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STABILITY AND DUCTILITY OF STEEL STRUCTURES 2019

PROCEEDINGS OF THE INTERNATIONAL COLLOQUIA ON STABILITY AND DUCTILITY OF STEEL STRUCTURES, PRAGUE, CZECH REPUBLIC, SEPTEMBER 11-13, 2019

Stability and Ductility of Steel Structures 2019

Editors František Wald & Michal Jandera Czech Technical University in Prague, Czech Republic

CRC Press/Balkema is an imprint of the Taylor & Francis Group, an informa business © 2019 Czech Technical University in Prague, Czech Republic Typeset by Integra Software Services Pvt. Ltd., Pondicherry, India All rights reserved. No part of this publication or the information contained herein may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, by photocopying, recording or otherwise, without written prior permission from the publisher. Although all care is taken to ensure integrity and the quality of this publication and the information herein, no responsibility is assumed by the publishers nor the author for any damage to the property or persons as a result of operation or use of this publication and/or the information contained herein. Published by: CRC Press/Balkema Schipholweg 107C, 2316XC Leiden, The Netherlands e-mail: [email protected] www.crcpress.com – www.taylorandfrancis.com ISBN: 978-0-367-33503-8 (Hbk) ISBN: 978-0-429-32024-8 (eBook) DOI: https://doi.org/10.1201/9780429320248

Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Table of contents

Foreword

xv

Committees

xvii

Keynote lectures Structural steel design by advanced analysis with strain limits L. Gardner, A. Fieber & L. Macorini

3

Cold-formed high strength steel RHS under combined bending and web crippling H.T. Li & B. Young

16

Stability design of steel structures: From members to plates and shells L.S. da Silva, T. Tankova & J.P. Martins

28

Advancements in the stability design of steel frames considering general nonprismatic members and general bracing conditions D.W. White, R. Slein & O. Toğay Design by advanced analysis–2016 AISC specification R.D. Ziemian & Y. Wang

42 55

Regular papers On the modal buckling of longitudinally stiffened plates S. Adany & M.Z. Haffar

71

Strength characterisation of a CFS section with initial geometric imperfections H.S.S. Ahmed & S. Ghosh

80

Shear behaviour of sandwich panel fasteners in fire T. Arha, K. Cábová, N. Lišková & F. Wald

88

Bracing details for trapezoidal steel box girders S.V. Armijos-Moya, Y. Wang, T. Helwig, M. Engelhardt, E. Williamson & P. Clayton

96

Behaviour of slender plates in case of fire of different stainless steel grades F. Arrais, N. Lopes, P. Vila Real & C. Couto

106

Numerical modelling of cold-formed steel members at elevated temperatures F. Arrais, N. Lopes, P.Vila Real & M. Jandera

115

Experimental study on the general behaviour of stainless steel frames I. Arrayago, E. Real, E. Mirambell & I. González de León

124

Experimental investigation of flexural buckling of sandwich panels with steel facings I. Balázs & J. Melcher

133

Nonlinear behavior and instability of deployable arches A.B.G. Barcellos, M.V.B. Santana & P.B. Gonçalves

139

Numerical advanced analysis of steel-concrete composite beams and columns under fire R.C. Barros, R.A.M. Silveira, P.A.S. Rocha, D. Pires & Í.J.M. Lemes

147

v

Buckling of spatial laced columns composed of built-up cold-formed channel members C.C.D.O. Bastos & E.M. Batista

155

Local-distortional buckling interaction of cold-formed steel columns design approach E.M. Batista, G.Y. Matsubara & J.M.S. Franco

164

Solutions to simplified von Karman plate equations J. Becque

173

Simplified method for lateral torsional buckling of beams with lateral restraints A. Beyer & A. Bureau

181

Buckling resistance of mono-symmetric I-/H-section members in biaxial bending, axial compression, and torsion A.-L. Bours, R. Winkler & M. Knobloch

189

Cyclic plastic behavior of steel material under uniaxial load paths V. Budaházy & L. Dunai

197

Simulation based imperfections and their effects on stability resistance V. Budaházy, D. Kollár & L.G. Vigh

205

Development of an innovative multi-performance system for LWS structures A. Campiche

213

Seismic design criteria for CFS steel-sheathed shear walls A. Campiche, S. Shakeel, L. Fiorino & R. Landolfo

221

Behaviour of a concrete slab in compression in composite steel-concrete frame joints P. Červenka & J. Dolejš

229

On the development of IoT platforms for the detection of damage in steel railway bridges R. Chacón, A. Rodriguez, P. Sierra, X. Martínez & S. Oller

235

Experimental and numerical studies on shear behaviour of stainless steel plate girders with inclined stiffeners X.W. Chen, H.X. Yuan, X.X. Du & E. Real

244

Ultimate strength analysis of steel-concrete cross-sections at elevated temperatures C.G. Chiorean, M. Selariu & S.M. Buru

253

Seismic design of two-storey X-bracings S. Costanzo, M. D’Aniello, G. Di Lorenzo, A. De Martino & R. Landolfo

262

Experimental tests on bolted end-plate connections using thermal insulation layer attached to steel structures M. Couchaux, A. Alhasawi & A. Ben Larbi

269

Assessment of Eurocode fire design rules for structural members made of high strength steels C. Couto & P. Vila Real

277

Numerical investigation on thin-walled CFS columns in fire H.D. Craveiro, J. Henriques, A. Santiago, L. Laím & J. Henriques

286

FEM analysis of the buckling behavior of thin-walled CFS columns. Part I—channel (C) and Double Channel (I) cross-sections H.D. Craveiro, L.S. da Silva & J.P. Martins & J. Henriques

295

FEM analysis of the buckling behaviour of thin-walled CFS columns. Part II—monosymmetric (R) and double symmetric built-up box cross-sections H.D. Craveiro, J. Henriques, L.S. Silva & J.P. Martins

303

vi

Comparison of two different innovative solutions for IPE beam to CHS column connections R. Das, A. Kanyilmaz & H. Degee Performances of moment resisting frames with slender steel and composite sections in low and moderate seismic areas H. Degée, Y. Duchêne & B. Hoffmeister Laser technology for innovative connections in steel construction: An overview of the project LASTEICON H. Degée, A. Kanyilmaz, C. Castiglioni, L. Calado, M. Couchaux & B. Hoffmeister & F. Morelli Analytical assessment of CFS wall-panels sheathed with MgO board A. Dewangan, G. Bhatt & C. Sonkar

312

321

329

337

Direct Strength Method (DSM) design of simply supported short-to-intermediate hot-rolled steel equal-leg angle columns P.B. Dinis & D. Camotim

345

Stability of ring stiffened steel liners under external pressure—comparison of the existing design concept with 3D-FEM analysis A. Ecker & H. Unterweger

354

Numerical investigation of steel built-up columns composed of track and channel cold-formed sections M.A. El Aghoury, E.A. Amoush, A.M. El Hady & S.M. Ibrahim

363

Fatigue failure of skew beam grid steel bridges—causes and assessment M.A. El Aghoury, I.M. El Aghoury & Amr M. El Hady

371

Lateral torsional buckling of hybrid steel–glass beams M. Eliasova & I. Pravdova

379

A yielding criterion for seismic gusset plates in tension M.D. Elliott & L.H. Teh

388

Experimental calibration of centrally loaded built-up battened compression members G.M. El-Mahdy

394

Stainless steel fillet weld tests N. Feber, M. Jandera, L. Forejtova & L. Kolarik

402

Experimental investigation of compressed stainless steel angle columns A. Filipović, J. Dobrić, Z. Marković, M. Spremić, N. Fric & N. Baddoo

409

Accurate and efficient account of geometrical imperfections in Koiter analysis of elastic solid-like shells G. Garcea, F.S. Liguori, L. Leonetti, D. Magisano & A. Madeo Ductility of different types of shear connectors—experimental and numerical analysis N. Gluhović, M. Spremić, B. Milosavljević, Z. Marković & J. Dobrić

417 427

GBT-based semi-analytical solutions for the elastic/plastic stability analysis of stainless steel thin-walled columns exposed to fire R. Gonçalves, R. Marçalo Neves & D. Camotim

434

A comparative analysis on the stability and ultimate strength of steel plated girders with planar and corrugated webs A. González, L. Vallelado & M.A. Serna

443

vii

Experimental testing of plastic buckling of moderately thick circular rings under uniform lateral loading F. Guarracino

452

Ultimate bending resistance of pipes: Testing arrangements and design approaches. A multi-year perspective F. Guarracino

460

On the GNI analysis of simple thin-walled beams with using linear buckling mode as geometric imperfection M.Z. Haffar, M.H. Taher & S. Ádány

468

U-shaped steel plate dissipative connection for concentrically braced frames J. Henriques, L. Calado, C.A. Castiglioni & H. Degée Critical buckling load on transversally and longitudinally stiffened steel plate girders subjected to patch loading J. Herrera & R. Chacón Buckling behavior and strength of corroded steel shapes under axial compression K. Hisazumi & R. Kanno Modal analysis of thin-walled members with transverse plate elements using the constrained finite element method T. Hoang & S. Ádány

476

483 491

499

Experimental verification of shear connection of thin-walled steel built-up members M. Horáček, J. Melcher & O. Ceh

508

Buckling analysis of circular arches with trapezoidal corrugated web J.R. Ibañez, R. Díez, C. López & M.A. Serna

515

Effect of stiffener position on buckling behavior of H-shaped steel beam with upper flange restraint N. Igawa & K. Ikarashi

523

Local buckling strength of vertical haunch H-shaped beam under shear bending W. Ishida & K. Ikarashi

531

Imperfection sensitivity of corrugated web girders subjected to lateral-torsional buckling B. Jáger, M. Kachichian, L. Dunai & C. Égető

539

Enhanced buckling capacity of axially compressed stiffened plates taking into account the shear-lag effect A. Jäger-Cañás

548

Axial buckling behavior of welded ring-stiffened shells A. Jäger-Cañás, Z. Li & H. Pasternak

556

Ductility assessment of structural steel and composite joints J.-P. Jaspart, A. Corman & J.-F. Demonceau

564

Study on the influence of reduced beam sections on the seismic behaviour of a moment resisting frame A. Jiménez, E. Mirambell & E. Real Stability of double-symmetric sections subjected to axial force, bending moments and torsion F. Jörg & U. Kuhlmann

viii

570 578

Distortional buckling of stiffeners in stainless steel profiled sheeting J. Jůza & M. Jandera Appropriate spring stiffness models for the end support of bolted single steel angle members in compression M. Kettler, H. Unterweger & T. Harringer

587

596

Global buckling strength of built-up cold-formed steel column under compression T. Kobashi & N. Shimizu

605

Coupled buckling of steel LC-beams under bending Z. Kołakowski, T. Kubiak & M. Kamocka

614

Crashworthiness performance of tubular energy absorbing structures with triggers M. Kotełko, M. Ferdynus & K. Okoń

622

Failure plastic mechanisms in TWCFS columns under eccentric compression M. Kotełko, V. Ungureanu & D. Dubina

629

Experimental study of beam-to-column connection with bolted joints Y. Koyama, A. Sato, H. Idota, Y. Sato, S. Yagi, S. Takaki & M. Kamada

638

Material strength statistics and reliability aspects for the reassessment of end-of-service-life steel bridges R. Kroyer & A. Taras The influence of stiffeners width on buckling modes of steel LC-beams subjected to bending T. Kubiak, Z. Kolakowski & F. Kazmierczyk

646 655

Design of slender compressed plates in structural steel joints by component based finite element method M. Kuříková, F. Wald & J. Kabeláč

664

Experimental verification of lateral-torsional buckling of steel I-beams with tapered flanges J. Kuś

673

Composite floor system with cold-formed trussed beams and pre-fabricated concrete slab L.A.A. de S. Leal, E. de M. Batista

682

Experimental investigation of stability behavior of members supported by sandwich panels at elevated temperature A. Lendvai & A.L. Joó

691

Ultimate shear resistance of cylindrically curved steel panels F. Ljubinković, J.P. Martins, H. Gervásio & L.S. Silva

699

Nonlinear finite element analysis of delta hollow flange girders subjected to patch loading N. Loaiza, C. Graciano & E. Casanova

708

Critical loads of semi-rigid columns subjected to non-linear temperature distributions T. Ma & L. Xu

717

The stability of semi-braced steel frames containing members with stepped segments T. Ma & L. Xu

727

A quasi-static nonlinear analysis for assessing the fire resistance of steel 3D frames exploiting time-dependent yield surfaces D. Magisano, F. Liguori, L. Leonetti & G. Garcea Fire design of class 4 tapered steel beams with the general method—a proposal É. Maia, C. Couto, P. Vila Real & N. Lopes

ix

735 744

Progressive collapse assessment of storage racks due to localized failures. Explicit consideration of dynamic effects I. Marginean, F. Dinu & D. Dubina

753

Elastic buckling strength of lipped channel section beam restrained on upper flange subjected to bending H. Masuda & K. Ikarashi

761

Experimental study on SCFs of empty SHS-SHS T-joints subjected to static out-of-plane bending F.N. Matti & F.R. Mashiri

770

Experimental study of cold-formed high strength steel circular hollow sections X. Meng & L. Gardner Towards automated identification of structural steel components from 3D-point clouds to subsequent GMNA-stability-analysis C. Merkl & A. Taras

778

787

Static effects of modular structures made of containers O. Miller, V. Křivý, D. Mikolášek, P. Pařenica & R. Cuřín

795

Calibration of European web-crippling equations for cross-sections with one web T. Misiek & A. Belica

804

Explanatory notes to buckling design of longitudinally welded aluminium compression members T. Misiek, B. Norlin & T. Höglund

813

Buckling of circular hollow section stainless steel columns in fire A. Mohammed & S. Afshan

821

The capacity of bolted cold-formed steel connections in bending S.M. Mojtabaei, J. Becque & I. Hajirasouliha

830

Study on the deformation and rotation capacity of HSS hollow sections A. Mueller & A. Taras

839

An analytical solution for the compressed simply-supported plate with initial geometric imperfections M. Nedelcu

847

Experimental study on the shear connections of composite girders with trapezoidally corrugated web G. Németh, B. Jáger, N. Kovács & B. Kövesdi

854

Numerical study of end-plate steel connections with two and four bolts-per-row D.L. Nunes & A. Ciutina

863

Experimental study on square steel tubular columns under compressive force with biaxial bending moment T. Onogi & A. Sato

872

Considering realistic weld imperfections in load bearing capacity calculations of ringstiffened shells using the analytical numerical hybrid model H. Pasternak, Z. Li, C. Stapelfeld & B. Launert, A. Jäger-Cañás

882

Seismic response of steel dual eccentrically braced frames with equal-strength joints Č. Penelov & N. Rangelov

x

890

Tests and design of built-up section columns D.K. Phan & K.J.R. Rasmussen

898

Buckling and strength of prestressed steel stayed columns R. Pichal & J. Machacek

906

Numerical simulation and analysis of axially restrained stainless steel beams in fire A. Pournaghshband, S. Afshan & M. Theofanous

911

Experimental and numerical investigations of unstiffened steel girders with non-rectangular panels subjected to bending and shear V. Pourostad & U. Kuhlmann

921

Studying bolt force distribution in ultra-large capacity end-plate connections A.A. Ramzi, I.M. El Aghoury, S.M. Ibrahim & A.I. El-Serwi Sensitivity of the stiffness reduction model used to analyze the ultimate load condition of steel frames B. Rosson, T. Villalon-Camacho, H. Gurneian & R. Ziemian Statistical evaluation of the bearing capacity of short polygonal columns G. Sabau, E. Koltsakis, O. Lagerqvist & P. Manoleas Direct Strength Method (DSM) design of end-bolted cold-formed steel columns failing in distortional modes W.S. Santos, A. Landesmann & D. Camotim

929

937 946

954

Design limitations for the steel beam-column to ensure full plastic moment A. Sato, M. Aoyama, K. Inden, T. Ono & K. Mitsui

963

Experimental study on LTB behaviour and residual stresses of welded I-section members L. Schaper, R. Winkler, F. Jörg, U. Kuhlmann & M. Knobloch

972

Behavior of column base plates under bi-axial bending moment L. Seco, M. Couchaux, M. Hjiaj & L.C. Neves

981

Quantifying the seismic ductility of lightweight steel lateral force resisting systems through procedures of FEMA P695 S. Shakeel Numerical modelling of a two storey LWS building braced with gypsum-based panels S. Shakeel, A. Campiche & R. Landolfo Elastic buckling strength of H-shaped beams subjected to end moment and uniformly distributed load D. Shinohara & K. Ikarashi

989 997

1005

On the incorporation of cross-section restraints in Generalised Beam Theory (GBT) T.G. da Silva, C. Basaglia & D. Camotim

1015

Stability interaction effects in 3D steel frames—a case study H.H. Snijder, L.H.J.D. Titulaer, P.A. Teeuwen & H. Hofmeyer

1025

Experimental investigation on the instability phenomenon in stainless steel connections—plate curling K. Sobrinho, A. Tenchini, M. Cordeiro, P. Vellasco, L. Lima & J. Henriques

1034

Experimental and analytical study of Cold-Formed Steel (CFS) single-stud walls sheathed with FCB, CSB and MgO under compression C. Sonkar, A. Kr. Mittal, S. Kr. Bhattacharyya, S. Kumar & A. Dewangan

1042

xi

Lateral-torsional buckling of stainless steel beams with slender cross section M. Šorf & M. Jandera

1051

Improve load capacity calculations by considering realistic imperfections induced by welding for plates and shells 1059 C. Stapelfeld, B. Launert, H. Pasternak, N. Doynov & V.G. Michailov Plastic collapse loads of rectangular plate assemblies with constant and linear load distribution S. Stehr & N. Stranghöner

1068

Stability of axially compressed cylindrical shells made of stainless steel for different imperfection patterns N. Stranghöner & E. Azizi

1077

Proposal for improving the consistency between Eurocode 3-1-8 and Eurocode 8-1 A. Stratan & D. Dubina

1086

Assumption of imperfections for the LTB-design of members based on EN 1993-1-1 R. Stroetmann & S. Fominow

1095

Welds on high-strength steels—influence of the welding process and the number of layers R. Stroetmann & T. Kästner

1103

Validation of the Overall Stability Design Methods (OSDM) for tapered members J. Szalai, F. Papp & G. Hajdú

1111

Stability design of cable-stayed columns T. Tankova, L.S. da Silva & J.P. Martins

1120

Influence of geometrical imperfection of rib stiffeners on beam-to-column joint behaviour R. Tartaglia, M. D’Aniello, G.M.Di Lorenzo & R. Landolfo

1128

The fire behaviour of extended stiffened joints designed for seismic actions R. Tartaglia, M. Zimbru, A. Linguiti, M. D’Aniello, R. Landolfo & F. Wald

1136

Buckling length assessment with finite element approach T. Tiainen, K. Mela & M. Heinisuo

1145

Experimental and numerical analysis of the local and interactive buckling behaviour of hollow sections A. Toffolon, A. Müller, A. Taras & I. Niko

1151

Proposal of a design curve for the overall resistance of cold-formed rectangular and square hollow sections A. Toffolon & A. Taras

1159

Behavior of extended end-plate connections under cyclic alternate loading I.C. Tomăscu, R.M. Bâlc

1167

Thermorheological testing and modelling of seismic bearing elastomers C. Treib, M.A. Kraus & A. Taras

1176

Built-up cold-formed steel beams with web openings V. Ungureanu, I. Both, C. Neagu, M. Burca, D. Dubina & A.A. Cristian

1185

Numerical investigation of built-up cold-formed steel beams with corrugated web V. Ungureanu, I. Lukačević, I. Both, M. Burca & D. Dubina

1193

Study on the out-of-plane stability of steel portal frames M. Vassilev & N. Rangelov

1201

xii

Warping transfer superelement model for bolted end-plate connections subject to 3D loads B. Vaszilievits-Sömjén, J. Szalai & R.M. Movahedi

1210

Tests of gusset plate connection under compression J. Vesecký, K. Cábová & M. Jandera

1218

Numerical modelling of gusset plate connections under eccentric compression J. Vesecký, M. Jandera & K. Cábová

1227

Buckling of columns during welding M. Vild & M. Bajer

1236

Design of gusset plate connection with single-sided splice member by component based finite element method 1243 M. Vild, J. Kabeláč, M. Kuříková & F. Wald Beam-to-column joints for slim-floor systems in seismic zones: Numerical investigations and experimental program C. Vulcu, R. Don & A. Ciutina

1251

A reexamination of high strength steel Q690 plasticity model Y. Wang, Y. Wang, G. Li & Y. Lyu

1260

Analysis of mechanical properties of cold formed high strength steel in weld area M. Werunský & J. Dolejš

1269

Degradation processes in normalized mild- and low-alloy steel building structures in service W. Wichtowski & J. Hołowaty

1275

Effect of the steel grade on equivalent geometric imperfections for lateral torsional buckling R. Winkler & M. Knobloch

1283

Influence of collision damage on load-carrying capacity of steel girder E. Yamaguchi, Y. Tanaka & T. Amamoto

1292

Modelling of one-sided unstiffened beam-to-column joint J. Zamorowski & G. Gremza

1300

Modelling of roof bracings of single-storey industrial buildings J. Zamorowski & G. Gremza

1309

Buckling assessment of cylindrical steel tanks with top stiffening ring under wind loading Ö. Zeybek & C. Topkaya

1317

Slim-floor beam bending moment resistance considering partial shear connection Q. Zhang & M. Schäfer

1326

Stainless steel SHS and RHS beam-columns B. Židlický & M. Jandera

1334

Calibration of parameters of combined hardening model using tensile tests C.I. Zub, A. Stratan & D. Dubina

1342

Author index

1351

xiii

Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Foreword

The series of International Colloquia on Stability and Ductility of Steel Structures have been supported by the Structural Stability Research Council (SSRC) for more than forty years and its objective is to present the progress in theoretical, numerical and experimental research in the field of stability and ductility of steel and steel-concrete composite structures. Special emphasis is laid on new concepts and procedures concerning the analysis and design of steel structures and on the background, development and application of rules and recommendations either appearing in recently published Codes or Specifications or about to be included in their upcoming versions. This International Colloquium series started in 1972 in Paris and its subsequent editions took place in different cities with the last five being held in: Timisoara, Romania (1999), Budapest, Hungary (2002), Lisbon, Portugal (2006), Rio de Janeiro, Brazil (2010) and Timisoara, Romania (2016). The 2019 edition of SDSS is organized by the Czech Technical University in Prague. The university held the second edition of the Eurosteel conference in 1999 and the first three editions of Applications of Structural Fire Engineering (ASFE) conference (2009, 2011 and 2013).

František Wald

Michal Jandera

xv

Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Committees ORGANIZING COMMITTEE K. Cábová J. Dolejš M. Eliášová M. Jandera M. Kuříková J. Macháček P. Ryjáček Z. Sokol M. Šorf F. Wald B. Židlický SCIENTIFIC COMMITTEE Chairman: František Wald Scientific Secretary: Michal Jandera G.A. Altay (Turkey) I. Balaz (Slovakia) C. Baniatopoulos (Greece) A. Bureau (France) E.M. Batista (Brazil) R. Beale (UK) R. Bjorhovde (USA) M.A. Bradford (Australia) B. Brune (Germany) L. Calado (Portugal) D. Camotim (Portugal) R. Chacon (Spain) S.L. Chan (Hong Kong, China) T.M. Chan (Hong Kong, China) R. Casciaro (Italy) K.F. Chung (Hong Kong, China) C. Chiorean (Romania) M. D’Aniello (Italy) H. Degée (Belgium) J.-F. Demonceau (Belgium) F. Dinu (Romania) J. Dobric (Serbia) D. Dubina (Romania) L. Dunai (Hungary) S. Easterling (USA) A. Elghazouli (UK) M. Fontana (Switzerland) D. Frangopol (USA) xvii

L. Gardner (UK) M. Garlock (USA) G. Garcea (Italy) P. Gonçalves (Brazil) F. Guarracino (Italy) J. Hajjar (USA) G.J. Hancock (Australia) M. Hjiaj (France) B. Izzuddin (UK) M. Knobloch (Germany) R. Landolfo (Italy) N. Lopes (Portugal) R. Leon (USA) J.R. Liew (Singapore) J. Loughlan (UK) J. Macháček (Czech Republic) M. Mahendran (Australia) F. Mazzolani (Italy) E. Mirambell (Spain) D. Nethercot (UK) J. Packer (Canada) J. Paik (South Korea) S. Pajunen (Finland) H. Pasternak (Germany) N. Rangelov (Bulgaria) K.J.R. Rasmussen (Australia) E. Real (Spain) B. Rossi (Belgium) F. Roure (Spain) A. Sato (Japan) R. Sause (USA) B.W. Schafer (USA) L.S. Silva (Portugal) N. Silvestre (Portugal) H. Snijder (Netherlands) R. Stroetmann (Germany) A. Taras (Germany) J.G. Teng (Hong Kong, China) V. Ungureanu (Romania) H. Unterweger (Austria) B. Uy (Australia) I. Vayas (Greece) P. Vellasco (Brazil) P. Vila Real (Portugal) A. Wada (Japan) Y.B. Yang (Taiwan, China) N. Yardimci (Turkey) B. Young (Hong Kong, China)

xviii

Keynote lectures

Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Structural steel design by advanced analysis with strain limits L. Gardner, A. Fieber & L. Macorini Imperial College London, London, UK

ABSTRACT: The design of steel structures traditionally involves two steps: first, a structural analysis is performed to determine the internal forces and moments; then, design checks are carried out to verify the stability of individual structural members. In design by advanced analysis, both material and geometric nonlinearities are captured during the analysis, hence eliminating the need for subsequent member checks. However, steel frames are typically analysed using beam elements, which cannot capture local buckling. Hence, steel design specifications use the concept of crosssection classification to limit the strength and deformation capacity of a cross-section. A more sophisticated approach is set out herein, whereby strain limits are employed to mimic local buckling. This allows cross-sections of all classes to be analysed in a consistent advanced analysis framework. The approach has been applied in this paper to individual members and indeterminate systems and shown to be more consistent and accurate than current steel design specifications.

1 INTRODUCTION The structural analysis of steel frames is typically performed using beam finite elements. These elements are able to accurately capture the overall elastic-plastic load-deformation response of structures composed of compact cross-sections. However, conventional beam elements do not account for local cross-section deformations (i.e. local instabilities in either the elastic or inelastic range). To overcome this limitation, steel design specifications, such as EN 1993-1-1 (2005), use the concept of cross-section classification, whereby class-dependent restrictions on the cross-section resistance and permissible analysis type (i.e. elastic or plastic) are specified. As a result, plastic analysis and design methods, which allow for the beneficial effect of force and moment redistribution, are restricted to structures composed of compact Class 1 cross-sections. Structures composed of more slender cross-sections (i.e. Class 2 to 4) must be analysed elastically and step-wise resistance functions are employed to limit the capacity of the cross-sections and members. Recently, a novel design approach based on geometrically and materially nonlinear structural analysis, also referred to as advanced analysis, using beam finite elements in conjunction with strain limits, has been proposed (Gardner et al., 2019). The benefits of design by advanced analysis are widely recognised (Liew et al. 2000; Chen 2008; Kim & Chen 1999; Trahair & Chan 2003; Buonopane & Schafer 2006; Rasmussen et al. 2016; Surovek 2012). Compared to the traditional approach to structural design, whereby the structural analysis is followed by individual member and cross-section checks, in advanced analysis global sway (P–Δ) and member (P–δ) instabilities are captured directly and the need for subsequent member checks is eliminated. In the proposed approach, the strain limits are determined from the continuous strength method (CSM) (Gardner 2008) and are used to mimic the effects of local buckling. Hence, a consistent advanced analysis framework using beam finite elements with strain limits can be employed to analyse and design structures comprising cross-sections of any class, with failure defined as the lower of (1) the peak load factor reached during the analysis or (2) the load factor at which the strain limit is first reached. In this paper, the proposed method of design by advanced analysis is briefly outlined, including a description of the quad-linear material model for hot-rolled steel (Yun & Gardner 2017) and the continuous strength method (CSM) strain limits. Then, the most recent developments of the 3

proposed design approach are presented, including its application to beam-columns consisting of hot-rolled steel I-shaped and SHS/RHS sections, continuous beams and a planar frame example.

2 TRADITIONAL STEEL DESIGN In traditional steel design, the treatment of material nonlinearity, namely the spread of plasticity and the redistribution of forces in a structure, is tied to the classification of the crosssections. Plastic design is allowed when the cross-sections have sufficient rotation capacity to enable plastic hinges to develop and to be sustained. Plastic design can account for the redistribution of forces as plasticity spreads and can thus offer more accurate capacity predictions when it can be used. Typically an elastic-perfectly plastic material model is assumed (i.e. strain hardening is ignored). Alternatively, elastic analysis can be used for all cross-section classes. The classification process is based on the width-to-thickness ratio of the critical isolated plate element within the cross-section, and currently does not account for any flange-web interaction. To facilitate the design process, EN 1993-1-1 (2005) defines four discrete cross-section classes, each representing an idealised cross-sectional response with a corresponding moment and rotation capacity. Class 1 sections can reach their full plastic moment capacity and have sufficient plastic hinge ductility to be used in plastic design. An idealized moment-curvature relationship with unrestricted rotation capacity is generally assumed in plastic design methods, while the beneficial effects of strain hardening are typically ignored. Class 2 cross-sections can also attain their full plastic moment capacity, yet are insufficiently ductile for plastic design purposes and consequently elastic analysis methods have to be used. Local buckling prevents Class 3 sections from developing their full plastic moment capacity and hence their resistance is limited to their elastic moment capacity. The design of Class 4 sections reflects explicit allowance for the occurrence of local buckling below the first yield resistance and generally involves somewhat lengthy calculations using effective section properties. After determining the member forces and moments from structural analysis, a series of design checks are performed to ensure strength and stability requirements are satisfied. Strength checks are applied to the most heavily stressed cross-section, with the bending resistance limited to the plastic, elastic or effective section moment capacity (Mpl, Mel and Meff respectively) and the compression resistance limited to the plastic or effective squash load (Npl and Neff respectively), depending on the cross-section class. However, experiments (Lay 1964; Gioncu & Petcu 1997) have shown that the maximum in-plane bending resistance of a member subjected to a moment gradient can exceed that under uniform bending; a moment gradient is of course present in most practical applications. While EN 1993-1-1 (2005) utilises equivalent moment factors to account for the influence of the bending moment distribution along the member length on member stability, it does not consider the beneficial effects of local moment gradients on local stability. The latest editions of many structural design codes permit the use of geometrically and materially nonlinear analysis, also known as advanced analysis, for the design of steel structures consisting of compact cross-sections. Examples include Section 5 of EN 1993-1-1 (2005), Appendix D of AS 4100 (1998) and Appendix 1 of AISC 360 (2016). Typically, any limit state that is not included in the structural analysis must be accounted for using appropriate design checks. For example, a linear (first order) analysis does not capture member buckling and thus a corresponding member buckling check is required. Advanced analysis methods reduce the number of required design checks by incorporating various limit states into the analysis itself.

3 DESIGN BY ADVANCED ANALYSIS WITH STRAIN LIMITS 3.1 Introduction Recently, a novel approach for structural steel design based on geometrically and materially nonlinear analysis including imperfections, also referred to as GMNIA or advanced analysis, 4

with strain limits has been proposed (Gardner et al., 2019). In the proposed approach, the continuous strength method (CSM) (Gardner 2008) strain limits are used to mimic local buckling in beam finite elements. In conjunction with a standardised material model for hot-rolled steel (Yun & Gardner 2017), it is hence possible to employ the same advanced analysis framework using beam elements for structures comprising cross-sections of all four classes since the strain limits control the degree of plastic redistribution in a rational manner. Failure is defined as the lower of (1) the peak load factor reached during the analysis or (2) the point at which the CSM strain limit is reached. Details of the proposed method of design by advanced analysis are outlined in the following subsections. 3.2 Material modelling The stress-strain relationship for structural carbon steels is often idealised by an elastic-perfectly plastic model, though this model fails to capture the characteristic strain hardening behaviour of carbon steels. Nevertheless, this simplified model generally forms the basis of the current design provisions in EN 1993-1-1 (2005). For advanced design methods, such as the continuous strength method (CSM) (Gardner 2008), an accurate representation of the stress-strain response of the material is important, particularly for stocky cross-sections that may benefit from strain hardening. A quad-linear material model, as illustrated in Figure 1 and described by Equation (1), has been proposed (Yun & Gardner 2017) to represent accurately the yield plateau and strain hardening behaviour of hot-rolled structural carbon steels. The material model has been calibrated against a large database of tensile coupon test results and depends only on three commonly available parameters: the Young’s modulus E, the yield stress fy and the ultimate stress fu. Two material coefficients (C1 and C2) are used in Equation (1): C1 defines a ‘cut-off’ strain to avoid over-predictions of material strength and is also included in the CSM base curve, as described in Section 3.3. and C2 is employed in Equation (2) to determine the strain hardening slope Esh. These two coefficients may be determined from Equations (3) and (4) respectively, which are expressed in terms of the strain hardening strain εsh and the ultimate strain εu. 8 E" > > < fy f ð"Þ ¼ f þ E ð"  " Þ y sh sh > > : f f fC1 "u þ "uu CC11 ""uu ð"  C1 "u Þ

Figure 1.

for "  "y for "y 5 "  "sh for "sh 5 "  C1 "u for C1 "u 5 "  "u

Quad-linear material model for hot-rolled structural carbon steel (Yun & Gardner 2017).

5

ð1Þ

Figure 2. Comparison of quad-linear material model with an experimental stress-strain curve for grade S355 steel tested by Chan and Gardner (Chan & Gardner 2008).

fu  fy C2 "u  "sh

ð2Þ

C1 ¼

"sh þ 0:25ð"u  "sh Þ "u

ð3Þ

C2 ¼

"sh þ 0:4ð"u  "sh Þ "u

ð4Þ

Esh ¼

The values of εsh and εu may be predicted from Equations (5) and (6), respectively. fy  0:055; but 0:015  "sh  0:03 fu   fy ; but "u  0:06 "u ¼ 0:6 1  fu

"sh ¼ 0:1

ð5Þ ð6Þ

A typical comparison between a measured stress-strain curve (Chan & Gardner 2008) and the quad-linear material model is shown in Figure 2. Further comparisons are presented and discussed by Yun and Gardner (2017). The model has recently been incorporated into the CSM for the design of hot-rolled steel cross-sections (Yun et al. 2018a; Yun et al. 2018b), showing improved accuracy over the EN 1993-1-1 design provisions. 3.3 Continuous strength method (CSM) and strain limits The continuous strength method (CSM) is a deformation based design approach that relates the cross-section deformation capacity to the cross-section slenderness (Gardner 2008). The CSM consists of two key components: (1) a base curve that defines the peak compressive strain εcsm that a cross-section can endure and (2) a material stress-strain model, which has already been introduced in Section 3.2. Combined, the CSM strain limit and the adopted material model define the stress distribution at failure, which in turn, when integrated over the depth of the cross-section, defines the CSM cross-section capacity. The CSM has been validated extensively and shown to be more accurate than traditional design methods for the structural design of stainless steel (Gardner 2008;

6

Figure 3.

Continuous strength method (CSM) base curve.

Zhao et al. 2016; Afshan & Gardner 2013), aluminium alloy (Ashraf & Young 2011; Su et al. 2016) and hot-rolled steel (Yun et al. 2018b; Yun et al. 2018a; Liew & Gardner 2015). The CSM base curve, shown in Figure 3, provides a continuous relationship between the
p and its deformation capacity, which is defined in norlocal slenderness of a cross-section l malised form as εcsm/εy, where εcsm is the maximum compressive strain a cross-section can sustain and εy is the yield strain. The base curve is split in two parts: Equation (7) applies to
p ≤ 0.68, which are referred to as Class 1 to 3 cross-sections in non-slender sections with l
p > 0.68, which EN 1993-1-1 (2005), and Equation (8) applies to slender cross-sections with l are currently referred to as Class 4 cross-sections. The cross-section slenderness is defined by Equation (9) and discussed further below.   "csm 0:25 "csm C1 "u ¼ 3:6 ; but  min ; ð7Þ for lp  0:68 "y "y "y lp ! "csm 0:222 1 ¼ 1  1:05 for lp 40:68 ð8Þ 1:05 "y lp lp Two upper limits are defined for the cross-section deformation capacity εcsm/εy in Equation (7). The first limit of Ω defines the permissible level of plastic deformation and may be defined on a project-by-project basis. For example, a high value of say Ω =30 may be specified where extensive plasticity is tolerable at ultimate limit state and a suitably ductile steel is being used. Conversely, a value of Ω =1 would be used when material yielding is not desirable, for example for serviceability limit checks. A value of Ω =15 is generally recommended to prevent excessive deformations and to ensure that the material ductility requirements from EN 1993-1-1 (2005) are satisfied. The second limit of C1εu/εy prevents over-predictions of material strength when the simplified CSM resistance functions are used (Gardner et al. 2017; Yun et al. 2018a). The accuracy of the method is dependent on how accurately the cross-section slenderness can be determined. The cross-section slenderness is a dimensionless parameter that quantifies susceptibility to local instability and is defined by Equation (9), where fy is the yield stress and σcr is the elastic critical buckling stress.

7

sffiffiffiffiffiffi fy lp ¼ cr

ð9Þ

Various methods are available to calculate the elastic critical buckling stress of a crosssection. Following Eurocode 3 (EN 1993-1-1 2005; EN 1993-1-5 2006), the buckling stress of the full cross-section may be taken as the buckling stress of the most slender plate element within the cross-section, though this approach conservatively neglects any element interaction. Full element interaction can be accounted for using numerical approaches (e.g. the finite strip method as implemented in CUFSM (Li & Schafer 2010)). Alternatively, approximate expressions calibrated from numerical studies (Seif & Schafer 2010) may be used to determine the elastic buckling stress of the full cross-section. While the expressions presented in (Seif & Schafer 2010) account for element interaction, they are only applicable to members subjected to pure compression and pure major/minor axis bending. To overcome this shortcoming, Gardner et al. (2019) recently developed predictive expressions for the elastic critical buckling stress of standard steel profiles subjected to compression, bending and combined loading, which have been used throughout the present study. 3.4 Application of CSM strain limits to advanced analysis Including the CSM strain limits in advanced analysis enables the CSM cross-section capacity to be computed directly since numerical integration is performed at each load increment of the analysis. In the calculation of the CSM cross-section resistance, the strain limit εcsm is applied to the peak compressive strain in the cross-section (i.e. the strain at the extreme fibre). The cross-section can withstand the required strain demand if the design strain εEd is less than the CSM strain limit εcsm, as given by Equation (10). "Ed  1:0 "csm

ð10Þ

Beam finite elements typically output strain values at the centreline of the wall thickness (i.e. at a distance of half the plate thickness away from the extreme fibre). It was found that the difference in capacity predictions is negligible when applying the strain limit to the compatible design strain at either the extreme fibre εEd or the centreline εEd,cl; this is because the slightly lower design strain at the centreline of the wall thickness is offset by the slightly lower strain limit εcsm,cl at this location. The application of the strain limit is shown schematically in Figure 4. The CSM base curve has been calibrated based on experimental results on cross-sections under uniform compression (i.e. stub columns) and uniform bending (i.e. the central region of laterally restrained beams under four-point bending). Thus, the influence of residual stresses and local geometric imperfections on the cross-section strength are directly accounted for. To account for the effects of initial imperfections (i.e. member out-of-straightness and residual stresses) on the member capacity, equivalent geometric imperfections are employed in the proposed approach (Lindner et al. 2018). In cases other than uniform compression and uniform bending, the local stability of the cross-section is enhanced by the presence of a strain gradient (i.e. the local moment gradient along the member length). To account for the beneficial effects of local moment gradients, the CSM strain limit can be applied to strains that are averaged over a characteristic length along the member, rather than simply (conservatively) to the peak strain. The length over which the strains are averaged in the proposed approach is the local buckling half-wavelength of the full cross-section Lb,cs, since local buckling is the limit state controlling the resistance of the crosssection in both the elastic and inelastic regimes. The local buckling half-wavelength is calculated from the predictive expressions presented in (Fieber et al., submitted), which requires

8

Figure 4. Application of strain limits to compatible design strains at either extreme fibre or centreline of wall thickness for (a) I-sections and (b) SHS/RHS subjected to major axis bending.

minimal additional calculation effort when the full cross-section local buckling stress is calculated from the expressions presented in (Gardner et al., 2019). The strain averaging approach is an extension to the cross-section strain check given in Equation (10). Instead of considering the peak compressive strain along the member, the strain contributions εi of all n elements located completely within the critical local buckling half-wavelength Lb,cs are considered. Assuming equal length elements, the strain averaging approach is expressed in Equation (11), where εEd,Lb is the averaged design strain. The weighted average strain must be determined in cases where the lengths of elements located within Lb,cs are not equal. The strain averaging approach can account for the beneficial effects of local moment gradients and reduces the sensitivity of strength predictions to the mesh density employed in the finite element model. "Ed;Lb  1:0 where "csm

"Ed;Lb ¼

n 1X "i n 1

and n  1

ð11Þ

The application of the strain averaging approach is shown schematically in Figure 5 for the case where n = 2. Assuming that first-order beam finite elements with a single integration point located at the centre of the element are used, the continuous strain distribution is approximated by the beam FE model in step-wise increments as shown in Figure 5. The considered member is able to withstand the applied design loads if the averaged strain εEd,Lb = ε1 + ε2 is less than the CSM strain limit εcsm. 4 APPLICATION OF PROPOSED APPROACH The accuracy of the proposed method of design by advanced analysis with strain limits is assessed in this section for isolated beam-columns, continuous beams and frames. The capacity predictions obtained from beam FE advanced analyses with strain limits are compared against those obtained from the benchmark shell FE models and conventional EN 1993-1-1 design. In the comparisons made herein, partial factors are set to unity. In practical design,

9

Figure 5. Schematic representation of the strain averaging approach for the case of n = 2. Note that the strain contribution ε3 is not included in the averaging approach since element 3 is not located completely within the local buckling half-wavelength Lb,cs.

the structural capacity obtained through the proposed design approach would be divided by the partial safety factor for the relevant limits state, typically γM1 4.1 Members subjected to combined compression and bending In this subsection, the proposed method of design by advanced analysis is applied to
y = 0.5, 1.0 or 1.5). a series of pin-ended beam-columns of varying member slenderness (l The complete loading range from pure compression to pure major axis bending was covered by varying the ratio of applied compression to bending. The capacity predictions of the proposed method of design by advanced analysis were compared against the results of shell FE modelling and two alternative EN 1993-1-1 (2005) design approaches: (i) a member check using the interaction factor kyy from Annex B of EN 1993-1-1 and (ii) a linear interaction cross-section check applied to a second-order elastic analysis with equivalent imperfections (i.e. geometrically nonlinear analysis with imperfections, denoted as GNIA in Figure 6). Note that the CSM strain limits were determined based on the first-order stress distribution in the critical cross-section and that the ultimate capacities from the proposed method were taken as the lesser of the peak load factor, indicated by empty circles in Figure 6, and the point at which the CSM strain limit was reached, indicated by the filled circles. The normalised moment-axial force interaction diagram for a series of RHS 120х60х5 beam-columns are shown in Figure 6. Overall, the proposed method of design by advanced analysis is able to accurately reflect the shell FE model behaviour, though in compression dominated cases, the assumed imperfection magnitude may result in slightly conservative capacity predictions. The beam-columns considered in Figure 6 were subjected to a uniform first-order moment gradient. Members under moment gradients are considered in Figure 7. As the level of firstorder moment gradient increases, the EN 1993-1-1 member check becomes increasingly conservative since the effects of local moment gradients and strain hardening are not considered. This may be seen in the normalised moment-axial force interaction diagram for a series of
y = 0.5 and 1.5 are HEB 100 beam-columns in Figure 7. For clarity, only the results for l shown. It can be seen that in compression dominated cases, failure is defined by the peak load factored reached during the analysis, while in bending dominated cases the strain limits 10

Figure 6. Normalised ultimate capacity of RHS 120 х 60 х 5 pin-ended beam-columns subjected to combined compression and major axis bending.

govern the member capacity. While the Eurocode 3 member checks become increasingly conservative in bending dominated cases, the proposed method of design by advanced analysis is able to accurately reflect the shell FE model behaviour. Furthermore, it can be seen that when the upper bound strain limit is relaxed to Ω = 30, the proposed method accurately captures the strain hardening behaviour of the shell FE model. Benefits of up to 25% may be achieved compared to EN 1993-1-1 capacity predictions. 4.2 Continuous beams The ultimate collapse load of a continuous beam may significantly exceed the load at which the first cross-section reaches its capacity, as shown in Figure 8, in which αel is the load factor at which the first cross-section reaches its elastic moment and αult is the ultimate load factor. The system capacity depends not only on the local cross-section strength, but is also influenced by the beneficial effects of local moment gradients and moment redistribution. Traditional design according to EN 1993-1-1 (2005) only permits full plastic moment redistribution in Class 1 cross-sections. This results in a discrete jump in capacity from plastic analysis for Class 1 cross-sections to elastic analysis for Class 2 to 4 cross-sections, which is clearly a simplistic representation of reality. As a result, the Eurocode 3 capacity predictions tend to be overly conservative for Class 2 and 3 crosssections. The significant capacity reserves due to strain hardening in stocky cross-sections
p < 0.4 are also not captured in Eurocode 3, as shown in Figure 8. with l Using a consistent materially nonlinear advanced analysis for all cross-section classes provides a more rational exploitation of moment redistribution, as well as strain hardening in stocky cross-sections. Figure 8 shows that the CSM strain limits are able to predict the appropriate levels of redistribution for each cross-section. Particularly in the typical hot-rolled slen
p = 0.35 to 0.55 there is an excellent match to the shell FE results. Below derness range from l
p = 0.32, the CSM limit of Ω = 15 prevents excessive deformations. However, a slenderness of l by increasing the strain limit to Ω = 30, the beam element models allow for additional redistribution and accurately follow the shell model trends again. 11

Figure 7. Normalised ultimate capacity of HEB 100 pin-ended beam-columns subjected to combined compression and major axis bending.

Figure 8. Normalised ultimate capacity of three span continuous beams with varying local cross-section slenderness.

12

4.3 Frames In this section, the proposed method of design by advanced analysis is applied to an asymmetric low-rise frame. The frame geometry and considered load case is shown in Figure 9. All members were made from grade S355 steel and the beam to column connections were modelled as fully fixed. The load was applied proportionally and the normalised loaddisplacement responses of the frame obtained using different analysis methods are shown in Figure 10. Failure in the shell FE model was governed by inelastic local buckling at the top of the column C4 and side-sway, at a maximum load factor of α = 1.02. The deformed shape of the critical region in the shell FE model at the peak load factor is shown in Figure 10. Following EN 1993-1-1 (2005), the columns are Class 3 cross-sections and the beams are Class 1 cross-sections. Hence, elastic analysis must be used. The critical elastic buckling load factor of the frame at the peak load factor of the shell FE model is αcr = 3.10; hence, the frame is sensitive to second-order sway effects. Various different design approaches are permissible according to EN 1993-1-1, namely: (1) a second-order elastic analysis with equivalent imperfections (GNIA) with cross-section checks, (2) a second-order elastic analysis (GNA) with member and cross-section checks using the non-sway effective length, (3) a linear elastic analysis (LEA) with amplified sway moments using kamp followed by member checks using nonsway effective lengths (with the non-sway effective length factor denoted keff), or (4) a linear elastic analysis (LEA) with member checks using the sway effective lengths (with the sway effective length factor denoted keff,sway) to account for second-order effects. All four Eurocode approaches correctly predict failure in column C4. Approaches (1) to (3) predict a similar load carrying capacity, with a maximum load factor of around α = 0.84, where the non-sway effective length of column C4 is taken as keff = 0.745. Accounting for second-order effects in a LEA through the sway effective length, taken as keff,sway = 2.1 for column C4, results in an overly conservative capacity prediction of α = 0.68. In the proposed method of design by advanced analysis, the strain limit for column C4 is εcsm = 1.67εy. Thus, in contrast to Eurocode 3, some plastification is allowed and failure is predicted at a load factor of α = 0.96, just 6% short of the shell FE model and 12% above the best Eurocode 3 prediction. Furthermore, the strain limits accurately mimic local buckling in the beam FE model, with the first strain limit reached at the top of column C4. Note that if no

Figure 9.

Asymmetric low-rise frame example: geomtry and applied loading.

13

Figure 10. Normalised load-deformation response of asymmetric low-rise frame example.

strain limits were applied to the beam FE advanced analysis, an unconservative peak load factor of α = 1.05 would be reached. 5 CONCLUSIONS The most recent developments of a new method of design by advanced analysis with strain limits are presented. The design method is based on a geometrically and materially nonlinear analysis with imperfections (i.e. advanced analysis) and employs the continuous strength method strain limits to mimic local buckling in beam finite elements. The proposed method is able to capture the beneficial effects of local moment gradients and predict realistic levels of force and moment redistribution. Application of the method is demonstrated for a series of beam-columns, continuous beams and a frame. It is shown that the proposed method is consistently more accurate compared to the current Eurocode 3 design approach and that advanced analysis with strain limits is a viable design tool for structures composed of crosssections of all classes. REFERENCES Afshan, S. & Gardner, L., 2013. The continuous strength method for structural stainless steel design. Thin-Walled Structures, 68, 42–49. AS 360–16, 2016. Specification for Structural Steel Buildings, American Institute of Steel Construction. AS 4100, 1998. Australian Standard. Steel Structures, Sydney: Standards Australia. Ashraf, M. & Young, B., 2011. Design formulations for non-welded and welded aluminium columns using Continuous Strength Method. Engineering Structures, 33, 3197–3207. Buonopane, S.G. & Schafer, B.W., 2006. Reliability of steel frames designed with advanced analysis. Journal of Structural Engineering, ASCE, 132(2), 267–276.

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Chan, T.M. & Gardner, L., 2008. Bending strength of hot-rolled elliptical hollow sections. Journal of Constructional Steel Research, 64, 971–986. Chen, W.F., 2008. Advanced analysis for structural steel building design. Frontiers of Architecture and Civil Engineering in China, 2(3), 189–196. EN 1993-1-1, 2005. Eurocode 3: Design of steel structures - Part 1-1: General rules and rules for buildings, Brussels: European Committee for Standardization. EN 1993-1-5, 2006. Eurocode 3: Design of steel structures – Part 1-5: Plated structural elements, Brussels: European Committee for Standardization. Fieber, A., Gardner, L. & Macorini, L., submitted for review. Formulae for determining elastic local buckling half-wavelengths of structural steel cross-sections. Journal of Constructional Steel Research. Gardner, L., Yun, X., Macorini, L. & Kucukler, M., 2017. Hot-rolled steel and steel-concrete composite design incorporating strain hardening. Structures, 9, 21–28. Gardner, L., Yun, X., Fieber, A. & Macorini, L., 2019. Steel design by advanced analysis: material modelling and strain limits. Engineering. 5(2), 243–249. Gardner, L., 2008. The continuous strength method. Proceedings of the Institution of Civil Engineers Structures and Buildings, 161(3), 127–133. Gardner, L., Fieber, A. & Macorini, L., 2019. Formulae for calculating elastic local buckling stresses of full structural cross-sections. Structures. 17, 2–20. Gioncu, V. & Petcu, D., 1997. Available rotation capacity of wide-flange beams and beam-columns Part 2. Experimental and numerical tests. Journal of Constructional Steel Research, 43, 219–244. Kim, S.E. & Chen, W.F., 1999. Design guide for steel frames using advanced analysis program. Engineering Structures, 21, 352–364. Lay, M.G., 1964. The experimental bases of plastic design. WRC, Bulletin No. 99, Publication No. 258. Li, Z. & Schafer, B.W., 2010. Buckling analysis of cold-formed steel members with general boundary conditions using CUFSM: Conventional and constrained finite strip methods. In Twentieth International Speciality Conference on Cold-Formed Steel Structures. Saint Louis, Missouri, USA. Liew, A. & Gardner, L., 2015. Ultimate capacity of structural steel cross-sections under compression, bending and combined loading. Structures, 1, 2–11. Liew, J.Y.R., Chen, W.F. & Chen, H., 2000. Advanced inelastic analysis of frame structures. Journal of Constructional Steel Research, 55, 245–265. Lindner, J., Kuhlmann, U. & Jörg, F., 2018. Initial bow imperfections e0 for the verification of flexural buckling according to Eurocode 3 Part 1-1 - additional considerations. Steel Construction, 11(1), 30– 41. Rasmussen, K.J.R., Zhang, H., Cardoso, F. & Liu, W., 2016. The direct design method for cold–formed steel structural frames. In Eighth International Conference on Steel and Aluminium Structures. Hong Kong: Hong Kong, 1–14. Seif, M. & Schafer, B.W., 2010. Local buckling of structural steel shapes. Journal of Constructional Steel Research, 66(10), 1232–1247. Su, M.N., Young, B. & Gardner, L., 2016. The continuous strength method for the design of aluminium alloy structural elements. Engineering Structures, 122, 338–348. Surovek, A.E., 2012. Advanced analysis in steel frame design. Guidelines for direct second-order inelastic analysis. A. E. Surovek, ed., Reston, Virginia: ASCE. Trahair, N. & Chan, S.L., 2003. Out-of-plane advanced analysis of steel structures. Engineering Structures, 25, 1627–1637. Yun, X. & Gardner, L., 2017. Stress-strain curves for hot-rolled steels. Journal of Constructional Steel Research, 133, 36–46. Yun, X., Gardner, L. & Boissonnade, N., 2018a. The continuous strength method for the design of hot-rolled steel cross-sections. Engineering Structures, 157, 179–191. Yun, X., Gardner, L. & Boissonnade, N., 2018b. Ultimate capacity of I-sections under combined loading – Part 2: Parametric studies and CSM design. Journal of Constructional Steel Research, 148, 265– 274. Zhao, O., Gardner, L. & Young, B., 2016. Behaviour and design of stainless steel SHS and RHS beam-columns. Thin-Walled Structures, 106, 330–345.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Cold-formed high strength steel RHS under combined bending and web crippling Hai-Ting Li School of Civil and Environmental Engineering, Nanyang Technological University, Singapore Department of Civil Engineering, The University of Hong Kong, Hong Kong, China

Ben Young Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hong Kong, China Department of Civil Engineering, The University of Hong Kong, Hong Kong, China

ABSTRACT: This paper presents experimental and numerical investigations of cold-formed high strength steel (CFHSS) rectangular hollow sections (RHS) under combined bending and web crippling. In the experimental investigation, 5 pure bending tests and 28 combined bending and web crippling tests were conducted on RHS with measured 0.2% proof stress in the flat portion of the section ranged from 679 to 971 MPa. The combined bending and web crippling tests were performed using the Interior-One-Flange loading condition that specified in the North American Specification for cold-formed steel structures. The specimens were tested at various lengths to examine the interaction relationship between bending moment and concentrated interior bearing load. Finite element (FE) models were developed and validated against the test results for members under combined bending and web crippling as well as pure bending. Upon validation of the FE models, a parametric study was performed using the validated models to generate further numerical data over a wide range of web slenderness ratio, bearing length to web thickness ratio and bearing length to web flat portion ratio. The ultimate strengths obtained from experimental and numerical investigations were compared with nominal strengths calculated using the European Code. It is shown that the codified bending and web crippling interaction formula can be used for the CFHSS RHS members, while more accurate predictions can be achieved by using the recently proposed web crippling design rules.

1 INTRODUCTION Cold-formed steel (CFS) rectangular hollow sections (RHS) are widely used in various engineering applications due to their structural efficiency and aesthetic appearance. The CFS rectangular (includes square in the context of this paper) hollow sections are often applied in construction without transverse stiffeners; therefore, the webs of the unstiffened CFS RHS may cripple when subjected to high local intensity of bearing forces or reactions. Moreover, the web crippling capacity of CFS RHS under local transverse forces may reduce noticeably due to the presence of bending moment, especially in the vicinity of the loading point within a span or at the interior support of a continuous beam. Therefore, combined bending and bearing check is crucial in designing CFS tubular structural members. The North American Specification AISI-S100-16 (AISI 2016), Australian/New Zealand Standard AS/NZS 4600 (AS/NZS 2005) and Eurocode EN 1993-1-3 (CEN 2006) provide design provisions for CFS members under combined bending and web crippling; the codified combined bending and web crippling design rules are generally empirical in nature. This is due to the fact that theoretical analysis of web crippling is rather complicated (Yu & LaBoube 16

2010) and the pure web crippling rules in the aforementioned codes of practice are empirically derived based upon experimental investigations conducted by researchers from the 1940s onwards. Therefore, the codified combined bending and web crippling provisions are only applicable for certain materials and cross-section profiles. High strength steel tubular members are increasingly attractive in construction owing to the structural and architectural advantages (Zhao et al. 2014, Ma et al. 2017). The applicability of the existing combined bending and web crippling provisions to cold-formed high strength steel (CFHSS) structural members shall be investigated. In this study, a test program on CFHSS RHS under combined bending and web crippling is presented. A total of 33 tests, including 5 pure bending tests as well as 28 combined bending and web crippling tests, were undertaken on RHS that had nominal yield strengths of 700 and 900 MPa. Four-point bending tests were conducted to obtain the moment capacities of the CFHSS RHS under constant bending moment. The combined bending and web crippling tests were carried out under the Interior-One-Flange (IOF) load case as per the AISI (2016); various specimen lengths were tested to study the interaction of bending moment and localized interior bearing load. The pure web crippling test results of IOF load case reported by the authors in Li & Young (2017a) are also used in this paper to non-dimensionalize the combined bending and web crippling test results. In addition, finite element (FE) models were developed and validated against the test results; upon validation, a parametric study was undertaken using the validated models to generate further numerical data. The results gained from the experimental and numerical investigations were compared against the nominal resistances as per the EN 1993-1-3 (CEN 2006) to assess the suitability of the codified combined bending and web crippling provisions to CFHSS RHS.

2 EXPERIMENTAL INVESTIGATION A test program was undertaken to study the CFHSS RHS under bending and web crippling. Three types of experiments were carried out: pure bending tests; pure web crippling tests; combined bending and web crippling tests. The RHS specimens of the abovementioned three types of tests were from the same batch of tubes, the material properties of which have been previously reported by the authors in Li & Young (2017a, 2018a). The measured static 0.2% proof stresses of the RHS gained from the tensile flat coupon tests varied between 679 and 971 MPa. Table 1 tabulated the mechanical properties gained from longitudinal tensile flat and corner coupon tests, namely, the Young’s moduli (E), static 0.2% proof stresses (σ0.2) and static tensile strengths (σu). 2.1 Pure bending tests The CFHSS RHS specimens of pure bending series had measured web heights H ranging between 50.1 to 160.1 mm, flange widths B ranging from 50.0 to 160.2 mm, thicknesses t ranging between 3.896 to 3.971 mm, and inside corner radii r ranging between 4.6 to 6.8 mm.

Table 1. Mechanical properties obtained from tensile flat and corner coupon tests. Corner coupon†

Flat coupon^ Section (H×B×t)

E (GPa)

σ0.2 (MPa)

σu (MPa)

E (GPa)

σ0.2 (MPa)

σu (MPa)

H80 × 80 × 4 H160 × 160 × 4 H50 × 100 × 4 V100 × 100 × 4

210.9 212.4 211.3 205.3

725 751 679 971

834 829 820 1079

214.2 216.2 207.2 219.6

877 899 860 1073

945 992 932 1175

Note: ^Conducted by Li & Young (2017a); †Conducted by Li & Young (2018a).

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Table 2. Measured dimensions and test/FE results of pure bending test specimens (Li & Young (2019b)). Specimen

H (mm)

B (mm)

t (mm)

r (mm)

L (mm)

MExp (kN·m)

MExp/MFEA

B-H80 × 80 × 4 B-H160 × 160 × 4 B-H50 × 100 × 4 B-H100 × 50 × 4 B-V100 × 100 × 4

80.1 160.1 50.1 100.4 99.9

80.1 160.2 100.3 50.0 100.2

3.896 3.971 3.931 3.930 3.946

4.7 5.1 4.6 4.6 6.8

1390 2188 1389 1389 1592

29.0 87.8 18.0 31.6 51.7

1.04 0.98 1.07 1.08 1.00

Mean COV

1.04 0.043

Figure 1.

Pure bending test setup.

The section H50 × 100 × 4 was bent about both major and minor axes. The specimen labels, as shown in Table 2, illustrated the test type, nominal yield strength and cross-sectional dimensions. For instance, the labels B-H100 × 50 × 4 and B-V100 × 100 × 4 defined the following specimens: the first letter B indicated they were pure bending specimens; the second letter indicated the material of the RHS, where H and V indicated the nominal yield strengths (0.2% proof stresses) of 700 and 900 MPa, respectively; the following symbols were the nominal cross-sectional dimensions, arranged as H × B × t, in millimetres, where the H, B and t are the height, width and thickness of the RHS, respectively. Four-point bending tests were undertaken by Li & Young (2019b) to obtain the moment capacities of the CFHSS RHS under constant bending moment. The four-point bending test setup is shown in Figure 1. Simply supported boundary conditions have been achieved by half-round and roller supports. The lengths of moment spans were 500 mm, except for the specimen B-H160 × 160 × 4, of which the moment span is 600 mm. Calibrated linear variable displacement transducers (LVDTs) were set at the bottom flange of the two load points and the specimen midspan. The curvatures of the RHS specimens were obtained from these LVDT readings. Compressive loads were imposed on the RHS specimens via displacement control through a servo-controlled hydraulic actuator; the applied load rate was 0.5 mm/min for all the tests. The moment capacities MExp gained from the pure bending experiments are reported in Table 2. 2.2 Summary of pure web crippling tests Pure web crippling tests of CFHSS RHS were conducted and have been previously reported by the authors (Li & Young, 2017a). The pure web crippling tests were performed under the 18

Figure 2.

Pure web crippling test setup.

four codified web crippling load cases in the CFS specifications, namely, the AISI (2016) and AS/NZS (2005); the pure web crippling specimen lengths were employed as per the AISI (2016). The loading or reaction forces were imposed to the RHS via bearing plates. Figure 2 shows the setup of a pure web crippling test under the Interior-One-Flange (IOF) load case. The web crippling capacities per web (PExp) of the IOF specimens of sections H80 × 80 × 4, H160 × 160 × 4, H50 × 100 × 4, H100 × 50 × 4 and V100 × 100 × 4 were employed in this study to non-dimensionalize the combined bending and web crippling test results, which will be described in Section 4 of this paper. Details of the pure web crippling tests are available in Li & Young (2017a). 2.3 Combined bending and web crippling tests The measured dimensions of the combined bending and web crippling specimens are reported by Li & Young (2019b) and shown in Table 3. The test specimens have measured H ranging from 50.0 to 160.1 mm, B ranging from 50.0 to 160.3 mm, t ranging from 3.902 to 3.979 mm, and r varying between 4.6 to 6.8 mm. The h/t ratios ranging between 8.3 to 35.8. In general, the lengths L of the combined loading series specimens equalled to 2a+90 mm, where a is the distance (in millimetres) from the support point to the midspan as shown in Figure 3. For specimens of the C-H160 × 160 × 4N150 series (i.e. bearing length N = 150 mm), the L were designed to be 2a+150 mm to avoid any possible failure at the end of the specimens. The distance a was varied to study the interaction relationship between bending moment and localized interior bearing load; this method has been previously employed by many researchers (e.g. Zhao & Hancock 1991, 1992, Young & Hancock 2002, Zhou & Young 2007). The distance a = kMExp/PExp, in which, MExp is the moment capacity from the four-point bending test; PExp is the experimental web crippling capacity per web under the IOF load case of the same RHS; and k is the interaction factor. The k values were selected to allow the interaction of bending moment and localized bearing load over a wide range; the selected k values varied between 0.7 to 1.8 in this test program. Test specimens with lower k values resulted in lower ratios of moment-to-localized bearing force, whilst specimens with higher k values led to higher ratios of moment-to-localized bearing force. The specimens, as shown in Table 3, were labelled so that the test type, nominal yield strength, cross-sectional dimensions, bearing length and interaction factor can be identified. For example, the label C-H80×80×4N50-k1.0-R defined the following specimen: the first letter C indicated it was a combined loading specimen; the second letter H showed the nominal 19

Table 3. Measured dimensions and test/FE results of combined bending and web crippling test specimens (Li & Young (2019b)). Specimen

H (mm)

B (mm)

t (mm)

r (mm)

L (mm)

PC,Exp (kN)

PC,Exp/PC,FEA

C-H80×80×4N90-k1.0 C-H80×80×4N90-k1.3 C-H80×80×4N90-k1.8 C-H80×80×4N50-k0.7 C-H80×80×4N50-k1.0 C-H80×80×4N50-k1.0-R C-H80×80×4N50-k1.5 C-H160×160×4N150-k0.8 C-H160×160×4N150-k1.1 C-H160×160×4N150-k1.6 C-H160×160×4N90-k0.7 C-H160×160×4N90-k1.0 C-H160×160×4N90-k1.5 C-H50×100×4N50-k0.9 C-H50×100×4N50-k1.2 C-H50×100×4N50-k1.7 C-H100×50×4N50-k0.7 C-H100×50×4N50-k0.7-R C-H100×50×4N50-k1.0 C-H100×50×4N50-k1.5 C-H100×50×4N30-k0.7 C-H100×50×4N30-k0.7-R C-H100×50×4N30-k1.0 C-H100×50×4N30-k1.0-R C-H100×50×4N30-k1.5 C-V100×100×4N50-k0.7 C-V100×100×4N50-k1.0 C-V100×100×4N50-k1.5

80.1 80.1 80.2 80.1 80.1 80.1 80.1 160.1 160.1 160.0 160.0 160.0 160.0 50.0 50.1 50.1 100.3 100.4 100.3 100.3 100.3 100.3 100.3 100.3 100.3 100.4 100.2 100.1

80.1 80.1 80.1 80.1 80.1 80.1 80.1 160.2 160.1 160.2 160.3 160.1 160.2 100.4 100.3 100.3 50.0 50.2 50.0 50.1 50.1 50.1 50.1 50.1 50.0 100.0 99.9 100.1

3.922 3.925 3.919 3.949 3.901 3.942 3.902 3.953 3.974 3.970 3.979 3.956 3.979 3.944 3.958 3.932 3.948 3.963 3.952 3.926 3.967 3.940 3.926 3.937 3.933 3.957 3.907 3.931

4.7 4.7 4.7 4.7 4.7 4.7 4.7 5.1 5.1 5.1 5.1 5.1 5.1 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 6.8 6.8 6.8

572 717 957 488 658 658 943 1053 1392 1957 1015 1411 2072 395 496 665 539 539 732 1053 617 617 843 843 1220 717 986 1434

111.4 90.9 66.1 99.4 81.5 81.1 60.3 144.9 119.0 89.7 121.6 97.7 74.4 96.3 78.6 57.8 95.2 93.7 77.9 57.8 77.4 75.3 62.9 61.3 46.8 95.5 77.2 59.0

1.00 1.02 1.01 1.00 1.02 1.00 1.02 0.95 0.93 0.93 0.98 0.95 0.94 1.10 1.11 1.09 1.01 0.98 0.99 0.99 1.02 1.00 1.02 0.98 0.99 0.94 0.94 0.95 1.00 0.048

Figure 3.

Specimen length design for combined bending and web crippling tests.

20

Figure 4.

Combined bending and web crippling test setup.

0.2% proof stress was 700 MPa; the following symbols showed the nominal H×B×t were 80×80×4 in millimetres; the N50 indicated the bearing length N=50 mm; the k1.0 indicated the interaction factor was 1.0; the R at the end (if any) indicated it was a repeated test. Regarding the combined bending and web crippling tests, various specimen lengths were employed using the IOF web crippling load case as per the AISI (2016). The combined bending and web crippling test setup is shown in Figure 4. A bearing plate (BP) was employed to transfer the bearing force to the RHS via a half round at the midspan. Two steel plates, supported by two rollers, were applied at the specimen ends to allow symmetric boundary conditions; the width of these two plates were 90 mm except for the specimens of C-H160×160×4N150 series, where 150 mm width plates were adopted. Similar to the IOF test setup previously detailed by the authors in Li & Young (2017a), steel stiffening plates were employed at the specimen ends. Displacement control mode was selected and the tests were conducted at a load rate of 0.5 mm/ min; this is the same as the pure bending tests and pure web crippling tests. The ultimate loads per web (PC,Exp) of the combined bending and web crippling specimens are reported in Table 3. The experimentally gained PC,Exp was utilized to obtain the ultimate moment of the RHS specimen by using MC,Exp = a PC,Exp. Out-of-plane deformation was not observed during testing for all the 28 combined bending and web crippling specimens.

3 NUMERICAL MODELLING In parallel to the experimental program, a numerical study (Li & Young 2019b) was conducted by using ABAQUS (2012). Finite element (FE) models were built to replicate the tests. Upon validation, a parametric study was undertaken to gain further numerical data on CFHSS RHS structural members under combined bending and web crippling. The numerical models were established based upon measured specimen geometries. The S4R element in the ABAQUS (2012), which has been successfully used in similar previous FE simulations by the authors (Li & Young 2017b, 2018a, b, 2019a, b), was employed herein to model the CFHSS RHS members. The applied meshes in the RHS flat regions varied between 4×4 to 12×12 mm, which depended on the RHS sizes; finer meshes at the RHS corners were adopted to accurately represent the corner regions. The nonlinearity of the CFHSS materials was incorporated based upon measured engineering stress-strain data gained from the flat and corner coupon tests; the engineering stress-strain data have been converted to gain the true stress and true plastic strain relationships prior to being put into the FE models. In this study, the corner properties were applied to the RHS corners with a 2t extension to the adjacent flat portions. 21

The boundary conditions were replicated according to the experiments. With regard to the pure bending tests, the loading points and supports were modelled using reference points, which were coupled to the contact surfaces between the specimens and load transfer plates. The half-round support was modelled by restraining the corresponding reference point against all degrees of freedom (DOF) except for rotation about the bending axis, whereas the roller support was simulated by allowing an extra DOF of longitudinal movement. The loads were imposed by applying axial displacements to the reference points that modelled the loading points. The FE modelling of the pure web crippling tests has been previously reported by the authors in Li & Young (2018a); regarding the combined bending and web crippling modelling, the same technique as employed by Li & Young (2018a) for the IOF load case was applied herein. The concentrated bearing forces were transferred to the RHS specimens by bearing plates (BP). The BP were simulated by discrete rigid 3D solid elements. The surface interactions of the BP and the RHS were defined using contact pairs. The loads were imposed by applying displacements to the BP, which is the same as the experiments using displacement control. The developed numerical models were validated against the experiments. For the CFHSS RHS under pure bending, the moment capacities obtained experimentally (MExp) and numerically (MFEA) were compared and shown in Table 2; the mean ratio of the MExp/MFEA equalled to 1.04 and the corresponding coefficient of variation (COV) was 0.043. The failure modes and moment-curvature curves derived from the finite element analyses (FEA) were also validated against their experimental counterparts. The numerical validation for IOF specimens undergoing pure web crippling was performed and has been reported by Li & Young (2018a). Regarding the CFHSS RHS members under combined bending and web crippling, the validation is carried out herein; the ultimate loads per web (PC,FEA) predicted by the FEA were compared to their experimental counterparts PC,Exp. The mean ratio of the PC,Exp/PC,FEA was 1.00 and the COV was 0.048, as tabulated in Table 3. Typical numerical failure mode, load-web deformation curves and load-midspan deflection curves derived from FEA were compared to their corresponding experimental ones, as illustrated in Figures 5–7. It is shown

Figure 5. Experimental and numerical failure modes for combined bending and web crippling specimen C-H160×160×4N90-k1.5. (a) Experimental failure mode. (b) Numerical failure mode.

22

Figure 6. Experimental and numerical load-web vertical deformation curves for combined bending and bearing specimens C-H80 × 80 × 4N90-k1.0, C-H80 × 80 × 4N90-k1.3 and C-H80 × 80 × 4N90-k1.8.

Figure 7. Experimental and numerical load-midspan deflection curves for specimens C-H80 × 80 × 4N90-k1.0, C-H80 × 80 × 4N90-k1.3 and C-H80 × 80 × 4N90-k1.8.

that the developed FE model successfully replicate the combined bending and web crippling tests, and therefore is deemed suitable to be used for parametric study. After validation of the FE models, a parametric study was undertaken to gain further numerical data. Similar as the test program, the nominal yield strengths of the RHS were 700 MPa for the H series and 900 MPa for the V series herein; the measured properties gained from material tests of the sections H160×160×4 and V100×100×4 were applied for the H and V series, respectively. An extensive range of 18 RHS was studied. These RHS had H ranging between 150 to 400 mm and t ranging between 2 to 8 mm; the h/t ratios ranging from 13.8 to 106.0. The bearing lengths N varying between 75 to 220 mm, and were chosen to be either N = B or N = 0.5B. The N/t ratios ranging between 9.4 to 110.0 and the N/h ratios varying between 0.3 and 1.4. In total, 18 parametric specimens were modelled under pure bending and 188 results were generated for specimens under combined bending and web crippling.

4 COMPARISON OF TEST AND NUMERICAL RESULTS WITH CURRENT DESIGN PROVISIONS In order to investigate the CFHSS RHS members under combined bending and web crippling, the pure bending and pure web crippling endpoints in the interaction curves are needed. The moment capacities Mu gained from the pure bending test and FE programs (Li & Young 2019b) were compared with nominal moment resistances obtained as per the Eurocode EN 23

Figure 8.

Comparison of pure bending test and FE results with nominal resistances predicted by EC3.

1993-1-1 (CEN 2014) (MEC3), as shown in Figure 8. It is shown that the MEC3 provided quite conservative predictions; the Mu/MEC3 had a minimum value of 1.01 and the mean ratio of the Mu/MEC3 was 1.14 with the corresponding COV of 0.087. The web crippling capacities per web Pu gained from the experimental (Li & Young 2017a) and numerical (Li & Young 2018a) studies have been compared to the codified nominal web crippling resistances, as shown in Figure 9. It has been demonstrated that the CEN (2006) predictions were overly conservative; therefore, improved design rules have been proposed for CFHSS RHS undergoing pure web crippling, as detailed in Li & Young (2018a). The Pu of the IOF specimens, which are required in this study to non-dimensionalize the combined bending and web crippling capacities, were compared against the nominal resistances per web based upon the CEN (2006) (PEC3) and Li & Young (2018a) (PL&Y) provisions. It is shown that the PL&Y, which has been proposed based upon modification of the AISI (2016) design rules, provided much-improved predictions than the current Eurocode predictions. The mean

Figure 9. Comparison of pure web crippling test and FE results with nominal resistances predicted by EC3 and Li & Young (2018a).

24

Figure 10. Comparison of combined bending and web crippling test/FE capacities with nominal resistances (non-dimensionalized with respect to PEC3 and MEC3).

ratio of the Pu/PL&Y was 1.07 with the corresponding COV of 0.099, while the Pu/PEC3 had a mean ratio of 1.76 with the corresponding COV of 0.098. The obtained combined bending and web crippling results (Li & Young 2019b) were compared to the nominal resistances calculated using the CEN (2006) and the bending and web crippling interaction equation in the CEN (2006) is illustrated in Eq. (1). 

PC PEC3





MC þ MEC3

  1:25

ð1Þ

in which, PC is the maximum concentrated interior bearing load per web in the presence of bending moment; MC is the maximum bending moment of the RHS. The PC and MC were non-dimensionalized with respect to the nominal web crippling resistances per web PEC3 and the nominal moment resistances MEC3. The comparisons with the CEN (2006) interaction curve were depicted in Figure 10, where great conservatism of the EC3 predictions is observed. This over conservatism is mainly due to the over-pessimistic predictions of the PEC3 endpoints. In this study, the PC were also non-dimensionalized with respect to the PL&Y; the comparisons with the CEN (2006) interaction curves by using PL&Y as pure web crippling endpoints in the interaction equation are shown in Figure 11, in which the vertical axis is still the ratio MC/MEC3 as its counterpart in Figure 10, but the horizontal axis is changed into PC/PL&Y. It is demonstrated that the CEN (2006) provided much-improved predictions when using the web crippling strengths predicted by Li & Young (2018a) as the web crippling endpoints. Hence, improved design rules for CFHSS RHS under combined bending and web crippling can be sought through the adoption of the PL&Y as the web crippling endpoints in the interaction curve. Furthermore, the experimentally and numerically derived PC and MC were also nondimensionalized with respect to the corresponding Pu and Mu that obtained from the pure web crippling specimens and pure bending specimens. This is to appraise the appropriateness of the CEN (2006) interaction coefficients for CFHSS RHS. The interaction equations in the CEN (2006) can be therefore expressed as Eq. (2).     PC MC þ  1:25 Pu Mu

25

ð2Þ

Figure 11. Comparison of combined bending and web crippling test/FE capacities with nominal resistances (non-dimensionalized with respect to PL&Y and MEC3).

Figure 12. Comparison of combined bending and web crippling test/FE capacities with EC3 interaction curve.

The PC/Pu and MC/Mu are plotted in Figure 12 with the codified interaction curve. As illustrated by Figure 12, the interaction curves in the CEN (2006) were conservative, although may slightly overestimate the strengths of a few specimens with high moment-to-concentrated load ratios. Overall, the codified interaction equations are deemed appropriate for the CFHSS RHS members under combined bending and web crippling.

5 CONCLUSIONS Design of cold-formed high strength steel (CFHSS) rectangular hollow sections (RHS) under combined bending and web crippling has been appraised. A test program was undertaken on cold-formed RHS of high strength steel with measured 0.2% proof stresses ranging from 679 to 971 MPa. The combined bending and web crippling tests were undertaken using the IOF web crippling load case. The specimens were tested at different lengths and various bending moment-to-concentrated bearing load ratios were achieved. Finite element models were built and validated with the experiments; a parametric study was performed thereafter and 188 parametric results were generated for specimens under combined bending and web crippling. 26

The experimentally and numerically obtained results were compared against the nominal resistances predicted by the CEN (2006). It has been demonstrated that the combined bending and web crippling resistances calculated from the CEN (2006) were overly conservative. It has been illustrated that the codified interaction equations can be used for the CFHSS RHS members under combined bending and web crippling, while improved predictions can be achieved through the adoption of the PL&Y as the pure web crippling endpoints in the interaction curve. ACKNOWLEDGEMENTS The authors are grateful to Rautaruukki Corporation for providing the test specimens. The research work described in this paper was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. 17209614). REFERENCES ABAQUS. (2012). Abaqus/Standard user’s manual volumes I-III and Abaqus CAE manual. Version 6.12. Hibbitt, Karlsson & Sorensen, Inc., Pawtucket, USA. AISI (American Iron and Steel Institute). 2016. North American specification for the design of cold-formed steel structural members. AISI-S100-16, Washington, DC, USA. AS/NZS (Australian/New Zealand Standard). 2005. Cold-formed steel structures. AS/NZS 4600, Sydney, Australia. CEN (European Committee for Standardization). 2006. Eurocode 3: Design of steel structures - Part 1-3: General rules - Supplementary rules for cold-formed members and sheeting. EN 1993-1-3, Brussels, Belgium. CEN (European Committee for Standardization). 2014. Eurocode 3: Design of steel structures - Part 1-1: General rules and rules for buildings. EN 1993-1-1:2005+A1:2014, Brussels, Belgium. Li, H.T. & Young, B. 2017a. Tests of cold-formed high strength steel tubular sections undergoing web crippling. Engineering Structures 141, 571–583. Li, H.T. & Young, B. 2017b. Cold-formed ferritic stainless steel tubular structural members subjected to concentrated bearing loads. Engineering Structures 145, 392–405. Li, H.T. & Young, B. 2018a. Design of cold-formed high strength steel tubular sections undergoing web crippling. Thin-Walled Structures 133, 192–205. Li, H.T. & Young, B. 2018b. Web crippling of cold-formed ferritic stainless steel square and rectangular hollow sections. Engineering Structures 176, 968–980. Li, H.T. & Young, B. 2019a. Behaviour of cold-formed high strength steel RHS under localised bearing forces. Engineering Structures 183, 1049–1058. Li, H.T. & Young, B. 2019b. Cold-formed high-strength steel tubular structural members under combined bending and bearing. Journal of Structural Engineering, ASCE. (In press) Ma, J.L. Chan, T.M. & Young, B. 2017. Tests on high-strength steel hollow sections: a review. Proceedings of the Institution of Civil Engineers170(SB9): 621–630. Young, B. & Hancock, G.J. 2002. Tests of channels subjected to combined bending and web crippling. Journal of Structural Engineering 128(3): 300–308. Yu, W.W. & LaBoube, R.A. 2010. Cold-formed steel design. Fourth Edition, New York: John Wiley & Sons. Zhao, X.L. & Hancock, G.J. 1991. T-joints in rectangular hollow sections subject to combined actions. Journal of Structural Engineering 117(8): 2258–2277. Zhao, X.L. & Hancock, G.J. 1992. Square and rectangular hollow sections subject to combined actions. Journal of Structural Engineering 118(3): 648–667. Zhao, X.L. Heidarpour, A. & Gardner, L. 2014. Recent developments in high-strength and stainless steel tubular members and connections. Steel Construction 7(2): 65–72. Zhou, F. & Young, B. 2007. Experimental investigation of cold-formed high-strength stainless steel tubular members subjected to combined bending and web crippling. Journal of Structural Engineering 133 (7): 1027–1034.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Stability design of steel structures: From members to plates and shells L. Simões da Silva, Trayana Tankova, & João Pedro Martins ISISE, Department of Civil Engineering, University of Coimbra, Portugal

ABSTRACT: This paper presents an overview of the current design procedures for the buckling design of members, plates and curved panels. It highlights recent developments in these fields, namely methodologies for the evaluation of the buckling resistance of generic beam-columns with variable cross section, loading and boundary conditions, the evaluation of the reliability of the Winter curve in slender webs and the development of design guidance for curved panels. Finally, the incorporation of some of these developments in the current revision of Eurocode 3 is discussed.

1 INTRODUCTION Nowadays, structural engineering is a highly standardized activity whereby current design procedures evolved over many years based on accumulated experience and a continuous effort to reflect a sound mechanical basis. Additionally, design standards in Europe are bound to product and execution standards which are aimed to guarantee that the adopted assumptions concerning material properties and the fabrication processes reflect reality. Historically, verification procedures were aimed at simple design processes, thus providing expressions suitable for hand calculations, very often empirical, calibrated to experimental tests whenever possible. More recently, there is a trend to transform the simpler expressions into design procedures that contain the mechanical background of the phenomenon under consideration, complemented by extensive numerical validation that in some cases leads to accurate design expressions that are statistically calibrated (Tankova et al, 2014). This is only made possible due to the tools available to the engineers – computers and sophisticated software. These advances have reached design codes as well. Both Eurocode 3 (2005) and AISC (2016) allow the design of steel structures using numerical analyses with different level of sophistication. There are three basic levels (Simões da Silva et al, 2016): i) global analysis which accounts for all imperfections and all 2nd order effects followed by cross-section verifications; ii) global analysis that partially accounts for the imperfections (only global) and 2nd order effects and followed by member and cross-section verification buckling length equal to the member geometrical length or the simplest version iii) basic cases where individual equivalent member checks are performed using buckling lengths corresponding to the global buckling mode of the structure/member and cross-section verifications. Both Eurocode 3 (2005) and AISC (2016) present fairly similar recommendations for the global analysis of the structure. Both design codes recommend the use of second order analysis for the determination of the design forces. In both cases, it is allowed to use fictitious horizontal forces in order to account for the global sway imperfections of the structure. In case of Eurocode 3, the designer has the choice of including member imperfections in the second-order analysis and further verifying only the cross-section resistance or neglecting the P-δ effects and accounting for them by further verification according to its specifications in section 6.3 for member stability using the geometrical member length (L) or the non-sway buckling length. In case of AISC (2016), second order analysis is always followed by a member verification; however, the member length is taken equal to the unbraced member length KL = L. 28

Even though it is conceptually simple, second order analysis with global and local imperfections is not yet the preferred method and it is yet to become the daily basis of design in a design office. This is due to the requirement of more in deep knowledge about member imperfections but also due to the software availability to apply the imperfections directly. Hence, the most common approach used by the engineers today is the second order global analysis with global imperfections and then followed by the member checks using as buckling length the member length. The member design rules in the modern codes also follow very similar concepts – for columns and beams the buckling curve approach, whereas beam-columns are designed using interaction equations which combine the utilization due to axial force and bending moments through calibrated factors. The verification format for the flexural buckling of columns is directly derived from the differential equations for a column with an initial imperfection. It is achieved by applying a first yield criterion at mid-span for a simply-supported column (i.e., equating the normal stress at the most compressed fiber to fy): N N^v0 ¼ fy þ A 1  N=Ncr;i

ð1Þ

which is transformed to χþη

χ 1  χλ

2

¼ 1:0

ð1Þ

where
λ is the normalized slenderness, χ is the buckling reduction factor and η is the generalized imperfection factor (accounting for out-of-straightness and residual stresses), with W denoting the elastic section modulus relative to the buckling axis. Eq.(1) is the buckling curve equation where the only unknown is the amplitude of the imperfection. This advantage of the Ayrton-Perry representation of the problem was used in the calibration of the European design rules. The buckling curves were established in the 1970’s, and their development was based on an extensive experimental programme carried out by the European Convention for Constructional Steelwork (ECCS) in several European countries Sfintesco (1970); on theoretical developments Beer & Schulz (1970) by thorough analysis of the experimental programme; and on reliability assessment by a Monte Carlo simulation Strating & Vos (1973). Finally, Maquoi & Rondal (1978) derived the analytical Ayrton-Perry format of the design verification and the curves were put into equation. Presently, the code proposes 5 buckling curves, a to d were originally calibrated on the basis of the ECCS experimental programme. The most relaxed curve a0, was added later to account for the more favourable properties of high strength steels (HSS). Similarly, the AISC design specification is based on SSRC experimental programme which initially specified 3 curves which were subsequently reduced to a single curve, SSRC curve 2P (Ziemian, 2010). Yet, this curve is divided into three regions – the inelastic and purely elastic which are calculated using three different expressions unlike the single expression of Eurocode 3. The stability of unrestrained beams can be verified according to two methods in Eurocode 3. The first approach, the so-called General case, assumes that columns and beam behave alike, and so the compression flange is considered as an equivalent column. This results in identical imperfection factors as for flexural buckling of columns. The method, however, adopts different splits according to the cross-section geometry, accounting for their different torsional rigidity. Furthermore, according to the European specifications for stability design given in ECCS No. 119, these curves were meant to be used with deep and slender sections which are outside of the scope of rolled sections. The method is of general application but too conservative when applied to members with variable bending moment diagram (Rebelo et al. (2009), Taras (2010)). 29

As an alternative for hot-rolled and equivalent welded sections, Eurocode 3 provides another set of buckling curves, the Special case. The method was calibrated on the basis of extensive numerical studies in the Research Project Lateral-torsional buckling of steel and composite beams (1993) and Salzgeber (2000). Additional calibration was carried out by Grainer & Kaim (2001) on the basis of experimental results by Janss & Maquoi. In order to justify, the plateau at 0.4, experimental results by Byfield & Nethercot (1998) were used in the assessment according to Annex Z of ENV 1993-1-1:1992. However, the method was shown to be unconservative when compared to numerical results Snijder & Hoenderkamp (2007), Rebelo et al. (2009). The European interaction formula for verification of members subject to bending and compression currently uses two sets of interaction coefficients: the “French-Belgian” team which was responsible for the interaction coefficients in Method 1 and the “Austrian-German” team responsible for the development of the coefficients associated with Method 2. The main difference between the two methods is the way of considering the various effects which affect the beam-column behaviour. The interaction coefficients associated with Method 1 were developed aiming to distinguish each structural effect in the interaction coefficient (plasticity, equivalent moment factors, lateral-torsional buckling), therefore laying the ground for any further modifications, if necessary, and to directly identify the impact of each physical phenomenon. Method 2, aimed at easier practical implementation, combines the non-linearities into global interaction factors kii and kij calibrated on the basis of a wide-range of numerical simulations. At cross-section level, in Eurocode 3, the ability of the cross-section to either develop its full plastic capacity, behave elastically or being unable reach the elastic limit due to local instability is distinguished by the cross-section classification. Depending on the cross-section class, the member stability design rules are applied with plastic, elastic or effective properties. In the specification of the effective properties for slender sections, it is considered that a redistribution of compressive stresses takes place due to reduction of stresses in the middlebuckled region which leads to increase of stresses near the plate edges (Beg et al., 2010). The non-linear stress distribution is simplified by either an effective width method (based on an appropriate cross-section reduction in central buckled part of the plate) or a reduced stress method (by calculating a reduced average stress). The concept of effective width was initially developed by von Karman (1932) by assuming that the effective cross-section works under the yield stress (fy). However, in this work, von Kaman did not consider the presence of initial imperfections and residual stresses. Later, Winter (1947, 1948) performed experimental studies on long plates stiffened along both longitudinal edges such as web of channels and I-sections. More than 100 tests under compression of thin walled cold-formed C-channels and I-sections with different b/t ratios and yield stress were performed. This experimental programme was the basis of the buckling curve considered in Eurocode 3-1-5 for the estimation of the effective width, it is often referred as the Winter curve. Since it was developed for plates that were supported at the edges (channels and I-sections) by normally thicker flanges, the level of rotational restraint provided is somehow incorporated in the design expression. This paper addresses the design procedures and their reliability for the stability design of members, plates and curved panels and discusses current developments and their incorporation in the current revision of Eurocode 3.

2 BEAM-COLUMN DESIGN Even though the development of stability design rules currently used and presented in this section were developed using analytical developments, experimental tests and statistical assessments, stability design continues to attract the attention of researchers. Nowadays, the efforts are focused in the incorporation of the enhanced material properties, new cross-section geometries and the possibility of computer aided calculations into the design process. In the 30

following, a brief overview of the recent developments in the scope of the European design rules for beams, columns and beam-columns is summarized. There are several works which build upon the Ayrton-Perry format for flexural buckling of columns, i.e. extending it to beam-columns, non-uniform members and members subject to arbitrary loading. It was succeeded by combining the first and second order effects corresponding to the case in consideration and calibration of overstrength factors which account for the variation of the most unfavourable cross-section depending on the loading. This concept was used to extend design verifications to flexural-torsional and lateral-torsional buckling once the correct analytical behaviour was considered by Taras (2010). In Naumes (2009), the equilibrium equation for the flexural buckling of tapered members was also established; in this derivation, the shape of the initial imperfection was considered eigenmode conform. It was shown that the Ayrton-Perry design format can be adopted for the design of non-uniform members. However, the proposed expressions are not applicable for practical design verification due to lack of recommendations for the determination of the design location. Furthermore, Taras (2010) offers the same type of model for flexural buckling of beam-columns which was combined with calibrated factors accounting for the plasticity effects at low slenderness. Marques (2012) provided design equations for flexural buckling of web-tapered columns and lateral-torsional buckling of web-tapered beams in the same Ayrton-Perry format which account in a systematic way for the position of the critical for verification design location. Simplifications of the current rules also exist, an approximate verification format for beamcolumns is proposed by Hoglund (2014) where the stress utilizations due to axial force and bending moments are specified as power functions. The method claims to provide better representation of the transition between Class 2 and Class 3 sections, which however are already implemented in the final draft of prEN 1993-1-1 (CEN/TC250, 2018) following the recommendations of the European project SEMICOMP+ (Greiner et al., 2011). It is also incorporated in Eurocode 9 for verification of aluminium beam-columns. Recent research by Szalai (2017) shows the extension of the Ayrton-Perry equation to prismatic simply supported members subject to arbitrary loading. The author does not provide calibration of the corresponding imperfection factors but shows that it is theoretically possible to achieve this format for various buckling modes. Based on this development, Szalai & Papp (2017) built their proposal for reformulation of the General method, by putting it into the derived Ayrton-Perry proposal for prismatic simply supported members subject to arbitrary loading, which is its major flaw, being unable to capture the specific aspects of non-uniform members. Even though the Ayrton-Perry based developments have the advantage of considering the first and second order effect in a consistent manner, in most of the cases the necessity of calibration additional factors was adopted, due to variation of the critical location along the member. This is a major drawback when non-conventional cases need to be considered. Driven by these limitations, there are a few developments supporting the design by use of numerical analyses. Their proposals were mainly focused on the definition of the equivalent geometrical imperfection to be considered in the design Chladný & Stujberová (2013a,b), Aguero et al. (2015a,b), Papp (2016) and a mixture between LBA conform imperfection and a reduction factor calculation Badari & Papp (2015). Recently, a new method, the so-called general formulation (GF) was proposed by (Tankova et al. (2018)). The proposed general formulation for the stability design of steel members comprises a linear interaction equation that combines the first-order normal stresses due to applied forces and the normal stresses due to second order forces. This linear interaction between first and second order stresses is consistent with the Eurocode 3 procedures where the reduction factor χ was also derived on the basis of a Ayrton-Perry design philosophy and assuming that the shape of the initial imperfection follows the same shape as the buckling mode. The proposed general formulation also adopts the buckling mode as the shape of the initial imperfection and the amplitude previously calibrated for the standard prismatic simplysupported columns and beams in Eurocode 3, while the amplitude for the standard prismatic 31

simply-supported beam-column buckling out-of-plane is obtained from recent work by Tankova (2018a) Since the terms concerning the stress utilization due to second order forces are amplified by the imperfection factors from Eurocode 3 they are consistent with the rules for prismatic members. Finally, the verification is implemented as a sequence of cross-section verifications along the member length. The fact that the stress utilizations due to first and second order forces are added at each location allows to avoid the calibration of additional factors for each specific case, in contrast with the case of web tapered members (Marques et al., 2012, Marques et al., 2013). where specific generalized imperfection factors had to be adjusted. The developed interaction equation needs to be applied for all potential failure modes. The utilization ratio of the generic single member may be expressed by equating the total longitudinal stress, due to first and second order forces, to the yield stress, fy: MyII ðxÞ My ðxÞ NðxÞ Mz ðxÞ M II ðxÞ MwII ðxÞ þ þ þ þ z þ  1:0 AðxÞ fy Wy ðxÞ fy Wz ðxÞ fy Wy ðxÞ fy Wz ðxÞ fy Ww ðxÞ fy

ð2Þ

where A(x) is the cross-section area, Wy (x) and Wz (x) are the section moduli relative to the yand z axes, respectively, and Ww (x) = Iw(x)/wmax(x) is the warping modulus at location x along the member, with wmax(x) = hb/4. It is noted that for section classes 1 and 2 the plastic section moduli should be used while for class 3 sections the semi-compact approach of Annex B of prEN 1993-1-1 may be used. Then, as long as the second order contributions can be determined, the buckling resistance may be verified for an appropriate number of locations along the member. The verification of a single member with variable geometry, boundary conditions, subject to arbitrary loading, is done by verifying Eq. (2) at a sufficient number of locations along the member. At each position, the respective values of the first order axial force, N(x), bending moments My(x), Mz(x), second order contributions obtained from the relevant buckling mode and cross-section properties are to be used. Figure 1 illustrates the procedure for a tapered single member:

Figure 1.

Schematic illustration of the verification procedure.

32

This verification shall be performed for the global buckling modes which may affect the studied member, for instance in and out-of-plane buckling. For instance, the verification of a beam-column shall be done by applying Eq. (3) for in-plane buckling and out-of-plane buckling. In addition, the requirement to check the cross-section resistance at the extremities of the member is automatically included, as the member is checked for a sufficient number of crosssections, including the end-sections, as explained above. Table 1 illustrates the calculation of the second-order forces in Eq. (3)

3 LOCAL BUCKLING 3.1 General remarks The stochastic nature of the governing parameters in the ultimate resistance of plated structures must be acknowledged if an accurate assessment of the actual reliability level is to be achieved. These parameters are on one hand geometrical (dimensional and geometric deviations) and material (yield stress, Young’s modulus and residual stresses). Several authors have drawn their attention to the reliability analysis of plated structures (Fukumoto & Itoh (1984), Duc et al. (2013), Rahman et al. (2017), Gaspar et al. (2015)). All of these authors performed their analyses by modeling the plate isolated from the rest of the structure, applying, therefore, idealized boundary conditions: in these studies, all edges are perfectly simply supported. Recently, Schillo (2017) collected several experimental results challenging the level of safety given by the effective width methodology (based on the well-known Winter curve) as it is presented in Eurocode 3-1-5. These experimental results are from mild and high strength steel welded class 4 box-sections. As a result of these and own experimental results, a recommendation for higher values of the partial factor or, alternatively, a new exponential curve to replace the Winter curve was proposed. Considering that the studies dealing with the stochastic nature of strength of plates are limited to perfect boundary conditions, and that there are cases where the current rules of the effective width methodology given by EC3-1-5 is apparently unsafe, Martins et al (2019) evaluated the reliability level of the referred methodology applied to welded I- cross-sections under compression. For this purpose, a Monte Carlo experiment was performed where the Latin Hypercube technique was applied to reduce the size of the sample. All simulations with the Monte Carlo experiment are evaluated using the finite element method and a normal distribution is fitted to the obtained resistance distribution. Finally, this normal distribution is compared to the nominal resistance provided by EC3-1-5 and partial factors are derived using the formulation given in EC0. These results are briefly described in the following sub-sections. 3.2 Scope of the analysis In this analysis, the following parameters are considered as having a stochastic nature: steel properties (S460 steel was considered; statistical data from prEN1993-1-1, see Table 2) and geometry of the cross-section (flange width and thickness, web height and thickness; statistical data from prEN1993-1-1). The remaining parameters, i.e. shape and amplitude of initial imperfections (where the first buckling mode and hw/200 are considered, respectively) and residual stresses (which are not considered explicitly) are taken as deterministic. In fact, these assumptions correspond to the equivalent imperfections option for the modeling of imperfections. In order to include a practical range of applications, 28 samples consisting in seven class 4 cross-sections (Table 3) with 4 values for the aspect ratio (1.00, 1.25, 1.50 and 1.75) were considered. These cross-sections are inspired in product catalogs (specifically, HE600B, HE800B, and HL1100B). Table 4 gives the nominal resistance for each cross-section.

33

  EIz ðxÞ v00 cr ðxÞ þ h2 θ00 cr ðxÞ þ θ0 cr ðxÞh0 ηðxÞ AðxÞfy ðαcr  1Þ

EIi ðxÞδ cr ðxÞ ηðxÞ AðxÞfy ðαcr  1Þ My jδcr ðxÞj EIi ðxÞδ00 cr ðxÞ ηðxÞ þ AðxÞfy ðαcr  1Þ Wy ðxÞfy ðαcr  1Þ

NðxÞ AðxÞfy NðxÞ Mi ðxÞ þ AðxÞfy Wi ðxÞfy Mi ðxÞ Wi ðxÞfy NðxÞ Mi ðxÞ þ AðxÞfy Wi ðxÞfy

FB (N) FB (N+M) LTB (M) LTB (N+M)

00

εII ðxÞ

εI ðxÞ

Second-order forces.

Mode

Table 1.

  αð
λð xÞ  0:2Þf ηδflcr ðxÞ

ηðxÞ

Ncr;z;eq   EIi ðxm Þ v00 cr ðxm Þ þ h2 θ00 cr ðxm Þ þ θ0 cr ðxm Þh0

αcr NðxÞ EIi ðxm Þjδ00 cr ðxm Þj αcr NðxÞ EIi ðxm Þjδ00 cr ðxm Þj

fη

Table 2.

Statistical data of S460 structural steel. fy [MPa]

E [MPa]

Structural Steel

Nominal

Mean

Co.V.

Nominal

Mean

Co.V.

S460

460

529

4.5%

210 000

210 000

3.0%

Table 3.

Geometrical statistical data of the analyzed cross-sections. hw [mm]

Welded cross-sections (hw × bf × tw × tf)

Mean

540 × 300 × 19.75 × 30 540 × 300 × 15.5 × 30 734 × 300 × 17.5 × 33 1028 × 300 × 20 × 36 1228 × 300 × 20 × 36 1528 × 300 × 20 × 36 1528 × 300 × 18.82 × 36

540 540 734 1028 1228 1528 1528

bf [mm] Co.V. Mean

0.9%

300 300 300 400 400 400 400

tw [mm]

tf [mm]

Co.V. Mean

Co.V. Mean

19.75 15.5 17.5 20 20 20 18.82

29.4 29.4 32.34 35.28 35.28 35.28 35.28

0.9%

2.5%

Co.V.

2.5%

3.3 Analysis of results Using a purely random sampling method leads to the need of a large number of simulations in a Monte Carlo experiment, otherwise the convergence of the method is questionable. Several authors suggest methods to calculate the required sample size: for example, while Mann et al. (1974) recommends a minimum of 10000 simulations for a 95% confidence limit, Melchers (1999) suggests, for the same confidence limit and for a failure probability equal to 10-3, around 3000 simulations. Additionally, the number of independent variables in each simulation may also influence the required sample size. Here, in order to decrease the size of the sample, the Latin Hypercube Sampling technique is applied. It was concluded that 5000 simulations for each sample were more than sufficient to achieve convergence. Specifically, convergence of the mean value and standard deviation (see example in Figure 2), as well as the probability of failure were achieved for all samples. The probability of failure (and the reliability index) is obtained for each aspect ratio and cross-section by fitting a normal distribution to the results and calculating the percentage of lower results than the nominal value of the resistance (see Table 3). It should be highlighted

Figure 2. α=1.

Convergence of mean value and standard deviation for cross-section 1528 × 300 × 18.82 × 36,

35

that the resistance statistical distribution passed the Kolmogorov-Smirnov test for the normal hypothesis. Therefore, the probability of failure is obtained by: 



PF ¼ P R5Rd;nom ¼

Rd;nom ð

N ðμR ; σR Þ

ð4Þ

∞

where μR and σR are the mean and standard deviation, respectively, for each sample. The reliability index is simply obtained by: β ¼ F1 ðPF Þ

ð5Þ

Finally, the partial factor is iteratively calculated until the reliability index of each resistance distribution equals β=3.04, see Eq. (6).     Rd;nom F βTARGET ¼ P R  γM

ð6Þ

3.3.1 Reliability index, partial factors and web reduction factor As already mentioned, all resistance distributions follow a normal distribution. In Figure 3a it can be seen that the highest value for the partial factor is equal to 1.05 for the stockiest crosssection. This conclusion indicates that only for cross-sections with lower web plate slenderness the partial factor is underestimated by EC3-1-5. On the other hand, for the remaining crosssections, the values of the partial factor are significantly lower than 1. This gives the indication that the estimative of EC3-1-5 is safe sided. Finally, the web reduction factor may be obtained by integration of stresses at ultimate load and calculating the equivalent width that would be yielded, or, alternatively (acknowledging the fact that the flanges are always fully effective) using the following equation: 

 Aw;i :ρ þ Af ;i fy;i ¼ NFEM

ð7Þ

where Aw,i is the gross area of the web, is the reduction factor, Af,i is the gross area of the flanges, fy,i is the yield stress and NFEM is the numerically obtained resistance.

Figure 3.

Partial factors and equivalent plate reduction.

36

Figure 4.

Ranges of parameters in existing bridge decks (Reis et al. (2017)).

Figure 3b plots the minimum and maximum values of the reduction factor (using equation (7)) against the Winter curve. Analyzing the results, it becomes clear the lower bound nature of the Winter curve. 3.4 General remarks Recently, a good amount of effort was carried to propose design rules for stiffened curved plates. An example of this effort is the RFCS project OUTBURST. In fact, curved steel plates in bridge decks are increasingly used in recent years (Reis et al. (2017)), both for aesthetic and structural demands (efficiency of load carrying capacity; high strength to weight ratio, added value from an architectural point of view, enhanced behaviour to wind loading). However, as pointed out by some researchers (Tran et al. (2013)), curved panels do not behave neither as a flat plate nor as a full revolution shell. This poses a problem for designers. Martins et al. (2018), give a detail revision on the available expressions to calculate the elastic critical stress and the ultimate strength of curved plates. Curved panels are usually defined by the global curvature parameter which is a nondimensional parameter given by the following expression Z¼

b2 Rt

ð8Þ

where b is the width of the plate, R is the radius of curvature and t is the thickness of the plate. This parameter may be also used to defined the local curvature of sub-panels (in which cases b is replaced by blocal, i.e., width of the unstiffened part of the plate). Figure 4 shows the values of global and local curvature, as well as values of the aspect ratio of the curved plates, of real curved plates found in bridge designs. 3.5 Elastic critical stress of unstiffened curved plates under compression and shear The first attempt to obtain the elastic critical stress of curved panels under compression is due to Redshaw (1935) who proposed a formula which was later modified by Stowell (1944). Later, Timoshenko (1961) based on approximate expressions of the displacement field proposed a different formula. Concerning the shear load case, the first known studies are due to Leggett (1937), Kromm (1939) and Timoshenko and Gere (1961). Subsequent works have been performed later by Batdorf (1947) and Schildcrout and Stein (1949). Based on the compilation of the work performed by these authors, in 1968 NASA (1968) publishes a document containing curves allowing to obtain the elastic critical coefficient of 37

Table 4. Elastic buckling coefficient for short curved panels under compression (Martins et al. (2013)) (10

23 < Z  100

0 > ψ > -1 A B

-1

þC ψþ D ψ

2

C ¼ c1 þ c2 Z þ c3 Z2 D ¼ d1 þ d2 Z þ d3 Z2

A ¼ a1 þ a2 Z þ a3 Z2 B ¼ b1 þ b2 Z þ b3 Z2

0 < Z  23

0

a1 ¼ 8:2 b1 ¼ 1:05 a2 ¼ 0:0704 b2 ¼ 0:0002 a3 ¼ 0:0163 b3 ¼ 0:0003

c1 ¼ 6:29 d1 ¼ 9:78 c2 ¼ 0:1971 d2 ¼ 0:2174 c3 ¼ 0:0004 d3 ¼ 0:0002 c1 ¼ 9:124 d1 ¼ 5:843 c2 ¼ 0:0646 d2 ¼ 0:0556 c3 ¼ 0 d3 ¼ 0:0002

a1 ¼ 3:214 b1 ¼ 0:961 a2 ¼ 0:5976 b2 ¼ 0:0104 a3 ¼ 0:0028 b3 ¼ 0

curved plates. These curves have been for very long time the most up-to-date methodology to obtain the elastic critical stress of curved panels. Recently, due the extraordinary increase in the computation capacity, the finite element method as enabled researchers to run large parametric studies consisting in linear buckling analysis and calibrate expressions which are more accurate. Within the framework of the above-mentioned research project, the formulas given in Tables 4 and 5 were developed to compute the elastic critical stress of curved panels under compression and under shear stresses, respectively.

4 CONCLUDING REMARKS: OUTLOOK OF FUTURE VERSIONS OF EUROCODE The first revision of the Structural Eurocodes is currently taking place, with a planned release in 2023. Concerning Eurocode 3 and the design rules related to the buckling resistance of

Table 5. Elastic buckling coefficient for curved panels under shear (Ljubinkovic et al. (2019a)) (1< Z < 100).  2 α≤1 α>1 kτ ¼ A þ B α1

C1

C2

C3

A ¼ 0:214Z þ 2:88 Z B ¼ 5:343  175:6

A ¼ 0:096Z þ 5:15 B ¼ 0:135Z þ 3:18

A ¼ 0:247Z þ 2:732 Z B ¼ 5:34  150:4 A ¼ 0:2734Z þ 2:794 Z B ¼ 5:33  127:2

A ¼ 0:124Z þ 4:94 B ¼ 0:137Z þ 3:756  2 Z þ 0:349Z þ 5:424 A¼ 28:86  2 Z B¼  0:0452Z þ 2:422 37:735

38

members, plates and curved panels, prEN 1993-1-1 (2018) and prEN 1993-1-5 (2019) already implement some of the aspects briefly presented and discussed above. Concerning the buckling resistance of members, prEN 1993-1-1 (2018) implements the Ayrton -Perry based developments for the buckling resistance of beams and extends the scope of the beam-column interaction equations to monosymmetric cross-sections. It further simplifies the interaction factors by adopting a single methodology for their evaluation. A more detailed selection of methodologies is also adopted concerning the choice of global analysis methods. Concerning the determination of the effective width of slender plates, a distinction is introduced in the buckling curves as a function of the degree of restraint along the longitudinal boundary conditions. Finally, it is planned to extend the scope to cylindrically curved panels. These developments, coupled with the possibility to implement design based on advanced finite element calculations will ensure that the Structural Eurocodes remain a leading code of practice that addresses current and future design challenges. REFERENCES Aguero A., Pallarés F.J., Pallarés L. (2015a). Equivalent geometric imperfection definition in steel structures sensitive to flexural and/or torsional buckling due to compression, Engineering Structures, 96, pp. 160–177. Aguero A., Pallarés F.J., Pallarés L. (2015b). Equivalent geometric imperfection definition in steel structures sensitive to lateral-torsional buckling due to bending moment”, Engineering Structures, 96, pp. 41–55. AISC (2016), American Institute of Steel Construction, Specification for Structural Steel Buildings, Chicago, Illinois, USA. Alinia M.M., Habashi H.R., Khorram A. (2009) Nonlinearity in the post-buckling behavior of thin steel shear panels. Thin-Walled Structures 47, 412–420. Amani, M., Edlund, B.L.O., Alinia, M. M. (2011) Buckling and post-buckling behavior of unstiffened curved plates under uniform shear. Thin-walled Structures, 49 (8), 1017–1031. Anonymous. Buckling of thin-walled truncated cones. NASA Space Vehicle Design Criteria (Structures), NASA SP-8019, September 1968. Badari B., Papp F. (2015) On design method of lateral-torsional buckling of beams: State of the art and a new proposal for a general type of design method” Periodica Polytechnica Civil Engineering, 59, pp. 179–192. Batdorf, S.B. (1947) A Simplified Method of Elastic Stability Analysis for Thin Cylindrical Shells. NACA Technical Report No. 847. Beer H., Schulz G. (1970). Bases Théoriques des Courbes Européennes de Flambement, In: Construction Métallique, no.3, pp. 37–57. Beg D., Kuhlamann U., Davaine L., Braun B. (2010)., Design of Plated Structures-Part 1-5: Design of Plated Structures, ECCS. Boissonnade N., Greiner R., Jaspart J.P., Lindner J. (2006). Rules for member stability in EN 1993-1-1, Background documentation and design guidelines. ECCS (European Convention for Constructional Steelwork) Publication no. 119, Brussels, 2006. CEN (2005). EN 1993-1-1, Eurocode 3: Design of steel structures - Part 1-1: General rules and rules for buildings CEN, Brussels. CEN/TC250 (2017). Eurocode 3: Design of steel structures - Part 1-1: General rules and rules for buildings, CEN/TC 250/SC 3 N 2532 - prEN 1993- 1-1- Final draft. Chladny´ E, Stujberová M. (2013),” Frames with unique global and local imperfection in the shape of the elastic buckling mode (part1)”, Stahlbau, 82.8, pp. 609–617. Chladny´ E, Stujberová M. (2013),” Frames with unique global and local imperfection in the shape of the elastic buckling mode (part2)”, Stahlbau, 82.9, pp. 684–694. Domb, M.M. & Leigh, R.L. (2002) Refined design curves for shear buckling of curved panels using nonlinear finite element analysis. 43rd AIAA/ASME/AHS/ASC Structures, Structural Dynamics and Materials Conference, Denver, U.S.A, Paper #2002-1257. ESCS Steel RDT Programme, Research Project: Lateral-torsional buckling in Steel and Composite Beams, No. 7210-PR-183, testing of 4 Tapered Steel Beams. (2002)

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Featherston, C.A. & Ruiz, C. (1998) Buckling of curved panels under combined shear and compression. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 212 (3), 183–196. Featherston, C.A. (2000) The use of finite element analysis in the examination of instability in flat plates and curved panels under compression and shear. International Journal of Non-Linear Mechanics 35, 515–529. Featherston, C.A. (2003) Imperfection sensitivity of curved panels under combined compression and shear. International Journal of Non-Linear Mechanics, 38, 225–238. Gerard, G. (1959), Handbook of Structural Stability: Supplement to Part III – Buckling of Curved Plates and Shells, NASA Technical Report D-163, Washington D.C. Greiner R., Kaim P. (2003a). Comparison of LT-buckling curves with test results, Supplementary Report, ECCS TC8, No. 2003-10, May 2003. Greiner R., Kettler M., Lechner A., Jaspart J.P. Weynand K. Ziller, C., Örder, R. (2011). SEMI-COMP+: Valorisation Action of Plastic Member Capacity of Semi-Compact Steel Sections – a more Economic Design, RFS2-CT-2010-00023, Background Documentation, Research Programme of the Research Fund for Coal and Steel – RTD. Hoglund T. (2014). A unified method for the design of steel beam-columns. Steel Construction, 7, pp. 230–245. Janss J., Maquoi R., Evaluation of test results on lateral torsional buckling in order to obtain strength function and suitable model factor, Background report to Eurocode 3. Johansson B., Veljkovic M., (2001) Steel Plated Structures. Progress in Structural Engineering and Materials, 3:1 2001. Kromm, A. (1939) The limit of stability of curved plate strip under shear and axial stresses. NACA. Technical note No.: 898. Leggett, D.M.A. (1937) The elastic stability of a long and slightly bent plate under uniform shear. Proc. R. Soc., A162, 62–83. Ljubinković, F., Martins, J.P., Gervásio, H., Simões da Silva, L. (2019a), “Eigenvalue analysis of cylindrically curved steel panels under pure shear”. Thin-Walled Structures 141, 447–459. Ljubinković, F., Martins, J.P., Gervásio, H., Simões da Silva, L. (2019b), “Ultimate load of cylindrically curved steel panels under pure shear”. Thin-Walled Structures 142, 171–188. Manterola, J. (2008) Pasarela de Peatones - Zaragoza EXPO 2008 [PowerPoint presentation]. Marques L., Taras A., Simões da Silva L., Greiner R., Rebelo, C. (2012), Development of a consistent design procedure for tapered columns, Journal of Constructional Steel Research, 72, pp. 61–74. Marques L., Simões da Silva L., Greiner R., Rebelo C., Taras, A. (2013). Development of a consistent design procedure for lateral-torsional buckling of tapered beams, Journal of Constructional Steel Research, 89, pp. 213–235. Martins, J.P., Simões da Silva, L., and Reis, A. (2013) Eigenvalue analysis of cylindrically curved panels under compressive stresses – Extension of rules from EN1993 1 5. Thin-Walled Structures, 68, pp. 183–194. Martins J, Simões da Silva L, Reis A. (2014) Ultimate load of cylindrically curved panels under inplane compression and bending – extension of rules from EN 1993- 1-5. Thin-Walled Structures 77, 36–47. Naumes J. (2009). Biegeknicken und Biegedrillknicken von Stäben und Stabsystemen auf einheitlicher Grundlage, PhD thesis, RWTH Aachen, Germany. Papp F. (2016). Buckling assessment of steel members through overall imperfection method. Engineering Structures, 106, pp.124–136. Pope G.G. (1965), “On the axial compression of long, slightly curved panel ”, Technical Report British Aeronautical Research Council; Ministry of Aviation, Reports and Memoranda No. 3392. Rebelo C., Lopes N., Simões da Silva L., Nethercot D., Vila Real P.M.M. (2009). Statistical Evaluation of the Lateral-Torsional Buckling Resistance of Steel I-beams, Part 1: Variability of the Eurocode 3 resistance model, Journal of Constructional Steel Research., 65, pp. 818–831. Reis, A. Pedro, J.O., Graça, A.B., Hendy, C., Romoli, P., Simões da Silva, L., Martins, J.P. (2017) Report on the characterization of relevant parameters of curved plated bridge structures and identification of bridge cases where they can be found. RFCS Research Project OUTBURST (RFCS-2015-709782): Deliverable 2.1. Redshaw, S.C. (1935), The Elastic Instability of a Thin Curved Panel Subjected to an Axial Thrust, Its Axial and Circumferential Edges Being Simply Supported, Report and Memorandum No. 1565, British Aeronautical Research Committee. Salzgeber G. (2000). LT-buckling curves, ECCS TC8, Report No. 2000-001, 20 March 2000.

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Schildcrout, M. and Stein, M. (1949) Critical combinations of shear and direct axial stress for curved rectangular panels. NACA. Technical Note No.: 1928. Schillo, N. (2017) Local and global buckling of box columns made of high strength steel. Ph.D. thesis, RWTH Aachen University. Sekine, H., Tamate, O. (1969), “Postbuckling behavior of thin curved panels under axial compression”, Bulletin of Japan Society of Mechanical Engineering, 12, pp. 415–420. Sfintesco D. (1970). Fondement Expérimental des Coubres Européennes de Flambement, Construction Métallique, no.3, pp. 5–12. Simões da Silva L., Simões R., Gervásio H., (2016). Design of steel structures. Eurocode3: Design of steel structures. Part-1-1 - General rules and rules for buildings. 2nd Edition. European Convention for Constructional Steelwork, John Wiley & Sons, 2016. Snijder H.H., Hoenderkamp J.C.D. (2007). Buckling curves for lateral torsional buckling of unrestrained beams, Rene Maquoi 65th birthday anniversary, 2007, Liège Belguim. Stowell, E.Z. (1943), “Critical compressive stress for a curved sheet supported along all edges and elastically restrained against rotation along the unloaded edges”, NACA War Report L-691. Strating J., Vos H. (1973). Simulation sur Ordinateur de la Coubre C.E.E.M de Flambement á l‘аide de la Méthode de Monte-Carlo, Construction Métallique, no.2, pp. 23–39. Szalai J. (2017). Complete generalization of the Ayrton-Perry formula for beam-column buckling problems. Engineering Structures, 153, pp. 205–223. Tankova, T, Simões da Silva, L., Marques, L., Rebelo, C. and Taras, A. (2014). Towards a standardized procedure for the safety assessment of stability design rules, Journal of Constructional Steel Research, 103, 290–302 (2014). Tankova T., Marques L., Simões da Silva L., Andrade A. (2017). Development of a consistent methodology for the out-of-plane buckling resistance of prismatic beam-columns,: Journal of Constructional Steel Research, 128, pp. 839–852. Tankova, T., Simões da Silva, L., Marques, L. (2018). Buckling resistance of non-uniform steel members based on stress utilization: general formulation, Journal of Constructional Steel Research, 149, 239–256 (2018). Taras A. (2010). Contribution to the development of consistent stability design rules for steel members” PhD Thesis, Technical University of Graz, Graz, Austria, 2010. Usami, T. (1993) Effective width of locally buckled plates in compression and bending. Journal of Structural Engineering, 119 (5), 1358–1373. Volmir, A. S. (1963), “Stability of Elastic Systems” (in Russian), Fizmatgiz, Moscow. (English translation: NASA Report AD 628508). Von Karman T., Sechler E.E., Donnell L.H., (1932). The strength of Thin Plates in Compression, Transactions of the American Society of Mechanical Engineers, vol. 54, p. 53, 1932. Winter, G. (1947) Strength of Thin Steel Compression Flanges, Transactions of American Society of Civil Engineers, 112, 527–554. Winter, G. (1948) Performance of Thin Steel Compression Flanges, Preliminary Publication, 3rd Congress IABSE, New York, N.Y., 137–148. Ziemian R.D. (2010). Guide to Stability Design Criteria for Metal Structures, Sixth Edition. John Wiley & Sons Inc.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Advancements in the stability design of steel frames considering general nonprismatic members and general bracing conditions D.W. White, R. Slein & O. Toğay Georgia Institute of Technology, Atlanta, GA, USA

ABSTRACT: This paper presents an innovative approach for design of planar steel frames composed of prismatic and/or nonprismatic members. The method uses an inelastic eigenvalue buckling analysis configured with column, beam and beam-column inelastic stiffness reduction factors derived from the ANSI/AISC 360-16 Specification provisions to evaluate the member overall buckling resistances. The resulting procedure provides a relatively rigorous evaluation of all member strength limit states accounting for moment and axial force variations along the member lengths, nonprismatic geometry effects, general out-of-plane bracing conditions, and beneficial end restraint from less critical adjacent unbraced lengths and/or from end boundary conditions. The approach uses a pre-buckling analysis based on the AISC Direct Analysis Method to estimate the in-plane internal forces, including second-order effects. Given these forces, a buckling solution is conducted to evaluate the overall member stability. Other limit states are addressed by cross-section strength checks given the computed internal second-order analysis forces. Calculations from this approach are compared with results from recent experimental tests.

1 INTRODUCTION In recent years, much progress has been achieved in the application of AISC and AASHTO design criteria toward the efficient design of steel frames using nonprismatic members. The current state of the art is captured in the second edition of AISC Design Guide 25 (DG25) (White & Jeong 2019). In addition to discussing more traditional elastic design methods and their associated “manual” calculations, DG25 provides guidance for application of inelastic nonlinear buckling analysis (INBA) procedures to isolated member unbraced lengths. However, further advantages can be realized by applying INBA tools to the assessment of entire planar frame structural systems. This paper provides an overview of the INBA calculations and illustrates the benefits of this “high end” application of the INBA procedures. Recommended INBA calculations are applied to isolated critical unbraced lengths as well as to the full test members from recent experimental tests conducted by Smith et al. (2013). Specifically, the recommended INBA approach accounts for the effects of: – Double- and single-symmetry of member cross-sections, – Single and multiple linear web taper, as well as general continuous variations in the crosssection dimensions along the member lengths, – Steps in the cross-section geometry, associated with changes in plate dimensions, – Any combination of compact, noncompact and slender flanges and/or webs, pertaining to member flexural resistance, – Any combination of slender and/or nonslender cross-section plate elements, pertaining to member axial resistance, – Any combination of equal or unequal spacing of out-of-plane lateral bracing on one or both flanges, as well as torsional bracing such as from diagonal members framed between the inside flanges of frame members and outset girts or purlins of wall or roof systems,

42

– End restraint in critical unbraced lengths due to continuity with adjacent less-critical unbraced lengths and/or due to physical boundary conditions, – The combined influence of flexure and axial loading, and – Load height of transverse loads applied along the member lengths. These INBA capabilities are implemented within the software system SABRE2 V2 (White et al. 2019). Tools such as SABRE2 eliminate the need for tedious and relatively approximate manual calculations of Cb factors, accounting for moment gradient and load height effects, and effective length factors, K, accounting for column and beam end restraint effects.

2 INBA METHODOLOGY The following sections explain the net stiffness reduction factors (SRF) employed within the recommended INBA approach. These factors, derived from the AISC member resistance equations, are summarized for the cases of axial compression only, flexure only, and combined flexure and axial compression. The corresponding equations are presented in the context of AISC Load and Resistance Factor Design (LRFD). These SRFs are applied cross section by cross section within a general-purpose frame finite element based on thin-walled open-section beam theory. The frame element has seven dofs per node – three translations, three rotations and one warping dof – and is formulated to address the influence of nonprismatic geometry (Jeong & White 2015). The reader is referred to White et al. (2016), Toğay & White (2019) and White & Jeong (2019) for further calculation details. 2.1 Stiffness reduction factor for axial compression only The stiffness reduction factor implicit within the AISC (2016) Chapter E axial compression strength curve may be written as SRF ¼ 0:877ϕc τa Ae =Ag

ð1Þ

where ϕc is the resistance factor for axial compression, taken as 0.9 in AISC LRFD,

τa ¼ 2:724

  Pu Pu ln ϕc Pye ϕc Pye

for

Pu > 0:390 ϕc Pye

ð2Þ

and τa ¼ 1:0

otherwise

ð3Þ

In these equations, Pu is a multiple of the member required LRFD axial resistance Pu, ϕc Pye is the factored yield strength of the effective cross-section under axial compression, Ae is the effective cross-section area based on the internal axial force ΓPu, and Ag is the cross-section gross area. As shown by White et al. (2016) and White & Jeong (2019), when the SRF given by Equation 1 is applied to the section rigidities and a buckling solution is obtained at a multiple of the applied load Γ, ΓPu for a member subjected to pure axial compression is in effect a rigorous calculation of the AISC factored design capacity ϕcPn. These solutions include basic prismatic simply supported columns, where ΓPu = ϕcPn reproduces the exact result from the column resistance equations with K = 1. In addition, they include more sophisticated solutions involving general end restraint conditions, continuity with less-critical adjacent unbraced lengths, variations in internal axial force along the member length, and any type of lateral and/or torsional bracing such as lateral bracing offset from the centroidal or shear center axis.

43

2.2 Stiffness reduction factor for flexure only The stiffness reduction factor implicit within the AISC Chapter F I-section flexural resistance equations may be written as SRF ¼ ϕb Rpg τltb

ð4Þ

where ϕb is the resistance factor for flexure (0.9 in AISC LRFD), Rpg is the bend buckling factor for slender-web members, equal to 1.0 if the web is compact or noncompact, and τltb is the base lateral-torsional buckling (LTB) stiffness reduction factor. The factor τltb may be expressed as τltb

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u Y 4X 2 F u i when m4 L ¼ th  2 F yc 6:76X 2 Fyc =E m2 þ 2Y 2

ð5Þ

for compact- and noncompact-web I-sections, where m = ΓMu/Myc, and where 2  Y ¼ m4

1  Rmpc



1  RpcFLFyc



Lr Lp  rt rt



3    Lp 5 Fyc 1 þ rt E 1:95

ð6Þ

and X 2 ¼ Sxc ho =J

ð7Þ

In the above equations, FL is the stress beyond which the inelastic LTB limit state applies under uniform bending, and Fyc is the yield strength of the flange in flexural compression. In addition, ΓMu is a given multiple of the required LRFD moment Mu, and Myc is the yield moment to the compression flange. The following terms are as defined in the AISC Specification: J = the crosssection St. Venant torsion constant; Lr = the limiting unbraced length for inelastic LTB under uniform bending; Lp = the limiting unbraced length corresponding to the LTB “plateau” strength (i.e., the compression flange yielding (CFY) strength) under uniform bending; Rpc = the web plastification factor; Sxc = the elastic section modulus to the compression flange; ho = the distance between the centroids of the I-section flanges; and rt = the effective radius of gyration for LTB. For slender-web I-sections, the following simpler form applies for the base LTB factor: 2

τltb

32 rffiffiffiffiffiffiffi  1 R  m h Fyc c Rpg FL Rpg m 4 c5  ¼ when m4 þ  FL F Fyc Rpg π π L Rh  Fyc

ð8Þ

where c is the coefficient in the equation for Lp, equal 1.1 for general welded I-section members in the current AISC Specification. For all I-section members, when m ≤ RpgFL/Fyc, the base LTB stiffness reduction factor is τltb ¼ 1:0

ð9Þ

As demonstrated by White et al. (2016) and White & Jeong (2019), when the SRF given by Equation 4 is applied to the section rigidities EIy, ECw and GJ and a buckling solution is obtained at a multiple of the applied load Γ, ΓMu for a member subjected only to bending is in effect a rigorous calculation of the AISC ϕcMn for LTB. This includes basic prismatic simply-supported beams subjected to uniform bending, where the buckling solution reproduces the exact result from the AISC LTB resistance equations for Cb = 1. In addition, it includes more advanced solutions involving general nonprismatic geometry, complex end restraint conditions, continuity with less44

critical adjacent unbraced lengths, general variations in internal moment along the member length, transversely applied loads at a specified height within the cross-section, and any type or combination of lateral and/or torsional bracing. In Equations 5, 6 and 8, the term m may be expressed as M M Mmax ¼ Rpg Rpc ϕb Myc ϕb Mmax Mmax:CFY

ð10Þ

Mmax ¼ min ðMn:CFY ; Mn:FLB ; Mn:TFY Þ

ð11Þ

m¼ where

is the maximum possible cross-section resistance based on the separate limit states of compression-flange yielding (CFY), flange local buckling (FLB), or tension flange yielding (TFY). The SABRE2 software (White et al. 2019) implements the cross section based CFY, FLB and TFY yielding checks from the Specification, in addition to the fundamental LTB resistance checks, which are captured via inelastic buckling analysis. If the maximum cross-section resistance ϕbMmax is reached at any location prior to the onset of LTB, the available member resistance is limited by this cross-section resistance. With the exception of checking ϕbMmax, the INBA algorithm for beams is essentially the same as that for the calculation of the column buckling load in Section 2.1. 2.3 Stiffness reduction factors for combined axial tension or compression and bending As discussed in the introduction, INBA methods can be applied to assess the strength of any type of I-section member subjected to in-plane bending and axial load, accounting for member overall stability limit states and their potential interaction with cross-section based limit states. The INBA procedures accomplish this in a more rigorous manner than can be achieved by routine application of Specification resistance equations. For members subjected to combined axial loading and flexure, this is achieved by a straightforward interpolation between the SRFs for axial loading discussed in Section 2.1 and the SRFs associated the AISC flexural resistance equations in Section 2.2. The equation for the interpolated beam-column inelastic stiffness reduction factor is  SRF ¼

   ζ Ae ζ þ 1  o ϕb Rpg τltb 0:877ϕc τa Ag 90o 90

ð12Þ

where 

Pu =ϕc Pye ζ ¼ atan Mu =ϕb Mmax

 ð13Þ

Furthermore, in the equations for τa and τltb, the unity check (UC) value from the following cross-sectional strength interaction equations is substituted for ΓPu/ϕcPye and ΓMu/ϕbMmax: – For cross sections in which all the plates are nonslender under axial compression and compact under flexural compression: UC ¼

Pu 8 Mu Pu þ for  0:2 ϕc Pye 9 ϕb Mmax ϕc Pye

ð14Þ

UC ¼

Pu Mu þ 2ϕc Pye ϕb Mmax

ð15Þ

45

otherwise

– For cross sections with slender plates under axial compression, and/or with noncompact or slender plates under flexural compression: UC ¼

Pu Mu þ ϕc Pye ϕb Mmax

ð16Þ

The net stiffness reduction factor from Equation 12 is applied to the rigidities EIy, ECw and GJ on a cross_section-by-cross_section basis. Toğay & White (2019) demonstrate the accuracy of the above interpolation for a comprehensive suite of prismatic I-section members. Further details regarding the corresponding INBA calculations are discussed in (White et al. 2016). 2.4 Rationale for the specific recommended INBA approach There are numerous ways to characterize the stiffness of steel structures for inelastic nonlinear buckling analysis (INBA). These range from refined plastic zone analysis, in which the detailed spread of plasticity is tracked through the member cross sections and along their lengths as the loads are increased, including consideration of residual stress and geometric imperfection effects, to other phenomenological approximations comparable to the SRFs discussed above. The INBA calculations using the above SRFs provide results that are fully consistent with the application of the AISC Direct Analysis Method and the Specification member resistance equations for basic prismatic members, and they extend the application of the AISC provisions to general member geometries, loadings, end restraints, and bracing conditions. Solutions employing refined plastic zone analysis arguably have the greatest level of rigor due to their ability to directly capture the influence of any specified member cross-section geometry, residual stresses and geometric imperfections. However, appropriate nominal residual stresses and geometric imperfections must be specified, and the results from plastic zone analysis never match precisely with predictions from the Specification equations in cases where a close or exact match might be expected or desired. The AISC Specification equations are a “codified” fit to member strengths considering these effects for a general range of steel structures. The above SRF values capture this fit for basic cases, and allow extension of the Specification rules to more general structures.

3 POTENTIAL IMPROVEMENTS IN AISC SPECIFICATION RESISTANCE EQUATIONS The AISC Specification resistance equations (AISC 2016) have many excellent qualities in terms of their ability to represent the strength limit states of steel I-section members and frames. However, potential improvements to these provisions may further enhance their ability to capture these strength limit states. These improvements are summarized below. Additional details are explained in Subramanian et al. (2018) and Toğay & White (2019). Since the SRFs within the INBA approach depend on the underlying Specification resistance equations, these potential improvements are important in demonstrating the merits of the approach. 3.1 Lateral-torsional buckling strength improvements For major-axis bending of welded I-section members, Subramanian et al. (2018) have demonstrated that the reliability index, estimated based on existing experimental data, is somewhat lower than the target value of β = 2.6. They recommend that the parameter FL, defined as the stress limit beyond which the inelastic LTB limit state applies under uniform bending, should be taken as FL ¼ 0:5Fyc

ð17Þ

for these member types. In addition, they recommend that the limiting unbraced length corresponding to the “plateau” strength under uniform bending should be taken as 46

Lp ¼ 0:8rt

pffiffiffiffiffiffiffiffiffiffiffiffiffi E=Fyc

ð18Þ

for routine design of welded I-sections utilizing the AISC resistance equations and assuming warping free and lateral-bending free conditions at the ends of the unbraced lengths, and that Lp ¼ 0:63rt

pffiffiffiffiffiffiffiffiffiffiffiffiffi E=Fyc

ð19Þ

should be employed when end restraint is considered explicitly in the LTB evaluation. Equations 17 and 18 are effectively the same as the original recommendations for slender-web plate girders by Cooper et al. (1978). It should be noted that in the recommended FLB calculations discussed below, the corresponding FL should still be taken as 0.7Fyc as in the current AISC Specification. 3.2 Web bend buckling strength improvements Subramanian et al. (2018) have also recommended that the noncompact-web limit in the AISC Specification, λrw, which establishes the transition between noncompact and slender web behavior, and influences the calculated values for the web plastification factor, Rpc, and the web bend buckling factor, Rpg, should be modified to λrw ¼ crw

pffiffiffiffiffiffiffiffiffiffiffiffiffi E=Fyc

ð20Þ

where crw = 3.1 + 5/aw, but not less than 4.6 nor larger than 5.7, aw = 2Dcy tw/bfc tfc, bfc and tfc are the width and thickness of compression flange respectively, Dcy is the depth of the web in compression at the nominal onset of compression flange yielding, and tw is the thickness of the web. Equation 20 recognizes that I-section members with relatively small compression flanges tend to exhibit a reduction in the effective noncompact web limit. 3.3 Improved characterization of compression flange local buckling resistance The AISC FLB provisions tend to underestimate I-section member flexural resistances when the compression flange becomes increasingly slender. This is because the AISC equations do not account for the reserve local post-buckling capacity. The following calculations consider an effective width of the compression flange to account for its local post-buckling strength. For sections with a slender compression flange in flexure: 1) The flange effective width is calculated directly given the flange elastic buckling stress Fel ¼

0:9Ekc ðbfc =2tfc Þ2

ð21Þ

and taking the compression flange stress within the effective width as Fyc at the flexural strength limit, where kc is the flange local buckling coefficient defined by the AISC Specification. The terms Feℓ and Fyc are substituted into Winter’s unified effective width equation  qffiffiffiffiffi qffiffiffiffiffi Fe‘ be ¼ bf 1  0:22 FFyce‘ Fyc

ð22Þ

2) The location of the effective cross section’s neutral axis at nominal initial yielding of the compression flange, measured relative to the inside of the compression flange, Dcye, and the corresponding yield moment, Myce, are determined.

47

3) The FLB resistance, considering the flange local post-buckling strength, is calculated as RpgMyce, where Rpg is less than 1.0 for slender-web sections but is equal to 1.0 for compactand noncompact-web sections. For sections having a noncompact flange in flexure: 1) The effective width reduction based on the noncompact flange slenderness limit, λrf, is applied to the compression flange, regardless of the actual flange slenderness, and the corresponding resistance Myce(λr) = RpgMyce(λr) is determined using the procedure explained above. This establishes an “anchor point” corresponding to λf = λrf. 2) A linear interpolation is then employed between (λpf, Mmax.FLB) and (λrf, Myce(λr)), where Mmax.FLB is the plateau resistance for FLB, equal to Mp for a compact-web section, RpcMyc for a noncompact-web section, and RpgMyc for a slender-web section, where Myc is the yield moment to the compression flange for the gross cross-section.

3.4 Improved handling of tension flange yielding When a singly-symmetric section with the larger flange in compression is subjected to flexure, the current AISC flexural resistance may be governed by tension flange yielding (TFY). If the section has a slender web, the TFY resistance is equal to the moment at the first nominal yielding of the tension flange, Myt. This estimate can be quite conservative. Sections with Myt < Myc, can have substantial inelastic reserve strength associated with distributed yielding in flexural tension. The conservative TFY calculation can be eliminated, and the Specification can be substantially shortened, by calculating Myc and Myce as the “true” yield moments to the compression flange, considering the early yielding in tension for these section types. It is recommended that these true yield moments to the compression flange should be used in the limit state calculations of the Specification. In addition, it is recommended that the depth of web in compression at the first nominal yielding of the compression flange, based on the gross-cross-section, Dcy, be used in calculating the slenderness of the web. That is, λw is defined as 2Dcy/tw. Figure 1 shows an example stress distribution at Myc for a homogeneous I-section of this type. For homogeneous cross-sections, relatively simple closed-form equations are available for Dcy and the “true” Myc. 3.5 Calculation of flexural resistance for members with unequal flange and web yield strengths Measured yield strengths generally can be different for both flanges and for the web in experimental tests. Measured yield strengths on thinner web material are often larger than the flange yield strengths. The measured yield strengths should be employed when comparing strength predictions to experimental test results. In addition, in bridge construction, it is common to use “hybrid” I-girders, having a lower-grade steel for the top flange and web combined with a higher-grade steel for the bottom flange. To accommodate all of these considerations, it is important to define the calculation of the flexural resistance for any combination of plate yield strengths. The recommended extensions to the AISC I-section member provisions are as follows: – The compression flange yield strength, Fyc, should be employed for Fy in the AISC provisions everywhere Fy appears either within the context of the compression flange, or within the context of assessing any aspects related to structural stability. This is an established precedent in (AASHTO 2017) and elsewhere. It should be noted that the flange in flexural compression depends on the sign of the bending moment. – The actual or specified yield strengths of the compression and tension flanges, Fyc and Fyt, and of the web, Fyw, should be employed in calculating the plastic moment, Mp, regardless of the relative magnitude of the different strengths, except Fyw should not be taken larger than 1.43 min (Fyc, Fyt). This is based on AASHTO (2017) Article 6.10.1.3 and is intended to avoid counting on web yield strengths beyond the limits that have been evaluated experimentally. – The “true” yield moments to the compression flange, Myc and Myce as applicable, should be calculated from a strain-compatibility analysis including any early yielding in the web or 48

Figure 1. Stress distribution associated with the “true” yield moment Myc and the corresponding depth of the web in compression, Dcy, for a homogeneous cross-section with Myt < Myc.

tension flange. The above 1.43 min (Fyc, Fyt) limit on the useable Fyw also should be applied in this calculation. These moments should be employed where the corresponding “true” yield moments appear within the calculations discussed above. Evaluation of the true Myc and Myce values is straightforward to program, and SABRE2 (White et al. 2019) implements this calculation. The algorithm sets the strain at the extreme fiber of the compression flange to Fyc/E and the section curvature is varied until a stress distribution is obtained for which the total cross-section axial force is zero.

4 COMPARISON OF INBA PREDICTONS TO EXPERIMENTAL RESULTS Smith et al. (2013) have conducted 10 experimental tests evaluating the LTB behavior of a range of web-tapered I-section members. The primary aim of these tests was to gain a better understanding of the cyclic LTB behavior of these types of members. However, all of the members were loaded past their flexural capacity within an initial monotonic half-cycle of the loading; therefore, these tests are also valuable for evaluating static monotonic strength predictions. Smith et al. provide an overall positive assessment of the ability of the first edition of DG25 (Kaehler et al. 2011) to predict the LTB resistance under static monotonic loading, contingent upon the consideration of end restraint effects from support conditions and less-critical adjacent unbraced lengths using elastic eigenvalue buckling calculations. The following discussions complement the assessments by Smith et al. by comparing INBA calculations based on the current AISC Specification, as well as the AISC provisions with the improvements discussed in Section 3, to the experimental results. The overall configuration of the experimental tests conducted by Smith et al. (2013) is illustrated in Figure 2. The specimens were tested in a horizontal orientation, simulating the rafter of a metal building frame, with moment applied at the north end of the specimen via an end-plate connection to a vertical loading column. The south end of the specimens was flexurally and torsionally simply supported, i.e. major- and minor-axis bending rotations, warping of the flanges, and longitudinal displacements were unrestrained, but torsional rotation and vertical and out-ofplane lateral displacements were prevented. Minor-axis bending and torsional rotations, warping of the flanges, and out-of-plane lateral displacements were effectively prevented at the north end of the specimens at the end plate connection to the loading column, and longitudinal and vertical displacements were restrained by a pin support below the knee at the bottom of the column. Flange-level out-of-plane lateral bracing was provided at different locations along the top and bottom flanges of the specimens. A typical conceptual arrangement of these lateral braces is indicated by the x symbols on the drawing. Two of the 10 tests included a constant axial load applied 49

Figure 2.

Test configuration, adapted from (Smith et al. 2013).

to the specimens. This was accomplished by tensioning of rods between the north side of the column at the knee of the frame and the south end of the specimen. Table 1 summarizes all the pertinent geometry and material attributes of the test specimens. Three groups of tests were conducted as denoted by the test names: 1) The CF tests had constant taper throughout the test length and the critical unbraced length for LTB was the first unbraced length adjacent to the column. 2) The CS tests had constant taper throughout the test length, but the critical unbraced length was the second unbraced length from the column. 3) The PF tests had a pinch point within the test length, and the critical unbraced length was the first unbraced length adjacent to the column. Test PF1 had a pinch point at the south end of its critical unbraced length while test PF2 had a pinch point at an intermediate location within its critical unbraced length. Clearly there is substantial complexity in the combined overall configuration of the member geometry and plate yield strengths, and the bracing and end restraint conditions in these tests. Table 2 summarizes the test to predicted strength ratios and the flexural failure modes identified from INBA solutions conducted using SABRE2. This table shows the analysis results using the AISC (2016) provisions as well as the AISC provisions with the potential improvements defined in Section 3. (The different plate yield strengths are included in the current AISC calculations as specified in Section 3.5, except that the AASHTO (2017) hybrid cross-section factor, Rh, is employed along with the current calculations as a commonly employed approximation, rather than calculate the “true” yield moments; the reader is referred to (AASHTO, 2017) for the specific equations.) In addition, the INBA calculations are performed in two ways: 1) The entire test specimen is modeled. This captures the influence of end restraint on the critical unbraced length from the less-critical adjacent lengths due to continuity across the braced points. In addition, the specified end conditions at the end-plate connection to the column (minor-axis bending and flange warping fixed) are modeled in these solutions. 2) Only the critical unbraced length is modeled. In this case, the common design assumption of torsionally simply-supported end conditions (minor-axis bending and flange warping free at both ends of the unbraced length) is employed in the SABRE2 solutions. This is the inherent assumption associated with the common implicit use of a LTB effective length factor K = 1, and the use of just the unbraced length Lb rather than a KLb < Lb for the critical unbraced length in design practice.

50

Table 1. Summary of specimens tested by Smith et al. (2013). Test^

bf tf (mm) (mm)

CF1

152

CF2 & CF2-A CS1 & CS1-A CS2 CS3

Flg.

Lb♣ * Web (m)

Fyf Fyw (MPa) (MPa) 431 414* 397

4.70

4.58 C♦

S♦

2.13 (1.07, 1.07), 1.22, 2.35

152

4.72

4.58 C

S

2.13 (1.07, 1.07), 1.22, 2.35

152

6.45

203

6.22

5.60 N♦

N

0.610, 3.05 (1.52, 1.52), 2.04 425

428

127 152

6.53 9.37 8.08 8.08** 7.80

305 305

4.70 4.70

4.58 N 4.58 C/

S S

0.406, 3.05 (1.52, 1.52), 2.25 481 0.914, 2.74 (1.37, 1.37), 2.04 431 481 462** 0.610, 3.05 (1.52, 1.52), 2.04 376

427 427

152

PF1## PF2♣♣

152 127

#

α* (°)

9.37 305 12.65♠ 9.40 305

CS4

^

h1# tw (mm) (mm)

9.37 7.87 6.20

N 305 356 457

4.58 N S 4.72 4.22^^ 4.70 9.46 C S 4.67 14.0 C/ S 3.81 N

2.45 (1.23, 1.23), 1.05, 2.20 2.44 (1.23, 1.21), 1.22, 2.04

431 383 468

427 496

496 460^^ 427 396 401

The nominal web depth at the end plate is the same, h2 = 762 mm, for all of the tests.

Web depth at the simple-support, at the right-hand end of the test. Taper angle ♣ The unbraced lengths for the top flange (in flexural compression) are the values not listed in parentheses. * The unbraced lengths for the bottom flange (in flexural tension), are the same as those for the top flange except for one segment where an additional intermediate brace is placed on the bottom flange. The brace spacings for the segment containing the additional bottom flange brace are listed in parentheses just after the corresponding top flange unbraced length. The corresponding top (compression) flange unbraced length is the critical one for lateral-torsional buckling of the members. ♠ CF1 is the only linearly-tapered member test that has nominally different top and bottom flange dimensions; the first and second values listed correspond to the top and bottom flanges respectively (the top flange is in flexural compression). The resulting singly-symmetric section has Myt > Myc, and therefore the nominal onset of yielding occurs first at the top (compression) flange. ♦ C indicates that the flange is compact within the critical unbraced length, N indicates that the corresponding flange or web plate is noncompact within the critical unbraced length, and S indicates that the web is slender within the critical unbraced length. ** CS3 has a flange splice in both flanges at 2.29 m from the end plate; the second and third reported values correspond to the top and bottom flange plates to the right of the flange splice. ^^ CS4 has a web splice at 0.610 m from the end plate; the second reported value corresponds to the web to the right of the web splice. ## PF1 has a linear taper from the end plate down to a pinch point at the brace location at 2.44 m from the end plate, then a constant web depth of 356 mm to the right of that location; the flange and web plates are the same on each side of this pinch point. ♣♣ PF2 has a linear taper from the end plate down to a pinch point at the brace location at 1.22 m from the end plate, then a constant web depth of 457 mm to the right of that location. The top flange plate is thicker within the tapered length of the member, resulting in Myt being (slightly) less than Myc and corresponding minor early yielding in flexural tension. Also, the web plate thickness is reduced to the right of the pinch point. The second value listed for the flange plate thickness corresponds to the flange plates other than the thicker top flange plate within the critical unbraced length, and the second value for the web plate thickness corresponds to the web plate to the right of the pinch point. *

The INBA solution considering the entire specimen and using the AISC provisions with the recommended potential improvements provides the best predictions of the test results, giving an average strength ratio of 1.07 along with the smallest coefficient of variation (COV = 0.06) and a minimum strength ratio of 0.98. In addition, this solution identifies the governing failure mode clearly as LTB in all of the tests except PF2. In PF2, FLB governs for the thinner flange just to the south of the pinch point. PF2 exhibited a local failure at the pinch point within the experimental test, due to the lack of sufficient capacity of the web to resist the concentrated 51

0.99 1.10 1.10 0.99 1.05 0.86 1.02 1.01 0.95 0.81

1.01 0.07 1.10 0.86

CF1 CF2 CF2-A* CS1 CS1-A# CS2 CS3 CS4 PF1 PF2♠

Avg. COV Max. Min.

CFY CFY CFY FLB FLB LTB FLB FLB CFY FLB

Failure Mode

1.07 0.06 1.16 0.98

1.05 1.15 1.16 1.02 1.11 1.00 1.08 1.07 0.98 0.79

Test/Pred. Strength LTB LTB LTB LTB LTB LTB LTB LTB LTB FLB

Failure Mode

1.25 0.15 1.62 1.06

1.10 1.21 1.22 1.26 1.52 1.62 1.13 1.15 1.06 0.81

Test/Pred. Strength LTB LTB LTB LTB LTB LTB LTB LTB LTB FLB

Failure Mode

1.39 0.11 1.68 1.19

1.26 1.37 1.38 1.40 1.62 1.68 1.31 1.34 1.19 0.83

Test/Pred. Strength LTB LTB LTB LTB LTB LTB LTB LTB LTB LTB

Failure Mode

633.4 639.3 611.9 424.3 429.9 333.2 536.7 413.0 477.0 187.9

Moment Capacity Mtest♣ (kN m)

CF2-A had a constant axial compression of 126 kN applied throughout the experimental testing. This load was applied at 318 mm above the bottom of the web at the left face of the column, and at 140 mm above the bottom of the web at the simply-supported end of the specimen. This corresponds approximately to the centroidal depth of the crosssection along the test length. # CS1-A had a constant axial compression of 185 kN applied throughout the experimental testing. The stated intent was to apply this load at the centroidal depth of the crosssection along the test length. However, the information provided by Smith et al. (2013) indicates that the resultant of the axial load was only at 50.8 mm above the bottom of the web at the simply-supported end of the test in CS1-A. The load was applied at 318 mm above the bottom of the web at the left face of the column. ♣ The moment capacities from the experimental tests reported here are the values at the deepest end of the critical unbraced length as identified by Smith et al. (2013) and by the SABRE2 full member solutions based on the AISC provisions with recommended improvements. The corresponding locations in the test specimens are at the deepest end of the top flange unbraced length adjacent to the column for the CF tests, and at the deepest end of the second top flange unbraced length from the column in the CS tests. In the PF tests, the “controlling cross-section” was identified by Smith et al. (2013) as the cross-section at the small end of the critical unbraced length for test PF1, and the cross-section on the thin-web side of the pinch point splice at the middle of the critical unbraced length for test PF2. In the SABRE2 solutions, these are the cross sections that have the largest internal moment relative to the cross-section flexural strength, and the corresponding smallest values of the SRF. ♠ The results for test PF2 are not included in the summary statistics since the strength in this test was governed by a local failure at the pinch point.

*

Test/Pred. Strength

AISC w/recom.

Current AISC

Current AISC

AISC w/recom.

Only critical unbraced length modeled

Entire member modeled

Test to predicted strength ratios, flexural failure modes identified by SABRE2, and moment capacities for the tests conducted by Smith et al. (2013).

Test/Sum. Stats.

Table 2.

transverse force caused by the change in angle of the top flange plates. It is expected that if a partial-depth bearing stiffener had been provided at the pinch point, the predictions would be accurate for PF2. The INBA solutions using the current AISC provisions give the best prediction of the tests on average, with a mean test/predicted strength of 1.01; however, they have a larger COV of 0.07 and they give a relatively small test/predicted strength of only 0.86 for test CS2. In addition, the INBA solution using the current AISC provisions predicts LTB as the governing failure mode only for test CS2. Based on the experimental results, LTB was clearly the dominant failure mode for all the tests, with the exception of PF2, as discussed above. The solutions based on both the current AISC provisions as well as the AISC provisions with the recommended potential improvements exhibit significant conservatism when applied only to the critical unbraced lengths, assuming torsionally simply-supported boundary conditions at the ends of these lengths. The current AISC provisions actually give the more accurate predictions in these cases, due to their tendency to predict larger LTB strengths in general for these tests. Figure 3 shows the buckling modes obtained from SABRE2 for tests CS2 and PF1, using the AISC provisions with the potential improvements. The darker arrows in the figure indicate the constraints from the end conditions and the intermediate lateral bracing. The light shaded circular arrow at the left-hand (north) end of the models shows the applied moment from the loading column. One can observe the influence of the warping and minor-axis bending restraint at the lefthand ends, as well as the effect of the close spacing of the first set of intermediate braces from the left-hand end in CS2 (preventing out-of-plane displacement and twist) on the buckled shape. Figure 4 plots the SRF and cross-section unity check (UC) values along the normalized specimen lengths for tests CS2 and PF1. The nonlinear variation of the UC for CS2, corresponding to the noncritical FLB limit state in this test, is due to the linear taper of the web depth. The smallest SRF values for CS2 are at the left-hand end where the cross-section is deepest. However, these SRF values occur within the short 0.406 m unbraced length adjacent to the loading column. The CS2 specimen is critical for LTB within the second unbraced length from the column. Due to sharper web taper between the left-hand support and the pinch point at the first brace from the column on the top flange, the SRF values for PF1 reduce to a minimum at the pinch point. The UC is equal to 1.0 at the pinch point in PF1, indicating that the CFY limit state (i.e. the plateau strength for LTB) is close to being reached at this location. The maximum UC and minimum SRF values tend to occur at the “governing cross-sections” in DG25 (White and Jeong 2019) strength checks based on more routine elastic LTB solutions (Slein and White 2019). However, the INBA solutions of the entire test members account for the “true” restraint from adjacent lesscritical unbraced lengths with better accuracy than can be achieved using elastic LTB solutions. The predicted capacities for the experimental tests discussed in this paper are affected only a minor extent by the improved handling of the FLB and TFY limit states discussed in Sections 3.3 and 3.4. Toğay and White (2019) show test simulation solutions that highlight the benefit of these improvements. Furthermore, there is evidence from these and other experimental tests that the LTB resistance of I-section members fabricated with minimal single-side welding of the flanges to the webs may be somewhat larger than characterized by the more generally applicable

Figure 3. Lateral-torsional buckling modes for tests CS2 and PF1 obtained from INBA solution based on the AISC Specification with the potential improvements discussed in Section 3.

53

Figure 4. Cross-section unity check (UC) and stiffness reduction factor (SRF) values versus the normalized position along the length at the LTB strength limit for tests CS2 and PF1, AISC Specification based INBA solution including potential improvements discussed in Section 3, ϕc and ϕb taken equal to 1.0.

recommendations by Subramanian et al. (2018) listed in Section 3.1. Additional research is needed to further evaluate rules for characterization of the LTB resistance of welded I-section members.

5 CONCLUSIONS Overall member stability can be influenced significantly by restraint from adjacent less-critical unbraced lengths as well as general member end restraint, bracing, and loading conditions that are difficult to capture within Specification resistance equations themselves. By implementing the Specification resistance equations as corresponding stiffness reduction factors within a generalpurpose buckling analysis based on thin-walled open-section beam theory, the impact of the above attributes can be captured by direct modeling of the actual structural conditions. REFERENCES AISC 2016. Specification for structural steel buildings, ANSI/AISC 360-16, Chicago: American Institute of Steel Construction. AASHTO 2017. AASHTO LRFD bridge design specifications, 8th Ed., Washington: American Association of State Highway and Transportation Officials. Cooper, P.B., Galambos, T.V. & Ravindra, M.K. 1978. “LRFD criteria for plate girders,” Journal of the Structural Division, ASCE, 104(ST9): 1389–1407. Jeong, W.Y. & White, D.W. 2015. “General nonprismatic frame finite element based on thin-walled open-section beam theory,” Research Report, School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA. Kaehler, R.C., White, D.W. and Kim, Y.D. 2011. Frame design using web-tapered members, Steel design guide 25, 1st Ed., Chicago: American Institute of Steel Construction. Slein, R. and White, D.W. 2019. “Streamlined design of nonprismatic I-section members,” Proceedings, Annual Stability Conference, Structural Stability Research Council, St. Louis, MO. Smith, M. D., Turner, A.K., & Uang, C.M. 2013. “Experimental study of cyclic lateral-torsional buckling of web-tapered I-beams”. Final Report to Metal Building Manufacturers Association, Department of Structural Engineering University of California, San Diego, La Jolla, CA. Subramanian, L., Jeong, W.Y., Yellepeddi, R., & White, D.W. 2018. “Assessment of I-section member LTB resistances considering experimental test data and practical inelastic buckling design calculations,” Engineering Journal, AISC, 55(1): 15–44. Toğay, O. & White, D.W. 2019. “Advanced design evaluation of planar steel frames composed of general nonprismatic I-section members”. Research report, Georgia Institute of Technology, Atlanta, GA. White, D.W., Jeong, W.Y. & Toğay, O. 2016. “Comprehensive stability design of planar steel members and framing systems via inelastic buckling analysis,” International Journal of Steel Structures, 16(4): 1029–1042. White D.W & Jeong, W.Y. 2019. Frame design using nonprismatic members, Steel design guide 25, 2nd Ed., Chicago: American Institute of Steel Construction (to appear). White, D.W., Toğay, O., Slein, R. & Jeong, W.Y. 2019. “SABRE2-V2”. .

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Design by advanced analysis–2016 AISC specification R.D. Ziemian Bucknell University, Lewisburg, Pennsylvania, USA

Y. Wang Cornell University, Ithaca, New York, USA

ABSTRACT: At the heart of the provisions for assessing structural stability within the American Institute of Steel Construction’s Specification for Structural Steel Buildings is the direct analysis method. The fundamental concept is that the more behavior that is explicitly modeled within the analysis, the simpler it is to define the Specification’s design requirements. In other words, the direct analysis method consists of calculating strength demands and available strengths according to a range of well-defined and fairly detailed analysis requirements. This paper begins with an overview of two logical extensions to AISC’s direct analysis method, both of which are now provided in the Specification’s Appendix 1 – Design by Advanced Analysis. In establishing these approaches, many systems were investigated and it was found that systems with beam-columns subject to minor-axis bending appeared to deserve additional attention. This paper presents a detailed study that investigates such members.

1 INTRODUCTION For the past sixty years, the Effective Length Method (ELM) has been a widely employed stability design method (Ziemian, 2010). By scaling actual unbraced lengths to effective lengths when calculating the available strengths of compression members, the effective length K-factor is assumed to account for most factors known to impact the stability of structural systems, including geometric system imperfections, stiffness reduction due to inelasticity, and to a much lesser degree uncertainty in strength and stiffness (AISC, 2016a). In 2005, design by the Direct Analysis Method (DM) first appeared in the American Institute of Steel Construction’s (AISC’s) Specification for Structural Steel Buildings. In DM, the available strengths of compression members are based simply on the unbraced length (K = 1), as long as system imperfections (but not member imperfections) and stiffness reduction due to inelasticity are represented in the structural analysis. Since then, many in the structural design profession have moved from employing ELM to DM. As a result, DM was relocated in AISC’s 2010 Specification from Appendix 7 to Chapter C, while ELM was relocated from Chapter C to Appendix 7. Both design methods rely on establishing the unbraced lengths of compression members, which in some cases may be difficult, if not impossible, to define. Examples include, but are not limited to arches, tree columns, and Vierendeel trusses. In response to this predicament, AISC introduced a Design by Advanced Elastic Analysis Method that appears in Appendix 1 of their 2016 Specification. In addition to the analysis modeling requirements of DM, the method further requires the direct modeling of member imperfections and, therefore the method is often represented by the acronym DMMI. In applying this approach, engineers can avoid the complexities of defining unbraced lengths, thereby being permitted to compute the nominal strengths of compression members as their axial cross-sectional strengths. This paper reports on an ongoing study to complement previous studies on systems (Nwe Nwe, 2014, Giesen-Loo, 2016) to evaluate the performance of DMMI, especially with an eye towards members that are subject to the combination of compression and minor-axis bending. Using AISC’s Design by Advanced Inelastic Analysis 55

Method, which is based on employing a rigorous geometric and material nonlinear analysis with imperfections (GMNIA), the accuracy of DMMI is assessed and further compared with the more traditional ELM and DM design methods. Additionally, the significances of thermal residual stresses, which are a consequence of uneven cooling of rolled cross-sections, and the axis of bending (minor versus major) are also explored. The paper begins by providing an overview of AISC’s ELM, DM, DMMI, and GMNIA methods along with details of the analysis method and interaction equation employed in each. Results of the study are then presented primarily in tabular format, which are followed by discussions of the effects residual stresses, axis of bending, and design method employed.

2 OVERVIEW OF DESIGN METHODS In this study, the ends of simply supported columns of various slenderness ratios are subjected to a wide range of combinations of applied axial force and end bending moments that are of equal magnitude and opposite direction (in the absence of axial force such moments would produce a uniform moment distribution). In all cases, the members are assumed to be fully braced out-of-the-plane of bending. To assess the LRFD strength of beam-columns based on an elastic analysis, the following interaction equation is provided in AISC’s Specification:   Muy Pu 8 Mux þ þ  1:0 Pn 9 Mnx Mny   Muy Pu Mux þ þ  1:0 2Pn Mnx Mny

for

Pu =Pn  0:2

ð1Þ

for

Pu =Pn 50:2

ð2Þ

where  ¼ 0:9, Pu is the required axial strength, Mu is the required bending strength, Pn is the nominal axial strength, and Mn is the nominal bending moment about either the major x- or minor y-axis. The analysis for the required axial strength Pu and flexural strength Mu should include second-order (geometric nonlinear) effects. The following design methods, including ELM, DM, and DMMI, are represented by Equation 1 with terms defined per that specific method. In all cases, the controlling combinations of axial force and bending moment are determined for each of these elastic design methods by iteratively solving for the maximum value of Mu for a given value of Pu that will satisfy Equation 1. For reference, Figure 1 shows the deflected shape of the beam-column. Equilibrium on the deformed shape is given by: Mu ðx; PÞ þ P  vðxÞ þ M ¼ 0

ð3Þ

where vðxÞ is the total lateral deflection as a function   of span length location x, and equals the sum of a geometric imperfection v0 ðxÞ ¼ δ0  sin πx L , and deflection vPM ðxÞ due to the applied combination of P and M. 2.1 Effective Length Method (ELM) In computing the nominal axial strength Pn from AISC’s column curve, the effective length factor of a simply supported beam-column is K = 1. In determining the required flexural strength Mu , equilibrium equations are defined on the deflected shape to account for secondorder effects. For a structural analysis associated with ELM, the beam-column is assumed geometrically straight, v0 ðxÞ ¼ 0, prior to any applied loading (AISC’s column curve accounts for member out-of-straightness). As a result, the P  δ effect in this method need only account for the interaction between the applied axial load and bending moments, and thereby is not influenced by the presence of an initial member imperfection.

56

Figure 1. Deflected shape of beam-column with second-order effects due to applied loading and geometric imperfection.

In establishing the design adequacy of this member, the required moment Mu ðx; PÞ is a maximum at mid-span because vPM ðxÞ takes on a maximum value when x ¼ L=2. Thus, the interaction equation only needs to be checked at mid-span, where the required strengths (terms in the numerators) are at a maximum. For an elastic analysis of a simply supported, originally perfectly straight, and prismatic member, the deflection and required flexural strength Mu at mid-span, which includes second-order effects, can be calculated as a function of the applied force P and moment M by the following “exact” equation (McGuire et al., 2000): jMu

mid j

¼

1  pffiffiffiffiffiffiffiffiffiffiffi M π cos 2 P=Pe

where Pe ¼ Euler buckling strength of the beam-column. With Pu ¼ P at mid-span, substitution of these terms for Pu and Mu ¼ jMu results in an interaction equation for ELM defined by:  p1 ffiffiffiffiffiffiffiffi M P 8 cos π2 P=Pe þ   1:0 fMn fPn 9

ð4Þ

mid j

in Equation 1

ð5Þ

In which specific to ELM, Pn ¼ Fcr Ag , Pe ¼ π2 EI=L2 , Mn ¼ Fy Z; where, Fcr is the critical buckling stress as defined by AISC’s column curve with K = 1 for the simply supported end conditions being investigated in this study, Ag is the gross area of the cross-section, E is the elastic modulus of the material, I is the moment of inertia, L is the unsupported length of the beam-column, Fy is the material yield stress, and Z is the plastic section modulus. In computing Mn , it is important to note that only members with compact sections are investigated, and any members subject to major-axis bending are assumed fully braced out-of-plane. 2.2 Direct Analysis Method (DM) Although DM permits the use of the unbraced length (K = 1), this provides no advantage over ELM for the specific end support conditions of the single beam-column investigated in this study. In fact, DM is somewhat penalized in this case by its required use of a stiffness reduction factor within the structural analysis. Although the equilibrium analysis is of the same form as that given for ELM, the Euler buckling strength Pe used in the analysis of the member is modified to represent the inelastic buckling strength of the member. As a result, 57

interaction equation Equation 1 for DM can be written as Equation 5, except Pe is defined as the inelastic buckling strength. Hence, Pn ¼ Fcr Ag , with Fcr defined by AISC’s column curve with no 0:8τb stiffness reduction on E, Pe ¼ π2 ð0:8τb E ÞI=L2 , and Mn ¼ Fy Z. According to the AISC Specification 

 (2016a)  and given that all sections are compact, τb is calculated as τb ¼ 4 P=Py 1  P=Py for P=Py 40:5, and τb ¼ 1:0 for P=Py  0:5 where Py ¼ Fy Ag . 2.3 Design by Advanced Elastic Analysis Method (DMMI) As described earlier, DMMI is an alternative design method that may be particularly useful for more complex structures in which the unbraced length is not discernable. By directly modeling member out-of-straightness and representing potential inelasticity through the use of the stiffness reduction strategy employed in DM, the nominal axial strength Pn of the member may be taken as its cross-section strength. The artificial and dramatic increase in axial strength Pn that appears in the interaction equation is compensated for by a larger required flexural strength Mu , which is obtained from an advanced elastic structural analysis that accounts for initial system and member imperfections, second-order (geometric nonlinear) effects, and stiffness reduction due to inelasticity. In contrast to the above analysis for determining strengths for ELM and DM, the analysis for DMMI must also include the direct modeling of member out-of-straightness. In this study, the shape of the initial imperfection is assumed a sine wave with an amplitude at mid-span of δ0 ¼ L=1000 per the AISC’s Code of Standard Practice for Steel Buildings and Bridges (AISC, 2016b). As such, the second-order P  δ effect needs to include both the impact of the applied axial force and bending moment as well as the initial imperfection. The solution to the governing differential equation (Eq. 3) at mid-span is given by: qffiffiffiffi qffiffiffiffi   P π P   sin 2sin Pe 2 Pe π L δ0 M  q ffiffiffiffi v þ ¼  2 P 1  PPe P π sin

ð6Þ

Pe

  With v L2 , equilibrium on the deformed shape at mid-span will result in a required moment strength of:   L jMu mid j ¼ M þ P  v 2

ð7Þ

Similar to DM, a stiffness reduction factor of 0.8τb should be applied to all the members of the system, which in this study means that all EI terms (within Pe) in the above equations should be 0.8τb EI. With values of Pu ¼ P and Mu ¼ jMu mid j as defined above, the interaction equation (Eq. 1) is expressed for DMMI as:   P 8 M þ P  v L2 þ   1:0 ð8Þ fMn fPn 9   In which specific to DMMI, v L2 is given by Equation 6, with δ0 ¼ L=1000, and Pe and τb as defined for DM, Pn ¼ Fy Ag , and Mn ¼ Fy Z. 2.4 Design by Advanced Inelastic Analysis Method (GMNIA) Since 2010, the Design by Advanced Inelastic Analysis Method has been provided in Appendix 1 of the AISC Specification. Given that this design method is based on a geometric and materially nonlinear analysis, it will be referenced by the acronym GMNIA. The second-order inelastic analysis routines used in this study are included in the finite element analysis software FE++ (Alemdar, 2001), in which a distributed plasticity model is employed. Each beam-column is modeled by 58

eight line elements, thereby permitting a sine wave member out-of-straightness of δ0 ¼ L=1000 to be directly modeled in the analysis. Residual stresses are represented by pre-stressing (compression or tension) the cross-section fibers that define the cross-section. The applied axial force P and bending moments M are applied simultaneously, and an incremental-iterative arc length solution scheme is employed until a limit point is achieved. Because of the relatively high accuracy of this analysis, the below error analyses of the above elastic design methods is based on the combinations of P and M that this inelastic design method would permit and still satisfy the provisions of Appendix 1 of AISC’s Specification. It is well known that partial yielding of the cross-section can have a significant effect on the stability of beam-columns. In cases where member out-of-straightness is not removed by processes, such as rotary straightening, this partial yielding can be accentuated by the presence of residual stresses. On the other hand, the use of such straightening processes can be shown to alleviate or even eliminate the presence of residual stresses (Ge and Yura, 2019). As result, ultimate strength combinations were determined for cases in which residual stresses are and are not included in the analysis. When residual stresses are taken into account, the Galambos and Ketter (1959) residual stress distribution was employed with a maximum compressive stress at the flange tips of Fy 0:3Fy . Additionally, the material elastic modulus E and yield stress Fy Fy are reduced by a factor of 0.9, per the requirements of Appendix 1 of the AISC Specification. An elastic-perfectly plastic material model is employed.

3 NORMALIZED P-M INTERACTION CURVES AND ERROR CALCULATIONS To compare the accuracy of each of the design methods, with special attention on DMMI, normalized P-M interaction curves of ELM, DM, DMMI, and GMNIA are first plotted. Data points are obtained by determining the maximum combination of axial load P and bending moments M that can be applied at the member ends such that the strength requirements of the design method would just be satisfied. Calculation of error values in the curves are then computed using the GMNIA curve as a basis. To further allow the errors to be comparable for the wide range of member slenderness ratios investigated, all axial forces and moments were normalized by the maximum GMNIA values, with PGMNIA being the maximum axial strength when the applied moment is M = 0, and with M GMNIA being the maximum moment strength when the applied axial force is P = 0 (which would equal 0.9FyZ for all members in this study). As an example, Figure 2 shows the normalized P-M interaction curves and a plot of the radial errors for a W12X120 member with an L/r = 90 that is subjected to minor-axis bending and with residual stresses included.  Using radial lines at 10 increments measured clockwise from the normalized P-axis to the M-axis, the intersections of the radial lines and the P-M curves are determined. It is noted that values at intersection points that lay between computed data points are obtained from a parabolic interpolation between the adjacent three data points. The percent errors of the design methods are then established by comparing their radial R-distances from the origin to the interaction curves according to: percent radial error ¼

RXXX  RGMNIA  100% RGMNIA

ð9Þ

where RXXX is the radial distance of the P-M curves for the elastic design methods (ELM, DM, and DMMI), and RGMNIA is the radial distance to the GMNIA P-M curve. As a result, error plots (Figure 2b) at different radial angles represent a comprehensive range of different combinations of applied axial force and moment. Points with positive percent errors are indicative of situations in which the elastic design method (ELM, DM, or DMMI) are unconservative when compared to design strengths determined by GMNIA. The legend within the rightward radial error graph (Figure 2b) contains information important to this study. Working from the top downward, rows within this legend represent results 59

Figure 2. For a W12X120 member with an L/r = 90 subject to minor-axis bending and with residual stresses included, (a) normalized P-M interaction curves of the four design methods, and (b) plots of percent radial errors. Table 1. W-shapes studied. W14

W12

W10 W8

W14X730 W14X398 W14X211 W14X109 W12X336 W12X170 W12X79 W10X112 W10X49 W8X67

W14X665 W14X370 W14X193 W14X82 W12X305 W12X152 W12X72 W10X100 W10X45 W8X58

W14X605 W14X342 W14X176 W14X74 W12X279 W12X136 W12X58 W10X88 W10X39 W8X48

W14X550 W14X311 W14X159 W14X68 W12X252 W12X120 W12X53 W10X77 W10X33 W8X40

W14X500 W14X282 W14X145 W14X61 W12X230 W12X106 W12X50 W10X68

W14X455 W14X257 W14X132 W14X53 W12X210 W12X96 W12X45 W10X60

W14X426 W14X233 W14X120 W14X48 W12X190 W12X87 W12X40 W10X54

W8X35

for the ELM, DM, and DMMI methods, respectively. The first two numbers in each row represent the error of each design method with an angle (θÞ that corresponds to where the DMMI error is at its maximum. The second two numbers correspond to the maximum error of each design method and the angle (θÞ where this maximum occurs. Points with positive percent errors are indicative of situations in which the elastic design method (ELM, DM, or DMMI) are unconservative when compared to design strengths determined by GMNIA.

4 CROSS SECTIONS INVESTIGATED As indicated in Table 1, this study investigated 65 wide flange W-shapes of A992 steel (E = 200 GPa and Fy = 345 MPa). These shapes are all of the compact sections that appear in the column design portion of the AISC Manual (AISC, 2016c), and their depth to width ratios are all less than 1.5. 5 RESULTS Interaction curves and plots of percent radial errors that correspond to the four different design methods (ELM, DM, DMMI, and GMNIA) were prepared (see for example, Figure 2 60

and Appendices 1 and 2) for all 65 W-shapes pffiffiffiffiffiffiffiffiffi over a range of member slenderness L/r ratios of 30, 60, 90, 120, and 150, with r ¼ I=A. With four cases, including minor- or major-axis bending and with or without residual stresses, this study evaluates 1,300 conditions, which are represented by a total of 57,200 analysis data points. A summary of the results for all members is provided in Table 2, in which the maximum, average, and median of all of the individual member maximum percent radial errors are reported. In general, the percent radial errors reported for the three design methods are fairly similar. The largest percent radial errors are always for the DMMI method, and the smallest percent radial errors are for the DM method. Given that the ELM and DM methods are essentially the same, except that DM requires the analysis to include the stiffness reduction 0:8τb , it is expected (and confirmed in Table 2) that DM will be more conservative (smaller radial errors) than ELM for all slenderness ratios. It is further noted that larger unconservative errors for DMMI for sections with residual stresses consistently occur when the applied loading combination is predominately axial force (θ = 10°), where in contrast the larger unconservative errors for ELM and DM occur when the loading is primarily bending (θ = 80°).

6 DISCUSSION As would be expected, not including a residual stress distribution increases the design capacities of the beam-columns per the GMNIA design method. As a consequence, and given that the GMNIA results form the basis for the error analysis, the unconservative percent radial errors for all three of the elastic design methods (ELM, DM, and DMMI) are significantly reduced. A representative example of this is shown in Figure 3, where the performance of the DMMI design method is significantly improved with much better agreement, smaller radial errors, with GMNIA. This increase in accuracy, however, is relatively pronounced where θ is small, when the axial load is more significant than the bending moment, and is less obvious when θ is large, a combination of a larger bending moment and a smaller axial force. Of course, this is expected because it is well known that such residual stresses rarely impact the strength of a member primarily subjected to a loading combination that is predominately bending (again noting that all members in this study are either subject to minor-axis bending or laterally braced when subject to major-axis flexure). The trend observed in Figure 3 is consistent for all shapes and design methods investigated in this study, regardless of the slenderness ratio or the axis of bending investigated. In general, the reduction in DMMI errors for sections without residual stresses is largest when the slenderness ratio is L/r = 60 for minor-axis bending and L/r = 90 for major-axis bending. The change in error is the smallest at the extreme slenderness ratios investigated, including the least-slender (L/r = 30) and most-slender (L/r = 150) members. It is further noted that the ELM and DM design methods are significantly more conservative when residual stresses are not present. With the exception of more-stocky members (L/r = 30), the percent radial errors for all three design methods, especially DMMI, are reduced when members are subject to major-axis bending instead of minor-axis bending. As further shown in Table 2, all three elastic design methods will produce unconservative errors when compared with GMNIA-based design. For the reasons given earlier, DM will always provide smaller percent radial errors when compared with ELM. It is important to note that this applies only for the simply-supported member explored in this study – for systems comprised of members with effective length K-factors exceeding 1.0, this will not necessarily be the case (Martinez-Garcia, 2006). The results for DMMI and ELM are not significantly different, with the largest differences occurring for members subject to minor-axis bending in the low- to mid-slenderness (L/r = 60 to 90). Knowing that ELM has been a well-established design method that has performed well in the U.S. since the early 1960’s, it is the authors’ opinion that the unconservative errors 61

reported in Table 2 for all three elastic design methods may not be reason for significant concern.

Table 2. Summary of percent radial errors for minor- and major-axis bending with and without residual stresses included in the GMNIA design. Minor-axis with residual stresses L/r = 30

L/r = 60

L/r = 90

L/r = 120

L/r = 150

DMMI Max = 3.0% Ave = 2.2% Median = 2.2% ELM Max = 3.2% Ave = 2.1% Median = 2.0% DM Max = 2.6% Ave = 1.5% Median = 1.5% DMMI Max = 14.8% Ave = 13.7% Median = 13.9% ELM Max = 9.7% Ave = 8.4% Median = 8.4% DM Max = 8.2% Ave = 7.3% Median = 7.3% DMMI Max = 15.8% Ave = 14.8% Median = 14.8% ELM Max = 13.0% Ave = 11.1% Median = 11.1% DM Max = 11.2% Ave = 9.6% Median = 9.6% DMMI Max = 15.3 % Ave = 14.2% Median = 14.1% ELM Max = 12.7% Ave = 11.3% Median = 11.3% DM* Max = 9.5% Ave = 8.0% Median = 8.0% DMMI Max = 14.0% Ave = 12.6% Median = 12.6% ELM Max = 12.4% Ave = 10.9% Median = 10.9% DM* Max = 9.0% Ave = 7.6% Median = 7.6%

Minor-axis without residual stresses

Major-axis with residual stresses

Major-axis without residual stresses

Max = 1.8% Ave = 0.5% Median = 0.4% Max = 2.5% Ave = 1.1% Median = 1.1% Max = 1.9% Ave = 0.6% Median = 0.5% Max = 7.3% Ave = 6.1% Median = 6.1% Max = 8.8% Ave = 7.5% Median = 7.6% Max = 6.7% Ave = 5.5% Median = 5.5% Max = 9.7% Ave = 8.2% Median = 8.2% Max = 11.3% Ave = 9.8% Median = 9.8% Max = 8.2% Ave = 6.7% Median = 6.7% Max = 11.0% Ave = 9.6% Median = 9.6% Max = 11.4% Ave = 9.9% Median = 9.9% Max = 8.1% Ave = 6.6% Median = 6.6% Max = 11.8% Ave = 10.4% Median = 10.4% Max = 11.2% Ave = 9.8% Median = 9.8% Max = 7.8% Ave = 6.4% Median = 6.4%

Max = 7.0% Ave = 6.5% Median = 6.6% Max = 6.9% Ave = 6.1% Median = 6.2% Max = 6.0% Ave = 5.1% Median = 5.2% Max = 10.5% Ave = 10.0% Median = 10.0% Max = 9.2% Ave = 8.5% Median = 8.6% Max = 6.4% Ave = 5.7% Median = 5.8% Max = 10.0% Ave = 9.2% Median = 9.2% Max = 7.6% Ave = 6.9% Median = 6.9% Max = 3.9% Ave = 3.2% Median = 3.3% Max = 7.1% Ave =6.2% Median = 6.2% Max = 5.8% Ave = 4.6% Median = 4.6% Max = 2.6% Ave = 1.5% Median = 1.4% Max = 5.6% Ave = 4.8% Median = 4.7% Max = 4.4% Ave = 3.4% Median = 3.4% Max = 1.1% Ave = 0.2% Median = 0.1%

Max = 5.9% Ave = 5.0% Median = 5.0% Max = 5.8% Ave = 4.7% Median = 4.6% Max = 4.9% Ave = 3.8% Median = 3.6% Max = 7.5% Ave = 6.7% Median = 6.6% Max = 6.1% Ave = 5.2% Median = 5.3% Max = 3.6% Ave = 2.9% Median = 3.0% Max = 5.4% Ave = 4.7% Median = 4.7% Max = 4.5% Ave = 3.5% Median = 3.5% Max = 1.5% Ave = 0.6% Median = 0.6% Max = 2.9% Ave = 2.2% Median = 2.2% Max = 2.7% Ave = 1.8% Median = 1.8% Max = n/a Ave = n/a Median = n/a Max = 2.1% Ave = 1.2% Median = 1.2% Max = 1.4% Ave = 0.5% Median = 0.5% Max = n/a Ave = n/a Median = n/a

DM* no unconservative errors are observed as indicated by n/a

62

Figure 3. Percent radial errors for a member of an L/r = 60 subjected to minor-axis bending comprised of a W12X120 section that (a) includes and (b) excludes residual stresses in the GMNIA-based design.

7 SUMMARY AND CONCLUSIONS This study evaluates three elastic design methods (ELM, DM, and DMMI) appearing in the 2016 AISC Specification by making comparisons with a fourth method (GMNIA) that some may consider the most “exact” and titled Design by Advanced Inelastic Analysis Method, which also appears in this Specification. With 1,300 conditions studied that required a total of 57,200 analyses, simply-supported beam-columns comprised of a fairly wide range of column W-sections and slenderness ratios are investigated for conditions of minor- or major-axis flexure that include or exclude the presence of residual stresses. In general, all three elastic design methods provide fairly similar results, with AISC’s relatively new Design by Advanced Elastic Analysis Method consistently indicating more strength (1% to 5%) than AISC’s Effective Length. Conditions of major-axis bending significantly improved the performance of all three elastic design methods. Regardless of the axis of bending, results are always improved when residual stresses are not present, a condition that is often the consequence of rotary straightening during the rolling process. REFERENCES AISC 2016a. ANSI/AISC 360-16 Specification for Structural Steel Buildings, American Institute of Steel Construction, Chicago, IL. AISC 2016b. Code of Standard Practice for Steel Buildings and Bridges, American Institute of Steel Construction, Chicago, IL. AISC 2016c. Steel Construction Manual, Fifteenth Edition, American Institute of Steel Construction, Chicago, IL. Alemdar, B.N. 2001. Distributed Plasticity Analysis of Steel Building Structural Systems. PhD dissertation, Georgia Institute of Technology, Atlanta, GA. Galambos, T.V. & Ketter, R.L. 1959. Columns Under Combined Bending and Thrust. Journal of the Engineering Mechanics Division, ASCE, 85(EM2): 135–152. Ge, X. and Yura, J. 2019. The Strength of Rotary-Straightened Steel Columns, Proceedings-Annual Stability Conference, SSRC, St. Louis, MO: 425–442. Giesen-Loo, E. 2016. Design of Steel Structures by Advanced 2nd-Order Elastic Analysis –Background Studies. Honors Thesis, Bucknell University, Lewisburg, PA. Martinez-Garcia, J.M. & Ziemian, R.D. 2006. Benchmark Studies to Compare Frame Stability Provisions. Proceedings-Annual Technical Session and Meeting, SSRC, San Antonio, TX: 425–442.

63

McGuire, W., Gallagher, R., & Ziemian, R. 2000. Matrix Structural Analysis, John Wiley & Sons, Inc., New York, NY. Nwe Nwe, M.T. 2014. The Modified Direct Analysis Method: An Extension of the Direct Analysis Method. Honors Thesis, Bucknell University, Lewisburg, PA. Ziemian, R.D. (ed.) 2010. Guide to Stability Design Criteria for Metal Structures. John Wiley & Sons, Inc., Hoboken, NJ.

Appendices A. Plots of interaction curves and percent radial errors As a complement to Figure 2, which is for L/r = 90, the remaining normalized P-M interaction curves and corresponding plots of percent radial errors that were studied for the specific case of a W12X120 member that includes residual stresses and subjected to minor-axis bending are provided.

Figure A1.

L/r = 30.

Figure A2.

L/r = 60.

64

Figure A3.

L/r = 120.

Figure A4.

L/r = 150.

B. Data for plots of percent radial errors The following tables provide numerical values for the data points appearing in the percent radial error plots given in Figure 2 and Appendix A. Table B1. L/r = 30 (values are percent radial errors). Minor-axis bending with residual stresses θ 

0  10  20  30  40  50  60  70  80  90

Minor-axis bending without residual stresses

Major-axis bending with residual stresses

Major-axis bending without residual stresses

DMMI ELM

DM DMMI ELM

DM

DMMI ELM

DM

DMMI ELM DM

2.1 -2.2 -5.3 -8.2 -10.4 -10.6 -8.5 -3.5 1.1 0.0

0.6 -6.0 -4.1 -6.7 -7.0 -6.9 -9.7 -9.6 -11.7 -12.1 -11.7 -12.9 -9.4 -11.1 -4.2 -6.0 1.2 0.4 0.0 0.0

-7.4 -8.4 -8.6 -11.1 -13.4 -14.1 -12.1 -6.7 0.4 0.0

2.1 3.8 5.1 5.8 6.2 6.5 6.3 5.9 2.9 0.0

-0.2 1.3 2.7 3.6 4.3 4.8 5.0 5.0 2.8 0.0

-0.8 -1.1 1.0 3.3 3.3 4.6 5.0 4.5 2.2 0.0

0.6 -2.8 -5.9 -8.8 -10.8 -10.8 -8.5 -3.4 1.8 0.0

-7.4 -7.2 -7.6 -10.2 -12.5 -13.2 -11.2 -5.9 1.1 0.0

65

-0.2 2.6 3.8 4.6 5.4 5.9 6.0 5.8 3.4 0.0

-3.1 -2.2 -0.2 2.2 2.4 4.0 4.6 4.4 2.8 0.0

-3.1 -3.5 -1.3 1.2 1.4 3.0 3.6 3.5 2.2 0.0

Table B2. L/r = 60 (values are percent radial errors). Minor-axis bending with residual stresses θ 

0  10  20  30  40  50  60  70  80  90

Minor-axis bending without residual stresses

Major-axis bending with residual stresses

Major-axis bending without residual stresses

DMMI

ELM DM DMMI

ELM

DM DMMI

ELM DM

DMMI ELM

DM

12.3 13.7 10.9 8.0 6.6 6.1 6.5 7.4 6.9 0.0

7.1 8.0 6.3 4.5 3.8 3.8 4.7 7.1 8.3 0.0

-9.1 -4.2 -3.4 -3.8 -3.6 -2.8 -0.7 3.2 7.7 0.0

-9.1 -7.0 -6.7 -7.2 -7.0 -6.0 -3.7 0.5 5.7 0.0

-2.6 0.3 2.5 4.0 5.3 6.8 8.1 8.5 6.3 0.0

-6.5 -2.0 1.0 3.2 4.5 5.3 6.2 7.0 3.8 0.0

-12.3 -10.7 -7.7 -4.8 -2.8 -1.3 0.5 2.7 3.0 0.0

7.1 4.6 2.6 0.8 0.2 0.4 1.7 4.5 6.4 0.0

-4.7 1.0 1.1 -0.2 -0.7 -0.4 1.2 4.2 6.3 0.0

3.8 6.8 8.1 8.5 9.0 9.8 10.5 9.5 5.1 0.0

-2.6 -2.7 -1.0 0.3 1.6 3.3 4.9 5.7 4.3 0.0

-12.3 -8.0 -4.4 -1.2 0.8 2.2 3.8 5.5 5.0 0.0

Table B3. L/r = 90 (values are percent radial errors).

θ 

0  10  20  30  40  50  60  70  80  90

Minor-axis bending with residual stresses

Minor-axis bending without residual stresses

Major-axis bending with residual stresses

Major-axis bending without residual stresses

DMMI ELM DM

DMMI ELM DM

DMMI ELM DM

DMMI ELM DM

0.4 4.1 5.4 6.6 8.0 9.2 9.4 8.0 4.8 0.0

-6.7 -0.8 0.5 2.6 3.6 4.0 4.6 4.4 1.9 0.0

9.7 14.7 14.9 13.9 13.2 12.8 12.1 11.0 9.5 0.0

10.5 9.1 8.9 8.2 7.8 8.0 9.0 11.0 10.8 0.0

10.5 2.2 2.1 1.8 1.9 2.5 4.2 6.9 8.0 0.0

-2.7 2.2 3.3 3.7 4.2 5.1 6.5 7.6 8.2 0.0

-2.0 -2.8 -2.1 -1.7 -0.9 0.4 2.9 6.7 9.7 0.0

-2.0 -8.8 -8.2 -7.4 -6.5 -4.8 -1.8 2.5 6.7 0.0

-1.0 -2.1 -0.9 0.5 2.1 3.9 5.4 6.9 6.1 0.0

-1.0 -8.1 -7.1 -5.4 -3.6 -1.5 0.5 2.7 3.2 0.0

-8.0 -6.7 -5.6 -3.4 -2.1 -1.2 0.4 2.8 3.2 0.0

-8.0 -12.4 -11.5 -9.2 -7.7 -6.4 -4.3 -1.3 0.2 0.0

Table B4. L/r = 120 (values are percent radial errors).

θ 

0  10  20  30  40  50  60  70  80  90

Minor-axis bending with residual stresses

Minor-axis bending Major-axis bending without residual stresses with residual stresses

Major-axis bending without residual stresses

DMMI ELM DM

DMMI ELM DM

DMMI ELM DM

DMMI ELM DM

-0.3 11.1 13.8 14.2 13.7 13.1 12.6 12.0 10.8 0.0

-5.6 0.9 3.2 4.9 5.8 6.7 7.7 8.7 9.4 0.0

-5.7 1.0 3.6 5.4 6.0 6.3 6.2 5.6 4.2 0.0

-7.1 -1.3 0.3 1.0 1.4 2.1 2.0 1.6 1.4 0.0

5.5 3.0 3.9 4.2 4.8 6.0 7.8 10.4 11.1 0.0

5.5 -4.9 -3.5 -2.8 -1.6 0.0 2.5 5.9 7.9 0.0

-0.1 -6.3 -5.7 -4.5 -2.7 -0.2 2.8 6.8 9.8 0.0

-0.1 -13.4 -12.5 -10.8 -8.7 -5.9 -2.4 2.3 6.5 0.0

66

-1.4 -7.0 -6.0 -4.6 -3.0 -1.1 0.9 3.4 4.4 0.0

-1.4 -14.0 -12.8 -10.9 -9.0 -6.8 -4.2 -1.0 1.2 0.0

-3.0 -9.1 -8.9 -8.6 -7.3 -5.0 -3.2 -0.6 1.6 0.0

-3.0 -15.9 -15.6 -14.7 -13.1 -10.5 -8.1 -4.9 -1.5 0.0

Table B5. L/r = 150 (values are percent radial errors).

θ 

0  10  20  30  40  50  60  70  80  90

Minor-axis bending with residual stresses

Minor-axis bending Major-axis bending without residual stresses with residual stresses

Major-axis bending without residual stresses

DMMI ELM DM

DMMI ELM DM

DMMI ELM DM

DMMI ELM DM

-3.0 8.6 11.6 12.5 12.7 12.6 12.6 12.5 11.5 0.0

-6.5 -0.2 2.4 4.2 5.7 7.1 8.5 9.7 10.2 0.0

-7.4 -0.8 1.6 2.8 3.6 4.3 4.7 4.7 3.9 0.0

-9.5 -4.8 -2.0 -0.1 0.6 0.0 0.4 0.9 1.2 0.0

0.5 -2.4 -0.8 0.3 1.6 3.3 5.7 9.0 10.9 0.0

0.5 -9.8 -8.0 -6.4 -4.7 -2.6 0.4 4.4 7.6 0.0

-3.2 -10.2 -9.0 -7.2 -4.8 -2.0 1.5 6.0 9.6 0.0

-3.2 -17.0 -15.6 -13.4 -10.8 -7.7 -3.6 1.5 6.2 0.0

67

-4.9 -11.3 -10.3 -8.9 -7.1 -4.9 -2.4 0.8 3.3 0.0

-4.9 -18.0 -16.8 -15.0 -13.0 -10.4 -7.4 -3.5 0.1 0.0

-7.2 -14.9 -13.4 -11.5 -9.9 -9.0 -6.5 -2.9 0.5 0.0

-7.2 -21.2 -19.7 -17.5 -15.7 -14.3 -11.4 -7.2 -2.7 0.0

Regular papers

Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

On the modal buckling of longitudinally stiffened plates S. Adany & M.Z. Haffar Budapest University of Technology and Economics, Budapest, Hungary

ABSTRACT: In this paper the elastic buckling of simple longitudinally stiffened plates are discussed. The special focus is on the buckling of the stiffeners that are assumed to have trapezoidal shapes. Stiffener buckling is frequently handled as a combination of plate buckling and column buckling in the design of orthotropic stiffened plates. On the other hand, stiffener buckling is handled as distortional buckling in the design of coldformed steel members. Here the buckling of stiffened plates is discussed by using the global, distortional and local deformation modes, by using the recently introduced constrained finite element method.

1 INTRODUCTION It is a usual engineering practice to apply stiffeners in thin plates to increase the resistance against plate buckling. In cold-formed steel structural members almost exclusively longitudinal stiffeners are applied. Since the stiffened plates are typically slender, buckling is always crucially important. It is also important to properly distinguish between the buckling of the plate panels between the stiffeners, and the buckling of the stiffeners, since these two kinds of buckling might have significantly different post-buckling reserve. When longitudinal stiffeners in cold-formed steel members are discussed, the buckling of longitudinal stiffener, i.e., when flexural deformations of the stiffener plus plate-like deformations of the plates are combined, is typically called distortional buckling. In design codes for cold-formed steel members, such as the North-American direct strength method, DSM (2006), or the European design method, CEN (2006a), distortional buckling is clearly distinguished from local-plate buckling and from global buckling, and though the design methods are different in the various design codes, some reduction factor is calculated from the elastic critical load to each of the buckling types. In the case of welded plate girders longitudinal stiffeners are widely employed, too, but their buckling is unusual to describe as distortional buckling. A possible design approach, which is included in the relevant part of the Eurocode, see CEN (2006b), is to interpret the stiffener buckling as a combination of so-called plate-like behavior and column-like behavior, and therefore to calculate reduction factor by interpolating from reduction factors defined for plate-like and column-like behavior. In the actual paper the buckling of simple, longitudinally stiffened plates is studied, however, by adopting the worldview of cold-formed steel design, by interpreting the deformations as the combination of global (G), distortional (D), local (L) and other (O) deformation modes. Linear elastic buckling behavior is discussed, considering various plate and stiffener geometries. The calculations are conducted by the constraint finite element method (cFEM), as in Ádány (2018a, 2018b) and Ádány et al. (2018), which can readily be applied for stiffened plates and can easily and objectively separate the global, distortional and local deformations.

71

2 THE CONSTRAINED FINITE ELEMENT METHOD The constrained finite element method (cFEM) is essentially a shell finite element calculation, but the method is developed so that modal decomposition would be possible .[In order to maintain constraining ability, the longitudinal shape functions of the employed finite elements are specially selected, but the shell elements can be used as any regular flat shell element. Separation of the behavior modes is realized by applying mechanical constraints. When a member is constrained into a deformation mode, it is enforced to deform in accordance with some mechanical criteria, characterizing for the intended deformation mode. The subsequent application of the properly selected mechanical criteria finally leads to an alternative basis system of the displacement field defined by the finite element nodal degrees of freedom: the practically useful feature of this alternative basis system is that the deformation modes are separated. An important feature of the finite element analysis by cFEM is that transverse and longitudinal directions are strictly distinguished. When deformations are constrained or decomposed, these are the cross-section deformations (i.e., transverse deformations and/or displacements) that are manipulated, practically independently of the employed longitudinal shape functions. Therefore the cFEM modes can also be interpreted, and will be referred here, as cross-section modes, similarly as it is done in the generalized beam theory. The modal decomposition might beneficially be used in two tasks. One is the calculation in a reduced (i.e., constrained) deformation space. For example, the linear buckling analysis can be done for the G deformations only, meaning that all the buckled shapes will satisfy the mechanical criteria of G deformations; in simple words, all the buckled shapes will be global (e.g., flexural, or lateral-torsional buckling). The calculation in a constrained deformation space is realized by selecting only those modal basis vectors that belong to the desired deformation mode space. The other task of cFEM is modal identification. The objective is to determine the participation of the characteristic deformations modes (G, D, L, etc.) in a deformed shape of the member. The deformed shape might come from a linear static analysis, or linear buckling analysis, etc. A special basis system is necessary in which the various deformation modes are separated. Such basis system is naturally provided by the cFEM basis functions. Mathematically this means that any deformation, defined by the displacement vector, can be expressed as the linear combination of the basis vectors. Once the combination factors are calculated, the magnitudes of the combination factors give the relative importance of the basis vectors in the displacement vector to be identified. Since the basis vectors are separated into G, D, L, etc. mode spaces, the relative importance of the G, D, L, etc. deformations can be calculated. For more information, see Ádány (2018a).

3 DEFORMATION MODES OF PLATES WITH TRAPEZOIDAL STIFFENERS In Figure 1 some characteristic deformation modes are shown for plates with three longitudinal trapezoidal stiffeners. The deformation modes can be interpreted as the modes of a beam or column member, where the direction of the stiffener identifies the longitudinal axis of the “beam”. As always, the deformation modes are independent of the supports, defined solely by the member (i.e., stiffened plate) geometry. In Figure 1 the deformation modes are shown by axonometric figures, by using a single half-sine-wave for the longitudinal distribution, but it is to emphasize that the longitudinal displacement distribution has no real effect on the displacement modes, so the half-sine-waves are used here just for illustration. The nature of the deformation modes is primarily determined by the number of stiffeners, though the exact shapes, or the order of the deformation modes are dependent on the geometric proportions (and in some cases on the discretization, too). Thus, the deformation modes presented here can be regarded as qualitatively representative for the given number of stiffeners. Global modes are characterized by rigid-body cross-section displacements. Usually 4 global cross-section modes are distinguished: axial mode, two bending modes in two perpendicular direction (e.g., principal axes, marked as GB1 and GB2), and torsional mode that involves 72

Figure 1.

Deformation mode samples for a plate with 3 stiffeners.

twisting rotation of the cross-section. In the case of plates with trapezoidal stiffeners the “cross-section” of the member has one or multiple closed cells (i.e., the stiffeners themselves), hence global torsion mode does not exist, since global modes are defined to be free of in-plane shear, and rigid-body torsion is not possible for closed cells without in-plane shear deformations. The number of D deformation modes is strictly determined by the number of stiffeners. If this latter one is denoted by nst, the number of distortional modes is 2 × nst-1. It is found that, from practical aspect, there are two major types of D modes. There are modes when the stiffeners are not (or hardly) distorted, but are displacing perpendicularly to the plate. Other D modes involve (mostly) the distortion of the stiffeners. When there is one single stiffener only, the D mode involves stiffener distortion. In the case of two stiffeners, the first D mode shows stiffener translation, the other two D modes show stiffener distortion. In the case of 3 73

stiffeners (Figure 1), the first two D modes show stiffener translation, while the remaining three D modes show stiffener distortion. For higher number of stiffeners roughly half of the D modes involve stiffener translation, the other half stiffener distortion. These are always the modes with stiffener translation that come first, meaning that the modes with stiffener distortion are associated with higher energy content, hence these seem to have smaller importance in practical problems. The number of primary L deformation modes (LP) is also defined by the number of stiffeners: it is 4 × nst + 4. The number of secondary L modes (LS) is dependent on the discretization. Figure 1 shows the first 12 LP modes for the plate with three stiffeners.

4 BUCKLING RESULTS In this Section buckling results are presented for one single plate, but with various numbers of stiffeners. In all these examples the plate is square with side length of 3000 mm. The plate thickness is 12 mm. The stiffener size is constant, with 75 mm depth and 6 mm thickness, while the widths are 60 and 90 mm. The material is steel, with Young’s modulus of 210000 MPa and Poisson’s ratio of 0.3. In all the examples the plate is supported along all its 4 edges by simple supports. These are realized by supporting the edges against translation perpendicularly to the plate only, while the edges of the stiffeners are left free. This can be considered as the possible least restrictive pinned support. First linear buckling analysis is performed, without any constraints, from which the first few critical stress values and corresponding buckling shapes are determined. The buckled shapes then are identified, by calculating the participations from the various deformations modes. 4.1 Plates in compression In Figure 2 selected results are shown when the plate is in uniaxial (i.e., longitudinal) compression. The load is transmitted to the structure as uniformly distributed over the end sections (including the stiffeners, too). As the results show, for the actual plate parameters the first mode always involve the buckling of the stiffener, which – from modal decomposition aspect – is always a combination of L and G/D deformations. The importance of G modes is remarkable, since global buckling alone is not possible due to the edge supports. The participations of G, D and L modes in the first buckling modes are shown in Figure 3 in the function of the stiffener number. The plot suggests that the importance of D modes is increasing, while the importance of G and L modes are slightly decreasing as the number of stiffeners is increasing. Some higher buckling modes are also shown in Figure 2. Among these higher modes there are modes that are dominantly L, e.g., mode #2 in the case of 1 stiffener, or mode #3 in the case of 3 stiffeners. Not surprisingly, these quasi-pure local-plate modes appear higher and higher as the distance between two neighboring stiffeners is getting smaller. 4.2 Plates in bending In Figure 4 selected results are shown when the plate is in bending, i.e., linearly varying distributed loads are applied over the end sections (including the stiffeners, too), varying along the plate width. In all the cases the compressed part of the plate buckles. As the results show, for the actual plate parameters, the first mode might be dominantly local-plate buckling when the number of stiffeners is small), or stiffener buckling (when the number of stiffeners is large enough). The participations of G, D and L modes in the first buckling modes are shown in Figure 5 in the function of the stiffener number. The identification results suggest that even if the buckling visually seems to be localplate buckling, there is non-negligible participation from the G modes, mostly due to the rigid-body-like twisting of certain segments of the member, which displacements are 74

Figure 2.

Buckling results samples: plate subjected to compression.

assigned to the G modes. Independently of the number of the stiffeners, there always exists a buckling mode with “classic” stiffener buckling, such as mode #4 for plates with 1 or 2 stiffeners, or mode #1 for plates with 3 or more stiffeners. In these cases the buckled shape is identified as the combination of L and G/D modes. As the number of stiffeners is increasing, the importance of D modes is increasing while that of G and L modes are slightly decreasing. However, the D modes have somewhat higher participations in general, compared to the compressed plate.

75

Figure 3.

G, D and L participations in the first buckling modes: plates subjected to compression.

4.3 Plates in shear Plates in shear has also been studied. The pure shear is realized by applying constant distributed load along all the 4 plate edges, parallel with the relevant edges. (The stiffeners are not loaded.) Selected results are shown in Figure 6. The participations of G, D and L modes in the first buckling modes are shown in Figure 7 in the function of the stiffener number. The plot shows tendencies similar to those observed in plates in compression and bending, however, the importance of D modes now is considerably higher, while the importance of G modes is smaller. As the results show, even in the case of pure shear, it is possible to have buckling modes without significant stiffener deformations: these modes are characterized by high L participations (though with the here-assumed geometric parameters pure L buckling is atypical, best approximated by mode #3 of the plate with 1 stiffener). For the actual examples the majority of shear buckling modes involves significant stiffener buckling, as a combination of L and G/D modes.

5 SUMMARY In this paper some examples for the modal buckling analysis of longitudinally stiffened plates are presented, by considering trapezoidal, i.e., hollow stiffeners. Examples are shown and discussed: unconstrained buckling problems are solved, the buckling shapes are determined, then the shapes are identified. The results prove that cFEM can successfully solve the modal decomposition for longitudinally stiffened plates: the modes can be determined by following the usual cFEM procedure, then the modes can be used as in any modal decomposition method. The here presented results show that modal decomposition leads to meaningful results even for plate-like members. Further aspects have also been studied, which are not reported here in detail. For example, plates with various widths have been considered. From modal identification aspect it is found that the wider the plate, the more importance the D modes might have. This is partially due to the fact that wider plates might include larger number of stiffeners, which increases the number of D modes, and finally the D participations might be higher. Other geometric parameters have also some influence, such as the thickness of the plate, or the size of the stiffener. The tendencies are mostly as-expected, e.g., thinner plate are more susceptible to local-plate buckling, or larger-size stiffeners increase the potential of having nearly-pure local-plate buckling. Moreover, cFEM can easily solve the buckling problem in a reduced deformation space, which can lead to pure global, pure distortional, or pure local buckling. Though for the considered edge supports pure global buckling does not exist, and pure distortional buckling is usually associated with fairly large critical stress, pure L, as well as G+D or G+D+L constraints lead to meaningful critical stresses and deformed shapes. It is also to mention that the support conditions have non-negligible effect. In the abovepresented examples the simple (i.e., pinned) restraints were realized in the possible least restrictive form. However, pinned supports could be slightly more restrictive, too, e.g., by supporting the 76

Figure 4.

Buckling results samples: plate subjected to bending.

edges of the stiffeners, too, or by supporting the plate edges against certain rotations or translation in the plane of the plate. Obviously, more restrictive pinned supports lead to higher critical stresses, but they influence the modal identification results, too. Though the effect of various supports were not shown in this paper, we add here just one comment. It was observed that the hereapplied pinned supports allow some in-plane movement of the plate edges, which is reflected in a few percentage of participation from other (i.e., mostly in-plane shear) modes; this contribution of other modes is reduced by the more restrictive pinned supports.

77

Figure 5.

G, D and L participations in the first buckling modes: plate in bending,

Figure 6.

Buckling results samples: plate subjected to shear.

78

Figure 7.

G, D and L participations in the first buckling modes: plate in shear,

It is believed that the introduced modal worldview can contribute in the understanding of the buckling behavior of stiffened plates. Moreover, modal decomposition technique can potentially lead to computerized design of stiffened plates, by developing an appropriate design method similar to the direct strength method. REFERENCES Ádány, S. 2018a. Modal identification of thin-walled members with and without holes by using CFEM, Proceedings of the ICTWS 2018 conference, July 24–27, 2018. Lisbon, Portugal. Ádány, S. 2018b. Constrained shell Finite Element Method for thin-walled members, Part1: constraints for a single band of finite elements, Thin-Walled Structures, Vol 128, July 2018, pp. 43–55. Ádány, S., Visy, D. & Nagy, R. 2018. Constrained shell Finite Element Method, Part 2: application to linear buckling analysis of thin-walled members, Thin-Walled Structures, Vol 128, July 2018, pp. 56–70. CEN. 2006a. EN 1993-1-3:2006 - Eurocode 3: Design of steel structures - Part 1-3: Supplementary rules for cold-formed members and sheeting, Brussels, Belgium. CEN. 2006b. EN 1993-1-5:2006 - Eurocode 3: Design of steel structures - Part 1-5: Part 1-5: Plated Structural elements, Brussels, Belgium. DSM. 2006. Direct strength method design guide. American Iron and Steel Institute. Washington DC, USA.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Strength characterisation of a CFS section with initial geometric imperfections Hashmi S.S. Ahmed Department of Civil Engineering, Marathwada Institute of Technology, Aurangabad, India

Siddhartha Ghosh Structural Safety, Risk & Reliability Lab, Department of Civil Engineering, Indian Institute of Technology Bombay, Mumbai, India

ABSTRACT: The strength of a cold-formed steel CFS member gets highly affected by the presence of any geometric imperfection (local or global). Moreover, these imperfections are the major source of uncertainty in the prediction of buckling strength (Pn) of a CFS member. Previous works on strength characterization of CFS members have shown the effects of considering the uncertainty in local imperfection for a member. However, the effect of uncertainties in global imperfections is still not addressed. The prediction of Pn in current design codes is based on some deterministic value of the global imperfection. Considering this, the current paper aims at the statistical characterisation of Pn taking into account uncertainties in both local and global imperfections. For this reason, a CFS member undergoing both local (Type 1) and global (flexuraltorsional) buckling is selected. However, the probabilistic assessment of Pn, in conjunction with finite element (FE) simulations becomes computationally expensive. Thus, to address this high computational cost, a metamodelling technique approximating the original simulation model with a simplified mathematical model is proposed. Characteristic strength (Pd) values are recommended for a selected range of non-dimensional slenderness and these values are then compared with current design recommendations for Pn. It is observed that at the higher range of nondimensional slenderness (λc ≥ 1), where the flexural-torsional buckling mode governs the performance of CFS members, the AISI prediction becomes conservative, while at the lower range (λc < 1), the AISI predictions are found to be close to uncertainty analysis based recommendations. 1 INTRODUCTION Inherent slenderness in the cross-section of cold-formed steel (CFS) members makes it susceptible to geometric imperfections. These geometric imperfections affect the strength of a (CFS) members significantly. The geometric imperfections (GI) can take any possible shape in a member. However, researchers have idealised them into two categories, namely i) sectional and ii) global imperfection. The sectional imperfection is again classified by Schafer & Peköz (1998) into two subcategories: 1. Type 1: Maximum local imperfection in a stiffened element, such as a web (and can be represented by Figure 1 (a)) 2. Type 2: Maximum deviation from straightness for a lip-stiffened or an unstiffened flange (and can be represented by Figure 1 (b)) Moreover, the global imperfections are further classified into three different groups by Zeinoddini & Schafer (2011): 1. Bow: Sweep about the minor axis of a member 2. Camber: Sweep about the major axis of a member 3. Twist: Sweep about the longitudinal axis of a member 80

Figure 1.

(a) Type 1 imperfection (b) Type 2 imperfection [Schafer & Peköz, 1998].

Figure 2. Deformed shapes associated with different global buckling modes in a CFS channel section, (a) bow, (b) camber and (c) twist mode.

Figure 2 shows the typical shape for bow, camber and twist in a CFS member. Importantly, these geometric imperfections (GI) are very random in nature resulting in large uncertainty in the prediction of buckling strength (Pn ). Thus, a probabilistic estimation of Pn becomes essential, as recommended by Schafer & Peköz (1998). In this line, Ahmed et al. (2017) presented a work on the statistical assessment of buckling strength of a CFS section with local Type 1 imperfection. In their work, they have shown the effects of considering the uncertainty into the local imperfection for a member undergoing local buckling. For this purpose, they have used statistical data on local geometric imperfection provided by Schafer & Peköz (1998). Similar to local GI, global GI also carries uncertainty in their magnitude and shape. However, the uncertainty imparted due to global imperfection is still not addressed. In fact, the buckling strength predictions in current design codes such as the AISI code are based on some deterministic value of the global imperfection. Many of the researchers including Zeinoddini & Schafer (2011), reported this deterministic value as L=960, where L represents the length of a member. However, a single GI amplitude does not give the realistic picture of a CFS member. Thus, considering the uncertainty involved in both local and global imperfections becomes very imperative in complete characterisation of Pn . The primary goal of the work presented here is a statistical characterisation of Pn considering uncertainties in both local and global imperfections. For this reason, a CFS member undergoing local (Type 1) and global (flexural-torsional) buckling is selected, as this kind of behaviour is typically exhibited by most commonly used CFS sections. However, the probabilistic assessment of Pn entails multiple simulations of a high fidelity FE model, making the process computationally expensive. To address this high computational cost, metamodelling technique in the form polynomial chaos expansion is employed in this work.

2 MODELLING OF GEOMETRIC IMPERFECTIONS While assessing the effects of GI on the performance of a CFS member, modelling of the GI becomes very important issue (Dinis & Camotim 2011). To this end, the conventional 81

approach of using critical buckling mode is found to be the simplest but effective method indeed (Schafer et al. 1998). Thus, in this work critical (first) governing buckling mode for that member, is used to perturb the perfect geometry. However, for the execution of probabilistic investigation, the statistical characterisation of GI is also essential. To this end, a probabilistic model of local GI was available, proposed by Ahmed et al. (2017). This probabilistic model on local GI can be expressed as, d1 =t LN ð0:50; 0:66Þ

ð1Þ

where d1 = magnitude of local GI; t = thickness of a CFS section. Using Equation 1 the magnitude of local GI are obtained. The statistics of global GI (G1 , G2 and G3 ) in the form of cumulative distribution function (CDF) are adopted from the work of Zeinoddini & Schafer (2012), as presented in Table 1. Based on the Kolmogorov-Smirnov goodness-of-fit test (K-S test), the lognormal model for G3 is found to be a better fit for the given statistics. The comparison between the two probability models with respect to the original data is provided through their empirical cumulative distribution functions (CDF) in Figure 3. Equation 2 shows the proposed probabilistic model for global GI.

Table 1. Statistics of global geometric imperfection.

Figure 3.

CDF

Bow, G1 (L=δ0 )

Camber, G2 (L=δ0 )

Twist, G3 (Deg/m)

0.25 0.50 0.75 0.75 0.75

4755 2909 1659 845 753

6295 4010 2887 1472 1215

0.20 0.30 0.49 0.85 0.95

Mean Std. dev.

2242 3054

3477 5643

0.36 0.23

CDF of the observed data and trial distributions for normalised imperfection.

82

θ ðdeg=mÞ LN ð0:36; 0:23Þ

ð2Þ

The scale factors (SF) are generated to scale the global ( flexural-torsional) mode using Equation 2. Using these factors, the desired twist imperfection is applied (about the shear centre) in the section along with the associated flexural component (camber) of the imperfection. In fact, it is observed that this camber part has an insignificant effect on the performance of the buckling strength of the selected section.

3 FINITE ELEMENT SIMULATION 3.1 Selection of the specimen A lipped channel section is selected for this work, which is designated as 400S200-68. This section has the following dimensions: depth of the web = 101.6 mm, width of a flange = 50.80 mm, thickness = 1.72 mm. The selection of section is done based on a preliminary buckling analysis using the software CUFSM (Schafer & Adany 2006), such that the global buckling mode ( flexural-torsional) governs the member behaviour at higher non-dimensional slenderness (i.e. λc  1), and the local buckling mode (Type 1) governs at lower nondimensional slenderness (i.e. λc 5 1). For this reason, five different lengths of the member are considered (1.42 m, 2.13 m, 2.82 m, 4.12 m and 5.38 m), which give non-dimensional slenderness (λc ) in the range of 0.65 to 2.20. 3.2 Finite element analysis of the imperfect member The general purpose finite element package Abaqus is used to find the critical buckling strength (Pn ) of the selected lipped CFS channel member. “Fixed-fixed” end boundary condition is applied by restraining all the degree of freedoms at both ends, except for the transnational degree of freedom in the axial direction at the ‘forcing end’. These boundary conditions as well as the axial compression force are applied at the centroid of the end crosssections. These are transferred to the member by connecting (all degrees of freedom of) the centroid with the nodes at the end section through “kinematic couplings”, as shown in Figure 4. The S4R element is used to model the member, with an elastic-perfectly plastic material model: elastic modulus ¼ 203 GPa; Poissons ratio = 0.3; and yield stress = 388 MPa. In order to understand the post-(local) buckling behaviour of the CFS section the Static, Riks iterative scheme in Abaqus is adopted. In order to have an imperfect member, the initial geometry of the section is perturbed with the governing buckling mode scaled to the desired amplitude of the GI. Figure 5 shows the members at failure at two different λc values (0.65 and 1.25). It is observed that the failure is predominantly due to the local buckling of the member at lower λc , while for member with higher λc the failure is totally governed by global buckling

Figure 4.

FE model of the CFS member with “kinematic couplings”.

83

Figure 5.

Deformations after buckling of section 400S200-68.

(flexural-torsional). The incremental axial compressive load and the axial deformation of the member are monitored throughout the incremental analysis. Figure 6 shows sample load-deformation curves for λc = 0.65 and λc = 1.25. The load-deformation curves presented here are for the CFS sections with and without geometric imperfection. For these cases, a magnitude of d1 =t = 3.676 corresponding to the local buckling mode and magnitude of θ = 1.351 =m corresponding to the flexural-torsional mode was used as the initial geometric imperfection to obtain the buckling load. A significant weakening effect of geometric imperfections is observed in both cases.

Figure 6.

Load deformation curves for the section 400S200-68 with d1 =t ¼ 3:676 and θ ¼ 1:351 =m:

84

4 STRENGTH CHARACTERISATION USING METAMODELLING APPROACH 4.1 Formulation of metamodel The probabilistic assessment of Pn by means of Monte Carlo simulation (MCS), in conjunction with finite element (FE) simulations becomes computationally expensive, since it involves a large number of FE model runs (Schenk & Schuëller 2003, Hashmi et al. 2014). To overcome this issue, the metamodelling technique of polynomial chaos expansion (PCE) is used in this work. The PCE based metamodel describes the uncertainty in a structural response parameter, using a set of orthogonal polynomial bases and associated coefficients (Spanos & Ghanem 1989). The general form of PCE can be represented as (Blatman & Sudret 2011) Y ¼ MðξÞ ¼

X α

aα α ðξÞ

ð3Þ

where, α are the multivariate orthonormal polynomials, and α are the multi-indices that map the multivariate α to their corresponding bases, which are denoted as the PCE coefficients aα . These associated coefficients in the PCE based metamodel can be interpreted in the form of mean, variance, and higher moments of the random output parameter of interest. Individual PCE based metamodels are created for each λc of the selected section. The metamodel contains polynomial based equations defining the response Pn as a function of the geometric imperfection magnitudes, which is treated as stochastic input variables. For this purpose, the PCE module of the MATLAB toolbox UQLab (Marelli & Sudret 2014) is been used, which facilitates the application of state-of-the-art algorithms for a non-intrusive approach of PCE metamodelling. As the imperfection magnitudes (both local and global) follow lognormal distribution, the Hermite polynomial basis is selected for the PCE metamodelling (Xiu & Karniadakis 2002). The coefficients aα are calculated, from the results obtained using a deterministic structural analysis model, at the selected experimental design points (i.e. λc ), using a non-intrusive approach. For this reason, least angle regression (LARS) method is adopted to calculate the PCE coefficients {a0 . . . a4 }. Based on the findings of Hashmi et al. (2017) only 50 runs of FE simulations are used to formulate a fourth-order (N ¼ 4) PCE metamodel. Hashmi et al. (2017) have shown that PCE response (with only 50 FE runs) for Pn of a CFS member with GI are sufficient to emulate (computationally-expensive) MCS response (of 1000 FE runs) with desirable accuracy. The truncated form of fourth order PCE metamodel (MPCE ) can be expressed as ffiffi þ a3 ðξ p3ξÞ ffiffi þ a4 ðξ Pn ’ M PCE ðξÞ ¼ a0 þ a1 ξ þ a2 ðξp1Þ 6 2 2

3

4

 6ξ 2 þ 3Þ pffiffi 2

ð4Þ

These coefficients, due to the prudent formulation of the (sparse) PCE, can be interpreted as the mean (μPCE ) and the variance (σ2 ) as follows (Blatman & Sudret 2011): μPCE ¼ E½MPCE ðXÞ ¼ a0 X σ2 ¼ a2α

ð5Þ ð6Þ

α

The PCE coefficients interpreted in the form of these two statistical parameters for the selected section are provided in the Table 2. 4.2 Strength characterisation based on PCE metamodel output In order to perform the strength characterisation of an imperfect CFS section, PCE based metamodels are created and the response statistics are obtained at each λc . A total of 250 “true model” simulations are performed, comprising of 50 different geometric imperfection realisations for each λc . PCE based metamodels are employed, however restricted to reduce the computational efforts only, to obtain the statistical parameters ( μPCE and σ2 ). These 85

Table 2. PCE coefficients (in kN). P Design point

a0 ¼ μPCE

λc λc λc λc λc

122.3 116.1 88.19 51.28 37.05

= 0.65 = 0.96 = 1.25 = 1.76 = 2.20

Table 3.

α

a2α ¼ σ2

5.035 5.787 3.385 0.5984 0.1793

PCE coefficients (in kN).

λc

Pd (kN)

Pn (kN)

Pd =Pn

0.65 0.96 1.25 1.76 2.20

112.4 104.8 81.56 50.11 36.69

112.9 94.99 74.72 43.98 28.53

0.9956 1.103 1.092 1.139 1.286

statistical parameters are reported in Table 2. The 5-percentile values are also obtained for each λc , which represents the characteristic value (Pd ) for the flexural buckling strength from a design specification perspective. This characteristic value (Pd ) includes the effect of geometric imperfection and associated uncertainty. Pd for a selected range of non-dimensional slenderness are compared with the current (AISI) design recommendations for Pn and are represented in Table 3. The Pn values recommended in the AISI specification are calculated for each value of λc as, Pn ¼ Fn Ae

ð7Þ

where, Fn ¼ 0:658λc Fy 2

¼

0:877 Fy λc 2

for λc  1:5 for λc 41:5

ð8Þ

where, Ae = effective area of the cross-section and Fy = yield stress of the material. Pd =Pn value in Table 3 shows the effect of consideration of uncertainty in geometric imperfections. At lower values of non-dimensional slenderness (λc 51), where failure is mainly governed by local buckling (Type 1), the Pd =Pn value is slightly less than 1, highlighting the fact that the AISI value is slightly unconservative, however negligible, at this level. Thus, the effect of uncertainty is found to be very low or almost negligible at this level. However, for the higher range of non-dimensional slenderness (λc  1), where failure is governed by global buckling ( flexural-torsional), this effect becomes more significant, as the Pd =Pn values are higher than 1.

5 CONCLUDING REMARKS The effect of uncertain (local and global) geometric imperfections on the strength of a lipped channel under compression is investigated here. This study involves a consideration of geometric imperfection and associated uncertainties for both the local and global GI. A framework of FE simulations within the loop of the uncertainty quantification is employed 86

in this work. Metamodelling technique in the form of polynomial chaos expansions is used here to avert the use of computation heavy Monte Carlo simulation. Eventually, the statistical estimate of buckling strength at different non-dimensional slenderness is obtained for the selected CFS section, using this approach. Probabilistic modelling of the global imperfection (twist component or G3 ) shows that the lognormal distribution is suitable fit for the given statistics. Based on the numerical investigations, it is observed that at the higher range of non-dimensional slenderness (λc  1), where the global ( flexural-torsional) buckling mode governs the performance of CFS members, a large and noticeable variation exist between Pd and Pn values. This shows that the AISI recommendation for design is very conservative at the higher range of non-dimensional slenderness (λc  1). In fact at very high value of non-dimensional slenderness (i.e. λc ¼ 2:2) the AISI prediction can be conservative up to 28.6%. While at the lower range (λc 51), the AISI recommendation is found to be close to the recommendations based on the uncertainty analysis. REFERENCES Ahmed, H.S.S., Ghosh, S. & Mangal, M. 2017. Probabilistic estimation of the buckling strength of a CFS lipped-channel section with Type 1 imperfection. In Thin-Walled Structures 119: 447–456. AISI. AISI-S100, North American specification for the design of cold-formed steel structural members 2007. American Iron and Steel Institute, Washington, DC: USA. Blatman G, & Sudret B. 2011. Adaptive sparse polynomial chaos expansion based on least angle regression. In Journal of Computational Physics 230(6): 2345–2367. Hashmi, S., Sable, S. & Ghosh, S. 2014. Probabilistic Capacity Estimation of CFS Channel Sections with Type-1 Imperfection. In The Twelfth International Conference on Computational Structures Technology. 106. Naples: Italy. Hashmi, SA., Subhamoy, S., & Ghosh, S. 2017. Prediction of the buckling strength of cfs members with local geometric imperfection using stochastic kriging. In The Twelfth International Conference on Structural Safety and Reliability. Vienna: Austria. Marelli, S. & Sudret, B. 2014. UQLab: a framework for uncertainty quantification in MATLAB. In Beer M, Au SK & Hall JW, (editors), Vulnerability, Risk Analysis and Management (ICVRAM2014). USA: American Society of Civil Engineers. 2554–2563. Schafer, B.W. & Adany, S. 2006. Buckling analysis of cold-formed steel members using cufsm: Conventional and constrained finite strip methods. In Eighteenth International Specialty Conference on ColdFormed Steel Structures. 39–54. Orlando: Florida. Schafer, B.W., Grigoriu, M. & Pekz T. 1998. A probabilistic examination of the ultimate strength of cold-formed steel elements. In Thin-walled Structures 31(4): 271–288. Schafer, B.W. & Pekz, T. 1998. Computational modeling of cold-formed steel: Characterizing geometric imperfections and residual stresses. In Journal of Constructional Steel Research 47(3):193–210. Schenk, C. & Schuller, G. 2003. Buckling analysis of cylindrical shells with random geometric imperfections. In International Journal of Non-Linear Mechanics 38(7): 1119–1132. Simulia 2013. Abaqus FEA Users Manual, Version 6.13. Vlizy-Villacoublay, France. Spanos, PD. & Ghanem, R.S 1989. Stochastic finite element expansion for random media. Journal of Engineering Mechanics 115(5): 103553. Xiu, D. & Karniadakis, GE 202. The wiener-askey polynomial chaos for stochastic differential equations. SIAM Journal on Scientific Computing 24(2): 619–644. Zeinoddini, V. & Schafer, B. 2011. Global imperfections and dimensional variations in cold-formed steel members. In International Journal of Structural Stability and Dynamics 11(05): 829854. Zeinoddini, V. & Schafer, B. 2012. Simulation of geometric imperfections in cold-formed steel members using spectral representation approach. Thin-Walled Structures 60: 105–117.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Shear behaviour of sandwich panel fasteners in fire Tesfamariam Arha, Kamila Cábová, Nikola Lišková & František Wald Czech Technical University in Prague, Czech Republic

ABSTRACT: In recent years the usage of sandwich panels as wall cladding and roofing has increased significantly. It has been shown that by using sandwich panels and trapezoidal sheeting as a stabilizing members, a considerable amount of savings of steel can be achieved for structural members at ambient temperature. Previous researchers has not covered this topic under elevated temperature and these stabilising effects may also help to achieve similar savings in case of fire. The behaviour of sandwich panel fasteners in fire is very important in order to predict and investigate the whole structure. Therefore, an experimental investigation was conducted to study the shear behaviour of sandwich panel connections in fire when loaded in shear under the diaphragm action. In this paper the experimental results of 16 sandwich panel fastener tests are presented which are carried out under the RFCS research project STABFI which is performing a research currently on the stabilization of a structural building using the cladding systems in fire. The results of the tests show that bearing failure of the inner steel sheet was the main failure mode. There was no failure of screws for all the tests at ambient and elevated temperatures except one. The results of the tests provided experimental data for the sandwich fasteners related to building stabilization in fire through the cladding systems which is under investigation of RFCS project STABFI.

1 INTRODUCTION Sandwich panels are increasingly being used as structural and non-structural components in buildings, such as wall and floor assemblies. Usually these comprise two layers of thin steel faces and a thick core layer of lightweight insulation material. The four most commonly used insulation materials are polyurethane foams (PUR), polyisocyanurate foams (PIR), extruded polystyrene, and mineral wool (MW) based products. The main advantages of such assemblies are the lightness, easy assembly and the highly efficient insulation characteristics. In recent years, the use of light-weight sandwich panel construction, for both industrial and civil buildings, is becoming more and more popular. When sandwich panels are used as a cladding wall, they contribute to structural behaviour of the building in addition to the aesthetic effects [1]. Recent RFCS projects [2,3,4,5] have shown that by using sandwich panels and trapezoidal sheeting as a stabilizing members, a considerable amount of savings of steel can be achieved for structural members at ambient temperature. These projects have not considered the fire limit state and the aim of this ongoing RFCS project STABFI is to produce new information to cover the fire situation. In the tests the translational displacement of the fastener was measured when the specimen was exposed to fire and then shear loaded mechanically using the displacement controlled machine. Also, the temperatures of the inner and outer steel sheet, insulation core and the supporting steel were measured. In addition to that, the failure mode of the specimen was also monitored and recorded.

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2 TEST PROGRAMME The main objective of the experimental testing was to determine translational stiffness of a sandwich panel connection with a supporting steel member at ambient and elevated temperatures. Testing was carried out at laboratory of Faculty of Civil Engineering, Czech Technical University in Prague, Czech Republic. The testing arrangement consists of a sandwich specimen of size 300 mm × 500 mm (width × length) connected to a steel plates of thickness 8 mm using a self-tapping stainless steel screw (SxC14–S19–5.5xL) of diameter 5.5 mm. The steel plates are considered equivalent to the flange thicknesses of the steel profiles to which the sandwich panels are connected in practice. Two types of sandwich panels: SPA E panels with mineral wool (MW) core and KS1000RW panels with PIR core were used for the tests. The tests include sandwich panel thicknesses of 100 mm (MW & PIR), 160 mm (PIR) and 230 mm (MW). The tests were conducted at elevated temperatures of 250°C, 300°C and 450°C for PIR panels and 300°C, 450°C and 600°C for MW panels. The material properties of the specimens are given in Table 1.

3 TEST SETUP Figure 1 shows the general scheme of the translational stiffness test. The sandwich panel was connected to the steel plates, which is equivalent to the flange thickness of the steel frames on which the sandwich panels are connected in reality. The boundary conditions and stiffness of the steel plates are considered to be similar to the flange of the steel supporting frames. The sandwich panel is connected using self-tapping screws from both ends. The lower end of the specimen was fixed on the testing machine with a pin while the upper end was free to move in the direction of loading (upward). For tests at elevated temperatures heating was applied using manning’s heating pads on one side of the sandwich panel and the supporting steel member close to the lower fastener, see Figure 2b. The sandwich panel was connected with a single screw at the lower connection where the elongation was measured using the optic extensometer, while it was connected with three screws on the other end in order to minimize the elongation of the fastener at that location. Displacement-controlled testing machine was used for the experiment. The displacement was increased monotonically until failure. A loading rate of 1 mm/min was used for the experiments according to the ECCS recommendation “The testing of connections with mechanical fasteners in steel sheeting and sections, ECCS No.124” [6]. The displacement of the fastening

Table 1.

Material properties and dimensions of the sandwich panel connection.

Parts

Material

Yield strength [MPa]

Thickness [mm]

SPA E panel (MW) Inner sheet Outer sheet Insulation

S280GD+Z S280GD+Z Mineral wool

280 280 0.086 (Tension)

0.5 0.6 100/230

SPA E panel (MW) Inner sheet Outer sheet Insulation

S250GD S250GD PIR

250 250 0.05 (Tension)

0.4 0.5 100/160

Stainless steel A2/AISI 304

450

Ø5.5

S355

355

8

Screw (SxC14–S19–5.5xL) Supporting steel

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Figure 1.

Scheme & experimental setup for shear resistance of sandwich panel joint.

Figure 2.

Experimental setup, a) Placement of optic sensors and b) Placement of manning’s heating pads.

was measured using an optical extensometer with optic sensors placed at the central axis of the connection as shown in Figures 1b and 2a. At elevated temperatures, the tests were carried out in two steps. Firstly, the specimens were heated to the designed temperatures by manning’s heating pads at the rate of 16.67°C per minute. Then the specimen was loaded at a rate of 1 mm/min and the temperature was kept constant until the end of the experiment. According to the “Preliminary European Recommendations for the Testing and Design of Fastenings for Sandwich Panels, ECCS No.127” [7] the maximum load is reached at a deformation of 3 mm for serviceability limit state. However, fire being an accidental action large deformation is an alternative mechanism to prevent the failure of the structure. In addition, catenary action of sheeting occurred in the 90

large deformation. Thus, the tests were stopped when a displacement of 20 mm was reached both at room temperature and at elevated temperatures.

4 RESULTS AND DISCUSSIONS 4.1 Sandwich connection failure modes Figure 3 shows a typical failure modes observed for the sandwich panel connection in the experimental tests both at ambient and elevated temperatures. For all the experiments the major failure was a bearing failure of the inner face of the sandwich panel near the hole of the screw connection. For most of the experiments at ambient temperature the tearing of the inner sheet was in a narrow path approximately equivalent to the screw diameter with a small stacking (folding) of the steel sheet during tearing off. However, for the experiments at elevated temperature the steel sheet teared off in a wider area and also experiencing stacking (folding) of the steel sheet near the end of the opening. This folding of the steel sheet strengthens the steel sheet further after the initial peak and enables it to carry more load which can be represented by the further peak strength values after the first peak, which can be seen in Figures 6 and 7. From all the 16 experiments there was only one specimen in which the screw failed in shear in addition to the bearing failure of the sheet, see Figure 4b. Generally the screws after the test showed no significant deformation for the PIR sandwich panels while it showed a small bending deformation with MW sandwich panels, Figure 4. 4.2 Heating and temperature distribution on a sandwich panel The specimens were heated to the designed temperatures by a system of ceramic heating pads at a rate of 16.67°C per minute. Then, the temperature was kept constant until failure. The mechanical loading of the specimen was started when the required temperature of the steel sheet of the sandwich panel was reached. During the experiment, temperature of different components of the testing specimen was recorded by coated thermocouples of type K diameter 2 mm. As a sample, the temperature distribution for the experiments at 300°C are given in Figures 5 and 6 for both SPA E (MW) and KS1000RW (PIR). When the obtained temperatures for the two core materials with the same thickness are compared, it was found that the temperature for PIR core was higher than that of MW core due to the fact that PIR has greater thermal conductivity than MW.

Figure 3.

Bearing failure of the inner steel sheet for MW_230 mm at: a) 20°C, b) 300°C, c) 450°C, d) 600°C.

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Figure 4.

Deformation of screws after the test for PIR and MW sandwich panels of 100 mm thickness.

Figure 5.

Temperature distribution at 300°C for SPA E (MW), a) 100 mm thick and b) 230 mm thick.

4.3 Force-displacement curves Displacement of the connection was measured using an optical extensometer in which the two sensors were placed on the central axis of the specimen close to the connection. It measures the relative movement of the two sensors using a beam of laser light. Because fire is an accidental action, the design requires that the structure will not collapse in case of fire. In addition, catenary action of sheeting occurred in large deformation. From these two points, the displacement was done up to 20 mm in the experiments.The applied force was also recorded from the testing machine, therefore the force-displacement curve was plotted for all the experiments in order to determine the initial translational stiffness and the ultimate shear strength. The force-displacement curves of the connections at ambient and elevated temperatures are shown in Figures 6 and 7. It can be seen that with the increase of temperature, both strength and stiffness of the connections are reduced. The stiffness of connection starts to reduce when the slip is initiated at

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Figure 6.

Temperature distribution at 300°C for KS1000RW (PIR), a) 100 mm thick, b) 160 mm thick.

Figure 7.

F-D curves at 300°C for KS1000RW (PIR), a) 100 mm thickness and b) 160 mm thickness.

inner steel sheet because of the spreading of yielding zone. At elevated temperatures, the stiffness reduced further due to the degradation of the material. It can be observed that the folding of the steel sheet after the first peak strengthens the steel sheet further and enables it to carry more load which can be represented by another peaks after the first one as shown in Figures 7 and 8.

Figure 8.

F-D curves at 300°C for SPA E (MW) a) 100 mm thickness and b) 230 mm thickness.

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4.4 Shear stiffness and resistance The maximum load and the shear displacement corresponding to the maximum load in the fastening shall be measured in order to determine the shear stiffness of the screw connection. According to the ECCS recommendation [7] the ultimate load is defined to be the smallest of: – the load corresponding to a displacement of 3 mm, if this occurs on the rising part of the load - deflection curve. – the maximum load recorded during the test. – the load at which the first decrease in load is observed in the load - deflection curve. According to ECCS recommendation [6], the shear flexibility (ch ) of a fastening should be determined by Eq. (1). The shear stiffness is obtained from the inverse of its shear flexibility. ch ¼

P

1 Rd= γ1

Rd ¼

:

ah n

ð1Þ

Rk γ2

ð2Þ

Where: ah ¼ the slip of the fastening at a load equivalent to Rd=γ . 1 Rd ¼ design resistance of the fastening γ1 ¼ an appropriate factor (1.5 for fastening sheets to substructure with the one-fastener test) n ¼ number of test specimens Rk ¼ is the characteristic resistance γ2 ¼ partial factor for resistance (1.25 according to EN 1993-1-3[8]) The recorded maximum load and its corresponding displacement at elevated temperatures are shown in Table 2. The initial shear stiffness calculated according to Eq. (1) is also given there.

Table 2. Initial stiffness and recorded maximum load during the test with its corresponding displacement. Test Identification No.

Maximum load [KN]

Maximum Elongation [mm]

Initial stiffness from Eq. (1) [KN/mm]

MW_100mm_8mm_20°C MW_100mm_8mm_300°C MW_100mm_8mm_450°C MW_100mm_8mm_600°C MW_230mm_8mm_20°C MW_230mm_8mm_300°C MW_230mm_8mm_450°C MW_230mm_8mm_600°C PIR_100mm_8mm_20°C PIR_100mm_8mm_250°C PIR_100mm_8mm_300°C PIR_100mm_8mm_450°C PIR_160mm_8mm_20°C PIR_160mm_8mm_200°C PIR_160mm_8mm_300°C PIR_160mm_8mm_450°C

1.616 1.708 1.532 0.692 2.324 1.28 2.196 0.888 1.948 1.312 0.924 0.808 1.8 1.6 1.128 0.725

1.308 1.48 2.176 2.792 4.38 4.116 8.032 9.616 6.132 7.48 2.664 7.84 2.44 3.684 6.516 5.014

2.15 1.34 0.862 0.488 0.863 1.076 0.527 0.405 0.667 0.96 0.322 0.195 1.23 0.745 0.19 0.17

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5 CONCLUSIONS The STABFI project is investigating the stabilising effect of sandwich panels and trapezoidal sheets for structural buildings in fire. Under the mentioned project this paper presented the experimental results on the shear behaviour of sandwich panel fasteners both at ambient and elevated temperature. The major failure was a bearing of the inner face of the sandwich panel. The bearing of the steel sheet at ambient temperature was narrow with small folding of sheet while at elevated temperature it teared off in a wider area with pronounced folding of the sheet. This folding of the steel sheet strengthens the steel sheet and enables to carry more load further. There was no shear failure of screws for all the 16 tests at ambient and elevated temperatures except one. With the increase of temperature, both the strength and stiffness of the connections are reduced. The obtained initial shear stiffness of the fasteners from the experiments will be used in the analysis of a whole building for further studies. REFERENCES [1] De Mattes. G, Landolfo. R. Structural behaviour of sandwich panel shear walls. An experimental analysis. Materials and Structures, Vol. 32, 1999, pp. 331–341. [2] EASIE. Ensuring advancement in sandwich construction through innovation and exploitation. Project final report, 2013. [3] ECCS/CIB. European recommendations on the stabilization of steel structures by sandwich panels. Publication 379, ECCS TC7 and CIB W056, 2013. [4] Hedman-Petursson, E. Column buckling with restraints from sandwich wall elements. Doctoral thesis, Steel Structure Division, Department of Civil and Mining Engineering. Lulea, Sweden, 2001. [5] Misiek, T., Käpplein, S., Saal, H. and Ummenhofer, T. Stabilization of beams by sandwich panels – Lateral and torsional restraint. EuroSteel, August 31-September 2, 2011. Budapest, Hungary. [6] ECCS. The Testing of Connections with mechanical Fasteners in Steel Sheeting and Sections, ECCS publication No. 124, 2009. [7] Preliminary European Recommendations for the Testing and Design of Fastenings for Sandwich Panels ECCS publication No. 127, 2009 [8] EN1993-1-2, Eurocode3. Design of steelstructures.Part1-2: General rules. Structural fire design, 2005.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Bracing details for trapezoidal steel box girders S.V. Armijos-Moya, Y. Wang, T. Helwig, M. Engelhardt, E. Williamson & P. Clayton The University of Texas at Austin, Austin, Texas, USA

ABSTRACT: Steel trapezoidal girders (tub girders) with a cast in-place concrete deck on top are a popular alternative for straight and horizontally curved bridges due to their high torsional stiffness and aesthetic appearance. However, steel tub girders possess a relatively low torsional stiffness during construction due to the thin-walled open section that is susceptible to stability issues. Top flange lateral bracing, in the form of a horizontal truss, is installed along the entire length of the steel tub girder to increase the torsional stiffness of the girder. Internal K-frames are placed to control cross-sectional distortion. This paper provides a summary of a research study focused on improving the efficiency of steel tub girders by investigating the impact of bracing details on the behavior of the girders. The study includes large-scale experimental tests and parametric finite element analytical studies. The goal of the study is to propose efficient details for trapezoidal steel girders to make them more cost-effective without undermining their structural performance.

1 INTRODUCTION Steel trapezoidal box girders have become a popular alternative for straight and curved bridges. The girders, often referred to as “tub girders”, consist of a single bottom flange, two sloping webs and two top flanges. The smooth profile of the girder provides an aesthetically appealing bridge that also possesses several structural advantages compared to other girder types. As a result of the large torsional stiffness, the girders are a popular choice in horizontally curved systems where the bridge geometry leads to large torsional moments. However, during construction the girders are an open section and generally require extensive bracing. The primary bracing systems consist of plate diaphragms at the supports, a top flange lateral truss, and intermediate internal and external K-frames (Figure 1). Though tub girders have mainly been used on horizontally curved bridges where concrete girders are not viable due to the longer span lengths or due to the curvature, steel tub girders have also been shown to be feasible for straight bridges with span lengths normally reserved for concrete girder systems. Relatively shallow straight steel tub girders were recently used by the Texas Department of Transportation in the Waco District in the U.S.A, what demonstrates that steel trapezoidal box girders offer a viable alternative for a wider variety of bridge applications. To augment the viability of the tub girders, improved girder geometries and bracing details may lead to improved economy and structural efficiency. Details investigated in this research study include the spacing between internal K-frames, the layout of the top lateral truss, and the cross-sectional geometry of the steel tub girders. Common geometrical practices for the tub girders consist of a 4V:1H web slope and the top flanges centered over the webs. A flatter web slope can lead to increased lateral coverage of a single girder and may eliminate a girder line, thereby improving economy. In addition, offsetting the top flanges towards the inside of the tub girder can provide increased efficiency with respect to connections to the bracing systems. To study the aforementioned-proposed details, three tub girders were fabricated for the experimental program. The experimental studies included loading the girders in vertical bending as well as in combined bending and torsion. Part of the experimental results are summarized in this paper. 96

Figure 1.

Bracing systems in twin tub girder during construction.

Finally, an analytical study to evaluate the torsional response in a continuous bridge under construction loads is presented. This paper focuses on the impact of improved bracing details on the torsional response of tub girders under demands expected on straight and horizontally curved systems.

2 TEST OF LARGE SCALE SPECIMENS 2.1 Description of specimens Three steel tub girders were designed and fabricated for the experimental stage of the study. First, the baseline girder has a web slope of 4V:1H with the flanges centered over the webs. An additional specimen also has a 4V:1H web slope with the top flanges offset towards the inside of the girders (offset top flange girder), while the final specimen has a web slope of approximately 2.5V: 1H and top flanges centered over the web (flatter web girder). All of the internal K-frames and top lateral truss members were bolted to facilitate variations of the bracing between tests. The baseline girder was designed and fabricated according to current engineering practices in the U.S.A. for straight and curved tub girders. The other two specimens were sized by conducting preliminary finite element analyses so that the girders are able to reach global lateral torsional buckling before any type of local buckling. The focus of this study is on both straight and horizontally curved girders. Though the research team considered fabricating horizontally curved girders, laboratory space limitations as well as the limitation of being able to test a single girder curvature was not desirable. Instead, the research team focused on a setup that allowed eccentric loading that can simulate the torsion from the horizontal curvature of the girder. With the ability to offset the load to achieve a torque, girder geometries from straight to a simulated curvature of approximately 600 ft. were possible. 2.1.1 Tub girder geometries The proportions of the girders were selected so that the girders would remain elastic during multiple bending and combined bending and torsion tests. The clear span L of the simply supported specimen was selected to be 84 ft., while the girder depth D was defined as 3 ft. (L/D = 28). A distance W equal to 5 ft. and 3 in. was selected as the separation of the top of the sloped webs (Figure 2). The resulting width-to-depth ratio (W/D) was 1.75, which is similar to values observed in current practice. The major difference between specimens is the thickness of the cross-section plates, the location of the top flanges with respect to the webs, and the slope of the webs. All the flanges and webs were fabricated with steel AASHTO M270 (ASTM A709), grade 50W. 97

Figure 2.

Specimen cross-section: a) baseline girder, b) top flange offset girder, c) flatter web girder.

The baseline steel tub girder was sized with webs sloped to 4V:1H (Figure 2a). The thickness of webs and flanges was set equal to 7/16 in., what is considerably smaller than commonly utilized in current bridge practice (1 in). However, this thickness was deemed necessary to obtain the elastic-buckling response of the system based upon finite element studies. This base line tub girder was built with two 12” wide top flanges which were centered to the center line of the sloped webs. The offset top flange girder was built with two 13” wide top flanges which were connected to the sloped webs at 1” from the edges, leaving 12” of unstiffened plate (Figure 2b). Finite element analyses were carried out to define the thickness of the top flanges for this second specimen to assure an elastic behavior of the girder during the tests. The top flange thickness was set equal to 9/16”. The bottom flange and sloped webs were sized with 7/16” thick plates. The flatter web girder, on the other hand, was fabricated with web slopes equal to approximately 2.5V:1H (Figure 2c), which exceeds the limits of AASHTO 2014. Similar to the baseline tub girder, the flatter web girder was built with top flanges centered over the webs. Webs and flanges were 7/16 in. thick. 2.1.2 Bracing geometry The spacing of the top lateral truss panel points was defined as 7 ft., generating 12 panels along the length of the beam (Figure 3). The internal K-frames and top lateral diagonals were able to be installed or removed as desired to study the behavior of the girders as the bracing is varied. In the cases where the internal K-frames or top lateral truss diagonals were removed, top lateral struts between the two top flanges were kept at a 7 ft. spacing to control separation of the top flanges. The single-diagonal type (SD-type) top truss was used as the top lateral system. This system is formed by single diagonals and struts connected to the tub girder top flanges. The top truss diagonals were WT5 x 22.5 members designed to be connected directly underneath the top flanges through three 3/4 in. high strength bolts. The strut cross-section was sized as a 2 in diameter x-strong pipe (2.375 in. outside diameter and 0.218 in. wall thickness). The struts were connected to a stiffener welded to the web of the tub girder through bolted connections made of 1/2 in. thick steel plates (ASTM A-36) and 7/8 in. high strength bolts. The vertical

Figure 3.

Example of bracing layout - half of baseline steel tub girder specimen - Plan view.

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eccentricity between the top flange and the centerline of the strut is 3.75 in. which is an acceptable value (Helwig and Yura 2012). The diagonals and pipes were designed and fabricated with steel ASTM A705, Grade 50 and ASTM A53, Grade B, respectively. Three diagonals were installed at each end of the steel tub girders to simulate partial lateral bracing of the top flange. Different cases of partial top lateral bracing were tested by removing diagonal members of the horizontal truss at each end (4 layouts). One strut (which is part of the top lateral truss) and two diagonals formed the internal K-frames. The section of the strut was sized for the top lateral bracing system, and the same section has been adopted for the K-frame diagonals (2 in. x-strong) for facility during fabrication. The K-frame bracing elements were fabricated with ASTM A53 – Grade B steel. Three different arrangements of internal K-frames were tested for each configuration of top lateral bracing. K-frame bracing at every 2, 4 and 6 panel points were evaluated during the experimental program. 2.2 Description of test setup The test setup (Figure 4a) consisted of two steel supports 84 ft. apart over which each specimen was tested as simply-supported straight girder under both pure positive bending and torsional loading conditions. Each steel support consists of three 12 ft. long W36 x 135 rolled beams stacked vertically so as to raise the elevation of the test girders above the loading system. The support located on the south side of the laboratory floor was supported laterally with two diagonal braces to stiffen the test setup and simulate “pinned conditions”. The opposing support consisted only of the stacked W36 x 135 sections and allowed some flexibility to simulate a “roller”. Two gravity load simulators (GLS) shown in Figure 4b were used to apply either pure bending or bending with torsion. Each GLS is able to apply vertical loads up to 160 kips, and to keep the load vertical even if the ram moved laterally up to 6 in. Consequently, the GLS provides minimal lateral restraint and essentially “simulates gravity load”. 2.3 Instrumentation and initial imperfections Two 100-kip load cells were used to measure the loads applied with the two GLS. Horizontal and vertical deflections of the specimens were measured at the third points along the tub length (28 ft. and 56 ft.) and at mid-span (42 ft.). The deflections at the third points were obtained with four string potentiometers; while at mid-span they were collected with two infrared cameras that were able to monitor the signal from LED markers attached to the tub girder with relatively high accuracy (error of approximately 0.01 mm). Rotations were calculated from the measured deflections.

Figure 4.

a) Test setup, b) Gravity load simulator (GLS).

99

Prior to testing, initial imperfections of each steel tub girder were measured. Two piano wires were extended between the test setup supports located 6 in. from both edges of the bottom flange. The taut wires served as reference point to measure lateral and vertical out-of-straightness of the tub girder. The baseline, top flange offset, and flatter web girders had an initial twist at midspan of 1.30, 1.60, and 2.30 degrees, respectively; and a maximum out-of-straightness on top flange of about L/1300 towards the east, L/750 towards the west, and L/500 towards the east, respectively. 2.4 Testing procedure Elastic-buckling tests were carried out by limiting the maximum loads applied to the specimen to keep stresses below 60% of nominal yield stress (30 ksi) to consider the impact of residual stresses and initial imperfections in the response, and to ensure that the girders remained elastic. Two types of loading conditions were studied: vertical positive bending and combined bending and torsion due to vertical eccentric loads (to simulate horizontal curvature). Two vertical loads were applied with gravity load simulators at approximately quarter points of the specimen (location denoted as “Pa” on Figure 3). Henceforth, the load on each GLS will be referred to as load “P”. The combined vertical bending and torsional demands were obtained by applying vertical eccentric loads at 8 in. and 16 in. from the shear center of the girders to simulate demands produced by curvature in tub girders with radii of curvature equal to 1260 and 630 ft., respectively. 2.5 Bracing configuration To measure the impact of bracing in the response of the specimens, different bracing layouts were tested on the each tub girder under the same loading conditions. Four configurations of top lateral bracing (zero, one, two, and three diagonals on each end of the simple supported girder) and three layouts of internal K-frames (frames at every two, four, and six panel points) were tested, which resulted in a total of 12 tests. These 12 configurations of top lateral and K-frame bracing were evaluated for the three cases of vertical loads (concentric, eccentric at 8 in., and eccentric at 16 in.) producing a total of 36 elastic tests performed on each specimen. The impact of each bracing layout in the response of the specimens was evaluated and is summarized in the next sections. 2.6 Experimental results 2.6.1 Impact of partial top lateral bracing distribution in stiffness To simulate the demands on straight tub girders, the GLSs were used to apply vertical concentric loads near quarter points. Figure 5 shows the total vertical load applied (2P) versus the twist angle of the baseline specimen at midspan (β), when the specimen was tested with zero, one, two, and three bracing diagonals at each end. K-frames were kept every 2 panel points. The tub girder without top lateral bracing presented a deformation curve that suggested the girder was approaching the elastic lateral torsional buckling limit during the test, which can be observed by the significant nonlinear response of the load versus deflection curve. The torsional stiffness of the specimen reduced significantly as the girder approached the lateral torsional buckling limit. The capacity to resist LTB is significantly improved with the addition of diagonals at the ends of the girder. The system without diagonals had a twist at midspan of 1.88 deg. at 70 kips of total load; while the specimen with 1 diagonal per end had a twist at midspan of 0.30 deg. at the same load step. This is a reduction of about 85% in the rotation of the cross-section and indicates that the torsional stiffness is highly improved with a single diagonal at each end. The baseline specimen with two truss diagonals per end presented a twist angle of 0.09 deg. showing a clear improvement in the torsional stiffness of the specimen. The tub girder with three diagonals per end did not show a significant improvement in torsional stiffness with respect to the previous case. Instead, the three diagonals per end produced a shift in the direction of lateral movement (shift of mode shape). The first diagonal on each end of the specimen produced the most significant improvement in the resistance to LTB, while additional

100

Figure 5.

Total GLS load vs twist angle at midspan – Baseline specimen (concentric loading).

diagonals were not as effective at improving the behavior. The experimental results shown that the effectiveness of the top lateral truss is lower with increasing distance from the girder ends. When applying eccentric vertical loads, similar trends to the concentric cases was observed. The specimens showed poor torsional resistance when no top lateral diagonals were installed. The torsional stiffness of the girders was enhanced when bracing diagonals were installed at each end because they restricted the warping deformations on the girders. Figure 6 plots the absolute values of torsional response of the baseline girder with zero and three diagonals per end subjected to concentric and eccentric loads. Similar to the concentric loading case, K-frames were maintained at every 2 panel points. As expected, in regards to the unbraced cases, the torsional stiffness of the baseline girder goes down when the torsional

Figure 6.

Total GLS load vs twist angle at midspan – Baseline specimen with 0 and 3 diagonals.

101

demands increase. The initial torsional stiffness of the unbraced concentric case is about 4 and 12 times higher than the cases with loads applied at 8 in. and 16 in. of eccentricity, respectively. Regarding the braced cases, with three diagonals, the specimen shows lower differences in the torsional stiffness between the three loading cases. The stiffness observed during the concentric test is about 1.5 and 4 times higher than the results obtained with eccentric loading at 8 in. and 16 in., respectively. As previously discussed, the addition of top bracing diagonals produced a high increment in the torsional stiffness. The torsional stiffness increased about 10 to 30 times after installing partial top lateral bracing with three diagonals. The horizontal truss diagonals are more effective at the supports where warping deformations are higher. Similar type of impact on the torsional behavior was observed in the other two specimens. 2.6.2 Impact of internal K-frame distribution in stiffness Figure 7 shows the total vertical load (2P) versus the twist at midspan (β) of the baseline girder with no top lateral bracing and three different configurations of internal K-frames subjected to concentric vertical loads. The tub girder with internal bracing at every four and six panels show the same response with no major variation in the torsional stiffness. Torsional demands in straight tub girders are small which implies that distortional effects are low. Even though the specimen with K-frames every 2 panels presents higher torsional stiffness for lower load levels, the impact on the response tended to be similar to the aforementioned two cases at higher loads. Elastic lateral torsional buckling was observed during the three tests. The relative insensitivity of girder response to the internal K-frame spacing is similar to previous observations in the case of the Marcy Pedestrian Bridge failure (Yura and Widianto 2005), as well as the system-buckling mode of narrow I-girder systems (Yura et al. 2008). Consequently, the K-frame bracing system becomes less effective in straight steel tub girders under pure positive bending, and there is no major change in its torsional behavior when the number of internal braces is reduced. Similar effect was observed in the other two specimens with partial top lateral bracing, including two and three diagonals at each end.

3 ANALYTICAL STUDY Parametric studies were conducted to further confirm the experimental findings and evaluate the effects of partial top lateral bracing on the stiffness of steel tub girders with different girder geometries and configurations. As part of the parametric study, the section in Figure 8 was evaluated with different amounts of partial-length top lateral bracing for both simply supported and continuous girder configurations. The simply supported tub girder was modeled as straight girder

Figure 7.

Total load vs midspan twist angle - Different K-frame layout (No TLB) – Baseline girder.

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Figure 8.

Prototype tub girder for analytical study.

with a clear span of 216 ft. (L/D ≈ 30), while the continuous tub girder was modeled as 2-equalspan horizontally curved system (radius of curvature = 2500 ft) with clear spans of 216 ft. as well. The cross-section dimensions of each prototype section were re-proportioned accordingly to satisfy the load demand, and the corresponding plate thicknesses for both systems are presented in Table 1. The number of top lateral truss panels for each girder was set at 16 panels per span with a panel length of 13.50 ft. Following current engineering practices, K-frames were placed every panel point for all the cases presented in this section. Different amounts of partial top lateral bracing were considered for both tub girders. The minimum levels of bracing for adequate performance during construction for both girder systems was then evaluated. Lateral displacements of the top flanges (δ) were examined to evaluate the load-deflection response of the prototype steel tub girders in the analytical study. The results were normalized by maximum allowable displacement and section twist defined for this study. In this study, a limitation of L/1000 is defined for lateral displacement of the top flange based on the maximum out-of-straightness fabrication tolerance specified in the AISC Code of Standard Practice (2005) for straight compression members. FEAs were first carried out on the straight girder case with various amount of top lateral bracing to identify the minimum required top lateral bracing to satisfy the admissible requirements described above. The layout of top lateral bracing varies from non-braced to fully braced cases. Figure 9 shows the sample lateral displacement responses of the simply supported girder with seven different layouts of top lateral bracing. (i.e., three diagonals (0.38 L) at each end, four diagonals (0.5 L) at each end and a fully braced system (1.0 L)). The X-axis of the plot shows the lateral displacement of top flange normalized by L/1000 (δmax = 2.60 in), while the Y-axis shows the bending moment normalized by an estimated moment produced by construction loads. In the plot, the different cases are labelled based on the percentage of braced length with respect to the girder span length. In Figure 9, the minimum amount of bracing required not to exceed the admissible lateral displacement (1.0 in the X-axis) is

Table 1. Plate thicknesses for prototype girder. Girder System

Simply Supported Continuous (2 Span)

bf (top)

tf (top)

tf (bot)

tw

in

in

in

in

20” 30”

2.5” 3.0”

2.0” 2.0”

0.81” 0.81”

103

Figure 9.

Bending moment vs lateral displacement of top flange – simply-supported girder.

50% of bracing (0.5 L) under the assumed construction demands (1.0 in the y-axis). Beyond this case, the girder starts to exhibit larger deformations. Similar FEAs were performed to examine the possibility of using partial top lateral trusses on horizontally curved tub girders. It was found that a radius of curvature equal to 2500 ft. was adequate to comply with the aforementioned requirements. Additionally, the top flanges within the unbraced length had to be resized because the lateral bending stresses induced in top flanges were higher that the yielding stress. The requirements were satisfied after resizing the top flanges for the unbraced length and choosing a curvature equal to 2500 ft. To resize the top flanges for the unbraced length, the lateral bending stresses in the top flanges were calculated considering the unbraced length of the top flanges between partial top lateral bracing with warping restrain at the transition points (panel point where the tub girder changes from top laterally braced to unbraced). After resizing the top flanges, the girders in this study were able to satisfy the response requirements for horizontal curvatures of 2500 ft or higher. Figure 10 shows the bending moment versus lateral displacement on the top flange of the critical span for the two span continuous girder with radius of curvature equal to 2500 ft. The most critical construction loading scenario was obtained when the construction loads were applied over the positive moment region of one span. These analyses indicate that 50% partial top lateral bracing is a reasonable minimum amount of bracing to keep maintain admissible girder deformations during construction for straight and mildly horizontally curved steel tub girders with minimum radius of curvature of 2500 ft. The horizontally curved girders in this study showed acceptable response up to lengths of 216 ft. with 50% partial bracing. Longer lengths required additional top lateral bracing (60%) to maintain acceptable behavior. Additionally, it was observed during the finite element analyses that when partial top lateral bracing is used, a K-frame has to be installed in the transition panel point (between the girder with top lateral diagonals and the unbraced girder) in order to avoid detrimental impact in the torsional stiffness of the girder. K-frames should be placed at the transition points and at midspan. It is recommended to use internal K-frames every 2 panel points and at transition zones for adequate torsional performance.

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Figure 10. Bending moment vs lateral displacement of top flange – continuous girder.

4 CONCLUSIONS – Experimental tests and analytical analyses showed that the top flange lateral bracing systems are more effective in the region near to the supports of straight girders where shear deformations are at the maximum. The LTB capacity of the straight tub girders was significantly improved by adding 50% of top truss diagonals at supports. The inclusion of subsequent diagonals resulted in significantly smaller increments in the performance as the distance to the supports increased. Thus, top lateral diagonals located near mid-span add little to no benefit in the LTB behavior and likely at increasing the torsional stiffness of the girder. – Internal K-frames provide minimal contribution to resist LTB in straight tub girders in comparison to top lateral bracing. Due to lower torsional demands, internal K-frames are less effective along straight tub girders. Thus, the number of K-frames, and their distribution along the straight girder, did not shown a significant impact in the torsional response of the girder. – When partial top lateral bracing is used, a K-frame has to be provided in the transition zone between braced and unbraced girder in order to maintain adequate torsional response.

REFERENCES American Association of State Highway Transportation Officials (AASHTO) (2014). “AASHTO LRFD Bridge Design Specifications, 6th Ed.” American Association of State Highway and Transportation Officials, Washington, D.C. Helwig, T. and J. Yura (2012). “Steel Bridge Design Handbook: Bracing System Design”, U.S. Department of Transportation Federal Highway Administration. 13. Yura, J. A., and Widianto, (2005), “Lateral buckling and bracing of beams – A re-evaluation after the Marcy bridge collapse”, Proc., Structural Stability Research Council, Montreal, April 7–9, pp 277–294 Yura, J., Helwig, T.A., Herman, R., and Zhou, C. (2008), “Global Lateral Buckling of I-Shaped Girder Systems,” ASCE Journal of Structural Engineering, Vol. 134, No. 9, pp. 1487–1494, September.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Behaviour of slender plates in case of fire of different stainless steel grades F. Arrais, N. Lopes, P. Vila Real & C. Couto RISCO—Civil Engineering Department, University of Aveiro, Portugal

ABSTRACT: Stainless steel has countless desirable characteristics for a structural material. Although initially more expensive than conventional carbon steel, stainless steel structures can be competitive because of their smaller or none need for thermal protection material and lower life-cycle cost, thus contributing to a more sustainable construction. Regarding structural fire resistance, in order to have a comprehensive understanding of the overall members’ resistance, it is important to first analyse the cross-section resistance, directly affected by local instabilities occurrence on the composed thin plates. This work presents a numerical study on the behaviour of isolated plates at elevated temperatures, corresponded to the web (internal element) and flanges (outstand element) of I-cross sections, comparing the numerically obtained ultimate load bearing capacities with simplified calculation formulae for the application of the effective width method. Comparisons between the numerical results and the EC3 formulae for determining the effective area of thin plates is also presented.

1 INTRODUCTION The application of stainless steel as a structural material has been increasing, due to a number of desirable qualities such as its durability, resistance to corrosion and aesthetic appearance (Gardner, 2005 & Euro Inox, 2006). Despite having a high initial cost, stainless steel can be a competitive material if life cycle cost analysis is considered, due to its low maintenance needs. Moreover, it has a higher fire resistance when compared to carbon steel (CEN, 2005b) allowing in some cases the absence of thermal protection. The austenitic stainless steels are generally the most used groups for structural applications but some interest has being recently shown for increasing the use of ferritic and austenitic-ferritic (Duplex) steels for structural purposes due to specific advantages. Some of those advantages are the very good resistance to wear and stress corrosion cracking of the duplex grade and the lower percentage of Nickel of the ferritic grade, which reduces its price. Regarding structural fire resistance, in order to have a comprehensive understanding of the overall members’ resistance, it is important to first analyse the cross-section resistance, which is directly affected by local instabilities occurrence on the composed thin plates. For structural design purposes, Eurocode 3 (EC3) (CEN, 2006a) considers that the walls slenderness determine the cross-section classification (from Class 1 - stocky sections to Class 4 - slender sections). Subsequently, the cross-section resistance is calculated considering plastic section properties for Classes 1 and 2 sections, elastic section properties for Class 3 sections and effective section properties, applying the effective width method, for Class 4 sections. In addition, for cross-section of Classes 1, 2 and 3 at elevated temperatures the strength at 2% total strain should be considered as the yield strength and for Class 4 cross-sections it should be applied the 0.2% proof strength (CEN, 2005b). Although the subject of local buckling at elevated temperatures has been studied by different authors (Couto et al., 2014, Couto et al., 2015, FIDESC4, 2014, Knobloch & Fontana, 2006, Maraveas et at., 2017, Quiel & Garlock, 2010), the mentioned research works only 106

address carbon steel elements and research of the local buckling effect on stainless steel sections at elevated temperatures is scarce and mostly focus on the member behaviour. According to Part 1-2 of EC3 (CEN, 2005b) design rules, stainless steel stress-strain relationships at elevated temperatures are characterized by having an always non-linear behaviour with an extensive hardening phase, when compared with carbon steel constitutive law. As existing fire design guidelines for stainless steel, such as in EN 1993-1-2 (CEN, 2005b), are based on the formulations developed for carbon steel members (CEN, 2005a, CEN, 2006b), in spite of their different material behaviour, it is still necessary to develop knowledge on stainless steel structural behaviour at elevated temperatures. This research work has the main objective of analysing the accuracy of EC3 present calculation proposals for stainless steel cross-sections in case of fire, subjected to compression or bending, by means of Geometrical and Material Non-linear Analysis with Imperfections applying the Finite Element software SAFIR (Franssen & Gernay, 2017). Plates behaviour at elevated temperatures is analysed considering compression or bending and different boundary conditions for modelling isolated outstand elements (flanges) and internal elements (webs), following the methodology used for the development of carbon steel design approaches (Couto et al., 2014, FIDESC4, 2014, CEN, 2006b). In this parametric study, as different stainless steel grades exibit different stress-strain relationships behaviours at elevated temperatures (CEN, 2005b), the following grades were considered: i) 1.4301 (Austenitic grade); ii) 1.4003 (Ferritic grade); iii) 1.4462 (duplex). Comparisons between the obtained numerical results, the EC3 design methods and a recent proposal for Class 4 carbon steel sections (Couto et al., 2015), are made, being concluded that new design expressions should be developed for the effective with method application on stainless steel I-sections subjected to fire.

2 SIMPLIFIED DESIGN RULES 2.1 Eurocode3 According to EN 1993-1-2 (CEN, 2005b), the section resistance of a stainless steel member in case of fire is calculated in the same way as for carbon steel, changing only the mechanical properties of the material to consider uniform elevated temperatures in the section. Regarding the cross-section classification, Equation 1 was used to determine the factor ε, a parameter necessary for the determination of the EC3 classification limits (Franssen & Vila Real, 2015). εθ ¼ 0:85

235 E fy 210000

0:5 ð1Þ

The design resistance value of axially compressed members of Class 1, 2 or 3 cross-sections with a uniform temperature θa is determined from Equation 2. Nfi;t;Rd ¼ A fy;θ =γM;fi

ð2Þ

For Class 4 sections, according to Annex E of EN 1993-1-2, the effective area (Aeff ), obtained from EN 1993-1-5 (CEN, 2006b), should be considered instead of the gross cross-section area A. In a fire situation higher strains are acceptable when compared to normal temperature design, therefore, instead of 0.2% proof strength usually considered at normal temperature, for crosssection of classes 1, 2 and 3 at elevated temperatures the stress corresponding to 2% of total strain should be adopted as the yield strength (CEN, 2005b).

107

fy;θ ¼ f2%;θ ¼ k2%;θ fy

ð3Þ

However, for Class 4 cross-sections, according to Annex E of EN 1993-1-2, the proof strength at 0.2% strain should be used, thus fy;θ ¼ f0:2p;θ ¼ k0:2p;θ fy

ð4Þ

The mentioned reduction factors are given on Annex C of EN 1993-1-2 for stainless steel at high temperatures for the different analysed stainless steel grades. In beams, the design value of the bending moment resistance of a cross-section with a uniform temperature θa is determined from: h i Mfi;θ;Rd ¼ ky;θ γM;0 =γM;fi Mc;Rd

ð5Þ

Being Mc; Rd for Classes 1 and 2 the plastic bending moment capacity, for Class 3 the elastic bending moment capacity, and for Class 4 sections the effective bending moment capacity, at normal temperature, determined with the effective section properties obtained from EN 19931-5. The effective area and effective section modulus (Weff ;y ) are determined through the application of the effective width method, considering the reduction of resistance due to local buckling effects (CEN, 2006b). On this regard, the EN 1993-1-4 (CEN, 2006a) provides specific equations for the determination of the plate reduction factors (ρ) to the width of elements composing the stainless steel sections, as presented in Equation 6 and Table 1. It can be observed that the reduction factor for internal elements do not depend on the stress distribution as proposed in carbon steel plates (CEN, 2005a). beff ¼ ρ:b

ð6Þ

The plate slenderness – λp – value is determined with Equation 7. sffiffiffiffiffiffi  b t fy pffiffiffiffiffi λp ¼ ¼ σcr 28:4ε kσ

ð7Þ

2.2 Proposal for class 4 carbon steel sections at elevated temperatures As mentioned before, recent research works (Couto et al., 2015, Knobloch & Fontana, 2006) proposed the use of the stress corresponding to 2% of total strain as the steel yield strength also for Class 4 cross-sections at elevated temperatures, as it is done for the remaining

Table 1. Reduction factor for stainless steel sections elements. Cross-section elements

Reduction factor

Welded outstand elements

ρ ¼ λ1p  0:242 1 λ2

Welded internal elements

ρ¼

108

p

0:772 λp

 0:125 1 2 λ

p

Table 2. Reduction factor proposed for carbon steel sections elements in case of fire (Couto et al., 2015). Cross-section elements Outstand elements Internal elements

Reduction factor ρ¼

1:2 ð
λp þ 1:10:52 ε Þ 0:188

 1:0 2:4 ð
λp þ 1:10:52 ε Þ 1:5 ð
λp þ 0:90:26 ε Þ 0:055ð3þψÞ  1:0 ρ¼ 3 ð
λp þ 0:90:26 ε Þ

sections, providing the plate reduction factors would be calculated as presented in Table 2. The accuracy of the application of this proposal for stainless steel sections is tested in this paper.

3 PLATES BEHAVIOUR 3.1 Numerical modelling Members composed of different cross-section shapes may exibit diferent plates behaviour. For instance in I-shape sections subjected to compression have both flanges and web in compression, whereas when the members are subjected to bending in the strong axis, a flange is in compression while the web is subjected to bending. Rectangular hollow sections subjected to compression will have only internal elements subjected to compression and when subjected to bending will have an internal element subjected to compression and others internal elements subjected to bending. To determine the ultimate load of rectangular plates the program SAFIR was used. Each shell element has four nodes with six degrees of freedom (three translations and three rotations). Simply supported conditions where applied to the plates by restraining the vertical displacements, in addition the rotations at the edges of the plate were also restrained to simulate the web-flange continuity. For the outstand elements, the vertical displacements were restrained on three sides while for the internal elements the vertical displacements were restrained in all four sides, this methodology follows the same principles as in Couto et al., (2014). Figure 1 presents the obtained deformed shapes of an outstand element subjected to compression, an internal element subjected to compression and an internal element subjected to bending. Geometric imperfections were introduced into the numerical model by changing the nodal coordinates affine to the buckling mode shapes obtained with the program CAST3M (CEA, 2012) and applying the interface RUBY (Couto et al., 2013). For the amplitude of the imperfections, it was considered 80% of b/50 for outstand elements and 80% of b/100 for internal elements, following the recommendations of EN 1090-2 (CEN, 2011). Plates of the stainless steel grade 1.4301, 1.4003 and 1.4462 (CEN, 2006a) subjected to four temperatures were considered 350ºC, 400ºC, 450ºC and 500ºC (common critical temperatures in slender sections). The nominal values applied of yield strength, ultimate strength and elastic modulus of stainless steel in the numerical models are presented in Table 3. These mechanical properties are reduced at elevated temperatures as presented in Figure 2, which vary for each grade. 3.2 Plates subjected to compression The results obtained for outstand and internal plate elements subjected to compression are here presented. At elevated temperatures, the equation to determine the reduction factor has to be adapted due to the transition that occurs from Class 3 to Class 4 section because of the change on the limit strength, leading to a discontinuity in the curve, as presented in Equation 8. 109

Figure 1. Deformed shapes (x5): a) outstand element subjected to compression; b) internal element subjected to compression; c) internal element subjected to bending.

Table 3. Nominal values for different stainless steel grades (CEN, 2006a). Type

Grade

Yield strength fy (MPa)

Ultimate strength fu (MPa)

Elastic modulus E (GPa)

Austenitic Ferritic Duplex

1.4301 1.4003 1.4462

210 280 460

520 450 660

200 220 200

Figure 2. Mechanical properties reduction at elevated temperatures (CEN, 2005b): a) yield strength retention; b) young modulus reduction.

ρθ ¼

fy;θ N c;Rd ¼ρ N Rd f2;θ

110

ð8Þ

Figure 3. Results for a) outstand elements and b) internal elements subjected to compression for austenitic stainless steel at elevated temperatures.

Figures 3 to 5 presents the comparisons between the ultimate load bearing capacities, for outstand and internal plate elements subjected to compression, obtained with EC3, the new proposal for Class 4 carbon steel elements (“CS New Proposal” in the chart) and SAFIR. For both outstand and internal elements subjected to compression, the proposal for carbon steel Class 4 sections (Couto et al., 2014) eliminates the un-conservative nature given by the plateau of EC3 for austenitic (Figure 3) and ferritic stainless steel (Figure 4). The results for austenitic-ferritic stainless steel (Figure 5) revealed that the rules are over conservatives. Nonetheless, the results highlight the need of improved design equations specifically developed for stainless steel plates subjected to compression at elevated temperatures, considering the stainless steel grade.

Figure 4. Results for a) outstand elements and b) internal elements subjected to compression for ferritic stainless steel at elevated temperatures.

Figure 5. Results for a) outstand elements and b) internal elements subjected to compression for austenitic-ferritic (Duplex) stainless steel at elevated temperatures.

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3.3 Plates subjected to bending The obtained results for internal plate elements subjected to bending are presented in Figure 6. The ultimate bending moments obtained in each plate for all methods were divided by the plastic bending moments. The different plateaus, in this figure, observed in Eurocode procedure of EN 1993-1-4 correspond to the transitions between Class 2 and Class 3 sections (from plastic to elastic resistance) and from Class 3 to Class 4 where at elevated temperatures the yield strength changes, as mentioned before. The curves from both proposals are over conservative when compared with the numerical results, which leads to conclude that specific formulae for stainless steel plates should be developed, which can be observed specifically for austenitic-ferritic stainless steel. 3.4 Statistical analysis The average value (μ) and the standard deviation (s) are important values to take into account in the statistical analysis for the different methodologies of EC3 and different Proposals. For each stainless steel grade and for each analysed curve it is possible to evaluate the ratio between the analytical value and the corresponding SAFIR (Figures 7 and 8). From Figures 7 and 8 and Table 4 it is possible to observe that the different design rules are not adapted for stainless steel thin plates with low average values and high standard deviations. The number of unsafe results is also relevant for the accuracy of these design rules.

Figure 6. Results for internal elements subjected to bending for a) austenitic, b) ferritic and c) austenitic-ferritic (Duplex) stainless steel at elevated temperatures.

112

Figure 7.

Comparison between EN 1993-1-4 and SAFIR – Grades a) 1.4301, b) 1.4003 and c) 1.4462.

Figure 8.

Comparison between CS New Proposal and SAFIR – Grades a) 1.4301, b) 1.4003, c) 1.4462.

Table 4. Statistical evaluation for different stainless steel grades at high temperatures Steel Grade 1.4301 1.4003 1.4462

Design rule

No. of simulations (n)

Average value (μ)

Standard deviation (s)

% Unsafe

EN 1993-1-4 CS New Proposal EN 1993-1-4 CS New Proposal EN 1993-1-4 CS New Proposal

283 283 302 302 276 276

0.8916 0.7568 0.9025 0.7391 0.8223 0.7576

0.1564 0.1798 0.1029 0.1915 0.1283 0.1262

14.1% 0.0% 17.2% 1.0% 1.4% 1.4%

4 CONCLUSIONS This work presented a numerical study regarding the plates’ behaviour of different stainless steel grades (austenitic, ferritic and austenitic-ferritic (Duplex) stainless steel), composing the cross-sections of members in fire situation. In order to better understand the behaviour of these stainless steel sections, thin plates at elevated temperatures were analysed. This study, on compressed outstand elements, compressed internal elements and internal elements subjected to bending, concluded that EC3 does not provide accurate and safe approximations to their numerically obtained counterparts regarding the ultimate load bearing capacities. Following this conclusion, a recent proposal for carbon steel plates (Couto et al., 2014) was also investigated. It was observed that using

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this proposal allowed to overcome the unsafety that was previously observed for Class 3 sections, but results remained too conservative for Class 4 sections. In summary, the prediction of the resistance of stainless steel members for the case of fire is still not completely understood, thus motivating and justifying the development of more studies with the objective to achieve more precise and safe formulations for these members. ACKNOWLEDGEMENTS This research work was performed within the framework of the project “Fire design of stainless steel members” - StaSteFi - POCI-01-0145-FEDER-030655, supported by the Operational Program “Competividade e Internacionalização”, in its FEDER/FNR component, and the Portuguese Foundation for Science and Technology (FCT), in its State Budget component (OE). REFERENCES CEA. 2012. CAST 3M research FEM environment. development sponsored by the French Atomic Energy Commission http://www-cast3m.cea.fr/. CEN European Committee for Standardisation. 2005a. EN 1993–1–1, Eurocode 3: Design of steel Structures – Part 1–1: General rules and rules for buildings. Belgium. CEN European Committee for Standardisation. 2005b. EN 1993–1–2, Eurocode 3, Design of Steel Structures – Part 1–2: General rules – Structural fire design. Belgium. CEN European Committee for Standardisation. 2006a. EN 1993–1–4, Eurocode 3: Design of steel Structures – Part 1–4: General rules – Supplementary Rules for Stainless steels. Belgium. CEN European Committee for Standardisation. 2006b. EN 1993–1–5, Eurocode 3: Design of steel Structures – Part 1–5: Plated structural elements. Belgium. CEN European Committee for Standardisation. 2011. EN 1090–2, Technical requirements for the execution of steel structures. Belgium. Couto, C., Vila Real, P. & Lopes, N. 2013. RUBY an interface software for running a buckling analysis of SAFIR models using Cast3M. University of Aveiro. Couto, C., Vila Real, P., Lopes, N. & Zhao, B. 2014. Effective width method to account for the local buckling of steel thin plates at elevated temperatures. Thin Walled Structures, 84, 134–149. Couto, C., Vila Real, P., Lopes, N. & Zhao, B. 2015. Resistance of steel cross-sections with local buckling at elevated temperatures. Journal of Constructional Steel Research, 109, pp. 101–114. Euro Inox, SCI, Steel Construction Institute. 2006. Design Manual for Structural Stainless Steel. 3rd ed. FIDESC4. 2014. Fire Design of Steel Members with Welded or Hot-Rolled Class 4 Cross-Section. RFCS-CT-2011-2014, Technical Report No. 5. Franssen, J-M. & Vila Real, P. 2015. Fire Design of Steel Structures. ECCS; Ernst & Sohn, a Wiley Company, 2nd edition. Franssen, J-M. & Gernay, T. 2017. Modelling structures in fire with SAFIR®: theoretical background and capabilities. Journal of Structural Fire Engineering. Knobloch, M., & Fontana, M. 2006. Strain-based approach to local buckling of steel sections subjected to fire. Journal of Constructional Steel Research, 62(1–2), 44–67. Gardner, L. 2005. The use of stainless steel in structures. Progress in Structural Engineering and Materials, vol 7, pp 45–55. Maraveas, C., Gernay, T. & Franssen, J-M. 2017. Amplitude of local imperfections for the analysis of thin-walled steel members at elevated temperatures, Applications of Structural Fire Engineering (ASFE’17), Manchester, UK. Quiel, S.E., & Garlock, M.E.M. 2010. Calculating the buckling strength of steel plates exposed to fire. Thin-Walled Structures, 48(9), 684–695.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Numerical modelling of cold-formed steel members at elevated temperatures F. Arrais, N. Lopes & P.Vila Real RISCO — Civil Engineering Department, University of Aveiro, Portugal

M. Jandera Czech Technical University, Prague, Czech Republic

ABSTRACT: Steel structural elements composed of cold-formed thin-walled sections have a high susceptibility to the occurrence of different buckling phenomena, particularly at elevated temperatures, such as local, global and distortional buckling. Due to the high costs associated with fire experimental tests, the evaluation of those cold-formed steel members’ resistance and behaviour has mainly been carried out through numerical studies, which should be duly validated. Hence, this work presents numerical simulations of experimental works performed by different authors, considering cold-formed simply supported beams and columns under fire situation, contributing to the establishment of corroborated procedures.

1 INTRODUCTION The cold-formed steel profiles can be applied to almost all existing buildings typologies. Currently, cold-formed profiles are commonly used in buildings due to its lightness and ability to support large spans, being quite common as roof or walls support elements (Silvestre and Camotim, 2010a). Besides the mentioned characteristics, other advantages are provided such as: the high strength and stiffness; the faster manufacturing process and with relatively light loads; the easy prefabrication and mass production; the favourable strength-to-weight ratios; the cross-section’s shape allowing the compact packaging; the economy in transportation, handling and the sustainability on construction (Yu, 2000; Silvestre et al., 2013). On one hand, compared with other materials, the combination of previous qualities can result in cost saving in construction. On the other hand, structural steel elements with thinwalled sections are characterized by being subjected to the possibility of different failure modes occurrence such as local, distortional and global (for example, lateral-torsional buckling on elements under bending and flexural buckling on elements under compression). The instability phenomena and its influence on the ultimate strength at normal temperature have been widely studied (Gonçalves and Camotim, 2007; Silvestre and Camotim, 2010b), but the corresponded behaviour under fire still requires further research. Thus, cold-formed steel profiles behaviour in fire has recently received more attention (Laím et al., 2015, Arrais et al., 2016, Muftah et al., 2016). In fact, the fire resistance evaluation of cold-formed profiles may have a major role in the design of these elements. According to Sidey and Teague (1988), the cold-formed steel strength can suffer a reduction of 10 to 20% higher when compared to hot rolled profiles due to the metallurgical composition. The thin walls of these members, together with the steel’s high thermal conductivity, are the reason for the great loss of strength and stiffness of these structural elements in fire situation. Due to the high costs, there is still a small number of fire experimental tests on cold-formed steel profiles, and the evaluation of their resistance has mainly been carried out through numerical studies, which should be duly validated.

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This work aims at contributing to the validation of the FEM numerical models, which is made against different experimental works performed by different authors, analysing different parameters that can be important on the numerical models conception, such as constitutive law at elevated temperatures and geometrical imperfections. Hence, experimental tests to members with lipped-channel (C), sigma (Σ) and zed (Z) cross-sections, from the literature, are here numerically modelled and analysed. Laterally unrestrained beams under bending moment about the strong axis (subjected to lateral torsional buckling) and axially compression columns, taken into account the buckling around the minor axis, are studied.

2 NUMERICAL MODELLING In this section, the main and general parameters considered for the numerical studies presented in this paper are described. Table 1 shows the chosen case studies, which were obtained from experimental fire tests found in the literature. In the finite element model, rectangular shell finite elements with four notes and six degrees of freedom (3 translations and 3 rotations) were used to reproduce possible local buckling phenomena, due to the walls high slenderness. The numerical analysis is performed using the software SAFIR (Franssen and Gernay, 2017). Local, distortional and global instability modes obtained in CAST3M (CEA, 2012) were used to define the geometrical imperfection shapes by applying the interface with SAFIR, RUBY (developed at the University of Aveiro, in Portugal) (Couto et al., 2013). According to Annex E of Part 1–2 of EC3, the steel stress-strain relationships at elevated temperatures of thin-walled cold-formed profiles should be the same as the one proposed for hot rolled sections (CEN, 2005b). However, the French National (FN) Annex of the same Part 1–2 of EC3 (CEN, 2007) proposes different reduction factors for the steel constitutive laws at elevated temperatures for cold-formed profiles. The FN Annex proposes lower values for these reduction factors, of the yield strength and Young’s modulus at high temperatures, for cold-formed profiles. The comparison of the different constitutive laws of Annex E and FN Annex from Part 1–2 of EC3 is presented in Figure 1. This last constitutive law was the one considered in the present research work for presenting better approximations to the experimental behaviour. The elevated temperatures were considered uniform throughout the cross-section due to the reduced thickness of the cross-sections’ walls. In the absence of information from the authors, the geometric imperfections adopted in all models were in accordance with EN 1090–2 (CEN, 2011). The values correspond to 80% of the geometric fabrication tolerances, following the recommendations from Annex C of EN1993–1–5 (CEN, 2006b), and described in section D.1 from Annex D of EN 1090–2+A1. The initial imperfections were combined considering EN1993–1–5 recommendations. This part states that in combining imperfections, a leading imperfection should be chosen and the accompanying imperfections may have their value reduced to 70%. The leading imperfection was chosen in function of the achieved lower resistances. Residual stresses and corner enhancement were not considered.

Table 1. List of analysed experimental tests. Element type

Cross-section

Reference

Columns

C

Mesquita et al. (2014)

Beams

Σ C Z

Laím et al. (2016) Laím et al. (2016) Jandera and Schwarz (2014)

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Figure 1.

Constitutive law for elevated temperatures according to EC3.

3 RESULTS AND DISCUSSION 3.1 Axially compressed columns The research work by Mesquita et al. (2014) presents an experimental study on simply supported cold-formed steel columns with lipped-channel cross-sections, comparing obtained fire resistance with Eurocode simplified models. From de different cases analysed by Mesquita et al. (2014), the lipped-channel element C_150×51×20×(1.5), reference C17 under ISO 834 fire curve and 1000 mm length, was considered. A transient condition was applied, the procedure of the fire test starts with the increasing of the load until a given load level, being afterwards kept constant during the fire action applied according to the ISO 834 fire curve. Other experimental data such as imperfections amplitude and mechanical properties can be found in Mesquita et al. (2014). The same procedure was applied on the numerical model considering the FN Annex constitutive law. The result from both experimental and numerical analyses are presented on Table 2 and in Figure 2. Figure 2 also shows the numerical model and the respective deformed shape at the collapse instance. The collapse temperature corresponds to the value obtained since the furnace was switched on until the collapse occurrence. From Table 2 and Figure 2, it is possible to observe differences of almost 12%, which is a reasonable approximation of the numerical model to the experimental results, providing also reasonable similar behaviour during the fire test. 3.2 Beams Laím et al. (2016) presented the results of an experimental study about cold-formed steel beams with different cross-section under fire conditions. Different cases of this research work, C_250 × 43 × 15 × (2.5) and Σ_255 × 70 × 25 × (2.5) with 3000 mm length, were considered. Again, the test was performed under transient test conditions. The maximum load applied was 50% of the design value of the load-bearing capacity of the element at normal temperature. When reached the required load, the element was subjected to the ISO 834 fire curve. Other experimental data such as imperfections amplitude and mechanical properties can be found in Laím et al. (2016).

Table 2. Experimental vs. Numerical results for C_150×51×20×(1.5).

Collapse Temperature

Experimental

Numerical

Difference

484 °C

429 °C

11.4%

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Figure 2. a) Shortening–time chart experimental and numerically obtained for the C_150×51×20×(1.5) model b) before and c) after the SAFIR analysis at high temperatures.

The temperatures obtained from the real fire test were applied on the numerical analysis. Again, the constitutive law from the FN Annex of EC3 Part 1-2 was used. The results from both analyses, experimental and numerical, and for the different models, are presented in Table 3, and Figure 3 presents the displacement-time chart for lipped-channel and sigma cross-sections with the experimental and the numerical analysis.

Table 3.

Experimental vs. Numerical results for high temperatures. Collapse temperatures

Section type

Experimental

Numerical

Differences

C_250×43×15×(2.5) Σ_255×70×25×(2.5)

718 ºC 681 ºC

709 ºC 686 ºC

1.3 % 0.8 %

Figure 3. a) Lipped-channel section and b) Sigma section displacement–time chart (experimental and numerical) at high temperatures.

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From Table 3 and Figure 3, it is possible to observe a good agreement between both analysis with differences less than 2%, with similar element behaviour during the fire test as shown in Figure 4 and 5. The sigma element fire test demonstrated to be in a better agreement with the numerical results and behaviour. Finally, experimental test on cold-formed Z purlins, performed in the Fire testing laboratory PAVUS by the Czech Technical University in Prague, in a horizontal furnace is here modelled. The tested profile had a Z section with a height of 200 mm and thickness of 1.5 mm. Different Z profiles were spanned for 6 m, inside the furnace, with an overhanging part (cantilever) of 2.5 mm on one side, outside the furnace (see Figure 6). The aim of the overhanging part (cantilever) was to simulate an internal support of a continuous beam. A sleeve system was adopted in this connection. Inside the furnace one element was used for the span, outside the furnace for the cantilevered part, and for the

Figure 4.

a) Numerical and b) experimental (Laím et al., 2016) C-beam after the analysis and the test.

Figure 5.

a) Numerical and b) experimental (Laím et al., 2016) Σ-beam after the analysis and the test.

Figure 6.

Template of the tested Z purlin (Jandera and Schwarz, 2014).

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connection a thicker element (Z purlin height 200 mm and thickness of 2.0 mm) was interconnecting both elements at the hogging moment area. All sections are S350GD steel grade. In the experimental test the connections were insulated. The loads were chosen in order to represent a realistic example of a roof structural element (purlins), based on the ultimate limit state combination, at normal temperatures. The dead weight of a sandwich panel and the snow load representing more than 90% use of the element in normal conditions were considered. This led to a load of 0.5 kN/m for the member inside the furnace, imposed by attaching it to the web in each 1 m of the span length, and a 1 kN at the end of the element outside the furnace was also imposed, at the top of the member. Comparisons between the standard curve and the measured temperature inside the furnace are presented on Figure 7. The element temperature was measured with welded thermocouples in three parts (upper flange, web and lower flange) of two different cross-sections (mid-span and at 500 mm from the support). The average temperatures at mid-span and at the support are presented on Figure 7. According to EN 1363–1 (CEN, 2012), for flexural loaded elements, the following criteria are presented (Table 4) where L is the span of the element and d is the distance from the extreme fibre of the cold design compression zone to the extreme fibre of the cold design tension zone of the structural section. However, the rate of the deflection criteria is not applied in the first 10 min of the fire test, since relatively fast deflection can occur until stable conditions are reached. Global geometric imperfections were taken into account in the numerical model. Local and distortional imperfections where not taken into account due to the long length of the element. In these long beams local and distortional imperfections have much lower influence on the results, as shown in Arrais (2016). The temperature considered in the Z purlin on the SAFIR numerical model was the one obtained on the experimental test (Figure 7). The constitutive law applied was the one proposed by the FN Annex of Part 1–2 of EC3. The numerical model with all the previous considerations is presented on Figure 8 where it is possible to compare the design drawing with the final numerical model adopted for the present research work.

Figure 7.

Comparison between the ISO 834 curve and the gas temperature, and element temperature. Table 4. Criteria for flexural loaded elements (CEN, 2012) and respective values for the present fire test. Limiting deflection D¼

L2 60002 ¼ ¼ 450 mm 400d 400  200

Limiting rate of deflection dD L2 60002 ¼ ¼ 20 mm  min1 ¼ 9000d 9000  200 dt

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Figure 8.

Numerical model from the Z purlin adopted for the SAFIR numerical model analysis.

Figure 9.

Criteria for flexural loaded elements reached in experimental and numerical tests (mid-span).

The obtained results from the experimental fire test and from the numerical analysis, considering the global imperfections, the FN Annex constitutive law and the loads applied on plates (Figure 8), are presented in Figure 9. From Figure 9 it is possible to observe the evolution of the deflection from the Z purlin, in function of time, at mid-span. The first limiting criterion to be reached was the limiting rate of deflection, after 17 min by the experimental fire test and after 22 min by the numerical analysis. As mentioned before, according to EN 1363–1, the rate of the deflection criterion is not applied in the first 10 min of the fire test, since relatively fast deflection can occur until stable conditions are reached as observed at the minute 6 of the fire resistance test. The second limiting criterion was reached by the fire test after 42 min and by the numerical analysis after 59 min. After 15 min it is possible to observe the lateral-torsional phenomenon occurring before the first criterion reached. Around the 20 min of fire test and 22 min of numerical analysis the phenomenon of catenary action is starting to develop. The Z purlin behaviour is similar to a tensile element (catenary action). The end of the experimental test and numerical analysis was considered after the catenary action occurs for a high deflection as observed. In Figure 10 and 11, it is visible the deformed shape of the entire element after the fire test and the numerical analysis, and the detailed buckling phenomenon observed next to the support. Considering the present numerical model it is visible a good agreement between the numerical analysis and the experimental test, in terms of resistance and overall element behaviour.

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Figure 10. a) Experimental (Jandera and Schwarz, 2014) and b) numerical purlin after the test.

Figure 11. a) Experimental (Jandera and Schwarz, 2014) and b) numerical purlin support after the test.

4 CONCLUSIONS In this work it is presented the numerical modelling of experimental tests to cold-formed steel profiles under fire exposure, from the literature. The consideration of initial geometric imperfections with the shapes of the critical buckling modes and the constitutive law at elevated temperatures proposed in FN Annex of Part 1–2 of EC3, resulted on reasonable approximations to the experimental results. In general, from the analysis, and in spite of the possible occurrence of different complex instabilities phenomena in cold-formed steel profiles (such as local, distortional and global buckling), a good agreement between the numerical analysis and the experimental tests, in terms of resistance capacities and structural behaviour in fire, could be observed. REFERENCES Arrais, F., Lopes, N., Vila Real, P. 2016. Behaviour and resistance of cold-formed steel beams with lipped channel sections under fire conditions, Journal of Structural Fire Engineering, Emerald Group Publishing Ltd, ISSN: 2040–2317, volume 7/4, pp. 365–387. CEA. 2012. CAST 3M research FEM environment. development sponsored by the French Atomic Energy Commission http://www-cast3m.cea.fr/. CEN European Committee for Standardisation. 2005a. EN 1993–1–1, Eurocode 3: Design of steel Structures – Part 1–1: General rules and rules for buildings. Belgium. CEN European Committee for Standardisation. 2005b. EN 1993–1–2, Eurocode 3, Design of Steel Structures – Part 1–2: General rules – Structural fire design. Belgium. CEN, European Committee for Standardisation. 2006a. EN 1993–1–3, Eurocode 3: Design of Steel Structures – Part 1–3: General rules - Supplementary rules for cold-formed members and sheeting. Belgium. CEN European Committee for Standardisation. 2006b. EN 1993–1–5, Eurocode 3: Design of steel Structures – Part 1–5: Plated structural elements. Belgium.

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CEN, European Committee for Standardisation. 2007. NF EN 1993–1–2, Eurocode 3: Calcul des structures en acier – Annexe Nationale à la NF EN 1993–1–2: Calcul du comportment au feu. Belgium. CEN European Committee for Standardisation. 2011. EN 1090–2, Technical requirements for the execution of steel structures. Belgium. CEN, European Committee for Standardisation. 2012. NP EN 1363-1, Ensaios de resistência ao fogo Parte 1: Requisitos gerais, Belgium. Couto, C., Vila Real, P. & Lopes, N. 2013. RUBY an interface software for running a buckling analysis of SAFIR models using Cast3M. University of Aveiro. Franssen, J-M. & Gernay, T. 2017. Modelling structures in fire with SAFIR®: theoretical background and capabilities. Journal of Structural Fire Engineering. Gonçalves, R.; Camotim, D. 2007. Thin-walled member plastic bifurcation analysis using generalised beam theory, Advances in Engineering Software. Elsevier. Vol. 38, n.º 8–9, p. 637–646. Jandera, M., Schwarz, I. 2014. Structural fire behaviour of Z purlins: Eurosteel 2014: 7th European Conference on Steel and Composite Structures, Naples, Italy. Laím, L., Rodrigues, J.P.C., Craveiro, H.D. 2015. Flexural behaviour of beams made of cold-formed steel sigma-shaped sections at ambient and fire conditions, Thin-Walled Structures. Vol. 87, p. 53–65. Laím, L., Rodrigues, J.P.C., Craveiro, H.D. 2016. Flexural behaviour of axially and rotationally restrained cold-formed steel beams subjected to fire, Thin-Walled Structures. Vol. 98, Part A, p. 39–47. Mesquita, L., Mendonça, M., Ramos, R., Barreira, L., Piloto, P. 2014. Thermomechanical analysis of cold-formed steel sections: 9º Congresso Nacional de Mecânica Experimental, Aveiro, Portugal. Muftah, F., Sani, M., Mohammad, S., Ngian, S., Tahir, M. 2016. Temperature development of cold-formed steel column channel section under standard fire, AIP Conf. Proceedings. Vol. 1774, n.º 1. Sidey, Teague. 1988. Elevated temperature data for structural grades of Galvanised steel: British Steel, Welsh Laboratories. Silvestre, N.; Camotim, D. 2010a. Construção em aço leve, Revista da Associação Portuguesa de Construção Metálica e Mista, ano 11, n.º 20, Março. Silvestre, N.; Camotim, D. 2010b. On the mechanics of distortion in thin-walled open sections, ThinWalled Structures - Elsevier. Vol. 48, n.º 7, p. 469–481. Silvestre, N.; Pires, J.; Santos, A. 2013. Manual de Conceção de Estruturas e Edifícios em LSF - Light Steel Framing: Associação Portuguesa de Construção Metálica e Mista. ISBN 9789899560581. Yu, W. W. 2000. Cold-Formed Steel Design - 3rd Edition: John Wiley & Sons.

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Experimental study on the general behaviour of stainless steel frames I. Arrayago, E. Real, E. Mirambell & I. González de León Department of Civil and Environmental Engineering, Universitat Politècnica de Catalunya, Barcelona, Spain

ABSTRACT: In recent years, a considerable am ount of research has been devoted to the understanding of the structural performance of single isolated stainless steel members. Notwithstanding, advances related to the analysis of more complex stainless steel structures, such as frames, are scarce. On this basis, a comprehensive experimental programme on sway and non-sway austenitic stainless steel frames with slender and stocky rectangular hollow sections subjected to static loading was carried out at Universitat Politècnica de Catalunya. The paper presents the preliminary experimental results on one non-sway and one sway frame as illustrative of the experimental programme. A more in deep analysis of these frames will contribute to the appraisal of existing design rules for stainless steel, which are based on carbon steel, in terms of predicted ultimate capacities, plastic design and second order effects, considering the particular non-linear behavior and strain hardening for stainless steels. 1 INTRODUCTION Mechanical properties such as high ductility, adequate toughness, considerable strain hardening and good fire resistance make stainless steel an excellent construction material for structures, especially those required to withstand accidental loading (Baddoo, 2008). During last decades, significant amount of research has been focused to the understanding of the structural performance of single isolated stainless steel members recently (Arrayago & Real, 2016; Afshan & Gardner, 2013; Arrayago et al., 2016; Huang & Young, 2013), but advances related to the analysis of more complex stainless steel structures, such as frames, are scarce (Arrayago et al., 2017; Walport et al., 2019). As a matter of fact, EN1993-1-4+A1 (2015) does not establish specific design rules associated with the global analysis of stainless steel frames and hence, provisions given for carbon steel in EN 1993-1-1 (2005) need to be adopted. Recent research on the behaviour of stainless steel frames by Walport et al., 2019 demonstrated that the degradation of stiffness – due to the nonlinear material response – considerably affects the characteristics of the structural system, causing greater deformations and increasing second order effects. This has a direct influence on the definition of non-sway – structures for which second order effects are negligible – and sway – structures for which internal force amplifications are relevant – frames. Thus, it was recommended that material nonlinearity should be considered in the global analysis of stainless steel frames, especially when the susceptibility to second order effects (through the αcr critical load factor) is considered. Moreover, the lack of guidance on plastic design in general and design of frames in particular is an obstacle to the optimal design of stainless steel structures considering the remarkable differences in this behaviour compared with carbon steel. On this basis, an extensive experimental programme on sway and non-sway austenitic stainless steel frames with slender and stocky Rectangular Hollow Sections (RHS) subjected to static loading has been conducted at the laboratory of the Department of Civil and Environmental Engineering “Luis Agulló” of the Universitat Politècnica de Catalunya. The experimental programme comprises several sub-programmes in which the performance of these 124

structures has been investigated at different levels – material characterization, cross-sections, members and frames (Arrayago et al., 2019a; Arrayago et al., 2019b). The final objective of this experimental programme and subsequent research is to assess the currently existing design rules, in terms of predicted ultimate capacities and second order effects, for stainless steel structures. In this context, this paper consists in a brief resume of the adopted experimental set-up for frame tests, as well as presenting the preliminary results on two of the tested austenitic stainless steel frames –including one non-sway and one sway frames. The comprehensive analysis of the tests through the measured data will be carefully analyzed in detail in the future.

2 FRAME DESCRIPTION Frame tests have been performed on single span and single height austenitic stainless steel frames with RHS, with a mid-section height (h) of 2 m and a span between columns (L) equal to 4 m. The connection between the beam and the columns were performed by welding an auxiliary steel plate with an inclination of 45º. Likewise, for the connections at supports, additional steel plates, which were provided with perforations to be screwed, were welded at the bottom of the columns. Hence, both fixed and pinned boundary conditions were allowed with the same general configuration. The experimental programme included a total of four frame tests from EN 1.4301 austenitic stainless steel with the same general geometry, but with different cross-sections. Table 1 summarizes the general definition of frame specimens, in which the overall geometries, the boundary conditions and cross-section shapes are reported. H corresponds to the height of the RHS, while B is the width and t is the wall thickness. In addition, αcr parameter values, which indicate the susceptibility of the frames to second order effects are also reported, along with the estimated column slenderness
λc , based on the effective length calculations for sway and non-sway frames. Likewise, the local slenderness of pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the cross-sections
λp ¼ σ0:2 =σcr;l is reported, with σ0:2 and σcr;l standing for the 0.2% proof stress and the local critical buckling stress, respectively. Note that, since according to EN19931-4+A1 (2015) local buckling effects appear beyond a slenderness value of
λp ¼ 0:65, Frames 1 and 2 correspond to stocky cross-sections in pure compression, while Frames 3 and 4 correspond to slender cross-sections. A preliminary finite numerical model was conducted by means for the advanced software ABAQUS and the critical buckling behavior of the frames was predicted according to the defined loading scheme. This preliminary analysis showed that Frame 1 and Frame 2 can be considered as non-sway frames, while Frames 3 and 4 are sway frames. Although the experimental programme comprised four different frame tests, in this paper only results corresponding to Frame 1 –a non-sway frame with a stocky sections- and Frame 3–a sway frame with a considerably slender section- are presented. 2.1 Preliminary tests and measurements: Material and cross-section response, imperfections For a correct analysis of the frame tests, several preliminary tests for the material characterization and for the determination of the cross-section and member resistances were conducted Table 1. General definition of frame specimens (nominal properties). Boundary h [mm] L[mm] conditions (supports) αcr Frame 1 – S1 Frame 2 – S2 Frame 3 – S3 Frame 4 –S4

2000 2000 2000 2000

4000 4000 4000 4000

Fix-end Fix-end Pin-end Pin-end

11.7 11.8 3.4 7.6

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Column H B t slenderness [mm] [mm] [mm]
λc

Local slenderness
λp

0.60 0.60 2.53 1.15

0.50 0.60 0.77 1.64

120 100 120 200

80 80 40 100

6 4 4 3

previously. Since a more in detail description of such tests is available in Arrayago et al., 2019a, only a brief summary of the most relevant parameters is provided in this paper. The stress-strain behavior of coupons extracted from flat (F) and corner (Co) sections of the RHS was determined by means of tensile tests. Key material parameters for sections S1 and S3 (comprising Frame 1 and Frame 3) are reported in Table 2, in which E is the Young’s modulus, α0.2 is the yield strength (proof stress for 0.2% plastic strain), αu and εu are the tensile strength and corresponding strain, and n, m are the optimized strain hardening parameters. Likewise, Table 3 summarizes the cross-section resistances to compression Nu and to bending Mu corresponding to sections S1 and S3, obtained from stub column and four-point bending tests, together with the cross-section classifications according to EN1993-1-4+A1 (2015). The actual initial geometry of the frames was carefully measured prior to testing by means of theodolites. The actual geometry of the columns was characterized through five different points along their height, while the position of additional five points along the beam length were also measured. This will allow introducing accurate initial imperfections into the future numerical studies, as well as evaluating the influence of local and global initial imperfections on the general behavior of the frames. In addition, several points of the frames were monitored during the tests and the movements were recorded using a Lidar system (see Figure 1). Table 2. Key material characterization parameters from tensile coupon tests. Specimen

E [MPa]

α0.2 [MPa]

αu [MPa]

εu [mm/mm]

n

m

S1-F S3-F S1-Co S3-Co

159642 210966 187795 186153

495 601 643 637

715 769 840 856

0.50 0.29 0.38 0.20

7.3 6.2 4.7 4.5

2.6 3.9 6.8 6.1

Table 3. Key experimental results for stub column tests and 4 point bending tests. Specimen

Cross-section class in compression

Nu [kN]

Cross-section class in bending

Mu [kNm]

S1 S3

1 4

1197.6 552.3

1 1

57.6 26.1

Figure 1.

General view of theodolite and Lidar system.

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3 FRAME TEST SET-UP In this paper the preliminary results of the experimental programme on austenitic stainless steel frames subject to static loading are presented. The loading scheme was carefully defined after an exhaustive study of different alternatives and options for vertical and horizontal loading schemes, since sway displacements difficult vertical loads to remain vertical (Wilkinson & Hancock, 1999, Blum & Rasmussen, 2018, Zhang et al., 2016, Avery & Mahendran, 2000). The adopted final loading to allow reproducing vertical – gravity- and horizontal loads consisted on a two-step equivalent loading scheme. During the first step, the frames were loaded vertically through two jacks applied on the beams up to a load value corresponding to the 6070% of the maximum vertical resistance of the frames. In the second step, while the vertical load was kept constant, a horizontal displacement was imposed into the column supports through a horizontal jack and a rigid beam. During these two steps, the top-right corner of the frames was tied to a reaction wall, restraining the in-plane horizontal movement of the beam, and thus, allowing the proposed loading scheme to work. A more exhaustive study of the loading options, adopted scheme and employed auxiliary elements is provided in Arrayago et al., 2019b, although the most relevant aspects are summarized in the following section, as well as the instrumentation considered during the tests. 3.1 General test layout The layout of the general test arrangement is presented in Figure 2, in which the most relevant parts are highlighted and described below: vertical loading sections, horizontal loading rigid beam, load cells, column supports and fixed horizontal point. 3.1.1 Vertical loading sections Vertical loads were introduced at a distance of 800 mm from each column through the use of two jacks, which were connected to guarantee that two identical point loads and were applied simultaneously while allowing different vertical displacements at these points once the horizontal loads were applied. In order to prevent local web failure at loading supports, guarantee a correct distribution of the loads and to contribute to the lateral stability of the frame, auxiliary elements provided by neoprene and Teflon plates and adjustable to the different crosssections were fabricated (see Figure 3). 3.1.2 Horizontal loading rigid beam and sliding supports As explained previously, horizontal loads were introduced by imposing a displacement at column supports once the vertical loads were applied. To ensure a uniform distribution

Figure 2.

General layout of stainless steel frame tests.

127

Figure 3.

Detail of the vertical loading section.

of displacement at both supports, a rigid beam was prepared. This beam was supported on two specially fabricated 500 mm long sliding supports, which allow a frictionless movement in-plane thanks to Teflon plates. In addition, the beam included a load cell that measured the horizontal reactions at the left supports, which together with the total applied load obtained from the horizontal jack, allowed a complete characterization of both supports. 3.1.3 Load cells and column supports Vertical load and bending moment reactions at supports were measured through two specially fabricated load cells, which consisted in four steel studs instrumented with strain gauges, and welded to two steel plates enabling the connection to the frame columns and the horizontal rigid beam (see Figure 4). These cells were calibrated prior to using them in the tests, as described in Arrayago et al., 2019b. In order to guarantee both fixed and pinned boundary conditions for the tested frames, a bolted connection through the columns and the load cells was adopted. While fix-ended boundary conditions were achieved by using 12 bolts between the steel plates welded to the columns and the load cell, for pin-ended conditions only four bolts were used. 3.1.4 Fixed horizontal point To restrain horizontal displacement of the frames, the top of the right column was tied to a reaction wall by two Macalloy bars, which also contributed to the lateral stability of the frames (Figure 1). 3.1.5 Adopted loading protocols The loading scheme adopted for the frame tests subject to static loading consisted of two loading steps, as described previously. Different vertical and horizontal loading rates were defined for each of the frame tests to ensure safety and a reasonable duration of the tests, which showed an approximate duration of the tests of 90 minutes. Both loading steps were performed under displacement control, although the hold in the vertical load during the horizontal loading step was load-controlled. Table 4 reports the adopted test rates for Frame 1 and Frame 3, as well as the values of the maximum applied total vertical loads.

128

Figure 4.

Detail of the horizontal jack, loading cell and column supports.

Table 4. Adopted test rates for vertical and horizontal loading steps in frame tests. Step 1: Vertical loading (30 min approx.) Test rate

Fv,u

Specimen

[mm/min]

[kN]

Frame 1 Frame 3

2.00 1.27

155.0 50

Proportion of Fv,max 65% 60%

Step 2: Horizontal loading (60 min approx.) Test rate [mm/min] 3.3 2.5

3.2 Instrumentation In this sub-section the instrumentation adopted in the frame tests is described. The measurement of different parameters is key for the accurate characterization of the performance of stainless steel frames and for guaranteeing the correct development of the test (see Figure 1). Actuators directly recorded the total applied vertical Fv,tot and horizontal Fh,tot loads, as well as their displacements. However, additional deflection measurement devices were adopted for the loading sections and the horizontal displacement of the rigid beam. Moreover, two laser devices were placed on the top of both columns to measure possible horizontal and out-of-plane displacements in an effort of tracking the adequacy and safety of the conducted tests. In addition, two inclinometers were placed to measure in-plane and out-of-plane rotations close to the support sections at frame columns and the strain gauges of the load cells provided the relevant information on the distribution of loads and bending moments in both supports. Strains at the most relevant sections of the frames were recorded by means of strain gauges, which will be used to estimate stress distributions and bending moments at these representative points. Finally, an alternative Digital Image Correlation (DIC) system was also employed to record the behaviour of the upper area of the right column in all frames, as these sections were identified as critical in the preliminary numerical analysis.

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Nevertheless, these measurements are still being processed and thus no preliminary results are available.

4 TEST RESULTS This section presents the preliminary test results from the conducted frame tests. As examples, results corresponding to one non-sway frame –Frame 1– and one sway frame –Frame 3– are provided. The recorded load-displacement paths for Frame 1 and Frame 3 are presented in Figures 5 and 6: while the responses corresponding to the vertical loading step are shown in Figure 5, the horizontal behaviour of such frames is illustrated in Figure 6. Figures show that the non linear behavior is more pronounced in the non-sway stocky frame (Frame 1). Achieved ultimate load and deflections for each of the loading steps are summarized in Table 5 for Frame 1 and Frame 3. Fv,u is the maximum vertical load applied and dv,u the vertical displacement when the vertical load is stopped and starts horizontal loading. Fh,v is the

Figure 5.

Vertical load-displacement.

Figure 6.

Horizontal load-displacement.

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Table 5. Summary of ultimate load and displacements. Specimen

Fv,u [kN]

dv,u [mm]

d*v,u [mm]

F,h,u [kN]

dh,u [mm]

Frame 1 Frame 3

157,7 56,9

71 48

81 55

39,3 18,4

83 82

Figure 7.

Deformed frames after vertical and horizontal loading.

maximum horizontal loading and dh,u the maximum horizontal displacement. Notice that vertical displacements also increase after the hold on the vertical loading and during the horizontal loading (d*v,u). Likewise, Figure 7 shows the final deformed configurations for Frame 1 and Frame 3 once the tests were completed.

5 CONCLUSIONS This paper presents a series of tests on sway and non-sway austenitic stainless steel Rectangular Hollow Sections frames as part of a vaster experimental programme devoted to the investigation of the performance of stainless steel structures under static loading. Experimental results corresponding to deflections of the beam caused by vertical loading and displacement of the supports caused by horizontal loading are presented for one stocky non-sway frame and for one slender sway frame. Furthermore, experimental results highlighted the non linear behavior of the frames and the more pronounced effect on the stocky non-sway frame.

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Once the experimental programme on sway and non-sway frames is finalized, efforts will be focused on analyzing in detail the obtained data to advance in the knowledge of stainless steel behaviour, which is currently scarce in plastic performance and frame performance, and to propose a specific guidance for the optimal design of stainless steel structures. ACKNOWLEDGMENTS This experimental programme was developed in the frame of the Project BIA2016-75678-R, AEI/FEDER, UE “Comportamiento estructural de pórticos de acero inoxidable. Seguridad estructural a acciones accidentales de sismo y fuego”, funded by MINECO (Spain). The last author would also like to thank the financial support provided by the Spanish Ministry for Science, Innovation and Universities for the deveolpemnt of her PhD. Authors would like to acknowledge the support, advice and suggestions by Professor K. Rasmussen from the University of Sydney, Professor Ben Young from the Hong Kong University and Professor Leroy Gardner from Imperial College London in the preparation of this experimental programme. Finally, the time and efforts from the staff at the Structures Laboratory at the Universitat Politecnica de Catalunya is much appreciated. REFERENCES Afshan, S. & Gardner, L. 2013. Experimental study of cold-formed ferritic stainless steel hollow sections. Journal of Structural Engineering (ASCE) 139(5), 717–728. Arrayago, I. & Real, E. 2016. Experimental study on ferritic stainless steel simply supported and continuous beams. Journal of Constructional Steel Research 119, 50–62. Arrayago, I., Real, E. & Chacón, R. 2019a. Experimental performance of austenitic stainless steel beams and columns. Proceedings of the 9th International Conference on Steel and Aluminium Structures (ICSAS-19). Bradford, UK. Arrayago, I., Real, E. & Mirambell, E. 2019b. Preliminary study and tests arrangements for experimental programme on stainless steel frames. Proceedings of the 9th International Conference on Steel and Aluminium Structures (ICSAS-19). Bradford, UK. Arrayago, I., Real, E. & Mirambell, E. 2016. Experimental study on ferritic stainless steel RHS and SHS beam-columns. Thin-Walled Structures 100, 93–104. Arrayago, I., Real, E., Mirambell, E. and Chacón, R. 2017. Global plastic design of stainless steel frames. Proceedings of the 8th European Conference on Steel and Composite Structures (Eurosteel 2017). Copenhagen, Denmark. Avery, P. & Mahendran, M. 2000. Large-scale testing of steel frame structures comprising non-compact sections. Engineering Structures 22, 920–936. Baddoo, N.R. 2008. Stainless steel in construction: A review of research, applications, challenges and opportunities. Journal of Constructional Steel Research 64(11),1199–1206. Blum, H.B. and Rasmussen, K.J.R. 2018. Elastic buckling of columns with a discrete elastic torsional restraint. Thin-Walled Structures 129, 502–511. EN 1993-1-1:2005. Design of steel structures. Part 1-1: General rules – General rules and rules for buildings. European Committee for Standardization Eurocode 3. Brussels, Belgium. EN 1993-1-4:2006 + A1:2015. Design of steel structures. Part 1–4: General rules. Supplementary rules for stainless steels. European Committee for Standardization Eurocode 3. Brussels, Belgium. Huang, Y. & Young, B. 2013. Tests of pin-ended cold-formed lean duplex stainless steel columns. Journal of Constructional Steel Research 82, 203–215. Walport, F., Gardner, L., Real, E., Arrayago, I. & Nethercot, D.A. 2019. Effects of material nonlinearity on the global analysis and stability of stainless steel frames. Journal of Constructional Steel Research 152, 173–182. Wilkinson, T. & Hancock, G.J. 1999. Tests of cold-formed rectangular hollow section portal frames. Research Report No. R783. Sydney: The University of Sydney. Young, B. & Rasmussen, K.J.R. 2003. Measurement techniques in the testing of thin-walled structural members. Experimental Mechanics 43, 32–38. Zhang, X., Rasmussen, K.J.R. & Zhang, H. 2016. Experimental investigation of locally and distortionally buckled portal frames. Journal of Constructional Steel Research 122, 571–583.

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Experimental investigation of flexural buckling of sandwich panels with steel facings I. Balázs & J. Melcher Institute of Metal and Timber Structures, Faculty of Civil Engineering, Brno University of Technology, Brno, Czech Republic

ABSTRACT: Sandwich panels are widely used in building industry particularly as members of cladding. Although they primarily resist transversal loads, axial forces may arise e.g. due to stabilizing function of panels that may prevent buckling of supporting members of steel load-bearing structure and transfer stabilizing forces. The paper focuses on problem of flexural buckling of sandwich panels with thin steel facings. The problem of stability of axially loaded sandwich panels is outlined. To verify the actual behavior of sandwich panels under axial load, a series of full-scale tests of flexural buckling of selected type of sandwich panels with thin steel facings and soft core was performed. Evaluation of the test results was performed using Southwell plot and statistical methods. The paper presents the utilized test setup, procedure of testing, failure modes and summarizes selected results of the tests. The findings obtained from the tests are discussed. 1 INTRODUCTION The sandwich panels are widely used in civil engineering as members of roof and wall cladding. They usually consist of thin metal facings and soft insulation core (various types of foams in most cases). It results from purpose of the panels that they are primarily loaded by transversal loads. The panels may also prevent buckling of supporting members of load-bearing structure (e.g. purlins) which results in stabilizing (axial) forces to be transferred by the panels. In some cases of small buildings, the load-bearing structure itself may consist of sandwich panels (with no substructure) which was investigated recently (Käpplein, Misiek, 2010). In that cases, the sandwich panels should be designed to resist axial forces. The assessment of the load-bearing resistance becomes complex especially in case of compressive axial force and related stability problems resulting from slenderness of the member. For sandwich panels, various materials of the facings and the core can be utilized. There is a wide range of types of these structural members available for building industry. Experimental analysis can be an effective way of investigation of the actual behavior of axially loaded sandwich members made of specific materials.

2 BUCKLING RESISTANCE OF AXIALLY LOADED MEMBERS 2.1 Slender metal members The behavior of slender members in compression was subject of number of theoretical and experimental studies in the past and now is good understood. It is necessary to distinguish between ideal member with no imperfections (structural, geometrical, constructional) and actual member. The problem of buckling of an ideal member in compression was first studied by Euler who established basis for mathematical concept of the theory of stability of slender members. The buckling (bifurcation of equilibrium) of an ideal member occurs when the value of the so called critical load Ncr is reached (Euler, 1744). The problem was defined using homogenous differential equation of second order and the critical load was found using solution of the eigenvalue problem of the equation. The theory was significantly developed and 133

extended by Vlasov who formulated general theory of slender members (Vlasov, 1962). In case of buckling of axially loaded members, the Vlasov’s theory takes into account also torsional and warping stiffness of the member. In case of actual member with unavoidable initial imperfections, the lateral deflection of the member increases with load (as opposed to an ideal member) and the bifurcation in terms of theory of stability of ideal member does not occur. The resistance of the member is therefore lower than the magnitude of critical load. Nevertheless, the value of the critical load is an important value necessary for evaluation of resistance of actual members. Within the frame of analysis and structural design of an actual member in compression, all types of imperfections are usually replaced by initial deflection (initial bow imperfection) of the member with the amplitude e0. A simplified comparison between behavior of ideal and actual member in form of relationship between load N and lateral deflection f is in Figure 1. 2.2 Sandwich panels The actual behavior of sandwich members under axial loads is very complex due to possible occurrence of stability problems (local buckling of thin facings and global buckling of the panel and their combination) and due to significant difference of material properties of the facings and the core. The theoretical problem of stability of sandwich structures was extensively developed on theoretical basis by Kovařík et al. (Kovařík, Šlapák, 1973). The work based on complex mathematical solutions of eigenvalue problems provided, among others, formulas for evaluation of global buckling of sandwich plates with rigid or soft core and also solution of local buckling of sandwich plates (wrinkling). Mechanical properties of both facings and core of the sandwich plate are taken into account within the calculation procedure. The theory of stability of sandwich members was later dealt with e.g. by Davies. Formula of the critical load of sandwich member was developed and results of few tests of axially loaded panels (with eccentricities of the applied normal force) were presented. The significant influence of the eccentricity of the applied load on the test results was highlighted (Davies, 1987). An extensive research of behavior of sandwich panels with metal facings and selected types of insulation core was performed in the frame of EASIE project (Käpplein, Misiek, 2010). Axially loaded panels were a subject of theoretical, experimental and numerical investigation. Within the project, design procedures for sandwich panels were developed as well. 3 EXPERIMENTAL INVESTIGATION 3.1 Object of the tests and preparation of the test setup To study the actual behavior of sandwich panels under axial compressive load, a series of fullscale tests was proposed as an effective method of investigation. It is assumed that response of the panels to applied load is influenced by large number of factors. The experimental analysis of such structural members using full-scale tests can bring valuable results and can be used for prospective

Figure 1.

Load-deflection relationship: ideal and actual member.

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verification of theoretical models (Melcher, 1997). A total of fourteen tests was completed in the testing laboratory. The boundary conditions of the tested specimens were proposed to comply with simply supported member as it is an elementary case when investigating buckling behavior of structural members. In the frame of four tests, the specimens failed by local collapse at the supports with no observation of global buckling. These tests were not considered for final evaluation of global buckling. For the following ten tests, the support area of the panels was therefore slightly modified to prevent this type of failure. The final test setup used for the test program is described below. 3.2 Specimens For experimental verification of buckling behavior, sandwich panels with slightly profiled steel facings of nominal thickness of 0.5 mm and polyisocyanurate (PIR) insulation core were used. Polyisocyanurate foam is one of the number of materials that can be used as insulation core of sandwich members. The foam itself is brittle (Gilbert, 2016). The benefit of use of polyisocyanurate is higher fire resistance in comparison with other types of foams, e.g. polyurethane. It can resist temperatures up to 150°C (Brydson, 1999). The length of the tested panels was 4.0 m, width 1.0 m and thickness of the core 80 mm. 3.3 Test setup and procedure of testing A special test device was prepared to perform the tests of buckling of sandwich panels. Simply supported member was assumed in the frame of the planning of the test program. The specimen was attached to a testing frame at both ends by pinned connections which enabled rotation of the specimen at the supports out of plane to comply with the assumed boundary conditions. The pins were greased to reduce friction. At one end of the specimen, hydraulic cylinder was attached to apply axial force. At the ends of the specimen, plywood plates were attached to prevent local failure at the supports. The plates did not prevent rotation of the specimen. The supports enabled lateral adjustment of the position of the specimen and therefore its precise positioning on the testing device. The test setup is in Figure 2. During each test, continuous measurement of applied force was performed. The out-ofplane displacement was measured using draw-wire sensors at midspan of the specimen. Strain gauges were attached at midspan of the specimen on both facings. Before applying the normal force using hydraulic cylinder, additional lateral force caused by a 120 kg concrete block (attached to the specimen via steel cable and pulley) was applied to the specimen at its midspan to cause an initial deflection. The deflection f0 caused by this lateral force was recorded. The specimen was then loaded by the hydraulic cylinder until the failure was reached.

Figure 2.

Test setup.

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3.4 Test results and their evaluation The evaluation of the test results focuses on specimens failed by global (flexural) buckling which was caused by failure of the compressed face of the members. The tests with local failure mode were excluded from further evaluation. Failure of the specimens by global buckling can be observed in Figure 3. Typical relationship between load and deflection (recorded during one of the tests) is in Figure 4. Initial deflection is not displayed in the chart (but was carefully recorded during each test). The evaluation of the tests primarily focused on experimental determination of the critical load of the tested panels. Measured data (axial force and lateral displacement) was used for evaluation. As the term “critical load” applies only for ideal members (with no initial imperfections), the critical load cannot be determined directly. For the evaluation, Southwell plot was used (Southwell, 1932). This method is suitable for evaluation of elastic critical load of compressed members using a chart with lateral displacement of the member f on horizontal axis and ratio between lateral displacement and actual applied force f/N on the vertical axis. It

Figure 3.

Failure of the specimen.

Figure 4.

Typical load-deflection curve.

136

has been found that this experimentally obtained relationship between f and f/N is linear only in the medium range (Březina, 1962). The critical load is then obtained as cotangent of the slope of the line passing through the medium range of the graphical relationship. Typical graphical interpretation of evaluation of this relationship is presented in Figure 5 (relationship between displacement and ratio between the displacement and applied force for one of the performed tests). The deflection starts at zero value (initial deflection is not considered in this graphical interpretation). The line is constructed in the linear range of the relationship. The slope of this line is quantified and used for calculation of the critical load. The initial deflection f0 caused by the additional lateral force at midspan (concrete block) was used to calculate bending stiffness of the panel using a simple formula for deflection caused by a point load (Equation 1). This bending stiffness (E·I)panel is used to calculate Euler critical load according to Equation 2. Table 1 summarizes results of the tests and calculations: initial deflection f0 caused by lateral force, bending stiffness of the panels (E·I)panel, maximum axial load Nmax applied using the hydraulic cylinder at failure of the specimen, appropriate displacement f (initial deflection is included), critical load determined using Southwell plot and Euler critical load determined using experimentally obtained bending stiffness for each test according to Equation 2. ðE  I Þpanel ¼ Ncr ¼

Figure 5.

F L3 48 f0

ð1Þ

π2  ðE I Þpanel

ð2Þ

L2

Southwell plot.

Table 1. Selected results of the tests and calculations. f0

(E·I)panel

Nmax

f

Ncr (Southwell)

Ncr (Euler)

Test No

mm

109 N·mm2

kN

mm

kN

kN

01 02 03 04 05 06 07 08 09 10

8.32 7.67 8.23 7.95 8.15 8.01 7.60 7.94 7.95 8.11

1.922 2.085 1.944 2.013 1.963 1.998 2.104 2.014 2.013 1.973

84.55 88.11 83.39 77.53 65.00 67.91 61.87 60.41 71.49 68.46

20.28 20.64 17.97 13.31 29.70 29.82 18.27 18.31 20.85 28.92

135.14 119.05 133.33 163.93 87.72 116.28 161.29 116.28 151.52 98.04

118.56 128.62 119.90 124.19 121.06 123.28 129.84 124.26 124.18 121.69

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The magnitudes of maximum loads at failure were statistically assessed in terms of EN 1990, Annex D (Eurocode, 2003). Assessment via the characteristic value is performed provided there is no prior knowledge of coefficient of variation of the tested quantities. The characteristic value of the axial load resistance is 53.83 kN for the tested specimens.

4 CONCLUSION The paper summarizes results of series of full-scale tests of global buckling of sandwich panels with steel facings and polyisocyanurate insulation core. The fundamental part of the paper is evaluation of experiments, especially experimental determination of critical load of the tested specimens. Comparison with Euler critical load determined using experimentally obtained values of bending stiffness of the specimens provided good agreement between results of both approaches. The research will be followed by experimental determination of mechanical properties of the facings and the core of the panels. The experimentally determined material properties can be then used within the frame of theoretical and numerical analysis of stability of sandwich member. They will enable accurate comparison between results of experiments and results of calculation of critical load according to theory of sandwich structures. ACKNOWLEDGMENT The paper was elaborated within the frame of the research program No FAST-S-18-5550 of the Faculty of Civil Engineering, Brno University of Technology and within the project No LO1408 AdMaS UP – Advanced Materials, Structures and Technologies, supported by the National Sustainability Program I of the Ministry of Education, Youth and Sports of the Czech Republic. REFERENCES Brydson, J.A. 1999. Plastics Materials. Boston: Elsevier. ISBN 978-0-7506-4132-6. Březina, V. 1962. Vzpěrná únosnost kovových prutů a nosníků (Buckling resistance of metal members). Prague: Czechoslovak Academy of Sciences Publishing. Davies, J.M. 1987. Axially loaded sandwich panels. Journal of Structural Engineering (United States) 113 (11): p. 2212–2230. ISSN 0733-9445. DOI: 10.1061/(ASCE)0733-9445(1987)113:11(2212). EN 1990 Eurocode: Basis of structural design. 2003. Euler, L. 1744. De curvis elasticis. Cited in: Březina, V. 1962. Vzpěrná únosnost kovových prutů a nosníků (Buckling resistance of metal members). Prague: Czechoslovak Academy of Sciences Publishing. Gilbert, M. 2016. Brydson’s Plastics Materials. Boston: Elsevier. ISBN 978-0-323-35824-8. Käpplein, S. & Misiek, T. 2010. EASIE – Ensuring Advancement in Sandwich Construction Through Innovation and Exploitation. Report No.: D3-2 – part 3 Tests on axially loaded sandwich panels. Karlsruhe: Karlsruher Institut für Technologie. Käpplein, S. & Ummenhofer, T. 2010. Axial beanspruchte Sandwichelemente in rahmenlosen Konstruktionen (Axially loaded sandwich panels in frameless buildings). Stahlbau 79 (10): p. 761–770. ISSN 0038-9145. DOI: 10.1002/stab.201001367. Kovařík, V. & Šlapák, P. 1973. Stabilita a kmitání sendvičových desek (Stability and vibration of sandwich plates). Prague: Academia – Czechoslovak Academy of Sciences Publishing. Melcher, J. 1997. Full-Scale Testing of Steel and Timber Structures: Examples and Experience. In Virdi et al. (ed.), Structural Assessment: The role of large and full-scale testing. London: E & FN Spon, p. 301–308. ISBN 0-419-22490-4. Southwell, R.V. 1932. On the Analysis of Experimental Observations in Problems of Elastic Stability. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 135 (828): p. 601–616. ISSN 1364-5021. DOI: 10.1098/rspa.1932.0055. Vlasov, V.Z. 1962. Tenkostěnné pružné pruty (Thin-Walled Elastic Bars). Prague: State Publishing of Technical Literature.

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Nonlinear behavior and instability of deployable arches Ana Beatriz G. Barcellos, Murillo V.B. Santana & Paulo B. Gonçalves Department of Civil and Environmental Engineering, Pontifical Catholic University of Rio de Janeiro, Rio de Janeiro, Brazil

ABSTRACT: Deployable structures consist of a group of structures capable of modifying their shape and volume in order to meet a range of conditions and needs. They are usually prefabricated structures consisting of straight or curved bars linked together in a compact bundle, which can then be unfolded into large-span, load bearing structural shapes. These structures have dual functionality since they act as mechanisms during its deployment and become immovable structures capable of supporting external loads during the service phase. In addition, they should be lightweight and compact to be easily transported and simple and quick to deploy. All these restrictions make it difficult to choose the best parameters regarding the shape and material of the structure, since many analyzes must be performed in order to find parameters that give lowest weight, highest stiffness, and that allow the structure to perform its two functions and ensure its reuse. Among the types of folding structure, those made of pantographic elements (scissors) have attracted great interest from engineers and architects in recent years. This study evaluates the geometric nonlinear behavior of plane arches constituted by two classic type of pantographic element, namely: polar and translational. For this, a detailed nonlinear geometrical analysis is conducted through a tailored corotational finite element software in order to evaluate the influence of the structure’s geometrical parameters, type of scissor units and supports on the nonlinear behavior and stability of the structure. The results obtained by our analyzes reveal, in most cases, a characteristic non-linear behavior of these structures with the nonlinear equilibrium path exhibiting several load and displacement limit points where jumps to remote and undesirable configurations may occur. Based on them, the influence of system parameters on the load carrying capacity of the arch is investigated.

1 INTRODUCTION Deployable structures are a class of structures that can be rapidly erected and easily folded for reuse. Moreover, they can change their topology in response to varying conditions and needs. Several transformable structures can be found in nature, like extensible worms, deployable leaves and certain insect wings [1], which serve as inspiration for engineering structures, including many civil engineering applications such as emergency shelters, exhibition halls, recreational structures and temporary buildings, among others. For this purpose, preassembled deployable scissor structures are highly effective: besides being transportable, they have the advantage of quick and ease deployment, while offering a huge volume expansion. The scissor mechanism was first presented in 1960 by the Spanish architect Emilio Perez Piñero. This mechanism uses the pantograph principle and consists of a bundle of scissor-like elements. These elements are comprised of two beams that are crosswise interconnected by a revolute joint, referred to as a pivotal connection, which introduces constraints of rotation normal to their common plane, as exemplified in Figure 1 where different scissor configurations are displayed. The units are connected to each other by joints at their end nodes that allow for inplane rotation. The upper and lower points of a scissor unit are connected by what is called unit lines [2]. 139

Figure 1. A scissor unit during deployment: (a) Completely deformed (b) partially deformed and (c) fully closed.

After Piñero, deployable scissor structures became more popular and attracted the interest of researchers who tried to improve their design and understand their mechanical behavior. Escrig [3], Gantes [4], Hoberman [5] and You and Pellegrino [6], among others, determined the geometric relations for different scissor configurations, proposed new types of scissor units and studied their structural response during deployment and under service loads in order to enhance their performance. New types of units were then obtained by changing the position of the pivot and the size and form of the beams, which have a profound influence on the final shape of the space enclosure and its deployment behavior. In spite of numerous new scissor elements configurations, two types of scissor units are more commonly used and are investigated in this work: translational units and polar units. Translational units are composed of two straight beams of same length with the pivot positioned in the same position in both beams, as illustrated in Figure 2(a), so that their axes remain parallel during deployment. In polar units, the pivot is shifted from the middle with a certain eccentricity which makes their axes intersect at a variable angle during deployment with respect to a fixed central point, as illustrated in Figure 2(b) [7]. Zeigler [8] improved Piñeros systems by introducing geometric incompatibilities between the member lengths of a scissor element thus creating self-locking and self-stiffening bistable structures which didn’t require any additional locking system. These bistable scissor structures are ideally geometrically compatible before and after deployment, but, during deployment, the large displacements combined with the geometric incompatibilities result in intended bending of some of the members. Consequently, these structures exhibit a snap-through behavior at sufficiently small loads during deployment to achieve a stable post-buckling configuration. The self-locking phenomenon can thus also be beneficial and desired because of the ease and speed of deployment.

Figure 2.

(a) Plane translational scissor unit; (b) plane polar scissor unit.

140

However, the structural response of these structures is complex, involving large displacements and rotations and a fully nonlinear analysis is necessary to evaluate the degree of nonlinearity and load carrying capacity. Plastic material behavior during deployment and use should be avoided since it would result in reduced load-bearing capacity in the deployed configuration and permanent strains. Due to this complexity, a nonlinear analysis requires the use of a set of advanced numerical tools. This research presents a detailed parametric analysis to evaluate the influence of the type of scissor unit (translational or polar), number of units and type of support on the structural behavior of scissor arches. For this, a tailored finite element software is used to conduct the geometric nonlinear analysis and stability of these structures [13].

2 DESIGN The realization and design of scissor like structures is hindered by their dual functionality: during deployment, it acts as a mechanism and afterwards, during the service phase, it becomes an immovable load-bearing structure. This increases the complexity of the design process where the relation between geometric design parameters, kinematic properties and structural response is crucial and important to take into account. Moreover, the designer has a large freedom when choosing design parameters for a scissor structure. With more structural insight into the geometrically nonlinear response of the scissor system, the engineer can make legitimate decisions related to these parameters in the design process. Langbecker and Albermani [9] and Mira et al. [10] investigated the effect of several design parameters on the structural behavior of barrel vault structures during its service phase when subjected to static loading. In addition, Gantes [11] and Arnouts [12] point out the importance of also evaluating the nonlinear behavior of scissor-like structures during their deployment and service life. The scissor elements are made of aluminum class A, EN-AW 6060, with modulus of elasticity equals to E = 70GPa, the same used by Mira et al. [10]. The bars of the translational units have square profiles of 100 x 3mm while the polar units have square profiles of 90 x 4mm. An arch with a span of 6m and height of 3m (height-to-span ratio of 0.5) when completely unfolded is considered and the number of units varies from 6 to 18. Five different types of support are considered, namely (a) pinned supports at one node of each end unit, (b) pinned supports at both nodes of the two end units, (c) pinned supports at the pivot of each end unit, (d) a fixed support at one node of each end unit and (e) fixed supports at both nodes of the two end units. The final configuration of the deployed arch depends on the number of units [1-12]. The design of the structures follow the methodology described in [7], where the lengths of the bars are calculated through a numerical method, the lengths varying with the number of elements. While in structures of polar units all the bars have the same length, in the ones with translational units the bars have different lengths, as illustrated for a structure with 10 units in Table 1. Table 1. Length of the bars of the arch with 10 units (translational and polar). Bar lenght (m) Bar Number

Tralational units

Polar units

1/20 2/19 3/18 4/17 5/16 6/15 7/14 8/13 9/12 10/11

1,48 0,54 1,42 0,6 1,3 0,7 1,2 0,82 1,06 0,94

0,98 0,98 0,98 0,98 0,98 0,98 0,98 0,98 0,98 0,98

141

3 RESULTS Due to their importance in the structural behavior, self-weight is taken into account in all analyses. In order to understand the arch nonlinear response, first the behavior of an arch composed of polar units with pinned supports at the end units of bars 1 and 20 (see Table 1) subjected to a concentrated load at the top none is investigated. Figure 3(a), where the load is plotted as a function of the vertical displacement of the top node, shows the nonlinear response of the arch for an increasing number of polar units. Structures composed of a small number of units (see results for 6 to 9 units) exhibit a strongly nonlinear behavior with several load and displacement limit points. Figure 3(b) shows in detail the nonlinear response for the structure composed of 7 units together with the deformed shape of the arch at each limit point. As observed here, the initial stiffness of the structure is very low leading to large displacements for small load levels. As the number of units increase, the nonlinearity decreases, as shown in Figure 3(c) for the structures composed by 10 and 11 units. In this case, the initial response is practically linear up to about 30kN. As the load increases beyond this point, the effective stiffness decreases and becomes zero at the first limit point where the structure jumps to a new stable position associated with large deflections, as illustrated in Figure 3(d) for the structure with 11 units, where the configuration of the structure before and after snap-through are displayed. This first limit point controls the load capacity of the deployed structure. Although not shown here for lack of space, structures composed of translational units exhibit larger deformations for the same number of units. The high flexibility of the structure is due to the type of support used in the model.

Figure 3. (a) Nonlinear equilibrium paths for structures composed by polar units pinned at one node of the end unit; (b) load-deflection curve for the structure with 7 units; (c) detail of nonlinear equilibrium paths for the structures composed of 10 and 11 units; (d) configuration of the structure composed by 11 units before and after snap-through.

142

In order to evaluate the influence of the supports and type of unit on the arch response, Figure 4(a) shows the nonlinear response for increasing number of polar units, while Figure 4 (c) shows the corresponding nonlinear response considering translational units. Figures 4(b) and (d) show a zoom of the initial behavior of the arch up to a vertical displacement of the top node displacement equal to 0.06m (arch span/100) for, respectively, the polar and translational units. Here, in both cases, the arch is pinned at the pivot node of the two end units, increasing thus its stiffness. By comparing the results in Figure 3(a) and Figure 4(a), both for polar units, the same overall behavior is observed. However, the load capacity of the structure is much higher when pinned at the pivot extreme nodes. There is no direct relation between the number of units and the first limit point load. First, for n=6 and 7 large deflections is observed for small load levels and no limit point is observed. For n=8 the path exhibits a flat region, indicating the appearance of a limit point. For higher number of unit, a limit point load appears and increases with the number of units. The highest load level is attained with 14 polar units, which presents a linear response up to a load level of 80kN. For n>14 the limit point load decreases with the number of units. By comparing the results in Figure 4(a, b) and Figure 4(c, d), the higher stiffness and load capacity of the arch with polar units becomes evident. As observed in Figures 4 (b) and (d), for the same vertical displacement, the load capacity of structures composed by polar units is at least five times higher than that for the structure composed by translational units. In addition, structures composed by translational units begin to display a nonlinear behavior at small load levels and vertical deflections, as shown in Figure 4(c, d). In order to understand the influence of the arch topology on its load capacity, Tables 2 shows the load level corresponding to a vertical displacement of the top node equal to 0.06m for the five types of support, considering an increasing number of polar units. Table 3 shows the same results for translational units. In all cases the highest load level occurs for the arch with clamped supports on both nodes of the two end units, as expected. The highest load level for the arch composed by polar units occurs with n=10 (71.03kN) and for the arch composed by translational units it occurs with n=8 (67.25kN), a decrease of about 5%. Having as

Figure 4. Nonlinear equilibrium paths for arches pinned on the pivot node of the end units: (a) arch composed by polar units; (b) zoom of Figure 4 (a); (c) arch composed by translational units; (d) zoom of Figure 4 (c).

143

Table 2. Load at a displacement equal to 0.06m for each support type and number of units of structures composed of polar units. Polar units Load at 0.06m (kN) Number of units

Support type (a)

Support type (b)

Support type (c)

Support type (d)

Support type (e)

n= n= n= n= n= n= n= n= n=

10.86 15.63 16.88 24.84 23.43 31.43 28.72 38.27 29.11

38.56 59.70 49.18 70.95 51.27 69.84 47.18 58.55 38.75

14.24 23.28 20.7 34.44 32.63 44.32 40.91 52.32 35.73

16.73 25.85 25.36 38.21 31.7 48.84 39.33 51.6 35.51

33.11 49.45 30.44 71.03 53.19 70.03 49.06 51.99 37.69

7 8 9 10 11 12 13 14 15

Table 3. Load at a displacement equal to 0.06m for each support type and number of units of structures composed of translational units. Translational units Load at 0.06m (kN) Number of units

Support type (a)

n= n= n= n= n= n= n= n= n=

8.05 9.15 7.09 7.59 6.25 6.02 4.99 0.9 0.01

7 8 9 10 11 12 13 14 15

Support type (b) 7.93 8.08 6.27 5.56 4.29 3.17 1,00 -0.2 0,00

Support type (c)

Support type (d)

Support type (e)

40.99 66.95 43.66 64.65 35.69 48.91 27.82 33.86 20.36

10.12 12.92 11.16 12.95 11.19 12.07 5.91 5.19 4.11

41.06 67.25 41.48 63.89 35.29 49.31 27.67 33.60 20.56

a design criterion the maximum deflection, no direct relation is observed between the load level and the type and number of units. Also the load capacity varies highly with the type of support for the same number of units. So, to obtain an optimal shape either a careful parametric analysis or an optimization algorithm should be employed by the designer. Figure 5 shows the nonlinear equilibrium paths for the arch with polar units and clamped supports at both nodes of the end units. Comparing with Figures 3(a) and 4(a), the more efficient behavior of this type of support becomes clear, corroborating their influence on the nonlinear response and load capacity. In the present case, the highest load level occurs for n=12 (around 225kN). Now the arch response under distributed load on the top nodes, as depicted in Figure 6, is investigated. Figure 7 shows the nonlinear equilibrium paths of the arch with clamped ends (case e) for selected values of scissor units considering both polar and translational units, while Figure 8 shows the results for the arch with pinned end (case c). The typical nonlinear behavior of a bistable structure is observed in both cases for most values of n, displaying an upper and lower limit point. Again the higher load capacity of the arch with polar units is observed for both types of support. For the clamped arch with polar units, the best structural response is observed for n=11. For n=8 a higher limit point load is observed but it is 144

Figure 5. Nonlinear equilibrium paths for arch composed of polar units and clamped supports at both nodes of the two end units.

Figure 6.

Arch with loads distributed along the top nodes.

Figure 7. Nonlinear equilibrium paths of the arch with loads distributed along the top nodes and clamped ends as a function of the number of scissor units. Load parameter versus central deflection.

associated with large deflections. Similar results are obtained for the pinned arch, as observed in Figure 8(b). For polar units, the highest load level at a deflection of 0.06m is the same for the two types of support and occurs for n=10 (35.3kN). For translational units, the load for clamped support is higher than that for pinned supports and occurs for n=8 (37.46kN). 145

Figure 8. Nonlinear equilibrium paths of the arch with loads distributed along the top nodes and clamped ends as a function of the number of scissor units. Load parameter versus central deflection.

4 CONCLUSION This work presents a detailed nonlinear analysis of deployable arches using a tailored finite element formulation. The influence of the type of scissor units (polar or translational), number of units and type of support is investigated. The results show that deployable arches present a highly nonlinear response with load and displacement limit points. The degree of nonlinearity and the load carrying capacity in terms of the first limit point load or a prescribed maximum deflection depends highly on the number and type of scissor units which also defines the form of the deployed arch. As expected, the arch load capacity is also highly dependent on the type of support. In all cases arches with polar units shows a higher load carrying capacity than those with translational units. So, to obtain an optimal shape either a careful parametric nonlinear analysis or an optimization algorithm should be employed by the designer to obtain the best structural configuration. REFERENCES [1] S. Pellegrino (Ed.). Deployable structures (Vol. 412). Springer, 2014. [2] K. Roovers and N. De Temmerman, “Deployable scissor grids consisting of translational units,” Int. J. Solids Struct., vol. 121, pp. 45–61, 2017. [3] F. Escríg, “Estructuras espaciales desplegables curvas,” Inf. la construcción, 1988. [4] C. Gantes, “A Design Metodology for Deployable Structures.” Ph. D. Thesis, Massachusetts Institute of Technology, 1991. [5] C. Hoberman, “Reversibly Expandable Doubly-Curved Truss Structures,” U.S. Patent n. 4,942,700, 1990. [6] Z. You and S. Pellegrino, “Foldable bar structures,” Int. J. Solids Struct., vol. 34, no. 15, pp. 1825–1847, 1997. [7] N. De Temmerman, “Design and analysis of deployable bar structures for mobile architectural applications,” Vrije Univ. Brussel. Ph. D. Thesis., 2007. [8] T.R. Zeigler, “Collapsible Self-Supporting Structures”. U.S. Patent n. 3,968,808, 1976. [9] T. Langbecker and F. Albermani, “Kinematic and non-linear analysis of foldable barrel vaults,” Eng. Struct., vol. 23, no. 2, pp. 158–171, 2001. [10] L.A. Mira, R.F. Coelho, A.P. Thrall, and N. De Temmerman, “Parametric evaluation of deployable scissor arches,” Eng. Struct., vol. 99, pp. 479–491, 2015. [11] C.Gantes, J.J. Connor, and R.D. Logcher, “Combining numerical analysis and engineering judgment to design deployable structures,” Comput. Struct., vol. 40, no. 2, pp. 431–440, 1991. [12] L.I.W. Arnouts, “Computational Investigation of the Structural Response of Bistable Scissor Structures,” M. Sc. Dissertation, ULB, Brussels, Belgique, 2017 [13] M.V.B. Santana, “Tailored Corotational Formulations for the Nonlinear Static and Dynamic Analysis of Bistable Structures”, Ph. D. Thesis, ULB & PUC-Rio, 2019.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Numerical advanced analysis of steel-concrete composite beams and columns under fire R.C. Barros, R.A.M. Silveira & P.A.S. Rocha Federal University of Ouro Preto, Ouro Preto, MG, Brazil

D. Pires Federal University of São João del-Rei, Ouro Branco, MG, Brazil

Í.J.M. Lemes Federal University of Lavras, Lavras, MG, Brazil

ABSTRACT: It is well known that high temperature causes changes in physical properties and mechanical strength of materials. In both steel and concrete, material properties deteriorate during the exposure to fire, and a steel, RC or composite structure load capacity and stiffness are reduced significantly as a consequence. The use of computational models as methodology of analysis/design has as advantage of capturing the strength and stability of the structural system and its members directly. This work shows the efficiency of the CS-ASA/FSA FE code in the analysis of insulated steel-concrete composite structural elements, such as beams and columns, in a fire situation. This approach performs the inelastic second-order analysis of composite structures under fire considering the refined plastic hinge method coupled to the strain compatibility method. Several composite structures in fire situation are analyzed using the developed tool.

1 INTRODUCTION The use of composite steel-concrete structures is increasing because of the many advantages that this combination of materials provides. This constructive system allows the two materials to be used together in beams, columns, frames and slabs in order to obtain a structure with high structural performance, geometric precision and low on-site waste. Despite of the significant increase in the use of steel-concrete composite structures, research on the behavior of these structures, especially in a fire situation, is still modest in Brazil. Thus, experimental studies and numerical models capable of simulating the behavior of structures in fire situation becomes extremely relevant. Espinos et al. (2016) presented the results of a numerical investigation on strategies for enhancing the fire behavior of concrete-filled steel tubular (CFST) columns by using inner steel profiles as circular hollow sections (CHS), HEB profiles or embedded steel core profiles. The ABAQUS commercial software was adopted, and a 3D finite element that allowed to compute different sections and nonlinear materials behavior under high temperatures was considered. Parametric studies were developed with the purpose of investigating the cross-section geometry influence on the resistant capacity of the columns under fire. Shallal et al. (2018) investigated the influence of high temperatures on continuous steelconcrete composite beams through an inelastic analysis. Their numerical models were based on an one-dimensional element of 4 nodes, with 3 degrees of freedom per node, including the partial interaction between the steel profile and the concrete slab. Good agreement was observed between their numerical results and the data from literature. Other recent works 147

involving composite structures numerical analysis in fire situations can be cited, i.e.: Zofrea (2014), Lai et al. (2017) and Pak et al. (2018). The objective of this contribution is to apply the in-house developed CS-ASA/FSA computational module (Computational System for Advanced Structural Analysis/Fire Structural Analysis) to the analysis of insulated steel-concrete composite structural elements, such as beams and columns, in a fire situation. This numerical module was developed for the inelastic second-order analysis of structures submitted to high temperatures. An approach based on the Strain Compatibility Method (SCM) is proposed to evaluate the cross-section strength level and the axial and bending stiffness of the composite structures under high temperatures. The construction of the moment-curvature relationship is essential for this evaluation. The tangent of the moment-curvature relationship, the stiffness, depends only on the chosen material behavior of the constituents. This methodology is coupled to the Refined Plastic Hinge Method (RPHM), in which the plasticity is evaluated only in the element nodal points using generalized stiffness parameters (Lemes 2018).

2 THERMAL ANALYSIS The thermal analysis is performed exclusively in the cross-sectional plane through finite element heat transfer models that allow the temperature distribution determination at different points in the cross-section. A time integration strategy based on the Finite Difference Method (FDM) is adopted for solving the discrete differential equations. The CS-ASA/FA module (Fire Analysis) also has two solvers: simple incremental and incremental-iterative (Picard or Newton-Raphson). It is important to mention that the steel and concrete thermal and mechanical properties in a fire situation are adapted according to the normative prescription of EN 1994-1-2 (2005). Further details of this computational module are available in Barros et al. (2018) and Pires (2018).

3 STRUCTURAL ANALYSIS The steel-concrete composite structures inelastic behavior in fire condition is captured using the RPHM and SCM coupling (Lemes 2018). The thermal action effects on the structure, i.e., the material stiffness and yield strength degradations, as well as the influence of the thermal strain on the element cross-section were considered within these numerical approaches. The following sections bring a summary of the FE formulation via RPHM, the SCM and how the bending moment-curvatures relationships are derived. Additional details of the structural solution strategy can be found in Pires (2018) and Barros et al. (2018). 3.1 Finite element formulation via RPHM The objective of the RPHM is to capture the evolution of the plastification at the nodes of the element, from the beginning of the yielding to its total plastification with a plastic hinge formation. The following main assumptions are considered in the model: all finite elements are initially straight and prismatic and their cross-sections remain plane after deformation; the steel profiles are compact; rigid body large displacements and rotations are represented; the shear deformation effects are ignored. The developed finite element is delimited by nodal points i and j, as illustrated in Figure 1. P, Mi, Mj and δ, θi, θj are the internal forces and associated displacements in the co-rotational system, respectively.

Figure 1.

Beam-column finite element.

148

The incremental equilibrium relationship of the finite element illustrated in Figure 1 is given by: 9 2 8 k11 < DP = DMi ¼ 4 0 ; : DMj 0

0 k22 k32

9 38 0 < Dδ = k23 5 Dθi ; : Dθj k33

ð1Þ

in which Δ denotes the increments of each quantity. The terms related to flexural stiffness in the matrix depend on geometric nonlinearity. A simplified second-order formulation, presented by Yang & Kuo (1994) was adopted here. Implementation details are available in Lemes et al. (2017) and Lemes (2018). 3.2 Strain compatibility method (SCM) When under external loads, a structure deforms to reach equilibrium. On the cross-section level, once the generalized internal forces equal the generalized external forces, the deformation stops (Lemes et al. 2017). This cross-section deformation is addressed in the SCM. For the application of this method, it is assumed that the strain field is linear and the section remains plane after deformation (Figure 2). This method couples the cross-section deformed configuration to the constitutive relationship of the material composing it. The evaluation of the axial and flexural stiffnesses is derived from the moment-curvature relationship and depends on the modulus of elasticity, which is obtained from the material (steel and concrete) constitutive relations (Pires 2018). 3.3 Moment-curvature relationship A discretization of the cross-section into fibers is performed with the objective of describing the deformation distribution using the axial strain (εi) in the plastic centroid (PC) of each fiber. From this the stress (σi) in each fiber is computed through the constitutive relationships of the materials. The axial strain in the ith fiber is given by: εi ¼ ε0 þ ϕyi þ εth;a þ ϕth yi

ð2Þ

where yi is the distance between the plastic centroids of the fiber analyzed and the crosssection; ε0 is the axial strain at the PC of the section; ϕ is the respective curvature; εth,a is the thermal axial strain; and ϕth is the curvature from the thermal strain, which is determined according to Pires (2018).

Figure 2.

Linear strain field.

149

The Newton-Raphson method is used at the cross-section level in order to obtain the moment-curvature relationship (M - ϕ). In matrix notation, the variables ε0 and ϕ are components of the generalized strain vector X = [ε0 ϕ]T. For computing the axial and flexural generalized stiffness, X = 0 is adopted in the first iteration, so that convergence is reached quickly (Chiorean 2013). It can be said that the cross-section equilibrium is obtained when the following equation is satisfied: FðXÞ ¼ f ext  f int ¼



Next Nint 5tol  Mext Mint

ð3Þ

where fext is the external force vector given by the axial force Next and bending moment Mext; the terms Nint and Mint are the components of the internal force vector, fint and tol is a tolerance. The internal forces are obtained from the deformed configuration of the cross-section through the expressions: ðð Nint ¼

ðð σa dA þ

Aa

ðð Mint ¼

Ac

σb dA ffi

nfib;a X

ðð σc ydA þ

σb ydA ffi

Ac

σai Aai þ

i¼1

Ab

ðð σa ydA þ

Aa

ðð σc dA þ

nfib;a X

i¼1

σai Aai yai þ

i¼1

Ab

nfib;c X

nfib;c X i¼1

σci Aci þ

nfib;b X

σbi Abi

ð4Þ

i¼1

σci Aci yci þ

nfib;b X

σbi Abi ybi

ð5Þ

i¼1

where nfib,a is the number of fibers; σa, σc and σb are the stress in steel, concrete and steel rebar, respectively; Ai and yai are the ith fiber area and its positions in relation to the Plastic Neutral Axis (PNA), respectively. For the following iteration, k+1, the strain vector is calculated according to (Chiorean 2013):  1   Xkþ1 ¼ Xk þ F0 Xk F Xk

ð6Þ

where F0 is the Jacobian matrix for the non-linear problems, i.e.: 0

F ¼



∂F ∂x

" ∂Nint

 ¼

∂ε0 ∂Mint ∂ε0

∂Nint ∂ϕ ∂Mint ∂ϕ

# ð7Þ

This numerical procedure is adapted and used for obtaining the N-M interaction curves as well. For a given axial force, the limit moment from the relationship of the moment-curvature is obtained, corresponding to the cross-section total plastification. This pair of forces define a point on the N-M interaction diagram. Noteworthy is the fact that the interaction curves are obtained independently the structural analysis, i.e. computed beforehand, in order to accelerate the execution of the structural simulations. More details on the computation of interaction curves, the structural analysis, as well as the thermal structural problem, can be found in Pires (2018), Barros et al. (2018) and Lemes (2018).

4 NUMERICAL ANALYSIS 4.1 Steel-concrete composite beams Caldas (2008) presented the numerical results of two simply supported steel-concrete composite beams in a fire situation. These beams were initially analysed by Huang et al. (1999) using the VULCAN software and experimental results were presented by Wainman & Kirby (1988). The beams cross-section is composed of a steel I profile 254 × 136 mm × 43 kg/m and 150

Figure 3. Room temperature structural analysis results of the composite beams. a) Test 15: deflection x temperature. b) Test 16: deflection x temperature.

a concrete slab with 624 × 130 mm. The reinforcing steel bars on the slab were neglected in the CS-ASA/FSA numerical simulations. The steel profile yield strength was considered equal to 255 MPa and the concrete had a compressive strength of 30 MPa. As shown in Figure 3a, four loads were applied in the experimental tests: 32.47 kN (Test 15) and 62.36 kN (Test 16). The room temperature structural analysis results, seeking to determine the beam critical load, as well as the interaction diagram between axial force and bending moment (N-M) for t = 0, are shown in Figure 3b. The steel-concrete composite beams were then exposed to standard fire curve (ISO 834-1 1999) on the steel profile bottom flange and the thermal properties were considered according to EC4 Pt.1.2 (EN 1994-1-2 2005). Eight linear finite elements were used to discretize the structural system and 346 linear quadrilateral finite elements (Q4) for the cross-section. Figures 4a and 4b present the mid-span deflection versus temperature of the bottom flange of the beam for the two tests, together with the numerical results of Caldas (2008) and Huang et al. (1999), and the Wainman & Kirby (1988) experimental results. 4.2 Steel-concrete composite columns Huang et al. (2008) presented a numerical study on fire resistance of embedded I-section composite columns through a FE program named FEMFAN-3D. The objective was to examine the effects of cross-sectional dimension and load level on the column fire resistance. Four groups of columns consisting of square cross-section were chosen for the study. Their crosssectional dimensions range from 250 × 250 to 400 × 400 mm2, respectively. These columns were subjected to axial compression forces and four-face uniform heating according to the standard fire curve (ISO 834-1 1999). All columns had 3 m length and were named as SZ1 to SZ4. The typical cross-section of columns with embedded I-sections and further details on the section’s dimensions are presented on Figure 5a. As in the previous example, Figure 5b shows the results of the structural analysis at room temperature and the interaction diagrams for the 4 cross-sections. Each column was modeled considering a mid-height initial deflection of L/300, where L is the length of the columns, and they were subjected to a centered axial load level of 0.5Py. The corner rebars were considered with yield strength equal to 460 MPa. For the concrete, a compressive strength of 42 MPa was adopted, while the steel profile yield stress was 330 MPa. Thermal and material properties of steel, concrete and rebar were taken from EC4 151

Figure 4.

Fire structural analysis results of composite beams.

Figure 5. Room temperature structural analysis results of the composite columns. a) Cross-section geometry. b) Equilibrium path and interaction diagram.

Pt.1.2 (EN 1994-1-2 2005). Concrete moisture content was set at 130 kg/m3 and relative emissivity was set at 0.5. The structural system was discretized with 10 linear finite elements and the cross-section with 240 linear quadrilateral finite elements. The thermo-structural analysis of the four columns are shown in Figure 6a, and Figure 6b brings the members’ last deformed configuration with the respective time intervals.

152

Figure 6. Fire structural analysis results of composite columns. a) Lateral displacement x time. b) Last deformed configuration of columns.

5 FINAL REMARKS The present study applied the CS-ASA/FSA computational module to steel-concrete composite structural elements in fire situation. This module efficiency had already been tested with steel and reinforced concrete structural elements in fire situations, where good results in both cases were observed (Barros et al. 2018, Pires 2018). The first numerical analysis consisted of composite beams subjected to thermal gradients. In general, the results obtained by CS-ASA/FSA showed good agreement with those found in literature, including experimental results. The numerical results presented for the steel-concrete composite columns were satisfactory as well. In this example it is worth pointing out a difference in numerical results and literature in the interval between 40 and 70 min. It may be associated to the representation of the degradation of the parameters of steel strength and stiffness, since a similar observation was made by Barros et al. (2018). As a final conclusion, it is possible to affirm that the computational implementations, specific for the thermo-structural analysis of steel-concrete composite structural elements under fire were successfully performed and yield satisfactory results, capturing well the behavior of the composite structures at high temperatures. ACKNOWLEDGMENT The authors would like to thank CAPES and CNPq (Federal Research Agencies), FAPEMIG (Minas Gerais Research Agency), Batir/ULB, Gorceix Foundation and UFOP/ PROPP for their support during the preparation of this research work. They also thank the prof. Péter Berke’ comments for the enhancement of this manuscript.

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REFERENCES Barros, R.C., Pires, D., Silveira, R.A.M., Lemes, I.J.M., Rocha, P.A.S., 2018. Advanced inelastic analysis of steel structures at elevated temperatures by SCM/RPHM coupling. Journal of Constructional Steel Research, vol. 145, pp. 368–385. Caldas, R.B., 2008. Análise numérica de estruturas de aço, concreto e mistas em situação de incêndio. Tese de Doutorado, Programa de Pós-Graduação em Engenharia de Estruturas, EE/ UFMG,Belo Horizonte, MG, Brasil (in Portuguese). Chiorean, C.G., 2013. A computer method for nonlinear inelastic analysis of 3D composite steel-concrete frame structures. Engineering Structures, vol. 57, pp. 125–152. Espinos, A., Romero, M.L., Lam, D., 2016. Fire performance of innovative steel-concrete composite columns using high strength steels. Thin-Walled Structures, vol. 106, pp. 113–128. European Committee for Standardization - EN 1994-1-2:2005. Eurocode 4: Design of composite steel and concrete structures, Part 1-2: General rules, Structural Fire Design. Huang, Z., Burgess, I.W., Plank, R.J., 1999. The influence of shear connectors on the behavior of composite steel-framed buildings in fire. Journal of Constructional Steel Research, 51, 219–237. Huang, Z.F, Tan, K.H., Toh, W.S., Phng, G.H., 2008. Fire resistance of composite columns with embedded I-section steel - Effects of section size and load level. Journal of Constructional Steel Research, 64, 312–325. ISO 834-1, 1999. Fire resistance tests - elements of buildings construction, Part 1: General requirements. ISO - International Organization for Standardization. Geneva. Lai, Z., Varma, A.H., Agarwal, A., 2017. Analysis of rectangular CFT columns subjected to elevated temperature. Proceedings of the Annual Stability: Conference Structural Stability Research Council, San Antonio, Texas, USA. Lemes, Í.J.M., 2018. Estudo numérico avançado de estruturas de aço, concreto e mistas. Tese de Doutorado, Programa de Pós-Graduação em Engenharia Civil, Deciv/EM/UFOP, Ouro Preto, MG, Brasil (in Portuguese). Lemes, Í.J.M., Silva, A.R.D., Silveira, R.A.M., Rocha, P.A.S., 2017. Nonlinear analysis of two-dimensional steel, reinforced concrete and composite steel concrete structures via coupling SCM/ RPHM. Engineering Structures, vol. 147, pp. 12–26. Pak, H., Kang, M.S., Kang, J.W., Kee, S.H., Choi, B.J., 2018. A numerical study on the thermo-mechanical response of a composite beam exposed to fire. International Journal of Steel Structures, vol. 18(4), pp. 1177–1190. Pires, D., 2018. Análise numérica avançada de estruturas de aço e de concreto armado em situação de incêndio. Tese de Doutorado, Programa de Pós-Graduação em Engenharia Civil, Deciv/EM/UFOP, Ouro Preto, MG, Brasil (in Portuguese). Shallal, M.A., Almusawi, A.M., 2018. Non-linear analysis of continuous composite beams subjected to fire. International Journal of Civil Engineering and Technology, vol. 9, pp. 521–532. Wainman, D.E, Kirby, B.R., 1988. Compendium of UK Standard Fire Test Data, Unprotected Structural Steel – 1, Rotherham (UF): Swinden Laboratories, British Steel Corporation, No. RS/RSC/ S10328/1/87/B. Yang, Y.B., Kuo, S.B., 1994. Theory and Analysis of Nonlinear Framed Structures. Prentice Hall. Zofrea, M., 2014. Behavior, analysis and design of concrete filled double skin tubular columns under fire. Tesi di laurea. Dipartimento ICEA, Corso di Laurea Magistrale in Ingegneria Civile, Padova, Italy.

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Buckling of spatial laced columns composed of built-up cold-formed channel members C.C.D.O. Bastos & E.M. Batista Civil Engineering Program, COPPE, Federal University of Rio de Janeiro, R.J., Brazil

ABSTRACT: This paper presents results of full-scale experimental tests carried out in spatial laced built-up columns, designed with lipped channel cold-formed steel members. The laced column chords, composed of two lipped channel members connected with selfdrilling screws, were previously analyzed using Generalized Beam Theory to identify its buckling loads and modes. Elastic buckling and nonlinear FEM analyses were performed with 3D bar elements for the laced column as a whole. Two analytical methods were applied to obtain global buckling load. The obtained results indicated the tested columns perform little influence of shear effect and critical buckling load may be accessed with the help of available analytical equations. Laced columns experimental performance and strength were recorded during the tests and compared with numerical results. Column collapse mechanism and ultimate strength proved to be strongly affected by local buckling effects. The obtained results indicate acceptable comparison between computed and experimental data.

1 INTRODUCTION Spatial laced columns composed of steel cold-formed steel members (CFS) are investigated in the present research, with the four chords connected with diagonal and bracing elements, conducting to trussed-type structural behavior. Hashemi and Jafari (2009, 2012), Bonab et al (2013), Kalochairetis et al (2014), Hashemi and Bonab (2013), have investigated the elastic critical load and strength of laced (or batten) columns through laboratory tests, but in all cases there were only two hot rolled chord members, connected by lacing bars. The behavior of a laced built-up column depends on its bending and shear stiffness, as well as on the connections stiffness. The effect of shear deformations may cause distortion of laced panels. Engesser (1891) was the first to consider the shear deformation effect on the elastic critical load of columns. Bleich (1952), Timoshenko and Gere (1963), and several other researchers followed the original findings of Engesser, such as: Gjelsvik (1991), Paul (1995), Aslani and Goel (1991) and Razdolsky (2005, 2010, 2011, 2014a and 2014b). The main objective of the present article is to present the results of full-scale experimental tests carried out in spatial laced built-up columns, designed with lipped channel cold-formed steel members. Four spatial laced columns were tested, with 12200 and 16200 mm length, 0.8 and 1.25 mm plate thickness, and 400 x 400 mm cross-section shown in Figure 1(a). The laced column longitudinal members (chords), composed of two lipped channel members connected with self-drilling screws, were previously analyzed by the generalized beam theory (GBT), in order to identify its buckling loads and modes. For the built-up column as a whole, elastic buckling and nonlinear FEM analyses were performed, with 3D bar elements. Two analytical methods were applied to obtain the global buckling load including shear effect, for pin-ended condition: Timoshenko (1963) and Eurocode 3 (CEN, 2006b).

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Figure 1. (a) Four chords laced built-up column cross-section (centroid of each chord is indicated with “+C”); (b) single lipped channel 88x86x40x42x12mm; (c) chords 2U88x86x40x42x12mm.

2 LACED BUILT-UP COLUMNS DESCRIPTION All tested columns have the same 400 x 400 mm cross-section shown in Figure 1(a). The laced column chords shown in Figure 1(c) were composed of two lipped channel members connected with 4.8 mm diameter self-drilling screws spaced at each 220 mm. Chords, diagonals and bracings are composed of lipped channel CFS with the measured dimensions (average) of single lipped channel cross-section 88 x 86 x 40 x 42 x 12 mm in Figure 1(b). Diagonals and bracings were connected to the chords with self-drilling screws. In order to evaluate the effect of global buckling, two lengths of columns were tested: 12200 and 16200 mm, as shown in Table 1. Laced columns were manufactured with two steel plate thicknesses t, for each column length: 0.8 or 1.25 mm. The chord’s edge stiffeners were cut off in the region of the arrival of diagonals and bracings, in order to enable connection. The laced columns were tested in horizontal position, as illustrated in Figure 2.

3 EXPERIMENTAL ANALYSIS OF THE LACED COLUMNS The built-up laced columns were tested in horizontal position, with compression load applied by a servo controlled hydraulic actuator. The actuator was placed at one end of the column, as shown in Figure 3, and a Dywidag steel rod crosses through the column length and is attached to a reaction stiff plate placed at the opposite end of the laced column. Loading condition was programed to smooth displacement control at 0.02 mm/s rate. The end condition for axial compression test was actually a semi-rigid type for flexural rotations (not perfect pined-pined). Rigid plate combined with spherical hinge was adopted at end section as shown in Figure 3. Semi-rigid behavior was imposed, since the spherical hinge is not able to allow actual free rotations for low loading condition. Displacement transducers (DT) were applied for minor and major axis deflection measurements, at mid-length and quarter-lengths of the column and close to the fixed end support. The minor inertia plane was placed in horizontal position for the tests. The strain gauge distribution included four sensors at each column end, placed externally in the web element of the chords. At the mid-length were placed eight strain gages, four in the internal side of the webs, thus performing four couples of strain gauges addressed to record the onset of local buckling deformation. The laced columns were manufactured with structural steel (commercial identification ZAR345) with nominal yield stress fy = 345 MPa. The steel mechanical properties were measured through standard tensile tests of 14 coupons extracted from the chord members (7 coupons for each thickness). The geometry of the coupons and the testing procedure were based on the guidance provided by ABNT ISO 6892 (2013) standard. The average result of yield stress fy, ultimate stress fu, Young modulus E and residual strain after failure εr are given in Table 2. 156

Figure 2.

Lateral views of the laced built-up column placed in horizontal position for the test.

Table 1. Built-up laced column IDs. Column ID

L (mm)

CFS

T12 x 0.8 T12 x 1.25 T16 x 0.8 T16 x 1.25

12200 12200 16200 16200

U88 x 86 x 40 x 42 x 12 mm lipped channel CFS, t = 0.8 mm U88 x 86 x 40 x 42 x 12 mm lipped channel CFS, t = 1.25 mm U88 x 86 x 40 x 42 x 12 mm lipped channel CFS, t = 0.8 mm U88 x 86 x 40 x 42 x 12 mm lipped channel CFS, t = 1.25 mm

During the tests of the laced columns, it was possible to observe the development of typical local buckling deformation mode along the webs of the chords before collapse. Although observed in almost all the tested columns, it was more visible for the case of 0.8 mm thick built-up laced columns, due to more slender CFS. Figure 4(a) shows the local buckling mode developing along the chords. Laced columns with nominal lengths 12200 and 16200 mm developed clear global buckling deformation with collapse mechanism at the mid length. It was possible to observe interaction between local (at the chord members) and flexural global buckling modes. Figure 5a) shows the horizontal displacements w recorded by displacement transducer at mid-length of column T16 x 1.25, indicating almost null flexural displacements until (aprox.) 110 kN. As referred, the spherical hinges were not able to allow free rotations for low loading

Figure 3.

Load application schematic arrangement.

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Table 2. Average results of the standard tensile tests. Thickness t (mm)

Yield Stress fy (MPa)

Ultimate Stress fu (MPa)

Young Modulus E (GPa)

εr %

0.8 1.25

370 375

477 479

198 215

20 18

Figure 4. Column T16 x 0.8, nominal length 16200 mm: (a) Local buckling deformation along the chords during the test, (b) collapse mechanism at mid length.

condition. Higher loading developed flexural buckling behavior which forced and liberated rotations at the ends of the laced column. The strain gauges registered linear (compression) behavior until the onset of local buckling. Figure 5(b) shows the strain gauges results for the built-up laced column T16 x1.25, where nearly linear behavior can be observed until (aprox.) 85 kN applied load. The beginning of local buckling effect was recorded by couple of strain gauges 12RI/12RE and 10RI/10RE. The registered collapse load was 196 kN. Finally, Table 3 shows a summary of the results for the tested built-up laced columns, including the records of (i) the collapse load (Puexp), (ii) the mid-length flexural displacement at the collapse load level and (iii) the loading level for which the onset of local buckling was identified (with the help of strain gage measurements).

Figure 5. Experimental results of column T16 x 1.25: (a) Load vs. horizontal displacement recorded at the column mid-length, (b) Load vs. strain measurements (με) at mid-length section of the laced column.

158

Table 3. Summary of the test results of the built-up laced columns. Buit-up laced column

Collapse load Puexp (kN)

Mid-length displacement (mm)

Local buckling load (kN)*

T12 x 1.25 T12 x 0.8 T16 x 1.25 T16 x 0.8

233 89 196 85

31 45 71 59

80 25 85 20

* Applied load related to the local buckling onset, according with strain gage couples measurements.

5 ELASTIC BUCKLING ANALYSIS OF THE LACED COLUMN In order to evaluate the global buckling of laced columns, numerical model was developed with the help of the finite element method with beam elements (BFEM). For this, SAP2000 (2011) computational program was achieved. The laced columns were taken as pined-pined and they were described by the centroid axis of the chords, as indicated in Figure 1(a). The first two buckling modes are flexural buckling around X and Y axes, the third buckling mode is torsional and the forth one is the second flexural buckling mode around X-axis. Buckling loads from numerical BFEM model are presented forward in the Table 4.

6 ANALYTICAL METHODS FOR GLOBAL BUCKLING In this section, the analytical values of the elastic critical loads for the built-up laced columns are computed according to methods proposed by Timoshenko (1963) and Eurocode (CEN, 2006b). The built-up laced columns were considered as pined-pined end condition. As previously reported, the steel properties are E = 198GPa and 215GPa, respectively for 0.8 and 1.25 mm steel plates. According with Timoshenko (1963), the critical load for laced column with “N” lacing, hinged ends and shear effect, is given by Equation 1: Pcr ¼

π2 EI l2

1   π2 EI 1 b 1þ 2 þ l Ad Esen cos2  aAb E

ð1Þ

where f is the angle between the diagonals and the batten bracings; Ad is the cross-section area of two diagonal members (one at each face of the laced column); Ab is the cross-sectional area of two batten bracings (one at each face of the laced column); b is the length of batten bracings and I is the moment of inertia of the cross-section of the laced column (considering the four chords).

Table 4. Global buckling critical loads: X- axis and Y- axis (Figure 10e) (kN). X - axis

Y - axis

Column ID

Eurocode

Timoshenko

BFEM

Eurocode

Timoshenko

BFEM

T6 x 1.25 T6 x 0.8 T12 x 1.25 T12 x 0.8 T16 x 1.25 T16 x 0.8

1454 (0.90) 862 (0.90) 410 (0.93) 243 (0.92) 236 (0.93) 140 (0.93)

1584 (0.98) 965 (1.01) 450 (1.02) 269 (1.02) 260 (1.02) 155 (1.03)

1618 959 442 263 254 151

2586 (0.96) 1533 (0.96) 749 (0.96) 444 (0.95) 434 (0.95) 257 (0.95)

2532 (0.94) 1584 (0.99) 762 (0.98) 459 (0.99) 445 (0.98) 266 (0.99)

2698 1600 777 465 456 270

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Eurocode EC3 Part 1-3 (CEN, 2006a), dedicated to cold-formed steel structures, does not present guidance to the design built-up laced columns. Therefore, Eurocode 3 Part 1-1 (CEN, 2006b) design procedure for this type of structural system will be considered. In this condition, the critical load for laced columns is given by Equation 2: Pcr ¼

π2 EIeff Ieff ¼ 0:5Ach h20 L2

ð2Þ

Ieff is the effective moment of inertia of laced built-up members; h0 is the distance between the centroids of the chords and Ach is the cross-section area of one chord. Equation 3 gives the effective critical load of built-up column considering shear effect (Pcr,v): Pcr;v ¼

1 1 1 þ Pcr Sv

Sv ¼

nEAd ah0 2

Ad h30 d3 1 þ Av d 3

ð3Þ

Sv is the shear stiffness of built-up columns with “N” lacing arrangement; “n” is the number of lacing plans; Ad is the diagonal cross-sectional area and Av is the horizontal bracings crosssectional area, of a single plan; “a” is the length of one panel; “d” is the diagonal length. Table 4 shows the computed results of the critical flexural buckling loads, for the laced columns, according with the proposed procedures from Timoshenko and Eurocode. Buckling loads from numerical BFEM model and the ratio between critical buckling loads related to BFEM results (inside parenthesis) are also included, from which one may observe quite similar results for both X and Y-axis. These results reveal the analytical method is able to provide accurate results in the present case, as well as the Eurocode procedure gives lower results if compared to analytical solution and numerical BFEM analysis. 7 COMPARISON BETWEEN BFEM MODEL AND EXPERIMENTAL COLLAPSE LOADS The laced column global buckling mode from the numerical buckling analysis was used as the initial geometric imperfection for the BFEM elastic geometrical nonlinear analysis. As recommended in Eurocode 3 part 1-1 (CEN, 2006b), the initial geometrical imperfection was adopted as L/500. The collapse of the built-up laced columns is recognized when one of the chord members reaches its axial compressive strength. So, the chord members compression forces, obtained from the BFEM elastic non-linear analysis, must be compared with the strength of one chord all along the loading path of the laced column. Additional experimental and numerical investigation of the axially compressed behavior of the built-up lipped channel chords with 480mm length (Figure 1(c)) was performed, in order to obtain strength of one chord. The results indicated that the direct strength method (DSM) proposed by Schafer and Peköz (1998) is able to estimate the ultimate strength of this type of built-up lipped channel section, for fully-composite and non-composite CFS condition. In the fully-composite section, the actual screws connection condition between the CFS members was admitted as fully effective and transformed in double thickness (2t) plate element in the portion of contact between web and flange of the channel elements, as shown at Figure 6(a). The non-composite condition considers isolated single channel section, as shown at Figura 6(c). As the members of the chords are connected with screws at discrete points, the actual condition is in between the non-composite and fully-composite values. Although the fully-composite CFS presented good agreement, the better results of the computed column strength was obtained for the non-composite built-up channel CFS. The analysis of the buckling modes and the corresponding critical buckling load Pcr is the first step for the design calculation of the chord using the DSM. Buckling analysis was performed taking into account fixed end condition, using generalized beam theory with the help 160

Figure 6. a) GBTUL fully-composite section (double thickness in the contact portion); b) local buckling deformation mode for fully-composite section; c) GBTUL non-composite section (members isolated); d) local buckling deformation mode for non-composite section.

of GBTUL computational program from Bebiano et al. (2018). Non-composite condition was admitted, considering the single section (members computed isolated), as it presented better experimental results. Both lipped and unlipped channel CFS chord member were considered, due to localized suppression of edge stiffeners at the connections. The Young modulus was assumed as E = 198 and 215 GPa, respectively for 0.8 and 1.25 mm steel plates, according with standard tensile test results of specimens obtained from the laced columns (Table 2). Poisson ration is admitted as 0.3. Figure 6 (a) and (b) shows the GBTUL fully-composite section and its local buckling mode, with double thickness at the contact. Figure 6 (c) and (d) shows the GBTUL non-composite section (members computed separately) and its local buckling mode. As the local buckling was predominant in the present case, Table 5 shows the values of PcrL for non-composite section, for single U. As the laced column has 400 mm modulation (see Figure 2), the compression strength of the chord was computed for constrained end condition and length L = 400 mm. Table 4 also presents the collapse load Pn obtained from DSM, for single U, for one chord (2U’s) and for four chords (8U’s). It was considered the measured yield stress from tensile tests fy = 370 and 375 MPa, for t = 0.8 and 1.25 mm respectively. Table 6 shows the experimental and numerical collapse loads for the tested built-up laced columns, considering the chord cross-sections with built-up lipped or unlipped channel members from Table 5. It is also included the ratio between numerical and experimental results, inside parenthesis. It is observed that the numerical results for unlipped channel chords presents adequate comparison with the experimental results. These results are sustained by the evidence of collapse mechanism developed at the cross-section with suppression of edge stiffeners. In addition, the last column of Table 6 displays the results of the laced column strength without global buckling effect, taking the axial compression strength of the chords from Table 5 (for built-up unlipped channel option). As expected, this solution gives very much inaccurate comparison with the experimental data.

Table 5. DSM column strength of the built-up chord members with constrained end condition, length L = 400 mm, considering non-composite behavior.

Built-up CFS Lipped channel U 88 x 86 x 42 x 40 x 12 Unlipped channel U 88 x 86 x 42 x 40 Lipped channel U 88 x 86 x 42 x 40 x 12 Unlipped channel U 88 x 86 x 42 x 40

Thickness GBTUL DSM (mm) Single U Single U 1 Chord (2U’s) 4 Chords (8U’s) Pn (kN) Pn (kN) Pn (kN) PcritL 0.80 0.80 1.25 1.25

13.4 7.0 54.0 26.8

161

28.8 20.6 63.4 44.9

57.5 41.1 126.9 89.8

230.1 164.5 507.4 359.0

Table 6. Numerical and experimental collapse loads for the tested built-up laced columns (kN). Built-up laced column

Experimental

BFEM Non-linear analysis

No global buckling

Column ID

Thickness (mm)

Experimental collapse load

Lipped channel

Unlipped channel

Unlipped channel *

T12 x 1.25 T12 x 0.8 T16 x 1.25 T16 x 0.8

1.25 0.8 1.25 0.8

233 89 196 85

311 (1.34) 152 (1.71) 207 (1.05) 115 (1.35)

248 (1.07) 118 (1.33) 186 (0.95) 98 (1.15)

359 (1.54) 164 (1.84) 359 (1.83) 164 (1.93)

* See last column of Table 4 for unlipped channel (4 chords).

8 CONCLUSIONS Elastic buckling analysis of steel laced column was performed with analytical and numerical solutions. The obtained results indicated the tested columns perform little influence of the shear effect and the critical buckling load may be accessed with the help of available analytical equations from Eurocode (CEN, 2006b) and Timoshenko (1963). As the collapse of the built-up laced columns is recognized when one of the chord members reaches its axial compressive strength, GBTUL and DSM were used to obtain the strength of the built-up CFS chord member. The DSM rules are able to estimate the ultimate strength of this type of built-up lipped channel section, adopted as chords of the laced columns. The better results of the computed chord strength were obtained for the non-composite built-up channel CFS. Experimental results clearly indicate the presence of global and local buckling during the tests of laced columns. Large elastic deformation developed before plastic localized collapse at the unlipped channel section (close to diagonal-chord connection). The tested columns, with nominal length of 12200 and 16200mm, developed important geometric nonlinear behavior from the flexural buckling mode, which was computed with the help of BFEM analysis. The laced column strength was finally computed taking the influence of the local buckling in the chord members, with the help of DSM equations. The obtained results indicate acceptable comparison between computed and experimental data with Pu/Puexp ranging from 0.95 to 1.33 (Table 6). It must be observed that the better results of the computed laced column strength was obtained for the unlipped built-up channel CFS, following experimental evidence of the collapse mechanism developed at these sections. ACKNOWLEDGEMENTS This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior- Brasil (CAPES) - Finance Code 001 and the National Council for Research and Technology, CNPq (Process 161975/2015-1). In addition, the authors would like to thank GYPSTEEL Company for the supply of the built-up laced columns. REFERENCES ABNT, 2013. ISO-6892: Materiais metálicos – ensaios de tração a temperatura ambiente. Associação Brasileira de Normas Técnicas, Rio de Janeiro, Brasil (in Portuguese). Aslani, F., Goel, S. C., 1991. An analytical criterion for buckling strenght of built-up compression members. Fourth quarter, Engineering Journal, AISC. Bleich, F., 1952. Buckling strength of metal structures, McGraw Hill, New York. pp. 174.

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Bonab, A.P., Hashemi, B.H., Hosseini, M., 2013. Experimental evaluation of the elastic buckling and compressive capacity of laced columns. Journal of Constructional Steel Research, vol. 86, 66–73. Bebiano, R., Camotim, D., Gonçalves, R., 2018. GBTUL 2.0 – a second-generation code for the GBTbased buckling and vibration analysis of thin-walled members, Thin-Walled Structures, 124, 235–253. CEN, 2006a. EN 1993-1-3:2006 - Eurocode 3: Design of steel structures. Part 1-3: General rules, supplementary rules for cold-formed thin gauge members and shetting. European Committee for Standardization, Bruxelas, Bélgica. CEN, 2006b. EN 1993-1-1:2006 - Eurocode 3: design of steel structures. Part 1-1: general rules and rules for buildings. European Committee for Standardization, Bruxelas, Bélgica. Engesser, F., 1891. Die knickfestigkeit gerader stabe (in german). Zentralbl bauverwaltung, Germany. Gjelsvik, A., 1991. Stability of built-up columns. ASCE Journal of Engineering Mechanics, vol. 117, 1331–1345. Hashemi, B.H., Jafari, M.A., 2009. Experimental evaluation of elastic critical load in batten columns. Journal of Constructional Steel Research; vol. 65, n. 1, 125–131. Hashemi, B.H., Jafari, M.A., 2012. Evaluation of Ayrton–Perry formula to predict the compressive strength of batten columns. Journal of Constructional Steel Research, vol. 68, n. 1, 89–96. Hashemi, B.H., Bonab, A.P., 2013. Experimental investigation of the behavior of laced columns under constant axial load and cyclic lateral load. Engineering Structures, vol. 57, 536–543. Kalochairetis, K.E., Gantes, C.J., Lignos, X.A., 2014. Experimental and numerical investigation of eccentrically loaded laced built-up steel columns. Journal of Constructional Steel Research, vol. 101, 66–81. Paul, M., 1995. Buckling loads for built-up columns with stay plates. ASCE, Journal of Engineering Mechanics, vol. 121, n. 11, 1200–1208. Razdolsky, A.G., 2005. Euler critical force calculation for laced columns. ASCE Journal of Engineering Mechanics, vol. 131, n. 10, 997–1003. Razdolsky, A.G., 2010. Flexural buckling of laced column with serpentine lattice. The IES Journal part A: Civil & Structural Engineering, vol. 3, n. 1, 38–49. Razdolsky, A.G., 2011. Calculation of slenderness ratio for laced columns with serpentine and crosswise lattices. Journal of Constructional Steel Research, vol. 67, n. 1, 25–29. Razdolsky, A.G., 2014a. Flexural buckling of laced column with fir-shaped lattice. Journal of constructional steel research, vol. 93, 55–61. Razdolsky, A.G., 2014b. Revision of Engesser’s approach to the problem of Euler stability for built-up columns with batten plates. ASCE Journal of Engineering Mechanics, vol. 140, n. 3, 566–574. SAP 2000 Version 15, 2011. Getting Started with SAP 2000 Linear and Nonlinear Static and Dynamic Analysis and Design of Three-Dimensional Structures. Computers and Structures Inc. Berkeley, California, USA. Schafer, B.W., Peköz, T., 1998. Direct strength prediction of cold-formed steel members using numerical elastic buckling solutions. In: 14th International Specialty Conference on cold-formed steel structures, St Louis, Missouri, USA, 69–76. Timoshenko, S.P., Gere, J.M., 1963. Theory of elastic stability. Mcgraw-Hill. New York, USA.

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Local-distortional buckling interaction of cold-formed steel columns design approach E.M. Batista & G.Y. Matsubara COPPE, Civil Engineering Program, Federal University of Rio de Janeiro, Brazil

J.M.S. Franco Urbanism and Architecture Department, Federal Rural University of Rio de Janeiro, Brazil

ABSTRACT: Local-distortional interaction buckling mode (LD) in steel cold-formed (CFS) columns is the objective of the present research. Experimental and FEM results are the basis of the original design proposition, based on both the direct strength method (DSM) and the Wintertype strength equation. The present results represents improvement of solution recently published by the authors, with modified equations bringing more simple formulation as well as keeping acceptable structural safety condition. The proposed solution offers strength surface bridging the widely accepted DSM solutions for local and distortional buckling. This concept is based on the main variable of LD interaction of lipped channel columns, the ratio between distortional and local buckling slenderness factor RλDL = λD/λL. After identification of the range of RλDL for which columns are significantly affected by the LD interaction, the Winter-type surface equation is calibrated with the help of experimental data and FEM results.

1 INTRODUCTION The design procedures of steel cold-formed section (CFS) members derives from buckling behaviour, because of its slenderness properties. Thin-walled CFS members develop local L, distortional D and global G buckling modes identified by their critical loading and buckling shape illustrated in Figure 1. In addition, buckling modes interaction (LG, DG, LD or LDG) produces strength reduction if compared to single mode buckling development, currently identified as the strength erosion of thin-walled CFS member. The combination of buckling modes with characteristic deformed shapes, as is the case of short L, long G and intermediate D semi-wave lengths, conducts to interaction behaviours with distinct severity, obliging definition of particular solutions regarding each type of interaction. The identification of the structural stability properties of a CFS column follows its “signature curve”, obtained with the help of numerical analysis, in order to identify the buckling loads associated to the column semi-wave lengths, Pcr vs. L. The signature curve reveals more than the half wavelength but the actual CFS buckling mode shape, allowing clear information about the column stability. The most relevant numerical methods to obtain the signature curve are (i) the finite strip (FSM) and (ii) the generalized beam theory (GBT) methods, available with the computational tools (free access) CUFSM, Li & Schafer (2010), and GBTUL, Bebiano at al. (2008), respectively. The first order linear elastic buckling analysis results configures the basis of the structural design of thin-walled steel columns and beams. The closer the buckling loads PcrL, PcrD and PcrG, respectively related to L, D and G modes, the more probable is the occurrence of buckling interaction, allowing stronger nonlinear post buckling behaviour of the column. In addition, there is no general rule to take into account all the possible buckling interaction effects of CFS columns, essentially because of the different nature of the modes interaction behaviour, by combining short-long or short-medium half wavelengths, LG and LD, for example. This is the reason LG buckling modes interaction 164

Figure 1. Buckling modes of lipped channel CFS columns: local L, distortional D, global flexural and flexural torsional G.

received safe and accurate design procedures, since much time ago, and only more recently LD, DG and LDG interactions are considered with more attention. The present research aims at contributing for design rules for lipped channel CFS columns experiencing LD interaction, considering almost null presence of global buckling. This condition is assured for columns with PcrG sufficiently higher than PcrL and PcrD or, in terms of slenderness coefficient, for λG lower enough than λL and λD, with λG ¼ (Py/PcrG)0.5, λL ¼ (Py/PcrL)0.5 and λD ¼ (Py/PcrD)0.5, Py ¼ Afy is the column squash load and fy the steel yielding stress. Based on FEM computation and available experimental data, the research methodology and results are thorough described in Matsubara et al. (2019). In the following, the sets of numerical and experimental data are resumed, the development of the design concept is explained and improved rule and equations for the calculation of lipped channel columns strength PnDL are presented. 2 EXPERIMENTAL DATA The following sets of experimental results of lipped channel columns displaying LD buckling interaction were considered: (i) Kwon & Hancock (1992) (5 columns); (ii) Young & Rasmussen (1998) (2 columns); (iii) Loughlan et al. (2012) (20 columns); (iv) Young et al. (2013) (26 columns) and (v) Salles (2017) (3 columns). The range of variation of the geometrical and material properties of the tested columns are indicated in Table 1, which includes the ratio between flange and web widths bf /bw and edge stiffener and web widths bs/bw, the column length L, fy and E, respectively yielding stress and Young elastic modulus of the steel and the recorded experimental column strength load Puexp. In addition, Table 2 includes the slenderness factors related to L, D and G buckling modes, respectively λL, λD and λG, as well as the ratio between distortional and local buckling slenderness factor RλDL ¼ λD/λL. Finally, only columns with cross-section geometry inside the followings ranges were taken into consideration: 0.4  bf /bw 5 1.0 and 0.1  bs/bw 5 0.3 (the only exception are the columns tested by Kwon & Hancock (1992), with bf /bw 5 0.1). This is so because (i) these are practical and usual geometries and (ii) out of these ranges, L and D modes exhibit inappropriate behavior inducing important loss of strength.

Table 1. Ranges of the geometric and material properties and experimental strength of the tested lipped channel columns. Reference

bf/bw

bs/bw

L(mm)

fy (MPa)

E (GPa)

Puexp (kN)

Kwon & Hancock (1992) Young & Rasmussen (1998) Loughlan et al. (2012) Young et. al (2013) Salles (2017)

0.75 0.5 0.35-0.63 0.45-1.00 1.0

0.04-0.06 0.12 0.07-0.12 0.1-0.3 0.1

400-800 616-1500 1000-1800 615-2500 2529

590 550 209 336-590 342-348

210 180 193 203-213 180

49-55 99-102 27-33 40-309 29-33

165

Table 2. Ranges of variation of the slenderness factors and DL slenderness factor ratio of the tested lipped channel columns. Reference

λL

λD

λG

λmáx

RλDL

Kwon & Hancock (1992) Young & Rasmussen (1998) Loughlan et al. (2012) Young et. al (2013) Salles (2017)

2.64 - 2.70 1.55 - 1.60 1.50 - 2.70 0.87 - 3.36 2.01 - 2.16

1.87 - 3.32 1.24 - 1.27 0.97 - 1.46 0.65 - 2.41 2.24 - 2.28

0.20 - 0.40 0.63 - 0.94 0.26 - 0.63 0.24 - 1.30 1.06 - 1.12

2.64 - 3.32 1.55 - 1.60 1.50 - 2.70 0.87 - 3.36 2.24 - 2.28

0.71 - 1.23 0.77 - 0.82 0.48 - 0.75 0.58 - 0.98 1.06 - 1.11

In particular, the experimental results from Salles (2017) were thoroughly described by Matsubara et al. (2019) and one may observe clear evidence of LD buckling interaction. These are lipped channel columns included in Tables 1 and 2, with fixed-fixed end condition, tested through displacement control hydraulic system. The length L was designed to give close local and distortional buckling loads, with the ratio between distortional and local buckling slenderness factor RλDL ¼ λD/λL ¼ 1.1, at the same time global buckling mode is much higher, with PcrG ≈ 4.2PcrD. Figure 2 shows local and distortional buckling development from column C2 from the tests performed by Salles, the former from couples of strain gages placed in the web and the last from displacement transducers, with the measuring devices placed at the mid length section of the column. These results show (i) the onset of local buckling is clearly observed at (approx.) 20kN load level (Figure 2(a)) and (ii) distortional buckling displaying typical inward deformation since the very beginning of the test in Figure 2(c) (observe the signs of displacement transducers measurements in Figure 2(d)). Buckling mode shapes L and D were visually observed and recorded during the tests.

3 FEM MODEL RESULTS Finite element SHELL181 from Ansys finite element package, Ansys (2009), was applied to perform computational analysis of thin-walled lipped channel columns. SHELL181 is suitable for thin-shell structures linear analysis, as well as geometric and material nonlinearity. The analysis was performed in full integration method, as indicated by Ansys (2013) theory reference. The von Mises isotropic hardening plasticity model considered in the present analysis is available in the FEM software package. FEM mesh is 5 mm width quadrilateral elements, according with previous results from Fena (2011) and Silvestre et al. (2012). Convergence tests confirmed the accuracy of the mesh. Material ductility was introduced with (i) bilinear ductile steel elastoplastic development or (ii) nonlinear ductile response based on experimental data from coupon tensile tests, from Salles (2017). The FEM results of the column behaviour with options (i) and (ii) display very similar results, with clear prevalence of elastic post buckling behaviour, which justify analogous results with options (i) or (ii). Inelastic effect of thin-walled structural members only arises at the very end of the loading, close to the ultimate limit load, with development of the final collapse mechanism. Initial geometrical imperfections were taken from the computed critical buckling mode shape, with maximum amplitude 0.1 of the cross-section thickness t. This means that for the FEM analysis of the experimental sets of columns described in Tables 1 and 2, the initial geometric imperfection follows distortional or local buckling shape, respectively for λD > λL or λL > λD. For the case of columns displaying combined LD critical buckling mode, with λD ≈ λL, this is the imperfection geometric shape adopted for numerical nonlinear analysis. Global buckling is never concerned in the sets of experimental results (global buckling slenderness factor λG is always much lower than λD and λL). In addition, Riks (1979) method was taken to access the ultimate limit loading, from now on nominated FEM-based column strength PuFEM. 166

Figure 2. Experimental evidence of LD interaction mode from tested lipped channel column C2, Salles (2017): (a) records of couple of strain gages E1-E2 show local buckling development, (b) couple of strain gages E1&E2 placed in the web, mid length of the column; (c) D4-D5 displacement transducers show development of inward distortional buckling, (d) displacement transducers D4&D5 at mid length of the column.

Column end conditions of the FEM model was fixed-fixed, according with fully restricted displacement experimental condition, as described by the authors of the sets of tests. For this purpose, stiff 25 mm thick plates were attached to the end sections of the columns and point compressive loads were applied at the centroid position of both end sections of the column. Figure 3 shows the shell finite element model as well as the cross-section typical geometry. Figure 4 shows typical example of results from the FEM model for lipped channel columns. Figure 4(a) displays (i) the evolution of the local buckling arising at the indicated load level I, (ii) elastic LD interaction developing from load level II and (iii) final collapse configuration at load level III. Figure 4(b) shows the FEM deformed shape at collapse of one of the columns tested by Young et al. (2013), with clear evidence of LD buckling interaction mode. Figure 5 shows complimentary comparison of FEM and experimental results from one of the columns tested by Salles (2017). In this case, FEM and experimental results of displacement transducers moving along the column length are in very good agreement, showing the evidence of LD interaction mode by displaying combined mode shape L+D with increased amplitudes for rising steps of the applied load. The developed FEM model reproduced typical LD interaction buckling results for the complete sets of experimental tests included in Tables 1 and 2, with very close agreement between 167

Figure 3.

Ansys (Shell 181) FEM model of fixed-fixed lipped channel CFS columns.

Figure 4. Examples of FEM results of a column tested by Young et al. (2013): (a) load vs. distortional buckling parametric displacement d/t at mid-length, complimented by the evolution of the deformed pattern showing local (point I) to distortional buckling (point III) modes; (b) deformed configuration at collapse.

numerical and experimental ultimate loads, PuFEM and Puexp, as can be accessed in Matsubara (2019). These results allowed additional computation of a planned set of 275 columns with 0.4 ≤ bf /bw < 1.0 and 0.1 ≤ bs/bw < 0.3 (usual lipped channel sections), designed for the occurrence of LD buckling interaction. The complete sets of experimental and numerical results permitted to develop and test analytical solution for the design of lipped channel columns under LD interaction, as presented in the following.

4 DIRECT STRENGTH METHOD BASED DESIGN The direct strength method DSM, Schafer (2006), is a widely accepted procedure to deal with the structural strength of CFS members. DSM solution for columns under L and D buckling are based on Winter-type equation, respectively Equations 1 and 2. Global buckling is addressed by the DSM with the help of widely accepted general solution of steel columns, by providing the reduction strength factor χ ≤ 1.0, according with the slenderness factor λG 168

Figure 5. Experimental and FEM results of tested column from Salles (2017). Displacements measured by transducers D2 and D4 (see Figure 2(d)): (a) FEM results and (b) experimental records of D2 confirm presence of L buckling mode; (c) FEM results and (d) experimental records of D4 shows evidence of D mode.

related to flexural or flexural torsional buckling. LG buckling interaction of CFS columns is a well stablished procedure solved by appropriate combination of χ factor with Equation 1. PnL ¼

! 0:15 Py 1  0:8 λL λ0:8 L

ð1Þ

PnD ¼

! 0:25 Py 1  1:2 λD λ1:2 D

ð2Þ

Concerning LD buckling interaction of CFS lipped channel columns, the most relevant propositions found in recent literature are those from Schafer (2002), Martins et al. (2017) and Matsubara et al. (2019). In the present research, the following conditions were considered to enable easy to apply and accurate proposition, in close agreement with Matsubara et al. (2019) findings. a. Column strength for LD buckling interaction is based on the Winter-type Equation 3. b. A and B parameters in Equation 3 are based on the main variables of the problem: (i) the slenderness ratio RλDL ¼ λD/λL and (ii) the column slenderness λmax ¼ max (λD; λL). c. The range of lipped channel columns affected by LD buckling was previously identified by Martins et al. (2017) and additionally confirmed by Matsubara et al. (2019). The range of LD interaction is defined by the slenderness ratio factor: 0.45 ≤ RλDL ≤ 1.05. d. For slenderness ratio RλDL < 0.45 local buckling L is dominant and no LD interaction should be considered. In this case the column strength takes into account Equation 1. e. For slenderness ratio RλDL > 1.05 distortional buckling D is dominant and no LD interaction should be considered. In this case the column strength takes into account Equation 2. f. Finally, global buckling is not the case for the examined columns, with PcrG /PcrD (as well as PcrG /PcrL) at least higher than 3.0, thus ensuring single LD buckling interaction mode.  PnLD ¼

 A Py 1 B B λ λ

169

ð3Þ

Expressions for the parameters A and B in Equation 3 were developed with the help of the above described and calibrated FEM model. For this, a representative set of 275 lipped channel columns was developed, covering the following conditions: (i) LD slenderness interaction factor RλDL ranging from 0.27 to 1.5; (ii) column slenderness λ ¼ λmax ¼ max(λD; λL) equal to 1.0, 1.5, 2.0 and 2.5; (iii) flange-web width ratio 0.4 ≤ bf /bw < 1.0; (iv) edge stiffener-web width ratio 0.1 ≤ bs/bw < 0.3. The columns’ properties to enable such representative selection were controlled by their cross-section geometry, column length L and material yielding stress fy. Original solution from Matsubara et al. (2019) implies polynomial A and B parameters with 3rd and 4th degree, respectively, resulting in very accurate results. Recent results of the investigation allowed improved solution with simpler equations and keeping acceptable accuracy of the method. Equations 4 and 5 show the proposed solution for A and B parameters, to be applied in the LD column strength Equation 3. A ¼ 0:40RλDL  0:17

ð4Þ

B ¼ 2:26R2λDL þ 4:06RλDL  0:57

ð5Þ

The proposed solution with Equations 3, 4 and 5 ensures the continuous strength surface illustrated in Figure 6 in parametric format PnLD /Py, based on the main variables RλDL and λ ¼ λmax ¼ max(λD; λL) and performing natural transition between accepted DSM solutions for single L and D buckling modes, Equations 1 and 2. The proposed surface includes usual ranges of the lipped channel cross-section geometry (bf /bw and bs /bw) stablishing a solution that includes acceptable dispersion of the results when compared with experimental and FEM sets. Original solutions reported by Matsubara et al. (2019) include not only a unique solution for A and B but also different equations for these parameters, addressed to restricted ranges of the flange-web width ratio bf /bw. Of course, the more tailored the mathematical model the more precise the comparison between analytical and reference results of the column strength. The choice of

Figure 6.

Proposed strength surface PnLD/Py for LD interaction buckling of CFS lipped channel columns.

170

Figure 7. Comparison between the proposed rule PnLD for CFS lipped channel columns and (a) the experimental results included in Tables 1 and 2; (b) the FEM results (those inside the range of validity of LD interaction buckling: 0.45 ≤ RλDL ≤ 1.05).

a single surface solution take into account simplicity for design purposes and acceptable solution for standards and codes of structural design practice. Figure 7(a) shows the comparison between the proposed solution PnLD and the set of experimental results included in Tables 1 and 2. Furthermore, Figure 7(b) illustrates the result of the comparison between the proposed solution and the set of FEM results, and one may confirm the accuracy of the proposition by the computed averages 1.08 and 0.99, and coefficient of variation 0.15 and 0.10, respectively for experimental and FEM sets of lipped channel columns.

5 FINAL REMARKS The presented results and comparisons between proposed formulation PnLD for LD buckling interaction of CFS lipped channel columns is an extension of previous results recently published by the authors in Matsubara et al. (2019). The proposed rule is essentially an integrated solution for the design of CFS columns including local and distortional buckling modes, by proposing the transition solution between DSM-based single L and D strength Equations 1 and 2. The proposed rule was tested and proved to be accurate by comparison with available experimental results as well as when submitted to comparison with large set of FEM results. Finally, the computation of the resistance factor on the basis of the load and resistance factor design method (LFRD) confirmed adequate performance of the proposed approach, with ϕ ¼ 0.83 (and γ ¼ 1/ϕ ¼ 1.21), in good agreement with the values recommended by both the North American standard AISI (2016) and the Brazilian code ABNT (2010), respectively ϕ ¼ 0.85 and γ = 1.2. ACKNOWLEDGEMENTS The first author acknowledges the financial support of CNPq, National Council for Scientific and Technological Research, through scholarship for his Master degree research. REFERENCES ABNT. 2010. NBR 14762 Design of cold-formed steel structures. Associação Brasileira de Normas Técnicas (in Portuguese). AISI. 2016. North American Specification for the design of Cold-formed Steel Structural Members – AISI S100, American Iron and Steel Institute, Washington, USA.

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ANSYS. 2009. Version 12: SAS – Swanson Analysis Systems Inc. ANSYS. 2013. ANSYS APDL Theory Reference, Release 15.0, Canonsburg, PA. Bebiano, R.; Silvestre, N.; Camotim, D. 2008. GBT theoretical background. Available at http://www.civil. ist.utl.pt/gbt/, (access on 8 February 2019). Fena, R.P.T. 2011. Interacção Local/Distorcional em colunas de aço enformadas a frio com secção em “Hat”. Master of Scienec dissertation. Technical University of Lisbon, Lisbon, Portugal. Kwon, Y. B.; Hancock, G. J. 1992. Tests of cold-formed channels with local and distortional buckling. Journal of Structural Engineering, v. 118: p. 1786–1803. Li, Z.; Schafer, B. W. 2010. Buckling analysis of cold-formed steel members with general boundary conditions using CUFSM: Conventional and constrained finite strip methods. In: Twentieth International Specialty Conference on Cold-Formed Steel Structures, St. Louis, Missouri, USA, p. 1–15. Loughlan, J.; Yidris, N.; Jones, K. 2012. The failure of thin-walled lipped channel compression members due to coupled local-distortional interactions and material yielding. Thin-Walled Structures, v. 61: p. 14–21. Martins, A.D.; Camotim, D.; Dinis, P.B. 2017. On the direct strength design of cold-formed steel columns failing in local-distortional interactive modes. Thin-Walled Structures, v. 120: p. 432–445. Matsubara, G.Y.; Batista, E.M.; Salles, G.C. 2019. Lipped channel cold-formed steel columns under local-distortional buckling mode interaction. Thin-Walled Structures, v. 137: p. 251–270. Riks, E. 1979. An incremental approach to the solution of snapping and buckling problems. Internacional Journal of Solids and Structures, v. 15: p. 529–551. Salles, G.C. 2017. Investigação analítica, numérica e experimental do modo de flambagem distorcional em perfis formados a frio. Master of Science dissertation. COPPE, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil. Schafer, B.W. 2002. Local, distortional and Euler buckling of thin-walled columns. Journal of Structural Engineering, v. 128: 289–299. Schafer, B.W. 2006. Designing cold-formed steel using the Direct Strength Method. In: Eighteenth International Specialty Conference on Cold-Formed Steel Structures, Orlando, FL., USA. Silvestre, N.; Camotim, D.; Dinis, P.B. 2012. Post-buckling behaviour and direct strength design of lipped channel columns experiencing local/distortional interaction. Journal of Constructional Steel Research, v. 73: p. 12–30. Young, B.; Rasmussen, K.J.R. 1998. Design of lipped channel columns. Journal of Structural Engineering, v. 124: p. 140–148. Young, B.; Silvestre, N.; Camotim, D. 2013. Cold-Formed Steel Lipped Channel Columns Influenced by Local-Distortional Interaction: Strength and DSM Design. Journal of Structural Engineering, v. 139: p. 1059–1074.

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Solutions to simplified von Karman plate equations J. Becque The University of Sheffield, Sheffield, UK

ABSTRACT: The non-linear von Karman equations describe the post-buckling behaviour of thin plates, but a general solution cannot be found. In this paper some simplifying assumptions are made which follow from a consistent interpretation of von Karman’s effective width concept. This reduces the coupled von Karman equations to a single equation and allows a Winter-type equation to be derived for the ultimate capacity of single plates, accounting for yielding and initial geometric imperfections. 1 SIMPLIFYING VON KARMAN’S EQUATIONS The von Karman equations (sometimes referred to as the Föppl-von Karman equations) comprise a system of two coupled differential equations which describe the non-linear postbuckling behaviour of thin elastic plates (Föppl 1907, von Karman 1910):

∂4 w ∂4 w ∂4 w þ 2 þ ¼ pz ∂x4 ∂x2 ∂y2 ∂y4 2 2

∂ ’ ∂ ðw þ w0 Þ ∂2 ’ ∂2 ðw þ w0 Þ ∂2 ’ ∂2 ðw þ w0 Þ þt  2 þ ∂y2 ∂x2 ∂y∂x ∂x∂y ∂x2 ∂y2

D

∂4 ’ ∂4 ’ ∂4 ’ þ 2 þ ¼E ∂x4 ∂x2 ∂y2 ∂y4



∂2 w ∂x∂y

2 

∂2 w ∂2 w ∂x2 ∂y2

∂2 w0 ∂2 w ∂2 w0 ∂2 w ∂2 w0 ∂2 w þ2   ∂x∂y ∂x∂y ∂x2 ∂y2 ∂y2 ∂x2

ð1Þ

ð2Þ

In these equations w is the vertical plate deflection, w0 is the initial geometric imperfection, ϕ is Airy’s stress function, E is the elastic modulus, t is the (constant) plate thickness, pz is the lateral pressure on the plate and D is given by: D¼

Et3 12ð1   2 Þ

ð3Þ

The x-y coordinate system is illustrated in Figure 1. The scope of this paper is limited to plates under uniform in-plane compression. Eqs. (1–2) account for imperfections, but not for material non-linearity. However, the biggest impediment to their practical application is that, while solutions exist for a few special cases (e.g. Levy 1942), a general solution is beyond reach. The von Karman equations consequently have little relevance in practical design, which relies instead on the empirical Winter equation (1970):   Pu 1 0:22 ¼ 1 Py λ λ

173

 1:0

ð4Þ

Figure 1.

Plate in axial compression.

where Pu is the ultimate capacity of the plate in compression, Py is the yield load and the slenderness λ is given by: sffiffiffiffiffiffi fy λ¼ σcr

ð5Þ

In the above equation fy is the yield strength and σcr is the critical elastic local buckling stress. The aim of this paper is to establish a link between the von Karman equations and a Winter-type design equation. In order to achieve this, it is clear that some additional assumptions leading to a simplification of Eqs. (1–2) are necessary. The inspiration for these is provided by the effective width concept, credited to von Karman et al. (1932). This concept is based on the observation that in the post-local buckling range the longitudinal stresses shift towards the longitudinal edges of the plate and can thus be idealized as being carried by two strips adjacent to those edges. The widths of these effective portions are obtained by equating the total stress in the actual and idealized distributions (Figure 1). Failure is assumed to occur when the effective strips yield. While this failure criterion will be employed later in the derivation, the major implication of this idealized stress distribution which proves useful for our objectives at this stage is that the longitudinal membrane stress σx is only a function of the transverse co-ordinate y and is constant along a ‘fibre’ in the longitudinal x-direction: σx ¼

∂2 ’ ¼ f ð yÞ ∂y2

ð6Þ

Integrating twice with respect to the y-coordinate yields the following form for Airy’s stress function: ’ ¼ gð yÞ þ y:cðxÞ þ d ðxÞ

ð7Þ

However, because of the symmetry of the problem, the mixed term in x and y has to vanish. Consequently, the shear stresses in the plate have to be zero: τxy ¼ 

∂2 ’ ¼0 ∂x∂y

ð8Þ

This is consistent with the view that each longitudinal fibre acts independently, carrying a constant stress along its length, while not partaking in any load sharing with its neighbours. The final implication is that σy is independent of y: 174

σy ¼

∂2 ’ ¼ hðxÞ ∂x2

ð9Þ

Two cases of boundary conditions are considered in this study. In both cases the loaded edges (x = 0 and x = L) remain straight in the post-buckling range. This corresponds to the practical case of a plate element in a long column, where ‘nodal lines’ develop in between buckled cells. For the longitudinal edges two cases are considered: case A, where the edges are free to pull in during the post-buckling stage (Figure 2), and case B, where the edges can move in while remaining straight (Figure 3). Case A is most representative of a plate (e.g. a web) in an actual column, given the limited out-of-plane bending stiffness of the adjacent plates (i.e. the flanges in this particular example). The elongation of a longitudinal fibre can be determined from the deflected shape as follows: 9 82LU sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3  ð   ð qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2 = < ð 0 1 1 1 ∂w 4 εx ¼ ds  L ¼ ðdxÞ2 þ ðdwÞ2  L ¼ 1þ dx5  L ; L L L: ∂x 0 9 82LU 3 2 3   ! ð 0 = 1 ðL ∂w2 1 0, respectively, are obtained, and the ratio of these coefficients with respect to C shown in Eq. (8). Furthermore, when the absolute value of the maximum moment in the span Mmax is larger than the absolute value of the left end moment Ml, C is corrected based on Mmax and is calculated by Eq. (9). C ¼ C1 ¼

Kcr ðMmax  Ml Þ Kcr0

ð8Þ

C ¼ C1 

Mmax ðMmax 4Ml Þ Ml

ð9Þ

First, C is calculated by only considering bending moments on both ends of the beam and without considering the uniformly distributed load (α = 0). C is thus approximated by Eq. (10).

Figure 3.

Relationship between Kcr0 and R (Warping restraint at both ends).

Table 4. Buckling length coefficient suggested value. Boundary condition

ku

kβ

Error from Kcr0 [%]

Simple support at both ends Fixed support at both ends Warping restraint at both ends Left end fixed support – Right end simple support Left end fixed support – Right end warping restraint Left end warping restraint – Right end simple support

1.00 0.500 0.933 0.696 0.659 0.957

1.00 0.500 0.463 0.696 0.464 0.638

0.0 0.0 3.2 0.0 4.5 2.3

1008

C¼

ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1 þ a4 β þ a5 β2 þ a6 β3 =ð1 þ a7 RÞ

ð10Þ

1 þ a1 β þ a2 β2 þ a3 β3

Table 5 lists the coefficients a1 to a7 of Eq. (10) obtained by the least squares method. C is evaluated based on each boundary condition and the influence of R. Table 5 also shows the difference between the value of C obtained from Eq. (10) and that obtained by the analysis. In this case, the error is found to be less than 7%; hence, the accuracy is assumed to be good. Next, C is calculated by considering a uniformly distributed load in addition to the bending moments. First, it is assumed that the uniformly distributed load acts on the shear center of the beam. This is investigated without considering the influence of twist. In this case, Reference [3] proposed an approximation formula for C in the case where the beam was simply supported at both ends, as shown Eq. (11). The equation does not include a factor indicating the cross-section index. Therefore, the influence of the cross-section index R is evaluated, as shown Eq. (12). 1 1 C ¼ pffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B 0:283ð1  βÞ2 þ 0:434ð1  αÞð1  βÞ þ ð0:283  0:869α þ 0:780α2 Þ

ð11Þ

1 C¼ pffiffiffi ξ ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

 1þða14 αþa15 α2 þa16 α3 Þ=ð1þa20 RÞþ a4 f1þða17 αþa18 α2 Þ=ð1þa21 RÞgβþa5 f1þa19 α=ð1þa22 RÞgβ2 þa6 β3 =ð1þa7 RÞ ð1þα8 αþa9 α2 þa10 α3 Þþa1 ð1þa11 αþa12 α2 Þβþa2 ð1þa13 αÞβ2 þa3 β3

ð12Þ The values of coefficients a1 to a7 listed in Table 5 are used. The coefficients a8 to a22 are determined by the least squares method and shown in Table 6. Figure 4 shows the results of the analysis together with Eq. (11) and Eq. (12). Although the errors are described later, the proposed approximation formula shows roughly good accuracy even though there are variations depending on the boundary conditions and α. Moreover, the influence of R is evaluated. Next, the approximation formula in the case where the beam is subjected to a uniformly distributed load on positions except the shearing center is calculated. In this case, C proposed in Reference [3] is given in Eq. (13). B in Eq. (13) is the one in Eq. (11), and S is expressed by Eq. (14) using the distance h between the point of application of the uniformly distributed load and the shear center, and bridge bw. In this study, C is evaluated by Eq. (15). r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 4α pffiffiffiffiffiffiffi S 4α pffiffiffiffiffiffiffi S þ þB π 2 1þR π2 1þR C¼ ð13Þ B Table 5. Suggested value of a1 to a7. Boundary condition

a1

a2

a3

a4

a5

a6

a7

Error from C

Simple support at both ends Fixed support at both ends Warping restraint at both ends Left end fixed support – Right end simple support Left end fixed support – Right end warping restraint Left end warping restraint – Right end simple support

−0.92 −0.90 0.89 −1.1

0.25 0.21 0.20 0.29

−0.0015 0.020 0.013 0.048

−0.0077 0.11 0.17 0.031

0.28 0.00077 −0.074 0.52

−0.12 0.018 0.063 −0.25

0.22 0.080 0.095 0.12

1.9 1.7 2.3 4.1

−1.1

0.34

0.018

0.13

−0.26

0.14

0.21

6.5

−0.78

0.044

0.065

0.038

1.0

−0.52

0.15

4.8

Units [%]

1009

Table 6. Suggested value of a8 to a22. Boundary condition

a8

a9 −11.7

Simple support at both ends

5.54

Fixed support at both ends

1.92

Warping restraint at both ends

13.8

Left end fixed support –

Right end simple support

Figure 4.

−27.1

a12

a13

a14

a15

5.54

6.69

−6.59

7.67

7.78

−1.17

4.26

3.11

−4.37

5.00

4.66

−0.0116

12.7

15.9

−15.2

19.6

17.4

a16

0.453 −0.00500

−2.64

1.10

a17

a18

−238 6.51 15.2

a19 −4.07

47.0 −2.22

a20

0.231

0.0163

0.714

0.00782 0.0948

27.3

0.00944

0.391

0.0182

−2.58

0.00558

0.142

0.305

2.87

3.43

−3.55

6.47

3.48

−0.758

0.428

2.69

−7.72

4.18

3.78

−4.38

5.41

5.30

−1.40

0.560

19.8

−4.40

4.49

0.0120

−0.0203

0.0449

1.01

−3.98

2.13

3.37

−3.57

3.73

−1.36

0.406

−2.28

10.2

−0.395

0.0160

−0.0236

0.0576

26.9

104

−28.4

a22

0.00501

−871

−5.76

a21

−5.42

Right end warping restrain Left end warping restraint –

−7.04

a11

1.82

Right end simple support Left end fixed support –

a10

Relationship between C and β.

2h bw vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 u0 u u αS αS A þξ a23 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ t@a23 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðku =kβ Þ þku2 R ðku =kβ Þ þku2 R S¼

C¼

ð14Þ

ð15Þ

ξ

The analysis results are shown in Figures 5 and 6, and Table 8 shows some of the errors in the value of C in each boundary condition. The position at which the uniformly distributed load acts has a great influence on the buckling strength. If the point of action is on the upper flange Table 7.

Suggested value of a23. a23

Boundary condition

S0

Simple support at both ends Fixed support at both ends Warping restraint at both ends Left end fixed support – Right end simple support Left end fixed support – Right end warping restraint Left end warping restraint – Right end simple support

0.392 0.303 0.338 0.227 0.331 0.246

0.307 0.175 0.245 0.135 0.188 0.169

1010

Figure 5.

Relationship between C and β (Simple support at both ends).

Figure 6.

Relationship between C and β (Left end warping restraint – Right end simple support).

(S is negative), then the strength is lower; if it is on the lower flange (S is positive), the strength is higher. Although there are cases where the error from the approximation formula is significant, as the evaluation was conducted on the relatively safe side, there is a trade-off in terms of accuracy. The approximate evaluation method for the elastic lateral buckling strength of H-shaped beams under various boundary conditions can be summarized as follows. First, determine the R of the beam to be studied from Eq. (4). Next, substitute ku and kβ and the moment gradient correction factor C into Eq. (6) to obtain Kcr. Then, substitute the obtained Kcr into Eq. (3) and evaluate the elastic lateral buckling strength. 3 ELASTIC LOCAL BUCKLING STRENGTH OF H-SHAPED BEAMS 3.1 Elastic local buckling strength estimation method of H-shaped beams In this section, the calculation method for elastic local buckling strength by the theoretical analysis based on energy method is described. The analysis model was assumed to be similar to that in Chapter 2. Figure 7 shows the bending shear stress distribution of the web and flange. Figure 8 shows the state of deformation of the beam during local buckling.

Table 8. Error of C between analysis value and approximate formula. α=0

α = 0.5

Boundary condition

S=0

S = −1.0

Simple support at both ends

1.9

7.0

7.5

12.0

5.8

7.8

37.0

Fixed support at both ends

1.7

10.6

11.1

17.7

7.8

17.4

40.0

Warping restraint at both ends

2.3

10.2

11.0

12.9

9.1

12.1

30.0

Left end fixed support –

4.1

10.4

19.3

33.0

12.5

13.8

11.0

12.0

21.4

6.8

10.6

21.1

29.9

10.8

Right end simple support Left end fixed support –

6.5

Right end warping restrain Left end warping restraint – Right end simple support

4.8

α = 1.0 S=0

S = 1.0

S = −1.0

α = 1.5 S=0

S = 1.0

S = −1.0

α = 2.0 S=0

S = 1.0

6.2

11.3

21.8

10.4

12.4

22.8

11.9

13.2

19.3

22.1

27.7

24.3

24.0

41.8

10.8

14.5

21.3

14.5

S = −1.0

S=0

S = 1.0

7.6

13.6

24.9

11.9

16.4

27.6

12.5

16.0

22.4

33.2

27.7

33.7

42.7

14.9

25.0

12.3

16.0

25.4

15.7

19.7

24.7

26.2

25.7

Units [%]

1011

Figure 7.

Figure 8. Deformation during local buckling

Stress distribution of plate elements.

The variables in Figure 7 and displacement functions are defined as follows.    4α 2 ð4α þ βÞ 2y wðx; yÞ ¼  2 x þ x1 1 l l bw   4α ð4α þ βÞ f ðxÞ ¼  2 x2 þ x1 l l     1 Af 8α 4α β γðxÞ ¼ þ þ1  xþ 6 Aw βl β λw

μm ¼ sin

ð17Þ ð18Þ

mπx ðSimple support at both endsÞ l

ð19Þ

ðm  1Þπx ðm þ 1Þπx  cos ðFixed support at both endsÞ l l

ð20Þ

μm ¼ sin μm ¼ cos

ð16Þ

mπx m ðm þ 1Þπx  sin ðLeft end fixed support  Right end warping restrainÞ l mþ1 l ð21Þ X X nπy a μ sin ð22Þ W¼ m n mn m bw X X z a μ nπcosnπ sin ð23Þ F1 ¼ m n mn m bw X X z a μ nπ sin ð24Þ F2 ¼ m n mn m bw

The strain energy increment ΔU1 in the web and the strain energy increment ΔU2, ΔU3 in the flange are as follows.  2 2 ) ð l ð bw ( 2 2  2 2 1 ∂ W ∂ W ∂2 W ∂2 W ∂W ΔU1 ¼ Dw dxdy ð25Þ þ þ 2 þ 2ð1   Þ 2 2 2 ∂y2 2 ∂x ∂y ∂x ∂x∂y 0 0 1 ΔU2 ¼ Df 2 1 ΔU3 ¼ Df 2

ð l ð bf ( 0 bf

ð l ð bf ( 0 bf

∂2 F2 ∂x2 ∂2 F1 ∂x2

2

2

 2 2 ) ∂ F2 dxdz þ 2ð1   Þ ∂x∂z

ð26Þ

 2 2 ) ∂ F1 dxdz þ 2ð1   Þ ∂x∂z

ð27Þ

1012

Figure 9.

Relationship between kσ and β.

The work of external force on the web ΔT1 and the work on the flange ΔT2 are as follows. 1 ΔT1 ¼ σcr tw 2

ð l ð b w ( 0 0

1 ΔT2 ¼ σcr tf 2

4α þ β 4α 1 x þ 2 x2 l l

ðl bðf ( 0 bf



2y 1 bw

4α þ β 4α 1 x þ 2 x2 l l

)   ∂W 2 ∂W ∂W þ 2γðxÞ dxdy ð28Þ ∂x ∂x ∂y

(

∂F1 ∂x

2



∂F2  ∂x

 2 )) dxdz

ð29Þ

The buckling conditional equation by the energy method is expressed by Eq. (30). This is a simultaneous linear equation with respect to amn, and it is calculated as an eigenvalue problem. Elastic buckling coefficient kσ can be calculated from the elastic buckling stress σcr using Eq. (31). ∂ðΔU1 þ ΔU2 þ ΔU3  ΔT1  ΔT2 Þ=∂amn ¼ 0 ðm ¼ 1; 2; . . . ; M; n ¼ 1; 2; . . . ; N Þ    12 1   2 bw 2 kσ ¼ σcr π2 E tw

ð30Þ ð31Þ

3.2 Analysis result In this section, the elastic local buckling strength results calculated using the analysis method from Section 1 are shown. Figure 9 shows an example of the results obtained from the analysis. The beam is the same as that used in Chapter 2, and there are three types of boundary conditions. From this result, it is considered that the moment gradient β does not have a considerable effect on the elastic buckling strength compared with elastic lateral buckling strength. The increase in the strength caused by uniformly distributed load is also small compared with the case of lateral buckling. The elastic local buckling was investigated with one type of beam; however, detailed investigation of other beams is given in other papers.

4 CONCLUSION This study proposed methods to evaluate the elastic lateral buckling strength of H-shaped beams subjected to bending moments and a uniformly distributed load. It also considered the elastic local buckling strength. The following conclusions were drawn. 1013

1. Displacement functions under various boundary conditions were proposed, and calculation methods for the elastic lateral buckling strength were obtained from a theoretical analysis by the energy method. 2. Approximate equations for the buckling length coefficients and the moment gradient correction factor at each boundary condition were proposed. In order to evaluate the influence of the cross-section on the buckling strength, an approximate expression considering the cross-section index R was proposed. 3. The elastic lateral buckling strength was particularly affected by twisting owing to the position at which the uniformly distributed load acted, and the more the point of action is on the upper part of the beam, the more the strength tends to be disadvantageous. 4. A displacement function under various boundary conditions was proposed, and calculation methods for the elastic local buckling strength were obtained from a theoretical analysis by the energy method. 5. The influence of the moment gradient on the local buckling strength is small compared with the case of lateral buckling. NOTATION α β l h

Coefficient indicating the magnitude of uniformly distributed load Moment gradient Length of H-shaped beam Distance from the shearing center of the beam to the point of application of uniformly distributed load (negative on the upper flange side) E Young’s modulus Iω, J Bending torsion constant, Torsion constant EIy Moment of inertia around weak axis EIω, GJ Warping rigidity, Torsion constant ku Buckling length coefficient for weak axial bending kβ Buckling length coefficient for warping moment Mmax Maximum moment in beam span Ml Left end moment of beam bw Width of web bf Half the width of flange tw Thickness of flange tf Thickness of flange ν Poisson's Ratio (=0.3) Dw Bending Stiffness of web Df Bending Stiffness of flange Aw Cross sectional area of web Af Cross sectional area of flange REFERENCES [1] [2]

[3]

[4] [5]

Salvadori, M.G. 1956. Lateral Buckling of Eccentrically Loaded I-Columns. In Italian. Trans. ASCE, Vol.121: pp.1163–1179. Ikarashi K., Tomo N., Wang T. 2011. Effects of Boundary Conditions and End Moment Ratio on Elastic Lateral Buckling Strength of H-Shaped Beams. In Japanese. Journal of Structural and Construction Engineering, Vol.670: pp.2173–2181. Nakamura T., Wakabayashi T. 1978. Approximate Solution to Correction Factor C of Elastic Lateral Buckling Moment of H-Shaped Beams - Design Formula. In Japanese. Summaries of technical papers of annual meeting, arch. Inst. of Japan: pp.1319–1320. Ikarashi K. 2003. Buckling Strength of Simply Supported Web Plate under The Action of Bending Shear Stress. In Japanese. Journal of Structural and Construction Engineering, Vol.565: pp.135–141. Architectural Institute of Japan. 1980. Recommendation for Stability of Steel Structures.

1014

Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

On the incorporation of cross-section restraints in Generalised Beam Theory (GBT) T.G. da Silva & C. Basaglia School of Civil Engineering, Architecture and Urban Design, University of Campinas, Brazil

D. Camotim CERIS, DECivil, Instituto Superior Técnico, University of Lisbon, Portugal

ABSTRACT: This paper reports the results of an investigation on the development, implementation and application of a Generalised Beam Theory (GBT) formulation for the local, distortional and global buckling analysis of thin-walled restrained members, based on a crosssection analysis procedure that incorporates elastic restraints and, therefore, requires less deformation modes to obtain accurate analytical and numerical buckling solutions. Its capabilities are illustrated by presenting and discussing numerical results concerning the buckling behaviour of cold-formed steel studs braced by sheathing and purlins restrained by sheeting. For validation and assessment purposes, some of these results are compared with values yielded by the GBT-based code GBTUL2.0 and/or ANSYS shell finite element analyses.

1 INTRODUCTION In Generalised Beam Theory (GBT), which incorporates genuine plate theory concepts (thus accounting for cross-section in-plane and out-of-plane deformations), the crosssection displacement field is expressed as a linear combination of deformation modes, making is possible to write the equilibrium equations and boundary conditions in an unique and very convenient format (e.g., Basaglia et al. 2015). The performance of a GBT structural analysis involves (i) a cross-section analysis (determination of the deformation modes and evaluation of the associated modal mechanical properties) and (ii) a member analysis (modal solution of the structural problem under consideration). In practical applications, thin-walled members are often continuously restrained along their lengths – e.g., purlin-sheeting system, steel-concrete composite members or sheathed wall stud systems. In order to incorporate such (elastic) restraints in a GBT analysis, two approaches can be followed: (i) incorporate the restraints in the cross-section analysis, taking them into account at the deformation mode determination stage (Schardt 1989 and Jiang & Davies 1997 adopted this approach for global and very specific distortional deformations) or (ii) include the restraints only in the member analysis, as constraint equations, which means that the deformation modes are calculated without considering the restraints (Camotim et al. 2008, Basaglia et al. 2013 and Bebiano et al. 2018, authors of the code GBTUL2.0, adopted this approach for arbitrary deformation patterns, which amounts to combining the conventional deformation modes at the member analysis stage. This last approach requires larger deformation mode sets and it may be argued that it somewhat “clouds” the structural interpretation of the results. The aim of this work to present and illustrate the implementation and application of a novel GBT formulation for the local, distortional and global buckling analysis of thinwalled restrained members, based on a cross-section analysis procedure that incorporates elastic restraints and, therefore, is capable of providing accurate buckling results with only a few deformation modes (often a single one). Moreover, it is also shown that the proposed 1015

formulation enables the development of analytical formulae to calculate critical buckling loads of restrained members. Its capabilities are illustrated through the presentation and discussion of numerical results concerning the buckling behaviour of cold-formed steel (i) studs braced by sheathing and (ii) purlins restrained by sheeting. For validation and assessment purposes, some of these results are compared with values yielded by the code GBTUL2.0 (Bebiano et al. 2018) and/or ANSYS (SAS 2013) shell finite element analyses.

2 GBT FORMULATION The modelling of the restraint provided by the sheeting, wall or slab to the member involves continuous translational and rotational elastic springs, which restrain the member transverse displacements and mid-width rotations. Figure 1(a) illustrates the case of a purlin-sheeting assembly: to simulate the restraint provided by the roof sheeting, continuous translational and rotational elastic springs (stiffness KT and KR, respectively) are continuously attached to the upper flange mid-points along the whole purlin length. Consider the prismatic thin-walled open cross-section member, formed by n distinct plate/wall elements, and restrained by continuous translational and rotational elastic springs, located, respectively, at points PT and PR indicated in Figure 1(b), where x, s and z are local coordinates along the member axis, cross-section mid-line and wall thickness, thus leading to member midsurface displacement components u(x,s), v(x,s) and w(x,s) expressed as uðx; sÞ ¼ uk ðsÞ’k;x ðxÞ

vðx; sÞ ¼ vk ðsÞ’k ðxÞ

wðx; sÞ ¼ wk ðsÞ’k ðxÞ

ð1Þ

where (.),x ≡ d(.)/dx, the summation convention applies to subscript k, functions uk(s), vk(s), wk(s) are mid-line “displacement profiles” and φk(x) is a dimensionless amplitude function defined along the member length – information on the derivation of these expressions can be found in Silvestre & Camotim (2002). The equations providing the member buckling behaviour, taking into account the presence of elastic constraints (springs), are obtained by imposing the stationarity of the total potential energy functional V¼

ð ð 1 1 σij εij dO þ KD2r dx 2 2 O

ð2Þ

L

in the close vicinity of the member fundamental equilibrium path (adjacent equilibrium) – (i) Ω is the member volume (n walls), (ii) σij and εij are the second Piola-Kirchhoff stress and Green-Lagrange strain tensors, respectively, both comprising pre-buckling and bifurcated components, (iii) L is the member length, (iv) K is the stiffness of a continuous (along a longitudinal axis r) spring, (v) Δ are spring generalised displacements (translation or

Figure 1. Prismatic thin-walled member (a) with continuous elastic restraints/springs and (b) local coordinate axes.

1016

rotation) and (vi) the summation convention applies to subscripts i and j. Then, the equilibrium equations defining the member buckling eigenvalue problem are obtained by (i) linearising the first variation (δ) of the total potential energy functional, at the fundamental equilibrium path, and (ii) discarding the pre-buckling strains (e.g., Camotim et al. 2008), thus yielding ð δV ¼



Cik ’k;xx δ’i;xx þ D1ik ’k;x δ’i;x þ D2ik ’k δ’i;xx

L

þ

D2ki ’k;xx δ’i

þ Bik ’k δ’i þ

σx λWj0 Xjik ’k;x δ’i;x

ð3Þ



dx ¼ 0

Wj0 ¼ Cjj ’0j;xx

ð4Þ

with ð Cik ¼ E t ui uk ds þ b

Bik ¼

ð E t3 wi wk ds 12ð1  2 Þ

ð5Þ

b

ð E t3 wi;ss wk;ss ds þ KT wi ðsPT Þwj ðsPT Þ þ KR wi;s ðsPT Þwj;s ðsPT Þ 12ð1  2 Þ

ð6Þ

b

ð

D1ik ¼

G 3 t wi;s wk;s ds 3

D2ik ¼

b

ð E t3 wi wk;ss ds 12ð1  2 Þ

ð7Þ

b

σx Xjik ¼

ð

E t uj ðvi vk þ wi wk Þds Cjj

ð8Þ

b

where (i) E, υ and G are the material Young’s modulus, Poisson’s ratio and shear modulus, (ii) Wj0 are stress resultant profiles and (iii) λ is the load parameter. The determination of the GBT deformation modes and evaluation of the corresponding cross-section modal mechanical properties require the performance of a cross-section analysis – in open-section members it involves a sequential procedure comprising the following major steps: (i) Cross-section discretisation into n + 1 natural nodes (ends of the n walls forming the crosssection) and m intermediate nodes (located within the walls). In open sections with branching natural nodes (nodes shared by more than two walls), the natural nodes must be still subdivided into independent and dependent (several subdivisions are possible). Since one must (i1) satisfy Vlasov’s null membrane shear strain assumption and (i2) ensure membrane transverse displacement compatibility at all branching nodes, the dependent node warping displacements cannot be chosen arbitrarily and must be appropriately selected (Silvestre & Camotim 2002, Dinis et al. 2006). (ii) Determination of the initial shape functions ui(s), vi(s) and wi(s), by sequentially imposing unit warping displacements (u = 1) at each independent natural node and flexural displacements (w = 1) at each intermediate node – concerning the imposition of unit displacements, the cross-section end nodes are treated as both natural (independent or dependent) and intermediate. Note that evaluating the flexural functions wi(s) involves solving a statically indeterminate folded-plate problem, a task carried out here by means of the displacement method – Figure 2 shows the geometry and a possible GBT discretisation of a lipped channel cross-section.

1017

Figure 2.

Geometry and possible GBT discretisation of a lipped channel.

(iii) Calculation of tensors (5) to (8), on the basis of the initial shape functions and applied loading. One obtains fully populated matrices (highly coupled equilibrium equations) whose components exhibit no obvious structural meaning. (iv) To uncouple the member equilibrium equation system as much as possible and, at the same time, have stiffness matrix components with clear structural meanings, the simultaneous diagonalisation of the linear stiffness matrices Cik and Bik, given in (5)-(6), is performed. This leads to the cross-section deformation modes (i.e., the final shape functions uk(s), vk(s), wk(s)) and to the associated cross-section modal mechanical properties – the new linear and geometrical stiffness matrix components, several of them with a clear/illuminating structural meaning. This procedure (the GBT “trademark”) makes it possible to express the equilibrium equations in modal form, thus leading to a considerable amount of interpretation and numerical implementation advantages. Unlike the above conventional GBT cross-section analysis (for unrestrained cross-sections), the determination of the GBT deformation modes (final base functions) is carried out through a procedure involving one to three auxiliary eigenvalue problems, which are addressed separately next: (i) Stage 1: Determination of the Distortional and Local Deformation Modes. Consider the auxiliary eigenvalue problem defined by ð½Bik  λk ½Cik Þf
ak g ¼ 0;

ð9Þ

which has n + m + 1 eigenvalues λk. Concerning this eigenvalue problem, it should be pointed out that: (i1) The null eigenvalues λk = 0 correspond to a subspace associated with cross-section rigidbody motions. Since every vector in this subspace is an eigenvector, no rigid-body deformation modes can be identified – this identification will be made in next stages. (i2) The positive eigenvalues λk > 0 correspond to deformation modes involving either wall transverse bending, warping displacements and fold-line motions (distortional deformation modes) or only wall transverse bending (local deformation modes). (i3) The base function change regarding the distortional and local deformation modes is
1 ], formed by the (often normalised) eigenvectors with positive defined by matrix [A eigenvalues. (i4) If λ4 = 0, the diagonalisation of Stage 2 must be performed, to obtain the torsion mode. (i5) If λ4 > 0, the Stage 2 diagonalisation no longer has to be performed, since the restraint precludes the occurrence of the torsion deformation mode – one then proceeds to Stage 3, in order to obtain the extension deformation modes, which are associated with λ1 and, if λ2 > 0 and/or λ3 > 0, also with the bending deformation modes. (ii) Stage 2: Determination of the Torsion Deformation Mode. This stage involves sub-matrices of [Cik] and [Dik], already expressed in terms of the eigenvector space base functions
1 ] and [D
1 ] (similar procedures are carobtained from the solution of (9) and denoted [C ik ik
1 ] and [X
1 ]). They are obtained as ried out for [B ik 1ik
1 ¼ ½Y
T ½Cik ½ Y


½C 1 1 ik

1018


1 ¼ ½Y
T ½Dik ½Y
1

½D 1 ik

ð10Þ


1 ] is a sub-matrix formed by the null eigenvectors of (9). These two sub-matrices where [Y
1 ] is are associated with the rigid-body (global) deformation modes, which means that [C ik fully populated (not diagonal). Then, a second auxiliary eigenvalue problem has to be considered, which is defined by  
1 Þ a
1 ¼ 0
1  λk ½C ð½D ik ik k

ð11Þ

Concerning this eigenvalue problem, it should be noted that: (ii1) The eigenvectors associated with the null eigenvalues defines a subspace associated with cross-section rigid-body translations (no rotation) – axial extension mode and, if λ2 > 0 and/or λ3 > 0, also the bending deformation modes. (ii2) The eigenvalue λ4 > 0 corresponds to a cross-section in-plane rigid-body rotation (torsion mode). (ii3) The base function changes regarding the distortional and local deformation modes, determined in Stage 1, and (iii2) the torsion mode, determined in Stage 2, are defined
2 ], obtained from [A
1 ] by adding the torsion mode eigenvector, norby sub-matrix [A malised to exhibit a unit rotation. (iii) Stage 3: Determination of the Axial Extension and Bending Deformation Modes. This
2 ] and [X
2 ], if λ4 > 0 in stage deals with sub-matrices of [Cik] and [X1ik], denoted [C ik 1ik 1 1

Stage 1, or [Cik ] and [X1ik ], if λ4=0 in Stage 1. These sub-matrices are obtained from
2 ¼ ½Y
T ½Cik ½Y


½C ik 1 1


2 ¼ ½Y
T ½X1ik ½Y


½X 1ik 1 1

if λ4 40 in Stage 1

ð12Þ


2 ¼ ½Y
1 ½ Y
2 T ½C


½C ik ik 2


2 ¼ ½Y
2 T ½X
1 ½ Y
2

½X 1ik 1ik

if λ4 ¼ 0 in Stage 1

ð13Þ


2 ] is a sub-matrix formed by the null eigenvectors of (11). Since these subwhere [Y
2] matrices are associated with the rigid-body (global) translation deformation modes, [C ik is fully populated (not diagonal). Then, a third auxiliary eigenvalue problem is required, defined by  2
2 Þ a
2  λk ½C
k ¼ 0 ð½X 1ik ik

ð14Þ

Concerning this eigenvalue problem, it should be noted that: (iii1) The null eigenvalue λ1 = 0 correspond to the axial extension deformation mode. (iii2) The positive eigenvalues correspond to the bending deformation modes. (iii3) The base function changes concerning the distortional, local and/or torsion deformation modes, determined previously, and the axial extension and bending modes, just
3 ], obtained from [A
1 ] or [A
2 ] by adding these determined, are defined by matrix [A eigenvectors. Finally, matrices [
uik ], [
vik ] and [
wik ], containing the various cross-section normalised deformation mode component functions, are given by
3 T ½uik ½A
3 ½
vik ¼ ½A
3 T ½vik ½A
3 ½

3 T ½wik ½A
3

½
uik ¼ ½A wik ¼ ½A

ð15Þ

After performing the cross-section analysis, one obtains the member GBT system of adjacent equilibrium equations, which (i) is expressed in modal form as 
ik ’
ik ’
ik ’
σx W
0’
k;xxxx  D
k;xx þ B
k  λ X
C ¼0 jik j k;x ;x

1019

ð16Þ

and, together with the adequate end support conditions, (ii) defines the buckling eigenvalue problem to be solved. In order to be able to carry out this task for members with arbitrary support, it is necessary to use a GBT-based beam finite element (e.g., that developed by Silvestre & Camotim 2003).

3 NUMERICAL RESULTS This section presents and discusses numerical results concerning the buckling behaviour of cold-formed steel (E = 210GPa and υ = 0.3) purlins restrained by sheeting and studs braced by sheathing. Figure 3(a) shows the dimensions of the two lipped channel cross-sections dealt with in this work (Sections A and B), which are continuously restrained by different spring arrangements (Constraintss I to IV), involving rotational and translational springs with stiffness KR and KT, repectively (values given in Figure 3(a)) – four combinations of cross-section dimensions and spring arrangement are considered. For each of them, Figure 3(b) displays the 8 most relevant deformation modes obtained by means of the proposed restrained crosssection analysis procedure (mode 1 stands for axial extension) – for comparison purposes,

Figure 3. (a) Restrained lipped channel cross-sections and (b) 8 most relevant deformation modes for each of them.

1020

Figure 4. Main features of the most relevant conventional lipped channel deformation modes yielded by GBTUL2.0.

Figure 4 depicts the first 9 (conventional) deformation modes yielded by code GBTUL2.0 for
ik ] (diagan unrestrained lipped channel cross-section. Table 1 show the components of the [C

onal), [Bik ] (diagonal) and [Dik ] (almost diagonal) stiffness matrices concerning deformations modes 2 to 9 of (i) Section A, unrestrained and with Constraint I, and (ii) Section B, unrestrained and with Constraint III. The observation of the results presented in these two figures and table prompts the following remarks: (i) While Constraint I does not restrain major and minor axis bending (modes 2 and 3), Con
22 and C
33 are equal for the sections straint III only does not restrain mode 2 – thus, C
22 are equal for the sections unrestrained and unrestrained and with Constraint I, and C with Constraint III.
ik tends to be “less diagonal” as the number of restraints increase – indeed, (ii) Matrix D
matrix Dik is nearly fully populated for Constraint III and still “almost diagonal” for Constraint I. Table 2 shows the critical buckling loads of uniformly compressed continuously restrained simply supported steel studs exhibiting Section A – the restraint provided by the sheeting to the studs is modelled through Constraint II. The results are obtained by means of GBT analyses with unrestrained (GBTUL2.0) and restrained deformation modes – for validation purposes, some values yielded by ANSYS SFEA are also presented. Comparing the two sets of buckling results leads to the following conclusions: (i) First of all, the critical loads yielded by the ANSYS SFEA and the two GBT analyses (including all deformation modes) are virtually coincident. (ii) The GBT analyses with restrained deformation modes provide accurate buckling results with only a single deformation mode – mode 7 for studs with lengths up to 250cm (local buckling) and mode 2 for longer studs (minor-axis flexural buckling). Naturally, the conventional GBTUL2.0 analyses must include more than one deformation mode to provide accurate buckling results – considering only the dominant mode leads to errors that may reach 19% (for short studs). Attention is turned next to simply supported steel purlins restrained by steel sheeting and subjected to uniform negative major-axis bending (bottom flange under compression) – the purlins exhibit Section B and the sheeting restraints are modelled through Constraint IV with fully prevented translation and rotation (springs with infinite stiffness). Figure 5 shows the signature curves of the unrestrained (KR = KT = 0) and fully restrained (KR = KT = ∞) purlins, providing the variation of the critical buckling moment Mcr and mode shape with the length L (logarithmic scale) – the buckling mode half-wave numbers are inside brackets. Besides the results obtained through restrained-mode GBT analyses (solid and dashed curves, and buckling mode shapes), the figure also display, for validation and comparison purposes, some critical buckling moment determined by means of GBTUL2.0 analyses (circles) – all analyses adopt longitudinal discretisations into 10 finite elements and include all deformation modes. The close observation of the buckling results displayed in this figure prompts the following remarks: (i)

As before, the buckling moments provided by GBTUL2.0 and those obtained with the proposed formulation virtually coincide (all differences are below 1.0%). 1021

0 0.19 0 2.23 0 -0.728 0 -0.159

-0.23 0 -6.91 0 2.24 0 -0.087 0

0 -0.25 0 -0.728 0 2.533 0 -1.169

0 0 61.93 0 -6.91 0 -2.62 0

0 0 0 0 -0.23 0 1.141 0

0 0 0 0.19 0 -0.25 0 0.28

0 0.25 0 0.779 0 2.537 0 -1.179

0 0 0.016 0 0.245 0 0 0 0.205 0 0.016 0 0.109 0 -0.16 0 0.205 0 2.305 0 0.245 0 -0.16 0 2.3 0 0.25 0 0.779 0 1.141 0 0.129 0 0.134 0 -0.28 0 0.231 0
ik Unrestrained Section A (GBTUL2.0) - D 1.141 0 -2.62 0 -0.087 0 9.601 0

1.141 0 0.129 0 0.134 0 9.604 0

141.8

22.68

0 0 0 2.967
ik × 103 Section A - Constraint I - D

6.474

142.0

0 0.28 0 -0.159 0 -1.169 0 8.481

0 -0.28 0 0.231 0 -1.179 0 8.457

172.0

172.4

0 0 0 0 -0.919 0 -6.078 0

0 0 0 -1.085 0 -1.431 0 -6.854

0 0 2355 0 53.01 0 32.01 0

0 -1.085 0 5.927 0 3.966 0 -2.714

-0.919 0 53.01 0 5.962 0 -3.279 0

0 0 0.106 -0.715 -0.261 0 0.0764 -0.083 0.224 0.035 0.106 -0.083 5.114 -0.817 4.395 -0.715 0.224 -0.817 3.738 1.262 -0.261 0.035 4.395 1.262 14.54 -0.018 0 -3.612 0.315 -14.51 6.083 -0.222 -0.430 2.480 1.815 -0.019 0.587 -2.657 -0.045 3.911
ik Unrestrained Section A (GBTUL2.0) - D

0 0 0 1.789 5.103
ik × 104 Section B - Constraint III - D

0 -1.431 0 3.966 0 15.65 0 4.966

-0.018 0 -3.612 0.315 -14.51 15.70 0.127 -5.019

31.62

0 0.004 2.162 3.625 374.4 34.16
kk × 10−2 Unrestrained Section B (GBTUL2.0) - B

0.043

0 0 0.092 6.863 11.21 22.76
kk × 10−2 Unrestrained Section A (GBTUL2.0) - B

2556 327.3 35163 1.038 1.135
kk × 10−2 Section B - Constraint III - B

0.014

0.010

852.4 216.9 3947 0.301 0.325
kk × 10−2 Section A - Constraint I - B

0.012

2556 2.797 1.013 0.905 0.594 0.046
kk × 104 Unrestrained Section B (GBTUL2.0) - C

0.014

852.4 216.9 9.514 0.309 0.334 0.010
kk × 103 Unrestrained Section A (GBTUL2.0) - C

0.012


kk × 104 Section B - Constraint III - C


kk × 103 Section A - Constraint I - C

-6.078 0 32.01 0 -3.279 0 68.95 0

6.083 -0.222 -0.430 2.480 1.815 0.127 68.97 0

278.8

278.9

0.047

0.047

0 -6.854 0 -2.714 0 4.966 0 158.2

-0.019 0.587 -2.657 -0.045 3.911 -5.019 0 157.8

858.7

856.9

0.048

0.048


kk , B
kk and D
ik stiffness matrices concerning modes 2-9 for unrestrained and restrained (Constraints I and III) lipped channel crossTable 1. Components of C sections A and B – dimensions in cm, Young and shear moduli in kN/cm2.

Table 2. Stud buckling results: ANSYS and GBT (GBTUL2.0 and restrained deformation modes) analyses. Conventional Modes (GBTUL2.0) ANSYS

All Modes

L

Pcr

Pcr

Δ (%)

(cm)

(kN)

(kN)

GBT/

Restrained Modes Single Mode

Modal participations

Pcr

Mode

(kN)

All Modes Pcr

Δ (%)

Single

(kN)

GBT/

All

ANSYS

Single Mode

Δ (%)

Modal participations

Pcr

Mode

(kN)

Δ (%) Single

ANSYS

All

10

25.04

26.22

4.71

83%(7) + 16%(9) + 1%

31.32

(7)

19.5

25.46

1.68

91.10%(7) + 3.5%(6) + 3.4%(5) + 2%

25.79

(7)

1.3

50

24.32

24.41

0.37

85%(7) + 14%(9) + 1%

27.54

(7)

12.8

23.99

-1.36

92.5%(7) + 2.2%(6) + 2.1%(5) + 3.2%

24.19

(7)

0.8

100

24.30

24.37

0.29

86%(7) + 14%(9)

27.72

(7)

13.7

24.06

-0.99

92.2%(7) + 2.6%(6) + 2.5%(5) + 2.7%

24.22

(7)

0.7

150

24.31

24.39

0.33

85%(7) + 14%(9) + 1%

27.79

(7)

13.9

24.08

-0.95

93.1%(7) + 2%(6) + 1.95%(5) + 2.95%

24.25

(7)

0.7

200

-

25.57

-

84%(7) + 15%(9) + 1%

29.35

(7)

14.8

25.09

-

92.8%(7) + 2.5%(6) + 2.3%(5) + 2.4%

25.30

(7)

0.8

250

25.70

26.46

2.95

82%(7) + 18%(9)

30.43

(7)

15.0

25.96

1.00

90.2%(7) + 4.3%(6) + 4.2%(5) + 1.3%

26.17

(7)

0.8

300

-

23.78

-

99.9%(3) + 0.10% (9)

23.79

(3)

0.0

23.78

-

99.5%(2) + 0.5%

23.79

(2)

0.0

350

-

17.47

-

99.9%(3) + 0.10% (9)

17.48

(3)

0.1

17.47

-

99.7%(2) + 0.3%

17.48

(2)

0.1

400

13.35

13.38

0.23

100%(3)

13.38

(3)

0.0

13.38

0.23

100%(2)

13.38

(2)

0.0

450

-

10.57

-

100%(3)

10.57

(3)

0.0

10.57

-

100%(2)

10.57

(2)

0.0

500

-

8.56

-

100%(3)

8.56

(3)

0.0

8.56

-

100%(2)

8.56

(2)

0.0

Figure 5.

Variation of the critical buckling moment and mode shape with the purlin length.

(ii) For L ≤ 252 cm, the buckling behaviour does not depend on the restraint conditions (the solid and dashed curves coincide). This is because the critical buckling modes combine local and/or distortional deformations that do not involve upper flange horizontal displacements or rotations. (iii) In the restrained purlins with L > 252 cm, the buckling mode shape is clearly lateraldistortional, due to the full restraint of the upper flange horizontal displacements and (mostly) rotations – this instability phenomenon has also been investigated by other authors (e.g., Hancock et al. 2001). (iv) In the unrestrained purlins with L > 252 cm, lateral-torsionalbuckling occurs and is associated with a very pronounced Mcr decrease. As shown by Basaglia et al. (2013), there is a minute horizontal displacement of the point where the translational restraint is located – see the buckling mode shape of the unrestrained purlin with L = 400 cm. Finally, for a few purlins buckling in distortional or lateral-distortional modes, Table 3 provides the critical buckling moments obtained by means of GBT analyses including all deformation modes and also either only the dominant (“restrained GBT”) or the two most relevant (GBTUL2.0) ones. It is observed that, once again, the GBT analyses including a single restrained deformation mode (3 or 4 – see Figure 3(b)) provide accurate buckling moments. Conversely, the inclusion of the two most relevant deformation modes in the GBTUL2.0 analyses still leads to inaccurate buckling moments.

1023

Table 3. Purlin buckling results: GBT (GBTUL2.0 and restrained deformation modes) analyses. Conventional Modes (GBTUL 2.0) All Modes L

Mcr

(cm)

(kNcm)

Restrained Modes Two Modes

Modal Participations

Mcr

All Modes Δ (%)

Modes

(kNcm)

Mcr(kNcm)

Single Mode Modal Participations

Two

Mcr

Mode

(kNcm)

Single

All 75

4933.78

52%(6) + 47%(5) + 1%

8608.13

(6) + (5)

74.5

Δ (%)

All 4933.88

95%(4) + 1.1%(3) + 3.9%

5028.73

(4)

1.9

150

4934.20

52%(6) + 47%(5) + 1%

8235.79

(6) + (5)

66.9

4934.30

95%(4) + 1.1%(3) + 3.9%

5029.17

(4)

1.9

224

4936.10

52%(6) + 47%(5) + 1%

7870.98

(6) + (5)

59.5

4936.19

95%(4) + 1.1%(3) + 3.9%

5031.76

(4)

1.9

400

3551.75

52%(3) + 47(4) + 1%

71389.38

(3) + (4)

1910.0

3554.28

99.5%(3) + 0.5%

3566.53

(3)

0.3

500

3863.29

52%(3) + 47(4) + 1%

71510.10

(3) + (4)

1751.0

3867.27

99.4%(3) + 0.6%

3888.06

(3)

0.5

800

3552.10

52%(3) + 47(4) + 1%

72237.60

(3) + (4)

1933.7

3554.63

99.5%(3) + 0.5%

3566.88

(3)

0.3

4 CONCLUSION The paper reported the results of an ongoing investigation on the development of a GBT formulation for the buckling analysis of restrained thin-walled members, differing from the conventional one in the cross-section analysis (it already incorporates the elastic restraints). Special attention was paid to the procedures involved in the determination of the restrained deformation modes. The application and capabilities of the above GBT formulation were illustrated through the presentation and discussion of numerical results concerning cold-formed steel (ii) studs braced by sheathing and (ii) purlins restrained by sheeting. For validation and assessment purposes, some results were compared with values provided by the codes GBTUL2.0 (conventional GBT) and/or ANSYS (SFEA). Besides the expected virtual coincidence of the results, it was found the proposed GBT formulation is much more efficient than the conventional one. In particular, it becomes possible to obtain accurate buckling results with a single deformation mode, which enables the development of analytical formulae to calculate critical buckling loadings of restrained members – this feature, currently being exploited by the authors, will be addressed in future works. ACKNOWLEDGMENTS The second author gratefully acknowledges the financial support of Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq – Ministry of Science, Technology and Innovation of Brazil), through project 308530/2016-0. REFERENCES Basaglia, C., Camotim, D. (2015). Buckling analysis of thin-walled steel structural systems using Generalised Beam Theory (GBT), International Journal of Structural Stability and Dynamics, 15(1), 1540004. Basaglia, C. Camotim, D. Gonçalves, R., Graça, A. (2013). GBT-based assessment of the buckling behavior of cold-formed steel purlins restrained by sheeting, Thin-Walled Structures, 72(November), 217–229. Bebiano, R., Camotim, D., Gonçalves R. (2018). GBTUL 2.0 – A second-generation code for the GBT-based buckling and vibration analysis of thin-walled members, Thin-Walled Structures, 124(March), 235–257. Camotim, D., Silvestre, N., Basaglia, C., Bebiano, R. (2008). GBT-based buckling analysis of thin-walled members with non-standard support conditions, Thin-Walled Structures, 46(7–9), 800–815. Dinis, P.B., Camotim, D., Silvestre, N. (2006). GBT formulation to analyse the buckling behaviour of thinwalled members with arbitrarily “branched” open cross-sections, Thin-Walled Structures 44(1), 20–38. Hancock, G.J., Murray, T.M., Ellifritt, D.S. (2001). Cold-Formed Steel Structures to the AISI Specification, New York: Marcel Dekker Inc. Jiang, C., Davies, J.M. (1997). Design of thin-walled purlins for distortional buckling, Thin-Walled Structures 29(1–4), 189–202. SAS (Swanson Analysis Systems Inc.) (2013). ANSYS Reference Manual (version 15). Schardt, R. (1989). Verallgemeinerte Technische Biegetheorie, Berlim: Springer Verlag. (German) Silvestre, N., Camotim, D. (2002). First-order generalised beam theory for arbitrary orthotropic materials, Thin-Walled Structures, 40(9), 755–789. Silvestre, N., Camotim, D. (2003). GBT buckling analysis of pultruded FRP lipped channel members, Thin-Walled Structures 81(18–19), 1889–1904.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Stability interaction effects in 3D steel frames—a case study H.H. Snijder Department of the Built Environment, Eindhoven University of Technology, Eindhoven, The Netherlands

L.H.J.D. Titulaer Ballast-Nedam, Nieuwegein, The Netherlands

P.A. Teeuwen Witteveen+Bos, Deventer, The Netherlands

H. Hofmeyer Department of the Built Environment, Eindhoven University of Technology, Eindhoven, The Netherlands

ABSTRACT: 3D steel frames are usually assessed by checking their columns and beams separately using design rules, e.g. EN 1993-1-1 (Eurocode 3). For this, the force distribution in the 3D steel frame is determined first, followed by cross-sectional resistance and member stability checks. When increasing the load on the frame, the first critical check defines the ultimate load. This paper presents a case study in which a 3D steel frame is assessed both by code checking and by more realistic numerical simulations by the finite element method (FEM). It shows that the code checking approach overestimates the ultimate load of the FEM approach. The FEM simulations show simultaneous failure of beams and columns and naturally take into account the mutual stiffness interaction of unstable beams and columns. As the latter is not the case for the design rules used in the code checking approach, the suggestion is made that the current design rules predict too high values for this reason. As similar findings are obtained for 2.5D and 2D frames, further research is needed, including full scale experiments.

1 INTRODUCTION In practice, 3D steel frames are usually analysed by assessing the different columns and beams separately using design rules, e.g. the ones of EN 1993-1-1 (Eurocode 3). The force distribution in the 3D steel frame is determined first, followed by cross-sectional resistance and member stability checks for the beams and columns. The load can be proportionally increased until one of the checks is violated, thus obtaining the ultimate load of the frame, which is expressed in the ultimate load proportionality factor (LPF) αult,DR. Using the FEM, 3D steel frames can also be analysed (A) as a whole, taking material (M) and geometric (G) non-linearities (N), and imperfections (I) into account, i.e. an assessment by GMNIA. In this way the ultimate load can also be expressed in terms of an LPF: αult,FEM. The first approach, called the code checking approach, is usually expected to be conservative in terms of ultimate load compared to the more sophisticated second approach, called FEM approach. However, in the code checking approach it is not obvious that all 3D instability effects are taken into account. Especially if two or more members fail simultaneously by instability and plasticity, stability interaction effects may cause the 3D frame to have a lower ultimate LPF αult,FEM than αult,DR. This would mean that the code checking approach would not be sufficiently conservative. The design rules for member stability used in the code checking approach have been derived for rather basic cases with standard boundary conditions. However, in a 3D frame, the boundary conditions for each member are complex and may change due to plasticity and stability effects with increasing LPF. Nevertheless, these design rules are used in practice, also if the members form part of 1025

Figure 1.

Geometry and loading of a 3D frame (left), a 2.5D frame (middle) and a 2D frame (right).

a 3D frame. It is expected that cases with columns and beams failing simultaneously may be critical. Therefore, a research project was carried out to investigate these cases (Titulaer, 2018). This paper presents the research for a case study where interaction effects are present, and compares different analysis strategies, which resulted in different ultimate load predictions.

2 STEEL FRAMES CONSIDERED The 3D steel frame as shown in Figure 1 (left) is used for the case study. Its dimensions equal 3m for all three directions. The columns are sections HEB140; the beams are sections IPE180. The fillets of these sections have been neglected. The steel grade is S235. Figure 2 (left) shows a connection between beams and column, which are welded and are assumed to be rigid and full-strength. The weld dimensions have been neglected. At each column base, the displacements in all three directions and the rotation about the longitudinal column axis are constrained. A vertical concentrated load acts on each column equal to Fv = 96 kN. Two horizontal concentrated loads act at the beam level: Fh = 8.35 kN. Finally, uniformly distributed loads act on each beam in vertical direction: qv = 13 kN/m. For comparison two other frames are considered. Figure 1 (middle) shows a 2.5D frame with similar properties as the 3D frame. At the top ends of the beam, the 2.5D frame is restrained against out-of-plane displacements, leaving other out-of-plane displacements and rotations possible. Figure 1 (right) shows a 2D frame with similar properties as the other two frames, but now all out-of-plane displacements and rotations are restrained. For a better comparison with the 3D frame, the vertical concentrated loads on the columns of the 2.5D and 2D frames have been increased to Fv = 115.5 kN to compensate for the uniformly distributed loads on the removed beams. However, the out-of-plane bending moments at the top of the columns of the 2.5D and 2D frame, associated with the removed loaded beams, have been neglected to be able to compare the results of the 2.5D and 2D frame better.

Figure 2. Rigid full-strength welded connection (left) and section representation by shell elements (right).

1026

3 FEM APPROACH FEM is used to predict a load-displacement curve for the frames considered (Figure 1) by GMNIA. Linearly increasing load control is applied, and for the solver the Riks (arc-length) method is used to predict the behavior after the ultimate load. The applied load is expressed as an LPF α with reference to the loads mentioned earlier. The ultimate load is characterized by the ultimate LPF αult,FEM. The finite element model has been developed in Abaqus and consists of S4R shell elements. A mesh density study and validations have been carried out in Titulaer (2018), which includes the selection of the elements. Figure 2 (right) indicates the shell element positions relative to the cross-section, and Figure 3 (left) gives an impression of the mesh density and layout. The material model for steel grade S235 is taken from EN 1993-1-5 (2006) and is shown in Figure 3 (right), both in terms of engineering stress and strain and true stress and strain, the latter used in the finite element model. Young’s modulus equals E = 2.1 x 105 N/mm2 and Poisson’s ratio equals ν = 0.3. Plasticity is described by a Von Mises yield surface with associated plastic flow and isotropic hardening Residual stresses are taken into account according to ECCS (1984), see Figure 4. For the IPE180 beam h/b = 180/91 = 1.98 > 1.2 so Figure 4 (right) is applicable, while for the HEB140 column h/b = 140/140 = 1.0 < 1.2 so Figure 4 (left) applies. Geometrical imperfections are taken into account using the Eigen buckling-mode method (EBM) as presented in Liu et al. (2014). In this method, several Eigen modes determined by a linear buckling analyses (LBA) are combined using a specific amplitude for each one. Using the first three Eigen modes leads to a good trade-off between practical use and accuracy of the EBM and the amplitudes of these Eigen modes (Aj) can be obtained as follows (Liu et al. 2014): Aj ¼ Cj  F  H

ð1Þ

where Cj is the contribution factor for buckling mode j (here C1 = 0.831, C2 = 0.132 and C3 = 0.037), F a factor depending on the number of Eigen modes considered (here

Figure 3.

Mesh (left) and material model (right) used.

Figure 4.

Residual stress patterns according to ECCS (1984) for rolled sections.

1027

Figure 5.

First three Eigen modes as used in the EBM to define the geometrical imperfection.

F = 0.00163 for three Eigen modes) and H the height of the frame (here H = 3000 mm). Slightly different numbers for F and Cj apply for 2D frames (Shayan et al. 2014). The first three Eigen modes of the 3D frame are shown in Figure 5. They are combined according to Equation 1 into the geometrical imperfection as used in the GMNIA. As an alternative, the imperfections have been used as given in EN 1993-1-1 (2006). In this code, so-called equivalent geometrical imperfections are given so residual stresses should not be taken into account separately. This is called the method of initial geometric imperfections (IGI) after Chan et al. (2005). Sway and bow imperfections should be combined such that they have the most unfavourable effect, considering the different directions as indicated in Figure 6 (translations in x and z-direction and rotation in the y-z-plane). If residual stresses are nevertheless taken into account explicitly, the bow imperfection may be reduced to 1/1000 of the member length. In conclusion, six different imperfection types have been investigated in the simulations: – – – – – –

No imperfection (No); EBM and residual stresses (EBM-RS); IGI in y-direction without residual stress (IGI-y); IGI in z-direction without residual stress (IGI-z); IGI in y-z-plane rotational direction without residual stress (IGI-yz); IGI in y-direction with residual stress (IGI-y-RS).

Figure 7 shows the imperfect 3D frame for GMNIA calculations IGI-y, IGI-z an IGI-yz. More detailed information on the imperfections is given in Titulaer (2018). Figure 8 shows the load-displacement curves for the six different GMNIA calculations. The horizontal axis shows the horizontal maximum displacement of the top of the front column (Figure 9). Regardless the imperfection type used, the ultimate LPF is about αult,FEM = 0.375, which is the value for EBM-RS. Since EBM-RS is a straightforward procedure to apply, this is the selected method for all further GMNIA simulations. Imperfections according to EN 1993-1-1 (2006) give similar results as shown in Figure 8. The deformed 3D frame at ultimate load is shown in Figure 9 (left), which also shows the deformed 2.5D and 2D frames at ultimate load (middle and right). The deformed frames indicate simultaneous failure of a beam and a column, further evidence of which is given in Titulaer (2018).

Figure 6.

Directions of imperfections according to EN 1993-1-1 (2006).

1028

Figure 7. Imperfect 3D frames for GMNIA calculations IGI-y, IGI-z and IGI-yz (scaled imperfections).

Figure 8.

Load-displacement curves for the 3D frame with different imperfections.

Figure 9.

Deformations (scaled) of frames at the ultimate load: 3D (left), 2.5D (middle) and 2D (right).

4 CODE CHECKING APPROACH Two different approaches are considered for checking the frames by the design rules of Eurocode EN 1993-1-1 (2006): the Sway-mode Buckling length Method (SBM) and the Amplified Sway moment Method (ASM). The SBM starts with a first-order linear elastic analysis of the sway frame, resulting in internal force distributions as shown in Figure 10. Then cross-sectional resistance and member 1029

Figure 10. Internal force diagrams: normal force N (left), bending moment My (middle) and Mz (right).

stability are checked. Sway stability is included by the use of sway buckling lengths in the column stability checks. The principle of the ASM is illustrated in Figure 11. It uses a first order linear elastic analysis in combination with an amplification of the sway moments to calculate the internal forces. The ASM is carried out by the following steps: – – – – – –

add a horizontal support to obtain a non-sway frame; carry out a linear elastic analysis of the non-sway frame to determine the internal forces; determine the horizontal reaction force at the support; remove the horizontal support and position the reaction force on the sway frame; execute a linear elastic analysis of the sway frame to determine the internal forces; amplify the bending moments from the previous step by the amplification factor β of Equation 2 below; – add these amplified bending moments to the internal forces in the non-sway frame (second step above). The amplification factor is: β ¼ αcr =ðαcr  1Þ

ð2Þ

where αcr is the elastic critical LPF at which the frame buckles elastically in its sway mode. The ASM uses the non-sway buckling lengths for the columns since the effect of sway instability is already included in the obtained internal forces, which are reported in Titulaer (2018). Once the internal forces according to the SBM and the ASM are known, cross-sectional resistance and member stability checks are carried out. As an example of a cross-sectional resistance check, Equation 3 is given for the most general case of a normal force N, shear forces Vy, Vz, and bending moments My, Mz simultaneously present in a cross-section. Equation 3 is taken from the Dutch National Annex (NA) to EN 1993-1-1 (2006).

Figure 11. Amplified sway-moment method.

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 β0

My;Ed My;N;V;Rd

α1

 þ β1

Mz;Ed Mz;N;V;Rd

α2

 1:0

ð3Þ

where α1, α2, β0 and β1 are coefficients, My.Ed and Mz,Ed are design bending moments about the y- and z-axis of the cross-section caused by the applied load and My.N,V,Rd and Mz,N,V,Rd are the design bending moment resistances about the y- and z-axis of the cross-section, taking the effects of the normal force N and the shear force V into account; see also the Dutch NA to EN 1993-1-1 (2006). As an example of a member stability check, the well-known Equation 4 is given for a beam-column under simultaneous compressive normal force and bi-axial bending. This design rule is obtained from EN 1993-1-1 (2006). NEd χY NRk γM1

þ kyy

NEd χz NRk γM1

þ kzy

My;Ed χLT

My;Rk γM1

My;Ed M χLT γ y;Rk M1

þ kyz

Mz;Ed

þ kzz

Mz;Ed

Mz;Rk γM1

Mz;Rk γM1

 1:0

ð4aÞ

 1:0

ð4bÞ

where for the meaning of the symbols, the reader is referred to EN 1993-1-1 (2006). In the member stability checks, e.g. Equation 4, the elastic critical buckling force Ncr and the elastic critical moment Mcr are needed to determine, amongst others, the reduction factors χy and χz for flexural buckling and χLT for lateral torsional buckling. The elastic critical buckling force can be determined by using the elastic critical buckling length in the well-known Euler buckling formula. The elastic critical buckling force and moment can be determined by either design rules (DR) as given in handbooks or via a FEM linear buckling analysis (LBA). The DR approach is elaborated in Titulaer (2018), where the design rules given in the Dutch NA to EN 1993-1-1 (2006) are used. The LBA approach is given by Equation 5. Ncr ¼ NEd  αcr

ð5aÞ

Mcr ¼ MEd  αcr;LTB

ð5bÞ

where NEd and MEd are the design compressive normal force in the column and the design bending moment in the beam or column respectively, and αcr and αcr,LTB are the elastic critical LPF’s for flexural buckling of the column and lateral torsional buckling of the column or beam, respectively. As an example, Figure 12 shows the lateral torsional buckling modes of the beam (left) and the column (right) for which the elastic critical LPF’s have been calculated by the LBA method as given by Equation 5 to evaluate the elastic critical buckling force and moment. The ultimate LPF based on a design rule αult,DR is defined as the load for which the first time somewhere in the frame one of the design rules, e.g. Equations 3 or 4, yields unity (1.0). Three different values of αult,DR are evaluated, namely for the:

Figure 12. Lateral-torsional-buckling modes for the beam (left) and the column (right).

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– SBM with design rules based elastic critical buckling forces and moments (SBM-DR); – SBM with LBA based elastic critical buckling forces and moments (SBM-LBA); – ASM with design rules based elastic critical buckling forces and moments (ASM-DR).

5 RESULTS, COMPARISONS, AND DISCUSSION Figure 13 shows the results for the 3D frame. The load-displacement curve as obtained with the FEM approach by a GMNIA with geometrical imperfections according to the EBM and with residual stresses (EBM-RS) is compared to the ultimate LPF’s according to the three code checking approaches. Figure 14 shows similar comparisons for the 2.5D and 2D frames. For the 3D frame, Figure 13 shows that none of the code checking approaches is conservative compared to the ultimate LPF αult,FEM = 0.375 of the FEM approach. Especially when the elastic critical force and moment are based on design rules (SBM-DR and ASM-DR), αult,FEM is significantly overestimated. However, the result is almost correct if the elastic critical force and moment are based on an LBA (SBM-LBA). This suggests that coupled linear elastic stability, as taken into account by an LBA but not by the design rules, is responsible for the differences. For the 2.5D frame, Figure 14 (left) shows similar results as for the 3D frame. Again, the result is better if the elastic critical force and moment are based on an LBA (SBM-LBA), but the result is more unconservative than for the 3D frame. All code checking approaches overestimate αult,FEM even more than for the 3D frame.

Figure 13. Load-displacement curve and ultimate LPF’s for the 3D frame.

Figure 14. Load-displacement curve and ultimate LPF’s for the 2.5D frame (left) and 2D frame (right).

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For the 2D frame, Figure 14 (right) shows that the SBM with the elastic critical force and moment based on an LBA (SBM-LBA) is not the code checking approach closest to the ultimate LPF αult,FEM. Note that this undermines the suggestion made for the 3D frame regarding coupled linear elastic stability being responsible for the differences. Maybe in this case plasticity effects are responsible. All code checking approaches overestimate αult,FEM even more than for the 3D and 2.5D frames. For all frames considered, the FEM approach (EBM-RS) predicts a column and beam failing simultaneously (Figure 9). The FEM approach is perfectly able to capture the associated effect of the reduced support stiffness of the buckled column (due to flexural buckling and plasticity) to the beam, and the reduced support stiffness of the buckled beam (by lateral torsional buckling and plasticity) to the column. This unfavorable elastic-plastic stability interaction effect may not be sufficiently covered by the code checking approaches.

6 CONCLUSIONS AND RECOMMENDATIONS In this paper, a case study of a 3D steel frame was presented to explore stability interaction effects. Also similar 2.5D and 2D frames were considered. Two code checking approaches, the sway-mode buckling length method (SBM) and the amplified sway-moment method (ASM), with their appropriate design rules for checking resistance and stability, were compared with the finite element approach taking into account geometrical and material non-linearities with imperfections (GMNIA). It can be concluded that the code checking approaches do not conservatively cover the ultimate load of the steel frames studied, in which the members fail simultaneously. However, using a linear buckling analysis (LBA) to determine the elastic critical buckling force and moment of members yielded less unconservative results. Extended research on stability interaction effects in steel frames is recommended since only one specific 3D sway frame and its associated 2.5D and 2D sway frames have been studied in this paper. Also research is needed on different geometries of non-sway frames. In general, only limited research has been carried out on the stability interaction of members in steel frames. Therefore, it is recommended to execute full-scale experiments, combined with finite element analyses, to obtain further insights into the complex stability interaction effects in steel frames. REFERENCES Chan, S. L., Huang, H. Y. & Fang, L. X. (2005). Advanced analyses of imperfect portal frames with semi-rigid base connections. Journal of Engineering Mechanics 131(6): 633–640. ECCS/TC8 ‘Stability’. 1984. Ultimate Limit State Calculation of Sway Frames with Rigid Joints. Brussels. ECCS. EN 1993-1-1. 2006. Eurocode 3: Design of steel structures- part 1-1: General rules and rules for buildings. CEN. Brussels. EN 1993-1-5. 2006. Eurocode 3: Design of steel structures- part 1-5: Plated structural elements. CEN. Brussels. Liu, W. & Rasmussen, K. J. R. & Zhang, H. (2014). On the modelling of geometric imperfections in 3D steel unbraced frames. In R. Landolfo and F.M. Mazzolani (eds.), Eurosteel 2014 – 7th European Conference on Steel and Composite Structures: 163-164 and 6 page paper on USB, Napoli, 10-12 September 2014. Brussels: ECCS. Shayan, S., Rasmussen, K. J. R., & Zhang, H. (2014). On the modelling of initial geometric imperfections of steel frames in advanced analysis. Journal of Constructional Steel Research 98, 167–177. Titulaer, L.H.J.D. 2018. Influence of stability interaction effects on the ultimate resistance of 3D steel frames. MSc thesis. Report no. O2018.249. Eindhoven: Eindhoven University of Technology. Department of the Built Environment.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Experimental investigation on the instability phenomenon in stainless steel connections—plate curling K. Sobrinho PGECIV – Post-graduate Program in Civil Engineering, State University of Rio de Janeiro, Brazil

A. Tenchini, M. Cordeiro, P. Vellasco & L. Lima Structural Engineering Department, State University of Rio de Janeiro, Brazil

J. Henriques CERG – Construction Engineering Research Group, University Hasselt, Belgium

ABSTRACT: The overlap bolted steel connection with thin-plates can be subjected to the occurrence of the phenomenon known as curling effect, which is able to influence the global behavior and decrease its ultimate strength. For stainless steel joints, the mechanism is even more important, because it is a material with high deformation capacity. Therefore, both experimental tests and numerical analyses are presented in this paper in order to investigate the influence of curling effect in overlap bolted connection. In addition, the lips were also studied because they increase the stiffness for the out-plane displacement. The outcomes shown that the curling effect reduced the ultimate bearing resistance. Comparing the actual codes, the design load provided by they are lower than those are obtained in both experimental and numerical tests. In addition, it can be reported the efficiency from the lips for the ferritic overlap connection.

1 INTRODUCTION The connection plays a very important role in structure in order to transfer the forces among the structural members and contribute in the global structural behavior. In nowadays, the most commonly used types of connections are welded and bolted. The rivet connections also were very common in the past. In details, the overlap bolted connections have been observed in several civil constructions due to easy applicability. In structural design of these connections type, it is important to define the possible failure modes associated to applied load type, geometric proprieties and material employed. It is recognized that these connection types can present the following failure modes corresponding to net rupture, bearing, bolted shear and yielding gross section. In addition, in thin-plates, there is another failure mode associated to high compression stress in near hole known as curling. In fact, the curling effect can occur due to compression deformation in the end-region of the connection, where the end-plate is fixed by both nut and bolt, and at the other there is no restriction resulting in out-plane deformation. This phenomenon has superior relevance in bolted connections with high distance between the hole and edge plate, as well as, in materials with high deformation capacity, such as, stainless steel (Henriques et al, 2018). In fact, the stainless steel provides large deformation capacity in comparison with the carbon one. Although there is still a limited examples of civil structures with stainless steel members, the application of stainless steel for structural has been used more frequently in recent years, mainly due to the increase of research on its use. This steel grade is recognized by the presence of the chromium and nickel, and may contain molybdenum, iron and other elements. In special, stainless steel should contain at least 10.5% chromium. The interesting in this 1034

steel grade from engineer and researcher is associated to excellent proprieties in comparison with traditional mild carbon steels, such as high corrosion resistance, durability, fire resistance and high aesthetic value. On the other hand, the material cost of the stainless profiles is very higher complicating the use in large scale in civil constructions. Recently, this idea was overcome through of the study performed by Silva et al (2016) when the structural design is addressed to assessment of the maintenance cost to be spent over the life of power transmission tower when there is comparison of mild carbon against stainless steel grades. Therefore, stainless steel can provide several advantages related to carbon steel, which over time can become a more economical solution, such as corrosion resistance, better behavior at high temperatures when compared to carbon steel, higher reuse capacity, ductility and impact resistance (Baddoo, 2008). Considering the promising use of stainless steel in civil structures, this paper aims to investigate the influence of this steel type in overlap shear connection using thin-plates when it is possible to observe the curling effect. Several studies can be found in literature about the structural behavior of stainless bolted connections. In details, Kim et al (2009) observed that in the bolted connections, the ultimate resistance increases in proportion to distance between hole and edge-plate until the appearance of the curling effect. The occurrence and magnitude of the influence of the curling effected the ultimate resistance being associated to thickness plate and distance from hole and edge-plate. For the described arrangements, this phenomenon reduced the connection ultimate resistance by 11%, 16% and 14% for the plates with 1.5 mm, 3.0 mm and 6.0 mm, respectively. In order to reduce the influence of the curling effect, Yancheng & Young (2014) investigated through experimental tests the behavior of the stainless bolted connections using lips. The idea was to assess the bolted connections capacity without the influence of the curling effect since the lips increase the stiffness for the out-plane displacement. The results shown that the nominal resistance obtained from actual design codes are generally conservative for both single and double shear bolted connections. On the other hand, the failure modes observed in experimental tests are close to those predicted by European code (EN 1993-1-4, 2006). Hence, the purpose of the present investigation is to contribute with further experimental and numerical analyses in order to investigate the curling effect in bolted stainless steel connections using lips. Hereafter, experimental tests are presented where it was carried out two stainless steel grades: Austenitic and Ferritic. In addition, a numerical model has been developed on basis of the experimental tests and the results are compared in terms load-displacement curves and failure modes.

2 STRUCTURAL DESIGN OF OVERLAP BOLTED CONNECTIONS 2.1 Eurocode According to EN 1993-1-4 (2006), the structural design of overlap bolted connections is addressed to verification of the plastic resistance of the both gross and net cross-section, bearing and limitation of the geometries on basis of the ultimate capacity. Considering the particular stainless behavior for gross cross-section, based on the yield stress, this criterion can control the structural design and limit the connection resistance. The gross cross-section resistance should be determined using the following equation: Npl;Rd ¼

A  fy γM0

ð1Þ

where, A corresponds to gross cross-section area, fy is the yield strength obtained from the stress-strain curve. The partial safety factor γM0 is equal to 1.1. On the other hand, if this expression is employed for the carbon steel connections with a distinct value equals to 1.0 is adopted.

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The net cross-section resistance of a plate with holes is prescribed in EN 1993-1-4 (2006) on basis of the following equation: Nu;Rd ¼

kr  Anet  fu γM2

ð2Þ

where, Anet is associated to net cross-section area, fu is the ultimate tensile strength of the material and γM2 is a partial safety factor equal to 1.25. In this case, there is no difference between the carbon or stainless steel connections for this partial safety factor. In contrast, the value of the kr is equal to 0.9 for the carbon steel. The bearing resistance is given in EN 1993-1-8 (2005) using a similar curve for carbon steel: Fb;Rd ¼

k1  αb  fu  d  t γM2

ð3Þ

where, d is the nominal bolt diameter, t corresponds to plate thickness, k1 depends on the edge distance and inner bolts and αb is minimum value of the following based in geometric relationships. For stainless steel connection, EN 1993-1-4 (2006) establishes that the ultimate strength, fu, should be reduced due to a hole elongation limitation under serviceability loads. This reduction is defined by combination of the yielding and ultimate strength of the material: fu;Red ¼ 0; 5fy þ 0:6fu

ð4Þ

Recently, it was published a new version of the Design Manual for Structural Stainless Steel developed by The Steel Construction Institute (SCI) of for stainless steel structures in which there is an important contribution in the bearing resistance of bolted connections. The SCI P413 (2017) establishes that the use of the reduced ultimate strength to be applied in bearing capacity is replaced by ultimate strength. In addition, this manual determinates that bolted connection design needs to be verified based on two limit states: serviceability and ultimate. This design strategy has been investigated in other study (Salih et al, 2011).

3 STUDY CASES 3.1 Experimental program In this section, it is presented the experimental program carried out, which consists of eight tests with and without lips being divided by stainless steel grades. For the first, two connections with two shear planes were investigated where the external plates are controlling the structural design. Here, the use of lips aim to mitigate the curling influence on the global behavior due to stiffness increasing for out-plane displacements. For the second one, tests were carried out on two shear planes with the central plate controlling the connection capacity. In these models, the main objective is to analyze the behavior of the central plate, which is prevented from occurring the curling effect. Therefore, with the tests carried out, it was possible to evaluate the influence of the curling effect on austenitic and ferritic bolted connections steels. A schematic drawing of the tests with and without lips is shown in Figure 1 where the nominal value of the length of the stiffener L2 is equal to twice the value of e1, and its height, h, equal to 10 mm. The plates have 3 mm thickness where the value of e1 is equal to 32 mm, w corresponding to 50 mm, hole diameter equal to 13 mm and total length is 40 mm. In addition, the bolted connections with lips are composed by mild carbon steel with 15 mm thickness being designed in order to avoid a premature failure. In order to identify the study cases, a nomenclature has been used being composed of a code with two parameters. The first represents if the bolted connection is composed by outer plates (OP) or inner plate (IN) to be considered as controlling member. The outer plates have lips in order to increasing the stiffness against out-plane displacement. Thus, the aim is 1036

Figure 1.

Geometries properties of the plates investigated.

Figure 2.

Universal machine and LVDT position used in experimental tests.

avoiding the influence of the curling effect on bolted connection behavior, which may influence its maximum resistance (Sobrinho et al, 2018). The last parameter of the code represents whether the bolted connections is with Austenitic (A) or Ferritic steel grades. The bolt used for all tests was M12 class 12.9 type. In order to perform the tests, the Losenhausen machine of 600kN was used, as shown in Figure 2. The tests were instrumented with a linear differential transformer (LVDT) for measuring the axial displacement of the tests and the curling displacement in the region between the hole and the end of the plate. All the instruments were connected to the system Quantum X-MX1615B universal data acquisition from HBM Test and Measurement. 3.2 Material characterization The stress-strain curves were obtained for both ferritic and austenitic stainless steel grades using the longitudinal tensile tests. The coupons tests were extracted considering the orientation of the batch of stainless steel plate. In detail, it was fabricated twelve coupons tests being considered the load axis for parallel, perpendicular and one direction corresponding to forty-five degrees of batch of the plate. Figure 3 illustrates the three stress-strain curves found in tensile coupon tests. In addition, the Table 1 reports the summary of main proprieties obtained from these curves. As can observed, there is a notorious difference observed in structural behavior from the both austenitic and ferritic stainless grades. The batch orientation has more influence for the ferritic plates in comparison with the austenitic. Comparing the outcomes with the EN 1994-1-4 (2006), it can be observed a good correlation in terms of σ0.2%. On the other hand, the ultimate strength from the Austenitic plate presented an higher difference. In general, the values found in longitudinal tensile tests are superior to reported in EN 1993-1-4 (2006). 3.3 Numerical modelling The finite element model used in this paper to investigate the tension capacity of overlap bolted connections was developed by of software Abaqus 6.14 (2014). This element finite 1037

Figure 3.

Stress-strain curves reported in tensile coupons tests.

Table 1. Main proprieties of coupon tests. EN 1993-1.4 (2006)

CP

E [GPa]

σ0,2% [MPa]

Ԑ0,2% [%]

σu [MPa]

Ԑu [%]

A00 A45 A90 F00 F45 F90

202 245 258 271 254 219

275 276 279 281 322 325

0.336 0.313 0.308 0.260 0.286 0.291

860 873 879 472 484 500

55.2 64.7 62.0 17.2 15.6 15.0

σ0,2% [MPa]

σu [Mpa]

230

540

260

450

program is recognized by powerful tool that can incorporate material, geometric and boundary non-linearity cause by nonlinear elasticity, plasticity, large displacement, contact problem, etc. The numerical models were implemented using solid elements C3D8R defined by eight nodes with three degrees of freedom per node: translations in the nodal x, y and z directions. Concerning to adopted mesh, it was chosen a distribution which the proportions and size to be adopted had the aim of avoid the numerical problems. In special, the mesh was refined locally near the bolt hole for improved resolution of stress and strain due to be a region with recognized high stress concentration. Contact surfaces were considered in bolt and plates to better fit the adopted mesh distribution. The load was applied by means of axial load plate displacements in reference node of the lateral face. In addition, all nodes of this face were constrained to reference node through of MCP-Tie. The bolt material was idealized as linear elastic with Young modulus of 210 GPa and 0.3 Poisson coefficient. This strategy was also used in the mild carbon steel plate. On the other hand, the stainless materials were modelled considering the stress-strain curves obtained in longitudinal tensile coupons tests for the same direction of the batch with load. Due to high strain capacity from stainless steel plates, the stress-strain curves were converted to true stress versus true strain where it is considered the large deformation observed in tensile coupons tests. Therefore, a full nonlinear analysis was performed in all numerical models. The material non-linearity was considered using a Von Mises yield criteria associated to a multi-linear stress-strain relationship. The geometrical non-linearity was introduced in the model by using an updated Lagrangean formulation. 4 RESULTS Figure 4 illustrates the load-displacement curves for the four experimental tests together with the numerical response. In addition, Table 2 reports the maximum load observed in both 1038

Figure 4.

Load-displacement curves from both experimental and numerical tests.

Table 2. Maximum load observed in both experimental and numerical tests. Test

Code

LoadEXP [kN]

LoadNUM [kN]

EXP/NUM

1 2 3 4 Média C.O.V

OP-A IP-A OP-F IP-F

71,33 70,11 53,93 48,15

68,31 67,89 53,5 49,15

1,04 1,03 1,01 0,98 1,02 0,03

experimental and numerical tests. As can be observed, it was possible to obtain a similar behavior for the numerical analyses in comparison with the outcomes observed in experimental tests. Thus, it is possible to conclude that finite element analysis can consistently represent the behavior of overlap bolted stainless steel connection submitted to shear. Another important observation is related to maximum resistance observed for Austenitic study cases. In fact, the ultimate strength observed in tensile coupon tests provided a significant increasing of the bolted connection capacity. This issue resulted in bearing resistance higher in compliance with the design code. In addition, the Austenitic bolted connection reported an high axial deformation. In this case, a possible design on basis of the serviceability 1039

or ultimate limit state should be taking into account this important difference reaching a value five times higher for the Austenitic study cases. Analyzing the maximum load obtained for the connection with lips, it can be mentioned that this system is more efficiency for case with ferritic stainless steel grade. There was a similar behavior comparing the bolted connection with and without lips for the study cases with Austenitic steel grade. There is no difference for the resistance observed in both experimental and numerical analyses. On the other hand, the case where the bolted connection with lips using Ferritic steel grade presented an increasing of the maximum resistance equal to 10%. Figures 5 and 6 show the deformed obtained in both analyses. As can be noted, there is an out-plane displacement in study case with Ferritic steel grade. It is possible to note that there is high stress concentration in near hole due to high deformation capacity from the Austenitic one. In fact, the Ferritic study case provided a better stress distribution along the distance between the end-plate to near hole. This fact is very important because the actual codes using similar formulas in order to determine the bearing capacity of the overlap bolted stainless steel connections. Comparing the outcomes with the expression given by EN 1993-1-4 (2006) and SCI P413 (2017), Table 3 reports the ratio experimental and code provisions. It can be noted that there is a better correlation in terms of maximum load capacity for the SCI P413 (2017). However, the value provided by both design code are very restrictive to provide an adequate safe level with economic aspects. In particular, EN 1993-1-4 (2006) does not distinguish for prediction of the maximum load of connections with two planes in shear with the central or end plate controlling the structural design. The EN 1993-1-4 resulted in maximum load equal to 48.26 kN and 31.29 kN for austenitic and ferritic steel, respectively. The SCI P413 (2017), which has an equation for each model, provides a load of 49.54 kN for the OP-A type connection and 63.51 kN for the IP-A. And for the ferritic study cases, values equal to 27.19 kN and 34.86 for OP-F and IP-F, respectively. In general, the results reported in Table 3 show that the use of the reduced ultimate resistance provided a conservative safe level for the bearing resistance. This issue is not

Figure 5.

Comparing the experimental and numerical responses of Austenitic study case with lips.

Figure 6.

Comparing the experimental and numerical responses of Ferritic study case with lips.

Table 3. Comparison with code provisions. EXP/EM 1993-1-4

EXP/SCI P413

OP-A

IP-A

OP-F

IP-F

OP-A

IP-A

OP-F

IP-F

1,43

1,40

1,72

1,52

1,39

1,07

1,96

1,37

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addressed to SCI P413 (20017) as was mentioned in previous sections. In contrast, this last code establishes a conservative value for the bolted connection when the outer plates are responsible by controlling of the structural design. In fact, the code is very conservative for the cases where stainless thin-plates can be susceptible to curling effect. Thus, the SCI P413 (2017) is not efficiency for the cases with Ferritic steel grade. Another issue is addressed to difference found among the Ferritic and Austenitic steel grades. In particular, the relation between the ultimate strength observed in tensile coupons tests is nor proportional to bearing resistance in experimental. This was confirmed in Table 3 where it is not linear correlation for both structural codes because the expression are proportional to ultimate strength from materials.

5 CONCLUSIONS Experimental and numerical analyses have been studied in order to evaluate the influence of the curling effect, as well as, the use of the lips in bolted connection to mitigate this effect. Thus, a set of four study cases were selected modifying the structural scheme in order to investigate the bearing resistance comparing both Austenitic and Ferritic stainless steel grades. Concerning to outcomes observed, it was possible to reported that the performance of the lips was more evident for the cases where Ferritic steel grade is employed. This fact is related to high deformation capacity from the Austenitic steel grade. In fact, the deformed observed in both experimental and numerical analyses provided a reduced out-plane displacement for the Ferritic study cases minimizing the curling effects. Comparing the outcomes with the expression given by design codes, it can be noted that the codes presented lower values for the determination of the bearing resistance. This is related to use of reduced ultimate strength from EN 1993-1-4 (2006) and a value very restricted for the overlap connection subjected to curling effect in SCI P413 (2017). Therefore, these issues should be investigated in further analyses with a large number of the variables. REFERENCES Abaqus 6.14 2014. “Theory Manual and Users Manuals”, Dassault Systèmes Simulia Corp. Baddoo, N.R. 2008. A review of research, applications, challenges and opportunities. Journal of Constructional Steel Research 64: 1199–1206. Cai. Y. & Young, B. 2014. Structural behavior of cold-formed stainless steel bolted connections. Thinwalled Structures 83: 147–156. EN 1993-1-4 2006. Eurocode 3 – Design of steel structures: Part 1–4: General rules – Supplementary rules for stainless steel. Brussels: Europen committee for standardization. EN 1993-1-8 2005. Eurocode 3 – Design of steel structures: Part 1–8: Design of joints. Brussels: Europen committee for standardization. Henriques, J., Batista, G. Tenchini, A., Vellasco, P., Lima. 2018. Overlap shear connections in bearing in thin-walled stainless steel structures. In Eighth International Conference on Thin-walled Structures; Lisbon, 24–27 July 2018. Kim, T.S., Kuwamura, H., Kim, S., Lee, Y., Cho, T. 2009. Investigation on ultimate strength of thin-walled steel single shear bolted connections with two bolts using finite element analysis. Thinwalled Structures 47: 1191–1202. Salih, E.L., Gardner, L., Nethercot, D.A. 2011. Bearing failure in stainless steel bolted connections. Engineering Structures 33: 549–562. SCI P413 2017. Design manual for structural stainless steel. 4rd Ed. The Steel Construction Institute, Building series. Silva, G., Silva, A., Vellasco, P., Lima. L. 2016. Structural and economic assessment of stainless steel power transmission towers. Metálica – Portuguese Steelwork Association Magazine 1: 12–18. Sobrinho, K., Rodrigues, M., Tenchini, A., Vellasco, P., Lima, L., Henrique, J. 2018. Avaliação do efeito curling em ligações aparafusadas em aço inoxidável submetida a esforço de tração uniaxial. Simpósio de mecânica computacional; Vitória, ES novembro de 2018.

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Experimental and analytical study of Cold-Formed Steel (CFS) single-stud walls sheathed with FCB, CSB and MgO under compression Chanchal Sonkar Academy of Scientific & Innovative Research (AcSIR), India CSIR-Central Building Research Institute, Roorkee, India

Achal Kr. Mittal CSIR-Central Building Research Institute, Roorkee, India

Sriman Kr. Bhattacharyya Department of Civil Engineering, IIT Kharagpur, West Bengal, India

Sachin Kumar CSIR-Central Building Research Institute, Roorkee, India

Abhinav Dewangan National Institute of Technology, Raipur, India

ABSTRACT: Cold-formed steel members are widely used in residential, industrial and commercial buildings as primary load bearing elements. It has been extensively used in lightweight framing of low and mid-rise residential constructions. Limited studies have been carried out on CFS wall panels sheathed with Fiber Cement Boards (FCB), Calcium Silicate boards (CSB), and Magnesium Oxide (MgO) Boards under compression. In the present study, CFS single-stud walls with FCB, Heavy Duty FCB (HDFCB), CSB, and MgO boards are experimentally tested under axial loading applied at a constant rate. Analytical modelling of CFS single-stud wall panels with the respective sheathing is carried out by calculating stiffness that the fastener-sheathing system supplies to the stud as bracing. Elastic buckling analysis of the stud with sheathing based springs is completed in CUFSM (Constrained and Unconstrained and Finite Strip Method) software version 4.05, consecutively axial load carrying capacity has been calculated using Direct Strength Method as per AISI S100. Analytical results seem to be in good agreement with the experimental results and the ratio between the two is varying from 0.84 to 1.06. There is significant amount of increment in the axial strength of CFS single-stud wall panels with the use of sheathing, which acts as a restraint. The maximum increase in axial strength is due to HDFCB boards, which is found to be 87% and 149% in one-sided and two-sided sheathed specimen respectively. The results obtained are interesting and useful for the research, academic and industrial community working in the area of CFS.

1 INTRODUCTION Cold-formed steel (CFS) wall framing systems, which have the advantage of being environmentally friendly, light-weight, aesthetically good and easy to construct. CFS are generally utilized as load-bearing structural components in low- and mid-rise CFS structures and non-load bearing structural components in other residential, commercial and industrial buildings. Previously, numerous studies are conducted for assessing the behavioral pattern of the CFS wall-panel (CFSWP) subjected to compression underneath axial loading, such as, Ye et al. 2016 investigated 1042

a complete of sixteen full-scale tests on CFS wall specimens with varied configurations of boards, including fire-retardant GB, MgO board, OSB and CSB. The influence of different sheathing boards, layer of sheathing boards, CFS stud section and spacing, and the joint detail of the CFSWPs were tested. Vieira et al. 2012 evaluated the CFS studs with sheathings under the axial load and found that that the load bearing CFS stud walls has positive impact on the stability and strength because of the bracing supplied by sheathing. Schafer et al. 2010 performed the buckling analysis of CFS steel members using the general boundary conditions in CUFSM. Vieira et al. 2012 gave the expression for the lateral stiffness of the sheathed of CFS wall-panels. Schafer et al. 2010, had investigated the rotational restraint and distortional buckling in the CFS framing system. Tian et al. 2004 had analyzed different screw connection for the CFS structures sheathed with the gypsum and cement-based boards. The objective of the paper is to study the effect of different types of sheathing on the axial strength of CFS single-stud wall panels as sheathing produces the confinement of the frame which gives rise to strength of the CFSWP. Four types of sheathing boards i.e. Fiber Cement Board (FCB), Heavy Duty Fiber Cement Board (HDFCB), Magnesium-Oxide Boards (MgO) and Calcium Silicate boards (CSB) has been used for the present study. Screw spacing of 300mm has been kept constant for all the specimens. The study has been validated by using analytical model utilized by AISI S-100 and Vieira & Schafer (2013).

2 EXPERIMENTAL STUDY A test set-up has been fabricated and installed on 300-ton Instron UTM available in Structural Engineering Laboratory, CSIR-CBRI for testing of CFS single-stud wall panels under compression. Tests set-up consists of 02 nos. of top and bottom plate girders designed using IS 800: 2007 for a load of 200 Tons as shown in Figure 1. 2.1 Specimen configurations Axial load tests are performed at CSIR-CBRI Roorkee. In this work nine CFS single-stud wall specimen configurations as discussed in Table 1 were prepared in the laboratory and tested under axial loading condition. CFS single-stud bare frame (W1) was fabricated using C-section (89 × 44 × 13 × 1 mm) and U-section (92 × 50 × 1 mm) as shown in Figure 2. One-sided and two-sided

Figure 1.

Axial Load Test Set-up. (a) Schematic Diagram of set-up. (b) Experimental set-up.

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Table 1. Specimen details. S. No. Specimen Configuration No. 1 2 3

4 5 6 7

8 9

Schematic Plan View

W1 CFS single-stud W2 CFS single-stud with onesided FCB (8mm) sheathing W3 CFS single-stud with onesided HDFCB (9mm) sheathing W4 CFS single-stud with onesided CSB (12mm) sheathing W5 CFS single-stud with onesided MgO (10mm) sheathing W6 CFS single-stud with twosided FCB (8mm) sheathing W7 CFS single-stud with twosided HDFCB (9mm) sheathing W8 CFS single-stud with twosided CSB (12mm) sheathing W9 CFS single-stud with twosided MgO (10mm) sheathing

Figure 2.

Sheathing Position Layer

Screw Spacing

—–

—–

—–

1-0

one-side

300

1-0

one-side

300

1-0

one-side

300

1-0

one-side

300

1-1

two-side

300

1-1

two-side

300

1-1

two-side

300

1-1

two-side

300

CFS element cross-section. (a) Stud Cross-section. (b) Track Cross-section.

sheathed specimens (W2-W9) included four different types of sheathing FCB, HDFCB, CSB and MgO boards as shown in Figure 3. The sheathing was kept 8.0 mm below the top track so that when the load is applied on the panel, the load doesn’t come directly on the sheathing and whole is restricted to the top track only. 2.2 CFSWP fastener configuration Two types of screws were utilized for fabrication of specimens as shown in Figure 4. The selfdrilling dry wall screws of diameter 4.0 mm and length 24 mm; used to assemble the steel studs with the top and bottom tracks. The self-drilling bugle head screws of diameter 4 mm and length 32 mm; used for assembling the CFS frame and sheathing. 1044

Figure 3. Specimen configuration details. (a) Isometric view single CFS single-stud; (b) Isometric view of one sheathed CFS single-stud frame; (c) Elevation & Plan of CFS single-stud frame.

Figure 4. Screw used for connecting CFS frame members; (a) Self-drilling dry wall screws (assemble the steel studs with the top and bottom tracks). (b) Self-drilling bugle head screws (assembling the CFS frame and sheathing).

2.3 CFSWP material configuration The CFS frame members are fabricated using a galvanized steel sheet of nominal yield strength of 350 MPa. The physical properties of the CFS material has been determined through tensile coupon test as per IS 1608-2005 is observed to be an average of 403.56 MPa for tensile strength and elastic modulus of 200.23 GPa . As per the literatures considered, assumed engineering properties of the sheathing boards are {1} CSB (12 mm thick) U = 0.16, E = 6500 {2} FCB (8 mm thick) U = 0.18, E = 5500 {3} HDFCB (9 mm thick) U = 0.18, E = 9000; {4} MgO (10 mm thick) U = 0.20, E = 4500; Where, U = Poisson’s ratio, and E = Elasticity Modulus.

3 LOADING PROTOCOL Loading was applied at a rate of 0.5 mm/min using 300 Ton Instron UTM available at Heavy Testing Laboratory at CSIR-CBRI Roorkee. Test set-up was designed for a uniformly distributed loading on the top track. The specimen was fixed at the base.

4 INSTRUMENTATION As shown in Figure 5, the compressive axial loading applied on the CFS single-stud walls with and without sheathing by 300 Ton Instron UTM controlled. The application loads are 1045

Figure 5. Instrumentation for experiment. (a) Laser Displacement Sensors (L.D.S.) placed to record the displacement. (b) Fixity of the CFS single-stud specimen at the bottom using MS plate and C-clamps.

measured by the load cells fixed with actuator. Laser displacement sensors were placed on the specimen in-plane displacements as shown in Figure 5(a). The bottom track of the specimens is fixed to the bottom plate girder with the help L-shape plates and C-clamps (Figure 5(b)).

5 ANALYTICAL STUDY OF CFSWP SUBJECTED TO COMPRESSION Analytical modelling of CFS wall panels with sheathing is carried out by the method described by calculating stiffness that the fastener-sheathing system supplies to the stud as bracing. Theoretically, the fastener-sheathing system supplies three translational and three rotational springs at every fastener location bracing the stud. Practically, a more limited set of springs in the plane of the cross section, consisting of lateral translation (kx ), which is in the plane of the sheathing, vertical translation (ky ), which is out of the plane of the sheathing, and rotational stiffness (k ), which is in the plane of the cross section, are the most important. The springs restrain against buckling modes associated with weak-axis flexure and torsion, while k springs restrain flange (AISI S-100 and Vieira & Schafer 2013). The expressions for determining these restraint parameters are: Lateral Translational Stiffness ðkx Þ (Vieira & Schafer (2013))

kx ¼ 1=ð1=kxl þ 1=kxd Þ

Out-of-Plane Translational Stiffness (ky ) (Vieira & Schafer (2013))

ky ¼

Rotational Stiffness (k ) (Vieira & Schafer (2013))

k ¼ 1=ð1=kc þ 1=kw Þ

ðEIÞW π4 df L4

ð1Þ ð2Þ ð3Þ

3Ed 4 t3 π (Green et al. 1947; Winter 4t2board ð9d 4 π þ 16tboard t3 Þ 2 π Gtboard df wtf ; kc ¼ 0:00035Et2 þ 75; 1960); kxd ðdiaphragm translational stiffness) ¼ L2 ðEIÞ kw ¼ df w ; E ¼ Young’s modulus of the steel stud; d = fastener diameter; t ¼ flange thickness; tboard ¼ board or sheathing thickness; G ¼ Shear modulus of the sheathing; wtf ¼ fastener tributary width; df ¼ distance between fasteners; L ¼ sheathing height; ðEI ÞW ¼ sheathing rigidity Where, kxl ðlocal translational stiffness) ¼

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5.1 Elastic stability and direct strength method The elastic stability of CFS stud is significantly altered by the presence of sheathing. Elastic buckling analysis of the stud with sheathing based springs is completed in CUFSM (Constrained and Unconstrained and Finite Strip Method) software version 4.05. The basic method proposed for strength determination is to correct the elastic buckling loads for the presence of the sheathing and then to use existing design expressions; either the Direct Strength Method (DSM) of AISI S-100, or the Effective Width Method (EWM) of the main specification of AISI S-100 to find the strength. Consecutively axial load carrying capacity has been calculated using Direct Strength Method as per AISI S100, according to which the elastic buckling loads; Local Buckling (Pcrl ), Distortional Buckling (Pcrd ), and Global Buckling (Pcre ) will be obtained as per the member strength determination procedure given by Vieira & Schafer (2013) using the product of elastic buckling load factors and the squash load, Py obtained from CUFSM analysis to get the elastic buckling loads. Then for predicting the nominal strength of the wall-panel by DSM as per section A1.2.1 of AISI S-100. The nominal strength of the wall-panel will be determined by equation number (7) which is finally termed as predicted ultimate load in this study. Global Strength,  Pne ¼

0:658λc Py 0:877Pcre 2

for λc  1:5 for λc > 1:5

qffiffiffiffiffiffi and λc ¼

Py Pcre

ð4Þ

Local-global interaction,

Pnl ¼

8 >

: Py Py

sffiffiffiffiffiffiffiffiffi Py and λd ¼ for λc 40:561 Pcrd

for λc  0:561

ð6Þ

Nominal Strength, Pn ¼ minðPne ; Pnl ; Pnd Þ

ð7Þ

6 RESULTS AND DISCUSSIONS The ultimate loading capacity of single-stud was 20.95kN. Flexural-torsional buckling was observed at failure in the W1 specimen (Figure 6(a)). The increase in the axial strength of onesided CFS single-stud wall panels due to FCB, HDFCB, CSB and MgO were basically 77.08%, 87.11%, 66.25 and 82.71% respectively. The major mode of failure in all the one-sided CFS singlestud wall panels was flexural-torsional buckling. In W2, W3, W4 and W5 location of buckling at 650 mm (Figure 6(c)), 950 mm (Figure 6(d)), 1100 mm (Figure 6(g, h)) and 800 mm (Figure 6 (j, k)) from the top track respectively. Local lip buckling was also observed at various locations in both W2 and W3. Screw pull-through was observed in W2 and W5 (Figure 6(l)). Board separation and cracking was observed at the peak load in W3 (Figure 6(e)) and W5 (Figure 6(l)) respectively. Local web buckling was observed in W4 at the top stud-track junction (Figure 6(f)). The increase in the axial strength of two-sided CFS single-stud wall panels due to FCB, HDFCB, CSB and MgO were basically 110.16%, 149.02%, 141.19%and 108.40% respectively. In W6 local flange buckling was observed at 600 mm (Figure 6(p)) from the top track. In W7, W8 1047

Figure 6.

Failure pattern observed in different specimen.

(a) W1: FT-failure (b) W2: LB-failure (c) W2: FT-failure (d) W3: FT-failure (e) W3: BS-failure (f) W4: WLB (g), (h) W4: FT-failure (i) W4: WLB (j), (k) W5: FT-failure (l) W5: BC & SPT- failure (m), (n) W7: DB-failure (o) W7: LB in Stud (p) W6: FB-failure (q), (r) W6: WLB (s) W6: BC-failure (t) W6: WLB (u), (v) W8: DB-failure (w), (x) W8: LLB (y) W9: DB-failure *note: FT: flexural-torsional; LB: local-buckling; BS: board-separation; WLB: web local-buckling; BC: boardcracking; DB: distortional-buckling; LLB: lip local-buckling; SPT: screw pull-through; FB: flange local-buckling.

and W9 distortional buckling occurred at location of 300 mm (Figure 6(m, n)), 400 mm (Figure 6 (u, v)) and 2050 mm (Figure 6(y)) from the top track respectively. Local lip buckling observed at several locations in W6 and W8 (Figure 6(w, x)). local buckling in stud (Figure 6(o)) at the bottom stud-track junction W7. Local buckling waves were also generated in all the specimens near the peak load. No screw pull-through and board cracking has been observed in W4 and W7. Among the non-heavy-duty one-sided sheathed specimens, axial load carrying capacity W5 is highest whereas in two-sided specimens axial load carrying capacity of W7 is highest. 1048

Table 2. Test results. Ultimate Load Predicted to Specimen Experimental Predicted Experimental test ratio Test (kN) (kN)

% increase in ultimate load w.r.t. W1 specimen

W1 W2 W3 W4 W5 W6 W7 W8 W9

77.08 87.11 66.25 82.76 110.16 149.02 141.19 108.40

20.95 37.10 39.20 34.83 38.29 44.03 52.17 50.53 43.66

22.20 39.45 39.53 33.83 39.03 43.78 43.81 43.36 43.02

1.05 1.06 1.00 0.97 1.01 0.99 0.84 0.86 0.98

Therefore, MgO board performs well in one-sided configuration and CSB board is performs better is two-sided configuration in terms of strength. The ultimate loading capacity increase in W3 and W7 HDFCB sheathed specimen are 87% and 149% in comparison with CFS singlestud. In this study the HDFCB sheathed specimen is found to be best performing sheathed specimen among all the sheathing boards as it is as compressed boards of higher density and elastic modulus. The major concern in this study behind fixing the length of all the specimen to a single length is to observe the influence of type of sheathing and its elastic modulus on the occurrences of failure pattern. The influence of asymmetrical and symmetrical boundary condition imposed by application of sheathing boards on either side of CFSWP.

7 CONCLUSIONS An axial compression test of 09 nos. of CFS single-stud walls was conducted to study the effects of sheathing types on axial strength. The ultimate load of the wall studs obtained from experiments was compared with that from analytical method discussed by Viera & Schafer 2013. It was observed that there is significant amount of increment in the axial strength of CFS single-stud wall panels with the use of sheathing. The following conclusions were drawn from this work: 1. The increase in the axial strength of one-sided CFS single-stud wall panels due to FCB, HDFCB, CSB and MgO were basically 77.08%, 87.11%, 66.25 and 82.71% respectively. 2. The common buckling failure patterns observed in CFS single-stud bare frame and onesided sheathed specimens is flexural-torsional failure. 3. In one-sided sheathed specimens, constraints are only applied in one of the flanges of the stud section, which gives origin for an eccentricity in the model, which leads to widening of flanges resulting into flexural torsional failure. 4. The increase in the axial strength of two-sided CFS single-stud wall panels due to FCB, HDFCB, CSB and MgO were basically 110.16%, 149.02%, 141.19%and 108.40% respectively. 5. The type of failure pattern observed in the both side sheathed specimen is the distortional failure because the stud has been constrained in both vertical direction and on both the flanges which gave rise for occurring failure as distortional buckling. 6. Analytical results seem to be in good agreement with the experimental results and the ratio between the two is varying from 0.84 to 1.06. It can be said that the discussed analytical method by Vieira & Schafer 2013 can be used for prediction of axial strength of sheathed CFSWP. 7. Among the non-heavy-duty boards, MgO and CSB boards carried maximum axial load in one-sided and two-sided CFS sheathed single-stud wall panels respectively. Whereas in the case of heavy duty FCB boards the axial strength approximately increased by 87% and 149% for single side and both sides sheathing respectively. 1049

REFERENCES AISI S100, 2016. North American Specification for Cold-Formed Steel Structural Members. Green, G. G., Winter, G., and Cuykendall, T. R. 1947. Light gage steel columns in wall-braced panels. Rep. 35, Pt. 2, Engineering Experiment Station, Cornell Univ., Ithaca, NY. IS 800 (Indian Standard). 2007. General Construction in Steel- Code of Practice. IS 1608 (Indian Standard).2005. MetallicMaterials-TensileTesting at Ambient Temperature. Li, Z. and Schafer, B.W. 2010. Buckling analysis of cold-formed steel members with general boundary conditions using CUFSM conventional and constrained finite strip methods. Schafer, B.W., 2008. The direct strength method of cold-formed steel member design. Journal of constructional steel research, 64(7–8), pp.766–778. Schafer, B.W., 2012. CUFSM 4.05–finite strip buckling analysis of thin-walled members. Baltimore, USA: Department of Civil Engineering, Johns Hopkins University. Schafer, B.W., 2013. Sheathing Braced Design of Wall Studs. Schafer, B.W., Vieira Jr, L.C., Sangree, R.H. and Guan, Y. 2010. Rotational restraint and distortional buckling in cold-formed steel framing systems. Revista Sul-americana de Engenharia Estrutural, 7 (1). Vieira Jr, L.C.M. and Schafer, B.W. 2012. Lateral stiffness and strength of sheathing braced cold-formed steel stud walls. Engineering Structures, 37, pp.205–213. Vieira Jr, L.C.M. and Schafer, B.W. 2012. Behavior and design of sheathed cold-formed steel stud walls under compression. Journal of Structural Engineering, 139 (5),pp.772–786. Vieira, L. C. M., Jr. 2011. Behavior and design of sheathed cold-formed steel stud walls under compression. Ph.D. thesis, Johns Hopkins Univ., Baltimore. Tian, Y.S., Wang, J., Lu, T.J. and Barlow, C.Y. 2004. An experimental study on the axial behaviour of cold-formed steel wall studs and panels. Thin-walled structures, 42(4),pp.557–573. Winter, G. 1960. Lateral bracing of beams and columns. J. Struct. Div. 102(1),77–92. Ye, J., Feng, R., Chen, W. and Liu, W. 2016. Behavior of cold-formed steel wall stud with sheathing subjected to compression. Journal of Constructional Steel Research, 116, pp.79–91.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Lateral-torsional buckling of stainless steel beams with slender cross section M. Šorf & M. Jandera Czech Technical University in Prague, Prague, The Czech Republic

ABSTRACT: The contribution shows experimental and numerical research of welded slender stainless steel I-section beams. Difference in behavior of stainless steel and common carbon steel members is generally known, but design of stainless steel members has been established mainly for hollow sections (CHS, SHS/RHS) as these are the typical stainless steel profiles. Currently, open sections are also being used in structures and the design rules for both local buckling of very slender sections as well as lateral torsional buckling reduction factors are based on very limited experimental and numerical research. For this reason new research covering these phenomena was started at the Czech Technical University in Prague. A numerical model was used for design of test arrangement, the geometrically and materially nonlinear analysis with imperfection was made in software Abaqus. The experimental program consisted of six stainless steel beam tests being used for a model validation. The tests employed two stainless steel materials (austenitic and ferritic steel), one section slenderness (Class 4 for web and flange) and three beam slenderness. A parametric study based on the validated numerical model will be used to compare existing design procedures and their possible refinement for Class 4 sections.

1 INTRODUCTION 1.1 Stainless steel Over the last several decades, popularity in use of stainless steel for structures increases. This material is a specific type of steel which is highly alloyed containing more than 10.5% of chromium. Austenitic, ferritic and duplex stainless steel are the most used stainless steel types in structural applications. These groups are different in strength, ductility, weldability, toughness and the ability to resist corrosive environment as result of using various alloying elements in varying amount. The main reason for limited use is the initial cost of the material, which differs significantly for each group of stainless steel and is much higher than for common carbon steel. The important difference from common carbon steel is the material stress-strain diagram behavior. Instead of typically linear behavior up to a visible yield strength for carbon steel, the stress-strain curve for stainless steel has a more rounded response with no clearly defined yield point. The material nonlinearity is the reason for the need of other design procedures for stainless steel structures as the stiffness is reduced by yielding below the 0.2 % (yield) proof strength and strain hardening is usually much higher. 1.2 Existing design procedure Generally, open cross sections subjected to bending around the major axis with unrestrained or partly restrained compressed flange or compressed web tend to fail with influence of lateral torsional (global) buckling. Whereas slender cross section resistance may be governed by plate (local) buckling. Both phenomena influence significantly design of steel beams.

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The resistance of beam subjected to lateral-torsional buckling should be determined according to EN 1993-1-4, where supplementary rules are given for structural stainless steel, as follows: Mb;Rd ¼ χLT Wy fy =γM1

ð1Þ

where Wy = Wpl, y for Class 1 or 2 section; Wy = Wel,y for Class 3; Wy = Weff,y for Class 4; fy = yield strength; χLT = reduction factor accounting for lateral torsion buckling: χLT ¼

1

2 LT þ ½LT 2 
λLT 0;5

1 2

LT ¼ 0; 5ð1 þ αLT ðλLT  0; 4Þ þ λLT Þ sffiffiffiffiffiffiffiffiffiffiffi Wy fy λLT ¼ Mcr

ð2Þ ð3Þ ð4Þ

where Mcr = critical moment; αLT = imperfection factor suggested as 0,34 for cold-formed and hollow sections and 0,76 for open welded or other sections. As far as the authors are aware, there were no experimental data for welded open sections when the procedure was codified. Beams of comparatively stocky sections were tested and modelled by (Wang et al. 2014) and (Yang et al. 2014). A wider test program is being prepared at KU Leuven, where preliminary results were published (Fortan et al. 2016).

2 NUMERICAL MODELLING A numerical model was primary used for the design of test arrangements and for the development of an experimental program. The geometrically and materially non-linear analysis with imperfection (GMNIA) was made in software Abaqus using shell element S4R (four-node shell element with reduced integration). The section resp. member imperfections were assumed by the lowest elastic buckling eigenmode (Figure 1) for local resp. global buckling. The geometric (local) and global imperfection were determined as 0.8 of the fabrication tolerance. Amplitude for local imperfection was considered as 0.8b/100, where b is the web height. For global imperfection, there was considered imperfection amplitude by value of 0.8L/750, where L is the distance between two cross-sections where lateral torsional buckling is prevented. The amplitude for eigenmode with greater value of αcr was reduced by 0.7 times as suggested in EN 1993-1-5.

Figure 1.

Eigenmodes for buckling – local (left), global (right).

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3 EXPERIMENTS 3.1 Experimental program The prepared experimental program consisted of 6 four point bending tests. There was selected only one cross section (Class 4) made of two stainless steel grades and with three different lengths (2.4, 3.8 and 5.4 m span). Summary of beams is given in Figure 2, where the section geometry, as well as the section and beam slenderness are given. It can be seen that the test arrangement was identical for all beams and similar to the used by (Prachař et al. 2016), with modification in the beam span and the distance between lateral restraints only. The beam was supported at its ends under the lower flange. Lateral restraints were used at ends and points of loading for both the upper and lower flange (Figure 3). 3.2 Material properties Furthermore, several material tests were also carried out, measured material properties are shown in Table 1. A two-stage Ramberg-Osgood material diagram (Gardner et al. 2004) was used for description of the stress-strain diagram and used for validation of numerical models. 3.3 Measurement of imperfections Before each test, the initial local and global imperfections were measured along three lines on the web and along both edges of the upper flange. An example of measured amplitudes for the longest beam is shown in Figure 4.

4 FE MODEL VALIDATION The geometrically and materially nonlinear numerical model with imperfections (GMNIA) in software Abaqus was used and validated on the tests. Section resp. member imperfections were assumed by the lowest elastic buckling eigenmode for local resp. global buckling. The amplitudes of imperfections were taken as measured for each specimen. The number of elements was selected as 10 per width of the flange and 33 per height of the web.

Figure 2.

Experimental program overview.

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Figure 3.

Test arrangement and beam failure modes.

Table 1. Material properties of tested specimens. Type of steel

Yield strength

Ultimate strength

Modulus of elasticity

R-O parameter

parameter n`0.2.1.0

[MPa]

[MPa]

[GPa]

[-]

[-]

1.4016 1.4016 1.4016 1.4301 1.4301 1.4301

307.016 306.642 297.787 297.345 298.834

306.83

297.99

430.698 428.029 637.413 622.003 638.079

429.36

632.50

201.371 204.343 195.693 197.339 195.188

202.86

196.07

8.10 7.70 5.60 5.50 5.80

7.90

5.63

1.80 1.77 2.30 2.35 2.37

1.79

2.34

In addition, the true stress-strain diagram was calculated from the measured material properties and implemented in the model. As no residual stresses were measured, the residual stress pattern for welded stainless steel open I sections proposed by (Yuan et al. 2014) was used in the model (Figure 5). Specimens after tests can be seen in Figure 6, load-deflection diagrams for each beam are shown in Figure 7. Table 2 shows the comparison of the beam resistance obtained from experiment, FE model and from the design procedure according to EN1993-1-4 with using measured material properties. Failure modes for test 1 and test 3 are shown in Figure 8. The load-deflection curve comparison of results from FE model and from the experiment for the shortest as well as for the longest beam is in Figure 9, 10, where the influence of imperfections and residual stresses is also demonstrated. Generally, the numerical model predicts the real behavior well as the predicted resistance is in average 11,8% lower. Especially for longer beams, the prediction was very accurate. For shorter beams, where the resistance is more influenced by local imperfections, the differences in beam resistance prediction are higher. The difference is contributed to the simplification in the local imperfection shape being considered by the first elastic buckling eigenmode. The real shape of imperfection is more favourable for the beam resistance. More precise imperfection modelling is planned in the future. Furthermore, can be stated that existing design procedure from EN 1993-1-4 provides safe but inaccurate results as can be seen in Table 2 and further improvement is possible. 1054

Figure 4.

An overview of imperfection measurement including the scheme of measured points.

Figure 5.

Implemented residual stresses.

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Figure 6.

Beams after test.

Figure 7.

Load-deflection diagrams for all tested beams.

Table 2. Beam resistance comparison. Test

Material

Li

LLTi

Mb Rd[KNm]

[-]

[-]

[m]

[m]

Test

1 2 3 4 5 6

1.4301 1.4301 1.4301 1.4016 1.4016 1.4016

5400 3800 2400 5400 3800 2400

2500 1500 1000 2500 1500 1000

76.1 89.8 90.8 83.3 98.7 97.4

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FE model 77.0 75.8 77.8 79.4 79.9 81.9

1.2% 15.7% 14.4% 4.6% 19.0% 15.8%

EN 1993-1-4 49.0 62.2 69.3 50.6 64.2 71.5

35.6% 30.8% 23.7% 39.3% 35.0% 26.6%

Figure 8.

Test and FE model failure mode for test 1 (right) and for test 3 (left).

Figure 9.

FE model comparison for test 1.

5 CONCLUSIONS Behavior of slender open section stainless steel beams loaded by major bending moment is investigated in the paper. Numerical models created in software Abaqus using GMNIA were validated based on experimental data. Furthermore, the comparison of results obtained according to the existing design procedure was made. 1057

Figure 10. FE model comparison for test 3.

The following research will focus on more precise imperfection modelling for greater match of results from experiment and FE model. Subsequently, the design procedure of stainless steel welded open cross section resistance in bending as well as the lateral-torsional buckling curves will be investigated in a numerical parametric study. Initial part of the study is expected for publication on the conference. ACKNOWLEDGEMENT The support of the GAČR 17-24769S “Nonlinear stability and strength of slender structures with nonlinear material properties” are gratefully acknowledged. REFERENCES Gardner, L., Nethercot, D. 2004. Experiments on stainless steel hollow sections - Part 1: Material and cross-sectional behavior. Journal of Constructional Steel Research Vol. 60: 1291–1318. Wang Y., Yang L., Gao B., Shi Y., Yuan H. 2014. Experimental Study of lateral-torsional buckling behaviour of stainless steel welded I-section beams. International Journal of Steel Structures 14, 411–420. H.X. Yuan, Y.Q. Wang, Y.J. Shi, L. Gardner. 2014. Residual stress distributions in welded stainless steel sections. Thin-Walled Structures: 79, 38–51. Yang L., Wang Y. Q., Gao B., Shi Y. J., Yuan, H. X. 2014. Two calculation methods for buckling reduction factors of stainless steel welded I- section beams. Thin-Walled Structures: 83, 128–136. Fortan M., Zhao O., Rossi B. 2016. Lateral torsional buckling of welded duplex stainless steel I section beams. Sixth International Conference on Structural Engineering, Mechanics and Computation (SEMC 2016), Cape Town, South Africa. Prachař, M., Hricák, J., Jandera, M., Wald, F., Zhao, B. 2016. Experiments of Class 4 open section beams at elevated temperature. Thin-Walled Structures: 98(1), 2–18.

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Improve load capacity calculations by considering realistic imperfections induced by welding for plates and shells C. Stapelfeld, B. Launert & H. Pasternak Chair of Steel and Timber Structures, Brandenburg University of Technology, Germany

N. Doynov & V.G. Michailov Department of Joining and Welding Technology, Brandenburg University of Technology, Germany

ABSTRACT: The topic of this paper is the application of an analytical numerical hybrid model for a realistic prediction of weld imperfections. At the beginning, the analytical model, its physical basis as well as the physical interrelationships are explained. This is followed by the explanation of the coupling procedure between the analytical model and the numerical calculation. The significance of this approach is proven by an application on a large welded structure with several welds and a comparison of the calculated distortions with measurements. Afterwards, the hybrid model is applied on the investigated stiffened structure for the determination of the weld imperfections. An ultimate load analysis gives information about the load carrying behavior under axial loading. The results are compared with the results of ultimate load analysis from a literature example assuming different eigenvalues and scaling. The comparison underlines the potential additional utilization of load bearing capacity by this new approach.

1 INTRODUCTION Welding distortions and residual stresses in terms of geometrical and structural imperfections can significantly affect the quality and the stability behavior of welded structures such as stiffened plates or shell. To simplify, in numerical strength calculations by the finite element method (FEM) both types of imperfections, the geometrical and structural ones, are typically combined being considered as equivalent geometrical imperfections. Values for standard cases are included in EN 1993-1-5 in case of for example plated structures (Eurocode, 2017). Additionally, global and local imperfections are to be distinguished. The global imperfections can be defined as a bending of the whole structure with a maximum depending on its dimensions. The local imperfections are mostly determined through a numerical eigenvalue analysis. The challenge of a numerical ultimate load analysis is the detection of the lowest ultimate load by the combination of different local and global imperfections. Numerous research works deal with the numerical load capacity calculation of stiffened plates and shells (Beg 2010, Manco 2016, Degée 2008, Ghavamia 2006, Tran 2014). However, it remains unclear to some extend how accurate these equivalent geometrical imperfections represent the actual deformations and residual stresses caused by welds, especially for complex cases. The significance of numerical load capacity calculations could be increased enormously if these imperfections were known more exactly and could be considered directly during the computation. Nowadays, the deformations and residual stresses can be determined by means of a thermomechanical FE-simulation. This approach contains two steps: the calculation of the transient temperature field followed by the calculation of the mechanics. Generally, such calculations can provide very realistic and accurate values. However, the effort for the calibration and validation of these simulations is huge and temperature-dependent thermophysical and thermomechanical material properties are required. In addition, relevant structures and weld 1059

length are very large what leads to enormous calculation times and a huge demand of storage capacity (Stapelfeld 2009). Simplified numerical approaches are available and able to remedy this situation. However, the application of these models partly requires more expertise than a conventional thermomechanical FE calculation [Duan 2007] or the simplifications are so extensive that the weld imperfections calculated by the approach partially loses their validity (Thikomirov 2008). To answer the industrial needs, different modeling approaches based on the inherent strain method exist. However, in the most works the leading hypothesis is that the inherent strain depends only on the welding process and the material. The influence of the structure on the inherent strain is seldom considered and limited to the structural strength (Ueda 1989, Ueda 1993, Mun 2011, Murakawa 2013). The analytical-numerical approach (hybrid model) realizes a strong coupling between the local and global mechanical effects (Michailov 2011). Thus, it can consider the interactions in the structure, like the welding sequence or the variation of the structural stiffness as well as the clamping conditions during the assembling process. Therefore, the model provides an uncomplicated and sufficient solution for calculating welding distortions. Currently the approach has been successfully applied to ship and railway carriage structures, considering welding and thermal straightening (Michailov 2011, Michailov 2014, Doynov 2018a, Doynov 2018b).

2 THE ANALYTICAL NUMERICAL HYBRID MODEL 2.1 General description of the approach The basic idea of the coupled analytical numerical shrinkage force model is the linking of the major advantages of both, analytical and numerical procedures. On the one hand, the matchless marginal calculation time of the analytical shrinkage force model and its simple application, and on the other hand, the possibility to conduct a FE simulation to calculate distortions and residual stresses at any location of complex welded structures. All the determining factors on quality and quantity of weld imperfections are initially passed to an analytical calculation program, capturing the mathematical approach of the shrinkage force model. The program calculates the shrinkage volume wpz, the inherent strain components εx, longitudinal and εy, transversal to the weld as well as their centroids zc,x and zc,y for every single weld. The calculated entities are then transferred to the global finite element model in order to predict the distortions of the structure after every welding operation (i.e., welds, assembly steps, changing of fixtures, etc.). The calculated residual stresses in the domain of the next weld caused from the previous welding operations serves as initial condition for the analytical calculation of the shrinkage volume. It allows simultaneous variation of the geometry and clamping conditions in the FE analysis. Thus, the welding sequence as well as the changing of geometry and clamping conditions can be taken into account. Finally, the results of the application of the hybrid model are superposed with additional fabrication tolerances followed by a numerical load capacity calculation (Figure 1). 2.2 The analytical model The analytical solution for the calculation of the inherent strains in longitudinal and transversal direction is derived from the shrinkage force model (Okerblom 1955, Kuzminov 1974, Gatovskii 1980). This model contains a one-dimensional problem formulation for the longitudinal strain and a two-dimensional plane stress problem formulation for the transverse one. The following assumptions are made: • • • •

The plane section hypothesis is valid; The width of the plastic zone is significantly smaller than the width of the plate; The welds are long (i.e., quasi-stationary temperature field); The welding speed is relatively high (i.e., fast moving heat source). 1060

Figure 1.

Schedule of the load capacity calculation taking into consideration realistic weld imperfections.

The weld imperfections depend significantly on the maximum temperatures that every point vertical to the weld direction is exposed to and the stiffness of the structure. Equations for the calculation of the maximum temperatures were derived by Rykalin (Rykalin 1957) constituting the basis of the shrinkage force model. Two border cases are considered: a fast moving line source in a thin plate with vertical isotherms penetrating the plate and a fast moving point source on a semiinfinite body with circular isotherms around the weld. The proportion of each of the two border cases to the resulting temperature field in the investigated weld structure depends on the heat input per unit length, the geometrical properties as well as the heat exchange with the environment and is calculated iteratively. Considering further influencing factors, an axial force Fx is calculated corresponding to the heat effect of welding (Kuzminov 1974): Fx ¼

rffiffiffiffiffi 2 α lnð2Þ qs EKχδ Kk Kσ ; πe cρ

ð1Þ

with the thermal expansion coefficient α, the specific heat capacity c, the density ρ and the Young’s modulus E. Kχδ, Kk and Kσ are capturing the temperature field in medium thick plates, the stiffness of the weld structure and the effect of existing stresses in the weld. Furthermore, the transversal shrinkage force Fy is calculated as follows: " !#    2  2 !   α lnð2Þ lnð2Þ 2 2 Fy ¼ q s E Kav K 1 þ Kμ Kδ  þ 1 þ 1 Kδ cρ e e 3π 3π εF  ½1 þ KC ð1 þ Kδ Þ KW Kδ þ qs ð1  Kδ Þ; θ

ð2Þ

where Kδ captures the degree of heating through the thickness, Kav captures the influence of stiffening cross-beams, Kμ determines the effect of longitudinal strains on the plastic transversal strains, Kv considers the stretching of the temperature field, Kc is the degree of excessive heat and Kw captures the effect of forced heat exchange. εF is the yield strain and θ is the proportionality factor between the heat input per unit length and the cross section of the weld. The axial shrinkage force Fx, eq. 1, is proportional to the width of the plastic zone: bPZ ¼

Fx : εm Eδ

ð3Þ

Here, εm is the averaged yield strain and δ is the plate thickness. The appropriate points of action zc are equal to the centre of the zone of plastic deformations. They are significantly influenced by the material and its properties as well as the heat input per unit length and the plate thickness. Depending on the points of action, an equivalent linear strain distribution or respectively stress distribution over the plates thickness can be calculated. Considering 1061

the point of origin in the centre of gravity of the plate’s cross section, the strain distribution ε(z) follows as: εðzÞ ¼ εm þ

12εm zc z: δ2

ð4Þ

2.3 The coupling procedure For the coupling of the analytical shrinkage force model with the FE simulation, different kinds of mechanical loads are available. The deformation state calculated by applying loads and appropriate points of actions or alternatively eccentric pressures matches well with experimental results [Stapelfeld 2016]. However, the calculated stress state in the structure is qualitatively wrong. Instead, specifying strains or stresses linearly distributed over the plates thicknesses, (Figure 2), leads to correct stresses with tension in the weld and balancing compressive stresses in the nearby regions. The procedure is already validated and verified with Ansys®, LS Dyna®, Sysweld® and Abaqus®. When loading the FE model with longitudinal and transversal stresses σ, the Poisson’s ratio ν must be considered: σx;y ¼

εx;y E þ εy;x E : 1  2

ð5Þ

2.4 Application of the hybrid model on real structures The hybrid model for calculating weld imperfections, mostly distortions, was already validated by means of numerous experimental welded butt and T-joints. The geometry, the material as well as the welding technique and welding parameters were varied during the experimental studies. The calculated distortions in the longitudinal and transversal direction as well as the bending and the angular distortions always showed a good agreement with the measured data (Stapelfeld 2016). For the demonstration of the feasibility of the suggested model, the calculation procedure is applied to a deck structure (20 m × 16 m) build from ship steel grade A (mild structural steel), (Figure 3). The structure is welded with 109 welds in three manufacturing steps. For each step measurements of the plane displacements at the plate edges were carried out. In the first manufacturing step, 24 deck beams with thicknesses of 6 and 7 mm are welded to a 5 mm thick base plate by Laser-MSG-Hybrid welding. The second manufacturing step involves the joining of perforated stringers by double-sided manual GMAW welding. To simplify, the perforation is modelled by reduced thicknesses of the stringers being 6.5 mm and 5.6 mm. In the

Figure 2.

Coupling procedure by means of linearly distributed strains (Stapelfeld 2016).

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Figure 3.

Sketch of a large welded ship structure.

Figure 4.

Scatter bands and averages of the experimental data compared with the calculations.

last manufacturing step, smaller parts and the walls are joined, again using manual GMAW welding. The process parameters for the welding processes were assumed from the welding procedure specifications provided. The investigated structure is almost symmetrical and thus the welding of each single beam as well as each single stringer has a similar effect on the measured welding distortions in longitudinal and transversal direction. Therefore, the results of the measurements alongside the seven paths in x direction and nine paths in y direction and the results of the hybrid calculation are summarized in scatter bands, (Figure 4). The real base plate is made up of several small plates welded together. The resulting stresses and strains are not documented, thus the calculated transversal distortions of manufacturing step 1 are too large. The results of the further steps shows a good agreement. 3 CALCULATION OF WELD IMPERFECTIONS OF THE CURVED STIFFENED STRUCTURE 3.1 The finite element model of the curved stiffened structure For the application of the hybrid model and a subsequent load capacity calculation, a panel similar to the panels of the Confluence bridge in France, introduced in [Tran 2014], was chosen, (Figure 5). For defining the boundary conditions, a cylindrical coordinate system was 1063

Figure 5.

Geometrical properties and boundary conditions of the curved panel.

created. The four edges of the panel are simple supported, uR ¼ 0. To restrain any movement or rotation of the structure, two nodes in the middle of the curved edges are fixed alongside θ and one node in the middle of the panel is fixed in the direction of the z coordinate. The steel grade is S355. Correspondingly, the Young’s modulus is E ¼ 210 GPa, Poisson’s ratio is  ¼ 0:3 and the yield strength is σF ¼ 355 MPa. A slope of E=100 is assumed for the hardening and the ultimate stress is assumed to be reached at 470 MPa. The structure is discretized with four node shell elements (S4R in Abaqus®) with three integration points over the elements thickness. A fine mesh with an approximated element edge length of 30 mm is used. Considering the different thicknesses, the panel and the stiffeners are loaded with different shell edge loads targeting a homogeneous axial pressure. For the verification of the loading as well as the boundary conditions, the curvature of the plate was removed and a load capacity calculation was carried out. The numerically calculated plastic normal force was about 27 MN and matches the analytical solution: F ¼ σF A ¼ 27:26 MN. For the determination of the lowest ultimate load, several load deformation calculations were then executed under the consideration of different buckle modes as well as combinations of them. Here, it turned out that the imperfection according to the first buckle mode leads to the lowest load capacity. 3.2 Application of the hybrid model for the calculation of weld imperfections The calculation of the mechanical loads with the analytical shrinkage force model requires information about the type of joint and its dimensions, the material data as well as the welding technique and the welding parameters. Here, the assumption is made that the stiffeners are only welded one-sided. The material data correspond to the material data of the steel S355J2. The welding technique is conventional MAG-welding with welding parameters targeting a fillet weld with a design throat thickness of about 6.6 mm, which resulted in a heat input per unit length of qs ¼ 2400 J/mm. In the example case, the linking between the analytical model and the subsequent numerical calculation was done by linearly distributed initial stresses, Table 1. For the proper loading of the FE model, two sections were created at each weld, representing the idealized zone of plastic deformations. The width of the sections is specified by the width of the analytically calculated plastic zone. When creating the single sections, the position of the welds is considered by an offset of the center of the idealized plastic zone of half of the stiffeners

Table 1. Analytically calculated stresses for the numerical calculation of weld imperfections. Curved panel Width of plastic pone bpz in mm Location of the integration points Longitudinal stresses σz in MPa Transversal stresses σ in MPa

Upper 849.6 1313

29.1 Middle 597.4 801

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Stiffener Lower 345.2 290

Upper 807.7 1341

29.1 Middle 593.8 789

Lower 379.9 237

Figure 6.

Assembly of the FE-Model in the region of the welds (a) and out of plane deformations (b).

Figure 7. Asymmetrical deformation state of weld scenario 1(a) and symmetrical deformation state of weld scenario 2 (b).

thickness, (Figure 6a). The initial stresses in longitudinal and transversal direction are defined as “initial state” in Abaqus®. Geometric nonlinear behavior is considered within the elastic calculation. The calculated deformation state indicates buckling of the curved panel in the positive z-direction and a tilt over of the stiffeners towards the welds, (Figure 6b). The calculation of the weld imperfections was done assuming two different positions of the welds. In the first scenario, all the welds were done on same side of the stiffeners which leads to an asymmetric deformation state, (Figure 7a). In the second scenario, all the left stiffeners are welded on the left side and the right ones on the right side, (Figure 7b). This weld sequence leads to a symmetrical deformation state.

4 RESULTS OF THE ULTIMATE LOAD CAPACITY CALCULATIONS The ultimate load calculations were done assuming three kinds of imperfections. The first case is a combination of the calculated geometrical and structural weld imperfections caused by weld scenario 1 and manufacturing tolerances captured by a global buckle of b=1000 ¼ 4:8 mm. In the second case, all imperfections in the structure are caused only by the eight welds being welded according to weld scenario 1. The same calculation is again done with weld scenario 2 as a third case. The asymmetric and symmetric initial deformation state leads to asymmetric and symmetric deformation states at ultimate load, (Figure 8). The comparison of the calculated ultimate loads with the results of a conventional calculation considering the scaled critical first buckling mode shows an increase in load capacity of 16 % up to 57 %, (Figure 8).

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Figure 8.

Out of plane deformation at ultimate load for weld scenario 1 (a) and weld scenario 2 (b).

Figure 9. Load displacements curves: three kinds of imperfections compared with the state of the art calculation.

5 CONCLUSIONS Numerous applications of the analytical numerical hybrid model for the calculation of weld imperfections indicate the significance of the results. In the case of consideration, the calculated weld imperfections correlate well with known empirical values. The use of calculated realistic weld imperfections instead of equivalent geometrical imperfections in the load capacity calculation of the stiffened curved panel results in an increasing of the ultimate load in the range of 16 % till 57 %. The approach gives the opportunity to analyze the influence of different weld scenarios on the load capacity. A general statement on the effects of weld imperfections in load capacity calculations cannot be given, because they are depending on numerous influencing factors. A realistic consideration will be only possible by the application of physically based approach of such type that has been presented in this paper. The future potential of this method is thus very high. However, the approach in combination with loading calculations has to be validated and calibrated by means of more experimentally determined load displacement curves. These works are ongoing at the moment. ACKNOWLEDGEMENTS These works are part of the IGF project No. 19173 BR of the German Research Association for Steel Application (FOSTA). This project is kindly funded by the German Federal Ministry of Economic Affairs and Energy (BMWi) by the AiF (German Federation of Industrial Research Associations) as part of the program for support of the Industrial Cooperative Research (IGF) on the basis of a decision by the German Bundestag.

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REFERENCES Beg, D., Kuhlmann, U., Davaine, L. et al. 2010. Design of Plated Structures, Ernst W. & Sohn Verlag, Berlin. Degée, H., Kuhlmann, U., Detzel, A. et al. 2008. Der Einfluss von Imperfektionen in dünnwandigen geschweißten Rechteckquerschnitten unter Druckbeanspruchung, Stahlbau 74 (4): 257–265. Duan, Y.G., Vincent, Y., Boilot, F. et al. 2007. Prediction of welding residual distortions of large structures using a local/global approach, Journal of Mechanical Science and Technologie 21 (10): 1700– 1706. Doynov, N., Michailov, V.G., 2018a. Distortion analysis of heat spot straightening thin-walled welded structures: part 1: formation of the plastic deformation zone, Int J Adv Manuf Technol, 94: 667–676. Doynov, N., Michailov, V.G., 2018b. Distortion analysis of heat spot straightening thin-walled welded structures: part 2: analytical-numerical approach’, Int J Adv Manuf Technol, 95: 469–478. Eurocode 3, 2017. Design and construction of steel structures (EN 1993-1-5). Gatovskii, K.M., Karkhin, V.A, 1980, Theoria svarochnih deformaciy i napriajeniy, LKI, Leningrad (in Russian). Ghavamia, K., Khedmatib, M.R. 2006. Numerical and experimental investigations on the compression behaviour of stiffened plates, Journal of Constructional Steel Research 62: 1087–1100. Kuzminov, S.A. 1974, Svarochnie deformacii sudoviech korpusniech konstrukcii. Sudostroenie, Leningrad (in Russian). Manco, T., Martins, J.P., Rigueiro, C. et al. 2016. Numerical Analysis of Stiffened Curved Panels under Compression, Eight International Conference on Steel and Aluminium Structures, Hong Kong. Michailov, V.G., Doynov, N., Stapelfeld, C. et al, 2011. Hybrid model for prediction of welding distortions in large structures’, Front. Mater. Sci., 5(2): 209–215. Michailov, V.G., Stapelfeld, C., Doynov, N. 2014. Upgrade of an analytic-numerical hybrid model for the distortion simulation of large structures, FOSTA VV mbH Düsseldorf (in German). Mun, H.S., Jang, C.D. 2011. Prediction of welding deformation of hull panel blocks using an advanced inherent strain analysis method considering the heat equivalent layer effect’, Met Mater Int 17(6): 993– 1000. Murakawa, H., Okumoto, Y., Rashed, S. et al. 2013. A practical method for prediction of distortion produced on large thin plate structures during welding assembly’, Weld World 57: 793–802. Okerblom, N.O 1955. Svarochnie napriajenia v metallokonstrukciiah, Moskau/Leningrad,Mashinostroenie (in Russian). Rykalin, N.N 1957. Berechnung von Wärmevorgängen beim Schweißen, VEB Verlag Technik, Berlin. Stapelfeld, C. 2016. Vereinfachte Modelle zur Schweißverzugsberechnung, PhD Thesis, Berichte des Lehrstuhls Füge- und Schweißtechnik der BTU Cottbus-Senftenberg, Band 10, Shaker Verlag, Aachen. Stapelfeld, C., Doynov, N. und Michailov, V. 2009. Hybride Berechnungsansätze zur Prognostizierung und Minimierung des Verzugs komplexer Schweißkonstruktionen. Sysweld Forum 2009: 91–105. Weimar. Thikomirov, D., Rietman, B., Kose, K. et al. 2008. Computing Welding Distortion: Comparison of Different Industrially Applicable Methods” SHEMET 11: 195–202. Tran, K.L., Douthe, C., Sab, K. et al. 2014. Buckling of Stiffened Curved Panels Under Uniform Axial Compression, Journal of Constructional Steel Research 103: 140–147. Ueda, Y, Kim, K., Yuan, M.G. 1989. A predicting method of welding residual stress using source of residual stress (report I), Transactions of JWRI 18 (1): 135–141. Ueda, Y, Yuan, M.G.1993. Prediction of residual-stresses in butt welded plates using inherent strains, Journal of Engineering Materials and Technology – Transactions of the ASME115 (4): 417–423.

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Plastic collapse loads of rectangular plate assemblies with constant and linear load distribution S. Stehr & N. Stranghöner Institute for Metal and Lightweight Structures, University of Duisburg-Essen, Essen, Germany

ABSTRACT: The storage content of a rectangular silo causes loads normal to its vertical walls which might lead to a plastic collapse of the structure in case of unfavourable design. The corresponding design standards are EN 1993-1-7 and EN 1993-4-1. In the frame of the current revision of EN 1993-1-7 it is envisaged to provide the designer with analytical expressions for the determination of the plastic collapse load of rectangular plate assemblies under transverse loads. For this reason, numerical investigations have been carried out at the Institute for Metal and Lightweight Structures of the University of Duisburg-Essen regarding the determination of plastic collapse loads of rectangular plate assemblies made of carbon steel. Herein, the plastic collapse loads were evaluated using the methods of the modified Southwell plot and the Convergence Indicator Plot. Within this contribution, first results are presented for a constant and a linear load distribution as the extreme load cases of the Janssen load distribution.

1 INTRODUCTION Silos can be built as with a circular or rectangular/square plan-form. Within this contribution, rectangular silos made of carbon steel are covered which have a square plan-form. Rectangular plate assemblies provide an easy mounting and are of advantage in case only a limited area is given. Silos may serve for storage of e.g. liquids or solids. However, depending on the storage content, the load distribution normal to the vertical walls of the silo differs. Loads caused by fluids may be represented by a distribution that corresponds to a linear slope over the filling height of the silo. In principle, the load resulting from a solid can be represented by the Janssen distribution as described for silos in EN 1991-4. The Janssen distribution is limited on the one side by a linear and on the other side by a constant load distribution, see Figure 1. These two limiting load cases are considered in this contribution. EN 1993-4-1 provides the principles for the structural design of steel silos of circular or rectangular plan-form. Besides this, EN 1993-1-7 should be used for the determination of the resistance of a silo that is loaded by out of plane actions. Herein, the design rules for unstiffened or stiffened plates as part of a plate assembly such as a silo are given. In this contribution, the ultimate limit state of plastic collapse (LS 1) is considered. EN 1993-1-7 defines this plastic collapse as the limit state where the plate assembly cannot longer resist loads without a plastic mechanism arises and furthermore, excessive plastic deformations occur. Hence, the maximum load that can be achieved is the plastic collapse load usually based on small deflection theory. In addition to a stress-based design by the Von Mises equivalent stress, supplementary rules for the design by global analysis are given. Using a materially nonlinear analysis, the load may be incrementally increased until the plastic collapse load is achieved. Additionally, internal stresses and deflections based on small deflection theory can be calculated for plates loaded by a uniform pressure or a central patch. In the frame of the current revision of EN 1993-1-7, numerical and analytical investigations have been carried out at the Institute for Metal and Lightweight Structures of the University of Duisburg-Essen regarding the determination of plastic collapse loads of rectangular plate assemblies made of carbon steel. The final objective is to derive analytical expressions for the 1068

Figure 1.

Linear, Janssen and constant load distribution in a rectangular silo made of a plate assembly.

determination of plastic collapse loads depending on the load distribution for implementation in EN 1993-1-7. Within this contribution, first numerical and analytical results are presented for constant and linear load distribution.

2 NUMERICAL MODEL 2.1 General All finite element investigations were carried out by materially nonlinear analyses (MNA) using four-node shell elements based on an ideal elastic-plastic stress-strain curve without any strainhardening defined by the following parameters: Young’s modulus E ¼ 210,000 N/mm², yield strength fy ¼ 235 N/mm² and Poisson’s ratio ν ¼ 0.3. From each FE simulation, firstly, (1) the maximum load factor was taken for the evaluation of the plastic collapse loads. Secondly, due to difficulties in accurately determining the largest load factor from the FE results, two additional methods have been applied: (2) modified Southwell plot (Holst et al. 2005) and (3) Convergence Indicator Plot (Doerich & Rotter 2011) whereby the two latter ones rely on (1), see details in section 3. 2.2 Load cases The simulated load cases result from possible storage loads on the vertical wall of a silo. As already mentioned, the first load case is a linear load distribution which represents liquids and the second load case is given by a constant load distribution. Thus, these two basic load cases cover the upper and lower limit of the Janssen pressure distribution which is usually used to determine storage loads caused by solids. Regardless of any geometrical conditions, the initial load p0 was chosen to 1⋅10-3 N/mm2 for each load case, see Figure 1. The self-weight of the plate assembly was considered in each simulation, although the effect of the self-weight on the results is not decisive. The resulting plastic collapse loads are given as a factor of the chosen load case. 2.3 Geometrical conditions Each numerical simulation was carried out using a rectangular plate assembly model consisting of four vertical rectangular plates each with the same thickness t1 ¼ t2 ¼ t3 ¼ t4 ¼ t, see Figure 1. Neither a top nor a bottom plate were considered. All plate assembly models have a square plan-form based on the horizontal dimensions d1 ¼ d2 and a height of d3.

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The first variable parameter is the height d3. Starting with a cubic plate assembly (d1 ¼ d2 ¼ d3), five different d3/d1-ratios with 0.5, 1.0, 1.5, 2.0 and 3.0 were investigated. Within these investigations, the dimension d1 always remained unchanged. The thickness t of the plates was set as the second variable parameter with d2/t-ratios from 50 to 500. 2.4 Boundary conditions Three different sets of boundary conditions according to the definitions of EN 1993-1-6 were investigated as illustrated in Figure 2. The first set of boundary conditions BC1f - BC1f includes BC1f on the bottom as well as on the top edges of the plate assembly. BC1f is defined by restrained displacements normal to the plate surface and in the plane of the plate surface but with free rotation. The second set of boundary conditions BC1f - BC2f differs from the first one in the chosen boundary condition on the top edges with BC2f (pinned), which is in contrast to BC1f defined by free displacements in the plane of the plate. The third set of boundary conditions BC1r - BC2f includes BC1r on the bottom edges and BC2f on the top edges of the plate assembly. BC1r (clamped) is defined by restrained displacements normal to the plate surface and in the plane of the plate surface and restrained rotation. In each case, the four vertical edges of the plate assembly were fixed to each other so that all internal forces could be transferred.

3 NUMERICAL DETERMINATION OF THE PLASTIC COLLAPSE LOAD FACTOR ξpl,FE In the frame of the numerical investigations, the load factor ξpl,FE has been determined in relation to the applied load p0 ¼ 1⋅10−3 N/mm2 describing the achievable load at which plastic collapse occurs. For this, first, numerical load factor-displacement curves were obtained from the finite element analyses at the point of extreme displacement at the height h and d2/2, see Figure 3a. Herein, the highest load factor is ξpl,FE. As not all load factor-displacement curves show a clear plateau, in some cases it is difficult to determine the largest load factor ξa ¼ ξpl,FE accurately. Due to a possible deviation in the accuracy of interpreting the results, the methods of the modified Southwell plot (MS) (Holst et al. 2005) and of the Convergence Indicator Plot (CIP) (Doerich & Rotter 2011) have been applied additionally. For the modified Southwell plot (MS) the gradient at each point of the load factordisplacement curve was plotted by ξa/δ2,x, see Figure 3b. This modification provides a clearer view on the elastic range and the plastic collapse load factor. The elastic range is presented by the vertical part on the right side followed by the elastic-plastic range in which the gradient ξa/δ2,x decreases. Where the gradient ξa/δ2,x reaches its minimum, the plastic

Figure 2.

Different sets of boundary conditions considered in the numerical investigations.

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Figure 3. Exemplary numerical load factor-displacement curve (a) and modified Southwell plot (b) for h/d3 ¼ 0.5, d2/d1 ¼ 1.0, d3/d1 ¼ 1.0, d2/t ¼ 500, BC1f - BC2f and constant load distribution.

plateau is indicated. The interception point of the tangent at this point with the vertical axis provides the value for the plastic collapse load factor ξpl,MS, see Figure 3b, which is more precise than the value of ξpl,FE. The Convergence Indicator Plot (CIP), illustrated in Figure 4a, is based on the previous explained modified Southwell plot. Following the MS-procedure, the value of ξpl,MS is determined for each load factor increment ξa. The CIP-curve indicates the plastic collapse load factor more accurate the closer ω, defined in Equation 1, approaches a value of zero.  ω¼

ξpl;MS  ξa ξa

½

ð1Þ

The interception point of the tangent at the lowest value of ξpl,MS with the vertical axis provides ξpl,CIP, see Figure 4b, which is the most precise result for the plastic collapse load factor. Values for ξpl,CIP were achieved for the constant and linear load distribution, for all three boundary condition sets, illustrated in Figure 2, for d3/d1-ratios from 0.5 to 3.0 and for d2/t-ratios from 50 to 500, always d2 ¼ d1, exemplary illustrated in Figure 5 for the constant load distribution, d3/d1-ratios of 1.0 and 2.0 and BC1f - BC2f.

Figure 4. Exemplary Convergence Indicator Plot for h/d3 ¼ 0.5, d2/d1 ¼ 1.0, d3/d1 1.0, d2/t ¼ 500, BC1f - BC2f and constant load distribution.

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Figure 5. Plastic collapse load factors ξpl,CIP for rectangular plate assemblies with BC1f - BC2f and constant load distribution.

4 ANALYTICAL EXPRESSIONS The final objective was to develop analytical expressions for the determination of the plastic collapse load for all geometries, boundary conditions, and load cases in the scope of the investigations. For this purpose, the numerically derived load factors ξpl,CIP have been used as the basis. For the development of the analytical expressions, in a first step, each load case and boundary condition have been treated independently. In a second step, a unified equation has been developed in which both load cases and three different boundary conditions are considered by simple factors. Exemplary, in the following, the stepwise procedure of the development of the analytical expression is presented for the constant load case and the boundary condition BC1f - BC2f. Based on the values of ξpl,CIP, a first simple equation has been approximated by using the least square method for each d3/d1-ratio to determine analytical load factors ξpl,calc, see Equation 2. In principle, the values of ξpl,calc are calculated conservatively compared to the values of ξpl,CIP. ξpl;calc ¼ k  106 

 2 t ½

d2

ð2Þ

where k is a factor depending on the d3/d1-ratio with k ¼ 3.9 for d3/d1 ¼ 0.5, k = 1.97 for d3/d1 = 1.0, k = 1.56 for d3/d1 = 1.5, k ¼ 1.39 for d3/d1 ¼ 2.0 and k ¼ 1.25 for d3/d1 ¼ 3.0. Based on these varying k-values, a more detailed equation has been approximated, again using the least square method, which covers all investigated d3/d1-ratios, see Equation 3: ξpl;calc ¼

 2  2 !  2 d3 d3 d1 t 6  10  k1 þ k2  þ k3  þ k4  ½

d1 d1 d3 d2

ð3Þ

where k1, k2, k3 and k4 are parameters depending on the boundary condition and load case, see Table 1. Equation 2 and Equation 3 are valid for S235 as the numerical simulation was based on fy ¼ 235 N/mm2. All variants of Equation 3 provide load factors ξpl,calc that are conservative compared to the values of ξpl,CIP obtained from the numerical analyses, see Figure 6. A statistical classification has been carried out by determining the mean value correction factor b as well as the estimated standard deviation sΔ according to EN 1990 on the basis of Equation 4 and Equation 5. Herein, all boundary conditions, geometrical variations and both load cases were considered.

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Table 1. Parameters k1, k2, k3 and k4 for the effect of boundary condition and load case on the plastic collapse load of a rectangular plate assembly. Boundary condition at each end

Parameters [-]

Load Case

Bottom end

Top end

k1

k2

k3

k4

Constant

BC1f BC1f BC1r BC1f

BC1f BC2f BC2f BC2f

BC1r

BC2f

1.53 1.53 1.47 3.0 3.02 3.37 3.03

−0.215 −0.215 −0.165 −0.546 −0.566 −0.782 −0.475

0.0322 0.0322 0.0227 0.0453 0.0638 0.0875 0.0418

0.6161 0.6161 0.9 1.19 1.19 1.68 1.89

Linear

d2/t 5 58.8 d2/t  58.8 d2/t 5 58.8 d2/t  58.8

Figure 6. Correlation of the load factors ξpl,CIP and ξpl,calc for rectangular plate assemblies with BC1f - BC2f. n  P

b ¼ i¼1

ξpl;CIP;i  ξpl;calc;i

n  P

ξpl;calc;i



2

ð4Þ

i¼1

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n  X 2 1 sΔ ¼ Δi  Δ  n  1 i¼1 where n is the number of ! numerical results, ξpl;CIP;i Δi ¼ ln is the logarithm of the error term and b  ξpl;calc;i Δ ¼ 1n 

n P

Δi is the estimated mean value of Δi.

ð5Þ

(6) (7)

i¼1

On this basis, for both load distributions, the mean value correction factor has been determined to b = 1.01 and the estimated standard deviation to zero which are extraordinary good results. 1073

Exemplary curves of the analytically calculated plastic collapse load factors ξpl,calc based on Equation 3 are illustrated in Figure 7 comparing both load distributions and Figure 8 comparing different boundary conditions. The values for the plastic collapse load factors decrease with increasing d2/t-ratios and increasing d3/d1-ratios. Furthermore, the plastic collapse load factors for the linear pressure distribution are much higher than for the constant load distribution, see Figure 7. The boundary conditions BC1f - BC1f and BC1f - BC2f yield to identical results for the constant load case, see Figures 7 and 8. However, the plastic collapse load factors for BC1r BC2f are slightly higher than those for BC1f - BC1f or BC1f - BC2f as shown in Figure 8.

5 PROPOSED DESIGN CONCEPT Based on Equation 3, which is approximated by the least square method, an analytical expression for the determination of the plastic collapse load can be derived for design against the plastic failure limit state (LS1). As the design is based on pressure values, Equation 3 has to be generalized taking into account the initial load p0, for which the load factors have been determined, to provide an analytical expression for the characteristic plastic reference resistance ppl,a,Rk for a rectangular plate assembly with d1 = d2 and variable yield strengths. Equation 8 is the proposed expression for the characteristic plastic reference resistance:

Figure 7. Plastic collapse load factors ξpl,calc for rectangular plate assemblies with BC1f - BC2f comparing different load distributions.

Figure 8. Plastic collapse load factors ξpl,calc for rectangular plate assemblies with constant load distribution comparing different boundary conditions.

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 2  2 !  2   fy d3 d3 d1 t 3  10  ppl;a;Rk ¼ k1 þ k2  þ k3  þ k4   d1 d1 d3 235 d2  2  2 ! 4:26  t2  fy d3 d3 d1  þ k4  , ppl;a;Rk ¼ k1 þ k2  þ k3  d1 d1 d3 d22

ð8Þ

with k1 to k4 according to Table 1. Herein, all investigated d3/d1-ratios, boundary conditions and both constant and linear load distributions are covered. Subsequently, the design value of the plastic reference resistance ppl,a,Rd of a rectangular plate assembly of square plan-form with γM0 ¼ 1.0 for the plastic failure limit state (LS1) results to: ppl;a;Rd ¼

ppl;a;Rk γM0

ð9Þ

6 SIMPLIFIED ANALYTICAL MODEL A simplification of Equation 8 is possible by relying the analytical expression on the plastic collapse of an individual rectangular plate. Herein, the boundary conditions of this reference plate are not of interest, but the correlation to the overall behaviour of the plate assembly. Exemplary, the simplified analytical model can be based on a reference plate with d2 ¼ d3 ¼ 1000 mm and boundary conditions BC1f on all four edges. The characteristic value of the plastic reference resistance of this individual rectangular reference plate is presented by Equation 10, which is based on the expression for internal stresses of plates according to EN 1993-1-7: ppl;p;Rk ¼ κ 

fy  t2 d2  d3

ð10Þ

where κ is a value depending on the load case with κ 5.9 for constant and κ ¼ 11.325 for linear load distribution. Based on Equation 10 the characteristic value of the plastic reference resistance of a rectangular plate assembly with d1 ¼ d2 can be determined by Equation 11: ppl;a;Rk ¼ β  ppl;p;Rk

ð11Þ

where β is the plastic reference resistance factor for a rectangular plate assembly, see Table 2. Herein, the values of β result from simple correlations of the analytically determined plastic

Table 2. Plastic reference resistance factors β for a rectangular plate assembly of square plan-form. Boundary condition at each end

β [−]

Load Case

Bottom end

Top end

d3/d1 = 0.5

1.0

1.5

2.0

3.0

Constant

BC1f BC1f BC1r

BC1f BC2f BC2f

2.81 2.81 3.6

1.42 1.42 1.61

1.12 1.12 1.21

1.0 1.0 1.05

0.9 0.9 0.92

Linear

BC1f

BC2f

BC1r

BC2f

2.82 2.81 3.65 3.89

1.39 1.39 1.64 1.69

1.06 1.07 1.22 1.18

0.9 0.92 1.02 0.97

0.71 0.76 0.82 0.75

d2/t 5 58.8 d2/t  58.8 d2/t 5 58.8 d2/t  58.8

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collapse load factors ξpl,a,calc of a rectangular plate assembly with the analytically determined plastic collapse load factors ξpl,p,calc of the individual rectangular reference plate, see Equation 12. β¼

ξpl;a;calc ξpl;p;calc

ð12Þ

7 CONCLUSIONS The design of rectangular silos in the LS 1 can be simplified using analytical expressions for the determination of plastic collapse loads. In this contribution, newly developed analytical equations are presented which have been derived on the basis of numerical simulations for constant and linear load distribution for rectangular steel silos of square plan-form. Further investigations into the Janssen pressure distribution are currently carried out at the Institute for Metal and Lightweight Structures at University of Duisburg-Essen in order to generate general analytical expressions for implementation in the ongoing revision of EN 1993-1-7. ACKNOWLEDGEMENTS The research activities were supported by the work and comments of the CEN TC250 SC3 Project Team PT 05 “Shells and Related Structures”. Special thanks apply to Prof. J. Michael Rotter (PT 05 Leader) and Chris J. Brown (PT 05 Member). REFERENCES Doerich, C. & Rotter, J.M. 2011. Accurate Determination of Plastic Collapse Loads From Finite Element Analyses. Edinburgh: Institute for Infrastructure & Environment. EN 1990:2002 + A1:2005 + A1:2005/AC:2010. Eurocode: Basis of structural design. EN 1991-4:2006. Eurocode 1: Actions on structures – Part 4: Silos and tanks. EN 1993-1-6:2007 + AC:2009 + A1:2017. Eurocode 3: Design of steel structures – Part: 1-6: Strength and Stability of Shell Structures. EN 1993-1-7:2007 + AC:2009. Eurocode 3: Design of steel structures – Part: 1-7: Plated structures subject to out of plane loading. EN 1993-4-1:2007 + AC:2009 + A1:2017. Eurocode 3: Design of steel structures – Part 4-1: Silos. Holst, J.M.F.G., Doerich, C. & Rotter, J.M. July 2005. Accurate determination of the plastic collapse loads of shells when using finite element analyses. Proc., Fourth International Conference on Advances in Steel Structures, ICASS’05: pp. 1798-1794. Shanghai.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Stability of axially compressed cylindrical shells made of stainless steel for different imperfection patterns N. Stranghöner & E. Azizi Institute for Metal and Lightweight Structures, University of Duisburg-Essen, Essen, Germany

ABSTRACT: The major difference between shells made of carbon steel and those made of stainless steel is the nonlinear stress-strain behaviour of stainless steels below the 0.2 % proof stress whereas carbon steel shows a typically linear elastic behaviour up to the yield stress. This nonlinear stress-strain behaviour of stainless steel effects the buckling resistance of medium slender shells. Furthermore, the shape of the imperfection has a significant influence on the buckling resistance of shells. For these reasons, finite element investigations have been carried out validated by experimental tests of axially loaded shells. Hereby, the parameters material nonlinearity and imperfection pattern have been considered with the imperfection types (i) weld depression pattern type A, (ii) sinusoidal (multiwave) imperfection and (iii) measured imperfections from test samples for shell buckling tests. This paper presents the results of these investigations.

1 INTRODUCTION Stainless steel shell structures are widely used in industrial applications with their key advantages in corrosion resistance and durability. The load carrying behaviour of stainless steel shell structures in pre- and post-buckling patch differs from those made of carbon steel due to their different material characteristics (Hautala & Schmidt 1999; Gorbachov et al. 2017; Stranghöner et al. 2019). While carbon steel can be characterised by a bilinear (elastic, perfectly plastic) stress-strain response, stainless steel is attributed by a rounded (nonlinear) stress-strain behaviour below the 0.2 % proof stress with no sharply defined yield point. EN 1993-1-6 as the current European standard for the design of shell structures includes design rules which were developed explicitly for carbon steel shell structures. Herewith, it does not provide any specific design rules for the determination of the buckling resistance of cylindrical shells made of stainless steel under meridional loading, but simply adopts the reduced tangent modulus method which leads to an over-conservative design for slender and stocky shells. Thin-walled shell structures most commonly fail by buckling under meridional compression (Rotter 2004). Hautala & Schmidt (1998, 1999) already proposed slenderness and temperature dependent buckling correction factors as a more efficient design approach specifically for medium slender shells made of austenitic stainless steel under axial compression. Further experimental, numerical and theoretical investigations into the shell buckling behaviour of meridionally loaded duplex and ferritic stainless steel shells were carried out in the frame of the RFCS research project BiogaSS (Thomas et al. 2017; Stranghöner & Gorbachov 2015; Gorbachov et al. 2017). The objective was to develop buckling correction factors for austenitic, duplex and ferritic stainless steel according to EN 1993-1-4 considering the different fabrication tolerance classes (FQCs) A to C according to EN 1993-1-6. For the numerical simulation of the experimental tests of BiogaSS, sinusoidal multiwave, single inward and outward imperfection patterns as well as measured imperfections of the test samples were considered. In these studies, it could be shown that multiwave imperfection pattern reflects the experimental behaviour of the tested shells best. On this basis, recent numerical investigations into the effect of the nonlinear material behaviour below the 0.2 % proof stress have been performed which showed that the imperfections of the tested shells partly superimposes the effect of the material nonlinearity on the buckling resistance so that the decrease 1077

due to the material nonlinearity became less with larger imperfections (Stranghöner et al. 2019). Especially, this behaviour becomes obvious for FQC C with the largest imperfections. However, the axisymmetric weld depression pattern type A as proposed by Rotter & Teng (1989) was not included in this study. This imperfection pattern is widely used as a suitable imperfection shape for exploring the imperfection sensitivity of cylindrical shells dominated by meridional compression. Herewith, the focus of the presented study is on the comparison of the effects of the weld depression pattern type A with the already investigated sinusoidal (multiwave) imperfection pattern as well as measured imperfections on the buckling resistance of meridionally loaded shells by numerical simulation of the experimental shell buckling tests of BiogasSS (Thomas et al. 2017).

2 IMPERFECTION SENSITIVITY OF AXIALLY COMPRESSED SHELLS Initial geometric imperfections are introduced into real shell structures (e.g. silos, tanks) during the fabrication or the manufacturing process. Since the real failure behaviour of shell structures under axial compression is strongly sensitive to imperfections, it is of exceptional importance to incorporate the imperfections into the design process and herewith into numerical simulation. EN 1993-1-6 offers two types of nonlinear numerical analyses taking imperfections into account, GNIA and GMNIA, either ignoring or including the influence of material nonlinearity. However, the selection of a suitable imperfection pattern is one of the biggest challenges for designers, because the imperfection sensitivity itself depends on the imperfection shape and this shape can be changed independently of the boundary and load conditions as well as the shape of the shell (Jansseune 2016). A huge number of theoretical (e.g. Koiter 1945, Tennyson & Muggeridge 1969, Hutchinson et al. 1971) and numerical (e.g. Rotter & Teng 1989; Schmidt & Winterstetter 2004) investigations were carried in the past, aiming to study the effect of various imperfection patterns on the buckling resistance of thin-walled shells. Since none of these studies were able to propose or identify a universal applicable imperfection pattern (Jansseune 2016), three main philosophies for classification and selection of an imperfection shape were proposed firstly by Schmidt (2000) and subsequently in the Buckling of Steel Shells European Design Recommendations (Rotter & Schmidt 2013) as follows: • Most realistic imperfection shape: geometrical imperfections are considered in the FE models as “realistic” as possible, whereby the deviation from the ideal-perfect shape of real shell structure (Arbocz 1982; Song 2004; Coleman 1992; Teng et al. 2005) or of laboratory shells should be measured for implementation into the numerical modeling. This approach is not very common especially for practical use, as the design is usually carried out before erection of a shell structure. Furthermore, it is highly time and cost consuming to extract measured data and to apply them meaningfully in a numerical model. • Worst possible imperfection shape: the imperfection shape can be adopted in simulations in principle in such a way that shell buckling failures are reduced as much as possible to provide a safe lower bound for design. However, several analyses with different shapes must be performed to compute and find the “worst possible” geometrical shape due to the fact that the worst imperfection form depends on the shell geometry and loading, see Koiter (1963), Greiner & Derler (1995). The application of this approach yields to an overconservative buckling strength, which cannot be observed in practice and is not addressed as an economical solution for stability problems of thin-walled shells, see the historical background of shell stability design rules in Schmidt (2018), Rotter (2017) and Rotter & Schmidt (2013). • Simple equivalent imperfection shape: A simple “equivalent” shape can be implemented into numerical shell models, which is perhaps not completely identical to the realistic imperfection pattern, but which is able to reduce the failure loads relatively in the same manner as realistic imperfections. An equivalent imperfection pattern can either be extracted from the failure modes of linear (e.g. Koiter 1945, Yamaki 1984) or nonlinear (e.g. Esslinger et al. 1972, Schneider 2005) analyses considering the pre- and postbuckling region, or can be assumed as a simple specific shape, e.g. sinusoidal (multiwave) or axisymmetric weld 1078

depression, which is representative for the surface deviations of real shell structures during fabrication or execution. The last approach has been widely used as the most suitable device to predict realistic failure loads by means of FE analyses. In these studies, the focus was more on slender shells and their imperfection sensitivity under meridional compression as they fail elastically and their stability is dominated by imperfections. However, covering shells made of stainless steel, additional attention is necessary regarding medium slender and slightly stocky shells, which fail by elastic-plastic buckling and where the consideration of the material nonlinearity in the design process of stainless steel shell structures is significant.

3 NUMERICAL MODELLING OF EXPERIMENTRAL STAINELSS STEEL SHELLS OF BIOGASS 3.1 Introduction In the frame of the presented study, the experimental shell buckling tests of BiogaSS were simulated numerically using GMNIA analyses considering the three aforementioned imperfection shapes: (i) weld depression pattern type A, (ii) the sinusoidal (multiwave) imperfection pattern and (iii) measured imperfections. In total, 12 shell buckling tests were carried out under meridional compression. The test samples were made of duplex (1.4462) and ferritic (1.4521) stainless steel (Thomas et al. 2017, Gorbachov et al 2017, Stranghöner & Gorbachov 2015). The finite element program ANSYS APDL v18.1 has been used for the numerical analyses. Assuming that a bifurcation point would stay undetected by applying an arc-length method for nonlinear analyses (Riks 1979), it was required to ensure that the first reported negative eigenvalues of the global stiffness matrix were detected in all geometrically nonlinear analyses. Thus, the nonlinear equilibrium path near failure was automatically followed, aiming to avoid that no bifurcation point with a negative global or local stiffness matrix stays undetected. 3.2 Geometry, material and boundary conditions The geometries of the tested shells varied between diameters of 300 mm and 400 mm, wall thicknesses of 0.5/0.6 mm, 0.6 mm, 1 mm and 3 mm, a shell height of 350 mm and resulting r/t-ratios between 50 and 400 (nominal dimensions), see Thomas et al. (2017), Stranghöner & Gorbachov (2015). All numerical simulations were carried out considering the measured geometrical dimensions and the measured true stress-train curves of the stainless steel plate materials which were used for fabrication of the test shells (Thomas et al. 2017). Simply supported boundary conditions of BC1f (all displacements restrained, but free to rotate about the circumferential axis) and BC2f (free to displace meridionally and rotate about the circumferential axis) were assumed for the bottom and top edge of the modelled cylindrical shells with BC1f and BC2f as defined in EN 1993-1-6. 3.3 Meshing details For the determination of the optimal mesh size, preliminary studies were performed on perfect cylindrical shells using a linear bifurcation analysis (LBA) as well as a geometrically nonlinear elastic analysis (GNA). For the justification of the element density, a mesh refinement has been progressively applied in the perfect length of the modeled shells, so that the LBAs or/and GNAs provide approximately same results (max. difference 1 %). As a significant higher mesh resolution is required to capture the local curvatures of a weld depression accurately, a mesh refinement (20time finer) was adopted in the vicinity and the local weld depression range. Four-node shell elements 181 with six degrees of freedom at each node were selected, whereby the element lengths in the perfect range of the shell in longitudinal direction was connected to the classic buckling halfwave and varied between 0.14 λcl to 0.4 λcl for r/t-ratios from 50 to 150 with 1079

λcl ¼

pffiffiffiffi π rt 0:25

½12ð1  2 Þ

pffiffiffiffi ≈ 1:728 rt

for  ¼ 0:3

ð1Þ

where λcl = classical axisymmetric buckle half-wavelength; υ = Poisson’s ratio, r = radius and t = thickness of the shell. 3.4 Imperfection shape EN 1993-1-6 gives only a general guide on the shape of imperfections. A sufficient number of imperfection patterns (eigenmode affine, weld depression, collapse affine, post-buckling affine, etc.) should be considered in a nonlinear buckling analysis of imperfect shells so that the worst case can be identified. The definition of the imperfection shape in a unique and repeatable manner is difficult, so that the judgement of the final calculation outcomes based on nonlinear buckling analyses for an imperfect shell (e.g. GMNIA) can be considerably complicated. Because of the above-mentioned reason, the eigenmode affine, collapse affine and post-buckling affine patterns, which can be obtained from an LBA, GNA and GMNA analysis respectively, were not considered in this numerical study. Since all other mentioned imperfection patterns seem to be suitable candidates for the nonlinear buckling analysis of real imperfect shell structures, the already mentioned imperfection shapes (i) weld depression pattern type A of (Rotter and Teng 1989), (ii) sinusoidal (multiwave) pattern and (iii) real measured imperfections of the buckling test specimens were considered. The initial amplitude and gauge length (wavelength) of the imperfections for the weld depression pattern type A and sinusoidal imperfection pattern were assumed as defined in EN 1993-1-6. EN 1993-1-6 distinguishes between three FQCs A to C with FQC A representing the least and FQC C the worst imperfections. The imperfections should be considered depending on the existing stress state which is in this case meridional compression. As the experimentally investigated cylindrical shells were almost perfect in the meridional direction due to their manufacturing process including subsequent rolling, FQC A was chosen for the numerical investigations for defining the size of imperfection for weld depression type A and sinusoidal imperfection. Besides this, additional numerical investigations have been carried out for FQCs B and C for weld depression type A and sinusoidal imperfection.

4 COMPARISON OF EXPERIMENTAL AND NUMERICAL BUCKLING STRENGTHS Exemplary, experimental and numerical relative load-deformation diagrams of the test specimens 150-3-D1 and 150-1-D2 are given in Figure 1 considering all three kind of investigated imperfections. Both shells were made of duplex stainless steel 1.4462. Test sample 150-3-D1 is described by a radius of r =151.04 mm, a wall thickness of t = 3.022 mm, a height of h = 349.31 mm, a r/ t-ratio of about 50, a relative slenderness of
λx = 0.56 and a measured 0.2 % proof stress of fy,exp,t-m = 789 N/mm2 in the transverse direction in the middle of the plate (measured values). Test sample 150-1-D2 is described by a radius of r =150.23 mm, a wall thickness of t = 1.068 mm, a height of h = 350.15 mm, a r/t-ratio of about 150, a relative slenderness of
λx = 0.91 and a measured 0.2 % proof stress of fy,exp,t-m = 759 N/mm2 in the transverse direction in the middle of the plate (measured values). From Figure 1 it becomes obvious that the numerical simulations based on all three kind of imperfection types simulate the ultimate experimental loads relatively good. Nevertheless, for both shells, the simulations based on the weld depression type A imperfection slightly overestimate the ultimate load and the sinusoidal imperfection leads to the lowest ultimate loads comparable to the experimental ones. Shell 150-3-D1 has a relative slenderness of 0.56 where an influence of the material nonlinearity on the buckling strength is given (Stranghöner et al. 2019). Even the simulation with measured imperfections yields to a higher ultimate load for shell 150-3-D1. This can be explained by the fact that using measured geometric 1080

Figure 1. Experimental and numerical relative load-deformation curves for the two duplex test specimens 150-3-D1 and 150-1-D2 with r/t = 50 and 150 respectively.

imperfections, material imperfections are not considered. Material imperfections are residual stresses caused by rolling, pressing, welding, straightening etc., inhomogeneities and anisotropies etc. EN 1993-1-6 specifies geometric imperfections for GMNIA and different FQCs which consider both geometric and material imperfections. Shell 150-1-D2 with a relative slenderness of 0.91 is more slender; in this case, the influence of the material nonlinearity is not significant, the shell behaves elastic-plastically in the transition range to elastic failure. In a first step, from Figure 1 it can be concluded that the sinusoidal imperfection type meets the ultimate loads best. Furthermore, the experimental load-deformation paths are not met by the numerical simulations as the numerical deformations are always smaller than the experimental ones. Figure 2 shows a diagram in which the relative experimental and numerical results are compared for all 12 test samples considering FQC A and measured imperfections. Table 1 summarizes the mean value correction factor b and estimated standard deviation sΔ which have been determined according to EN 1990 with n   P Fx;exp  Fx;FEM

b¼

i¼1

n  2 P Fx;FEM

ð2Þ

i¼1

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n  X 2 1 sD ¼ Di  D  n  1 i¼1

ð3Þ

where n is the number of experimental results,   Fx;exp Di ¼ ln b  Fx;FEM n 1 X D¼  Di n i¼1

is the logarithm of the error term and

ð4Þ

is the estimated mean value of Di :

ð5Þ

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Figure 2. Experimental and numerical relative buckling strengths based on FQC A and measured imperfections.

Table 1. Statistical evaluation acc. to EN 1990 for numerical prediction of ultimate relative buckling strengths based on imperfections for FQC A and measured imperfections. Imperfection pattern

Mean value correction factor b [-]

Estimated standard deviation sΔ [-]

Weld depression type A Sinusoidal (mulitwave) imperfection Measured imperfection

0.94 1.01 0.96

0.125 0.132 0.145

Both parameters, the mean value correction factor b and estimated standard deviation sΔ, provide values of a comparable order of magnitude. The resulting mean value correction factors show that a sufficient correlation can be achieved between the actual resistances of the tested shells and the numerical predictions by adopting the considered imperfection patterns. On the basis of these results, neither of the two investigated imperfection types can be claimed to be the preferred one for the simulation of the experimental tests; on the contrary, both seem to be equally well suited. Nevertheless, a mean value correction factor smaller 1 demonstrates that the buckling strengths achieved by numerical simulation are not conservative. This means that using a weld depression type A as an imperfection pattern for FE simulations might yield to slightly unconservative values at least in the presented study for the experimentally tested shells. Figure 3 presents the buckling correction factors ψ as defined in Hautala & Schmidt (1998, 1999) and Rotter & Schmidt (2013) as the ratio of two numerically determined buckling strengths using the measured nonlinear stress-strain curve of the investigated stainless steel shells and using an ideal elastic-plastic stress-strain response on the basis of the measured 0.2 % proof stress. The resulting buckling correction factors ψ show that the effect of the material nonlinearity on the load capacity of duplex and ferritic stainless steel shells is reduced by increasing the influence of imperfections from FQC A to C, see also Stranghöner et al. (2019). Furthermore, it can be seen that there is nearly no influence of the imperfection type on the buckling correction factor visible. Weld depression type A shows slightly smaller buckling corrections factors compared to the sinusoidal imperfection but with deviations of about only 1082

Figure 3. Influence of the imperfection type on the buckling strength considering a bilinear or multilinear stress-strain relationship for FQC A to C according to EN 1993-1-6.

max. 0.3 %. Together with the fact that the sinusoidal imperfection yields to smaller ultimate buckling strengths compared to the weld depression type A, the influence of the imperfections pattern on the buckling correction factor has to be taken as fully negligible. Besides this, the numerical ultimate loads of cylindrical shells with sinusoidal imperfection pattern are more sensitive to wavelength and amplitude variations. Based on these results, it can be concluded that the buckling resistance of cylindrical shells with weld depression type A is slightly more sensitive to the material nonlinearity in comparison to that with sinusoidal imperfection pattern although the effect is – for design purpose – negligible. A comparison between the relative experimental and numerical failure loads taking into account FQC A to C for all investigated shells is plotted in Figure 4. Both diagrams show that FQC A meets mostly the experimental results best and FQC C yields to considerably lower ultimate buckling strengths especially for those shells with higher relative buckling strengths. On the basis of the presented study it can be concluded that in general both imperfection types, weld depression type A and sinusoidal imperfection according to EN 1993-1-6, are suited to simulate at least the experimental shell tests with a preference to the sinusoidal imperfection as this type of imperfection leads to a slightly better fit. Nevertheless, this study does not give an answer to the question which imperfection type reflects best real shell structures with their tolerances due to fabrication and erection. But it confirms the use of the weld depression type A as a suitable imperfection in principle which is assumed to reflect best the imperfections in real structures, see e.g. Teng & Rotter (1992).

5 CONCLUSIONS The main objective of the presented study was to evaluate the effect of the imperfection types weld depression type A and sinusoidal imperfection on the numerical determination of buckling strengths of meridionally loaded shells made of stainless steel using GMNIA analyses. For this reason, 12 experimental shell tests made of duplex and ferritic stainless steel have been used as a basis for simulation taking into account their measured true stress-strain curves and three different types of imperfections: measured imperfections, weld depression type A and sinusoidal imperfection, the latter ones according to EN 1993-1-6 for the relevant FQC A and furthermore, also for FQCs B and C. 1083

Figure 4.

Experimental and numerical relative buckling strengths for different imperfection types.

Generally, both imperfection types according to EN 1993-1-6 are well suitable for simulating at least the experimental tests with a small preference of the sinusoidal imperfection as it yields to a slightly better fit looking at the mean value deviation. As the weld depression type A is commonly assumed to reflect best the imperfections in real structures, it can be concluded on the basis of this study, that this imperfection pattern can be used for the numerical determination of the buckling strength of meridionally loaded shells made of stainless steel. However, it has to be kept in mind that weld depression type A imperfection might yield to higher ultimate buckling strengths than the sinusoidal imperfection. REFERENCES ANSYS Mechanical APDL, Version 18.1. Arbocz, J. 1982. The imperfection data bank, a mean to obtain realistic buckling loads. In Buckling of Shells, Springer: 535–567. Coleman, R., Ding, X., Rotter, J.M. 1992. Measurement of imperfections in full-scale steel silos, In 4th International Conference on Bulk Materials 2 (92(7)): 467–472. Storage, handling and Transportation Seventh International Symposium on Freight Pipelines. Institution of Engineers: Australia. EN 1990:2002 + A1:2005 + A1:2005/AC:2010, Eurocode – Basis of structural design. EN 1993-1-4:2006 + A1:2015, Eurocode 3: Design of steel structures – Part 1-4: General rules- Supplementary rules for stainless steels. EN 1993-1-6:2007 + AC:2009 + A1:2017, Eurocode 3: Design of steel structures – Part 1-6: Strength and stability of shell structures. Esslinger, M., Geier, B. 1972. Gerechnete Nachbeullasten als untere Grenze der experimentellen axialen Beullasten von Kreiszylindern. Der Stahlbau 41 (12): 353–360. Gorbachov, A., Stranghöner, N., Rotter, J.M. 2017. Buckling behaviour of axially compressed cylindrical shells made of stainless steel, EUROSTEEL 2017, 13-15 September 2017: 828–837. Copenhagen: Den-mark. Greiner, R., Derler, P. 1995. Effect of imperfections on wind-loaded cylindrical shells. Thin-Walled Structures 23 (1-4): 271–281. Hautala, K.T., Schmidt, H. 1998, Buckling Tests on Axially Compressed Cylindrical Shells Made of Various Austenitic Stainless Steels at Ambient and Elevated Temperatures. Universität – GH Essen, Forschungsbericht aus dem Fachbereich Bauwesen 76. Universität – GH Essen.

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Hautala, K.T., Schmidt, H. 1999, Buckling of axially compressed cylindrical shells made of austenitic stainless. Light-Weight Steel and Aluminum Structures: 233–240. Amsterdam/New York: Elsevier Science 1999. Hutchinson, J.W., Tennyson, R.C. Muggerldge, D.B. 1971. Effect of a Local Axisymmetric Imperfection on the Buckling Behavior of a Circular Cylindrical Shell under Axial Compression. In Reprinted from American Institute of Aeronautics and Astronautics (AIAA) Journal 9 (1): 48–52. Jansseune, A., De Corte, W., Belis, J. 2016. Imperfection sensitivity of locally supported cylindrical silos subjected to uniform axial compression. International Journal of Solids and Structures 96: 92–109. Koiter, W.T. 1945, On the Stability of Elastic Equilibrium (in Dutch). Ph. D. Thesis: Delft University. Koiter, W.T. 1963. The effect of axisymmetric imperfections on the buckling of cylindrical shells under axial compression. In Proc Koninklijke Nederlandse Akademie van Wetenschappen 66: 265–279. Riks, E. 1979. An incremental approach to the solution of snapping and buckling problems. In International Journal of Solids and Structures 15: 529–551. Rotter J.M., Teng J.G., 1989, Elastic stability of cylindrical shells with weld depression. ASCE Journal of Structural Engineering 115 (5): 1244–1263. Rotter J.M. 2004, Cylindrical shells under axial compression. Buckling of Thin Metal Structures: 42–87, In J.G. Teng and J.M. Rotter (ed.). Spon: London. Rotter, J.M., Schmidt, H. 2013 (eds.). Buckling of steel shells, European design recommendations, 5th Edition, Revised 2nd impression. ECCS - European Convention for Constructional Steelwork. Rotter, J.M. 2017. Shell buckling transformed: mechanics, design processes and their interrelation. Stahlbau 86 (4): 315–324. Schmidt, H. 2000. Stability of steel shell structures: general report. Journal of Constructional Steel Research 55 (1-3):159–181. Schmidt, H., Winterstetter, T.A. 2004. Cylindrical shells under combined loading: axial compression, external pressure and torsional shear. Buckling of Thin Metal Shells. IN J.G. Teng and J.M. Rotter (ed.). Spon: London. Schmidt, H. 2018. Two decades of research on the stability of steel shell structures at the University of Essen (1985–2005): Experiments, evaluations, and impact on design standards, Advances in Structural Engineering: Special Issue for Professor Rotter: 1–29. Schneider, W., Timmel, I., Höhn, K., 2005. The conception of quasi-collapse-affine imperfections: A new approach to unfavourable imperfections of thin-walled shell structures. Thin-Walled Structures 43: 1202–1224. Song, C.Y., Teng, J.G., Rotter, J.M. 2004. Imperfection sensitivity of thin elastic cylindrical shells subject to partial axial compression, International Journal of Solids and Structures 41: 7155–7180, December 2004. Stranghöner, N., Gorbachov, A. 2015. Experimentelles Tragverhalten axialgedrückter Kreiszylinderschalen aus nichtrostenden ferritischen und Duplex-Stählen (Experimental load bearing behaviour of axially compressed cylinders made of ferritic and duplex stain-less steels), Stahlbau 84 (4): 275–284. Stranghöner, N., Azizi, E., Gorbachov, A. 2019, Influence of material nonlinearity on the buckling resistance of stainless steel shells. Journal of Constructional Steel Research, accepted for publication in February 2019 (in press). Teng, J.G., Rotter, J.M. 1992. Buckling of presurrized axisymmetrically imperfect cylinders under axial loads, Journal of Engineering Mechanics 118 (2): 229–247, February 1992. Teng, J.G., Lin, X., Rotter, J.M., Ding, X.L., 2005. Analysis of geometric imperfections in full-scale welded steel silos, Engineering Structures 27 (6): 938–950. Tennyson, R.C., Muggeridge, D.B. 1969. Buckling of Axisymmetric Imperfect Circular Cylindrical Shells under Axial Compression. In American Institute of Aeronautics and Astronautics (AIAA) Journal 17, (11):2127–2131. Thomas, E., Soccol, D., Romero Barragan, M., Matres, V., Ohligschläger, T., Säynäjäkangas, J., Stranghöner, N., Gorbachov, A., Stehr, S., Brunstermann, R., Brinkmann, B., Widmann, R., Baddoo, N., Aggeloppoulos, E., Tholen, R. 2017. In: Innovative and competitive solutions using SS and adhesive bonding in biogas (BiogaSS); Final Report, RFSR-CT-2012-00035, European Commission, Research Fund for Coal and Steel: Luxembourg, Publications Office of the European Union. Yamaki, N., 1984, Elastic Stability of Circular Cylindrical Shells. Elsevier Science. North-Holland.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Proposal for improving the consistency between Eurocode 3-1-8 and Eurocode 8-1 A. Stratan & D. Dubina The Politehnica University of Timisoara, Romania

ABSTRACT: This paper makes on overview of provisions for design of joints in steel structures according to EN 1993-1-8 and EN 1998-1 and discusses the relationship between the two codes. EN 1993-1-8 provides detailed design criteria and guidance on modelling of joints for global structural analysis, being limited to joints subjected to predominantly static loading. On the other hand, EN 1998-1 provides additional requirements for seismic design of joints in steel structures, totally relying on design tools from EN 1993-1-8. Several inconsistencies between the two codes concerning calculation of design resistance, modelling and strength classification are discussed in this paper, and possible improvements are proposed.

1 INTRODUCTION This paper discusses the application of two complementary Eurocodes for the design of momentresisting beam-to-column joints: EN 1993-1-8, addressing the design of joints in steel structures and EN 1998-1, addressing the seismic design of structures. Reference is made to the current version of the codes (EN 1993-1-8, 2005) and (EN 1998-1, 2004). A brief overview of the two codes is given in the next sections and it is shown that seismic design of beam-to-column joints according to EN 1998-1 heavily depends on the design tools in EN 1993-1-8. However, there are several inconsistencies between the two codes which makes it difficult for the engineer to find a rational way around. Possible improvements concerning the calculation of design resistance, modelling and strength classification are proposed in this paper. Hopefully, this proposal may contribute to improvements in the next generation of structural Eurocodes, which are currently in the process of revision by CEN/TC 250 as part of the EC Mandate M/515 (NEN, n.d.).

2 OVERVIEW OF CODE PROVISIONS 2.1 EN 1993-1-8 EN 1993-1-8 provides rules for the design of joints in steel structures subjected to predominantly static loading. It covers aspects related to the calculation of structural properties of joints (resistance, stiffness and, to a certain extent, ductility), as well as rules for classification of joints by stiffness and strength, and guidance on modelling of joints for global structural analysis. While the application of basic principles would allow calculation of structural properties for joints subjected to arbitrary loading (bending moments, axial and shear forces), detailed guidance is given to moment-resisting joints only. This is especially true in what concerns classification by stiffness and by strength, as well as guidance on modelling of joints for global structural analysis. It is worth noting the meaning of terms “connection” and “joint” as used in EN 1993-1-8: – The connection is defined as the location at which two or more elements meet. For design purposes, it is the assembly of the basic components required to represent the behaviour during the transfer of the relevant internal forces and moments at the connection.

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– The joint is the zone where two or more members are interconnected. A beam-to-column joint consists of a web panel and either one connection (single sided joint) or two connections (double-sided joint). In principle, EN 1993-1-8 provides two approaches for modelling of beam-to-column joints: – A “general model”, in which the web panel in shear and the connections are modelled separately, Figure 1a. – A “simplified alternative” in which the connection and the column web panel (CWP) are considered being lumped in a single component, Figure 1b. Consequently, a double-sided beam-to-column joint configuration has two moment-rotation characteristics, one for the right-hand joint and another for the left-hand joint. However, EN 1993-1-8 provides application rules for obtaining structural characteristics of beam-to-column joints only for the “simplified alternative”. The code uses the component method (Jaspart and Weynand, 2016) for deriving the moment resistance and stiffness of rotational springs used to represent the joints in the global structural analysis. Joints are classified by strength in EN 1993-1-8 as follows: – Full-strength. A full-strength joint is defined as having a design moment resistance larger than the one of the members it connects. – Partial strength. A joint which does not meet the criteria for a full-strength joint or a nominally pinned joint is classified as a partial-strength joint. – Nominally pinned. This classification is used to identify whether the effects of joint behaviour on the global structural analysis need be taken into account. According to Table 5.1 in EN 1993-1-8: – If a joint is full-strength, it may be considered as continuous, i.e. the behaviour of the joint need not be taken into account in case of global plastic analysis. – If a joint is partial-strength, it may be considered as semi-continuous, i.e. the behaviour of the joint shall be taken into account in the case of global plastic analysis.

2.2 EN 1998-1 EN 1998-1 applies to the design and construction of buildings in seismic regions. As stated by the code, “attention must be paid to the fact that for the design of structures in seismic regions the provisions of EN 1998 are to be applied in addition to the provisions of the other relevant EN 1990 to EN 1997 and EN 1999”. EN 1998 contains only those provisions that, in addition to the provisions of the other relevant Eurocodes, must be observed for the design of structures in seismic regions. It complements in this respect the other Eurocodes.

Figure 1. Modelling of beam-to-column joints in EN 1993-1-8 (2005): general model (a) and “simplified alternative”(b).

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Design of earthquake-resistant steel buildings maybe accomplished following one of the following concepts: (1) low-dissipative structural behaviour; (2) dissipative structural behaviour. Design using the concept of low-dissipative structural behaviour is the preferred approach in low-seismicity regions and/or for structures with a small mass. Typical examples of such structures are industrial steel halls. According to EN 1998-1, the resistance of members and connections of steel structures designed using this approach should be performed in accordance with (EN 1993-1-1, 2005) and (EN 1993-1-8, 2005), without any additional requirements. On the other hand, design using the concept of dissipative structural behaviour is the rational approach to the design of multistorey buildings. In this concept, the capability of parts of the structures (dissipative zones) to dissipate seismic action through plastic deformations is considered. The approach generally leads to more economical design, as it allows structures to be designed for seismic forces considerably smaller than those applied in the case of low-dissipative structural behaviour. The design may be accomplished based on a plastic structural analysis (static or dynamic), which is the straightforward approach, as it models explicitly the plastic response of dissipative zones and allows determining strength demands in non-dissipative ones. However, since plastic structural analysis is time-consuming and requires specialized knowledge, in practice, it is used only for the design of more important structures or in research. In all other situations, the design is accomplished based on elastic structural analysis, with the design seismic action reduced using the behaviour factor q. The overall ductility of the structural system is enforced by observing a set of rules provided by EN 1998-1 at the level of material, cross-section, members, connections and structural system. EN 1998-1 allows that dissipative zones be located in the structural members or in the connections. If dissipative zones are located in the structural members, the non-dissipative parts and the connections of the dissipative parts to the rest of the structure shall have sufficient overstrength to allow the development of cyclic yielding in the dissipative parts. Conversely, if dissipative zones are located in the connections, the connected members shall have sufficient overstrength to allow the development of cyclic yielding in the connections. The adequacy of design (strength and ductility) of members and their connections in dissipative zones should be supported by experimental evidence, in order to conform to the specific requirements for each structural type and structural ductility class. This applies to partial and full-strength connections in or adjacent to dissipative zones (EN 1998-1, 2004). In beam-to-column joints of moment resisting frames, the column web panel should be designed to resist the action effects corresponding to the development of plastic resistance in the adjacent dissipative zones in beams or connections.

3 RELATIONSHIP BETWEEN EN 1993-1-8 AND EN 1998-1 As stated in EN 1993-1-8, it provides rules for the design of joints in steel structures subjected to predominantly static loading. It covers aspects related to the determination of structural properties of joints (resistance, stiffness and, to a certain extent, ductility), as well as rules for classification of joints by stiffness and strength, and guidance on modelling of joints for global structural analysis. In most practical situations, structures subjected to static loading, (persistent design situation in (EN 1990, 2002)) are designed using elastic global analysis. Consequently, EN 1993-1-8 provides only limited guidance on ductility (plastic rotation capacity) of joints. On the other hand, even if guidance on elasto-plastic analysis exists, it is based on elastic-perfectly plastic behaviour of members and joints, neglecting such effects as strain hardening. EN 1998-1 applies to the design and construction of buildings in seismic regions, but it provides only additional rules to those of the other relevant Eurocodes. In relation to the design of joints in steel structures, EN 1998-1 complements the rules in EN 1993-1-8. Consequently, EN 1998-1 relies heavily on design rules in EN 1993-1-8. Conceptually, seismic design of steel structures (seismic design situation in EN 1990) is different from the non-seismic one (persistent design situation according to EN 1990) to the extent to which the former exploits inelastic response of the structure. Irrespective of the type of global structural analysis employed (elastic or plastic), a set of specific rules should be followed in order to 1088

guarantee a ductile response of the structure under seismic action. Conceptually, predetermined dissipative zones should be designed to be ductile, while the non-dissipative ones should be designed to withstand the maximum forces and moments generated by the yielded and fully strain-hardened dissipative zones undergoing cyclic plastic deformations. The latter requirement is known as “capacity design” (Landolfo et al., 2017). It is to be underlined that the fundamental principles necessary to model a structure for plastic analysis are the same in the case of seismic and persistent design situations. The distinctive features of seismic plastic analysis are the cyclic response of structural components and the dynamic nature of the seismic action. With respect to the design of connections in steel structures, there are several ways in which EN 1998-1 interrelates with EN 1993-1-8 in accomplishing a ductile structural behaviour: – EN 1998-1 references EN 1993-1-8 for determining the resistance and the stiffness of connections. This is quite straightforward, as the cyclic response in the inelastic range would rather affect the plastic deformation capacity and strain hardening, but not too much the initial stiffness and resistance at yielding of connections. – EN 1998-1 limits the categories of connections allowed for seismic applications, with respect to those available in EN 1993-1-8. For example, bolted connections in shear of bearing type (category A) and non-preloaded bolted connections in tension (category D) are not allowed for seismic applications in dissipative zones. – EN 1998-1 specifies ductility criteria under cyclic loading that should be fulfilled by connections in dissipative zones. For example, beam-to-column connections in moment-resisting frames should have a rotation capacity θp not less than 0,035 rad for structures of ductility class DCH and 0,025 rad for structures of ductility class DCM designed for q > 2. Due to insufficiently developed and unreliable analytical tools for assessing the cyclic response of connection in the plastic range, EN 1998-1 requires that the rotation capacity is demonstrated experimentally.

4 INCONSISTENCIES AND PROPOSAL FOR IMPROVEMENT 4.1

Modelling of beam-to-column joints

Both dissipative and non-dissipative beam-to-column connections are allowed by EN 1998-1. However, non-dissipative connections are preferred in practice, due to the difficulty of proving the ductility of dissipative connections under cyclic loading, and requirements of more refined global plastic analysis. To accomplish a ductile response of the joint region, a hierarchy of resistance of the beam end, connection and column web panel is required by EN 1998-1, see Figure 2. In the case of non-dissipative (brittle) connections, plastic deformations should occur at the beam end

Figure 2.

Hierarchy of resistance in the beam, connection and column web panel according to EN 1998-1.

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(Figure 2a), while the connection should have sufficient overstrength (Figure 2b) to prevent its failure under the maximum moment that can develop at the plastic hinge location under cyclic loading. Limited plastic deformations are also allowed to occur in the column web panel (Figure 2c). The “simplified” joint model currently adopted by EN 1993-1-8 is not able to provide an implementation of the requirements in EN 1998-1, as the three parts of the joint (connection, connected member and column web panel) are merged into a single spring. Given the complexity of the component method rules in EN 1993-1-8, engineers would usually rely on commercial software for obtaining joint properties (resistance and stiffness). This makes it very difficult, if impossible, to apply the requirements in EN 1998-1 using the tools in EN 1993-1-8. Therefore, independent resistance checking of the connection and the column web panel is necessary. In principle, this is possible to be accomplished using the “general approach” of modelling beam-to-column joints in EN 1993-1-8. However, the lack of application rules for the general approach makes it mostly unusable in practice. Consequently, providing more detailed guidance on the application of the “general approach” for modelling beam-to-column joints in EN 19931-8, as well as the application of the component method, in this case, would be of great benefit in achieving consistency between EN 1998-1 and EN 1993-1-8. The “general approach” of modelling beam-to-column joints using distinct springs for the column web panel and the connection was discussed by (Jaspart, 1991), (Charney and Downs, 2004) and (da Silva et al., 2016), among others. In addition to providing more accurate results, the refined modelling of joints using the “general approach” brings the following benefits: – Avoid the necessity of analysing distinct structural models for each load combination (da Silva et al., 2016). – Straightforward calculation of the design shear force in the column web panel by subtracting shear force in the columns in equation 5.3 from EN 1993-1-8 (2005), which leads to more economical design. – Design verification efforts are reduced for double-sided beam-to-column joints. Firstly, connections are designed to resist the applied bending moment. The same connection characteristics may be appropriate for internal and external joints. Secondly, the column web panel is checked for internal and external joints, under different load cases (gravity only or gravity + wind, gravity + seismic loading, or any other relevant combination). It leads to easy identification of the load combinations which would require strengthening of the column web panel, a common design problem. – The correct approach in modelling of varying moment distribution acting on the column web panel in case of global plastic analysis with non-proportional loads (e.g. in case of nonlinear static analysis). Though refined modelling of joints offers many benefits, it is strictly necessary only in case of double-sided joints when the loading is non-proportional throughout the analysis. The simplified model is not appropriate in such cases, as the stiffness and strength of the springs in the simplified model (Figure 1b) depend on the ratio of bending moments in the beams, and would need to be updated (re-computed) at each step of the analysis. For all other cases, the simplified model may be used, even if independent resistance checks are performed for the connection and the column web panel. 4.2 Classification by strength 4.2.1 General In EN 1993-1-8 a joint is considered full-strength if it fulfils the following requirement: Mj;Rd  Mpl;Rd which assumes zero post-yielding stiffness of the plastic hinge in the beam.

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ð1Þ

The real response of members (Galambos, 1968) and joints (Weynand et al., 1996) is characterised by a significant amount of strain-hardening. Ignoring strain-hardening in a plastic global analysis significantly underestimates the forces and moments that cause brittle failure, which may lead to unsafe design. Variability of steel yield strength is another issue that might lead to unsafe design. Statistical data obtained recently by (da Silva et al., 2017) and (Badalassi et al., 2017) shows that: – mean yield strength of European steels is significantly larger than nominal values; – mean yield strength is larger for lower steel grades than for higher-strength grades. The last observation is particularly important for bolted beam-to-column connections in the case of global plastic analysis, since: – a beam fabricated from lower steel grades (S235/S275) may be characterised by a large over-strength (ratio between mean and nominal strength), and consequently generate a maximum moment in the plastic hinge significantly larger than the nominal one, while – the bolted connection, and more specifically high-strength bolts have a significantly lower over-strength. Accounting for the effects of strain hardening and material variability, EN 1998-1 requires that the resistance of non-dissipative (full-strength) joints Mj,Rd should be larger than the resistance of the connected dissipative (ductile) member Mpl,Rd, amplified by material overstrength factor (γov = 1.25) and a factor accounting for strain hardening (1,1): Mj;Rd  1;1γov Mpl;Rd

ð2Þ

A similar requirement exists in EN 1993-1-8: Mj;Rd  1;2Mpl;Rd

ð3Þ

which should be fulfilled if the “rotation capacity of the joint need not be checked”. The condition (3) from EN 1993-1-8 is a simplified form of the requirement (2) from EN 19981, with a single factor (1,2) accounting for strain hardening and material overstrength. Definition (1) of a full-strength joint in EN 1993-1-8 allows a continuous model to be adopted (the behaviour of the joint need not be taken into account in the global plastic analysis), and is formally correct for elastic-perfectly plastic model. The continuous model implies that plastic deformations occur exclusively in the connected member. Which is confusing, because according to the same code, this is assured if the joint resistance is at least 1,2 times stronger than the connected member, according to expression (3). 4.2.2 Full-strength macro-components In the case of global plastic analysis, it should be shown that the structure has sufficient plastic deformation capacity. This can be accomplished by designing the structure such that: – plastic deformations occur in structural components which are ductile, and – design resistance of brittle components is not exceeded under forces that cause full yielding and strain hardening of ductile components. To improve the consistency within EN 1993-1-8 and between the two codes, it is proposed that in EN 1993-1-8 the definition of full strength macro-components (1) and the criteria for disregarding the check of the rotation capacity of the joint (3) be merged into a single one, and explicitly accounting for the effects of material over-strength and strain hardening. Thus, for plastic analysis, non-ductile macro-components (connection or column web panel) should be full-strength, by fulfilling the following criterion: Rd;fs  γsh  γov  Rm

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ð4Þ

where Rm is the plastic resistance of the connected member obtained without considering the partial factor for material, evaluated in the section where plastic hinge is expected to occur; Rd,fs is the design resistance of the non-ductile macro-component; γsh is the strain hardening factor of the ductile macro-component (member); γov is the material overstrength factor of the ductile macro-component (member) and represents the ratio between the mean and nominal values of yield strength. Expression (4) is defined in general terms, and generally refers to bending moment but may apply to other relevant action effects (shear or axial force). In principle, it is appropriate for the cases when the ductile and non-ductile macro-components are adjacent to each other. When the ductile and non-ductile macro-components are located at a non-negligible distance (e.g. in case of beam-to-column joints with haunches), a translation term would have to be used. In the case of elastic-perfectly-plastic global analysis, it is convenient designing the fullstrength non-ductile macro-component for the maximum forces and moments obtained from the analysis, multiplied by γshγov. Values of γsh depend on the type of loading. The recommended value for members subjected to static monotonic loading is γsh = 1,1. Evidence of larger values of the strain hardening γsh factor exists, even under static monotonic loading (Faella et al., 2000), (Mazzolani and Piluso, 1996). Even if potentially unconservative in some cases, the value of γsh = 1.1 is retained as a simplification. Larger strain hardening would be required in the case of cyclic loading in EN 1998-1. Tentative values are γsh = 1,20 for bending moments, γsh = 1,50 for shear forces, γsh = 1,10 for axial forces. Based on the SAFEBRICTILE project (da Silva et al., 2017), values for γov may be taken as follows: γov = 1,45 for S235, γov = 1,35 for S275; γov = 1,25 for S355, γov = 1,2 for S460. These values are representative of the modern European steel market. 4.2.3 Equal-strength macro-components To cover the gap between full-strength and partial-strength macro-components, it is proposed introducing a new strength classification: equal-strength. The design resistance of an equalstrength macro-component (connection or column web panel) Rd,es, should be larger than or equal to the plastic resistance of the member Rm: Rd;es  Rm

ð5Þ

Due to the variability of material characteristics and strain hardening in the macro-components of a joint, plastic deformations may occur in both the member and in the “equalstrength” macro-component. Therefore, both macro-components shall be ductile if the global plastic analysis is used, and their resistance shall be modelled in the global analysis (semi-continuous model). 4.2.4 Partial-strength macro-components A macro-component (connection or column web panel) is partial-strength if it limits the design resistance of the structural system. Partial-strength macro-components have a design resistance Rd,ps which is smaller than the one of the connected member Rm: Rd;ps 5Rm

ð6Þ

A partial-strength macro-component shall be designed for the action effects in the relevant design situation. If the plastic analysis is used, it should be shown that the partial-strength macro-component have the necessary deformation capacity, and it shall be modelled as semicontinuous. If the structure is designed to develop plastic deformations exclusively in the partialstrength macro-component, the adjoining member shall be designed for the relevant forces that can be developed in the partial-strength macro-component, including the effects of material overstrength and strain hardening: 1092

Rd  γsh  γov  Rps

ð7Þ

where Rd is the design resistance of the connected member or other non-ductile macrocomponent; Rps is the resistance of the partial-strength macro-component, obtained without considering the partial factor for the material; γsh is the strain hardening factor of the partialstrength macro-component; γov is the material overstrength factor of the partial-strength macro-component. Values of γsh depend on the type of loading. In lieu of more precise data, the same values as for members may be used. However, it is known that certain types of connections (e.g. bolted end-plate connections) may develop substantially larger hardening.

5 CONCLUSIONS EN 1993-1-8 provides comprehensive procedures for the design of joints in steel structures subjected to predominantly static loading. EN 1998-1 covers the design of joints in seismic-resistant steel structures but relying heavily on the design tools from EN 1993-1-8 and providing only additional requirements targeting a ductile response. Therefore, the consistency of the design provisions in the two codes is highly important. Areas in EN 1993-1-8 that may be improved include modelling of beam-to-column joints and classification by strength. The authors suggest that three macro-components are explicitly considered in the design: the connection, the column web panel and the connected member. For global analysis, the “general approach” which explicitly considers the three macro-components, is more correct and avoids tedious iterations in the design process. Of course, appropriate tools should be available in structural analysis software. For the design process, independent modelling of the three macro-components allows establishing the hierarchy of resistance necessary for obtaining a ductile response of the joint. Another possible improvement concerns the classification of macro-components by strength, and it is suggested that three categories be defined: full-strength, equal-strength and partial strength, and accounting explicitly for material overstrength and strain hardening. ACKNOWLEDGEMENTS The research leading to these results has been funded from the European Community’s Research Fund for Coal and Steel (RFCS) under grant agreement no 754048 RFCS-2016/ RFCS-2016 EQUALJOINTS - PLUS. This support is gratefully acknowledged. REFERENCES Badalassi, M., Braconi, A., Cajot, L.-G., Caprili, S., Degee, H., Gündel, M., Hjiaj, M., Hoffmeister, B., Karamanos, S.A., Salvatore, W., Somja, H., 2017. Influence of variability of material mechanical properties on seismic performance of steel and steel–concrete composite structures. Bulletin of Earthquake Engineering 15, 1559–1607. https://doi.org/10.1007/s10518-016-0033-2. Charney, F.A., Downs, W.M., 2004. Modeling procedures for panel zone deformations in moment resisting frames, in: Proceedings of the Fifth International Workshop, Connections in Steel Structures V. Amsterdam, The Netherlands, pp. 121–130. da Silva, L.S., Simões, R., Gervásio, H., 2016. Design of steel structures: Eurocode 3: Design of steel structures. Part 1-1, General rules and rules for buildings, 2nd edition. ed, ECCS Eurocode Design Manuals. ECCS - European Convention for Constructional Steelwork. da Silva, L.S., Tankova, T., Marques, L., Kuhlmann, U., Kleiner, A., Spiegler, J., Snijder, H. H., Dekker, R., Taras, A., Popa, N., 2017. Safety assessment across modes driven by plasticity, stability and fracture. ce/papers 1, 3689–3698. https://doi.org/10.1002/cepa.425 EN 1990, 2002. Eurocode - Basis of structural design. European Committee for Standardization (CEN).

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EN 1993-1-1, 2005. Eurocode 3: Design of steel structures - Part 1-1: General rules and rules for buildings. European Committee for Standardization (CEN). EN 1993-1-8, 2005. Eurocode 3: Design of steel structures - Part 1-8: Design of joints. European Committee for Standardization (CEN). EN 1998-1, 2004. Eurocode 8: Design of structures for earthquake resistance - Part 1: General rules, seismic actions and rules for buildings. European Committee for Standardization (CEN). Faella, C., Piluso, V., Rizzano, G., 2000. Structural steel semirigid connections: theory, design, and software. CRC Press, Boca Raton. Galambos, T.V., 1968. Deformation and energy absorption capacity of steel structures in the inelastic range. Steel Research and Construction Bulletin No. 8. Jaspart, J.-P., 1991. Etude de la semi-rigidité des noeuds poutre-colonne et son influence sur la résistance et la stabilité des ossatures en acier (PhD Thesis). Université de Liège, Belgium. Jaspart, J.-P., Weynand, K., 2016. Design of joints in steel and composite structures, ECCS Eurocode design manuals. ECCS/Ernst & Sohn. Landolfo, R., Mazzolani, F., Dubina, D., Silva, L.S.D., D’Aniello, M., 2017. Design of Steel Structures for Building in Seismic Areas. ECCS, Ernst & Sohn, Berlin. Mazzolani, F., Piluso, V., 1996. Theory and Design of Seismic Resistant Steel Frames, 1 edition. ed. CRC Press, London. NEN, n.d. Call for Tender for the development of the second generation of Structural Eurocodes [WWW Document]. URL https://www.nen.nl/Normontwikkeling/Eurocodes-2020.htm (accessed 12.14.17). Weynand, K., Jaspart, J.-P., Steenhuis, M., 1996. The stiffness model of revised Annex J of Eurocode 3, in: Connections in Steel Structures III. Elsevier, pp. 441–452. https://doi.org/10.1016/B978-0080428215/50100-0.

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Assumption of imperfections for the LTB-design of members based on EN 1993-1-1 Richard Stroetmann Institute of Steel and Timber Structures, Technical University of Dresden, Germany

Sergei Fominow Institute of Numerical Methods and Structural Analysis, TH Mittelhessen University of Applied Sciences, Giessen, Germany

ABSTRACT: The stability design of members and structures by a geometrically non-linear analysis with equivalent geometric imperfections (GNIA) is a commonly used method. In the case of lateral torsional buckling (LTB) of members, the rules in EN 1993-1-1 are unsatisfactory. With the current code but also with the new draft prEN1993-1-1 results are achieved, some of which are significantly on the safe side but some also on the unsafe side. The reasons for this are the limitation to bow imperfections e0 out of plane, an inappropriate differentiation related to the cross-section shape and yield strength, as well as neglecting the influence of the moment distribution over the member lengths. Parameter studies have shown that a differentiation is required due to the different structural behavior of members loaded with pure bending, pure compression, or a combination of bending and compression. In this article current research results are presented, which consider the case of pure bending My of I and H sections. Dependencies on the shape of sections, cross-sectional resistance models and the influence of steel grades are analyzed with respect to the assumption of equivalent geometrical imperfections.

1 INTRODUCTION The design process by a geometrical nonlinear analysis using equivalent initial imperfections (GNIA) will be carried out by applying equivalent initial geometric imperfections, determination of internal forces using geometrical non-linear analysis and verification of the cross-section resistance at the most unfavorable position. The LTB resistance according to this design method depends on the size and shape of the geometrical imperfection, the cross-sectional shape, the moment distribution and possible additional compression forces, the resistance model and the steel grade. In EN 1993-1-1 imperfections for the LTB-design of members are defined in chapter 5.4.3. The shape is specified as an initial bow imperfection out of plane. The amplitude is given as k · e0, where k = 0,5 is the recommended value and e0 is defined in table 5.1 and 6.2 of the code. For rolled I-sections the relevant buckling curve for flexural buckling depends on the h/b ratio, the plate thickness and steel grade. For slender cross-sections with h/b > 1.2 this results in smaller imperfection amplitudes than for compact cross-sections with h/b ≤ 1.2. Kindmann & Beier-Tertel 2010 have shown that this assumption is not suitable for lateral torsional buckling. Cross-Sections with h/b ≤ 2.0 are more favorable than those with h/b > 2. This is also considered in the selection of the LTB curves according Table 6.5 of EN 1993-1-1. In the German National Annex of EN1993-1-1 other amplitudes of imperfection are specified which are derived from the assignment to the LTB-buckling-curves. The k-factor is given as 1.0 in the medium range of slenderness (0,7 ≤ λLT ≤ 1,3). This rules are adopted modified in the latest version of prEN1993-1-1 by considerffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pand ing the material parameter ε ¼ fy =235 as an amplifier. The shape of the equivalent geometrical imperfection is still specified as a bow-imperfection out of plane (Table 1).

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Table 1. Initial bow-imperfections e0 for rolled I-sections for the LTB-design of members by GNIA. current EN 1993-1-1 elastic analysis e0 =L

Elastic crosssection verification ε  e0 =L

Plastic crosssection verification ε  e0 =L

plastic analysis e0 =L

Limits

h=b  1:2 1/400

1/400

h=b  2:0 1/250

1/200

h=b41:2

1/300

h=b42:0

1/150

Limits

Figure 1.

prEN 1993-1-1

1/500

1/200

Shapes of imperfections – bow- and mode-imperfections.

Investigations by Ebel 2014 on different structural systems and loads have shown, that the structural behavior may be better reflected by considering the shape of the first LTB buckling mode for the equivalent geometrical imperfection. Snijder, van der Aa, Hofmeyer & van Hove 2018 also considered imperfection shapes based on LTB modes and developed a proposal for geometric and material nonlinear analysis. In Hajdú, Papp & Rubert 2017 a proposal is presented, which considers the combination of compression and bending. The imperfection shape also based on the relevant elastic critical LTB mode. In the following parameter studies, both imperfection shapes, the sinusoidal bow imperfection out of plane (IMP 1) and the first eigenmode for LTB (mode-imperfection, IMP 2) are investigated (Figure 1). In this context, a comparison of the results is made, which also enables a comparison with the rules in EN1993-1-1. For both imperfection shapes, the amplitude e0 corresponds to the maximum amplitude. The aim of the present investigations is to derive imperfection approaches for the LTB design by GNIA, which take into account the essential influences of cross-sectional shape, yield strength, moment distribution and model for the cross section resistance. For reasons of simplification, only linear interaction formulas are used to determine the cross-section resistance. – Interaction formula 1 (I-1) Elastic verification of the cross-sectional resistance. The limit criterion is fulfilled, if the von Mises stress σv reaches the yield strength fy at the most critical point of the cross-section. σV ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ x 2 þ 3  τ 2  fy

ð1Þ

– Interaction formula 2 (I-2) Linear plastic interaction. The influence of shear may lead to a reduction of the full plastic cross section resistances. 1096

My N Mz B þ þ  1:0 þ A Mpl;y Mpl;z Bpl

ð2Þ

– Interaction formula 3 (I-3) Linear plastic interaction with limitation. The full plastic cross section resistances are limited to 1.25 times of the elastic values. Background of the limitation is that in the case of LTB the cross-section is usually not completely plasticized when the structural resistance is reached. Usually the plastic zones reach only over a part of the flanges but not the entire cross section. The new design rules for flexural buckling in prEN1993-1-1 (see Lindner, Kuhlmann & Jörg 2017) consider also this limitation. For LTB of I- and H-sections the limit is relevant for Mz and B. My N Mz B þ þ  1:0 þ A Mpl;y 1:25  Mel;z 1:25  Bel

ð3Þ

2 PARAMETER STUDIES FOR PURE BENDING 2.1 Calibration of imperfections by comparison of results of GMNIA and GNIA calculations In a comprehensive parametric study, numerous LTB resistances of hot-rolled I-sections considering a wide range of member lengths, section shapes and load types were calculated by GMNIA with ANSYS. For this purpose, a hybrid shell-beam model was created modeling the I-section as a combination of shell and beam elements. The shell elements were located in mid-plane of the flanges and the web. The beam elements were arranged to consider the rolled radii at the intersection of the flange and the web. Initial geometric imperfections were implemented into the FE models considering the first buckling mode from an elastic buckling analysis. The amplitude was scaled to L/1000. For the residual stress patterns, the recommendations of the ECCS were considered. The magnitudes of the stress patterns depending on the height-to-width (h/b) ratio of the cross-section and are independent of the yield stress. The values of 0.5 · 235 MPa for h/b ≤ 1.2 and 0.3 · 235 MPa for h/b > 1.2 were assumed with a triangulated distribution over the width of the flanges and the web. Further information are given in Stroetmann & Fominow 2018. Based on LTB resistances according to GMNIA, equivalent geometric imperfections were derived using geometric nonlinear analysis (GNIA). A single span beam with fork end conditions consisting of open I- and H-sections was considered as the structural system (Figure 2). The calculation of the required amplitude of imperfection e0 was an iterative process with the aim of achieving the same LTB resistance as with GMNIA. They are defined in relation to the member length L as a non-dimensional j-value (Eq. (4)). j¼

L e0

ð4Þ

In the parameter study, the maximum beam length was limited to exclude irrelevant lengths in the calibration of imperfections. In general the limitation was set to L/h = 50 and for very slender cross-sections to L/h = 40. In the parameter studies presented here, the limitation can be recognized by the fact that the j-values end within the slenderness range shown. This is the case if the beams are only slightly prone to lateral torsional buckling, e.g. those with HE200B sections. 2.2 Influence of the cross-sectional shape and interaction formulas The required imperfection amplitudes e0 are significantly influenced by the cross-sectional shape and the model for determining the resistance. In the parameter study, the dependencies 1097

Figure 2.

Structural system and loads.

Table 2. Ratios h/b and Iy/IT of the investigated I- and H-sections. Section

h/b

Iy/IT

IPE80 IPE200 IPE500 IPE600 HE200A HE360A HE400A HE550A HE800A HE1000A

1.74 2.00 2.50 2.73 0.95 1.17 1.30 1.80 2.63 3.30

114.8 277.9 539.8 558.1 175.7 222.1 238.5 317.9 508.2 649.4

Section HE200B HE360B HE400B HE800B HE1000B HE360AA HE400AA HE600AA HE800AA HE1000AA

h/b

Iy/IT

1.00 1.20 1.33 2.67 3.33 1.13 1.26 1.90 2.57 3.23

96.1 147.9 162.0 379.6 515.8 324.5 369.0 613.3 813.4 1007.5

Section HE200M HE320M HE340M HE400M HE650M HE800M HE1000M

h/b

Iy/IT

1.07 1.16 1.22 1.41 2.19 2.69 3.34

41.1 45.4 50.6 68.9 178.3 268.2 424.9

of the h/b- and Iy/IT-ratio were examined. For this purpose, 27 rolled I- and H-sections were considered, which are listed in Table 2. Figure 3 shows the minimum j-values for the described range of beam lengths. From the evaluation for the h/b-ratio it can be seen that at h/b = 1.2 there is a strong increase in the minimum j-values due to the different assumptions for the residual stresses in the GMNIA. However, the h/ b-ratio is generally not significant as the minimum j-values for cross-sections are similar for h/b ≈ 1.2 and h/b > 3. The dependence on the Iy/IT ratio is pronounced for the assumption of mode affine imperfections IMP-2. This applies in particular to beams subjected to transverse loads. If the cross-sectional resistance is determined elastically, significantly lower amplitudes are required with respect to the linear plastic interaction formula. However, the reduction of the full plastic moment resistance causes only a slight reduction of the required amplitudes of the imperfections. Concerning the imperfection shape, pure bow-imperfections require significantly higher amplitudes than mode-imperfections. 2.3 Influence of the steel grade Figure 4 shows the course of the required j-values for a compact (HE200B), a medium (HE400A) and a slender cross-section (HE1000A) in the steel grades S 235, S 355 and S 460 considering the elastic and the plastic cross-sectional resistance. In most cases, the beams of the higher steel grades require slightly larger amplitudes e0. There are exceptions for the slender cross-sections in the medium slenderness range, where larger amplitudes are required. The reason for this is that the occurring shear stresses reduce the load-bearing capacity calculated by GMNIA. For slendernesses around 1.0, beams with cross-sections HE1000A have relatively short lengths. For higher steel grades the beam length decreases for the same slenderness

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Figure 3. Minimum j-values for sections in the steel grade S355 according to Table 2, depending on the interaction formulae, imperfection shapes as well as ratios h/b and Iy/IT.

and the influence of shear stress increases. The superposition of the influences of shear, bending and torsion leads to a lower load-bearing resistance at GMNIA. 2.4 Influence of the bending moment distribution In a further parametric study different moment distributions for bending about the major axis have been analysed, see Table 3. They were generated by end moments and uniform distributed or concentrated loads, acting on the top chord of the cross-sections, see Figure 2. Figures 5 and 6 show the j-values for the cross-sections HE200B and HE1000A considering the elastic and plastic cross-sectional resistances and imperfections IMP-1 and IMP-2. The significant dependence on the moment distribution is recognizable. The approach of mode imperfections leads to larger j-values and thus to smaller amplitudes e0. While the imperfection form IMP-2 is significantly influenced by the moment distribution My, the bow imperfection IMP-1 is independent. If the bending moment My at mid span of the beam is small, the design point relevant for the cross-sectional check is asymmetrical. To achieve the same design result with IMP-1, the amplitude e0 must be increased. As illustrated in Figure 6, the load cases LC 1 and LC 2 require the largest imperfection amplitudes when the mode-imperfection shape IMP-2 is applied. The moment distributions caused by uniform loads (LC 3, LC 6 and LC 7) require slightly less imperfections. Comparatively small values resulting for concentrated loads (LC 8, LC 9 and LC 10). 3 CONCLUSION AND OUTLOOK The stability design of steel structures using equivalent geometric imperfections is frequently used in the case of flexural buckling, but is rather used for lateral and flexural torsional 1099

Figure 4. S 460.

j-values for sections HE200B, HE400A and HE1000A in the steel grades S 235, S 355 and

Table 3. Overview of the investigated load cases for bending about the major axis.

1100

Figure 5. Required j-values for bow-imperfections IMP-1 for sections HE200B and HE1000A in S 235, considering interaction formulae I-1, I-2 and various moment distributions.

Figure 6. Required j-values for mode-imperfections IMP-2 for sections HE200B and HE1000A in S 235, considering interaction formulae I-1, I-2 and various moment distributions.

1101

buckling. This is partly due to the demand of more complex calculation software, but partly also due to missing or unclear specifications for imperfection assumptions and cross section resistances to be used for the calculations. As imperfection assumptions, sinusoidal bow-imperfections out of plane of the members and the scaled mode-imperfections are common. The amplitudes of the equivalent geometric imperfections, which have to consider, depending on many influences. These include the cross-sectional shape, the loads and its distributions, the structural system and the boundary conditions, but also the type of structural mechanic calculations and the model for the crosssection resistance. Thus, for the elastic stress analysis and the linear plastic sectional interactions, different assumptions of imperfections are necessary in order to achieve equal load bearing capacities from more accurate calculations. In this article, parameter studies are presented in which imperfection measures e0 for the LTBdesign of members with I- and H-sections under pure bending My were derived. The values are based on the results of more detailed structural analysis according to the GMNIA. The imperfection amplitudes e0 were derived by a comparison of the results of a linear elastic calculation of internal forces and the assessment of the sectional resistance with three different interaction formulas. The parameter studies show, that the mode-imperfection approach leads to significant smaller imperfection amplitudes e0 than the approach of sinusoidal bow-imperfections in y-direction. The higher the cross-section can be used by selecting the interaction formula for the crosssectional resistance, the greater the amplitude e0 of the imperfection must be applied. The parameter studies on the influence of the cross-sectional shape of hot rolled sections reveal that the ratio Iy/IT better describes the tendency for the imperfection size e0 than the ratio of h/b. Discontinuities arise in that, from h/b > 1.2 the assumed residual stress decreases from 0.5 · fy,S235 to 0.3 · fy,S235. The studies of the effect of the yield strength related to the value e0 show that this is only slight and is overestimated with the provisions in the current draft prEN1993-1-1. The shape of the moment distribution has a very strong influence on the imperfection assumption e0. For strongly asymmetric moment diagrams with alternating signs, the approach of sinusoidal bow imperfections e0 is unsuitable. With the assumption of mode imperfections it is possible to react to the special bearing conditions of a structural system and its moment distribution. The results of the parameter studies show the way forward for calibrating imperfections for a codification approach based on EN1993-1-1. The evaluation of the data, in combination with the necessary simplifications for the design practice, leads to corresponding definitions of imperfection values e0 and the necessary differentiations. REFERENCES ANSYS, Finite Element Software Package, Version 18, ANSYS Inc., Canonsburg, PA, USA. EN 1993- 1-1. 2005. Design of steel structures – Part 1-1: General rules and rules for buildings, 2005-05. prEN1993-1-1. 2018. Design of steel structures – Part 1-1: General rules and rules for buildings, final draft, 2018-07. Ebel R. 2014. Systemabhängiges Tragverhalten und Tragfähigkeiten stabilitätsgefährdeter Stahlträger unter einachsiger Biegebeanspruchung. In: Dissertation, Ruhr-Universität Bochum, Schriftenreihe des Instituts für Konstruktiven Ingenieurbau, Heft 2014-03. Hajdú G., Papp H. & Rubert A. 2017 Vollständige äquivalente Imperfektionsmethode für biege- und druckbeanspruchte Stahlträger. In Stahlbau 86 (2017), P. 483–496. Berlin: Verlag Ernst & Sohn. Kindmann R. & Beier-Tertel J. 2010. Geometrische Ersatzimperfektionen für das Biegedrillknicken von Trägern aus Walzprofilen – Grundsätzliches. In Stahlbau 79 (2010). P. 689–697. Berlin: Verlag Ernst & Sohn. Lindner, J., Kuhlmann, U. & Jörg, F. 2017: Initial bow imperfections e0/L of Table 5.1 of EN 1993- 1-1. Doc. CEN-TC 250-SC3-WG1 N0155 (2017). Snijder, H.H., van der Aa, R.P., Hofmeyer, H. & van Hove B. 2018. Lateral torsional buckling design imperfections for use in non-linear FEA. In: Steel Construction 11 (2018), No. 1. Stroetmann R. & Fominow S. 2018. Imperfections for the LTB-design of members by geometrical nonlinear analysis. Eighth International Conference on THIN-WALLED STRUCTURES - ICTWS 2018, Lisbon, 24–27 July, 2018.

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Welds on high-strength steels—influence of the welding process and the number of layers R. Stroetmann & T. Kästner Technische Universität Dresden, Dresden, Germany

ABSTRACT: In addition to the development of a new design approach for welded joints of high-strength steels, the influence of the welding process and the number of weld layers was examined within the ongoing AiF-FOSTA research project P1020. Hereby, the load-bearing characteristics of fully mechanized welds was compared to manual produced welds. Furthermore, the influence of different number of weld layers was examined considering two filler-metals. These examinations include tensile tests and hardness measurements in two different levels of the weld.

1 INTRODUCTION The mechanical properties of welds are influenced by a variety of factors. As part of the AiFFOSTA research project P1020 the influence of the manufacturing process of the highstrength steels and the welding process on the mechanical properties of welds was examined. For this purpose, experimental tests were carried out to study the influence of the automation level of the welding process and number of layers. Within this examination, under- and overmatching material combinations were considered. The aim was to quantify these influences on the strength and ductility of the welds.

2 CHARACTERIZATION OF THE BASE AND FILLER METALS For the examinations two thermomechanical rolled fine grain steels S500ML (EN 10025-4) and S700MC (EN 10149-2) as well as two quenched fine grain steels S690QL and S960QL (EN 10025-6) were selected. All base materials had a plate thickness of 20 mm. The thermomechanical rolled fine grain steels (TM-steels) have a lower carbon equivalent than the quenched fine grain steels (QT-steels). The difference results mainly from the significantly lower carbon contents. Lower carbon equivalent results in a lower preheat temperature for the same process parameters. Thus, the TM steels can be welded at lower preheat temperatures than the QT steels. For TM-steels, however, care must be taken to minimize the energy input during the welding process, as the mechanical properties of these steel grades are largely achieved by hot rolling in the not re-crystalized austenite and the transition temperature from austenite to ferrite. Higher heating than the transformation temperature Ac1 leads to secondary crystallization processes that affect the mechanical properties. In (Stroetmann et. al. 2018a), the authors examined the influence of the peak temperature and cooling time t8/5, representative for the areas of the heat-affected zone and the used process parameters, on the tensile strength of the base metals. It could be shown that the tensile strength of the S700MC steel for each peak temperature and cooling time was below the strength of the unaffected base metal. A reduction in the strength in the area of the heataffected zone took place through the welding process. For the other base metals, in some cases significantly higher strengths were achieved in the area of the heat-affected zone than for the unaffected base material.

1103

Table 1. List of filler metals. Filler metal

Designation

Standard

Union K 40 Union K 52 Union MoNi Union X 85 Union X 90

G 35 A M23 Z2Si1 G 42 4 M21 3Si1 G 62 5 M21 Mn3Ni1Mo G 79 5 M21 Mn4Ni1,5CrMo G 89 6 M21 Mn4Ni2CrMo

EN ISO 14341-A EN ISO 14341-A EN ISO 16834-A EN ISO 16834-A EN ISO 16834-A

The choice of welding consumables was made taking into account the mechanical properties of the base metals. Only solid wire electrodes for metal active gas welding according to EN ISO 14341 and EN ISO 16834 were used for the investigations carried out. The welding consumables used in the research project and their complete designation are given in Table 1.

3 EXAMINATIONS OF THE MECHANICAL PROPERTIES OF WELDS Within the ongoing AiF-FOSTA research project P1020 a new test specimen was developed to determine the mechanical properties of welds. A flat tensile test specimen based on EN ISO 6892-1 and EN ISO 4136 with a centric hole in the weld root area was used. In conjunction with the contour of the test specimen, a clear definition of the weld area and the notch effect at the root of the weld as well as the transition to the base metal takes place. Therefore, test results are comparable and reproducible. Furthermore, the failure of the test specimen occurs in the weld itself. This is also the case for overmatching welds where the strength of the welds is higher than that of the base metal. This failure is necessary for a quantitative determination of the mechanical properties of the weld. Figure 1 shows the geometry of the test specimen and the weld area before and after the test procedure. The specimen is part of a new test procedure for welded joints, designed as a comparative test. Using the new design approach, the mechanical properties of the welds can be transferred to the component level taking into account the type of joint and the dimensional ratios (Stroetmann et. al. 2018a, 2018b). To determine the mean value of the weld stress the force (F) of the testing machine will be divided by the net section area of the test specimen (S0) as shown in equation 1. The yield and tensile strengths determined this way are test specific mechanical properties of the weld. The mean strain will be derived from the ratio of the elongation (ΔL) to the initial gauge length (L0) (see equation 2). This is not a local but a mean value over the measuring range. It serves as a comparative value for assessing the ductility of the welds.

Figure 1.

Geometry of the test specimen and range of initial gauge length before and after the test.

1104

σ w ¼ F = S0

ð1Þ

ε ¼ ΔL = L0

ð2Þ

3.1 Test program The mechanical properties of welds are influenced in different ways. From interest are the combination of base and filler metal, the execution parameters and the number of layers of the welds. These influences are partly dependent on each other. Therefore, several parameter studies were performed to examine the influences on the mechanical properties within the framework of the AiF-FOSTA research project P1020. In addition to the determination of the mechanical properties, metallographic examinations were carried out. These include the preparation of macro-sections of each weld, microstructure analyses and hardness measurements. Furthermore, miniature tensile specimens were taken from the area of the base material, the heat-affected zone and the weld metal. 3.2 Manufacturing of the test specimens The welds of the samples were produced under laboratory conditions fully mechanized at the Professorship for Welding Technology of the TU Chemnitz as well as manually by industrial partners. The fully mechanized specimens were welded at a specially designed test setup including a six-axis welding robot and a synchronized rotatable table, where they were stored at certain points. For reasons of comparability, the electrical power of the welding process was chosen on an almost constant basis for all examinations. During the complete welding process, the execution parameters and the distortion forces were measured and recorded. The desired cooling times t8/5 were achieved by varying the welding speed, preheating and interpass temperatures. By applying thermo-elements in the liquid weld metal, the cooling time was measured for each welding bead. In addition, macro-sections of each weld were prepared and metallographic examinations were carried out. The microstructure of selected test specimens was compared with the results of the quenching tests to evaluate the measuring systems and the configuration of the test setup. To produce the flat tensile specimens from the welded plates, thin segments were cut out and the contour of the sample were milled. Then, the hole was made in the area of the weld, taking into account the structure of the macro-sections. During production, the temperatures of the specimens were limited so that no microstructural changes occurred. The opening angle of the welds was for all specimens 90°. Two different geometries were used. A reduction of the specimen width (b0) for the plates with the two-layer welds was necessary due to the smaller weld volume compared to the three-layer welds. The dimensions of the two-layer specimen, including the diameter of the hole (d), were scaled accordingly and the initial gauge length was reduced based on the change in the net cross-sectional area. The specimen dimensions are summarized in Table 2. Thereby the terms and formula symbols of EN ISO 6892-1 were used. The test of the specimens was performed strain-controlled with a displacement rate of 0.5 mm/min.

Table 2.

Specimen dimensions and initial gauge lengths in mm (see Figure 1).

Specimen geometry

b0

t

d

L0

Geometry A Geometry B

15.0 13.5

5.0 4.5

5.0 4.5

25.0 20.0

1105

3.3 Influence of the welding process For the assessment of the influence of the welding process, the base metals S500ML, S700MC and S690QL were welded with filler metals G42, G62 and G79. The process parameters were selected in a way that a cooling time t8/5 of almost 12 seconds was achieved for each material. For the assessment of the influence of the welding process, the base metals S500ML, S700MC and S690QL were welded with filler metals G42, G62 and G79. The process parameters were selected in a way that a cooling time t8/5 of almost 12 seconds was achieved for each material combination. The test were carried out with the specimen geometry A. Figure 2 shows the macro-sections of a fully mechanized (left) and a manually (right) produced weld. The fully mechanized produced weld has an approximately symmetric structure in which the penetration of the roots of the individual weld beads is low. In contrast to this, the weld structure of the manually produced weld is more asymmetrical and there is a clear penetration of the individual weld beads. In the visible areas of the heat-affected zones, there are clear differences in the geometric characteristics and overlapping areas. The volume of the manually produced weld is larger than that of the fully mechanized process. Representative for the test with the filler metal G79, the stress-strain curves of the combination with the steel S690QL are shown in Figure 3. An influence of the manufacturing process on the mechanical strength properties of the weld can be recognized only to a small extent. The average tensile strength of manually produced welds is 3% lower than for fully mechanized produced welds. For the yield strength, however, the difference is 2% and therefore negligible. There are slight differences in the ductility of the welds and the associated elongation at break. One reason for this may be the described differences in the geometry of the welds. For this purpose, numerical investigations are currently carried out to assess the influence of local strengths and strains within the weld cross-section and the heat-affected zone. For the undermatching material combination S690QL-G42 the stress-strain curves are shown in Figure 4. The behaviour with regard to the strength properties is similar to that of the overmatching combination S690QL-G79. The mean tensile strength of the manual produced welds is about 1% below the fully mechanized produced specimens. For the yield strength the difference is 3%. It is slightly larger than for the overmatching material combination. Further differences lie in the ductility of the manually produced welds, which on average exceeds the values of the fully mechanized produced welds. Table 3 shows the averaged tensile strengths of all tests depending on the filler metal. It can be seen that the tensile strengths of the manually produced welds are always slightly lower than those of the fully mechanized produced welds. However, the differences are within 3.8% of the standard deviation for the tensile strength. The influence of the degree of automation on the

Figure 2.

Macro sections of welds produced fully mechanized (left) and manually (right).

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Figure 3. Stress-strain curves of the material combination S690Q – G79 for fully mechanized and manually produced welds.

Figure 4. Stress-strain curves of the material combination S690Q-G42 for fully mechanized and manually produced welds.

Table 3. Average tensile strength in N/mm² and tensile strength ration of the manually and fully mechanized produced welds. Filler metal

G42

G62

G79

fwu - manual fwu - full mechanized Tensile strength ratio

638 646 0,99

777 806 0,96

857 884 0,97

strength characteristics of the welds can be considered as low for the same cooling time t8/5. The influence of the weld geometry on the ductility will be examined in more detail.

1107

3.4 Influence of the number of weld layers The base materials S700MC and S690QL were welded with the filler metals G42 and G79 to examine the influence of the number of weld layers on the mechanical properties of welds. In addition, the material combination S960QL–G42 was examined. The specimens were welded fully mechanized with a cooling time t8/5 of 12 seconds. Figure 5 shows the macro sections of welds with three (left) and two layers (right). For the 3-layer welds a total of 8 weld beads were introduced, for the 2-layer welds 4 weld beads. With the large opening angle and the chosen cooling time, two weld beads were needed to get a complete flange connection in the 3rd layer of the weld. The stress-strain curves of the material combination S700MC-G42 shown in Figure 6 are exemplary for the behaviour of under-matching welds. The tensile strength of the weld with three layers is 4 %, the yield strength 7 % lower than for the weld with two layers. In addition, the ductility of the welds with two layers is slightly higher. This behaviour can also be observed for overmatching welds. In contrast to the under-matching weld, the characteristic strength values are at a similar level. Table 4 summarizes the average values of the yield and tensile strength as well as the corresponding ratio values of the welds with three and two layers. For under-matching welds, there is a slight influence of the number of layers on the strength. However, in the case of overmatching

Figure 5.

Macro section of a weld with three (left) and two layers (right).

Figure 6.

Stress-strain curves of the material combination S700M–G42 for two- and three-layer welds.

1108

Table 4. Average strengths in N/mm² for welds with three and two layers and their ratio values. Filler metal

G42

G79

Filler metal

G42

G79

fwu – 3 layer fwu – 2 layer Tensile strength ratio

658 687 0.96

884 885 1.00

fwy – 3 layer fwy – 2 layer Yield strength ratio

541 584 0.93

752 749 1.01

welds this was not observed. One reason for this behaviour are the differences in the alloying concepts of the filler metals. The G79 filler metal used is highly alloyed and has, among other things, a fin-grain-forming alloying element in titanium. When the weld metal will be heated above the Ac3 temperature, such as in the case of multilayer welds, a finer-grained microstructure will be produced, which leads to higher strengths. The filler metal G42 does not contain any fine-grainforming alloying elements, multilayer welds result in locally coarser microstructures and a drop in strength. The mixing of liquid filler metal and molten base material during the welding process can also have an influence on the results. With a lower number of weld layers, the proportion of melted flank material is higher than for more layers. The distribution of these proportions in the weld metals depends on the dominant melt flow. For undermatching welds, the influence of the molten flank material can be higher as for overmatching welds were the strength differences between base and filler metal are small. However, the test results do not provide any unambiguous information for this assumption. In addition to the tensile tests carried out, hardness measurements were performed on welds with three layers in two planes. The tests were carried out in accordance to EN ISO 9015-1 using the Vickers hardness test method with a test load of 9.81 N. The results and the position of the measuring points for the hardness measurement on the weld S700MC–G42 with three weld layers and a cooling time of 20 seconds is shown in Figure 7. An analysis of the hardness values in the heat-affected zone shows that the values are below the hardness of the base material. This can be seen in the upper (layer 1) and lower measuring series (layer 2). It can be stated that the maximum hardness values of the heat-affected zone at the lower measurements are lower than those of the upper measurements due to the repeated thermal influence. In addition, the expected drop in hardness between the heat-affected zone and the weld metal is clearly shown in both measurement series. The hardness values of the upper measurements are higher in the edges than in the middle area. This results from welding of the cover layer, which was carried out with two welding beads. From this, multiple microstructural

Figure 7.

Hardness curves of the material combination S700M-G42 for a cooling time of 20 seconds.

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transformations and tempering effects were happening in this area. This is also indicated by the increasing tendency of the hardness from left (first cover layer) to the right (second cover layer) in the middle of the weld metal. Quantitatively, differences in hardness of up to 35 HV1 can be observed in this area. The hardness values in the lower measuring range are above those of the centre of the weld metal in the upper measuring range, as these have been subjected to less thermal influence. Taking into account a correlation between tensile strength and hardness, the series of hardness measurements confirm qualitatively the results of the tensile tests.

4 SUMMARY In this paper, experimental investigations on the influence of the welding process and the number of layers on the strength and ductility of the welds are presented. The investigations, which include undermatching and overmatching welds, have been carried out with a new test procedure, which is part of a new design approach for welded joints in the frame of the ongoing AiF-FOSTA research project P1020. It could be determined with the scope of the examinations carried out, that the influence of the base material on the mechanical properties of the welds is negligible. The filler metal is decisive for the load-bearing capacity of the weld. This behaviour was observed for welds produced fully mechanized under laboratory conditions and in welds produced manually by industrial partners. A comparison of the degree of automation of the welding process has shown that with manually produced welds approximately the same strength values can be achieved as with welds produced fully mechanized. Studies on the influence of the number of layers on the strength and ductility of welds have shown that a differentiation must be made between the individual welding consumables. While no influence of the number of layers on the strength characteristics was found for the high-alloyed filler G79, the tensile and yield strength decreased for the low alloyed filler G42 with higher number of layers. One reason for this lies in the different alloying concepts of the welding consumables and the associated chemical composition of the weld metal. Fine grain alloying elements have a positive influence on the strength of multi-layer welds. In addition, there are slight differences in the ductility of the welds depending on the number of layers. Among other things, these can be attributed to the structure of the weld metal. The assumption, which takes into account a mixing of the molten base material and liquid filler metal during the welding process, could neither be confirmed nor denied by the experimental tests. For this reason, further investigations are necessary taking into account a lager selection of mismatching ratios and filler metals with different alloying concepts. The results of the presented investigations will be considered in the development of the new design approach for welds. Detailed knowledge about various influencing parameters on the strength and ductility of welds allow an economical design of welded joints. ACKNOWLEDGMENT The IGF-project 19043 BR/P1020 “Economic welding of high-strength steels” of the FOSTA – Forschungsvereinigung Stahlanwendung e. V. Düsseldorf, is encouraged within the program for the promotion of Industrielle Gemeinschaftsforschung (iGF), funded by the Federal Ministry of Economics and Energy on the basis of a resolution of the Deutscher Bundestag. The authors of this paper would like to thank for this funding and the participation of the industry partners. REFERENCES Stroetmann, R. & Kästner, Th. & Hälsig, A. & Mayr, P. 2018a. Influence of the cooling time on the mechanical properties of welded HSS-joints. Steel Construction 11 (2018), Nr. 4, S. 264–271. Stroetmann, R. & Kästner, Th. & Werner, L. 2018b, Welds for high-strength steels – Development of new design rules, 40th IABSE Symposium – Tomorrow´s Megastructures 2018, Nantes.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Validation of the Overall Stability Design Methods (OSDM) for tapered members J. Szalai ConSteel Solutions Ltd., Budapest, Hungary

F. Papp & G. Hajdú Faculty of Architectural, Civil and Transportation Engineering, Széchenyi István University, Győr, Hungary

ABSTRACT: Two new stability design methods are demonstrated and validated: the Overall Strength Reduction Method (OSRM) and the Overall Imperfection Method (OIM). Both methods are based on the linear buckling analysis (LBA) of global structural models and use the standard reduction curves. The OSRM is formulated in the classic way using generalized slenderness and reduction factors while the OIM uses equivalent amplitude for the buckling mode based geometrical imperfection. These new design methods cover all types of buckling modes, which can be calculated by LBA of structural models composed of tapered members with arbitrary support conditions and subjected to any complex loading. This paper clarifies the mechanical interpretation and proper calculation of all the components of the two methods in case of tapered members with arbitrary support conditions. The validation is performed on GMNIA results for several different buckling situations of tapered members proving the accuracy of the OSDM.

1 INTRODUCTION In Szalai & Papp (2019) a new design methodology is presented which utilizes the overall Linear Buckling Analysis (LBA) results of any structural model. The Overall Stability Design Methods (OSDM) have two equivalent alternatives based on the same underlying mechanics: the Overall Strength Reduction Method (OSRM) which uses the traditional reduction factor for the calculation of the design buckling resistance and the Overall Imperfection Method (OIM) which calculates the equivalent amplitude for the buckling mode based geometrical imperfection. Both methods are based on an essential underlying assumption which states that any complex global buckling mode calculated by the LBA can be classified into finite number of fundamental buckling mode types which are significantly different in terms of various mechanical characteristics (loading, mode shape displacement components etc.). It is also assumed that the well-known, calibrated standard buckling curves are solutions for some of these fundamental buckling modes (for flexural buckling or LTB of doubly symmetric cross-sections etc.) which can be used within the OSDM to ensure their reliability level. Accordingly the methodology consists of two basic steps: (1) a universal transformation method which converts the real structural model with a certain complex buckling problem into a properly defined equivalent reference member which is a prototype model of the corresponding fundamental buckling mode type – this is the ultimate generalization of the effective length (or equivalent member) method to any buckling problem (2) a closed-form analytical solution for the reference member which is based on the standard buckling curves corresponding to the equivalent fundamental buckling mode type – this is the ultimate generalization of the beam-column buckling strength interaction equations. The calculation scheme of the OSDM with this two basic steps is illustrated in Figure 1. In this paper the concrete calculation steps for both methods are presented with a benchmark example on a tapered member, then the results of the numerical validation is shown. 1111

Figure 1.

Calculation scheme of the OSDM.

2 STEPS OF THE OSDM In this section the calculation steps of the OSDM are presented based on Szalai & Papp (2019) where a bit more detailed background is given for the formulas. In this paper for the sake of simplicity and brevity the following assumptions are made: 1. The pre-buckling deformations from initial loading are negligible so the second order effects are due entirely to the imperfections. The handling of second order effects due to loading is described in Szalai (2017) and Szalai (2018). 2. The real structural model is a single web tapered member with doubly symmetric I crosssection and arbitrary load and support condition. Application of the OSDM to plane frames is demonstrated in Szalai & Papp (2019).

2.1 Numerical calculations on the real structural model 2.1.1 Step 1: Structural analysis Two numerical structural analysis results are required for the OSDM performed on the perfect real structural model: linear elastic analysis (LA) and linear buckling analysis (LBA). The following results are used further (x denotes the longitudinal coordinate along the reference line of the tapered member):

 – first order internal forces and moments: S I ðxÞ ¼ N I ðxÞ; M I v ðxÞ; M I z ðxÞ; BI ðxÞ – elastic critical buckling load factor: αcr – buckling mode shape: ηcr ðxÞ ¼ ½0; wcr ðxÞ; vcr ðxÞ; ’cr ðxÞ

2.2 Forward model transformation In this phase an equivalent reference member (ERM) is defined by the previous results obtained on the real structural model. The ERM in general is a straight, prismatic, simply supported member subjected to uniform compression force and/or bending moments. For the proper definition of the ERM the following data should be determined: – geometry: cross-section and member length – member loads (causing uniform compression force and/or bending moments) – buckling mode type (one of the fundamental buckling modes) These set of data is defined through a suitably selected equivalent point (ep) along the real structural model where the second order internal stress utilization effect from the buckling 1112

mode shape is the highest (or the second order flexural curvature of the compressed flange from the buckling mode is the highest). 2.2.1 Step 2: Determination of the equivalent point (ep) The recovery of the equivalent point is practically done by calculating the internal force and moments due to the deformation of the buckling mode shape along the longitudinal member axes of the real structural model:

 S cr ðxÞ ¼ 0; M cr y ðxÞ; M cr z ðxÞ; Bcr ðxÞ

ð1Þ

and calculating the corresponding cross-section resistances (which is varying along the tapered member axis) using the appropriate classes:

1 1 1 RðxÞ ¼ Nsec ðxÞ My;sec ðxÞ Mz;sec ðxÞ

1 Bsec ðxÞ

ð2Þ

then the linear utilization function form these internal force and moments can be finally determined: Usec;cr ðxÞ ¼ RT ðxÞS cr ðxÞ ¼

1 αsec;cr ðxÞ

ð3Þ

where αsec;cr ðepÞ is the corresponding linear load multiplication factor (LMF). The equivalent point x = ep is where the utilization function of Eq. (3) takes the highest value: Usec;cr ðepÞ ¼ max Usec;cr ðxÞ ! min αsec;cr ðxÞ ! x ¼ ep

ð4Þ

From the equivalent point the following data can be received directly: – the cross-section of the ERM and its class and resistances – the uniform compression force and/or bending moments acting on the ERM 2.2.2 Step 3: Buckling mode classification through the equivalent point For the calculation of the length of the ERM the proper buckling mode type should be defined, since the obtained cross-section and loads can generally lead to various buckling mode types. The classification into a fundamental buckling mode class (BMC) is done by the buckling mode shape components and the loading at the equivalent point of the tapered member according to Table 1. 2.2.3 Step 4: Equivalent length of the reference member The length of the ERM (Leq) can be calculated from the well-known analytical formulae for the proper BMC where the only unknown parameter is the length (Table 2.). 2.3 Analytical solution of the ERM Once the ERM is fully defined it is solved based on the generalized Ayrton-Perry formulation defined by Szalai (2017). 2.3.1 Step 5: Equivalent imperfection factor For the different fundamental BMCs the equivalent imperfection factors are shown in Table 3. For the pure modes these are the calibrated standard imperfection factors of the Eurocode 3, and for the coupled mode it is based on the interpolated formulation derived in Szalai (2017).

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Table 1. The fundamental Buckling Mode Classes (BMC). BMC

Cross-section at Active load Buckling mode shape the ep components at the ep component(s) at the ep Buckling mode type fwcr g fvcr g f’cr g fvcr ; ’cr g fvcr ; ’cr g

NI NI NI MI y N I ; MyI

BMC_01 BMC_02 BMC_03 doubly BMC_04 symmetric BMC_05

strong axis flexural buckling weak axis flexural buckling torsional buckling lateral-torsional buckling coupled lateral-torsional buckling

Table 2. Equations including the Leq equivalent length which can be expressed for the ERM. αcr N I ¼ Ncr;y ¼ π2 EIy =L2 αcr N I ¼ Ncr;z ¼ π2 EIz =L2  αcr N I ¼ Ncr;x ¼ 1=r2 GIT þ π2 EIw =L2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi αcr MyI ¼ r Ncr;z Ncr;x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi αcr MyI ¼ r Ncr;z  αcr N I Ncr;x  αcr N I

BMC_01 BMC_02 BMC_03 BMC_04 BMC_05

Table 3. The equivalent imperfection factors.   BMC_01 ηERM ¼ αy λy  0; 2 Ny BMC_02

ηERM ¼ αz ðλz  0; 2Þ Nz

BMC_03

¼ αz ðλT  0; 2Þ ηERM Nx

BMC_04

2 ηERM My ¼ ðλLT =λz Þ αLT ðλz  0; 2Þ

BMC_05

ηERM coupled

¼

NI Nsec NI Nsec

MI

y þ My;sec

ηERM N

þ

MyI My;sec μ I MyI N Nsec þ My;sec

ηERM My ¼

αsec;a ERM αsec;a ERM η þμ η αsec;N N αsec;My My

2.3.2 Step 6 in OIM: Second order effect on the reference member From this point there are different calculation steps for the OIM and OSRM. In case of the OIM the basic result of the ERM which is used for the backward transformation is the proper value of the second order effect which can be calculated from the equivalent imperfection factor. It is done in the form of linear load multiplication factor as defined in Szalai & Papp (2019) using the general form of the imperfection factor as derived in Szalai (2017) as follows: αII;ERM sec;cr

  αsec;a 1 ¼ ERM 1  η αcr

ð5Þ

2.3.3 Step 6 in OSRM: Equivalent reduction factor The base for the backward result transformation in the case of OSRM is the reduction factor of the ERM. It is a straightforward calculation using the generalized slenderness pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (λERM ¼ αsec;a =αcr ) and the equivalent imperfection factor in the well-known Ayrton-Perry based reduction formula: χERM ¼

1 1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where  ¼ 1 þ ηERM þ λERM 2 2  þ 2  λERM 1114

ð6Þ

2.4 Backward result transformation When the ERM has the proper solution its results should be transformed back to the real structural model to reach the final buckling solution. The steps are now going again on two paths for the OIM and the OSRM. 2.4.1 Step 7 in OIM: Equivalent geometrical imperfection In the OIM the correct design amplitude is to be determined for the applied equivalent geometrical imperfection with the shape of the calculated buckling mode of the tapered member. Important to note that the buckling mode shape (ηcr ðxÞ) can have arbitrary amplitude when calculating S cr ðxÞ of Equation 1 and it is linearly dependent on the actual amplitude. Accordingly the proper amplitude scale factor (δeq ) can be calculated from the equality of the second order II II;ERM effect of the real structural model at the equivalent point (Usec;cr ðepÞ) and the ERM (Usec;cr ): II II;ERM ðepÞ ¼ Usec;cr ! δeq Usec;cr

1 ηERM 1 ¼ αsec;cr ðepÞ αcr  1 αsec;a 1  α1 1

ð7Þ

cr

The correct amplitude of the equivalent geometrical imperfection of the real structural model which after some arrangement takes the following form: ηcr;eq ðxÞ ¼ δeq ηcr ðxÞ ¼ ηERM

αcr αsec;cr ðepÞηcr ðxÞ αsec;a

ð8Þ

The final step of the OIM is running a second order analysis and a cross-section check on the real structural model with the equivalent geometrical imperfection. 2.4.2 Step 7 in OSDM: Non-uniform equivalent reduction factor Since by definition the reduction factors includes all the second order effects due to the proper equivalent geometrical imperfection in the OSRM the equivalency relationship means the equality of the reduction factors of the ERM (χERM ) and the real model at the equivalent point (χea ðepÞ). Important to note that the equivalent reduction factor of the real structural model is not constant in general but its distribution follows the distribution of the second order effects defined by Equation 3. Accordingly after some manipulation – described in Szalai (2018) – the final from of the equivalent non-uniform reduction factor along the real structural model can be written as follows: χeq ðxÞ ¼ χERM mðxÞ ¼ χERM

χERM

1 αsec;cr ðepÞ αsec;a ðxÞ þ ð1  χERM Þ αsec;cr ðxÞ αsec;a ðepÞ

ð9Þ

The final check of the buckling resistance along the real tapered member takes the following form: 1 1 χeq ðxÞαsec;a ðxÞ=γM1

ð10Þ

3 ILLUSTRATIVE EXAMPLE The calculation steps of the OSDM is demonstrated on a tapered member (Figure 1) with S235 material, symmetric I section of flanges 300 mm/14 mm and web 290-435 mm/8 mm, span of 8629 mm, simple supports at the ends and a lateral restraint at the top flange at a distance of 4314 mm from the deeper section end. The member is subjected to uniform compression (N = 379,64 kN) and uniform bending moment (My = 233,36 kNm). The dominant LT buckling mode of the tapered member is shown in Figure 2. All the necessary calculations are performed in ConSteel 12 (2018) software using 7 DOF beam-column element accounting for the tapering effect, the steps are shown in Table 4. 1115

Figure 2.

Geometry, support and load condition of the tapered member example.

Figure 3.

LTB mode shape of the tapered member example.

Table 4. General steps of OIM and the OSRM for tapered member. Designations of parameters Step 1.1 Linear elastic analysis (LA) – Axial force – Bending moment Step 1.3 Linear buckling analysis (LBA) – Elastic critical load factor – Amplitude of buckling shape on the beam centroid Step 2.1 Internal forces and moment due to buckling mode shape – Bending moment in the equivalent point – Bimoment in the equivalent point Step 2.2 Cross-section resistance – Axial compression – Bending around major axis – Bending around minor axis – Bimoment Step 2.3 Linear load multiplication factor in the equivalent point Step 2.4 Location of the equivalent point Step 3.1 Buckling-active internal forces and moments

Notation

Dimension

Value

NI MIy

kN kNm

379.64 233.36

αcr vcr.max

mm

4.45 9.698

Mcrz(ep)

kNm

21.26

Bcr(ep)

kNm2

2.86

Nsec(ep) My,sec(ep) Mz,sec(ep) Bsec(ep) αsec,cr(ep)

kN kNm kNm kNm2

2530,2 346,9 14,91 2,293 1.085

ep

mm

6615

(Continued )

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Table 4. (Continued ) – Axial force – Bending moment around the major axis Step 3.2 Classification of the buckling mode Step 4 Equivalent length of the reference member Step 5 Equivalent imperfection factor – Imperfection factor for FB – Reduced slenderness for FB – Imperfection factor for FB – Imperfection factor for LTB – Reduced slenderness for LTB – AP imperfection factor for LTB – Modifying factor – Equivalent imperfection factor Step 6.1 Buckling-active linear multiplication factor (in equivalent point) Step 6.2 Second order effects linear multiplication factor Step 7 Equivalent scale factor Final check – Second order internal forces and moments due to equivalent geometric imperfection (in critical point) ● Axial compression ● Bending around major axis ● Bending around minor axis ● Bimoment Step 8 Maximum cross-section utilization (in critical cross-section) Ultimate load factor Step 6.1 Buckling-active linear multiplication factor (in equivalent point) Step 6.2 Equivalent reduced slenderness Step 6.3 Equivalent reduction factor Step 7.1 Non-uniform load effect modification factor (in critical cross-section) Step 7.2 Non-uniform reduction factor (in critical cross-section) Step 8 Maximum buckling utilization (in critical cross-section) Ultimate load factor

Na(ep) May(ep) BMC_05 Leq

kN kNm

379.64 233.36

mm

4170

αz λz ηN αLT λLT ηMy μ ηeq αsec,a(ep)

0.490 0.584 0.188 0.378 0.516 0.113 0.979 0.125 1.226

αIIeq,cr

7.608

δeq

0.492

NII,imp MII,impy MII,impz BII,imp Umax

kN kNm kNm kNm2

379.6 243.8 10.06 1.320 0.985

αu αsec,a(ep)

1.008 1.226

λERM χERM mcr

0.678 0.832 1.023

χeq,cr

0.850

Umax

0.990

αu

1.002

4 VALIDATION In order to validate the accuracy of the proposed OSDM in the buckling design of tapered members numerical GMNIA performed on three different support and load configurations: Case 1: simple support, uniform compression and bending moment Case 2: simple support, uniform compression, linear bending moment Case 3: simple support with an intermediate compressed flange restraint, uniform compression and bending moment In each cases several different cross-section types, tapering ratio and slenderness values are considered. The detailed description of the numerical model can be found in Hajdú (2019).

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The normalized buckling resistance results from the GMNIA calculations (re) and from the proposed OSDM (rt) is plotted in Figures 4–6.

Figure 4.

Validation plot for Case 1.

Figure 5.

Validaqtion plot for Case 2.

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Figure 6.

Validation plot for Case 3.

5 CONCLUSIONS Two new buckling design methods are presented for the buckling verification of tapered members. The Overall Imperfection Method (OIM) uses directly a properly scaled buckling mode based equivalent imperfection and the Overall Strength Reduction Method (OSRM) generalized the classical buckling design based on reduction factors. The calculation steps of the methods are presented in general and illustrated on a benchmark example with a tapered member with intermediate restraint. Finally the methods are validated against GMNIA results performed on tapered members with different cross-sections, slenderness, support and load conditions. The methods are suitable for software implementation providing a fully automatic and economic way of buckling design for any tapered members. REFERENCES ConSteel 12 2018. Structural analysis and design software, www.consteelsoftware.com. Hajdú, G. 2019. The Validity of the Universal Transformation Method in Global Buckling Design. The 14th Nordic Steel Construction Conference, September 18–20, 2019, Copenhagen, Denmark. Papp, F. & Szalai, J. & Movahedi, R. M. 2019. Out-of-Plane Buckling Assessment of Frames through Overall Stability Design Method. The 14th Nordic Steel Construction Conference, September 18–20, 2019, Copenhagen, Denmark. Szalai, J. & Papp, F. 2019. New stability design methodology through overall linear buckling analysis. The 14th Nordic Steel Construction Conference, September 18–20, 2019, Copenhagen, Denmark. Szalai, J. 2017. Complete generalization of the Ayrton-Perry formula for beam-column buckling problems. Engineering Structures 153:205–223. Szalai, J. 2018. Direct buckling analysis based stability design method of steel structures. Ninth International Conference on Advances in Steel Structures (ICASS2018) 5–7 December, 2018, Hong Kong, China.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Stability design of cable-stayed columns T. Tankova, L. Simões da Silva & J.P. Martins Institute for Sustainability and Innovation in Structural Engineering, University of Coimbra, Portugal

ABSTRACT: Pre-stressed stayed columns offer several advantages among traditional ones: they have enhanced buckling resistance in comparison to conventional columns provided by the pre-stressed cables and cross-arms; they are also aesthetically appealing, combining a strong architectural effect with a top notch of engineering solution. In this paper, design rule for stability design of pre-stressed cable-stayed columns is presented. It is based on the wellknown Ayrton-Perry format, i.e., on a combination of first and second order effects. It is consistent with the design rules for uniform columns in Eurocode 3.

1 INTRODUCTION The design of long span structures implies using slender columns whereby buckling dominates the load-carrying capacity. In such cases, using high strength steel has little, or no effect, on the load carrying capacity and so the common solution is to increase the cross-section dimensions. However, improvement of the stability resistance can be achieved by the adoption of structural solutions or configurations which increase the buckling resistance, such as cable-stayed columns. In fact, adding stays and cross-arms to slender columns can improve both the elastic buckling load and the load carrying capacity as they provide translational and rotational restraint along the length. Cable-stayed columns have been studied in the past by several researchers. First attempts are found by Chu & Berge (1963) and Mauch & Felton (1967) in pursuit of the buckling load of cable-stayed columns. Many studies were developed in the 70s, starting with the work by Smith et al. (1975) whereby expressions for the determination of the maximum critical load in symmetric and asymmetric buckling shapes were proposed. Numerical solutions of the problem can be found by Khosla (1975) and Hathout (1977). A significant advance in the knowledge of the behaviour of cable stayed columns is the paper by Hafez et al. (1979) who proposed expressions for the optimum pre-stressing force. Furthermore, the effect of initial imperfections on the buckling load was investigated based on experimental tests by Wong & Temple (1982) and later extended using FE simulations, Temple et al. (1984). Smith (1985) provided analytical studies to distinguish between buckling modes. Later on, the second-order instability of stayed columns was derived including initial imperfections by Chan et al. (2002). The advantage of using split up cross-arms was studied numerically and compared with other type of cross-arms by Van Steirteghem et al. (2005). A series of full-scale tests were performed in Araujo et al. (2008, 2009), which were used for FE simulations and an extensive parametric study was performed to assess the effect of different parameters on the buckling behaviour. The post buckling of the stayed column was assessed by the application of the Rayleigh-Ritz method and the results were compared with numerical simulations in Saito & Wadee (2008, 2009a, 2009b). Osofero et al. (2012a, 2012b) summarized the assessment of the effects of imperfection shape, orientation and magnitude on the buckling behaviour of columns. The sensitivity of the load-carrying capacity to the geometry of the stayed column, the initially applied pre-stress level within the stays and the initial global imperfection was investigated numerically. A general design procedure for pre-stressed stayed columns with a single cross-arm system was proposed (Wadee et al., 2013). Finally, Serra et al. (2015) and 1120

Martins et al. (2016) presented the results of an experimental study on 12 and 18 m pre-stressed stayed column with single and double cross-arms. In this paper, a design approach in an Ayrton-Perry format is presented and explained. Firstly, the essential background for the buckling resistance of columns is briefly summarized. Furthermore, the necessary adjustments to account for the enhanced structural behaviour are explained. The proposed expressions are calibrated against the experimental results obtained in Serra et al. (2015). 2 BUCKLING RESISTANCE: BACKGROUND 2.1 Introduction In this section the theoretical background behind the adopted approach is briefly presented. It is based on the current design format of Eurocode 3 as a reduction factor to the plastic resistance of the column. For that, the background of the design rules for flexural buckling of prismatic columns is presented first. Then a short discussion on the possible buckling modes of stayed columns is given. Finally, the assumptions behind the proposed model are presented. 2.2 Flexural buckling of simply supported columns The behaviour of a perfect simply supported compressed column with length L is described by the following differential equation and boundary conditions EIv00 þ Nv ¼ 0 with vð0Þ ¼ vðLÞ ¼ 0 and v00 ð0Þ ¼ v00 ðLÞ ¼ 0

ð1Þ

The solution of Equation (1) yields the elastic critical load and mode shape: Ncr ¼

π2 EI πx with vðxÞ ¼
v sin L2 L

ð2Þ

In a similar way, it is also possible to describe the imperfect simply supported columns (Figure 1): EIv00 þ Nv þ Nv0 ¼ 0 with vð0Þ ¼ vðLÞ ¼ 0 and v00 ð0Þ ¼ v00 ðLÞ ¼ 0

ð3Þ

which leads to relationship between the amplitude of the lateral displacement and the amplitude of the initial imperfection:
v ¼

N
v0 ðNcr  NÞ

ð4Þ

The design equation is established on the basis of linear yield criterion for the imperfect column. It accounts for the axial load and second order bending moment that arises due to initial imperfections: N M II þ ¼ 1:0 Afy Wfy

Figure 1.

N N ð
v þ
v0 Þ þ ¼ 1:0 Afy Wfy

Column buckling: deformed configuration.

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ð5Þ

where N is the applied axial force, MII is the bending moment due to second order effects, Afy is the column’s plastic axial capacity and Wfy is the column’s elastic bending. Inserting Equation (4) into Equation (5) it is possible to express the equation as a function of the initial imperfection:
v0 N N ¼ 1:0 þ Afy Wfy ð1  N=Ncr Þ

ð6Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N Afy =Ncr and χ ¼ Afy

ð7Þ

 
v0 A χ χ χ ¼ 1:0 ! χ þ η ¼ 1:0 ! χ þ α λ  0:2 ¼ 1:0 2 2 2 Wz ð1  χλ Þ ð1  χλ Þ ð1  χλ Þ

ð8Þ

Defining λ¼ it is possible to conclude that χþ

where χ is the reduction factor of the column’s axial capacity, η is the generalized imperfection factor as defined in EC3, which was calibrated to account for initial out-of-straightness and residual stresses. 2.3 Flexural buckling of stayed simply supported columns Stayed columns have the advantage of increased buckling load due to the restraint provided from the cross-arms and the prestressed cables. The elastic buckling load for these columns was derived by Smith (1975) and was further developed by Hafez et al. (1979) as mentioned earlier. Generally, two buckling modes are possible: i) symmetric, when the restraint provided by the cross-arms and stays is not completely sufficient to prevent the displacement at midspan (Figure 2, left); and ii) asymmetric, when the restraint at midspan is sufficient and the buckling is also accompanied by buckling of the cross-arms (Figure 2, right). The earlier studies on stayed columns report that the predominance of one mode or another depends only on the geometrical characteristics of the stayed column such as ratio between lengths of the crossarms and column and diameter of stays. The application of the linear yield criterion, as it was shown in the previous section, requires the consideration of the different buckling modes, where the resistance is given by the lowest. If Mode 1 (Figure 2, left) is considered, the linear yield criterion can be expressed by Equation (9), where the additional term accounting for the restraint introduced by the stays is accounted by an extra bending moment and the critical buckling force calculated using linear buckling analysis in the presence of the prestressing force (Ncr,a).

Figure 2.

Stayed columns buckling modes (Smith et al., 1975).

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N M II Q1 L þ  ¼ 1:0 Afy Wfy 2Wfy

ð9Þ

This moment is calculated using the Equation (42) from Smith et al. (1975) using the reaction Q1 given by: Q1 ¼ 2Ks sin2 ðαÞδc ¼ 2Ks sin2 ðαÞ
v

ð10Þ

Hence, the retraining bending moment also depends on the lateral displacement at midspan, so that Equation (9) becomes
v0 N N Ks sin2 ðαÞ
vL   þ ¼ 1:0 Afy Wfy ð1  N Ncr;f Þ Wfy

ð11Þ

After straight-forward transformations, Equation (11) yields: ! Ks sin2 ðαÞL ¼ 1:0 χþη 1 2 Ncr;f ð1  χ
λ Þ χ

ð12Þ

whereby the slenderness and the reduction factor are defined as: λ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Afy =Ncr;f and

χ¼

N Afy

ð13Þ

with Nf ¼ Na þ 4Tf sin α ¼

Na þ 4Tini cos α C2

Ncr;f ¼ Ncr;a þ 4Tf sin α ¼

Ncr;a þ 4Tini cos α C2

ð14Þ ð15Þ

where Na is the applied force, Tini is the initial prestress in the cables, Ncr,a is numerically obtained from a LBA (linear buckling analysis) and C2 is given by (Hafez et al. (1979)): 2cos2 α

C2 ¼ 1 þ Kc χ¼

1 2sin2 α þ Ks Kca

!

 1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with  ¼ 0:5 1 þ ηtot þ
λ 2  þ 2 
λ

ð16Þ

ð17Þ

The last equation constitutes the proposed method to compute the buckling resistance of prestressed stayed columns corresponding to Mode 1. Since buckling Mode 2 is characterized by a complete restraint at midspan, therefore, the application of Equation (12) should be corrected with the bending moment arising from the reaction Q2.

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3 VALIDATION 3.1 Introduction The proposed procedure is based on the Ayrton-Perry format currently used for the stability design of columns according to EN 1993-1-1. It makes use of the critical buckling force calculated using linear buckling analysis in the presence of the prestressing force (Ncr,a). Furthermore, the imperfection factors to be used were calibrated based on an experimental investigation (Serra et al. (2015) and Martins et al. (2016)). In this section, the proposed approach is validated against experimental results by Serra et al. (2015). Firstly, the parameters of the experimental programme are presented. The additional calibration of imperfection factors is performed and finally, the results obtained using the proposed method and method by Wadee et al. (2013) were compared. 3.2 Experimental programme The experimental programme covered 10 columns, using three different cross-section geometries, two different cable diameters were tested for each column. The experimental programme covers steel grades S355 and S690; two lengths 12 and 18m; and also, for each column 5 levels of initial prestress were tested. The summary of these parameters is given in Table 1 and more details can be found in Serra et al. (2015). In this study only the specimens with single cross-arms were used. 3.3 Calibration Based on the analytical proposal from Section 2.3, the imperfection ηtot was calibrated against the numerical results using Equation (18). It was split into three terms: i) ηEC3 which is the generalized imperfection directly taken from Eurocode 3 curve a0 (αEC3 = 0.13) for S690 steels and from curve a (αEC3 = 0.21) for S355 steels; ii) Cpr, which accounts for the cable stiffness; and iii) ηpr, additional imperfection. These parameters are aggregated in one global parameter following Equation (12).

Table 1. Experimental programme. Column length Code

12m

C01-C1 C01-C2 C02-C1 C02-C2 C03-C1 C03-C2 C04-C1 C04-C2 C05-C1 C05-C2 C06-C1 C06-C2 C07-C1 C07-C2 C09-C1 C09-C2 C10-C1 C10-C2

X X X X X X X X

Column cross-Section 18m

CS1

CS2

Number of cross-arms CS3

4

X X X X

X X X X X X X X X X X X X X X X X X

X X X X X X X X X X X X X X X X X X

X X X X X X

Steel 8

where: CS1 – CHS 101.6 x 8.0; CS2 – CHS 139.7 x 6.3; CS3 – CHS 177.8 x 6.3.

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S355

Cables S690

X X

10mm

13mm

X X X X

X X

X X X X

X X X X X X

X X X X X X

X X X X

X X X X

X X

X X

χexp þ ηtot

χexp ¼ 1:0 2 1  χ
λnum

ð18Þ

exp

ηtot ¼ ηpr  ηEC3  Cpr

ð19Þ

where Cpr ¼

! Ks sin2 ðαÞL 1 Ncr;f

ð20Þ

  ηEC3 ¼ α
λ  0:2

ð21Þ

In Equations (18) and (19) the only unknown is ηpr:  ηpr ¼

1:0  χexp



1  χexp
λnum

χexp  Cpr  ηEC3

2

ð22Þ

The parameter ηpr was chosen according to Equation (20). It is set to unity once the prestress is equal to 0 and thus allowing for direct correspondence with the Eurocode 3 expression for columns without stays. Once the prestress is non-zero, it depends on two coefficients pi and qi, which in this case were chosen according to the steel grade. Nonetheless, global coefficients could be also adopted which would lead to slightly different accuracy.   1 Ti ¼ 0   p1 Ti =As þ p2 ð23Þ ηpr ðTi Þ ¼  2 ð T =A  i s Þ þ q2 Ti =As þ q2  Ti =As in MPa The application of the proposed design procedure is summarized in Figure 3. It starts with the value of Ncr,a. Next Ncr,f is calculated in step 2 (during this step the values of axial stiffness of the column, cables and cross-arms are also necessary to calculate the value of C2). In step 3

Figure 3.

Application of the method.

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Figure 4.

Comparison of different methods against experimental results.

Table 2. Comparison of results.

Average St. Dev. CoV R2

Pu,exp/Nu,Wadde (L/1000)

Pu,exp/Nu,Wadde (L/400)

Pu,exp/Nu,Wadde (L/200)

Pu,exp/Nu,proposed

0.99 0.26 26% 0.68

1.23 0.40 32% 0.60

1.57 0.33 21% 0.83

1.02 0.19 19% 0.87

the slenderness is evaluated and in step 4 the equivalent imperfection is calculated according to the level of initial prestress in the cables. With this step concluded, it is possible to calculate the reduction factor χ in step 5 and the maximum applied load in step 6. 3.4 Comparison Using the method that results from the calibration explained above, it is possible to calculate the values of the ultimate loads. Figure 4 summarizes the scatterplots of experiments and numerical analysis and compares them with the now calibrated and with the method proposed by Wadee et al. (2013) for an amplitude of the geometrical imperfection of L/1000. The scatter of results is presented in Figure 4 and Table 2 show a better correlation (higher value of r-square and lower value of CoV) for the proposed method.

4 CONCLUSIONS In this paper, a methodology for the design of cable stayed columns was presented. It is based on the Ayrton-Perry format and thus consistent with the current Eurocode 3 prescriptions for flexural buckling. The method can distinguish between buckling modes and it is easily applied using a step-by-step procedure as summarized by the flowchart (Figure 3). The calibration of the method against experimental results was also demonstrated. It was done only for Mode 1, because of the available experimental results. When the method was compared to other design proposal from the literature, it was shown to provide better results in terms of mean value and standard deviation. Nevertheless, the results presented in this paper are based on limited number of experimental results. Therefore, the concept can be further extended on the basis of advanced numerical simulations and the investigation may be deepened in order to provide expressions which are applicable to a larger range of cases. ACKNOWLEDGEMENTS This work was partly financed by: 1126

– the Research Fund for Coal and Steel under grant agreement RFSR-CT-2012-00028 (HILONG) – FEDER funds through the Competitively Factors Operational Programme - COMPETE and by national funds through FCT – Foundation for Science and Technology within the scope of the project POCI-01-0145-FEDER-007633.

REFERENCES Araujo, R.R., Andrade, S.A.L., Vellasco, P.C.G.d.S., da Silva, J.G.S., Lima, L.R.O. 2008. Experimental and numerical assessment of stayed steel columns, Journal of Constructional Steel Research, 64, 1020–1029. Araujo, R.R. 2009. Comportamento Estrutural de Colunas de Aço Estaiada e Protendida, Civil Engineering Department, Pontifical Catholic University of Rio de Janeiro, Brazil. Chan, S.-L., Shu, G.-P. Lü, Z.-T. 2002. Stability analysis and parametric study of pre-stressed stayed columns, Engineering Structures, 24, 115–124. Chu, K.H., Berge, S.S. 1963. Analysis and design of struts with tension ties, Journal of Structural Division, 89, 127–163. Gkantou, M; Tran, A; Martins, J. P; Ellen, M; Koltsakis, E; Afshan, S; McCormick, F; Veljkovic, M; Manoleas, P; Baniotopoulos, C; Remde, C; Baddoo, Nancy; Simões da Silva, Luís; Chen, A; Theofanous, M; Herion, S; Gardner, L; Aggelopoulos, Eleftherios; Fleischer, O; Cederfeldt, L. (2017). High Strength Long Span Structures (HILONG), Grant agreement RFSR CT 2012-00028, final report, ISBN 978-92-79-65601-9. Hafez, H.H., Ellis, J.S., Temple, M.C. 1979. Pre-tensioning of single cross-arm stayed columns, Journal of the Structural Division, 105, 359–375. Hathout, I.A.-S 1977. Stability analysis of space stayed columns by the finite element method, University of Windsor. Khosla, C.M. 1975. Buckling loads of stayed columns using the finite element method, University of Windsor. Martins, J.P., Shahbazian, A., Simões da Silva, L., Rebelo, C. R., Simões, 2016. Structural behaviour of prestressed stayed columns with single and double cross-arms using normal and high strength steel. Archives of Civil and Mechanical Engineering, 16, 618–633. Mauch, H.R., Felton, L.P. 1967. Optimum design of columns supported by tension ties, Journal of Structural Division, 93, 210–220. Osofero, A.I., Wadee, M.A., Gardner, L. (2012a). Experimental study of critical and post-buckling behaviour of prestressed stayed columns, Journal of Constructional Steel Research, 79, 226–241. Osofero, A.I., Wadee, M.A., Gardner, L. (2012b). Numerical Studies on the Buckling Resistance of Prestressed Stayed Columns, Advances in Structural Engineering, 16 487–498. Saito, D., Wadee, M.A. 2008. Post-buckling behaviour of prestressed steel stayed columns, Engineering Structures, 30, 1224–1239. Saito, D., Wadee, M.A. 2009a. Buckling behaviour of prestressed steel stayed columns with imperfections and stress limitation, Engineering Structures, 31, 1–15. Saito, D., Wadee, M.A. 2009b. Numerical studies of interactive buckling in prestressed steel stayed columns, Engineering Structures, 31, 432–443. Savin, I.V. 1977. Prestressed Load -Bearing Metal Structures, MIR Publishers, Moscow. Serra, M., Shahbazian, A., Silva, L. S., Marques, L., Rebelo, C., da Vellasco, P. C. G. S. 2015. “A full scale experimental study of prestressed satyed columns”, Engineering Structures, 1000, 490–510. Simulia (2014) ABAQUS FEA (version 6.14). Smith R., Mc Caffrey G.T., Ellis J.S. 1975. Buckling of a single cross-arm Stayed Column. Journal of the Structural Division, 101, 249–268. Smith, E.A. 1985. Behaviour of Columns with Pretensioned Stays, Journal of Structural Engineering, 111, 961–972. Temple, M., Prakash, M., Ellis, J. 1984 Failure Criteria for Stayed Columns, Journal of Structural Engineering, 110, 2677–2689. Van Steirteghem, J., De Wilde, W.P., Samyn, P., Verbeeck, B.P., Wattel, F. 2005. Optimum design of stayed columns with split-up cross arm, Advances in Engineering Software, 36, 614–625. Wadee, M.A., Gardner, L., Osofero, A.I. 2013. Design of prestressed stayed columns, Journal of Constructional Steel Research, 80, 287–298. Wong, K.C., Temple, M.C. 1982. Stayed columns with initial imperfection, Journal of the Structural Division, 108, 1623–1641.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Influence of geometrical imperfection of rib stiffeners on beam-to-column joint behaviour R. Tartaglia, M. D’Aniello, G.M.Di Lorenzo & R. Landolfo Department of Structures for Engineering and Architecture, University of Naples “Federico II”, Naples, Italy

ABSTRACT: The introduction of rib stiffeners in the extended end-plate joints can guarantee a beneficial increase of both the strength and ductility of the connection. Indeed, the presence of the rib influences the yield line distribution in the tensile zone of the connection, the depth of internal lever arm as well as the resistance of compression components of the joint. However, unexpected phenomena can develop if the stiffeners are not properly aligned with the beam web. Therefore, in this work, a parametric finite element (FE) analyses was conducted to investigate the influence of the rib constructional imperfection on both the local and global joint response. In particular, the variation of the compression center position and the development of additional internal actions in the bolt rows was monitored for five different rib misalignments. The results show that some geometrical imperfections could have a beneficial effect on the joint capacity limiting the beam out-of-plane mechanism when the plastic hinge develops at the beam extremity.

1 INTRODUCTION Steel rib plates are commonly used to increase the strength and the stiffness of steel beam– to-column joints in moment resisting frames (Giordano 2017, Montuori 2015, 2016, 2017a, and b, Nastri 2018). Among the wide variety of stiffened bean-to-column connections, extended end-plate stiffened by rib plates on both the tension and compression side, are commonly adopted for seismic applications because they are less expensive than haunched end-plate connections and characterized by symmetric hogging and sagging behaviour. Notwithstanding the key role of the stiffeners, their design, verification and fabrication is considered troublesome by European engineers and constructors due to the limited guidance given by the current Eurocodes and the increase of constructional costs as respect to unstiffened connections. The nonlinear monotonic and cyclic behaviour of stiffened connections can be predicted using experimental tests (D’Aniello et al. 2018), analytical approaches and sophisticated finite element models, methods which can be impractical and inconvenient in current design practice. With this regard, Kurejková and Wald (2017) recently presented a promising method to predict the response of steel joints with different types of stiffeners (i.e. rib plates and haunches) based on the “research finite element model” (RFEM). However, as highlighted by the same Authors, the RFEM is still less applicable in current practice owing to the difficulties in setting the geometrical and mechanical imperfections in advanced numerical models as well as the time-consuming calculations. Recently, a European seismic pre-qualification procedure of ESEP joints has been developed within the RFCS EQUALJOINTS project (Landolfo et al 2018) on the basis of both experimental tests and finite element simulations. In the framework of this project, the present paper describes and discusses the influence of the rib stiffeners focusing on their constructional imperfections for different beam-to-column assemblies. Indeed, although the imperfections are generally considered as highly affecting the response of slender systems as light weight structures (Fiorino et al. 2016, 2017, 2018, Pali et al. 2018), they can also affect the local response of heavy steel systems. Following this consideration, in this study the yield line 1128

distribution and the relevant variation of internal forces into the rib and the bolt rows are monitored. These results allow to characterize the evolution of the internal lever arm, the outof-plane bending and torsional moment developing when the connected beam experiences large plastic rotations as well as the forces acting on lateral torsional restraints. The paper is organized in two main parts, as follows: the investigated parameters and the design assumptions of the joints are briefly presented in the first part; the results of the FE parametric study are presented and critically discussed in the second part.

2 FRAMEWORK OF THE STUDY 2.1 The investigated joint assemblies The beam-to-column assemblies constituting the examined joints have been extracted from a set of reference buildings, designed according to EN1993:1-1 (2005) and EN1998-1 (2005). The selected beam-to-column assemblies are the following: • beam IPE360 – column HEB280 (hereinafter corresponding to the joints labelled as “ES1”); • beam IPE450 – column HEB340 (hereinafter corresponding to the joints labelled as “ES2”); • beam IPE600 – column HEB500 (hereinafter corresponding to the joints labelled as “ES3”). All the joints have been designed to guarantee the activation of the plastic hinge at the beam extremity, preserving the connection and the column web panel that should remain in elastic range. The joints were designed according to the procedure described within EQUALJOINTS project; to fulfil the capacity design requirements the Eq. (1) is satisfied: Mwp;Rd  Mc;Rd  Mc;Ed ¼ α  ðMB;Rd þ VB;Ed  sh Þ

ð1Þ

Mwp;Rd ¼ Vwp;Rd  z

ð2Þ

In Equation 1, Mwp,Rd is the flexural strength corresponding to the capacity of column webpanel (see Eq. (2), Vwp,Rd is the column web shear resistance, z is the internal level arm, Mc,Rd is the flexural strength of the connection zone, Mc,Ed is the design bending moment at the column face, MB,Rd is the design bending strength of the beam, VB,Ed is shear force corresponding to the occurrence of the plastic hinge in the connected beam, sh is the distance between the applied shear and the column face. α depends on the design performance level and it is given by γsh × γov. The geometrical features of all the investigated joints are summarized in Table 1 and Figure 1. 2.2 Investigated parameters The construction imperfections of the stiffeners, namely misalignment of the rib with respect to the beam and the connection (Figure 2), was investigated. These constructional faults can easily occur in ordinary production conditions, and when recognized during the quality

Table 1. Joints geometrical dimensions. End-Plate

Rib

Bolts

Continuity plates

Supplementary web plate

Joint ID

hEP mm

bEP mm

tEP mm

hRib mm

LRib mm

d mm

e mm

w mm

p1 mm

p2 mm

bCP mm

tCP mm

Side -

tSWP mm

ES1 ES2 ES3

760 870 1100

260 280 280

25 25 30

200 210 250

235 250 295

30 30 36

50 50 55

150 150 160

75 75 95

160 180 210

222 234 232

14 15 20

2 2 2

8 10 15

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Figure 1.

Joint geometrical features.

Figure 2.

Position of the rib stiffeners for the investigated joint configurations.

control of the execution they have to be rectified process resulting in an increased unitary costs. Therefore, in order to understand how this type of defect can compromise both the global and the local joint behaviour, four configurations of constructional faults were investigated. As shown in Figure 2, the stiffeners (both on tension and compression side) were shifted from the vertical symmetry axis of a distance equal to ± their thickness (that is constant and equal to 20mm for all the investigated beam-to-column assembly).

3 FE MODELLING DESCRIPTION The finite element models (FEMs) were developed using ABAQUS 6.14 (2014). All modelling assumptions and their validation against experimental tests carried out within EQUALJOINTS project are as described by the Authors in previous publications (Tartaglia et al., 2018a and 2018c). Therefore, for the sake of brevity only the main features of the models are summarized hereinafter. The beam-to-column joints were modelled considering a sub-assemblage obtained by extracting the beam and the column at the inflection points of the bending moment diagram induced by shear type lateral loads on the reference MRFs (Nastri 2017, 2018). The boundary conditions and assumed length of the members are reported in Figure 3a. 1130

Figure 3.

Boundary condition and element mesh dimension.

All elements were discretized using C3D8I solid element type (i.e. 8-node linear brick, incompatible mode), while the mesh dimension change in function of the model parts. According to a sensitivity analysis performed by the Authors and described in previous research (Tartaglia et al. 2018d), the adopted mesh dimensions are 12.5, 10 and 15mm respectively for bolt, end-plate and steel profiles (beam and column) with at least two elements through the element thickness. The beam, the column and all the plates were considered as made out of European S355 steel with the average yield strength equal to γov × fy. The von Mises yield criterion was adopted to model steel yielding; plastic hardening was simulated using both nonlinear kinematic and isotropic hardening law on the basis of the data provided by Dutta et al. (2010). The material of the welds was modelled with an elastic perfectly plastic constitutive law, with yield stress set equal to 460MPa. The non-linear response of the adopted high-strength pre-loadable bolts was modelled according to D’Aniello et al. (2017). The pre-tensioning was modelled using the “Bolt load” option available in ABAQUS and the clamping force as recommended by EN1993:1-8 (2005). The geometrical imperfections of beam profiles due to mill tolerances given by EN 10034 (1993) were accounted for. Contacts were modelled considering both the normal and the tangential behavior. Hence, “surface-to-surface” interactions were used to model the contacts between (i) end-plate and column flange, (ii) bolt head and end-plate and nut and column flange, (iii) shank and the corresponding surface of the holes. Both fillet and full-penetration welds were connected to the corresponding parts by means of “Tie” constraints. 4 RESULTS The influence of constructional imperfections on the global and local response of the joints was investigated by monotonic analyses with respect to a references structure called “C0” where the rib both on the tension and compression side are perfectly aligned with the beam axis. The influence of the imperfections on the joint behaviour is not function of the assembly dimensions reason why in the following only the results of the intermediate joint (ES2-Full strength) are described. 1131

Figure 4a depicts the monotonic moment-rotation curves that are almost overlapped up to 3.5% of chord rotation. For larger imposed rotation, the results are apparently surprising. Indeed, the constructional imperfections C3 and C4 exhibit larger capping rotation than the reference “perfect” configuration C0. The local behaviour of the connection is more affected by the imperfections, which induce non-symmetrical distribution of internal forces and deformations. Indeed, except for the configuration C1, all the other cases with imperfection have the compression center closer to the beam flange than the reference assembly C0 (see Figure 4b), with a consequent decrease of the internal level arm. The different types of behaviour mostly depend on the position of the stiffener on the compression side. Indeed, only the C1 configuration has the rib in compression perfectly aligned with the beam web, thus being effective to transfer the compression forces. In addition, the eccentricity of the rib on the tension side provides a beneficial effect because it restrains the beam flange of the plastic hinge. In all the other cases, the eccentricity of the rib in the compression side impairs the effectiveness of the transfer mechanism of compression forces, thus the compression center moves from the rib web to the beam flange. However, some differences can be recognized between the C2, C3 and C4 configurations. The latter cases show a similar variation of the compression with respect to the reference joint C0, while the C2 configuration is the closest joint C0. These differences are due to the position of the stiffener on the tension side. In the C3 and C4 configurations both the ribs in the tension and compression side are not aligned with the beam web; contrariwise in C2 configuration the rib in tension is perfectly centered with the beam. The misaligned stiffeners restrain the buckling of the beam flange more effectively than the cases with aligned ribs, thus providing larger bending strength and rotation capacity to the plastic hinge of the beam of C3 and C4 configurations, despite the smaller internal lever arm of the connection (see Figure 4). This latter consideration is better clarified analysing the distribution of contact forces (CPREES) between the end-plate and the column (see Figure 5). For C1, the CPREES distribution shows that the rib web can transfer compression forces increasing the imposed rotation up to 6% of rotation, where the internal distribution of contact forces changes due to the buckling of the beam flange (see Figure 5a). In the case of C2 (where the rib on the compression is closer to a bolt alignment) the contact forces are almost equally distributed in the beam and in the rib for small rotations (e.g. 2%), but increasing the imposed rotation the compression forces mostly concentrate in the beam (see Figure 5b) with asymmetric distribution into the beam flange due to its plastic buckling, whose deterioration effects are magnified by the eccentricity of the stiffener on compression side. A similar trend can be also observed for the C3 and C4 configurations (see Figures 5c and d, respectively). Even on the tension side, the constructional imperfections induce non-symmetric distributions of bolt forces, especially in the outer bolt rows adjacent to the rib stiffener. Figure 6 shows the evolution of bolt forces in the four upper bolts of the ES2-F assemblies. As it can

Figure 4.

Results ES2-F joints under monotonic loads.

1132

Figure 5.

CPREES distribution for all the investigated joint configurations.

Figure 6.

Bolt internal forces.

be recognized, the larger differences occur in the outer bolt row (i.e. bolts 1 and 2), while the bolt row close to the beam flange shows negligible differences. Indeed, the first bolt row is generally subjected to forces smaller than those acting in the second row: The latter is directly subjected to the larger tensile force transferred by the beam flange which additionally restrains 1133

the yield line pattern. Hence, the additional effects due to the imperfections are more evident in the bolts 1 and 2 (compare Figure 6a,b to Figure 6c,d). The variation of the internal forces in the bolts differs with the type of imperfection. If the rib on tension is aligned to the beam web (e.g. C2 configuration), the bolt forces are almost symmetrically distributed regardless of the misalignment of the stiffener in the compression side. In the cases with misaligned rib on the tension side, the equivalent T-Stub per bolt row is non-symmetric and the bolt forces increase in the bolt closer to the stiffener. For instance, in the C1 configuration the rib is closer to the bolts on the right side of the connection and, comparing the results of Figures 6a and b, it can be observed that the force in the relevant bolt 2 is larger owith about 30kN than the paired bolt 1. The same trend can be also observed for the C3 and C4 configurations where the ribs are closer to bolt 1 that experiences forces 7% larger than its paired bolt 2 in both cases.

5 CONCLUSIONS The constructional imperfections of the rib stiffener may unexpectedly have beneficial influence on the global response curves of full strength joints. Indeed, the eccentricity of the stiffener allows restraining the buckling of the beam flange and the torsional deformations of the plastic hinge, thus increasing the capping rotation and the overall ductility. However, the bolt forces increase due to the additional secondary effects induced by the rib eccentricity. Indeed, the internal forces could overcome the bolt’s resistance and reduce the joint ductility; this could be very dangerous in the case of partial strength connections where an accurate estimation of the bolt’s internal forces is crucial to avoid a brittle failure and dissipate the energy in the end-plate. REFERENCES D’Aniello, M., Cassiano, D., Landolfo, R. 2017. Simplified criteria for finite element modelling of European preloadable bolts. Steel and Composite Structures, 24(6),643–658. D’Aniello, M., Tartaglia, M., Costanzo, S., Campanella, G., Landolfo, R., De Martino, A. 2018. Experimental Tests on Extended Stiffened End-Plate Joints within Equal Joints Project. Key Engineering Materials, 763, 406–413. Dassault (2014), Abaqus 6.14 - Abaqus Analysis User’s Manual, Dassault Systèmes Simulia Corp. Dutta, A., Dhar, S., Acharyya, S.K. 2010. Material characterization of SS 316 in low cycle fatigue loading. Journal of Materials Science, 45, 1782–1789. EN 10034, 1993, Structural Steel I and H Sections: Tolerances on Shape and Dimensions. European Committee for Standardization; Brussels, Belgium. EN 1993 1-8 2005, Eurocode 3: Design of Steel Structures. Part 1–8: Design of Joints. European Committee for Standardization; Brussels, Belgium. EN 1998-1 (2005), Design of Structures for Earthquake Resistance - Part 1: General Rules, Seismic Actions and Rules for Buildings. CEN. EN 1993:1–1 (2005), Design of Steel Structures - Part 1–1: General rules and rules for buildings. CEN. EQUALJOINTS – European pre-QUALified steel JOINTS: RFSR-CT-2013-00021. Research Fund for Coal and Steel (RFCS) research programme. Fiorino, L., Terracciano, M.T., Landolfo, R. 2016. Experimental investigation of seismic behaviour of low dissipative CFS strap-braced stud walls. Journal of Constructional Steel Research, 127, 92–107. Fiorino, L., Macillo, V., Landolfo, R. 2017. Shake table tests of a full-scale two-story sheathing-braced cold-formed steel building. Engineering Structures, 151, 633–647. Fiorino, L., Pali, T., Landolfo, R. 2018. Out-of-plane seismic design by testing of non-structural lightweight steel drywall partition walls. Thin-Walled Structures, 130, 213–230. Giordano, V., Chisari, C., Rizzano, G., Latour, M. 2017. Prediction of seismic response of moment resistant steel frames using different hysteretic models for dissipative zones. Ingegneria Sismica - International Journal of Earthquake Engineering, 34(4),42–56. Kurejková M., Wald, F. 2017. Design of haunches in structural steel joints. Journal of civil engineering and management, 23(6): 765–772.

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Landolfo et al. 2018. European pre-QUALified steel JOINTS – EQUALJOINTS: final report. European Commission Research Programme of the Research Fund for Coal and Steel, Technical Group: TG S8. Montuori, R., Nastri, E., Piluso, V. 2015. Advances in theory of plastic mechanism control: Closed form solution for MR-Frames. Earthquake Engineering and Structural Dynamics, 44(7),1035–1054. Montuori, R., Nastri, E., Piluso, V., Troisi, M. 2016. Influence of the cyclic behaviour of beam-tocolumn connection on the seismic response of regular steel frames. Ingegneria Sismica - International Journal of Earthquake Engineering, 33(1),91–105. Montuori, R., Nastri, E., Piluso, V. 2017. Influence of the bracing scheme on seismic performances of MRF-EBF dual systems. Journal of Constructional Steel Research, 132, 179–190. Montuori, R., Nastri, E., Piluso, V., Troisi, M. 2017. Influence of connection typology on seismic response of MR-Frames with and without ‘set-backs’. Earthquake Engineering and Structural Dynamics, 46(1),5–25. Nastri, E. 2018. Design and assessment of steel structures in seismic areas: Outcomes of the last Italian conference of steel structures. Ingegneria sismica - International Journal of Earthquake Engineering, 35(2),1–4. Pali, T., Macillo, V., Terracciano, M.T., Bucciero, B., Fiorino, L., Landolfo, R. 2018. In-plane quasi-static cyclic tests of non-structural lightweight steel drywall partitions for seismic performance evaluation. Earthquake Engineering & Structural Dynamics, 47(6),1566–1588. Tartaglia, R., D’Aniello, M. De Martino, A. 2018a. Ultimate performance of external end-plate bolted joints under column loss scenario accounting for the influence of the transverse beam. Open Construction and Building Technology Journal. 12, 132–139. Tartaglia, R., D’Aniello, M., Zimbru, M., Landolfo, R. 2018b. Finite element simulations on the ultimate response of extended stiffened end-plate joints. Steel and Composite Structures, 27(6),727–745. Tartaglia, R., D’Aniello, M., Landolfo, R., 2018c. The influence of rib stiffeners on the response of extended end-plate joints. Journal of Constructional Steel Research, 148, 669–690.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

The fire behaviour of extended stiffened joints designed for seismic actions R. Tartaglia, M. Zimbru, A. Linguiti, M. D’Aniello & R. Landolfo Department of Structures for Engineering and Architecture, University of Naples “Federico II”, Naples, Italy

F. Wald Department Steel and Timber Structures, Czech Technical University in Prague, Prague, Czech Republic

ABSTRACT: The structural behaviour of steel moment resisting frames (MRFs) is strongly dependent on the beam-to-column joint behaviour. The role of the joints is crucial especially under accidental natural and human induced actions, as in the cases of earthquake and fire scenarios, which can occur subsequently after severe seismic events in urban areas. The study summarized in this paper aims at investigating the fire behaviour of seismically designed extended stiffened end-plate joint by means of finite element analyses (FEAs). The joint performance was investigated considering two scenarios: (i) in the first case the assemblies were subjected only to the fire action; (ii) in the second scenario the fire actions were applied to seismically damaged joints. The numerical results show that the fire action changes the restraining capacity of the joint, and local failure can also occur, especially when fire occurs after severe seismic damage.

1 INTRODUCTION During the second part of the 20th century, the necessity of integrating Fire Engineering in the design of buildings heightened because of the countless losses of human lives and destruction of material goods. In this field, important results come from the Cardington experimental campaign (Newman et al., 2004). Steel beams subjected to the fire action develop large deflections as a consequence of progressive loss of stiffness ultimately behaving like cables hanging from the joints. As observed during the Cardington fire test (Newman et al., 2004), the beams can reach high temperatures with large deformations if the connection can withstand the significant catenary actions that develop. The significant axial forces developed must be resisted by the joint simultaneously with design levels of shear force and bending moment while the local ductility demand is increased as well (Wald et al. 2009a). The current European fire design regulations were put to test in experimental campaigns on structures (Wald et al., 2006) and several studies investigated the behaviour of traditional bolted steel connections under fire actions using both experimental and numerical approaches (Liu, 1998, Al-Jarbri et al. 2008, Qian et al. 2008, Yu et al. 2009b, c, Dai et al 2009, Strejček et al. 2010, Garlock & Selamet 2010, Wang et al., 2011, Wang & Wang 2013). Studies show (Wald et al 2009b) that, if not properly designed, there is a clear possibility of the fracture in the joint, which could lead to both the fire spreading and the building’s progressive collapse. Several of the shear and partial-strength connection typologies investigated proved to have poor behaviour at elevated temperatures (Yu et al. 2009b, c). The traditional end-plate connections (flush, extended, flexible and spaced) were thoroughly investigated by Wang & Wang (2013) who proposed the use of reduced beam section or channel extended end-plate to improve their fire performance. The Authors highlighted the limited capacity of these configurations and pointed out that by upgrading the joint, its response under fire is 1136

significantly improved. In European framework, Wald et al. (2009b) demonstrated, based on the Cardington tests that, with regard to bolted joints, the Eurocode design prescriptions are conservative. The recently developed extended stiffened end-plate (ESEP) joints were not investigated under fire action. The joint configurations designed for seismic applications proved to have superior strength and stiffness, when compared with gravity load designed end-plate connections, while the ductility is improved as well. An important aspect regarding the structural robustness is related to the fire resistance of damaged joints. Indeed, differently from light weight cold formed structures (Fiorino et al. 2016, 2017a,b, Macillo et al. 2014), the steel frames under fire may guarantee sufficient robustness provided that the joints do not fail prematurely. Nowadays, the possibility of fire occurring after an earthquake has been investigated by research ventures like PEER in California (USA). In Europe, experimental and numerical investigations carried out by Pucinotti et al. (2011) on welded beam to column composite joints subjected to fire after seismically induced damage, showed the limited impact of the plastic deformations on the fire performance of the joint. The main objective of the current work is to examine the robustness of ESEP joints designed according to newly proposed design strategy, under accidental fire action. An important aspect analysed is the impact the seismic damage has on the joint capacity when the fire occurs. The joints designed within the framework of the EQUALJOINT research project (Landolfo et al. 2018) focused on three design criteria: full, equal and partial strength joints, out of which the first two were considered for the current investigation. In the framework of the research carried out in the past years (Yu et al. 2009a, Garlock & Selamet, 2010, Wang & Wang, 2013, Shakil et al., 2018), the numerical models were often used and continually improved providing significant help both in the interpretation of the experimental results and in performing additional investigations with good accuracy of results. In this paper the fire performance of extended end-plate joints was investigated considering two scenarios. In the first case the fire action was applied on the undamaged joints while in the second case, the joints were pre-damaged by subjecting them to a cyclic loading protocol. The second scenario is representative of the fire occurring after a seismic event and the selected damage levels correspond to varying levels of imposed chord rotation. The beam-tocolumn assemblies were investigated assuming both the case of protected (P) and nonprotected (NP) joints.

2 FRAMEWORK The joint assemblies have been extracted from reference buildings conforming with the current state-of-practice in Europe which are detailed with moment resisting frames (MRF) in one direction and concentric braced frames (CBF) transversally. The structure plan is square (3 x 3 bays) with uniform span lengths in both principal directions and the storey height was considered 4,5 m for the ground storey and 3,5m for all upper storeys. The structures were designed assuming medium and high seismicity level (PGA = 0.25g and 0.35g, respectively) and soil Type C according to the definition of EN 1998-1 (2005). The vertical loads have been selected according to the type of use (residential/office) and have values of 5 and 3 kN/m2 respectively for permanent and live loads. Steel S355 has been used for all frame elements. Depending on the geometry of the connected members (i.e. beams and columns), the analysed set of beam-to-column assemblies includes three types of bolted extended stiffened end-plate joints: EXS1 (with a IPE 360 beam and a HE 280 B column), EXS2 with a IPE 450 beam and HE 340 B column) and EXS3 with a IPE 600 beam and HE 500 B column). The geometrical configuration of the joints and the components are detailed in Figure 1 and Table 1. All the assemblies have been designed as both full and equal strength joints. As explained in more detail by Tartaglia et al. 2018a, the full-strength joints allow the formation of the plastic hinge at the beam extremity, leaving the joint in elastic range. On the other hand, the equal 1137

Figure 1.

Plan layout of structure and joint characteristics.

strength joints are designed to dissipate the seismic energy in both the beam and the connection. The fire response of these two joint typologies is comparatively analysed in this work. Ideally, in case of a seismic event, the moment resisting frame (MRF) is expected to develop plastic hinges at the beam extremities and achieve a global plastic mechanism. The damage accumulated in the plastic hinge can compromise the joint behaviour if the fire is onset after a strong earthquake, and the fire resistance is affected in two ways: (i) reduced capacity of the joint and (ii) loss of effectiveness of fire protection. To have a comprehensive understanding of the joint fire behaviour, two scenarios have been considered for the fire action onset moment and both protected (P) and non-protected (NP) joints were analysed for the second scenario. The investigation key aspects are hereinafter described.

3 FINITE ELEMENT SIMULATIONS Finite element analyses (FEAs) were carried out using the finite element software package ABAQUS 6.14 (Dassault Systemés, 2015); in the following paragraph the main model features were described, while the modelling procedure and the hypothesis made are presented in a previous study of the joint seismic behaviour (Tartaglia et al. 2017a, and 2018c). On the other hand, all the modelling assumptions regarding the fire action and the variation of the material properties in function of the temperature are in line with the EN1993-1-2 (2005) assumptions and the procedure described by Yu et al. (2009b) and Qiang et al. 2014a and b. The geometric properties of the investigated EXS joints are summarized in Table 1. The beams and the columns have the same length of 3.4m and were extracted from the MRF using the sub-structuring methodology. The beam local imperfections, due to the mill tolerance allowed by EN 10034 (1993) were accounted for by an initial buckling analysis, hence the most severe out-of-square buckling modes were selected as proposed by Tartaglia et al (2018b). All assemblies are made by the European S355 steel with the exception of Grade 10.9 for the pre-loadable bolts. The materials’ elastic and plastic properties, at ambient temperature (20°C), were obtained from a set of coupon tests performed within the EQUALJOINTs research project. In the FE model, the Von Misses criteria and the combined (i.e. isotropic and kinematic) plastic hardening was introduced (in line with Dutta et al. 2010). The bolts’ behavior was modelled by multilinear force-displacement curve described by D’Aniello et al. (2017). To model the shank necking and the fracture in the threaded area the ductile damage was introduced in the model as proposed by Pavlovic et al (2015). The material of the welds was modelled by an elastic perfectly plastic constitutive law, with the yield stress equal to 460 MPa, which corresponds to an electrode grade A46 (as given by EN ISO 2560, 2009). For increasing temperature, the material properties were evaluated according to EN1993-1-2 (2005) and are reported in Figure 3a. Hence, the reduction coefficient k was used for both the structural steel and bolts materials, for the material strength and Young modulus reduction. An example of the stress strain relationship varying the temperature was reported in Figure 3b. 1138

Full St. Equal Str. Full Str. Equal Str. Full Str. Equal Str.

Joint ID

EXS1-F EXS1-E EXS2-F EXS2-E EXS3-F EXS3-E

760 600 870 770 1100 1100

mm

hEP

End-Plate

260 280 280 300 280 300

mm

bEP

Features of the designed joints.

Performance level

Table 1.

25 18 25 20 30 22

mm

tEP

200 120 210 160 250 250

mm

bRib

Rib

235 140 250 190 295 295

mm

aRib

12 8 12 8 12 8

-

n°

Bolts

30 27 30 30 36 36

mm

d

50 50 50 55 55 55

mm

e

150 160 150 160 160 160

mm

w

75 160 75 200 95 95

mm

p1

160 180 180 260 210 210

mm

p2

222 222 234 234 232 232

mm

bCP

14 14 15 15 20 20

mm

tCP

Continuity plates

2 1 2 1 2 1

-

Side

8 8 10 8 15 15

mm

tSWP

Supplementary web plate

The interaction of the joint assembly parts in contact was modelled considering “Hard Contact” for the normal behaviour and “Penalty”, with a friction coefficient equal to 0.3, for the tangential behaviour. “Tie” constraints were assumed in place of the parts welded together. The boundary conditions (BC) are modelled to mimic as closely as possible the interaction with the structure the assembly is extracted from. At the beam end, the conditions are dependent on the step and the actions applied; hence for the step where the fire action and gravitational loads are applied. a double pendulum constraint is placed to simulate the beam continuity, while only a vertical restraint is considered for the application of the cyclic loading. The torsional restraints out of the length of plastic hinge are placed to simulate the restraining conditions imposed by the slab. The spacing of lateral torsional restraints was taken according to the lateral-torsional stable length segment proposed by EN 1993-1, clause 6.3.5.3. The columns are pinned at the bottom (all DOF blocked except in plane rotation) and have a roller at the top (free in plane rotation and displacement along the column axis). In order to assess the joint behaviour under varying levels of internal forces, four alternatives were investigated. In the first case, the external loads were evaluated compliant to the structural scheme and the fire loading combination (FLC) and for the other three cases, the vertical loads were assumed equal to 25, 50 and 75% of the beam plastic shear resistance (identified hereinafter as 0.25Vpl,B ed, 0.5Vpl,B and 0.75Vpl,B respectively). The clamping force simulating the tightening of bolts was evaluated depending on the bolt diameter according to EN 1993-1-8 (2005) and applied in the middle face of the bolt shanks using the “Bolt Load” command. The cyclic loading protocol given by AISC 341 (2016) for the qualification of beam-to-column connections was applied. The uniform distributed load q [kN/m2], evaluated as specified in the previous chapter, was applied on the surface of the upper beam flange. Finally, the application of the fire load consisted in varying the temperature of the structure’s elements not protected against fire. The temperature distribution in time for beams and columns are those evaluated following the prescription of Chapter 4.2.5.1 of EN 1993-1-2 (2005), which is based on the nominal fire curve ISO834. According to Ding and Wang (2009), for the investigated joint configuration, the temperature can be considered uniform in all the elements of the connection and the magnitude is a factored value of the bottom beam flange temperature (the reduction was assumed as equal to 80%). Three sides were considered exposed to fire for the non-protected beams, the slab protecting the top side, and for the nonprotected columns, all sides were considered exposed to fire. By way of example, the temperature-time fire curves are shown in Figure 5b and c respectively for an IPE360 and an HE280B member. The models were discretized using C3D8I solid element type (i.e. 8-node linear brick, incompatible mode). Based on the mesh sensitivity study performed by Tartaglia et al. (2018b) the maximum elements size used in the end-plate, bolts and beam elements is respectively equal to 12.5, 10 and 15mm.

4 NUMERICAL RESULTS In this chapter the results of the two investigated scenarios were reported in terms of temperature-rotation curves, equivalent plastic strain distribution (PEEQ) and the time–temperature curves to define the joint fire resistance. 4.1 The behaviour of undamaged seismic joints The first scenario concerns the study of the beam–to-column joints subjected directly to the fire action. Figures 2 a and b show for EXS2-F and E the results in terms of temperaturerotation curves. Independent from the design criteria (full or equal) and from the assembly size, the joint performance is strongly influenced by the shear demand, showing a large 1140

Figure 2.

Temperature rotation curve and PEEQ for EXS2 full (a) and equal strength joint (b).

decrease of the maximum temperature achievable for increasing values of shear actions. Indeed, the increase of vertical loads in the EXS1-F joint for instance, leads to a decrease of the joint maximum temperature equal to 24% (from 774 to 591°C). Once the maximum temperature is reached, the joints behave as a hinge, showing a constant increase of rotation (the horizontal plateau). As it can be noted in Figure 2 for EXS2-F at 600°C, a large amount of shear plastic deformation concentrates at the beam extremity on the joint side and at the beam tip, where the symmetry constraint prevents the free rotation, in line with the sub-structuring hypothesis. The connection parts (bolts, end plate, rib stiffeners) do not develop plastic deformations nor in the case of full or equalstrength joints, the damage remaining concentrated in the beam. The analyses of equal and full-strength joints show no appreciable differences (the temperature-rotation curve, the PEEQ). Indeed, the largest differences in terms of maximum resisting temperature is of about 3%. This result is explained by the identical shear resistance of the two joint configurations, which despite having different connections (i.e. number and dimensions of bolts) have a capacity governed by the connected beam. An important and well-known difference between the full and equal strength joint is the different connection masses associated with a larger steel consumption for the former. As reported in Table 1, the full-strength joints are heavier compared to the equal strength, especially for the EXS1 and EXS2 cases where also the number of bolts changes. The larger massiveness of full-strength joints should influence the temperature propagation in the connection, and therefore, the joint behaviour. Figure 3 summarizes the shear and tensile forces in the first bolt line for EXS2 F and E. The shear force in the bolts linearly increases during the loading step, while in the second step, when the external load is maintained constant, the shear action increases slowly, reaching a value 20% larger compared to the one from the previous step. Therefore, the fire action implies a larger shear demand on the bolts. However, in all the examined cases, the maximum increase for the full-strength joint is equal to 5% and it never leads to the bolts’ shear failure. The resilience of the joint bolts is a consequence of the local resistance checks introduced in the seismic design requirements which are aimed to avoid brittle failures in the joint. The normal force distribution in the bolts for the full strength and equal strength joints analysed in the four cases of external force applied show the same initial equal clamping forces and the degradation in the force when the thermal action is applied in the second step. The bolts practically lose their clamping due to relaxation and start working as non-preloaded bolts, the tensile forces in the bolts decreasing and reaching a limit value function of the vertical loads applied. Owing to the lower temperatures in the connection area, the material 1141

Figure 3.

Shear and tensile forces in the first bolt row.

preserves sufficient resistance to carry the demand from the external forces applied and as already shown, no plastic deformations were observed. The results presented for the EXS2 joints are representative of what was observed in the EXS1 and EXS3 assemblies hence, the latter will not be presented. 4.2 Influence of the cyclic damage on the joint fire behaviour The second investigated scenario concerned the study of the joint fire behaviour subsequent to the application of a cyclic loading, simulating thus fire after a damaging earthquake. From the point of view of joint fire protection, two cases were investigated: (i) non-protected joints (NP), for which the fire load was applied on all joint components and, (ii) protected joints (P) for which all the joint components are protected against the fire, with the exception of the zones where the plastic deformations occur during the seismic events. Figure 4 shows comparatively the results of the protected (P) and non-protected (NP) configurations of both full and equal-strength joints. Similar to the non-protected joints previously presented, assuming fire protection is used, the joint maximum capacity is not influenced by the cyclic plastic

Figure 4.

Temperature-rotation curves for NP and P joints.

1142

damage imposed. Independently from the seismic design criteria, the influence of the fire protection is strongly depended from the joint dimensions. Hence the effectiveness of the protection is more evident for the EXS1 joint where the ultimate temperature increases from 800°C for the NP model to 1000°C for the P model, respectively. Increasing the assembly dimensions, the differences between the protected and non-protected joint is reduced becoming negligible for EXS3. This difference is mainly due to the expected type of failure. Indeed, as shown for the undamaged models, all the joints show a shear failure concentrated in the beam at the rib extremity. Therefore, considering that all joints are subjected to the same external action, the stress ratio of the smaller beam is higher, hence, the degradation of the steel properties during the fire leads to a poorer joint performance.

5 CONCLUSIONS Based on the results hereby presented the following conclusions can be drawn: • The joint resistance degradation and the maximum resisting temperature are function of the shear action on the beam. Indeed, an increase of the vertical loads leads to a strong degradation of the maximum resisting temperature. This resistance degradation is independent from the joint dimensions and from the seismic design performance investigated (full and equal strength). • Differently from what was observed in literature, all the investigated joints show a shear failure mode mainly concentrated in the beam, leaving the bolts almost elastic. The investigated joints were designed to resist the seismic action, and particular attention was given to the local hierarchy between the end-plate and bolts. This local capacity design leads to slightly oversized bolts, perfectly able to resist to the catenary action actin developing under the fire action. • The joint pre-damage does not influence the joint ultimate fire capacity; however, plastic cumulated damage localized in the connection or at the beam extremity (in case of equal and full-strength joints, respectively) imply a reduction of the structure stiffness, with a consequent increase of beam deflection. This phenomenon is more evident when a large shear is applied on the frame. • The effectiveness of the fire protection is function of the joint stress ratio. In the investigated cases, keeping constant the vertical loads, the fire protections give beneficial effects only for the smaller assemblies, while the heavier joints did not show any significant differences between the protected and non-protected configuration.

REFERENCES Al-Jabri, K.S., Davison, J.B. Burgess, I.W. 2008. Performance of beam-to-column joints in fire-A review. Fire Safety Journal 43: 50–62. Dai, X.H. Wang, Y.C. Bailey, C.G. 2009. Effects of partial fire protection on temperature developments in steel joints protected by intumescent coating. Fire Safety Journal 44: 376–386. D’Aniello, M., Cassiano, D., Landolfo, R. 2017. Simplified criteria for finite element modelling of European preloadable bolts. Steel and Composite Structures 24(6): 643–658. Dassault Systèmes - Simulia Inc. 2015. Abaqus analysis 6.14 user’s manual. Ding, J. Wang, Y.C. 2009. Temperatures in unprotected joints between steel beams and concrete-filled tubular columns in fire. Fire Safety Journal 44: 16–32. Dutta, A. Dhar, S. Acharyya, S.K. 2010. Material characterization of SS 316 in low-cycle fatigue loading. Journal of Material Science 4: 1782–1789. Fiorino, L., Terracciano M.T., Landolfo, R. 2016. Experimental investigation of seismic behaviour of low dissipative CFS strap-braced stud walls. Journal of Constructional Steel Research 127: 92–107. Fiorino, L. Macillo, V., Landolfo, R. 2017a. Shake table tests of a full-scale two-story sheathing-braced cold-formed steel building. Engineering Structures 151: 633–647.

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Fiorino, L., Shakeel S., Macillo, V., Landolfo, R. 2017b. Behaviour factor (q) evaluation the CFS braced structures according to FEMA P695. Journal of Constructional Steel Research 138: 324–339. Garlock, M. E. and Selamet, S. 2010. Modeling and Behavior of Steel Plate Connections Subject to Various Fire Scenarios. Journal of Structural Engineering 136(7): 897-906. Landolfo, R. et al. 2018. European pre-QUALified steel JOINTS – EQUALJOINTS: final report. European Commission Research Programme of the Research Fund for Coal and Steel, Technical Group: TG S8. Liu, T.C.H. 1998. Effect of connection flexibility on fire resistance of steel beams. Journal of Constructional Steel Research 45: 99–118. Macillo, V., Iuorio, O., Terracciano, M.T., Fiorino, L., Landolfo, R. 2014. Seismic response of Cfs strap-braced stud walls: Theoretical study. Thin-Walled Structures 85: 301–312. Newman G.M., Robinson J.T., Bailey C.G. 2004. Fire safety design: a new approach to multistorey steel-framed buildings. The Steel Construction Institute, UK. Pavlovic, M., Heistermann, C., Veljkovic, M., Pak, D., Feldmann, M., Rebelo, C., Da Silva, L.S. 2015. Connections in towers for wind converters. Part I: Evaluation of down-scaled experiments. Journal of Constructional Steel Research 115: 445–457. Pucinotti, R., Bursi, O.S., Demonceau, J.F. 2011. Post-earthquake fire and seismic performance of welded steel–concrete composite beam-to-column joints. Journal of Constructional Steel Research 67: 1358–1375. Qian, Z.H. Tan, K.H. Burgess, I.W. 2008. Behavior of Steel Beam-to-Column Joints at Elevated Temperature: Experimental Investigation. Journal of Structural Engineering 134: 713–726. Qiang, X. Bijlaard, F.S.K. Kolstein, H. Jiang, X. 2014a. Behaviour of beam-to-column high strength steel endplate connections under fire conditions - Part 1: Experimental study. Engineering Structures 64: 23–38. Qiang, X. Bijlaard, F.S.K. Kolstein, H. Jiang, X. 2014b. Behaviour of beam-to-column high strength steel endplate connections under fire conditions - Part 2: Numerical study. Engineering Structures 64: 39–51. Shakil S., Lu, W., Puttonen, J. 2018. Response of high-strength steel beam and single-storey frame in fire: Numerical simulation. Journal of Constructional Steel Research 148: 551–561. Strejček, M. Wald, F. Sokol, Z. 2010. Column web panel at elevated temperature. Fire Technology 46: 37–47. Tartaglia, R., D’Aniello, M. 2017a. Nonlinear performance of extended stiffened end plate bolted beam-to-column joints subjected to column removal. The Open Civil Engineering Journal 11: 369–383. Tartaglia, R., D’Aniello, M., Landolfo R. 2018a. The influence of rib stiffeners on the response of extended end-plate joints. Journal of Constructional Steel Research, 148: 669–690. Tartaglia, R., D’Aniello, M., Zimbru, M., Landolfo, R. 2018b. Finite element simulations on the ultimate response of extended stiffened end-plate joints. Steel and Composite Structures 27(6): 727–745. Tartaglia, R., D’Aniello, M., Rassati, G.A., Swanson, J., Landolfo, R. 2018c. Influence of composite slab on the nonlinear response of extended end-plate beam-to-column joints. Key Engineering Materials, 763, 818-825. Wald, F., Simoes da SIlva, L., Moore, D.B., Lennon T., Chladna, M., Santiago A., Benes, M., Borges, L., 2006. Experimental behavior of a steel structure under natural fire. Fire Safety Journal 41: 509–522. Wald, F. Sokol, Z. Moore, D. 2009a. Horizontal forces in steel structures tested in fire. Journal of Constructional Steel Research 65: 1896–1903. Wald, F., Chlouba, J., Uhlir, A., Kallerova, P., Stujberova, M. 2009b. Temperatures during fire tests on structure and its prediction according to Eurocodes. Fire Safety Journal 44: 135–146. Wang, Y.C. Dai, X.H. Bailey, C.G. 2011. An experimental study of relative structural fire behaviour and robustness of different types of steel joint in restrained steel frames. Journal of Constructional Steel Research 67: 1149–1163. Wang, M., Wang, P. 2013. Strategies to increase the robustness of endplate beam-column connections in fire. Journal of Constructional Steel Research 80: 109–120. Yu, H. Burgess, I.W. Davison, J.B. Plank, R.J. 2009a. Tying capacity of web cleat connections in fire, Part 1: Test and finite element simulation. Engineering Structures 31: 651–663. Yu, H. Burgess, I.W. Davison, J.B. Plank, R.J. 2009c. Experimental investigation of the behaviour of fin plate connections in fire. Journal of Constructional Steel Research 65: 723–736.

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Buckling length assessment with finite element approach T. Tiainen, K. Mela & M. Heinisuo Tampere University, Finland

ABSTRACT: In the design of steel frames, the consideration of stability and buckling is an important issue. It can be done in multiple ways. If the concept of buckling length is used, widely used procedure is to calculate the eigenmodes and corresponding eigenvalues for the frame and by using them define buckling length of the members with the well-known Euler’s equation. However, it maybe difficult to tell, which eigenmode should be used for the definition of the buckling length of a specific member. Conservatively, the lowest positive eigenvalue can be used for all members. In this contribution, two methods to define the buckling length of a specific member are considered. The first one uses geometric stiffness matrix locally and the other one uses strain energy measures to identify members taking part in a buckling mode. Compared to simplified approaches presented in literature the approaches based on the finite element discretization have certain advantages. First, the method is applicable to any kind of distributed loading. Secondly, also tapered members can be handled with the technique. Moreover, the out-of-plane buckling behavior and with suitable element the lateral buckling loads can be also be assessed. The applicability and features of the methods are shown in a numerical 3D example. Both methods can be relatively easily implemented into automated frame design procedure. This is essential when optimization of frames is considered.

1 INTRODUCTION In design of skeletal steel structures, stability is in important role. According to European design standards EN 1993-1-1 (2006), either effective length approach or approach based on geometrically non-linear analysis can be used. As the analysis software and sufficient computational power have become available, the latter option has become more and more tempting. However, if automated design procedure such as optimization is used, usually the analysis will have to be performed dozens, hundreds or even thousands of times and to keep the needed computational effort at an acceptable level. Thus effective length approach may be more efficient. In certain structures, such as tubular trusses, buckling length factors are given by EN 19931-8 (2006). For frames, multiple methods have been proposed for finding the effective length of a single member. Widely used simplified approach has been presented by Dumonteil (1992). In his contribution, the transcendental stability equation is solved approximately with simplified formulas. Multiple extensions for this work have been carried out by several other authors. For example, semi-rigid joints have been considered by Maigerou et al. (2006). Webber et al. (2015) have proposed an extension to cover the effect of axial force in columns adjoining the considered member as well as the effect of axial force in other columns in the same floor. In the examples, it is demonstrated that this approach gives very accurate values in comparison to results given by a finite element software. Even if the presented simplified methods can be considered accurate enough to be applicable with design codes, they do not necessarily fit well in integrated design systems. For example, in approach proposed by Webber et al., the user needs to identify other columns in the floor which is not always straightforward task in a complicated structure. 1145

In this contribution, the approach based on finite element discretization (Tiainen & Heinisuo, 2018) is adopted. The reference describes application and performance of two different methods in planar structures. In this contribution, the two methods are extended to cover 3D structures. The performance of the proposed extension is illustrated in a simple 3D frame structure calculation example.

2 FINITE ELEMENT BASED METHODS The basis of both methods are presented in the reference Tiainen & Heinisuo (2018). Both methods rely on the well-known finite element approach where basis is the eigenvalue problem 

 K þ λKg q ¼ 0

ð1Þ

where K is the stiffness matrix, Kg is the geometric stiffness matrix, λ is the eigenvalue and q is the eigenvector representing the buckling mode. When using the finite element model to assess the buckling length of a certain member, the designer needs to manually scroll the eigenmodes until the mode containing buckling of the particular member. When the correct mode is found, the buckling length around local axis y can be calculated as

Lcr;y

sffiffiffiffiffiffiffiffiffiffiffiffi EIy ¼π λj jNi j

ð2Þ

where E is the Young’s modulus for the material, Iy is the second moment of area of the crosssection, and Ni is the axial force in the member i. Similarly, if the correct eigenpair is found for buckling around the local z axis the buckling length can be written as

Lcr;z

sffiffiffiffiffiffiffiffiffiffiffiffi EIz ¼π λj jNi j

ð3Þ

2.1 Local geometric stiffness approach In the local geometric stiffness approach, the geometric stiffness matrix is only applied to elements belonging to the member whose buckling length is being assessed. This can be expressed as  K þ λKig q ¼ 0

ð4Þ

where Kig ¼

X

Keg

ð5Þ

where the sum is taken over the element belonging to member i. The extension to 3D can be done by applying the geometric stiffness matrix partly only including those elements of the matrix being part of buckling behavior in the plane in question. This means that for each member, two limited eigenvalue problems need to be solved.

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2.2 Strain energy based method The strain energy based method supposes that in the relevant buckling mode, the compressed member will have substantial share of total strain energy. By identifying in-plane and out-of-plane elements in the geometric stiffness matrix, for each member in-plane and out-of-plane buckling loads can be obtained in similar manner. In the well-known linear finite element framework, the element strain energy is calculated as 1 E e ¼ qT ke q 2

ð6Þ

where ke is the element stiffness matrix and q is the vector of displacements. The member strain energy can be calculated as 1 X e kq E m ¼ qT 2

ð7Þ

where sum is taken such that elements belonging to the member in question are taken into account. Respectively, for the whole structure, the total strain energy can be calculated when the global stiffness matrix K is used: 1 E ¼ qT Kq 2

Figure 1.

ð8Þ

The example 3D frame topology, dimensions, loads and column numbering.

Table 1. Buckling length factors both proposed approaches and manual assessment. Energy based

Local kg

Manual

Columns

y

z

y

z

y

z

1 2 3 4 5 6

0.66 0.66 0.65 0.65 0.66 0.66

0.61 0.61 0.70 0.70 0.61 0.61

0.64 0.64 0.97 0.97 0.64 0.64

1.39 1.39 0.98 0.98 1.39 1.39

0.64 0.64 0.97 0.97 0.64 0.64

1.39 1.39 0.98 0.98 1.39 1.39

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Figure 2.

First eight buckling modes for the example frame.

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The ratio for each member in a deformed shape can be thus calculated as Rm ¼

Em E

ð9Þ

In 3D case, the member stiffness matrix has to be replaced by a partial stiffness matrix including only elements needed for flexural behavior in the plane in question. The criterion for a member to take part in a buckling mode is proposedly proportional to it’s share of the total energy. The original proposal by Tiainen & Heinisuo (2018) was Rm 

1 n

ð10Þ

where n is the number of members in the structure. 3 NUMERICAL EXAMPLE Consider a 3D frame in Figure 1. All the members are of cold-formed rectangular hollow section with outer dimension 100 mm and wall thickness 5 mm. All joints are supposed ideally rigid but the ends of diagonal braces are supposed hinged. The task is to find buckling length for each column in both planes. The material Young’s modulus is 210 GPa. The well-known Euler  Bernoulli beam theory assumptions and corresponding finite elements are used. Each member is modeled with five elements. The local axes of the columns are defined such that buckling in the plane where the frame is braced is considered the local z axis and the other main axis is the z axis. Both of the methods were applied to the problem. The resulting buckling length factors can be seen in Table 1. Moreover, in Table 1 there are the results for each column obtained by manual evaluation inserting λ values to Eqs. 2 and 3. For the manual evaluation, consider the 8 lowest modes shown in Figure 2. Clearly, in the first mode all the columns buckle simultaneously in a sway mode around their z axis. In the second mode, the middle columns buckle around their y axis. The corner columns’ lowest buckling mode around their y axis is the 8th mode. As both the structure and loading are symmetric, these modes cover all the relevant modes for the columns. Both methods seem to capture the non-sway mode (mode 8 where corner columns buckle around their local y axis). However, the local geometric stiffness matrix based method seems to fail in modes exhibiting sway of the whole frame (modes 1 and 3 in Figure 2). The strain energy-based method seems to perform well in this example finding the correct eigenmodes for every column. The criterion of a member to take part in a buckling mode is the same proposed by Tiainen & Heinisuo (2018). However, as found in Tiainen & Heinisuo (2018), the criterion is not general. The strain energy shares are seen in Tables 2 and 3.

Table 2. Shares of strain energy in y axis buckling for eight first modes. Mode Column

1

2

3

4

5

6

7

8

1 2 3 4 5 6

0 0 0 0 0 0

0 0 23 23 0 0

0 0 0 0 0 0

0 0 42 42 0 0

0 0 0 0 0 0

0 0 42 42 0 0

0 0 0 0 0 0

21 21 1 1 21 21

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Table 3. Shares of strain energy in y axis buckling for eight first modes. Mode Column

1

2

3

4

5

6

7

8

1 2 3 4 5 6

10 10 17 18 10 10

0 0 0 0 0 0

8 8 13 13 8 8

0 0 0 0 0 0

1 1 39 41 1 1

0 0 0 0 0 0

1 1 42 39 1 1

0 0 0 0 0 0

4 CONCLUSIONS In this paper, the extensions from 2D to 3D for two finite element based buckling length assessment methods are presented. Based on the example 3D frame it can be said that the results obtained in 2D apply also in 3D. The approach based on local geometric stiffness matrix may fail if the relevant buckling mode is a sway mode. In non-sway modes, the approach is rather accurate. The energy-based method seems to correctly predict the buckling lengths in the example. However, the question of a general criterion for the energy share present in 2D remains unanswered. Both of the methods contain several benefits in comparison to many approaches found in the literature. Both can be used with non-prismatic members or distributed axial loads. Also, different beam behaviour assumptions may be used such as the Timoshenko beam theory. Moreover, in future research on the topic, the torsional modes in axially loaded members as well as critical bending moment resulting in lateral torsional buckling could be considered. REFERENCES Dumonteil, P. 1992. Simple equations fo effective length factors. Eng J AISC: 29(3):111–115. EN 10219-2, Cold formed welded structural hollow sections of non-alloy and fine grain steels. Part 2: Tolerances, dimensions and sectional properties. CEN, 2006. EN-1993-1-1. Eurocode 3: Design of steel structures. Part 1-1: General rules and rules for buildings. CEN, 2006. EN-1993-1-8. Eurocode 3: Design of steel structures. Part 1-8: Design of joints. CEN, 2006. Mageirou, G.E. and Gantes, C.J. 2006. Buckling strength of multi-story sway, non-sway and partially-sway frames with semi-rigid connections. Journal of Constructional Steel Research: 62 (9):893–905. Tiainen, T. and Heinisuo, M. 2018. Buckling length of a frame member. Journal of Structural Mechanics: 51(2):49–61. Webber, A., Orr, J., Shepherd, P., and Crothers, K. 2015. The effective length of columns in multi-storey frames. Engineering Structures: 102:132–143.

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Experimental and numerical analysis of the local and interactive buckling behaviour of hollow sections A. Toffolon, A. Müller & A. Taras Institute of Structural Engineering, Bundeswehr University Munich, Germany

I. Niko Department of Steel and Timber Structures, Slovak University of Technology, Slovakia

ABSTRACT: Inadequate knowledge regarding specific local and interactive behavior of slender high-strength steel (HSS) hollow sections presents an obstacle in implementing these sections in the construction practice. The current approach in the standard is simplified and offers overly conservative results. Dealing with non-standard cross-sections only expands on these difficulties. This paper contributes to the on-going RFCS project “HOLLOSSTAB”, which has been dealing with the aforementioned issues. The project design proposal is based on the “Overall interaction concept” (OIC), and a new set of design rules for hollow sections is currently being developed. In this concept, linear buckling analysis (LBA) is used to obtain the slenderness of the member, and Geometrically and Materially Non-linear Imperfection Analysis (GMNIA) is used to determine an “overall” buckling reduction factor. Extensive experimental tests are used to validate the method, by correlating the experimental results with numerical test results (GMNIA-real) and statistically analyzing them. The focus of the tests in this paper is the use of high-strength steel, ranging from S500 to S890, used for cylindrical hollow-sections, rectangular hollow-sections and hexagonal hollow-sections. The state-of-the-art measuring tools available offer the possibility of precise reverse-engineering process, creating a numerical model of experimental test. It is possible to reach an accurate prediction of ultimate load capacity of the specimen. The buckling shape can also be checked, using digital image correlation (DIC), comparing numerical models with real shape of tested specimen. The paper aims to validate the numerical models, as well as validity of assumptions on imperfection amplitude.

1 INTRODUCTION Manufacturers of racking systems for large logistics centers face the challenge of producing very light-weight, economical structures that nevertheless meet very high standards with respect to stiffness and strength. In order to meet this challenge, in the market an increasingly number of highstrength steel profiles and stiffened cold formed cross-sections are found. However, current design standards, such as the Eurocode 3, do not sufficiently address the resistance of such sections – as well as more common square and rectangular hollow sections - against local (L) and lip-stiffener = “distortional” (D) buckling modes, especially for general combinations of loading (compression and mono- or biaxial bending). This paper describes an experimental and numerical study of the local buckling behavior of a wide variety of hollow sections, with the aim of calibrating a numerical campaign within the framework of the on-going RFCS project “HOLLOSSTAB”. Cold-formed sections with and without lip stiffeners, circular hollow sections, hexagonal hollow sections both mild-steel and high strength steel are tested in pure compression, bending and bending + compression, with various slenderness ranges for local buckling. Finally, the results of the parametric study will be used for the development of appropriate design rules for these sections, with a continuous representation of strength throughout slenderness ranges, using the “Overall Interaction Concept” (Boissonnade et al. 2017) as a conceptual basis for the development of the design rules. This concept – which is similar to the Direct Strength Method (DSM) 1151

(Schafer 2008) used in North America for the design of cold-formed steel open cross-sections makes use of the results of (numerical) linear buckling analyses (LBA) for the whole member to determine the slenderness and consequently an “overall” buckling reduction factor. 2 SCOPE AND METHODOLOGY As stated above, the strength and stability of hollow sections of various shapes and steel grades are being studied within the scope of the European research project HOLLOSSTAB by means of an extensive experimental and numerical test campaign and parametric study. Among the various section types considered, unstiffened and stiffened, double- and mono-symmetrical cold-formed sections are being tested at the Chair of Steel Structures of Bundeswehr University Munich. A review of international tests on high-strength cold-formed hollow sections was recently given in (Jia-Lin Ma et al. 2015), (Jia-Lin Ma et al. 2017), (Gardner et al. 2017), (Wang et al. 2010), (Kim DK et al. 2014), (Schillo & Feldmann 2015) and (Toffolon & Taras 2017). This section of the paper describes the scope and methodology of this series of tests and numerical analyses. 2.1 Types of studied cross-section and materials Several distinct types of cold-formed and hot rolled cross-section are the subject of study during the HOLLOSSTAB project and are illustrated in Figure 1: i. rectangular (SHS, RHS) hollow sections produced in accordance with the European fabrication standard EN 10219, which applies to cold-formed sections, with steel grades from (normal strength) S355 to (high strength) S700. ii. cold-formed and welded sections of bespoke, stiffened shape, produced primarily for storage racking applications by voestalpine Finaltechnik Krems GmbH in Krems, Austria. The two shapes studied within the scope of this paper are designated VHPS (“S” for “stiffened”) and VHPT (“T” for T-shaped) and are shown in Figure 1a and b.

Figure 1. Stiffened section types studied at Bundeswehr University Munich within the scope of the RFCS-HOLLOSSTAB project: a) “VHPS”; b) “VHPT”; c) Hexagonal hollow section; d) Rectangular hollow section, e) Square hollow section, f) Circular hollow section.

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iii. circular hollow sections produced in accordance with the European fabrication standard EN 10219 and EN 10210, with steel grades from (normal strength) S355 to (high strength) S890. iv. hexagonal hollow sections produced in accordance with the EN10210 with S355 steel grade.

2.2 Experimental methodology and test campaign A total of 68 full-scale tests presented on cold-formed members and hot rolled members of short length (L = 800 mm) and longer tests (L = 2000 mm), with varying wall thicknesses and load eccentricities. These tests were carried out in the 10MN 4-column test rig in the laboratory of Bundeswehr University Munich. Stub column tests were carried out with a centric load application (in the centroid axis, determined analytically in the case of the VHPT section). The bending moment was introduced by adding eccentricity to the test specimen by employing a stiff lever arm plate. Figure 2b and c illustrate the employed experimental loading scheme. The overarching aim of the experimental test campaign is to obtain a reliable basis for the calibration of numerical (FEM-based) simulations of an even wider set of load parameters and cross-sectional configurations. The simulation of the experiments thus employs a process of reverse engineering which is based on the real geometry of the specimen, as well as the measured stress-strain curve of the material, in the FEM modelling of the experimental tests. To facilitate this, each specimen’s geometric shape and the shape deviations from the ideal geometry have been measured with a 3D scanning system made by Zeiss ©. 3D spline curves were laid over the point cloud obtained from the 3D scan and imported into the finite element simulation. Furthermore, a DIC (Digital Image Correlation) measurement system was employed, see Figure 2a. At each test time step (consistent with the experimental test duration) two pictures were taken with GOM Aramis high-resolution cameras, as the basis for the derivation of the deformations and local strains in a randomly applied speckle field on the specimens, see Figure 3b. 2.3 Numerical methodology and campaign The mentioned reverse engineering process consists in replicating the experimental test in a “numerical test”, i.e. in a geometrically and materially non-linear analysis on the imperfect geometry (GMNIA), with the highest possible accuracy. A GMNIA analysis with the measured geometrical shape of the sections and material law obtained from tensile coupon tests can lead to minimum ( Tg) to cold circumvents physical ageing effects. At each new temperature step the temperature equilibrating time is 3 min to avoid a temperature gradient in the specimen. Table 1 summarizes the settings of the experiments:

Table 1. Test Parameters of the DMTA – Temperature-Frequency Sweeps. Sample

Bearing elastomer NR

Bearing elastomer CR

Sample geometry [mm]

Rectangle: L × W × T: 30 × 10 × 2

Rectangle: L × W × T: 30 × 10 × 2

Test mode

Temperature- Frequency-Sweeps in Tension

Temperature program [°C] Temperature steps [°C] Frequency program [Hz] Logarithmically equally spaced frequencies [-] Contact Force [N] Static strain [%] Dynamic strain [%] Setting Number

+50: -80 2 1 to 20

+50: -30 5 1 to 20

+80: -50 2 1 to 20

+50: -30 5 1 to 20

6

10

6

10

0,15 1.00 0.01 (1)

10.00 0.10 (2)

0,15 1.00 0.01 (4)

10.00 0.10 (5)

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7.00 (3)

7.00 (6)

4 RESULTS AND DISCUSSION 4.1 Master Curves according to GUSTL Having obtained the stiffness-temperature-frequency raw data with the settings as described in Sec. 3 now Master Curves can be calibrated by using the method GUSTL, cf. Sec. 2.3. 4.2 Dissipated heat Using the formulas by Bergström (2015) as presented in Sec. 2.4 together with the experimentally obtained Loss Moduli for NR and CR for different strain amplitudes, temperatures and frequencies, the potential energy for dissipation can be computed. For reasons of brevity, within this paper only some graphs can be shown, Figures 6 and 7 show the loss energy for the different experimental settings Both materials show a peak energy loss, as can be expected, close to their Tg but at the lowest tested frequency. At higher temperatures frequency dependent behavior changes to the opposite, though, the amount of energy loss decreased significantly. Comparing the results of setting (2) and (3) respectively (5) and (6), a remarkable shift of the maximum loss from below 30°C to about 15°C is recognized. Considering the same settings (2), (3) and (5), (6) both an amplitude and temperature dependent increase in the area enclosed by the hysteresis graph can be recognized, cf. Figure 8. Whereas temperature rise lowers the energy loss, an increase in strain amplitude supports energy dissipation. Another material property, the storage modulus, reacts in a similar manner with respect to temperature and amplitude or rather strain rate. Lower temperatures as well as higher amplitudes steepen the inclination of the ellipsis as more Maxwell elements in the Prony-series are active, which is more pronounced for the case of CR compared to NR for the small strain experiments. In general CR reacts more sensitive with respect to temperature changes, which may be caused by the proximity of the investigated temperature range to the Tg of CR. While for the small strain experiments (2) and (5) ellipses can be recognized in Figure 8, for the cases (3) and (6) mild nonlinearities can be seen in the hysteresis plots. This gives rise to investigations to a greater degree in that direction with even higher strain amplitudes in order to assess the further evolution of the nonlinearities in that region. 4.3 Approximate computation of increase in temperature In order to determine the scale of the increase in temperature in the center of an elastomeric bearing due to dissipation, the temperature rise of the specimen is approximately computed by Eq. (8) under neglection of further heat fluxes to the outside. For any material the inner energy dU and the volume work dQ relate according to:

Figure 5.

Master Curve of the Storage Modulus regarding setting (1), left and (4), right.

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Figure 6.

Calculated energy loss (uloss) setting (1), (2) and (3) for natural rubber.

Figure 7.

Calculated energy loss (uloss) for the settings (4), (5) and (6) CR.

Figure 8.

Exemplary hysteretic behavior for settings (2), (3), (5) and (6).

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Table 2. Specific heat capacity and mass of specimens. Sample

Bearing elastomer NR

Bearing elastomer CR

Setting Number

(2)

(3)

(5)

(6)

Mass [g] Specific heat capacity [J/gK]

0.845 1.91-2.08

0.833

0.913 2.20

0.916

dQp ¼ dU þ dQ ¼ dH

ð6Þ

This enthalpy change requires an amount of heat being supplied: ð Qp ¼ m 

cp ðT ÞdT

ð7Þ

Without heat transport over the surface and neglecting the volume work as well as the temperature dependence of the specific heat capacity cp the increase in temperature of a specimen is calculated accordingly to: ΔT ¼

uloss with Qp ¼ cp  m  ΔT ¼ uloss cp  m

ð8Þ

Within this paper, only exerpts of all conducted computations can be shown for reasons of brevity. For a specimen of setting (3) with 7% dynamic strain, the energy loss within one load cycle theoretically generates a rise of about 0.015°C for ambient temperatures below -15°C (Figure 9) and even less for higher temperatures. Since the energy loss calculates per volume, it is possible to refer the temperature rise to e.g. liter (Figure 9) which is easier to compare to the size of a real bearing.

5 CONCLUSION AND OUTLOOK In the present paper, Prony Series’ parameters and associated TTSP for small and large strains are determined and discussed. Considering the low temperature rise due to dissipative heating in the specimen used in DMTA analysis, there is no dissipation induced error in these parameters. They will be incorporated into Finite-Element-Software to numerically investigate the experimental findings or else serve as validation base to other authors. Further analyzing the hysteretic behavior, the dissipated energy maximum regarding temperature scale is dependent

Figure 9. Approximated temperature rise of specimen setting (3) and the temperature rise per liter calculated from the setting (3).

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on frequency and strain amplitude. Dissipative heating should not be neglected in the design process regarding the increase in damping at higher frequencies as well as the temperature rise per volume. As could be shown, the temperature rise inside the bearing can be approximated through Prony Series’ parameters based on DMTA experiment data making use of the TTSP and the assumption that no energy is lost at the surface. In order to be able to include dissipative heating in simulations of elastomeric bearings with high accuracy, several further investigation steps are recommended. First it is suggested to assess dissipative heating by a thermocouple inside the specimen. Then a possible increase in nonlinearity at higher strain rates compared to the experiments shown but within the limit of applications, needs to be checked. In addition, the influence of the chemical composition of the elastomers should be further investigated. Finally, in order to verify the simulated dissipative heating in an elastomeric bearing, tests on an instrumented real product are planned. REFERENCES Bergström, J. 2015. Mechanics of Solid Polymers- Theory and Computational Modeling. PDL Handbook Series. USA: FluoroConsultants. Block, T. 2010. Verdrehwiderstände bewehrter Elastomerlager. Dissertation. Ruhr-Universität Bochum. Dippel, B. & Johlitz, M. & Lion, A. 2015. Thermo-mechanical couplings in elastomers - experiments and modelling. ZAMM Zeitschrift für Angewandte Mathematik und Mechanik 95 (11): 1117–1128. Findley, W.N. & Davis, F.A. (revised ed.) 2013. Creep and Relaxation of Nonlinear Viscoelastic Materials. New York: Courier Corporation. Guo, Q. & Zaïri, F. & Ovalle Rodas, C. & Guo, X. 2018. Constitutive modeling of the cyclic dissipation in thin and thick rubber specimens. ZAMM - Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik (series number if necessary) 98 (10): 1878–1899. Halm, H. & Deckmann, H. Dynamic-Mechanical Thermal Analysis of Polymers and Solids. Johlitz, M. & Dippel, B. & Lion, A. 2016. Dissipative heating of elastomers: a new modelling approach based on finite and coupled thermomechanics. Continuum Mechanics and Thermodynamics 28 (4): 1111–1125. Köppl, J. & Kraus, M.A. & Mangerig, I. 2018 Thermorheological Testing and Modeling of a Bridge Slide-Bearing Elastomer. Proceedings of the 12th Japanese German Bridge Building Symposium. Kraus, M.A. & Niederwald, M. 2017. Generalized collocation method using Stiffness matrices in the context of the Theory of Linear viscoelasticity (GUSTL). Tech. Mech. (37) 1: 82–106. Lion, A. 1997. A physically based method to represent the thermo-mechanical behaviour of elastomers. Acta Mechanica 123 (1–4): 1–25. Marques, S. & Creus, G. 2012. Computational Viscoelasticity. Berlin: Springer. Okui, Y. & Nakamura, K. & Sato, T. & Imai, T. 2019. Seismic response of isolated bridge with high damping rubber bearings: self-heating effect under subzero temperatures. Steel Construction 12: 2–9. Ovalle Rodas, C. & Zaïri, F. & Naït-Abdelaziz, M. 2014. A finite strain thermo-viscoelastic constitutive model to describe the self-heating in elastomeric materials during low-cycle fatigue. Journal of the Mechanics and Physics of Solids 64 (1): 396–410. Schneider, J. & Kuntsche, J. & Schula, S. & Schneider, F. & Wörner, J.-D. (vol. 2) 2016. Glasbau – Grundlagen, Berechnung, Konstruktion. Springer. Schwarzl, P.D.F.R. 1990. Polymermechanik. Berlin, Heidelberg, New York: Springer. Tschoegl, N. 1989. The Phenomenological Theory of Linear Viscoelastic Behavior - an Introduction. Berlin: Springer. Wrana, C. 2014. Polymerphysik. Springer Spektrum.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Built-up cold-formed steel beams with web openings V. Ungureanu Politehnica University of Timisoara, Timisoara, Romania Timişoara Branch, Romanian Academy, Romania

I. Both, C. Neagu & M. Burca Politehnica University of Timisoara, Timisoara, Romania

D. Dubina Politehnica University of Timisoara, Timisoara, Romania Timişoara Branch, Romanian Academy, Romania

A.A. Cristian Technical University of Civil Engineering of Bucharest, Bucharest, Romania

ABSTRACT: Cold-formed steel profiles are lightweight elements which can be assembled in a numerous variety of shapes considering either truss structures or corrugated web beams. Especially in residential or office buildings, adjustments are required for the service installations. The web openings represent a weak point in a beam and special attention must be considered to maintain the initial capacity. Previous experimental tests were performed on built-up beams with lipped channel sections as flanges and trapezoidal corrugated steel sheets as web. The connection of the beam components was performed by two methods, i.e. resistance spot welding and MIG brazing. The paper presents the experimental investigations on two full-scale beams with different strengthening solutions for the web openings, in function of the welding technique, i.e. a reinforcing steel plate was spot welded to the corrugation of the web and a border type frame was MIG brazed on the opening perimeter. A lesser influence of the web opening was observed for the beam connected by MIG brazing, a superior bearing capacity being obtained. 1 INTRODUCTION The results of a previous experimental investigations on built-up corrugated web beams (CWB), entirely made of cold-formed elements connected by resistance spot welding or by MIG brazing (Ungureanu et al., 2018a, 2018b), suggested the possibility of using these solution at a larger scale having a high bearing capacity and a ductile response. Due to their strength-to-weight ratio, the corrugated web beams represent an attractive solution for residential/office buildings, but the building services sometimes interfere with the structural elements requiring adjustments of the elements. Web openings are a common solution for multi-storey steel structures. The castellated beams are known for their high material saving and the possibility for large spans as well as trusses. The solution of using cold-formed steel elements for trusses has the disadvantage of part joining which represents a time consumer. An automated process may be used in the case of welded elements, but light gauge steel elements are difficult to be welded. Still, the advances in the automotive industry allow the connection of thin steel plates either by resistance spot weld (SW) or by MIG brazing. The sensitivity of thin-walled cold-formed steel elements and structures to imperfections is very well known (Blum & Rasmussen, 2018; Cardoso et al., 2019; Dubina, 2008) and the web opening in a CW beam certainly reduces, even more, its bearing capacity.

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Two solutions for reinforcing the web opening of the corrugated web beam are investigated in the paper, in function of the connecting technique. For the case of the resistance spot welding, a plane steel sheet positioned in the web plane can be conveniently welded while for the case of MIG brazing, a plate perpendicular to the web plane is more suitable to be used. Based on the experimental results, the behaviour of these beams is presented by their bearing capacity and deformation, as well as by the evolution of the components instabilities. 2 EXPERIMENTAL PROGRAM In a previous experimental program, 5 built-up corrugated web beams (2 using resistance spot welding and 3 using MIG brazing) were tested with various distributions of the web panels function of the sheet thickness (Ungureanu et al., 2018a, 2018b). The current investigations highlight the effect of a web opening in two CWB built-up beams, one using the spot welding (CWB-SW) and the second one using MIG brazing (CWB-CMT). 2.1 The test setup The built-up beams were tested in a planar rigid frame with both ends fixed to the frame. A 500 kN actuator loaded the beam through a leverage system that uniformly distributed the load in 4 points in order to simulate a uniform distributed load. Restrictions for the out-ofplane displacements were applied by a separate structure in two locations. Figure 1 presents the setup of the specimens’ 6 point bending test. In order to simulate the quasi-static loading regime, a rate of 2 mm/s was applied by the actuator. The force was recorded by the actuator load cell, while the vertical displacements were monitored at each quarter of the span by wire linear transducers (see Figure 2a). The relative displacement between the flanges and the end-plates and, the deformation of the endplate was recorded by linear displacement transducers, as shown in Figure 2b. 2.2 The specimens Compared to the previous built-up CWB without web openings, for which the web consisted of individual corrugated panels of approximately 1 m, the web of the current specimens was made of a single piece, i.e. a continuous web. Thus, the built-up of the beams consisted in three stages: 1) connecting the shear panels to the corrugated web, 2) connecting the flanges to the web and 3) connecting the supporting device to the beam. The final operation consisted of machining the web opening. A gauge nibbler was used to cut the perimeter of the web opening after performing 4 holes in the corners of the web opening. Although the connection between the parts of the beam was made by two different welding techniques, the manufacturing process is presented in Figure 3 for the spot-welded specimen only. Having the same base material as for the built-up beams

Figure 1.

Test setup of the built-up beams.

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Figure 2.

Monitoring displacement and deformations: a) vertical displacement, b) beam ends.

Figure 3.

Stages of the built-up process.

without web opening, the mechanical properties are consistent with the values determined in (Ungureanu et al., 2018a) for the corresponding steel sheet thicknesses. The 1.0 mm sheet for the corrugated web is classified as S250GD+Z while the 1.2 mm and the 2.0 mm thicknesses for the shear panel and for the flanges respectively have a yielding characterised by S350GD+Z steel. The beams consisted of the following components: (1) corrugated steel sheet for the web 1.0 mm; (2) additional shear panels - flat plates of 1.2 mm; (3) two back-to-back lipped channel sections for flanges - 2 × C120/2.0; (4) reinforcing profiles U150/2.0 used under the load application points; (5) bolts M12 grade 8.8 for flange to end plates connections, as presented in Figure 4. It must be mentioned that the corrugation height of the SW beam was 60 mm, while the corrugation height in the case of the MIG brazed beam was 45 mm. The web opening dimension considered a reasonable height for the service installations while the length was limited by the distance between the corrugations, such that an optimised position of the spot welding to be applied, i.e. a minimum distance to the edge of the corrugation to be assured. In the case of MIG brazed specimen, the length of the opening is not limited by a minimum distance to the edge of the corrugation, but it was chosen for the similarity to the SW beam specimen. The position of the web opening was chosen to avoid the maximum bending moment and shear force, close to the supports or the middle of the beam, respectively. Similar welding techniques used for the built-up beam were considered also for the reinforcing solution of the web opening. In order to apply the welding, different configurations were selected. For the beam built-up by spot welding, a 2.0 mm thick flat steel plate was welded on the contour of the opening, with the dimensions given in Figure 5. Only one reinforcing plate was used due to the limited possibility of welding another plate on the opposite side of the corrugation. The reinforcing plate was also bent to 90° on both sides parallel to the flanges. These lips were also

Figure 4.

Components of CWB specimens (all dimensions in mm).

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Figure 5.

Reinforcing the web opening of the CWB-SW-WO specimen.

Figure 6.

CWB-SW with web opening specimen.

spot welded to the flanges. The undeformed shape of the built-up beam in the experimental stand is presented in Figure 6. A convenient configuration for the reinforcing of the web opening was performed for the CWB specimen built-up using MIG brazing. A 1.2 mm steel plate was bent in order to take the shape of the web opening. Since the corrugation height of the web was 45 mm, a wider plate was necessary to allow the MIG brazing. Also, to facilitate the insertion, the reinforcing plate was conceived by two U shaped pieces of 80 mm width. For the side parallel to the flanges, the brazing was performed alternatively on each corrugation, while for the vertical sides the brazing was applied as intermittent segments Figure 7 presents the solution for the web opening reinforcing. The fullscale beam specimen of the built-up beam connected by MIG brazing is presented in Figure 8.

Figure 7.

CWB-CMT with web opening specimen.

Figure 8.

CWB-CMT with web opening specimen.

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2.3 Results The response of the beam is assessed not only by the bearing capacity but also the failure mechanism that conducted to the collapse. Starting with the general view, Figure 9 presents the deformed shape of the beam specimen built up by spot welding. A significant deformation is noticed on the left side of the web opening. On single parts, the deformation and failure observation were recorded in the following order: (1) shear buckling of the shear panels, (2) deformations of the corrugated web in the corner of the opening, (3) distortions of the web corrugations close to the end of the beam, (4) shear buckling of the corrugations, (5) shear buckling of the corrugated web sheet (connecting the shear buckling of the corrugations), 6) failure of the spot welds after increase of the previous deformations and (7) buckling of the flanges under the load application point. The first four instabilities occurred in the first part of the capacity degradation, as shown in Figure 10, while (5), (6) and (7) were noticed after large deflection of the beam with plastic deformations (see Figure 11). The force-displacement curve is compared in Figure 12 with the curves obtained from the CWB without web opening (Ungureanu et al., 2018a). It must be mentioned that, although the general configuration is similar, in the previous tests the web was composed of corrugated web plates with approximately 1 m, with thicknesses varying along the beam. Therefore, a direct specification of the force reduction is incorrect. Qualitatively, it can be observed the tested beam has a stiffness of 12106 N/mm, of the same order as the previously tested one. On the other hand, a smaller capacity was expected compared to the previously tested beams, where the thickness of the corrugated web panels near support was higher, i.e. 1.2 mm. The second beam specimen, built-up by using MIG brazing, presented also an increased deflection in the web opening side but only in the last stage of the testing, see Figure 13.

Figure 9.

Deformed shape of the CWB-SW beam with web opening.

Figure 10. Instabilities of the CWB-SW beam during plastic response initiation.

Figure 11. Deformations of the CWB-SW beam with plastic deformations.

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Figure 12. Force-displacement curve for the specimens using spot welding.

Figure 13. Deformed shape of CWB-CMT beam with web opening.

However, during testing, the instabilities and local failures were very symmetric as presented in a sequence of intermediate deformed shapes of the beam depicted in Figure 14. The reinforcing of the web opening together with the MIG brazing lead to rigid components of the corrugated beam, above and below the web opening. The degradation of the elastic response was initiated by a limited shear buckling of the shear panels, less obvious than the shear buckling of the corrugation of the web. Ultimately, the increase of displacement led to linking the individual buckling into the shear buckling of the corrugated web (see Figure 15). During the expansions of the shear buckling of the web to the web opening, the buckling of the shear plate and buckling of the flanges were observed, as shown in Figure 16.

Figure 14. Evolution of the deformed shapes of the CWB-CMT specimen.

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Figure 15. Instabilities of the CWB-CMT beam during plastic response initiation.

Figure 16. Deformations of the CWB-CMT beam in the plastic stage.

Compared to the results obtained for the built-up CWB beams without web openings (Ungureanu et al., 2018b), the force-displacement curve of the current test suggests a minimum effect of the web opening on the bearing capacity. Both the initial stiffness, 27746 N/mm, and the ductility of the beam is comparable to the previously tested beams, as presented in Figure 17. Similar to the case of the spot welded beam, the two cases (specimens with and without web opening) were not identical, but the bearing capacity of the CWB beam with web opening is situated between the specimen CMT 1 having both the corrugated web and shear panels thicknesses of 1.2 mm, and specimen CMT 2 with the steel sheets thickness of the corrugated web and shear panel of 1.0 mm. Since the currently tested specimens were identical from the configuration and components’ thicknesses point of view, except the connecting techniques and the reinforcing solution, the responses of the two beams can be compared. A higher rigidity can be seen in the MIG brazed specimen, along with an increased ductility (see Figure 18). The increased rigidity of the MIG brazed specimen is due to the restraint of the corrugations against distortions. The spotwelded specimen allows distortion of the corrugation, the initial shape of the corrugation being deformed in early stages of the loading, thus reducing the initial rigidity.

Figure 17. Force-displacement curve for the MIG brazed specimens.

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Figure 18. Comparison of CWB with web openings.

3 CONCLUSION The use of the web openings in corrugated web beams made of cold-formed steel elements is possible only by strengthening the affected area. Using a similar joining technique for the strengthening as the ones used in the fabrication of the beam, an adequate bearing capacity was obtained. In the case of spot-welding technology, a steel plate parallel to the web plane can be used as reinforcement, surrounding the web opening. Combined with small rigidity of the corrugations due to the discrete connection of the spot weld, only 61% of the bearing capacity of the MIG brazed specimen was reached. The less stable web of the SW beam, in the opening area, leads to a weak point which constitutes the main deflection source of the beam. Connecting the corrugated web to the flanges by MIG brazing increases the rigidity of the beam as well as the rigidity of the corrugations. Together with the border type reinforcement of the web opening, the remaining of the web in the opening area can transmit the shear force without large deformations leading to a bearing capacity of similar magnitude as the previously tested specimen without web opening. In terms of ductility, the MIG brazed specimen allows a higher deflection of the beam, while the SW specimen failure modes concentrate the damage in the web opening area, reaching the collapse at smaller deflection. ACKNOWLEDGEMENT This work was supported by a grant of the Romanian Ministry of Research and Innovation, project number 10PFE/16.10.2018, PERFORM-TECH-UPT - The increasing of the institutional performance of the Politechnica University of Timișoara by strengthening the research, development and technological transfer capacity in the field of “Energy, Environment and Climate Change”, within Program 1, Subprogram 1.2. REFERENCES Blum, H.B. & Rasmussen, K.J.R. 2018. Experimental investigation of long-span cold-formed steel double channel portal frames. Journal of Constructional Steel Research 155: 316–330. Cardoso, F.S., Zhang, H., Rasmussen, K.J.R. & Yan, S. 2019. Reliability calibrations for the design of cold-formed steel portal frames by advanced analysis. Engineering Structures 182: 164–171. Dubina, D. 2008. Structural analysis and design assisted by testing of cold-formed steel structures. ThinWalled Structures 46: 741–764. Ungureanu, V., Both, I., Burca, M., Grosan, M., Neagu, C. & Dubina, D. 2018a. Built-up cold-formed steel beams using resistance spot welding: experimental investigations. Proc. of the Eight International Conference on Thin-Walled Structures - ICTWS 2018, Lisbon 24-27 July 2018 (e-Proceedings). Ungureanu, V., Both, I., Tunea, D., Grosan, M., Neagu, C., Georgescu, M. & Dubina, D. (2018b). Experimental investigations on built-up cold-formed steel beams using MIG brazing. Proc. of the Eight International Conference on Thin-Walled Structures - ICTWS 2018, Lisbon 24–27 July 2018 (e-Proceedings).

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Numerical investigation of built-up cold-formed steel beams with corrugated web V. Ungureanu Politehnica University of Timisoara, Timisoara, Romania Timişoara Branch, Romanian Academy, Timisoara, Romania

I. Lukačević University of Zagreb, Zagreb, Croatia

I. Both & M. Burca Politehnica University of Timisoara, Timisoara, Romania

D. Dubina Politehnica University of Timisoara, Timisoara, Romania Timişoara Branch, Romanian Academy, Timisoara, Romania

ABSTRACT: Built-up corrugated web beams (CWB) represent an assembly of multiple coldformed steel components of various thicknesses connected by means of screws or welding. Recently, tests on such built-up beams have been performed within the CEMSIG Research Center of the Politehnica University of Timisoara, in which the connections between the components were made by spot welding. Following the validation of the numerical model, the paper investigates the influence of several parameters, i.e.: the thickness of the flanges, the thickness of the corrugated web, the thickness of the shear panel, the magnitude of the initial imperfections and the number and position of the spot welds. The parametric study was conducted on a beam with the same global dimensions as the tested one. 1 INTRODUCTION Built-up cold-formed steel elements are efficient structural elements, very attractive due to the material savings, but also for ease of construction. Moreover, such built-up beams add another advantage to the list, namely the ease of handling due to the low weight of the components. The connection between the built-up beam components can be easily obtained by screws, but the developments in the welding process also led to other solutions like spot welding. A new technological solution of such a built-up beam was proposed at the Department of the Steel Structures and Structural Mechanics of the Politehnica University of Timisoara, consisting mainly of the lipped channel profiles as flanges, corrugated sheets for the web and flat shear panels at both ends (Dubina et al., 2013). The parts were connected using self-drilling screws, a common practice for creating built-up elements. An improvement for the initial solution is represented by the use of spot welding as a connecting technique, which eliminates the screws and reduces the manpower. Previous numerical studies, considering appropriate imperfections, material parameters, mesh size and element type, have shown that finite element (FE) models can be used to accurately predict the load carrying capacity and post-buckling behaviour of built-up cold-formed steel beams (Dubina et al., 2013; Ungureanu et al., 2018a). Different consideration of the initial imperfection on the numerical analyses on corrugated web beams are presented in the literature leading to more or less accurate results (Elgaaly & Seshadri, 1998; Gil et al., 2005; Nie et al., 2013). The paper presents a calibrated numerical model based on the experimental results and the influence of several parameters i.e.: (1) the initial imperfections, (2) the number and distance 1193

between spot welds on flanges, (3) the thickness of the flanges, (4) the thickness of the corrugated web and (5) the thickness of the shear panel. From the parametric study, it results that the bearing capacity of the corrugated web beams made of cold-formed steel components is highly affected by the stability of the components and less affected by the configuration and the number of spot welding. Especially in terms of rigidity, improvements were obtained if the spot welds are positioned in a stabilizing array. 2 EXPERIMENTAL DATABASE The experimental results used for the validation of the numerical model were considered from the tests performed at the Politehnica University of Timisoara (Ungureanu et al., 2018b) where two CWBs were tested, having different thicknesses of the shear panel and a different configuration of the spot welding fastening of the corrugated web panels. A 6-point bending test configuration was used in order to simulate a uniform distributed load in a planar rigid frame with both ends fixed to the frame. A 500 kN actuator loaded the beams through a leverage system that evenly distribute the load in 4 points. A separate structure constrained the beam to vertical displacements in two positions. The database of the testing includes the load force recorded by the actuator and the vertical displacements at each quarter of the span monitored by LVDTs. The built-up corrugated web beams have a span of 5157 mm, a beam height of 600 mm and a corrugation height of 60 mm. The beams consisted of the following components: (1) corrugated steel sheet for the web, with 0.8 mm at the mid-span and 1.2 mm at the beam ends; (2) additional shear panels, i.e. flat plates of 1.0 mm (CWB SW-1) and of 1.2 mm (CWB SW-2); (3) two back-toback lipped channel sections for flanges, 2 × C120/2.0; (4) reinforcing profiles U150/2.0 used under the load application points and (5) bolts M12 grade 8.8 for flange to end plates connections, as shown in Figure 1. The mechanical properties of base material have been tested and presented by Ungureanu et al. (2018a). The 1.0 mm sheet for the corrugated web is classified as S250GD+Z, while the 1.2 mm and the 2.0 mm thicknesses for the shear panel and for the flanges respectively are classified as S350GD+Z steel. Figure 2(a) presents the force-displacement curves of the two tested beams, while Figure 2 (b) shows the qualitative deformations of the beam. The deformations include the buckling of the shear panels, distortion of the web corrugations and failure of the spot welds. 3 NUMERICAL ANALYSIS 3.1 Validation The general-purpose finite element program ABAQUS/CAE v.6.14 (Dassault Systemes, 2014), was used to carry out geometric and material non-linear analyses including the effects

Figure 1.

Components of CWB specimens.

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Figure 2.

(a) Force-displacement curves of the beams, (b) Failure modes during the experiment.

of initial imperfections (GMNIA). Finite Element (FE) models have been calibrated in accordance with the experimental results based on the characteristics determined experimentally such as material and lap joint specimen tests. Initial geometric imperfections, mesh size and element type have been calibrated in accordance with the experimental results of tested beams. Each part of the built-up beam was defined as a 3D shell element extruded according to the shape of the part. Rectangular 4-node doubly curved thin or thick shell, reduced integration, hourglass control, finite membrane strains (S4R) were used to model the thin-walled components. The global mesh size of 15 mm was used for the web, flanges and shear panels, and 25 mm was used for the reinforcing profiles under the load application, as shown in Figure 3. The mesh size was further reduced around the bolt holes to connect the shear panels to the endplates. General contact with the following parameters was used: normal direction - Hard Contact, transversal direction - a friction coefficient of 0.1 and separation was allowed after the general contact takes place. While the support conditions were defined on the nodes of the holes provided for the bolts that connect the beam to the end plate assembly as null displacements and rotations, the loading of the beam was defined as a vertical displacement in a set of multipoint constraints MPC that forms a leverage system able to transmit the deflection to the 4 loading points (see Figure 3). The link between the control points and the pressure surface was defined by a Kinematic coupling constraint for all DOFs. RB3D2 elements were used as a rigid body for load transfer and multi-point constraint beam (MPC beam) for DOF coupling between groups of specified nodes. The spot welds between different parts of the built-up beams were defined depending on the tensile-shear tests of the simple specimens as follows. Attachment points were defined on each part where SW was applied. The connection between the attachment points was defined using Point Based Fasteners with the Connector response of the SW initially calibrated from the results of the tensile-shear tests. The connector was considered by the Elasticity, Plasticity, Damage and Failure parameters. Bushing connector elements were used to model the spot welds. This type of the elements provides a connection between two nodes that allows independent behaviour in three local Cartesian directions that follow the system at both nodes

Figure 3. Thicknesses of the CWB SW-1 beam by components: blue 2.0 mm (flanges), grey 1.0 mm (shear panels), red 1.2 mm (webs near supports) and green 0.8 mm (mid-span webs).

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a and b and that allows different behaviour in two flexural rotations and one torsional rotation (Dassault Systemes, 2014). In order to obtain realistic results from the finite element nonlinear analyses, plastic strains were included in the material definition, according to Annex C of EN1993-1-5 (2006). The measured stress-strain curves based on tensile tests on coupons cut from the cross-section of component elements were included in the model. In the plastic analysis, the static engineering stress-strain curves obtained from tensile coupon tests were converted to true stress vs logarithmic true plastic strain curves. The true stress and true plastic strain were calculated using Equations (1) and (2) as follows:   σtrue ¼ σengineering 1 þ εengineering   εtrue ¼ ln 1 þ εengineering

ð1Þ ð2Þ

where σtrue = true stress; σengineering = engineering stress; εtrue = true strain; and εengineering = engineering strain. For large deformations, the stress-strain points past yield were input in the form of true stress and logarithmic plastic strain. The logarithmic plastic strain has been calculated with Equation (3) as follows: εplastic ¼ εtrue  ln

σtrue E

ð3Þ

where εlnplastic = logarithmic plastic strain, σtrue/E is the elastic strain and E is Young’s modulus. The numerical model consists of two steps. In the first step, the initial imperfections are modelled by performing static nonlinear analysis with the target displacements which results in desired imperfection. In the second step the dynamic, explicit analysis is used to run the load-displacement analysis of the beam based on geometry from previous static analysis for imperfections and with all contacts and material nonlinearity included. The calibration of the numerical models is affected by the magnitude of the initial imperfections. The influence of the initial imperfections is investigated with three out of web global imperfections with magnitudes of L/500, L/1000 and L/1500 (L - the span of the beam) and local imperfections with a magnitude of approximately equal the sheet thickness t. Figure 4(a) compares the FEM and experimental load-displacement curves for CWB SW-1 beam, whereas Figure 4(b) compares the FEM and experimental load-displacement curves for CWB SW-2 beam. As shown in Figure 4, the bearing capacity of the beam is limited affected by the imperfection and, according to Nie et al. (2013), it is concluded that the shear buckling strength is smaller than the shear yield strength. Comparing the load-displacement curves of the experiments and the numerical analyses a good correlation is observed. For an initial imperfection of magnitude L/1000, the deformed shape of the numerical model, Figure 5, replicates the phenomena encountered during the experiments i.e. shear panel buckling, distortion of the corrugated web and the local buckling of the flange in the load application points. 3.2 Parametric study Considering the same beam dimensions, a parametric study was performed to investigate the influence of the following parameters: the number, the position and the distance between the spot welds on flanges, the thickness of the flanges, the thickness of the corrugated web and the shear panel. 1196

Figure 4.

Experimental vs. FEM load-displacement curves: (a) CWB SW-1, (b) CWB-SW-2.

Figure 5.

Qualitative deformation of the specimen.

3.2.1 Influence of the number of the spot welds In order to investigate different arrangements of the SW, the model was modified to simulate the effect of three spot welds per corrugation (instead of 2 as in case of experimental tests). Additionally, the curve CWB SW-1 - FEM - SF in Figure 6, represents a reduced distribution of spot welding in the mid-span, according to the shear force distribution, i.e. has been investigated with 2SW at every two corrugations in this area. As shown in Figure 6, the reduced number of SW in the second third of the beam results in a decreased capacity and stiffness. Without a significant influence on the capacity, the number of SW in the midspan can be reduced at the cost of losing the initial rigidity. The results from Figure 6 show a small influence of 3SW in comparison with 2SW. As observed during the experimental tests, one of the instabilities that occurred during loading was the distortion of the corrugation. The phenomenon occurred since the discrete connection of the spot weld has an axis parallel to the corrugation direction. A more rigid connection is achieved if the spot welds axis is horizontal, constraining the corrugation against distortion. Figure 6 shows the increase in rigidity offered by this configuration of the spot welds. It is to be mentioned that the capacity is less than in the initial case. In order to achieve a capacity of similar magnitude as the initial case, four spot welds on two rows (horizontal spacing of 40 mm, vertical spacing 50 mm) can be assigned, leading to increased rigidity of the element. 1197

Figure 6.

The influence of the number of spot welds between the flange and the web.

3.2.2 Influence of the distance between spot welds on the flanges Influence of the distance between SW on flanges has been investigated for the beam CWB SW-1, by changing the distance between SW on flanges according to Figure 7. Although this is unrealistic to be done in practice for this particular case, the authors wanted to see the influence of a larger distance between spot weldings. Consequently, the distance increases from 50 mm used in the experiment to 100 mm just to emphasise if such influence exists. The curves in Figure 7 shows a negligible influence. 3.2.3 Influence of the flange thickness By considering the flange thicknesses of 1.2, 1.5, 2.0 (experiment) and 2.5 mm, the influence of the flange thickness was assessed. A relatively small reduction in the thickness of the flange profile (from 2.0 to 1.5 mm) causes a significant reduction in the resistance of the entire system. The relative increase of the flange sheet thickness compared to the reference case CWB SW-1 (from 2.0 to 2.5 mm) results in a small increase of the resistance as depicted in Figure 8. It must be mentioned that the analysis shows that the flange thickness has a high impact also in the initial rigidity of the beam. 3.2.4 Influence of corrugated web thickness As the CWB can be formed of a single corrugated web, another case of analysis was set by the fixed thickness of the flanges, 2.0 mm, and the shear panels, 1.2 mm, while the corrugated web varies between 0.8, 1.0, 1.2 and 1.5 mm, having one thickness for the entire length of the beam. The results are shown in Figure 9(a). Although for small thicknesses, i.e. 0.8 mm, both the rigidity and the yielding are dependent on the web thickness, for the thickness of the web above 1.0 mm the yielding is very similar but the ultimate force is different. For the 1.5 mm corrugated web, the force-displacement curve shows a continuous increasing force in the post-elastic stage, similar to the catenary effect. According to Nie et al. (2013), this thickness can provide a shear buckling strength larger than the shear yield strength. 3.2.5 Influence of the shear panel thickness As observed from the experimental investigations, the thickness of the shear panel is a parameter that may influence the rigidity of the specimen. In this direction, the analyses were performed also for the case where the flange and the corrugated web are kept at the same thickness as for the CWB-SW-1 specimen, while the shear panel thickness took the values of 0.8 mm, 1.2 mm, and 1.5 mm. The parametric numerical analyses showed that the influence in the rigidity is not significant, but the resistance is affected by approximately 15%, (see Figure 9(b)). Over the entire range of thicknesses, the influence of the shear panel thickness is smaller than in the case of the web thickness from the previous case. 1198

Figure 7.

The influence of the distance between spot weldings on flanges.

Figure 8.

The enhancement of the flange thickness.

Figure 9.

a) The influence of corrugated web thickness, b) The influence of the shear panel thickness.

4 CONCLUSIONS Based on experimental results, the lightweight steel structures represent a sustainable solution in structural engineering due to their material saving and ease of manipulation. For the built-up 1199

cold-formed elements with corrugated webs, the number of parameters is very large due to the multiple parts involved and the number of possible configurations of the beam. Among the parameters that may influence the response of the beam are the position of the spot welding, the flange thickness, the web thickness and the shear panel thickness, all investigated in the present paper. Their contribution influences the capacity and the rigidity at different shares. The studied parameters which have a small influence on the results are the number of spot welds on the flange and the distance between the spot welds. From the numerical analyses of the studied beam, it results that the bearing capacity of the corrugated web beams made of cold-formed steel components is highly affected by the stability of the parts and less affected by the configuration and number of spot welds. Nevertheless, the increasing thickness of the parts does not necessarily mean an increase of the resistance but the most unstable parts limit the maximum force. A numerical model considering an alternative arrangement of the spot welds on the flange, i.e. aligned horizontally, can improve the response of the beam in terms of rigidity. Overall, the rigidity of the beam is mostly influenced by the thickness of the flanges and by the arrangement of the spot welds, while the contribution of the shear panels and the corrugated web is small except for the very thin web. ACKNOWLEDGEMENT This work was supported by a research grant of the Romanian National Authority for Scientific Research and Innovation, CNCS/CCCDI-UEFISCDI, project number PN-III-P22.1-PED-2016-1684/WELLFORMED - Fast welding cold-formed steel beams of corrugated sheet web and by a grant of the Romanian Ministry of Research and Innovation, project number 10PFE/16.10.2018, PERFORM-TECH-UPT - The increasing of the institutional performance of the Polytechnic University of Timișoara by strengthening the research, development and technological transfer capacity in the field of “Energy, Environment and Climate Change”, within Program 1 - Development of the national system of Research and Development, Subprogram 1.2 - Institutional Performance - Institutional Development Projects Excellence Funding Projects in RDI, PNCDI III. REFERENCES Dassault Systemes 2014. Abaqus 6.14 Documentation (Providence, RI, Simulia Systems). Dubina, D., Ungureanu, V. & Gîlia, L. 2013. Cold-formed steel beams with corrugated web and discrete web-to-flange fasteners. Steel Construction 6: 74–81. Elgaaly, M. & Seshadri, A. 1998. Depicting the behavior of girders with corrugated webs up to failure using non-linear finite element analysis. Advances in Engineering Software 29: 195–208. EN 1993-1-5: 2006, Eurocode 3: Design of steel structures - Part 1-5: Plated structural elements, CEN, Brussels. Gil, H., Lee, S., Lee, J. & Lee, H.E. 2005. Shear buckling strength of trapezoidally corrugated steel webs for bridges. In Transportation Research Board - 6th International Bridge Engineering Conference: Reliability, Security, and Sustainability in Bridge Engineering, 473–480. Nie, J.-G., Zhu, L., Tao, M.-X. & Tang, L. 2013. Shear strength of trapezoidal corrugated steel webs. Journal of Constructional Steel Research 85: 105–115. Ungureanu, V., Both, I., Burca, M., Tunea, D., Grosan, M., Neagu, C. & Dubina, D. 2018a. Welding technologies for built-up cold-formed steel beams: experimental investigations. Proc. of the International Conference on Engineering Research and Practice for Steel Construction 2018 (ICSC2018), 5-7 September, Hong Kong, China (e-Proceedings). Ungureanu, V., Both, I., Burca, M., Grosan, M., Neagu, C. & Dubina, D. 2018b. Built-up cold-formed steel beams using resistance spot welding: experimental investigations. Proc. of the Eight International Conference on Thin-Walled Structures - ICTWS 2018, 24-27 July 2018, Lisbon, Portugal (e-Proceedings).

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Study on the out-of-plane stability of steel portal frames M. Vassilev & N. Rangelov Department of Steel and Timber Structures, UACEG, Sofia, Bulgaria

ABSTRACT: Within the current EN 1993-1-1 the verification of rafters of portal frames for lateral-torsional buckling in the haunched portions loaded by hogging bending moments is not clarified. Recently the authors have conducted an extensive theoretical analysis programme on lateral stability of steel portal frames of hot-rolled profiles. Special software has been developed for automatic modelling and applying both the GMNIA and the general method for lateral and lateral-torsional buckling. To validate the numerical simulations, an experimental programme has been conducted on three typical portal frames of hot-rolled profiles. It has been proven that the results by GMNIA comply well with the tests, therefore is adopted as benchmark method of numerical analysis. An extensive parametric study is then conducted on thousands of portal frames composed of hot-rolled profiles. An important finding based on the comparative analysis is that the application of the general method to whole portal frames appears non-conservative and therefore its use shall be limited to single members. Based on the obtained results, simple design rules for practical application are suggested by adapting the well-known equivalent compressed strut model.

1 INTRODUCTION Despite the extensive application of steel portal frames for single-storey buildings, there are still some aspects of their stability that require additional clarification. No codified practical method is specified in EN 1993-1-1 for lateral-torsional buckling verification of rafters in the haunched portions loaded by hogging bending moments. It seems that, within the code, there are only two possible approaches: the general method for lateral and lateral torsional buckling (§6.3.4 of EN 1993-1-1) and geometrically and materially nonlinear analysis with imperfections (GMNIA). However, both methods seem quite complicated and cumbersome for practical use. The general method is clarified in details in Simões da Silva et al. (2010). The method uses a Merchant-Rankine type of empirical interaction expression to uncouple the in-plane effects and the out-of-plane effects, and has been successfully applied to single tapered members (Marques et al. 2013, Marques et al. 2014). However, the lateral-torsional stability of the frame rafters seems to be even more complicated problem, taking into account the haunched portions, the negative (hogging) bending moments and the specific restraint conditions with lateral restraints at the top (tensile) flange only (rafter ‘fly bracing’ is not considered herein). Therefore it seems more appropriate to consider the portal frame stability as a whole. However, though the general method is declared applicable to plane frames, very few applications are available in literature, e.g. Papp & Szalai (2011). In this paper is summarised the research carried out by the authors in the above context on lateral stability of single storey single span steel portal frames of hot-rolled profiles. The general method for lateral buckling is discussed with emphasis on the specific issues of its application to frame lateral stability. The application of the geometrical and material nonlinear analysis with imperfections (GMNIA) is also discussed. The above methods are confronted with a well-known equivalent compressed strut model for the haunch, with a view to propose simple and reliable design rules for practical use. To validate the numerical simulations, an experimental programme 1201

has been conducted on three typical portal frames of hot-rolled profiles, which is also summarised in the paper. Special software has been developed for automatic modelling and applying the above methods. An extensive parametric study is then conducted on thousands of portal frames composed of hot-rolled profiles. Based on the obtained results, a modification of the equivalent compressed strut model is proposed for simple practical applications. An important finding based on the comparative analysis is that the application of the general method to whole portal frames appears non-conservative and therefore its use shall be limited to single members.

2 NUMERICAL ANALYSIS The scope of the study is limited to single-span portal frames composed of hot-rolled profiles. Purlins are adopted equally spaced at 1,5 to 2,0 m, and hinged lateral restraints only to the top flange are considered at each purlin. The haunches are assumed with the same section as that of the rafter, with length considered 10%, 15% and 20% of the frame span. At the ends of the haunches, stiffeners to the rafter are always present. A typical portal frame of consideration is illustrated in Figure 1. Three types of analysis are performed. The first one is the conventional design approach based on linear elastic analysis and the design methods of EN 1993-1-1. The well-known equivalent compressed strut model (Koschmidder & Brown 2012) is used for the haunch buckling verification (Figure 2). The equivalent strut is defined in Section 1 with cross-section composed of the haunch flange and 1/3 of the compressed portion of the web. The axial compression force for the strut is obtained based on the bending moment and normal force in Section 1. Though no out-of-plane restraints are considered at the bottom flange, in this analysis the out of plane buckling length, Lcr,z, is assumed equal to the actual length of the haunch flange. The second analysis is based on the general method for lateral and lateral-torsional buckling (§6.3.4 of EN 1993-1-1). The critical point when applying the method is the determination of

Figure 1.

Typical portal frame made of hot-rolled profiles with points of lateral restraint.

Figure 2.

The equivalent compressed strut model for the haunch.

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Figure 3.

Typical frame failure mode obtained by GMNIA.

two load multipliers, αult,k and αcr,op, the former being the minimum load amplifier to reach the characteristic resistance of the most critical cross-section, and the latter – the minimum load amplifier to reach the elastic critical load with regards to lateral or lateral torsional buckling without accounting for in-plane flexural buckling. To obtain αult,k, an in-plane geometrically nonlinear analysis with imperfections is carried out. The frame is modelled with beam/frame elements. The imperfection pattern is according to §5.3.2 of EN 1993-1-1 and includes both local bow member imperfections, scaled appropriately, and a global frame imperfection (1/200 of column height). Since the analysis is nonlinear, αult,k cannot be refined by simply scaling the load. Therefore an iterative procedure is developed to find precisely the critical cross-section and the relevant load multiplier. For αcr,op another model is automatically generated with shell elements to perform linear buckling analysis. An algorithm is developed to exclude in-plane and local buckling modes and to identify the first out-of-plane mode and the corresponding load multiplier αcr,op. Thus the global slenderness can be determined. Finally, instead of one reduction factor χop, the reduction factors χz and χLT are used as suggested by Marques et al. (2008) using the relevant buckling curves for the critical cross-section. The third type of analysis, GMNIA, is also carried out automatically. The model with shell FE is generated and linear buckling analysis is initially performed. The first overall out-ofplane buckling mode is used to obtain the initial imperfections pattern, scaled according to §5.3.4 of EN 1993-1-1. A revised model is thus generated. Material nonlinearity is based on bilinear constitutive law with isotropic strain hardening. The load-carrying capacity of the frame is assumed to correspond to the ultimate state criterion ‘attainment of the maximum load’. The stressed state and the failure mode are also analysed. The software used is ABAQUS nonlinear FE software (Abaqus 2016). A typical picture at limit state is illustrated in Figure 3.

3 EXPERIMENTAL STUDY To calibrate and validate the models used in GMNIA and to demonstrate the actual behaviour of the studied frames under monotonic loading, a test programme has been conducted. 3.1 Scope and test set-up Three representative test frames have been designed with 7.0 m span as illustrated in Figure 4. The cross-sections are selected to represent some stiffness variety. Care has been taken to avoid cases in which the joint web panel governs the frame capacity. The load application points approximately correspond to the purlin locations of an actual frame; lateral supports are provided at those points, while the out-of-plane displacements are measured at the midpoints. The test specimens, the loading system and the lateral restraints are carefully designed to correspond to the model assumptions. Column bases are detailed as perfectly hinged.

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Figure 4.

Dimensions and cross-sections of the tested portal frames.

Figure 5.

Loading system.

Figure 6.

General overview of the test setup (a) and load application and lateral restraint devices (b).

An interesting loading system has been developed to apply monotonic gradual forcecontrolled loading. The latter is provided by a suspended water tank and is applied through a system of rope and pulleys, thus guaranteeing a uniform load distribution (Figure 5). The detailing provides centric and ‘hinged’ load application. The bracing system is also designed to correspond to model assumptions. Lateral restraints are provided at the loading points, which at the same time can move vertically. Care is taken for proper detailing, as illustrated in Figure 6. A total of 13 inductive displacement transducers are installed to record the global behaviour of the frames. Seven transducers measure vertical displacements, four ones measure the lateral displacements of the top flange, and two transducers measure the lateral displacements at the bottom flanges near the haunch, where lateral-torsional buckling phenomena are expected. The load is continuously measured by a commercial water meter with ordinary precision that has additionally been verified by geometrically measured volume of the water in the tank. Such a precision is found sufficient for the tests. To synchronise the load measurement in time, a computer is used to video-record the water meter with its clock synchronised with the clock of the data-recording computer. Since the steel strength appeared well above the nominal, in addition to the water, test weights are added to increase the experimental load. 1204

Figure 7.

Steel test specimens and measured mechanical properties.

3.2 Material mechanical properties The nominal steel grade of all profiles is S275. To estimate the real mechanical properties of the steel, a total of 18 (3 from each profile) standard specimens has been made by water jet cutting to avoid any additional thermal or mechanical influence to the steel (Figure 7). The results for different profiles appear variable, especially for IPE80 and IPE120. Moreover, the steel cut from IPE80 exhibited no yield plateau. The actual mechanical properties were used to re-calculate the test frames and to estimate the expected ultimate load. 3.3 Behaviour of the tested frames and observed failure modes All tested frames reached ultimate limit state by lateral-torsional buckling of the rafters at the haunch ends, as predicted by GMNIA analysis. Accordingly, largest lateral displacements were recorded there, depending on the actual initial imperfections of the test frames, which unfortunately were not measured and recorded. In the test of Frame 1, the lateral restraint mechanism exhausted its run and leaned on the bracket. However, this took place well after reaching the limit state and did not affect the final result. For Frame 2 the restraint detail was altered to provide larger displacement capacity of bracing bracket. The observed ultimate behaviour was quite similar. For the test of Frame 3, the restraint details were additionally improved to avoid any malfunction at large displacements. Together with that, the loading water tank was modified to accommodate more test weights due to the expected higher load capacity of this frame. Some illustrative photos representing the observed failure behaviour are shown in Figure 8 for Frame 3. 3.4 Comparison with numerical analysis To prove and validate GMNIA as benchmark analysis method, the three tested frames were modelled with measured material properties of the material. Since the actual initial

Figure 8.

Observed behaviour at failure of Frame 3.

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Figure 9.

Experimental and analytical vertical and lateral displacements.

imperfections were not recorded, initially the analysis was performed using the most unfavourable imperfection patterns based on properly scaled first lateral buckling mode. The displacements obtained at the gauge locations were monitored and plotted against the experimental data in Figure 9, where the numerical results are plotted by dotted lines. The comparison in terms of displacements shows a very good agreement between the analytical and the experimental results. As opposed to the numerical analysis, where a clear limit state criterion is available, the determination of the experimental load-bearing capacity is based on a careful examination of the recorded data to identify the moment when the rate of lateral displacement increments at the relevant transducer (8 or 13 as indicated) becomes ‘significant’. The predicted by GMNIA behaviour corresponds well to the experimentally observed one. However, the theoretical ultimate loads appear somewhat lower by 24% for Frame 1, 33% for Frame 2 and 18% for Frame 3. Indeed, these results provide a comfortable safety for the numerical analysis and prove its adequacy, but an explanation of the discrepancies seems noteworthy. In this regard, an additional numerical study on the effect of imperfections has been carried out.

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Figure 10. The effect of imperfections to the numerical load-carrying capacity.

3.5 Effect of initial imperfections To evaluate the imperfection sensitivity, the following approach is adopted. On one hand, two boundary imperfection levels are adopted: ‘standard’ level with amplitude equal to 0,5e0,d, as per §5.3.4 of EN 1993-1-1, ‘minimum’ imperfection level with 100 times smaller amplitude, and the case without imperfections. On the other hand, various imperfection patterns corresponding to different overall lateral buckling modes have been considered. The analysis demonstrates a substantial effect of the initial imperfections as illustrated in Figure 10 in terms of the obtained numerically total load-carrying capacity with various imperfections in comparison with the experimental capacity attained. It is well seen in Figure 10 that the case of ‘standard’ imperfection level with imperfection pattern corresponding to the first lateral-torsional buckling mode appears always on the safe side. Thus, on one hand, a reasonable explanation for the discrepancies between the experimental and GMNIA results is pointed out, and on the other hand, the safety of the adopted numerical analysis procedure is proven. Indeed, these conclusions are drawn from only three case studies, however, the authors believe that similar conclusion can be generalised. All the above results validate the analytical model and prove GMNIA as a benchmark method of analysis.

4 PARAMETRIC ANALYSIS AND RESULTS Based on the procedure described in Section 2, using the specially developed software, an extensive parametric study is carried out. Several thousands of frames are automatically analysed, varying the column and rafter sections, frame span (12 to 24 m), haunch length and number of lateral restraints at the rafter top flange. By varying the different hot-rolled

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Figure 11. Comparison between the results of the general method with GMNIA.

profiles, a vast variation of flexural and torsional stiffness ratios is covered. The most representative stiffness parameter for a member appears to be the following: K¼

qffiffiffiffiffiffiffiffiffi π2 EIw GIt L2

ð1Þ

where Iw is the cross-section warping constant, It is the St. Venant torsional constant, L is the length of the member, E is the modulus of elasticity and G is the shear modulus of the steel. For a frame, more important appears the ratio: ω ¼ KC =KR

ð2Þ

where KC and KR are the values of K for the column and for the rafter, respectively. Since the research is aimed at studying essentially the rafter out-of-plane stability, all irrelevant cases have been disregarded (e.g. when in-plane buckling or joint shear panel resistance is governing). Generally, the analyses are grouped into series. The main objective of the parametric study is to compare the results of the general method (§6.3.4) and the simplified equivalent compressed strut model with those from GMNIA. The comparative results are presented in terms of utilisation (demand-to-capacity) ratio FEd/FRd, FEd representing the total load and FRd being the frame resistance according to the method used. The comparison between the results obtained by the general method and those from GMNIA appears interesting and astonishing. It seems that if the general method is applied to the whole frame, the obtained results appear non-conservative, i.e. on the side of unsafety. This is illustrated in Figure 11, where the utilisation ratio is plotted against KR for a typical series of frames. On the contrary, the results obtained by the simplified equivalent compressed strut approach as described in Section 2 appear always more or less conservative, as illustrated in Figure 12 for two typical series of frames. The analysis proves that the conservatism of the equivalent compressed strut method mostly depends on the ratio ω, therefore, for the most typical case of 10% haunch length, the following correction factor is proposed: η ¼ 0:015ω þ 1:1:

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ð3Þ

Figure 12. Comparison between the results by the simplified approach with GMNIA.

Thus the design resistance of the equivalent compressed strut may be obtained as: Nb;Rd ¼ ηAfy χz =γM1

ð4Þ

with χz obtained for the out-of-plane buckling length as shown in Figure 2. In the above equation A is the area of the cross-section of the strut (Figure 2) and the rest of notation is obvious.

5 CONCLUSIONS An extensive numerical and limited experimental research on steel portal frames of hot-rolled profiles is presented, aimed at clarifying the out-of-plane stability of rafters in the haunched portions loaded by hogging bending moments. The numerical analysis is carried out by specially developed software which generates the models and controls the analysis by three methods: GMNIA, the general method of §6.3.4 of EN 1993-1-1 and the conventional design approach combined with the equivalent compressed strut model. The comparative analysis shows that if the general method is applied to the whole frame, the obtained results appear well on the side of unsafety. Therefore, the use of this method shall be limited only to single members. The simplified equivalent compressed strut model is proven to be suitable and conservative even without restraints to the bottom flange of the rafter. A correction to this method to reasonably minimise its conservatism is proposed that fits well to the numerical results. This approach is both simple and provides a good base for engineering judgement. REFERENCES Simões da Silva, L., Simões, R., Gervásio, H. 2010. Design of Steel Structures. ECCS. Marques, L., Simões da Silva, L., Greiner, R., Rebelo, C., Taras, A. 2013. Development of a consistent design procedure for lateral-torsional buckling of tapered beams. Journal of Constructional Steel Research 89: 213–235. Marques, L., Simões da Silva, L., Rebelo, C., Santiago, A. 2014. Extension of EC3-1-1 interaction formulae for the stability verification of tapered beam-columns. Journal of Constructional Steel Research 100: 122–135. Papp, F., Szalai, J. 2011. Theory and application of the general method of Eurocode 3 Part 1-1. Eurosteel 2011, Budapest, 31.08-02.09.2011. Koschmidder, D.M., Brown, D.G. 2012. Elastic design of single-span steel portal frame buildings to Eurocode 3, SCI Publication P397, Ascot: SCI. Marques, L., Simões da Silva, L., Rebelo, C. 2008. Numerical validation of the general method in EC3-1-1: Lateral and lateral-torsional buckling of non-uniform members. Eurosteel 2008, Graz, 03-05.09. 2008. Abaqus, 2016. Dassault Systems/ Simulia, Providence, RI, USA.

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Warping transfer superelement model for bolted end-plate connections subject to 3D loads B. Vaszilievits-Sömjén KÉSZ Holding Ltd., Budapest, Hungary

J. Szalai ConSteel Solutions Ltd., Budapest, Hungary

R.M. Movahedi Faculty of Architectural, Civil and Transportation Engineering, Széchenyi István University, Győr, Hungary

ABSTRACT: A simple beam element based modelling technique has been developed which makes possible to analyze frames made of I sections with column-rafter bolted end-plate connections, subject to 3D loads, compatible with the thin walled beam theory with 7DOF beam elements. The model previously developed by the same team for welded connections has been ex-tended with the addition of linear spring elements to model the bolts located at the upper and lower beam flange level. The spring stiffnesses are calculated based on the extension of the Eurocode component method and verified by simulations performed with FEA software Abaqus. 1 INTRODUCTION The goal of this research was to demonstrate how the out-of-plane flexibility of a bolted corner connection can be considered in structural analysis based on 7DOF beam elements. A superelement modelling concept has been developed and presented by the same team Vaszilievits-Sömjén & Szalai (2019) and has been validated for welded connections. This modelling technic has been extended with the out-of-plane flexibility resulting from the usage of bolts with typical bolt patterns, instead of assuming a welded connection. As the present edition of Eurocode 3 (EN 1993-1-8) doesn’t provide rules to calculate outof-plane stiffness of bolted connections, a simple extension of the rules will be recommended. 2 OUT-OF-PLANE STIFFNESS OF BOLTED CONNECTIONS Eurocode 3 doesn’t propose rules to calculate stiffness of bolted-connections subject to out-ofplane effects. Several researchers have proposals to handle this question. Neumann & Nuhic (2011) and Neumann & Buzaljko & Thomassen & Nuhic (2012) have proposed a model on the bases of existing Eurocode method for in-plane stiffness calculation and extended with additional modes for cases specific for out-of-plane s tiffness problem. Romero (2010) has proposed for I sections a simple 50%-50% division of standard T-stub stiffnesses to come to a practical value applicable for a half T-stub consisting of 1 bolt only. Couchaux & Rodier (2016) have proposed additional component to model the flexibility of compressed side of such connections. All these researches have calculated a unique stiffness value corresponding for the whole I section dealing mainly with symmetrical bolt patterns. Only the presence of Mz (out-of-plane, weak axis) bending moment has been assumed, without any interaction of other internal forces acting on the same bolted connection. Additionally, they have assumed, that as a result of the application of the out-of-plane effects, the flanges of the connection will “close” at the most compressed end. This assumption is valid if there is only Mz moment applied on the connection. In 1210

Figure 1.

Effect of Mz and B shown with concentrated forces.

case of general 3D loading case this isn’t necessarily true. It depends on the ratio of all involved internal force components (N, My, Mz and B, namely axial force, in+out-of-plane bending, bimoment) and rigidity of the elements of the connection itself. The stiffness of the connection will strongly depend on the way how contact works between opposite elements. The proposed method of stiffness calculation for I sections eliminates some of the limitations and extends the applicability to a wider range of connections and more general 3D load application. It is proposed to “cut” the I section into a separate upper and lower flange zone (see Figure 1) and calculate separate stiffness values for these flange zones, respectively, by considering the bolt rows located directly above and below the corresponding flanges only. Other bolts will be disregarded by this model. This separation makes the approach also compatible with an application with the superelement introduced to handle welded corner connections in Vaszilievits-Sömjén & Szalai (2019). This proposed separation of flanges also allows a straightforward consideration of Mz + B interaction as these effects on flange level can be added together. For sake of simplicity we concentrate on 2D frames made of I sections. We assume that such a frame is loaded dominantly by in-plane gravity forces. There are usually no direct out-ofplane loads on such a frame, such effects would be mainly eventual second order consequences of applied imperfections with out-of-plane deformation components, using for example a properly scaled eigenshape with out-of-plane displacement components (indirect loads). For the interaction of opposite side flange zones, we define 3 possible simple scenarios: In case of a dominant My bending moment, one of the flanges (on the compressed side) will be “closed” and the other flange will be “open” or “partially open”. This assumption can be simply checked by calculating the stress distribution from the actual 3D loading case. If there are no direct out-of-plane loads and stresses available, this check is not possible. A safe assumption might be in this case to assume an “open” condition for the flange zone in tension. For each 3 scenario we define appropriate elastic springs values, to be applicable for a flange zone. – If a flange remains closed, there will be no elongation of the bolts and no resulting deformation of the end-plates. For such case we assume that the elastic spring can be assumed as perfectly rigid.

Figure 2.

Basic scenarios considered.

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– If a flange is partially open, we refer to the calculation method presented by Neumann & Nuhic (2011) and Neumann & Buzaljko & Thomassen & Nuhic (2012) and Romero (2010). – For the case of “open” flanges we propose a new calculation model 2.1 Proposed stiffness calculation method for the “open” scenario As stated above, the upper and lower flange zones will be handled separately. The proposed model allows to calculate stiffness value for a flange zone considering an “open” scenario. As it will not be assumed that the connection will “close” it is applicable to the general case when the effect of in-plane bending moment has a dominant effect on the connection. Such a situation exists when the tension force resulting from in-plane bending moment and/or other axial force is large enough compared to the out-of-plane combined flange bending moment Mz + B, so the end-plates around this bolt row are not in contact. In this case forces are acting only through the bolts. The rotational stiffness of a bolt row S(j,z) in this state can be calculated from analyzing a unit Mz bending moment and the caused φ rotation angle. Sj;z ¼

Mz ’

ð1Þ

A mechanical model can be made where the stiffnesses of all participant components are represented with a spring stiffness denoted by keq at both bolts. This equivalent spring stiffness keq represents the stiffnesses of the following components: endplate in bending in one side of the connection, bolt under longitudinal loading and the end-plate in bending on the other side of the connection. The stiffness coefficients of the individual components can be calculated based on Eurocode EN 1993-1-8. From the stiffness coefficients spring stiffnesses can be calculated by multiplying with the elastic modulus, therefore the equivalent spring stiffness is: keq ¼

E 1 k5;1

þ k110 þ k15;2

ð2Þ

where k5,1: stiffness coefficient for end-plate in bending in one side of the connection k10: stiffness coefficient for bolt under longitudinal loading k5,2: stiffness coefficient for end-plate in bending on the other side of the connection Applying an Mz out-of-plane bending moment on this model translates into a compression and a tension force FMz on these springs. Mz ¼ FMz  z The forces cause a δ deformation in both springs.

Figure 3.

Mechanical model for the “open” scenario.

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ð3Þ

δ¼

FMz keq

ð4Þ

Since the model and the forces are symmetrical the center point of the rotation angle will be midway in between the springs. Based on this the rotation angle can be calculated. If we assume that ’ is small, then FMz

’¼

δ k ¼ eq z=2 z=2

ð5Þ

Substituting back into the formula for the rotational stiffness yields: Sj;z ¼

Mz keq  z2 ¼ ’ 2

ð6Þ

3 VERIFICATION OF THE PROPOSED STIFFNESS VALUE In order to verify the correctness of the proposed analytic solutions for “open” and “partially open” scenarios, we built numerical models using Abaqus FEA software. Although we target the analysis of 2D portal frames, the correctness of the corner superelement for welded connection has already been validated in Vaszilievits-Sömjén & Szalai (2019). In order to work with simpler models, we created beam-to-beam models with 500-500 mm lengths with an assumed bolted connection between them. The end of one of the beams has been fixed against all displacements and at the other end of the other beam we applied loads appropriate for the different scenarios and we checked the resulting force-displacement diagrams for the upper and lower flange zones separately. For the models we used solid elements of C3D8R type. We considered non-linear material models. We used the built-in Risk analysis based on arc-length method. We considered 2 different bolt patterns (4 bolts and 6 bolts) with M16 8.8 bolts, without pre-stressing. We considered 15 mm end-plate thicknesses. In order to test the analytic spring value proposed by Neumann & Nuhic (2011) and Neumann & Buzaljko & Thomassen & Nuhic (2012) and Romero (2010) for the “partially open” we made a simulation with Mz bending moments incrementing up to failure. Mz is applied as 2 pieces of concentrated bending moments shared equally between the flanges. We plotted the forcedisplacement diagrams of the upper flange, considering both bolt patterns and compared with the initial stiffness calculated with the proposed analytic method. Diagrams obtained with Abaqus are drawn with intermittent lines and marked as N=0. Continuous lines are the analytic values, calculated based on methods by Neumann & Nuhic (2011) and Neumann & Buzaljko & Thomassen & Nuhic (2012) marked as “Neu” and by Romero (2010) marked as “Hei”. Regarding the test for the proposed new “open” condition, we applied in step 1 a preload of 100 kN on both flanges to create tension along the flanges (no contact) and started with the same incremental Mz application in step 2. The obtained initial stiffness values are compared with the calculated analytic spring values. Diagrams obtained with Abaqus are drawn with intermittent lines and marked as N = 100 kN tension. Continuous lines are the analytic values. Two different bolt patterns have been considered as shown on Figure 5. We can observe that the analytic values show agreement in tendency with the simulation based results but give generally higher initial stiffness. The agreement in case of the six bolts pattern is good, but for the four bolts pattern the analytic value clearly overestimates the simulated stiffness. This might come from the fact, that in case of the four bolts configuration the end-plate has a visible rotation and doesn’t start with a vertical tangent, as would be supposed by Eurocode. In case of six bolts this assumption seems to be correct, as the bolt rows above and below the upper flange result a vertical tangent of the end-plate. The tendency of Eurocode T-stub model to result higher stiffness than the real one has also been reported by Wald (2016). 1213

Figure 4.

Application of loads for “open” and “partially open” scenarios.

Figure 5.

Four and six bolts patterns.

On the diagrams corresponding to the “open” condition there are points marked as “scenario changing points”. They correspond to the load level of to the monotonic increasing outof-plane bending moment where it closes the gap at the compressed side and the scenario changes to a “partially open” one. For practical case we assume, that it is a generally acceptable safe side approach to consider the “open” condition for a flange zone which is subjected to tension along the full width. For flanges under fully compressed condition infinitely rigid spring values are used which is equivalent with a continuous connection as appropriate for a welded connection. In intermediate cases, where the “partially open” condition is deemed as justified, the corresponding higher connection stiffness can be used. 4 EXTENSION OF THE PROPOSED SUPERELEMENT MODEL TO INCORPORATE LINEAR SPRINGS The corner superelement model presented in the previous paper was valid for welded connections. In case of bolted connections, the flexibility of the connection must be considered. Following the logic of Eurocode, this will be handled by the application of linear springs which represent the flexibility of the connection. As written before the I section will be cut into flange zones and use two appropriate spring values for the out-of-plane stiffness of the flange zones, instead of the usual one spring value used by Eurocode to model the in-plane stiffness of bolted connections, as shown on Figure 8.

5 APPLICATION OF THE EXTENDED MODEL The application of the model extended with linear springs to represent the out-of-plane stiffness of bolted corner connection is demonstrated on the following example. For simplicity it has been assumed that the connection can be considered as rigid for in-plane deformations and therefore no additional springs should be considered in this direction. 1214

Figure 6.

Force-displacement diagram for the 4 bolts pattern.

Figure 7.

Force-displacement diagram for the 6 bolts pattern.

Figure 8.

Extension of the corner model with springs for each flange.

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We assume a simple 2D portal frame with 12 m span with 300 mm deep, symmetric welded I sections. The proposed analysis steps are carried out using ConSteel 3D structural design package. It can be done in other software as long it is able to handle 7DOF beam elements considering warping deformations for global structures: 1. Calculate internal forces assuming first welded corner models, without the application of out-of-plane springs. 2. Calculate stresses from all 4 components of internal forces (N, My, Mz and B) at both flanges. Assign the appropriate spring values for open or partially open cases or keep the continuous connection for the closed case. 3. Rerun the analysis with the properly set springs. 4. Calculate stresses again from all 4 components (N, My, Mz and B) at both flanges and recheck the validity of the assumption. If due to changed static system the stress distribution falls into a scenario different from originally assumed, change the spring and repeat step 3. 5. First order displacements and internal forces are available.

5.1 Analysis models In this paper we compare the results obtained from different analysis 1. First order analysis with using hinge at the tensioned flange as a conservative approach, marked as “Model 1”. At the compressed side rigid connection is considered. 2. First order analysis with using hinge at the both flanges as a very conservative approach, equivalent with the Eurocode 3 proposed approach of using isolated members with warping and out-of-plane moment releases. This is marked as “Model 2”. 3. The proposed method. First order analysis using the linear spring values at the tensioned flange as proposed above, marked as “Proposed model”. At the compressed side rigid connection is considered. Calculated spring values corresponding to four and six bolts pattern are 1122 kNm/rad and 1632 kNm/rad, respectively. 4. First order analysis assuming that connections are fully rigid out-of-plane, marked as “Model 4”. This corresponds to a welded connection.

Figure 9.

Loads on 2D frames analyzed, with lateral supports at top flange level.

Table 1. Lateral displacements of the midspan section for different models and different bolt patterns. Type of model

4 bolts pattern

Model 1 Model 2 Proposed model Model 4

50.7 mm 52.8 mm 37.7 mm 36.7 mm

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6 bolts pattern

37.4 mm

Table 2. Normal stresses in the beam flanges. Four bolts pattern

Six bolts pattern

Type of model

Top flange [MPa]

Bottom flange [MPa]

Model 1 Model 2 Proposed model Model 4

147.96 to 153.98 147.90 to 154.05 76.85 to 225.09 70.97 to 230.98

−178.85 to −140.87 −157.46 to −162.27 −104.07 to −215.65 −97.86 to −221.87

Top flange [MPa]

Bottom flange [MPa]

75.19 to 226.75

−102.08 to −217.64

5.2 Results of displacements Displacements of the centerline of the middle cross section from the given loads are shown below for all four static systems. Visibly the hinged models (Model 1 and Model 2) result excessive lateral displacements. On other side, considering the out-of-plane stiffness of any of the considered bolt patterns results displacements very close to a welded connection assumption (Proposed model vs Model 4). 5.3 Results of stresses The normal stresses of the first beam cross section right after the bolted connection are drawn for all three static systems. Normal stresses are calculated from N, My, Mz and B internal force components. The stresses along the web are the same in cases therefore are not shown. Stress distribution along the flanges are very different between the first two cases (hinged models) and the second two cases (continuity assumed) The very conservative modelling proposed by Eurocode 3 gives an un-safe stress reduction from 225.09 MPa down to 153.98 MPa (30%!) 6 CONCLUSIONS We have presented a simple method to estimate out-of-plane stiffness of bolted connections, suitable for typical application in frames. With the help of the spring values obtained for the upper and lower flanges we extended our corner superelement model and we demonstrated its application for a 2D frames. The obtained results are in line with the expectations and based on engineering judgements seem to be realistic. A proper validation of this method is under preparation. The first results show that a connection designed to have enough stiffness for inplane bending moment demand might have enough stiffness for out-of-plane effects, therefore an automatic consideration of a hinge at such locations seems to be excessively conservative. REFERENCES Vaszilievits-Sömjén, B. & Szalai, J. 2019. Simple superelement model of warping transfer in moment connections between I sections Ninth International Conference on Advances in Steel Structures (ICASS2018) 5–7 December, 2018, Hong Kong, China Neumann, N. & Nuhic, F. 2011. Design of structural joints connecting H or I sections subjected to in-plane and out-of-plane bending. Eurosteel Conference 2011, Budapest, 2011 Neumann, N. & Buzaljko, M. & Thomassen, E. & Nuhic, F. 2012 Verification of design model for out-of plane bending of structural joints connecting H or I sections. Nordic Steel Construction Conference (NSCC 2012), Oslo, 2012 Couchaux, M. & Rodier, A. 2016. Behavior of bolted end-plate connections of beams subjected to biaxial bending moments and axial forces, Eurosteel Conference September 13–15, 2017, Copenhagen, 2016 Romero, E. 2010 Finite element simulation of a bolted steel joint in fire using Abaqus program, MSc Thesis, Tampere University of Technology, 2010 Wald, F. & al 2016. Benchmark Cases for Advanced Design of Structural Steel Connections, CVUT September 2016 ConSteel 12 2018. Structural analysis and design software, www.consteelsoftware.com

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Tests of gusset plate connection under compression J. Vesecký, K. Cábová & M. Jandera Faculty of Civil Engineering, Czech Technical University in Prague, Prague, Czech Republic

ABSTRACT: Over the past years, research has shown, that behavior of connections with gusset plates subjected to compression represents a complicated task. There has been developed a few analytical methods since then, some of which are used in standards procedures, mainly in Australia, New Zealand and Canada. However, in European standards, no design procedure is described and EN 1993-1-5 offers only general rules for buckling of steel plates. This paper introduces four analytical models used for gusset plates design – Whitmore, Thornton, Modified Thornton and model proposed by Khoo, Perera and Albermani. Furthermore, results of new experimental study on six full scale specimens are presented. Among other measuring methods, 3D digital image correlation was used for detection of planar strain of plates. Finally, experimental results are compared to predictions of analytical models. It is shown, that all methods based on dispersion angle are completely inappropriate for this type of connections.

1 INTRODUCTION Gusset plates represent frequently designed component, which are used for joints of steel structures. They can be typically found in trusses, connecting their individual members together. Simultaneously, they are commonly used for connection of bracings to main loadbearing elements. Gusset plate connections are mostly subjected to axial force (tensile or compressive), or sometimes to cyclic loading (dynamic and earthquake forces). Gusset plate can be connected to structural member either centrically (e.g. by doubled cleat plate or by doubled structural members with open section) or eccentrically (by slotted cleat plate). While concentric connection is better from statics point of view, it is also more expensive and difficult for fabrication. Therefore, eccentric connection is often used instead. New research, conducted at Faculty of Civil Engineering of Czech Technical University in Prague, was focused just on eccentric connections. An experiment was executed on six full scale specimens, representing bracing composed of CHS member with connection on both sides consisting of cleat and gusset plate bolted together. Axial and lateral deflection of specimens and plane deformation of plates, using 3D digital image correlation (DIC) were measured during experiment. Results were later used for validation

Figure 1.

Specimen C2 (before testing) with nomenclature used in this paper.

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of numerical model which was then used for parametric study (Vesecký et al., 2019). Measured load bearing capacity was compared to calculated capacity using four different analytical models. It was shown, that older analytical models based on dispersion angle greatly overestimate capacity of tested connections. Therefore, these analytical models should not be used in these cases.

2 ANALYTICAL MODELS Even though gusset plates are used from the end of 19th century (i.e. Eiffel Tower), the research of their behavior has begun much later, in the middle of 20th century. Whitmore (1952) defined in his work concept of dispersion angle and effective width, based on his observations of stress flow in plates. Thirty-two years later, Thornton (1984) improved his work by introducing buckling length and critical load for gusset plate connections. Yam and Cheng (2001) later proposed modification of Thorntons method by taking plasticity into account. Finally, new series of tests on 12 specimens was conducted by Khoo et al. (2009). They have proposed completely new analytical model, based on observed collapse mechanism. In this paper, design procedures for each analytical model are just briefly introduced, all equations can be found in specific papers mentioned above or together in (Vesecký, 2019). 2.1 Whitmore method Method is based on assumption, that stress is distributed under 30° angle from point load. Therefore, Whitmore proposed that dispersion line should be considered from the first row of bolts up to last row of bolts. Effective width is then determined as width of dispersed stripe (see Figure 2). Load bearing capacity of connection is then simply calculated by equation (1), while stability, effect of gradual yielding and bending moment caused by eccentricity are not taken into account. Nu ¼ beff  t  fy

ð1Þ

where beff = the Whitmore width; t = the gusset plate thickness; and fy = the yield strength of steel. 2.2 Thornton method Thornton proposed a procedure considering the effect of buckling into account by introducing buckling lengths. He defined three nominal lengths (L1, L2 and L3) on the Whitmore section and average value (Lavg). Based on specific gusset plate configuration, one of these lengths is used. Buckling length is then simply calculated as nominal length multiplied by buckling length coefficient (βcr). Appropriate values have been proposed by Dowswell (2006), see part 2.5.

Figure 2.

Examples of Whitmore effective section beff (grey area) for corner gusset plates.

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Figure 3.

Left: Nominal lengths of gusset plate defined by Thornton. Right: Modified Thornton method.

With known buckling length, relative slenderness (
λ) is calculated. It is then used for calculation of critical stress (σcr). Load bearing capacity is defined by equation: Nu ¼ beff  t  σcr

ð2Þ

2.3 Modified Thornton method Yam and Cheng (2001) proposed modification of Thornton method, taking plasticity into account. They have suggested using 45° dispersion angle instead of 30°, see Figure 3. For connection with one bolt in each row it means 73 % increase in effective width and therefore load bearing capacity. All equations from Thornton method remain unchanged. 2.4 Khoo, Perera, Albermani (KPA) proposal Khoo et al. (2009) observed collapse mechanism with two plastic hinges (see Figure 4) during their experiment on specimens with eccentrically connected gusset plates. Based on such observation, they have proposed a new analytical model. It takes into account stability, plasticity as well as bending moment due to eccentricity. Load bearing capacity is computed in two steps, using principle of virtual works. Original analytical model with limitation on gusset and cleat plates with the same rectangular shape and same thickness has been later generalized (Vesecký, 2019) by replacing plate width with new one – length of plastic hinge (Lpl), which can be determined graphically as the shortest link between side edges of plates outside the overlapping part.

Figure 4.

Specimen at the end of test and analytical model KPA with two plastic hinges.

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Figure 5. Basic gusset plates configurations. From the left: Compact, Noncompact, Extended, Single-Brace, Chevron-Brace.

Table 1. Critical length coefficients (βcr) and nominal lengths (L0) for different gusset plate configurations. Gusset plate configuration

βcr

L0

Compact Noncompact Extended Single-Brace Chevron-Brace

* 1,00 0,60 0,70 0,75

* Lavg L1 L1 L1

*compact gusset plates should never fail in buckling mode.

In the first step, collapse load (NI) can be calculated by equating external work (Wext = N·e·θ) and internal work (Wint = (Mgpl,I +Mcpl,I)·θ): NI ¼

g c Mpl;I þ Mpl;I

e

ð3Þ

where superscript “g” indicates a gusset plate; and “c” indicates cleat plate. Then conservative estimate of squash load (Ny,min) and moment of inertia (Ic,min) is calculated, followed by elastic buckling load (Ncr), relative slenderness (
λ) and critical buckling load (Nc). The first step of the procedure is finished by calculation of ultimate load: Nu;I ¼

Nc 1 þ Nc =NI

ð4Þ

In the second step, plastic moments are reduced, taking ultimate load and critical buckling load from the first step into account. New collapse load (NII) is calculated and then, using equation (4), the ultimate load (Nu,II) is determined. Ny, Ncr,
λ and Nc remain unchanged from the first step. 2.5 Nominal lengths and buckling length coefficients Dowswell (2006) collected and evaluated results of all available experiments and numerical studies focused on connections with gusset plates. Then he recommended nominal lengths and buckling length coefficients for specific configurations of gusset plates (see Figure 5 and Table 1).

3 EXPERIMENTS Behavior of connections under compression has been investigated on six full-size experimental specimens, representing bracing member connected on both sides by cleat and gusset plates.

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Figure 6. General geometry of specimen connections (type C and type D bottom supported end); × indicates position of displacement sensors with corresponding number.

Table 2. Average material properties obtained by tensile coupon tests. component

t [mm]

E [MPa]

fy [MPa]

fu [MPa]

A [%]

εy=fy/E [-]

gusset/cleat plate CHS 102/4

8,05 3,65

199 900 201 100

405,0 358,2

610,9 598,4

24,2 23,0

0,0020 0,0018

Two types of specimens were prepared – type C with perpendicular end of gusset plate and type D with oblique end of gusset plate. Geometry is shown in Figure 6. 3.1 Tensile coupon tests Three tensile tests for 8 mm thick plate and two tensile tests for 4 mm thick plate (made out of CHS 102/4) were conducted, to measure real material properties of major specimen components. Measured stress-strain diagrams were transformed onto true stresses (σtrue) and true strains (εtrue). Values of modulus of elasticity (E), yield strength (fy), ultimate strength (fu) and ductility (A) were evaluated. Table 2 summarizes obtained results. 3.2 Connection tests Objectives of experiments were determination of load-deflection diagrams, ultimate loads, development of lateral deflection of specimens, planar deformation of plates and mode of failure. All specimens were made out of grade S355 steel, composed out of CHS 102/4, 8 mm thick cleat and gusset plates and 20 mm thick end plate. Cleat plate was embedded into prepared longitudinal groove at the end of CHS and then welded by 4 mm thick fillet weld. Cleat and gusset plates were bolted by M20 8.8 bolts. Connections were situated symmetrically on both ends of specimens. Figure 7 gives details of the tested specimens. Free length of gusset plate (Lfree,g), overlapping length (Lol) and number of bolts (always situated in two rows) were chosen as variable parameters of specimens. Table 3 summarizes values of each parameter. There was one test for each specimen carried out. Loading was displacement controlled. Deformation speed -0,3 mm/min was selected as appropriate. The test took about 30 minutes on average per one specimen (ultimate load was reached after about 18 minutes). Descending branch of the load-deflection diagram was recorded for all specimens. Eight displacement sensors were used to record lateral deflection of predefined points on specimens (seven situated at both connections, one in the middle of CHS). Planar deformation of plates was recorded by two pairs of cameras. Recording was later evaluated by DIC software. Prior to test, plates were painted in white and then sprayed in black to create speckle pattern. 1222

Figure 7. Left: Frontal and side view of test set-up. Middle: Photo of specimen C2 attached to hydraulic jack. Right: Close up photo of top connection of specimen D4 after test with sensors on the right.

Table 3. Geometry and parameters of specimens (nominal values). α

Lc

Lol

Lfree,g

Lfree,c

LCHS

Specimen

[°]

[mm]

[mm]

[mm]

[mm]

[mm]

Number of bolts

C1 C2 C3 D1 D2 D4

90 90 90 45 45 45

210 245 210 248 298 333

135 170 170 135 135 170

55 55 20 93 143 143

20 20 20 20 20 20

2000 2000 2000 2000 2000 2000

4 2 2 4 4 2

4 TEST RESULTS Simply by comparison of measured ultimate loads with number and position of bolts, it’s obvious, that their placement has no visible effect on capacity of connection (as long as bolts does not fail on their own prior to plates buckling). On the other hand, with longer total length of connection (Lc) and longer free length of gusset plate (Lfree,g) the ultimate load is decreasing as expected. 4.1 Collapse mechanism During testing of specimens, total of four phases of behavior were observed, differentiated by sudden change in stiffness as can be seen in Figure 8. In the first phase, ideally elastic behavior was observed. When axial load reached around 20 to 30 kN, bolt slip (within the hole tolerance) has occurred, followed by immediate drop in stiffness. Later on, when bolt reached edge of the hole, stiffness increased again, almost to the same value as prior to the slip. Loading force raised up to the ultimate load, which occurred 1223

Figure 8.

Load-deflection (axial/lateral) diagrams of all specimens. Significant bolt slip for specimen C3.

between 90 to 110 kN. Finally, descending branch was recorded, creating approximately hyperbolic shape on the load-deflection diagram. It should be noted, that for type D specimens, bolt slip was eliminated before tightening. Collapse mechanism with two plastic hinges, located at the free lengths of gusset and cleat plate, was observed for all six specimens. Plastic hinges always occurred exactly when ultimate load was reached. In all cases, only one connection buckled (Figure 9), while other one and CHS member remained fully elastic during the whole loading process. There was even visible decrease in lateral deflection of non-buckled connections, after reaching ultimate load of specimens. For type C specimens, buckling always occurred at the bottom, supported end, while for type D specimens, buckling always occurred at the top, loaded end. Observed collapse mechanism is in agreement with the results obtained by Khoo et al. (2009). 4.2 Comparison of measured and calculated ultimate load Using analytical models described in parts 2.1 to 2.4, predicted ultimate load was calculated for all six specimens. For Thornton and Modified Thornton method, it was considered βcr = 0,70 and L0 = L1 (see Table 1). For KPA method, buckling length factor was considered βcr = 1,20, according to (Khoo et al., 2009) and nominal length as length of the connection L0 = Lc. Plastic hinge lengths (Lgpl and Lcpl) were determined graphically as the shortest link between side edges of plates. All calculations considered real material properties, see Table 2. Following table summarizes and compares reached results.

Figure 9.

All six buckled connections. Right to left: C1, C2, C3, D1, D2, D4.

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Table 4. Comparison of measured and calculated ultimate loads of the specimens. Test

Whitmore

Spec.

Nu,exp [kN]

Nu,W [kN]

ρ [-]

Thornton Nu,T [kN]

ρ [-]

Mod. Thornton Nu,MT [kN]

ρ [-]

KPA Nu,KPA [kN]

ρ [-]

C1 C2 C3 D1 D2 D4

106,2 93,3 112,5 102,8 101,7 92,1

400,2 261,9 261,9 400,2 400,2 261,9

3,77 2,81 2,33 3,89 3,93 2,84

372,7 240,1 251,9 347,3 307,2 195,2

3,51 2,57 2,24 3,38 3,02 2,12

512,9 415,8 436,4 478,0 422,8 338,0

4,83 4,45 3,88 4,65 4,16 3,67

85,8 68,5 87,0 66,1 49,1 40,7

0,81 0,73 0,77 0,64 0,48 0,44

where Nu,i = measured or calculated ultimate load; ρ = calculated to measured resistance ratio.

It’s evident, that analytical models based on dispersion angle and effective width (Whitmore, Thornton, Modified Thornton), are significantly overestimating the resistance of eccentric gusset plate connections. They give unsafe results by hundreds of percent. The greatest source of error can be associated with false assumption about influence of specific placement and number of bolts. Other source of error is definitely neglection of bending moment arising from eccentricity. The question remains, whether it’s appropriate to consider coefficient βcr as low as 0,70, which is value for fixed-pinned member. On the other hand, KPA method, based on collapse mechanism with two plastic hinges, is suitable for such type of connections. Even this method could be improved as it gives conservative results, especially for specimens with oblique end of gusset plate (type D). 4.3 DIC measurement Deformation of connections was recorded by two pairs of cameras and then evaluated by 3D DIC software VIC-3D. Accuracy of DIC was compared to values measured by displacement sensors (LDVTs and laser sensors), see Figure 10. The chart shows a very good agreement of the results. The same figure also shows deformed shape and isolines of the principal strain for one specimen. By comparison of the maximal measured strain (0,0069) with yield strain (0,0020) calculated in Table 2 it is clear, that gusset plate undergoes plastic deformation in the area of free length.

Figure 10. Left: Lateral deflection of bottom connection C1 measured by sensors and DIC (sensors position is shown in Figure 6). Middle and right: Final deformed shape and the principal strain (e1) for the C2 bottom gusset plate as measured by 2D and 3D DIC. Red color: e1 = 0,0069; violet color: e1 = -0,0002.

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5 CONCLUSION AND FINAL RECOMMENDATIONS New tests of six specimens, representing bracing with eccentric gusset plate connections subjected to compression were described in the paper. This type of connection is still widely used, despite its disadvantages (additional bending moment due to the eccentricity). The main reason for use is simplicity of fabrication when compared to more sophisticated concentric connections. For the first time, configuration with oblique end of gusset plate (under 45° angle) was tested. Collapse mechanism with two plastic hinges on buckled connection was observed during all six tests. Other parts (second connection and CHS member) undergone purely elastic deformation. Both connections were captured by cameras during tests and recordings were later evaluated by 3D DIC, to obtain planar strain and spatial deformation of plates. Comparison with values measured by sensors shows great accuracy of DIC. Results of test revealed, that placement and number of bolts (resp. number of parallel rows) has no visible influence on the resistance of connections. As expected, lower resistance was measured for specimens with larger total length or greater gusset plate free length of connection. Four analytical models were described in this paper. Three of them (Whitmore, Thornton and Modified Thornton), based on dispersion angle and effective width, were proven insufficient for eccentric gusset plate connections, giving greatly unsafe results (by hundreds of percent). Only the most recent model KPA (Khoo et al., 2009), based on true collapse mechanism, gives realistic predictions. However, it can be improved in the future as it tends to be conservative for gusset plates with oblique end. Improvement can be done in more accurate calculation of critical buckling load (moment of inertia, buckling length), in consideration of rotational stiffness of bracing member or in inclusion of second order influence. Despite the fact, that aforementioned procedure allows simple calculation of required thickness of specific gusset plate, a correct design approach should follow these steps: 1. 2. 3. 4.

the shortest possible bolted joint between gusset and cleat plate; minimization of free lengths for gusset and cleat plate; calculation of required gusset and cleat plate thickness using KPA method; decision about alternative ways to increase buckling resistance (e.g. stiffeners) when thickness calculated in step 3 is too big; 5. if so, update of gusset and cleat plate thickness taking increased stiffness into account. ACKNOWLEDGEMENT Authors of this article would hereby like to thank to Technological Agency of Czech Republic for its support of research by grant n. TJ01000045 “Advanced procedures of steel and composite structure connection design and production”. REFERENCES Dowswell, B. 2006. Effective Length Factors for Gusset Plate Buckling. Engineering Journal (New York). 43: 91–101. EN 1993- 1-5. 2013 Eurocode 3: Design of steel structures – Part 1-5: Plated structural elements. European Committee for Standardization (CEN). Brussels. Khoo, X.E & Perera, M. & Albermani, F. 2009. Design of eccentrically connected cleat plates in compression. Advanced Steel Construction. 6(2): 678–687. Thornton, W. 1984. Bracing Connections for Heavy Construction. Engineering Journal. 21: 139–148. Vesecký, J. 2019. Buckling Resistance of Gusset Plates. Diploma thesis. Czech Technical University in Prague. Supervisor Jandera M. Vesecký, J. & Jandera, M. & Cábová, K. 2019. Numerical modelling of gusset plate connections under eccentric compression. (in press). Whitmore, R.E. 1952. Experimental Investigation of Stresses in Gusset Plates. Knoxville: Engineering Experimental Station, University of Tennessee. Yam, M.C.H & Cheng, J.J.R. 2001. Behavior and design of gusset plate connections in compression. Journal of constructional steel research. 58: 1143–1159.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Numerical modelling of gusset plate connections under eccentric compression J. Vesecký, M. Jandera & K. Cábová Faculty of Civil Engineering, Czech Technical University in Prague, Prague, Czech Republic

ABSTRACT: Recently, new experimental study which was focused on behavior of bolted connections with gusset plate under eccentric compression was carried out at CTU in Prague. Buckling resistance and longitudinal and lateral deflection was measured for six specimens. The specimens represented full scale tubular bracing connected to rigid load-bearing construction on both ends. Progressive 3D digital image correlation technology was used, to record planar strain of plates. This paper describes complex numerical models based on aforementioned physical specimens. General FEA software Abaqus including geometric and material nonlinearities (large deformations, plasticity, imperfections) was used. Numerical models were validated on results measured during experiments. Later, an extensive parametric study was performed with nearly 50 numerical models. Influence of total of eight free parameters was examined. Besides other results, parametric study confirmed previous findings, that analytical models based on dispersion angle are generally not accurate for design of eccentric gusset plate connections.

1 INTRODUCTION New experimental study focused on connections with eccentric gusset plates under compression is described in detail in paper (Vesecký et al., 2019). Based on these experiments a numerical study had followed. Its aim was creation of accurate complex numerical model (see Figure 1 and chapter 2) which was later successfully validated on results of aforementioned experiment. An influence of input properties on bearing capacity of numerical models was studied in a parametric study. Following parameters were considered in the study: inclination angle of gusset plate side edges; inclination angle of gusset plate end edge; free length of gusset plate; free length of cleat plate; thickness of plates; length of bracing member; number of bolts; steel grade. Result of the parametric study are presented in chapter 4.

2 NUMERICAL MODELS General FEA software Abaqus/CAE 6.14 was used to create complex numerical models. Real dimensions of plates and tube measured prior the tests were used. Sensitivity study was performed for the first numerical model to optimize accuracy and complexity of calculation. Besides other findings, sensitivity study confirmed, that geometric and material nonlinearities must be always taken into account. This finding was first published years ago by Chakrabarti (1987). Analysis was performed in two steps. In the first one, normalized linear elastic buckling eigenmodes were calculated. These were used in the second step as imperfections for nonlinear static analysis. Amplitude of imperfection was determined in two ways – the first as a portion of geometric fabrication tolerances according to EN 1993-1-5 (Annex C) and the second as actually measured initial deformations by digital image correlation (DIC).

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Figure 1. Specimen C2. Top left: Test specimen. Top right: Numerical model with nomenclature. Bottom left: First elastic buckling eigenmode. Bottom right: Deformed geometry with von-Mises stress. Parameters of all six specimens (dimensions, number of bolts etc.) can be found in (Vesecký et al., 2019).

2.1 Model assumptions Although no numerical models can fully capture reality, a maximal effort was made during its creation to ensure good accuracy. Nevertheless, it should be noted, that few simplifications were assumed to reduce computational complexity: • • • • • •

general stress-strain diagram was replaced by trilinear elastoplastic one; bolt holes were modeled with the same diameter as bolts (i.e. 20 mm); no axial force (related to tightening) was applied to bolts; screw shank was modeled as completely smooth, without thread; end plates were replaced by boundary conditions (fixed connection); self-weight wasn’t considered (it was less than 0,5 % of the bearing capacity).

As the consequence of the listed simplifications, it is obvious, that bolt slip wasn’t simulated. 2.2 Sensitivity study Specimens C2 and D4 were chosen for sensitivity study. The results for the investigated geometries show that: • • • •

plasticity has to be included, otherwise great computational errors occur; effect of hardening after yielding is completely negligible; when large deformations are not considered, results are about two times overestimated; bearing capacity of model without imperfections was about 20 % higher than the capacity measured during tests; • strength and stiffness of bolts have negligible effect on results; • it’s necessary to model bolts with head and nut and perfectly hard contact with plates; • a compromise between accuracy and calculation speed was achieved by using four node shell elements with 8 mm mesh size for plate and 16 mm mesh size for bracing member and by using four node tetrahedron elements with 4,5 mm mesh size for bolts. Influence of material and geometric nonlinearities is shown in Figure 2.

Figure 2. Numerical models of specimens C2 and D4. Left: Influence of large deformations and imperfections on ultimate load. Right: Influence of material nonlinearities (plasticity) on ultimate load.

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2.3 Description of numerical models 2.3.1 Model parts All models were assembled from four basic parts – gusset plate, cleat plate, CHS and bolt. Plates were created as 3D planar shell, CHS as 3D extrusion shell and bolts as solids, see Figure 3. 2.3.2 Material properties Real material properties, obtained by tensile coupon tests (see Vesecký et al. 2019), were used for numerical models. Although sensitivity study showed hardening as negligible parameter, trilinear elastoplastic stress-strain diagram was chosen to substitute general diagrams from tests. Unlimited plastic strain was assumed after ultimate stress was reached, but during loading of all models, such condition did never occur. Material parameters of bolts 8.8 were assumed according to EN 1993-1-8. All three assumed stress-strain diagrams and material properties are summarized in Figure 4 and Table 1.

Figure 3. Individual parts of numerical models (different scale). Left to right: Gusset plate, cleat plate, bracing member (CHS 102/4) and bolt (M20).

Figure 4.

Stress-strain diagrams used for numerical models.

Table 1. Material properties used for numerical models. Property

Symbol

Unit

plates

CHS 102/4

bolt 8.8

Young’s modulus Poissons ratio Yield strength Ultimate strength Plastic strain at ultimate strength

E ν fy fu εpl

MPa MPa MPa -

199 000 0,3 405,0 610,9 0,242

201 100 0,3 358,2 598,4 0,230

210 000 0,3 640,0 800,0 0,250

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2.3.3 Sections Gusset and cleat plates and CHS were all modeled by homogeneous shell section with real thickness measured during tests. Simpsons integration was used with 11 points for plates and 5 points for tube. Section integration was performed for each step of the analysis. 2.3.4 Constraints and contacts Fillet weld connecting cleat plate to CHS was replaced by simple rigid connection, due to its four times higher capacity than gusset plate connection itself. Contact between gusset and cleat plate and between head/nut of bolt and plates was considered completely frictionless in tangential direction and perfectly hard in normal direction. Compressed parts of bolt hole were connected to shank by “shell to solid” constraint. 2.3.5 Boundary conditions There were boundary conditions applied to the end edges of both gusset plates. Non-loaded end was considered fixed in all directions, while loaded end was considered fixed with exception of free axial movement. 2.3.6 Finite elements and mesh size Sensitivity study showed, that plates and CHS can be discretized by general shell finite elements S4R with four nodes, reduced integration, hourglass effect control and linear approximation. Mesh size was set to 8 × 8 mm for plates and CHS end parts resp. 8 × 16 mm for the central part. Bolts were discretized by solid tetrahedron finite elements C3D4 with linear approximation. Average mesh size was set up to 4,5 mm to create three layers of FE for head and nut. 2.4 Analysis of numerical models The calculation was performed in two steps. The first, linear buckling analysis was used to obtain eigenmodes, which were then used as imperfections into nonlinear static analysis. 2.4.1 Buckling analysis Unitary axial load was imposed on numerical models for linear buckling analysis. Critical loads and corresponding eigenmodes were calculated by subspace iteration with limit of 30 iterations with 18 vectors. Only the five lowest eigenmodes were calculated, as higher eigenmodes represent shapes, that can never occur in reality (e.g. torsional failure of gusset plate). Because of linearity of the problem, analysis was very fast, lasting approximately 1 minute. 2.4.2 Static analysis During geometric and material nonlinear analysis (GMNIA) numerical models were loaded by prescribed increments of axial deformation. That way, the execution of the physical experiments was exactly simulated. Also, full Newton-Raphson method could be used to record whole load-deflection diagram, including descending branch. Imperfections were introduced as linear combination of few lowest eigenmodes so that initial deformation was applied for both connections and bracing member. Amplitude (e0) of imperfection was determined in two ways: 1) based on measured initial deformation of plate using DIC, see Figure 5; 2) as geometric imperfection according to EN 1993-1-5 – for plates e0 = L/50 (L is the connection length), for CHS as 80 % of fabrication tolerances (EN 1090-2), thus e0 = 0,8·L/1000 = L/1250 (L is the bracing member length). It took approximately 31 minutes for each nonlinear static analysis of numerical models.

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Figure 5. Initial shape of gusset plates as measured by DIC. Left: Bottom gusset plate of specimen C2. Right: Bottom gusset plate of specimen D1.

3 RESULTS AND VALIDATION ON EXPERIMENT All numerical models reached the same collapse mechanism – buckling of one connection with two plastic hinges occurring at the free lengths of gusset and cleat plate. This collapse mechanism is in complete agreement with the one observed during physical tests (see Vesecký et al., 2019, Khoo et al., 2009 and Figure 6). When plastic hinges occurred during static analysis, decrease in maximal stress of nonbuckling connection and CHS was observed. Development of plastic hinges continued, which was signalized by exceeding of the yield strength (fy = 405,0 MPa), see Figure 7. Axial deformation was limited to 12 mm, while buckling usually took place at around 2 to 4 mm. Ultimate strength of steel was never reached even at the maximal deformation (for comparison, see Figure 7: fu = 610,9 MPa, σmax ≤ 500,0 MPa).

Figure 6.

Buckled connections of physical specimens. Left to right: C1, C2, C3, D1, D2, D4.

Figure 7. Deformed shape and von Mises stress for buckled connection of C2 (left) and D4 (right). First picture: State at the ultimate load. Second picture: State at the maximal deformation. Yielding in orange and red areas.

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Figure 8. Comparison of experimental and numerical load-deflection diagrams for specimens C2 (left) and D4 (right). Diagrams shifted so that the ultimate load is reached at the same axial deflection.

Figure 9.

Comparison of experimental and numerical ultimate loads for all six specimens.

3.1 Load-deflection diagrams As stated before, bolt slip wasn’t considered in numerical models and some other simplifying assumptions were made. Therefore, load-deflection diagrams obtained from numerical analysis did not exactly fit the ones measured during tests. In all six cases, numerical models had higher stiffness prior to buckling than physical specimens (see Figure 8). Descending branch of diagrams corresponded perfectly. Diagrams for all six specimens can be found in (Vesecký, 2019). 3.2 Ultimate load Very good agreement between the test and the numerical results was achieved for the ultimate load of specimens (see Figure 9). When imperfections determined from DIC measurements were considered, the ultimate load of numerical models was approximately 8 % higher compared to the tests. For imperfections according to EN 1993-1-5, the ultimate load of the numerical models was approximately 7 % lower than for the tests. It can be assumed, that the real amplitude of imperfections lies somewhere between those two limit values. However, both predictions were very close to the test results and the model is accurate enough for its use in a parametric study.

4 PARAMETRIC STUDY A main goal of the parametric study was to determine the degree of influence of various input values on the final load capacity and the first elastic critical load. Less attention was dedicated to the load deflection behavior. Therefore, loading by force in combination 1232

Figure 10. Schematic connection drawings of specimens C2 and D4.

with full Newton-Raphson method was chosen because of its time savings. Consistency with loading by deformation (until the limit point) was successfully verified. Specimens C2 and D4 were chosen (see Figure 10) as reference models. All other models with altered parameters were created by their modifications. Unlike for numerical models described in chapter 2, material parameters for parametric study models were determined according to EN 1991-1-1. Steel grade was S355 unless otherwise stated. Material parameters were: E = 210 000 MPa, ν = 0,30, fy = 355 MPa, fu = 490 MPa, εpl = 0,25. Imperfections were determined according to EN 1993-1-5 and EN 1090-2 with amplitude e0 = L/1250 for CHS and e0 = L/50 for plates that form connection. Examined parameters are summarized in chapter 1. Always only one parameter was changed, while all others remained constant. 4.1 Charts

Figure 11. Results of parametric study. Green line = ultimate load; red line = critical load.

4.2 Influence of individual parameters From the results presented above in form of charts, it can be derived that: • inclination of side edges of gusset plate has significant influence on the both ultimate and the critical load for small angles with diminishing influence for greater angles. Decline can be explained by just partial yielding of wide gusset plates (see Figure 12).

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Figure 12. Partial plastic hinges (in blue) for wide gusset plates at ultimate load.

• Growing inclination of end edge of gusset plate causes small decrease in the ultimate and critical load, which then stops and starts growing for greater angles. • Growing free lengths of both gusset and cleat plate causes (for examined interval) approximately linear decrease in the ultimate load and hyperbolic decrease in the critical load. • Ultimate load grows quadratically and critical load grows in cubic for growing thickness of gusset and cleat plates. • Growing length (and thus slenderness) has negligible influence on the ultimate load but significant influence on the critical load. For high slenderness, buckling of bracing member becomes decisive collapse mechanism. • Number and placement of bolts have almost no influence on the ultimate and critical load. • Growing yielding strength of steel causes almost linear increase in ultimate load.

5 SUMMARY AND CONCLUSIONS 5.1 Conclusions for the numerical models Gusset plate connections under eccentric compression can be modeled by shell elements (gusset plate, cleat plate and bracing member), solid elements must be used for bolts. It is necessary to perform geometrically and materially nonlinear analysis (GMNIA), so that large deformations and plasticity are all accounted. Hardening of steel has no effect on bearing capacity and can be neglected (thus replaced by limitless plastic strain when yielding occurs). For the examined models, neglection of the imperfections lead to 10 % to 20 % overestimation of actual bearing capacity. It is appropriate to implement imperfections as combination of eigenmodes obtained by linear stability analysis. Amplitude of imperfection can be conservatively determined as value for geometric imperfection according to EN 1993-1-5, or as 80 % of fabrication tolerances according to EN 1090-2. It’s convenient to load models by prescribed displacement increments. Then Newton-Raphson solver can be used to get the whole load-deflection diagram including descending branch. For better agreement with real load-deflection diagrams, it might be necessary to include bolt slip, and bolt thread in the numerical model. 5.2 Conclusions for the parametric study Parameters that increase bearing capacity: • growing inclination of side edges of gusset plate; thickness of plates, grade of steel. Parameters with little to no effect on bearing capacity: • inclination of end side of gusset plate; length (slenderness) of bracing member; number and placement of bolts. Parameters that reduce bearing capacity: • growing free length of gusset plate, growing free length of cleat plate. 1234

Results of parametric study confirmed, that analytical models based on dispersion angle and effective width (Whitmore, Thornton, Modified Thornton) does not correspond with real behavior of the examined connections. Number of bolts and its placement does not influence the bearing capacity at all (as long as bolts does not fail on their own). On the other hand, bearing capacity keeps rising even when gusset plate is expanded outside the effective width (beff). Although length of a bracing member does not influence the bearing capacity (for examined interval), buckling of bracing member is likely to prevail for greater member slenderness. 5.3 Practical application Complex numerical models can very accurately simulate real behavior of gusset plate connections. However, their use is usually limited to scientific purposes, because of its timeconsuming process (preparation and computational difficulty). In everyday practice, when designing structures, engineers are facing a lot of other problems, that need to be solved. As result, they need much faster and simpler tools, yet sufficiently accurate. One such solution is combination of analytical procedures (component method) with finite elements method – CBFEM. Software IDEA Statica utilizes those principles. Its use for design of gusset plate connections is described in detail by Vild et al. (2019). Results obtained by complex numerical models (like ones described in this paper) can be used as source of improvement for CBFEM analysis as well as for analytical models. ACKNOWLEDGEMENTS Authors of this article would hereby like to thank to Technological Agency of Czech Republic for its support of research by grant n. TJ01000045 “Advanced procedures of steel and composite structure connection design and production”. REFERENCES Abaqus 6.14 Documentation. Available in: http://ivt-abaqusdoc.ivt.ntnu.no:2080/texis/search/?query=wet ting&submit.x=0&submit.y=0&group=bk&CDB=v6.14 Chakrabarti, S. K. 1987. Inelastic Buckling of Gusset Plates. Disseration. The University of Arizona. EN 1090-2. 2018 Execution of steel structures and aluminum structures – Part 2: Technical requirements for steel structures. European Committee for Standardization (CEN). Brussels. EN 1993- 1-1 2nd ed. 2013 Eurocode 3: Design of steel structures – Part 1-1: General rules and rules for buildings. European Committee for Standardization (CEN). Brussels. EN 1993- 1-5. 2013 Eurocode 3: Design of steel structures – Part 1-5: Plated structural elements. European Committee for Standardization (CEN). Brussels. EN 1993- 1-8 2nd ed. 2013 Eurocode 3: Design of steel structures – Part 1-8: Design of joints. European Committee for Standardization (CEN). Brussels. Khoo, X. E & Perera, M. & Albermani, F. 2009. Design of eccentrically connected cleat plates in compression. Advanced Steel Construction. 6(2): 678–687. Vesecký, J. 2019. Buckling Resistance of Gusset Plates. Diploma thesis. Czech Technical University in Prague. Supervisor: Jandera M. Vesecký, J. & Cábová K. & Jandera M. 2019. Tests of gusset plate connection under compression. (in press). Vild, M. & Kabeláč, J. & Kuříková, M. & Wald, F. 2019 Design of gusset plate connection with single-sided splice member by component based finite element method. (in press)

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Buckling of columns during welding M. Vild & M. Bajer Faculty of Civil Engineering, Brno University of Technology, Brno, Czech Republic

ABSTRACT: The paper presents experiments of columns loaded by compressive force during welding along their longitudinal axis. A weld bead was being laid in the corner of the crosssection from the bottom of the column to its mid-height at constant load. Then, still during welding, the load was increased until the column failed by flexural buckling. Manual gas metal arc welding with carbon dioxide was used. All the columns had the height of 3 m. The tested cross-sections were HEA 100, IPE 120, IPE 160, SHS 100 × 5, and CHS 76 × 4. The measured values were applied load, column deflections and changes in length, welding voltage, current and speed, and temperatures. The experiments were performed at the laboratory of Department of Metal and Timber Structures of Faculty of Civil Engineering of Brno University of Technology. Material properties of steel are temporarily decreased by the high temperature caused by the welding process. Analytical method is offered to design the buckling resistance of a column during welding. The cross-sectional properties are assumed to be reduced at the most dangerous height and the column is treated as stepped. The welds should be placed in short segments symmetrically. Otherwise, the deflection caused by the shrinkage of asymmetrically placed welds must be also taken into account. Keywords: column buckling, experiment, steel, welding

1 INTRODUCTION Welding under load is often performed without consideration of load resistance of a member. Welding of a gusset plate or a stiffener to an existing column might serve as an example. Such column has temporarily decreased material properties due to the high temperatures in the vicinity of the weld. While welding longitudinally to the column axis is relatively safe, transverse welding is dangerous because a large portion of the column cross-section is weakened. Another example, which is the core of the authors’ main research, is the strengthening under load, i.e. welding of strengthening plates to an existing column without unloading, Vild & Bajer (2016). In this case, long longitudinal welds are applied to transfer the shear flow between an existing column and strengthening plates that are welded in different states of stresses. The shear flow is not very large so there is no need for transverse welds. The weld is formed by inducing very high temperatures to fuse two steel parts together, usually with added welded material. The steel near the welding rod is in a molten state. Gradually, both in distance from the welding rod and in time, the temperature decreases. The temperature distribution can be determined by simple analytical procedures, e.g. method of moving heat source; see Figure 1, Masubuchi (1980), Rosenthal (1941), Rosenthal (1946).

2 ANALYTICAL METHOD The analytical method was developed with a simple assumption that the steel heated to temperature higher than 500°C is ineffective. At this temperature, according to EN 1993-1-2:2006, the steel yield strength is reduced by the factor 0.78, proportionality limit by 0.36 and modulus 1236

Figure 1.

Temperature field behind the welding point.

of elasticity by 0.6. This temperature was chosen as a safe value corresponding to more detailed analysis in general cases, Huenersen, Haensch & Augustyn (1990). Welding and especially manual welding is full of uncertainties and more complicated assessment is not worth the gain. The buckling resistance of a simply supported column is the smallest when the heat source is in the mid-height of the column. The part of the cross-section is weakened and the column is treated as stepped column. The critical buckling load is the function of moments of inertia of full cross-section and weakened cross-section, the length of the column, and the length of the weakened part – ČSN 73 1401 (1998), Trahair & Kitipornchai (1971). The weakened cross-section is determined by taking into account only the part where the temperature is below 500°C. The temperature distribution can be calculated by Rosenthal equations – Rosenthal (1941, 1946). The decrease in the buckling resistance of a weakened stepped column compared to full, existing column was determined by numerical simulations in ANSYS software. The results of numerical analysis were approximated by a formula of increased buckling length Lcr;e of the stepped column:     Ltemp 0:6 Lcr;e I0 Ltemp 0:6 ¼1 þ  Lcr L Itemp L where Lcr ¼ buckling length; Ltemp ¼ length of the weakened part; L ¼ column length; I0 ¼ moment of inertia of the full cross-section; Itemp ¼ moment of inertia of the weakened cross-section. Due to the increased buckling length, buckling load Ncr;e is decreased and equivalent initial imperfection etemp increased: Wel etemp ¼ α  ð
λtemp  0:2Þ  A0 where α ¼ imperfection factor;
λtemp ¼ relative slenderness of the stepped column; Wel ¼ section modulus of the full cross-section; A0 ¼ area of the full cross-section. Second order theory is used to determine the resistance of the column stability during welding. Therefore, the stepped column deflection, wtemp , is necessary: wtemp ¼

1 5 N1  Δw  L2  etemp þ Δw þ  N1 E  I0 48 1  Ncr;e

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Figure 2.

Comparison between the numerical simulation and proposed analytical procedure.

where N1 ¼ preload under which the welding is performed; Δw ¼ difference between the positions of centres of gravity of the full cross-section and the weakened cross-section. Finally, maximum normal stress in the weakened cross-section σx;1 at mid-height is determined as: σx;1 ¼

wtemp  N1 N1 þ  fy Atemp Wel;temp

where Atemp ¼ area of the weakened cross-section; and Wel;temp ¼ elastic section modulus of the weakened cross-section. The maximum normal stress must not exceed yield strength fy . The procedure was verified by another set of numerical simulations; see Figure 2.

3 EXPERIMENTAL PROGRAM There are very few experiments in the literature. The most relevant paper is by Suzuki & Horikawa (1984) comprising 8 specimens susceptible to buckling. Therefore, a new experimental program was performed in the laboratory of Department of Metal and Timber Structures of Faculty of Civil Engineering, Brno University of Technology. The tested cross-sections were HEA 100, IPE 120, IPE 160, SHS 100 × 5, and CHS 76 × 4. There were three specimens for each cross-section except for the CHS which contained only 2 specimens. All columns had the length of 3 m. The boundary conditions were knife-edge bearings; pinned around the weaker axis and fixed around the stronger axis. The column was inserted into the loading frame and all the measuring devices were activated. The column was loaded by a high compressive force near its limit calculated by the proposed analytical method. Then a weld bead was laid at the corner/edge of the cross-section from the bottom to the mid-height. After the weld reached the mid-height, the load was gradually increased while welding continued, until flexural buckling of the column around the weaker axis occurred. The temperature was measured by thermocouples in two height levels and in different distance from the weld bead. There were seven thermocouples in total, labeled T1–T7. The lateral displacements were measured by four draw-wire sensors in three height levels, labeled wl, wm1, wm2, and wu. At mid-height, there were two sensors to measure also possible rotation of the column. The column was loaded by loading cylinder from the bottom. The force and the displacement in longitudinal direction were measured by load cell and by LVDT, respectively. The position of sensors is shown in Figures 3 and 4.

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Figure 3.

Selected cross-sections of tested specimens with the placement of measuring devices.

Figure 4. Test set-up: Column IPE160-2 after failure by buckling (left), measuring devices across the column height (right).

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Table 1. Welding speed v, heat input q, and preload magnitude N1 . HEA1

HEA2

HEA3

SHS1

SHS2

SHS3

IPE120-1

v [mm/s] q [J/mm] N1 [kN]

2.05 878 160

2.73 661 160

3.85 469 160

2.7 668 290

2.92 618 290

2.54 710 290

3.45 464 48

v [mm/s] q [J/mm] N1 [kN]

IPE120-2 3.45 464 40

IPE120-3 3.3 553 37

IPE160-1 3.75 458 100

IPE160-2 2.90 577 100

IPE160-3 2.26 741 100

CHS1 2.50 730 80

CHS2 2.14 852 70

Manual gas metal arc welding with CO2 as a shielding gas was used. The parameters of electric current, voltage and welding speed were designed for the rate of cooling from 800 to 500°C equal to 20 s to achieve high-quality weld – EN 1011-2:2003. However, due to manual welding, the speed of welding varied; see Table 1.

4 RESULTS Measured temperature distribution was slightly lower compared to the calculation according to Rosenthal equations of the moving heat source. It seems that the calculation is the safe assumption. The applied weld was long and eccentric to the column axis. The heat of the welding is causing the steel to expand and then, when the steel cools, to shrink. Huenersen, Haensch & Augustyn (1990) suggest that the column should first deflect in the direction to the weld as the steel expands and then in the direction away from the weld as the steel shrinks. However, the first phase – expanding – was not observed. Column deflection related to assumed weld length can be seen in Figure 5. The subsequent increase in load, during which the deflection soared,

Figure 5.

Lateral displacement at mid-height during welding.

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Figure 6.

Lateral displacement at mid-height during welding.

is not shown. The specimens IPE120-1 and CHS2 are not shown because they buckled prematurely during welding and the deflection caused by buckling and welding cannot be distinguished. The expanding was effectively held by the column bending stiffness. On the other hand, the weld shrinkage was significant and affected the buckling resistance. The column started to deflect at the length of the weld from 300 to 600 mm corresponding to time of welding from 90 to 280 s. The columns with lower area of the cross-sections and lower moment of inertia around the weaker axis started to deflect sooner. The examples of load-displacement diagrams can be seen in Figure 6. Specimens with crosssections IPE 160 (labeled IPE1–3) and SHS 100 × 5 (labeled SHS1–3) are compared to the analytical method (labeled An). The initial deflections are included. The initial deflections of specimens were determined using Southwell plot – Southwell (1932). The analytical method disregarded the weld shrinkage and it can be seen that for such a long weld, the weld shrinkage should be taken into account to keep the method safe. Preferably, it is convenient to divide the welds into segments up to the length of 300 mm and place them so that the weld shrinkage is in opposite directions.

5 CONCLUSION The analytical method to calculate the buckling resistance of compressed column during welding was outlined. The steel in the vicinity of the weld heated over 500 °C is assumed ineffective. The column is treated as stepped column. The lateral deflection is determined. Second order theory is used to calculate the normal stress at the weakened cross-section and this normal stress must not exceed the yield strength. The method was validated by experimental program comprising cross-sections HEA 100, IPE 120, IPE 160, SHS 100 × 5, and CHS 76 × 4. The columns were 3 m long and pinned around the weaker axis. Gas metal arc welding with carbon dioxide as shielding gas was used. The shortcoming was that manual welding was used and the welding properties, especially welding speed, was not constant. The single pass weld starts to shrink after 300 mm of the weld length or 90 s of welding. If longer weld is applied, the shrinkage of the weld should be taken into account. If symmetrical welds are designed, the welds should be divided into segments with the length of up to 300 mm and the welding sequence planned to place weld segments symmetrically to balance the shrinkage. If unsymmetrical welds are designed, the full shrinkage should be added to the lateral deflection in the calculation of the column buckling resistance.

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ACKNOWLEDGEMENT The financial support of projects No. TH02020301 and No. LO1408 “AdMaS UP – Advanced Materials, Structures and Technologies”, supported by the Ministry of Education, Youth and Sports under “National Sustainability Programme I” is gratefully acknowledged. REFERENCES ANSYS Academic Research Mechanical APDL, Release 16.2, Help System, ANSYS, Inc. ČSN 73 1401: Design of Steel Structures. Prague, 1998 (in Czech). EN 1011-2 Welding – Recommendations for welding of metallic materials – Part 2: Arc welding of ferritic steels. 2003. EN 1993-1-2 Eurocode 3: Design of steel structures – Part 1-2: General rules – Structural fire design. Prague: CNI, 2006. Huenersen, G., Haensch, H. & Augustyn, J. 1990. Repair welding under load. Welding in the World 28 (9): 174–182. Masubuchi, K. 1980. Analysis of Welded Structures: Residual Stresses, Distortion, and Their Consequences. Pergamon Press, 642 pp. ISBN-13: 978-1483172620. Rosenthal, D. 1941. Mathematical theory of heat distribution during welding and cutting. Welding Journal 20: 220–234. Rosenthal, D. 1946. The theory of moving sources of heat and its application to metal treatments. Trans. ASME 48: 848–866. Southwell, R.V. 1932. On the Analysis of Experimental Observations in Problems of Elastic Stability. Proc. Roy. Soc. London, Series A, 135: 601–616. Suzuki, H. & Horikawa, K. 1984. Welding to Pipe Column under Axial Compressive Load. Transactions of JWRI 13: 151–159. Trahair, N.S. & Kitipornchai, S. 1971. Elastic Lateral Buckling of Stepped I-beams. Journal of the Structural Division, Proceedings of the ASCE. Vild, M. & Bajer, M. 2016. Strengthening of Steel Columns under Load: Torsional-Flexural Buckling. Advances in Materials Science and Engineering 2016(1). ISSN: 1687-8434. DOI: 10.1155/2016/2765821.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Design of gusset plate connection with single-sided splice member by component based finite element method M. Vild & J. Kabeláč Brno University of Technology, Czech Republic

M. Kuříková & F. Wald Czech Technical University in Prague, Czech Republic

ABSTRACT: This paper describes the design of the single-sided gusset plate connection of a truss steel member by a component based finite element method. The elements are analyzed by geometrically and materially non-linear analysis. The proper behavior of components, e.g. of bolts, anchor bolts, welds, is treated by introducing components representing its behavior in term of initial stiffness, ultimate resistance and deformation capacity. Research oriented finite element model is validated on experiments. The models are analyzed by geometrically and materially non-linear analysis with imperfections. The research oriented model is compared with simplified design one, which includes only the joint and equivalent horizontal disruptive force. Contribution shows the current trends in advanced modelling of connection components and differences of the research oriented and design oriented finite element models. The upcoming models with coupling of member and its joints are shown on this single-sided gusset plate connection.

1 INTRODUCTION Typical case which combines the behavior of member and connection and is difficult to design separately is the single-sided gusset plate connection of diagonal steel member. Kitipornchai (1993) predicts three types of collapse of eccentrically connected diagonals. The first possible failure is the collapse of the rod itself, which deflects in a sinusoidal shape. The other two modes of collapse are the failure of the joint plates without swinging and the failure of the joint plates with swinging, Figure 1. The Kitipornchai’s approaches were applied in models studied by Wilkinson et al. (2010), who proposed modifications in the load capacity calculation. The analytical design methods developed based on this principle by Khoo (2010) and Lutz (2005) are conservative, but served as fast and relatively accurate in current practice and are commonly used (Dowswell, 2005), (Rodier & Couchaux, 2014), (CDS348:2014) and Bardot et al. (2017). Fang et al. (2014) studied yielding patterns and failure modes for different plate geometries and improved the design approach also for HSS. The global analyses of steel structures are today carried out by finite element analysis (FEA) and all the traditional procedures are not used any more. Currently, fast development of software ability of connection design by FEA and thousands of experiments are available for the validation process. In such situation, the verification process performed through benchmark tests gains crucial importance. The source and the extent of such benchmark tests for the field of structural connections is yet to be established. To achieve this goal a set of small benchmark tests that can be used as a reference in the verification process of simulations was developed. The recommendation for design by advanced modelling in structural steel is ready to be used in Annex C of EN 1993-1-5:2005. Development of modern general-purpose software and decreasing cost of computational resources facilitate this trend. The FEA of structural joints is the coming step in structural steel design. As the computational tools become more readily 1243

Figure 1.

Failure mode of a single-sided gusset plate connection (Vesecký, 2019).

available and easier to use even by relatively inexperienced engineers, the proper procedure should be employed when judging the results of computational analysis. FEA for connections is used from 70s of the last century as research-oriented FEA (ROFEA). Their ability to express real behavior of connections is making numerical experiments a valid alternative to testing and source of additional information about local stresses. Material model for FEA uses true stress-strain diagram. Validation & Verification (V&V) process of models is integral part of the procedure (Wald et al. 2014), and the FEA studies are based on the researcher’s own experiments. During preparation of CM for EN19931-8:2006, all basic components were deeply modelled (Bursi & Jaspart, 1997). Special attention was given to modelling of the T-stub, which represents the end plate connections of beam to column joints, beam splices and column bases (Virdi et al.1999). Last generation of FEA models of connections is utilized in studies focused on application of high strength steel (Coelho, 2013) and bolts in the connections (Moze & Beg, 2011). Prediction of hollow section joints is based on experimental evidence approved by FEA numerical experiments; see e.g. Partanen et al. (2001). Due to large variety of geometry, some types were studied only numerically, Fleischer et al. (2010). V&V of the FEA design models (DFEA) of steel connection design is native part of its preparation (Wald et al, 2014). The detailed procedure for verification of CBFEM was prepared (Wald et al, 2019). 2 COMPONENT BASED FINITE ELEMENT METHOD The collapse mechanism by two plastic hinges at a gusset plate and a connecting plate well represent the situation, when the member buckling resistance is higher than the resistance of the connection, see Figure 1. The CBFEM model comprises only the joint with stubs of connected members (Legner, 2019). The rotations and torsion of the member are restrained. The member is loaded by normal force N and to simulate the bending of the gusset and connected plates, eccentricities,

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Figure 2. CBFEM model of a single-sided gusset plate connection with supports and schema of bending moment diagram.

Figure 3.

Von Mises stress on specimen C2 and plastic strain on specimen D4, deformation scale 3.

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Figure 4.

Specimens of circular hollow sections connected with eccentric gusset plates (Vesecký, 2019).

and second order effects, shear force is added, see Figure 3. The applied shear force is in the position of the center of bolt group, see Figure 2. The disruptive force has the magnitude of V ¼ N=10

ð1Þ

Design resistances calculated by CBFEM were validated to experimental results (Cábová et al, 2019) and research oriented finite element model (Jandera et al, 2019). The members CHS 102 × 4 were connected by single-sided gusset plates and connecting plates with the thicknesses of 8 mm; see Figure 4 (Vesecký, 2019). The research oriented finite element model was made in Abaqus software and validated on the experiments. ROFEM - DIC is using the measured imperfections obtained by digital image correlation to apply the real geometrical imperfections; ROFEM - EN is using imperfections according to EN 1993-1-5:2005. The analytical models are labelled KPA1 and KPA2. Model KPA1 is Khoo-Perera-Albermani (Khoo et al. 2010) with minimal lengths of plastic hinges and minimal moments of inertias of gusset and connecting plates; model KPA2 is using average moments of inertia. The calculated and measured resistances are summarized in Figure 5. The ROFEM was further used for parametric study. The variables were free length of the connecting plate, thicknesses of gusset and connecting plate, number of bolts, and steel grade. The comparison of load resistances using ROFEM and CBFEM is plotted in Figure 6. The CBFEM is usually conservative except for specimens D2 and D4. However, for these specimens, the ROFEM is conservative compared to the experiments. The CBFEM model is much simpler and uses geometrically linear analysis but the accuracy of its results is comparable to ROFEM and exceeds analytical models.

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Figure 5. Validation of resistance by ROFEM - DIC (measured imperfection) and EN (standard imperfection), CBFEM, and analytical models KPA1 and KPA2 to experiments.

Figure 6.

Verification of CBFEM to ROFEM.

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Figure 7. a) Joint-member-joint subsystem with loads and supports; b) the modelled single-sided gusset plate connection.

Figure 8.

The shape of the first and second failure modes.

3 COUPLING OF A MEMBER AND ITS JOINT DESIGN The single-sided gusset plate connection is a typical joint which should not be checked individually but as a part of the joint-member-joint subsystem. The coming solution is the FEA modelling of complex model of member with its joints – see Figure 7. Figures 8 and 9 shows the influence of the thickness of the gusset plates on the buckling mode shape in CBFEM model including the member. Members of cross-sections CHS 114.3 × 5.0 and CHS 1248

Figure 9.

Influence of the gusset and connecting plate thicknesses on the first and second failure modes.

76.1 × 4.0 have theoretical length of 3 m and are connected at both sides by the single-sided gusset plates with the thickness of 4, 6, 8, and 10 mm to beam of cross-section HEB 200. Four bolts M16 are used. The lengths of the gusset plate and the connected plate are 130 mm with the gap of 20 mm. In the case of slender gusset plates, the member remains undeformed and failure occurs at the connections. Only when the thickness of the gusset plates is greater, the member buckles with the expected sinusoidal deflection shape. The compressive load resistance of such subsystem is increasing with the plate thickness until it reaches the buckling resistance of the member. Then it is climbing only very slowly due to slight decrease in buckling length.

4 SUMMARY AND ACKNOWLEDGMENT The conservative analytical models are developed and used for joints. They do not fully represent the influence and limits of interaction of a joint and a member. The CBFEM model of connection taking into account the eccentricity may include the geometry of joint and angle of a diagonal. The disruptive force with the magnitude of V = N/10 represents the imperfections and second order effects. The compressive resistance of the subsystem of the member including its joints is more precisely determined by modelling of the whole subsystem and reveals the behavior of the subsystem as a whole. The work was prepared under the R&D project supported by Technology Agency of the Czech Republic, project Advanced procedures of steel and composite structure connections design and production, No TJ01000045. REFERENCES Bardot, C. Cábová, K. Kurejková, M. Wald, F. 2017. Behaviour of a gusset plate connection under compression. Civil Engineering Journal: 26 (1). Bursi, O. S. & Jaspart, J. P. 1997. Benchmarks for Finite Element Modeling of Bolted Steel Connections, Journal of Constructional Steel Research: 43 (1–3), 17–42. Cábová, K., Jandera, M. Vesecký, J. 2019. Tests of gusset plate connection under compression, in printing, in SDSS 2019: International Colloquium on Stability and Ductility of Steel Structures, Prague.

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CDS348. 2014. Design of eccentric gusset plate connections. SCI. Calculation sheet. Couchaux, M. & Rodier, A. 2014. Eccentric bolted gusset plate of tube, Models for resistance in compression, in Proceedings of Eurosteel, Naples, 125–129. Dowswell, B. 2006. Effective Length Factors for Gusset Plate Buckling. Engineering Journal, 2, 91–102. EN 1993- 1-8. 2006. Eurocode 3: Design of steel structures – Part 1-8: Design of joints, CEN, Brussels. EN 1993-1-5. 2007. Eurocode 3: Design of steel structures - Part 1-5: Plated structural elements, CEN, Brussels. Fang, Ch. Yam, M.C.H., Cheng, R. J.J., Zhang, Y. 2015. Compressive strength and behaviour of gusset plate connections with single-sided splice members, Journal of Constructional Steel Research, 106, 166–183. Fleischer, O. Puthli, R. Wardenier, J. 2010. Evaluation of numerical investigations on static behaviour of slender RHS K-gap joints, in 13th International Symposium on Tubular Structures, Hong Kong, 75–83. Jandera, M. Cábová, K. Vesecký, J. 2019. Modelling of the eccentric gusset plate connection, in printing, in SDSS 2019: International Colloquium on Stability and Ductility of Steel Structures, Prague. Khoo, X. Perera, M. Albermani F. 2010. Design of eccentrically connected cleat plates in compression, Advanced Steel Construction, 6(2),678–687. Legner, Š. 2019. Eccentric gusset plate connection of diagonal steel member, in Czech, diploma thesis, ČVUT, Prague. Lutz, D.G. & Laboube, R.A. 2005. Behavior of thin gusset plates in compression, Thin-Walled Structures, 43(5),861–875. Moze, P. & Beg, D. 2011. Investigation of high strength steel connections with several bolts in double shear, Journal of Constructional Steel Research, 67/3, 333–347. Partanen, T. Niemi, E. Liukku, H. et al. 2001. Transverse and axial load capacities of the chord in X-joints of square hollow sections due to the interaction of brace and chord loads, in 9th International Symposium on Tubular Structures, Karlsruhe, 195–201. Vesecký, J. 2019. Buckling resistance of gusset plates, in Czech, diploma thesis, ČVUT, Prague. Wald, F. et al. 2019. Benchmark cases for advanced design of structural steel connections, Prague, Česká technika. Wald, F. Kwasniewski, L. Gödrich, L. Kurejková, M. 2014. Validation and verification procedures for connection design in steel structures, in Proceedings of Conference on Steel, Space and Composite Structures. Singapore, 111–120. Wilkinson, T. Stock, D. Hastie, A. 2010. Eccentric cleat plate connections in hollow section members in compression. In Tubular Structures XIII. CRC Press, 197–203.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Beam-to-column joints for slim-floor systems in seismic zones: Numerical investigations and experimental program C. Vulcu, R. Don & A. Ciutina Politehnica University of Timisoara, Romania

ABSTRACT: The slim-floor building system is attractive to constructors and architects due to the integration of steel beam in the overall height of the floor, which leads to additional floor-to-floor space, used mostly in acquiring additional stories. The concrete slab offers natural fire protection for steel beams, while the use of novel corrugated steel sheeting reduces the concrete volume, and can replace the secondary beams. Currently the slim-floor solutions are applied in non-seismic regions, and there are fe w studies that consider continuous or semi-continuous fixing of slim-floor beams. The current ongoing study was carried out with the aim to develop reliable end-plate bolted connections for slim-floor beams, that can be applied to buildings located in seismic areas. For this purpose, a numerical program was carried out in two stages: (i) calibration of a FE model based on a four point bending test of a slim-floor beam; (ii) case study performed for the investigation of beam-to-column joints with moment resisting connections between slim-floor beams and columns. The current paper presents the main findings of the study, an overview of the experimental program, the main conclusions and future research activities.

1 INTRODUCTION The structural solutions provided by the usage of composite elements are regarded as an effective method of enhancing structural performance. A series of advantages emerge as concrete, steel and additional components are integrated into a more resistant and ductile member – Arcelor-Mittal (2016). In particular, the slim-floor building system is attractive to constructors and architects due to the integration of steel beam in the overall height of the floor, which leads to additional floor-to-floor space, used mostly in acquiring additional stories. The concrete slab offers natural fire protection to the steel beams, while the use of novel corrugated steel sheeting reduces the concrete volume, and replaces the secondary beams (for usual spans). The slim-floor solutions are currently applied mostly in non-seismic regions – Hauf (2010), Braun et. al (2014) – and there are few studies that consider continuous or semi-continuous fixing of slim-floor beams. It was shown by Malaska (2000), that the semi-continuous joining of slim-floor beams improves the flexural stiffness of the slim-floor beams and allows the use of shallower beam and floor sections, and better performance of beams in service conditions by reducing cracking, deflections and vibrational problems. Wang et. al (2009) and Bernuzzi et. al (1995) showed that in case of increasing gravitational loads the continuous fixing of the slimfloor beams can lead to ductile plastic hinges in both beam-ends and middle spans. In contrast, the usual seismic behaviour rely on increased frame lateral stiffness and failure mechanisms by dissipation of seismic input energy by plasticization of dissipative elements or connections. Consequently, in case of Moment-Resisting-Frames (MRF) or dual frame configurations considering MRF contribution, the beams or the beam-to-column joints of MRF will dissipate energy through plastic hinges. Therefore, the application of slim-floor beam systems in seismic zones

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Figure 1.

Slim-floor system.

should consider moment-resisting connection with columns, thus developing hogging bending capacity, too. However, certain aspects characteristic for slim-floor systems should be considered (see Figure 1): – the concrete slab encases the top steel flange and requires one layer of hogging reinforcement; – the natural bonding and/or the concrete dowels contribute significantly to steel-toconcrete connection. In many cases there is no need for additional connectors; – bottom part of steel profile is larger than the top flange in order to accommodate the concrete supporting system: shallow decking or precast concrete slabs. The present study investigates the possibility to develop reliable connections for slim-floor beams, in view of application to buildings located in areas with seismic hazard. The paper presents the finite element numerical investigations and the outcomes of the study. In a first stage, a finite element numerical model was calibrated based on a four point bending test of a simply supported slim-floor beam. Further, a case study was developed in view of investigation of continuous slim-floor beam-to-column connections under both sagging and hogging bending. 2 FEM CALIBRATION OF A SLIM-FLOOR BEAM The current numerical program on slim-floor beam-to-column joints – was initiated through the calibration of a finite element (FE) model based on the experimental investigation as detailed in Hauf (2010) on a four-point bending test of a slim-floor beam. The numerical study validated the accuracy of the FEM models used for materials, contacts and boundary conditions further used in modelling the beam-to-column connection models. Thus, the information on the behaviour of a composite element, the steel-to-concrete friction coefficient, modelling procedure, importance of concrete dowel connectors and reinforcement, meshing and interactions were derived through calibration. Detailed outcomes of the study can be found in Vulcu et. al (2017, 2018). 3 PRE-TEST NUMERICAL INVESTIGATIONS OF SLIM-FLOOR BEAM-TOCOLUMN CONNECTIONS 3.1 Configuration of the slim-floor beam-to-column joint The joint assembly was conceived as an extended end-plate connection and reduced beam section. The reinforced concrete slab assures the continuity of the reinforcement by extension beyond the column. Figure 2 shows the configuration of the investigated external beam-to-column joint assembly as well as the joint components, concrete slab reinforcement and steel elements. The current technical solution resulted based on the outcomes of the previous study - Vulcu et.al (2017). The steel column is a HEB340 profile, while the steel beam is composed by a bottom steel plate (PL-20×380 mm) welded to half of an IPE600 profile. The column length is of 3930 mm, while the beam length is 2680 mm. A supplementary web plate and continuity plates were considered in the column web panel in order to limit the panel deformation. The connection between the slim-floor beam and the column is realized as bolted extended end-plate connection using four bolt rows of M36-HR.10.9 (see Figure 2). Within the lower steel plate, 1252

Figure 2. Configuration of the beam-to-column joint model: overall joint, slab reinforcement, bolt rows and dog bone in lower steel plate.

a reduced cross-section was considered with the aim to force the development of the plastic hinge in the beam and assure a prevailing elastic response of the connection. The concrete slab, integrating the steel beam and the reinforcement (transversal, longitudinal and inclined), was considered with a width of 1500 mm and a height of 145 mm. The effective width computed according to the norm EN 1994-1 (2004) was of 1200 mm. In order to assure a continuous reinforcing over the connection zone, the slab was extended beyond the column with 600 mm. Additional to transversal and longitudinal bars, inclined reinforcement was used for the concrete slab (see Figure 2). The continuity of the longitudinal reinforcing bars is assured around the column. The reinforcement bars considered in the analyses satisfy the reinforcement connection conditions required in the Annex C of EN 1998-1-1 (2004). Consequently, the longitudinal reinforcing bars are included to contribute to the negative bending moment capacity within the connection zone. Concrete in the bottom troughs has been ignored in the analysis and consequently not modelled as additional FEM analyses have shown that its influence is insignificant in the overall resistance of joint. Concrete dowels were considered, i.e. reinforcement of 12 mm diameter passing through 40 mm holes in the beam web. The centre-to-centre distances of the perforations is 125 mm. 3.2 Modelling procedure The numerical investigations of the slim-floor beam-to-column joint assembly (see Figure 2) were performed by using Abaqus v6.13 software (2013). Finite beam elements were used for the reinforcement, and solid elements for other components (bolts, plates, concrete, etc.). The material characteristics were defined for the following: concrete (C30/37), structural steel (S355), bolts (HR.10.9), and reinforcement bars (S400), considering both elastic and plastic properties. Figure 3a illustrates the true stress - true strain curves (excepting elastic deformation) for bolts (HR.10.9), reinforcement (S400) and structural steel (S355). The material model for bolts was defined based on a previous calibration of a T-stub characterized by failure mode 3 (i.e. bolt failure), see Dubina et. al (2015). For all steel elements, the elastic modulus for steel was taken as 210 MPa, and the Poisson coefficient was 0.3. The input for concrete material model is detailed in Vulcu et. al (2017), considering: Young modulus E = 32.5 MPa, and Poisson coefficient ν = 0.2. The global mesh size was adapted to different FE: reinforcing bars/20 mm; concrete/18 mm; steel profile/14 mm; bottom steel plate/ 15 mm; column/13 mm; end plate/10 mm; column web plate and stiffeners/12 mm; bolts/ 8 mm. The discretization of the beam-to-column joint assembly and its components are illustrated in Figure 3b. The boundary conditions for the column and beam considered: (i) at the 1253

Figure 3. (a) True stress-strain curve: steel (measured S355 –), reinforcement (nominal S400), bolts (Gr.10.9); (b) discretization of the beam-to-column joint assembly/connection.

top and bottom end of the column - a simple and respectively a fixed support; (ii) at the tip of the beam the load was applied in displacement control, inducing sagging/hogging bending moment within the connection. 3.3 Numerical results and parametric study The numerical models of the beam-to-column joint assemblies were subjected to negative and positive bending moment. Table 1 presents the overview of the studied joint models. The graphic response, respectively the stress and strain distribution of the numerical investigations are presented for the following configurations: (i) reference model (see Figures 4a–7a); (ii) joint assembly without dog-bone in the lower steel plate (see Figures 4b–7b). The results are presented in terms of moment-rotation curves under positive and negative bending (Figure 4a and 4b) and stress and plastic strain response (Figures 5–7). As can be observed, the reference model presents a balanced behaviour under hogging and sagging (see Figure 4a) both in terms of moment resistance and initial stiffness, with a plastic descending behaviour but proving an important ductility. The failure mechanism was characterized by the formation of the plastic hinge in the steel beam under both sagging and hogging moment. On the other hand, in case of the configuration without dog-bone (model M2), a small increase of capacity was evidenced under both positive and negative bending moment (see Figure 4b). However, under positive bending moment a brittle failure mode was recorded by breaking of the bolts in tension (see Figure 7b), reducing the overall joint rotation capacity. In order to optimise the response of the joint, several parameters were considered, as described in Table 1. The importance of each parameter is highlighted below:

Table 1. Investigated numerical models. Model

Description

Loading

M1 M2 M3 M4 M5 M6 M7

Reference model Influence of dog-bone, i.e. joint model without dog-bone Partial interaction - reduction of the concrete dowels Influence of the end plate type, i.e. flush plate at the top Influence of the reinforcement amount (6/20 mm diameter) Influence of the concrete class (C20/25; C40/45) Influence of the concrete slab, i.e. joint model without slab (only steel)

M+ M+ M+ M+ M+ M+

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M– M– M– M– M– M– M–

Figure 4. (a) Reference model M1: moment-rotation curves; (b) Comparison of M1 and M2 models – illustrating the influence of the reduced section in the lower steel plate of the beam.

Figure 5. (a) Reference model M1: response under negative bending moment (stress/plastic strain); (b) Model without dog-bone M2: response under negative bending moment.

Figure 6. (a) Reference model M1: response under positive bending moment (stress/plastic strain); (b) Model without dog-bone M2: response under positive bending moment.

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Figure 7. (a) Reference model M1: failure mechanism under negative and positive bending moment; (b) Model without dog-bone M2: failure mechanism under negative and positive bending.

– the influence of the reduced section in the lower steel plate – proved to be positively improve the joint behaviour, allowing the formation of the plastic hinge in the slim-floor beam under both sagging and hogging bending moment; – the steel-to-concrete connection degree was studied by lowering the considered number of concrete dowels. Thus, the influence of the number of concrete dowels proved to be negligible. The reduction of the concrete dowels (by 50% and 66%) did not affect the momentrotation curves for the considered planar T-shape joint assembly with slim-floor beam. However, in the case of a full length beam with end-supports, this observation is expected to be different; – the influence of the end-plate typology is important, as can be noticed in Figure 8. Through the use of a flush plate at the top, the joint resistance to hogging bending was directly influenced by the absence of the bolt-row in the extended end-plate (model M4),

Figure 8. Influence of the end plate type: (a) reference model with extended end-plate vs. joint model with flush plate at the top; (b) plastic strain recorded for M4 joint model illustrating the failure mode.

Figure 9. Influence of reinforcement amount: (a) reference model (longitudinal reinforcement as: 2Φ20mm rebars adjacent to the steel beam +, 10Φ20mm rebars towards slab borders) vs. joint model with 12Φ6mm longitudinal rebars; (b) reference model vs. joint model with 12Φ20mm longitudinal rebars.

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Figure 10. Influence of the concrete class: (a) reference model (with C30/37 concrete) vs. joint model with C20/25 concrete class; (b) reference model vs. joint model with C40/45 concrete class.

Figure 11. Influence of the presence of the concrete slab in hogging (a), and respectively in sagging (b).

and lead to a reduction of 30% in capacity. The stiffness was slightly affected as well. The failure mechanism consisted in the brittle failure of the bolt row in tension; – the influence of the longitudinal reinforcement diameter was observed to slightly affect the response under negative bending moment (Figure 9). In particular, a reduction of the reinforcing diameter to 6mm was accompanied by a resistance reduction of 7%, while the increase of the bar reinforcing diameter from 12 to 20 mm, lead to an increase of resistance of 3%; – the influence of the concrete class – was observed to be insignificant under both positive and negative bending moment (see Figure 10), as the global joint response was not affected (differences were less than 2%); – the presence of the concrete slab – was observed to be important for the response of the beam-to-column joint assembly as shown in Figure 11. Under both, positive and negative bending moment, the moment-rotation curves evidenced a significant reduction of the capacity (25%) and stiffness (40%) – in case the concrete slab was not present. In spite of this, the moment-rotation curves from the reference model and the bare-steel joint seemed to converge for rotations higher than 0.11 radians. The gain in hogging resistance is mainly due to the contribution of the reinforcing bars to the moments, while in sagging the gain is due to the increased level-arm by shifting the centre of compression into concrete.

4 EXPERIMENTAL PROGRAM 4.1 Overview: Specimen configuration, test set-up, instrumentation, loading protocols The experimental investigations are in preparation, thus only a brief overview is offered in current paper. Similar to the FE models, the specimens are planar T-shape beam-to-column 1257

Figure 12. (a) Bare steel joint assembly; (b) overview of the reinforcement arrangement; (c) overview of the specimen configuration and the experimental test set-up.

joints (see Figure 12). The technical solution adopted for the connection between beam and column represents the model M1 and is shown in Figure 12a. The reduced section in the lower steel plate is aimed for the development of the plastic hinge in the beam. A top view of the slab’s reinforcement arrangement is shown in Figure 12b. The following reinforcing bars were set: (i) longitudinal rebars (continuous around the steel column); (ii) transversal rebars (placed on the upper beam flange); (iii) inclined rebars that assure the connection between slab and the concrete region between steel flanges; (iv) rebars passing-through the beam, forming the concrete dowels and assuring the composite action between steel beam and concrete slab. Figure 12c illustrates the testing set-up and of the beam-to-column joint specimen. The load will be applied at the tip of the column through a hydraulic actuator. A pinned support was considered at the base of the column, respectively a pendulum with pinned supports was adopted for the beam. In order to avoid the out of plane deformations during the test, an out of plane system is conceived. Global as well as local instrumentation will be considered in order to characterize the response of the joint assembly during the tests in order to assure the measurement of: ▪ force in the actuator; ▪ displacement at the tip of the column; ▪ displacement at the supports as well as ▪ deformation of the column web panel; ▪ deformation in the dissipative zone (measurement in the reduced beam section and concrete slab); ▪ relative displacement between steel beam and concrete slab; ▪ force in the bolts through the use of strain gages placed in the bolts (in drilled holes of 3 mm diameter). In order to have both monotonic and cyclic response of the joints, two specimens are considered i.e. one monotonic and one cyclic. The cyclic loading procedure will consider increasing amplitude cycles.

5 CONCLUSIONS The presented study was performed with the aim to develop reliable connections between slim-floor beams and columns – for application in steel structures located in seismic zones. In a first stage, a numerical model was calibrated based on a four point bending test of a slimfloor beam. In a second step, a case study was performed for the investigation of slim-floor beam-to-column joint configurations. The FEM investigations lead to the following conclusions: – in seismic regions it is possible to rely on the full or semi-continuity of joints in the global failure mechanism of MRF or dual steel structures with slim-floor systems; – the failure mode of the joint configuration with reduced beam-section in the lower steel plate is characterized by a ductile formation of the plastic hinge in the beam. In contrast, 1258

the configuration without dog-bone lead to the failure of bolt rows in tension (brittle failure). It is to be stressed that the design avoided a connection typology characterized by brittle failure of the bolts. Consequently, the adopted failure criterion and the modelling of the post-failure behaviour – developed based on the outcomes of Dubina et. al (2015) – are not relevant. In this view, the bolts should be characterized by an overall elastic response; – the influence of concrete slim-floor slab is effective in sagging bending as it contributes to the global increase of both stiffness and bending capacity. In hogging its influence is less important and the connection characteristics are mainly based on steel components including reinforcement; – the presence of the reinforced concrete slab lead to an increase of capacity and stiffness. The inclined reinforcement and the concrete dowels contributed to the load transfer mechanism by connecting the concrete slab to the concrete within flanges. A significant increase of longitudinal reinforcement will lead to higher capacity under negative bending moment; – the slope of the moment-rotation curves in the post-peak range slightly decreases due to the reduction of the lever arm in the connection zone due to concrete crushing (i.e. change of the compression centre location). Based on the current investigation, it is proven that the slim-floor beams can be adapted to seismic-resistant structures and the key aspect is related to the behaviour of slim-floor beamto-column joints. The ongoing and future research activities will involve: (i) experimental investigations; (ii) calibration of numerical models and parametric study for improved slimfloor configurations and (iii) structural numerical analyses for improving the applicability of such systems. REFERENCES Abaqus v6.13 [Computer software]. Dassault Systèmes, Waltham, MA. ArcelorMittal, 2016. Slim floor – An innovative concept for floors. Luxembourg: ArcelorMittal. Bernuzzi, C., Gadotti, F., Zandonini, R. 1995. Semi-continuity in slim floor steel–concrete composite systems. 1st European Conference on Steel Structures. Eurosteel 1995. Braun, M., Obiala, R., Odenbreit, C., Hechler, O. 2014. Design and application of a new generation of slim-floor construction. 7th European Conf. on Steel & Composite Structures, Naples, Italy. Dubina, D., et al. 2015. High strength steel in seismic resistant building frames. Final Report. Grant No. RFSR-CT-2009-00024, RFCS Publications, European Commission, Brussels, Belgium. EN 191994-1-1, 2004. Eurocode 4: Design of composite steel and concrete structures – Part 1: General rules and rules for building. Brussels, Belgium. EN 191998-1-1, 2004. Eurocode 8: Design of structures for earthquake resistance – Part 1: General rules, seismic actions and rules for buildings. Brussels, Belgium. Hauf, G. 2010. Trag- und Verformungsverhalten von Slim-Floor Trägern unter Biegebeanspruchung. Malaska, M. 2000. Behaviour of a semi-continuous beam-column connection for composite slim floors. Ph.D. Thesis, Espoo, Finland, ISBN 951-22-5224–4. Vulcu, C., Don, R., Ciutina, A., Dubină, D. 2017. Numerical investigation of moment-resisting slimfloor beam-to-column connections. 8th International Conference on Composite Construction in Steel and Concrete (CCVIII 2017), July 30 – August 2, Spring Creek Ranch in Jackson, Wyoming (USA). Vulcu, C., Don, R., Ciutina, A. 2018. Semi-continous beam-to-column joints for slim-floor systems in seismic zones. 12th International Conference on Advances in Steel-Concrete Composite Structures (ASCCS 2018). Universitat Politècnica de València, València, Spain, June 27–29. (Doi: http://dx.doi. org/10.4995/ASCCS2018.2018.7199). Wang, Y., Yang, L., Shi, Y., Zhang, R. 2009. Loading capacity of composite slim frame beams. Journal of Constructional Steel Research 65.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

A reexamination of high strength steel Q690 plasticity model Yuanzuo Wang, Yanbo Wang, Guoqiang Li & Yifan Lyu Tongji University, Shanghai, China

ABSTRACT: The most generally used yield criterion of typical metals, the von Mises yield criterion, shows that the effects of stress triaxiality and Lode angle on the steel plasticity model are negligible. However, recently numerous experimental researches have shown that the influence of the stress triaxiality and Lode angle on the plasticity model of some typical metals should be considered. In the present paper, experiments and finite element analyses are carried out to evaluate the effect of stress triaxiality and Lode angle on the plastic behavior of Chinese high strength steel (HSS) Q690, and the application of von Mises yield criterion of the is reexamined. Results demonstrate that an appropriate yield criterion of HSS Q690 should consider effects of the Lode angle and the influence of stress triaxiality is negligible. Based on experimental and numerical results, a new yield function of HSS is proposed, to essentially simulate the experimental results. Keywords: High strength steel, Yield criterion, Stress triaxiality, Lode angle

1 INTRODUCTION The plastic behavior of metals is characterized by the constitutive model which is composed of yield function, flow criterion and hardening rule. The importance of yield function for actual engineering computations has been well noted. The von Mises yield criterion is generally adopted to describe the plasticity of typical metals. There is one basic tenet of von Mises yield model: the influences of hydrostatic pressure and Lode angle on metal plasticity are negligible. Based on this assumption of the von Mises yield criterion, it is generally accepted that the yield and flow stresses are identical in compression and tension. However, in a number of previous publications [1–10], the yield strength of some real metal materials, is greater in compression than in tension at the given strain, which is termed as strength differential effect. The strength differential effect is induced by the dependence of the yield condition on the hydrostatic pressure and the Lode angle. Therefore, the von Mises yield function is fail to describe this phenomenon and it has been well demonstrated that the dependency of plasticity model of metals on the hydrostatic pressure and Lode angle should be reexamined [11–15]. The influences of the stress triaxiality and Lode angle on plasticity model of soil and rocks has been demonstrated in literatures [16–18]. Generally, the stress triaxiality controls the shape of the yield locus on the meridian plane and the Lode angle controls the shape on the deviatoric plane. For typical metals, Spitzig and Richmond first found the plastic behavior of aluminum alloys is influenced by hydrostatic pressure, especially under high confining pressure. Richmond et al. [19] found that the yield strength of four types of steels (4330, 4310, maraging steel and HY80) is a linear function of hydrostatic pressure. Moreover, Richmond and Spitzig demonstrated that a classical yield function proposed by Drucker and Prager is applicable for iron-based materials. The classical plasticity model of geomaterials has long recognized the Lode angle effect and Tresca yield function is a typical one. Recently a number of experiments have shown that the influence of the Lode angle on the plasticity model of some typical metals should be considered. The high strength steel (HSS) with a nominal yield stress not less than 460 N/mm2 has been increasingly used in engineering structures, especially in high-rise buildings and bridges. Therefore,

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an accurate description of the HSS plastic behavior is significant for computational simulation and structure design. In the present paper, the applicability of von Mises yield criterion for Chinese Q690 HSS is examined firstly. Based on experiments on five types of testing specimens, it is found that the von Mises yield function, calibrated from the smooth round bar tensile test, fails to simulate the real response of some other experiments. Results demonstrate that an appropriate yield function of the HSS should consider effects of the Lode angle and the influence of hydrostatic pressure is negligible. A new yield function of Chinese HSS Q690 are proposed and calibrated.

2 FUNDAMENTAL DEFINITION The stress state can be described in terms of three stress tensor invariants and principal stresses (σ1 , σ2 , σ3 ) are often adopted. An arbitrary stress state can be represented geometrically in the Cartesian coordinates which take principal stresses as the axis, as shown in Figure 1. The Cartesian coordinates can also be transferred to cylindrical coordinates (σm , θ, σeq ) which defined by σm ¼ I1 =3 ¼ ðσ1 þ σ2 þ σ3 Þ=3 h i θ ¼ 1=3  arccos ðr=
σ Þ3 σ
¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h iffi 1=2 ðσ1  σ2 Þ2 þ ðσ2  σ3 Þ2 þ ðσ3  σ1 Þ2

ð1Þ ð2Þ ð3Þ

where r ¼ ½27=2ðσ1  σm Þðσ2  σm Þðσ3  σm Þ 1=3

ð4Þ

The parameter θ (0  θ  π=3) is often referred to as the Lode angle and σ
is the von Mises equivalent stress. The Cartesian coordinates can also be transferred to spherical coordinates which adopt (η, θ, σ
) as axis. The parameter η is often adopted to describe triaxiality of stress state, defined by η ¼ σ m =
σ

Figure 1.

Geometry representation of Cartesian, cylindrical and spherical coordinates.

1261

ð5Þ

3 APPLICABILITY OF VON MISES YIELD CRITERION FOR HSS Q690 The HSS Q690 is investigated in the present study. The experimental program, as summarized in Table 1, is conducted to examine the applicability of von Mises yield criterion for HSS. In addition, the corresponding initial values of stress triaxiality and Lode angle at the crack initiation set are calculated using Bai’s formulas and listed in Table 1. All experiments were conducted in Tongji University, Shanghai, China. 3.1 Smooth round bar tensile test For the von Mises yield function, there is only hardening function σ
ð
εÞ needed to be calibrated. In this study, because two different thicknesses (10mm and 30mm) of steels are used to manufacture specimens, we design two types of uniaxial tensile tests (flat specimen with rectangular cross-section for 10mm steel plate and smooth round bar tensile test for 30mm steel plate) to obtain the hardening functions. The dimensions of the flat specimen and smooth round bar are shown in Figure 2. All uniaxial tensile tests are performed on a MTS machine with the loading speed is 0.3mm/min to achieve the quasi-static loading condition. The extensometer with length of 20mm is assembled on the specimen to measure the elongation δ. The typical load-elongation curves of Q690 steel are shown in Figure 3. The engineering stress-strain curve is given by σE ¼ P=A0

ð6Þ

εE ¼ δ=δ0

ð7Þ

Table 1. Overview of experimental specimens. Loading type

Monotonic uniaxial tension

Monotonic uniaxial compression

Figure 2.

Specimen type

ID

Repeat

η0

θ0

Smooth round bar Flat specimen

SRB FS NRB-1 NRB-2 NRB-3 PS CS

3 3 3 3 3 3 3

1=3 1=3 1.026 0.739 0.556 0 -1=3

0 0 0 0 0 π=6 π=3

Notched round bar Pure shear specimen Cylinder specimen

Dimensions of flat specimen and smooth round bar.

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Figure 3.

A comparison of load-elongation curves between experiments and FE analyses.

where A0 is the initial area of the minimum cross-section and δ0 is the initial gauge section length. Based on engineering stress-strain curve, the true stress-strain can be calculated by σT ¼ σE ð1 þ εE Þ

ð8Þ

εT ¼ lnð1 þ εE Þ

ð9Þ

Because these two transformation equations are based on incompressible assumption, Eq. (8) and Eq. (9) are no longer valid in the post-necking phase in which severe non-uniform deformation occurs in the necking region. In order to solve this problem, a number of correction methods are proposed in previous publications. In the present study, a power function is used to fit the true stress-strain curve obtained from pre-necking phase, and then the approximate true stress-strain curve after necking can be obtained by extrapolating the fitting curve. The true stress strain curves for two different thickness steel plates are plotted in Figure 4. The finite element analyses are carried out to verifies the hardening function by examining the discrepancy between experimental data and simulation results of tensile tests. Due to the symmetrical shape of flat specimen and axisymmetric shape of smooth round bar, only 1/8 of the

Figure 4.

True stress-strain curves of HSS.

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Table 2.

Figure 5.

Dimensions of notched round bar (Q690).

ID

R0 (mm)

D0 (mm)

L0 (mm)

NRB-1 NRB-2 NRB-3

2.5 5.0 10.0

10.03 10.04 10.03

5.02 8.61 13.27

Dimensions of notched round bar.

full flat specimen is modeled using solid elements (C3D8R) and a quarter of full smooth round bar is modeled using axisymmetric elements (CAX8R). After comparing different mesh density, a fine mesh size is selected and the necking region is further refined with the minimum mesh size of 0.2mm and 0.05mm, respectively. As illustrated in Figure 3, the load-elongation curves of smooth round bar simulation agree with the corresponding experimental results well, which implies the calibrated mechanical properties of HSS Q690 are valid. 3.2 Notched round bar tensile test A group of notched round bar tensile tests are conducted to investigate the plastic behavior of HSS Q690 in different triaxiality stress states. It is worth mentioned that the Lode angle θ of notched round bar is identical with that of smooth round bar, but the corresponding ranges of triaxiality are different. The range of triaxiality depends on the radii of the notch of the notched round bar. In present study, three types of radii of the notches are machined: 2.5mm, 5.0mm and 10.0mm. Actual dimensions of notched round bars are listed in Table 2 and the other dimensions are identical to Figure 5. These notched round bars are all manufactured from steel plates with 30mm thickness. Due to the symmetrical shape of the smooth round bar, only a quarter is modeled by using axisymmetric elements (CAX8R) with finer mesh in the notched region with the minimum mesh size of 0.05 mm. The extensometer is assembled on the specimen to measure the elongation δ of the gauge section with 20mm length. The typical load-elongation curves of Q690 steel are shown in Figure 6. It can be observed that the analysis results by using von Mises criterion gives a well prediction of the actual load-elongation response of the notched round bar tensile tests. The stress triaxiality on the minimum cross-section of the notched round bar tends to be higher than that of the smooth round bar, but there is no distinct discrepancy appears between the analysis results by using von Mises criterion and the actual responses. It indirectly implies the influence of stress triaxiality on the plasticity model of Q690 steel is negligible. 3.3 Cylinder specimen compression test The von Mises yield criterion assumes that there is no difference between compressive and tensile yield strength at the given strain. In order to check whether the Q690 steel shows strength differential effect, another group of cylinder specimen compression tests are conducted. The cylinder specimen compression test corresponds to stress state of θ ¼ π=3 and η 5 0. In order to prevent buckling failure of the specimen, the ratio of height H to diameter D of the specimen is set as 2.0 as shown in Figure 7. Lubricant is applied to the machinespecimen interfaces to reduce the friction effect. The extensometer with length of 10mm is assembled on the specimen to measure the elongation δ. The FE model of the cylinder is 1264

Figure 6.

Notched round bar load-elongation curves for Q690.

Figure 7.

Dimension of cylinder specimen.

discretized using 4-node axisymmetric elements with the minimum mesh size of 0.05mm. The numerical simulation results are compared to the experimental results in Figure 8. It is observed that the FE results using von Mises yield function underestimates the experimental response with 10% error, which means the strength differential effect exists for Q690 steel but von Mises yield function fails to characterize it. The strength differential can arise from the dependence of yield condition on stress triaxiality or Lode angle. Because it has been demonstrated the stress triaxiality independence of HSS Q690 based on notched round bar tensile tests, the discrepancy between von Mises result and experimental result implies the yield strength of the Q690 steel is greater in uniaxial compression with θ ¼ π=3 than in uniaxial tension with θ ¼ 0 at the given strain. 3.4 Pure shear test The specimen configuration for pure shear loading is shown in Figure 9. During the loading process, an upward boundary condition is applied at the top and the bottom is fixed. It is observed that severe deformation occurs in the gauge section and the crack initiates at the pure shear section. The extensometer is assembled on the specimen to measure the elongation δ of the gauge section with 20mm length. Corresponding numerical simulations, the pure shear specimen is modeled using 8-node solid elements with finer mesh at the pure shear section. As shown in Figure 10, it can be observed that the simulation results by using von Mises criterion overpredicts the actual load-elongation response of the pure shear test with 15% error. It is worth mentioned that the Lode angle on the pure shear section of the pure shear specimen (θ ¼ π=6) is different with that of the smooth round bar (θ ¼ 0). Because it has been demonstrated that the influence of the stress triaxiality on plasticity model is negligible, the reason of this discrepancy is differences between pure shear strength (θ ¼ π=6) and tensile strength (θ ¼ 0), which means that the influences of Lode angle on the plasticity model of HSS Q690 is not negligible. 1265

Figure 8.

Cylinder specimen load-elongation curves for Q690.

Figure 9.

Dimension of pure shear specimen.

Figure 10. Pure shear specimen load-elongation curves for Q690.

4 A NEW PLASTICITY MODEL OF HSS Q690 According to the experimental results, it has been demonstrated that the Lode angle effect should be considered in the plasticity model of HSS Q690 and influence of stress triaxiality is negligible. Therefore, a new yield function which considers the Lode angle effect, is adopted to predict plastic behavior of HSS Q690 and defined as 1266

 h π πi  f ¼ σ
 σ
ð
εÞ Cs þ ðCθ  Cs Þ  θ   6 6 8 π > < 1 for 0  θ  6 Cθ ¼ > : Cc for π < θ  π: 6 3

ð10Þ

ð11Þ

There are two parameters Cc = 0.92 and Cs = 1.08 in the new proposed yield function. According to results of five types experiments mentioned in section 3, these parameters are calibrated using the inverse method to minimize discrepancy between experimental and simulated loadelongation curves. The numerical results using the new proposed plasticity model are plotted in Figures 11 and 12. By examining the discrepancy between experimental data and simulation results, the new plasticity model gives better prediction of plastic behavior of HSS Q690 with an error less than 2%.

Figure 11. Cylinder specimen load-elongation curves.

Figure 12. Pure shear specimen load-elongation curves.

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5 CONCLUSION Based on a series of experiments with 4 specimen geometries, the plasticity model has been reexamined in case of Q690 steel. According to the experimental and numerical results, the following conclusions can be drawn: The von Mises yield function, a classical metal plasticity model which assumes the stress triaxiality and Lode angle have no effect on yield strength, is no longer applicable well for high strength steel Q690, especially in shear and compressive condition. The plasticity model of HSS Q690 should considers effects of the Lode angle and the influence of hydrostatic pressure is negligible. The new proposed yield function with consideration of Lode angle effect can give more accurate prediction of the actual response. ACKNOWLEDGMENTS Financial support by the National Key Research and Development Program of China (Project No. 2018YFC0705505) is greatly acknowledged. REFERENCES [1] Chait, R., Factors influencing the strength differential of high strength steels. Metallurgical & Materials Transactions B, 1972. 3(2): p. 369–375. [2] Spitzig, W.A., R.J. Sober and O. Richmond, Pressure dependence of yielding and associated volume expansion in tempered martensite. Acta Metallurgica, 1975. 23(7): p. 885–893. [3] Meyer, L.W. and S. Abdel-Malek, Strain rate dependence of strength-differential effect in two steels. Journal De Physique IV, 2000. 10(PR9): p. 63–68. [4] Holmen, J.K., et al., Strength differential effect in age hardened aluminum alloys. International Journal of Plasticity, 2017. [5] Singh, A.P., et al., Strength differential effect in four commercial steels. Journal of Materials Science, 2000. 35(6): p. 1379–1388. [6] Spitzig, W.A., R.J. Sober and O. Richmond, The effect of hydrostatic pressure on the deformation behavior of maraging and HY-80 steels and its implications for plasticity theory. Metallurgical Transactions A, 1976. 7(11): p. 1703–1710. [7] Spitzig, W.A. and O. Richmond, The effect of pressure on the flow stress of metals ☆. Acta Metallurgica, 1984. 32(3): p. 457–463. [8] Rauch, G.C. and W.C. Leslie, The extent and nature of the strength-differential effect in steels. Metallurgical Transactions, 1972. 3(2): p. 377–389. [9] Chait, R., The strength differential of steel and Ti alloys as influenced by test temperature and microstructure. Scripta Metallurgica, 1973. 7(4): p. 351–354. [10] Leslie, W.C. and R.J. Sober, The strength of ferrite and of martensite as functions of composition, temperature and strain rate. ASM-Trans, 1967. 60(3): p. 459–484. [11] Wilson, C.D., A Critical Reexamination of Classical Metal Plasticity. Journal of Applied Mechanics, 2002. 69(1): p. 63–68. [12] Voyiadjis, G.Z., S.H. Hoseini and G.H. Farrahi, A Plasticity Model for Metals With Dependency on All the Stress Invariants. Journal of Engineering Materials & Technology Transactions of the Asme, 2013. 135(1): p. 279–284. [13] Yoshida, K., A. Ishii and Y. Tadano, Work-hardening behavior of polycrystalline aluminum alloy under multiaxial stress paths. International Journal of Plasticity, 2014. 53(53): p. 17–39. [14] Campanelli, F., A J2–J3 approach in plastic and damage description of ductile materials. International Journal of Damage Mechanics, 2016. 25(2): p. 89–91. [15] Keralavarma, S.M., A multi-surface plasticity model for ductile fracture simulations. Journal of the Mechanics & Physics of Solids, 2017. 103: p. 100–120. [16] Bardet, J.P., Lode Dependences for Isotropic Pressure-Sensitive Elastoplastic Materials. Journal of Applied Mechanics, 1990. 57(3): p. 498. [17] Menetrey, P.H., Triaxial Failure Criterion for Concrete and Its Generalization. Aci Structural Journal, 1995. 92(3): p. 311–318. [18] Bigoni, D. and A. Piccolroaz, Yield criteria for quasibrittle and frictional materials. International Journal of Solids & Structures, 2004. 41(11): p. 2855–2878. [19] Richmond, O. and W.A. Spitzig, PRESSURE DEPENDENCE AND DILATANCY OF PLASTIC FLOW. 1980.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Analysis of mechanical properties of cold formed high strength steel in weld area Martin Werunský & Jakub Dolejš Faculty of Civil Engineering, CTU in Prague, Prague, Czech Republic

ABSTRACT: This research deals with mechanical properties of cold formed area of high strength steel STRENX S960E, which was subjected to welding. Recent standard EN 1993-1-8 deals with welding of high strength steel up to the grade S700 and with a welding in a cold formed area for only the mild steels. Main purpose of this paper is description of toughness, hardness, stress-strain and metallography of 10 mm thick steel sheet made of high strength steel STRENX S960E, which had been cold formed (bent) and subsequently steel sheet of the same material has been welded to the cold formed area by MIG welding procedure. All above mentioned properties will be investigated with respect to bending radius, welding parameters and filler material matching grade.

1 INTRODUCTION In civil engineering steels with a yield strength higher than 550 MPa and tensile strength higher than 700 MPa are considered as high-strength steels (HSS). Thanks to a specific manufacturing process and an exact amount of alloys added in the steel during this process, a fine microstructure of the HSS is achieved. Thereby it is possible to produce HSS with high strength, excellent toughness and adequate formability and weldability. However, weldability and formability decrease with increasing strength of the steels generally. Utilization of HSS allows to construct members with less dead weight, higher load capacity and thereby with lower purchase cost. In Europe, HSS with yield strength up to 700MPa are used for long span bridges especially and for high buildings. However, new design codes based on a recent research for HSS (460-700 MPa) were published recently only. Before that, same mechanical properties of mild steels were considered also for HSS. Therefore, there was no motivation for their application in civil engineering, because it was not possible to fully utilize their potential and thereby cost of the structures could not be sufficiently reduced. One can conclude, that HSS (even with yield strength higher than 700 MPa) will be widely used in civil engineering in Europe in near future, which proofs many intensive researches all around Europe, which aim is to supress negative properties of HSS, in order to update recent Eurocodes so that a full potential of HSS in civil engineering can be utilized. Logically, next grade of HSS that is going to be implemented in Eurocodes is S960 grade steel.

2 STATE OF ART Unlike for welding of conventional steels, welding of high strength steels requires to follow strict manufacturing rules. Non-compliance of those rules, which are sometimes contraindicating each other, leads to a deterioration of mechanical properties of welded joint or sometimes to its complete failure. Final mechanical properties and microstructure of weld and heat affected zone (HAZ) depends primarily on an amount of heat input, t8/5 cooling time, amount of added alloys, on consumable matching, pre-heat temperature and on a welding process. All these parameters relate to each other and cannot be determined separately (Tornlom, 2007; Brtnik, 2016). 1269

Generally, HSS is necessary to weld with the lowest possible heat input. Welding with the heat input higher than prescribed by manufacturer leads to prolonging of t8/5 cooling time, thus forming undesirable microstructure such as bainite or ferrite in the weld or/and in HAZ and to its coarsening which causes the deterioration of mechanical properties – strength, microhardness and toughness (Cui, 2016). High heat input also leads to extending the width of HAZ and soft zone (Rodriguez, 2004; Erns, 2013). However, low heat input can cause lack of fusion and initiation of hot cracking. It is worth to mention that the higher strength of the HSS, more strict rules need to be satisfied (Brtnik, 2016). In civil engineering the emphasis is put on the ratio of the yield strength of the base material (BM) and the filler material (FM) the so called evenmatch, overmatch, undermatch. Utilization of the filler material with the lower yield strength than the yield strength of the base material (undermatch), leads to a deterioration of the overall strength of the welded joint (Brtník, 2016; Gunter, 2013; Kuhlmann, 2014). On the other hand, its ductility increases, which is crucial to fulfil the strict requirements of the Eurocodes for a plastic design. Utilization of undermatching filler material has also a beneficial effect on residual stresses caused by welding, because the lower the yield strength of the filler material is, the lower residual stresses are present in the weld joint. Based on number of researches, it was determined that if multipass undermatch welds are made, it is appropriate to make root welds with lower yield strength than the filler weld, which leads to overall better ductility of the weld joint, but to its less strength limitation (Wang, 2017). If welding HSS another important aspect that needs to be taken to account is an area of the lowest hardness (strength) in HAZ called “soft zone”. The soft zone is defined as the width of the HAZ where hardness values are up to 90% of the unaffected parent material and is created during the welding process by tempering the martensitic microstructure in sub-critically HAZ (SCHAZ), which was thermally affected by welding. However, some researches came up with a finding, that for the specific ratio of the width of the soft zone and thickness of the steel sheet, negative properties of the soft zone are negligible because of the constraint effect (Rodriguez, 2004; Erns, 2013). Despite the fact HSS perform lower elongation compared to mild steels, it is possible to subject them to cold forming. Formability of HSS depends more on local elongation than on general elongation, when maximal gradients of strain spread to adjacent areas (Ruoppa 2015, Ruoppa 2017). During the free-bending a plastic deformation develops form the neutral axis to the edges of the steel sheet and thereby to strain hardening. The degree of strain hardening is possible to measure by hardness test, when the most strain hardened area indicates the highest hardness value (Ruoppa 2015). After unloading an elastic stress is relieved due to springback, which causes a zigzag shape of the residual stress along the thickness of the steel sheet (Weng, 1990; Anis, 2012).

3 EXPERIMENTAL STUDY During the cold forming of the steel sheet, particularly on the inner and outer edges an alteration of mechanical properties occurs, such as strain hardening and a deformation of the grains of the microstructure. If a cold formed area is subsequently exposed to a welding process, another alteration of mechanical properties and microstructure occurs. Main goal of this experimental study is to determine toughness, hardness, alteration of the microstructure and stress-strain properties of twice altered cold formed and welded area. High strength steel STRENX S960E was chosen as a default material. Its mechanical and chemical properties are in detail described in Tables 1 and 2. Steel sheet with dimension 10 × 2000 × 6000 mm was cut by laser to smaller sheets with dimensions 10 × 250 × 500mm (S1) and 10 × 125 × 500 mm (S2). S1 were subsequently cold formed by free bending with the inner radius of 30 mm (29 pieces) and 45 mm (29 pieces). For the bending of both radii, a force of 120 tons had do be developed. Two different inner radii were manufactured so that an effect of the inner radius on the properties of the steel sheet can be determined. In the middle of the cold formed area on the outer side of the bend, a S2 steel will be welded by MIG welding (see Figure 1). 1270

Table 1.

Mechanical properties of STRENX S960E.

fy,Rp0.2

fu

Elongation A5

Toughness

960MPa

980 – 1150MPa

min. 12%

40J/ −40˚C

Table 2. Chemical composition of STRENX S960E [%]. C

Si

Mn

P

S

Cr

Cu

Ni

Mo

B

0.20

0.5

1.60

0.02

0.01

0.80

0.30

2.00

0.70

0.005

OK Aristorod 69 and OK Aristorod 89 were chosen as an undermatching filler materials. Two different filler materials were chosen in order to investigate a dependency of the welded joint on their mechanical and chemical properties (refer to Table 3 and refer to Table 4). Multipass welds with different/same filler material for root welds and filler welds will be made so that material properties of a such made weld joint could be investigated. Different levels of heat input, t8/5 cooling times, preheat temperatures and combination of filler materials will be chosen so that a wide spectrum of results will be achieved of which a suitable welding procedure will be determined for welding in cold formed area of HSS. 10 mm wide test samples will be removed from finished welded members, on which Vickers HV 0.1 hardness test will be performed according to EN ISO 9015-2 and EN ISO 6506-1 for the weld material, HAZ and cold formed area.

Figure 1.

Scheme of manufacturing of a test member (mm).

Table 3.

Chemical composition of filler materials.

Filler material

Mn

Mo

C

Si

Ni

Cr

Aristorod 69 Aristorod 89

1.54 1.75

0.24 0.533

0.089 0.081

0.53 0.80

1.23 2.22

0.26 0.41

Table 4.

Mechanical properties of filler materials.

Filler material

fy

fu

Elongation A5

Toughness

Aristorod 69 Aristorod 89

715 MPa 920 MPa

805 MPa 940 MPa

min. 17% min. 18%

60J/ −40˚C 47J/ −40˚C

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Figure 2. Left: complete test member with inner radius 30mm, right: complete test member with inner radius 45mm.

Another test samples will be prepared so that a metallography according to ISO 17639 will be performed with Carl Zeiss AxioObserver microscope with 500 x zoom in order to investigate a microstructure of weld the material, HAZ and cold formed area. 2 cylinder test samples with 10 mm radius will be milled form the middle of the bent area on which stress-strain properties will be tested according to EN ISO 6892-1. Similar way will be used to extract two test samples for toughness tests. Toughness tests will be performed in −20˚C and −40˚C. Aforementioned test samples will be extracted from only cold formed steel sheet and form cold formed and welded steel sheet so that a comparison of their mechanical properties could be made. Until today, two members were welded, one with inner radii 30 mm, second with inner radii 45 mm (see Figure 2). S2 steel sheet was welded with 3 multipass welds on both sides, when filler and root welds were made with Aristorod 69 filler metal. Welding was performed in a room temperature with parametres given in Tables 5 and 6. Heat input was then determined according to equation (1), t8/5 time was then determined according to equation (2). Table 5. Welding parametres for 30˚ bent steel sheet. Weld

Material

U [V]

A [I]

v [mm/min]

Q [kJ/mm]

t8/5 [s]

L - Root weld L - Filler weld L - Filler weld R - Root weld R - Filler weld R - Filler weld

Ar69 Ar69 Ar69 Ar69 Ar69 Ar69

240 235 235 240 235 235

29,1 28,8 28.9 29.1 28.9 28.8

424.4 474.1 365.1 459.6 518.1 395.9

0,6 0.54 0.7 0.55 0.49 0.64

18,3 15.0 25.3 15.9 12.5 21.5

Table 6. Welding parametres for 45˚ bent steel sheet – left welds. Weld

Material

U [V]

A [I]

v [mm/min]

Q [kJ/mm]

t8/5 [s]

L - Root weld L - Filler weld L - Filler weld R - Root weld R - Filler weld R - Filler weld

Ar69 Ar69 Ar69 Ar69 Ar69 Ar69

210 210 210 210 210 210

25.2 25.2 25.2 25.2 25.2 25.2

461.5 587.7 524.5 400.0 507.4 412.5

0.73 0.55 0.62 0.84 0.64 0.79

27.5 15.9 20.0 36.6 21.4 32.4

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Equation for the determination of heat input: Q¼

k:U:I:60 v:1000

ð1Þ

where Q = heat input [kJ/mm], U = Voltage [V], I = Current [A], v = welding speed [mm/min], k = thermal efficiency [-] Equation for the determination of t8/5 time: t8=5

Q2 ¼ ð4300  4:3T0 Þ  10  2  d 6

"

1 500  T0

2



1  800  T0

2 #  F2

ð2Þ

where t8/5 = time t8/5 [V], T0 = air temperature [˚C], d = plate thickness, F2 = 0,45 (shape factor for two – dimensional heat flow) After welding, test samples for hardness testing were extracted from both the pure cold formed (see Figure 3) steel sheets and from welded members (see Figure 4).

Figure 3.

Extracted test samples of only cold formed steel sheets (without welding).

Figure 4.

Extracted test samples of completely welded members.

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4 CONCLUSION Execution of all aforementioned tests on more than 40 test samples will take part in April and May 2019. Test results will be used for validation of numerical model. Validated numerical model will be used for a parametric study, in which different inner bend radiuses and different materials (stronger) will be used. Results will serve as the support for an effective design of structural members in civil engineering. ACKNOWLEDGMENT Current research is supported by grant TAČR – TJ01000045 „Advanced procedures of steel and composite structure connections design and productions“. REFERENCES Anis A., Bjork, T., Heinilla S. 2012. Prediction of residual stresses in cold formed corners. Journal of Advanced Science and Engineering Research Vol.2, No.4 December (2012) 252–264. Brtník, T. 2016. Svary prvků z vysokopevnostních ocelí. Katedra ocelových a dřevěných konstrukcí, ČVUT v Praze. Cui, B. 2016: Effect of heat input on microstructure and toughness of coarse grained heat affected zone of Q890 steel. ISIJ international, Vol.56 (2016), No. 1, pp. 132–139, 2016. Erns, W., Vallant, R, Lozinger, N. 2013. Influence of the soft zone on the strength of welded modern HSLA steels. Welding in the World, Le Soudage Dans Le Monde, 2013. EN 1993-1-12: Eurocode 3: Design of steel structures – Part 1-12: Additional rules for extension of EN 1993 up to steel grades S700, European Standard, CEN, Brussels, 2005. Günter, H.P., Rasche, C. 2008. High-strength steel fillet welded connections. Stuttgart University, Germany. Kuhlmann, U. 2014. Load bearing capacity of fillet welded connections of high strength steels. Stuttgart University, Germany. Rodrigues, D.M. 2004. Numerical study of the plastic behaviour in tension of welds in high strength steels. International Journal of Plasticity 20, pp. 1–18. Ruoppa, R. 2015. Bendability tests for ultra-high-strength steels with optical strain analysis and prediction of bending force, Finland Ruoppa R. 2017. Bending tests of very thick plates with advanced research equipment and techniques. METNET Annual Seminar, Cottbus, 11th–12th October, 2017. Törnlom, S. 2007. Undermathching butt welds in high strength steel. Departmens of civil and enviromental engineering, Lulea University of technology. Wang, Z. 2017. Effect of strength matching on mechanical properties of WELDOX 960 steel welded joint. IOP Conf. Series: Materials Science and Engineering 248 (2017) 012018. Weng, C. C., White, R. N. 1990. Residual stresses in cold-bent thick steel plates. Journal of Structural Engineering, ASCE, 116: 1, 24–39.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Degradation processes in normalized mild- and low-alloy steel building structures in service W. Wichtowski & J. Hołowaty West Pomeranian University of Technology, Szczecin, Poland

ABSTRACT: Material and strength tests on normalized steels from two building structures are described; these are a railway bridge constructed from mild steel which has been in service for over 80 years, and a complex of five cylindrical tanks of 1500 m3 capacity each, constructed from lowalloy steel, in service for 45 years. Chemical and mechanical tests were performed on a range of micro and macro aged and normalized steels from these structures. The ageing and degradation tests carried out on such steel grades have likely not yet been described in the technical literature.

1 INTRODUCTION About 80 % of steel is presently cast using the continuous casting process. Output from this casting method is up to 95% while casting into ingot moulds gives 80% (Blicharski 2002). Continuous-cast steel gives a better quality, higher cleanness and a more advantageous microstructure than mould-casted steel (Tasak 2002, Rykaluk 1999). These steels are totally killed. To obtain such a steel about 0.03% aluminum is added to give a fine-grained steel, or silicon is added to give a course-grained steel. Killed steels usually contain 0.15-0.50% Si, whose main feature is to deoxidize the steel. In order to obtain a variety of steel plates with specific properties, a number of control rolling processes were designed and carried out (Figure 1). Standard control rolling uses steel heated up to austenitic structure and rolling at different temperature (ranges I-III in Figure 1). This thermo-mechanical process is described in detail in (Tasak 2002), as well as the SHT (Sumitomo High Toughness) process of control rolling for low temperature service steels. According to the literature, the pioneers of new structural steels were the German, French, English and American steel industries. The development and production of structural steels required guaranteed and uniform material properties. As early as 1894-1895, the Wuppertal bridge

Figure 1.

Comparison of classical and control rolling with microstructure change after rolling.

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Figure 2.

Railway bridge of Thomas steel across the Wupper Valley near Müngsten (1897).

near Müngsten was constructed from early mild steel (Figure 2). The bridge was an arch construction with a span length of 170 m and height of 107 m. The steel grade later was designated as St37. The minimum tensile strength of the steel was 37 kg/mm2 (370 MPa). The steel came into use in 1895 and later became the main steel material for the manufacture of steel bridges. Later, steel St37∙12 was standardized by DIN 1612 (Schaper 1947, Mang 1977). Due to its manufacturing processes, it was called Thomas-Stahl and SM-Stahl. It should be mentioned that in Germany, Thomas convertors had been in use since 1880 and Siemens-Martin furnaces since 1864. The first DIN 1612-09 was published in September 1924, but patents for structural steels and methods of production to improve their properties also appeared. In 1928, “a way of steel production with lowered possibility for cracking due to brittleness” was patented (Figure 3). The invention involved the cold mechanical treatment of steel and then annealing at a temperature of 700-900ºC. The heat treatment of steel is used to change the microstructure and give improved properties. Annealing involves refining steel grains while increasing yield strength and simultaneously lowering brittle fracture transition temperature. Professor Gottwald Schaper (1949) defined the

Figure 3.

Patent for production of normalized steel.

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Table 1. Properties of normalized steels St37·12 and S235. Chemical composition, % Steel

C

Thomas St37·12 SM St37·12

max 0.16 max 0.20 max 0.17

S235J2

Mn 0.4÷0.5 0.4÷0.5 max 1.40

Si

P

S

P+S

~ 0.01 ~ 0.01

max 0.09 max 0.06 max 0.025

max 0.06 max 0.06 max 0.025

max 0.13 max 0.10 max 0.05

–

ReH

Rm

MPa

MPa

max 230

370÷450

max 235

360÷510

properties of early mild steel St37∙12 (Normalgüte) according to the German standard DIN 1612. For comparison, data for the contemporary non-alloy steel S235J2 according to EN 10025-2:2007 are also given. 2 PROPERTIES OF STRUCTURAL STEEL Precise values for structural steel properties are taken directly from product standards or, with some simplification, from Table 3.1. of EN 1993-1-1:2005. Over their service life, steel structures are influenced by functional and material degradation processes. The material undergoes changes in mechanical properties caused by operational factors in the structure, chemical processes in steel, atmospheric factors and so on. Assessing a structure for a further operational period requires appropriate tests and precise analysis in order to characterize the material and the magnitude of changes over a long service period (Hołowaty & Wichtowski 2013). This paper gives a fragmentary analysis of the following normalized steels: – St37∙12 mild steel, of German origin, from a railway bridge constructed in 1938, – 18G2A low alloy steel, of Polish origin, from the sheeting of five cylindrical tanks of V = 1500 m3 each, constructed in 1973. Chemical and mechanical tests are described on a range of micro and macro aged and normalized steels from these structures. Ageing and degradation tests on such steel grades are very rare and they may be expected to model the ageing processes in contemporary European normalized steel grades. Testing steels from these structures before their refurbishment gave some astounding fracture property test results. The Charpy impact energy results measured are given in Figure 4, with the results the average values from three specimens. The tests were conducted on test bar specimens of 10 × 10 × 55 with V-notch machining according to EN ISO 148-1:2017. The test specimens were machined from samples taken from the web of the first bridge span of 26 m length and from a 12-mm sheeting plate on the tank wall. The fracture properties were assessed on two types of test pieces: – naturally-aged A without any heat treatment, – normalized N, annealing at temperature 930 ºC for an hour and then cooled in air. The aged specimens with additional normalizing (N) are characterized by fine grain size and mechanical properties similar to the properties at delivery. The “cosmic” values of KV for N specimens ensured that the steels were normalized and additional microstructure tests were undertaken. The results in Figure 4 show an exceptionally large ageing effect on the steel from the railway bridge which has been in service for almost 80 years. An ageing factor equivalent to the ratio of impact energy KV after ageing to impact energy KV of a non-aged (normalized) piece, at temperature -40 ºC and -20 ºC is: 0.056 and 0.054. Such low ageing ratio values had not previously been detected, even in bridges with a much longer service period. 1277

Figure 4.

Charpy impact energy for bridge steel (B) and tank steel (T).

3 THE STRUCTURES, AND THE MECHANICAL PROPERTIES OF THEIR STEELS The single-track railway bridge spanning the Warta River is located in Gorzów Wielkopolski. The bridge’s structural system and a current view is shown in Figure 5. The structure includes eight riveted spans of length from 26.00 to 95.80 m with total bridge length 315.57 m. The bridge was constructed for the German Railways in 1938. Low carbon mild steel designated as St37 was made for the span structures. Here we are concerned with the seven riveted plate girder through spans (Wichtowski & Woźniak 2015). General views of the five cylindrical tanks and their geometric structure are shown in Figure 6. The cylindrical tanks with a capacity of 1500 m3 were constructed for the storage of phosphoric acid for a facility in Szczecin. For this reason, the tank sheeting and roof covers were protected on the inside with a rubber lining of 5 mm thickness. The tank bases were covered with protective acid-resisting slabs. Due to a change in technology production at the facility, the tanks have been used to store garden liquid fertilizer for the past 45 years. The steel walls of the tank of diameter DI = 13.0 m are constructed from six lower rings of height 1.75 m and a top ring of height 1.50 m. The two lower rings utilize 12 mm steel plates, the next two (III and IV) 10 mm plates and the three top rings (V, VI and VII) 8 mm plates. The roof cover is constructed from 24 conical plates of thickness 6 mm strengthened by external roof rafters. 4 STEEL PROPERTIES FOR PLATE GIRDER BRIDGE SPANS AND TANK WALLS As mentioned above, the non-standard results of the Charpy impact tests required additional explanation and further extended tests were undertaken, including: – – – –

chemical composition, static tensile tests on rounded specimens, Brinell hardness, microstructure examination.

Chemical composition was identified by optical emission spectroscopy. Three samples from the bridge structure were tested and one sample t = 12 mm from the tank wall. The bridge samples 1278

Figure 5.

Railway bridge structure and its current side view.

are taken from the web of the first span of L = 26.0 m, from a vertical angle in the ballasted deck of this span and from a plate in the top flange of the last span of L = 27.0 m. The chemical composition for the 9 basic alloyed elements is given in Table 2. For comparison, the chemical composition for mild steel used in the period 1888-1930 for bridge constructions as well as contemporary non-alloy steel of grade S355NL according to EN 10025-3 are also given in Table 2. Analysis of the structural steel chemical composition shows that the plate girder bridge spans were manufactured from low-carbon mild steel St37∙12 according to the German standard DIN 1612. It was the only normalized annealing steel of this standard, with ultimate

Figure 6.

View of five tanks complex and tank schematic structure.

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Table 2. Chemical composition for plate girder bridge spans and tank walls. Element content, % Type of steel

Bridge

web1) L-deck top flange

Tank

wall plate 18G2A

mild steel

S355NL 1)

C

Mn

Si

P

S

Cu

Cr

Ni

Al

0.084 0.031 0.035 0.030 ÷0.35 0.166 max 0.18

0.461 0.309 0.351 0.040 ÷0.75 2.055 0.90÷ 1.65

0.045 0.043 0.042 0.00 ÷0.18 0.395 max 0.50

64 ppm 0.036 0.027 0.004 ÷0.16 0.035 max 0.03

0.026 0.044 0.013 0.004 ÷0.12 0.027 max 0.025

0.156 0.021 0.045 0.110 ÷0.14 0.066 max 0.55

0.018 0.071 0.012 0.007 ÷0.01 0.066 max 0.30

0.031 0.016 0.030 0.030 ÷0.04 0.019 max 0.50

0.035 0.007 0.010 0.010 ÷0.02 0.083 min 0.02

all the presented test results are from the sample

strength Rm = 370 - 450 MPa and min A5 = 18-25 %. The characteristic feature of the steel is a low level of carbon. This is not a good feature, however, as ageing speed increases when carbon content is lower than 0.10 %. The chemical composition of the steel from the tank wall is that of a low-alloy steel with higher strength, grade 18G2A (ReH = 355 MPa, Rm = 490÷650 MPa) according to the former Polish standard. It is the equivalent of the current steel S355NL. A much higher content of manganese (2.055 %) negatively influences the steel weldability. At the same time, the 0.083 % aluminum content and 0.395 % silicon content increase ageing resistance. This is shown in Figure 4. Static tensile tests allowed the mechanical properties of the steel in the structures to be assessed. The test were carried out on rounded, fivefold specimens with a base diameter of 10 mm (bridge steel) and 8 mm (tank steel). The tests refer to: – six specimens from steel St37∙12, including three specimens in a naturally-aged condition (A) and three following normalized annealing (N) at a temperature of 930 ºC for an hour, and air cooled, – five specimens from low-alloy steel 18G2A, from the tank wall, including three in naturally-aged conditions (A) and two specimens after normalization (N). Figure 7 shows a sample diagram of the tensile tests on steel 18G2A, and in Table 2 the mechanical properties obtained are given. These are average values from the three samples except for Re = min ReH.

Figure 7.

Tensile curves for specimens A and N from tank wall.

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Table 3. Mechanical properties for bridge and tank steels. ReH

ReL

Rm

A5

Z

Brinell hardness

Structure (steel)

Type of steel

[MPa]

[MPa]

[MPa]

[%]

[%]

HB5

RmB [MPa]

α

ReB [MPa]

Bridge (St37·12) Tank (18G2A)

S N S N

220 275 361 387

212 269 366 380

342 376 536 526

36 41 29 32

69 67 64 72

107 106 163 165

361 357 547 553

0.70 0.70 0.65 0.65

252 249 355 359

When there is no possibility to obtain tensile specimens for testing, then the ultimate tensile strength Rm may be determined from hardness test results. This is an approximate method, but usually acceptable. For comparison purposes, hardness measurements of the steels were also carried out. The results are given in the final columns of Table 3. Hardness was measured using the Brinell method and a B3C-type tester. An indentation steel sphere of 5 mm diameter was used with a testing weight of 7350 kN for a period of 15 s according to EN ISO 6506-1:2014. The coefficient values α = ReB/RmB are taken as 0.70 and 0.65 (Hołowaty & Wichtowski 2017). The obtained external difference for Rm and RmB are +5.0 % and -5.0 %. The differences for ReH and ReB values are +14.5 % and -9.5 %. Microstructural examination was also carried out under two conditions: naturally-aged A and normalized N. The microstructures obtained are typical ferrite – pearlite phases for structural steel. Under low magnification (×10), for the tank steel, a band structure is visible which appeared during the plate rolling process (Figure 8). This is caused by dendrite segregation of Mn with a high content of 2.055 % during steel solidification. The microstructure consists of two phases: dark pearlite and light ferrite grains in different shadows of grey. At the same time, in a wider range, an analysis of impurities and inclusions in microsections of aged – (A) and normalized (N) specimens from the bridge steel were carried out. The nontreated structure and microstructure of specimens A and N using magnification ×70, ×350 and ×700 were observed (Wichtowski & Jasiński 2016).

Figure 8.

Microstructure of aged (A) and normalized (N) tank steel (×10).

Figure 9.

Microstructure of bridge steel: aged (A) and normalized (N).

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Grain size was determined by comparative method (using a master scale) according to the requirements of EN ISO 634: 2013. As per metallographic investigations, grain size was determined for the steel specimens under two conditions, A and N. The grain size for bridge steel in naturally-aged conditions S is suited to the G5 master with average grain diameter of dm = 0.0625 mm (Figure 9), while for tank steel it is the G9 master with average grain diameter of dm = 0.0156 mm. Under normalized annealing conditions N, the grain size is smaller, for the bridge steel it is the G7 master with grains dm = 0.0312 mm (Figure 9), and for the tank steel it is the G10 with grains dm = 0.0110 mm. 5 SUMMARY The steel from the railway bridge constructed in 1938 is a mild steel designated as St37∙12. The steel is a fine-grained material with G5 grain size number that has an average grain diameter of dm = 0.0625 mm. The current steel yield strength is ReH = 220 MPa and ultimate strength Rm = 340 MPa. These values are only 4.3 % and 8.1 % lower than the minimal values recommended by the standard DIN 1612 for this steel grade. Degradation changes arising during service were found in fracture properties (Figure 6). Their contribution to safety assessment and the calculation of structure durability is impossible to determine. The wall steel of tanks with a capacity of 1500 m3 which have been in service since 1973 is a low-alloy steel, normalized with subgrade E. It is fine-grained steel with actual yield strength ReH = 360 MPa and ultimate strength Rm = 535 MPa. The values are analogous to normative values for this steel grade. The actual grain size for this steel is equivalent to the G9 scale with an average grain diameter of dm = 0.0156 mm. The degradation changes of tank wall steel: – lower the values of ReH from 390 MPa to 360 MPa, that is by 7.7 %, with a small increase in of 1.9 % Rm, – lower the fracture properties – the Charpy impact energy KV; at temperature -40 ºC from value 84 J to 29 J, that is by 65.5 %, and at temperature range from -20 ºC to +20 ºC by a similar value of 51 J, that is by 34.5 %. It is the authors’ opinion that these tests on such steel grades and their results are likely to be the first published. There exists a significant probability that similar degradation processes might appear in currently-manufactured normalized structural steels over their service life. REFERENCES Blicharski, M. 2004. Material engineering. Steel. Warszawa: WNT (in Polish). Tasak, E. 2002. Weldability of steel. Kraków: Fotobit (in Polish). Rykaluk, K. 1999. Cracks in steel structures. Wrocław: DWE (in Polish). Schaper, G. 1949. Stählerne Brücken. Band I. Berlin: Verlag von Wilhelm Ernst & Sons. Mang, F. 1977. Stähl in Altbau und Wahnungsbau. Stuttgart: Frauhofer IRB Verlag. Albrecht, R. 1975. Richtlinien zum Brückenbau. Band 1. Stählerne Brücken einschließlich Stahlträger in Beton und Verbund-Konstruktionen. Wiesbaden und Berlin: Bauverlag GmbH. Hołowaty, J. & Wichtowski, B. 2013. Properties of structural steel used in earlier railway bridges. Structural Engineering International 23 (4): 512–518. Wichtowski, B. & Woźniak, Z. 2015. Properties of a normalized mild steel of a railway bridge after 75 years in service. Welding Technology Review 87 (5): 110–114(in Polish). Wichtowski, B. & Jasiński, W. 2015. Microstructural degradation processes in normalized mild steel from a railway bridge. Welding Technology Review 87 (5): 94–99 (in Polish). Hołowaty, J. & Wichtowski, B. 2017. Remarks on the material testing of historical railway bridges. EUROSTEEL 2017, Copenhagen, 13–15 September, 2017.

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Effect of the steel grade on equivalent geometric imperfections for lateral torsional buckling R. Winkler & M. Knobloch Ruhr-Universität Bochum, Bochum, Germany

ABSTRACT: The equivalent geometric imperfections of EN 1993-1-1:2005 were developed based on studies considering steel grades up to S460. For flexural buckling an extension of the scope of the verification method to higher steel grades up to S700 is currently being evaluated in the context of the revision and further development of Eurocode 3. For LTB, however, similar studies, are still lacking. This paper presents a comprehensive study on the development of equivalent geometric imperfections for high-strength steel members subjected to lateral torsional buckling. The study focuses on members with hot-rolled double symmetric I-/H-cross sections of steel grade S700. The proposed imperfections and their application to verification methods are compared to existing design approaches. In addition to normative rules applied in Europe, a novel design approach is also taken into account, which is currently discussed as part of the further development of Eurocode 3. Furthermore the paper provides a new proposal for LTBimperfections and the additional consideration of steel grades up to S700 using the partial internal forces method. This proposal leads to design results that fit well with the results of numerical simulations.

1 INTRODUCTION The equivalent geometric imperfection method can easily be applied within the framework of a computer-based analysis of steel structures using state-of-the-art computational tools. Internal forces are computed according to 2nd order theory considering the effects of equivalent geometric imperfections. These internal forces are used for a cross-section verification. The equivalent imperfections take into account, among other things, the effects of geometrical imperfections, residual stresses as well as stiffness degradation and load-redistribution due to partial yielding of the structural members. The method is already often applied in design practice for a straightforward assessment of steel members and structural systems that may fail due to flexural buckling. Increasingly computer tools are provided which facilitates the application of the method also for the assessment of lateral-torsional buckling, e.g. [1]. The framework for the structural analysis provided in prEN 1993-1-1:2018 also facilitates the application of the equivalent geometric imperfection method for lateral-torsional buckling verification. In the context of the revision of EN 1993-1-1: 2010, Lindner et. al. [1] recommended to critically reflect the existing approaches [2, 3] for equivalent geometric imperfections. The cause and background of this discussion were among other things: (i) the equivalent bow imperfections were specified for a member that is subjected to a compression force only [4]. For flexural buckling, however, the extended yielding range and stiffness reduction due to additional bending moments should be taken into account. In case of lateral torsional buckling the divergent stability behaviour between a member subjected to uniform compression and lateral torsional buckling ought to be considered. (ii) As part of the further development of EN 1993-1-1, structural steels with steel grades up to S700 are considered. Therefore, novel studies concerning equivalent geometric imperfections should include these steel grades. 1283

Lindner et al. [5-7] published new studies on equivalent bow imperfections for the stability case flexural buckling. The results were included in the proposal of the prEN1993-1-1: 2018 [8] (so-called Final Document). The equivalent bow imperfections are determined as a function of the buckling curve and the steel grade. In addition, a distinction is made between bending about the strong and weak cross-sectional axis. While any plastic cross-sectional interaction formula can be used for buckling about the weak axis (taking into account the limitation of the plastic bending resistance to Mpl = 1.25 Mel), the cross-sectional check for buckling about the strong axis is limited to the plastic linear plastic interaction. Winkler et al [9, 10] extended the application of the approach for equivalent bow imperfections for flexural buckling to the partial internal forces method (PIM) [11]. The extension enables a consistent design approach for cross-section and stability failure. In addition, the method is commonly used in design practice and easy to apply within the framework of the RUBSteEl-tools [12-14]. Lindner [15] proposed equivalent bow imperfections for the stability case lateral torsional buckling that have been considered in [8]. These imperfections are based on a study carried out by Kindmann et al. for steel grades S235 and S355, [16]. The results of this study were considered in the German National Annex of EN 1993-1-1 [3]. The classification uses the ratio h/b ≤ 2.0. The equivalent bow imperfection is calculated using a reference relative bow imperfection βLT and the material parameter ε to consider the steel grade. The reference relative bow imperfection is 1/200 for h/b ≤ 2.0 and 1/150 for h/b > 2.0 and a plastic cross-section verification. This paper presents a proposal for equivalent bow imperfections for lateral torsional buckling of a member in bending. The proposal is based on the results of a numerical simulation study and avoids a limitation of the cross-section verification method. In conjunction with imperfections for flexural buckling presented in [10, 9] a full set of equivalent geometric imperfections for second order analysis taking account of flexural and lateral torsional buckling is available.

2 EQUIVALENT GEOMETRIC IMPERFECTION METHOD 2.1 Aim of equivalent bow imperfections The simplified assessment approach “equivalent imperfection method” is based on the 2nd order theory. The equilibrium is formulated on the deformed system, taking equivalent geometric imperfections into account and the internal forces are determined according to 2nd order theory. With these internal forces an elastic or plastic cross-section verification is performed. Thus, the load bearing capacity according to this method is strongly affected by the size and shape of the equivalent geometric imperfections as well as the cross-section verification approach. A key objective for determining the amplitude of equivalent bow imperfections is that the structural analysis according to second order theory and the subsequent cross-section verification should cause the same design results as sophisticated numerical simulations. 2.2 Plastic cross-section interactions Applying the equivalent geometric imperfection method for design purposes, the imperfections are linked to the cross section verification method. For lateral torsional buckling, the cross-section verification must capture biaxial bending moments My and Mz axial forces N and warping moments Mω. For this, the linear plastic interaction and the Partial Internal forces Method (PIM) [11] are suitable. The partial internal forces method can be used for the assessment of the cross section capacity based on the plastic theory. A main advantage of the method is that it can easily take into account all internal forces and moments of steel members of arbitrary cross-sections. In principle, the allocation of the internal forces and moments on the individual partial elements of the cross-section takes place, taking into account the equilibrium conditions. This results in partial internal forces and moments. By considering the plastic capacity of the partial elements 1284

and interaction relationships, the load-carrying capacity can be verified. The method is based on mechanical principles, promotes a fundamental understanding and is therefore very popular in education and engineering practice. The implementation of the method in software tools, i.e. [12, 14, 13], also encourages its dissemination. 2.3 Method A key objective of the paper is to develop required equivalent bow imperfections using the partial internal forces method for rolled double-symmetric I/H cross-sections and to compare them with existing approaches. The results are based on numerical simulations. For this purpose geometrically and materially nonlinear investigations on imperfect members were performed (GMNIA). The numerical simulations considered both structural and geometrical imperfections with the common values and distributions (geometric imperfection v0 of the compressed flange v0 = L/1000 and residual stresses according to ECCS No. 33 [20]). The imperfections for the analyzed stability case lateral torsional buckling included prerotations and predeformations. This is the common approach but differs from the study made in [16]. The residual stresses were assumed to be 0.3 and 0.5 × 235 N/mm², respectively, irrespective of the grade of the considered steel grade, [20]. For members with predefined LTB-slenderness the maximum bending capacities were determined. This capacities were used in a next step to determine the required equivalent bow imperfections. The RUBSteEl eEducation tool FE-STAB [14] was used to perform a 2nd order theory analysis. For this the numerical determined capacities were applied and the bow imperfections e0,LT were increased to reach a cross section utilization of 100 %. The required equivalent bow imperfections are analyzed as j-values (j = L/e0,LT) to allow a direct comparison with the standard approaches. The strict application of the described method resulted in disproportionately small j values for specific cases. In order to avoid this, steel members outside the range of practical application were excluded by means of a limit of rotation. The rotation according to 2nd order theory was limited to ϑlim = 0.3 rad [16]. If the rotation limit was decisive, the j-value was set to 1000. The numerical study considered various doubly symmetric cross-sections; and the parameter field was selected based on the objectives of the analysis: – For an analysis of the geometric bow imperfections of members made of high-strength steels, in addition to the steel grades S235 and S355, steels with the higher steel grades S460 and S700 were included. – The h/b ratios of the considered cross-sections covered a large range of the commonly used hot-rolled doubly symmetric cross-sections. The ratio was used for the classification of the European buckling curves and thus enabled a comparison to common approaches found in the literature. – LTB slenderness ratios between 0.25 and 3.0 have been considered to check the consistency of the approaches for compact to slender members.

2.4 Bending moment distribution First, the influence of the applied moment distribution on the required equivalent bow imperfections was analyzed. The decisive load case was determined for further investigations. For this purpose, Figure 1 shows the required equivalent bow imperfections for five different bending moment distributions (LC 1 to 5) and three different cross sections (HEB 200 (left), HEB 600 (middle) and HEAA 1000 (right)). Using the partial internal forces method the imperfections are presented in terms of the j-value as a function of the LTB slenderness. The constant moment distribution (LC 1, blue) and the parabolic moment distribution (LC 4, purple) are decisive in terms of minimal j-values for the stability case lateral torsional buckling. The following analyses are exclusively presented, based on the constant moment distribution. The slenderness ratio influences the minimal j-values. For all cross sections the smallest value result for a slenderness ratio of 0.75. For more slender members the j- value distinctively increases.

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Figure 1. Influence of the moment distribution on the j-values for the partial internal forces method for an HEB 200, HEB 600 and HEAA 1000 with a steel grade S235 as a function of the slenderness ratio.

2.5 Cross section verification and steel grade To analyze the influence of the cross section verification method on the required equivalent bow imperfections, the j-values are determined using the partial internal forces method and the linear plastic interaction. Figure 2 exemplarily shows the resulting j-values for the cross section HEB 200 as a function of the LTB slenderness. The j-values for the partial internal forces method and the linear plastic interaction differ significantly. The minimal j-values according to PIM are approximately 130, while the use of the linear plastic interaction lead to minimal j-values of 300. The j-values partially correct the failure if the linear plastic interaction. Figure 2 additionally shows the influence of steel grade on the equivalent bow imperfections. The various colors display the influence of the steel grade. The analysis considers the steel grades S235 (dark blue), S355 (blue), S460 (green) and S700 (orange). The influence of the steel grade differs with the considered cross section verification. Using the partial internal forces method the influence is marginal. The minimal j-values were between 132 (S235) and 150 (S700). For the linear interaction the influence is more pronounced. The minimal j-values were between 300 (S235) and 423 (S700). In an approach for

Figure 2. Effect of the steel grade on the j-values as a function of the LTB slenderness ratio (PIM left and linear interaction right) for a HEB 200 section and a constant moment distribution.

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Figure 3. Effect of various limits on the j-values for an HEA 1000 section with a steel grade S235 and a constant moment distribution.

equivalent bow imperfections using the partial internal forces method the consideration of the steel grade is less decisive. 2.6 Sensitivity study The displayed results so far consider a rotation limit of ϑ < 0.3 rad and a cross section utilization of 100%. The limitation of the rotation ϑ was chosen according to the calculations described in [16]. The limit of 0.3 rad is equal to 17.4° and considers the practical application range. Lindner et al. introduce two more limit criteria to avoid disproportional small j-values for the imperfections for flexural buckling [6, 1]. The first criteria allows to exceed the crosssection utilization by 3 %. For the stability case flexural buckling this small overestimation leads to useful results, [1, 9, 10]. The second criteria limits the plastic shape coefficient αpl,z to 1.25. This limitation is decisive for buckling about the weak cross section axis and with this also for lateral torsional buckling and was implemented in [21]. Both criteria build the basis for the proposed bow imperfection approach for flexural buckling in [8]. Figure 3 shows the j-values for an HEAA 1000 section as a function of the relative LTB slenderness ratio. The grey curve shows the results without any limitation. In this case the limitation (continuous black curve) does not influence the minimal j-value. For compact cross sections of slender members this criterion helps to consider only the practical application range, [22] and thus it builds the basis for the present study. With the overestimation of the cross section utilization as well as the limitation of the plastic shape coefficient the j-values result higher. Without additional criteria a minimal j-value of 126 result. Considering 103% for the cross section utilization the value slightly increase to 134 and with an additional limitation of αpl,z the minimal j-value result to 160. The influence is less distinctive than for flexural buckling. To reach a consistent approach for all stability cases both criteria are considered in the development of a new approach for equivalent bow imperfections for lateral torsional buckling.

3 PROPOSAL AND COMPARISON The results presented in the last section are the basis for the development of an easy applicable and consistent design approach based on the equivalent imperfection method using the partial

1287

Table 1. Values for initial bow imperfections, LC 1.

Cross section

αpl,z

Approach L [mm] for NA Steel
λLT ¼ 1:0 BC EC3 EC3

HEM 200

235 355 1,533 460 700

14596 9766 7628 5188

HEB 200

235 355 1,527 460 700

9264 6434 5195 3774

IPE 80

235 355 1,576 460 700

1818 1286 1051 778

HEB 600

235 355 1,542 460 700

9732 7243 6102 4706

HEA 800

1,557

235 355 460 700

8147 6329 5445 4308

235 355 IPE 600 1,577 460 700

5684 4448 3840 3050

235 355 1,604 460 700

6885 5490 4779 3833

HEAA 1000

c

300

b

400 -

c

300

b

400 -

b

400

a

500 -

b

400

a

500 -

b

400

a

500 -

b

400

a

500 -

b

400

a

500 -

Required equivalent bow imperfections based on numerical determined resistances 100% CS prEN utilization

103% CS utilization

103% and αpl,z < 1.25

200

200 163 143 116

148 148 149 152

196 211 218 222

236 254 262 266

200

200 163 143 116

132 141 150 148

150 164 177 179

175 197 213 214

200

200 163 143 116

124 126 126 119

140 144 145 146

161 173 173 176

200

200 163 143 116

140 138 137 135

151 154 157 159

181 185 189 191

150

150 122 107 87

134 130 129 123

145 145 145 140

174 174 175 169

150

150 122 107 87

130 124 122 115

141 137 136 130

169 164 163 156

150

150 122 107 87

126 115 110 101

134 1251 120 113

161 150 144 135

internal forces method. In addition, existing approaches are compared with the analysis results and j-values. Table 1 shows the resulting j-values under consideration of the limit criteria described in section 3.4 (PIM = 103% cross section utilization and αpl,z < 1.25). The j values for a constant moment distribution (LC 1) are shown separately for all cross sections (sorted according to their h/b ratios). In addition, the investigation was separately carried out for the different steel grades. In addition, for a direct comparison, the table includes the j-values of the existing standards and approaches described in section 2.3. Both, the limitation of the plastic shape coefficient and a permitted cross section utilization of 103 % lead to larger j values for all cross-sections than the analysis without limit criteria. For compact cross-sections (h/b < 2.0) the j-values decrease slightly with increasing steel grade, for slender sections the j-values increase. This results from two different effects. With increasing slenderness of the cross section the influence of the steel grade on the member length decreases (Tab. 1) which directly affects the stability risk of the member. The second effect is the influence of the residual stresses. With increasing steel grade the influence of the residual stresses decreases. Overall, however, the influence of the steel grade on the j-values is 1288

Figure 4.

Comparison of different approaches for equivalent bow imperfections with FEM-results, LC 1.

low. A direct dependence on the j-values of the buckling curves (BC) as well as the h/b-ratios cannot be determined. The classification of the cross sections using the h/b-value leads to minimal j-values for both groups and each considered limit criteria. Considering a cross section utilization of 103 % for compact cross sections with h/b ≤ 2.0 a minimal j-value of 140 result. With the same limit criteria for slender cross sections j = 113. With an additional consideration of the limit for αpl,z the minimal values result to 161 (h/b ≤ 2.0) and 135 (h/b >2.0). In the following, two approaches for members under LTB are proposed and compared with the results of the numerical simulation study. Both approaches consider the minimal determined j-values with the cross section classification using the h/b-ratio and are set to fix values without consideration of the steel grade. 1) j = 140 (h/b ≤ 2.0) resp. j = 113 (h/b > 2.0) The first approach takes a cross section utilization of 103 % into account. 2) j = 160 (h/b ≤ 2.0) resp. 135 (h/b > 2.0) The second approach additionally limits the plastic shape coefficient αpl,z to 1.25. Figure 4 compares the limit load capacities in order of the maximum capacities per approach My,EI,i in the related form my,i,lim = My,EI,i/My,pl (horizontal axis) to the numerical limit load capacities My,FEM in related form my,FEM,lim = My,FEM/My,pl (horizontal axis). In 1289

addition to the proposals described above, the limit carrying capacities resulting from the approach in prEN 1993-1-1 are compared with the carrying capacities from the numerical analysis. The two proposals correctly record the load capacities and lead to conservative design results with little scatter. The value Δm describes the absolute mean deviation of the load capacities according to the equivalent imperfection method compared to the numerical simulations over the entire parameter range. Δm,S235 contains only the results for the steel grade S235. The lowest mean deviation result for the imperfection approach according to prEN 1993-1-1 in combination with the partial internal forces method. However, the bow imperfections lead to a few slightly non-conservative results for steel grade S235 (dark blue symbols). The prEN 1993-1-1 proposal is based on calculations described in [16]. The geometric imperfection of the numerical model did not consider a combination of predeformations and prerotations. Considering both leads to smaller capacities and to smaller required j-values. Both proposals in this publication lead to conservative design results and are consistent to the concepts for flexural buckling according to [8] and [9, 10].

4 CONCLUSIONS Initial bow imperfections for doubly symmetric I-/H- cross sections subjected to lateral torsional buckling have been determined based on a numerical simulation study. These bow imperfections consider the well-suited partial internal forces method (PIM) for the crosssection verification. For simplification, the bow imperfections have been proposed to be independent of the steel grade and bending moment distribution for design purposes. A comparative study has shown that the proposed bow imperfections lead to suitable design results compared to numerically determined member capacities. Together with the previously established equivalent geometric imperfections for flexural buckling, the proposal of this paper forms a consistent approach for the structural design of compact and slender steel members. REFERENCES [1] Lindner, Joachim; Kuhlmann, Ulrike; Just, Adrian. Verification of flexural buckling using initial bow imperfections e0/L of table 5.1 of EN 1993-1-1, CEN/TC250/SC3/WG1 N131, Meeting of the Working Group 1-1 of CEN/TC250/SC3, Berlin, 12 October 2016. [2] EN 1993-1-1:12.2010. EN 1993-1-1: 2010 Design of steel structures - Part 1.1: General rules and rules for buildings”. [3] EN 1993-1-1/NA:08/2015. Nationaler Anhang – National festgelegte Parameter – Eurocode 3: Bemessung und Konstruktion von Stahlbauten – Teil 1-1: Allgemeine Bemessungsregeln und Regeln für den Hochbau”. [4] Lindner, J.; Scheer, J.; Schmidt, H. 1998. Stahlbauten, Erläuterungen zu DIN 18800, Teil 1 bis Teil 4. Steel structures, Commentary to DIN 18000 Part 1 to 4, Bd. 67. Beuth Verlag GmbH; Ernst & Sohn. [5] Lindner, Joachim. 2017. “Repräsentative Vorkrümmungen e 0 für das Biegeknicken - Ergänzende Untersuchungen”. Stahlbau 86, Heft 8, S. 707–715, doi: 10.1002/stab.201710512. [6] Lindner, Joachim; Kuhlmann, Ulrike; Just, Adrian. Nachweis des Biegeknickens auf der Grundlage von Vorkrümmungen e0/L nach Tabelle 5.1 in DIN EN 1993-1-1. In: NA 005-08-16 AA. [7] Lindner, Joachim; Kuhlmann, Ulrike; Just, Adrian. 2016. “Verification of flexural buckling according to Eurocode 3 part 1-1 using bow imperfections”. Steel Construction 9, Heft 4, S. 349–362, doi: 10.1002/stco.201600004. [8] prEN 1993-1-1:2018:2018. Eurocode 3 - Design of steel structures - Part 1-1: General rules and rules for buildings”. [9] Winkler, Rebekka; Niebuhr, Martin; Knobloch, Markus. 2017. “Geometrische Ersatzimperfektionen für Biegeknicken um die starke Querschnittsachse unter Berücksichtigung des Teilschnittgrößenverfahrens”. Stahlbau 86, Heft 11, S. 961–971, doi: 10.1002/stab.201710545.

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[10] Winkler, Rebekka; Knobloch, Markus. 2018. “Geometrische Ersatzimperfektionen zur Anwendung des Teilschnittgrößenverfahrens für Biegeknicken um die schwache Querschnittsachse”. Stahlbau 87, Heft 4, S. 308–322, doi: 10.1002/stab.201810590. [11] Kindmann, Rolf; Frickel, Jörg. 2017. Elastische und plastische Querschnittstragfähigkeit; Grundlagen, Methoden, Berechnungsverfahren, Beispiele. Online-Auflage 2017. http://www.ruhr-unibochum.de/stahlbau/publikationen/buecher.html.de. [12] QST I-plastisch. RUBSteEl eEducation Tool, Ruhr-Universität Bochum; Lehrstuhl für Stahl-, Leicht- und Verbundbau. Version 10.2014. . [13] QST-Kasten. RUBSteEl eEducation Tool, Ruhr-Universität Bochum; Lehrstuhl für Stahl-, Leichtund Verbundbau. . [14] FE-STAB. 2012. RUBSteEl eEducation Tool, Ruhr-Universität Bochum; Lehrstuhl für Stahl-, Leichtund Verbundbau. Version 7.2012. Bochum. . [15] Lindner, Joachim. 7.3.3.2 second order analysis for lateral torsional buckling, CEN/TC250/SC3/ WG1 N240, Meeting of the Working Group 1-1 of CEN/TC250/SC3, Berlin, 21 March 2018. [16] Kindmann, Rolf; Beier-Tertel, Judith. 2010. “Geometrische Ersatzimperfektionen für das Biegedrillknicken von Trägern aus Walzprofilen – Grundsätzliches”. Stahlbau 79, Heft 9, S. 689–697, doi: 10.1002/stab.201001347. [17] Winkler, R.; Kindmann, R.; Knobloch, M. 2017. “Lateral Torsional Buckling Behaviour of Steel Beams – On the Influence of the Structural System”. Structures, Heft 11, S. 178–188, doi: 10.1016/j. istruc.2017.05.007. [18] Snijder, H.; van der Aa, R. P.; Hofmeyer, H., van Hove, B. Design imperfections for steel beam lateral torsional buckling. In: Proceedings of The International Colloquium on Stability and Ductility of Steel Structures. Timisoara, Romania 2016. [19] Snijder, H. H.; van der Aa, R. P.; Hofmeyer, H.; van Hove, B.W.E.M. 2018. “Lateral torsional buckling design imperfections for use in non-linear FEA”. Steel Construction 11, Heft 1, S. 49–56, doi: 10.1002/stco.201810015. [20] European convention for constructional steelwork. ECCS TC 8. Ultimate Limit State Calculation of Sway Frames with Rigid Joints, 1984. [21] DIN 18800-2:11.2008. Stahlbauten - Teil 2: Stabilitätsfälle - Knicken von Stäben und Stabwerken”. [22] Ebel, Rebekka. 2014. Systemabhängiges Tragverhalten und Tragfähigkeiten stabilitätsgefährdeter Stahlträger unter einachsiger Biegebeanspruchung. System-dependent bearing characteristics and load bearing capacities of steel beams under uniaxial bending. Dissertation, Ruhr-Universität Bochum.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Influence of collision damage on load-carrying capacity of steel girder E. Yamaguchi, Y. Tanaka & T. Amamoto Kyushu Institute of Technology, Kitakyushu, Japan

ABSTRACT: An accident that a truck running underneath collides against an overpass bridge happens occasionally. The influence of the damage on the safety of the bridge must be judged right away. Yet it is not always an easy task, since the load-carrying capacity of a damaged girder has not been studied much. The first author has been involved in the evaluation of a steel girder overpass bridge damaged by collision. Based on the data obtained from this bridge, the loadcarrying capacity of the deformed girder is investigated numerically in the present study. To be specific, the deformation of the main girder due to collision is reproduced by the finite element analysis and the deformed steel girder is loaded to evaluate the load-carrying capacity. The result indicates that as far as the damage is confined to the deformation of the girder, the collision does not threaten the safety of the bridge even when the deformation is quite large.

1 INTRODUCTION A truck running on a highway collides with an overpass bridge across the highway occasionally. It is required to evaluate the influence of the accident on the bridge safety. However, study on such influence appears very limited. In Japan, the construction of railways preceded that of highways. Therefore, the clearances under quite a few railway bridges do not satisfy the current requirement, which leads to collisions of trucks against railway bridges. Some technical reports on collision damage in railway bridges are available in the literature, for example Nieda & Suzuki (2000), Suginoue et al. (2006) and Nakayama et al. (2008) among others. Most of those technical papers describe only the damage and the first-aid measure employed without touching on the safety issues such as the influence on the stiffness and the load-carrying capacity. The investigation by Nakayama et al. (2008) is one of a very few studies on the influence of collision damage on the load-carrying capacity of the damaged main girder. Nakayama et al. (2008) first studied the characteristics of collision damage in the steel railway bridge. They stated that out of 14 damaged bridges, eight bridges were subjected to severe damage such as the deformation of track, the fall from the bearing and crack in the main girder, which have resulted in the immediate closure of those railway bridges. The remaining six girders underwent only the deformation of the main girders. The major damages of those six girders were classified into three groups: local upward deformation of the lower flange, horizontal deformation of the lower flange and the combination of the two. Focusing on the damages of the first two groups, Nakayama et al. (2008) have investigated the load-carrying capacity of the deformed girder experimentally and numerically. To that end, they prepared three girder specimens. The span of each girder was 5360 mm long and the difference between the three girder specimens lay in the initial deformation: one had no deformation, another girder had a locally upward displacement of the lower flange up to 78 mm and the lower flange of the other girder was displaced horizontally up to 27 mm. These initial deformations were decided referring to the maximum values they observed in the actual railway bridges damaged by the collision. Their research results have revealed that the damages they considered have 1292

Figure 1.

Damaged overpass bridge.

Figure 2.

Bridge cross section.

insignificant influence on the load-carrying capacity. They have observed the tendency that the collision increased the capacity. They stated that a possible reason for this phenomenon was the strain-hardening due to the deformation caused by the collision. The first author has been involved in the safety evaluation of a steel overpass bridge damaged by collision. The damage of the main girder was severer than the one investigated by Nakayama et al. (2008). Yet no cracks in the girder and no significant damage around the bearings were found. The main concern was therefore the load-carrying capacity of the damaged girder. The present paper deals with this issue.

2 OVERVIEW OF BRIDGE The damaged overpass bridge to be studied is a single-span steel girder bridge (Figure 1). The bridge is 29.8 m long and the span is 29.0 m long. The superstructure consists of two main girders, G1 and G2, and orthotropic deck. The main girder has transverse stiffeners while the orthotropic deck has longitudinal stiffeners and cross girders. Lateral struts are also installed to support two pipes. The dimensions of the cross section are shown in Figure 2.

3 DAMAGE Only one of the main girders, G2, was found damaged. This is because the expressway below the damaged part of the bridge is uphill. The lower flange and the web were displaced 1293

Figure 3.

Schematic of deformation due to collision.

Figure 4.

Measured residual displacement of lower flange in horizontal direction.

outward, as is shown schematically in Figure 3. The cross girders set in the upper part of the web helped restrict the web deformation only to the lower part. The residual horizontal displacements of the lower flange of G2 were measured, the results of which are presented in Figure 4. 29 transverse stiffeners are welded to each web in addition to those at the locations of the bearings. The cross section having the transverse stiffener is given the number; the section closest to the abutment A2 is Section 1 and the section closest to the abutment A1 is Section 29. The circled numbers in Figure 4 correspond to those section numbers. Figure 4 indicates that the largest horizontal displacement was caused around Section 9, which is 8.8 m away from the A2 bearing. A truck must have collided against around Section 9. The displacement at Section 9 is 186 mm, which is 1/156 of the span length. Note that the displacement of the girder studied by Nakayama et al. (2008) is 1/199 of the span length and that the Japanese design specifications (Japan Road Association 2012) requires the initial deflection to be less than 1/1000 of the member length. The deformation of the main-girder web and the lower flange are not the only damage in this bridge. In addition to them, some transvers stiffeners were bent and/or buckled; some welded connections between the transvers stiffeners and the web were fractured, separating the transvers stiffeners from the web; and some bolted connections between the transvers stiffeners and the lateral struts were broken, separating the lateral struts from the transverse stiffeners. The photos of those damages are presented in Figure 5.

4 COLLISON LOAD It was observed that the lateral strut at Section 11 was separated from the transverse stiffener and that it was held between the two main girders. Since the residual horizontal displacement at Section 11 is 148 mm, the phenomenon can be created if and only if the maximum horizontal displacement of the lower flange at Section 11 due to the collision had been equal to or greater than 148 mm while the residual displacement would have been smaller than 148 mm if not for the lateral strut at Section 11. Herein the static load that can cause the above situation is estimated by the finite element analysis that takes into account the material and geometrical nonlinearities. This load has the effect virtually equivalent to the collision and so it is called the collision load in this study. For this analysis, ABAQUS (Dassault Systemes Simulia Corp. 2008) is used. Shell elements are employed for all the members except for the lateral struts that are modelled by beam elements. 1294

Figure 5.

Damage in girder.

Since major deformation of the web is in the lower part and the primary role of the orthotropic deck is to provide the main girders and the cross girders with the clamped edge, the deck is modelled as a very stiff plate for the sake of simplicity. Steel used for the bridge is SM490Y specified in Japan Industrial Standard. Young’s modulus E is 205 GPa and Poisson’ ratio 0.3. Yield stress is 365 N/mm2 for the plate thickness t less than 16 mm and 355 N/mm2 for 16 mm ≤ t < 40 mm. Beyond the yield stress, the material stiffness is assumed to be E/100. The elastic-plastic behavior of von Mises type with the kinematic hardening rule is assumed. As noted earlier, the largest horizontal displacement was found around Section 9 (Figure 4). The coating on the bottom surface of the lower flange was scratched out near Section 9. It is then legitimate to conclude that the collision had happened in this portion. In the present study, based on the observation of the scratch on the coating of the lower flange, the collision load is assumed to be the uniformly distributed load over 400-mm range, as the red arrows in Figure 4 indicate. With this load, various analyses are conducted and it is found that once the load is increased up to 2145 N/mm, the total removal of the load leaves the displacement of 148 mm. The load-displacement curve A in Figure 6 represents this mechanical behavior. The curve A also shows that the displacement is 148 mm at 1650 N/mm. Therefore, the collision load must be between 1650 N/mm and 2145 N/mm for the lateral strut at Section 11 to be held between the main girders. The lateral strut at Section 11 was found bent by 8 mm. The lateral strut is 3200 mm long and the cross section is shown in Figure 7. The finite element analysis of this member under axial load is conducted using shell elements and the axial load that causes the 8-mm deflection is obtained as 45 kN. The bridge is then analyzed again together with the point load of 45 kN, the reaction from the lateral strut, at the lower flange of Section 11. It is found that the collision load of 2120 N/mm leaves the horizontal displacement of 148 mm with the presence of the point load: the 1295

Figure 6.

Horizontal load-displacement curve at Section 11.

Figure 7.

Cross section of lateral strut.

load-displacement curve B in Figure 6 is the result with the point load of 45 kN at the lower flange of Section 11. The computed horizontal residual displacement of the lower flange is plotted together with the measured displacement in Figure 8. Fairly good agreement is observed, confirming the validity of the present analysis.

5 INFLUENCE OF COLLISION DAMAGE 5.1 Load-carrying capacity of damaged girder To evaluate the influence of the damage caused by the collision, the load-carrying capacity of the damaged girder by itself is obtained by the nonlinear finite element analysis. The girder to be analyzed here is taken out of the bridge deformed by the collision load in the previous section.

Figure 8.

Residual displacement of lower flange in horizontal direction.

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Figure 9.

Applied load.

Figure 10. Influence of damage.

The girder is simply supported. The out-of-plane displacement, the rotation around the vertical axis and the rotation around the girder axis (torsion) are constrained at the top of the girder. As shown in Figure 9, two point-loads are applied at the top ends of Sections 8 and 10, as the severest damage is around Section 9. Point C is located at the bottom of the web at the center between Sections 8 and 10. To quantify the influence of the damage, the intact girder is also analyzed. The numerical results in terms of the load P and the vertical displacement at Point C are presented in Figure 10. The figure shows that while the stiffness is reduced by about 8%, the 3% increase in the maximum load, from 243.0 kN to 253.0 kN, is observed. The intact girder (NO DAMAGE) exhibits little nonlinear behavior, but the damaged girder (DAMAGED) undergoes nonlinear response rather extensively. Nakayama et al. (2008) conducted the similar study and observed the same phenomenon of the increase in the load-carrying capacity due to the collision damage. It is noted that the damage in the present study is 1.4 times larger than that assumed by Nakayama et al. (2008) although their damage is just about the maximum they observed in their investigation of the damaged railway bridges. 5.2 Influence of damage Nakayama et al. (2008) have suggested that the increase in the load-carrying capacity is caused by the strain-hardening associated with the plastic deformation due to collision. To study the effect of strain-hardening, the numerical analysis is conducted, using different material stiffness beyond the yield stress: in addition to E/100, 0 and E/50 are employed for the second slope in uniaxial material behavior. Figure 11 shows that the load-carrying capacity of the damaged girder becomes larger with the increase of the second slope. But the load-carrying capacity has increased from 245.0 kN to 246.7 kN even when no strain-hardening is assumed. This result indicates that the phenomenon of the increase in the load-carrying due to damage cannot be explained well from the viewpoint of the strain-hardening. The difference between the intact girder and the damaged girder lies in the residual stress and the geometrical configuration. To see the influence of the configuration change solely, the residual stress is eliminated from the damaged girder and then analyzed. The results are indicated by GEOM in Figure 11. 1297

Figure 11. Influence of strain-hardening and configuration change.

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Regardless of the magnitude of the second slope, the difference in the initial stiffness between DAMAGED and GEOM is very little. The change in the configuration is therefore the main cause for the reduction in the stiffness. In the results of GEOM, the increase in the load-carrying capacity is clearly observed, but the magnitude of the second slope doesn’t make any difference. It can be confirmed that the configuration change must be responsible also for the increase in the load-carrying capacity. The residual stress influences the plastification quite much. With the presence of the residual stress, plastic deformation starts in smaller load while the maximum load is reached with little plastic deformation if not for the residual stress, which explains little influence of the second slope on the load-carrying capacity in the results of GEOM.

6 CONCLUDING REMARKS The influence of the deformation of the steel bridge girder due to collision was studied. Through the nonlinear finite element analysis, the collision load was identified, which gives the horizontal residual displacement of the lower flange caused by the collision fairly well. Using the damaged girder reproduced by the finite element analysis, the influence of the damage due to the collision was investigated. The result indicated that while the initial stiffness was reduced by 8%, the bending strength became 3% larger. This phenomenon of the larger strength has been found attributable to that the damaged girder behaved in a quite different way from that of the original girder because of the configuration change. From the present study, it may be concluded that the deformation of the girder doesn’t necessarily threaten the safety of the bridge. However, it needs be noted that the other damage such as crack, if present, could lead to considerable reduction in safety. ACKNOWLEDGEMENTS The financial support for the present study, Grant-in-Aid for Scientific Research (C) (KAKENHI, No. 15K06184), is gratefully acknowledged. REFERENCES Dassault Systemes Simulia Corp. 2008. User’s Manual, ABAQUS Ver. 6.8. Japan Road Association 2012. Specifications for Highway Bridges: Part 2 Steel Bridges. Tokyo: Maruzen. Nakayama, T., Kimura M. et al. 2008. Residual load carrying capacities of riveted steel girders subjected to collision deformation. Structural Engineering, JSCE 54A: 68–79. Nieda, H. & Suzuki, H. 2000. The pier overturning accident of inspect and restoration on the steel railway bridge. Proceedings of Annual Conference of Japan Society of Civil Engineers 55(IV): 604–605. Suginoue, T., Inoue, E., Imai, T. & Oka, Y. 2006. First-aid measures of steel girder deformed by collision. Proceedings of Annual Conference of Japan Society of Civil Engineers 61(IV): 331–332.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Modelling of one-sided unstiffened beam-to-column joint J. Zamorowski The University of Bielsko-Biała, Bielsko-Biała, Poland

G. Gremza Silesian University of Technology, Gliwice, Poland

ABSTRACT: In unstiffened beam-to-column joints and in end-plate joints of beams with a few rows of bolts below the tension flange, as in (Kawecki et al. 2013, Kawecki & Kozłowski 2017), the load capacity of the bolts in the row below the tension flange of the beam may be exceeded at the value of the bending moment Mj,Ed ≤ Mj,Rd, where Mj,Rd – design moment resistance of these joint determined according to the formula (6.25) in EN 1993-1-8. In such cases, there is a need to reduce the design moment resistance according to point 6.2.4.2 of this standard, so that the forces in the internal row of bolts or in the group of rows do not exceed the capacity of the T-stub which imitate this row or group of rows. This work presents the full calculation model of the unstiffened beam-to-column joint, together with numerical example, in which the need of reduction of design moment resistance of such joint, which is not taken into account in the examples published in available literature as well as in computer programs, was demonstrated.

1 INTRODUCTION In case of the group of bolts consisted of three or more rows number, the capacity of the equivalent T-stub flange that imitate the inner row may be smaller than the capacity of flange for the end rows. This may happen when the failure of the T-stub is determined by the load capacity of the column flange or the end-plate of the beam. The difference in this capacity may be substantial when a mechanism of failure with non-circular pattern of yield lines occurs, and smaller when the mechanism with circular pattern is decisive. Such a variation of the capacity is caused by the different effective length of the equivalent T-stub that imitates (represents) the bolt rows –end and inner – see Figure 1. In turn, the difference of the joint stiffness in the area of the inner and the end rows is substantially smaller. This is due to the fact, that the flexibility of the joint is determined by all its components, that is: unstiffened web of column in the tension zone, column flange in bending, end-plate in bending and bolts in tension. Change of the flexibility of one of these components affects the rigidity to a much lesser extent than in a case of the load capacity of the row. Conducted numerical and experimental tests indicate that in the case of the end-plate joint of a beam to an unstiffened column with several bolt rows, the load capacity of the bolts in the row below the tension flange of the beam may be exceeded at the value of the bending moment Mj,Ed ≤ Mj,Rd, where Mj,Rd – design moment resistance of these joint determined according to the formula (6.25) in EN 1993-1-8. Similar phenomenon may occur in the case of the end-plate joint of beams with many rows of bolts – Kawecki et al. (2013) and Kawecki & Kozłowski (2017). This occurs when the resistance of the inner bolt row is determined by the resistance of the column flange or the endplate in bending, the effective length of which is small compared to the effective length of the flange in the area of the end bolt rows.

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Figure 1.

Effective lengths of the equivalent T-stub: a) non-circular pattern b) circular pattern.

The point 6.2.4.2 (3) of the standard EN 1993-1-8 contains a provision for an equivalent T-stub, in which the designer is committed to ensure that the forces transmitted by the individual bolt rows and groups of these rows do not exceed the design load capacities of the row and the group of bolt rows. Therefore it should be interpreted, that in a case when it turns out that the force in any row or group of rows exceeds the computational load capacity of this row or group, then the load capacity of the joint should be appropriately reduced. However, the standard does not contain a description of the joint’s calculation model, on the basis of which one could determine the forces in individual bolt rows. Such models are described in literature, e.g. Coelho et al. (2005). Figure 2 shows a mechanical model of the beam-tocolumn joint, in which the stiffness coefficients as in table 6.10 of EN 1993-1-8 were taken into account. The stiffness of the individual bolt rows in the tension zone is the inverse of the sum of flexibilities: the web of the column in tension, the flange of the column in bending, end-plate in bending and the bolts in tension. According to the standard, in this zone the flexibility of the beam web in tension is omitted. In compression zone, the influence of the flexibility of the web panel in shearing conditions and the influence of the flexibility of the column web in compression were taken into account. In the work an example of the unstiffened beam-to column joint, in which the need to reduce the load capacity of this joint, according to the point 6.2.4.2 (3) w EN 1993-1-8, was shown. In this example it turned out, that the forces in the second row of bolts, which were determined on the basis of the calculation model as in Figure 2, are higher than the design

Figure. 2. Mechanical model of a joint with symbols like in EN 1993-1-8.

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load capacity of the T-stub imitating row 2 from the group of rows 1-2, at the design resistance of node Mj,Rd determined from formula (6.25) in EN 1993-1-8. Most of the examples in the literature present calculation algorithms for the joint of the beam to the stiffened column, with two or three bolt rows in the tensile zone. In such joints the need to reduce their load capacity usually does not occur, because the lengths of the equivalent T-stub are comparable for all rows in the groups. If the joint is shaped this way, creation of the groups of rows is clear. The rows placed above the column’s stiffener and above the top flange of the beam in tension form one group of rows, and below the stiffener and the beam flange - the second group. Difficulties in grouping the rows of bolts appear in the case of the unstiffened beam-to-column joint. In this case, groups 1-2 and 1-2-3 are created for the column, and for the beam, the row located above the flange forms one group, and rows 2-3 forms the second group - see Figure 3 and Publication P398 (2013). In the case of such grouping (creation) of rows, the load capacity of rows 2 from group 1-2 is determined for the T-stub of the column flange. In turn, to determine the load capacity of row 3, the group of rows 1-2-3 for the column and group 2-3 for the beam should be considered. The load capacity of row 2, in case of the column, is the difference between the load capacity of the group of rows 1-2 and the load capacity of row 1. In turn, to calculate the load capacity of the row 3, the load capacities of the rows 1 and 2 from the load capacity of the group 1-2-3 set for the column should be subtracted. Then, from the load capacity of the group 2-3 for the end-plate of the beam, the load capacity of the row 2 should be subtracted. As a result, the value of the load capacity of the row 3 is assumed equal to the smaller value from the load capacities calculated this way (Publication P398 2013).

2 CALCULATIONAL MODEL OF AN UNSTIFFENED BEAM-TO-COLUMN JOINT 2.1 General description of the model Model based on the component method provided by EC3, described in the literature, e.g. Kozłowski et al. (2009) and Publication P398 (2013), which is adopted for assessing the moment resistances of joint, consists of several stages of calculation. At the first stage, a minimum resistance of the joint components in compression and the column web panel in shear is evaluated. The second stage consists of determining the resistances of the rows, as an individual or as a part of the group. Next, the sum of the resistances obtained in the rows from 1 to r is compared with the minimum load capacity of the joint components in the compression zone and in the horizontal shear zone. If this sum is greater than the minimum resistance of the tension zone and the compression zone, the resistance of the row r should be reduced. Further reduction takes place in connections subject to impact loading and vibration. In the third stage, the design moment

Figure. 3. Groups of bolt-rows: a) in column, b) in end - plate.

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resistance of the joint from the standard formula (6.25) in EN 1993-1-8 is determined, which resistance is considered reliable. In the algorithm included in the presented model, there is no reference to the point 6.2.4.2 (3) of EN 1993-1-8, in which a requirement for the T-stub was introduced that the forces transferred by the individual bolt rows and the groups of rows should not exceed design resistances of these rows and groups of rows. Therefore, to the computational model of the moment resistance of the joint the next stage should be introduced, that include determining the values of forces in individual bolt rows and groups of these rows, at the load of the joint Mj,Ed = Mj,Rd. Then, the obtained values of forces should be compared with the design resistance of the rows and the groups of rows. In case it turns out that the forces in the bolt rows or groups of these rows are greater than their design resistance, it would be necessary to make the appropriate reduction of the load capacity of the joint defined by the formula (6.25) in EN 1993-1-8. This standard does not contain a computational model of the joint, on the basis of which one could determine the forces in individual rows of bolts. Therefore, it is necessary to use the models included in the literature, e.g. Coelho et al. (2005). For the purpose of the example contained in this work, the model presented in Figure 2 was used. It was necessary to determine the appropriate joint stiffness coefficients, in accordance with Table 6.10 in EN 1993-1-8, and, on the basis of them, the flexibility of individual bolt rows and the flexibility of the compression zone as well as the column web panel in shear. The detailed algorithm of the computational model of the load capacity of the joint, together with the results of calculations that were obtained for the joint presented in Figure 4, is included in the following tables in the section 2.3. 2.2 Characteristics of the analyzed joint The analysis was performed on the example of a joint of a beam I 500 PE to a column HE 300 B, with the end-plate and pre-tensioned bolts M20-10.9. Geometrical characteristics are shown in Figure 4. All joint components except the bolts are made of S235 steel. The analysis included determining the load capacity and stiffness of the joint by use of the component method according to EN 1993-1-8.

Figure. 4. Analyzed beam-to column joint with bolts M20-10.9.

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2.3 Algorithm and results of calculations of the joint capacity acc. to (6.25) in EN 1993-1-8 2.3.1 Stage 1 – design resistances of basic components in compression and shear Table 1. Stage 1 - design resistances of basic components in compression and shear. Column web panel in shear acc. to 6.2.6.1

Column web in transverse compression acc. to 6.2.6.2

Beam flange and web in compression acc. to 6.2.6.7

Minimum value

Fc;v;min;Rd 8 > < Vwp;Rd =β ¼ min Fc;wc;Rd > : Fc;fb;Rd

Vwp,Rd/β

Fc,wc,Rd

Fc,fb,Rd

579.4 kN

588.1 kN

1068 kN

579.4 kN

2.3.2 Stage 2– design resistances of bolt-rows Table 2. Stage 2.1 - design resistance of the bolt-row 1 considered individually. Column web in transverse tension acc. to 6.2.6.3

Column flange in transverse bending acc. to 6.2.6.4

End-plate in bending acc. to 6.2.6.5

Design resistance* acc. to 6.2.7.2(6)

Ft1;Rd Ft1,wc,Rd

Ft1,fc,,Rd

Ft1,ep,,Rd

411.0 kN

266.5 kN

264.3 kN

8 < Ft1;wc;Rd ¼ min Ft1;fc;Rd : Ft1;ep;Rd

264.3 kN

* Limiting the design resistance due to the compression and shear of the joint parts, acc. 6.2.7.2(7):

Ft1,Rd = 264.3 kN ≤ Fc,v,min,Rd = 579.4 kN.

Table 3. Stage 2.2 - design resistance of bolt-row 2 considered individually. Column web in transverse tension acc. to 6.2.6.3

Column flange in transverse bending acc. to 6.2.6.4

End-plate in bending acc. to 6.2.6.5

Beam web in tension acc. to 6.2.6.8

Design resistance of the row* acc. to 6.2.7.2(6)

Ft2;Rd ;ind Ft2,wc,Rd

Ft2,fc,,Rd

Ft2,ep,,Rd

Ft2,wb,,Rd

411.0 kN

266.5 kN

323.1 kN

624.2 kN

266.5 kN

* Limiting the design resistance due to compression and shear of the joint parts:

Ft2,Rd,ind = 266.5 kN ≤ Fc,v,min,Rd – Ft1,Rd = 579.4 – 264.3 = 315.1 kN

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8 Ft2;wc;Rd > > < Ft2;fc;Rd ¼ min Ft2;ep;Rd > > : Ft2;wb;Rd

Table 4. Stage 2.3 - design resistance of bolt-row 2 as a part of group of bolt rows 1-2. Column web in transverse tension acc. to 6.2.6.3

Column flange in transverse bending acc. to 6.2.6.4

Design resistance of groups of rows*

Ft,1-2,wc,Rd

Ft,1-2,fc,,Rd

Ft,1-2,Rd = min (Ft,1-2,wc,Rd, Ft,1-2,fc,,Rd)

506.6 kN

348.5 kN

348.5 kN

* Limiting the design resistance of the bolt-row due to design resistance of the group of bolt-rows accord-

ing to 6.2.7.2(8): Ft2,Rd = min(Ft2,Rd,ind; Ft2,Rd,group), where Ft2,Rd,group = Ft,1-2,Rd – Ft1,Rd Ft2,Rd,group = Ft,1-2,Rd – Ft1,Rd = 348.5 – 264.3 = 84.2 kN, Ft2,Rd = min(266.5, 84.2) = 84.2 kN. If in connections exposed to dynamic actions and vibrations Ft1,Rd > 1.9 Ft2,Rd then Ft2,Rd ≤ Ft1,Rd · h2/h1 – acc. to 6.2.7.2 (9) + polish National Annex The connection is not exposed to impact and vibration.

Table 5. Stage 2.4 –design resistance of bolt-row 3 considered individually. Column web in transverse tension acc. to 6.2.6.3

Column flange in transverse bending acc. to 6.2.6.4

End-plate in bending acc. to 6.2.6.5

Beam web in tension acc. to 6.2.6.8

Design resistance of the row* acc. to 6.2.7.2(6)

Ft3;Rd ;ind Ft3,wc,Rd

Ft3,fc,,Rd

Ft3,ep,,Rd

Ft3,wb,,Rd

411.0 kN

266.5 kN

303.7 kN

537.6 kN

8 Ft3;wc;Rd > > < Ft3;fc;Rd ¼ min > Ft3;ep;Rd > : Ft3;wb;Rd

266.5 kN

* Limiting of the bolt-row resistance due to components in bending and shear:

Ft3,Rd ≤ Fc,v,min,Rd – (Ft1,Rd + Ft2,Rd) – acc. to 6.2.7.2(7). Ft3,Rd ≤ 579.4 – (264.3 + 84.2) = 230.9 kN, therefore Ft3,Rd = 230.9 kN. Table 6. Stage 2.5 –design resistance of bolt-row 3 as a part of group of bolt-rows 1-3 and 2-3. Column web in transverse tension acc. to 6.2.6.3

Column flange in transverse End-plate in bending acc. bending acc. to 6.2.6.4 to 6.2.6.5

Beam web in tension acc. to 6.2.6.8

Design resistances of groups of rows*

Ft13;Rd ¼ min Ft,1-3,wc,Rd

Ft, 1-3,fc,,Rd

Ft, 2-3,ep,,Rd

Ft, 2-3,wb,,Rd

688.1 kN

569.2 kN

538.0 kN

768.0 kN

n

Ft13;wc;Rd Ft13;fc;Rd

569.2 kN

Ft23;Rd ¼ min

n

Ft23;ep;Rd Ft23;wb;Rd

538.0 kN

* Limiting of resistance due to resistance of the group of rows – acc. 6.2.7.2(8):

Ft3,Rd = min(Ft3,Rd,ind, Ft3,Rd, group,c, Ft3,Rd, group,b), where Ft3,Rd, group,c = Ft1-3,Rd – (Ft1,Rd + Ft2,Rd), Ft3,Rd, group,b = Ft2-3,Rd –Ft2,Rd. Ft3,Rd, group,c = 569.2 – (364.3 + 84.2) = 220.7 kN, Ft3,Rd, group,b = 538.0 – 84.2 = 453.8 kN Ft3,Rd = min(230.9, 220.7, 453.8) = 220.7 kN. If in connections exposed to dynamic actions and vibrations Ft1,Rd > 1.9 Ftx,Rd, where x = 1,2 then Ft3,Rd ≤ Ftx,Rd · h3/hx – acc. to 6.2.7.2 (9) + polish National Annex The connection is not exposed to impact and vibration.

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2.3.3 Stage 3 – design resistance of the joint in bending Mj,Rd acc. to formula (6.25) Distance of rows r from the compression zone are (see Figure 4): h1 = 526 mm, h2 = 442 mm and h3 = 382 mm. According to the formula (6.25) given in EN 1993-1-1, the design moment resistance is: Mj,Rd = Σ Ftr,Rd · hr = 264.3 · 0.526 + 84.2 · 0,442 + 220.7 · 0.382 = 260.5 kNm. 2.4 Checking the requirements contained in point 6.2.4.2 of EN 1993-1-8 In order to determine the values of forces in the individual bolt rows, a mechanical model of the joint as in Figure 2 was used. Values of stiffness coefficients ki of the basic components of the joint are collected in Table 7. Due to the fact that the resistance of the row 2 as a part of the group of rows 1-2 was determined by plasticization of the column flange in bending, for that row an elastic-plastic material model with a plastic plateau corresponding to their resistance was adopted (Ft2,Rd = 84,2 kN). In turn, the resistance of the row 1 was determined by the end-plate, which plasticization were associated with the destruction of the bolts, as in the failure mode 2 of the T-stub. For this reason, for the row 1 the elastic model was adopted, because after breaking the bolts, they will not affect the joint. The obtained values of forces in the individual bolt rows and in the groups of these rows, under the load MEd = Mj,Rd = 260,5 kN, are collected in column 2 of Table 8. Column 3 of this table shows design resistances of these rows and their groups, and column 4 shows the relation of the resistance to the force. As can be seen, the load capacity of the groups 1-2 and the row 1 has been exceeded. According to point 6.2.3.2 (3) of EN 1993-1-8, reduction of the design resistance of the joint becomes necessary. The largest deficiency of the resistance was found for row 1, where wi = 0.915. Therefore, the reliable load capacity of the joint should be Mj,Rd, red = 260.5 . 0.915 = 238.4 kNm. A similar issue occurred during investigation on end-plate connections of beams subjected to bending (Kawecki et al. 2013) and (Kawecki & Kozłowski. 2017), conducted in Rzeszow University of Technology. As a result of the performed tests and calculations a significant underestimation of the bolts resistance in the row near the flange and a significant overestimation of the resistance of the further bolt rows and the whole joint was found.

Table 7. Stage 4.1 – determination of the force values in the individual bolt rows. Basic component

Parameter

Bolt-row 1

Bolt-row 2

Bolt-row 3

Column web in transverse tension Column flange in bending End-plate in bending Bolts in tension The effective stiffness coefficient The equivalent lever arm Column web panel in shear Column web in compression

k3  103 ½m

k4  103 ½m

k5  103 ½m

k10  103 ½m

keff ;r  103 ½m

zeq ½m

k1  103 ½m

k2  103 ½m

4.620 44.683 25.129 6.374 2.296

2.116 20.466 17.328 6.374 1.359

4.267 41.272 13.810 6.374 2.050 0.463 3.894 8.655

Table 8. Stage 4.2 – Comparison of the values of forces with the load capacity of the rows. Bolt-rows

Force [kN]

Design resistance [kN]

wi = (3)/(2)

1 1 2 3 1-2 1-3 2-3

2 288.7 84.2 187.0 288.7 + 84.2 = 372.9 372.9 + 187.0 = 559.9 84.2 + 187.0 = 271.2

3 264.3 266.5 230.9 348.5 569.2 538.0

4 0.915

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0.935

2.5 Rotational stiffness of the joint The reduction of the design load capacity of the node also affects the rotational stiffness of the joint – the relationship M – ϕ. According to EN 1993-1-8, at the range of the moment value in node Mj,Ed ≤ 2/3 Mj,Rd, the initial stiffness Sj,ini, is applied, and after exceeding this value, the stiffness Sj. The ratio of these stiffnesses is expressed in the standard using the coefficient μ.. The value of this coefficient is determined by the normative formulas (6.28 a and b). Using the normative formula (6.27) together with formulas (6.28a) and (6.28b) the relationship M – ϕ, was determined, where ϕ, = M/Sj for the ranges: 0 ≤ Mj,Ed ≤ Mj,Rd,red – continuous line and 2/3Mj,Rd,red ≤ Mj,Ed ≤ Mj,Rd – dashed line – see Figure 5. When the reduction of the analysed joint load capacity is included, the stiffness Sj of that joint for M = 2/3Mj,Rd is equal to approximately 78% of the joint stiffness without this reduction.

Figure. 5. The relationship M - ø: 1 – for Mj,Rd, 2 – for Mj,Rd,red.

3 CONCLUSIONS It is commonly recognized/accepted, that the fulfillment of the condition (6.23) in EN 1993-1-8, with Mj,Rd calculated from the formula (6.25) is sufficient to ensure the safety of the end-plate beam-to-column joint. At the same time, the provision included in point 6.2.4.2 (3) of this standard, which introduces the requirement that the forces in the individual bolt rows and groups of these rows should not exceed the load capacity of rows and groups of these rows, is omitted. In case of unstiffened beam-to-column joint, as a result of large differences in effective lengths of the equivalent T-stub of the outer and inner bolt rows (located outside and below the upper flange of the beam), plasticization of the column flange in the area of the inner row may develop. Hence, the resistance of the external bolt row may be exceeded. This results from the relatively small differences in the stiffness of the individual bolt rows compared to the differences in their design resistances. This paper presents a full algorithm for calculating of an unstiffened beam to column joint, which also includes the provisions contained in point. 6.2.4.2 (3) of EN 1993-1-8. This algorithm is illustrated by the results of calculations of such a joint, in which the need to reduce the joint’s load capacity as a result of non-fulfillment of the conditions contained in these provisions was taken into account. The values of forces in individual bolt rows were determined using the mechanical model of joint shown in Figure 2, in which the stiffness coefficients of the components of the joints according to the standard provisions were used. In case of inner bolt-row, taking into consideration that its resistance was determined by the plasticization of the column flange, an elastic-plastic model with a plastic plateau at the load-bearing level of this row was adopted. The obtained results indicate the urgent need to perform wider experimental tests of joints with internal bolt rows in groups. This applies both to the beams-to-column joints as well as 1307

to the end-plate beams joints. Until such tests will have been carried out, as a result of which the need for such a reduction of load capacity would be confirmed or ruled out, in the design process of such nodes the calculation algorithm presented in this paper should be used. REFERENCES Coelho, A.M. & Silva, L.S. & Bijlaard, F.S.K. 2005. Ductility analysis of end plate beam-to-column joints. In Hoffmeister B. & Hechler O. (ed.), Eurosteel 2005; Proc. 4th European Conference on Steel and Composite Structures – Eurocodes – Practice, Maastricht 8-10 June 2005. Aachen: Mainz. EN 1993-1-8: Eurocode 3: Design of steel structures. Part 1.8: Design of joints. Kawecki, P. & Łaguna, J. & Kozłowski, A. 2013. Analiza nośności doczołowego styku belki dwuteowej z wieloma szeregami śrub. Czasopismo Inżynierii Lądowej, Środowiska i Architektury. Journal of Civil Engineering, Environment and Architecture. JCEEA 60(2): 117 –136. Kawecki, P. & Kozłowski, A. 2017. Badanie rozkładu sił wewnętrznych w zginanych wielośrubowych stykach doczołowych blachownic. Inżynieria i Budownictwo 73(6): 309 –312. Kozłowski, A. & Pisarek, Z. & Wierzbicki, S. 2009. Projektowanie doczołowych połączeń śrubowych według PN-EN 1993- 1-1i PN-EN 1993- 1-8. Inżynieria i Budownictwo 65(4): 193 –204. Publication P398. (2013). Joints in Steel Construction: Moment-Resisting Joints to Eurocode 3. Ascot: The Steel Construction Institute & London: The British Constructional Steelwork Association.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Modelling of roof bracings of single-storey industrial buildings J. Zamorowski The University of Bielsko-Biała, Bielsko-Biała, Poland

G. Gremza Silesian University of Technology, Gliwice, Poland

ABSTRACT: The rules included in the standards EN 1090-2 and EN 1993-1-1 are not fully compatible in terms of the industrial buildings roof bracings modelling, therefore they also do not guarantee a reflection of the actual behavior of these bracings in proposed, normative calculation model. This work, by example of single-bay industrial building (hall) with vertical roof bracings, presents the results of elastic spatial analysis of the structure with geometrical imperfections, obtained using various calculation models. These results were referred to the reference model with global and local imperfections of the roof structure of the hall and compared with state of forces that can be obtained from the model given in standard rules. This allowed to conclude about the deficiencies of this model and to point out the need of verification of the standard provisions in the field of roof bracing.

1 INTRODUCTION Utilization ratio of the roof bracings in spatial industrial halls is determined by loads and geometrical imperfections of the roof structure introduced during fabrication of steel members and their assembly. The fabrication of the structure and its assembly should comply with the requirements of EN 1090-2. This standard defines the permissible values of two types of deviations: essential important due to the load-bearing capacity and stability of the structure, and functional - important due to the possibility of its fit up or appearance. Moreover, according to the point 11.1 of EN-1090-2, special tolerances in terms of their type or value may be defined in specification. Due to the distribution of forces in the roof structure of the hall, amongst the manufacturing tolerances listed in EN 1090-2, the following essential deviations may be significant: from straightness of a press braked profile or a flange of a welded profile Δ ¼ ±L/1000 (where L – distance between restraints), the deviations of nodal (panel) points relative to the designed straight line in lattice girders Δ ¼ ±L/500 but not less than /Δ/ ¼ 12 mm as well as deviation of bracing from straightness Δ ¼ ±L/1000 but not less than /Δ/ ¼ 4 mm. In turn, amongst the essential erection tolerances, the straightness of member subjected to bending or compression on the length between restraints Δ ¼ L/750 as well as an angular misalignment Δθ ¼ ±1/500 rad in a full contact end bearing are important. The range and the values of functional deviations presented in the EN 1090-2, that may determine the fabrication and the assembly of the roof structure of the hall, are practically analogous to those in the case of basic deviations, with one exception - in the case of functional assembly deviations, a deviation from the straightness of the beam in the plan with permitted value in class 1 L/500 additionally occur. If the transverse roof bracings are considered as lattice girders, then for the upper chords of roof trusses being the part of these bracings an initial deflection out of vertical plane with a value Δ ¼ ±L/500, but not less than /Δ/ ¼ 12 mm, may be permitted. This deviation may be caused by the initial deflection of the chord and/or the angular misalignment in the full contact end bearing, at Δθ  ±1/500 rad. If under the transverse roof bracings the horizontal bracings between lower 1309

chords of roof trusses were installed, than these chords could be mounted with the same deviation as the upper chords, wherein their initial deflection may have the opposite direction in relation to the deflection of the upper chords. In turn, when horizontal bracings in the plane of the lower chords are not installed, then the lower chords of all roof trusses may be kinked at the full contact end bearing, at Δθ ≤ ±1/500 rad. Moreover, in case of lack of vertical bracings, if compression in the lower chords might occur, then chords of all trusses might be deflected along their entire length L, with a maximum value Δ ¼ ±L/750. In turn, if roof vertical bracings were present, the lower chords could be deflected on the length between bracings with a deviation Δ  ±L/750, where L - distance between these bracings. The upper chords of the remaining roof trusses could be kinked in full contact end bearing, at Δθ  ± 1/500 rad. In turn, these chords could not be deflected along the entire length, because EN 1090-2 does not foresee such a case. The standard EN 1993-1-1 contains provisions for the calculation of bracings that ensure lateral stability of beams or compressed members. In this provisions initial bow imperfections e0 ¼ αm·L/500, at α ¼ [0,5(1+1/m)]1/2 have been taken into account, where m - the number of members to be restrained, as well as the possibility of kinking of the compressed members in the joint due to angular misalignment, assuming local impact on the bracing about the value αm·NEd/100. This value results from kinking of a single chord of truss girder by an angle Δθ ¼ 1/ 200 rad. This deformation can be taken into account omitting initial bow imperfection e0. On the basis of the calculation model of roof bracings contained in the standard EN 1993-1-1, the internal forces in the upper chords of trusses, purlins and diagonals of bracings can be determined. However, this model does not provide an opportunity to assess the impact of initial deformation of the roof on the state of forces in roof vertical bracings as well as in the lower chords, struts and diagonals of roof trusses. This model also does not capture the influence of the number and location of vertical roof bracings on the utilization ratio of the transverse roof bracings members. In general, industrial halls with latticed roof structure are calculated by use of various models based on the of the 1st or 2nd order theory. In the second order calculation models, in the equilibrium equations of compressed rod, the following influences are usually included: initial and elastic bow deformation (P – δ effect) on their flexural rigidity and bending on an axial stiffness (P – κ effects) and the influence of the change in the rod’s chord inclination on the value of nodal forces (P – Δ effect for the rod). The change of axial stiffness of the initially and elastically deflected, compressed rod results from its additional deformations along its chord due to the bending caused by axial force and transverse load. It is usually included in calculations by introducing substitute (reduced) axial stiffness (EA)z, which is changed during the iterative solution depending on the total deflection and axial force, wherein the total deflection results also from the value of the bending moments in the nodes at the rod ends. The reduction of the bending stiffness of the initially and elastically deflected rod, which is subjected to compression, results from the additional bending deformations due to the axial force acting on the eccentric. In the solution algorithm based on the displacement method, this reduction is taken into account by use of an additional factor with hyperbolic functions that results from such a rod description. In the case of the finite element method, construction of the equilibrium equation depends on the simplifications made in the tensors of strains and stresses. In turn, in the case of the influence of a change of the rod’s chord inclination on the values of forces in nodes, the influence of the axial force on the value of transverse force is usually taken into account, as in the case of the standard effect of P – Δ within the frames (Zamorowski 2013). The above information indicate that the above-mentioned influences cannot be directly taken into account in the linear model, however, it is possible to take into account the influence of the P – Δ effect iteratively by adding to the co-ordinates of the nodes their displacements in subsequent iterative steps. Nevertheless, it is not possible to assess precisely the influence of elastic deflection of the analyzed rod, e.g. by dividing it into several elements and adding elastic displacements to coordinates of intermediate nodes, because these displacements would be obtained assuming the initial bending and axial stiffness of these rod. Therefore, the method of calculating of the equivalent load of bracings proposed in standard may give slightly underestimated results. In the calculations of roof bracings, similarly as in trusses, different methods are used to take geometric imperfections into account. These methods are mainly based on either the direct introduction of changes in the geometry of the system and deformation of the rods, so-called IGI 1310

models - modeling of Initial Geometric Imperfections, or on the application of equivalent loads, so called NHF models - application of Notional Horizontal Forces (Piątkowski 2017). In the case of IGI models, it is necessary to consider all possible combinations of initial bow bending of rods, what may be tedious in calculation. In order to limit the number of these combinations, scaling of the first form of the loss of stability is used here, by the EBM method - scaling of first Eigen Buckling Mode. Theoretically, the method of randomly selecting of initial deflection combinations from the allowable solution set could also be used, but in this case it would be necessary to search the entire set. During selection of the combinations of the rods’ initial bow imperfections, one must remember that it is necessary to choose their arrangement in which the bracings are most susceptible to flexural deformation as a truss. For example, a cross-section with different slenderness in the transverse bracings plane and in the truss plane was chosen for the compressed chord: higher susceptibility of that chord will be obtained assuming the initial bow deformation in the plane of its bigger slenderness. Technical literature describes various models of bracings based on an equivalent load, in which this load is either determined on the basis of internal forces in the chords of the roof trusses and stiffness of the bracings expressed by elastic deflections (EN-1993-1-1, Biegus & Czepiżak 2017, Giżejowski et al, 2008), or on the basis of external loads and tilting of the truss – initial and elastic, similarly to the P – Δ effect in frame systems (Niewiadomski & Zamorowski 2017). Each of these models may be more accurate if the solution is based on the second order theory. In this case, during additional loads determination, only initial deformations should be taken into account. There are also some publications in which both aforementioned models are used simultaneously, whereby the obtained effects of global of bow imperfection in roof bracings are overvalued. In the real structure of the hall of the imperfect type, additional forces resulting from deformation and loads appear not only in the transverse roof bracings but also in the vertical bracings as well as in diagonals and struts of the roof trusses. At the same time, bending of the lower chord of the truss out of its plane occur. The mechanism of creating these forces is well explained at work (Niewiadomski 2003). Configuration of the initial deflections of rods and related change in rigidity of the side restraints of the roof trusses may also affect the coefficients of buckling lengths of the chords of the trusses subjected to compression when buckling out of the truss plane (Krajewski & Iwicki 2015).

2 DESCRIPTION OF ANALYSED HALL Spatial industrial hall consisting of seven transverse systems spaced at 6,0 m, with gable walls located 1.2 meters away, were analyzed (Figure 1). The height of the main columns (IPE 360) was 8.3 m, and their axial spacing – 24.0 m. Trusses were designed with a height of 3.9 m with endplate joints in the middle of their span and a relatively large slope of the roof at 10°. The location of the hall in the third snow and wind zone in Poland was planned. Chords of the trusses were designed from ½ HEB 220, posts and diagonals from RHS 100x100x4 profiles. For purlins, IPE 180 is provided. Vertical roof bracings and wall bracings in the columns line were designed as a cross made of L90  90  6 and 2L60  60 6 respectively. Transverse roof bracings, also type X, were designed from L60 x 60 x 6. For the bottom horizontal struts of roof vertical bracings 2L60606 were adopted. The load-bearing element supporting the roof covering is a trapezoidal sheet protecting the purlins from lateral torsional buckling and the purlins near eaves - additionally against flexural buckling. The total weight of the roof covering was 0.45 kN/m2.

3 CALCULATION MODELS OF THE HALL The work analyzed the spatial structure of the hall as in Figure 1. In the analysis, three nonlinear models with geometric imperfections in the roof area and four models based on the first order theory were used. As a reference model (model 1N), the system with global imperfections of the all trusses chords and with local imperfections of compressed rods was adopted. Global imperfections were adopted in accordance with EN 1090-2, in particular (Figure 2): bow imperfections of the 1311

Figure 1.

Scheme of the analysed building (compressed rods of cross-bracings were removed).

Figure 2. Global imperfections according to EN 1090-2: a) identical to EC-3, b) considered differently than in EC3 c) not included in EC3.

upper chords of the trusses being simultaneously the chords of the transverse bracings (e0 ¼ L/500), deviations of the angle misalignment in ridge joints of the upper chords of the remaining trusses (Δθ ¼ ±1/500 rad), deviations resulting from the angle misalignment in the joints of the lower chords of all trusses (Δθ ¼ ±1/500 rad). In this model, local geometric bow imperfections of the compressed rods were also introduced according to the buckling curves defined in the EN 1993-1-1. An equivalent to this model (equivalent reference model) would be the perfect model with lateral forces applied in all nodes of the upper and the lower chords of all trusses, resulting from the change of the inclination of the rod chords relative to the vertical planes of the trusses and axial forces in these rods, i.e. PΔ effects for rods - see Figure 3. In case of the 2nd order analysis, in the equivalent perfect model, the lateral forces calculated this way should be taken into consideration, at the displacement of the nodes resulting only from the initial global imperfections of the truss chords. Analogous forces and displacements are then obtained as in the reference model. In the second model (model 2N), in comparison to the reference model 1N, global imperfections of the lower chords of all trusses were omitted; these chords were straightened. In the third model (model 3N), the global imperfections of the upper and the lower chords were replaced by the ΔHg forces applied in the nodes of the upper chords. These forces were calculated on the basis of the load of these nodes and tilting of posts out of the plane of trusses, analogous to the of 1312

Figure 3.

Applying of equivalent forces in equivalent reference model and in model 3L.

P – Δ effect in frames. The arrangement of local bow imperfections remained as in the reference model. In comparison to the equivalent reference model, in model 3N the lateral forces ΔHd in nodes of the lower chords that results from the initial tilting of the rods out of the plane of the truss due to the global imperfections of the chords, were not taken into account. Also not included, in the upper nodes, the influence of horizontal components of forces in rods at initial lateral displacements of nodes. Only the influence of vertical components of forces in the rods and lateral displacement of the chords on lateral forces was taken into account. In linear modeling, four cases were considered: model of a spatial hall without geometrical imperfections and normative stabilizing loads qd – model 1L, model with stabilizing load qd applied to the nodes of the upper chords of the end trusses in axes 1 and 7 and with reactions ΣQd/2 that balance load qd in the end nodes of bracings – model 2L, model with a normative stabilizing load qd distributed over the upper nodes of all trusses and the opposite directed equivalent load distributed over the nodes of the lower chords of all trusses – model 3L – see Figure 3 as well as in-plane 2D normative model, in which the force values introduced in the upper chords of girders from the model 1L were obtained – model 4L. 4 ANALYSIS OF CALCULATION RESULTS The obtained results of calculations are presented in Table 1 (markings of the rods are shown in Figure 4). Columns 3 to 9 of the Table 1 present values of the axial forces obtained for individual members of the roof structure by use of the calculation models mentioned in section 3: in trusses, transverse roof bracings and vertical roof bracings. In case of the trusses, besides the extreme value of the force that was chosen for the rod from the seven roof trusses, after the slash, the axis number of the truss in which it occurred, was also given. In turn, in case of rods in roof bracings, also after the slash, the side of the hall at which the extreme force value occurred – windward - n or leeward - z, was given. 1313

Table 1. Values of maximum forces (in kN) in members of the roof structure. Members of Rod the roof marking 1N

Nonlinear models

Linear models

2N

3N

1L

2L

3L

4L

3

4

5

6

7

8

9

Truss upper 5 chords 6 7 8

-149,9/6 -210,2/5 -220,6/5 -204,4/5

-149,0/6 -210,2/5 -220,6/5 -204,4/5

-149,1/6 -209,7/5 -220,1/5 -203,9/5

-148,1/6 -209,1/5 -220,9/5 -203,4/5

-151,0/6 -209,1/5 -220,9/5 -203,4/5

-148,1/5 -209,1/5 -220,9/5 -203,4/5

-152,1/6 -209,1/5 -220,9/6 -203,4/5

Truss lower 14 chords 15 16

154,2/4 215,1/4 227,7/4

154,2/4 215,0/4 227,7/4

154,3/4 215,0/4 227,7/4

153,8/4 215,1/4 227,9/4

153,8/4 215,1/4 227,9/4

153,8/4 215,1/4 227,9/4

-

Truss 21 diagonals 23 25 27

177,9/4 77,0/4 18,6/2 -27,6/6

177,9/4 77,0/4 18,6/2 -27,6/6

178,0/4 77,0/4 18,5/2 -27,3/6

178,2/4 77,0/4 19,5/7 -27,6/6

178,2/4 77,0/4 20,2/7 -28,2/6

178,2/4 77,0/4 21,1/7 -28,9/6

-

Truss struts

-89,6/4 -45,8/4 -11,9/2 39,8/6

-89,6/4 -45,8/4 -11,9/2 39,5/6

-89,6/4 -45,7/4 -11,8/2 39,6/6

-89,3/4 -45,8/4 -12,2/7 38,7/5

-89,3/4 -45,8/4 -12,7/7 39,6/6

-89,3/4 -45,8/4 -13,2/7 38,7/5

-

Roof braces 121 123

38,27/n 14,61/z

37,72n 14,24/z

37,48/n 14,43/z

35,33/n 14,16/z

38,36/n 15,12/z

35,88/n 12,97/z

49,30/n 17,49/n

Purlins

-13,48/n 3,49/n -1,82/n -28,61/n -9,72/z

-13,31/n 3,55/n -1,81/n -28,36/n -9,68/z

-13,57/n 3,60/n -1,74/n -28,36/n -9,87/z

-13,09/n 0,34/z -2,73/n -27,73/n -16,83/z

-13,07/n 1,24/z -3,67/n -29,02/n -15,97/z

-13,23/n 0,32/z -2,62/n -27,33/n -16,80/z

-46,02/n -0,35/n

5,84/z -24,10/n -7,08z

6,13/z -23,94/n -7,31/z

5,87/z -24,02/n -6,95/z

9,09/z -7,46/n 5,51/n

7,66/z -8,03/n 5,72/n

10,30/z -23,38/n 3,92/n -7,13/z 15,05/z -2,68/n 2,92/n -0,48/n

-23,74/n -0,55/z -

8,34/z -8,10/n 5,76/n

9,88/z 9,18/z -24,23/n -24,67/n 7,27/n 6,61/n -5,59/z -5,66/z 10,31/z 10,04/z -4,73/n -5,00/n 3,71/n 3,82/n

1

2

22 24 26 35

449 450 451 452

Vertical bracings

453 611 610 609

-33,77/n -0,46/n

-

The calculation results indicate that the values of the forces in the upper chords of the trusses obtained from the in-plane 2D (flat) calculation model provided by the standard (col. 9) are in practical accuracy consistent with the values of forces obtained using the reference model (col. 3). Similar accuracy was obtained in case of the linear 2L model. A slightly better coincidence of results in the upper chords of trusses was obtained using the non-linear model with ΔHg forces

Figure 4.

Markings of the rods in the model.

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Table 2. Extreme values of bending moments in the lower chords (in kN). Calculation models nonlinear

linear

Truss in axis 1

1N 2

2N 3

3N 4

1L 5

2L 6

3L 7

1 2 3 4 5 6 7

0,75 3,78 0,78 0,80 0,80 0,82 4,06

0,76 3,65 0,88 0,90 0,91 1,70 3,92

1,07 3,10 0,66 0,68 0,69 1,55 3,63

0,88 3,15 0,49 0,51 0,52 0,81 3,42

0,95 3,33 0,51 0,52 0,52 0,91 3,52

2,16 4,10 1,41 1,41 1,42 2,42 4,37

(col. 5), as well as the linear spatial model of the hall with the normative stabilizing loads qd applied to the nodes of the upper chords of trusses in axes 1 and 7 and the equivalent load in the form of ΣQd/2 forces in the nodes on the ends of bracings (col. 7). Practical agreement of extreme values of axial forces, for all calculation models, also occurred in the rods of the lower chords, in diagonals of the last and penultimate panels of trusses, as well as in struts, except the middle and one the closest to the middle. As the calculations shows, a lateral flexural deformation of the lower chords of the trusses occurred in all spatial non-linear and linear models. This deformation is accompanied by bending moments with values as shown in Table 2. The order of models in this table follows the order in the Table 1. It can be noted that safe (higher) values of the bending moments in comparison to the reference model (col. 2) were obtained for the linear model 3L (col. 7). In this model, a stabilizing load Qd,t was applied to the nodes of the upper chords of all trusses, where a – spacing of the nodes in the projection. This load have been balanced by the opposite directed load Qd,b, applied in the lower nodes of the all trusses – see Figure 3. Values of these loads were obtained by adding the stabilizing load applied to the upper chords of trusses and then dividing them into 7 trusses and spreading to 7 nodes of the lower chord, Qd,b ¼ Σqd.a/7, where a – spacing of nodes along the bottom chord. The sum of these loads is zero. In the actual roof structure, the nodes of the lower chords will be affected by the forces resulting from the effect of P – Δ for the rods leaned out of vertical plane of the truss - see description of the equivalent reference model. In the model 3L, the extreme force in the strut below the ridge is about 3% smaller in relation to the force from the reference model and in the posts near to the symmetry axis of the hall - higher about 11%. This model also underestimates the extreme forces in diagonals of the transverse bracings up to 11.2%, in the purlins up to 4.5% and overestimates the values of the tension forces in the vertical bracings. Due to its simplicity, the model 3L can be used to estimate the forces in the bracings of the roof structure. Flat (2D) calculation model may be useful only for determining the forces in diagonals and purlins of the transverse roof bracings (being the members of these bracings). The results best fitted to the reference model in all rods were obtained using the non-linear model 3L, in which the global chord deformations were replaced by ΔHg forces resulting from tilting of truss and nodal loads.

5 CONCLUSIONS In engineering practice, various types of calculation models are used to design the roofs of industrial halls. The simplest of them, the standard in-plane 2D model, only allows to estimate the forces in members of the lateral transverse bracings, while the forces in vertical braces will not be 1315

known. The influence of global imperfections of trusses’ chords on the utilization ratio of their bottom chords, diagonals and struts will also not be determined. In turn, when the normative stabilizing load qd according to the EN 1993-1-1 would be introduced into the spatial model of the hall calculated according to the 1st order theory, then the influence of tilting of the rods axes out of the planes of the trusses on the loading of their lower chords will not be taken into account. To eliminate this deficiency in the spatial model of a hall calculated according to the first order theory, besides the stabilizing load qd applied in the upper nodes of the end trusses (or distributed over all the trusses), equivalent to them lateral load should be introduced to the lower chords of all trusses – see Figure 3. The total value of this load results from general equilibrium conditions, and the sum of stabilizing loads in the nodes of the upper chords and the equivalent load in the nodes of the lower chords should be zero (in known solutions presented in the literature, equivalent load is applied at the end nodes of bracings). The value of the equivalent load applied to the nodes of the lower chords may be approximated by summation the stabilizing load from the nodes of the upper chord and then dividing them by the number of trusses and the number of internal nodes of their lower chords. It is also possible to calculate the spatial hall using the second order (or non-linear) theory and stabilizing loads qd in the upper nodes of the end trusses (or in the upper nodes of all trusses), and equivalent load in the bottom nodes of all the trusses. In such a case, these loads should be determined from the standard formula (5.13) given in EN 1993-1-1, assuming δq = 0. In order to more accurately estimate the value of the nodal equivalent load for a nonlinear model or a model based on the second order theory, the out of the truss plane tilts of all the rods converging at the node should be determined. Next, the values of lateral actions on the nodes ΔHdi = Σϕi·NED,i should be added together ϕi = (uki – upi)/li, whereby uki, upi – horizontal displacement of the end-and-start nodes of the i-th rod, and NED – axial force in the rod). The most accurate calculation model is the model of the spatial hall with geometric imperfections – the global of the whole chords and the local of the individual rods. The limit values of global initial imperfections that may occur in the properly assembled hall are defined in EN 1090-2. In turn, the EN 1993-1-1 standard provides equivalent values of geometric imperfections - local and global, which should be taken into account when using a non-linear or second order theory. There is no full compatibility between these standards. It would be advisable to specify which global imperfections and their values should be included in the spatial calculations of the halls according to the second order theory. There are also no guidelines regarding the values of equivalent (stabilizing) loads, which should be introduced to the lower nodes of trusses when the hall is treated as a spatial structure and calculated according to the 1st order theory. REFERENCES Biegus, A. & Czepiżak, D. 2017. Obciążenia imperfekcyjne elementów wytężonych znakozmienną wzdłużnie siłą osiową. Czasopismo Inżynierii Lądowej, Środowiska i Architektury Journal of Civil Engineering, Environment and Architecture. JCEEA 64(3/I): 371–386. Giżejowski, M. & Barszcz, A. & Ślęczka L. 2008. Projektowanie stężeń stalowych układów konstrukcyjnych według PN-EN 1993- 1-1. Inżynieria i Budownictwo 64(11): 614–621. Krajewski, M. & Iwicki, P. 2015. Analysis of brace stiffness influence on stability of the truss. International Journal of Applied Mechanics and Engineering 20(1): 97–108. Niewiadomski, L. 2003. Ocena wpływu imperfekcji ściskanych pasów dźwigarów kratowych na siły wewnętrzne w stalowych elementach konstrukcji dachu. Zeszyty Naukowe Politechniki Śląskiej. Seria Budownictwo 101. 287–294. Niewiadomski, L. & Zamorowski, J. 2017. Wstępne imperfekcje łukowe w analizie połaciowych stężeń poprzecznych. Zeszyty Naukowe Politechniki Częstochowskiej. Seria Budownictwo 23: 231–244. Piątkowski, M. 2017. Metody uwzględniania imperfekcji geometrycznych w kratownicach stalowych. Czasopismo Inżynierii Lądowej, Środowiska i Architektury. Journal of Civil Engineering, Environment and Architecture. JCEEA 64 (4/I): 229–243. Zamorowski, J. 2013. Przestrzenne konstrukcje prętowe z geometrycznymi imperfekcjami i podatnymi węzłami. Gliwice: Wydawnictwo Politechniki Śląskiej.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Buckling assessment of cylindrical steel tanks with top stiffening ring under wind loading Ö. Zeybek Department of Civil Engineering, Middle East Technical University, Ankara, Turkey Department of Civil Engineering, Muğla Sıtkı Koçman University, Muğla, Turkey

C. Topkaya Department of Civil Engineering, Middle East Technical University, Ankara, Turkey

ABSTRACT: A stiffening ring is commonly used at the top of the tank wall to increase its strength against external pressure instability. Traditional design treatments generally consider cylindrical storage tanks under uniform external pressure for sizing of the top ring. However, cylindrical steel tanks under non-uniform wind loading have rather different and complex buckling behaviour from those of tanks subjected to uniform external wind loading. In this study, the buckling resistance of the cylindrical steel storage tanks with top stiffening ring under wind loading is investigated using finite element analyses. The changes in the buckling capacity are studied in light of the proposed stiffness ratio for a particular harmonic of wind loading. The results revealed that the changes in the buckling capacity are closely related to the shell-top ring stiffness ratio. Furthermore, a generalized solution that shows buckling pressure ratio (qcr,w/qcr,D) is then developed. 1 INTRODUCTION Cylindrical ground-supported storage tanks are widely used to store a great variety of liquids for both short and long term purposes. A very common failure mode of such a storage tank is under wind loading, where the tank wall may have the insufficient strength and stiffness to resist external pressure. The wall thickness of a storage tank is normally chosen to resist only the internal pressure from the stored product. Storage tanks are susceptible to buckling under wind load when they are empty or at a low-level of filling (Ansourian 1992, Flores & Godoy 1998, Maraveas et al. 2015). One way to strengthen the tank wall is to use a stiffening ring placed at the top or near the top of the tank (Figure 1). This ring also plays an important role in maintaining circularity when the tank is subjected to wind loads. The circularity is particularly important when a floating roof is used within the tank, and this condition also means that there is no structural roof to resist buckling displacements at the top of the wall. Because the tank is very thin, it is very susceptible to buckling under external pressure, and under wind this is exacerbated because the pressure varies significantly around the circumference, flattening the wall locally and inducing stresses in different directions (Maher 1966). Thus, the specific pattern of pressure variation around circumference is of great importance in design against wind. Traditional design treatments generally consider cylindrical storage tanks under uniform external pressure for the sizing of the top ring. However, cylindrical steel tanks under non-uniform wind loading have rather different and complex buckling behaviour from those of tanks subjected to uniform external wind loading (Chen & Rotter 2012). The stiffening ring should be designed against buckling under external pressure. For this purpose, this ring must have an adequate stiffness to fulfil its function. Buckling of ring stiffened cylinders under non-uniform wind loading is a rather difficult problem in shell analysis. Blackler (1986) proposed an expression for the minimum stiffness requirement of top stiffening ring under uniform external pressure as follows: 1317

Ir;min ¼ 0:048Lt3

ð1Þ

where L = length of the cylindrical shell; t = thickness of the shell wall, Ir,min = minimum moment of inertia of the ring. Schmidt (1998) also suggested a minimum moment of inertia for the top ring based on the post-buckling behaviour under external pressure. Ir;min ¼ 0:5Lt3

ð2Þ

Greiner & Guggenberger (2004) identified a different requirement for the limiting stiffness, which this value was about 10 times the proposal of Blackler (1986). Ir;min ¼ 0:48Lt3

ð3Þ

Eurocode EN 1993-4-1 (2007) requires both a strength and a stiffness requirement for the top ring. It has a moment capacity requirement to address the effect of uniform external pressure on a ring that is imperfectly round, as well as a further requirement for wind conditions. A further requirement is for the flexural rigidity of the ring, intended to ensure that the ring does not participate in the buckling mode if the shell wall buckles. The expressions for the flexural rigidity of the ring about its vertical axis were given as follows: EIr;min ¼ k1 ELt3 pffiffiffiffiffiffi EIr;min ¼ 0:08Cw Ert3 r=t

ð4Þ ð5Þ

where Cw = the wind pressure distribution coefficient, r = radius of the shell, E = modulus of elasticity, k1 = 0.1 is recommended. According to this specification, the bending stiffness of the girder should be sufficiently large to restrict the out-of-round displacement under wind action of the ring stiffened crosssection to 2% of the radius (ECCS 2014). The above recommendations present a wide range of required minimum stiffness values for the ring size. There are two potential requirements for the top stiffening ring: strength to ensure that the ring does not yield under the unsymmetrical loads that will be applied to it under wind, and the stiffness to ensure that the buckling assessment of the tank wall under wind can be based on complete radial restraint at the top. This paper addresses the latter using a more thorough analysis than any found in previous studies. The buckling resistance of the cylindrical steel storage tanks with top stiffening ring under wind loading that varies along the circumference is investigated using finite element analyses. A shell-ring stiffness ratio is devised algebraically from the relative radial stiffnesses of the ring and the cylindrical shell under each harmonic of wind loading. The changes in the buckling capacity are studied in light of the proposed stiffness ratio.

Figure 1.

Rendering of tank and stiffening ring system.

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2 WIND PRESSURE ON CIRCULAR CYLINDERS The wind pressure distribution around a cylindrical shell has been studied extensively using wind tunnel tests (Maher 1966, Purdy et al. 1967, Resinger & Greiner 1982, MacDonald et al. 1988, Uematsu et al. 2018) where significant amounts of experimental data have been collected, leading to the characteristic pattern. The wind distribution for an isolated cylindrical structure depends on many parameters like the Reynolds number of the wind flow, the cylinder aspect ratio and potentially the shape of the roof (Maher 1966). The wind pressure varies both up the height and around the circumference of a cylindrical shell structure. However, the vertical pressure variation on the cylinder is usually assumed to be constant for tank structures because the aspect ratio is relatively low, leading to a relatively small vertical variation in the pressure (Bu & Qian 2016, Shokrzadeh & Sohrabi 2016) when compared with the major circumferential variation. Circumferential distribution of wind pressure on a circular tank structure can be reasonably approximated by a Fourier harmonic cosine series of the form as follows (EN 1993-4-1 2007): Cp ðθÞ ¼

4 X

ai cosðmθÞ ¼ 0:54 þ 0:16ðdc =LÞ þ f0:28 þ 0:04ðdc =LÞg cos θ

m¼0

þ f1:04  0:20ðdc =LÞg cos 2θ þ f0:36  0:05ðdc =LÞg cos 3θ  f0:14  0:05ðdc =LÞg cos 4θ

ð6Þ

where θ = the circumferential angle measured from the stagnation meridian, dc = the diameter of the cylindrical shell, ai = coefficients of each harmonic, m = the harmonic number. The range of wind pressure distributions considering different tank aspect ratios given in the Eurocode standard EN 1993-4-1 (2007) is shown in Figure 2. The expression given in Equation 6 considers significant effects of the aspect ratio of the cylindrical structures. Small changes in the aspect ratio may considerably alter the wind pressure profile. But, this pressure distribution changes little as the length increased for intermediate and tall cylindrical structures.

3 SHELL-RING STIFFNESS RATIO The wind loading is principally resisted by membrane shear in the shell which transmits the translational force to the support. But because it contains loading components in higher harmonics than m=1, it also induces bending in the top ring and in the shell according to their relative stiffnesses. The ring must have an adequate stiffness to fulfil its function. A test for its adequacy was developed by comparing the relative stiffnesses of the ring and the shell under non-uniform load.

Figure 2. Wind pressure distribution around circumference for circular structures with different aspect ratios (EN 1993-4-1, 2007).

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χ¼

Kshell qr ðθÞ=ur;shell ur;ring ¼ ¼ Kring qr ðθÞ=ur;ring ur;shell

ð7Þ

where Kshell and Kring are radial stiffnesses of the shell and the top ring respectively; ur,ring and ur,shell are radial displacements of the ring and shell respectively, qr(θ)= the circumferential variation of the non-uniform line load considering wind pressure (
q) as defined by Equation 8. q  L  ai cos mθ qr ðθ Þ ¼


ð8Þ

For any single harmonic loading, closed-form expressions were obtained for the radial displacement of the shell and stiffening ring. 3.1 Ring beam stiffness The Vlasov curved beam differential equations (Vlasov 1961, Heins 1975) were used to study the response of the top ring. The equilibrium equations were first derived for the curved beam element shown in Figure 3, where three orthogonal internal forces and three internal moments develop at each cross-section. The six basic equilibrium equations can be expressed as follows:

1 dQr 1 dQx þ Qθ þ qr ¼ 0 þ qx ¼ 0 ð9Þ r dθ r dθ

1 dQθ  Qr þ q θ ¼ 0 r dθ 1 dMx þ mx þ Qr ¼ 0 r dθ



1 dMr þ Tθ  Qx þ mr ¼ 0 r dθ

ð10Þ



1 dTθ  Mr þ mθ ¼ 0 r dθ

ð11Þ

where Mr = bending moment in the ring about a radial axis; Mx = bending moment in the ring about a transverse axis; Tθ = torsional moment in the ring; qx, qθ, qr = distributed line loads per unit length in the transverse; circumferential and radial directions respectively; mx, mθ, mr = distributed applied torques per unit circumference about the transverse, circumferential and radial directions respectively; Qθ = circumferential force in the ring; Qx, Qr = shear forces in the ring in transverse and radial directions respectively. The differential equation for bending of the ring in its own plane can be uncoupled from the other two. For the case of wind loading, only radial loads qr are needed (i.e. qθ = qx = mr = mθ = mx = 0), and the uncoupled differential equation of equilibrium becomes:

Figure 3.

Differential curved beam element and sign conventions.

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1 d 3 Mx dMx dqr þ ¼ dθ dθ r2 dθ3

ð12Þ

Equation 12 can be solved for the loading condition (qr) defined in Equation 8 to arrive bending moment in the ring about a transverse axis: Mx ðθÞ ¼ 

ai
q L r2 cos mθ  1Þ

ðm2

ð13Þ

The following force-deformation expression was used to obtain radial displacement (ur): Mx ¼

  EIx d 2 ur þ u r r2 dθ2

where Ix= bending moment of inertia of the ring about transverse The radial displacement of the ring can be found using Equation 13 as follows: ur ðθÞ ¼


q L r4 cos mθ 2 ðm2  1Þ EIx ai

ð14Þ axis.

ð15Þ

3.2 Shell stiffness The radial shear loading (qr(θ)) applied to the edge of the shell shown in Figure 4 is carried by what Calladine (1983) terms an edge-string. This non-symmetric load can be transformed into a tangential shear loading (Nxθ) at the top edge of the shell as shown in Figure 4. The membrane theory of shells (Flügge 1973, Calladine 1983, Rotter 1987, Ventsel & Krauthammer 2001) was adopted to obtain the radial displacements of the shell under circumferential shear loading using the edge-string treatment. Considering the cylindrical shell element shown in Figure 5, the membrane theory of equilibrium equations (Rotter 1987, Ventsel & Krauthammer 2001) are: ∂Nx 1 ∂Nxθ þ þ px ¼ 0 ∂x r ∂θ

Figure 4. 1983).

∂Nxθ 1 ∂Nθ þ þ pθ ¼ 0 Nθ þ rpn ¼ 0 ∂x r ∂θ

ð16Þ

Transferring applied non-uniform radial edge loading to tangential shear loading (Calladine

1321

where r = middle surface radius; Nx, Nθ, Nxθ = axial, circumferential, and shear membrane stress resultants, respectively; and px, pθ, pn = external distributed pressures in the axial, circumferential and radial directions, respectively. The membrane stress resultants (Ventsel & Krauthammer 2001) were given as: Z Nθ ¼ rpn Nxθ ¼ 

pθ þ

  Z 1 ∂Nθ 1 ∂Nxθ dx þ f1 ðθÞ Nx ¼  dx þ f2 ðθÞ px þ r ∂θ r ∂θ

ð17Þ

where f1(θ), f2(θ) = unknown functions of θ to be determined from two boundary conditions. The displacements were found considering strain-displacement relationships as: Z Z Z Et ∂ux dx þ f4 ðθÞ Etux ¼ ðNx  Nθ Þdx þ f3 ðθÞ Etuθ ¼ 2ð1 þ Þ Nxθ dx  ∂θ r Etur ¼ Et

∂uθ  rðNθ  Nx Þ ∂θ

ð18Þ ð19Þ

where ux, uθ, ur = displacements in the axial, circumferential, and radial directions, respectively; ν = Poisson’s ratio; f3(θ), f4(θ) = additional functions to satisfy the boundary conditions on the edges x = constant. The initial algebraic treatment here involves no surface loading on the shell (px = pn = pθ = 0) but involves only tangential shear loading Nxθ at the top (Figure 4). Considering appropriate boundary conditions which were given in Figure 4, the radial displacement can be found as follows at the top of the shell (x = L). ur ðθÞ ¼ m2 ai
q L2

2  3r ð þ 2Þ þ L2 m2 cos mθ 3Etr2

ð20Þ

3.3 The shell-ring stiffness ratio The ratio of the stiffness of the shell to the ring (χ) was found by combining the above expressions. Inserting Equations 15 and 20 into Equation 7 yields the shell-ring stiffness ratio (χ) which can be expressed as: χ¼

Figure 5.

ur;ring 1 3r6 t ¼ ur;shell m2 ðm2  1Þ2 LIx ½3r2 ð þ 2Þ þ m2 L2

Loading, displacements and stress resultants in an element of the cylindrical shell.

1322

ð21Þ

4 BUCKLING BEHAVIOUR OF THE CYLINDRICAL TANK STRUCTURES 4.1 A brief assessment of each harmonic wind loading The effect of each harmonic of loading recommended by EN 1993-4-1 (2007) was investigated separately. A unit pressure was applied at the stagnation point. The commercial finite element program ANSYS v12.1 (2010) was used to perform these numerical analyses. Numeric studies used a constant radius-to-thickness (r/t=1000) ratio and height-to-diameter (L/dc) ratios of 0.25, 0.5, 0.75, and 1.0 considering different size annular plate ring. The uniform buckling pressure was determined from Donnell theory as follows. qcr;D ¼ 0:92E

ffi  t 2 pffiffiffi rt r

L

ð22Þ

From each analysis, the stagnation pressure at buckling (qcr,w) was extracted using linear bifurcation analysis. Then, these values were normalised by the buckling uniform pressure obtained from Donnell theory (qcr,w/qcr,D). The buckling pressure ratios for each harmonic of loading were plotted in Figure 6 as a function of the shell-ring stiffness ratio. The comparison shows that the dominant harmonic term is m = 2 for all cases. 4.2 Linear bifurcation analysis for cylindrical steel tanks with stiffening ring under wind loading The linear buckling behaviour of the cylindrical steel storage tanks with top stiffening ring under wind loading was investigated for the same geometries as in the previous part, but r/t changes from 500 to 2500 for all cases. From each analysis, the stagnation pressure at buckling (qcr,w) was normalised by uniform pressure obtained from Donnell theory (qcr,w/qcr,D). The buckling pressure ratios for wind loading are plotted in Figure 7 as a function of the shell-ring stiffness ratio considering dominant harmonic term of m = 2. The relationship between buckling pressure ratio and stiffness ratio can be represented by: qcr; w ¼ f ðχÞ qcr;D

Figure 6.

Buckling pressure ratios with different aspect ratios under each harmonic wind loading.

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ð23Þ

Figure 7.

Buckling pressure ratios with proposed equations under wind loading.

Table 1. Coefficients used in the Equation 24. L/dc

C1

C2

1.0 0.75 0.50 0.25

1.71 1.66 1.58 1.52

0.07 0.15 0.36 0.87

where f(χ) is a function that can be approximated by curve fitting to the data (Figure 7) as f ðχÞ ¼ C1 

0:8   r r 2 1 þ Cχ2 L t

ð24Þ

where the C1 and C2 = the coefficients given in Table 1. The proposed expressions considering the shell-ring stiffness ratio are also shown in Figure 7. Obviously, the recommendations given by Schmidt (1998) and Greiner & Guggenberger (2004) are fairly close to each other for design of stiffening rings, but differ from results obtained by Blackler (1986) and EN 1993-4-1 (2007).

5 CONCLUSIONS This paper has presented a new stiffness requirements of the stiffening ring at the top of the wall of a ground-supported cylindrical tank under wind loading. The buckling resistance of the cylindrical steel storage tanks with top stiffening ring under wind loading that varies along the circumference is investigated using finite element analyses. The changes in the buckling capacity are studied in light of the proposed stiffness ratio that represents the ratio of stiffnesses of the shell and the top ring for a particular harmonic 1324

of wind loading. The results revealed that the changes in the buckling capacity are closely related to the shell-top ring stiffness ratio. Furthermore, a generalized solution that shows buckling pressure ratio (qcr,w/qcr,D) is then developed as a function of the shell-top ring stiffness ratio. REFERENCES ANSYS, Version 12.1 On-Line User’s manual, 2010. Ansourian, P. (1992) On the buckling analysis and design of silos and tanks, Journal of Constructional Steel Research., 23, pp. 273–294. Blackler, M.J. (1986) “Stability of Silos and Tanks under Internal and External Pressure”, PhD Thesis, School of Civil and Mining Engineering, University of Sydney, Australia. Bu, F., and Qian, C. (2016) On the Rational Design of the Top Wind Girder of Large Storage Tanks, Thin-Walled Structures, 99, pp. 91–96. Calladine, C.R. (1983) Theory of shell structures. Cambridge University Press, U.K. Chen, L. & Rotter, J. (2012) Buckling of anchores cylindrical shells of uniform thickness under wind load, Eng. Struct. 41, pp. 199–208. EN 1993-4. (2007), Eurocode 3: Design of steel structures, Part 4.1: Silos, Eurocode 3 Part 4.1, CEN, Brussels. Flores, F.G. & Godoy, L.A. (1998) Buckling of short tanks due to hurricanes, Engineering Structures, 20 (8), pp. 752–760. Flügge, W. (1973) Stresses in Shells. Springer-Verlag, Berlin. Greiner, R. & Guggenberger, W. (2004) “Tall cylindrical shells under wind pressure”, in Buckling of Thin Metal Shells, eds J.G. Teng & J.M. Rotter, Spon, London, pp 198–206. Heins, C.P. (1975) Bending and torsional design in structural members. Lexington Books, Lexington, Massachusetts. MacDonald, P.A., Kwok, K.C.S. & Holmes, J.D. (1988) Wind Loads on Circular Storage Bins, Silos and Tanks: I. Point Pressure Measurements on Isolated Structures, Journal of Wind Engineering and Industrial Aerodynmamics, 31, pp 165–188. Maher, F.J. (1966) Wind loads in dome-cylinder and dome-cone shapes, ASCE Journal of the Structural ivision, Vol 92 (5), pp. 79–96. Maraveas, C., Balokas, G.A. & Tsavdaridis, K.D. (2015) Numerical evaluation on shell buckling of empty thin-walled steel tanks under wind load according to current American and European design codes, Thin-Walled Structures, 95, pp. 152–160. Purdy, D.M., Maher, F.J. & Frederick, D. (1967) Model studies of wind loads on flat-top cylinders. ASCE Journal of the Structural Division, Vol. 93, pp. 379–398. Resinger, F. & Greiner, R. (1982) Buckling of Wind Loaded Cylindrical Shells — Application to Unstiffened and Ring-Stiffened Tanks. In: Ramm E. (eds) Buckling of Shells. Springer, Berlin, Heidelberg. Rotter, J.M. (1987) “Membrane Theory of Shells for Bins and Silos”, Transactions of Mechanical Engineering, Institution of Engineers, Australia, Vol. ME12 No. 3 September, pp 135–147. Rotter, J.M. & Schmidt, H. (2014) Buckling of Steel Shells: European Design Recommendations. Brussels: European Convention for Constructional Steelwork (ECCS). Schmidt, H. Binder, B. & Lange, H. (1998) Postbuckling strength design of open thin walled cylindrical tanks under wind load, Thin Walled Structures, 31, pp. 203–220. Shokrzadeh, A.R., and Sohrabi, M.R. (2016) Buckling of Ground Based Steel Tanks Subjected to Wind and Vacuum Pressures Considering Uniform Internal and External Corrosion, Thin-Walled Structures, 108, pp. 333–350. Uematsu, Y., Yamaguchi T. & Yasunaga J. (2018) Effects of wind girders on the buckling of open-topped storage tanks under quasi-static wind loading, Thin-Walled Structures, Vol. 124, pp. 1–12. Ventsel, E. & Krauthammer, T. (2001) Thin plates and shells: theory, analysis and applications, Marcel Dekker, NY. Vlasov, V.Z. (1961) Thin-walled elastic beams. National Science Foundation, Washington, D.C.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Slim-floor beam bending moment resistance considering partial shear connection Q. Zhang & M. Schäfer Research Unit in Engineering Science, FSTC, University of Luxembourg, Luxembourg

ABSTRACT: Having the advantage of flat lower surface, high stiffness and integrated fire resistance, slim-floor composite beams are widely used and favoured in many design solutions. The current bending design methods are mainly derived from plastic design methods for classical composite beam with consideration of the special features for slim-floor beams such as the transverse bending of bottom flange when used as support for slabs. Alternatively, more advanced strain-limited design method or FE-method can be used. In the case of full shear connection, with deep position of neutral axis and great compression zone height, there is a risk that plastic design method may overestimate the bending resistance of the cross section compared to the strain-limited design method. In the case of the partial shear connection, shear design diagram for slim-floor beams obtained by means of the strain-limited design can also differ significantly from the one obtained by plastic design method, thus further research on slim-floor beams is still necessary.

1 INTRODUCTION Slim-floor beam (shallow floor beam or slim-floor construction) refers to a construction type where the steel beam is partially or fully embedded into the concrete slab (Figure 2). Usually prefabricated slab systems or special profile sheetings are directly supported on the bottom steel flange of the asymmetric section to simplify the construction process. Compared to traditional composite beam systems, in buildings with slim-floor beams structural height is greatly reduced, allowing the installation of technical services without any beam openings due to their flat lower surface. Composite behavior between the steel section and the concrete slab can be attained by shear connectors in forms of headed studs or concrete dowels. Due to the integration of the steel beam into the concrete slab, only the bottom flange of steel section is directly exposed to fire. In the case of fire, this bottom flange can be substituted by longitudinal reinforcement bars, thus achieving high fire resistance without any passive protection. Because of their high bearing resistance and stiffness, composite slim-floor sections can attain economical span length, e.g. 8-14m. If dry construction is employed by using prestressed concrete hollow core slabs, the slabs are normally supporting span length of 6 to 12m while the slimbeam is supporting the shorter span length of 5 to 7.5m (Schäfer et al. 2018).

2 SLIM FLOOR BEAM TYPES AND DEVELOPMENT Composite slim-floor system originated from Scandinavia region in the 1970s and continuous developed afterwards, the development are also summarized in Schleich 1997, ECCS 1995, Baehre & Pepin 1995, Lawson 1999, Schäfer & Braun 2019 and other documents. Initially the precast concrete hollow core slab was used to reduce the slab self-weight and to allow fast erection. Through the use of asymmetric steel sections with larger bottom flange, the support of the pre-fabricated concrete elements became possible. One of the first development is the THQ profile, consisting of a welded box section. Some of the variations of this system are the NSQ-profile, the TBB-Hut profile and the SWT-Profile. Since the 1990s, a new type of slim-floor system called SFB (Slim Floor Beam) has been developed. SFB has 1326

a standard rolled profiled steel section welded with an additional wider steel plate at bottom plate. Subsequently, the IFB Profile (Integrated Floor Beam) entered the market - it was made by welding a half of the hot rolled profiled steel section onto a steel plate. Initially, the composite action was not considered, with profiles being designed merely as pure steel beams, which led to conservative results. Since then, composite effects have been more closely researched and applied in order to attain economically optimal design. The special rolled asymmetric steel beams (ASB) with wider bottom flange have special embossment of the top flange surface with the purpose of activating the composite behavior between concrete slab and steel beam through friction mechanism. Several new systems such as the FEDU profiles (Tschemmernegg 1996), UPE-profiles (Fries 2001) and DELTA sections have also been developed since. These systems use different shear connectors: the FEDU beam uses punched steel ribs as shear connector, the UPE beam uses headed studs and the DELTA beam originally used the concrete dowels in the circular holes of the web to transfer longitudinal shear forces, while the modern DELTA® Beam uses additional reinforcement. Today also SFB sections are used with vertical or horizontal headed studs in order to active composite effect (Figure 1). Compared to the open steel sections, the box-sections (hat-profiles) provide higher torsional stiffness, resulting in the better structural performance in the cases of asymmetric loading and edge beams. For traditional composite beam, the headed studs are most commonly placed on the top of steel beam, while for slim-floor beams this layout usually faces difficulties due to the limited concrete cover. As a result, many innovative shear connectors have been developed. Nowadays many new types of slim-floor system solutions are commercially developed by companies. Some examples in Europe are: the USFB® (Ultra Shallow Floor Beam) from Westok Limited, the DELTABEAM® slim floor structure from Peikko Group and the CoSFB (Composite Slim-Floor Beam) from ArcelorMittal. The steel section of the USFB® is fabricated by welding together the two different halves of the hot-rolled steel sections with circular web-openings. CoSFB uses CoSFB-Betondübel as shear connector (Braun 2018) to improve the composite behaviour and also the section resistance. These three types of beam are shown in Figure 2.

Figure 1.

Development of part of European composite slim-floor systems (also see: Schäfer & Braun 2019).

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Figure 2.

Examples of modern Slim-Floor beam systems (Zhang 2016).

3 PLASTIC BENDING RESISTANCE OF SLIM-FLOOR BEAMS 3.1 Design with full shear connection Current Eurocode 4 (EN1994-1-1 2004) does not provide specific design rules for slim-floor beams. The design methods applied for slim-floor beams are mostly derived from the plastic design methods for traditional composite beams with additional consideration of their special features. Plastic design is based on the assumption that sections have sufficient rotation capacity, implying that plastic strains can be reached at each fiber of the section. For the calculation of the plastic moment resistance Mpl;Rd full interaction is assumed between structural steel reinforcement and concrete (Johnshon 2004) meaning that the section is assumed to remain plain. Full shear connection is reached when an increase in the number of shear connectors within the critical length does not lead to further increase in the moment resistance. Additional design rules to Eurocode 4 are given for hot-rolled and box-sections in many research works, some examples are Bode et al. 1997, Lawson & Brekelmans 1999, Hauf 2010, Lam et al. 2015, Leskelä et al. 2015, Schäfer et al. 2018, much more other literature related to different topics are to be found, here it is not possible to list all of them. The general design principles are summarized as follow: In the case of the full shear connection (Figure 3) and sagging bending moment, the compression force consists of the integral of concrete stress bock ð0:85fcd Þ and the compression blocks in the steel part (fyd ), while concrete in tension is neglected. For the concrete inside a box-section fcd can be used, due to the better hardening conditions and confinement effects. Due to the interaction of vertical shear and normal stress from bending moment, according to EN1994-1-1 section 6.7.3.2, the steel strength of the steel web is to be reduced to ð1  ρÞfyd if Va;Ed 40:5Vpl;a;Rd (Figure 3). In the case of important web openings, secondary bending moments due to Vierendeel-effects are to be considered for moment resistance and vertical shear design (Schäfer 2015b). Furthermore, the effects from torsional moments can lead to additional reduction of the moment and shear capacity. When the extended bottom flanges are

Figure 3. Plastic moment design of slim-floor section based on full shear connection (also see: Schäfer et al. 2018).

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Figure 4.

Plastic moment design of slim-floor section based on partial shear connection.

used as supports for the concrete slab, significant amount of transverse local bending moment can be generated. The interaction of the compression stress and tensile stress in longitudinal direction reduces the design resistance. Thus, the design strength of the bottom flange needs be reduced to fyd;eff (Schäfer 2015a). For the open sections, some researchers (Bode et al. 1997) have suggested using a reduced steel plate thickness to reflect this impact. 3.2 Design with partial shear connection Where the shear connector arrangement is controlled by detailing, such as the geometry of profile sheeting, or the proof in SLS become decisive, often partial shear connection is realized. In partial shear connection, the bending moment resistance MRd is limited by the longitudinal shear resistance Nc . The Plastic Neutral Axis (PNA) in (reinforced) concrete part and the PNA in the steel beam are separated. Their locations are controlled by the longitudinal shear force transferred by shear connectors and the stress distribution. The bending moment resistance of partial shear connection can be calculated by solving the equilibrium equation according to plastic theory. A more practical way is to use the partial shear diagram provided in Eurocode 4 (Figure 4). This diagram is developed using the plastic design method, for which a nonlinear ABC curve or a simplified linear line AC can be adopted (A: no shear connection, B: zone of partial shear connection, C: full shear connection). Here the degree of shear connection is defined as η ¼ ðNc  Ns Þ=Nc; f where Nc is the total concrete normal force, and Nc;f is the value with full shear connection. Design moment resistance in the case of partial shear connection can be then easily obtained from the previously calculated plastic bending moment of composite section Mpl;Rd and the pure steel section Mpl;a;Rd . If solid slab with big amount of reinforcement is used the contribution of slab should also be considered at least in SLS (Hauf, 2010). Considering the limitation of shear connector deformation capacity, a minimum degree of shear connection is required, depending on the relation of Mpl;Rd to Mpl;a;Rd , ductility of shear connectors, loading and overall moment allowance. 4 STRAIN-LIMITED DESIGN RESISTANCE OF SLIM-FLOOR BEAMS According to plastic design method, concrete stress is represented by a rectangle stress block with compression stress of 0:85fcd . For traditional composite beams with small compression zone height, the plastic design is on the safe side (Hanswille 1996, Schäfer et al. 2019). However, for slim-floor beams the position of neutral axis is deeper due to the non-symmetric steel profile with enlarged bottom flange. The raised compression zone height usually brings reduction of the rotation capacity and the bearing behavior. Premature concrete crushing failure may happen before yielding of the steel section. In this case the plastic method can overestimate the design resistance. This effect was also mentioned in many former research works, such as in Ansourian 1984, plastic design required conditions related to rotation capacity and others were analysed and a compression zone height smaller than 16% of cross-section height was suggested. For a bigger 1329

compression zone height until xpl =h ¼ 0:3, in Hanswille & Sedlacek 1996 the reduction of plastic bending resistance for steel Grade S420 and S460 was analysis. Consequently, in the design procedure provided in the EN1994-1-1 section 6.2.1.2 (2) full plastic bending moment is limited by a β factor in the cases when steel grade S420 or S460 is employed and when the relation of xpl =h is over 0.15. However, this problem does not occur only when the steel grade is greater than S420 - it is more a question of the cross-section shape and the position of the plastic neutral axis. Therefore a reduction function β 0 ¼ Msl;Rd =Mpl;Rd between plastic bending moment resistance Mpl;Rd and strain-limited bending moment resistance Msl;Rd was defined by Schäfer et al. 2019, expanding the design procedure to other steel grades as well. Schäfer et al. 2019 pointed out the limitations of the plastic design for composite sections, by comparing this approach to the strain limited design procedure. In strain limited method, the slim-floor section follows Bernoulli hypothesis, for which a linear strain distribution in full interaction is assumed. Considering the shear-lag effects in concrete slab, a simplification with effective width can be applied, similar to the procedure used in the design of classical composite beams. The resistance is reached when the ultimate strain in any fibre reaches its strain limitation. Similar to the the bending design of reinforced concrete beams according to Eurocode 2 (EN1992-1-1 2004), a parabolic-rectangle stress-strain model can be used. The strain limits of structural steel can be calculated according to stress-strain relationship of EN19931-5, Annex C (EN1993-1-5 2006) considering strain-hardening. However other suitable material models are not excluded. 4.1 Design with full interaction Figure 5 illustrates the strain-limited design principal on an example of a slim-floor beam under sagging bending moment. For determination of strain-limited bending resistance (Msl;Rd ) full interaction is assumed, implying that there is no slip at steel-concrete interface. With full interaction, the strain limits can be either the compression strain-limit of concrete (εcu;2 ) at top fiber, the steel tensile strain-limit (εau ) at the bottom fiber (concrete in tension is neglected) or the strain-limit of the reinforcement (εsu ). Due to the fact that steel is a ductile material, in most cases the concrete strain will be the governing factor. To calculate the location of neutral axis and strain stress distribution, the following two requirements must be met: ● at least one critical fiber of the section should reach its strain limit, and all other fibers

stay below their limits. For sagging bending moment, it is usually the fiber located at concrete upper surface (εc ¼ εc;u ). ● in the case of P pure bending, the sum of normal forces of each fiber inside a cross-section shall be zero ( Ni0 ¼ 0); By setting a strain-limit point and rotate the strain curve based on the point, equilibrium can be found, further the cross-section resistances can be acquired. For many finite difference method based numerical procedure, in order to calculate beam deflection, moment

Figure 5.

Strain-limited design of slim floor beam based on full shear connection.

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redistribution or longitudinal shear force, the Moment-curvature (M-k) curve is important. Such procedures can be found in Fabbrocino G. et al. 2000, Fries 2001 and many other publications. To acquire the M-k curve, the strain-limits can be step-wisely changed from zero to the limiting values, and calculate out each moment and curvature value. 4.2 Design with partial interaction In the case of sagging moment and cross-section class 1 and 2, with ductile shear connectors, the theoretical partial interaction can be applied. In reality there will always be slip at the interface of concrete and steel element as typical shear connector needs to experience some deformation in order to attain its design resistance. In the case of partial interaction, it is usually assumed that the concrete and steel part have the same curvature at each cross-section, which can be represented by pair of parallel strain curves (similar to the widely accepted Newmarkmodel (Newmark 1951), however the assumption of elastic shear connector is not used here, as this paper mainly discuss on cross-section level). Within the small deformation domain, these assumptions are reasonable. Due to slip there will be two neutral axes for concrete and steel section each, (while for full interaction there is only one neutral axis). The neutral axes of the two parts separate with distance of slip-strain εslip (Figure 6). The slip in the composite joint depends on the flexibility of shear connectors and degree of shear connection and the longitudinal shear force. This model was widely applied for tradtional composite beams for example it was used in the work of Fabbrocino et al. 2000 to develop the numerical calculation method of continuous composite beams consider partial interaction. General theoretical basics are also explained in Oehlers et al. 1995 for the definition of degree of interaction. When the slip occurs, critical fibers, with regard to the imposed strain-limits, are no longer only the out-most fibers of the section. Due to the slip, the strain in the internal fibers may increase, which can cause the steel section to reach its yield strains - reinforcement or steel upper flange can attain their failure strain first. This happens especially in the cases when the degree of shear connection is very low. In the case of partial shear interaction, there are three unknown factors in the strain diagram: the strain limits at critical fibres εi;u , the slip-strain value εslip and the curvature χ (Figure 6). Therefore, at least three equilibrium equations need to be established: ● at least one critical fiber of the section should reach its strain limit (εi ¼ εi;u ), and all other

fibers stay below their limits;

● in the case of P pure bending, the sum of normal forces of each fiber inside a cross-section 0

shall be zero (

Ni ¼ 0);

● the integral of longitudinal shear force within the critical length ensured by the provided

shear connectors R should be 0equal to 0 the normal forces inside either the steel or concrete part by value ( VL dx ¼ ηNc;f ¼ Na ). In order to calculate the bending resistance for different degrees of shear connections η by strain-limited design method, a numerical program is necessary, the steps outlined in the flow-chart in Figure 6 can be followed. In general, the curvature can be first calculated

Figure 6. Strain-limited design of slim floor beam based on partial shear connection (also see: Schäfer & Zhang 2019).

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Figure 7.

Comparison of partial shear diagrams.

based on internal force ηNcf0 given by the shear degree η and afterwards the stain distribution over the whole cross-section and resistances can be acquired. 5 PARTIAL SHEAR DIAGRAM OF SLIM-FLOOR BEAMS The partial shear diagram, based on the plastic design method presented in Eurocode 4, is explained in section 3 of this paper. Similarly, we can generate the partial shear diagram by strain-limited method or FE-method. The strain-limited bending resistance and maximum longitudinal shear are different to the values obtained by plastic design. Thus, longitudinal shear resistance obtained by plastic design method should be replaced with the strain limited resistance in order to calculate the degree of shear connection η. Figure 7 shows the partial shear diagrams according to the plastic method (ABC), the simplified method diagram (AC), as well as by strain-limited method. For comparison purposes, the partial shear diagram of strain-limited design is presented in relative values, by dividing the full shear connection resistance values with the values obtained from plastic design. For classical composite beams with small compression zone height (xpl =h50:15), strain-limited design without steel strain-hardening gives similar results as plastic design, as shown in Figure 7a. Parameter studies on more than 5,000 cross-sections confirmed this conclusion. Comparison of partial shear diagrams for a slim-floor cross-section is shown in Figure 7b. It shows considerable difference in the results between these two methods of design: At full interaction, the maximum bending moment resistance by strain limited design (b) is around 7% smaller than the plastic bending resistance (a). The longitudinal shear force is also around 17% smaller than plastic design suggests. From the stress curves for strain-limited design and plastic design at full interaction we can notice important differences. In the case of the strain limited design, the reinforcement is not likely to yield nor the steel bottom flange and a great part of the steel web. This is due to the fact that concrete compression strain is the governing factor here, with limited strain in the steel due to the deep neutral axis. Thus the maximum total longitudinal shear force is much smaller than full plastic resistance - the same applies for the bending resistance. The difference is more significant when compression zone height is greater and high strength steel is used. Similar phenomenal happens to over-reinforced concrete beams or in the case of high strength reinforcement. Therefore, from all of the above, it can deducted that in the cases when slim-floor beams are being used, due to the relatively deep neutral axis position, plastic design method and partial shear diagram provided in the current Eurocode 4 is not always suitable for safe use. 6 CONCLUSIONS In the recent years the use of the composite slim-floor beams is becoming more popular as they are expanding the areas of their applications. However, there are still no universal 1332

standards for their design. Many of the current design methods are based on plastic theory, which may overestimate the bending resistance in the cases of deep compression zones. In these cases strain-limited design is a preferable method of analysis as it offers more accurate results. With the aim of reducing the difficulty of design procedure and in order to allow for more economical design solutions, research on new theoretical design approaches and partial shear diagrams for slim-floor beams is currently in the process. This paper presents some of its early results. Further investigation in the rotation capacity, moment redistribution and other related topics of slim-floor beam system are still required to reach a safe and economic design. REFERENCES Ansourian, P. 1984. Beitrag zur plastischen Bemessung von Verbundträgern, Bauingenieur 59: 267–272. Baehre, R & Pepin, R. 1995. Flachdecken mit Stahlträgern in Skelettbauten, Bauingenieur (Springer-Verlag): 70. Bode, H. et al. 1997. Untersuchungen des Tragverhaltens bei Flachdeckensystemen mit verschiedener Ausbildung der Platte und verschiedener Lage der Stahltrager, Forschung fur die Praxis: 261. Braun, M. 2018. Investigation of the Load-Bearing Behaviour of CoSFB-Dowels. Dissertation, University of Luxembourg ECCS. 1995. Multi-story building in steel-Design Guide for Slim Floors with Built-in Beams, Report No.83 Paris. EN 1992- 1-1. 2004. Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules for buildings. EN 1993- 1-5. 2006. Eurocode 3: Design of steel structures - Part 1-5: Plated structural elements. EN 1994- 1-1. 2004. Eurocode 4: Design of composite steel and concrete structures Part 1-1: General rules and rules for buildings. Fabbrocino G. et al. 2000. Analysis of continuous composite beams including partial interaction and boud. Journal of structural engineering 126(11): 1288–1294. Fries, J. 2001. Tragverhalten von Flachdecken mit Hutprofilen. Dissertation, University of Stuttgart. Johnson, R.P. 2004. Composite Structures of Steel and Concrete - Beams, Slabs, Columns, and Frames for Buildings, Blackwell Publishing, third edition: 24–25. Hanswille G. & Sedlacek, G. 1996. The Use of Steel Grades S460 and S420 in Composite Structrues, ECCS-EUROFER improvements by TC 11 to EUROCODE 4 report. Hauf, G. 2010. Trag- und Verformungsverhalten von Slim-Floor Trgern unter Biegebeanspruchung. Dissertation. Unveristy of Stuttgart. Lam, D. et al. 2015. Slim-floor construction - design for ultimate limit state. Steel Construction 8(2): 79–84. Lawson, R.M. & Brekelmans, J.W.P.M. 1999. Design recommendations for shallow floor construction to Eurocodes 3 and 4. European Commission technical steel research report: 11–12. Newmark, N M. 1951. Test and analysis of composite beams with incomplete interaction. Proceedings of society for experimental stress analysis 9(1): 75–92. Oehlers, D.J, Bradford, M.A. 1995. Composite steel and concrete structural members: fundamental behaviour. Oxford: Pergamon Press: 23–26. Schäfer, M. 2015a. Zur Biegbemessung von Flachdecken in Verbundbauweise, Stahlbau - Ernst & Sohn 84(4): 231–238. Schäfer, M. 2015b. Schubtragfäigkeit und M-V-Interaktion von Flachdecken mit integrierten hohlkastenfömigen Stahlprofilen, Stahlbau - Ernst & Sohn 84(5): 314–323. Schäfer, M. et al. 2018. Flachdecken in Verbundbauweise-Bemessung und Konstruktion von Slim-FloorTrägern, StahlBau Kalender - Ernst & Sohn: 631–741. Schäfer, M. et al. 2019. Plastic design for composite beams - are there any limits? 9th International Conference on Steel and Aluminium Structures, Bradford,UK. Schäfer, M. & Braun, M. 2019. Entwicklung der Slim-Floor Bauweise in Europa, Special edition Slimfloor beams Stahlbau - Ernst & Sohn (88) 7. Schäfer, M & Zhang, Q. 2019. Zur Momententragfhigkeit von Flachdecken in Verbundbauweise Stahlbau - Ernst & Sohn (88) 7. Schleich, J.B 1997. Slim floor construction: why?, Outstanding Composite structures for Buildings, Composite Construction Conventional and Innovative, Innsbruck IABSE reports, Zurich: 53–64. Tschemmernegg, F & Huber, G. 1996. Flachdecken mit Stanzdubeln Bauingenieur, Springer-Verlag 71. Zhang, Q. 2016. CET-Report I, Internal research report for Research on Longitudinal shear in Composite Structures (unpublished). University of Luxembourg.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Stainless steel SHS and RHS beam-columns B. Židlický & M. Jandera Czech Technical University in Prague, Prague, Czech Republic

ABSTRACT: Presented research deals with stainless steel slender square hollow and rectangular hollow section (SHS and RHS) members loaded by combination of uniaxial bending moment and compressive axial force. Four tests of eccentrically loaded stainless steel pin-ended columns were conducted. Data obtained from the tests were used for the numerical model validation. A comprehensive numerical parametric study was made using finite element software Abaqus employing geometrically and materially non-linear analysis with imperfections (GMNIA). All three main stainless steel groups were considered in study, namely austenitic, ferritic and duplex. The investigated variables were mainly cross-section slenderness, member slenderness and ratio between compressive force and bending moment. Only uniform bending moment along the member length was considered. Based on the numerical parametric study results a new design approach for stainless steel SHS and RHS beam-columns has been derived. It is described and evaluated in this paper together with recently published design procedure developed by Zhao.

1 INTRODUCTION Stainless steel is a highly alloyed steel exhibiting high corrosion resistance together with great mechanical properties, easy maintenance and aesthetic appearance. Many procedures given by the stainless steel codes were adopted from the procedures for carbon steel. As the Eurocode was developed almost 15 years ago, some of the rules for stainless steel were based on much lower number of tests and numerical models than available today. This paper is focused on stainless steel square and rectangular hollow section (SHS and RHS) beam-columns. Only uniform uniaxial bending moment along the member length is considered. There is a codified design approach in the Eurocode EN 1993-1-4 (2006) for stainless steel, see Equations 1 and 2. N Ed N b;Rd;y

Ed þky MMb;Rd;y  1:0

N Ed ky ¼ 1:0 þ 2 ðλ
y  0:5Þ N b;Rd;y

but

1:2  ky  1:2 þ 2

ð1Þ N Ed N b;Rd;y

ð2Þ

where NEd ¼ the axial compressive force; MEd ¼ the maximal I. order bending moment; Nb,Rd,y ¼ the flexural buckling resistance, Mb,Rd,y ¼ the bending moment resistance; ky ¼ the interaction factor considering compression and bending interaction; and
λy ¼ the member slenderness. However, this procedure exhibits some shortcomings, namely: it does not consider moment distribution along the member length which leads to over-conservative results in non-uniform bending moment cases; the lower bound of the interaction factor ky = 1.2 makes the procedure over-conservative in cases where bending moment is dominant. Many investigations have been carried out in order to make the beam-column design both accurate and safe. Most of them were focused on the improvement of the interaction factor calculation. Evaluation made by Jandera et al. (2017) shown that all of them have some inaccuracies and a new method should be developed. A new approach was recently developed by Zhao et al. (2016). It considers the same interaction formula 1334

Table 1.

Interaction factor coefficients Di.

Stainless steel group

D1

D2

D3

Austenitic Ferritic Duplex

2.0 1.3 1.5

0.30 0.45 0.40

1.3 1.6 1.4

as EN 1993-1-4 (2006), see Equation 1, with flexural buckling resistance Nb,Rd,y calculated considering new flexural buckling curves derived by Afshan et al. (2017), bending resistance Mb,Rd,y,CSM calculated according to CSM method developed by Gardner and published e.g. in Afshan & Gardner (2013) and interaction factor kCSM calculated according to a new formula, see Equation 3. kCSM ¼ 1 þ D1 ðλy  D2 Þ nb  1 þ D1 ðD3  D2 Þnb

ð3Þ

where Di = the interaction factor coefficients given by Table 1; and nb = the compressive force NEd to flexural buckling resistance Nb,Rd,y ratio. As could be seen, there are several coefficients regarding the stainless steel group. The procedure was presented as a safe and accurate for the design of stainless steel SHS and RHS members loaded by compression and bending combination. The evaluation of this approach on numerical results is given in this paper below.

2 EXPERIMENTAL STUDY Two sets of experiments were conducted. First consists of four SHS specimens carried out for a numerical model validation. The recently conducted second set of 12 specimens extended the previous tests for more slender Class 4 SHS members as well as both stocky and slender crosssection RHS members. Data obtained from the second set of the tests will be presented at the conference. 2.1 Tensile tests In order to obtain material properties, tensile coupon tests were conducted. It was necessary to made tensile tests for both flat and corner part of the cross-section as the corner material properties are significantly higher due to cold-rolling. Figure 1 presents location of the coupons. 2.2 Experiments Together 20 experiments were carried out, namely 12 SHS members and 8 RHS members. Specimens were loaded by an eccentrically applied compressive force. The eccentricity was the same for both ends of the member, causing uniaxial uniform bending moment in the tested member, see Figures 3 and 4. Pin-ended supports were made from two steel plates on both ends of the specimen. One plate was equipped by the wedge and the second by the V-shaped notch which ensured deflection in appropriate plane and the eccentricity was easily adjusted, see Figure 2. Both global and local imperfections of all specimens were measured manually before the test. Specimen information are summarized in Table 2.

3 NUMERICAL STUDY Numerical model was created in finite element software Abaqus in order to obtain behaviour of the tested members. 3D shell element model using GMNIA analysis (Geometrically and 1335

Figure 2.

Pin-ended support.

Figure 3.

Test set-up and photo of a tested member.

Table 2. Tested member geometry. Crosssection

Measured dimensions

Wall Member Member Global Local thickness length slenderness Eccentricity Imperfection Imperfection

mm

mm

mm

mm

-

mm

mm

mm

80  3 80  3 80  5 80  5

79.74  79.74 79.74  79.74 Not measured Not measured

2.8 2.8 5.07 5.07

2610 2620 2625 2575

1.28 1.29 1.46 1.43

20 40 20 40

0.833 0.767 1.3 1.233

0.01 0.0125 0.0125 0.01

Materially Non-linear Analysis with Imperfections) was used. Both global and local imperfection amplitudes were introduced through the appropriate (global and local buckling) eigenmodes. Material properties for both flat and corner part of the cross-section were obtained from the tensile tests. The corner region with increased strength was considered by the area of the corner itself with no extension to the flat part. 3.1 Numerical model validation The numerical model prediction was compared with the experimental data, namely the relationship between compressive loading force and deflection at the mid-span of the member length. The comparison is given by Figure 4 and Table 3. 1336

Figure 4.

Comparison between the numerical model (ABQ) and experiments (EXP).

Table 3. Comparison between the numerical model (ABQ) and experimental (EXP) ultimate forces. Cross-section

Eccentricity

FEXP/FABQ

mm

mm

-

80  3 80  3 80  5 80  5

20 40 20 40

0.911 0.988 1.005 0.967

Average Standard deviation

0.968 0.035

As could be seen, the results obtained from the numerical model calculations are in good agreement with the experimental data. There are some differences in the ultimate loading forces, however, it is less than 10% in all cases. Therefore, the numerical model was considered as accurate enough suitable for subsequent numerical parametric study. 3.2 Numerical parametric study The comprehensive numerical parametric study was made using the numerical model described above. In the numerical parametric study, the global imperfection amplitude was considered as L/1000 (where L is the member length) and local imperfection amplitude was calculated considering Dawson & Walker (1972) with Gardner and Nethercot´s (2004) modification for stainless steel. The main three stainless steel groups common in structures were considered, namely austenitic, ferritic and duplex. Young modulus E0 = 200 000 MPa was considered for all materials. One grade of each stainless steel group was considered, however, with two values of strain hardening factor n (representing variation among the grades and degree of cold-forming). Regarding the appropriate choice of the stainless steel grades, materials with low yield strength fy and low ultimate strength fu (ferritic grade), high fy and high fu (duplex grade) and the greatest ratio between fy and fu (austenitic grade) were considered. Material properties are summarized in Table 4. Both SHS and RHS cross-sections were investigated. Dimensions were considered as 80 mm for SHS and 100  40 mm for RHS crosssections (representing the highest h/b ratio common for sections). Thickness of the wall was 1337

set as a variable parameter and calculated regarding the considered material in order to cover cross-section Classes 1 and 4 of SHS and cross-section Classes 1, 3 and 4 of RHS crosssections. Moment to axial force loading ratio was established with the idea of reaching some levels of nb = NEd =Nb;Rd;y for the whole slenderness range. Section bending resistance for the estimation of the loading was calculated according to Zhao et al. (2016) proposal using CSM. The investigated nb ratios and member slendernesses are shown in Table 5. 3.3 Comparison of Zhao’s procedure In this chapter, the design approach developed by Zhao et al. (2016) and described above is compared with the results of the numerical parametric study. The comparison was done using flexural buckling resistance according to Afshan et al. (2017) and bending resistance proposed by Zhao et al. (2017) as for the planning of the study. The evaluation is made in terms of the product of the whole interaction formula, where the results greater than unity indicates safe predictions whereas lower than unity unsafe ones. Figures 5 and 6 provide comparison of the approach as dependent on the member slenderness and nb ratio, respectively. As could be seen in Figures 5 and 6, the approach developed by Zhao (2015) provides generally good prediction with some scatter and conservativeness for Class 4 result and slight unconservativeness in Class 1 cross-section cases. Very limited number of results for Classes 3 and 4, namely with increasing influence of the bending moment became little un-conservative for austenitic and ferritic stainless steel grades. On the other hand, the procedure gives safe results for duplex stainless steel. It should be noted that Zhao et al. (2016) procedure was developed based on resistances calculated according to Afshan et al. (2017) and Afshan & Gardner (2013). Therefore a scatter in the comparison, could be in some cases result of the scatter in section or column resistance prediction. 3.4 New proposal A new procedure for the stainless steel SHS/RHS beam-column design described in this chapter was developed based on the numerical parametric study. Both flexural buckling resistance Table 4. Material properties considered in the parametric study. E0

fy

fu

n

Material

MPa

MPa

MPa

-

Austenitic Austenitic Duplex Duplex Ferritic Ferritic

200,000 200,000 200,000 200,000 200,000 200,000

210 210 480 480 220 220

380 380 660 660 520 520

4.5 14 4.5 14 4.5 14

Table 5. Member slenderness and nb = NEd/Nb,Rd values investigated in the parametric study. SHS members

RHS members

Member slenderness

nb = NEd/Nb,Rd

Member slenderness

nb = NEd/Nb,Rd

0.2 0.3 1.0 1.5 2.0 3.0

0.05 0.3 0.5 0.7 0.8

0.5 0.8 1.0 1.5 2.0

0.05 0.1 0.2 0.5 0.8

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Figure 5.

Comparison for the Zhao et al. (2016) procedure in terms of nb = NEd/Nb,Rd ratio.

Figure 6.

Comparison for the Zhao et al. (2016) procedure in terms of the member slenderness.

and bending resistance were also obtained from the numerical model. The same interaction formula as given by EN 1993-1-4 (2006) (Equation 1) was considered, but with different formulae for the interaction factor itself. It is given by Equations 4 and 5. ky;New ¼ 1:0 þ 1:5λy nβb for λ
y  1:0 0:8 ffi ky;New ¼ 1:0 þ 1:5λy nβb p
ffiffiffiffiffiffiffiffiffiffiffi

λy 0:36

 2 el β ¼ MMb;Rd

for λ
y > 1:0

ð4Þ ð5Þ ð6Þ

where Mel = the elastic bending moment resistance. The comparison of the New proposal is made through the whole interaction formula calculation again and given by Figures 7 and 8. Furthermore, comprehensive statistical evaluation 1339

Figure 7.

Comparison for the New proposal in terms of nb = NEd/Nb,Rd ratio.

Figure 8.

Comparison for the New proposal in terms of the member slenderness.

according to Afshan et al. (2015), using resistances calculated according to EN 1993-1-4 (2006) considering both nominal and mean values of input parameters, including also a proposal for the safety factor is provided by Table 6. As could be seen in Figures 7 and 8, the New proposal predictions are safe and consistent for both SHS and RHS stainless steel beam-columns. It is a general procedure for all stainless steel groups with no dependency on member slenderness, cross-section slenderness (Class of the cross-section) and loading case (compressive force to flexural buckling resistance nb ratio). The accuracy is confirmed by statistical evaluation given by Table 5. The results exhibit low scatter with slightly conservative results in the average. However, the average value conservativeness is caused by the complex statistical evaluation according to Afshan et al. (2015) condition where the partial safety factor γM should be lower than partial safety factor γM1 value given by stainless steel code EN 1993-1-4 (2006) which is recommended as 1.1. Based both on the graphical comparison and statistical evaluation, the New proposal is very general and accurate procedure for both SHS and RHS stainless steel beam-column design. 1340

Table 6. Statistical evaluation of the New proposal. Average value Standard deviation γM

1.096 0.069 1.086

4 CONCLUSIONS Presented paper is focused on behaviour of SHS and RHS stainless steel members loaded by compression and bending moment combination. There are several approaches for stainless steel beam-column design with a proposal of Zhao et al. (2016) describing the behaviour quite accurately. However, a new proposal was developed and presented in this paper. A 3D shell element model using GMNIA was created in finite element software Abaqus simulating the stainless steel beam-column behaviour. Validation of the numerical model was successfully made based on the data obtained from experiments. A comprehensive numerical parametric study was made subsequently. The recent procedure developed by Zhao et al. (2016) was evaluated first in detail. It was shown that the approach exhibits slightly scattered results with slight un-conservativeness for Class 1 cross-section cases. A new proposal was presented and evaluated. The procedure provides accurate and consistent predictions with similar safety and scatter for various member and cross-section slenderness, material properties and compressive force to flexural buckling ratio. Furthermore, statistical evaluation was presented and confirmed the accuracy of the new proposal. ACKNOWLEDGEMENT The support of the Czech Science Foundation grant 17-247695 “Nonlinear stability and strength of slender structures with nonlinear material properties” is gratefully acknowledged. REFERENCES Afshan, S. & Gardner, L. 2013. The continuous strength method for structural stainless steel design. Thin-Walled Structures. 68: 42–49. Afshan, S., Francis, O., Baddoo, N. & Gardner, L. 2015. Reliability analysis of structural stainless steel design provision. Journal of Constructional Steel research. 114: 293–304. Afshan, S., Zhao, O. & Gardner, L. 2017. Buckling curves for stainless steel tubular columns. Eurosteel 2017. Copenhagen, Denmark. Dawson, R.G. & Walker, A.C. 1972. Post-buckling of geometrically imperfect plates. Journal of the Structural Division ASCE. 98: 75–94. EN 1993- 1-4. 2006. Eurocode 3: Design of steel structures – Part 1-4: General rules – Supplementary rules for stainless steels. European Committee for Standardization (CEN). Brussels. Gardner, L. & Nethercot, D. A. 2004. Numerical Modeling of Stainless Steel Structural Components-A Consistent Approach. Journal of Structural Engineering, ASCE. 130: 1586–1601. Jandera, M., Syamsuddin, D. & Židlický, B. 2017. Stainless steel beam-column behaviour. Open Civil Engineering Journal. 11: 358–368. Zhao, O., Afshan. S. & Gardner, L. 2017. Structural response and continuous strength method design of slender stainless steel cross-sections. Engineering Structures. 140: 14–25. Zhao, O., Gardner, L., & Young, B. 2016. Behaviour and design of stainless steel SHS and RHS beam-columns. Thin-Walled Structures. 106: 330–345.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Calibration of parameters of combined hardening model using tensile tests C.I. Zub, A. Stratan & D. Dubina Department of Steel Structures and Structural Mechanics, Politehnica University of Timisoara, Timisoara, Romania

ABSTRACT: When performing finite element simulations on structural elements made of mild carbon steel proper modelling of the cyclic response of material allow for numerical simulations with a high level of reliability. Good predictions can be obtained using the combined isotropic-kinematic hardening model to simulate the plastic behaviour of steel. As recommended by Abaqus, the finite element software used within these simulations, the calibration of the material parameters of the combined model requires cyclic test data. In comparison to tensile tests, the cyclic tests are more difficult to be performed. Therefore, a simplified calibration procedure of the material parameters using only tensile tests is presented in this paper. The material parameters obtained in this way allow for predictions with an acceptable level of correlation, for both monotonic and cyclic loading. These predictions are validated against experimental tests performed on S355 mild carbon steels.

1 INTRODUCTION When performing finite element simulations on structural elements made of mild carbon steel proper modelling of the response of the material is necessary to obtain reliable simulations. While for the simulations involving monotonic loading simple material models can be used, in the case of the simulations under cyclic loading (e.g. low-cycle fatigue simulations) more complex material models are necessary. This will allow to properly capture the behaviour of structural components (e.g. buckling restrained braces) undergoing large plastic deformations (Zub et al. 2018). It was shown by Lemaitre & Chaboche (1990) that proper modelling of metal plasticity under cyclic loading can be achieved using the combined isotropic-kinematic hardening material model. The combined model, consisting of a kinematic (main) and isotropic (optional) component, was implemented in the Abaqus finite element package (Dassault, 2014). Also, calibration procedures were proposed to obtain the input parameters for both hardening components. As described in Dassault (2014), the implemented combined model has several limitations which relate to its capability of simulating the metal plasticity under arbitrary loading history with a high level of accuracy. However, Abaqus allows for advanced users to define their own material models via UMAT/VUMAT subroutine. The combined model was used by many researchers in their numerical studies that involved metal components. Among others, Nip et al. (2010) used the built-in combined model from Abaqus and the calibration procedure described in Dassault (2014) to perform numerical simulations on structural components made of carbon and stainless steel under cyclic loading. Zub et al. (2018) used the combined model to simulate the cyclic response of buckling restrained braces (BRB) under quasi-static loading regime (low-cycle fatigue simulations). The combined model showed good correlation with the experimental results, although the BRBs developed large plastic deformations. On the other side, there are some researchers who developed their own material models (UMAT for Abaqus Standard and VUMAT for Abaqus Explicit) in order to obtain more accurate numerical results on metal plasticity. Among 1342

others, Ucak and Tsopelas (2011) developed a material model (UMAT) to simulate the cyclic behaviour of structural mild carbon steels (with yield plateau). Shi et al. (2012) developed a UMAT for high-strength steel. Also, Hu et al. (2016) developed a UMAT, which will be further discussed. In all cases additional software (Fortran) and programming skills are required in addition to a good understanding of the theory of plasticity. According to the current implementation of the combined model in Abaqus, for the calibration of the parameters of the model experimental uniaxial cyclic tests on coupon specimens are required. In comparison to the more common monotonic tensile tests, the cyclic tests are more difficult to perform, as buckling of the specimen should be prevented. As an alternative to performing cyclic tests, Hu et al. (2016) developed a user material subroutine (UMAT) for the implicit integration procedure in Abaqus/Standard that requires only tensile test data for the calibration. The objective of this paper is to provide a calibration procedure for the parameters of the combined hardening model, available in both Abaqus/Standard and Abaqus/Explicit packages, that uses only tensile test data. The approach is based on the procedure developed by Hu et al. (2016), adjusted to the requirements of the built-in models in Abaqus. The material parameters calibrated with the proposed procedure can be used for both monotonic and cyclic loading regimes.

2 PROPOSED CALIBRATION PROCEDURE Considering a monotonic tensile test as schematically presented in Figure 1 a), the proposed calibration procedure uses as input the following mechanical properties, expressed as engineering values: yield strength fy, ultimate strength fu, rupture strength fr, yield strain ey, end of plateau strain esh, ultimate strain eu, rupture strain er. For simplification reasons and based on experimental observations, several assumptions were made: the rupture strength is fr = 0.8fu, the yield strain is ey = fy/Es (Es = 210000 N/mm2 is the elastic modulus of steel), the strain corresponding to ultimate strength is eu = 0.55er, the Poisson’s ratio is ν = 0.3. As Abaqus requires the input parameters of the material model to be expressed as true (Cauchy) stress σ and logarithmic strain ε, therefore, the engineering stress ( f ) and strain (e) values must be transformed accordingly using the following formulas (Dassault 2014): σ ¼ f ð1 þ eÞ

ð1Þ

ε ¼ lnð1 þ eÞ

ð2Þ

In the procedure proposed by Hu et al. (2016) several assumptions are made, and additional simulations and calibrations of specific parameters are required. In this paper, all the

Figure 1. Schematic representation of the response of steel under monotonic tensile loading a) engineering stress-strain curve and b) true stress-strain curve.

1343

parameters describing the plastic behaviour of steel are determined based on the input data, therefore no additional simulations or empirical assumptions are required. The first assumption is the weighted average factor w (Figure 1 b), which in the procedure of Hu et al. (2016) is determined after several iterations (simulations). Within the proposed procedure, the w factor is obtained directly as a function of rupture strain er. Although equation (3) was obtained by train and error, it shows to be consistent for the study cases presented in section 3 of this paper. w¼1

0:1 er

ð3Þ

σj0 ¼ σy C1 ¼

ð4Þ

σu  fy ð1 þ esh Þ γ1  3 2

γ1 ¼ 10  w  γ2 C2 ¼

ð6Þ

σu  fy ð1 þ esh Þ γ2   1:8 3 2

σu  fy ð1 þ esh Þ 1  5ðεu  εsh Þ 2  σu  fy ð1 þ esh Þ σu C3 ¼  C4 εu   εsh  γ2  1:66 E 2 γ2 ¼

γ3 ¼ C4 ¼

ð5Þ

ð7Þ ð8Þ ð9Þ

esh ey

ð10Þ

w  σu  1:2 u 1  wσ E

ð11Þ

γ4 ¼ 0:075  γ3

ð12Þ

where: σj0 is the true stress at zero plastic strain, Ck are the kinematic hardening moduli and γk are their corresponding decreasing rate with respect to increasing plastic deformation. The previously obtained kinematic parameters must be introduced in Abaqus using the “Parameters” option of the combined hardening model:

 KIN ¼ σj0 ; C1 ; γ1 ; C2 ; γ2 ; C3; γ3 ; C4 ; γ4

ð13Þ

The evolution of the isotropic component is more difficult to be determined due to: (1) specific features (yield plateau, Bauschinger effect, cyclic hardening) of the mild carbon steel under monotonic and cyclic loading (Zub et al. in press.); (2) limitations of the Abaqus built-in cyclic (isotropic) hardening component, as the loading history dependence (Dassault 2014). To capture the main features of the cyclic response of mild carbon steel (yield plateau, Bauschinger effect, cyclic hardening), it is convenient defining the isotropic component using tabular data type (Zub et al. in press.). The evolution of the size of the yield surface (σ0 ) over the entire loading history can be specified directly by providing data pairs (σ0j ; εjpl ). The calibration of the input data pairs for the isotropic component can be performed considering a fictional tensile test with a large plastic strain range (e.g. Δεpl = 2.0). The equivalent stresses, σ0j , corresponding to the equivalent plastic strains, εjpl , can be obtained by using an incremental procedure ( j = current increment):

1344

σ0j ¼ σ0j1 ;

if εjpl ¼ 0

σ0j ¼ σ0j1 þ ðRj  Rj1 Þ;

if εjpl 40

ð14Þ

where: Rj is the amount of isotropic hardening at increment j, for j ¼ 1; σ00 ¼ σj0 . The variation of Rj with respect to the equivalent plastic strain εjpl is obtained using the superposition of several isotropic hardening rules (as in the case of the kinematic component) to properly capture the behaviour of the mild carbon steel under monotonic and cyclic loading: Rj ¼ R1j þ R2j þ R3j þ R4j þ R5j

ð15Þ

where: R1j – is the very-short-range nonlinear isotropic softening parameter in the plateau region, activates immediately after yielding. R2j – is the short-range nonlinear isotropic softening parameter in the plateau region, activates on the entire plateau region. R3j – is the short-range nonlinear isotropic hardening parameter in hardening region, activates up to εu . R4j – is the additional short-range or long-range (depending on the case of fitting) nonlinear isotropic hardening parameter. R5j – is the long-range linear isotropic hardening parameter. This hardening rule is used to provide the combined hardening model with isotropic hardening even at large values of equivalent (cumulative) plastic strain (e.g. ε pl ¼ 2:0). For the isotropic softening rules (Rjk ¼ 1;2 ), the following formulas (Lemaitre & Chaboche 1990) are used to evaluate the amount of reduction of the yield surface at each increment j:  h  i pl ; Rkj ¼ Qk þ Rkj1  Qk  exp bk εjpl  εj1

if εjpl  εsh if εjpl 4εsh

¼ Qk ½1  exp ð bk  εsh Þ ;

ð16Þ

(for j ¼ 1; Rkj ¼ 0) For the nonlinear isotropic hardening rules (Rkj ¼ 3;4 ), the following formulas (Lemaitre & Chaboche 1990) are used to evaluate the amount of increase of the yield surface at each increment j: Rkj ¼ 0;

 h  i pl ; ¼ Qk þ Rkj1  Qk  exp bk εjpl  εj1

if εjpl  εsh if εjpl 4sh

ð17Þ

(for j ¼ 1; Rkj ¼ 0) For the linear isotropic hardening rule (Rkj ¼ 5 ), the following formula is used to evaluate the amount of increase of the yield surface at each increment j: if εjpl  εsh

Rkj ¼ 0;

if εjpl 4εsh

¼ Q0  εjpl ;

ð18Þ

(for j ¼ 1; Rkj ¼ 0) The parameters used by the hardening/softening isotropic laws are defined below. Some of the parameters (Qk and bw, with k = 1.3) have the same meaning as the one from Hu et al. (2016) but computed using other formulas which were determined by trial and error and 1345

showed proper fitting to the experimental study cases. Other parameters (Qk and bw, with k = 0.4) were also considered for closer fitting with the experimental results, and the proposed formulas were determined by trial and error.   Q3 εsh  εy Q ¼ 3  w  εsh

ð19Þ

b1 ¼ 0:5  b2

ð20Þ

1

Q2 ¼  0:7  Q1

ð21Þ

b2 ¼

Q1 1  εu  εsh 2:8

ð22Þ

Q3 ¼

σu  fy ð1 þ esh Þ 2:2

ð23Þ

b3 ¼

b2  εsh  1:5 εu þ εsh

ð24Þ

Q4 ¼ additional ðuser inputÞ   b4 ¼ additional user input or ¼ 2b1 Q0 ¼ Q1

2 3

ð25Þ ð26Þ ð27Þ

where Qk , bk are material parameters and represents: Q1 and b1 – are the maximum decrease in size of the yield surface and its corresponding rate, respectively, at very small plastic strains ð εsh ÞÞ. Q2 and b2 – are the maximum decrease in size of the yield surface and its corresponding rate, respectively, at plastic strains smaller than εsh . Q3 and b3 – are the maximum increase in the size of the yield surface and its corresponding rate, respectively, at plastic strains larger than εsh but smaller than εu (the range of equivalent plastic strain εpl is not strictly defined). Since the isotropic component is loading history dependent, two values can be used for b3 depending on the of  type  loading (calibration): for 3 2 monotonic loading, the authors recommend that b ¼ b  ε =ðεu þ εsh Þ  1:5, while for sh   cyclic loading b3 ¼ b2  εsh =ðεu þ εsh Þ  1:0. Q4 and b4 – have similar meanings as Q3 and b3 and they are additionally used in cases where using only Q3 and b3 the calibration is not properly achieved over the range εsh 5εpl  εu . Q0 – is the slope of the equivalent stresses σ0i on the hardening region. 3 APPLICATION The above-presented calibration procedure was validated against experimental tests performed for characterization of cyclic response of steel within the frame of research projects IMSER (Zub et al. in press.) and EQUALJOINTS (Both et al. 2017). Mild carbon steels with yield plateau (steel grade S355) were used for assessing the capability of the proposed analytical formulas in providing input parameters for the combined hardening material model that yields reliable numerical predictions. The mechanical properties obtained from the tensile tests are summarized in Table 1. Figure 2 presents the influence of considering or not the isotropic component of the combined hardening material model in predicting the monotonic and cyclic behaviour of 1346

Table 1. Mechanical properties from tensile tests of steels used for the validation of the proposed calibration procedure.

fv , N/mm2 fu , N/mm2 fr , N/mm2 ev , mm/mm esh , mm/mm eu , mm/mm er , mm/mm

test-1-S355

test-2-S355

test-3-S355

test-4-S355*

test-5-S355*

363 525 420 0.0017 0.0140 0.1650 0.3000

345 522 417 0.0016 0.0170 0.1746 0.3880

349 534 374 0.0017 0.0150 0.1655 0.3246

398 509 331 0.0019 0.0180 0.1634 0.3230

398 509 331 0.0019 0.0180 0.1634 0.3230

*

test-4-S355 and test-5-S355 characterizes the same material. Since two coupon specimens were cyclically tested, therefore, individual IDs were assigned to each specimen/cyclic test.

Figure 2. FEM predictions using parameters calibrated with the proposed procedure: a) kinematic parameters only, b) kinematic and isotropic parameters.

mild carbon steel with yield plateau. From Figure 2 a) it can be observed that using only the kinematic component the material model cannot predict the monotonic or the cyclic behaviour of steel that was experimentally obtained, test-1-S355 (Zub et al. in press.). For the monotonic case, it can only accurately predict the yield and the ultimate strength, while for the cyclic loading the model underestimates the stress level and the capacity of energy dissipation. 1347

Figure 3. FEM predictions using the combined hardening material model with parameters calibrated using the proposed procedure.

1348

Using both isotropic and kinematic components of the combined model acceptable predictions can be obtained for both monotonic and cyclic cases (Figure 2 b). For the monotonic tensile simulation, the prediction is accurate up to the ultimate strength. Beyond this point, the model cannot accurately predict the failure (necking region) due to the fact that the calibration of the input parameters was performed to properly predict the cyclic response. In the case of the cyclic test, close predictions are obtained at all strain ranges. If the monotonic case is the target for the calibration, then the describing the isotropic component   parameters  should be modified, as follows: b3 ¼ b2  εsh =ðεu þ εsh Þ  1:5 and Q0 ¼ Q1  x=3, where x can take values from x ¼ 0:002  0:2. In Figure 3 are presented the stress-strain predictions for the other tensile and cyclic tests which were obtained within the frame of EQUALJOINT project (Both et al. 2017). Considering the same observations as in the case of the test-1-S355, in all cases an acceptable level of correlation is observed with respect to the experimental results. Based on these results it can be concluded that the simplified calibration procedure proposed in this paper can yield numerical predictions with an acceptable level of accuracy for both tensile and cyclic loading. The great advantage of this calibration procedure is that it uses only tensile tests results for the calibration of input parameters which can be used for both monotonic and cyclic loading histories. To allow for developing of a database with input parameters for different steel grades (to be used for FEM simulations), the procedure should be also validated against different European steel grades which exhibits yield plateau (S235, S275, S460) or not (S690).

4 CONCLUSIONS A procedure for the calibration of the parameters of the combined isotropic-kinematic hardening model is proposed in this paper. The procedure requires only uniaxial tensile test data and the parameters obtained can be used for both monotonic and cyclic simulations. The combined model is implemented in both Abaqus Standard and Abaqus Explicit finite element packages, thus both static and dynamic analyses can be performed using the calibrated parameters. The combined material model can be used to simulate the structural components (i.e. buckling restrained braces BRBs) under low cycle loading regime, which usually involves the development of large plastic deformations. The performance of the proposed calibration procedure is validated against experimental results obtained on the structural mild carbon steel S355 with yield plateau. Both monotonic and variable cyclic uniaxial tests were performed and predictions with an acceptable level of correlation were obtained. Proper calibration of the parameters defining the kinematic component of the combined model should provide simulations with similar yield and ultimate strength as in the experimental uniaxial monotonic tensile test. For close fitting, the calibration of the isotropic component should be performed considering the type of loading (monotonic or cyclic) that is applied to the structural element (i.e. BRBs cyclically loaded) which is simulated using the combined model. This is due to the fact that the current implementation of the combined model is loading history dependent. The authors propose different formulas for some isotropic parameters (b3 and Q0 ) depending on the type of loading (monotonic or cyclic). Further investigations are required to validate the proposed calibration procedure for other types of European structural mild carbon steels such as S235, S275 and S460, but also for high strength steels S690. ACKNOWLEDGEMENTS The research leading to these results has received funding from the MEN-UEFISCDI grant Partnerships in priority areas PN II, contract no. 99/2004 IMSER: “Implementation into Romanian seismic resistant design practice of buckling restrained braces.” 1349

Partial funding was received from the European Community’s Research Fund for Coal and Steel (RFCS) under grant agreement no RFSR-CT-2013-00021 “European pre-qualified steel joints (EQUALJOINTS)”. This support is gratefully acknowledged. REFERENCES Both, I., Zub, C., Stratan, A. & Dubina, D. 2017. Cyclic behaviour of European carbon steels. CE/ Papers, Ernst& Sohn/ Wiley, Vol.1, Issue 2-3, 3173–3180. Dassault 2014. Abaqus 6.14 - Abaqus Analysis User’s Manual, Dassault Systèmes Simulia Corp. Hu, F., Shi, G. & Shi, Y. 2016. Constitutive model for full-range elasto-plastic behavior of structural steels with yield plateau: Calibration and validation, Eng. Structures, 118, 210–227. Lemaitre, J. & Chaboche, J.L. 1990. Mechanics of Solid Materials, UK: Cambridge University Press. Nip, K.H., Gardner, L., Davies, C.M. & Elghazouli, A.Y. 2010a. Extremely low cycle fatigue tests on structural carbon steel and stainless steel. J Construct Steel Res, 66(1), 96–110. Nip, K.H., Gardner, L. & Elghazouli, A.Y. 2010b. Cyclic testing and numerical modelling of carbon steel and stainless steel tubular bracing members. Eng Struct, 32(2), 424–441. Shi, G., Wang, M., Bai, Y., Wang, F., Shi, Y., & Wang, Y. 2012. Experimental and modeling study of high-strength structural steel under cyclic loading. Engineering Structures, 37, 1–13. Ucak, A. & Tsopelas, P. 2011. Constitutive model for cyclic response of structural steels with yield plateau. J Struct Eng, 137(2), 195–206. Zub, C.I., Stratan, A. & Dubina, D. (in press). Modelling the cyclic response of structural steel for FEM analyses. Proc. 1st Int. Conf. on Computational Methods and Applications in Engineering, Timisoara, Romania, May (acc. for publication). Zub, C.I., Stratan, A., Dogariu, A. & Dubina, D. 2018. Development of a finite element model for a buckling restrained brace, Proc. of the Romanian Academy, series A, no. 19(4), 581–588.

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Stability and Ductility of Steel Structures 2019 – Wald & Jandera (Eds) © 2019 Czech Technical University in Prague, Czech Republic, ISBN 978-0-367-33503-8

Author Index

Afshan, S. 911 Ahmed, H.S.S. 80 Alhasawi, A. 269 Amamoto, T. 1292 Amoush, E.A. 363 Aoyama, M. 963 Arha, T. 88 Armijos-Moya, S.V. 96 Arrais, F. 106, 115 Arrayago, I. 124 Azizi, E. 1077 Ádány, S. 71, 468, 499 Baddoo, N. 409 Bajer, M. 1236 Balázs, I. 133 Barcellos, A.B.G. 139 Barros, R.C. 147 Basaglia, C. 1015 Bastos, C.C.D.O. 155 Batista, E.M. 155, 164, 682 Becque, J. 173 Ben Larbi, A. 269 Beyer, A. 181 Bhatt, G. 337 Bhattacharyya, S. Kr. 1042 Bâlc, R.M. 1167 Both, I. 1185, 1193 Bours, A.-L. 171 Budaházy, V. 197, 205 Burca, M. 1185, 1193 Bureau, A. 181 Buru, S.M. 253 Calado, L. 329, 476 Camotim, D. 345, 434, 954, 1015 Campiche, A. 213, 221, 997 Casanova, E. 708 Castiglioni, C.A. 311, 484 Cábová, K. 88, 1218, 1227 Ceh, O. 508 Chacón, R. 235, 483 Chen, X.W. 244

Chiorean, C.G. 253 Ciutina, A. 863, 1251 Clayton, P. 96 Cordeiro, M. 1034 Corman, A. 564 Costanzo, S. 262 Couchaux, M. 269, 329, 981 Couto, C. 106, 277, 744 Craveiro, H.D. 286, 295, 303 Cristian, A.A. 1185 Červenka, P. 229 D’Aniello, M. 262 D’Aniello, M. 1128, 1136 Das, R. 312 Degée, H. 321, 329, 476, 312 Demonceau, J.-F. 564 Dewangan, A. 337, 1042 Díez, R. 515 Dinis, P.B. 345 Dobrić, J. 409, 427 Dolejš, J. 229, 1269 Don, R. 1251 Doynov, N. 1059 Du, X.X. 244 Dubina, D. 629, 753, 1086, 1185, 1193, 1342 Duchêne, Y. 321 Dunai, L. 197, 539 Ecker, A. 354 El Aghoury, I.M. 371, 929 El Aghoury, M.A. 363, 371 El Hady, A.M. 363, 371 Eliasova, M. 379 Elliott, M.D. 388 El-Mahdy, G.M. 394 El-Serwi, A.I. 944 Engelhardt, M. 96 Égető, C. 539

1351

Feber, N. 402 Ferdynus, M. 622 Fieber, A. 3 Filipović, A. 409 Fiorino, L. 221 Fominow, S. 1095 Forejtova, L. 402 Franco, J.M.S. 164 Fric, N. 409 Garcea, G. 417, 735 Gardner, L. 3 Gervásio, H. 699 Ghosh, S. 80 Gluhović, N. 427 Gonçalves, P.B. 139 Gonçalves, R. 434 González, A. 443 González de León, I. 124 Graciano, C. 708 Gremza, G. 1300, 1309 Guarracino, F. 452, 460 Gurneian, H. 937 Haffar, M.Z. 71, 468 Hajdú, G. 1111 Harringer, T. 596 Heinisuo, M. 1145 Helwig, T. 96 Henriques, J. 286, 295, 303, 476, 1034 Herrera, J. 483 Hisazumi, K. 491 Hjiaj, M. 981 Hoang, T. 499 Hoffmeister, B. 321, 329 Hofmeyer, H. 1025 Hołowaty, J. 1275 Horáček, M. 508 Ibañez, J.R. 515 Ibrahim, S.M. 363, 929 Idota, H. 638 Igawa, N. 531

Ikarashi, K. 523, 531, 761, 1005 Inden, K. 963 Ishida, W. 531 Jandera, M. 115, 402, 587, 1051, 1218, 1227, 1334 Jaspart, J.-P. 564 Jáger, B. 539 Jäger-Cañás, A. 548, 556, 882 Jiménez, A. 570 Joó, A.L. 691 Jörg, F. 578, 972 Jůza, J. 587 Kabeláč, J. 664, 1243 Kachichian, M. 539 Kamada, M. 638 Kamocka, M. 614 Kanno, R. 491 Kanyilmaz, A. 312, 329 Kazmierczyk, F. 655 Kettler, M. 596 Knobloch, M. 189, 972, 1283 Kołakowski, Z. 614 Kobashi, T. 605 Kolakowski, Z. 655 Kolarik, L. 402 Kollár, D. 205 Koltsakis, E. 946 Kotełko, M. 622, 629 Koyama, Y. 638 Kraus, M.A. 1176 Kroyer, R. 646 Kästner, T. 1103 Kuś, J. 673 Kubiak, T. 614, 655 Kuhlmann, U. 578, 921, 972 Kumar, S. 1042 Kuříková, M. 664, 1243 Lagerqvist, O. 946 Laím, L. 286 Landesmann, A. 954 Landolfo, R. 221, 262, 997, 1128, 1136 Launert, B. 1059 Leal, A.S. 682 Lemes, Í.J.M. 147 Lendvai, A. 691 Leonetti, L. 417, 735 Li, Guoqiang 1260

Li, Hai-Ting 16 Li, Z. 556 Liguori, F. 735 Liguori, F.S. 417 Lišková, N. 88 Lima, L. 1034 Linguiti, A. 1136 Ljubinković, F. 699 Loaiza, N. 708 Lopes, N. 106, 115, 744 Lorenzo, G. Di 262 Lorenzo, G.M. Di 1128 López, C. 515 Lukačević, I. 1193 Lyu, Y. 1260

Pichal, R. 906 Pires, D. 147 Pournaghshband, A. 911 Pourostad, V. 921 Pravdova, I. 379

Ma, T. 717, 727 Machacek, J. 906 Macorini, L. 3 Madeo, A. 417 Magisano, D. 417, 735 Maia, É. 744 Manoleas, P. 946 Marçalo Neves, R. 434 Marković, Z. 409, 427 Martino, A. De 262 Martins, J.P. 28, 295, 303, 699, 1120 Martínez, X. 235 Matsubara, G.Y. 164 Mela, K. 1145 Melcher, J. 133, 508 Michailov, V.G. 1059 Milosavljević, B. 427 Mirambell, E. 124, 570 Mitsui, K. 963 Mittal, A.K. 1042 Müller, A. 1151 Morelli, F. 329 Movahedi, R.M. 1210

Okoń, K. 622 Oller, S. 235 Ono, T. 963

Sabau, G. 946 Santana, M.V.B. 139 Santiago, A. 286 Santos, W.S. 954 Sato, A. 638, 872, 963 Sato, Y. 638 Schaper, L. 972 Schäfer, M. 1326 Seco, L. 981 Selariu, M. 253 Serna, M.A. 443, 515 Shakeel, S. 221, 989, 997 Shimizu, N. 605 Shinohara, D. 1005 Sierra, P. 235 Silva, L.S. 303, 699 Silva, da, L.S. 295 Silva, da, T.G. 1015 Silveira, R.A.M. 147 Simões da Silva, L. 28, 1120 Slein, R. 42 Snijder, H.H. 1025 Sobrinho, K. 1034 Sonkar, C. 337, 1042 Šorf, M. 1051 Spremić, M. 409, 427 Stapelfeld, C. 1059 Stehr, S. 1068 Stranghöner, N. 1068, 1077 Stratan, A. 1086, 1342 Stroetmann, R. 1095, 1103 Szalai, J. 1111, 1210

Papp, F. 1111 Pasternak, H. 556, 882, 1059 Phan, D.K 898

Taher, M.H. 468 Takaki, S. 638 Tanaka, Y. 1292 Tankova, T. 28, 1120

Neagu, C. 1185 Neves, L.C. 981 Niko, I. 1151 Nunes, D.L. 863

1352

Ramzi, A.A. 944 Rangelov, N. 1201 Rasmussen, K.J.R. 898 Real, E. 124, 244, 570 Real, P.Vila 106, 115, 277, 744 Rocha, P.A.S. 147 Rodriguez, A. 235 Rosson, B. 937

Taras, A. 646, 787, 839, 1151, 1159, 1176 Tartaglia, R. 1128, 1136 Teeuwen, P.A. 1025 Teh, L.H. 388 Tenchini, A. 1034 Theofanous, M. 911 Tiainen, T. 1145 Titulaer, L.H.J.D. 1025 Toğay, O. 42 Toffolon, A. 1151, 1159 Tomăscu, I.C. 1167 Topkaya, C. 1317 Treib, C. 1176 Ungureanu, V. 629, 1185, 1193 Unterweger, H. 354, 596

Vallelado, L. 443 Vassilev, M. 1201 Vaszilievits-Sömjén, B. 1210 Vellasco, P. 1034 Vesecký, J. 1218, 1227 Vigh, L.G. 205 Vila Real, P. 106, 744 Vild, M. 1236, 1243 Villalon-Camacho, T. 937 Vulcu, C. 1251 Wald, F. 664, 1136, 1243 Wald, F. 88 Wang, Y. 55, 96 Wang, Y. 1260 Wang, Y. 1260 Werunský, M. 1269 White, D.W. 42 Wichtowski, W. 1275

1353

Williamson, E. 96 Winkler, R. 189, 972, 1283 Xu, L. 717, 727 Yagi, S. 638 Yamaguchi, E. 1292 Young, B. 16 Yuan, H.X. 244 Zamorowski, J. 1300, 1309 Zeybek, Ö. 1317 Zhang, Q. 1326 Ziemian, R. 937 Ziemian, R.D. 55 Zimbru, M. 1136 Židlický, B. 1334 Zub, C.I. 1342