in between columns).

Assembly method Viking Piper (GustoMSC).

499

Figure 7.

Final transverse brace lay-out.

Figure 8.

Setting a brace assembly (GustoMSC).

Figure 5. Bracing system in feasibility stage (GustoMSC).

Figure 6.

Sketch of brace system at contract signing.

At contract signing the diagonal braces B and C already were omitted resulting in a contract brace system as shown in Figure 6. During preparation of the preliminary design the longitudinal member at point A of Figure 6 was left out. Considering the force equilibrium at nodal point B in Figure 6 it was concluded that the vertical member at that point necessarily had to be a zero-member and thus could be left out. Thereafter it was concluded that at point A any vertical force from the large diagonal braces would probably be transmitted to the upper pontoon directly through the vertical member at A rather than through the shorter diagonal members at that point. Those members thus were left out. As no role was seen for the horizontal member through A and B that could be left out as well. The result was a transverse brace lay-out as shown in Figure 7. On the other hand it was felt that raising the lower horizontal member connecting the two floats

Figure 9. Lay-out horizontal braces in feasibility study (GustoMSC).

offered some advantages. First it would mean that during tow (at a design draught some 0.25 meters less than the depth of the floats) the braces could be above the still water line. This would reduce the resistance during transit. Secondly by doing so all bracing members between two columns could be pre-assembled and lifted by the dedicated jack-up Assembler I as one unit (Fig. 8) making assembly of the vessel easier. During the feasibility study the horizontal braces were laid out as in Figure 9. In the preliminary design stage it was suggested to replace the horizontal X-braces with K-braces. This would make the braces more efficient (a better

500

Figure 10. Necessity for additional transverse brace at stern.

Figure 12.

Figure 11. Final (GustoMSC).

lay-out

horizontal

braces

Transverse braces only on Balder (left; keppelverolme.nl) and Hermod (right; Bart Boon).

angle for accommodating any horizontal shear force between the two floats) and result in less complex nodal points. This was rejected by client and management “because everybody knows X’s are more efficient than K’s”. In combination with the transverse brace lay-out of Figure 7 the diagonal horizontal braces were laid-out as in Figure 10. Note that raising the lowermost braces made the horizontal diagonal braces less efficient due to their direction. Because in the final stages of the feasibility study the work deck had been shifted forward relative to the floats an additional set of vertical braces and a horizontal one were installed at frame 204 (position A in Figure 10; see also Figure 1). At the aft end of the vessel brace B was the last one. The finite element analysis made by Bureau Veritas showed that in a splitting load condition the force in this latter brace was about double of that in all other braces. The transverse deformation of the float in that situation was as sketched in Figure 10. This indicated the necessity to add an extra horizontal brace at the stern of the vessel (position C in Fig. 10). Figure 11 shows the final lay-out. Note that the additional brace is located above the float leading a rather special attachment node.

Today most likely the vessel would have only transverse horizontal braces. That system was developed by Gusto when designing the semi-submersible crane vessels Balder and Hermod (Fig. 12) and later Micoperi 7000 (now SAIPEM 7000). MSC used the system on the Smit Semi 1 and 2. Both vertical and horizontal diagonal braces can be done without. The system was developed by Gusto as a further development along the line of thinking originating with Viking Piper. It may be seen as an optimum trade-off between integrated box-structures and braces. Today this transverse-braces-only system is quite common in semisub design. 6

STEEL TYPE

Of course the vessel was constructed from normal strength steel grade A. The only exception are the braces and their nodal points. Because in the 1970s some fatigue problems showed up in nodal points of fixed North Sea platforms it was decided to minimise the risk of fatigue. Normal strength steel thus was selected and the best quality of steel available, which meant that grade E was selected both for

501

the brace tubes as well as for the plates in the nodal points. Those steels came from different suppliers. All welds of the braces and the nodes were fully penetrated as this again was supposed to minimise the risk of fatigue. It was chosen to let the plate material be continuous and the tube material discontinuous (Fig. 13). Unfortunately this “best” plate material was found to possess very limited through thickness properties. Extensive lamellar tearing was the result. Proper repair meant gouging out and nearly replacing all plate material in between the tubular material by weld material. Further buttering was applied. And all nodes were heat treated in large protection tents. The latter measure was considered by the specialists to be hardly effective, but management decided that “we must show the client that we did everything within our possession to obtain the best quality possible”. Certainly in a costly way finally very good quality for the nodes was achieved. Note that in this particular case making the tubular material rather than the plate material continuous probably would have been beneficial. But this is due to the coincidental material properties only. Moreover using fillet welds or partial weld penetration would have reduced the risk of lamellar tearing albeit at an increased risk of fatigue starting from the weld root (which in this case with welds mainly transferring in-line shear loads probably would have been minimal). Today because of better steel fabrication the risk of lamellar tearing has decreased significantly. Special steel with through-thickness properties, Z-steel, is readily available. That steel contains very little sulphur and phosphor by vacuum-degassing during production. Classification societies give rules for fabrication, testing and marking of such

Figure 13. Lamellar tearing under full penetration weld in brace nodes.

materials. Their application is generally restricted to “structural details subject to strains in the through thickness direction in order to minimize the possibility of lamellar tearing during fabrication” (ABS 2012). Many designers interpret this as meaning that Z-steel must be used when there are significant tensile (operational) stresses through the thickness of a plate. The effect of weld shrinkage during fabrication is often underestimated. The material today is better, but it is doubtful whether the design efforts to avoid lamellar tearing are much better than in the 1970s. 7

BRACE NODAL POINTS

As today also in the 1970s nodal joints of fixed platforms often consisted of tubulars joined together with or without local reinforcements. The larger diameters of braces in semi-submersibles often quite different in size (for instance columns and braces) meant that the nodes nearly always consisted of a combination of plates (brackets and gussets) and tubes. The actual structure sometimes was concealed inside the visible part of the tubulars and in other cases visible from the outside. Given the assembly method of VP with units of connected braces (such as in Figure 8) it was clear that it would be beneficial to have the entire nodal points outside the column structure. According to the contract brace lay-out the centre lines of the transverse braces were in line with the end shells of the columns. This automatically led to the choice of large brackets at the centre of the brace tubes. In order to make load transfer to the columns more smooth, it was considered beneficial to have horizontal brackets at the brace tube centre lines as well. General engineering knowledge indicated that very gradual transition from the tube structure to the brackets would make the unavoidable stress concentration as small as possible (Fig. 14). For the same reason the end of the brace tubes as well was foreseen with long bracket-like transitions. This principle was also applied at the top boxes connecting the braces to the upper pontoon (Fig. 15). Even the additional brace at frame 12 was connected to the float in the same way. For the latter it may be wondered whether for instance the horizontal brackets at some 1.5 meters above the float deck fulfilled any function. This was done as well during the design. However, as there was no way to quickly analyse this (see the finite element analysis described hereafter) it was decided that adhering to the same principle in the design was the most secure way to go. After the concept design for the brace nodes was made, it was decided that it would be worthwhile

502

Figure 14.

Figure 15.

Horizontal section over brace centre lines (GustoMSC).

Brace top box in 2012 (Bart Boon).

to analyse one detail using finite elements. This was quite new in those days and had to be performed by an outside company, i.e. KSEPL (Shell) in Rijswijk, Holland. A very detailed analysis confirmed the original design ideas. The analysis was very time consuming and could be made for one node only. Under the assumption that the nominal stress in the brace tubes would be near the allowable, the target of the finite element analysis was to keep any stress concentration factor below 1.3.

The calculated stresses as such were not directly assessed. Note that the smallest elements in the FEM analysis had dimensions in the order of magnitude of 0.05 meters. The other nodes had to be based upon the assumption that similar stress concentration factors would apply. It was found that notwithstanding the extreme gradual introduction of the brackets, still some serious stress concentration occurred at the bracket toes. As this meant high stresses at a point of a non-stiffened round plate (a fundamental horror to shipbuilders in view of fatigue initiation) it was decided to provide ring stiffeners on the outside and on the inside of the tubes at the bracket toes. Further FEM analysis still showed quite high stresses, now in bending of those rings. Providing very small triangular brackets in line with the large brackets could do away practical all those stresses. Of course those small brackets again ended on an unstiffened plate, but this was considered acceptable in view of their only 15 mm thickness and the fact that they were fillet welded rather than the full penetration of all other welds. The fact that today there is not the slightest indication of any fatigue cracking at these points (Fig. 16) shows how effective the original design was. The other type of nodal point is the crossing of the two horizontal diagonal braces. The basis for its design was continuity of material. From a practical fabrication point of view this was considered to be impossible with the two braces having the same diameter.

503

modern design may quite possibly have a shorter fatigue life than the VP nodes Giving one of the braces a larger diameter connected to the original diameter by conical transitions solved this fabrication problem. Providing locally 40 mm instead of 25 mm thickness reduced all stress concentrations to an very acceptable level. 8 Figure 16. Boon).

Nodal points on Castoro 7 in 2012 (Bart

COLUMN AND FLOAT CORNERS

The choice of position and shape of the columns was based upon transparency of the vessel side view in order to minimise wave excitation and at the same time optimisation of vessel stability in order to maximise the allowable pipe load on the deck. These considerations led to rectangular columns sitting on the outside of the floats (Fig. 18). This configuration at the same time made it easy to achieve structural continuity by aligning column shell with float shell and with longitudinal and transverse bulkheads in the floats. The orthogonal stiffening system more or less automatically led to square corners for the columns and floats. At that time this was completely new for semi-submersibles. Most had completely round columns and often floats or at least corners with large radii. The arrangement of Viking Piper obviously not only offered optimal structural alignment, but at the same time was most fabrication cost efficient. From client side objections to this solution were made, in particular because they were afraid of corrosion fatigue in such corners. Although such sharp corners underwater indeed were not very common in the offshore industry or shipbuilding, some special vessel types, such as dredgers, had shown that the detail performed completely satisfactory. Actual behaviour of the VP after four decades proved this

Figure 17. Node at crossing horizontal diagonal braces (GustoMSC).

Today probably no longer nodes would be designed with such an amount of external brackets. A transition from tubular to square as in Figure 12 is the more likely design. Yet with the same boundary conditions as during the construction of the VP a similar nodal point still might be a very good solution. Easy to use finite element analysis methods allow a much better optimisation and adaptation of designs in different locations. This probably would result in some less structural elements and smaller scantlings. However, optimising for a required fatigue life might easily lead to a solution which accepts higher stress concentrations than the minimisation of those that was strived for in the original VP design. As a consequence a

Figure 18. Columns set using floating sheerlegs (GustoMSC).

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assumption to be completely correct (see Figure 16 left to the far right of the photo). Another special feature of VP was the lack of any brackets in the transition of columns to floats (Figure 1). Again it was felt that this was justified in view of the good structural continuity at this point. Although it was recognised that some stress concentration might be caused in for instance the plane of the outer float and column shell, it was also expected that any bracket in the corner would make the stress transfer between column transverse shell and bulkhead in the floats, less effective. Note that at the upper end of the columns large box-type brackets are provided in the transition to the upper deck structure (Fig. 18). Partly it was recognised that the upper pontoon would be more flexible than the floats in view of their smaller depth (5.9 meters as opposed to 8.25 meters). The advantage of providing brackets for that reason was considered to be somewhat more at the top column end than at the lower end. But the main reason to provide these box brackets was supporting during fabrication the deck blocks in between the columns for as long as those blocks were not yet rigidly welded to the blocks above the columns. Figure 18 illustrates this situation. Objections to this arrangement were raised. It was actually stated that such transition should look like the transition of a bough to a tree trunk; “that is the natural stress flow and thus will provide the most smooth transition”. VP designers rejected this strongly with the argument that in trees this concerns the transition of a full 3-D structure, not that of a thin plate arrangement where continuity requirements led to quite different solutions from that found in nature. Lack of any fatigue cracks at these transitions after nearly forty years proves the correctness of the assumptions made then. Today extensive finite element analyses replace the partly intuitive understanding of structural behaviour that was the basis of the designs in former days. Rectangular columns and floats are well accepted now. Some rounding of the corners is often applied in view of diminishing somewhat the hydrodynamic forces exerted, safeguarding ropes and wires that now and then may run around the corners and reducing somewhat the risk of damage in case of small collisions (both to the semisubmersible as well as to the colliding vessel). The actual transition itself is often fully based upon rectangular structural elements. As a consequence transition pieces must be provided between the part with the square and the part with the rounded corner (Fig. 19). Note in Figure 19 the horizontal transverse braces only. Two between the columns means that some amount of redundancy is provided which is a new requirement in many cases

Figure 19.

DSS21 Maersk Developer (GustoMSC).

and something that was not taken into account forty years ago. 9

CONCLUSIONS

In the 1970s Gusto yard designed and built the semi-submersible pipe lay barge Viking Piper. Using common sense and a first principles approach it was possible to develop a vessel that even after 40 years of operation performs completely satisfactory. Modern computer analyses might have optimised the design as far as weight is concerned without altering fundamentally the safety of the vessel. ACKNOWLEDGEMENTS The late Bart Jan Groeneveld was project manager during the design of Viking Piper and the present author gratefully acknowledges the intensive cooperation with him in those days. But also the input of several other former colleagues gave their input in writing this paper is gratefully acknowledged. REFERENCES ABS 2012 Rules for Materials and Welding, Part 2. Groeneveld, B.J. 1973 Revised provisional design Third Generation Lay Barge Schiedam, IHC Gusto.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

A new generation of offshore structures F.P. Brennan School of Engineering, Cranfield University, Cranfield, UK

C.M. Rizzo DITEN, University of Genoa, Genova, Italy

ABSTRACT: Current offshore structural design guidance, knowhow and standards are based on Oil & Gas industry research conducted mainly in the 1970s and 1980s. Since that time, European Steel Fabricators have made significant advances in volume manufacturing processes and these along with improvements in steel performance, Quality Assurance and advanced structural analysis techniques promise lower cost, reliable steel structures for use in Wind, Wave and Tidal power. In addition, there are several fundamental differences between conventional offshore Oil & Gas structures and those used in the emerging offshore renewable energy sector, not least; the very large volume of similar structures, the unmanned nature of installations and the cost margins involved. These combined with improved capabilities of modern offshore fabrication and design demand a fresh perspective on the Guidance and Standards available to developers. This paper reviews the guidance available and identifies in particular fabrication, material improvement techniques and modelling approaches that may be useful in the design, operation and maintenance of Offshore Structures for Renewable Energy. 1

INTRODUCTION

Offshore wind is increasingly becoming the driver for Britain’s wind power. Statistics released by the European Wind Energy Association (EWEA) in February 2011 confirm that the UK is consolidating its position as the world leader in the offshore wind sector, with 1.3 GW installed, or 45% of the EU total of 3 GW, compared to 854 MW for Denmark, 249 MW for the Netherlands and 195 MW for Belgium. The current figures for installed capacity at the time of writing (November 2012) is 1.86 GW, and a further 665 turbines in construction, totalling 2.4 GW. In May 2010, a ground breaking report by the Offshore Valuation Group, a coalition of government and industry organisations, showed that the country’s offshore renewable resource matches North Sea oil and gas in equivalent barrels of oil. Using less than a third of the resource, the renewables sector could generate the energy equivalent of 1 billion barrels of oil per year, create 145,000 new jobs, provide the Treasury with £28bn in tax receipts and allowing Britain to become a net exporter of electricity by 2050, whilst reducing carbon emissions by 30% compared to 1990 levels. Carbon Trust puts the figure at 230,000 jobs by 2050 in the Offshore Wind Sector alone and that the UK could capture a 10% share of the global offshore wind market, which it estimates, could be worth up to £170bn/year by 2050.

Design standards and guidance for offshore steel structures have evolved from the first comprehensive rules in the early 1970s primarily by API (American Petroleum Institute, 2002) and the UK Department of Energy which were developed specifically for the Oil & Gas industry. Although the Department of Energy guidance notes are no longer maintained (by the UK HSE) and were revoked in the late 1990s, these contain valuable knowledge which is largely preserved through ISO, HMSO OTO and certification authority documents in addition to API and certain other useful publications. In parallel, structural design methodologies based on risk/reliability approaches have developed over the same period (e.g. International Organization for Standardization, 2008); the best examples encapsulated within the Eurocode series of standards. As a result, whereas structural standards and guidance exist to apply the most advanced risk based design methodologies, there is an absolute deficit in contemporary supporting information to allow these to be implemented for maximum advantage. For example, all standards allow the use of higher strength weldable steels (>500 MPa) but without appropriate component S-N design curves the option of utilising these is beyond the capability of any single design organisation and/or fabrication yard. Similarly it seems incomprehensible that in the year 2012, post-weld heat treatment,

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weld toe improvement, peening and other fatigue alleviation techniques are not commonly exploited in design of offshore steel structures, all because of the lack of basic design information. The emerging offshore renewables sector is unlike the Oil & Gas industry in that structures are unmanned and the commercial reality is that it cannot afford the luxury of designing for unrepresentative reliability levels; the only way in which it can make a step change to significantly reduce steel structure costs is to embrace modern steel and fabrication techniques coupled with advanced reliability based design approaches. This paper focuses on Tubular Steel structures which are fundamental to larger wind turbine jacket structures and potentially for tidal stream and wave power installations. The following sections detail the background to current guidance and highlight areas where there needs to be better information to support designers and operators of Offshore Renewable Energy installations.

2 2.1

CONVENTIONAL S-N DESIGN GUIDANCE T curves

The first recommendations for the design of tubular joints against fatigue based on the use of S-N curves were by the American Petroleum Institute (API) and the American Welding Society (AWS) in 1972 (Austin, 1994). The first S-N curve based on joint hot-spot stress was referred to as the X Curve and was recommended by API RP2A in 1972. The data used to obtain the S-N curve was obtained from small T and K joint specimens tested in air under constant amplitude (Austin 1994, Health & Safety Executive, 1999). The 2nd edition of the UK Department of Energy Guidance Notes guidance (1977) recommended the Q Curve, which is based on the data generated by the API and the AWS (Myers, 1998). Since this curve was published, two major revisions have taken place. The first revision was in 1984, the results from the UKOSRP and the European Coal and Steel Community (ECSC) sponsored research programme highlighted that the Q curve could be unconservative under certain conditions. The S-N curve recommended in this revision became known as the T curve. The second revision was in 1996 incorporating a significant amount of new data which became available on the fatigue behaviour of welded tubular joints (Health & Safety Executive 1999). The new curve for tubular joints was designated the T’ curve (Chang 1997). Figure 1 summarizes this development.

Figure 1.

Evolution of the T’ curve.

The Health & Safety Executive (1999) explains that a total of 64 T-, X- and K-joint test results were used to obtain the T-Curve. More specifically, the curve was formulated from the mean 32 mm chord wall thickness data curve less two standards deviation based on the 16 mm chord wall thickness data. It was recommended for joints in air or seawater where adequate protection against corrosion has been provided. A ‘Thickness Correction’ is applied where the thickness is anything other than 32 mm; a value of 22 mm imposed for calculating fatigue lives of joints with chord wall thicknesses less than 22 mm. A total of 59 T-, Y-, X- and K-joints of 16 mm were used to obtain the T’ Curve and this represents the largest subset of data with the widest range of joint geometries and loading modes. A simple assessment of the data showed that the slope (m) of the mean log10N vs. log10S line had a value which was very close to 3, and therefore a fixed value of m equal to 3 was retained for consistency with earlier Guidance. A new thickness correction based on deviation from 16 mm wall thickness was introduced. It should be noted that the tests that make up the design curve were in the main carried out over twenty-years ago, many of which were completed thirty-years ago under very different quality and welding control process that we apply today. No pre- or post-weld heat treatment, weld toe grinding/repair/profile improvement was carried out and fatigue test controllers and data acquisition systems were very different to what we expect today. 2.2

Local stress analysis of tubular joints

Fatigue is a local (mesoscopic) phenomenon and the type of fatigue experienced by large offshore structures will normally be dominated by the effects of local stress features. In tubular joints, stresses can be classified into three groups (Austin 1994): • Nominal stresses, which represent the structural response of individual members to applied loads; • Geometric (or Structural) stresses, which are developed due to the differences in deformation of brace and chord under applied loads; • Notch stresses, which occur in the weld toe region caused by the extra stiffening effect of the weld on the tube walls.

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The presence of notches, shoulders, etc., result in modifications of the simple stress distribution. This local high stress is termed a stress concentration and is described by the Stress Concentration Factor (SCF); which may be obtained analytically from elasticity theory, computationally from finite element methods, and experimentally using methods such as strain gauges (Pilkey, 2008). One of the most important terms to appreciate when utilising the S-N approach for tubular joints is the “Hot-Spot” Stress. Figure 2 shows the stress field in the vicinity of a weld toe and is characterised by a nominal (or remote) stress unaffected by the joint geometry but increases steeply as the toe is approached. For convenience a fictitious “Hot-Spot” Stress is frequently defined as a linear extrapolation of the stresses at two defined points some distance from the weld toe. The definition of the Hot Spot Stress for tubular joints was originally drafted by the review panel of the United Kingdom Offshore Steels Research Programme (UKOSRP) and adopted by the UK Department of Energy guidance notes (Department of Energy, 1984) and stated that it was: “the greatest value around the brace-chord intersection of the extrapolation to the weld toe of the geometric stress distribution near the weld toe. This hot spot stress incorporates the overall effects of joint geometry (i.e. relative size of brace and chord) but omits the stress concentrating influence of the weld itself which results in a local stress distribution”. There is not a general agreement for the calculation of the hot spots stresses (Berge, 1985), but the most common is the extrapolation region defined in the CIDECT (Zhao, 2001) design guide. The Hot-Spot Stress was developed at a time when the only way of reliably measuring stress was through Electrical Resistance Strain Gauges on a physical model (Fig. 3). Nowadays with Finite Element Analysis, DIC and other methods of measuring notch stresses, the Hot Spot Stress could be seen as a hindrance.

Figure 2. Nominal and hot-spot stresses (Bureau Veritas 2000).

Figure 3. Electrical Resistance Strain Gauge Rosettes for the measurement of Chord and Brace Hot Spot Stresses in a Tubular T-Joint.

The S-N curves used in all the aforementioned standards and guidance documents incorporate the Hot Spot Stress which omits the influence of the weld geometry. This means that there has been little incentive to improve weld toe geometry, a detail that if improved could have a significant beneficial effect on fatigue life. 3

3.1

SIGNIFICANT DEVELOPMENTS SINCE THE ORIGINAL FATIGUE DESIGN GUIDANCE Linear elastic fracture mechanics

Linear Elastic Fracture Mechanics (LEFM) has developed significantly in the past forty years and its understanding and application for damage tolerant structures is now widespread. Designers sometimes misunderstand its potential and very often LEFM is only used after damage has been detected. The Offshore Oil & Gas industry has now very much embraced LEFM particularly for structural life-extension activities supporting service inspection planning, reliability assessments based on “what-if ” scenarios etc., In many ways the traditional S-N approach shackles the quality fabricator and designer from implementing good fabrication practice as the Hot-Spot Stress S-N design curves give no benefit to weld toe improvement. Design Standards do in some cases give limited benefit for toe grinding and peening e.g. DNV will allow up to a factor of 3.5 on life for certain S-N Curves but these are very crude and conservative measures that are likely to significantly underestimate the enormous potential of using surface treatment techniques for fatigue life improvement that can be modelled and predicted with a combination of notch stress crack initiation understanding along with LEFM crack propagation techniques. LEFM is the understanding and modelling of stable crack growth. The stress intensity factor (K)

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(Etube et al, 2000) is a crucial parameter in this subject area and is a function of the Notch Stress in addition to the component geometry and loading mode. If the stress intensity factor in a potential crack path is known, then the rate at which this crack will grow can be predicted from the Paris Law, along with the loading and certain material properties. Fracture Mechanics also allows the calculation of the effects of beneficial residual stresses due to peening on fatigue life and a real opportunity exists to combine Fracture Mechanics understanding alongside S-N if it is considered properly at minimal additional cost from the outset of any new test programme. Fracture Mechanics standards exist through BS7910 (British Standards Institution, 2005) and API 579 (API/ASME, 2007) for geometries and welded components of the type used and likely to be used for Offshore Wind. Often LEFM is criticised as too sensitive to the quality of input variables to be relied upon for design purposes. Rizzo (2007) carried out a sensitivity analysis of LEFM calculations of typical welded joints by applying reliability approaches. Parameters describing loading conditions introduced the larger uncertainties whilst, among strength parameters, the initial crack size dominates rather than material or geometrical parameters. Clearly good quality input data and suitable calibration against experimental testing are always necessary to obtain valuable results as in any other engineering analysis. An alternative to LEFM is represented by the application of approaches based on the Notch Stress Intensity Factor (N-SIF) concept, like the Strain Energy Density approach (SED) or the Peak Stress Method (PSM), see Figure 5, such methods were recently revived in scientific literature and application in everyday working practice appears promising in the near future. A comprehensive literature review of such methods is reported in Rizzo (2011)

Figure 4. LEFM prediction of wind turbine monopile crack growth (stress range 60 MPa, section thickness 85 mm, 355 D Steel).

Figure 5. Notch Stress Intensity Factor (N-SIF) based approaches included in the classification of fatigue assessment approaches by Radaj, Sonsino & Fricke (2006).

and a useful application to a typical ship structural detail is detailed by Fischer et al. (2011). In short, N-SIF represents an extension of LEFM to notches whereas LEFM can only deal with cracks. It is therefore the parameter describing the singular stress field at a notch tip analogous to the stress intensity factor in fracture mechanics. While very refined meshes are necessary to numerically assess the N-SIF using finite element analysis, it has been demonstrated that under certain conditions the N-SIF is directly related to the SED averaged in a defined control volume or to the peak value of the principal stress. Such approaches overcome the problem of singular stress fields and implementation in numerical analysis appears less cumbersome than for LEFM. These therefore are promising approaches to account for the effects of the weld and other local effects. 3.2

Weld toe improvement methods

It is well known that surface geometry and other improvement methods can greatly increase the fatigue strength of structural details. Indeed as a mesoscopic phenomenon, fatigue is initially governed by very local parameters which describe the conditions of the material/geometry at the point where the crack is likely to start. Weld toe fatigue cracks initiate from undercuts, cold laps or sharp crack-like imperfections which are an inherent feature of most welding processes. Although quantitative assessment of weld improvement can be difficult to define, in general a multiplicative correction factor is applied to S-N life predictions whose value is very often empirically derived depending on the applied improvement method. A general reference concerning fatigue strength improvement methods for weldments is provided by the International Institute of Welding (IIW, Hobbacher et al, 2008). In addition to methods for protection against environmental conditions

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(e.g. painting and resin coating) two main categories of improvement methods are considered: a. Methods for improvement of weld profile: machining or grinding of weld transition at the toe, remelting of the weld toe by TIG-, plasma or laser dressing; b. Methods for improvement of residual stress conditions: peening (hammer-, needle-, shot-, brush-peening or ultrasonic treatment), overstressing (proof testing), stress relief. The effects of all improvement techniques are sensitive to the technique used and the applied loading and are all most effective in the low stress high cycle regime. They may also depend on the material, the structural detail, the applied stress ratio and the dimensions of the welded joint. Consequently, determination of their effectiveness can be rather difficult. The aim of grinding is to remove defects and imperfections and to create a smooth transition between weld and parent material, reducing the stress concentration. By weld dressing, the weld toe is remelted in order to remove the toe imperfections and again creating a gradual profile from the weld to the component surface. In both cases IIW restricts the benefit applied to the fatigue strength of the corresponding non improved joint to a factor of 1.3. By hammer, shot or needle peening, the material is plastically deformed at the weld toe in order to introduce beneficial compressive residual stresses. An improvement factor of 1.3 on strength is also considered for mild steel while for high strength steel (yield > 355 MPa) or aluminum, a factor of 1.5 can be applied within limits. The DNV guidelines for offshore wind turbine support structures (2010) account for improved fatigue performance of welded structures by grinding and provide detailed qualitative specifications. Improved S-N curves can be applied to girth welds if grinding is carried out according to rules’ specifications. In total an improvement in fatigue life by a factor of 3.5 can be obtained for tubular joints following the aforementioned guidance and recommendations. Furthermore, the S-N curve slope can also be reduced having a significant beneficial effect on fatigue life predictions. A complete section of the DNV Recommended Practice for offshore structures (2010) is devoted to improvement methods. General reference is made to IIW provisions on post weld improvement and to the fact that improvement of the toe will not improve the fatigue life if cracking from the root is likely. The weld profiling effect by machining and grinding is taken into account in structural analysis by reducing the stress concentration factor at the

Figure 6. Example weld toe profiling descriptions reported in DNV RP (2010).

hot spot. It is worth noting that very detailed and quantitative descriptions of weld toe profiling are provided (see Fig. 6), including even the specification of appropriate grinding tools as well as prescriptions for surface roughness. A strong message from this paper is that there is a significant increase in fatigue strength when appropriate weld toe improvement methods are applied according to specifications detailed in existing rules and guidance and that considerable further improvement is likely following research and development efforts specifically for marine and offshore renewable energy applications. 3.3 Monitoring, inspection, repair and maintenance Developments in inspection and maintenance of ship and offshore structures in general have been reported extensively e.g. by Rizzo (2008a, 2008b) and Rizzo (2011). Section 13 of the DNV offshore wind guidelines (2010) deals with in-service inspection, maintenance and monitoring of offshore wind farms, following the experience and knowledge learned over decades concerning ships and offshore structures by classification societies, rules and international conventions. During the assumed 20 years lifetime, periodical inspection consists of three levels of inspection, i.e. general visual inspection, close visual inspection and nondestructive examination. However, contrary from ships, inspection for fatigue cracks at least every-year/five-years may be waived in an offshore wind turbine structure depending on which design philosophy has been used for the structural detail in question: when the fatigue design of the structural detail has been carried out by use of safety factors corresponding to an assumption of no access for inspection, then there is no need to inspect for fatigue cracks, while if smaller safety factors have been used for the fatigue design, inspections then need to be carried out. In general, the lower the safety factor, the shorter the interval between consecutive inspections.

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Reliable inspection, such as an inspection by eddy current or a magnetic particle inspection can grant a reduced inspection interval calculated on the basis of the safety level to be achieved. It is interesting to note that DNV Recommended Practice for offshore structures (2010) explicitly provides a target safety level in terms of failure probability as follows (Fig. 7): • If a fatigue crack is without substantial consequence an accumulated probability of 10−2 may be considered acceptable and a larger inspection interval allowed; • If the consequence of a fatigue crack is substantial, the accumulated probability of a fatigue failure should be less than 10−4 and inspections should take place at shorter intervals. In fact, the time interval to the next inspection is estimated based on fracture mechanics and probabilistic analysis taking the uncertainty in

Figure 7. Accumulated probability of fatigue crack as function of service life for 20 years calculated fatigue life, DNV (2010).

Figure 8. Failure probability of typical welded joints assuming no detection or perfect repair at each 5th year inspection (Rizzo, 2007).

the inspection method into account. It was demonstrated (see Fig. 8 (Rizzo, 2007)) that only the last inspection results significantly contribute in the updating of the model, the effect of the previous inspections in the updated life predictions is negligible. Therefore, differently from ships requiring fixed inspection intervals because of operational reasons, the optimization of the inspection interval (based on no inspection carried out until a target reliability level is reached) has real potential to optimise Capex and Opex for offshore wind structures.

4

SUMMARY AND CONCLUSIONS

This paper summarised in broad terms the origin of the fatigue guidance currently applied to offshore tubular joints, monopoles are not specifically dealt with but have similar characteristics. The paper highlights the fact that whereas significant strides an understanding fatigue behaviour in the offshore environment certain historical practices are today hindering the effective optimization of a new generation of offshore structures for wind and marine energy application. The following conclusions are emphasised: • Hot-Spot stress based S-N curves do not allow the flexibility for designers to properly benefit from weld toe improvement techniques; • High strength weldable steels may benefit designers but there is inadequate information available to allow their wide spread use; • Thickness correction in modern weldments is likely to be overly conservative (volumetric effect); • Environmental Reduction Factors are generally based on fatigue crack propagation studies and do not represent short crack or a crack initiation phase; • Overall, there is a need to update the fatigue database for modern welded steel components under representative loading and environmental conditions embracing notch stress and fracture mechanics based approaches in order to reduce the costs of offshore renewable energy structures whilst increasing reliability and structural performance. • Notch stress intensity factor and energy based approaches are exciting developments that needs to be developed further if designers are to benefit from these emerging concepts; • Reliability based design and structural assessment allows both inspection and maintenance optimisation according to appropriate target reliability and an understanding of the degree of confidence in life predictions.

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ACKNOWLEDGEMENTS The background to the T-Curves is based largely on a much larger yet unpublished review conducted by Estivaliz Lozano Minguez at Cranfield University. REFERENCES American Petroleum Institute/American Society for Mechanical Engineers, 2007. API 579-1/ASME FFS-1 2007, Fitness-For-Service. American Petroleum Institute, 2002. API RP 2A-WSD: Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms, Twenty First Edition. Austin, J.A. 1994. The role of corrosion fatigue crack growth mechanisms in predicting the fatigue life of offshore tubular joints, PhD Thesis, University of London. Berge, S. 1985. On the effect of plate thickness in finite of welds. s.l.: Engineering Fracture Mechanics. Vol. 21, No. 2. British Standards Institution, BS 7910:2005, Guide to methods for assessing the acceptability of flaws in metallic structures. Bureau Veritas 2000. BV Rules for the Classification of Steel Ships, Paris, Marine Department. Chang, E. 1997. Parametric Study for Non-Destructive Fatigue Strength Evaluation of Offshore Tubular Welded Joints, PhD Thesis, University of London. Department of Energy, 1984. Offshore Installations: Guidance on Design and Construction, London, HMSO. DNV 1992. Classification Note 30.6. DNV, 2010. Offshore Standard DNV-OS-J101, Design of offshore wind turbine structures. DNV, 2010. Recommended Practice DNV-RP-C203 Fatigue Design Of Offshore Steel Structures. Etube L.S., Brennan F.P. and Dover W.D. 2000. A New Method for Predicting Stress Intensity Factors In Cracked Welded Tubular Joints, International Journal of Fatigue, Vol. 22, Issue 6, pp. 447–456. Fischer, C., Rizzo, C.M., Fricke, W. 2011. Fatigue assessment of hopper knuckles according to N-SIF based approaches, IMAM 2011, The International Congress of International Maritime Association of the Mediterranean, 13–16 September 2011, Genova, Italy, in Sustainable Maritime Transportation and Exploitation of Sea Resources—Rizzuto & Guedes Soares (eds), Taylor & Francis Group, London, ISBN 978-0415-62081-90.

Health and Safety Executive, 1999. Background to new fatigue guidance for steel joints and connections in offshore structures, Offshore Technology Report— OTH 920390. Hobbacher A. et al 2008. IIW Fatigue Recommendation, document IIW-1823-07, ex XIII-2151r4-07/ XV-1254r4-07, International Institute of Welding, Paris. International Organization for Standardization, 2008. ISO 2394:2008 General principles on reliability for structures. Myers, P.T. 1998. Corrosion fatigue and fracture mechanics of high strength jack up steels, PhD Thesis, University of London. Pilkey, W. 2008. Peterson’s stress concentration factors, New Jersey and Canada, John Willey and Sons. Rizzo, C.M. 2007. Application of Reliability Analysis to the Fatigue of Typical Welded Joints of Ships, Schiffstechnik/Ship Technology Research 54, 89–100. ISSN 0937-7255. Rizzo, C.M. 2011. Application of advanced notch stress approaches to assess fatigue strength of ship structural details: literature review. Report 655, Schriftenreihe Schiffbau, Technische Universität Hamburg-Harburg, Germany. (59 pp.), ISBN 978-3-89220-655-2. Rizzo, C.M. 2008. Ch. 13—Inspection of aged ships and offshore structures. In: Condition assessment of aged structures, Eds. Paik, J.K. and Melchers, R.E. (Woodhead publishing Ltd.), 367–406. ISBN 978-1-84569 334-3. Rizzo, C.M. 2008. Ch.15—Maintenance of aged ships and offshore structures. In: Condition assessment of aged structures, Eds. Paik, J.K. and Melchers, R.E. (Woodhead publishing Ltd.), 430–458. ISBN 978-1 84569-334-3. Rizzo, C.M. 2011. Life cycle of ships and offshore structures-inspection and survey of ship structures: an introduction. A Rational Overview of Ships Inspection and Maintenance Regime. In Ships and Offshore Structure, [Ed. Jeom Kee Paik], in Encyclopedia of Life Support Systems (EOLSS), Developed under the Auspices of the UNESCO, Eolss Publishers, Oxford, UK, [http://www.eolss.net], www.eolss.net/outlinecomponents/Ships-Offshore-Structures.aspx. UEG Offshore Research 1985. Design of tubular joints for offshore structures, Volume 2, Norwich. Zhao, X.L. et al. 2001, Design guide for circular and rectangular hollow section joints under fatigue loading, TUV-Verlag, Germany, CIDECT publication.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Multi-objective optimisation of ship hull structure by genetic algorithm with combined fitness function Z. Sekulski West Pomeranian University of Technology, Szczecin, Poland

ABSTRACT: This paper presents an evolutionary algorithm for multi-objective optimisation of the structural elements of the large spatial sections of ship hulls. The evolutionary algorithm where selection takes place, based on the aggregated objective function combined with domination attributes is proposed and applied to solve the problem of optimizing structural elements of the high speed vehicle-passenger catamaran hull with respect to the weight and surface area. The results of the numerical analysis with the use of the developed algorithm are presented. 1

INTRODUCTION

Due to high complexity, in spite of rising research and computational resources the multi-objective optimisation of the seagoing ship hull structures is still held back by a number of obstacles hindering its application in practice, and the attempts to solve the problem can be judged as marginal. Most authors assume that the outcome of the multiobjective optimisation task is a set of the Paretooptimal solutions. Evolution-based algorithms allow for determination of this set in the single algorithm run thanks to the fact that they process not single solutions but usually the large set of potential solutions which in the consecutive steps gradually evolve to a Pareto-optimal set. The evolutionary algorithm based on the genetic algorithm was proposed in the paper to the optimisation of the seagoing ship structure using in the process of the selection the combined fitness function including in one mathematical expression: (1) optimisation criteria, (2) penalty function for constraints violation, and (3) domination attributes dominance rank as well as dominance count. A practical example is presented, featuring the multi-objective optimisation of the structure of fast passenger-vehicle ferry named Auto Express 82 developed by Austal. 2

SOLVING OF MULTI-OBJECTIVE OPTIMISATION PROBLEM

Classical methods used for the solving of multiobjective optimisation problems based primarily on the aggregation of vector objective functions are easy to implement but ineffective in many cases. However, evolutionary multi-objective optimisation algorithms based on the domination relation

have been proven to be highly effective (Deb 2001, Osyczka 2002, Sarker and Coello Coello 2002, Abraham et al. 2005, Coello Coello et al. 2007). However the researchers have reported for several years that if the number of the optimisation criteria is greater than 3, the evolutionary algorithms based on the domination relation formulated by Pareto (1996) turn to be ineffective since together with the increase of the number of optimisation criteria the number of non-dominated variants decreases reducing the effectiveness of the selection operator (Hughes 2003, Purshouse and Fleming 2003, Jaszkiewicz 2004, Hughes 2005). The evolution-based algorithms for multiobjective optimisation feature both the fitness function and the selection process taking into account a number of criteria which are included in the single fitness function. From this point of view these methods can be divided with respect to the type of the fitness function used for calculations as follows: a) selection with respect to the scalar objective function with fixed weights of optimisation criteria, b) selection with respect to the scalar objective function with random weights of optimisation criteria, c) division of the variant set into sub-sets and selection in each of them with respect to single criteria, d) selection with respect to domination attributes where selection of individuals is based on the location of the current individual in the set of feasible solutions. 3

LOOKING INSIDE THE FEASIBLE SOLUTIOS SET: DOMINANCE RANK AND DOMINANCE COUNT

Assignment of the two-argument dominance attribute (0 or 1) to the feasible variants allows

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Figure 1. Graphical illustration of basic concepts used for taking into account of variant domination in the multiobjective optimisation algorithms.

for dividing the feasible variants set into two subsets (Fig. 1a): (0) subset of dominated variants and (1) subset of non-dominated variants. As it can be seen, this information is quite general and does not refer to the internal structure of the feasible set. Goldberg (1989) proposed for ordering the feasible solutions depending on the depth of consecutive fronts of non-dominated solutions— Figure 1b. More detailed analysis of the structure of feasible set allows for the use of following concepts as dominance attributes: (a) rank of feasible variants, dominance rank—Figure 1c, and (b) feasible variant evaluation dominance count— Figure 1d. The dominance rank for a given variant is proportional to the number of feasible variants dominated by a given variant. The dominance count of a feasible variant is proportional to the number of other feasible variants dominating a given variant. 4

CALCULATION TOOL FOR EVOLUTIONARY MULTI-OBJECTIVE OPTIMISATION OF SHIP HULL STRUCTURE

The most important point of calculation tool for evolutionary multi-objective optimisation of ship hull structure is appropriate formulation of fitness function which governs the optimisation process. The simplest concept is the introduction of

objective function F(x) as a linear combination of no partial objective criteria fs(x): no

F ( )= ) ∑

f (x )

(1)

s s

s =1

where ws are coefficients determining the weights assigned to partial optimisation criteria. Developing fitness function in the form of the scalar substitute optimisation criteria is a commonly accepted practice. Partial optimisation criteria, employed in Eq. 1, were replaced by properly formulated utility functions of these criteria: fs(x) → us(fs(x)): ui uj

⎛ fi (x ) ⎞ ⎝ fi,max ⎠ ⎛f ⎝

fi ( ) → max! f (x ) ⎞

f j,max

⎠

(2a)

fi ( ) → min! (2b)

where fi,max and fj,max are the greatest values of respective optimisation criteria anticipated in computations. After assuming utility function in form of Eq. 2 the scalar substitute optimisation criterion can be written in the form:

516

no

( )= ∑ s =1

s s

(x )

(3)

The calculation tool developed for use in optimisation of ship hull structure should allow to account for a series of constraints imposed by design, local strength and overall strength. Observing that the genetic algorithms require neither continuity nor existence of the derivative functions, an external penalty function has been used: no

f ( )= ∑

nc

s s

s =1

(x)+ )+∑ wk Pk ( )

The strategies for dominance ranking and the dominance count the feasible variants proposed by the author allows for their inclusion directly in the earlier formulated (Eq. 4) extended objective function of unconstrained maximization problem f(x): f

no

∑

s s

s =1

(4)

wrank R f i ( ) nc

(7)

+ wcount C fi ( ) + ∑ wk Pk (x )

k =1

k =1

where: Pk(x)—component of penalty function for the violation of k-th constraint, wk—penalty coefficient for the violation of k-th constraint, nc—number of constraints. As we already know the scheme of multiobjective optimisation proposed in Eq. 4 allows only for rough differentiation of feasible solutions with regard to domination relation in the Pareto sense, Figure 1a. For the solving of the mentioned problem the author proposed a scheme in which the feasible solutions are ranked by the number of other solutions dominated by them, relative to the number of feasible solutions in the current population. Therefore, dominance rank Rfi of i-th feasible solution is specified by the equation: R fi

∑ (i ) =

N fi j 1, j i

dm( j )

N fi

(5)

where dm(i, j) = 1 when i dominates j, and dm(i, j) = 0 in other cases, i, j—indices of verified feasible solutions, Nfi—number of feasible solutions in the current population—Figure 1c. In such a case selection is going to promote feasible solutions located close to the Pareto front, which is a numerical realization of selection pressure exerted on the solutions located close to the Pareto front. Similarly, feasible solutions may be classified by the number of solutions dominating them, relative to the number of feasible solutions. Thus, evaluation dominance count Cfi of i-th feasible solution is expressed by the formula: N fi

C fi (i ) =

∑j

1 j ≠i 1,

dm(j )

N fi

(6)

where dm(j, i) = 1 when j dominates i, and dm(j, i) = 0 otherwise, i, j—indices of verified feasible solutions, Nfi—number of feasible solutions in the current population—Figure 1d. In such a case selection is going to promote feasible solutions located far from the Pareto front, which is a numerical realization of selection pressure exerted on solutions located far from the Pareto front.

where: Rfi(x)—dominance rank of feasible variant, wrank—dominance rank weight coefficient, Cfi(x)— dominance count of feasible variant, wcount—dominance count weight coefficient. As combined objective function f(x) expressed by Eq. 9 is: (1) well defined, (2) single-valued, (3) ascending, having real values and positive in the search space, it has been adopted directly as combined fitness function F(x). The proposed combined fitness multi-objective evolutionary algorithm (CFMOEA) which uses combined fitness function F(x) in the proposed form (Eq. 9), includes instruments which guarantee effective solution of the constrained multi-objective optimisation ship hull structure problem. 5 5.1

SHIP HULL STRUCTURE MODEL FOR MULTI-OBJECTIVE OPTIMISATION General

Effectiveness of the developed evolutionary algorithm for the multi-objective optimisation of seagoing ship hull structures has been verified solving the multi-objective optimisation problem for the midship segment based on the Austal Auto Express 82 design—Figure 2. The structural material is aluminium alloy. The 5083-H111 aluminium alloys are used for plates elements while 6082-T6 aluminium alloys are used for bulb extrusions. The plate thicknesses and the bulb and T-bulb extruded stiffener sections are assumed according to the commercial standards. Bulb extrusions are used as longitudinal stiffeners while T-bulb extrusions are used as web frame profiles. The strength criteria for calculation of plate thicknesses and section moduli of stiffeners and web frames are taken in accordance with the UNITAS (1995) classification rules. It was assumed that bottom, wet deck, outer side and superstructure I and II are subject to the pressure of water dependant on the speed and navigation region. The main deck was loaded by the weight of the trucks transmitted through the tires, mezzanine deck the—weight of

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Figure 2.

Assumed model of craft—midship block-section, frame system and structural regions.

the cars while the upper deck—weight of equipment and passengers. Pressures were calculated according to the procedures of the classification rules. A minimum structural weight (volume of structure) and total of outer area of structural elements for maintenance (cleaning, painting, etc.) were taken as criteria in the study and were introduced in the definition of the objective functions and constraints defined using the classification rules. 5.2

Taking into consideration the operational loads as well as the constraints imposed on the design variables especially resulting from conditions of local and global strength formulated in the approved rules the substitute scalar objective function can be expressed as an augmented objective function of the unconstrained minimization problem: f

k =1

Multi-objective optimisation model of ship hull structure

nc

= w1 f1 (x ) + w2 f2 ( ) + ∑

For the multi-objective optimisation problem a substitute scalar objective function can be formulated, with the use of the components of the vector objective function, in the following form: F(x) = F(f1(x), f2(x)) = w1f1(x) + w2f2(x) → min!

nc

F ( ) + ∑ wk Pk (x ) k

(x )

min! (9)

where all symbols are described before. The augmented objective function (Eq. 11) has been extended by the components corresponding to the dominance attributes:

(8)

f

w1u1 ( ) + w2 u2 ( ) + wrank R fi ( ) nc

where f1(x) is a structural weight of midship blocksection taken to optimisation, f2(x) is an area of the outer surface of structural members subjected to cleaning and painting operations (surface area for maintenance) in the section, w1 and w2 are weight coefficients used for partial optimisation criteria.

k

k =1

+ wcountC fi (x ) + ∑ wk Pk (x )

(10)

k =1

where wrank is a dominance rank coefficient, wcount is a dominance count weight coefficient, u1 is an utility function for structural weight and u2 is an

518

utility function for the area of the outer surface of structural members: ⎛f f1 (x ) ⎞ u1 ( ) = ⎜ 1, max ⎟ → max! f1, max ⎝ ⎠

(11a)

⎛f f2 (x ) ⎞ u2 ( ) = ⎜ 2, max ⎟ → max! f ⎝ ⎠ 2 , max

(11b)

where f1(x) is a current value of the first optimisation criteria, f1,max is a maximum value of the first criteria, f2(x) is a current value of the second optimisation criteria, f2,max is a maximum value of the second criteria and all the remaining symbols are as outlined before. In the present formulation a set of 37 design variables is applied—Figure 3. Introduction of design variable representing the number of transversal frames in a considered section: x4, and numbers of longitudinal stiffeners in the regions: x5, x9, x13, x17, x21, x25, x29, x33, x37 enables simultaneous optimisation of both topology and scantlings. 5.3

Genetic model of the ship hull structure

5.3.1 Chromosome structure In this paper, the ship hull structure is modelled by 37 design variables xi described above, each of them being represented by a string of bits.

Figure 3.

5.3.2 Fitness function As combined objective function f(x) expressed by Eq. 12 is: (1) well defined, (2) single-valued, (3) ascending, having real values and positive in the search space, it has been adopted directly as the combined fitness function: F( )

w1u1 ( ) + w2 u2 ( ) + wrank R fi ( ) nc

+ wcountC fi (x ) + ∑ wk Pk (x )

(12)

k =1

Eq. 14 makes it possible to include the domination attributes to the process of selection of trial solutions. This concept is the key point of the proposed Combined Fitness Multi-Objective Genetic Algorithm (CFMOGA). As it has already been formulated in Section 2, three aggregation-based multi-objective evolutionary strategies for taking account of the partial optimisation criteria f1(x) and f2(x) are used in the scalar objective function (Eq. 10) calculation, and therefore also in the fitness function value calculation (Eq. 14): – selection of variants using the scalar objective function (Eq. 10) with the fixed values of weight coefficients w1 and w2 (w_strategy = 2), – selection of variants using the scalar objective function (Eq. 10) with the values of weight coefficients w1 and w2 randomly and independently generated in the range [0, 1] (w_strategy = 4),

Assumed model of ship hull structure—specification of design variables.

519

– selection of variants using the random selected single partial optimisation criteria F(x) = w1f1(x) or F(x) = w2f2(x) (w_strategy = 3) which is implemented by the random selection of a single nonzero weight criterion. Additionally, it is also possible to have: – selection of variants without (wrank = 0) or with (wrank ≠ 0) (Eq. 14) taking into account the dominance rank of feasible solutions, – selection of variants without (wcount = 0) or with (wcount ≠ 0) (Eq. 14) taking into account the dominance count of feasible solutions. 5.3.3 Genetic operators The basic genetic algorithm (Simple Genetic Algorithm—SGA) produces variants of the new population using three main operators that constitute the GA search mechanism: selection, mutation and crossover. The algorithm in present work was extended by introduction of elitism and updating. Many authors described the selection operators which are responsible on chromosome selection due to the value of their fitness function (Goldberg and Deb 1991, De Jong 1995, Back 1996, Michalewicz 1996). After the analysis of the selection operators a roulette concept was applied for the proportional selection. The mutation operator which introduces a random changes of the chromosome was also described (Back 1996, Michalewicz 1996). Mutation is a random modification of the chromosome. It gives new information to the population and adds diversity to the mate pool (pool of parents selected for reproduction). The crossover operator combines the features of two parent chromosomes to create new solutions. The crossover allows to explore a local area in the solution space. Analysis of the features of the described operators (Goldberg and Deb 1991, Back 1996, Michalewicz 1996) led to developing a n-point random crossover operator. The crossover parameters in this case are: the lowest n_x_site_ min and the greatest n_x_site_max number of the crossover points and the crossover probability pc. The operator works automatically and independently for each pair being intersected (with probability pc), and it sets the number of crossover points n_x_site. The number of points is a random variable inside the set range [n_x_site_min, n_x_site_max]. The effectiveness of the algorithm was improved applying an additional updating operator as well as introducing the elitist strategy. The elitist strategy mitigates the potential effects of loss of genetic material copying certain number of the best adapted parental individuals to the progeny generation. The algorithm selects

fixed number of parental individuals np having the greatest values of the fitness function and the same number of descendant individuals having the least values of the fitness. Selected descendants are substituted by selected parents. In this way the operator increases exploitation the of searching space. Update operator with fixed probability of updating pu introduces an individual, randomly selected from the parental population, to the progeny population, replacing a descendant less adapted individual. 5.4.4 Control parameters Single program run with the defined genetic model is characterized by values of seventeen control parameters. In this case the set of genetic model parameters set for each simulation run signed as symi includes 17 components: symi = (ndv, lch, ng, ni, np, pm, pc, c_strategy, n_x_ site_min, n_x_site_max, pu, elitist, w_strategy, w1, w2, wrank, wcount) where ndv—number of design variables (number of genes), lch—chromosome length (number of bits), ng—number of generations, ni—size of population, np—number of pretenders, pm—mutation probability, pc—crossover probability, c_strategy— denotation of crossover strategy (0 for fixed, 1 for random number of crossover points), n_x_site_min—the lowest number of crossover points, n_x_site_max—the greatest number of crossover points, pu—update probability, elitist— logical variable to switch on (elitism = yes) and off (elitism = no) the pretender selection strategy, w_strategy—denotation of strategy for aggregation of objective function, w1—weight coefficient for weight of structure, w2—weight coefficient of surface area of structural element for cleaning and painting, wrank—weight coefficient of solution dominance rank, wcount—weight coefficient of individual dominance count.

6 6.1

COMPUTATIONAL INVESTIGATIONS Computational investigations program

In order to verify the suitability of the proposed method and the computer code developed for the seeking of Pareto-optimal solutions a number of calculation experiments have been carried out, Table 1, using the ship structure models earlier formulated and discussed in Section 5. From multi-objective optimisation point of view, the aim of the simulation was searching nondominated variants with respect to two optimisation criteria with varying strategies for setting the

520

Table 1.

Control parameters of computational experiments.

No.

Designation symi

Series 1. 1. 2.

sym1-1 sym1-2

3.

sym1-3

Series 2. 4. 5.

sym2-1 sym2-2

Specification (ndv, lch, ng, ni, np, pm, pc, c_strategy, n_x_site_min, n_x_site_max, pu, elitist, w_strategy, w1, w2, wrank, wcount) (37, 135, 10,000, 5,000, 10, 0.086, 0.800, 1, 1, 7, 0.33, yes, 2, 0.5, 0.5, 0.0, 0.0) (37, 135, 10,000, 5,000, 10, 0.086, 0.800, 1, 1, 7, 0.33, yes, 4, random in [0,1], random in [0,1], 0.0, 0.0) (37, 135, 10,000, 5,000, 10, 0.086, 0.800, 1, 1, 7, 0.33, yes, 3, random 0 or 1, random 0 or 1, 0.0, 0.0) (37, 135, 10,000, 5,000, 10, 0.086, 0.800, 1, 1, 7, 0.33, yes, 1, 0.0, 0.0, 3.0, 0.0) (37, 135, 10,000, 5,000, 10, 0.086, 0.800, 1, 1, 7, 0.33, yes, 1, 0.0, 0.0, 0.0, 3.0)

values of weight coefficients for various criteria as well as dominance attributes: 1. Series 1.: simulations marked with symbols sym1-1, sym1-2 and sym1-3. In the simulation marked as sym1-1 fixed values of weight coefficients are used for whole simulation: w1 = 0.5 and w2 = 0.5; what refers to the classical method of weighted criteria. In the simulation marked as sym1-2 the values of weight coefficients w1 and w2 were generated by the computer code as random variables in the range [0, 1], which was done independently for each variant whenever the value of fitness function is calculated. In the simulation marked as sym1-3 the values of weight coefficients w1 and w2 were generated by the computer code as random variables equal to either 0 or 1, which was done independently for each variant whenever the value of fitness function is calculated; the value of 1 was used only for one, randomly selected criterion, with the remaining ones equal to 0. 2. Series 2.: simulations marked with symbols sym2-1 and sym2-2. Search for non-dominated variants while excluding the optimisation criteria from the process of variant selection w1 = w2 = 0.0 (w_strategy = 1) which was governed in particular simulations only by: (i) the value of the dominance rank of feasible solution, wrank = 3.0, wcount = 0.0, in the simulation marked as sym2-1, (ii) the value of feasible variant dominance count, wcount = 3.0, wrank = 0.0, in the simulation marked as sym2–2. The purpose of this simulation series was to find out whether the developed tool is effective in case of evolution being governed only by (i) dominance rank, or (ii) dominance count. This refers to the modern algorithms of the evolutionary multi-objective optimisation, where the evolution is governed only by the dominance attributes.

In all simulations the functions of penalties im posed for the violations of constraints were active, wk ≠ 0, k = 1, 2, …, nc. 6.2 Results of computational investigations 6.2.1 Results of computational investigations— Series 1: sym1-1, sym1-2 and sym1-3 Figures 4a and 4b present a detailed structure of the non-dominated solution set of the last generation reached in sym1-1. It can be seen that the set of nondominated solutions including 14 variants of ship hull structure was found during the simulation. The designer may select for further development one of these variants or a group of them considered as the best. Optimisation objectives values for the variant closest to the origin of coordinate system which was found in 5116 generation are given in Table 1. Figures 5a, 5b, 6a and 6b present a detailed structure of the non-dominated solution set of the last generation in sym1-2 and sym1-3. The set of the non-dominated solutions including 8 variants of the ship hull structure has been found during simulation sym1-2 as well as 15 variants during simulation sym1-3. Optimisation objectives values for the variant closest to the origin of coordinate system which was found in 6145 generation (simulation sym1-2) and in 7611 generation (simulation sym1-3) are given in Table 1. 6.2.2 Results of computational investigations— Series 2: sym2-1 and sym2-2 Figures 7a and 7b present a detailed structure of the non-dominated solution set of the last generation in sym2-1. The set including 10 variants of ship hull structure has been found during the simulation. For the variant closest to the origin of coordinate system, which was found already in 196 generation, the distance equal 1.064 in the normalized objective space, the objective functions values are given in Table 1.

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Figure 4. Results of evolutionary multi-objective optimisation of ship hull structure realised in the simulation sym1-1.

Figure 5.

Results of evolutionary multi-objective optimisation of ship structure realised in the simulation sym1-2.

Figure 6. Results of evolutionary multi-objective optimisation of ship hull structure in realised in the simulation sym1-3.

522

Figure 7.

Results of evolutionary multi-objective optimisation of ship hull structure realised in simulation sym2-1.

Figure 8.

Results of evolutionary multi-objective optimisation of ship hull structure realised in simulation sym2-2.

the distance equal 1.047 in the normalized objective space, the values of optimisation criteria are given in Table 2.

Table 2. Specification of “best” solutions obtained in individual simulations. No.

Specification of “best” solutions

Series 1. 1. 2. 3.

f≈sym1-1 = [1086.28 kN 7422.10 m2]T f≈sym1-2 = [1113.66 kN 7361.45 m2]T f≈sym1-3 = [1153.68 kN 7381.57 m2]T

Series 2. 4. 5.

f≈sym2-1 = [1105.95 kN 7345.11 m2]T f≈sym2-2 = [1192.04 kN 7327.41 m2]T

7

Figures 8a and 8b present a detailed structure of a non-dominated solutions set of a last generation in sym2-2. The set including 13 variants of ship hull structure has been found during the simulation. For the variant closest to the origin of coordinate system, which was found in 5533 generation,

ANALYSYS OF RESULTS, PERFORMANCE ASSESSMENT AND CONCLUSIONS

Despite the fact that the evolutionary algorithms for the multi-objective optimisation do not guarantee identification of the Pareto-optimum compromises they can help identifying a satisfactory approximation, i.e. a set of solutions expected to be not too far distant from the searched front of the optimum solutions. However in this case methods are necessary to evaluate how good produced solutions of formulated problems are. And this leads to the question: how to compare effectiveness of different algorithms? In the

523

context of the present paper the question may be formulated as follows: how to compare effectiveness of the studied evolutionary multi-objective optimisation of ship hull structure, assumed for different evaluation strategies of fitness function. In the case of the evolutionary algorithms for multi-objective optimisation, statistical in their nature, evaluation of the obtained results and comparison of effectiveness of optimisation algorithms implementing different strategies is a very difficult task, resulting in controversy and misunderstanding. Whereas visual and qualitative comparison of the sets approximating Pareto front is commonly used for deduction of quality of the evolutionary multi-objective optimisation, in the case of quantitative methods the searching proper standards is just under way (Knowles et al. 2006). Zitzler et al. (2002b), Zitzler et al. (2002a), Zitzler et al. (2003), Fonseca et al. (2005), Knowles et al. (2006) presented the most extensive review of the problems related to evaluation of effectiveness of the randomized multi-objective optimisation algorithms and proposed a mathematical basis for studying effectiveness of the multi-objective optimisation algorithms. In particular Zitzler et al. (2002b) showed that it is impossible to formulate in a close and exact quantitative way the supremacy of one set of nondominated solutions over the other, therefore it is impossible to prove the supremacy of one algorithm over the other. Choice of one of them is determined by the efficiency in every specific case. Computer simulations confirmed efficiency of the developed computational algorithm and computer code for solving the formulated problem. As the result of calculations, approximations of the set of Pareto-optimum solutions were obtained which contain from several to a dozen of non-dominated solutions. The results do not allow to determine superiority of any examined strategies of the fitness function evaluation unequivocally. To formulate more detailed quantitative conclusions further systematic statistical studies, performed on much larger number of samples, are necessary. REFERENCES Abraham, A., Jain, L. and Goldberg, R., 2005. Evolutionary Multiobjective Optimization. Springer. Back, T., 1996. Evolutionary Algorithms in Theory and Practice. New York: Oxford University Press. Coello Coello, C.A., Lamont, G.B. and Veldhuizen, D.A., 2007. Evolutionary Algorithms for Solving Multi-objective Problems. New York: Springer. De Jong, K., 1995. On Decentralizing Selection Algorithms. In Proceedings of the Sixth International Conference on Genetic Algorithms. San Francisco: Morgan Kaufmann Publishers.

Deb, K., 2001. Multi-Objective Optimization using Evolutionary Algorithms. Chichester: John Wiley & Sons, Ltd. Fonseca, C.M., Knowles, J.D., Thiele, L. and Zitzler, E., 2005. A Tutorial on the Performance Assessment of Stochastic Multiobjective Optimizers. Invited talk. In Evolutionary Multi-Criterion Optimization Conference (EMO 2005), Guanajuato, Mexico. Goldberg, D.E., 1989. Genetic Algorithms in Search, Optimization and Machine Learning. Boston: Addison-Wesley Pub. Goldberg, D.E. and Deb K., 1991. A Comparative Analysis of Selection Schemes Used in Genetic Algorithms. In Foundations of Genetic Algorithms. San Mateo: Morgan Kaufmann Publishers. Hughes, E.J., 2003. Multiple Single Objective Sampling. In Proc. of 2003 Congress on Evolutionary Computation. Piscataway: IEEE. Hughes, E.J., 2005. Evolutionary Many-objective Optimization: Many Once or One Many? In Proceedings of 2005 Congress on evolutionary Computation. Edinborough: IEEE Press. Jaszkiewicz, A., 2004. On the Computational Efficiency of Multiple Objective Metaheuristics: The Knapsack Problem Case Study. European Journal of Operational Research, 158:418–433. Knowles, J.D., Thiele, L. and Zitzler, E., 2006. A tutorial on the Performance Assessment of Stochastic Multiobjective Optimizers. Computer Engineering and Networks Laboratory, ETH Zurich, Switzerland, TIK-Report No. 214. Michalewicz, Z., 1996. Genetic Algorithms + Data Structures = Evolution Programs. Berlin-Heidelberg: Springer-Verlag. Osyczka, A., 2002. Evolutionary Algorithms for Single and Multicriteria Design Optimization. Heidelberg: Physica-Verlag. Pareto, V., 1896. Cours D’Economie Politique, Volume 1. Lausanne: F. Rouge. Purshouse, R.C. and Fleming, P.J., 2003. Evolutionary Many-Objective Optimization: An Exploratory Analysis. In Proceedings of 2003 Congress on Evolutionary Computation. IEEE Press. Sarker, R. and Coello Coello, C.A., 2002. Evolutionary Optimization, Chapter 7, Assessment methodologies for multiobjective evolutionary algorithms. In R. Sarker, M. Mohammadian, X. Yao (eds.) Evolutionary Optimization. Boston: Academic Publishers. UNITAS, 1995. Rules for the Construction and Classification of High Speed Craft. Zitzler, E., Laumanns, M. and Bleuler, S., 2002a. A tutorial on evolutionary multiobjective optimization. In Workshop on multiple objective metaheuristics (MOMH 2002). Berlin: Springer-Verlag. Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M. and Grunert da Fonseca V., 2002b. Performance Assessment of Multiobjective Optimizers: An Analysis and Review. TIK-Report No. 139, Swiss Federal Institute of Technology (ETH) Zurich, Switzerland. Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M. and Grunert da Fonseca V., 2003. Performance Assessment of Multiobjective Optimizers: An Analysis and Review. IEEE Transactions on Evolutionary Computation, 7(2):117–132.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Optimization of structural design to minimize lifetime maintenance cost of a naval vessel Dylan Temple & Matthew Collette University of Michigan, Ann Arbor, Michigan, USA

ABSTRACT: In the design of most naval combatants, the internal structure is designed to minimize the overall weight of the vessel while meeting certain constraints introduced by minimum strength requirements or regulations and standards. This approach to design, however, does not take into account the maintenance that will be necessary over the course of the vessel’s life and the costs that these repairs can cause the ship’s owner to incur. This work assesses year-by-year costs of the naval vessel over her service life to determine a life-cycle maintenance cost value for the ship. The model captures the logistical aspects of structural repair costs by utilizing a maintenance schedule over the vessel’s life. This model is integrated with a multi-objective genetic algorithm to find the trade-offs between the structural weight and life-cycle maintenance costs. 1 INTRODUCTION As current naval fleets age, and governments are forced to extend the service life of vessels beyond what they were designed for, maintenance cost is becoming a massive monetary burden. One of the root causes of this is that the internal structure for most vessels is designed with an emphasis on minimizing the total weight. This can reduce ownership costs through lower steel costs during production and reducing fuel costs through lowering resistance. However, this approach does not take into account the maintenance that will be necessary over course of the vessel’s life. In fact, many leastweight design strategies utilized for naval combatants implicitly contain in their formulations major overhauls or repairs as part of the ship’s service life in an effort to minimize the weight of the vessel. This can lead not only to higher maintenance cost over the expected life of the vessel, but also large increases in cost for extending the service life or utilizing the vessel for unexpected missions. These expenses make it necessary to design the structure of future naval combatants with the lifetime maintenance in mind. This focus can include the costs to maintain the vessel throughout its expected life, as well as ensuring that the costs to extend the operation are minimized. In order to design for these aspects it is also important to understand the trade-offs between the structure’s weight and the life-cycle maintenance costs. This work uses a lifetime structural model that captures both the physical process that degrades the internal structure, as well as the logistical aspects of

repair. Using these aspects of ship maintenance the year-by-year costs for a naval vessel can be assessed over the entire course of her service life. Corrosion and fatigue are both used to assess the damage to each element of the hull’s structure as the combatant ages. In order to capture the inherent uncertainty in the process, a probabilistic method of determining the repair costs due to fatigue damage is used in the analysis. The model captures the logistical aspects of structural repair costs by utilizing a maintenance schedule over the vessel’s life that includes major repairs at set intervals as well as dry-docking the vessel at specific times in her service life. There has been work done on modeling and optimization of the life-cycle maintenance costs for vessels and examining the importance of using them as a driver for designs. Life-cycle maintenance cost due to fatigue damage was used in (Kwon & Frangopol 2012, Okasha & Frangopol 2009) to optimize the inspection and maintenance strategies for structural components. Life cycle cost optimization, based on the structural weight, including maintenance costs was performed on a chemical tanker in (Turan et al. 2009) and shown to aid in the minimization of the total owner ship costs for vessel operators. The importance of designing naval vessels to minimize the lifetime maintenance expenses has been described by (Bankes & Spicknall 1991) where the author concluded that these costs can outweigh many other expenses the vessel will incur over its service life. The monetary impacts of using these costs as a driver in preliminary design has also be discussed in (Gratsos et al. 2009, Rigo 2001).

525

Work has also been done on modeling the damage due to fatigue and corrosion a ship’s structure experiences as it ages. Paik has done work developing models to estimate corrosion rates over the course of a vessel’s life and determining the effect of the corrosion on the strength of the structural components in (Paik et al. 2004, 2003a, b). Work has been done to model fatigue damage to ship’s structures as well. Expressions for fatigue damage were formulated in (Wirsching 1984) using a reliability-based approach. The use of a probabilistic fatigue damage model was proposed in (Collette 2011) based on these formulations and is also utilized in this work. This work uses models for both fatigue and corrosion to assess year-by-year costs due to maintenance over the course of the entire service life of a naval combatant. The lifetime model includes scheduled maintenance, unplanned-maintenance, and costs associated with dry-docking the vessel to perform the necessary repairs. This model is then linked with a genetic algorithm to optimize the internal structure of a vessel’s midship section for lifetime cost and weight so that the trade-offs between these two design objectives can be compared. In the remaining sections the corrosion and fatigue models utilized by the model will be presented followed by the architecture to use them in analyzing the lifetime maintenance costs over an entire service life. The optimization problem formulation will then be presented followed by an example in which the internal structure for the midship section of a nominal DTMB-51 is optimized for both lifetime maintenance cost and weight. 2

CORROSION MODEL

In order to estimate the corrosion damage in each year of the vessel’s service life a time-dependent model proposed in (Paik et al. 2003b) was utilized.

This model is based on the statistical analysis of corrosion data collected from 230 crude and product oil tankers. The model breaks the vessel’s structure into various functional locations for which corrosion occurs at different rates. For each functional location the corrosion of the plate, the stiffener web, and the stiffener flange are modeled independently. This work utilized certain functional locations presented by Paik to define the corrosion rates across a naval combatant’s midship sections. The functional locations are assigned designating codes and specific corrosion rates based on the model. These functional locations, their meaning, and their corrosion rates can be seen in Table 1. In Table 1 a repair type group is also listed, these are explained in further detail in subsequent sections. At the start of the lifetime analysis each grillage within a vessel’s structure is assigned one of the functional location codes. Each year damage, according to the corresponding rate, is applied. 3

FATIGUE MODEL

The fatigue model for this work utilizes a probabilistic formulation of a classical S-N approach (Collette 2011). In the S-N approach the number of cycles until a fatigue crack appears, NI, is given as a function of the stress range acting on a given structural detail and experimentally determined constants, as can be seen in equation (1). NI =

DCR A m km f Δσ

(1)

In equation (1) Δσ is an equivalent stress range acting at the location of a specific fatigue detail, kf is a modeling uncertainty term for the stress, and DCR is the cumulative damage index from the

Table 1. Functional locations used for modeling corrosion rates, their location code via Paik’s definitions, and the corresponding corrosion rates. Functional location

Location code

Corrosion rate (m/year)

Repair type

Bottom shell plating (segregated ballast tank) Bottom shell longitudinals (web) Bottom shell longitudinals (flange) Side shell plate below draft line Side shell longitudinals in ballast tank (web) Side shell longitudinals in ballast tank (flange) Side shell plating above draft line Deck plating (segregated ballast tank) Deck longitudinals in ballast tank (web) Deck plating

B/S-H BSLB(W) BSLB(F) B/S-V SSLB(W) SSLB(F) A/B-V A/B-H DLB(W) A/O-H

0.0597 × 10−3 0.1367 × 10−3 0.1127 × 10−3 0.0622 × 10−3 0.1413 × 10−3 0.0882 × 10−3 0.06613 × 10−3 0.1084 × 10−3 0.2403 × 10−3 0.0523 × 10−3

Dry-dock Dry-dock Dry-dock Dry-dock Dry-dock Dry-dock Pier-side Pier-side Pier-side Pier-side

526

Palmgren-Miner cumulative damage rule, which must be determined empirically. A and m are constants which must also be found empirically. However, in this work the value of NI is treated as a random variable. This aids in modeling the inherent uncertainty in the fatigue process and allows the probability of a crack each year to be easily determined. If it is assumed that A, DCR, and kf follow a log-normal distribution then it can also shown that NI follows a log-normal distribution with the parameters, λI and ζI, shown in equations (2) and (3).

λI

λDCR + λA ζ D2 CR

ζI

λk f + ( (Δσ )))

+ζ A2

ζ kf )

2

(2)

CT (3)

In equations (2) and (3) λDCR , λA, λk f , ζ DCR , ζA, and ζ k f are the parameters for the log-normal distributions of DCR, A, and kf. Using these parameters the probability density function, p(t), and cumulative probability function, P(T) that there is a fatigue crack in a given detail at time t can be defined by equations (4) and (5). p(t ) =

⎛( 1 exp p⎜ ζ I 2π t ⎝

P (T ) =

∞

∫0

λ I )2 ⎞ ⎟ 2ζ I2 ⎠

p(t )dt

(4)

(5)

If it is then assumed that the detail is inspected at evenly spaced time-intervals, I, the probability of detecting a crack initiation in interval In is defined by equation (6). Pcrack ( I n ) = P ( I n ) − P ( I n 1 )

framework which can be linked to an optimization algorithm. In the current scope this is performed on a nominal midship structure for a naval combatant by discretizing the section into individual panels with longitudinal and transverse stiffeners. Each of these grillages is considered an independent panel that experiences corrosion and fatigue damage according to the models presented in the previous sections. The total lifetime maintenance cost for the vessel, CT, is considered to be a sum of four different values as seen in equation (8)

(6)

CF + CC

CS + CFR

(8)

In equation (8) CF is the costs due to fatigue damage, CC is the total cost due to corrosion damage, CS is the costs associated with scheduled maintenance, and CFR is the total costs charged for the method used to perform any repairs (i.e. maintenance done at dry dock versus that done while pier side). Each of these costs is a sum of the corresponding yearly costs over the service life of the vessel. The individual costs are explained below and the entire process is outlined in Figure 1. 4.1

Fatigue costs

At the initialization of the analysis the T-Panels defining the hull are discretized further into fatigue details centered around the intersection of each longitudinal stiffener with a transverse stiffener. During each year of the vessel’s service life the costs due to fatigue damage are assessed based on the incremental probability of detecting a fatigue crack as seen in equation (7). The yearly fatigue cost, Cfi is found using equation (9). m

This can be extended to allow a fatigue detail to require successive maintenance after an initial repair in a previous time interval by using equation (7) Pcrack ( I n ) = P ( I n ) − P ( I n 1 ) + n −1

∑ [ P(Ii ) i =1

P ( I i −1 )][ )][P [ P(I n i ) P(I n

i −1 )]

(7) During each year of the vessels service life the probability that a fatigue crack has been found can be determined using equation (7). 4

LIFETIME MAINTENANCE COST

This work utilizes the above structural damage models to create a lifetime maintenance cost

∑ Pcrack d =1

d

( I n )(Ccrac crack k)

(9)

In equation (9) Ccrack is the cost to repair a fatigue crack in a panel, m is the total number of fatigue details used to define the structure, and Pcrackd ( I n ) is the probability of a fatigue crack in the current time step for fatigue detail d. In this work it assumed that Ccrack is 100,000 USD. 4.2 Corrosion costs Corrosion costs for the vessel are incurred in two circumstances: costs to replace a panel due to local corrosion alone and costs to replace panels due to degradation in the overall global strength of the structure. When the lifetime maintenance analysis is initialized thresholds for the plate thickness, longitudinal web thickness, longitudinal flange

527

Is there maintenance scheduled?

Begin year = 0

Replace corroded panels and incur cost

yes

year += 1

yes Forecast panels that will need to be replaced before next maintenance cycle due to local corrosion

no Are there any panels below thresholds?

yes

no

no Incur pier-side repair charge

year=service life?

Perform maintenance and incur cost

Did the repair require a dry dock? Incur unplanned dry dock charge

no

Incur planned dry dock cost Ultimate moment < moment threshold?

Determine probability of fatigue crack in each panel

yes

End

Incur fatigue costs

no

yes Has the vessel been dry docked already? no Incur cost of panels + pier-side repair cost

Figure 1.

Replace panels such that yes no dry dock is necessary until moment > threshold

Can the moment threshold be reached without drydocking?

yes Replace panels no until moment > moment threshold

Flow chart for lifetime maintenance framework.

thickness, transverse web thickness, and transverse flange thickness are defined. These thresholds set a portion of the original thickness that each component is allowed to degrade before it is required to be replaced. In each year of the service life every panel is assessed individually and if it has corroded below the given thresholds it is replaced and a cost is incurred. In this work the threshold is set at 75% of the original thickness. At the outset of the analysis, in addition to setting limits on the local thicknesses, global moment limits in both hogging and sagging, LMS and LMH, are defined. During each year the total ultimate moment in both hogging and sagging is assessed for the current structure, given any corrosion that has occurred. This is done using a Smith-type progressive collapse analysis (Smith 1977) on the midship section. If the ultimate moment in either direction is found to be less than the limits set upon initialization of the analysis then panels are replaced until the limit is reached. Panels are chosen for replacement based on the repair method necessary to perform the maintenance and the total increase in moment of inertia the structure will gain if the panel is replaced. The cost of replacing a panel is calculated using Rahman and Caldwell’s cost equations for stiffened panels (Rahman & Caldwell 1995). This cost model takes into account specifics such as labor costs, material costs, and fabrication costs. 4.3

Incur cost of panels + unplanned dry-dock cost

Scheduled costs

At the start of the life-cycle analysis a maintenance schedule is assigned to the structure dictating years

in which it will receive maintenance. In each of these years, the frame-work forecasts to the year containing the next scheduled maintenance cycle and determines which panels, if any, will fail due to local corrosion in that time interval. All of those panels are then replaced, and the associated costs are incurred. In this work a fixed interval maintenance cycle strategy is used with intervals of five years. 4.4

Flat rates

Whenever a panel is replaced it is determined if the repairs can be done pier-side or if they must be done at dry-dock. This is determined based on their functional location code as defined in Table 1. If any unscheduled repairs must be done at dry-dock then an emergency dry-docking fee is incurred, which in this work is set to 500,000 USD. If all of the necessary replacements can be performed pierside then an pier-side repair fee of 100,000 USD is charged. During scheduled maintenance cycles a planned dry-docking fee of 250,000 USD is incurred by the vessel. 5

OPTIMIZATION

In order to optimize a structure for both minimum weight and lifetime maintenance costs, as defined in the section above, a Multi-Objective Genetic Algorithm (MOGA) was used. The specific algorithm utilized is the Non-dominated Sorting Genetic Algorithm II (NSGAII). This algorithm, developed

528

by Deb, has been shown to be effective at developing Pareto fronts for difficult or discontinuous optimization problems (Kalyanmoy et al. 2002). 5.1

Application problem

This optimization algorithm was used to minimize both the structural volume and the lifetime maintenance cost for the midship section of a nominal DTMB-51 hull form. The structure that was used, (Ashel et al. 2009), is shown in Figure 2. Each of the stiffened T-panels defining the structure is assigned a functional location from Table 1, and the scantlings for the panels in different groups of functional codes are used as the design variables for the problem. This optimization problem can then be written as shown in equation (10).  minimize : f ( x ) (CT , S )      by varying: x ( p , tw , t f , hw , f ) subject b to : ( LMS × 1.. ) (10) S M H ( MH .1) SM MP ≥ ( ) poriginial

In equation (10) the objective functions, CT and WS, are the total lifetime maintenance cost and the weight of the structure. The design variables, tp, tw, tf, hw, and bf are the thicknesses of the plate, web, and flange for the stiffened panel as well as the height of the web and breadth of the flange. Note that each of the design variables is itself a vector as they are defined differently depending on the location code designated for the panel. In order to group these variables different design groups were defined. Design group 1 are bottom shell and ballast tank plates, design group 2 are side shell plates below the draft line contained in ballast tanks, group 3 are side shell plates below the draft line in the double bottom, design group 4 is side shell plating above the draft line, group 5 is deck plating, design group 6 represents the weather decks, and group 7 are plates comprising the inner hull. These design groups were used to formulate the design vector, details of which can be seen in Table 2. The minimum values are based on the corresponding scantlings in the original midship section. Note that this table also shows the values for four different designs along the final Pareto front as seen in Figure 3 and will be discussed in detail in the subsequent section. The first two constraints ensure that the as-built ultimate moments in both hogging and sagging, MH and MS, are greater or equal to 110% of the respective moment limits, LM H d LMS . The third constraint ensures that the local strength of each panel, p, is no less then the strength of the panel in the original structure. 6

Figure 2. Internal structure DTMB-51 midship section.

Table 2.

for

the

nominal

RESULTS

Using the problem formulation in equation (10) the structure was optimized using 80 individuals evolved over 80 generations. The resulting Pareto front can be seen in Figure 3. Figure 3 shows that there is a clear trade-off between the maintenance cost and structural weight. Note that the structural weight for the designs on the Pareto front is never below that of

Detailed description of the design vector for the problem formulation seen in equation (10).

Variable

x1

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

x12

x13

Symbol Groups Min (mm) Max (mm) Design a Design b Design c Design d

tp 1 3 12 3.11 6.84 7.42 5.79

tp 2,3,4 4 14 8.86 8.85 4 4.00

tp 5 4 15 4.00 4.00 4.02 4

tp 6 6 15 12.84 8.47 6 6

tp 6 3 10 3.94 3.06 3.02 3.02

tw 1–5 3 10 7.81 8.44 5.86 3.001

tw 6,7 3 15 12.44 6.95 8.04 8.13

tf 1–5 3 15 6.27 7.49 5.97 6

tf 6,7 3 30 20.58 11.60 19.79 19.56

hw 1–5 100 300 288.17 268.11 285.97 297.14

Hw 6,7 70 500 454.40 499.82 499.87 499.88

bf 1–5 70 150 73.36 70 75.82 76.31

bf 6,7 70 200 199 165.72 70.53 70

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Figure 3. Pareto front between lifetime maintenance and structure weight both normalized by the values for the original hull form.

Figure 4.

the original structure. This is due to the fact that the panels are bundled into design groups in order to reduce the number of design variables as seen in Table 2. Were the groups to be discretized down to individual panels the original structural design could be fully realized and lighter variants could be formed. In order to further examine the results four individuals along the front (marked a through d) are examined in further detail. The fatigue costs, scheduled maintenance costs, and the unplanned corrosion costs for each year of the design’s service life are shown in Figure 4. Figure 4 shows that as the designs become lighter the fatigue costs increase from near zero to having a significant impact on the lifetime maintenance cost for the vessel. This highlights one of the problems with designing the structure of naval vessels for least weight; as the structure ages, even

Yearly costs for four different designs in the Pareto front.

530

with proper maintenance, fatigue can become a large monetary burden. If the service life of the vessel is extended beyond what it was originally designed for this effect could become increasingly worse. The heaviest structure in Figure 4a, on the other hand, incurs only small costs from fatigue even towards the later end of its life. However, the weight is over 170% that of the original design, which is most likely unacceptable. It is also important to note that none of the structural designs experience unplanned maintenance costs due to corrosion since the scheduled repair process assumes 100% accurate forecating. Designs b and c represent structural designs showing balance in the two objective functions. Design c has less than a 20% increase in weight over the original midship structure; however, also sees a significant decrease in fatigue costs over the course of its service life. This design shows that, by using the lifetime maintenance cost as a design driver, significantly more economically robust structures can be designed with only a small increase in weight. Design b highlights that as the structure weight is allowed to increase more, around 35% higher than the original, considerably more savings can be realized over the course of the vessel. The fatigue costs are moderately lower than for c and the structure requires much less periodic replacement in order to maintain local thicknesses in the midship section over the course of its life. Depending on the mission of a vessel and its expected fuel consumption this increase in weight could definitely be worth the significant decrease in maintenance costs; especially if there is a high likelihood of the operational life being extended. It is also interesting to note the design variables that the optimizer manipulated to arrive at these designs. The heaviest and cheapest variant, as expected, has larger panel components than any of the other midship sections. However, in areas such as the deck plates, the bottom shell plates, and the thickness of most of the flanges, where the corrosion damage is less, the thicknesses have decreased significantly. Similarly; in the lightest variant places such as the height of the web and breath of the flange on the weather deck have increased size in order to maintain overall strength requirements. Designs b and c have scantlings closer to the middle of the allowable variable ranges; however, in places with significantly higher corrosion the sizes are increased and vice versa with panels in functional locations with lower corrosion rates. This shows that the optimizer is using the information supplied by the designer to intelligently size the structure for the given goal.

7

CONCLUSION

A frame-work has been presented to analyze the lifetime maintenance costs of a naval vessel that undergoes both fatigue and corrosion damage over the course of its service life. Using this model the internal structure of a nominal warship midship section has been optimized to find the trade-offs between the weight and lifetime maintenance costs of the section. The trade-spaces that are developed give insights into the use of lifetime maintenance cost as a driver for structural design and show that the relationship between weight, cost, and specific scantlings is not always intuitive. Utilizing this framework allows a designer to better incorporate these costs into the initial design of the structure and minimize them over the operational life of the vessel. In order to expand on this work more advanced structural models that take into account operational specifics such as location of service and the expected nature of the vessel’s mission could be integrated into the model. Models could be also be incorporated into the cost model that analyze the fuel consumption as the vessel ages; which could drive the designs further from the heavier structural variants showing the lowest maintenance costs. This could be intelligently integrated with the scheduled maintenance cycles through the reduction of resistance seen when the hull is cleaned during drydocking. ACKNOWLEDGEMENTS This works was supported by a U.S Department of Defense SMART PhD Scholarship. Additional support was provide by Dr. Paul Hess, ONR Code 331 under grant N00014-11-1-0135 and the Naval Engineering Education Center (NEEC) under NAVSEA Contract N65540-10-C-0003. REFERENCES Ashel, G. et al. 2009. ‘Naval Ship Design’. In 17th International Ship and Offshore Structures Congress, vol. 2. Bankes, F. & Spicknall, M. 1991. ‘Importance of Considering Life-Cycle Maintenance and Modernization Costs in the Design of Navy Ships’. Journal of Ship Production 7(4):227–233. Collette, M. 2011. ‘Hull Structures as a System: Supporting Life-cycle Analysis’. Naval Engineers Journal 123(3):37–47. Gratsos, G. et al. 2009. ‘Life Cycle Cost of Maintaining the Effectiveness of a Ship’s Structure and Environmental Impact of a Ship Design Parameters: an Update’. In RINA Conference on the Design and Operation of Bulk Carriers, Athens, Greece.

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Kalyanmoy, D. et al. 2002. ‘A Fast and Elitist Multiobjective Genetic Algorithim: NSGA-II’. In IEEE Transactions on Evolutionary Computation, vol. 6. Kwon, K. & Frangopol, D. 2012. ‘Fatigue Life Assessment and Lifetime Management of Aluminum Ships Using Life-Cycle Cost Optimization’. Journal of Ship Research 56(2):91–105. Okasha, N.M. & Frangopol, D. 2009. ‘Lifetime-Oriented Multi-Objective Optimization of Structural Maintenance Considering System Reliability, Redundancy and Life-Cycle Cost using GA’. Structural Safety 31:460–474. Paik, J.K. et al. 2004. ‘Ultimate Shear Strength of Plate Elements with Pit Corrosion Wasteage’. Thin-Walled Structures 42:1161–1176. Paik, J.K. et al. 2003a. ‘Time-Variant Ultimate Longitudinal Strength of Corroded Bulk Carriers’. Marine Structures 16:567–600. Paik, J.K. et al. 2003b. ‘A Time-Dependent Corrosion Wastage Model for the Structures of Single- and Double Hull Tankers FSOs and FPSOs’. Marine Technology 40(3):201–217.

Rahman, M.K. & Caldwell, J.B. 1995. ‘Ship Structures: Improvement by Rational Design Optimisation’. International Shipbuilding Progress 42(429). Rigo, P. 2001. ‘Least-Cost Structural Optimization Oriented Preliminary Design’. Journal of Ship Production 17(4):202–215. Smith, C. 1977. ‘Influence of Local Compressive Failure on Ultimate Longitudinal Strength of a Ship’s Hull’. In Proceedings of the International Symposium on Practical Design in Shipbuilding, pp. 73–79. Turan, O. et al. 2009. ‘Maintenance/Repair and Production-Oriented Life Cycle Cost/Earning Model for Ship Structural Optimisation During Conceptual Design Phase’. Ships and Offshore Strucutres 4(2):107–125. Wirsching, P.H. 1984. ‘Fatigue Reliability for Offshore Structures’. Journal of Structural Engineering 110(10):2340–2356.

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Structural reliability and risk models

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Hull girder reliability assessment of FPSO Nian-Zhong Chen, Ge Wang & James Bond ABS, Houston, USA ABS, Singapore

ABSTRACT: The present paper aims to establish a rational reliability assessment procedure for hull girder ultimate strength assessment of ship-shaped FPSOs. The hull girder ultimate strength of FPSOs is calculated by a rigorous progressive collapse analysis using Smith method. The stochastic model of Still-Water Bending Moment (SWBM) is established based on the loading conditions from the operation manual of FPSOs. A stochastic model for the extreme value of Vertical Wave-Induced Bending Moment (VWBM) is proposed based on the long-term distribution of VWBM and the extreme value theories. Hull girder reliability is measured by a first-order reliability method. Four ship-shaped FPSOs are utilized to demonstrate the capability of the hull girder reliability assessment procedure. The effects of return period of VWBM, environment severity factor, and corrosion effects on hull girder reliability index are investigated. A sensitivity analysis for each random variable is also conducted. 1

INTRODUCTION

Floating, Production, Storage and Offloading systems (FPSOs) have been widely constructed in offshore oil and gas fields. FPSOs are operated at specific locations and normally it is unlikely for them to avoid the worse weather conditions during their service life. The Structural Reliability Approach (SRA) has demonstrated that it has the potential to take into account various uncertainties associated with structural degradation (Chen and Wang, 2009; Chen et al., 2011a) and loading effects (Chen et al., 2011b). The structural strength assessment and especially the reliability-based hull girder ultimate strength assessment for FPSOs under the severe sea state induced in the worst weather condition during their service life are of vital importance. The early attempt to conduct hull girder reliability assessment was made by Abrahamsen et al. (1970), Mansour (1972), and Mansour and Faulkner (1973) in which only one load and one strength variable are considered and they are described by appropriate probability distributions and the measure of safety was provided by the probability of failure. Later, Mansour (1974), Faulkner and Sadden (1979) improved the hull girder reliability approaches based on first-order second-moment methods in which the safety of hull girder is evaluated by reliability index. After that, considerable work has been conducted on hull girder reliability assessment in last four decades. The computational methods and stochastic models for hull girder strength and load effects and as well as the reliability methods

have been significantly improved. The work on hull girder reliability has been not only limited to the time-independent reliability problems (e.g., Guedes Soares, 1984; Mansour et al, 1997; Chen et al., 2003; Moan et al, 2006; Chen and Guedes Soares, 2007a; Hørte et al., 2007; Harada et al., 2010; Gaspar and Guedes Soares, 2012) but also extended to the time-dependent reliability problems to account for structural degradation due to fatigue damage and corrosion effects (e.g., Guedes Soares and Ivanov, 1989; Wirsching et al, 1997; Guedes Soares and Garbatov, 1999; Akpan et al., 2002; Paik et al, 2003; Sun and Bai, 2003; Ku et al., 2005). However, among the previous work on hull girder reliability assessment, it is noted that the hull girder ultimate strength is usually calculated by simplified formulae because hull girder reliability analysis will be a time-consuming task if the hull girder ultimate strength in reliability analysis is evaluated by the rigorous progressive collapse analysis using Smith method. In addition, load effects normally have significant impacts on reliability index. But unfortunately, there are still no unified stochastic models for still-water and vertical wave-induced bending moments. This paper aims to establish a rational reliability assessment procedure for hull girder ultimate strength assessment of ship-shaped FPSOs, in which the hull girder ultimate strength of FPSOs is predicted by a rigorous progressive collapse analysis using Smith method and appropriate stochastic models are set up for hull girder ultimate strength and primary load effects. A case study is performed to demonstrate the capability of the established

535

reliability assessment procedure. The effects of the return period of the Vertical Wave-Induced Bending Moment (VWBM), environment severity factor, and corrosion effects on hull girder reliability index are investigated. A sensitivity analysis for each random variable is conducted as well. 2

HULL GIRDER ULTIMATE STRENGTH

The first attempt to predict the hull girder ultimate strength was made by Caldwell (1965). He introduced the fully plastic bending moment of a cross section considering the influence of yielding of all structural members. However, the strength reduction in individual members after they have attained their ultimate strength locally as well as the time lag in collapse of individual members was not considered. This problem was solved by Smith (1977), who proposed a method in which the cross section is divided into a set of elements composed of a stiffener and attached plating, and a progressive collapse analysis is performed based on the assumption that the cross section remains plane after deformation and each panel behaves according to its load/average stress—average strain relationships. After Smith, a lot of research papers were published to develop accurate load/average stress— average strain relationships of stiffened panels forming a hull cross section. For instance, Gordo et al. (1996) modeled the relationship of stiffened steel panels with simple analytical formulas while Chen and Guedes Soares et al (2008) modeled the load—average strain relationship of stiffened composite panels using a nonlinear finite element method (Chen and Guedes Soares, 2007b). The Finite Element Method (FEM) is also a powerful tool to perform the progressive collapse analysis for predicting the hull girder ultimate strength. For example, ABS (Chen et al. 1983) and DnV (Valsgaard et al. 1991) have adopted the FEM to carry out the collapse analysis. The Idealized Structural Unit Method (ISUM) originally proposed by Ueda and Rashed (1984) is another effective method to perform the progressive collapse analysis for hull girder. However, as indicated in Yao (2003), the hull girder is in general too large to perform the progressive collapse analysis using the common FEM, and ISUM still needs to develop more appropriate elements. The Smith method is in general the most effective method among the established known methods for predicting the hull girder ultimate strength. Therefore, the Smith method is used herein to calculate the hull girder ultimate strength of FPSOs. For the limited space, the details of how to perform the rigorous progressive collapse analysis

for predict hull girder ultimate strength can be found in ABS FPI Guide (2009). In applying the calculation procedure in ABS FPI Guide (2009) for predicting hull girder ultimate strength, the following assumptions and limitations should be noted: 1) Geometry symmetry: the mid-ship cross section is symmetric; 2) Material symmetry: material over the mid-ship cross section is symmetric; 3) The hull girder of FPSO is only subjected to a vertical bending moment. If any of the above conditions are not satisfied, the neutral axis of the mid-ship cross section may be rotated and not necessary to be parallel to the baseline of the cross section and the plane of the applied bending moment. 3

STILL-WATER BENDING MOMENT

The Still-Water Bending Moment (SWBM) on hull girder results mainly from the action of the ship’s lightweight, cargo, personnel and buoyancy. The distribution and weight of the personnel and cargo may be the major contributors to variability in SWBM. The first publication on probabilistic presentation of SWBM was made by Trafalski (1967). Truhin (1970) (river going ships), Lewis et al. (1973) (tankers), and Ivanov (1973) (general cargo ships and bulk carriers) stated that there is a need to model the SWBM as a random parameter and a proposal for its presentation with normal distribution was made. Later, Ivanov and Madjarov (1975) investigated 8 cargo ships and a normal distribution was used to fit the SWBM with periods from 2 to 7 years for full and partial load conditions. Mano et al. (1977) addressed that the SWBM approximately follows the normal distribution on the basis of the investigation of 10 container ships and 13 tankers. Guedes Soares and Moan (1988) performed an extensive and systematic study on SWBM. They analyzed about 100 ships with 2000 voyages. The study covers different types of ships belonging to 39 ship owners in 14 countries. Their study shows that the normal distribution might be appropriate to represent the statistical variability of the SWBM on various sections along the ship. A further study was also carried out by Guedes Soares and Dias (1996), in which the SWBM of 40 forty containerships in a total of about 3500 voyages were analyzed, and the resulting descriptive statistics of mean and standard deviation agree well with previously published data. Based on the previous work on probabilistic presentation of SWBM, it is assumed herein that the SWBM of FPSOs follows a Normal distribution.

536

Moan et al. (2006) and Hørte et al. (2007) both indicated that the mean value and the standard deviation of SWBM for tankers under sagging condition may be assumed to be 70% and 20% of the maximum value in the loading manual, respectively. From the point of view of the structure type, ship-shaped FPSOs are similar to tankers. Hence, the mean value and the standard deviation of the SWBM of FPSOs are also assumed herein to be 70% and 20% of the maximum value of the operation manual of FPSOs. In addition, an uncertainty factor ηsw will be introduced to multiple with SWBM for accounting for the model uncertainty of SWBM. ηsw is defined as a normally distributed random variable with a mean value of 1.0 and a coefficient of variation of 0.1. 4

∑ ∑ ∑ ∑ n pi p j pk pl f* ( x )

VERTICAL WAVE—INDUCED BENDING MOMENT

fM w x ) =

The Vertical Wave-Induced Bending Moment (VWBM) is normally calculated by means of: 1) Rule value, which is normally specified by Classification Societies, e.g., ABS, BV, CCS, DnV, GL, LR, NK, IACS etc; 2) direct calculation based on the wave scatter diagrams and Response Amplitude Operators (RAOs). 4.1

Class rule value

The ABS rule value (ABS FPI Guide, 2009) for VWBM of FPSOs at mid-ship, expressed in kN ⋅ m, is given by: Mw Mw

published on probabilistic presentation of the vertical wave-induced bending moment. Only a few pioneering works are mentioned here, such as those of Jasper (1956), Lewis (1957), Bennet et al. (1962), Nordenstrom (1964) etc, that provided a solid base for application of the probabilistic methods in hull girder loads calculations. VWBM is usually described in two ways, either short-term or long-term statistics. The amplitude of VWBM within a short-term duration (normally several hours) corresponding to a steady sea state is usually considered to follow a Rayleigh distribution. The probability density function of the amplitude of long-term VWBM may be obtained by the weighted short-term probability density functions as follows (Ochi, 1978):

2 kV VBM BM C1L B (Cb + 0.7 ) kVBMC1L2 BC Cb

fM w M w ) =

1.5

⎛ 300 − L ⎞ C1 = 10.75 − ⎜ for 90 m ≤ L 300 m ⎝ 100 ⎟⎠ = 10.75 for 300 m ≤ L 350 m 1.5 ⎛ L − 350 ⎞ = 10 10.75 75 − ⎜ for 350 m ≤ L ≤ 500 m ⎝ 100 ⎟⎠

Long-term probabilistic presentation of VWBM

After the revolutionary publication of St. Denis and Pierson (1953), hundreds of papers have been

l

j

k

(3)

l

k ⎛ Mw ⎞ λ ⎜⎝ λ ⎟⎠

k−1

⎡ ⎛ M ⎞ k⎤ exp ⎢ − ⎜ w ⎟ ⎥ ⎢⎣ ⎝ λ ⎠ ⎦⎥

(4)

where k and λ are scale and shape parameters of the distribution, respectively. Accordingly, the long-term VWBM of FPSOs is assumed herein to follow a two-parameter Weibull distribution. 4.3

(2)

k

where f*(⋅) is the short-term probability density function, n* the average number of responses per unit time of short-term response, pi the weighting factor for sea condition, pj the weighting factor for wave spectrum, pk the weighting factor for heading to waves in a given sea, pl the weighting factor for speed in a given sea and heading. Extensive studies on the long-term VWBM (Jensen, 2001) show the long-term probability density function of VWBM fMw() may be well approximated by a two-parameter Weibull distribution as

for Sagging for Hogging

where L, B, and Cb are length, width and block coefficient of a FPSO, respectively. kVBM is the environmental severity factor and C1 is given by

j

∑ ∑ ∑ ∑ n pi p j pk pl i

(1)

4.2

i

Extreme value of VWBM

A stochastic model is proposed herein to represent the extreme value of VWBM Mw with a certain return period. If the long-term VWBM Mw is assumed to follow a two-parameter Weibull distribution, in accordance with the extreme value theories, the extreme value x of Mw within a return period of T years is reasonable to be assumed to follow a Gumbel distribution with a probability density

537

function fexe() and a cumulative density function Fexe() given as ⎧ 1 ⎡ (x u) ⎤ ⎡ ( x u ) ⎤⎫ exp ⎢− ⎥ exp ⎨ exp ⎢− σ ⎥⎬ σ σ ⎣ ⎦ ⎣ ⎦⎭ ⎩ ⎧ ) ⎤⎫ ⎡ ( Fexe ( x ) eexp xp ⎨− exp ⎢− ⎬ σ ⎥⎦⎭ ⎣ ⎩ fexe x ) =

(5) where u and σ are the location parameter and scale parameter of the Gumbel distribution, respectively. If the number of wave cycles is N during the return period of T years and Mw,c is the value of Mw corresponding to the exceeding probability of 1/N, according to the extreme value theories, the mean value and the coefficient of variation of the extreme value of Mw can be derived as: exe µM = M w ,c + w

COVMexew =

0.2124 N(

6

N −1

N) fM w u ) 0.4719

0.2124 + Mw,c N (1 1 / N )

N −1

6.1 fM w u ) (6)

Since the ABS rule value for VWBM was calibrated based on the linear strip theory, an uncertainty factor ηw is introduced to multiple with VWBM to take into account the uncertainty induced by linear response calculation and nonlinear effects. ηw is defined as a normally distributed random variable with a mean value of 1.0 and a coefficient of variation of 0.1. Hull girder ultimate strength of FPSOs in hogging condition is normally much higher than that in sagging condition and the failure mode of a hull girder is usually governed by the sagging failure. Therefore, the hull girder reliability in this paper is only evaluated based on the sagging condition. 5

Moan & Jiao method (Moan and Jiao, 1988), peak coincidence method, etc, were developed for engineering applications. These methods have been applied or modified to predict the maximal value of the total vertical bending moment of a vessel and the corresponding loading combination factors. The numerical comparison of these methods can be found in the work of Guedes Soares (1992), Wang and Moan (1996), Huang and Moan (2008), Teixeira and Guedes Soares (2010), Chen et al.(2011b) etc. In this paper, Turkstra’s rule is used to account for the load combination between SWBM and VWBM. SWBM is modelled as a random variable with a Normal distribution as defined in section 3. VWBM is modelled as a random variable, as defined in section 4, with a Gumbel distribution which represents the probabilistic characteristics of the extreme value of VWBM with a return period considered.

LOAD COMBINATION

SWBM and VWBM are two different stochastic load processes that vary with time. It is practically impossible for both maxima of SWBM and VWBM happen simultaneously. In order to predict the maximal value of the combined two stochastic processes, methods, e.g., square root of the sum of squares (Goodman et al., 1954), Turkstra’s rule (Turkstra, 1970), Ferry Borges—Castanheta model (Ferry Borges and Castanheta, 1971), load coincidence method (Wen, 1977), Söding method (Söding, 1979), pointcrossing method (Larrabee and Cornell, 1981),

RELIABILITY ANALYSIS Limit state function

Structural reliability assessment traditionally considers the limit state to define a failure event. A limit state is a state of the structure including its load at which the structure is just on the point of not satisfying the requirement. The limit state function for hull girder reliability assessment of FPSOs is defined herein as g

uMu

− ηsw M sw

ηw Mw

(7)

where Mu is the hull girder ultimate strength, Msw is the SWBM, Mw is the VWBM, ηu, ηsw, and ηw represent the model uncertainty factors of Mu, Msw, and Mw, respectively. 6.2

Stochastic models

Yield stress of material σy, ηu, ηsw, Msw, ηw, and Mw are considered herein as random variables. Previous studies, e.g. Mansour et al. (1984), has shown that the data of σy is well fitted by a lognormal distribution and thus σy is considered herein to follow a lognormal distribution. ηu, ηsw, and ηw are assumed to follow a Normal distribution. As discussed in sections 3 and 4, Msw and Mw are considered to follow a Normal distribution and a Gumbel distribution, respectively. 6.3 Reliability analysis A First Order Reliability Method (FORM) (Chen and Guedes Soares, 2007c) is used to predict the reliability index. The fundamental idea of the

538

FORM is to find the point on the limit state surface with the minimum distance β to the origin in the standard normal space. This point is traditionally called the design point and β the reliability index. Once the design point is found, the probability of failure Pf is given by Pf = Φ (− β )

(8)

where Φ(⋅) is the standard normal cumulative distribution function. 7

CASE STUDY

Four ship-shaped FPSOs are utilized for the case study. The principal dimensions of the four FPSOs are given in Table 1. Probabilistic characteristics of random variables used in hull girder reliability assessment are listed in Table 2. In this case study, the effects of the return period of VWBM, environment severity factor, and corrosion effects on hull girder reliability index are investigated and the results are shown in Figures 1–3. A sensitivity analysis for each random variable is also conducted and the results are given in the Figure 4. 7.1 Return period of VWBM The relationship between hull girder reliability index β (& probability of failure Pf ) of FPSOs and Table 1.

7.2

Environment severity factor

The VWBM, which are defined and specified by Classification Societies such as ABS FPI Guide (2009), are usually used for design purpose. While FPSOs are normally operated at specific locations, the design VWBM needs to be adjusted for the specific locations. This can be achieved by calculating the environment severity factors applying the service conditions that represent the effect of wave conditions of specific sites. The relationship between hull girder reliability index β (& probability of failure Pf) of FPSOs and environment severity factor is shown in Figure 2. Figure 2 shows that β decreases and Pf increases dramatically with the increase of the environment severity factor, which means β and Pf of a FPSO are sensitive to the specific service condition. It is thus important to take the specific location where a FPSO is located into account in the reliability assessment.

Principal dimensions of FPSOs.

FPSOs

1

2

3

4

Length overall (m) Beam (m) Depth (m)

330 58 29.7

274 48 23.2

366 57 31.5

322 56 29.5

Table 2. variables.

the return period of VWBM is shown in Figure 1. Figure 1 shows that β and Pf are sensitive to the variation of the return period of VWBM since β decreases and Pf increases significantly with the increase of the return period of VWBM from 25 to 100 years. This is because the extreme value of the VWBM increases with the increase of the return period and this leads to the significant increase of Pf and thus the significant decrease of β.

Probabilistic

characteristics

of

random

Symbol

Mean

COV

Distribution

ηu σy ηsw Msw ηw Mw

1.05

0.1 0.089 0.1 0.286 0.1 Eq. (6)

Normal Lognormal Normal Normal Normal Gumbel

1.2 σ yABS 1.0 max 0.7 M sw 1.0 Eq. (6)

where σ yABS is the yield stress given in ABS Rules and max M sw is the maximum value in the operational manual of FPSOs.

Figure 1. Relationship between (a) hull girder reliability index and the return period of VWBM; (b) hull girder probability of failure and the return period of VWBM.

539

Figure 2. Relationship between (a) hull girder reliability index and environment severity factor; (b) hull girder probability of failure and environment severity factor.

Figure 4. The percentages of the sensitivity factors of random variables (a) FPSO 1; (b) FPSO 2; (c) FPSO 3; (d) FPSO 4.

7.3 Figure 3. Relationship between (a) hull girder reliability index and year of service; (b) hull girder probability of failure and year of service.

Corrosion effects

The effects of the degradation of hull girder ultimate strength due to corrosion on hull girder reliability index and probability of failure are investigated. The average coating life is assumed

540

herein to be 3 years and the average wastages of plate thickness at different locations are calculated based on the values specified by ABS FPI Guide (2009). The relationship between hull girder reliability index β (& probability of failure Pf ) of FPSOs and the year of service is shown in Figure 3. It can be seen from the Figure 3 that the corrosion effects on β and Pf are evident. Figure 3 shows β decreases and Pf increases steadily with the increase of year of service. As a result, the corrosion effects need to be considered in the hull girder reliability assessment for aging FPSOs. 7.4

Sensitivity analysis

Sensitivity analysis is an important part of structural reliability assessment. The analysis can not only identify the random variables that have important effects on the reliability estimates but also those variables that are not necessary to be considered as random variables in reliability assessment. Traditionally, the sensitivity factor αi of the random variable i is defined as

αi =

∂β ∂yi

A case study was performed to demonstrate the capability of the established reliability assessment procedure. The effects of the return period of VWBM, environment severity factor, and corrosion effects on hull girder reliability index are investigated. A sensitivity analysis for each random variable is conducted as well. ACKNOWLEDGEMENTS Many thanks to Dr. L.D. Ivanov for his valuable discussion and Mr. J. Speed for improving the manuscript. The views expressed in the paper are those of the authors and do not necessarily reflect those of ABS. REFERENCES

(9) y*

where y* is the design point in the standard normal space. The value of αi is a measure of the sensitivity of the reliability index β to inaccuracies in the value of yi at the design point. The percentages of the sensitivity factors of random variables σy, ηu, ηsw, Msw, ηw, and Mw are shown in Figure 4. Figure 4 shows that the highest sensitivity factor is that of ηu, which means the hull girder reliability index β is most sensitive to the variation of ηu. Figure 4 shows that β is also sensitive to the variation of σy, Msw, ηw, or Mw, but β is not sensitive to the variation of ηsw. This indicates that σy, ηu, Msw, ηw, and Mw should be regarded as random variables, however, ηsw can be regarded as a deterministic variable in the hull girder reliability assessment. 8

• A stochastic model is proposed to represent the probabilistic characteristics of the extreme value of vertical wave-induced bending moment VWBM based on the long-term distribution of VWBM and the extreme value theories. • Hull girder reliability is measured by a first-order reliability method.

CONCLUSIONS

A reliability assessment procedure for hull girder ultimate strength assessment of ship-shaped FPSOs was established in the paper: • Hull girder ultimate strength of FPSOs is predicted by a rigorous progressive collapse analysis using Smith method. • Stochastic model of Still-Water Bending Moment (SWBM) is established based on the loading conditions from the operation manual.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

A study on random field model for representation of corroded surface M.M. Htun, Y. Kawamura & M. Ajiki Department of System Design for Ocean-Space, Yokohama National University, Yokohama, Kanagawa, Japan

ABSTRACT: This paper presents a study on random field model for representation of corroded surface. In this study, as an alternative to the traditional uniform corrosion model, the surface geometry of corroded plate is represented by the random field model (Kerhunen-Loeve Expansion Method). The effect of plate size and the correlation parameter on the random properties of the generated surface is studied. Capability of the random field model to use reduced number of random variables and the effect of reduced random variables on the accuracy of generated random field are examined. And the random characteristics of the real corroded surface are investigated based on the measurement of the real corroded specimen obtained from real ship structure. Then the stochastic property of the minimum cross sectional area of corroded plate generated by random field model is estimated as a reference index of the ultimate strength of the corroded plate. 1

INTRODUCTION

Corrosion is a process of uncertain nature, and the estimation of the strength of the corroded plate is very important for safety assessment of ship structures. Because of uncertain nature governed by many variables, only probabilistic models can describe the corrosion process itself and its effect on the strength of structural components. Many researchers proposed the uniform corrosion wastage models to predict the random property of corrosion wastage of structural components. Hart et al (1986) considered the probabilistic modeling of the ultimate strength by using the uniform corrosion wastage model represented by exponential distribution. Yamamoto & Ikegami (1998) proposed a corrosion model by assuming the general corrosion as the results of degradation of paint coatings, generation of pitting point and progress of pitting point. Paik et al (2003a) suggested a linear corrosion model by three phases: durability or life of the coating, a transition period, and corrosion progression. Guo et al (2008) represented a non-linear formula which defines the time-variant corrosion rate. However in the real field, the corroded surfaces do not have uniform thickness but having fluctuated geometry. This may affect to the ultimate strength estimation. For this reason, it is important to consider the geometry of the corroded surface to estimate its effect on the strength reduction. Some studies have been carried out in the past to predict the ultimate strength of the corroded plate considering the geometry of the surface with pitting corrosion (Paik et al 2003b, Rabuil & Sumi 2011).

Recently, Teixeira & Guedes Soares (2008) modeled the corroded surface by a random field model where the spatial variability of corrosion wastage is represented. And the ultimate strength of the plate is assessed by non-linear finite element analysis. They showed that the strength of the plate with a spatial distribution of corroded thickness represented by random field is usually lower than the one obtained for uniform corrosion. In this study, we investigated the availability of the random field model by using Karhunen-Loeve (K-L) expansion method (Ghanem & Spanos 1991) to represent the stochastic property of the corroded plate as an alternative to the traditional uniform corrosion model. By using this model, the hypothetical corroded plates are generated numerically, by which the random characteristics of minimum cross sectional area of corroded plate are evaluated as a reference index of the strength reduction. Also in this study, the random characteristics of the real corroded surface are investigated using the measurement data of real corroded specimen. 2 2.1

RANDOM FIELD REPRESENTATION OF CORRODED SURFACES Probabilistic model of thickness reduction due to corrosion

In this section the probabilistic characteristic of corrosion wastage is calculated for the targeted ship age which will be used in random field representation of corroded surfaces. Many publications have proposed the corrosion wastage model by which corrosion degradation at

545

a certain ship age can be estimated and they proved that corrosion degradation is following a nonlinear trend as a function of time (Yamamoto & Ikegami 1998, Paik et al 2003a and Guo et al 2008). But for the targeted plate with the relatively high age, it is conservative to use linear corrosion model where the corrosion degradation increase linearly regardless of time. Therefore in this study, linear corrosion wastage model proposed by Paik et al (2003a) is used and according to this model, the thickness reduction of the plate due to corrosion can be estimated as a function of time (t) by: r

C1(T

Tc )

(1)

Figure 1. Schematic diagram of random field of corroded plate.

random variables, ξi using K-L expansion method as in Equation 2 and Figure 1. N

where tr is thickness reduction at a certain ship age. C1 corresponds to the annualized corrosion rate. T is the ship age, and Tc is coating life. From the statistical analysis of measurement data of corrosion, Paik et al (2003a) obtained the probabilistic characteristics of C1 for plate elements in different location of single hull tankers. By using these data for bottom plate, the probabilistic characteristic of thickness reduction for 27 years old (T) is estimated as Table 1 by assuming coating life (Tc) to be 7.5 years for simplicity. 2.2

Random field representation

In this section the hypothetical surfaces with random field of corrosion are generated based on K-L expansion method. It is assumed that the probabilistic nature of the corroded surface is to follow normal distribution having a mean and standard deviation calculated in Table 1 of previous section (mean is 1.184 mm (µorg) and variance is 0.953 mm (varorg) assuming the corroded surface has average corrosion condition). It is noted that this assumption is adopted from the observation of real corroded plate as shown in section 3. Also it is assumed that covariance function, C(x1, x2) is known for the corroded surface which represents the correlation property between two points, x1 and x2. Then the random field of that corroded surface can be represented by a set of uncorrelated Table 1. Probabilistic characteristics of thickness reduction due to corrosion for 27 years old tanker. Corrosion state

Mean

Standard deviation

Variance

Severe Average

3.465 mm 1.184 mm

0.751 mm 0.976 mm

0.564 mm 0.953 mm

Note: Average means the corrosion condition that represents to the wide range of ship so that standard deviation is larger than sever corrosion state.

v = µ + ∑ ξ i λi ϕ i

(2)

i =1

where, v is a vector of random variables having size N (the numbers of discretized points). μ is the mean vector of the original random field in which all of the components have same constant value, µorg. λi and ϕi are eigenvalues and normalized eigenvectors of the covariance matrix C(x1,x2). ξi is a set of N uncorrelated random variables having zero mean and unit variance. In Figure 1, x1, x2, …, xN are the discretized points of random field and v1,v2, …, vN are the random variables at each discretized point which can represent the fluctuation of surface around the mean value. The correlation of these random variables can be specified by the covariance matrix C(x1,x2) in which its components can be obtained from the assumed covariance function, C(x1,x2 ). It is noted that though Figure 1 is drawn as one dimensional (x) for explanation purpose, we discretized two-dimensional (x-y) surface into square grid. Choice of covariance function is a practical consideration and the accuracy of the random characteristics of generated surface depends on the form of covariance function. In this study, based on the measurement of real corroded plate shown in section 3, the covariance function of random field is assumed to be a function of the distance between two points and to follow the exponential form as follow. C ( 1,,xx 2 )

2

⎡ d ⎤ exp ⎢ − ⎥ ⎢⎣ lc ⎥⎦

(3)

where, σ is the standard deviation of the random field. d is the distance between x1 and x2. lc is a parameter commonly known as the correlation length. If lc is small, only closely adjacent points are correlated, and for large lc, correlation exists to a large distance. In Figure 2, the chosen covariance function for different lc is shown.

546

Figure 2. Exponential autocovariance function with different lc.

It is noted that lc should be appropriate one in order for the generated random field to be the representative of the realistic situations. In the present paper, the surfaces are generated by using different lc and the random properties of those surfaces are investigated. The following procedure is used for generation of surfaces which can represent the random properties of corroded surface. 1. Firstly the targeted surface is discretized into N points using 1 mm mesh. 2. Secondly, the covariance matrix for the discretized field is obtained with the assumed correlation function as Equation 3. 3. Thirdly eigenvalues (λi) and eigenvectors (ϕi) of the covariance matrix are computed. 4. Finally, uncorrelated random variables (ξi) are generated and multiplied to the eigenvalues and eigenvectors as in Equation 2 to generate random surface (v). According to Equation 2, when all of the eigenvalues and eigenvectors (full set of random variables (N)) are used, the generated random field can accurately represent to the original random properties. However when the random field model is applied to the simulation to estimate the strength, the computational effort is proportional to the numbers of random variables to be used. Therefore, it is desirable to use small numbers of random variables to represent the random field of corroded surface which corresponds to fewer terms in K-L expansion equation. So, in this study, the surfaces are also generated by the reduced numbers of uncorrelated random variables (M) out of N, which is corresponding to the largest eigenvalues and the accuracy of those surfaces are investigated. According to the procedure explained above, a series of hypothetical corroded surfaces (Ns = 500) having dimension of 105 × 30 mm are generated for each condition of lc and M. Total numbers of discretized points, N, is 3286.

Figure 3. Representation of plate thickness reduction due to corrosion by using different numbers of random variables (lc = 200 mm) (a) M = 3286 (b) M = 1000 (c) M = 30 and (d) M = 10.

1. Six lc, 200 mm, 105 mm, 60 mm, 30 mm, 9.6 mm and 5 mm are used to study the effect of lc on the random properties of the generated surfaces. 2. For each lc, M = 1, 2, 3, 4, 5, 10, 20, 30 and N (3286) numbers of random variables are used to study the effect of reduced numbers of random variables on the accuracy of the generated surfaces. Some of the generated surfaces which represent the reduction of plate thickness due to corrosion process represented by the random field are shown in Figure 3. From this figure we can see that the high locally fluctuation of the generated surface occurs when all random variables are used. 2.3

Effect of plate size and correlation length

In order to study the effect of plate size (a) and correlation length (lc) on the random properties of generated surfaces, the statistical characteristic of the generated surfaces for different a/lc are investigated based on the parameters defined in Equations 4–7; where, a is plate width (30 mm). Firstly statistical characteristics (mean and variance) of each generated surface are calculated by using Equation 4 and 5. N

∑v

ij

vj = varrj

i =1

N

,

E[( E [(vij

(j 12 v j )2 ],

Ns ) (j 12

(4) Ns )

(5)

where, vij is the corrosion diminution of the generates surface at ith discretized point of j th surface. Ns is a number of generated surfaces. Then the mean of these two terms for all surfaces in each case are defined as mean(µsurface) and mean(varsurface) and calculated using Equation 6 and 7.

547

Figure 5. Figure 4. Size effect on the generated surface for different a/lc (all random variables (M = 3286) are used). Ns

mean( µsurfac f e) =

∑v

j

j =1

(6)

Ns Ns

∑ varr

Accuracy vs M for different a/lc.

And then the mean of total discretized points are defined as mean(µgrid) and mean(vargrid) and they are calculated using Equation 10 and 11. These parameters represent the average stochastic property of each grid point and different from Equation 6 and 7. These indexes should be equal to µorg and varorg regardless of the ratio a/lc. N

j

mean(varrsurf surfac face ) =

j =1

(7)

Ns

These parameters represents average stochastic property of one generated surface and it can be considered that the effect of a/lc on the random characteristic of the generated surfaces can be evaluated by these computed parameters. It might be expected that these two parameters are equal to µorg and varorg when all random variables are used. The calculated mean(µsurface) is nearly the same with µorg for all a/lc as shown in Figure 4. However, we can see in Figure 4 that the variance of generated surfaces is smaller than the varorg for small a/lc even though all of the random variables are used. With increasing a/lc, the variances become close to the varorg. From that point we can say that when lc is large compared to the plate size (a), the generated small surface cannot represent to the original random properties and it will have smaller variance. 2.4

Effect of reduced numbers of random variables used in K-L expansion

In this section, the effects of reduced numbers of random variables on the accuracy of the generated surfaces are studied. Firstly, the mean and variance of each discretized point, i (i = 1, 2, ..., N) for all generated surfaces in each case are estimated using Equation 8 and 9. Ns

∑ vij vi = varri

j =1

Ns

i

mean( µ grid ) =

E [(vij

(i 1, 2, , N ) vi )2 ],

(i 1, 2, , N )

(10)

N

∑ varr

i

mean(varrgrid ) =

i −1

N

(11)

Here also, it is found that the computed mean of each grid point (mean(µgrid)) is nearly the same as µorg for all conditions. So that the accuracy of generated surfaces are also discussed by comparing the mean(vargrid) with varorg using the normalized value as shown in Figure 5. The calculated variance of the generated surfaces using all random variables become the same as the varorg for all a/lc (even though it is not shown in Figure 5 for all random variables). However the accuracy of the generated surfaces becomes lower as decreasing M. For the same M, the accuracy decreases as increasing a/lc. And also we can see in Figure 5 that for small a/lc in which lc is large compared to the plate size, the accuracy of the generated surfaces by using all random variables and reduced random variables are not so different. For example if a/lc = 0.2, the normalized variance is close to 1.0 if the number of random variables (M) is greater than 5. From this fact, we can conclude again that for the plate having large lc compared with the plate size (a), we can use small numbers of random variables to represent the original random properties.

(8) (9)

i −1

N

3 ,

∑v

MEASUREMENT OF REAL CORRODED SURFACE

In this section, random characteristics of corroded surface are investigated from real corroded speci-

548

Figure 6.

Yamamoto (2008), the probability density function follow normal distribution as corrosion proceeds to general corrosion. Therefore, we will mainly discuss about the random characteristics of the surface B which has normal distribution. The standard deviation of large surface and mean of that of 9 small surfaces on Surface B are shown in Table 2. We can see the standard deviation of small surface is smaller than that of large surface.

Picture of specimen (surface B).

3.2 Autocovariance function of corroded surface

Figure 7.

Autocovariance function for the specimen is calculated in this section. It is assumed that the height of two points at same distance (d) has same correlation and autocovariance function can be defined as shown in Equation 12;

Specimen size and measurement image.

men, which is part of a ship NAKHODKA (Yao et al. 1998) as shown in Figure 6. The specimen has general corrosion and the size of corroded area is 105 mm × 250 mm. Each side of the specimen are named Surface A and Surface B. Corroded surfaces are measured by using laser sensor, LJ-G080 (Keyence) and desktop NC milling machine, SG01, which is used only to move specimen. The measurement was divided into 9 areas as shown in Figure 7. The measurement intervals are set as 0.14625 mm both on X and Y directions. Random characteristics of different plate sizes (small (105 mm × 30 mm) and large (105 mm × 250 mm)) are calculated from the measurement data. Small one represents the data from one of the 9 small surfaces, and large one includes whole corroded area made by combining of 9 small surfaces. It should be noted that the original thickness of this specimen is unknown. So the height in our measurement data does not mean the thickness reduction but the geometry of the corroded surface because the height will change depending on the position of laser sensor.

ρ( ) =

E [(v( 1 ) − µ )( ( σ2

2

) µ )]

(12)

where, v(xi) is the height at xi, d is the distance between x1 and x2, µ and σ are the mean and standard deviation of the height of a surface.

Figure 8. Probability density functions of large surface. Table 2. Comparison of standard deviation between large and small surface (surface B).

Standard deviation

Large surface

Mean of small surfaces

0.4299 mm

0.3643 mm

3.1 Probabilistic characteristics of corroded surface In general, corrosion condition changes (from pitting to general corrosion) as the ship ages. As progress of corrosion, the shape of probability density function of the corrosion diminution varies from exponential to normal distribution (Yamamoto 2008). As shown in Figure 8, we obtained the probability density function of the height for two surfaces, A and B of the corroded specimen. It can be seen that the probability density function of Surface B well agrees to have normal distribution. According to

Figure 9. Calculated autocovariance function and estimated lc.

549

The computed autocovariance function for large surface and mean of 9 small surfaces on Surface B are shown in Figure 9 as dotted line. Then the correlation length (lc) is estimated by least square method so that Equation 3 fit to the obtained autocovariance function. In Figure 9, the two solid lines are the exponential autocovariance functions (Equation 3) using estimated lc for large surface and small surfaces. We can see lc calculated from large surface is larger than that calculated from small surfaces. From this result, we can again conclude that there is some possibility that lc calculated from such small specimen to be smaller than that of real corroded panel on ship. 4

Table 3.

Real correlation length estimation from viewpoint of variance

For severe corrosion condition, by using Equation 5, variances of the generated surfaces for each assumed lc is calculated. The calculated variances of 500 surfaces have distribution as shown in Figure 10 as an example. The variances at 5% of the distribution, at mean and at 95% of distribution are plotted in Figure 11 for each assumed lc. The computed variance of the real specimen (0.3643 mm) shown in Table 2 is also plotted in the same figure. When the real corroded specimen has the random properties of variance at 5% of the distribution, at

Figure 10. Probability density function of variance of 500 generated surfaces for correlation length, 9.6 mm.

Range of real correlation length, lc (mm).

Range of real correlation length(mm)

ESTIMATION OF REAL CORRELATION LENGTH

In this part of the study, the correlation length, lc of the real large plate in actual field is estimated based on the calculated lc and variance of real corroded specimen in the previous section. In order to estimate the real lc of the corroded plate in actual field, the random corroded surfaces having the size 105 mm ×30 mm generated in section 2.2 for different lc with 3286 numbers of random variables are used. It is assumed that the targeted plate has the random characteristics of corrosion diminution as shown in Table 1 for severe and average corrosion condition. 4.1

Figure 11. Estimation of real correlation length(lc) from variance of generated surfaces (severe case).

From viewpoint of variance From viewpoint of lc

Severe corrosion

Average corrosion

8.5-28-107.7

25-73.6-215

7.22-17-439.4*

7-15-2000*

*Extrapolated value of correlation length.

mean and at 95% of distribution, the real correlation length is estimated to be 8.5 mm, 28.3 mm and 107.7 mm respectively. Therefore we assume that the range of estimated real correlation length is 8.5–107.7 mm as shown in Figure 11. The same procedure is used for average corrosion condition. The estimated range of correlation lengths are shown in Table 3: around 8-30-108 mm for severe corrosion and 25-70-215 mm for average corrosion condition. When average corrosion is considered, distant points has correlation while in severe corrosion condition, only close points has correlation. It is observed that this effect well agrees to the physical phenomenon of corrosion. 4.2

Real correlation length estimation from viewpoint of correlation length

For the generated surfaces for each assumed lc, the correlation length of each generated surface is calculated by using the same method as section 3.2 and the calculated lc for 500 surfaces also has distribution such like Figure 10. For this case also, the calculated correlation lengths at 5% of the distribution, mean and 95% of distribution are plotted in Figure 12 for each assumed lc. The lc of the real corroded specimen calculated in section 3.2 is plotted also in the same figure. With assumption that real corroded specimen to have within these three conditions of lc, the range of real lc is estimated as shown in Figure 12 and Table 3. From viewpoint of correlation length, the range is 8-17-450 mm and 7-15-2000 mm for severe and

550

And the random characteristic of minimum cross sectional area of the corroded plate is investigated. Based on the estimated real correlation length in previous section, lc is taken as 30 mm for severe corrosion.

Figure 12. Estimation real correlation length from correlation length of generated surfaces (severe case).

average corrosion condition. We can see that the range is larger comparing with the estimation from viewpoint of variance. 5

ESTIMATION OF THE RANDOM CHARACTERISTICS OF MINIMUM CROSS SECTIONAL AREA OF THE CORRODED PLATE

5.1

Random characteristic of minimum cross sectional area of corroded plate

0.73

σ xu ⎛ Amin ⎞ = σ xuo ⎜⎝ Ao ⎟⎠

(13)

where, Ao is the original cross-sectional area, Amin is minimum cross-sectional area at the most heavily pitted location. σxu and σxuo are the ultimate compressive strength of the pitted and intact plate. Rabuil & Sumi (2011) also used this equation to estimate the tensile strength of the pitted plate. Based on the above consideration, in this study it is assumed that the ratio of minimum cross sectional area can be regarded as a reference index of the ultimate strength reduction of the plate with random field of corrosion as Equation 14. ⎛A ⎞ R = ⎜ min ⎟ ⎝ Ao ⎠

Table 4 shows the values of parameters used in this section. The distribution of minimum cross sectional area of the corroded plate is shown in Figure 13 and we can see that it has variation from mean value which might cause the strength reduction to be larger than the mean value. 5.2 Effects of numbers of random variables used

Considering the fluctuated geometry of the surface, the cross sectional area of the corroded plate has distribution and the distribution is different for different surfaces. And also the minimum cross sectional area of different surfaces might have a distribution. When considering tensile strength, it seems to be reasonable to assume that ultimate strength is proportional to the minimum cross sectional area. Moreover, Paik (2003b) proposed a closed form formula to estimate the ultimate compressive strength of the plate by pitting corrosion where the estimation is based on the minimum cross sectional area of the corroded plate as Equation 13. R=

1. Firstly, corroded surfaces for the targeted plates are generated by the proposed random field model. 2. Secondly, random characteristics of minimum cross sectional area of the generated corroded plates are calculated.

(14)

In estimation of the response or strength by timeconsuming analysis such as non-linear FEM, it is desirable to use smaller numbers of random variables. However as we have seen in section 2.4, by using reduced numbers of random variables (M), the accuracy of random characteristics becomes lower. Therefore the effects of M on the accuracy of mean and standard deviation of the maximum cross sectional area of thickness reduction are Table 4. Values of parameters for targeted corroded plate. Parameters

Values

Plate dimension Original thickness, to Numbers of discretized points, N Correlation length, lc a/lc Mean, µorg Variance, varorg Numbers of random variables, M

(40 × 100 × to) mm 10, 15, 20 mm 4141 30 mm 1.33 3.4652 mm 0.5638 mm 10, 20, 40, 4141

Figure 13. Probability density function of minimum cross sectional area ratio (mean = 0.596, standard deviation = 0.049) for to 10 mm, M = 10.

551

Table 5. Mean and standard deviation of maximum cross-sectional area of thickness reduction by Montecarlo simulation. Number of random variables (M)

Mean (mm2)

Standard deviation (mm2)

10 20 40 4141 (all random)

161.55 164.16 165.36 167.60

19.52 19.82 19.46 19.63

Original thickness (mm)

Mean of R

Mean of R, *corrected

Standard deviation of R

**Mean

10 15 20

0.596 0.731 0.798

0.581 0.721 0.791

0.049 0.033 0.024

0.653 0.769 0.827

*Corrected value to that of using all random variables. **Mean value considering uniform thickness reduction.

Figure 14. Estimation of maximum cross-sectional area of thickness reduction using all random variables (a/lc = 1.33).

investigated in this section as shown in Table 5. As we expected that by using M random variables, the mean values are smaller than that of using all random variables. However standard deviation is nearly same. By using the data of Table 5, the formula to estimate the exact accuracy with M random variables can be proposed by the fitted curve in Figure 14. Horizontal axis is the accuracy of standard deviation of random field with M random variables (square root value of accuracy of variance shown in Fig. 5). Vertical axis is the normalized value of the mean of maximum cross sectional area of thickness reduction using M random variables to that of using all random variables. The application of the fitted curve to estimate mean value of the minimum cross sectional area using all random variables from that of using M random variables is discussed in next section. 5.3

Table 6. Mean and standard deviation of the ratio of minimum cross sectional area.

Effects of original thickness on the random characteristics of minimum cross sectional area

In this section, the effects of plate original thickness on the random characteristics of the ratio of the minimum cross sectional area is investigated by changing the original thickness of the targeted plate used in Section 5.1. As comparison, the ratio of the mean cross sectional area is also investigated (considering as uniform thickness reduction).

The mean and standard deviation of the minimum cross sectional area using M = 10 random variables are firstly calculated and the mean values are corrected to the values using all random variables as shown in Table 6. Correcting procedure is as shown in Equation 15. It is noted that the fitting curve shown in Figure 14 is only applicable for a/lc=1.33. For different a/lc, it is necessary to reproduce the fitted curve like in Figure 14. From Table 6, we can conclude that as thickness becomes small, strength reduction by corrosion becomes larger. Moreover variability becomes larger. We can also confirm that the mean of strength reduction (R) is large compared with considering as uniform thickness reduction as shown in the last column of Table 6. Amax( M )

A

= Ami min( n( all l ) →

6

→A Amin( alll )

a ( all )

A0

Amax( alll ) (15)

CONCLUSIONS

In this study, the availability of the random field model by using the K-L expansion method to represent the corroded surface is investigated. By generating the surfaces numerically based on the random field model, the probabilistic property of the corroded plate is investigated. 1. The size effect on the random properties of generated surfaces is studied. When the correlation length, lc, is large compared with the plate size (a), the generated surface cannot represent the original random characteristics. It will have smaller variation than the original one. 2. The effect of a reduced number of random variables in the K-L expansion equation on the accuracy of the random properties of generated surfaces is studied. For same accuracy, the plate having the smaller a/lc (small plate size compared with correlation length) can be represented by a smaller number of random variables. This effect

552

is favorable to the reliability analysis of corroded plate in which the computational effort can be minimized by using a few random variables. On the other hand, if a reduced number of random variables are used to estimate the random property of large plates, direct application of the random field model may cause inaccurate results. 3. The random characteristics of the real corroded surface are investigated by measurement of the real corroded specimen. Then, the range of real lc for corroded plate in actual field is estimated by using the random field model. If severe corrosion condition is considered, the correlation length becomes smaller. 4. The random characteristic of minimum cross sectional area of the corroded plate is investigated. From the proposed relation of the accuracy of random field and the estimated maximum cross sectional area of thickness reduction, the accurate value of strength reduction can be calculated using a reduced number of random variables. 5. The estimation of random characteristics of ultimate strength of the corroded plate with random field by non-linear finite element method will be the remaining part of this study. ACKNOWLEDGEMENTS This work is supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (B) 23360383 and (A) 22246109. The authors are grateful for these supports.

REFERENCES Ghanem, R.G. & Spanos, P.D. 1991. A stochastic finite element: A spectral approach (Rev. Ed). Newyork: Springer Verlag. Guo et al. 2008. Time-varying ultimate strength of aging tanker deck plate considering corrosion effect. Marine structures 21:402–419. Hart, D.K. 1986. Structural reliability analysis of stiffened panel. Trans. on. Inst. Nav. Architects (RINA) 128:293–310. Islam, M.R & Sumi, Y. 2011. Geometrical effects of pitting corrosion on strength and deformability of steel rectangular plates subjected to uniaxial tension and pure bending. J. of the JASNAOE 14:9–17. Paik, J.K. et al. 2003a. Time-dependent risk assessment of aging ships accounting for general/pit corrosion, fatigue cracking and local denting damage. ABS technical papers: 219–255. Paik, J.K, Lee, J.M & Ko, M.J. 2003b. Ultimate compressive strength of plate elements with pit corrosion wastage. Proc. Instn Mech. Engrs 217:185–200. Teixeira, A.P & Soares, C.G. 2008. Ultimate strength of plates with random fields of corrosion. Structure and Infrastructure Engineering 4(5):363–370. Yamamoto, N. & Ikegami, K.1998. A study on the degradation of coating and corrosion of ship’s hull based on the probabilistic approach, J. Offshore Mechanics and Arctic Engineering 120:120–127. Yamamoto, N. 2008. Probabilistic model of pitting corrosion and the simulation of pitted corroded condition; Proc. ASME 27th International Conference on Offshore Mechanics and Arctic Engineering OMAE200857623. Yao, T & Sumi, Y, et al. 1998. Analysis of the accident of MV Nakhodka. Part 2. Estimation of structural strength. J. of the Marine Science and Technology 3:181–193.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Lifecycle structural optimization of mid-ship of double hull tanker based on holistic risk evaluation Y. Kawamura, Y. Ohba & Y. Kaede Yokohama National University, Yokohama, Japan

ABSTRACT: Recently, risk-based methods have become widely used in the design of ships and marine structures. Moreover it is considered that sustainability of ships and marine structures should be evaluated based on the assessment of sustainability when the structure is constructed, which includes not only the assessment of the risk of accidents, but also the assessment of environment, economy, social responsibility and etc (ISSC, 2009 and ISSC, 2012). Based on this concept, we have tried to carry out lifecycle structural optimization of the mid-ship section of a double hull tanker in this study. The following five optimization problems are solved, (1) Minimization of construction cost, (2) Maximization of Life Cycle Benefit (LCB) considering the risk of oil outflow, (3) Maximization of LCB considering the risk of CO2 emissions, (4) Maximization of LCB considering the risk of failure, and (5) Maximization of LCB considering all of the risks, which we call the holistic risk. Finally, the optimum solutions of these five problems are compared to discuss about the advantage of the design of ship structures based on the holistic risk evaluation. 1

INTRODUCTION

Table 1.

Because failure of ship structures may cause a serious disaster such as loss of life or environmental destruction, a lot of study about the evaluation methods of structural integrity of ships has been carried out. Recently, instead of the traditional deterministic approach, the risk-based methods have become widely used for evaluation of safety of ships. For example, IMO introduced the FSA (Formal Safety Assessment) for rule-making process, whereas GBS (Goal Based Standard) is developed in which safety level is defined by the risk as a quantitative index. Also in recent years, the concept of risk-based optimization (Papanikolaou, 2010) is studied where multi-objective assessment, such as the weight of the structure (building cost) and risk of oil-outflow, is considered. Moreover, in the committee IV.1 (Design Principle and Criteria) of the International Ship and Offshore Structural Congress (ISSC), it has been discussed that sustainability of ships and marine structures should be evaluated based on the lifecycle assessment of various losses when the structure is constructed, which includes not only the loss by the failure of the ship, but also the losses related to assets (operating cost, building cost), environment (GHG emissions, oil outflow) or human (fatality, injury and etc.) as shown in Table 1 (ISSC, 2012). From the above background, we have tried to carry out lifecycle structural optimization

Systemic P=1 Random P EEDIREQ10% We also assume that the penalty for unsatisfactory of the requirement is imposed if the calculated EEDI value for the target design is greater than the required value. The penalty (RPenal) is assumed as 100 times of the price of CO2 emissions trading (RCO2Em).

(

CO2 navi = RPPenal × EEDI − EEDI R REQ10 × drp r DWT × 10 −6

(baseline

and

required

(10%

(13)

It is note that the risk of CO2 emissions in the lifecycle of the ship is then calculated by using the service period (tlife) and a number of voyages per year (NR) as follows. CO2ope = tlife f

4.3

NR × CO 2 navi

(14)

Evaluation of risk of failure (CRFailure)

4.3.1

Evaluation of CRFailure by structural reliability analysis The authors have proposed the method for maintenance planning of ship structures (Kawamura et al, 2009) in which structural reliability analysis for longitudinal strength of a bulk carrier is carried out. In this study, by using the similar structural reliability analysis, the probability of failure in one voyage (pfvoyage(i)) is computed by using the following limit state function, g (t ) = xu M u − ( xSW MSW

xW MW ),

(15)

where Mu is the longitudinal ultimate strength, MSW is the static bending moment and MW is waveinduced bending moment. And xu, xSW and xW are the modeling uncertainties. Table 4 shows the definition of the probability variables in the above limit state function which are decided by reference to Kawamura et al (2009). Then, the annual failure probability (pfyear(i)) is computed from the pfvoyage(i), and the risk of failure Table 4.

Figure 4. EEDI reduction)).

)

Probabilistic variables.

Variable

Mean

E [GPa] σY [MPa] r (t) [mm] MSW (sagging) [GNm] MW (sagging) [GNm] xu xSW xW

205.8 15.435 411.8 20.59 See 4.3 (2) 4.065 0.678 9.985 0.808 1.0 0.15 1.0 0.05 1.0 0.15

559

St. dev.

Distr. type Normal Lognormal Weibull Normal Frechet Normal Normal Normal

can be defined as the product of the annual failure probability and the amount of the damage (CFfailure). Finally, the lifecycle risk of failure (CRFailure) is simply evaluated as follows, tlife f

CRFailure

∑ ( K × p fyear(i )

CFf Ffailure failure )

(16)

i=1

where the factor, K, is assumed as 50.0 which is introduced to estimate the probability of all failure mode. 4.3.2 Effect of corrosion wastage to the strength Generally, the strength of a ship is decreased as the ship ages because of structural deterioration such as corrosion or fatigue. In the evaluation of the lifecycle risk of failure, such deterioration effect should be considered. In this study, we introduced the corrosion wastage model where the thickness diminution, r(t), at the ship-age, t, is expressed as follows. ⎧0 r(t ) = ⎨ β ⎩α (t t0 )

(t t0 ) (t t0 ) (t

(17)

In this equation, t0 is the life of coating, and α and β are the parameters related to the corrosion rate. Guo et al (2008) proposed the non-linear timedependent corrosion wastage model where β < 1.0, and indicated two equations to predict the “mean” and the “mean+standard-deviation” of r(t) at the deck plate of tankers at a certain ship age, t. Based on this concept, we constructed the estimating equations in Figure 5 and Table 5 which are determined by reference to the data by Paik et al (2003). From these equations, the mean (µ) and the standard deviation (σ) of r(t) at the ship age (t) is computed, and the probability distribution of r(t) can be defined as Weibull distribution by using the µ and the σ. It is noted that the corrosion wastage does not proceed if the ship-age (t) is less than t0 or if the elapsed year after the re-coating is less than t0. Based on this corrosion wastage model, the thickness of the plate or stiffeners are reduced with r(t) when the ultimate strength Mu in equation (15) is evaluated for a certain ship year (t), so that the strength reduction by the corrosion wastage is accounted for. 5

COMPARISON OF OPTIMUM SOLUTION

Table 6 shows the results of optimization of 5 different strategies solved by the GA (Genetic Algorithm). 5.1

Figure 5. Classification of structural location for corrosion wastage model.

Table 5.

Minimization of construction cost (1)

Firstly, convergence of optimization by Genetic Algorithm is confirmed by using this problem. As shown in Figure 6, convergence is achieved at about 200 generations. In this optimization, we regard the solution at 500th generation as the optimum solution, whereas the solution at 1000th generation is used as the optimum solution in the other cases. It is noted that, in the developed GA program, the

Corrosion wastage model for each location. Corrosion wastage (r(t))

Location

Mean(µ)

Mean + Std (µ + σ)

µ2 = 2 × µ1 µ3 = 3 × µ1

[µ + σ]2 = 2 × [µ + σ]1 [µ + σ]3 = 3 × [µ + σ]1

µ1

[µ + σ]1

µ2

[µ + σ]2

Longitudinals DLC, LBLC SSLB, BSLB DLB, LBLB Plate A/O-H, A/B-V,B/S-V, BLGB, O/O-V, B/S-H A/B-H, O/B-V, B/B-H

560

Table 6. strategy.

Optimum solution for each optimization

Figure 6. Convergence of genetic algorithm for minimization of construction cost.

population size (the number of individuals) in one generation is set as 40, the mutation rate is set as 0.05, and the two point crossover method is used. As shown in the optimum solution for minimization of construction cost ((1) Min. CI in Table 6), thickness of the plate and the longitudinals is comparatively small (t = 19, tw = 10, tf = 13). This may cause reduction of the construction cost. Also, it seems that the numbers of longitudinals (Ns and Nb) are small in order to reduce the construction cost. On the other hand, the web-height is slightly large (hw = 820) compared with other cases, maybe because the constraint of the strength of longitudinals should be satisfied. As for the width and height of the double hull (Y and Z), the optimum values are not at the lower limit of MARPOL convention (2000 mm). This is different from the following optimization results, because this optimization does not account for the operational revenue. 5.2

of the ship becomes smaller the necessary engine power becomes smaller in the estimation process of CO2 emissions described in the section 4.2 (2). Next, it is noted that width and height of the double hull (Y and Z) has the value of lower limit of MARPOL convention (Y = Z = 2000 mm). Generally, it is possible to say that the risk of oil outflow becomes larger as the width (height) of the double hull becomes smaller. Even though, the value of Y and Z takes the smallest value in this optimum solution. Figure 7 shows the relationship between the oil outflow and the LCB for the sample design points which is generated during the optimization by the Genetic Algorithm. As shown in this figure, the optimum solution has not only the largest LCB but also the largest oil outflow. It can be concluded that, in this formulation of optimization, the increase of revenue (increase of the volume of the cargo hold) by the reduction of the width (height) of the double-hull has superiority over the increase of the risk of oil outflow. 5.3

Maximization of LCB considering risk of CO2 emissions (CRCO2)

Table 6(3) shows the optimum solution for maximization of LCB considering the risk of CO2 emissions((3)Max. LCB (CRCO2)). The solution of this optimization also has similar tendency to the result of the optimization (2) (Max. LCB (CRoil)). Figure 8 shows the computed EEDI for each sample point generated during the genetic algorithm with the change of DWT. As shown in this figure, EEDI decreases as the DWT becomes larger. This means that the larger ship with larger cargo holds is much better than the small ships from the viewpoint of CO2 emissions. Also in this figure, the blue curve shows the required curve of the EEDI used in this study. It can be observed that a large number of sample points generated during the genetic algorithm are located below the curve. This is because the penalty is imposed if the

Maximization of LCB considering risk of oil outflow (CRoil)

Table 6(2) shows the optimum solution for maximization of LCB considering the risk of oil outflow ((2)Max. LCB(CRoil)). In this solution, similar to the previous optimum solution ((1)Min. CI), the thickness of the plate and longitudinals is also small (t = 19, tw = 10, tf = 14), and the height of the web is relatively large (hw = 760). It seems that small construction cost is achieved satisfying the constraint of strength. Also in this optimization, small structural dimension may affect not only to the reduction of the construction cost, but also to the reduction of the operational cost (fuel cost), because it is assumed in this study that as the weight

Figure 7. Relationship between LCB and the oil outflow for all sample points during the genetic algorithm.

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Figure 8. Computed EEDI for each sample point generated by GA with the curve of the required EEDI.

EEDI value of a sample point becomes larger than the required value. It is also observed that the optimum solution is at the lower right position of the all generated points. This means that the optimum solution has largest DWT with minimum EEDI. However, if the curve of the required EEDI is assumed as the green dotted curve in Figure 8, the optimum solution may not have small EEDI because the possible design with large DWT might be located above the required curve. It is noted that the required EEDI curve should be properly defined so that the efficient ship design with large DWT is not precluded by this requirement. 5.4

Maximization of LCB considering risk of failure (CRFailure)

Table 6(4) shows the optimum solution for maximization of LCB considering the risk of failure ((4) Max. LCB (CRFailure)). The solution in this optimization is different from the previous results in that the solution has larger structural dimensions because the longitudinal strength of the ship is considered for this optimization. Especially, the thickness of plate (t = 27 mm), the thickness and the width of the flange of longitudinals (tf = 32, bf = 120) and the number of longiudinals (Nb = 34) take the large values. As for the double hull, both the width and the height takes the value of the lower limit (Y = Z = 2000 mm). Similar to the optimum solution of the section 5.2 and 5.3, by taking the small width (height) of the double hull, it is possible to make the cargo tank with larger volume so that the operational revenue becomes larger. In addition, the design variable, M, which means the interval of maintenance (re-coating), becomes M = 18 in this optimum solution, though M = 0 (no maintenance) is obtained in the previous optimization where the structural deterioration is not considered. Figure 9 shows the relationship between the probability of failure and the initial cost for the sample design points generated during the optimization. It is observed that as the probability of failure becomes smaller, the initial cost becomes

Figure 9. Initial cost vs probability of failure at 30 ship-year.

larger. The optimum solution seems to be located at the well-balanced point that has the proper initial cost and the probability of failure. 5.5

Maximization of LCB considering holistic risk

Table 6(5) shows the optimum solution for maximization of LCB when all risks (oil outflow, CO2 emissions and structural failure) are considered ((5)Max. LCB (holistic)). Similar to the optimum solution when the risk of failure is considered, the thickness of the plate (t), the number of longitudinals (Nb) and the thickness of the web (tw) take the large values. On the other hand, the thickness of the flange (tf), the height of the web (hw) and the breadth of the flange (bf) take the smaller values compared with the solution of the “(4)Max. LCB (CRFailure)” shown in the previous section. It seems that the solution in this section is the intermediate between the solution of the “(4)Max. LCB (CRFailure)” and that of the “(2) Max. LCB (CRoil)” or “(3) Max. LCB (CRCO2)” shown in the section 5.2 and 5.3. As for the double hull, both the width and the height takes the value of the lower limit (Y = Z = 2000 mm) which is similar to other optimum solutions with maximization of LCB. Similar to the previous optimization ((4)Max. LCB (CRFailure)), the design variable, M, is also not 0. By increasing the frequency of the maintenance, the failure probability (the risk of failure) seems to be reduced. However, the value of the design variable, M = 8, is not practical in this optimum solution. It seems that there is a problem in the balance of weighting between the risk of failure and the maintenance cost in the objective function of our formulation. In order to get reasonable results, calibration of these assessments might be necessary. 6

CONCLUSION

The conclusions of this study are summarized as follows.

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1. In this study, we considered the concept of risk-based optimization framework for the support of design of ship structures. The LCB (lifecycle benefit) is used as the objective function of the optimization in which the various risks are evaluated as a part of the lifecycle cost. 2. As a simple application of the proposed concept, optimization is carried out for the midship cross section of a double hull tanker by using the Genetic Algorithm (GA). We solved five optimization problems, that is (1) minimization of construction cost, (2) maximization of Lifecycle Benefit (LCB) considering risk of oil outflow, (3) maximization of LCB considering CO2 emissions, (4) maximization of LCB considering the risk of failure, and (5) maximization of LCB considering all of the risks which we call the holistic risk. 3. The following observation is made for the obtained optimum solutions. a. In the first to third optimization ((1) Min. Construction Cost, (2) Max. LCB considering Risk of Oil Outflow or (3) CO2 Emissions), the size and the number of longitudinals of the optimum solution is relatively small so that the construction cost and the operational cost becomes lower. b. In the maximization of LCB, the width (height) of the double hull of the optimum solution is minimal in the constraint condition for all cases. This is because the increase of revenue (increase of the volume of the cargo hold) by the reduction of the width (height) of the double-hull has superiority over the increase of the risk of oil outflow in the formulation of this study. c. In the maximization of LCB considering the risk of failure, the thickness of the plates or longitudinals and a number of longitudinals become relatively large in order to increase the strength and to decrease the risk of failure. d. In the maximization of LCB considering the holist risk (all of the risk), the optimum solution is intermediate between the solution considering the risk of failure and the solution considering the risk of CO2 emissions or oil outflow. 4. Because we simplified the problem so as to carry out the optimization easily by the genetic algorithm, some of the assumptions in the optimization in this paper cannot be applied directly to the practical problem. For example, the assumption that all of the longitudinals in the mid-ship section has same geometry is not practical. Also, the concept of the penalty when evaluating the EEDI cannot be adopted in the real regulation at present. Though the above problems exist for practical application of this

concept, it can be concluded that, by using the risk-based optimization framework, it is possible to support the well-balanced design of ship structures in which the specific risks are considered. ACKNOWLEDGEMENTS This work is supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (B) 23360383. Also, this work is supported by Kawasaki Heavy Industries, Ltd. The authors are grateful for these supports. REFERENCES Class NK. 2008. Rules and Guidance for the Survey and Construction of Steel Ships. Guo, J., Wang, G, Ivanov, L., Perakis, A.N. 2008. Timevarying ultimate strength of aging tanker deck plate considering corrosion effect, Journal of Marine Structures, 21(4): 402–419. Hirooka et al. 2005. Application of LCA to the shipbuilding, Report of the National Maritime Research Institute, 2(5): 23–132. (In Japanese). ISSC. 2009. Committee Report, IV.1 Design Principle and Criteria, Proc. of the 17th ISSC: 587–688. ISSC. 2012. Committee Report, IV.1 Design Principle and Criteria, Proc. of the 18th ISSC: 435–506. Kawamura, Y. and Miyazaki, M. 2011. Structural optimization of the hold frame of a bulk carrier considering lifecycle risk, In Guedes Soares & Fricke (eds), Advances in Marine Structures (Proc. Marstruct 2011): 691–698. Kawamura, Y., Nishimoto, M., & Sumi, Y. 2009. A study on a method for maintenance of ship structyres considering remaining life benefit, In Guedes Soares & Das (eds). Analysis and Design of Marine Structures (Proc. Marstruct 2009): 279–289. MARPOL, 2007. International Convention for the Prevention of Pollution from Ships (Consolidated Edition). Ohtsubo, S. 2009. Regulation of NOx, SOx and CO2 Emissions for Ships, Seminar of Japan Ship Technology Research Association: 31–62. (in Japanese). Paik, J.K., et al. 2003. Time-dependent risk assessment of aging ships accounting for general/pit corrosion, fatigue cracking and local denting damage, Trans. SNAME, 111: 159–197. Papanikolaou, A. 2010. Risk-based Tanker Design, Conference Proceedings, The Japan Society of Naval Architects and Ocean Engineers, 10: 363–366. Tsukii, S., & Kawamura, Y. 2009. Assessment of environmental load for ships and marine structures—Calculation of CO2 emission for different type of tankers by LCA-, 21st Ocean Engineering Symposium (JFOES, JASNAOE): OES21-116 (CD-ROM). (In Japanese). Yamada, Y. 2009. The Cost of Oil Spills from Tankers in Relation to Weight of Spilled Oil, Marine Technology, 46(4): 219–228.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Probabilistic pit depth corrosion model of subsea gas pipeline M. Hairil Mohd, D.K. Kim, D.W. Kim & J.K. Paik The Ship and Offshore Research Institute (The Lloyd’s Register Educational Trust Research Centre of Excellence), Pusan National University, Busan, Korea

ABSTRACT: Precisely estimating the reliability of aging structures is crucial, espescially in the oil and gas area where inaccurate estimation of structural behaviour may lead to significant harmful consequences. Related to this issue, it is essential to judiciously predict the corrosion behavior of the gas pipeline structure used in the production of gas in subsea area. This study aims to develop a time-dependant corrosion wastage model for aging subsea gas pipeline. The empirical model is formulated by applying the effect of exposure time to probability density distribution parameters using the 3-Parameter Weibull distribution function. The developed empirical model will give simplification on the prediction of the pit depth of a gas pipeline at any given age by manipulating the scale, shape and location parameter of the probability density distribution with respect to time. The results and outcomes of the present study will be useful for predicting the corrosion pit depth of gas pipeline structures and can be used in the design stage of a new gas pipeline structure. 1

INTRODUCTION

Oil and natural gas have long been used to meet the continuous and increasing demand for energy. In recent years, the hydrocarbon assets in shallow waters have been largely depleted and the oil and gas industries have been compelled to move into more demanding environments such as deeper waters and harsher metocean conditions. They will continue to explore and exploit these deeper waters as long as the demand for oil and gas keep on the increase. Most of the equipments currently used in the offshore oil and gas industries is approaching the end of its useful life. At this stage, the possibility of equipment failing without any significant warning is possibly high. Recent oil spills and equipment failures have demonstrated this danger. Various factors may contribute to these incidents such as human error, lack of advanced knowledge, corrosion effects, and many more. Generally, aging is one of the dominant lifelimiting factor for any structure, and corrosion is one of the most serious features of aging (Nakai and Yamamoto, 2008). It is well known that corrosion is a very complex process by nature, particularly in marine environments where many environmental and material factors significantly affect corrosion (Paik and Thayambali, 2007; Little and Lee, 2007; Palmer and King, 2007). Corrosion problems may occur in numerous subsystems within the offshore oil and gas production system, including the gas pipeline. Therefore, it is essential to ensure that the

gas pipeline structure is always running in safe and controlled environments. Other structures, such as ships can be repaired and maintained in a variety of ways. Ships may have dry docking procedures for maintenance and inspection. There are no such procedures for the maintenance and repair of gas pipeline structures at subsea and if have it may involve high cost for maintenance service. Therefore, corrosion tolerance must be carefully considered in the design of gas pipeline structures. A schematic figure of a subsea system, including pipeline is shown in Figure 1. This study is concerned with modeling the timedependent pit depth corrosion (the amount of wall thickness reduction in term of pit depth penetration)

Figure 1. Schematic view of a subsea system (Bai and Bai, 2010).

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of gas pipeline structures. Previous studies have assessed the importance of corrosion damage evaluation to numerous structures, including gas pipeline and offshore structures as well as its mathematical models. One of the widely used mechanistic models for CO2 corrosion was proposed by Waard and Milliams (De Waard and Milliams, 1975a; De Waard and Milliams, 1975b). They concluded that Based on the assumption of direct reduction of H2CO3 and their own glass cell experiments, the authors presented a correlation for the corrosion rate which is a function of the velocity and temperature. As for electrochemical corrosion model is concern, Gray et al. (Gray et al., 1989; Gray et al., 1990) presented a more complete electrochemical model as a part of their experimental study of CO2 mechanisms. In this study, a number of mechanisms for the electrochemical reactions occurring at the metal surface were adopted from the literature and included into an overall model. In the scope of semi-empirical model especially in CO2 corrosion, De Waard and collaborators (1993; 1995) made some modification to the original model by recalibrating the model constants against the emerging factors to the original correlation to account for various complicating effects. Many studies have been conducted on the corrosion of pipeline structures, especially on pipeline pitting corrosion (Igor and Melchers, 2011: Caleyo et al., 2009), failure pressure prediction on corroded pipelines (Xu and Cheng, 2012; De Leon and Flores Macías, 2005; Sadeghi Meresht et al., 2011) as well as general overviews of pipeline corrosion (Bai and Bai, 2010). Some research has also been conducted on corrosion modeling. Paik et al. (1998; 2003a; 2003b; 2004) developed time-dependent corrosion wastage models for ship structures. Using the statistical analysis of corrosion data, they proposed a mathematical function to define a time-dependent corrosion wastage model. Recently, Paik and Kim (2012) and Hairil and Paik (2012) developed an advanced method for developing a time-dependent empirical corrosion wastage model that applies the probability density parameter against the age of a structure. Melchers et al. (2010) statistically characterized corroded steel surfaces exposed to marine environments. They found that the considerable differences of corrosion loss between different exposure zones are statistically dependent on surface topography. Melchers also reviewed the fundamental research on physical corrosion modeling in marine environments (Melchers, 2011; Melchers, 2008; Melchers, 1997; Melchers, 2003a; Melchers, 2003b; 23–27). Chernov (1990), and Chernov and Ponomarenko (1991) developed a corrosion model that takes account on the effect of the environment

on corrosion. Numerous other empirical models of ship corrosion wastage exist (Paik et al., 2004; Paik et al., 1998; Paik et al., 2003b; Paik et al., 2003a; Yamamoto and Ikegami, 1998; Soares et al., 2006; Qin and Cui, 2005). Previous research on corrosion wastage models is limited to certain types of structures such as ships and floating structures. To the authors’ knowledge, there are no corrosion wastage models for gas pipeline structures at subsea. Therefore, in the present study, the progress of corrosion in gas pipeline structures is statistically defined and the corrosion damage is analyzed in terms of gas pipeline age. The method for the development of an empirical model to predict time-dependent corrosion wastage suggested by Paik and Kim (2012) is applied. The results and the formula developed to describe the progression of the corrosion damage (pit depth) will be useful for designing reasonable corrosion allowances for a gas pipeline and for predicting time dependent corrosion damage loss. 2

TARGET GAS PIPELINE

Four gas pipelines, ranging in age from 8.0 to 29.0 years, were examined in this research. All of the pipelines are located in offshore of Terengganu, Malaysia (South China Sea). Details of each gas pipeline are given in Table 1. The principle view of gas pipeline corrosion is shown in Figure 2. Generally based on Table 1, the longer the distance of gas pipeline, the larger number of measurements will be.

Table 1. Measurements for thickness reduction of gas pipeline. Age Distance I.D (years) (km) Grade

Thickness (mm) Total

A B C D

25.1 25.1 17.1 17.1

8 12 19 29

129.2 129.2 25.4 14.0

API 5L X-65 API 5L X-65 API 5L X-42 API 5L X-60

870 1308 662 150

Figure 2. Principle view of gas pipeline corrosion (PETRONAS, 2011).

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3

DATA COLLECTION OF GAS PIPELINE CORROSION

In a real practice, it is hard to get the corrosion measurement of the gas pipeline. However, with the advancement of technology, the corrosion measurement of gas pipeline can be obtained by having an internal pipeline inspection using Magnetic Flux Leakage (MFL) intelligent pig tool. Measurements of the instances and penetration depth of corrosion on the inner side of gas pipelines from four operating gas pipeline were collected. A total of 2990 measurements are available for the present study. The internal surface of a gas pipeline is normally well coated and corrosion does not occur until the coating has become ineffective. Methods such as internal surface coating are an important part of gas pipeline corrosion protection and the breakdown of the coating will result in the start of corrosion. The sample of corrosion view is illustrated in Figure 3. 4

this statistic will be. From the goodness of fit test, it can be concluded that 3-Parameter Weibull distribution is the reliable distribution function to represent the corrosion characteristics of gas pipeline structures.. Figures 4(a) to (g) show the typical goodness of fit test (Anderson-Darling) for 12 years and 29 years gas pipeline corrosion data respectively. As of goodness of fit test is concerned, it is found that 3-Parameter Weibull distribution function is the reliable distribution function to represent corrosion characteristic of gas pipeline structures. Therefore, the method developed by Paik and Kim (2012) was adopted to develop a time-dependent corrosion wastage model for gas pipeline structures. A statistical analysis is performed to the collected corrosion data to formulate the probability density distribution of pit depth. Figure 5 shows the step-by-step procedure used to develop an empirical time-dependent corrosion wastage model for gas pipeline structures. In the first step, corrosion measurement data for each gas pipeline is collected. The methods used to

DEVELOPMENT OF TIME-DEPENDENT CORROSION MODEL

Different target structure will have different characteristics of its corrosion behaviour. As seawater ballast tank structures of ship corrosion model by Paik et al. (2004) is concerned, this target structure is best to model with exponential distribution. The recent trend of corrosion models by Paik and Kim (2012) and Hairil and Paik (2012) show that the Weibull distribution is also one of the best distribution functions to represent corrosion model. In this study, before certain distribution function is chosen, goodness of fit test is tested again selected distribution functions. It is done by applying the Anderson-Darling (Anderson and Darling, 1954) goodness of fit test statistics on each years gas pipeline corrosion data. This AndersonDarling statistic goodness of fit test will measure how well the data follow a particular distribution. The better the distribution fits the data, the smaller

Figure 3. The typical view of gas pipeline corrosion (PETRONAS, 2011).

Figure 4. Typical view for selected distribution function goodness of fit test (Anderson-Darling) for 12 years corrosion data.

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Figure 6.

Pit depth versus gas pipeline age.

Figure 5. Procedures to develop time-dependent gas pipeline corrosion model.

collect this corrosion data have been discussed in the previous section. Based on the data collected, the relevant statistical analyses are carried out. Figure 6 shows the corrosion damage (pit depth) as a function of time (gas pipeline age) and its relative frequency for each year. From this figure, it shows that the distribution of the relative frequency of corrosion loss (pit depth) is scattered. However, it is found that the relative frequency (probability) distribution of the corrosion damage tends to follow a 3-Parameter Weibull distribution as proof by Anderson-Darling goodness of fit test. In the second step, the corrosion data measurements are statistically analyzed with respect to the age of each gas pipeline. Particular care needs to be taken when performing statistical analysis where the effects of interval (bin width) have significant outcomes on the mean and Coefficient Of Variation (COV) values. In this step, two steps were used to determine the best possible interval for each year corrosion data. First, in order to estimate the total number of intervals for certain years of corrosion data, Doane’s formula (Doane, 1976) was applied. Then, the best possible interval size was calculated based on the method proposed by Paik et al. (2004). They have stated that the best interval value is based on the criteria of the highest mean and the lowest COV. Therefore, a parametric study was first conducted to find the best interval width with respect to the highest mean and lowest COV for each age of the gas pipeline. Figures 7 (a)

Figure 7. Example for the effect of histogram interval (bin width) on mean pit depth characteristics for 8 years and 19 year of gas pipeline structure.

to (b) show the sample of best interval width for 8 years and 19 years corrosion data respectively. For the third step, the 3-Parameter Weibull distribution function is formulated based on the plotted histograms created in the previous step. The best interval value (bin width) is then used to construct histograms of corrosion wastage (pit depth) against the probability density of corrosion damage. In general, a 3-Parameter Weibull function is considered the most suitable for the assessing the progress of corrosion damage (pit depth)

568

of gas pipeline structures, as described in previous goodness of fit test. Eq. 1 shows the 3-Parameter Weibull function itself. The plotted histograms in Figures 8(a) to (d) are then used to obtain the best fit of the 3-Parameter Weibull distribution function. f x) =

α 1

α ⎛x γ ⎞ β ⎜⎝ β ⎟⎠

⎡ ⎛ x − γ ⎞α ⎤ ⋅ exp ⎢ − ⎜ ⎟ ⎥ ⎢⎣ ⎝ β ⎠ ⎥⎦

(1)

where, α = shape parameter, β = scale parameter, and γ = location parameter. In 3-Parameter Weibull formulation, location parameter is not equal to zero. Therefore, compare to Hairil and Paik (2012), this location parameter will remain in the equation and considered as one of the formulated function instead of shape and scale factor. Eq. (1) can be rewritten for corrosion loss at any exposure of time, as follows. fc =

α ⎛ Ye β ⎜⎝ β

⎞ ⎟⎠

⎡ ⎛Y − γ ⎞ α ⎤ ⋅ exp ⎢ − ⎜ e ⎟ ⎥ ⎢⎣ ⎝ β ⎠ ⎥⎦

α 1

(2)

the statistical characteristics of corrosion progress may vary with respect to time. The plotted graph in Figures 11(a) to (c) shows the relationship between the shape, scale and location parameters with the age of the structures. It is found that a second order polynomial is the best fit function to represent the distribution of α, β and γ. Proof of the best fit polynomial also based on the value of coefficient of determination R2. The closer the R2 approaching a value of 1, the better this best fit function will be. Both shape and scale parameters may be approximated as a continuous function of the gas pipeline age, as follows.

α = 00.003337 003337Y Ye 2 − 0.130420 0 130420Ye + 2.4557

(3a)

β = − 0.000997Ye 2 + 0.013425 0 013425 0.0 3 5Ye + 1.58201

(3b)

γ = 0.0003455Ye + 0.062137 0 062137Ye − 0.365129

(3c)

2

The fifth and final step is the development of the corrosion model by inserting the coefficient of α, β and γ into the 3-Parameter Weibull function.

where fc = function of corrosion damage depth (pit depth); Ye = age of the gas pipeline in the years after the breakdown of coating, which can be given as Ye = Y −Yc; Yc = coating life; and Y = age of the gas pipeline structure. The coating life for the present study is unknown and assumed to be zero in the present study. In the fourth step, coefficients α, β and γ are formulated as a function of gas pipeline age. As plotted in Figures 8(a) to (d), it is apparent that

Figure 9. Best fit of Weibull parameter against gas pipeline age.

Table 2. Mean, S.D. and COV value of pit depth for different gas pipeline.

Figure 8. Best fit of the probability density distribution for each year gas pipeline structure.

Target gas pipeline

Age (years)

Pit depth mean (mm)

S.D. (mm)

COV

A B C D

8.0 12.0 19.0 29.0

1.674 1.772 2.407 2.735

0.901 1.128 1.146 0.709

0.538 0.637 0.476 0.259

Note: S.D. = standard deviation, COV = coefficient of variation.

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The time-dependent corrosion wastage (pit depth) model for gas pipeline can then be predicted by the following equation, where: = 0.003337Ye 2

0 130420Ye + 2.4557 0.130420

β = − 0.000997Ye 2 + 0.013425 e + 1.58201

(4)

γ = 00.0003455 0003455Y Ye + 00.062137 062137Ye − 0.365129 2

Eq. 4 can also be used to calculate the corrosion rate in any given year by dividing the related pit depth with respect to the age of gas pipeline Y. Table 2 summarizes the mean value, standard deviation and COV of pit depth for each age of gas pipeline in this study. 5

APPLICATION OF DEVELOPED FORMULA FOR GAS PIPELINE CORROSION PREDICTION

to (c) show comparison values for the shape parameter, scale parameter and location parameter of measured and approximation data. For the comparison of mean and standard deviation, it can be referred to Figures 11(a) and (b). These figures explain that if the plotted value be positioned close to a straight line (y = x), good agreement are achieved between measured and approximate data. Therefore, based on Figures 11(a) and (b) observation, it can be observed that almost every point of comparison lies very close to a straight line (y = x) resulting a good model have been achieved. Examination of Figures 12(a) and (b) also suggest that the developed formula provides approximations with values close to the measured data. It is found from Figures 12(a) and (b) that the pit depth

The developed formula, shown in Eq. 4 is now tested against the available data of gas pipeline pit depth corrosion. This step to make sure that the developed formula using 3-parameter Weibull is capable enough to represent the corrosion characteristic of aging subsea gas pipeline. The differences in the mean and COV values of the measured and theoretically derived pit depth formulae are compared and illustrated in Table 3 and Table 4. Figures 10(a) Table 3. Comparison of shape, scale and location parameter of measured data and approximations value. Measured data

Approximate value

Age (years)

α

β

γ

α

β

γ

8.0 12.0 19.0 22.9

1.644 1.336 1.204 1.475

1.613 1.624 1.462 1.136

0.232 0.280 1.033 1.707

1.626 1.371 1.182 1.480

1.626 1.600 1.477 1.133

0.154 0.430 0.940 1.727

Figure 10a–c. Comparison of shape, scale and location parameter values between measured and approximations data.

Note: α = shape parameter, β = scale parameter, γ = location parameter. Table 4. Comparison of mean and standard deviation of measured data and approximations value. Measured data

Approximate value

Age (years)

Mean (mm)

S.D (mm)

Mean (mm)

S.D (mm)

8.0 12.0 19.0 22.9

1.674 1.772 2.407 2.735

0.901 1.128 1.146 0.709

1.610 1.893 2.335 2.752

0.918 1.080 1.184 0.704

S.D. = standard deviation.

Figure 11a&b. Comparison of mean and standard deviation (S.D.) of pit depth values between measured data and approximations.

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inconsistent increases with time and its corrosion progress can be modeled by 3-Parameter Weibull distribution. Recognizing that the characteristics of pit depth progress are irregular with time, it can be concluded that the formulation proposed here characterizes this effect in a more refined way than has been previously achieved. ACKNOWLEDGEMENTS This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No.: K20903002030-11E0100-04610). The study was undertaken at The Lloyd’s Register Educational Trust (The LRET) Research Centre of Excellence (The Ship and Offshore Research Institute) at Pusan National University, Korea. The LRET funds education, training and research programmes in transportation, science, engineering, technology and the safety of life, worldwide for the benefit of all.

REFERENCES Figure 12. (a) Comparison of pit depth and mean values and its maximum deviation between measured data and approximations. (b) Comparison of pit depth progress with time characterized using a 3-Parameter Weibull type probability density as determined using measured and approximate data.

of gas pipeline tends to have different characteristics of each year of corrosion data. 6

CONCLUDING REMARKS

General overview of gas pipeline corrosion characteristics has been presented in this paper starting from the method on corrosion data collection and its corrosion growth with increasing age. Then, a statistical approach is used in this study to develop a time-dependent mathematical model for the pit depth corrosion of gas pipeline structures. The method involves six steps, as depicted in Figure 5. The corrosion measurements are used to formulate a 3-Parameter Weibull-type probability density for pit depth at various ages. In the probability density, shape, scale and location parameters are empirically formulated as a function of time. It is found that the pit depth of gas pipeline structure at subsea condition significantly

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Renewable energy devices

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Structural design of a floating foundation for offshore wind turbines in red sea K.R. Hussein, A.W. Hussein, E.H. Hegazy & A.A. Amin Naval Architecture and Marine Engineering Department, Faculty of Engineering, Port Said University, Egypt

ABSTRACT: WindFloat concept is a recent innovation that appears beside the main categories of offshore wind turbines. It consists of three columns forming an equilateral triangle; one of these columns is supporting the tower with the turbine blades. The structural design of such a structure is not explicitly given in the classification societies rules. This paper gives a methodology to calculate the scantlings of the floating foundation of the WindFloat to perform a detailed design using a combination of the available guidelines and rules for wind turbines. A computer program has been developed to calculate the scantlings of the floating foundation according to DNV rules. A 3D finite element analysis is performed using ANSYS software for two loading conditions to check the adequacy of the calculated scantlings taking into account the wind axial force, the tower weight and all environmental loads consisting of wave, wind, current and sea level. These loads are calculated by using a developed program. To assess the significance of the environmental loads in the Red Sea, a Finite Element Analysis is made for each load individually. The analysis of each load showed that the hydrostatic pressure results in the highest stresses on the columns. The results obtained for both loading conditions considered have been used to identify the critical areas in the column supporting the tower, and hence determine the structural enhancement required to avoid any undesirable response. 1 1.1

INTRODUCTION Wind energy

Wind energy is a renewable and clean source of power that may provide electricity from other types of power plants and thus reduce greenhouse gases which produce global warming. There are two types of wind turbines: onshore and offshore. Offshore wind turbines usually generate more energy than onshore turbines because coastal wind energy is usually much more reliable and of greater force than inland wind energy due to the open spaces increasing the ability to use wind. Offshore wind turbines are gaining attention for their ability to capture the immense wind resources available over coastal waters. There are two types of offshore wind turbines: Fixed and floating offshore turbines. The former are limited in water depth to approximately 30∼50 m and the latter are extended in water depth to approximately 60∼900 m as mentioned in Musial et al. (2006). There is a good opportunity in Egypt to install floating offshore wind turbines in the Red Sea, precisely in the Gulf of Aqaba and Suez, since the wind speed there can reach 30 m/sec at 50 m height above the sea level; in this region the average water depth is about 220 m. There are a number of offshore wind turbine floating foundation concepts in various

stages of development. They fall into the main categories shown in Figure 1 which represent (A) the Spars concept, (B) Tension Leg Platform (TLP) and (C) the Hybrid spar/TLP (single tendon). In 2009, a new concept was developed by Marine Innovation & Technology called WindFloat. Roddier et al. (2009) had explained that the WindFloat foundation is a semisubmersible attached with 4–6 mooring lines, and can withstand up to 10 MW wind turbine. Waves and wind induced motions are not the only parameters to consider in the floater type. Economics play a significant role. They have reported that the WindFloat is completely installed onshore and towed out to its position fully commissioned. It has simplicity in the design when compared to other concepts. WindFloat is a floating foundation for large wind turbines based on a

Figure 1.

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Floating wind turbine concepts.

main environmental conditions for offshore wind turbines which may contribute to structural damages are mainly wave, wind, current, ship and ice impact, earthquakes, temperature, tides, and wake turbulence. They also include the variation in hydrostatic pressure and buoyancy on members caused by changes in water level due to waves and tides. In this paper the environmental loads which are taken into account are: wave, current, hydrostatic pressure and wind. 2.1 Figure 2. Main components of the WindFloat as described by Roddier et al. (2009).

small column-stabilized semi-submersible platform with one column supporting the tower for a large wind turbine and the other two stabilized column are spread out so as to form an equilateral triangle between the three column centers. These columns are connected to each other with a truss structure composed of main horizontal members connecting columns and bracings as shown in Figure 2. A horizontal Water Entrapment Plate (WEP) is located at the base of each column to provide additional hydrodynamic inertia to the structure due to the large amount of water displaced as the platform moves. Roddier et al. (2010) have reported that the permanent water ballast, inside the bottom of the columns is used to lower the platform to its target operational draft. An active ballast system, which is located in the upper half of each column, moves water from column to column to make the WindFloat in an upright position. The objective of this paper is to review different codes to find the structure scantlings of the WindFloat which can withstand the extreme Red Sea conditions. A program was developed to perform a detailed design of a WindFloat according to DNV guidelines by using the commercial software called MATLAB. It was necessary to develop a Finite Element (FE) model to check the rule-based design by Finite Element Analysis (FEA) and take into account the environmental loads such as: wave, current, wind, sea level and axial force on the rotor blades. For this purpose a computer program was developed to calculate the different environmental loads which will be used in the FEA. 2

ENVIRONMENTAL LOADS

As reported by GL (2005), the design of an offshore wind turbine is based on the environmental conditions to be expected at a proposed site over the project’s lifetime (typically 20 or more years). The

Wave loads

There are two types of waves: regular and irregular waves. Regular waves may be described by deterministic waves which are idealistic. The corresponding theories include Airy wave theory, second-order stokes wave theory, fifth-order stokes wave theory and the stream function theory. Irregular wave theories are described by energy density spectra, (e.g. JONSWAP and Pierson-Moskowitz spectra) as introduced by Dean & Dalrymple (1984). Tempel (2006) has explained that the first step to calculate the wave loads is to convert the spectrum back into individual sinusoids. The sinusoids have amplitude and frequency that can be derived from the energy density given by the spectrum. As reported in GL (2005), Airy wave theory is applicable to define the wave kinematics parameters for deep and transitional water waves. Semi empirical formulae such as Morison’s equation is used only for determining the horizontal wave loads acting on a vertical cylinder having a diameter less than 20% of the wave length as mentioned in the ABS (2010) and the DNV (2007): Fwave

1 π CD ⋅ Dc ρwater | u | u CM ⋅ ρwater ⋅ Dc2 a 2 4 (1)

where Fwave = wave load, N/m; CD = drag coefficient; ρwater = water density, N.s2/m4; Dc = column diameter, m; u = water particle velocity, m/s; CM = inertia coefficient; and a = water particle acceleration, m/s2. For vertical cylinders which have diameters greater than 20% of the wavelengths the incident flow field, diffraction forces and the hydrodynamic interaction of structural members are to be accounted in the design. Linear wave theory is valid only up to the still water level, then the water particle velocity (u) and acceleration (a) are computed by using the formulae of linear wave theory corrected with the Wheeler stretching formulation as follows (Tempel, 2006): u ( x, z , t ) = H π w ⋅

576

cosh ⋅ k ( z d ) cos( kx k − 2π w wt ) sinh( k d ) (2)

H ( a( x, z, t ) = 2H

w)

2

cosh ⋅ k ( z

d)

sinh( k d )

cos(( kx k − 2π w wt )

(3) where u = water particle velocity, m/s; a = water particle acceleration, m/s2; k = wave number; d = mean water depth, m; z = distance from sea water level, positive upward, m; x = distance of propagation, m; w = wave frequency, sec−1; t = time, sec; H = significant wave height, m. Figure 3 shows the flowchart that is used to calculate the wave loads and hydrostatic pressure. 2.2 Current loads Currents are very important in the design of offshore structures because they affect the forces acting on the structure. Several categories of currents are described by GL (2005), but the main category is the wind generated current. The current speed varies with depth of water and the current profile can be obtained by the formula given in GL (2005): ⎛ z d⎞ Uc ( z ) = Uc0 .⎜ ⎝ d ⎟⎠

sea water level, positive upward, m; αcur = current exponential and usually equal 1/7. The current load is given by: FD ( z ) = CD qD ( z )D( z )

(5)

where qD (z ) = design sea current pressure at elevation (z); D(z) = column diameter at elevation z, m. 2.3

Hydrostatic pressure

Hydrostatic loads act in a direction normal to the contact surface; they may be external due to the surrounding water or internal due to the ballast water which is located into each column. Each column is divided into 4 separate tanks by one horizontal and one vertical bulkhead; the lowest tank is a static ballast tank and the upper one is to maintain the WindFloat in a stable condition to withstand any loading condition during the installation, operation and maintenance. The design hydrostatic pressure to be used is calculated according to API (2000): Phyd

α cur

(4)

where Uc0 = wind induced current speed at sea surface, m/s; d = mean water depth, m; z = distance from

N/m

(6)

H z ρwater

where Phyd = Hydrostatic pressure, kg/m2; ρwater = water density, kg/m3; Hz = hydrostatic head, m. Hz

d+

H ⎛ cos k (d d ) ⎞ 2 ⎜⎝ cosh( kd ) ⎟⎠

(7)

where d = depth below still water surface including tide, positive downward, m; d = mean water depth, m; H = wave height, m. 2.4

Wind load

Wind speed varies with time. It also varies with the height above the sea surface. For these reasons, the average time for wind speed and the reference height must always be specified. The wind shear (the increase of mean wind speed with height) and wind turbulence intensity (fluctuations in wind speed on a relatively fast time-scale) are dependent on the wind turbine class and the design wind condition. The wind shear profile is calculated by (ABS, 2010): V ( h ) =Vhub . ⎛ h h ⎞ ⎝ hub ⎠ Figure 3. Flowchart to calculate the wave loads and hydrostatic pressure.

α

(8)

where Vhub = wind speed at hub height, m/s; h = distance from sea water level, m; hhub = hub height above sea water level, m.

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The wind acts on three main areas as follows:

3

1. On the air gap of the WindFloat columns. 2. On the mast of the wind turbine along its height above the sea level. 3. On the turbine blades. Wind load on the first two areas defined above is given by (ABS, 2010): Fw

ρaair . C s ⋅ A V ( h )2 2

N

(9)

where ρair = air density, N ⋅ s2/m4; Cs = shape coefficient; A = projected area, m2; V(h) = wind speed at height h above sea level, m/s. The axial force acting on the turbine blades is given by (Burton et al. 2001): Faxial

2A Ad ρaair V 2 fa

fa )

N

(10)

where ρair = air density; Ad = projected area; V = wind speed; fa = axial flow induction factor (Burton et al. 2001). A sub-program was developed to evaluate the wind force acting on the subjected area and the rotor blades as mentioned above. The flowchart of the sub-program is shown in Figure 4.

The structural design of WindFloat structure is not given directly in the classification societies rules. This section demonstrates the rules and the guides which may be used to determine the minimum required section modulus of the floating structure of the WindFloat, as follows: − DNV-OS-J101, Design of Offshore Wind Turbine. (Det Norske Veritas). DNV (2007) is applicable to the design of complete structures, including substructures and foundations, excluding wind turbine components such as nacelles and rotors. It gives the impact of environmental effects on offshore wind turbine and how to calculate the loads generated from these effects. Formulae for calculation of the required section modulus of different components are given. − DNV (2002), Buckling strength of shells— Recommended practice for planning, designing and constructing floating production system. This guide reports the different buckling modes for stiffened cylindrical shell, and the required geometry of the stiffeners and their proportions. − ABS (2004), Guide for buckling and ultimate strength assessment for offshore structure (American Bureau of Shipping). This guide gives the criteria for calculating the buckling limit state of orthogonally stiffened cylindrical shell subjected to axial loading, bending moment, radial pressure or a combination of these loads. It also gives the geometry and the scantlings proportions required for designing a cylindrical shell after calculating the minimum required moment of inertia I which is based on the axial and circumferential load acting on shell. − API (1997), Recommended Practice for planning, designing and constructing fixed offshore platforms (American Petroleum Institute). This gives the allowable stresses for cylindrical members and the sequence for calculating the circumferential ring size as well the effective width of the shell. 4

Figure 4.

Flowchart of wind loads calculation.

STRUCTURAL DESIGN CODES

RULE-BASED STRUCTURAL DESIGN OF WINDFLOAT

The design basis for the WindFloat and the requirements that must be addressed by design teams in this new technology is explained by Roddier et al. (2009). Stiffened cylindrical shells are used in the fabrication of the floating structure for WindFloat. The columns are orthogonally stiffened by

578

a stringer and a ring frame as shown in Figure 5 as described by API (2000). General types of stiffener profiles, such as flat bar, unsymmetrical T-bar, angle and bulb plate, may be used. The geometry of the stringer and ring frame are as shown in Figure 6 and Figure 7 to prevent the local instability as described in ABS (2004). A MATLAB Program has been developed to calculate the structural scantlings of the floating foundation according to the rules and guidelines available. The procedure is explained in the flowchart shown in Figure 8.

Figure 5.

Orthogonally stiffened cylinder.

Figure 8. Flowchart for the developed program for scantlings calculation.

Figure 6.

Figure 7.

Stringer stiffened shell.

Ring stiffened shell.

5

CASE STUDY

The application is performed on an existing WindFloat that can withstand up to 10 MW wind turbine as proposed by Cermelli et al. (2009). The developed programs have been applied to carry out the structural design of the floating foundation of WindFloat in Red Sea. Cermelli et al. (2009) have mentioned the main dimensions of the floating foundation that are summarized in Table 1. The input given in Table 2 defines the average sea state for a 100 year return period as given by the Authority (2005). It is to be noted that the operating draft given in Table 1 corresponds to 2917 tonnes light weight plus 4134 tonnes of ballast distributed in all columns of WindFloat; this condition is defined as 100% ballast condition. The relation between the shell thickness (tshell), effective breadth (beff) and the angle between stringers (Θ) is plotted in Figure 9 using the devel-

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Table 1. Main dimensions of the floating foundation of the WindFloat (Cermelli et al. 2009).

Table 3. Scantlings of the main components of the floating foundation.

Items

Dimensions (m)

Items

Outputs

Column diameter Pontoon diameter Bracing diameter Length of heave plate edge Column center to center Operating draft Air gap l (distance between rings)

10.7 1.8 1.2 13.7 56.4 22.6 10.7 3.36

Column thickness Horizontal members thickness Bracing thickness Heave plate thickness Top plate thickness Required section modulus for ring frame Required section modulus for stringer

25 mm 10 mm 8 mm 20 mm 15 mm 6.4E+5 mm3 2.3E+6 mm3

Table 2. Average sea state for a 100 year return period for Red Sea (Authority, 2005). Average depth (m) Current speed (m/sec) Wind speed at 10 m above sea level (m/sec) Peak period (T) (sec.) Significant wave height (H) Tide range (m)

Figure 9.

220 1 10.7 12 8 ±1

Table 4. Stiffener dimensions of the column (mm), tshell = 25mm. Items

Stringer

Ring frame

Cross section

Unsymmetrical T-section – 934 394 16 16 170 130

Flat bar 1500 – 394 34 – 95 –

1.568

1

Leff beff = s Web depth (h) Web thickness Flange thickness Flange width (b) Over hanging part of the flange Ratio (Zcalc//Zreq)

Shell thickness vs. angle between stringers.

oped program at constant design pressure and constant diameter. To have an effective breadth (beff) of the stringer of not more than 1 m as conventionally adopted is ship structures as shown in step 1 in Figure 9, an angle of 10 degrees is selected as shown in step 2 in Figure 9. Consequently, the thickness of the shell is 25 mm as shown in step 3 in the same Figure. Table 3 summarizes the results obtained from the equations that are used to calculate the thicknesses of the main components of the floating structure of the WindFloat which are shown in Figure 2, and the minimum section modulus (Z) of the stiffeners required by the DNV rules.

Many thicknesses are assumed in the design of the column stiffeners (Ring and stringer) in the range between 0.5 and 1.5 times the column shell thickness to obtain the minimum section modulus as required by the rules; the type of the cross section used in the column stiffening is then selected. After many trials using the developed computer program the flat bar is used in the ring frame stiffener and the unsymmetrical T-section that is shown in Figure 6 is used in the stringer stiffener. According to DNV (2002) the depth of each stiffener is determined according to the buckling requirements giving the maximum value of the web depth. Table 4 summarizes the dimensions of each stiffener used. The effective length of the stringer (Leff) is calculated according to the DNV and API recommendations. The effective breadth of the ring frame (beff) is assumed to be equal to the frame spacing (s) according to the same recommendations. The depth of the ring frame and stringer (hw) are calculated to give minimum section modulus. Table 4 shows the ratio between the section modulus calculated (Zcal) according to the selected dimensions, and the section modulus required (Zreq) by the rules. If this ratio equals to 1, this

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Table 5. Stiffener dimensions of top plate and bottom plate (mm), ttop plate = 15mm, tbottom plate = 25 mm. Items

Top plate

Bottom plate

Cross section

Unsymmetrical T-section 4550 950 361 10 10 92 57

Unsymmetrical T-section 4550 950 499 13 13 136 97

1

1

Unsupported span s Web depth (h) Web thickness Flange thickness Flange width (b) b2 (over hanging part) Ratio (Zcalc//Zreq)

means that the corresponding dimensions are the minimum dimensions that can be used. It is clear from the results given in Table 4 that the ratio between the calculated section modulus and the required section modulus is equal to 1.568 for the stringer and equal to 1 for the ring frame. The ratio of 1.568 for stringer is due to considering the depth of the stringer equal to the depth of the ring frame web to facilitate the construction and welding process. The stiffeners of the top and closed plates are calculated as shown in Table 5 following the same procedures. 6

FINITE ELEMENT ANALYSIS

6.1

Loading conditions

All loads previously presented acting on the WindFloat are applied to perform the FEA for the two loading conditions shown in Table 6, where the loading condition 1 (LC 1) represents the installation condition of the WindFloat with 25% ballast water in the columns; this generates a draft equal to 14.3 m. Loading condition 2 (LC 2) represents the operating condition of the WindFloat with 100% ballast water in the columns which generates a draft equal to 22.6 m. For each loading condition the different environmental loads are applied. These loads are used in the FEA as well as the weight of the tower and rotor appointed 854 tonnes acting on the top of one column, namely column A as shown in Figure 2 as given by Roddier et al. (2009). As the ballast is increased, the hydrostatic pressure at the base of all the columns decreases from 0.104 MPa in LC 1 to 0.048 MPa in LC 2. 6.2

Results and discussion

FEA was performed and five vertical paths on each column were specified to calculate the von Mises stress on the WindFloat as shown in Figure 10. 6.2.1

Effect of each load acting individually on windfloat FEA was performed for each load act individually on the WindFloat to estimate the predominant Table 6.

A static strength assessment is performed by means of 3D FEM for the floating structure of WindFloat using the commercial software ANSYS version 13.0. The element SHELL281 is adopted. It consists of eight nodes. Element COMBIN14 is used for modeling the springs that are used circumferentially under each column to simulate the boundary conditions under the column that represent the water effect, this element is a spring damper element that has no mass and is an uniaxial tension-compression element with up to three degrees of freedom at each node. Zero linear motion in x, y, and z directions were applied as boundary conditions circumferentially at each node in the end of the spring element. Normal tensile strength steel with yield strength equal to 235 MPa, modulus of Elasticity (E) equal to 210000 MPa and Poisson’s ratio 0.3 is used. A static analysis was performed to calculate the von Mises stress at each point. Von Mises stress allows any arbitrary three dimensional stress states to be represented by a single positive stress value.

Loading conditions.

Ballast

Height of ballast water (m)

Draft (m)

Turbine status

LC 1 LC 2

25% 100%

4.48 17.94

14.3 22.6

Installation Operating

Figure 10.

Sketch for all paths.

Loading condition

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load acting on the WindFloat. Table 7 shows the maximum value of von Mises stress (σmax) for each load individually and its position from WEP (y). The response of each of the applied loads has been studied individually to examine the importance of each. The most significant load is found to be the hydrostatic pressure since it results von Mises stress equal to 127.1 MPa at a position between the WEP and the lower horizontal member in the path aligned with connection between the bracings, horizontal members and columns (path 2A). The stress due to the wind axial force and moment due to operation of the wind turbine does not exceed 6% of the resultant stress in column A. The stress due to the weight of the tower and blades results in 21% of the resultant stress in column A. it has been noticed that columns B and C are not significantly affected by the tower weight; the effect is only seen around the connection of the bracing and horizontal members.

Figure 11.

von Mises stress for the WindFloat in LC 2.

6.2.2

Effect of simultaneous action of different loads on windfloat FEA was performed for all loads acting simultaneous on the WindFloat in the installation condition (LC 1) and the operating condition (LC 2). The results obtained for the von Mises stress in LC 1 and LC 2 are not realistic as shown in Figure 11, since the values obtained are between 230 and 666 MPa in LC 1 and between 230 and 645 MPa in LC 2 which is unacceptable. Critical areas showing unacceptable stress values require structural enhancement to withstand the load conditions under consideration; these are the top plate of column A, the top plate’s stiffeners of column A, outer shell of the top part of column A, the stiffeners located in the middle area of column and the

Table 7. Maximum value and position of von Mises stress for each load individually. Hydrostatic Waves & pressure wind Path σmax

y

Axial force

Tower weight

σmax

y

σmax y

σmax

y

68.2 32 68 32

A 2A 5A

127.1 1.7 123.7 1.7

6 2

16 18

3.1 8.8

32 33.6

B 4B 5B

124.8 1.7 123.7 1.7

5.6 2.1

16 16

1.4 0.3

17 32

4 16 1.4 32

C 4C 5C 6C

89.7 1.7 123.7 1.7 124.8 1.7

1.4 3.9 3.5

13.4 16 16

0.7 3 0.5

20.1 16 23.5

1.1 13.4 4 16 1.4 32

Figure 12. von Mises stress in the top plate of column A in LC 2 after reinforcement.

joint between the lower horizontal member and the bracing of column A respectively. No critical areas have been detected in the other two columns B and C. The proposed enhancement consists of increasing the thickness of the top plate and the upper part of column A supporting the tower from 23 to 46 mm, increasing thickness of the brackets and stiffeners from 20 mm to 25mm and increasing thickness in the middle area of the column supporting the tower from 23 to 46 mm. The range of von Mises stress has greatly improved after enhancement as shown in Figure 12.

7

CONCLUSION

This paper has discussed the detailed structural design of a Floating Foundation for Offshore Wind Turbine in Red Sea. The scantlings of the structure are calculated to comply with the DNV rules and satisfy the buckling requirements. A computer program was developed to carry out the detailed structural design of a WindFloat taking into account the buckling

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requirements given by the rules. A 3D FEM is developed using ANSYS to perform a FEA for the static strength assessment of the structure in the installation and operation conditions. From the analysis given in this paper the following conclusions can be reached: • The range of stresses in the column supporting the tower (column A) is higher than the ranges in the other columns in both studied loading conditions. • The hydrostatic pressure is the predominant load that generates a high stress level in the WindFloat. • The weight of the tower results in 21% of the resultant stress in the column supporting the tower. • The wind axial force and the weight of the tower subjected to the column supporting the tower have a very small effect on the stress level in the other column. • The top part of the column supporting the tower is subjected to unacceptable von Mises stress in both operating conditions, whereas the two other columns are safe. • The middle part of the column supporting the tower is a critical area in the operating condition. • A structural enhancement consisting of an increased thickness in the top plate of the column supporting the tower and in its stiffeners is proposed to obtain acceptable stress values. • The ranges and distribution obtained of von Mises stress in the identified paths in both loading conditions are approximately similar. • The suitability of the WindFloat in the Red Sea should be studied from the economical point of view. REFERENCES

API 1997. Recommended Practice 2A-LRFD for Planning, Designing, and Constructing Fixed Offshore Platforms. American Petroleum Institute. API 2000. Recommended practice 2A-WSD for planning, designing and constructing fixed offshore platforms-working stress design. American Petroleum Institute. Aubault, A., Cermelli, C. & Roddier, D. 2009. WindFloat: A Floating Foundation For Offshore Wind Turbines. Part III: Structural Analysis. OMAE 2009. Hawaii, USA. Authority, E.M., 2005. http://www.ema.gov.eg/. Burton, T., Sharp, D., Jenkins, N. & Bossanyi, E. 2001. Wind Energy Handbook. England: British Library Cataloguing. Cermelli, C., Roddier, D. & Aubault, A. 2009. WindFloat: A Floating Foundation For Offshore Wind Turbines. Part II: Hydrodynamics Analysis. OMAE 2009, Hawaii, USA. Dean, R.G. &. Dalrymple, R.A. 1984: Water Wave Mechanics for Engineers and Scientists. Prentice Hall, Inc. DNV 2002. Buckling Strength of Shells, Recommended Practice DNV-RP-C202. Norway: Det Norske Veritas. DNV 2007. Design of Offshore Wind Turbine Structures, Recommended Practice DNV-OS-J101. Norway: Det Norske Veritas. GL 2005. Germanischer Loyd, Guideline for the Certification of Offshore Wind Turbines. Musial, W., Butterfield, S. & Ram, B. 2006. Energy from Offshore Wind. Offshore Technology Conference. Houston, Texas. Roddier, D., Cermelli, C., Aubault, A. & Weinstein, A. 2010. WindFloat: A floating foundation for offshore wind turbines. Journal of Renewable and Sustainable Energy. Roddier, D., Cermelli, C. & Weinstein, A. 2009. WindFloat: A Floating Foundation For Offshore Wind Turbines. Part I: Design Basis and Qualification Process. OMAE 2009, Hawaii, USA. Technology, M.I.A. 2009. Principle power WindFloat Brochure. Tempel, J.V.D. 2006. Design of Support Structure for Offshore Wind Turbines. Master Thesis. Delft University.

ABS 2004. Guide for Buckling and Ultimate Strength Assessment for Offshore Structures. American Bureau Veritas. ABS 2010. Guide for Building and Classing Offshore Wind Turbine Installation. American Bureau Veritas.

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Analysis and Design of Marine Structures – Guedes Soares & Romanoff (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00045-2

Performance of tethered floating breakwater supporting small scale wind turbine of 750 kw R. Sundaravadivelu, V.K. Gopikrishnan & C. Yuvraj Department of Ocean Engineering, Indian Institute of Technology, Madras, India

ABSTRACT: Agathi Island, a part of Lakshadweep islands in India is subjected to high waves. To protect this island and to harvest energy, a model of Hegushi floating breakwaters are proposed which proves more effective among breakwater models. In this paper the benefit of mounting wind turbine on top of breakwater is studied. The experiments made are presented wherein a small wind turbine of power capacity 750 kw is scaled and mounted on top of the model. Experimental studies on transmission coefficient (KT), reflection coefficient (KR) and motion response (heave and roll) of the taut moored tethered floating platform with and without wind turbine are presented. The experimental results are compared with numerical results obtained using WAMIT software. This new type of floating breakwater is expected to act as both breakwater and as floating platform supporting wind turbine under different wind and wave loading conditions. Moreover the performance of floating platforms under different drafts is studied and its influence is presented. This paper is aimed at developing the feasibility of floating breakwaters supporting small wind turbines. 1

INTRODUCTION

Floating breakwaters are commonly used to protect shorelines, marine structures, moored vessels, marinas and harbors from wave attacks. Compared with permanently fixed breakwaters, floating breakwaters have superiority in terms of environmental friendliness, low cost, flexibility and mobility. They are especially competitive for coastal areas with a high tidal range or deep water depth. Moreover, they may even be the only viable option for locations with poor bottom foundation. The hydrodynamic interactions between the incoming waves and the floating breakwater are complex with the wave energy being partially reflected, partially transmitted beneath the breakwater and partially dissipated. Meanwhile, the incident waves excite the motion responses of the breakwater. A different idea of mounting a small wind turbine model aimed at power production is proposed where the purpose of breakwater is not compromised. To accommodate this new idea a breakwater with moon pool that efficiently reduces the motions. 2

HEGUSHI MODEL

The resonance type of single floating breakwater SFB used in Hegushi, Japan (Fig. 1) has worked effectively for 10m water depth. The cross section

Figure 1.

Side view of the SFB.

of SFB has a lot of structural damping than other types. The twin model will perform better than single model for large wave period and also will provide stability to mount a wind turbine on top. The twin floating breakwater TFB will have reduced oscillations. The experimental studies on 1:30 scale model is carried out and the reflection coefficient, transmission coefficient and motion responses of TFB are presented. In addition to this a wind tower model has been scaled to suit the model dimensions and is mounted on top of this TFB. Experimental results obtained with the wind tower are compared with TFB without tower model and discussions are made.

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Table 1. Comparative statement between the model and the prototype of floating breakwater. Parameter

Model

Prototype

Scale Plan dimension Height Weight Draft Natural Material Water depth Mooring length Wave height Wave period

1:30 1.92*0.5 m 0.3 m 41.65 kg 0.15 m 1.07 s Aluminium 1m 1 16 cm 2.2 s

– 60*15 m 9m 1125T 6m 5.9 s Steel 30 m 30 4.8 m 12 s

Table 2. Comparative statement between the model and the prototype of wind turbine. Parameter

Model

Prototype

Hub height Head mass Total mass of turbine Tower mass Material

1.67 m 1.48 kg 3.98 kg 2.18 kg Acrylic

50 m 39989 kg 107498 kg 58740 kg Steel

Table 3.

3

Box plan of the floating breakwater.

Figure 3.

Sectional views of floating breakwater.

Figure 4.

Experimental setup view in wave flume.

Wave flume details.

Length

30 m

Width Depth Water depth Wake maker Flume

2m 2m 1m Piston type Wave (regular & random) cum current flume

WAVE FLUME DETAILS

The following are the details of the 2 m wave flume used for the experiment. 4

Figure 2.

EXPERIMENTAL SETUP

The experiments were done in the 2 m flume in the Department of ocean engineering, IIT Madras. The water depth in the model can be varied as per our requirements. The model is set up at a distance of 10 m from the wave maker. A wave probe is fixed at a distance of one wavelength from the wave maker to get incident heights. A movable trolley is kept at some distance from the model to measure the reflected wave heights. To measure the

transmitted wave heights a wave probe is fixed at 2 m from the model on the lee side. The floating structure is taut moored with the help of mooring lines whose length is 1m each to the anchor blocks exactly located inside the water. The mooring lines are of 4 mm diameter and are made of steel wire rope. There are wave probes, two in the front and one at the rear sides of the model to measure incident, reflected and transmitted wave heights. The experiments were done for periods from 0.8 s to 2 s. The duration of a single trial is 45 s. The following sketches show the different views of breakwater and arrangementes in the flume. The box plan and the sectional views of the breakwater model are showin in Figures 2 & 3 respectively. Figure 4 shows the experimental setup in the wave flume. The Data Acquistion System (DAQ) shown above records all the signals from wave probes and accelerometers. Wave probes are used to measure incident, reflected and transmitted waves.

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Figure 5.

Plan view.

Figure 8.

Data Acquistion System.

Figure 6 shows accelerometers mounted on front and back of the breakwater. Figure 7 shows the turbine tower clamped to the top of breakwater. 5 Figure 6. Snap shot of BW without wind turbine model model during experiments.

RESULTS OF TFB

The TFB which is taut moored has a draft of 20 cms.The wave periods used for the study vary from 0.5 s to 2 s. Here the initial draft of TFB is increased from 15 cm to 20 cm. The increase in draft accounts for pretension in the taut mooring. It is shared by 4 mooring lines. The TFB is tested for regular waves in beam sea condition varying the wave periods from 0.5 to 2 s. The predominant motions of the structure are sway, roll & heave. The reflection coefficient, KR is defined as KR =

Figure 7.

Breakwater with scaled wind tower on top.

Table 4. Wave constants.

probe

calibration

Wave probes

Calibration constant

WP1 WP2 WP3

4.41 4.43 4.20

Table 4 lists the value of the calibration constants for the wave probes used. Prior to running experiments in wave flume both wave probe and accelerometers are calibrated and the calibration constant thus obtained are used in the WS4 software.

HR HI

where, HR = reflected wave height, HI = incident wave height. For the transmission coefficient (KT), simply replace HR by HT, where the HT is the transmitted wave height. The reflection coefficient, KR is calculated using moving wave probe method experimentally. In single (moving) probe method, the trolley onto which the wave probe is fitted is moved from one end of the flume to the other end to capture the maximum and minimum wave heights. The KT and KR for TFB without turbine model mounted on top are shown in Figures 7 & 8 respectively. The legends 1 and 2 indicate breakwater transmission coefficient with and without tower model on top. It can be seen that when breakwater is mounted with wind turbine model on top of it, the transmission of waves has considerably reduced

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constants of accelerometers the heave and roll are calculated. 7

Figure 9. Transmission coefficient KT vs nondimensional Lw/B.

Figure 10. Lw/B.

Wamit is a panel program designed to solve the boundary-value problem for the interaction of water-waves with prescribed bodies in finite- and infinite-water depth. For the analysis of new type of structures wamit is highly helpful and it saves considerable time and efforts and experimental test runs. Viscous effects of the fluid are not considered throughout and thus the flow field is potential without circulation. The modelling of breakwater is done in rhinoceros software and it is imported in wamit and the analysis is done for different wave periods and the responses (mainly heave and roll) are compared with experimental values. In Figure 11 and Figure 12 the responses obtained from wamit is compared with experimental results.

Reflection coefficient KR vs nondimensional

as the draft increased due to additional weight of wind tower model mounted on top. The effect of slack and taut mooring on KT is shown in Figure 10. It is evident that taut moored breakwater reduces the transmission of waves more effectively than slack moored condition. The effectiveness of breakwater and the range of frequencies over which it can be operational is found to be considerably increased. 6

HYDRODYNAMIC ANALYSIS USING WAMIT

Figure 11. Transmission coefficient KT with and without tower.

MOTION RESPONSES

Two accelerometers are placed on to the edges of the TFB such that when the body starts to oscillate because of wave, heave and roll responses are measured. Data Acquistion System (DAS) that include oscilloscope is used to collect the response signals from the accelerometers and using the calibration

Figure 12. Transmission coefficient KT comparison chart.

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mainly intended to predict the feasibility of breakwater to support the wind turbine in the presence of wave alone. The peak in roll response is almost same in both the cases but it occurs at different wave periods. The heave response of TFB with tower model is small and very less noticeable difference in response comparitively which makes it feasible in reality and gives a futuristic view in implementation.

REFERENCES

Figure 13.

Heave comparison chart.

Figure 14.

Roll comparison chart.

8

CONCLUSIONS

From the experimental studies concluded it is observed that the taut mooring is more effective compared to slack mooring for less transmission of waves. The moonpool in the design of this breakwater helps in structural damping and the responses recorded are less comparitively. As draft increases, the KT reduces for a given wave period and KR increases consequently. The heave responses recorded for different drafts is said to vary inversely with increase in draft. The shift in peak value of heave response has occurred because of the increase in draft. Same pretension is maintained in both the set of experiments. The presence of wind will surely influences the responses of the whole system. But this study is

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