Practical Design of Ships and Other Floating Structures: Proceedings of the 14th International Symposium, PRADS 2019, September 22-26, 2019, Yokohama, Japan- Volume II [1st ed.] 9789811546716, 9789811546723

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Lecture Notes in Civil Engineering

Tetsuo Okada Katsuyuki Suzuki Yasumi Kawamura   Editors

Practical Design of Ships and Other Floating Structures Proceedings of the 14th International Symposium, PRADS 2019, September 22–26, 2019, Yokohama, Japan- Volume II

Lecture Notes in Civil Engineering Volume 64

Series Editors Marco di Prisco, Politecnico di Milano, Milano, Italy Sheng-Hong Chen, School of Water Resources and Hydropower Engineering, Wuhan University, Wuhan, China Ioannis Vayas, Institute of Steel Structures, National Technical University of Athens, Athens, Greece Sanjay Kumar Shukla, School of Engineering, Edith Cowan University, Joondalup, WA, Australia Anuj Sharma, Iowa State University, Ames, IA, USA Nagesh Kumar, Department of Civil Engineering, Indian Institute of Science Bangalore, Bengaluru, Karnataka, India Chien Ming Wang, School of Civil Engineering, The University of Queensland, Brisbane, QLD, Australia

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Tetsuo Okada Katsuyuki Suzuki Yasumi Kawamura •

•

Editors

Practical Design of Ships and Other Floating Structures Proceedings of the 14th International Symposium, PRADS 2019, September 22–26, 2019, Yokohama, Japan- Volume II

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Editors Tetsuo Okada Faculty of Engineering Yokohama National University Yokohama, Kanagawa, Japan

Katsuyuki Suzuki School of Engineering The University of Tokyo Bunkyo-ku, Tokyo, Japan

Yasumi Kawamura Faculty of Engineering Yokohama National University Yokohama, Japan

ISSN 2366-2557 ISSN 2366-2565 (electronic) Lecture Notes in Civil Engineering ISBN 978-981-15-4671-6 ISBN 978-981-15-4672-3 (eBook) https://doi.org/10.1007/978-981-15-4672-3 © Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

The first PRADS (International Symposium on Practical Design of Ships and Other Floating Structures) was held in Tokyo in 1977 in celebration of the 80th anniversary of the Society of Naval Architects of Japan (later the Japan Society of Naval Architects and Ocean Engineers), where 245 delegates from 26 countries attended and 56 papers were presented. In the preface of the first PRADS proceedings, the chairman advocated two important principles; one is the emphasis on the practical application to ship design, and the other is the promotion of international mutual understanding through discussions and exchange of information on practical problems in shipbuilding. Since then, 42 years have passed, and our circumstances and environment experienced tremendous changes, for example, rapid development of information technology and computer science, expansion of offshore development, and more and more emphasis on green shipping and renewable energy. Accordingly, the topics covered by the subsequent PRADS symposia evolved, but we believe that the first two principles have been kept in mind throughout by all the participants like a basso continuo. Respecting the traditions of these past successful PRADS symposia, we are proud to organize the 14th PRADS Symposium in Yokohama. Yokohama, located just south of Tokyo, is the second largest city in Japan by population. The city has been Japan’s major gateway for international transportation and communication. Yokohama is also a center of excellence for shipbuilding and offshore engineering, with many leading universities, research institutes, and shipping companies. Yokohama is also a tourist’s destination with excellent accessibility from all over the world. More than 220 abstracts were accepted. The full papers were peer-reviewed by two reviewers for each paper, and about 170 papers were finally selected for presentation. The topics cover hydrodynamics, ship dynamics, structures, machinery and equipment, design and construction, navigation and logistics, and ocean engineering.

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Preface

On this opportunity, we would like to express our sincere appreciation to the main sponsor, ClassNK, and other sponsors, Mitsui O.S.K. Lines, Ltd., Mitsubishi Shipbuilding Co., Ltd., Namura Shipbuilding Co., Ltd., Japan Marine United Corporation, IHI Corporation, and Sumitomo Heavy Industries Marine Engineering Co., Ltd. In addition, our gratitude is extended to the worldwide reviewers of the full papers, without whose excellent voluntary works, our high-quality symposium was not even made possible. Special thanks are also extended to the local organizing committee members, who devoted themselves to manage everything. Last but not least, we would like to thank all the delegates for their participation and invite them to enjoy the symposium as well as this exciting city. Tetsuo Okada Katsuyuki Suzuki Yasumi Kawamura

Organization

Standing Committee Seizo Motora (Honorary Chairman) Yasumi Kawamura (Chairman of Standing Committee) Tetsuo Okada (Chairman of PRADS 2019) Alan J. Murphy Bas Buchner Enrico Rizzuto Ge (George) Wang Ilson P. Pasqualino Patrick Kaeding Quentin Derbanne Seung-Hee Lee Sverre Steen Ulrik D. Nielsen Xiaoming Cheng

Professor Emeritus, The University of Tokyo, Japan Yokohama National University, Japan

Yokohama National University, Japan

Newcastle University, UK MARIN (Maritime Research Institute Netherlands), the Netherlands University of Genoa, Italy Seastel Marine System (USA) LLC, USA Federal University of Rio de Janeiro, Brazil University of Rostock, Germany Bureau Veritas Marine & Offshore, France Inha University, Republic of Korea Norwegian University of Science and Technology, Norway Technical University of Denmark, Denmark China Ship Scientific Research Center, China

Local Organizing Committee Tetsuo Okada (Chairman) Katsuyuki Suzuki (Vice Chairman)

Yokohama National University, Japan The University of Tokyo, Japan

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Yasumi Kawamura (Vice Chairman) Daisuke Yanagihara Kazuhiro Aoyama Kazuhiro Iijima Kentaro Kobayashi Koji Gotoh Kunihiro Hamada Motohiko Murai Takanori Hino Tsutomu Fukui Yasuhira Yamada Yasuyuki Toda Yoshiaki Hirakawa Yukitaka Yasuzawa

Organization

Yokohama National University, Japan Kyushu University, Japan The University of Tokyo, Japan Osaka University, Japan Japan Society of Naval Architects and Ocean Engineers, Japan Kyushu University, Japan Hiroshima University, Japan Yokohama National University, Japan Yokohama National University, Japan ClassNK, Japan National Maritime Research Institute, Japan Osaka University, Japan Yokohama National University, Japan Kyushu University, Japan

Honorary Committee Shinjiro Mishima Hiroyuki Yamato Takanori Kunihiro Yukito Higaki Takashi Nakabe Yoshinori Mochida Takashi Ueda Kotaro Chiba Tetsushi Soga Hideshi Shimamoto Sachio Okumura Kensuke Namura Tetsuro Koga Koji Okura Yasuhiko Katoh Junichiro Ikeda Koichi Fujiwara Masahiko Fujikubo Shotaro Uto Tetsuo Okada Katsuyuki Suzuki

Japan Society of Naval Architects and Ocean Engineers, Japan National Institute of Maritime, Port and Aviation Technology, Japan IHI Corporation, Japan Imabari Shipbuilding Co., Ltd., Japan Onomichi Dockyard Co., Ltd., Japan Kawasaki Heavy Industries, Ltd., Japan Sanoyas Shipbuilding Corporation, Japan Japan Marine United Corporation, Japan Shin Kurushima Dockyard Co., Ltd., Japan Sumitomo Heavy Industries Marine & Engineering Co., Ltd., Japan Tsuneishi Shipbuilding Co., Ltd., Japan Namura Shipbuilding Co., Ltd., Japan Mitsui E&S Shipbuilding Co., Ltd., Japan Mitsubishi Shipbuilding Co., Ltd., Japan The Shipbuilders’ Association of Japan, Japan Mitsui O.S.K. Lines, Ltd., Japan ClassNK, Japan Osaka University, Japan National Maritime Research Institute, Japan Yokohama National University, Japan The University of Tokyo, Japan

Contents

Structures Full-scale Measurement Statistical Characteristics of Whipping Response of a Large Container Ship Under Various Sea States and Navigational Conditions Based on Full-Scale Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tetsuji Miyashita, Tetsuo Okada, Yasumi Kawamura, Noriaki Seki, and Ryo Hanada Practical Investigation on Hull Girder Response for a Large Container Ship with Direct Load Structure Analysis and Full-Scale Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Katsutoshi Takeda, Tsutomu Fukui, and Toshiyuki Matsumoto Three-Dimensional Shape Sensing by Inverse Finite Element Method Based on Distributed Fiber-Optic Sensors . . . . . . . . . . . . . . . . . . . . . . . Makito Kobayashi, Takuya Jumonji, and Hideaki Murayama

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Slamming Research on the Strengthening of Double Bottom Floors or Girders Against Bottom Slamming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wenbo Zhu, SuSu Zhou, Jiameng Wu, Fan Zhang, and Shijian Cai A Study on Dynamic Response of Flat Stiffened Plates to Slamming Loads Considering Fluid-Structure Interaction . . . . . . . . . . . . . . . . . . . Dac Dung Truong, Beom-Seon Jang, Han-Baek Ju, Sang Woong Han, and Sungkon Han

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Ultimate Strength Scantling Evaluations of Plates and Stiffeners Based on Elasto-Plastic Analysis Under Axial Loads and Lateral Pressures . . . . . . . . . . . . . . . . 100 Yoshiaki Naruse, Masato Kim, Rikuto Umezawa, Kinya Ishibashi, Hiroyuki Koyama, Tetsuo Okada, and Yasumi Kawamura ix

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Contents

A Study on Progressive Collapse Analysis of a Hull Girder Using Smith’s Method – Uncertainty in the Ultimate Strength Prediction . . . . 128 Akira Tatsumi, Kazuhiro Iijima, and Masahiko Fujikubo Verification of an Automated Structural Design Procedure Using Ultimate Limit States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Nathaniel Cope and Joshua Knight Structural Behavior Application of Automatic FE-Modeling Using 3D CAD . . . . . . . . . . . . . 155 Hyokwon Son, Junghyuk Chun, and Ohseok Kwon Design Contours for Complex Marine Systems . . . . . . . . . . . . . . . . . . . 168 Harleigh C. Seyffert, Austin A. Kana, and Armin W. Troesch Potential of Homogenized and Non-local Beam and Plate Theories in Ship Structural Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 Jani Romanoff, Bruno Reinaldo Goncalves, Anssi Karttunen, and J. N. Reddy Structural Design of Hinge Connector for Very Large Floating Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Ye Lu, Bei Teng, Yikun Wang, Ye Zhou, Xiaoming Cheng, and Enrong Qi Structural Design and Strength Estimation of Energy Saving Y-FIN by Using Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Wen-Huai Tsou, Pai-Chen Guan, Wen-Hsuan Chang, and Chao-Jieh Chen Sustainable Sandwich Panels for Use in Ship Superstructures . . . . . . . . 232 Jeanne Blanchard and Adam Sobey Design Principle of Wheel Patch Loaded Ship Plating . . . . . . . . . . . . . . 246 Ling Zhu, Wei Cai, Yuansong He, and Youjun Wu Numerical Simulations of Grounding Scenarios–Benchmark Study on Key Parameters in FEM Modelling . . . . . . . . . . . . . . . . . . . . . 257 Lars Brubak, Zhiqiang Hu, Mihkel Kõrgesaar, Ingrid Schipperen, and Kristjan Tabri Development of Analytical Formulae to Determine the Response of Submerged Composite Plates Subjected to Underwater Explosion . . . 270 Ye Pyae Sone Oo, Hervé Le Sourne, and Olivier Dorival Stress Concentration Factors of Damaged Ship Side Panels . . . . . . . . . . 291 Cristian Lombardi, Bianca Pinheiro, and Ilson Paranhos Pasqualino Vibration and Noise A Procedure for the Identification of Hydrodynamic Damping Associated to Elastic Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 Daniele Dessi, Edoardo Faiella, and Filippo Riccioli

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Vibration Damping of Large Containership in Operation . . . . . . . . . . . 321 Saeed Shakibfar, Ingrid M. V. Andersen, and Anders Brandt Study on the Effect of Liquid in Tanks on the Hull Girder Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 Hiroyuki Takahashi and Yukitaka Yasuzawa Calculation of Structural Damping of the Global Hull Structure from In-Service Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 Remco Hageman and Ingo Drummen Vibration Characteristics of Separated Superstructure of a Ship . . . . . . 365 Yukitaka Yasuzawa, Akihiro Morooka, and Kazuyuki Tanimoto Ship Vibration and Noise Reduction with Metamaterial Structures . . . . 377 Deqing Yang Experimental and Numerical Investigations About Hydraulic Top Bracings and Its Influence on Engine and Vessels Superstructure Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 Michael Holtmann and Martin H. Frandsen Design Procedure to Estimate the Mechanical Behaviour of Resilient Mounting Elements for Marine Applications . . . . . . . . . . . . . . . . . . . . . 408 Jacopo Fragasso and Lorenzo Moro Fatigue and Life Evaluation Experimental and Simulate Research on Fatigue Crack Propagation Behavior Under Varying Loadings in High Strength Steel of Marine Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 Jingxia Yue, Wenjie Tu, Hao Xie, Ke Yang, and Weiguo Tang The Effect of Loading Sequence on Fatigue Crack Growth of a Ship Detail Under Different Load Spectra . . . . . . . . . . . . . . . . . . . 442 Xiaoping Huang, Honggan Yu, and Dimitrios G. Pavlou Study on Fatigue Strength of Welded Joints Subject to Intermittently Whipping Superimposed Wave Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 Naoki Osawa, Luis De Gracia, Kazuhiro Iijima, Norio Yamamoto, and Kyosuke Matsumoto Determination of a Methodology for the Fatigue Strength Evaluation of Transverse Hatch Coaming Stays on Container Ships . . . . . . . . . . . . 473 Chi-Fang Lee, Yann Quéméner, Po-Kai Liao, Kuan-Chen Chen, and Ya-Jung Lee An Investigation of Fatigue and Long-Term Stress Prediction for Container Ship Based on Full Scale Hull Monitoring System . . . . . . 492 Chong Ma, Masayoshi Oka, and Hiroshi Ochi

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Contents

Evaluation of Long-Term Corrosion Fatigue Life of Ship and Offshore Structural Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 Won Beom Kim Statistical Modelling and Comparison of Model-Based Fatigue Calculations and Hull Monitoring Data for Container Vessels . . . . . . . . 517 Erik Vanem, Lars Holterud Aarsnes, Gaute Storhaug, and Ole Christian Astrup Optimization of Superstructure Connection Design Base on Fatigue Strength Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 Xianyin Chen, Wenyuan Zeng, and Shuo Li Assessment of CALM Buoys Motion Response and Dominant OPB/IPB Inducing Parameters on Fatigue Failure of Offshore Mooring Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548 Ceasar Edward and Dr Arun Kr. Dev Comparison of Spectral Fatigues Methodologies Using Equivalent Stresses Obtained by Mises and Battelle Applied to a Semisubmersible Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580 Jonas Haddad Bittar Filho, Marcelo Igor Lourenço de Souza, and Ilson Paranhos Pasqualino Predicting Crack Growth in Multiple Degradation Experiment with Dynamic Bayesian Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608 Kaihua Zhang and Matthew Collette Utilization of Structural Design Models in Operation to Monitor Fatigue Strength Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622 Lars Holterud Aarsnes, Gaute Storhaug, and Maciej Radon Structural Integrity Development of Practical Fatigue Strength Evaluation Method for Weld Root Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 Shinsaku Ashida, Tomohiro Sugimoto, Kinya Ishibashi, and Norio Yamamoto Effect of Residual Stress on Hydrogen Diffusion in Flat Butt Welding Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 Liangbi Li, Qianqian Jia, and Zhengquan Wan Coating Condition Monitoring and the Practical Application . . . . . . . . . 666 Martijn Hoogeland, T. Van Dijk, and A. W. Vredeveldt Stress Concentration Factors on Welded Tubular Joints . . . . . . . . . . . . 681 Nathalia Paruolo, Bianca Pinheiro, Thalita Mello, and Ilson Paranhos Pasqualino

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Predicting Ductile Fracture in Maritime Crash with a Modified Implementation of BWH Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701 Burak Can Cerik, Sung-Ju Park, and Joonmo Choung Safety Evaluation of the High Manganese Steel LNG SPB® Tank . . . . . 715 Takashi Takeda, Takashi Hiraide, and Keiji Ueda Inexpensive Fracture Toughness Testing of Welded Steel . . . . . . . . . . . 729 Carey L. Walters, Okko J. Coppejans, and Martijn Hoogeland Structural Safety Assessment of Lift Apparatus on Jack-Up Barge for Eco-Friendly Offshore Installation . . . . . . . . . . . . . . . . . . . . . . . . . . 740 Chang Yong Song, Chun-Sik Shim, Ha-Cheol Song, Doo-Yeoun Cho, Sol Ha, Ho-Kyung Kim, Woo-Chang Park, Dong-Joon Lee, Jeong-Wook Yang, and Tae-Yang Moon Machinery and Equipment An Innovative Machine Learning System for Real Time Condition Monitoring of Ship Machinery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753 Stylianos Oikonomou, Iraklis Lazakis, and George Papadakis Statistical Analysis of Installed Power on Board Modern Cruise Ships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769 Luigia Mocerino and Enrico Rizzuto Evaluating LIDAR Sensors for the Survey of Emissions from Ships at Harbor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784 Antonella Boselli, Carmen de Marco, Luigia Mocerino, Fabio Murena, Franco Quaranta, Enrico Rizzuto, Alessia Sannino, Nicola Spinelli, and Wang Xuan Application Research of Slow Steaming in IMO Sulfur Emission Control Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797 Zhihu Zhan Study on the Dynamic Unbalance of Turbocharger Rotors Effecting to the Operation of Marine Diesel Engine . . . . . . . . . . . . . . . . . . . . . . . 807 Cao An Truong Phan and Van Quan Phan Natural Gas Engine Thermodynamic Modeling Concerning Offshore Dynamic Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 820 Sadi Tavakoli, Eilif Pedersen, and Jesper Schramm Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843

Structures

Statistical Characteristics of Whipping Response of a Large Container Ship Under Various Sea States and Navigational Conditions Based on Full-Scale Measurements Tetsuji Miyashita1(B) , Tetsuo Okada2 , Yasumi Kawamura2 , Noriaki Seki3 , and Ryo Hanada4 1 Technical Research Center, Japan Marine United Corporation, Tsu, Mie, Japan

[email protected] 2 Faculty of Engineering, Yokohama National University, Yokohama, Japan

{okada-t,kawamura-yasumi-zx}@ynu.ac.jp 3 Design Division, Japan Marine United Corporation, Yokohama, Japan

[email protected] 4 College of Engineering Science, Yokohama National University, Yokohama, Japan

[email protected]

Abstract. The recent increase in the size of container ships has resulted in lower natural frequency of the hull girder vertical bending and torsional vibrations; this has caused great concern regarding excessive whipping in these responses to incident waves. To accurately account for whipping in ship design, it is important to understand the relationship between the whipping response and sea states in which the ship operates. However, there has been little reported to date on this specific topic. In this study, we quantitatively analyze the influence of various sea states and navigational conditions on the whipping response in hull girder bending vibration and torsional vibration. Full-scale measurements of an 8,600 TEU container ship were taken over four years and two months. Longitudinal stresses in way of four corners at the mid-ship section were measured and then decomposed into four components: hull girder vertical bending stresses, horizontal bending stresses, warping stresses, and axial stresses. Furthermore, components of the high-frequency vibrational responses and wave frequency wave responses were decomposed, and the probability distribution of peak values of each component was calculated. Then, the values corresponding to 1/1000 expected maximum stresses were obtained, allowing for statistical investigation. Moreover, a whipping factor was obtained in order to evaluate the correlation with sea states, navigational conditions, etc. By providing quantitative analysis of the relationship between whipping and sea states/navigational conditions, our study will contribute to future ship design to ensure better operation safety and longer hull life. Keywords: Full-scale measurement · Hull vibration · Large container ship · Ship design · Statistical characteristics · Whipping factor

© Springer Nature Singapore Pte Ltd. 2021 T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 3–24, 2021. https://doi.org/10.1007/978-981-15-4672-3_1

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T. Miyashita et al.

1 Introduction A recent increase in the size of container ships has resulted in lower natural frequency of the hull girder vertical bending and torsional vibrations; therefore, the influence of hydro-elasticity on the hull structure has become even more important for hull strength as well as ship operation. In particular, the lower natural frequency of the larger hull resonates with a wave encounter; this phenomenon is called springing. In addition, hull girder vibration caused by an impact such as slamming is also important; this is generally called whipping. Whipping is a transient response lasting several tens of seconds, which is likely to occur under severe sea conditions. Recently, there has been great concern over excessive whipping with regard to hull girder torsional vibration as well as vertical bending vibration. From the viewpoint of hull structure design, great effort has been put into determining whether hydro-elasticity should be considered in ship design to ensure appropriate hull strength and safety margins. This concern has been extensively reported in the existing literature. For example, Okada et al. [1] conducted full-scale measurements of a 6,700 TEU Post-Panamax container ship between 1999 and 2002. Subsequently, Toyoda et al. [2] conducted full-scale measurements of a 6,400 TEU between 2002 and 2004. These studies have reported that the fatigue damage approximately doubles in the presence of whipping stresses. Based on measurements performed for two container ships (4,600 TEU and 14,000 TEU), Rathje et al. [3] discovered that whipping caused an increase of 2.36 times the normal rate of fatigue damage. Thereby the fatigue life when high frequency components are included is 15.9 and 13.8 years, respectively, according to the cumulative fatigue damage calculation with the rain flow count and the FAT 56 S-N curve. When only the low frequency component is counted, the values become 24.4 and 32.7 years, and the effect of the high frequency components is calculated as 35% and 57%, respectively. Other previous research has suggested that the influence of whipping on fatigue strength was clearly observed. However, some uncertainties on the influence of whipping and springing on fatigue strength and extreme stress still remain. Nielsen et al. [4] proposed a means of estimating the fatigue damage including hull girder flexibility effects using a spectrum-based procedure, and showed that the fatigue accumulation was well predicted. One of the uncertain factors in the early studies was the resonance with torsional vibration. Therefore, full-scale measurements focusing on torsional vibration have been carried out since 2015. For example, Ki et al. [5] calculated section load from local stresses using full-scale measurements of a 14,000 TEU container ship. They calculated the extreme values of the vertical bending and torsional moments and then compared them with the ship design values. It was shown that the hull girder moment was less than approximately 80% of the design value; in contrast, horizontal bending moment and torsional moment were larger than 80% in some cases. Kim et al. [6] investigated the influence of torsional vibration on the fatigue damage of the upper deck longitudinal stress for a 13,000 TEU container ship. It was found that the vertical bending stress was dominant, and the horizontal bending stress and warping stress had a small influence. Storhaug et al. [7] conducted full-scale measurements on 8400 TEU and 8600 TEU container ships, where they investigated the influence of torsional vibration on fatigue strength and extreme stress at the upper deck. It was shown that the influence of torsional vibration on fatigue damage was 14%–16% on the 8600 TEU container ship. In contrast,

Statistical Characteristics of Whipping Response

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while measurement data was collected for only 10 months on the 8400 TEU container ship, the influence of torsional vibration on fatigue damage increased to approximately 52%–55%. Resonances at 0.5 Hz of the hull 2-node vibration and 0.48 Hz of torsional vibration were observed on the 8400 TEU container ship. These results indicate that the vertical bending stress was dominant on the fatigue strength of the upper deck. However, clarifying the specific influence of torsional vibration on fatigue damage was still necessary, which is the motivation of our investigation. Therefore, to accurately account for whipping responses in ship design, it is important to understand the relationship between whipping responses and the conditions in which the ship is operating. These conditions include: sea state information (i.e., wave height, wave period and direction), ship navigational information (i.e., ship speed and course), etc. Unfortunately, little has been reported in detail on these relationships to date. In this study to cope with the problems mentioned above, we quantitatively analyze the influence of various sea states and ship conditions on the whipping response in hull girder bending vibration and torsional vibration. Full-scale measurements of an 8,600 TEU container ship were carried out for four years and two months; the results were then analyzed. Longitudinal stresses in way of the four corners at the mid-ship section were measured and then decomposed into four components: hull girder vertical bending stresses, horizontal bending stresses, warping stresses, and axial stresses. Furthermore, the high-frequency vibrational responses and wave frequency wave responses were decomposed using a low-pass filter. The probability distribution of peak values of each component was calculated at an hourly interval. Then, the values corresponding to 1/1000 expected maximum stresses were obtained by curve fitting to the Weibull distribution with the least-squares method. Thus, the statistical characteristics of each stress component were investigated. Moreover, the whipping factor, or the stress increase ratio due to the superposition of the whipping responses on the wave responses, was obtained from the results of the statistical analyses on hull girder vertical bending stresses. Finally, the correlation between the whipping factor and various sea states and navigational conditions were investigated, and statistical table of the whipping factor is proposed as anew index for quantitative evaluation.

2 Full-Scale Measurement 2.1 Ship Characteristics, Measurement Period, Travel Route, and Sensor Arrangement The basic dimensions of the 8600 TEU container ship in this study are given in Table 1. Measurements were performed over four years and two months that included seven voyages between Asia and Europe, four voyages from Asia to the East Coast of North America via the Mediterranean, one voyage from Asia to the East Coast of North America via the Pacific Ocean, and three voyages from the East Coast of North America to Asia via the Cape of Good Hope. These trading routes are shown in Fig. 1. Twelve optical fiber strain gauges were installed into the hull structure to measure hull girder stress. The strain gauges were arranged in 3 sections of the container ship: fore (FR139-140), mid (FR115-117), and aft (FR98-100). The layout of the strain gauges is shown in Fig. 2, and the names of the gauges are listed in Table 2.

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T. Miyashita et al. Table 1. Basic characteristics of the container ship used in this study. LOA

334.5 m (approx.)

Breadth

45.6 m

Depth

24.4 m

Design draft

14.0 m

Gross tonnage

97,000 ton (approx.)

Max. service speed 24.5 knot

Fig. 1. Ship trading route during the measurement period.

Fig. 2. Sensor arrangement (profile and mid-ship section).

Table 2. Strain gauge sensor name and location list. DAP/DAS

deck at aft port/starboard

BAP/BAS

lower longitudinal bulkhead at aft port/starboard

DMP/DMS deck at mid-ship port/starboard BMP/BMS lower longitudinal bulkhead at mid-ship port/starboard DFP/DFS

deck at fore port/starboard

BFP/BFS

lower longitudinal bulkhead at fore port/starboard

Statistical Characteristics of Whipping Response

7

2.2 Sea State Data To investigate the relationship between the ship structural response and sea states, it is necessary to first collect wave height, period, and direction information. Various wave estimation methods exist with pros and cons summarized in Collette et al. [8]. Among these are: (1) installing a wave radar system on the ship, (2) wave forecast data, (3) wave hindcast data and (4) estimation using the ship responses to wave [9]. The wave radar is thought to accurately represent the encountering sea state. However, the radar system installed onboard the tested ship was missing a lot of data; therefore, the available data were not sufficient for use in the statistical analysis. Hindcast data during the measurement period was also difficult to obtain due to the high cost. Therefore, we needed to investigate the practicality of using wave forecast data by determining if it were as accurate as wave radar measurements. These forecast data were provided by the Japan Meteorological Agency. Figure 3 shows the Asia–North America route from 2015/07/05 (year/month/day) to 2015/07/31 and the North America–Asia route via Cape of Good Hope from 2015/08/01 to 2015/08/31. Figure 4 shows a comparison between the wave radar measurements and the forecast for the significant wave heights at hourly intervals; the mean wave periods are also compared in Fig. 5. We can observe that there is general agreement between the forecast and the wave radar values. Therefore, the use of the wave forecast data is considered to be sufficient for investigating the relationship between the ship structural response and the sea states.

6

Forecast

Measurement (Radar)

4 2 8/30/2015

8/20/2015

8/10/2015

7/31/2015

7/21/2015

7/11/2015

0 7/1/2015

Wave height [m]

Fig. 3. Asia–North America route from 2015/07/05 to 2015/08/31.

Fig. 4. Comparison between the wave radar and forecast for the significant wave height.

T. Miyashita et al.

8/30/2015

8/20/2015

7/21/2015

7/11/2015

Measurement (Radar)

7/31/2015

Forecast

8/10/2015

18 15 12 9 6 3 0 7/1/2015

Wave period[sec]

8

Fig. 5. Comparison between the wave radar and forecast for the mean wave period.

Number of sea state

5075 3528

head sea

4491 3664

3139

forward beam sea aft oblique following sea sea oblique sea

Wave height frequency

In this study, 19,897 sea states were used and required that the ship speed be over 0.1 knot. Sea states where the course or ship speed changed drastically within one hour were removed to assure statistical stationarity of the data. The frequency distribution table for the relative direction is shown in Fig. 6, where the direction of the head sea, the forward oblique (quartering) sea which the wave comes from bow, the beam sea, the aft oblique (quartering) sea which the wave comes from stern, and the following sea is defined as 180° ± 22.5°, 135° ± 22.5° or 225° ± 22.5°, 90° ± 22.5° or 270° ± 22.5°, 45° ± 22.5° or 315° ± 22.5°, and 0° ± 22.5°, respectively. Frequency distribution tables for significant wave height, mean wave period, and ship speed are also shown in Figs. 6 and 7, respectively. Encounter with head sea is observed to be relatively more frequent than other wave encounter angles, while encounter with the beam sea less frequent. 91% of the encounters had a low wave height of 3 m or less, and 69% of the encounters had an average wave period of 8 to 12 s. From these observations, it is inferred that the encountered sea state was generally calm. 7915 6119 4145

1247

0.1-1

1-2

2-3

3-4

384

87

0

4-5

5-6

6-

Wave height [m]

Fig. 6. Frequency distribution table of encountered wave direction (oblique and beam sea covers wider range of angle including waves from both sides of the ship) and that of significant wave height.

Mean wave period frequency

6063

176 0-2

4529 3436

3111 1175

836 2-4

4-6

517

54

6-8 8-10 10-1212-1416-16 16-

Ship speed frequency

Statistical Characteristics of Whipping Response

9

11447

6492

50 0.1-5

Wave period [sec]

1616

292 5-10 10-15 15-20 Ship speed [knot]

20-

Fig. 7. Frequency distribution tables for mean wave period and ship speed.

3 Mathematical Decomposition of Hull Stress from Full-Scale Measurements 3.1 Procedure for Mathematical Decomposition Assuming that the ship hull is regarded as a beam, longitudinal stress in the mid-ship section can be expressed as the sum of four components: 1) vertical bending stress σ V 2) horizontal bending stress σ H 3) warping stress due to torsion σ W , and 4) axial stress σ A . Thus, the stress of DMP σDMP is formulated as Eq. (1). The relationship between the stress of DMS σDMS and the four components can be formulated as Eq. (2), because the strain gauges at DMP and DMS are arranged symmetrically. Using the coefficients, α and β, the relationship between the stresses at BMP or BMS and the four components can be formulated as Eqs. (3) and (4), because the strain gauges at BMP and BMS are arranged symmetrically. We then have, σDMP = σV + σH + σW + σA ,

(1)

σDMS = σV − σH − σW + σA ,

(2)

σBMP = −ασV + σH − βσW + σA ,

(3)

σBMS = −ασV − σH + βσW + σA ,

(4)

where, derivation of the coefficients α and β is explained below. Expressing Eqs. (1)–(4) in matrix form and taking the inverse matrix, each stress component can be obtained from the four sensor data as shown in Eq. (5), ⎤ ⎡ ⎤−1 ⎡ σDMP 1 1 1 1 σV ⎢ σH ⎥ ⎢ 1 −1 −1 1 ⎥ ⎢ σDMS ⎥ ⎢ ⎢ ⎥ ⎢ ⎣ σW ⎦ = ⎣ −α 1 −β 1 ⎦ ⎣ σBMP σA σBMS −α −1 β 1 ⎡

⎤ ⎥ ⎥. ⎦

(5)

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In this study, α is defined as the ratio of the distance of the sensors from the neutral axis using beam theory. The distances from the neutral axis to the strain gauge DMP and BMP are 11.72 m and 7.19 m, respectively. Therefore, α is calculated to be 0.6135. β is obtained from finite element (FE) vibration analysis using a whole-ship model, so that accurate warping stress distribution can be found along the mid-ship section. Figure 8 shows the 3-D finite element model of the whole ship comprising of 57,180 mesh elements and 32,947 nodes. The containers on the upper deck are modeled with solid elements arranged along the ship length and connected to the hull using multiple point constraints. The containers under the upper deck are modeled with the mass properties of cargo tank elements, representing a fully loaded cargo condition. The added water mass on the hull surface of the ship was considered, representing the wet condition. The vibration analysis is performed to derive the vibration mode related to the vertical bending and torsion. Figure 9 shows that the torsional mode twisted at the mid-sections and at the fore and aft sections. Figure 10 shows the 2-node and the 3-node vertical bending mode and these natural frequencies, respectively. From the torsional vibration modes shown in Fig. 9 on the left, the stresses of DMP and BMP are calculated to be 0.000116 and −0.000109, respectively. Therefore, the coefficient β is calculated to be 0.9397.

Fig. 8. 3-D finite element model of the entire ship.

Fig. 9. Torsional mode (0.327 Hz) and combined mode of torsion and horizontal bending (0.518 Hz), respectively.

The procedure to calculate the whipping factors is as follows: (1) The measured time series data are divided into 60 min intervals. The mean values of the time series are removed to obtain only the dynamic values. (2) The time series stresses at the four sensors are decomposed into four components: vertical bending stresses, horizontal bending stresses, warping stresses, and axial stresses, using the decomposition procedure shown in Sect. 3.1. (3) The raw data of the four components are decomposed into wave components and whipping components using a low-pass filter with a cutoff frequency of 0.3 Hz.

Statistical Characteristics of Whipping Response

11

Fig. 10. Vertical bending mode (2-node: 0.475 Hz and 3-node: 0.932 Hz, respectively).

(4) The amplitude extremum of the wave component during each wave cycle is counted using the zero-up crossing method; the whipping component (raw data) is also counted during each wave cycle. (5) The exceedance probability distribution of the wave component and the raw data at hourly intervals is calculated. (6) The values corresponding to the exceedance probability of 1/1000 (1/1000 expected maximum values) are calculated by curve fitting to the Weibull distribution. To estimate the Weibull parameters, the least-squares method is applied using the top 20% of the obtained peak values. (7) Finally, the whipping factor is calculated as the ratio of the 1/1000 expected maximum stress of the raw data (including the whipping components) to that of the wave components. 3.2 Verification of Hull Stress Decomposition Without the Whipping Response In this section, the hull girder stress of the wave component without the whipping response is discussed. Table 3 shows the top five sea states causing significant vertical bending stress. For example, large vertical bending stress was observed in head seas where wave direction was in the range of 178.6° to 210.2°. The mean wave period in these sea conditions was 13.08 to 13.20 s, which corresponds to a wave length of approximately 270 m assuming a regular wave on an infinite water depth. Thus, all of these conditions are believed to have led to the large vertical bending stress due to the wave length close to the ship length. Table 4 shows the top five sea states causing significant warping stress. The warping stress was large in oblique seas, where wave direction was in the range of 40.29° to 65.09°, except on the date of 2014/01/29. Table 5 shows the top five sea states causing significant horizontal bending stress. The horizontal bending stress was high in oblique seas or beam seas, except on the date of 2014/01/29. From these results, we can presume that the measured stresses were successfully decomposed into each component. As stated above, an unusual phenomenon was observed on 2014/01/29, when the vertical bending stress, the horizontal bending stress, and the warping stress increased simultaneously. In order to confirm the cause of this phenomenon, the time history of the vertical bending stress, the horizontal bending stress, and the warping stress were compared, as shown in Fig. 11. It can be observed that the vertical bending stress was the largest and dominant. However, the horizontal bending stress and the warping stress were observed to increase during the period between 800 and 1000 s. Figure 12 is a magnified plot of this stress–time series for the period from 700 to 1200 s. Both the horizontal

12

T. Miyashita et al. Table 3. Top five sea conditions causing large vertical bending stress.

Date/Time (JST)

Latitude

Longitude

1/1000 expected stress

Wave direction

Wave period

Wave height

2014/01/29 23:00

37.07

−9.38

56.01

210.20

13.18

5.18

2014/01/29 22:00

36.88

−9.31

55.32

184.29

13.15

5.28

2014/01/29 21:00

36.75

−9.17

53.39

178.60

13.08

5.31

2014/01/30 01:00

37.46

−9.51

50.96

209.74

13.20

4.95

2014/01/30 00:00

37.26

−9.44

49.93

208.40

13.20

5.06

Table 4. Top five sea states causing large warping stress. Date/Time (JST)

Latitude

Longitude

1/1000 warping stress

Wave direction

Wave period

Wave height

2016/01/10 19:00

39.34

−49.58

23.09

60.81

9.52

5.63

2016/01/11 09:00

39.74

−55.26

21.91

22.23

10.70

4.05

2016/01/11 01:00

39.52

−51.97

21.38

41.40

9.38

5.45

2016/01/10 20:00

39.38

−49.93

20.98

59.40

9.34

5.69

2014/01/29 21:00

36.75

−9.17

20.89

178.60

13.08

5.31

bending stress and the warping stress can be seen to fluctuate for a long period of 20 s or more. Considering that the roll period of the ship is about 20 s, it is probable that a nonlinear phenomenon such as “parametric rolling” may have occurred, when the ship was navigating in head seas.

4 Statistical Analysis of Whipping Factor 4.1 Influence of the Whipping Response on the Decomposed Stress From this analytical process, the exceedance probabilities for the total stress at DMP, the vertical bending stress, the horizontal bending stress, the warping stress, and the

Statistical Characteristics of Whipping Response

13

Table 5. Top five sea states causing large horizontal bending stress. Date/Time (JST)

Latitude

2014/01/29 21:00

36.75

2016/01/10 17:00

Longitude

1/1000 expected stress

Wave direction

Wave period

Wave height

−9.17

14.55

178.60

13.08

5.31

39.26

−48.88

13.30

65.09

9.61

5.45

2016/01/11 20:00

39.83

−59.11

11.85

276.75

8.68

5.03

2016/01/10 16:00

39.22

−48.50

11.55

66.93

9.37

5.32

2016/01/10 20:00

39.38

−49.93

11.19

59.40

9.34

5.69

Fig. 11. Stress time history of DMP at 21:00-22:00 on 2014/01/29 (JST).

Fig. 12. Magnified stress time history of DMP from 700 s to 1200 s.

axial stress were calculated and are shown in Figs. 13–14. These figures also show the stress without the whipping component. As shown in these figures, the maximum value of the total stress was 100 MPa. The rule wave induced bending moment (hogging) stipulated in IACS-UR-S11A is 105 MPa at the sensor location. It was, therefore, confirmed that the bending stress including high frequency component was close to, but

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T. Miyashita et al.

less than the design value. And the vertical bending stress accounts for 81% of the total stress. Because the maximum value of the vertical bending stress was 81 MPa during the measurement period, it was confirmed that the stress corresponding to the upper deck longitudinal stress is dominated by the vertical bending stress. Moreover, superposition of the whipping response was highly noticeable in the vertical bending stress when the exceedance probability is less 10−3 . In contrast, superposition of the whipping response was hardly observed with respect to the horizontal bending stress or warping stress. Figures 13 and 14 show that the warping stress is larger than the horizontal bending stress. In the future, it would be interesting to further investigate into how the tendency varies for the three different measured cross section versus the anticipated wave profile. As for the axial stress, the whipping factor was large presumably because of the vertical bending component mixed into the axial stress due to insufficient decomposition of components. However, because the axial stress is relatively small, its influence is also small. Therefore, it is important to get reliable vertical bending stress measurements, with whipping response, when evaluating the influence of whipping response on the stress fatigue or ultimate strength of container ships. For reference, the fatigue damage during the measured term was 0.009 for the low frequency components, and 0.030 for the raw data according to the linear accumulative fatigue damage calculation with the rain flow count and the FAT 65 S-N curve.

Fig. 13. Comparison of exceedance probability between total stress and vertical/horizontal bending stress during the measurement period.

4.2 Distinction Between Whipping and Springing Response The previous section shows that whipping responses were observed in certain sea states and dominated in the vertical bending stress component. Therefore, to understand the whipping response more quantitatively, the relationship between the whipping factor and the sea state was investigated, focusing on the vertical bending stress. To find these relationships, it is necessary to distinguish whipping response from springing response, even though it is often difficult to make a clear distinction. There are two ways to distinguish between whipping and springing; one is by simply using the encountering wave height, and the other is to use the standard deviation of the wave and vibrational components [4]. If the standard deviation of high frequency response is smaller than that of wave frequency, this phenomenon is defined as a whipping response.

Statistical Characteristics of Whipping Response

15

Fig. 14. Comparison of exceedance probability between total stress and warping/axial stress during the measurement period.

Figure 15 shows the relationship between the standard deviation of the wave frequency as it affects vertical bending stress and that of high frequency response, distinguishing between whipping and springing for all sea states. These distinctions were validated by observing high frequency stress and wave frequency stress of the vertical bending stress at DMP in several types of sea states. In Figs. 16-18, stress time series data in the three different head sea states as tabled in Table 6 are shown. Figure 16 shows a portion of the time series data for the sea state 1 where the standard deviation of high frequency response is greater than that of wave frequency response; the wave component is depicted by a red line, and high frequency response is depicted by a blue line. The maximum stresses for both wave and high frequency response were small due to a low wave height of 0.785 m. By observing high frequency response component, this response is considered to be springing because it continues almost without interruption or indication of a strong impact. Therefore, if the ratio of standard deviation between high frequency response and wave frequency response is greater than 1.0, we can assume that springing has occurred. Next, Fig. 17 shows a portion of the time data for the sea state 2, where the standard deviation of high frequency response is approximately equal to that of wave frequency response. The wave height and the wave period of this sea state were 3.724 m and 8.6 s, respectively. By observing high frequency response component, this example is considered to be whipping due to an impact. Moreover, springing is also observed, indicating that whipping and springing co-occurred. Therefore, it is difficult to strictly distinguish in a scenario where the ratio of standard deviation is close to 1.0. Finally, Fig. 18 shows a portion of the time data for the sea state 3, where the standard deviation of high frequency response is less than that of wave frequency response. The wave height and the wave period of this sea state were 5.164 m and 12.82 s, respectively. By observing high frequency response component, an impact response like whipping was often observed, although the maximum stress amplitude of high frequency response component was smaller than that of wave frequency response due to the larger wave height. Springing could be also observed, but it was not dominant because its amplitude value was small value against the design value. Thus, in case of smaller wave height, the hull girder bending stress is very small; therefore, the energy of the continuous springing vibration can become dominant. In contrast, with larger wave height, the hull girder bending stress is large. Therefore, while a strong impact can cause a large whipping response, the relative energy of the whipping response is less dominant than that of the

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T. Miyashita et al.

wave response. Thereby, we have confirmed that the ratio of standard deviation method can effectively distinguish between whipping and springing.

Fig. 15. Relationship between standard deviation of wave frequency response and that of high frequency response as it affects vertical bending stress; distinguishing between whipping and springing.

Fig. 16. Stress time history of DMP in the sea state 1 where the standard deviation of high frequency response is greater than that of wave frequency.

In order to investigate the phenomenon at the boundary between whipping and springing, only the data with their wave height of 3 m or more are plotted in the left of Fig. 19. According to this Figure, 41 sea states were determined to be included in the springing zone; this is for a ship speed of 15 knots or more in head or forward oblique seas. As the encounter wave period and wave length become shorter, vertical bending stress becomes smaller, but a large springing response is induced. Conversely, most of the data categorized as springing disappeared, showing that they were in the wave height of 3 m or less. Then, Fig. 20 shows the relationship between the standard deviation of the high-frequency response and the significant wave height, distinguishing between whipping and springing in head seas and following seas, respectively. In addition, note that the color is muted for a wave height of 3 m or less in this figure. We can observe that the larger wave height caused a stronger whipping response in head seas, whereas smaller wave height caused a springing response. In the following seas, significant hull

Statistical Characteristics of Whipping Response

17

Table 6. Typical head sea states observing stress time history of high frequency response (HF) and wave frequency response (WF). Sea state

Date/Time [JST]

Wave height [m]

Wave period [s]

Ship speed [knot]

Standard deviation of HF stress [MPa]

1

2013/08/09 13:00

0.785

6.0

19.8

1.87

2

2013/01/09 6:00

3.724

8.60

17.2

3

2014/01/29 9:00

5.164

12.82

12.6

Standard deviation of WF stress [MPa]

Ratio (HF/WF)

Whipping factor

0.80

2.34

2.83

6.16

6.14

1.00

1.73

5.43

11.32

0.48

1.37

Fig. 17. Stress time history of DMP in the sea state 2 where the standard deviation of high frequency response is approximately equal to that of wave frequency.

Fig. 18. Stress time history of DMP in the sea state 3 where the standard deviation of high frequency response is less than that of wave frequency response.

girder vibration is not observed even in the larger wave height. It is difficult to distinguish the frequency responses in cases with smaller wave heights; thus, it would also be difficult to distinguish between whipping and springing using the wave height method. Therefore, we have confirmed that the ratio of standard deviation method is useful for

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T. Miyashita et al.

distinguishing clearly between whipping and springing. For another perspective, Fig. 21 shows the voyage route during the measurement period, distinguishing between whipping and springing. According to this figure, the whipping response primarily occurred in the open sea, whereas the springing response primarily occurred in the Mediterranean Sea and other bodies of water close to land, channels, etc. Figure 22 shows the histogram of the whipping factor in the vertical bending stress. As a result of these observations of 19,897 sea states in total, we concluded that springing can exacerbate the whipping factor due to its low level wave response, but the large whipping factors due to short term strong wave impacts are limited.

Fig. 19. Relationship between standard deviation of the wave frequency and that of the highfrequency response as it affects vertical bending stress; distinguishing between whipping and springing and filtering by a wave height of 3 m or more.

Fig. 20. Relationship between standard deviation of high frequency and significant wave height in head seas and following seas respectively.

Considering these observations, there is still a lot of uncertainty about the relationship between sea states and whipping. Therefore, it is necessary to investigate the correlation between whipping factor and ship and wave information, i.e., ship speed, wave height, wave period, the encountering wave direction, etc. This analysis is discussed in the next section.

Statistical Characteristics of Whipping Response

19

Fig. 21. Voyage routes; distinguishing between whipping and springing responses.

Fig. 22. Histogram of whipping factor in the vertical bending stress.

4.3 Statistics Table of Whipping Factor First, to observe the distribution of the whipping factor of the data categorized as whipping by the ratio of standard deviation method, box-and-whisker plots were created as shown in Table 6. The whipping factor can be classified according to ship and wave conditions; i.e., significant wave height, mean wave period, and relative wave direction. This information is given in Table 7, where the blue dotted points show whipping factors at an hourly interval, and the red and pink boxes indicate the middle 50% of the data. It can be seen from this table that the whipping factors classified by ship condition vary widely; however, it is difficult to find a clear relationship between the whipping factor and sea state. Therefore, whereas the data was not sufficient to be able to provide whipping factor correlation with specific sea state parameters, the average value of the whipping factor was calculated. These values are shown in Table 8, called “the statistics table of whipping factor”. From this table, it can be observed that there is a tendency that the whipping factor in a head sea and forward oblique sea becomes larger than that in a following sea; in a head sea, the whipping factor tends to increase as the wave height increases, due to the increase in the impact pressure acting on the bow. Therefore, to operate a large container ship to effectively avoid the whipping response, it is better to position the ship direction for following seas versus head seas.

20

T. Miyashita et al. Table 7. Statistics table of the whipping factor in each sea condition.

Table 8. Statistics table of the whipping factor in each sea condition.

Statistical Characteristics of Whipping Response

21

Second, Table 9 shows the statistics table of whipping factor classified by ship speed was investigated, focusing on the head sea. Due to the restrictions on the steepness of waves, larger wave height tends to appear in association with larger mean wave period.

Whipping factor [-]

Table 9. the statistics table of the whipping factor classified by ship speed in the head sea.

1.8

H=2-3m

1.6

H=3-4m H=4-5m

1.4 1.2 1.0 0.8

5-10

10-15 15-20 Ship speed [knot]

20-

Fig. 23. Relationship between whipping factor and ship speed when the wave period is between 8 and 10 s in head seas.

Figure 23 shows the relationship between the whipping factor and the ship speed when the wave period is from 8 to 10 s. Focusing on the higher wave heights, it was confirmed that a faster ship speed caused a larger whipping factor for ship speeds in the range of 0 to 20 knots. Therefore, reducing ship speed is also effective for avoiding the

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T. Miyashita et al.

whipping response. However, in the case of ship speed of 20 knots or more, a tendency was observed that the whipping factor became smaller. Moreover, reducing the ship’s speed did not appear to reduce the whipping factor in the highest sea states. Because there are only few sea states on ship navigation in high speed or high wave condition, it is difficult to draw a conclusion here. Therefore, it will be necessary in the future to validate these observations with sufficient data.

5 Conclusions In this study, we quantitatively analyzed the influence of various sea states and ship conditions on the whipping response in hull girder bending vibration and torsional vibration. Full-scale measurements of an 8,600 TEU container ship were carried out and the results were analyzed. Longitudinal stresses were decomposed into four components: hull girder vertical bending stresses, horizontal bending stresses, warping stresses, and axial stresses. Furthermore, components of the high-frequency vibrational responses and wave frequency wave responses were decomposed, and the probability distribution of peak values of each component was calculated. The whipping factor, or the stress increase ratio due to the superposition of the whipping responses on the wave responses, was obtained from the results of statistical analyses on hull girder vertical bending stresses. Lastly, the relationship between the whipping factor and various sea states, ship operating conditions, etc. were investigated. The findings drawn from this study are as follows: 1. Although there is some variability case by case, the wave height and wave period obtained through the wave radar agree with that of the wave forecast. Based on this result, it is appropriate to use the wave forecast data to investigate the relationships between the ship structural response and the wave information. However, these relationships are highly dependent on the actual sea state. Therefore additional verification and validation of the wave forecast data are needed. 2. As for upper deck longitudinal stress, it was confirmed that the vertical bending stress is dominant and accounts for 81% of the total stress. Moreover, superposition of whipping response was observed in vertical bending stress. In contrast, superposition of the whipping response was hardly observed with respect to horizontal bending stress and warping stress. Thus, the whipping response on the upper deck longitudinal stress is influenced largely by the vertical bending stress. In addition, the warping stress was found to be greater than the horizontal bending stress. It should be studied further in the future how these tendencies vary in the fore and aft sections. 3. There is a possibility that a nonlinear phenomenon such as “parametric rolling” may have occurred during the measurement term, where both the warping stress and horizontal bending stress were large. 4. Although the whipping factors vary widely, the statistical table of whipping factor classified by ship information indicated that the whipping response on the vertical bending stress was highly noticeable in head seas, while it was not observed in beam seas or following seas. Therefore, to operate a large container ship to effectively avoid the whipping response, it is better to position the ship direction for following seas versus head seas. However, this may not be always possible in practice.

Statistical Characteristics of Whipping Response

23

5. Furthermore, the statistics table of the whipping factor classified by ship speed shows that a higher ship speed will cause a higher whipping factor in head seas. Therefore, reducing the ship speed is also effective for avoiding the whipping response. However, for ship speeds over 20 knots, there was a tendency for the whipping factor to become smaller. Because data is lacking for ship navigation with high speed, a sufficient number of data is needed in the future to draw better conclusions. Future tasks are to clarify if the torsional whipping response may become dominant in any specific wave and ship conditions, although as a whole the torsional whipping is insignificant. In addition, based on these achievements, further studies must be taken up to correlate the results with numerical simulation, such as those employed in Refs. [10–12] to establish practical strength assessment method. Acknowledgement. The authors wish to gratefully acknowledge Kawasaki Kisen Kaisha, Ltd. and Nippon Kaiji Kyokai (ClassNK) for providing data concerning the large container ship considered in this study.

References 1. Okada, T., Takeda, Y., Maeda, T.: On board measurement of stresses and deflections of a post-panamax containership and its feedback to rational design. Mar. Struct. 19, 141–172 (2006) 2. Toyoda, M., Okada, T., Maeda, T., Matsumoto, T.: Full Scale Measurement of Stress and Deflections of Post-Panamax Container Ship. Design and Operation of Container Ships. RINA, London (2008) 3. Rathje, H., Kahl, A., Schellin, T.E.: Semi-empirical assessment of long-term high-frequency hull girder response of containerships. Int. J. Offshore Polar Eng. 23(4), pp. 292–297 (2013) 4. Nielsen, U.D., Jensen, J.J., Pedersen, P.T., Ito, Y.: Onboard monitoring of fatigue damage rates in the hull girder. Mar. Struct. 24, 182–206 (2011) 5. Ki, H.G., Park, S.G., Jang, I.H.: Full scale measurement of 14 k TEU containership. In: Proceedings of 7th International Conference on Hydroelasticity in Marine Technology, Split, Croatia (2015) 6. Kim, Yooil., Kim, Byung-Hoon., Park, Sung-Gun., Choi, Byung-Ki, Malenica, Sime: On the torsional vibratory response of 13000 TEU container carrier - full scale measurement data analysis. Ocean Eng. 158, 15–28 (2018) 7. Storhaug, G., Kahl, A.: Full scale measurements of torsional vibrations on Post-Panamax container ships. In: Proceedings of 7th International Conference on Hydroelasticity in Marine Technology, Split, Croatia (2015) 8. Collette, M., Zhan, Z., Zhu, L., Zanic, V., Okada, T., Arima, T., Skjong, R., Jeong, H.K., Egorov, G.: Committee IV.1 design principles and criteria. In: Proceedings of the 20th International Ship and Offshore Structures Congress (ISSC 2018), pp. 549–607 (2018) 9. Nielsen, U.D.: A concise account of techniques available for shipboard sea state estimation. Ocean Eng. 129, 352–362 (2017) 10. Kawasaki, Y., Okada, T., Kobayakawa, H., Amaya, I., Miyashita, T., Nagashima, T., Neki, I.: Influence of hull girder flexibility to whipping response of an ultra large container ship. In: The 30th Asian–Pacific Technical Exchange and Advisory Meeting on Marine Structures (TEAM2016), Mokpo, Korea, pp. 124–131 (2016)

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11. Kawasaki, Y., Okada, T., Kobayakawa, H., Amaya, I., Miyashita, T., Nagashima, T., Neki, I.: A study on forced vibration of double bottom structure due to whipping on an ultra large container ship. In: Proceedings of the ASME 2017 36th International Conference on Ocean, Offshore and Arctic Engineering (OMAE2017), Trondheim, Norway, Paper No. OMAE201761149 (2017) 12. Kawasaki, Y., Okada, T., Kobayakawa, H., Amaya, I., Miyashita, T., Nagashima, T., Neki, I.: Strength evaluation of containerships based on dynamic elastic response calculation of hull girder 2nd report - influence of hull girder rigidity and correlation between double bottom bending and hull girder bending. J. Jpn. Soc. Naval Architects Ocean Eng. 25, 191–203 (2017). in Japanese

Practical Investigation on Hull Girder Response for a Large Container Ship with Direct Load Structure Analysis and Full-Scale Measurements Katsutoshi Takeda(&), Tsutomu Fukui, and Toshiyuki Matsumoto Nippon Kaiji Kyokai (ClassNK), Chiyoda-ku, Tokyo, Japan [email protected]

Abstract. A consistent approach combined with a seakeeping analysis code and finite element analysis (FEA) tool, so-called direct load structure analysis is one of the practical methods for the structural strength assessments such as yield strength, buckling strength and fatigue strength assessments. It is considered that this method makes it possible to take into account the details of ship characteristics by directly using wave loads simulated by seakeeping analysis and is often applied to a new design or larger ship. Generally, there are various uncertainties derived from design assumption for the structural strength assessments. In classification societies’ rules the assumption of design loads on a hull is usually based on the operation in the North Atlantic Ocean for 25 years. However, sea states encountered by a ship would depend on an operation route. In terms of more reliable and rational strength assessments, a ship route is considered to be one of the critical factors in contributing to the evaluation of structural strength. Therefore, this paper focused on the effect of ship routes with numerical calculation by the direct load structure analysis and statistical analysis using data from full-scale measurements for a large container ship navigating in several routes. Finally, the short-term and the long-term hull girder stress are estimated with a statistical approach using the simulation results and measured data. Comparing with both results, it is found that the direct load structure analysis tends to give conservative evaluation for all ship routes. Keywords: Direct load structure analysis route
Container ship


Full-scale measurements
Ship

1 Introduction In classification societies’ rules or guidelines as well as the IACS CSR (Common Structural Rules), direct strength calculation by FEA is commonly required as one of the criteria for structural strength assessments, in which dominant design loads are described by simplified formulae with parameters expressing characteristic of a ship. Currently, increase in size of ships and design optimization have been advanced to save fuel consumption and achieve more efficiency in service. To ensure further safety of hull structures and enhance the rules for a new design or larger ships, it is necessary © Springer Nature Singapore Pte Ltd. 2021 T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 25–39, 2021. https://doi.org/10.1007/978-981-15-4672-3_2

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to constantly develop more reliable and appropriate approaches to estimating waveinduced loads and hull girder responses as well as evaluating structural strength in realistic sea environments. Direct load structure analysis is well known as one of the practical methods for the structural strength assessments. The advantage of this method is to make it possible to take into account an individual ship characteristic and directly estimate hull girder responses, combining with seakeeping analysis and FEA. In particular, the transfer function of hull stress range obtained by the direct load structure analysis is often used to estimate fatigue damage, combining with a wave spectrum, wave scatter diagram and S-N curve, which is so-called full spectral analysis. Some studies focused on the investigation on the validity of fatigue damage analysis including the combined method of seakeeping analysis and FEA by comparing data from full-scale measurements. For example, Li et al. (2014) [1] carried out a comparative study on various direct calculation methods and full-scale measurements to investigate the factors that influence fatigue loads on ship structures and evaluate the uncertainties associated with the direct calculation procedures. Thompson (2016) [2] validated the combined method of a linear frequency-domain hydrodynamic code with a finite element solver for spectral fatigue analysis using full-scale measurements at sea trial of a naval vessel and investigated the influence on fatigue damage due to four different wave spectrum models based on measured wave spectral data. Generally, there are various uncertainties derived from design assumption for the structural strength assessments. In classification rules the assumption of design loads on a hull is usually based on the operation in the North Atlantic Ocean for 25 years. However, sea states encountered by a ship depend on an operation route. In terms of more reliable and rational strength assessments, ship route is considered to be one of the critical factors that have the influence on the evaluation of structural strength. For example, Petricic et al. (2011) [3] investigated the ship route effect on the values of the long-term correlation coefficient between wave-induced sectional forces. Furthermore, ship route especially plays an important role in fatigue damage. Mao et al. (2012) [4] demonstrated the possibility and benefits of ship route planning which lead to a reduction in fatigue damage accumulation. The results show that the fatigue damage of the studied container ship structures could be decreased if awareness and knowledge of fatigue in ship route planning are employed. The main motivation of this work is to investigate the effect of ship routes on hull girder responses based on both of numerical calculation by the direct load structure analysis and data from full-scale measurements for the 8,600 TEU container ship. As representative routes where the ship actually navigated, the South China Sea (SCS), the North Indian Ocean (NI), the Mediterranean Sea (MS) and the North Atlantic Ocean (NA) are selected. Finally, the short-term and long-term hull girder stress in all routes are estimated with a statistical approach using simulation results and measured data.

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2 Full-Scale Measurements 2.1

Measured Data

Many full-scale measurements campaigns were widely performed in the past with great effort in an attempt to collect data of hull girder responses in actual sea states and validate design loads. Currently, hull stress monitoring system that can continuously monitor the real time status of a ship in a voyage and the technology for ship-to-shore telecommunication have been remarkably progressing in development and operation, and it is expected that these stored data are effectively utilized to enhance safer operation and improve the ship design and rules. In this study, measured hull girder stress data onboard for the 8,600 TEU container ship are available for about 4 years. During the full-scale measurements project, the ship was operated between East Asia and Europe through the Suez Canal in the first 2.5 years and then operated through the Cape of Good Hope for only route from Europe to East Asia in the remaining a half year. After that, the ship was launched to East Asia - North America route through the Suez Canal and the Cape of Good Hope in about 1 year. The main particulars of the ship are shown in Table 1. Hull girder stresses were measured with optical strand monitoring system (OSMOS) sensors. A total of twelve sensors were installed on the deck and lower part of the longitudinal bulkhead on both port and starboard sides at fore part, midship part and aft part of the ship, as illustrated in Fig. 1. Hull stress data in time series was continuously taken at sampling rate of 20 Hz. The navigational data including position, ship heading and ship speed were also recorded. Table 1. The main particulars of the 8,600 TEU container ship Length overall (LOA) Breadth Depth Design draft Gross tonnage

No.9(S) / 10(P) No.11(S) / 12(P)

abt 334.5 m 45.6 m 24.4 m 14.0 m abt 97,000 GT

No.5(S) / 6(P) No.7(S) / 8(P)

No.10/12

No.6/8

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No.1(S) / 2(P) No.3(S) / 4(P)

No.2/4

No.1/3 OSMOS sensor

Fig. 1. The positions of OSMOS stress sensors

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Sea States and Operational Conditions

The ship which is used in this study was equipped with a wave radar system so that wave conditions such as the wave headings, wave periods and significant wave heights can be measured in operation. However, it is considered that the wave radar system gives less reliable significant wave heights and an appropriate calibration is necessary if data of the significant wave heights observed through the wave radar is used (e.g. Andersen (2014) [5] and Stredulinsky et al. (2011) [6]). Actually, the wave radar system installed on the ship apparently provided overestimated significant wave heights from experiences. Accordingly, in this study hindcasted data provided by Japan Weather Association (JWA) were applied to investigate the relation between hull girder stress and sea states encountered by the ship. The following representative voyage routes encountered relatively harsh sea states by the ship were determined in a part of measurements periods. —SCS route —NI route —MS route —NA route

: : : :

3 2 3 3

voyages voyages (from East Asia to the Suez Canal) voyages (from the Suez Canal to the Strait of Gibraltar) voyages (from the Strait of Gibraltar to North America)

However, NI, MS and NA routes for wave hindcasting contain only one way from East Asia though the Suez Canal and the Strait of Gibraltar to North America. With these data for each route, the frequency of the significant wave heights and relative wave headings to the ship are shown in Figs. 2 and 3, respectively. According to hindcasted data, the North Atlantic Ocean is naturally the harshest sea area as well known. The ship has operated at over 6 m of significant wave height for a few times in NA route. In SCS route, it also presents the relatively high significant wave height. Relative wave headings are divided into eight parts at 45 degrees increments, that is 0 ± 22.5 (following seas), 45 ± 22.5 (quartering seas from starboard side), 90 ± 22.5 (beam seas from starboard side), 135 ± 22.5 (bow seas from starboard side), 180 ± 22.5 (head seas), 225 ± 22.5 (bow seas from port side), 270 ± 22.5 (beam seas from port side) and 315 ± 22.5 (quartering seas from port side) degrees. Overall, the frequency of head seas is relatively high in all routes. NI route significantly concentrated on 225 ± 22.5 degrees representing bow seas from the port side. This is considered to be due to including only one way voyage from East Asia to the Suez Canal. Although the number of hindcasted data may not be sufficient to comprehend the trend of wave directions in each route, it is found that wave headings encountered by the ship seem to be typically dependent on the ship routes.

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Statistical Analysis with Measured Data

Hull stress data in time series mainly contains low frequency component corresponding to wave frequency (WF) and high frequency (HF) vibration caused by transient hydroelastic responses like whipping and springing. The high frequency response that would have the most significant influence on the hull occurs at about 0.5 Hz, which is the natural frequency dominated by the 2-node vertical bending moment. The filtering of raw data in time series was carried out by the Fast Fourier Transform (FFT). However, this comparative study deals with only the WF response, so the HF component over 0.3 Hz is excluded by low-pass filtering. Figure 4 presents an example of the hull stress in time series data. 40 WF WF+HF

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Furthermore, the data with the ship speed less than 1.0 knot were excluded, regarding as operating near ports. With these data only including WF component, the hull stress range per wave was obtained by the zero-up crossing method and the hourly standard deviation of hull stress was calculated. To predict the long-term extreme value for waves or ship responses, Weibull distribution is typically used. The cumulative function of Weibull distribution is given by 
  x b F ð xÞ ¼ 1  exp  a

ð1Þ

where a and b represent scale parameter and shape parameter, respectively. These Weibull parameters were estimated by fitting data based on the least squares method. Figure 5 plots an example of Weibull fitting with WF component of hull stress. The statistical analysis was conducted for all available measured stress data in all routes, that is SCS, NI (East Asia - the Suez Canal), MS and NA route. However, in the longterm prediction for NA route also includes data operating along the European coastal. The results are presented in Sect. 4.

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3 Direct Load Structure Analysis 3.1

Seakeeping Analysis

A seakeeping analysis code based on linear potential flow theory in frequency domain can provide response amplitude operators (RAOs) of ship motions, dynamic wave pressures on a hull and so on. The seakeeping analysis was conducted with a linear strip theory code. Although several loading conditions should be determined to comprehensively investigate the influence on structure analysis due to different loading conditions, a full loading condition at the design draft was used as a representative condition for this study. The analysis condition of ship speed, wave lengths and wave headings are shown as follow: —Ship speed —Wave headings pffiffiffiffiffiffiffiffi —Wave lengths ( L=k)

: 5 knots : from 0 to 330 degrees at 30 degrees increments : from 0.5 to 2.0 at 0.05 increments

where L and k represent ship length and wave length, respectively. Ship speed for seakeeping analysis was determined based on the assumption of operational condition under the extreme sea state in design. 3.2

Structure Analysis

In this study, structure analysis was carried out to obtain the transfer functions of hull stress range using the full ship model in floor-span mesh size as illustrated in Fig. 6. Loads on a FE model contain static loads determined from the loading condition and dynamic loads obtained by the seakeeping analysis. Dynamic loads for FEA contain dynamic wave pressures and inertial loads due to ship motions and accelerations. Dynamic wave pressures and hydrostatic pressures are mapped on the surface of the hull. However, a linear theory in frequency domain cannot provide the dynamic pressures on the wetted area above the waterline. Therefore, in the case of positive pressure at the waterline, it is assumed that pressures are linearly distributed from the

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waterline to the top of the wetted area, which is determined by converting the pressure on the waterline into the water head. While, in the case of negative pressure at the waterline, the combined pressures of static and dynamic pressures are not allowed to be taken as negative values below the waterline. The internal pressures acting on tanks are not applied to the model, because the mass of full ship model contains those of ballast water, fuel oil and other constants by adjusting material properties. In addition, container cargos on deck and hatch covers were modeled with solid elements and shell elements, respectively. The loads of container cargos only in holds are applied to the model as nodal forces.

Fig. 6. Full ship FE model for structure analysis

3.3

Short-Term and Long-Term Predictions

In order to compare with the statistical data from full-scale measurements, the shortterm and long-term predictions were carried out by the use of the hull stress transfer functions at the same positions as stress sensors onboard. The short-term prediction has been widely used as a typical approach to estimate the statistical properties expressed by the standard deviation of various ship responses in irregular waves on a short-term sea state. The standard deviation of hull stress range per significant wave height can be obtained based on linear superposition principle with transfer functions at all regular wave headings and the Pierson-Moskowitz wave spectrum described in IACS Recommendation No. 34 [7] given as follows: ( )     Hs2 2p 4 5 1 2p 4 4 Sð x Þ ¼ x exp  x p Tz 4p Tz

ð2Þ

where Hs , Tz and x represent significant wave height, zero-up crossing wave period and wave frequency, respectively. The hourly standard deviation of hull stress range was calculated based on the sea states obtained by hindcasted data. For long-term prediction during lifetime of a ship, the probability of occurrence of ship responses can be estimated using standard deviation on sea states and a wave scatter diagram based on British Marine Technology (BMT) Global Wave Statistic data [8], which is described by the joint probabilities of occurrence of significant wave height and wave period. Sea areas in the worldwide are divided into 104 areas as illustrated in Fig. 7.

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Fig. 7. Definition of the extent of the North Atlantic [7]

Sea areas covering actual ship operation routes were determined as shown in Table 2. The wave scatter diagram for each area was generated by averaging the joint probabilities of occurrence, assuming that the occurrence probability is equal for each area. With these wave scatter diagram, the long-term extreme values at all headings was estimated. Table 2. Combined sea area in this study Route South China Sea (SCS) North Indian Ocean (NI) Mediterranean Sea (MS) North Atlantic Ocean (NA)

Area No. 40, 62 37, 50, 60, 61 26, 27 8, 9, 15, 16, 17, 24, 25

4 Results and Discussions First of all, the relations between the significant wave heights given by wave hindcasting and the hourly standard deviation of hull stress range are investigated in representative voyages encountered relatively harsh sea states by the ship during a part of measurements periods (see Sect. 2.2). As an example, Fig. 8 plots the results of No. 5 and 6 sensors installed on the deck of the midship part on starboard and port side, respectively. In NA route there seem to be more variations of hull stress with regard to significant wave height. Similarly, the relations of the standard deviation of stress range to the relative wave headings are plotted in Fig. 9. In NA route, there were considerable variations in the relative wave heading. In contrast the relative wave headings are concentrated on a part of range in other routes. For example, in NI route, hull stresses occurred at bow seas and beam seas from port side of 225 to 270 degrees. Actually, the highest frequency of the relative wave heading is 225 ± 22.5 degrees in NI route (see

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Fig. 3). This results in relatively higher stress on No. 6 sensor installed on the port side. Hull stresses on the weather side where waves approach seem to be higher than that on the other side. Figure 10 plots the hourly standard deviation of hull stress range by the numerical calculation and those estimated from measured data in each route. It is obvious that numerical calculation gives conservative results in comparison with measured data. Although various uncertainties can be considered in the process of short-term prediction, estimating wave loads is one of the most important factors in generating discrepancy between both the results. A linear strip method is known to provide larger values than experimental data for vertical bending moment (e.g. Miyake et al. (2002) [9]). In the direct load structure analysis as well, hull girder stresses due to the vertical bending moment would be particularly overestimated since the wave loads simulated by the seakeeping analysis are directly applied to the structure analysis. Besides, the uncertainties of hindcasted data and ISSC wave spectrum also may have influences on estimation of hull girder responses in irregular waves on a short-term sea state. Figure 11 presents the distribution of exceedance probability of hull stress range and Weibull fitting curves at No. 5, 6, 7 and 8 OSMOS sensors of the midship part in the same representative route, using all available data. Compared with these results, it is found that the ship is subjected to the severest wave load in NA route, and higher stress was especially measured at No. 5 and No. 6 sensors on the deck. It can be also seen that there are the slightly asymmetrical distributions between both sides in three routes except for NA route.

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Fig. 9. The relation between the relative wave heading and the standard deviation of hull stress range at No. 5 and No. 6 sensors

Finally, the long-term extreme values at exceedance probability of 10−8 at all headings are calculated in accordance with the procedure using the direct load structure analysis as stated in Sect. 3. On the other hand, with full-scale measurements data the values of exceedance probability are also estimated by Weibull fitting. In order to investigate the influence on hull girder stresses due to ship routes and the tendency by sensor positions, both results at all sensor positions were compared in each route. Figure 12 presents the ratio of the long-term extreme values at exceedance probability of 10−8 simulated by the direct load structure analysis to those from measured data. On the assumption of all headings, long-term prediction basically provides symmetrical extreme values at port and starboard sides. Depending on the difference of a ship route, measured data at both sides results in the asymmetrical extreme values. Actually, there seem to be the slightly asymmetrical distributions between both sides in three routes except for NA route. Especially, the hull stress on the position of lower part of the longitudinal bulkhead such as No. 7 and No. 8 sensors is more likely to be asymmetric because lower structures close to bilge

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part are more subjected to horizontal bending moment and torsion caused by oblique waves than deck sides. If relative wave headings to a ship are concentrated on one side, it is expected that asymmetric stresses occur at both sides especially in lower structures close to bilge part. More measured data may be needed for more accurate estimate on the long-term extreme values, however the long-term prediction on each ship route based on BMT Global Wave Statistic data results in larger values than statistical analysis with measured data. As previously known, the encountered wave height distribution is lower than the distribution provided by classification societies’ rules due to weather routing (e.g. Olsen et al. (2005) [10]). Accordingly, this would be because the effect of weather routing to avoid severe sea conditions is not included in the long-term prediction of hull girder responses although there are various reasons such as the effect of seakeeping analysis and Weibull distribution.

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5 Conclusions In this study, measured data onboard for the 8,600 TEU container ship are available for about 4 years. As representative routes where the ship actually navigated, four routes (SCS, NI, MS and NA route) are selected. First, the relations between the sea states and the hourly standard deviation of hull stress range are investigated with hindcasted data in representative voyages encountered relatively harsh sea states by the ship. In the short-term prediction of hull stress range, the numerical calculation provides larger results than statistical analysis with measured data in all routes. One reason for the discrepancy between both the results is that the seakeeping analysis would overestimate wave loads. The long-term extreme values at exceedance probability of 10−8 at all headings are calculated with the wave scatter diagram on each ship route based on BMT Global Wave Statistic data. On the other hand, with full-scale measurement data the long-term extreme values at the exceedance probability of 10−8 are also estimated by Weibull fitting. Finally, both results at all sensor positions were compared in each route. Measured data may not be enough to estimate more accurately the long-term extreme values in each route, however the statistical predictions using the direct load structure analysis tend to give conservative evaluation at all measurement points. Although there are various reasons such as the effect of seakeeping analysis and Weibull distribution, this would be also because the effect of weather routing is not included in the long-term prediction of hull girder responses. This study focused on the influence on the difference of a ship route which is one of the critical factors in terms of the structural strength assessments. However, it is important to comprehend various uncertainties for an appropriate and reliable assessment. Therefore, future work will focus on the effect of seakeeping analysis codes and design assumptions. Acknowledgements. The authors would like to gratefully acknowledge Kawasaki Kisen Kaisha, Ltd and Japan Marine United Corporation for providing full-scale measurement data and the cooperation associated with this work.

References 1. Li, Z., Mao, W., Ringsberg, J.W., Johnson, E., Storhaug, G.: A comparative study of fatigue assessments of container ship structures using various direct calculation approaches. Ocean Eng. 82, 65–74 (2014) 2. Thompson, I.: Validation of naval vessel spectral fatigue analysis using full-scale measurements. Mar. Struct. 49, 256–268 (2016) 3. Petricic, M., Mansour, A.E.: Long-term correlation structure of wave loads using simulation. Mar. Struct. 24, 97–116 (2011) 4. Mao, W., Li, Z., Ringsberg, J.W., Rychlik, I.: Application of a ship-routing fatigue model to case studies of 2800 TEU and 4400 TEU container vessels. Proc. Inst. Mech. Eng. Part M J. Eng. Marit. Environ. 226(3), 222–234 (2012)

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5. Andersen, I.M.V.: Full scale measurements of the hydro-elastic response of large container ships for decision support. Ph.D. thesis, Technical University of Denmark, Denmark (2014) 6. Stredulinsky, D.C., Thornhill, E.M.: Ship motion and wave radar data fusion for shipboard wave measurement. J. Ship Res. 55(2), 73–85 (2011) 7. International Association of Classification Societies (IACS): Standard Wave Data. Recommendation No. 34 (2001) 8. British Marine Technology: Global Wave Statistics. Unwin Brothers Limited (1986) 9. Miyake, R., Zhu, T., Kagemoto, H.: On the estimation of wave-induced loads acting on practical merchant ships by a Rankine source method. ClassNK Tech. Bull. 20, 81–93 (2002) 10. Olsen, A.S., Schrøter, C., Jensen, J.J.: Wave height distribution observed by ships in the North Atlantic. Ships Offshore Struct. 1(1), 1–12 (2005)

Three-Dimensional Shape Sensing by Inverse Finite Element Method Based on Distributed Fiber-Optic Sensors Makito Kobayashi1(B) , Takuya Jumonji2 , and Hideaki Murayama2 1 School of Engineering, The University of Tokyo, Tokyo, Japan

[email protected] 2 Graduate School of Frontier Science, The University of Tokyo, Tokyo, Japan

Abstract. Shape sensing, identifying the shape of the structure based on sensor, which was installed on the structure, is an important technology for structural health monitoring (SHM). It contributes monitoring the condition of the structure, feedback control, and fatigue estimation. By installing fiber-optic sensor on the structure like nerves, the deformed shape can be identified based on inverse finite element method (iFEM) from the measurement strain. The measurement error and the placement of fiber-optic sensor affect the result of shape sensing. Here we show the shape sensing of a bended stiffened panel in the simulation and verify the influence of the measurement error and the sensor placement. The deflection of the plate is identified with an error less than 1% of maximum deflection. The mean percent difference (MPD) calculated from the measurement strain giving Gaussian noise with a standard deviation of 5% of the maximum strain was 2.86%. Furthermore, with simple bending deformation, it was found that shape sensing can be performed with less sensor placement. Our results demonstrate that the realistic sensor performance and sensor placement enable shape sensing of local structure of ship structure such as stiffened panel. Keywords: Shape sensing · Inverse finite element method · Fiber-optic sensor · Structural health monitoring

1 Introduction Shape sensing, identifying the shape of the structure based on sensor, which was installed on the structure, is an important technology for structural health monitoring (SHM). The deformation of whole structure is critical information to monitoring the condition of structures. It enables feedback controls to keep small deformation and the history of deformation is useful to estimate fatigue damage accumulation especially for the structures frequently deformed by external force such as ships or other marine structures. Fiber-optic sensor is a sensor which can measure distributed strain in longitudinal direction along the fiber. Its measurement time, measurement range and accuracy are improved in recent years. Fiber-optic sensor can measure strain fast and accurately at the multiple points on the structure and is a promising sensor for shape sensing in real © Springer Nature Singapore Pte Ltd. 2021 T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 40–48, 2021. https://doi.org/10.1007/978-981-15-4672-3_3

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environment. Installing fiber-optic sensors on the structure surface like nerves can make it possible to identify the mechanical state of the structure [1]. Although Strain, which is a quantity which represents local deformation state is important itself for SHM, the displacement is calculated by integrating it. Inverse finite element method (iFEM) is applied to SHM by Tessler and Spangler [2], which is used to reconstruct displacement field form discrete strain based on finite element model. iFEM has a high robustness to erroneous input strain and can be applied to the problem with complicated geometry or boundary conditions. Kefal et al. applied iFEM to the bulk career model and simulated the bending deformation [3]. They estimated the deformation with high accuracy from noisy strain limited to only 2.26% of the total elements. The combination of fiber-optic sensor and iFEM is a strong method for shape sensing of rigid structure. Kobayashi and Murayama applied this to buckled pipe in the simulation [4]. The deformation of buckled pipe was identified using iFEM and fiber-optic sensors installed along four lines in longitudinal direction in simulation. The influence of performance of fiber-optic sensor on shape sensing was also investigated. Stiffened panel, which has high rigidity and strength with reduced weight by attaching stiffeners to the plate, is one of the typical local structures in ship structures [5]. In this study, shape sensing for a bending stiffened panel was simulated using iFEM based on strain measured by fiber-optic sensor. In the iFEM, only the plate part was modeled briefly. The influence on shape sensing of the placement of fiber-optic sensor and error of measurement strain was evaluated. The deformation was identified successfully with proper finite element model and sensor performance.

2 Methods 2.1 iQS4 Formulation iQS4, a four-node quadrilateral inverse-shell element, was adopted as element in the iFEM. iQS4 was developed by Kefal et al. [6] and has six displacement degrees-offreedom (DOF) including hierarchical drilling rotation. In each element, in-plane strain is expressed using membrane strain e and bending curvature k as ⎫ ⎧ ⎨ εx ⎬ (1) = e + zk = Bm ue + zBb ue , ε ⎩ y ⎭ γxy and out-of-plane shear strain g is 

γyz γxz

= g = Bs ue ,

(2)

where z is a thickness coordinate of the element, ue is a nodal displacement vector, and Bm , Bb and Bs are parts of strain-displacement matrix, which was shown in Kefal et al. [6].

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In-situ membrane strain eε and bending curvature kε are calculated from measurement strain on both top and bottom surface of each element. ⎫ ⎧ top ⎪ εx + εxbottom ⎪ ⎬ ⎨ 1 top εy + εybottom (3) eε = 2⎪ ⎭ ⎩ γ top + γ bottom ⎪ xy xy ⎫ ⎧ top ⎪ εx − εxbottom ⎪ ⎬ ⎨ 1 top εy − εybottom , kε = (4) 2h ⎪ ⎭ ⎩ γ top − γ bottom ⎪ xy xy top

top

top

where εx , εy , and γxy are measurement in-plane strain on top surface, bottom are on bottom surface, and h is a height of sensor from εxbottom , εybottom , and γxy neutral surface. Let gε be in-situ out-of-plane shear strain. 2.2 iFEM Formulation iFEM is formulated based on weighted least squares method, minimizing the following objective function φ e , φe =

3 i=1

wim ϕim + ¨

ϕim

= ¨

ϕib = ¨ ϕis =

3 i=1

wib ϕib +

2 i=1

wis ϕis

(5)

2 e ei u − eiε dxdy

(6)

e 2 ki u − kiε dxdy

(7)

e 2 gi u − giε dxdy,

(8)

where ei , ki , and gi are the components of strain function e, k, and g respectively, eiε , kiε , and giε are the components of in-situ strain eε , kε , and gε respectively, and wm , wb , and ws are the weight coefficient of each strain components and they are positive values. Next, find optimal total nodal displacement vector u that minimizes the objective function φ e over whole structure. This means to determine the total nodal displacement that minimizes the difference from the measurement strain. The nodal displacement vector ue that minimizes the objective function φ e satisfies the following linear equation, ∂φ e = Ke ue − f e = 0, ∂ue

(9)

where Ke is a matrix calculated by taking the sum of second-order integral of the product of parts of strain-displacement matrix, and f e is a vector function of in-situ strain. Although this equation optimizes objective function φ e of each element, in order

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43

to optimize the nodal displacement in the entire structure, the equation of each element is merged, which the element node number is replaced with the entire node number. Let K and f be merged Ke and f e respectively. Finally, total nodal displacement vector u is calculated all at once by solving following linear simultaneous equations Ku = f.

(10)

Since the number of sensors to be installed and the surface of structures on which sensors can be installed are limited, most elements cannot be given some or all the in-situ strain components practically. By setting the ungiven strain component to zero and its weight coefficient to very small value (not zero) in such an element, the deformation of whole structure can be estimated from limited measurement strain. 2.3 Evaluation Index of Shape Sensing To evaluate identified shape, mean percent difference (MPD) was used.   1 N Xiest − Xi  × 100, MPD = i=1 max(X) N

(11)

where N is the number of nodes, X is true nodal displacement and X i is its component, and Xiest is corresponding estimated nodal displacement. MPD represents the estimated displacement error of the entire structure.

3 Shape Sensing of Bending Stiffened Panel 3.1 Overview of the Simulation Figure 1 shows the overview of simulation. First, deformation and strain of bending stiffened panel was simulated based on Finite Element Analysis (FEA) using ANSYS®, a commercially available FEA program. Next, strain on the placement of the fiber-optic sensor was extracted and Gaussian noise was given. Then, deformation was calculated from strain based on iFEM. In-house iFEM code was used. Finally, results of iFEM were compared to true deformation based on FEA and the error was evaluated. 3.2 Problem Setting Figure 2 shows a stiffened panel which is simulated in this work. The size of plate is 6 meters in height and 2.4 meters in width and the thickness is 10 mm. Three stiffeners are attached in longitudinal direction at intervals of 0.8 meters, and two stiffeners in transverse direction intervals of 3 m. The height of longitudinal stiffeners is 0.2 meters, that of transverse stiffeners is 0.4 meters, and the thickness of both stiffeners is 20 mm. The plate and stiffeners are made of structural steel. The stiffened panel was fixed in short side as a cantilever plate and bent in the positive direction of the z axis by the equally distributed load on the surface. When adding the equivalent distribution load of 100 Pa, the maximum displacement is about 1.64 mm. Fiber-optic sensor was installed on both sides of the plate along the red line A and B in Fig. 2 in the simulation.

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Fig. 1. The overview of simulation.

Fig. 2. The design of stiffened plate (Units: mm). Fiber-optic sensor was installed on the both sides of plates along two red lines. (Color figure online)

3.3 The Model of Stiffened Panel The finite element model of stiffened panel used in FEA is shown in Fig. 3. The model has 13271 nodes and 12450 elements, and it consists of all shell elements. On the other hand, iFEM was applied to a model consisting only of plates without reinforcement (see Fig. 4). The model has 697 nodes and 640 elements. The displacements of these nodes were optimized in iFEM. 3.4 Input Strain and Setting of Weight Coefficients The deformed shape was calculated from three cases of input strain. Case1 : All strain including out-of-plane shear strain γyz and γxz for all elements, top

Case2 : Strain εx and εxbottom on the line A and B,

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Fig. 3. The model used in FEA. The edge on the green line was fixed. (Color figure online)

Fig. 4. The model used in iFEM. The element includes red lines were given the measurement strain. The edge on the green line was fixed. (Color figure online) top

Case3 : Strain εx and εxbottom on the line A. It is difficult to install the sensor as in Case 1, but it is important to know the applicability of the method. In Case 2 and Case 3, the Gauss noise whose standard deviation is set to 0, 1, 5 or 10% of maximum strain was added (see Fig. 5). Strain was given only to elements of 17.4% in Case 2 and 8.70% in Case 3. All weight coefficients were set to 1 in Case 1. On the other hand, the weight coefficients w1m and w1b of the element to which the sensor belongs were set to 1, and the others were set to 10−5 in Case 2 and Case 3 depending on the lack of input strain. 3.5 Results In all input strain cases (Case 1, Case 2, and Case 3 without Gaussian noise), the result of iFEM corresponds reasonably well with the true deflection based on FEA (see Fig. 6). Figure 7 shows the deflection error of the center line of the plate in the longitudinal direction estimated based on iFEM without noise under the equivalent distributed load of 100 Pa. In all cases the deformed shape is identified with an error less than 1% of maximum deflection.

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Fig. 5. The example of input strain on the line A.

Fig. 6. The result of iFEM under the equivalent distributed load of 100 Pa.

Fig. 7. The deflection errors.

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The following Table 1 gives a comparison of MPD including input strain with error. MPD under noise is mean and standard deviation of 100 samples. As the noise increased, the mean and standard deviation of MPD increased in Case 2 and Case 3. The difference of mean of MPD with 10% noise between Case 2 and Case 3 was not statistically significant. Table 1. Comparison of MPD mean and standard deviation (in brackets) Case

No noise

1% noise

5% noise

10% noise

1. All strain

0.209%

–

–

–

2. Strain on the line A and B

0.298%

0.795% (0.340)

3.68% (1.97)

6.89% (3.72)

3. Strain on the line A

0.514%

0.805% (0.404)

2.86% (1.84)

5.46% (4.24)

3.6 Discussions In this study, iFEM was applied to a simple model (see Fig. 4) with only a plate, but the shape could be estimated with high accuracy. Since iFEM only uses the strain-displacement relationship, a model with a detailed structure is considered to not significantly affect the results. In Case 2 and Case 3, it can be seen from the Fig. 7 that the error of deflection of the center line of the plate changes obviously around the position that x is 1.5 m and 4.5 m where transverse stiffeners exist. This change of deflection error can be attributed to the modeling of these stiffeners, but the results do not change much. With simple bending deformation, it was found that the shape can be identified with less sensor placements. It is necessary to verify the influence of sensor placement also in the case where shape sensing is performed for the entire ship structure instead of the local structure, or shape sensing for the structure with complex load such as torsion load and biaxial in-plane load in addition to bending deformation.

4 Conclusion We simulated shape sensing for a bending stiffened panel using iFEM based on strain measured by fiber-optic sensor and evaluate the influence on shape sensing of the placement of fiber-optic sensor and error of measurement strain. In the iFEM, only the plate part was modeled briefly, and fiber-optic sensor was installed on both sides of the plate in the longitudinal direction. The deflection of the plate was identified with less than 1% error of the maximum displacement. The mean percent difference (MPD), which represents the estimated displacement error of the entire structure, calculated from the measurement strain with Gaussian noise whose standard deviation was set to 5% of the maximum strain was 2.86%. Furthermore, with simple bending deformation, it was found that small sensor

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placement provides that shape sensing with high accuracy. Our results suggest high feasibility of shape sensing of local structure of ship structure such as stiffened panel based on realistic sensor performance and placement.

References 1. Murayama, H.: Structural health monitoring of composite materials using distributed fiberoptic sensors. In: Rajan, G., Prusty, B.G. (eds.) Structural Health Monitoring of Composite Structures using Fiber Optic Methods, pp. 105–156. CRC press (2016) 2. Tessler, A., Spangler, J.L.: Inverse FEM for full-field reconstruction of elastic deformations in shear deformable plates and shells. In: 2nd European Workshop on Structural Health Monitoring (2004) 3. Kefal, A., Mayang, J.B., Oterkus, E., Yildiz, M.: Three dimensional shape and stress monitoring of bulk carriers based on iFEM methodology. Ocean Eng. 147, 256–267 (2018) 4. Kobayashi, M., Murayama, H.: Shape sensing for pipe structures by inverse finite element method based on distributed fiber-optic sensors. In: 26th International Conference on Optical Fiber Sensors (2018) 5. Cui, J., Wang, D., Ma, N.: Case studies on the probabilistic characteristics of ultimate strength of stiffened panels with uniform and non-uniform localized corrosion subjected to uniaxial and biaxial thrust. Int. J. Nav. Archit. Ocean Eng. 11(1), 97–118 (2019) 6. Kefal, A., et al.: A quadrilateral inverse-shell element with drilling degrees of freedom for shape sensing and structural health monitoring. Eng. Sci. Technol. Int. J. 19(3), 1299–1313 (2016)

Research on the Strengthening of Double Bottom Floors or Girders Against Bottom Slamming Wenbo Zhu1,2 , SuSu Zhou1 , Jiameng Wu1,2,3,4(B) , Fan Zhang1 , and Shijian Cai1,2 1 Marine Design and Research Institute of China, Shanghai, China

[email protected] 2 Shanghai Key Laboratory of Ship Engineering, Shanghai, China 3 State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, China 4 Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration,

Shanghai, China

Abstract. Bottom slamming is normally considered as an impact scenario and calculated by quasi-static approach for scantlings in structural rules. Most studies for slamming are focused on local supporting members (LSM) including plates and stiffeners. However, for primary supporting members (PSM) such as double bottom floors and girders, the requirements and calculation results by different Class Rules for strengthening against slamming loads are of great difference, which is worthy of further study. Prescriptive requirement, known as rule formula based method and direct calculation are commonly carried out for the scantling of PSMs against bottom slamming. The purpose of the paper is to clarify some details in the strengthening of double bottom PSMs, so that the relevant calculation methods can be correctly used in practical design work. By comparing 4 combinations of load mode and boundary condition, we proposed a formula for coefficient distribution of maximum shear force. Furthermore, direct calculation method is used for calibration by case studied, which shows that the modified distribution factor in this paper is more reasonable than that required in CSR. Finally, a practical way of applying the patch loads to the finite elements concerning the real structural arrangement and mesh size. Keywords: Primary supporting member · Bottom slamming · CSR · Prescriptive requirement · Direct calculation

1 Introduction Slamming load is a local dynamic load with high peak value and short duration, which makes it uneconomical to carry out dynamic response analysis for ship designers. Therefore, generally reduction factors such as dynamic load factor are adopted to simplify the slamming load as static value for industrial application. Common Structural Rules for Bulk Carriers and Oil Tankers (CSR) [1] consider slamming as an impact scenario and still takes quasi-static approach for scantlings. It was found by many calculations that for plates and stiffeners, although different requirement for slamming loads and criteria, similar scantling results will be gained based on different Class Rules. However, © Springer Nature Singapore Pte Ltd. 2021 T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 49–74, 2021. https://doi.org/10.1007/978-981-15-4672-3_4

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for primary supporting members (PSMs) such as double bottom floors and girders, the requirements and calculation results by different Class Rules for strengthening against slamming loads are of great difference, which is worthy of further study. Hu [2] also have the same conclusion. For the scantling calculation of PSMs, prescriptive requirement and direct calculation are commonly needed. Prescriptive requirement, known as rule formula based method, is only applied for PSM with single span, where the grillage effect may be ignored. For complex arrangement of PSMs, only direct calculation method, including double bottom grillage analysis and finite element analysis, could be carried out to evaluate the strength. Whichever the method is, there are two ways to deal with the slamming load that PSM suffers: One is using uniformly distributed slamming load; while the other is using patch load to simulate the slamming load. Up to now, classification societies do not have a consistent approach. In this paper, we proposed a practical way of applying the patch loads to the finite elements concerning the real structural arrangement and mesh size. In CSR, patch loads for slamming are used for the strengthening of bottom PSM against slamming impact with prescriptive requirement and/or direct calculation method. However, it is verified by the Shipbuilder Industry that it is quite conservative in prescriptive CSR requirement (only for single span load model) for one factor f dist for the greatest shear force distribution along the span of PSM. By theoretical deduction for single span load model, the maximum value of f dist is modified considering the relationship between the shear span of PSM and the patch load scope. Furthermore, direct calculation method is used for calibration by case studied, which shows that the modified distribution factor in this paper is more reasonable than that required in CSR.

2 Background and Potential Problems About PSM in CSR Generally, the class societies have two approaches of dealing with the slamming force on bottom PSM, which are distributed load considering a reduction coefficient and patch load considering the local slamming. For example, in DNVGL’s structural rules [3] the effective pressure acting on primary supporting member (PPSM ) equals to 0.4 times slamming pressure (PSL ) for bottom PSM strengthening. Meanwhile, in CSR patch load is adopted. It is similar with CSR-OT [4] and CSR-BC [5] with some small difference, as shown in Table 1. However, it is found by the Shipbuilding Industrial that the requirement for strengthening of PSM against slamming in CSR is too conservative, i.e. after direct strength calculation of bottom PSM against slamming according to CSR a large margin is found for yield utilization (proposed by KOSHIPA 2007) [6]. With a further study, f dist should take the main responsibility.

3 Study on the Shear Force Distribution of PSM Against Bottom Slamming 3.1 Greatest Slamming Force on a PSM with Single Span Due to the requirement of structural arrangement, in some cases the span of bottom floor (L fl ) will be much shorter than span of bottom girder (L gir ), to be more specific L fl /L gir

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51

Table 1. Rule requirement of PSM strengthening against slamming CSR Patch Ideal load area

ASL =

CSR-OT 1.1LBCb 1000

Aslm =

Practical ASL = bSL lSL area √ Length lSL = ASL , but no more than 0.5l shr √ Width bSL = ASL , but no more than S  3  2 +2 Correction fpt = 0.5 fSL − 2fSL coefficient where f = 0.5b /S SL

Coefficient distribution of maximum shear force along the span

SL

1.1LBCb 1000

ASL = bslm lslm √ lslm = Aslm , but no more than l shr √ bslm = Aslm , but no more than S  3  2 +2 fpt = 0.5 fslm − 2fslm where fslm = 0.5bslm /S

CSR-BC / A = Sl l S CA = 3/A, and 0.3 ≤ CA ≤ 1.0 0.5

L: Rule length of ship, m. B: Moulded breadth of ship, m. C b : Block Coefficient. l shr : Effective shear span of the PSM, m. S: Spacing of the primary support members, m.

≤ 0.6, in which the bottom grillage can be considered mainly supported by the floors. In these cases, the grillage effect may be ignored and the bottom floors can be treated as single-span beam. Generally, this will suit for oil tanks and fore and aft cargo hold areas of bulk carriers. For PSM with single span, the slamming force is transmitted from longitudinals. In CSR, the slamming force bore by PSM with single span is treated as a patch load. Considering the longitudinals under patch load will transmit the slamming load to the floors on the two ends, the center of patch load should be placed on the floor in order to distribute greatest slamming force to the floor, as shown in Fig. 1. X represents longitudinal direction. Y represents transverse direction. Longitudinals are rigidly fixed at the floors. The impact load that each longitudinal undertakes is PSL l SL /n as shown in Fig. 2. n = l SL /s. R is the reacting force of the floor supporting one longitudinal. The support of floor provided by bottom side tank or longitudinal bulkhead is between simply support and fixed support, as shown in Fig. 3. Z represents vertical direction.

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Fig. 1. Model of bottom beam system

Fig. 2. Mechanical model of bottom longitudinal

Fig. 3. Mechanical model of bottom floor

According to beam theory, the slamming load on the floor transmitted by the longitudinals on one side can be expressed as:      0.5bSL 3 PSL lSL bSL 0.5bSL 2 · Rone−side = −2 +2 (1) 2 2 S S Where, PSL is bottom slamming pressure, i.e. patch load. lSL is length of the patch load. bSL is width of the patch load.

Research on the Strengthening of Double Bottom Floors or Girders

53

So, the greatest slamming force carried by the floor is: Rtotal = 2Rone−side = FSL · fpt

(2)

 3  bSL 2 +2 ,f Where, FSL = PSL lSL bSL , fpt = 0.5 fSL − 2fSL SL = 0.5 S . F SL , f pt and f SL are the same as which are in CSR. S is PSM spacing. As a result, the shear force calculation of PSM with single span is transferred as the shear force calculation of a beam under variable concentrate or distributed load, with both supports between simply and fixed support. 3.2 Simulation with Different Combination of Load and Boundary Conditions Distributed Load and Simply Supported Boundary Condition According to beam theory, the shear force along a beam with distributed load and simply supported boundary condition can be expressed as:

QSL−sim

  a lSL for 0 ≤ y ≤ a − 0.5 QSL−sim = FSL fpt 1 − lshr lshr   a lSL lshr y − a for a < y ≤ a + lSL = FSL fpt 1 − − 0.5 − · lshr lshr lSL lshr   a lSL QSL−sim = FSL fpt − for a + lSL < y ≤ lshr − 0.5 lshr lshr

(3) (4) (5)

Where, a represents the position of starting point of slamming load. lshr is shear span. Consider a as variable, it can be found that for a given position y along the beam, QSL-sim reaches its maximum at a = y. Here a = a − l SL /2. a is the distance between center of load and beam ends, as shown in Fig. 3. Further, we can get: QSL−sim = FSL fpt fdist−sim

(6)

Where, the shear force distribution coefficient f dist-sim is: fdist−sim = 1 −

y lshr

− 0.5

lSL lshr

(7)

When lshr  l SL, it can be simplified as l SL /l shr = 0. In the meantime, the distributed force turns to a concentrated load: y fdist−sim = 1 − (8) lshr  Considering the practical design and rule requirement, we have: 0.2 ≤ lSL lshr ≤ 0.5. Table 2 gives the values of shear force distribution coefficient f dist-sim for certain cases, while Fig. 4 display f dist-sim along the span with different lSL/ l shr respectively.

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W. Zhu et al. Table 2. Calculation of shear force coefficient 1 y/l shr l SL /l shr 0.0 0.2

0.3

fdist−sim = 1 − l

0.4 y shr

0.5

− 0.5 llSL

shr

0.0

1.00 0.90 0.85 0.80 0.75

0.05

0.95 0.85 0.8

0.1

0.9

0.8

0.75 0.7

0.65

0.2

0.8

0.7

0.65 0.6

0.55

0.3

0.7

0.6

0.55 0.5

0.45

0.4

0.6

0.5

0.45 0.4

0.35

0.5

0.5

0.75 0.7

0.4 0.35 0.3 0.25  For 0.5 < y lshr ≤ 1.0, the results  are symmetric about y lshr = 0.5.

Fig. 4. Display of shear force coefficient 1

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Distributed Load and Rigidly Fixed Boundary Condition According to beam theory, the shear force along a beam with distributed load and rigidly fixed boundary condition can be expressed as: ⎡

QSL−fix

2 lshr −a−0.5lSL a+0.5lSL SL + 2 0.5l · · l l l shr shr ⎢

2

shr 3 ⎢ Rtotal ⎢ − 2 0.5lSL · a+0.5lSL · lshr −a−0.5lSL − 2 0.5lSL · lshr −a−0.5lSL lshr lshr l l lshr = lshr ⎢  shr  shr ⎢ lSL  





3 ⎣   y−a y−a−lSL SL SL − + 2 0.5l · a+0.5l + lshr lshr lshr  a+l  a lshr SL 2 0.5l lshr ·

lshr −a−0.5lSL lshr

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

SL

(9) Same as in above sub section, maximum shear force for y ≤ a and y ≥ a + lSL are symmetric. The value depends on a and lSL /l shr . For a given y, it reaches its maximum at a = y. Equation (9) can be simplified as: QSL−fix = FSL fpt fdist−fix fdist−fix

        y lSL 3 y lSL 2 y lSL = 0.5 + 2 −1 +3 −1 lshr lshr lshr lshr lshr lshr 3 2   y y +2 −3 +1 (10) lshr lshr

Table 3 gives the values of shear force distribution coefficient f dist-sim for certain cases, while Fig. 5 display f dist-fix along the span with different lSL /l shr respectively. Table 3. Calculation of shear force coefficient 2 y/l shr l SL /l shr 0.0

0.2

0.3

0.4

0.5

fdist−fix 0.0

1.00 0.96 0.92 0.87 0.81

0.05

0.99 0.93 0.88 0.82 0.76

0.1

0.97 0.89 0.83 0.77 0.70

0.2

0.90 0.78 0.71 0.64 0.57

0.3

0.78 0.65 0.57 0.50 0.50

0.4

0.65 0.50 0.50 0.50 0.50

0.5

0.50 0.50 0.50 0.50 0.50  For 0.5 < y lshr ≤ 1.0, the results  are symmetric about y lshr = 0.5.

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Fig. 5. Display of shear force coefficient 2

Concentrated Load and Simply Supported Boundary Condition Simulate the slamming force on PSM with single span as a group of concentrated forces, i.e. assuming all the slamming forces are transmitted from longitudinals to floors. The amount n of concentrated force is decided by lSL and longitudinal spacing s. Considering the less concentrated force the greater the shear force will be, n and concentrated load P for each concentrated load are defined as: n = int(lSL /s) FSL · fpt Rtotal = n n According to beam theory, the shear force can be expressed as:       a n−1 s   QSL−sim = nP 1 − −  P − ... −  · − P   a+(n−1)s lshr 2 lshr a P=

(11) (12)

(13)

For a given position y, it reaches its maximum at a = y, which can be simplified as: QSL−sim = FSL fpt fdist−sim

(14)

Where, fdist−sim = 1 −

y lshr

−

n−1 s · 2 lshr

(15)

As expected, for single concentrated load case i.e. n = 1, Eq. (15) turns to (8). Further, substituting n = l SL /s, get: fdist−sim = 1 −

y lshr

− 0.5

lSL s + 0.5 · lshr lshr

(16)

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When s < 0 2n lshr lshr 2n lshr

Eq. (17) − Eq. (7) = (1 −

i.e. Case 3 always have larger f dist at y/l shr = 0 than case 1. Substitute y/lshr = 0 into Eq. (20)–Eq. (10):       lSL 2 (n − 1)(2n − 1) lSL 3 (n − 1)2 · − · +1 Eq. (20) − Eq. (10) = 0.5 lshr n2 lshr 2n2       lSL 3 lSL 2 − 0.5 − +1 lshr lshr    3n − 1 2n − 1 lSL lSL 2 = − 2n2 2n2 lshr lshr Given lSL /l shr < 1, get   lSL 2 3n − 1 2n − 1 lSL − 2n2 2n2 lshr lshr    lSL 2 3n − 1 2n − 1 > − >0 2n2 2n2 lshr 

Eq.(20) − Eq.(10) =

i.e. Case 4 always have larger f dist at y/l shr = 0 than case 2. Accordingly, we give the following suggestion for practical design: • Take 0.5 as the coefficient for mid span

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Fig. 12. Comparison between rule requirement and the four combination cases

• Take f dist-max as the coefficient for the position no more than one longitudinal spacing from the ends. f dist-max equal to the larger of case 3 and case 4, which covers all four cases as verified.  2 3  n(n−1)2 (n−1)(2n−1) s s s fdist−max = max 1 − n−1 · , 1 − + 2 l 2 l 2 l shr

shr

• Using linear interpolation for the rest of the span as shown in Fig. 13.

Fig. 13. Proposed distribution of shear force coefficient f dist

shr

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4 Study on the FEM Analysis of Shear Force Distribution of PSM Against Slamming Direct analysis for strength evaluation of bottom PSMs against slamming is allowed in CSR, but it does not give a clear requirement about FEM method, position of load, boundary condition, accuracy of slamming force modelling etc. It may lead to a misunderstanding by industrial community and plan approval engineers. Once wrong method is used, the evaluation results will possibly be distorted, which may cause safety problem and economic loss. In this chapter, the foremost cargo hold of a 320,000 DWT VLCC (without back brackets of vertical web frames on longitudinal bulkheads) is adopted for case study. 4.1 Study on Load Position In order to get the maximum shear force, how to simulate the load area is studied. According to Sect. 3.2, for a given position x, the maximum shear force can be acquired by taking a = x, as shown in Fig. 14. However, when doing 3D FEM calculation, we can only read the shear stress instead of shear force of the section. Moreover, shear stress depends on the results of element center and element geometry. A floor with 13 sections is adopted as case study. The mesh size equals to longitudinal spacing as shown in Fig. 15. According to what is discussed in chapter 3, it can be expected that both loading approach in Fig. 15 will lead to a slight underestimated of shear force.

Shear force distribution

Fig. 14. Schematic diagram of load position

A three cargo hold structure as shown in Fig. 16 is used for FEM verification. Out shell, inner bottom, bulkheads, floors and platforms are modeled with shell elements. Longitudinals are model with beam elements. The results for section 11 and 2 under 1 Section 1 represents the closest section with the end of floor. P_n represents the load starts from

section n.

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various load position with two boundary conditions are given in Table 10 and 11 respectively. Boundary condition 1 represents the boundary condition applied for three cargo hold analysis (see Part 1 Chapter 7 Section 2.5 in [1]). Boundary condition 2 represents fixed condition at the connection between hopper and bottom side holds of both port and starboard. The slamming loads extends four longitudinal spaces which will be further discussed in next section. For load conditions P_11, P_12 and P_13 partial of the loads acts on the elements outside the boundary, which is beyond longitudinal bulkhead.

Shear force distribution

Shear force distribution

Fig. 15. Case study

It is found that shear stress is larger under boundary condition 2. For section 2, load position 3 (the next section of target section) gives the greatest shear stress for both boundary conditions. For section 1, in boundary 1 same phenomenon is found. Figure 17 shows that with load case P_1 the maximum shear stress occours at the element closest to bottom shell. With load case P_2, section 1 has its greatest shear stress at the element in middle, with average value larger than load case P_1. We think P_2 leads to more accurate results as results of P_1 may be affected by concentration effect caused by the fixed boundary condition. So we suggest apply the load from the section next to the section which is about to check.

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Fig. 16. Half FEM model for case study (structures below hopper areas shown only)

Table 10. Results for boundary condition 1 P_1 P_2 P_3 P_4 P_5 P_6 P_7 P_8 P_9 P_10 P_11 P_12 P_13 Section 1 109 (MPa)

119 111 96.8 81.3 67.2 54.7 43.8 34.4 26.2

Section 2 71.5 102 109 97 (MPa)

79.9 64.5 51.8 41

31.8 24

19.5

14.6

11.1

17.8

13.2

9.9

Table 11. Results for boundary condition 2 P_1 P_2 P_3 P_4 P_5

P_6

P_7

Section 1 156 151 130 111 92.8 76.5 62 (MPa) Section 2 102 133 139 120 100 (MPa)

P_8

P_9

P_10 P_11 P_12 P_13

49.3 38.2 28.7

82.2 66.6 52.9 41.1 30.9

21

15.4

11.4

22.7

16.7

12.4

4.2 Study on Load Area √ In our case, ASL = 4.1844 m but 0.5lshr = 3.6 m, so l SL is set equal to 3.6 m. Mesh size equals to longitudinal spacing, which means theoretically patch load should be applied to 4.235 elements. Practically, we have to apply the load to 4 four 5 elements while keeping the total force unchanged, the specific values are shown in Table 12. Figure 18 demonstrates the comparison of shear stress between the two approaches. As expected, the more concentrated load (approach 1) leads to more conservative results.

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Fig. 17. Shear stress of section 1 for load case P_1 (top) and P_2 (down)

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Table 12. Patch load adjustment Theoretical value

Approach 1

Approach 2

Length

3.6 m

3.4 m (4 elements)

4.25 m (5 elements)

Width

4.1844 m

3.23 m (4 elements)

4.85 m (6 elements)

PSL (KN/m2 )

978.72

1342.50

715.26

4.3 Study on Load Mode For elements that are not square, we also have to adjust the patch load. In this case, we use a group of concentrated load applying to nodes on the floor while keeping the total force unchanged. The result is shown in Fig. 19. Comparing the result in Sect. 4.2, the maximum shear stress is increased from 119 MPa to 123 MPa. The difference is under 5%, so when restricted by the element shape equivalent concentrated force can be used to check the strength of the floor resisting slamming loads. 4.4 Study on Coefficient Distribution of Max. Shear Force of PSM In order to verify the coefficient distribution proposed in chapter 3 and study the influence of boundary conditions, case studies with five different boundary conditions are carried out. The five boundary conditions are shown in Table 13. The double bottom structure is shown in Fig. 20. Table 14 displays the results for case 2. The results are further discussed in Sect. 4.5. 4.5 Comparison Between FEM Results and Theoretically Value Figure 21 gives the maximum shear stress distribution along the span of floor. From the figure, we can tell that: • Under bhd without B condition, the stress is symmetric and has its maxima at both ends. The trend is same as CSR rule. So, both left and right end structures provide almost same support for the floor. • Results of knu & Lbhd fix without B condition is similar as bhd without B condition, except its shear force at ends are about 30 MPa higher which is caused by the fixed condition on hopper and longitudinal bulkhead. • knu fix without B, knu & CL fix without B and knu fix, CL sym. without B result in nearly same results. The difference of symmetric point is caused by the different boundary conditions. Most importantly, all the results by FEM analysis are cover by the distribution we proposed which is lower than the rule requirement.

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Fig. 18. Shear stress of section 1 for approach 1 (top) and 2 (down).

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Fig. 19. Shear stress under concentrated loads.

Table 13. Boundary conditions Case Boundary condition

Description

1

bhd without B

Apply boundary condition on fore and aft end of model as the strength evaluation of fore cargo hold

2

knu fix without B

Beside the boundary condition on model ends, the model is fixed at hopper knuckle

3

knu & Lbhd fix without B

Beside the boundary condition on model ends, the model is fixed at hopper knuckle and longitudinal bulkhead (y = 11050 mm)

4

knu & CL fix without B

Beside the boundary condition on model ends, the model is fixed at hopper knuckle and centerline girder

5

knu fix, CL sym. without B Beside the boundary condition on model ends, the model is fixed at hopper knuckle. Symmetric boundary condition is applied on centerline girder

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Fig. 20. Double bottom structure

Fig. 21. Comparison between FEM results and theoretically value. Horizontal axis represents the position along the floor with 1 and 14 representing left (22100 off C.L in Fig. 20) and right (11050 off C.L in Fig. 20) end of the floor respectively.

−47.6

−59.5

−13.9

−22.1

28.1

48.5

45.3

40

35.5

32.6

30.3

28.9

29.5

32.9

Section 3 (MPa)

Section 4 (MPa)

Section 5 (MPa)

Section 6 (MPa)

Section 7 (MPa)

Section 8 (MPa)

Section 9 (MPa)

Section 10 (MPa)

Section 11 (MPa)

Section 12 (MPa)

Section 13 (MPa)

42.8

39.1

38.6

40.6

44.1

48.9

54.7

58.6

38.9

−95.3

−107

−75.5

Section 2 (MPa)

53.2

49.4

48.9

52

57.5

63.4

67.7

48.6

24.2

−115

−117

−124

−113

Section 1 (MPa)

P_3

P_2

P_1

−46.3

−9.67

64.7

60.8

60.6

65.6

72.2

76.7

77

73.5

74.4

80.5

85.7

66.8

90.8

88.5

89.6

94.8

75.9

30.2

−65.7

−55.5

21.2

−18.8

57.8

106

105

106

85.4

39.5

120

117

96.9

49.3

6.43

−61.2

−55.3

−74.6

−64.5

−69.8

−83.5

−27.8

−50.4

−47.8

−48

−50.9

P_8

−73.7

12

−63.9

−59.5

−59.3

−62.2

P_7

−37.2

−78.8

−73.5

−72.1

−74.6

P_6

−93.2

−88.8

−87

−88.3

P_5

−83.7

−104

−103

−103

P_4

−19.5 −22.9

−27.5 −10.9 126

113

77.4

79.4

40.7

−16.3 109

61.4

24

−26.4

−37

56.8

15.2

−17.1

−14.4

−12.7 −26.3 15.5

−22.5

−19.3 −34

−29.7

−11.7

−11

−10.4

−10

−9.86

−10.2

−11.4

P_13

−46.6

−17.4

−16

−15

−14.3

−14

−14.3

−15.9

P_12

−43.3

−25.9

−23.5

−21.8

−20.6

−20

−20.3

−22.2

P_11

−36.8

−38.2

−33.8

−31

−29

−28

−28.3

−30.6

P_10

−56.5

−52.6

−46.9

−42.1

−39.1

−37.5

−37.7

−40.4

P_9

Table 14. Maximum shear stress for case 2: knu fix without B Research on the Strengthening of Double Bottom Floors or Girders 73

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5 Conclusion and Further Study To conclude, the CSR rule formula will lead to a too conservative result for bottom SPM resisting slamming loads. The proposed coefficient distribution for maximum shear force is verified and suits engineering application. FEM analysis is also carried out and suggestions are made for choosing load position, load area, load mode. The influence of boundary condition for FEM analysis is also discussed. In future, more case studies should be carried out to summarize the actual rigidity of the support by FEM analysis. Such that we can reduce the weight of PSM while assuring the safety of the structure.

References 1. IACS: Common Structural Rules for Bulk Carriers and Oil Tankers, 01 January 2018 2. Hu, W., Zhu, C., Wu, D.: On the greatest shear force of primary supporting members due to bottom slamming loads of CSR. Ship Ocean Eng. 47(2), 6–10 (2018). (in Chinese) 3. DNVGL Rules for classification Ships, July 2018. http://www.dnvgl.com 4. IACS: Common Structural Rules for Double Hull Oil Tankers, July 2012 5. IACS: Common Structural Rules for Bulk Carriers, July 2012 6. Shipbuilders’ Experience & Feedback on IACS Common Structural Rules. The Korea Shipbuilders’ Association. Tripartite Meeting (2007)

A Study on Dynamic Response of Flat Stiffened Plates to Slamming Loads Considering Fluid-Structure Interaction Dac Dung Truong1,2 , Beom-Seon Jang1(B) , Han-Baek Ju3 , Sang Woong Han1 , and Sungkon Han1 1 Research Institute of Marine Systems Engineering, Department of Naval Architecture

and Ocean Engineering, Seoul National University, Seoul, Republic of Korea {ddtruong,seanjang,playda,sungkon.han}@snu.ac.kr 2 Department of Mechanics Engineering, Nha Trang University, Nha Trang City, Khanh Hoa Province, Vietnam 3 Department of Naval Architecture and Ocean Engineering, College of Engineering, Seoul National University, Seoul, Republic of Korea [email protected]

Abstract. The slamming phenomenon commonly exists in many practical engineering areas. When slamming occurs, structures are likely subjected to slamming load characterized by high pressure within a short time duration. This load can cause damages to structures. In this study, slamming loads on flat stiffened aluminum plates of high-speed vessels and its dynamic response are investigated numerically. The numerical method is validated against the relevant experimental data from the open literature. The explicit finite element with Multi-Material Arbitrary Lagrangian-Eulerian solver is adopted to simulate the slamming impact of flat stiffened plates. Bilinear strain hardening with no strain-rate hardening was considered for the numerical simulation of the aluminum plate models. The effect of the heat-affected zone (HAZ) on the response of flat stiffened aluminum plate is also numerically evaluated and found to be considerable. The effect of structural flexibility, impact velocity, and air cushion are investigated through parametric studies. The results showed that the impact pressure and deflection of the flat stiffened aluminum plate increased with the impact velocity, while higher rigidity of the plate resulted in a less deflection and higher pressure as well as shorter impact pressure duration. In addition, the air cushion effect was found to be significant which reduced peak impact pressure, deflection of the aluminum plate model and lengthened the impact pressure duration. The discussion of predicted results of slamming pressure and deflection of the stiffened plate model are presented in detail. Keywords: Slamming pressure · Deflection · Wet drop test · Fluid-structure interaction · Flat stiffened aluminum plate

© Springer Nature Singapore Pte Ltd. 2021 T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 75–99, 2021. https://doi.org/10.1007/978-981-15-4672-3_5

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1 Introduction Marine structures are commonly subjected to impact pressures such as slamming loads when hitting the water during its service. These slamming loads defined as high pressure in a very short time duration. A slamming phenomenon occurring on a flat plate of marine structures are typically detected on a small high-speed vessel which has a lightweight like aluminum ships and with large ships as well. This load can cause local and global damages to structures. In this paper, slamming loads acting on flat stiffened aluminum panels of high-speed vessels and its dynamic response are studied numerically. Many studies on slamming pressure on a rigid body using water entry models at a prescribed velocity have been reported. Von Karman [1] analytically studied water entry problems based on potential theory. Wagner [2] proposed a new approach to the study of the water entry of two-dimensional bodies with small local dead-rise angles under the assumption of the potential-flow problem without gravity and the absence of air cavity formation during impact. Later, based on Wagner’s theory other researchers [3–6] have studied slamming problems. Many types of impacted body shapes relating to engineering practices, such as the wedge, sphere, cone, cylinder, and flat plate, have been employed to study water impact characteristics [7–25] with various attack angles. It is found that the shape of the impacted body surface strongly affects slamming loads in terms of pressure peak, impact duration, and pressure profile. Nevertheless, many studies published, for example Refs. [7, 8, 11, 14, 18, 20] have analyzed water entry problems in cases where the impacting body considered is rigid or elastic and, accordingly, there is no presence of permanent deformation due to water impact. In reality, water impact on the flat plate more frequently occurs in ships and offshore applications and can cause structural damage due to the high peak and long duration of the impact loads. Impact pressure can be influenced by structural deformation during the slamming event. Several researchers have investigated the slamming response of deformable structures considering the hydroelasticity effect [8–10, 12, 13, 15–17, 19, 21– 25]. Regarding stiffened plate, Chuang [8] experimentally and analytically investigated the water entry problem for plates, wedges, and ship bottoms using elastic, elastic-plastic and rigid materials; He found that hydroelasticity and air cushion effects on slamming response were significant, showing a decrease and an increase in the maximum impact pressure and impact duration of water entry, respectively; Mori and Nagai [9] and Mori [10] investigated the response of a bottom plate to slamming by performing the series of wet drop tests on full- and small-scale stiffened aluminum models of the bottom plates of high-speed crafts. The theoretical results of the structural response obtained using the finite strip method, which agreed well with the test results, were provided; Cheon et al. [22] investigated numerically how slamming pressure acts on a deformable flat stiffened steel plate; Recently, Sun and Wang [24] experimentally and numerically investigated hydrodynamic impacts on the stiffened side of elastic stiffened steel plates. So far little effort has been focused on the prediction of slamming loads acting on flat aluminumalloy structures of light-weight ships which are being increased in size and speed. In this study, the slamming pressure on flat stiffened aluminum plates of bottom part of high-speed vessels and its structural response were studied numerically in consideration of deformable behavior of structures.

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77

In addition to the experimental and analytical methods, numerical calculations based on coupled FEM, Smooth Particle Hydrodynamics (SPH), Boundary Element Method (BEM), Computation Fluid Dynamics (CFD) and Arbitrary Lagrangian–Eulerian (ALE) have been developed and successfully applied to solve slamming problems in marine engineering applications. Among them, the SPH and ALE methods, as pointed out by Anghileri et al. [26], are two of the most common choices in the latest studies, since these methods had fewer limitations and cost less CPU time when compared to others. Both the coupling parameters in the ALE method and the particle quantities in the SPH method, however, should be properly determined prior to any analysis. Many ALEbased numerical studies on the water impacts on elastic structures have been reported. Many researchers studied the slamming problems of marine panels based on the ALE method [11–17, 20–25]. It can be concluded that these methods are generally capable of predicting slamming loads on various impacted 2D and 3D surface shapes and their dynamic responses, provided that relevant parameters are correctly defined. However, there are still challenges in applying numerical methods to water impact problems. Also, these methods require very careful selection of mesh size and contact parameters, not to mention high computational effort, in order to archive accurate estimations. In this paper, the slamming phenomenon acting on a flat stiffened aluminum plate and its structural response considering the coupling between fluids and structures based on the ALE formulation and penalty contact method are investigated numerically. The developed numerical method is validated with the relevant experimental data from the open literature by comparison of slamming pressure and deflection data. Air cushion effect is also highlighted. Based on the validated model, a parametric study is carried out on the stiffened aluminum plate to clarify the various effects of impact water velocity and structural flexibility on slamming loads and its structural responses as well.

2 Brief Description of Test Data In this paper, wet drop tests on flat stiffened aluminum plates, conducted by Mori [10], will be employed for validation of the present numerical simulations. Wet drop tests of two stiffened aluminum-alloy plates representing the bottom of a high-speed craft on the water surface conducted by Mori [10] are shown in Fig. 1. The dimensions of the two stiffened aluminum alloy plates, along with the points at which the water pressure, strain, and deflection were measured, as shown in Fig. 1b. Model No. 1 represents the bottom of an actual high-speed craft, while model No. 2 was scaled down so that its stiffener rigidity was half of that of model No. 1. The mechanical properties of the aluminum alloy material of the models are summarized in Table 1. Table 1. Mechanical properties of aluminum model material [10]. Material

Yield strength, σY [MPa]

Young’s modulus, E [MPa]

Hardening modulus, E h [MPa]

Density, ρ [kg/m3 ]

Poisson’s ratio, ν

Al 5083-O

134.4

68,700

2,750

2,700

0.3

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The drop test was repeated 19 times for model No. 1, while changing the drop height from 0.3 to 1.6 m, and it was repeated seven times for model No. 2, while changing the drop height from 0.3 to 1.0 m. The water pressure, strain, and deflection of the models were measured by the gauges attached to the models, as shown in Fig. 1b. The total mass including hanging jig and test models was 800 kg. Detailed information regarding the experiment can be found in Ref. [10]. It should be noted that only the first drop of model No. 2 was utilized in this study in order to eliminate the effect of repeated impacts on the subsequent drops, as presented in Table 2. Table 2. Test conditions of aluminum model No. 2 for the first drop. Model

Drop height, h [mm]

Theoretical impact velocity, V [mm/s]

Actual impact Total mass [kg] velocity, V i (assumed as 0.95% of V ) [mm/s]

No. 2

300

2,426.11

2,298.25

800

903 256.5

31 TYP

256.5

246

100

856

1000

120 P1

A

156.3 TYP

TYP 31

30 D1 D2 S3 S1,S2 P2

B 72

t = 6 mm

Model No. 1

A : Hanging jig B : Test specimen

(a)

Model No. 2

56 S3 6 S1 S2

6

40

6

74

TYP

94

140.16

St. PL12

6 S1 S2

(b)

Fig. 1. (a) Schematic of drop test set-up (reproduced from Ref. [9]), (b) geometry of flat stiffened aluminum plates (units: mm) (reproduced from Ref. [10]).

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3 Numerical Modeling The wet drop test on stiffened aluminum plate No. 2 described above was numerically simulated using the ALE method with a penalty algorithm. The numerical pressure and deflections were compared with the relevant test results to validate the nonlinear explicit codes, in order to investigate impact pressures and dynamic structural response of aluminum plate, then utilized in the parametric study that is presented in a later section. To gain accurate simulation results, structural geometry, material properties, and impact conditions should be simulated as closely as possible to the experimental conditions. The FSI model consists of two parts, fluid including air and water, and a stiffened plate. The fluid was modeled using solid elements that are defined according to the MultiMaterial Arbitrary Lagrangian-Eulerian (MMALE), which is the most versatile and widely used 1-point ALE multi-material element; the stiffened plate was modeled by shell elements by the classical Lagrangian approach using the default Belytschko-Tsay element formulation. The MMALE solver in the finite element analysis codes LS-Dyna is available to solve the mathematical formulations for slamming problems. In the present study, the commercial software LS-DYNA version 971 R7.1.1 with single precision was employed for the simulations. The effect of gravity was taken into account. 3.1 Fluid-Structure Interaction This section describes the relevant equations constituting this method. The ALE formulation can be derived from the relation between the time derivative of the material and that of a reference geometrical configuration. For the MMALE method, equations can be solved in two steps. The Lagrange approach is carried out in the first step. No material flows through the element boundary, so mass conservation is satisfied. The equilibrium equations of the velocity and energy can then be computed. In the second step, the mass, momentum, and energy of materials are transported through the boundaries of the element. The velocity and displacement of each node are then updated. 3.2 Model Description The explicit finite element analysis is based on MMALE solver with a penalty coupling method. This method has been successfully used to study water impacts and events involving fluid-structure interaction as mentioned previously. The fluid is described as a Eulerian formulation while the stiffened panel is described by a Lagrangian approach. It should be noted that in the present study, water surface tension effects were not considered. To reduce time consumption, according to the symmetry conditions of the model, a quarter FSI model with symmetric boundaries on the X-0-Z and Y-0-Z planes was analyzed, as shown in Fig. 2. Structure Model. To accurately reflect the structural behavior, the actual material properties used in the tests were applied to the present numerical model. Structure modeling is based on the assumptions of no deformation for support plates used in Mori’ tests, while the deformable material model is assumed for stiffened aluminum plates. Shell elements and material types of elastic-plastic were employed to define the materials of

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stiffened aluminum plates. Five integration points were defined through thickness and element formulation of Belytschko-Tsay (default) for the stiffened aluminum plate. It should be noted that for simplicity the support plates and hanging jig were not modeled. To reflect the weight of all of the drop components, the total mass given in Table 2 was assigned to the all nodes of edge plates of the aluminum model which were shown in blue (see Fig. 2) in the simulation, through using the mass_node_set option in LS-DYNA. It was noted that only ¼ total mass (200 kg) was set for the quarter plate model. All these nodes of the edge plates were clamped so as to move freely only in the z-direction.

Sym. Y

Sym. X Air

Fixed in x-direction Free in z-direction

Water Non-reflection

Fig. 2. Boundary conditions applied to present numerical simulations of the aluminum model.

To define the material of the stiffened aluminum plate, material model from the LSDyna library, *024-Mat_Piecewise_Linear_Plasticity, was selected. This material model allows for the definition of a true stress-strain curve as an offset table. The mechanical properties of the model materials applied to the finite element models for Mori’s tests are listed in Table 1. A simple linear hardening expression for the true stress-strain relation, proposed by Park and Cho [27], was employed using Eq. (1) with an isotropic hardening model. The justification for applying this hardening model for the numerical simulation was confirmed by Truong et al. [28]. It should be noted that no strain-rate effect was applied to Mori’s test model since aluminum is well-known as an insensitive material. σtr = σY +

E × Eh εpl,tr E − Eh

(1)

In addition, for the welded aluminum plate model, the HAZ needed to be investigated. As stated in Ref. [29, 30], the HAZ causes strength reduction of welded aluminum structures due to the reduction of stiffness of material at the welded parts. However, the

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600

True stress [MPa]

500 400 300 Al 5083-O

200

Al 5083-O_0.9HAZ

100

Al 5083-O_0.67HAZ

0 0

0.025

0.05

0.075

0.1

0.125

0.15

True plastic strain [-] Fig. 3. True stress-plastic strain curves applied to present numerical simulations of aluminum models.

level of reduction depends on the weld method and type of aluminum material. For 5xxx series material, the knockdown factors of 0.9 and 0.67 (reduction of material strength of 10% and 33%, respectively) and no HAZ were adopted for the simulations. Each knockdown factor was applied to both yield strength and the entire true stress-plastic strain curve of Al 5083-O, implying that the strain hardening characteristics of Al 5083O are maintained in the HAZ. The true stress-strain curves for the base and the HAZ material applied to the numerical simulations are shown in Fig. 3. The HAZ regions are shown in red in Fig. 4b. The width of the HAZ from the welding center line was taken to be 20 mm for this study, following the recommendation of Ref. [30]. Fluid Domain. Fluids including water and air were modeled, for 8-node brick elements, with the Solid164 element from the LS-DYNA material library. They were assigned with *Mat_Null with no shear stiffness or yield strength, which imparts fluid-like behavior. The equation of state (EOS) has to be used to define the pressure in a fluid-like material [31]. The EOS of the water was defined by the Gruneisen model, which defines pressure for compressed materials as in Eqs. (2) and (3), while for the air state, a perfect gas model, one with zero shear strength as indicated in Eq. (4) [31], namely a linear polynomial, was employed.   ρ0 C 2 μ 1 + 1 −

 γ0  a 2 2 μ − 2μ

p=  μ2 1 − (S1 − 1)μ − S2 μ+1 − S3

μ3 (μ+1)2

2 + (γ0 + aμ)Ei

(2)

and for expanded materials as

where μ =

p = ρ0 C 2 μ + (γ0 + aμ)Ei

(3)

 p = C0 + C1 μ + C2 μ2 + C3 μ3 + C4 + C5 μ + C6 μ2 Ei

(4)

ρi ρ0

−1

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where C, S 1 , S 2 , S 3 , and a are the constants; C i (i = 1 to 6) is the hydrodynamic constant; ρ0 is typically the density in the nominal or reference state, usually a non-stress or nondeformed state, and ρi is the current density; ρi /ρ0 is the ratio of current density to reference density, which equals the current normalized volume, and E i is the internal energy per unit reference volume calculated by LS-DYNA. The initial internal energy E 0 and initial relative volume V 0 were set at t = 0 moments. The material properties of the fluid are important in the FSI simulations. The suitable set of fluid properties may be validated through several case analyses. In this study, however, as the present study the fluid properties values following the suggestions of Ref. [32] was applied, which showed reasonably good representation fluid. Detailed definitions of the parameters in Eqs. (2) to (4) along with their values for water and air are shown in Table 3. The fluid domain in a water impact problem is an infinite or very large domain. Usually, to save time computation the fluid domain is reduced. The limitations of the fluid domain should then be defined in order to prevent the boundary effect. In this study, typically the width and length of the fluid domain were about three times the width and length of the structure model, respectively, and the depth of the fluid domain was more than three times the structure height. This observation was based on the set of preliminary analysis results and was also consistent with Ref. [33]. The detailed dimensions of the fluid domain are given in Table 4. Figure 4 shows the quarter numerical model of the Mori’s tests including the mesh regions of the fluid domain with an initial gap between the water surface and plate as well as the meshed aluminum models. The penalty-based coupling treats the FSI problem between a structure (Lagrangian formulation) and a fluid (ALE formulation). It allows fluid to flow around the contact surface of a Lagrangian structure but not penetrate the contact surface. And at the same time, the ALE algorithm will search for the elements intersected or overlapped between the Lagrangian parts and the ALE multi-material groups. Then, in the remap step, it computes the penetration distance of the Lagrangian surface across the ALE material surface. The penalty coupling behaves like a spring system, and its magnitude is calculated proportionally to the penetration and spring stiffness as in Eq. (5). At the coupling interface, both master and slave nodes are subjected to the coupling force F, however evaluation of k d in coupling problem is by user via defining factor f . F = ks d = f

KA d V

(5)

where k s is the critical contact stiffness; d is the penetration distance; K is the bulk modulus of the fluid element; f represents the user-supplied scale factor for the interface stiffness; A is the average area of structure element, and V is the volume of the fluid element that contains the master fluid node. The coupling mechanism between the MMALE and the stiffened plate model is controlled by the keyword *Constrained_Lagrange_In_Solid, and its parameters are manually defined to properly reflect the physical behavior of interactions. The penalty factor f , in Eq. (5), is a multiplier of the contact stiffness between the materials in contact. To clarify the effect of this factor, values of f of 0.05, 0.1 and 0.5 were selected for the analysis model. It is observed that the difference in calculated pressures and deflection of plates with penalty factors 0.05, 0.1 and 0.5 was minor, as shown in Fig. 5, which

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Table 3. EOS coefficients of fluid models. *EOS_Linear_Polynomial Parameter

*EOS_Gruneisen Air

Parameter

Water

Density, ρ0 [kg/m3 ]

1.225

Density, ρ0 [kg/m3 ]

1,000

C 0 [MPa]

0.0

Sound speed of fluid, C [m/s]

1,647

C 1 [MPa]

0.0

S 1 [-]

1.921

C 2 [MPa]

0.0

S 2 [-]

−0.096

C 3 [MPa]

0.0

S 3 [-]

0.0

C 4 [-]

0.4

γ0 [-]

0.35

C 5 [-]

0.4

First-order volume correction, a [-]

0.0

C 6 [-]

0.0

Initial internal energy, E 0 [MPa]

0.2895

Initial internal energy, E 0 [MPa]

0.25

Initial relative volume, V 0 [-]

1.0

Initial relative volume, V 0 [-]

1.0

Table 4. Dimensions of fluid domain (units: mm). Fluid (air and water) domain No. 2

Initial gap

L1

L2

L3

L4

L5

L6

L7

L8

hi

900

1,400

1,000

1,500

300

1,000

200

300

100

Remark Quarter model

consistent with the observation of Luo et al. [14] and Cheon et al. [22]; thus a penalty factor f of 0.1, as the default, was used for the present simulations. Because the mesh size of the structure and the impact fluid domain are identical, the number of coupling points distributed over each coupled Lagrangian surface segment was set by default to 2. A normal direction with compression and tension for the coupling direction was defined. The minimum volume fraction was set as zero, and as the water velocity was relatively low, and so no leakage control was employed. In order to avoid reflection of the impulse wave at the boundary, all sides of the fluid except for the free surface of air were defined as non-reflecting boundaries, as shown in Fig. 2. To save computation time, the initial position of the stiffened plates was located on the water surface with an initial gap much less than the drop height in the tests, as illustrated in Fig. 2. Meanwhile, the initial vertical velocity was assigned to the model. It is noteworthy that the initial gap had to be large enough to impose the air cushion effect. Based on a set of preliminary analyses, an initial gap hi of 100 mm was selected for the simulations of the aluminum models. It should also be noted that due to the effect of gravity, the impact velocity at the contact moment would be higher than at the initial

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L7

L8

L5

hi

L2

L6

L3

L1

L4

(a)

HAZ region

(b) Fig. 4. (a) Quarter FSI model for Mori’s tests including mesh regions, (b) close-up view of the meshed model (air is not shown) with HAZ regions.

stage (at 100 mm of drop height); thus, adjustment of the initial falling velocity was made through several analysis cases to achieve the same impact velocity V i as in the tests. The initial velocity was then found to be roughly 1950 mm/s which yielded the impact velocity of 2298.25 mm/s. The computation time was set as 0.05 s and output time interval was averaged as 0.002 s. It should be noted that there were significant noises with narrow duration as well as higher peak pressure value when output time interval became smaller. Since the slamming pressure was not predicted well, the averaged pressure over several elements was applied. This method was based on the previous observation of the previous study [22]. In present study, the pressures were obtained by averaging pressure over roughly 100 × 100 mm area at P1 and P2 locations.

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3.5 D2_EXP D2_f0.05 D2_f0.1 D2_f0.5

Deflection, D2 [mm]

3 2.5 2 1.5 1 0.5 0 0

-0.5

0.01

-1

0.02

0.03

0.04

Time [s] (a)

0.2

P2_EXP P2_f0.05 P2_f0.1 P2_f0.5

0.15

Pressure, P2 [MPa]

0.05

0.1 0.05 0 0

0.01

-0.05

0.02

0.03

0.04

0.05

Time [s] (b)

Fig. 5. Effect of penalty factor (a) on a deflection and (b) on an impact pressure.

3.3 Mesh Convergence Study It is known that numerical results are sensitive to the ALE mesh refinement in finite element analysis including FSI simulations. The meshes of regions of interest such as the interaction area or impact region need to be fine enough to cause no numerical problems and result in accurate results, whereas coarser meshes far from regions of interest may be favorably considered to remarkably reduce computation time. In order to find the best compromise between accurate results and computational time, a mesh sensitivity study of the impact region was first conducted. For this, an aluminum model (without HAZ) with dropping at 0.3 m height was employed to examine the effect of mesh size on the simulation results. Three cases with mesh sizes of 20 mm, 10 mm and 6 mm (the same as the plate thickness), were uniformly generated for the stiffened plate and fluids impact domain (L1 × L3 × (L5 + L7)), while bias meshes with 1/5, 1/10 and 1/15 outwards for case 1, case 2 and case 3, respectively, also were considered. For better coupling simulation, the mesh size of the structure was generated the same as that of the fluid in the impact domain, as suggested by Wang and Guedes Soares [20]. In the

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simulations, the value of penalty factor f in Eq. (5) was set to 0.1. A server workstation comprising 16 CPUs (2 × Intel Xeon E5-2650 v3, 8-core 2.00 GHz) with 128 GB memory banks (DDR3 1600 MHz), running the CentOS Linux release 7.2 operating system of 64 processors, was employed to execute the simulations. Table 5 lists the main parameters of the convergence tests for the three cases of the aluminum model. Table 5. Parameters of convergence tests for aluminum model. Quantity

Case 1

Case 2

Case 3

Mesh size

20 mm

10 mm

6 mm

Number of shell elements (stiffened plate)

694

2,699

7,549

Number of solid elements (fluid)

135,300

799,680

3,116,820

Total nodes

144,228

829,192

3,190,930

CPU time

0 h 57 min

1 h 5 min

6 h 18 min

Figure 6 shows the impact pressure at P1 and the deflection time histories at D2 for the models with different mesh sizes together with the experimental measurements for comparison. As seen in the figure, the three cases show that the results agreed relatively well with the experimental data in terms of the peak value of pressure. However, a better prediction of maximum deflections was achieved for case 2 (10 mm). The case 3 with mesh size of 6 mm resulted in the impact pressure result close to the test data. However, the overestimation result of deflections was observed for case 3, as shown in Fig. 6, while the great time-consumption required for this computation case (see Table 5). Also, when the mesh size was 20 mm, the predictions underestimated the deflection measurements. It was concluded that the model with the 10 mm mesh size is best suited to capturing the time history of impact pressure and deflection on the stiffened aluminum plate and requires a reasonable computation time. Therefore, the mesh size of 10 mm is to be used for subsequent simulations of the aluminum models in this study.

4 Validation of Numerical Simulation and Discussion 4.1 Validation of Numerical Simulation The predicted deflection and impact pressure of the flat stiffened aluminum model, as compared with the test results, are shown in Fig. 7. Good agreement of deflection and impact pressure was achieved between the numerical analysis and the experimental results, except for the pressure at the plate center of the aluminum model. The numerically observed oscillations in the pressure and deflection after the first peak of slamming impact were however somehow less than those of the test results. There was also an overestimation of the slamming pressure and deflection, as well as underestimation of impact duration. The possible cause of the difference between the predicted deflection and the test results could be attributed to the actual material of the HAZ. Since there was no data on the

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Deflection, D1 [mm]

2 D1_20 D1_10 D1_6 D1_EXP

1.5 1 0.5 0 0

0.01

0.02

0.03

0.04

-1

Time [s] (a)

5

D2_20 D2_10 D2_6 D2_EXP

4

Deflection, D2 [mm]

0.05

-0.5

3 2 1 0 0

0.01

0.02

0.03

0.04

0.05

-1 -2

Time [s] (b)

0.2

P1_20 P1_10 P1_6 P1_EXP

Pressure, P1 [MPa]

0.15 0.1 0.05 0 0

0.01

0.02

0.03

0.04

0.05

-0.05 -0.1

Time [s] (c)

Fig. 6. Deflection time history at (a) D1, (b) D2, and (c) impact pressure time history at P1 with different mesh sizes for a drop height of 0.3 m.

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Deflection, D1 [mm]

1.5 1 0.5 0

0

0.01

0.02

0.03

0.04

0.05

-0.5 -1

Time [s]

3.5

D2_EXP D2_NUM_No HAZ D2_NUM_0.9HAZ D2_NUM_0.67HAZ

Deflection, D2 [mm]

3 2.5 2 1.5 1 0.5 0 -0.5

0

0.01

-1

0.02

0.03

0.04

0.05

Time [s] (a)

0.2 P1_EXP P1_NUM_No HAZ P1_NUM_0.9HAZ P1_NUM_0.67HAZ

Pressure, P1 [MPa]

0.15 0.1 0.05 0

0

0.01

0.02

0.03

0.04

0.05

-0.05 -0.1

Time [s]

Fig. 7. Comparison of prediction and test results for (a) deflection and (b) impact pressure time histories of aluminum model No. 2.

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Pressure, P2 [MPa]

0.2

89

P2_EXP P2_NUM_No HAZ P2_NUM_0.9HAZ P2_NUM_0.67HAZ

0.15 0.1 0.05 0

0

-0.05

0.01

0.02

0.03

0.04

0.05

Time [s] (b) Fig. 7. (continued)

strength reduction of the HAZ region, the assumption of using some knockdown factors following the literature recommendations was applied to clarify the effect of the HAZ. As seen in Fig. 7, with consideration of the HAZ, the predicted deflection became greater and increased when the smaller knockdown factor was used, implying that the HAZ makes a model weaker. Additionally, the effect of the HAZ on slamming pressure in this particular case was found to be negligible. However, reasonable agreement between the prediction of pressure P1 and the experimental data was achieved, whereas the predicted P2 was somehow overestimated. Nevertheless, the predictions of both positions P1 and P2 were generally correlated with the test results in terms of the initial stage of the impact process and the impact duration as well. The other reason possibly causing the differences in test results and numerical predictions may be attributed to the definition of boundary conditions for the stiffened aluminum plate. In the test, the stiffened plate was mounted on the hanging jig (see Fig. 1a), so it was presumed that there were possibly small movements of the plate during impact, implying that the boundary conditions were somehow different from the constraints used ideally in the numerical model, as illustrated in Fig. 2. It is, however, premature to draw any conclusions on the absolute reason for the bias. More tests seem to be necessary to clarify this matter. All in all, considering the large uncertainties inherent in predicting slamming problems, these predicted results are reasonably acceptable. Therefore, it may be concluded that the FSI modeling strategy developed in this study is able to accurately simulate the slamming phenomenon for marine structures. For a better understanding of key parameter such as air cushion that affects the slamming phenomenon, the following discussion is presented. 4.2 Effect of Air Cushion During the process of slamming on a flat stiffened plate with zero dead-rise angles, the impact pressure acting on the plate is mainly induced by the compressed air cushion

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Deflection, D2 [mm]

6 5 4 3 2 1 0 -1

0

0.01

-2

0.02

0.03

0.05

Time [s] (a)

0.3

P2_EXP P2_NUM P2_NUM_No air

0.2

Pressure, P2 [MPa]

0.04

0.1 0 0

0.01

0.02

0.03

0.04

0.05

0.03

0.04

0.05

-0.1 -0.2

Time [s] (b)

-1600

Velocity [mm/s]

-1700

0

0.01

0.02

-1800 -1900 -2000 -2100 -2200 -2300 -2400

Impact Time [s]

Without air With air

(c) Fig. 8. (a) Deflection, (b) impact pressure and (c) impact velocity time histories at plate center for an aluminum model with and without air at a drop height of 0.3 m.

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rather than by direct water slamming. Here, to clarify the air cushion effect on the slamming phenomenon, a comparison between the simulation of the model with and without air is presented. The air material was replaced by *Mat_Vacuum in LS-Dyna to reflect the void condition of the air. The water and structure remained the same as in the original models, and the drop height was 0.3 m for the aluminum models. The deflection and impact pressure time histories at the plate center in the new model were compared with those of the original model and together with the experimental results for validation, as illustrated in Fig. 8. Shown in Fig. 8 are the deflection and impact pressure time histories at the plate center of the aluminum model and impact velocity time histories of the plate. Under the void condition, due to the absence of the air cushion effect, there was a sharp pressure peak with a higher magnitude and a shorter duration compared with those of the original model. It is noted that the initial velocity assigned to the plates is adjusted in order to yield the identical impact velocities while keeping the same initial gap for both cases (see Fig. 8c). It was apparent that the slamming response (deflection and pressure) started relatively later than in the case having air, as there was no coupling between the structure and the air. In the case of the model with air, as the water entered the cells adjacent to the structure, the mixture of air and water contacted the structure, and coupling between the two started. The coupling lasted until the cells were filled with more water and as the entrapped air, in turn, was compressed as much as possible. This made the pressure peaks

Air Plate

No air layer

Air layer

Water

at max. impact pressure event

at max. impact pressure event

Air Plate

Water

No air layer

at the end of the simulation (a)

Air layer

at the end of the simulation (b)

Fig. 9. Interaction between water and structures max. impact pressure and end of simulation event for model No. 2 (a) without air and (b) with air resulted in air layers.

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smooth. On the other hand, for the model without air, the coupling began immediately after the cells were fully filled with the water, and the duration of the coupling was quite short. This made the pressure peak steeper and sharper than in the model with air. These observations are consistent with those from Chuang [8], Luo et al. [14], Cheon [23] and Sun and Wang [25]. It was concluded that the air cushion equivalently resulted in lower impulsive pressure, and subsequently led to less damage to the structures. The contact between fluid and structure is visually clear in Fig. 9, which displays the interaction between water and structure at maximum impact pressure event and end of the simulation event with and without air. As seen in Fig. 8a, no air layer/pocket was formed (due to the absence of air), while with the presence of air, the air was locally entrapped between the structure and the water in the middle of the plate, resulting in a long water jet and higher pile-up water, as shown in Fig. 9b.

5 Parametric Study The validated model was employed to perform a parametric study to examine the effect of the change in water impact velocity and the rigidity of the bottom surface of the stiffened plate (the change in thickness of the bottom plate) on the slamming load and structural response using the simulations for the model No. 2 as a basis model. At the max. impact pressure event

At the end of the impact pressure event

(a)

(b)

(c)

Fig. 10. Von-Mises stress distribution of quarter of model No. 2 with different impact velocities (a) V i = 2.3 m/s, (b) V i = 3.26 m/s and (c) V i = 4.21 m/s (unit of stress: MPa).

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5.1 Effect of Impact Velocity The model No. 2 was employed to investigate the effect of impact velocity on the slamming load acting on the stiffened plate and its structural response considering air cushion effect. Two cases with impact velocities of 3.26 m/s and 4.21 m/s were simulated. The von-Mises (effective) stress distribution of a quarter of model No. 2 at the maximum impact pressure event and end of the simulation event with different impact velocities is shown in Fig. 10. Significant stress was distributed at the end of stiffener and edges as well as near the junction of the stiffener and the plate. Lower stresses were located in the middle region of the quarter model. Stress level at maximum impact pressure event was greater than that at the end of the simulation event. These stress distributions of impacted plate increased when the impact velocity was increased, as expected. Also, as can be seen in Fig. 11, the deformation gradually increased when the impact velocity was increased. The slamming pressures at P1 and P2 against the impact velocity is plotted in Fig. 12. The pressure oscillated widely with high impact velocity. It can be seen from 4.5 D1_V2.3 D1_V3.26 D1_V4.21

Deflection, D1 [mm]

4 3.5 3 2.5 2 1.5 1 0.5 0 -0.5 0

0.01

0.02

0.03

0.04

0.05

Time [s] (a)

8

D2_V2.3 D2_V3.26 D2_V4.21

Deflection, D2 [mm]

7 6 5 4 3 2 1 0 -1 0 -2

0.01

0.02

0.03

0.04

0.05

Time [s] (b)

Fig. 11. Deflection time histories at (a) D1 and (b) D2 of the stiffened plate with various impact velocities.

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the figure that the maximum pressure increased when the impact velocity was linearly increased, regardless of the calculation locations. Meanwhile, the maximum pressure P1 was nearly the same with P2, when increasing impact velocity. Moreover, impact duration increased with an increase in impact velocity, as revealed in Fig. 12. 0.3 P1_V2.3 P1_V3.26 P1_V4.21

Pressure, P1 [MPa]

0.25 0.2 0.15 0.1 0.05 0 -0.05

0

0.01

-0.1

0.02

0.03

0.05

Time [s] (a)

0.3

P2_V2.3 P2_V3.26 P2_V4.21

0.25

Pressure, P2 [MPa]

0.04

0.2 0.15 0.1 0.05 0 -0.05

0

0.01

-0.1

0.02

0.03

0.04

0.05

Time [s] (b)

Fig. 12. Time history of slamming pressures at (a) P1 and (b) P2 of the stiffened plate with various impact velocity.

5.2 Effect of Plate Rigidity In addition to the effect of impact velocity, the effect of structural rigidity on the slamming load acting on flat stiffened aluminum plates was investigated numerically. The FSI model No. 2 (plate thickness of 6 mm) with an impact velocity of about 2.3 m/s was again utilized as the basic model. And two more models having plate thicknesses of 8 mm and 10 mm were analyzed in order to identify the effect of plate rigidity on the impact pressure characteristics. It should be noted that the added mass was adjusted to yield

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the same total weight as that in the basic model No. 2 (800 kg). The deformation of the stiffener at D1 (center of the tip of the flange) and center of the plate D2 for the models having the plate thicknesses of 6 mm, 8 mm and 10 mm are shown in Fig. 13. It can be seen that the case with a bottom plate thickness of 6 mm had the largest out-of-plane deformation due to the slamming load. It was evident that this magnitude of deflection decreased as the rigidity of the plate increased. Figure 14 plots the impact pressure time histories at the center of the plate P2 with different plate thicknesses. It is apparent that the impact pressures predicted from the model with 10 mm of plate thickness were greater than those from the other models having thinner bottom plates (6 and 8 mm) while similar pressure progress was found at the early impact stage for all cases. It is presumed that increasing thickness of plate, i.e. the plate is stiffer (higher rigidity), leads to smaller deformation, which caused the decrease of the air cushion thickness (air layer) faster, and subsequently gave higher impact pressure and faster increase in pressure for thicker plate model. The results indicated that the hydro-elasticity effect is significant for 1.4 D1_t6 D1_t8 D1_t10

Deflection, D1 [mm]

1.2 1 0.8 0.6 0.4 0.2 0 -0.2

0

0.01

0.02

-0.4

0.03

0.04

Time [s]

(a)

3.5

D2_t6 D2_t8 D2_t10

3

Deflection, D2 [mm]

0.05

2.5 2 1.5 1 0.5 0 -0.5 -1

0

0.01

0.02

0.03

0.04

0.05

Time [s] (b)

Fig. 13. Deflection time histories of the bottom plate at (a) D1 and (b) D2 of the stiffened plate for various plate thicknesses.

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slamming load prediction, especially for peak impact value and the later impact stages. It was also noted that, as pointed out by DNV [34], hydrodynamic loading is affected by structural deformation or vibrations of structures resulting from slamming loads. Additionally, as for the particular cases in this study, the impact duration was slightly shorter for the latter case having a higher stiffness (thicker plate thickness) than the basis case. 0.25 P2_t6 P2_t8 P2_t10

Pressure, P2 [MPa]

0.2 0.15 0.1 0.05 0 0

0.01

0.02

0.03

0.04

0.05

-0.05 -0.1

Time [s]

Fig. 14. Impact pressure time histories at the center of plate P2 with different bottom plate thicknesses.

6 Conclusions The aim of this study was to assess the validity of the proposed numerical FSI simulation in order to investigate slamming pressure acting on a flat stiffened aluminum plate and its structural response (including effect of the heat-affected zone) of bottom plate of high-speed vessels. Investigations of the effects of plate rigidity, impact velocity, and air cushion on the slamming load characteristics and its structural response of aluminum plates were performed. Based on the results of the present study, the following conclusions can be drawn: The proposed FSI simulation procedure is capable of predicting the slamming pressure on flat stiffened aluminum plates with a reasonable degree of accuracy. Moreover, with the presented numerical analysis methodology, the experimental test scenarios can be reproduced satisfactorily and conveniently varied with any scantlings and/or test conditions for benchmarking studies. Also, the numerical method developed in this study can be used for further parametric studies of slamming problems acting on actual marine structures. It is observed that due to the zero dead-rise angle of water entry, the presence of air caused the reduction of the slamming impact pressure peak, the deformation of models, and the lengthening of the impact duration as well. As it is known that the ratio of load duration to natural period is a key parameter in structural responses under dynamic

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loadings, this phenomenon, the so-called air cushion effect, could lead to totally different responses given its longer load duration. Therefore, the air cushion effect should be considered for slamming phenomena occurring on marine structures with zero or small dead-rise angles. The effect of impact velocity on the slamming loads acting on flat stiffened aluminum plates was investigated. In the results, a higher peak of impact pressure was observed regardless of location when the impact velocity was increased. Additionally, with higher impact velocity, the impact duration decreased and the impact pressure oscillation increased. The effect of structural rigidity on slamming loads and deflection was investigated numerically by varying the thickness of the plate. It was found that when the thickness of the plate was increased (thus imparting higher rigidity to the plate), the peak impact pressure gradually increased while the impact duration slightly decreased. As expected, the maximum deflection of the plate was reduced when the plate thickness was increased, and the permanent deflection was decreased accordingly. The results of this study can be considered to be useful for validation of estimations of water impact pressure acting on various structures and its structural response as well, and consequently, they can help to enhance the structural designs of lightweight structures fabricated from aluminum alloy against slamming loads. Based on the results of the present study, for more advanced and optimal structural designs, further rigorous benchmarking studies on actual scantlings of high-speed vessels and other marine structures fabricating from aluminum alloy materials and empirical design formulations for prediction of slamming loadings will be performed subsequently in the near future. Acknowledgments. The authors gratefully acknowledge the financial support provided by a Korea Research Foundation (BK21 Plus) grant funded by the Korean Ministry of Education (No. 5266-20180100). The authors also acknowledge the financial support from the project “Development of lightweight structure of membrane-type cargo containment system for high value-added LNG ship, Project no.10077592” supported by “Korea Ministry of Trade, Industry and Energy”.

References 1. Von Karman, T.H.: The impact on seaplane floats during landing. National Advisory Committee for Aeronautics. Technical note No. 321. Washington, DC (1929) 2. Wagner, V.H.: Über Stoß-und gleitvorgänge an der oberfläche von flüssigkeiten. zeitschrift für angewandte mathematik und mechanik 12(4), 193–215 (1932) 3. Dobrovol’skaya, Z.N.: On some problems of similarity flow of fluids with a free surface. J. Fluid Mech. 36, 805–829 (1969) 4. Hirano, Y., Miura, K.: Water impact accelerations of axially symmetric bodies. J. Spacecraft Rockets 7(6), 762–764 (1970) 5. Cointe, R., Armand, J.L.: Hydrodynamic impact analysis of a cylinder. J. Offshore Mech. Arct. Eng. 109(3), 237–243 (1987) 6. Zhao, R., Faltinsen, O.: Water entry of two dimensional bodies. J. Fluid Mech. 246, 593–612 (1993) 7. Chuang, S.L.: Experiments on flat-bottom slamming. J. Ship Res. 10, 10–17 (1966)

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8. Chuang, S.L.: Investigation of impact of rigid and elastic bodies with water. Report for Department of the Navy. Report No. 3248. United States Department of the Navy, Washington, DC (1970) 9. Mori, K., Nagai, T.: Response of stifened plates under impulsive water pressure. J. Soc. Naval Archit. Jpn. 140, 165–173 (1976). In Japanese 10. Mori, K.: Response of the bottom plate of high-speed crafts under impulsive water pressure. J. Soc. Naval Archit. Jpn. 142, 297–305 (1977). In Japanese 11. Stenius, I., Rosén, A., Kuttenkeuler, J.: Explicit FE-modelling of fluid-structure interaction in hull–water impacts. Int. Shipbuilding Prog. 53, 103–121 (2006) 12. Stenius, I., Rosén, A., Kuttenkeuler, J.: Explicit FE-modelling of hydroelasticity in panelwater impacts. Int. Shipbuilding Prog. 54, 111–127 (2007) 13. Luo, H., Hu, J., Guedes Soares, C.: Numerical simulation of hydroelastic responses of flat stiffened panels under slamming loads. In: 29th International Conference on Ocean, Offshore and Arctic Engineering, pp. 373–381. ASME, Shanghai, China (2010) 14. Luo, H.B., Wang, H., Guedes Soares, C.: Numerical prediction of slamming loads on a rigid wedge subjected to water entry using an explicit finite element method. In: Advances in Marine Structures. Taylor & Francis Group, London, ISBN 978-0-415-67771-4, pp. 41–47 (2011) 15. Luo, H.B., Wang, H., Guedes Soares, C.: Comparative study of hydroelastic impact for one free-drop wedge with stiffened panels by experimental and explicit finite element methods. In: 30th International Conference on Ocean, Offshore and Arctic Engineering, pp. 119–127. ASME, Rotterdam, The Netherlands (2011) 16. Stenius, I., Rosén, A., Kuttenkeuler, J.: Hydroelastic interaction in panel-water impacts of high-speed craft. Ocean Eng. 38, 371–381 (2011) 17. Yamada, Y., Takami, T., Oka, M.: Numerical study on the slamming impact of wedge shaped obstacles considering fluid-structure interaction (FSI). In: 22nd International Offshore and Polar Engineering Conference, pp. 1008–1016. International Society of Offshore and Polar Engineers (ISOPE), Rhodes, Greece (2012) 18. Alaoui, A.E.M., Neme, A., Tassin, A., Jacques, N.: Experimental study of coefficients during vertical water entry of axisymmetric rigid shapes at constant speeds. Appl. Ocean Res. 37, 183–197 (2012) 19. Borrelli, R., Mercurio, U., Alguadich, S.: Water impact tests and simulations of a steel structure. Int. J. Struct. Integrity 3(1), 5–21 (2012) 20. Wang, S., Guedes Soares, C.: Numerical study on the water impact of 3D bodies by an explicit finite element method. Ocean Eng. 78, 73–88 (2014) 21. Wang, S., Karmakar, D., Guedes Soares, C.: Hydroelastic impact of a horizontal floating plate with forward speed. J. Fluid Struct. 60, 97–113 (2016) 22. Cheon, J.S., Jang, B.-S., Yim, K.H., Lee, H.S.D., Koo, B.-Y., Ju, H.: A study on slamming pressure on a flat stiffened plate considering fluid–structure interaction. J. Mar. Sci. Technol. 21(2), 309–324 (2015). https://doi.org/10.1007/s00773-015-0353-y 23. Yoshikawa, T., Miyake, R., Yoshida, T., Maeda, M.: Numerical simulation of structural response under bow flare slamming load. J. Soc. Naval Archit. Jpn. 26, 267–276 (2017). In Japanese 24. Sun, H., Wang, D.Y.: Experimental and numerical analysis of hydrodynamic impact on stiffened side of three-dimensional elastic stiffened plates. Adv. Mech. Eng. 10(4), 1–23 (2018) 25. Seo, B.S., Truong, D.D., Cho, S.R., Kim, D.J., Park, S.K., Shin, H.K.: A study on accumulated damage of steel wedges with dead-rise 10 degree due to slamming loads. Int. J. Nav. Architect. Ocean Eng. 10, 520–528 (2018)

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26. Anghileri, M., Castelletti, L.M.L., Francesconi, E., Milanese, A., Pittofrati, M.: Survey of numerical approaches to analyse the behavior of a composite skin panel during a water impact. Int. J. Impact Eng 63, 43–51 (2014) 27. Park, B.W., Cho, S.R.: Simple design formulae for predicting the residual damage of unstiffened and stiffened plates under explosive loadings. Int. J. Impact Eng 32, 1721–1736 (2006) 28. Truong, D.D., Shin, H.K., Cho, S.-R.: Permanent set evolution of aluminium-alloy plates due to repeated impulsive pressure loadings induced by slamming. J. Mar. Sci. Technol. 23(3), 580–595 (2017). https://doi.org/10.1007/s00773-017-0494-2 29. Zha, Y., Moan, T.: Ultimate strength of stiffened aluminium panels with predominantly torsional failure modes. Thin-Wall Struct. 39, 631–648 (2001) 30. Sensharma, P., Collette, M., Harrington, J.: Effect of Welded Properties on Aluminum Structures. Ship Structure Committee SSC-4 (2010) 31. Livermore Software Technology Corporation: LS-DYNA theoretical manual. Livermore Software Technology Corporation (2006) 32. Olvsson, L., Souli, M., Do, I.: LS-DYNA – ALE Capabilities (Arbitrary-Lagrangian-Eulerian) Fluid-Structure Interaction Modeling. Livermore Software Technology Corporation (2003) 33. ABS: Guide for slamming loads and strength assessment for vessels (2011) 34. DNV: Recommended Practice DNV-RP-C205. Environmental conditions and environmental loads. Det Norske Veritas, October (2010)

Scantling Evaluations of Plates and Stiffeners Based on Elasto-Plastic Analysis Under Axial Loads and Lateral Pressures Yoshiaki Naruse1(B) , Masato Kim2 , Rikuto Umezawa2 , Kinya Ishibashi3 , Hiroyuki Koyama3 , Tetsuo Okada2 , and Yasumi Kawamura2 1 Design Department, Onomichi Dockyard Co., Ltd. (Temporarily affiliated with ClassNK),

Onomichi, Japan [email protected] 2 Department of Systems Design for Ocean-Space, Yokohama National University, Yokohama, Japan 3 Hull Rules Development Department, ClassNK, Tokyo, Japan

Abstract. This paper presents a method for deriving the practical collapse strength against lateral pressure of hull local members such as plates and stiffeners which makes use of non-linear FEA calculations. Unlike the collapse strength against compressive axial loads, the collapse strength against lateral pressure cannot be clearly identified because such structures are capable of withstanding lateral pressure caused by membrane stress even though they may deform plastically. This paper, therefore, defines “collapse” to be the condition in which residual deflection develops up to a defined criterion after the unloading of both axial loads and lateral pressures, where the maximum lateral pressure that does not cause “collapse” is regarded to be the practical collapse strength. Utilizing this criterion, the practical collapse strength against lateral pressure of various types and scantlings of plates and stiffeners is investigated and compared with the assessment formulae for hull local members specified in the International Association of Classification Societies’ (IACS) Common Structural Rules (CSR). Additionally, the effects of the axial loads acting together with lateral pressures are also studied. The results of the study show that compressive axial loads are more critical than tensile axial loads with respect to collapse strength. For thinner plates and smaller stiffeners, tensile axial load gains their collapse strength. The out-plane component of internal force due to the axial load is considered to cause these phenomena. Keywords: Plate and stiffener · Lateral pressure · Axial load · Elasto-plastic analysis · Collapse strength · Residual deflection

1 Introduction 1.1 Background Hull local members such as plates and stiffeners are the most fundamental components of a ship’s hull structure. The arrangements and scantlings of these members are basically © Springer Nature Singapore Pte Ltd. 2021 T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 100–127, 2021. https://doi.org/10.1007/978-981-15-4672-3_6

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determined so as to ensure that they are capable of withstanding lateral pressures such as water pressure or cargo pressure. In addition, hull girder bending moment acts upon these local members as axial load. Thus, the combination of axial load and lateral pressure simultaneously acting upon them needs to be considered. In order to ensure sufficient structural safety for ships, the structural rules specified by classification societies typically require two types of strength evaluations be carried out for hull local members, and these are as follows: • strength evaluation against axial compression load, and • strength evaluation against lateral pressure in consideration of strength reductions due to axial loads. The former is based upon elastic/plastic buckling theory for both plates and stiffeners and is related to the prevention of local buckling, whereas the latter is based upon elastic beam theory for stiffeners and rigid-plastic analysis for plates and is related to determining whether there is sufficient strength against lateral pressure. In recent years, the use of non-linear FEA which can take into account large displacement and elasto-plasticity has become increasingly popular. The knowledge obtained from non-linear FEA results has not only revealed the collapse behavior of plates and stiffeners subjected to a combination of high axial compression loads and low-level lateral pressures (hereinafter referred to as the “axial load dominant condition”), but has also made possible the development of a number of closed-form simplified formulae which can be used to estimate the collapse strength of plates and stiffeners. Classification societies have incorporated these simplified closed-form formulae into their class structural rules as a way of improving the rationality of their strength evaluations, which in turn leads to greater ship safety. With regard to the strength assessments for loading conditions in which a combination of high-level lateral pressures and low-level axial loads is acting (hereinafter referred to as the “lateral pressure dominant condition”), the class structural rules of most classification societies still use conventional formulae based upon elastic beam theory and rigid-plastic analysis. Taking into account the long track record of their successful application to huge number of ships, these conventional formulae are deemed to ensure sufficient safety for hull local member design. For many years now, rigid-plastic analysis has been the standard way of assessing the collapse strength of plates or stiffeners against lateral pressures, and conventional rigid-plastic analysis considers the collapse mechanism shown in Fig. 1 (a). When a section in a beam entirely reaches its yield stress σY due to bending stress (Fig. 1 (b)), this point becomes “plastic hinge” that cannot withstand larger bending moments. When a collapse mechanism such as that shown in Fig. 1 (a) develops, rigid-plastic analysis regards the structure to be “collapsed”. When the axial load for the beam is applied, the area around plastic neutral axis is occupied by axial load. The neutral axis in a section, therefore, changes (Fig. 1 (c)) and the maximum capable bending moment (fully plastic moment) decreases.

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σY

σY

Lateral pressure

Plastic hinge (a)

Axial load

-σY

-σY (b)

(c)

Fig. 1. Conventional rigid-plastic analysis concept

Past studies into the collapse strength and plastic deformation behavior of stiffened or unstiffened plates against lateral loads have been carried out by many researchers. Jones [1] proposed a theoretical rigid-plastic analysis for beams and plates considering the effect of small axial displacement at the boundary, while Manolakos [2] presented a method to estimate collapse strength of grid structures using the end-fixity coefficient and Schubak [3, 4] developed a rigid-plastic model of stiffened plates subjected to uniformly distributed blast loads. For blast loads, Louca [5] predicted the dynamic response of unstiffened plate using Lagrange’s equation and carried out non-linear FEA, whereas Hong [6] utilized plastic yield line theory to propose a “Double-diamond” collapse model of plates for a laterally patch load. With regard to experimental study, Shanmugam [7] performed experiments for stiffened plates subjected to axial compression and lateral pressure, and then compared the results with those obtained using with non-linear FEA, while Yu [8] proposed a method to estimate resistance-deformation curve of stiffened plates subjected to lateral loads which is capable of taking into account stiffness for the inward motion of stiffener ends. Past studies on the collapse strength in lateral pressure dominant condition, however, did not consider the idea of “residual deflection” after unloading. There is, therefore, a possibility that significant residual deflection will occur in a hull local members even though the local member does not reach its collapse strength. The authors consider that any residual deflection which causes strength degradation should be avoided so as to preserve ship hull structural safety. Additionally, in most past studies, the difference between tensile and compressive axial loads acting together with lateral pressure is not discussed sufficiently. In this paper, to improve the assessment method based on conventional rigid-plastic analysis theory and to make it possible to give hull local members more rational scantling, the collapse strength from the perspective of residual deflection of various types and scantlings of plates and stiffeners in the lateral pressure dominant condition is derived by using nonlinear FEA. And the effects of axial loads for the collapse strength are studied. In addition, the strength criteria specified in CSR [9] are studied in comparison to derived values of collapse strength.

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1.2 Deriving Collapse Strength Against Lateral Pressure Using Non-linear FEA When hull local members are in the axial load dominant condition, an abrupt drop in the load-displacement curve (Fig. 2(a)) is observed after the yielding or buckling, and the maximum axial load value can be defined as collapse strength against axial load; however, when such members are in the lateral pressure dominant condition, deflection by the lateral pressure continues to increase (Fig. 2(b)) without any abrupt drop, even though they start to deform plastically, and the moment of collapse cannot be clearly identified by observing deflection-lateral pressure curve in cases where the curve is derived from non-linear FEA without considering fractures of material. This behavior is known to be caused by the membrane stress effect. Criteria for Residual Deflection

Lateral Pressure

Axial Compression

Collapse Strength

In-plane Displacement (a) Axial load dominant condition

Referred to as “Collapse”

Out-plane Deflection (b) Lateral pressure dominant condition

Fig. 2. Typical collapse behavior of hull local members

This paper defines “collapse” to be the condition in which residual deflection develops up to a defined criterion after the unloading of both axials load and lateral pressures. This assumption is based upon actual cases where excessive permanent deformation is considered to decrease strength against future severe loading conditions and is regarded as the damage to be repaired. Using this criterion, the highest lateral pressure which does not induce collapse can be examined from a series calculation of non-linear FEA in which the maximum values of lateral pressure are changed. The criterion for residual deflection after unloading is determined by the initial deflection considered in typical buckling analysis such as plate breadth/200 or 0.1% of longitudinal span. These values are similar to “Ship building quality standard for new construction” such as IACS Rec 47 [10] and generally recognized as the sufficiently small deformations that do not affect for the strength of hull local members.

2 Non-linear FEA Model and Analysis Conditions This paper derives practical collapse strength based upon the residual deflection criterion described in Sect. 1.2 using the commercial FEA software LS-DYNA. Section 2.1

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describes the conditions and procedures for deriving the collapse strength of plates, while Sect. 2.2 does the same for the collapse strength of stiffeners. The study uses LS-DYNA implicit method, and convergence studies are performed before the analysis so as to derive the collapse strength of hull local members. This is done to determine proper model mesh size and load increment interval of the non-linear FEA, these convergence studies compare past experiments carried out by Fujii [11] and Tanaka [12], which respectively observed the collapse behavior of plates and stiffened plates. Fujii [11] carried out experiments in which plates are subjected to lateral pressure until the plates fracture, and then observed the development of deflection and the residual deflection after unloading for several lateral pressures. Figure 3 shows one of the comparisons of deflection during loading and residual deflection after unloading between the experiment carried out by Fujii [11] and the non-linear FEA that produced the experiment. It can be seen that FEA results are in good agreement with the experimental results. Therefore, the mesh size and load incremental intervals of non-linear FEA that give good agreement with experimental results are adopted. 60 FEA value of deflection FEA value of residual deflection Experimental value of deflection Experimental value of residual deflection

Lateral pressure [kgf/cm2]

50 40 30 20 10 0

0

5

10

15

20

25

Deflection [mm] Fig. 3. Comparisons of experimental results from prior studies to non-linear FEA results

2.1 Plates FEA Model. As shown in Fig. 4, four plates which are separated by stiffeners and transverse girders are modelled by shell elements. Only plates are modelled, with stiffeners and girders being expressed by constraining the deflection of the plates. The specifications and material parameters of the plates are shown in Table 1 and Table 2, respectively.

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Boundary Conditions. This paper uses the periodic boundary condition proposed by Fujikubo [13] for the model edges. This boundary condition can realize that the modeled plate infinitely continues in both longitudinal and transverse directions. This assumption is consistent with a practical ship hull structure that is composed by large number of continuous stiffened plates. By using this boundary condition for four plates shown in Fig. 4, both any plate buckling mode and lateral deformation due to lateral pressure can be considered. The details of the boundary conditions used in the plate analysis are shown in Table 3. Analysis Method and Load Conditions. To investigate residual deflections after unloading and deriving collapse strength, the three-step analysis shown in Table 4 is carried out. Axial loads are applied to the model edges in Step 1; then, as shown in Fig. 4, the transverse axial load case (axial load for the long edge of the plate) and longitudinal axial load case (axial load for the short edge of the plate) are considered. Both compressive and tensile axial loads are also considered for both direction loads, and the magnitudes of the axial loads are shown in Table 5. Since the axial loads for hull local members are generated by the hull girder longitudinal bending moment, the cases where transverse axial load is applied assume a transverse framing structure; others cases, on the other hand, assume a longitudinal framing structure. Definition of Collapse Strength. The “collapse” of plates in non-linear FEA, which is described in Sect. 1.2, is defined as the condition in which residual deflection after the unloading of both axial and lateral pressures is developed up to “plate breadth/200 [mm]” (in this paper, 800 mm/200 = 4.0 mm). The maximum lateral pressure that does not induce “collapse” is derived for each plate thickness and load case through iterative changes to maximum lateral pressure, and a flow chart of the process for deriving collapse F

G

Plate

Longitudinal axial load

E

Plate D

H 2400 mm Plate

A

Plate

B

Y

X

Transverse axial load

Fig. 4. Analysis model and FEA boundary conditions for plates

C

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2400 [mm]

Plate breadth

800 [mm]

Plate thickness 6–30 [mm] (2 mm step interval) Element size

40 mm × 40 mm (shell element)

Table 2. Material parameters for non-linear FEM analysis Material parameters Yield Stress (σY )

315 [MPa]

Young’s Modulus (E)

206000 [MPa]

Poisson’s Ratio (ν)

0.3

Strain-hardening Coefficient E/ 65 [MPa]

Table 3. FEA boundary conditions for plates x-disp. (u)

y-disp. (v)

Edge AC Edge CE

= uC + uAG = vG + vAC

Edge EG

z-disp. (w)

Rot-x

Rot-y

Rot-z

Fix

= EG

= EG

= EG

Fix

= AG

= AG

= AG

Fix

Edge AG

Fix

Line BF

Fix

Line DH Point A Point C

Fix Fix

Fix

Fix

Fix

Fix

Table 4. Three-step analysis for deriving collapse strength Step 1

Apply axial load up to defined value

Step 2

Apply lateral pressure with keeping axial load that is applied in Step 1

Step 3

Axial load and lateral pressure that are applied in Step 1 and Step 2 are unloaded

strength is shown in Fig. 5. For plate analysis, the magnitude of deflection is measured at the center of the plate.

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Table 5. Axial load values for plates Magnitude of axial load [MPa] Transverse Axial load case Compressive/tensile

50, 100, 150, 200 (about 16%, 32%, 48%, 63% of σY )

Longitudinal Axial load case Compressive/tensile

50, 100, 150, 200 (about 16%, 32%, 48%, 63% of σY )

Start analysis

Step 1 Apply axial load Step 2 Apply lateral pressure

Step 3 Unload

Change lateral pressure value

Defined Criteria = Residual deflection?

NO

YES Obtain collapse strength Fig. 5. Collapse strength derivation process flow chart

An example of the procedure for deriving collapse strength is shown in Fig. 6. This figure shows the deflection history of 20 mm plate thickness for some cases with different maximum lateral pressure. In Step 3 of Fig. 6, unloading the lateral pressures and axial loads, the deflection decreases and the residual deflection can be observed at the last point of Step 3. In this case, a value 1.155[MPa] is used for the maximum lateral pressure so

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that the residual deflection does not to develop up to 4 mm. Thus, the collapse strength of a 20 mm plate can be determined to be 1.155 MPa. This method was applied to each load case and each plate thickness and the collapse strength and the effects of axial loads on collapse strength are investigated. 16 Step 1

Deflection change [mm]

14 12

0.575[MPa]

10

1.155[MPa]

4

Step 3

1.232[MPa]

8 6

Step 2

Residual deflection criterion (4mm)

2 0

Analysis steps Fig. 6. Example of history of deflection to derived the collapse strength

2.2 Stiffeners FEA Model. A stiffened plate with a T-type stiffener is modelled by shell elements. 2-longitudinal space (1/2 + 1+1/2) - 1-transverse space (1/2 + 1/2) model is used as shown in Fig. 7. The longitudinal span of the model is 4,000 mm, and the spacing between stiffeners is 800 mm. The dimensions of stiffeners considered in this paper are shown in Table 6, and the plate thickness is assumed to be 20 mm for all models. In the model, longitudinal span is divided into 80 elements and transverse space is divided into 16 elements, with the stiffener web being modelled by shell elements of 50 mm in height; for example, a stiffener web of 150 mm in height is to be divided into three elements in the height direction. The face plate is divided into two elements in the width direction, and the material parameters are same as those used for plate analysis shown in Table 2. Boundary Conditions. The boundary conditions for the stiffeners are shown in Table 7. The symmetry condition is used for all model edges, with the x-directional displacement between the nodes being fixed for edge AD (the edge subjected to the axial load. shown in Fig. 7) so that it remains straight during analysis, and the y-directional displacement between the nodes being fixed for edge AB. At the lines of the transverse members in Fig. 7, the z-directional displacement is fixed, and y-directional displacement between nodes of the web is fixed to prevent any tripping of the stiffener.

Scantling Evaluations of Plates and Stiffeners Based on Elasto-Plastic Analysis

Trans. girder

Trans. girder

Trans. girder

Y

Trans. girder C Longi.

D

800m m

109

stiffener

A

B

Longi. stiffener Longi. stiffener

4000mm

X Fig. 7. Analysis model for FEA of stiffeners

Table 6. Dimensions of stiffeners Model

Web height [mm]

Web thickness [mm]

Face breadth [mm]

Face thickness [mm]

Section modulus (Face) [cm3 ]

1

150

9.0

70

11.0

180

2

200

9.0

75

12.0

328

3

250

10.0

90

14.0

508

4

300

11.0

105

17.0

756

5

350

11.5

100

18.5

1033

6

400

12.0

110

18.5

1377

7

450

12.0

115

19.5

1712

8

500

12.5

120

21.0

2148

9

550

12.5

130

21.0

2562

10

600

13.0

135

22.5

3111

Size of attached plate is 4000 × 800 × 20 mm for all models

Analysis Method and Load Conditions. The same three-step analysis used for plate analysis described in Table 4 is also adopted for stiffener analysis. As shown in Table 8, the axial loads for stiffeners are applied so that the average stress is between 10% and 70% of yield stress σY at 10% increments, and both tensile and compressive loads are considered respectively. Additionally, lateral pressures from the plate side and stiffener side are also considered. Definition of Collapse Strength. The “collapse” strength of stiffener is derived the same way as is done during plate analysis as shown in Fig. 5. The criterion of residual deflection after unloading for stiffeners is defined as “0.1% of transvers space” (in this paper, 0.1% of 4000 mm = 4 mm). In the same manner is done for plate analysis,

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y-disp. (v)

Edge CD Edge BC

z-disp. (w) Rot-x Rot-y Rot-z

Fix

Fix

Maintain straight line

Fix

Fix

Fix

Edge AB Edge AD Maintain straight line Trans.

Fix Fix Fix Fix

Fix

Maintain straight line Fix (Stiffener web only)

Table 8. Axial load values for stiffeners Lateral pressure (from plate side)

Lateral pressure (from stiffener side)

Compressive axial load

10%–70% of σY

10%–70% of σY

Tensile axial load

10%–70% of σY

10%–70% of σY

maximum lateral pressures that do not induce collapse are derived for each load condition and dimension shown in Table 6. For stiffener analysis, the magnitude of deflection is measured at point E in Fig. 7, with the point being at the mid span of stiffener and the cross point of the stiffener and attached plate.

3 Collapse Strength of Plates Derived by Non-linear FEA 3.1 Collapse Strength of Plates As an example of the results of plate analysis, a von Mises stress contour plot of 20 mm plate thickness in the pure lateral pressure case is shown in Fig. 8. In Fig. 8 (a), when maximum lateral pressure 1.155 MPa is applied (at the end of step 2 described in Fig. 5), the stresses at the center and transverse edge of plates where large bending moment acts reach yield stress. In Fig. 8 (b) residual stress and residual deflection is observable when lateral pressure is completely unloaded (at the end of Step 3). A comparison of collapse strength values of plates derived by the procedure described in Fig. 5 and some analytical methods in pure lateral pressure case is shown in Fig. 9. As the analytical methods, we employed 3-point plastic hinge theory and roof shaped collapse theory. A unit width of plates was considered in the application of 3-point hinge theory (i.e., aspect ratio was assumed infinite). It can be observed that the collapse strength derived by FEA is similar to the roof shaped collapse theory. The collapse strength values of plates derived by the procedure described in Fig. 5 considering axial load are shown in Figs. 10–13. In addition, the criteria required by the assessment formulae of CSR [9] are also plotted in these graphs. The details of the CSR formulae are described in the appendix.

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Displacement Scale Factor: 20

(b) At the end of Step 3

(a) At the end of Step 2

Fig. 8. Von Mises stress contour of 20 mm plate thickness in pure lateral pressure case

35 Plate Thickness[mm]

30 25 20 15 FEA(Residual Deflection)

10

3-Point Plastic Hinge Collapse

5

Roof Shaped Collapse

0 0.0

0.5

1.0

1.5

2.0

Collapse Strength[MPa] Fig. 9. Comparison of collapse strength of plates in pure lateral pressure case

A comparison to the collapse strength derived by FEA in transverse axial load cases (Fig. 10 and Fig. 11) shows that large decreases in collapse strength due to compressive axial loads are observable, while the effects of tensile axial loads upon collapse strength are smaller than those of compressive axial loads. Moreover, the effect of axial load is observed to be smaller than that of transverse axial load (Fig. 10 and Fig. 11) with respect to the longitudinal axial load case (Fig. 12 and Fig. 13). A comparison of the pure lateral pressure case for FEA with that for the CSR formulae in Figs. 10–13 (i.e. the same values are plotted in these graph with regard to pure lateral pressure case) shows the values of collapse strength derived by non-linear FEA are at least 40% higher than the CSR criteria for each plate thickness. In addition, the collapse strength derived by non-linear FEA in all cases with axial loads acting is higher than the CSR criteria, with the ratio between the two varying according to the axial load condition. These comparison results prove that the CSR formulae do avoid residual deflection for each plate thickness and each load conditions with a sufficient margin of safety. It is noted that cases where collapse caused by only axial compression (i.e. buckling collapse) occurs are not plotted in these graphs; for example, the plot of “Axial load 200

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40

Plate thickness [mm]

35 30 25 20

Axial load 0[MPa] Axial load 50[MPa] Axial load 100[MPa] Axial load 150[MPa] Axial load 200[MPa] CSR Axial load 0[MPa] CSR Axial load 50[MPa] CSR Axial load 100[MPa] CSR Axial load 150[MPa] CSR Axial load 200[MPa]

15 10 5 0 0.0

0.2

0.4

0.6

0.8 1.0 1.2 1.4 Collapse strength [MPa]

1.6

1.8

2.0

Fig. 10. Collapse strength of plates of compressive-transverse axial load case (transverse framing structure)

40

Plate thickness [mm]

35 30 25 20

Axial load 0[MPa] Axial load 50[MPa] Axial load 100[MPa] Axial load 150[MPa] Axial load 200[MPa] CSR Axial load 0[MPa] CSR Axial load 50[MPa] CSR Axial load 100[MPa] CSR Axial load 150[MPa] CSR Axial load 200[MPa]

15 10 5 0 0.0

0.2

0.4

0.6

0.8 1.0 1.2 1.4 Collapse strength [MPa]

1.6

1.8

2.0

Fig. 11. Collapse strength of plates of tensile-transverse axial load case (transverse framing structure)

[MPa]” in Fig. 10 starts from a plate thickness of 26 mm, with the collapse strength at plate thicknesses 6 mm to 24 mm not being considered. This is because buckling collapse occurs within this range of plate thicknesses at the beginning of Step 2.

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40

Plate thickness [mm]

35 30 25 20

Axial load 0[MPa] Axial load 50[MPa] Axial load 100[MPa] Axial load 150[MPa] Axial load 200[MPa] CSR Axial load 0[MPa] CSR Axial load 50[MPa] CSR Axial load 100[MPa] CSR Axial load 150[MPa] CSR Axial load 200[MPa]

15 10 5 0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Collapse strength [MPa] Fig. 12. Collapse strength of plates of compressive-longitudinal axial load case (longitudinal framing structure)

40

Plate thickness [mm]

35 30 25 20

Axial load 0[MPa] Axial load 50[MPa] Axial load 100[MPa] Axial load 150[MPa] Axial load 200[MPa] CSR Axial load 0[MPa] CSR Axial load 50[MPa] CSR Axial load 100[MPa] CSR Axial load 150[MPa] CSR Axial load 200[MPa]

15 10 5 0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Collapse strength [MPa] Fig. 13. Collapse strength of plates of tensile-longitudinal axial load case (longitudinal framing structure)

3.2 Discussions on the Reduction of Collapse Strength for Plates In order to discuss the effects of axial loads on collapse strength, collapse strength reduction for the pure lateral pressure case is investigated. Figures 14–17 show the reduction

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ratio of collapse strength for some representative plate thickness cases, including that of the CSR.

Collapse strength reducon rao

1.4 1.2 1 0.8

6mm 10mm 14mm 18mm 22mm 26mm 30mm CSR

0.6 0.4 0.2 0 0

0.2

0.4 0.6 Axial stress/σY

0.8

1

Fig. 14. Reduction of collapse strength of plates of compressive-transverse axial load case (transverse framing structure)

Collapse strength reducon rao

1.4 1.2 1 0.8

6mm 10mm 14mm 18mm 22mm 26mm 30mm CSR

0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

Axial stress/σY Fig. 15. Reduction of collapse strength of tensile-transverse axial load case (transverse framing structure)

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Collapse strength reducon rao

1.4 1.2 1 6mm 10mm 14mm 18mm 22mm 26mm 30mm CSR

0.8 0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

Axial stress/σY Fig. 16. Reduction of collapse strength of plates of compressive-longitudinal axial load case (longitudinal framing structure)

Collapse strength reducon rao

1.4 1.2 1 0.8

6mm 10mm 14mm 18mm 22mm 26mm 30mm CSR

0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

Axial stress/σY Fig. 17. Reduction of collapse strength of plates of tensile-longitudinal axial load case (longitudinal framing structure)

In the compressive axial load cases shown in Fig. 14 and Fig. 16, collapse strength is reduced at a regular rate in proportion to the magnitude of the compressive axial load and

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the reduction in the transverse axial load case is more significant than that in longitudinal axial load case. In the tensile axial load cases shown in Fig. 15 and Fig. 17, on the other hand, the tendency of reduction depends upon not only the direction of axial load, but also plate thickness. For thinner plates such as 6 mm and 10 mm, opposite effect with respect to collapse strength is observed (i.e. the tensile axial load leads to increases in collapse strength depending upon load magnitude). Even though tensile axial loads slightly reduce the collapse strength of thicker plates, the reduction effect is quite smaller than that in compressive axial load cases. It can be also observed that the effects of axial loads in the transverse axial load case are larger than those in the longitudinal axial load case. Neither conventional theory nor past studies (as previously discussed in Sect. 1.1) have been able to sufficiently reproduce the above-mentioned phenomena, especially those cases where the tensile axial loads leads to increases in the collapse strength of plates. The authors assume that the “axial load effect on the deflection” is one of the factors of above phenomena. Figure 18 shows that a difference in the direction between the internal forces acting upon both sides is developed by transitioning the deflection angle into the small element “dx” of a plate which has deflection w with the axial load σ x t p being applied. This means that the distributed load q expressed by Eq. (1) occurs as the out-plane component of the internal force of the plate.

Fig. 18. Plate which has deflection with axial load acting

q = σx tp

∂ 2w ∂x2

(1)

In the compressive axial load cases shown in Fig. 18 (a), the distributed load q is superimposed upon the lateral pressure, which in turn increases its deflection. In the tensile axial load cases, on the other hand, the distributed load q cancels the lateral pressure as shown in Fig. 18 (b), which in turn decreases its deflection. Since the distributed load q is related to the curvature of the plate (second derivatives of deflection w), the effect of axial load on collapse strength becomes large when the deflection angle changes sharply. To study the magnitude of the effects of distributed load q for each plate thickness, the deflection of plates in pure lateral pressure cases where the maximum lateral pressure is applied at the end of Step 2 is shown in Fig. 19. It is, of course, observed that the deflection becomes large for thinner plates, such as those who thickness is 6 mm; this

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means that curvature, therefore, becomes larger as a result, and the distributed load q greatly impacts the collapse strength for these thinner plates. This mechanism can explain the reason why the tensile axial loads increases the collapse strength of the thin plates shown in Fig. 15 and Fig. 17. When plate thickness increases, the effect of the distributed load q gets smaller due to the slight deflection that develops as a result of its large flexural stiffness. It can, therefore, be assumed that the collapse strength of thicker plates is reduced by axial loads in accordance with the conventional theory that the axial load reduces the fully plastic moment of plates as described in Sect. 1.1.

Deflection [mm]

20 15 10 5 0 0

5

10

15

20

25

30

35

Plate thickness [mm] Fig. 19. Deflection of plates of pure lateral pressure case when maximum lateral pressure is applied

Deflection angle for transverse axial load direction Deflection angle for longitudinal axial load direction Transverse axial load X Y

Longitudinal axial load

Fig. 20. Difference of deflection angle between transverse and longitudinal directions

Focusing on the difference between the longitudinal and transverse axial load cases, the deflection angle and curvature in the longitudinal direction of a plate caused by lateral pressure must be smaller than their corresponding equivalents in the transverse

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direction as shown in Fig. 20. It follows, therefore, that the effect of the distributed load q in longitudinal axial load cases should be relatively smaller than that in transverse axial load cases. With regard to the CSR formulae, the reduction ratio for the CSR formula is similar to that of compressive-transverse axial load case (Fig. 14) where axial loads most significantly reduce collapse strength. It shows that the CSR formula considers the worst case scenario for the effects of axial loads for safety’s sake.

4 Collapse Strength of Stiffeners Derived by Non-linear FEA 4.1 Collapse Strength of Stiffeners The collapse strengths for stiffeners derived by the procedure described in Fig. 5 are shown in Figs. 21–24. Additionally, the criteria required by the assessment formulae of CSR [9] are also plotted as is done for plates. It is noted that the stiffener criteria required by CSR are based upon the initial yield strength of the stiffeners. Details of the CSR formulae are described in Appendix. 3500 3000

Secon modulus [cm3]

2500 2000 1500 Axial load 0% Axial load 30% Axial load 50% Axial load 70% CSR Axial load 0% - 70%

1000 500 0 0

0.2

0.4

0.6

0.8

1

1.2

Strength [MPa] Fig. 21. Collapse strength (FEA) and initial yield strength (CSR) of stiffeners (tensile load-plate side pressure)

With regard to the collapse strength derived by FEA, it decreases in proportion to the magnitude of the axial load in “compressive load-plate side pressure case” (Fig. 22) and the “tensile load-stiffener side pressure case” (Fig. 23), but only decreases in “compressive load-stiffener side pressure case” (Fig. 24) when a large axial load (axial stress = 70% of yield stress) is applied. In the “tensile load-plate side pressure case” (Fig. 21), on the other hand, the effect of the axial load is relatively smaller than other cases.

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3500 3000

Secon modulus [cm3]

2500 2000 Axial load 0% Axial load 30% Axial load 50% Axial load 70% CSR Axial load 0% CSR Axial load 30% CSR Axial load 50% CSR Axial load 70%

1500 1000 500 0 0

0.2

0.4

0.6

0.8

1

1.2

Strength [MPa] Fig. 22. Collapse strength (FEA) and initial yield strength (CSR) of stiffeners (compressive loadplate side pressure)

3500 3000

Secon modulus[cm3]

2500 2000 1500

Axial load 0% Axial load 30% Axial load 50% Axial load 70% CSR Axial load 0% CSR Axial load 30% CSR Axial load 50% CSR Axial load 70%

1000 500 0 0

0.2

0.4

0.6

0.8

1

1.2

Strength [MPa] Fig. 23. Collapse strength (FEA) and initial yield strength (CSR) of stiffeners (tensile loadstiffener side pressure)

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3500 3000

Secon modulus [cm3]

2500 2000 1500 1000

Axial load 0% Axial load 30% Axial load 50% Axial load 70% CSR Axial load 0% - 70%

500 0 0

0.2

0.4

0.6 Strength [MPa]

0.8

1

1.2

Fig. 24. Collapse strength (FEA) and initial yield strength (CSR) of stiffeners (compressive loadstiffener side pressure)

A comparison of the collapse strength derived by non-linear FEA and the CSR formulae in the pure lateral pressure case (Figs. 21–24) reveals that the collapse strength derived by non-linear FEA is anywhere from 25% to 80% higher than the CSR criteria for each section modulus. In addition, the FEA results are higher than the CSR criteria in all cases where axial loads are applied, except for only the “compressive load-stiffener side pressure case” (Fig. 24). When an axial load of 70% yield stress is applied in this case, the CSR formulae overestimates its strength compared to the collapse strength derived by FEA for some stiffener models; CSR, however, may cover this load condition through buckling assessment. 4.2 Discussions for the Reduction of Collapse Strength for Stiffeners Figures 25–28 show the reduction in collapse strength for some stiffener models described in Table 6. The CSR formulae consider the superimposing or cancelling of bending stress and axial load at the stiffener face of its span end, where the largest bending stress occurs in typical uniform stiffeners. For example, in compressive axial load-plate side pressure case shown in Fig. 29, compressive bending stress at the stiffener face of the span end is superimposed on the compressive axial load. Therefore, in the “compressive load-plate side pressure case” (Fig. 26) and the “tensile load-stiffener side pressure case” (Fig. 27), the CSR formulae take into account the axial load as it reduces the initial yield strength of stiffeners. On the other hand, when the bending stress at the point is canceled by the axial load, the CSR formulae do not take into account the effect of the axial load at all in

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1.4

Strength reducon rao

1.2 1 0.8 Model 1 (150x9+70x11) Model 5 (350x11.5+100x18.5) Model 10 (600x13+135x22.5) CSR

0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

Axial stress / σY Fig. 25. Reduction of stiffener strength (tensile load-plate side pressure)

1.4 Model 1 (150x9+70x11) Model 5 (350x11.5+100x18.5) Model 10 (600x13+135x22.5) CSR

Strength reducon rao

1.2 1 0.8 0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

Axial stress/σY Fig. 26. Reduction of stiffener strength (compressive load-plate side pressure)

the “tensile load-plate side pressure case” (Fig. 25) and the “compressive load- stiffener side pressure case” (Fig. 28). With regard to the collapse strength derived by FEA, even though the tensile axial load increases the collapse strength of Model 1 remarkably in the “tensile load-plate

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1.4

Model 1 (150x9+70x11) Model 5 (350x11.5+100x18.5) Model 10 (600x13+135x22.5) CSR

Strength reducon rao

1.2 1 0.8 0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

Axial stress/σY Fig. 27. Reduction of stiffener strength (tensile load-stiffener side pressure)

1.4

Model 1 (150x9+70x11) Model 5 (350x11.5+100x18.5) Model 10 (600x13+135x22.5) CSR

Strength reducon rao

1.2 1 0.8 0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

Axial stress/σY Fig. 28. Reduction of stiffener strength (compressive load-stiffener side pressure)

side pressure case” (Fig. 25), the collapse strength of other the models is not affected by axial load; however, in both the “compressive load-plate side pressure case” (Fig. 26) and the “tensile load-stiffener side pressure case” (Fig. 27), collapse strength decreases in proportion to the magnitude of axial load, while it is also observed that the reduction

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Tensile Compressive

Compressive axial load

Compressive Tensile

Z Plate side lateral pressure X

Fig. 29. Bending stress direction of stiffened plate to which plate side lateral pressure with compressive axial load is applied

of collapse strength in Fig. 26 is larger than that in Fig. 27. Finally, the collapse strength of Model 1 decreases remarkably by compressive axial load in the “compressive loadstiffener side pressure case” (Fig. 28), but the collapse strength of other cases decrease when axial load exceeds about 30% of the yield stress. The authors assume that axial load effect on the deflection considered in Sect. 3.2 is one of the factors of these phenomena just as is assumed with respect to plates. The common point of the conditions in Fig. 26 and Fig. 27 is the superimposition of bending stress and axial load at the stiffener face of span end as shown in Fig. 29. Since the distributed load q developed by compressive axial load is superimposed on lateral pressure, it can be assumed that the reduction of collapse strength shown in Fig. 26 becomes larger than that shown in Fig. 27 due to the distributed load q; moreover, it is also assumed that the increase in collapse strength of Model 1 in Fig. 25 and the remarkable decrease in strength of Model 1 in Fig. 28 are also caused by the distributed load q. This is because the flexural stiffness of Model 1 is small and thus the effect of the distributed load q is large. With regard to stiffeners, additional investigation should be carried out to consider their respective collapse mechanisms.

5 Conclusions The conclusions of this paper are summarized as follows. 1. A method for assessing the practical collapse strength of hull local members that are subjected to combination of high-level lateral pressures and low-level axial loads through the use of a residual deflection criterion is developed. The criterion is based upon actual cases where excessive permanent deformation is considered to cause a decrease in strength against future severe loading conditions and is regarded as damage requiring repair. Using this method, the collapse strength of rectangular plates and stiffeners in various load conditions are derived using the non-linear FEA software LS-DYNA. 2. With regard to plates, the results and discussions are summarized as follows:

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• Collapse strength values derived by residual deflection criterion are similar to the values derived by roof shaped collapse theory in the pure lateral pressure case. • Collapse strength derived by the residual deflection criterion is larger than the criteria required by the CSR formulae both in the pure lateral pressure case and in the combined lateral pressure and axial load cases. • With regard to the collapse strength derived by the residual deflection criterion, it is observed that the effect of an axial load on collapse strength is relatively small when the axial load is acting along the longer edge of plate. On the other hand, in cases where an axial load is acting along the shorter edge of plate, a significant axial load effect is observed. It is also observed that a compressive axial load is generally more critical than a tensile axial load with respect to collapse strength, but that, tensile axial load increases the collapse strength of thinner plates. • It can be assumed that the reason for the above-mentioned phenomena is related to the fact that a fully plastic moment is reduced by axial load in accordance with conventional theory and as a result an “axial load effect on deflection” exists. This effect is caused by the distributed load q, which is the out-plane component of internal force of the plates that is subjected to axial load and lateral pressure. When a compressive axial load is applied, the distributed load q is superimposed onto lateral pressure which in turn increases the deflection. On the other hand, when a tensile axial load is applied, the distributed load q cancels lateral pressure and, therefore, decreases the deflection. Since the distributed load q greatly affects collapse strength when the curvature of the plate is large, plates with small flexural stiffness are greatly affected by the distributed load q. Additionally, the effects of distributed load q become smaller when an axial load is acting along the longer edge of plate because the deflection angle and curvature caused by the lateral pressure must be smaller than those along the shorter edge of plate. 3. With regard to stiffeners, the results and discussions are summarized as follows: • When an axial load is superimposed onto the bending stress acting on the stiffener face of span end, the collapse strength derived by the residual deflection criterion decreases in proportion to the magnitude of the axial load. In such cases, the compressive axial load is more critical than the tensile axial load; in other cases, collapse strength of stiffener which has small flexural stiffness is increased by the tensile axial load and decreased by the compressive axial load. • The reason for these phenomena can be considered that the distributed load q affects the collapse strength of stiffeners in the same manner as it affects the collapse strength of plates. Further investigation should be carried out to consider the collapse mechanisms of stiffeners.

Appendix In 2014, IACS published its “Common Structural Rules for Bulk Carriers and Oil Tankers (CSR)” [9] for bulk carriers 90 m or longer in length and for oil tankers 150 m or longer

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in length. In Sect. 4 of Chapter 6 in Part 1 of CSR, the common requirement formulae for any plating are specified. The formula of requirement thickness for plates is as follows.  P 0.0158αp b (2) [mm] χ Ca ReH where b: Breadth of plate panel [mm] P: Design pressure [kN/m2 ] χ: Coefficient of flooding condition. (In this paper, to be taken as 1.0) ReH : Specified minimum yield stress [N/mm2 ] α p : Correction factor for panel aspect ratio to be taken as follow but not to be taken greater than 1.0 αp = 1.2 −

b 2.1a

a: Length of plate panel [mm] C a : Permissible bending stress coefficient for plate taken equal to: Ca = β − α

σhg , not to be taken greater than Ca−max ReH

β: Coefficient as defined in Table 9 Table 9. Definition of CSR coefficients for plates β

α

C a-max

Longitudinal strength member of longitudinally stiffened plating

1.05

0.5

0.95

Longitudinal strength member of transversely stiffened plating

1.05

1.0

0.95

Note: Values of AC-SD acceptance criteria in CSR are used.

Table 10. Definition of C s Sign of axial stress

Lateral pressure acting on

Coefficient C s

Tension

Stiffener side

Compression

Plate side

Cs = βs − αs R hg eH But not to be taken greater than C s-max

Tension

Plate side

Cs = Cs−max

Compression

Stiffener side

β s : Coefficient as defined in Table 11. α s : Coefficient as defined in Table 11. C s-max : Coefficient as defined in Table 11. Other symbols: refer to the formulae for plates.

  σ 

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αs

C s-max

Longitudinal strength member 1.0 1.0 0.9 Transverse or vertical member 0.9 0

0.9

Note: Values of AC-SD acceptance criteria in CSR are used.

α: Coefficient as defined in Table 9 C a-max : Maximum permissible bending stress coefficient as defined in Table 9 σ hg : Axial stress [N/mm2 ] In Sect. 5 of Chapter 6 in Part 1 of CSR, the common requirement formulae for any stiffeners are specified. The formula of requirement section modulus is as follows. 2 Pslbdg

fbdg χ Cs ReH



cm3

 (3)

where s: Spacing of frames [mm] lbdg: Effective bending span [m] f bdg: Bending moment factor taken as follows: 12 for horizontal stiffeners and upper ends of vertical stiffeners 10 for lower ends of vertical stiffeners C s: Permissible bending stress coefficient as defined in Table 10

References 1. Jones, N.: Influence of in-plane displacements at the boundaries of rigid-plastic beams and plates. Int. J. Mech. Sci. 15, 547–561 (1973) 2. Manolakos, D.E., Mamalis, A.G.: Limit analysis for laterally loaded stiffened plates. Int. J. Mech. Sci. 30, 441–447 (1988) 3. Schubak, R.B., Olson, M.D., Anderson, D.L.: Rigid-plastic modelling of blast-loaded stiffened plates—Part I: One-way stiffened plates. Int. J. Mech. Sci. 35, 289–306 (1993) 4. Schubak, R.B., Olson, M.D., Anderson, D.L.: Rigid-plastic modelling of blast-loaded stiffened plates—Part II: Partial end fixity, rate effects and two-way stiffened plates. Int. J. Mech. Sci. 35, 307–324 (1993) 5. Louca, L.A., Pan, Y.G., Harding, J.E.: Response of stiffened and unstiffened plates subjected to blast loading. Eng. Struct. 20, 1079–1086 (1998) 6. Hong, L., Amdahl, J.: Plastic design of laterally patch loaded plates for ships. Mar. Struct. 20, 124–142 (2007) 7. Shanmugam, N.E., Zhu Dongqi, Y.S. Choo, M.: Arockiaswamy: experimental studies on stiffened plates under in-plane load and lateral pressure. Thin-Walled Structures 80, 22–31 (2014)

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8. Zhaolong, Yu., Amdahl, J., Sha, Y.: Large inelastic deformation resistance of stiffened panels subjected to lateral loading. Mar. Struct. 59, 342–367 (2018) 9. Kyokai, N.K.: Rules for the Survey and Construction of Steel Ships. Part CSR-B&T, Tokyo (2018) 10. International Association of Classification Societies: Rec 47 Shipbuilding and Repair Quality Standard - Rev., 8 October 2017, London (2017) 11. Fujii, T., Uchino, K.: Strength of panels of plating loaded beyond the elastic limits. Ishikawajima-Harima Eng. Rev. 2(4), 84–93 (1962) 12. Tanaka, H., Umezaki, K., Terada, K.: Experimental research on stiffened panel loaded beyond the elastic limit. J. Kansai Soc. Naval Architects, Japan 112, 23–30 (1964) 13. Fujikubo, M., Uda, S., Tatsumi, A., Iijima, K.: Finite element modeling of a continuous stiffened panel under combined inplane shear and thrust, In: Proceedings of the 27th Asian-Pacific Technical Exchange and Advisory Meeting on Marine Structure (TEAM’2013), Keelung (2013)

A Study on Progressive Collapse Analysis of a Hull Girder Using Smith’s Method – Uncertainty in the Ultimate Strength Prediction Akira Tatsumi(B) , Kazuhiro Iijima, and Masahiko Fujikubo Osaka University, 2-1 Yamadaoka, Suita, Osaka, Japan [email protected]

Abstract. Smith’s method has been widely used for the progressive collapse analysis of a ship’s hull girder under longitudinal bending. In the Smith’s method, a hull girder’s cross section is divided into the plate and stiffened-panel elements which are given a priori an average stress-average strain relationship under uniaxial tension/compression considering buckling and yielding. This implies that pure axial load is applied to each element while no curvature effect over the element span due to hull girder bending is considered. In addition, it is assumed that the cross section remains in a plane during hull girder bending and that all elements deform independently with no interaction between adjacent elements. The objective of this study attempts to investigate the influences of these assumptions that are used in the Smith’s method. Progressive collapse behavior of a 14,000TEU container ship is analyzed by the Smith’s method using the average stress-average strain relationships of the plate and stiffened-panel elements estimated by the nonlinear finite element method (NFEM). On the other hand, the progressive collapse behavior of the hull girder is directly analyzed by NFEM using 3D-shell model. Comparing both results, the model uncertainty of the Smith’s method due to its basic assumptions is discussed. It is found that global hull-girder curvature, usually neglected in the Smith’s method, significantly affects collapse behavior of the stiffened panels. Keywords: Progressive collapse analysis · Smith’s method · Model uncertainty · Container ship

1 Introduction It has been more than 40 years since Smith [1] proposed a method of progressive collapse analysis of a hull girder under pure longitudinal bending. The so-called Smith’s method has been widely used as a practical method to obtain ultimate capacity of a hull girder’s cross section [2–5], and also incorporated into Common Structural Rules of Bulk Carriers and Oil Tankers (CSR B & T) [6]. On the other hand, the nonlinear finite element method (NFEM) is increasingly used to estimate the hull girder ultimate capacity [7–13]. One of the advantages of NFEM compared to Smith’s method © Springer Nature Singapore Pte Ltd. 2021 T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 128–144, 2021. https://doi.org/10.1007/978-981-15-4672-3_7

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is that NFEM can simulate progressive collapse behavior of hull girders under combined load, which cannot be basically analyzed by the Smith’s method. To overcome this disadvantage of the Smith’s method, some of the authors have extended the Smith’s method so that it can deal with combined load conditions, e.g. combined bending and torsion [14] and combined global bending and local lateral loads [15]. Recently, the authors have simulated a dynamic collapse response of a hull girder while considering an interaction between structural deformation and hydrostatic restoring force by using a beam FE model in which progressive collapse behavior is considered according to the Smith’s method [16]. The Smith’s method still has large potential to practically analyze progressive collapse behavior/response of hull girders under combined static/dynamic loads. In the Smith’s method, a hull girder cross section is idealized as an aggregation of plate/stiffened panel elements whose average stress-average strain curves under uniaxial tension/compression considering effect of buckling and yielding are prepared. By giving incremental curvature to the cross section considering a change of a position of a neutral axis, bending moment–curvature relationship of the cross section is calculated, and the hull girder ultimate strength is obtained as the peak value of the bending moment. The Smith’s method makes several assumptions as; (i) No curvature effect on behaviors of the plate/stiffened panel elements due to hull girder bending is considered; (ii) All elements deform independently and the interactions between different elements are neglected. (iii) The cross section remains plain and keeps the original shape during process of progressive collapse. These can be regarded as model uncertainties of the Smith’s method. Whereas the assumptions of the Smith’s method definitely affect estimation of the hull girder ultimate capacity, the effect has not been sufficiently investigated quantitatively. In the preset research, the assumption (i) mentioned above are addressed. Progressive collapse behavior of a 14,000 TEU class container ship under pure longitudinal bending is analyzed by using the Smith’s method and nonlinear finite element method (NFEM) so as to make the effect of the assumption in the Smith’s method clear. In addition, effect of constraint by a transverse floor against in-plane deformation of the plate is discussed.

2 Progressive Collapse Analysis of a Hull Girder 2.1 Subject Ship A 14,000 TEU class container ship is taken as a subject ship in this study. Length, breadth and depth are 366 m, 52 m and 30 m, respectively. Design draft is 14 m. A midship section is shown in Fig. 1. High tensile strength steels are used around the deck part. Material yield stresses in the regions 1, 2 and 3 are 352.8, 392.0 and 460.6 MPa, respectively. The materials are assumed to be elasto-perfectly plastic. 2.2 Analysis by Smith’s Method In the Smith’s method, a hull girder’s cross section is idealized as an aggregation of plate/stiffened panel elements as shown in Fig. 1, and the average stress-average strain relationships under uniaxial tension/compression considering effect of buckling and

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3 2 1

Fig. 1. Midship section of the subject ship

yielding are given to the plate/stiffened panel elements. By giving incremental curvature to the cross section while change of a position of a neutral axis is taken into consideration, bending moment–curvature relationship of the cross section can be calculated. The Smith’s method assumes follows; (i) Pure axial load is applied to each element while no curvature effect over the element span due to hull girder bending is considered (see Fig. 2). (ii) The stiffened plates deform independently, that is interaction between stiffened panels are neglected. (iii) The cross section remains plane and keeps its original shape during process of hull girder collapse. In the present study, effects of the assumption (i) is investigated. Progressive collapse of the container ship is analyzed by the Smith’s method and NFEM, and both results are compared each other. The curvature due to the hull girder bending which is ignored in the assumption (i) may affect the collapse behavior of the stiffened panels and increase or decrease their ultimate compressive strength, leading to overestimation or underestimation of the hull girder ultimate capacity. In addition, effect of constraint by a transverse floor against in-plane deformation of the plate is discussed. When a stiffened panel is subjected to in-plane compression in the longitudinal direction, tensile deformation is induced in the transverse direction owing to Poisson’s effect. The transverse floor restrains the in-plane deformation of the plate due to Poisson’s effect, which may increase or decrease the ultimate longitudinal strength of the stiffened panels.

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Hull girder in hogging condition

Pure compression Idealization Stiffened panel Compression and bending Fig. 2. Curvature due to hull girder bending

Average Stress-Average Strain Relationship. The average stress-average strain relationships of the stiffened panels subjected to uniaxial compression is calculated by NFEM. Based on an assumption of a continuous stiffened panel, a 1/2 + 1 + 1/2 floor space model is used for the FE analysis. A periodic boundary condition [17] is applied to the cross section at both ends. An advantage of the periodic boundary model is its capability of simulating buckling of both even and odd numbers of waves under the assumption of the continuous stiffened panel. In the transverse direction, 1/2 + 1/2 stiffener space is modeled and a symmetric condition is imposed on the longer edges when stiffener cross section is a tee or a flat type. In case of an angle type, 1/2 + 1 + 1/2 stiffener space is modeled and the periodic boundary condition is imposed in the transverse direction. Floors are not directly included in the FE model and deflection in the vertical direction is fixed along the floors. Initial deflections of buckling mode are added to the FE model. Local buckling mode of the plate w0p , flexural buckling mode of the stiffened panel w0s and torsional buckling mode of the stiffener v0t are taken into account as shown in Fig. 3, by equations; πy mπ x sin a b πx w0s = As sin a z πx v0t = At sin hw a

w0p = Ap sin

(1) (2) (3)

where a is floor space, b stiffener space, hw web height and m the number of buckling half wave. Ap , As and At are amplitudes of each initial deflection, and assumed to be 0.05β 2 t p , a/1000 and a/1000 respectively. β is slenderness ratio of the plate and t p is plate thickness.

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Fig. 3. 1/2 + 1 + 1/2 floor space model and initial deflections

Effective Section Area of a Floor. When a stiffened panel is subjected to in-plane compression in the longitudinal direction, tensile deformation is induced in the transverse direction owing to Poisson’s effect. Since the transverse floor restrains this deformation, the plate is subjected to bi-axial compression which may enhance plate buckling. On the other hand, as the out-plane deformation of the plate is developed after the buckling, edges of the plates are drawn inside due to large-deflection effects. This in-plain displacement is also restrained by the floor in transverse direction, which may result in increase of ultimate compressive strength. In order to discuss these restraint effect of the transverse floor, it is necessary to perform ultimate strength analysis of a stiffened panel model including the floor. In the present study, the floor is idealized by a bar having effective cross-sectional area. It means that constraint by the floor on the in-pane deformation is considered while that on the out-plane deformation of the plate is neglected. When a plate to which a bar is attached in transverse direction is considered as shown in Fig. 4 and the plane stress condition is assumed for the plate, the relationship between longitudinal stress σ x and transverse stress σ y in the elastic regime can be derived as;

Fig. 4. Poisson’s effect and idealization of a plate with a floor

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σy =

νAe σx Apl + Ae

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(4)

where Apl is the longitudinal cross-sectional area of the plate, Ae the effective crosssectional area of the floor, and σ the Poisson ratio. When σ x and σ y are obtained, Ae can be calculated. σ x and σ y can be estimated by stress analysis of a hull girder by FEM. The FE analysis of a hull girder is described in the next section. 2.3 Nonlinear Finite Element Analysis Progressive collapse analysis of the subject ship is performed using NFEM to investigate the influences of the assumption in the Smith’ method on the collapse behavior and the hull girder ultimate strength. Since the investigation into the assumption (iii) described in 2.2 is out of scope of this research, the cross sections at positions of the transverse frames must be constrained so that they remain plane during the process of hull girder collapse. On the other hand, continuity of structure and deformation should be satisfied on boundaries for accurate estimation of the collapse behavior. In addition, computational effort should be reduced as much as possible. To meet these needs, a hull girder model as shown in Fig. 5 is adopted. The region of 1 + 1 floor spaces in the longitudinal direction and a half the full breadth in the transverse direction is modeled, while imposing symmetric boundary conditions on the longitudinal cross section along the center line of the hull girder. The symmetric condition is herein applied only to the inner-and-outer bottom plate and the floors as shown in Fig. 5. The constraint equation for the plane cross section at the positions of the transverse frames is given by; ⎫ ⎧ ⎫ ⎧ ⎨ uxi ⎬ ⎨ uxl + hi θyl ⎬ = u + hj θym (5) u ⎭ ⎩ xj ⎭ ⎩ xj uxk uxk + hk θyn where u and θ show translational and rotational displacements. h is vertical distance from master nodes. The subscript xyz represents coordinate axes which the displacements refer. The subscript lmn represents the master nodes at aft, center and middle cross sections and ijk nodal points except the master nodes at each cross section as indicated in Fig. 5. Periodic boundary condition on fore-and-aft end cross sections can be expressed by the following equation; ⎫ ⎧ ⎫ ⎧ ⎪ ⎪ ⎪ ⎪ uyk uyi ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪u ⎪ ⎪ ⎪ ⎪u ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ zk ⎬ ⎨ yi = θxk (6) θxi ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θ − θ θ − θ yi yn ⎪ yl ⎪ ⎪ yk ⎪ ⎪ ⎪ ⎪ ⎩θ ⎭ ⎪ ⎭ ⎩θ zi z θyl and θyn represent rotation angle around the y axis of the hull girder cross sections at the fore-and-aft ends, and the fourth equation imposes the condition that relative rotation angles around the y axis are equal between the fore-and-aft ends. By Eqs. (5) and (6), the continuity of the deformation in the longitudinal direction is satisfied while considering that the cross sections at the position of the transverse frames remain plane.

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Fig. 5. 1 + 1 floor space model of a hull girder

The hull girder is simply supported at nodal point B on the bottom plate. Axial displacement ux is fixed at the master node l on the aft end section while allowed at the master node n at the fore section. The progressive collapse analysis under the pure vertical bending can be carried out by giving forced rotational displacement around the y axis to the master nodes l and n in the opposite direction incrementally. The 1 + 1 floor space with the periodic boundary condition Eqs. (5) and (6) can reasonably consider effect of restraints by adjacent structures under the assumptions that the structural configurations and the deformations are repeated in the longitudinal direction and the cross section at the positons of the transverse frame remain plane. On the other hand, when a real ultimate capacity of the hull girder is assessed, a longer model such as a several-hold model should be used in order to reduce effect of the boundary conditions. The initial deflections of the local buckling mode of the plate, the flexural buckling mode and the torsional buckling mode of the stiffened panel are given to inner-andouter bottoms and girders as shown in Fig. 6 and not to the bilge corner. These initial deflections are same as the stiffened panel model in Fig. 3.

3 Results and Discussions Progressive collapse behavior of the 14,000TEU container ship in pure hogging condition is analyzed by both the Smith’s method and NFEM. Longitudinal bending moment– curvature relationships obtained by the two methods are compared in Fig. 7. The hull girder ultimate strength estimated by the Smith’s method is smaller than that by NFEM.

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Fig. 6. Initial deflection for a hull girder model

In the case of the Smith’s method, stiffened panels in the outer bottom collapse and in the deck side yield around point A. Thereafter, the capacity of the hull girder cross section keeps approximately constant, and it reaches ultimate strength at point B where stiffened panels in the inner bottom collapse. In the case of NFEM, the tensile yielding is occurred in the stiffened panels in the deck side and the buckling and yielding spreads in the outer bottom around point C. The capacity of the hull girder is slightly increased after point C and it reaches ultimate strength at point D where the outer bottom plate and some of girders have collapsed. Figure 8 shows deformation and distribution of equivalent plastic strain at the hull girder ultimate strength. Plastic strain due to the tension is observed in the deck side. The buckling and the yielding is happened in the outer bottom plate and the girders. Although the plastic strain also takes place in the inner bottom plate, it is not large. 3.1 Effect of Global Curvature In order to consider a reason why there is the difference of the hull girder ultimate strength between the Smith’s method and NFEM, the collapse behavior of a stiffened panel in the outer bottom is investigated. Deformation and equivalent plastic strain of the stiffened panel at a post-collapse point are compared in Fig. 9. Figure 9(a) shows the deformation and the distribution of the 1/2 + 1 + 1/2 floor space stiffened panel model used in Smith’s method. The result of (b) is taken from the 1 + 1 floor space hull girder model. In the stiffened panel analysis of (a), plate buckling (plate induced failure) takes place in 1/2 + 1/2 floor space while lateral buckling of the stiffener (stiffener induced failure) in the other 1 floor space. On the other hand, in the result of the hull girder analysis (b), the plate induced failure is predominant and lateral buckling of the stiffener cannot be observed. Figure 10 shows a comparison of average compressive stress-average compressive strain relationship of the stiffened panel in the outer bottom. The stiffened panel in the hull girder model has larger capacity especially in post-collapse range, which results in the larger hull girder ultimate strength compared to the Smith’s method.

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Longitudinal bending moment [GNm]

3.5E+01 3.0E+01 2.5E+01

D

C A

B

2.0E+01 1.5E+01 1.0E+01 5.0E+00

Smith's method NFEM

0.0E+00 0.0E+00 5.0E-05 1.0E-04 1.5E-04 2.0E-04 2.5E-04 Curvature [1/m] Fig. 7. Longitudinal bending moment–curvature relationships

Fig. 8. Distribution of equivalent plastic strain at hull girder ultimate strength

One of the reasons of the difference regarding the collapse mode of the stiffened panel is effect of curvature due to the hull girder bending. Figure 11(a) shows bending deflection of the plate along the stiffener taken from the hull girder analysis when εx /εy = 0.4. Herein, the initial deflection is eliminated. The bending deflection consist of local bending deflection of the stiffened panel and global bending deflection of the hull girder. Subtracting the hull girder deflection which is estimated by beam theory from Fig. 11(a), the local deflection of the stiffened panel is given by the solid curve in Fig. 11(b). The dashed curve shows the bending deflection of the 1/2 + 1 + 1/2 stiffened panel model in the Smith’s method when εx /εy = 0.4. For the comparison with the result of the 1 + 1 floor space model of the hull girder, the deflection of the stiffened panel model in 1/2 + 1 + 1/2 floor space is translated to the deflection in 1 + 1 floor space. In

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Floor

a/2

a

Floor Floor

a Floor

a

a/2

Floor

(a) Stiffened panel analysis

(b) Hull girder analysis

Fig. 9. Deformation and equivalent plastic strain of a stiffened panel in the outer bottom at a post-collapse point

Fig. 10. Average stress-average strain relationships of a stiffened panel in the outer bottom

the stiffened panel analysis, the deflection of beam-column mode grows due to the compression, which is larger than that in the hull girder analysis. It implies that the curvature due to the hull girder bending restrains the growth of the deformation of the stiffened panel as a beam-column. Therefore, deformation of the stiffener buckling is not significant as shown in Fig. 9(b) and its compressive capacity in the post-collapse range is probably increased. Consequently, the ultimate bending capacity of the hull

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girder obtained by NFEM is larger than that by the Smith’s method in which the average stress-average strain relationships under pure axial compression are used.

(a) Deflection taken from the hull girder analysis.

(b) Deflection excluding the hull girder bending and comparison with the stiffened panel analysis

Fig. 11. Deflection along a stiffener of the stiffened panel in the outer bottom when εx /εy = 0.4

The deflection angle is almost zero at the positions of the transverse frames, 0 mm, 3625 mm and 7250 mm (which is equivalent to 0 mm under the periodical boundary condition), as shown in Fig. 11(b). This may be because the transverse floors restrain out-plane deformation of the outer bottom plate due to the hull girder bending. Further investigation into the constraint effect of the transverse floors on the out-plane deformation is needed.

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3.2 Effect of Transverse Floors To investigate the effect of transverse floors on the ultimate strength of a longitudinal stiffened panel, the 1/2 + 1 + 1/2 floor space model with bars representing the floors shown in Fig. 12 is analyzed under the pure compression. The effective section area Ae of the bar is calculated by Eq. (4) where the longitudinal compressive stress σ x and the transverse compressive stress σ y is estimated by the FE analysis of the 1 + 1 floor space hull girder model mentioned in the Sect. 3.1. Fig. 13 shows history of the bi-axial stress in the outer bottom stiffened panel during the process of the hull girder collapse under the pure hogging. Although the bi-axial stress field is generated because of the constraint by the transverse floors, the transverse stress is much smaller than the longitudinal one. Ae is derived from σ y /σ x in liner elastic region of Fig. 13, which is approximately a half of total section area of the floor. This is because the transverse floor constrains the in-plane displacement of the inner bottom as well as the outer bottom.

Floor (u z = 0)

a/2 Floor (u z = 0) a z Bar element b

a/2

y

x

Fig. 12. 1/2 + 1 + 1/2 floor space model including transverse floors

Figure 14 shows average stress-average strain relationships of the stiffened panel with/without the transverse floors obtained by NFEA. Since there is no difference between the two curves, it can be concluded that the constraint by the transverse floor on the in-plane displacement of the plate does not affect the collapse behavior of the stiffened panel under the longitudinal compression. Therefore, the constraint effect of the floors can be neglected in the assessment of hull girder ultimate strength of the subject ship under the pure bending. On the other hand, the transverse floor also restrains out-plane deformation of the stiffened panel and may increase its ultimate compressive strength. Investigation on this effect still remains future work. 3.3 Smith’s Method According to CSR B & T The progressive collapse analysis for the subject ship is performed according to CSR B & T in which average stress-average strain relationships of stiffened panels are calculated by

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Fig. 13. Bi-axial stress in the outer bottom during process of hull girder collapse

Fig. 14. Effect of transverse floors on average axial stress-average axial strain relationship of a stiffened panel under longitudinal compression

Gordo-Soares’s formulae [18]. The so-called hard corner element which is typically used in intersection areas between girders and platings is replaced by a normal plate element because reduction of load-carrying capacity was observed in such areas in the authors’ previous study [19, 20]. The longitudinal bending moment–curvature relationship is shown by the long dashed dotted curve in Fig. 15. The Smith’s method according to CSR B&T gives a larger estimation of the hull girder ultimate strength compared to the case using the average stress-average strain relationships estimated by NFEM, even though the effect of the curvature on the collapse behavior of stiffened panels is not taken into account. This is because the CSR formulae overestimate load-carrying capacity of stiffened panels especially in post-collapse as shown in Fig. 16. It results in the large discrepancy of hull girder bending capacity after the ultimate strength in Fig. 15.

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Longitudinal bending moment [GNm]

3.5E+01 3.0E+01 2.5E+01 2.0E+01 1.5E+01 1.0E+01

NFEM Smith's method (NFEM)

5.0E+00

Smith's method (CSR) 0.0E+00 0.0E+00 5.0E-05 1.0E-04 1.5E-04 2.0E-04 2.5E-04 Curvature [1/m]

Fig. 15. Comparison of longitudinal bending moment–curvature relationships

1.0

σx / σY

0.8 0.6 0.4 NFEM Smith's method (NFEM) Smith's method (CSR)

0.2 0.0 0.0

1.0

2.0

3.0

εx / εY Fig. 16 Comparison of average stress-average strain relationships of a stiffened panel in the outer bottom

Even when the average stress-average strain relationships under the pure compression are accurately estimated for example by NFEM, the Smith’s method using their average stress-average strain relationships underestimates hull girder ultimate strength because of the disregards of the curvature effect. Although the Smith’s method using the CSR formulae gives a better estimation, it is because of their overestimation of load-carrying capacity in the post-collapse range. It means that the overestimation of load-carrying

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capacity fortunately cancels the underestimation due to the disregard of the curvature effect. Ship designers should understand these effects due to model uncertainties of the Smith’s method.

4 Conclusions Progressive collapse behavior of a 14,000TEU container ship under pure hogging is analyzed by Smith’s method and NFEM to investigate influence of a few assumptions in the classic Smith method. The following conclusions may be drawn; 1. Since the curvature due to the hull girder bending restrains deflection of stiffened panels as a beam-column, the stiffener buckling does not significantly appear and the plate buckling is predominant. It results in higher load-carrying capacity of the stiffened panels compared to that under pure axial compression. 2. The Smith’s method using average stress-average strain relationships of stiffened panels under pure longitudinal compression underestimates the hull girder capacity because of the disregard of the curvature effect. 3. The Smith’s method using the average stress-average strain relationships of CSR BC & OT gives a larger ultimate strength of the hull girder compared to using those estimated by NFEM, even though the effect of curvature is not taken into account. This is because CSR BC & OT formulae themselves tend to overestimate the load-carrying capacity of the stiffened panels in the post-collapse regime. 4. The constraint by the transverse floors on the in-plain displacement of the plates does not affect collapse behavior of the stiffened panels in the hull girder subjected to pure bending. Therefore, there is no need to include transverse floors into the Smith’s method. In order to improve the Smith’s method, an analytical model to estimate average stress-average strain relationships considering the curvature effect due to the hull girder bending should be developed. In addition, it is necessary to investigate effect of the other assumptions of Smith’s method that a plane cross section remains plane and that all plate/stiffened panel elements deform independently with no interaction between adjacent elements. The restraint effect by transverse floors on out-plane deformation of stiffened panels should be also investigated. These still remain future works. Acknowledgements. It is acknowledged that the present work has been partially supported by Japan Society for the Promotion of Science (JSPS) KAKENHI, Grant Number 16K18321. The authors would also like to acknowledge that the present work has been conducted as partial fulfillment of bachelor course study of Ms. Megumi Iwama at Department of Naval Architecture and Ocean Engineering, Osaka University, 2018.

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Verification of an Automated Structural Design Procedure Using Ultimate Limit States Nathaniel Cope1(B) and Joshua Knight2 1 Navatek, Ltd., South Kingstown, RI, USA

[email protected] 2 Navatek, Ltd., Arlington, VA, USA

[email protected]

Abstract. Ultimate limit state-based iterative structural analysis methods hold great potential to ease early-stage ship design. Such methods must be verified, however, before their results can be trusted. A key way to do so is to examine their behavior in known cases; that is, to set them working on well-known and thoroughly investigated problems to see if they produce the expected results. “Brokkr,” one such method, was tested in just such a way. First, a test hull was chosen, and the loading set used to develop it was researched. An analysis was then performed to determine the maximal stress levels induced in the hull by that loading set, revealing the safety factors the hull enjoys. A basic scantling model was then developed with the same overall size and material characteristics as the test hull, and was fed into Brokkr and iteratively modified to remain as light as possible by the same margin of safety and under the same loading conditions as the test hull. The design, generated using one overall safety factor, succeeded in maintaining the same margin of safety, but was significantly lighter. A second design created using a set of partial safety factors matched the test hull very closely in both mass and margin of safety. These results represent a successful verification and suggest that Brokkr has great utility for early-stage ship design. Further work is needed to additionally verify this code and to extend its use to more numerous and novel hull designs. Keywords: Automation · Verification · Structural design · Early-stage design

1 Introduction Keeping a ship weight- and cost-efficient while also keeping it strong enough to withstand extreme loads is by no means a straightforward task. Currently, during the early-stage design process, engineers attempt to approximate an optimal ship structure through the combination of established rules, anecdotal best practices, and analyses of past proven designs. Though this effort has yielded great results, it also has drawbacks. For one, the achieved outcomes can vary widely based on which rules and practices are employed. For another, because design rules are often structured conservatively to ensure safe application across many varied cases, they may be overly conservative and therefore suboptimal in specific applications. A ship designed under too conservative a rule set © Springer Nature Singapore Pte Ltd. 2021 T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 145–154, 2021. https://doi.org/10.1007/978-981-15-4672-3_8

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could, for example, end up slightly heavier than needed, causing higher fuel usage and greater magnitudes of forces experienced, which in turn result in higher operating costs and a shortened lifespan. Furthermore, approaches based on allowable stresses and other rules-based design schemes naturally lose much of their efficacy when it comes to the analysis of new and innovative ships and ship structures for which there does not exist a large body of known data. This paper presents a potential solution: the use of ultimate limit states as opposed to allowable stresses in an automated iterative design process, to both address the potential costs of structural over-design and to avoid the dangers of applying allowable stress rules to novel designs where they have not been tested. This paper presents the preliminary results of a monohull midship section redesign executed as a verification of one particular such ultimate limit state method known as Brokkr. Brokkr is a medium-fidelity, computationally-efficient, hierarchical ship structural design program that applies physics-based solutions for structural strength through the direct estimation of ultimate limit states. Brokkr iterates on a base design to produce a final result that can withstand a given loading set while also minimizing a given objective characteristic (e.g., total mass, fabrication cost). This physics-based approach (sometimes called an approach from “first principles”), with its focus on specific and isolated structure analysis, can mitigate many of the drawbacks displayed by other, rules-based strategies. A derivation of the ultimate limit state calculations and iteration procedures employed by Brokkr is beyond the scope of this paper, but can be found in prior work [7, 10]. Before the results of such a new tool can be trusted, however, it is imperative that the tool be tested and verified. The key test in this verification process is the comparison of designs generated by Brokkr with a collection of known and vetted baseline designs. If the ultimate limit state design process represented by Brokkr is sound and of commensurate quality with other contemporary methods, a designer should be able to use it to take a base design, a loading set, and a set of appropriate partial safety factors to produce structural designs of similar weights and arrangements to vetted, baseline designs generated by other means (e.g., allowable stress approaches). The demonstration of this capability would affirm the utility of ultimate limit state-based design and build confidence in its capabilities and in their potential application to more novel hull forms and structural topologies. Therefore, the testing of Brokkr’s results against benchmark designs is of utmost interest. Section 2 of this paper summarizes the ultimate limit states used. Section 3 discusses selection, analysis, and preparation of a suitable test case for this endeavor, including calculation of the test hull’s design loading and resulting ultimate limit states and safety factors. Section 4 shows results of Brokkr’s ultimate limit state-based design procedure to generate a new midship scantling from a base design using the loading and safety factor sets derived from the test hull. Section 5 compares the Brokkr-produced midship scantling with the test hull’s midship scantling and offers an analysis of what those results mean about the efficacy of the utilized ultimate limit state-based procedure. Section 6 summarizes the project at large, considers the strengths and weaknesses in the testing procedure, and suggests potential areas for future work.

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2 Ultimate Limit State-Based Design The assessment of structural strength should be based, wherever possible and practical, on first-principles. It is the authors’ opinion that this requires transitioning from common allowable stress approaches to the direct estimation of ultimate limit states (ULS). Recent theoretical advances make it possible to estimate the membrane stresses in loaded plating under large-deflection (i.e., nonlinear) conditions either analytically or semi-analytically. These membrane stresses then allow for the prediction of structural failure and collapse by comparison of localized von Mises stresses with material yield criteria. The membrane stress-based approach provides a closer approximation to the true physics than the allowable stress criteria which are calibrated against historical data and subject matter expert opinion. A detailed derivation of all the limit states used by Brokkr is beyond the scope of this paper, but key failure modes are summarized in Table 1, along with references for the interested reader to explore in greater depth. One of the powerful features of these membrane stress-based ULS is that one can analyze the structural response to a combination of lateral pressure, biaxial in-plane loads, and edge shears. This is significant for novel hull forms, especially multihulls, where structural failure can result from complex loading scenarios. Previous work has validated the ultimate limit state formulations and made improvements for plate collapse (mode 2) under simultaneous lateral pressure load [10]. Where such prior research had focused on validating individual failure mode formulations, this paper focuses on verifying the behavior of an automated design procedure that uses these ultimate limit states as constraints on the design space. Table 1. Summary of ultimate limits state equations for design. Mode Description

References

1

Intra-frame collapse of plating and stiffeners as one unit.

Paik et al. (2001) [9]

2

Collapse of plating without failure of stiffeners.

Smith et al. (2018) [10], Paik et al. (2001) [9], Paik and Duran (2004) [8]

3

Beam-column type collapse of a stiffener Faulkner et al. (1973) [5], Hughes and Paik with its attached width of plating. (2010) [6], Paik and Duran (2004) [8]

4

Local buckling of the stiffener web.

5

Flexural-torsional buckling or tripping of Hughes and Paik (2010) [6], Faulkner the stiffeners. (1987) [4], Paik and Duran (2004) [8]

6

Gross yielding.

Hughes and Paik (2010) [6]

Collette (2011) [3], Hughes and Paik (2010) [6]

3 Establishing a Test Case The U.S. Navy DTMB 5415 [Fig. 1] was selected as the verification hull against which Brokkr would be tested. The 5415, representative of the U.S. Navy’s Arleigh Burke class

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destroyer, is a well-known and well-studied hull created using the historical best practices common to modern ship design. As such, it makes a fine candidate for a benchmark test of Brokkr’s ultimate limit state-based methods. In order to iterate upon a basic structure in the attempt to independently reproduce the 5415, however, Brokkr’s methods require details about the loading case the design must survive and about the safety factor or factors (calculated from large deflection membrane stress solutions for plate and beam bending) that determine how wide that margin of survival should be.

Fig. 1. U.S. Navy DTMB 5415 hull.

As such, it was first necessary to determine the loading case under which the 5415 was originally designed. Following U.S. Navy standard practice, these loads can be subdivided into environmental (such as wave slamming or ship bending), operational (such as equipment and cargo), or combat (such as weapon firing) loads; and further subdivided into ever-more specific categories. For the purposes of this study, environmental loads were identified as primary in design importance and were therefore taken as the only loads of interest. These environmental loads were further broken down into two major groups: primary bending loads and the secondary hydrostatic loads. According to U.S. Navy practice for early-stage ship design, the primary bending load calculation consists of two bending loads at midship, along the bow-stern dimension in the positive and negative directions (hogging and sagging). U.S. Navy practice is to then apply a stress distribution for certain structural elements such that the stress at the neutral axis is equal to one half of the maximum stress in the farthest fiber of the cross section. This is in line with the established allowable stress approach to design. However, it is effectively blending multiple load effects into a single applied stress distribution, which we prefer to avoid in our application of an ultimate limit state approach to design. As an alternative, port-starboard bending loads were calculated such that, when they were applied, the furthest fibers in the 5415 hull from the port-starboard neutral axis would experience half the amount of stress as the furthest fibers from the bow-stern neutral axis would under the largest magnitude bending loads, resulting in both a positive and a negative port-starboard bending load [Eq. 1]. Mmax port−starboard =

Mmax vertical ymax vertical Ihorizontal 2yhorizontal Ivertical

(1)

Where M’s are bending moments, y’s are distances from the respective neutral axis, and I’s are moments of inertia about the respective axis. Another set of bow-stern bending loads was then calculated such that, when applied simultaneously to the port-starboard bending loads, they would cause a maximum stress equal to that felt in the pure vertical bending case, per Navy specifications, [Eq. 2]. σmax pure vertical = σmax horizontal + σmax partial vertical

(2)

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Pairing these two component bending loads with the port-starboard bending loads produced four loading cases, which in addition to the two pure hog-sag bending moments made six total loading cases [Table 2]. The secondary hydrostatic loads were the same across all loading cases, reflecting the passing wave, green seas, and dead load conditions. All values shown are normalized by the pure hogging moment value due to the sensitivity of certain U.S. Navy calculations. Interested readers may contact the authors directly to request more information. Table 2. Summary of calculated load cases Load description

Vertical bending component (normalized)

Horizontal bending component (normalized)

Pure hogging bending (+)

1

–

Pure sagging bending (−)

−0.601

–

Horizontal bending (+) with hogging component (+)

0.784

0.223

Horizontal bending (+) with sagging component (−)

0.784

−0.140

Horizontal bending (−) with hogging component (+)

−0.471

0.223

Horizontal bending (−) with sagging component (−)

−0.471

−0.140

With the six bending and hydrostatic loading cases established, the analysis of safety factors could begin. Unfortunately, because the 5415 was designed under a best-practices method, no explicit safety factor was used. Thus, it was necessary to execute an analysis of the 5415 under the determined loadings rather than simply look up the used safety factor values. A Brokkr model of the 5415 was created and subjected to the determined design loads. Brokkr’s iterative design capabilities were then disabled, so that it would simply produce an ultimate limit state-based analysis of the given design instead of producing an entirely new design. Brokkr generated the fraction of the ultimate limit state reached by each deck section under each failure mode and each load case. This set of fractions was then analyzed to obtain an overall safety factor for the entire ship and a set of partial safety factors for each individual deck type (strength deck, bottom shell, internal bottom shell, side shell, and internal decks) and failure mode. The overall safety factor was calculated from the maximum ultimate limit state fraction generated, while the series of partial safety factors was calculated from the maximum ultimate limit state fractions within each deck type and mode. Some of the partial safety factors resulting from this process were unreasonably high, but with good explanation. The 5415 ship structure is weaker to failure in some modes and deck sections than it is in others. Because it must be designed robustly enough that these weak sections survive the worst potential loading cases it might experience, the hull will naturally seem overly strong in its least-vulnerable sections. This could lead to the false perception that very high safety

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factors were used in the design of those deck sections. As such, unreasonably high safety factor outliers were replaced with a chosen maximum reasonable value (an ultimate limit state fraction of 0.45 ultimate limit state (for a safety factor of 2.22). Additionally, all ultimate limit state fractions were rounded to the nearest 0.05 before safety factors were calculated [Table 3]. Table 3. Summary of safety factors Structural type

Failure mode

Partial safety factors

Overall safety factor

Strength deck

1 2 3 4 5 6

2.22 1.05 2.00 1.33 1.25 2.22

1.20

Bottom shell

1 2 3 4 5 6

2.22 1.05 2.00 1.33 1.25 2.22

1.20

Internal bottom shell

1 2 3 4 5 6

2.22 1.82 2.22 2.22 2.00 1.82

1.20

Side shell

1 2 3 4 5 6

2.22 1.82 2.22 2.22 2.00 2.22

1.20

Internal deck

1 2 3 4 5 6

2.22 1.82 2.22 2.22 2.00 2.22

1.20

4 Results A basic scantling was then created that would serve as the starting point for Brokkr’s design iterations. The topology, location, and material type of all decks, bulkheads, and

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longitudinal girders were set to match the original 5415 values and held constant, while the details of spacing and sizing for each deck section’s internal components (stiffeners, frames, girders, plates, etc.) did not match those found in the baseline 5415 and were treated as design variables by the design algorithm. The range for each design variable was limited to within twenty percent of the values present in the 5415, a margin that covered most of the feasible design space. The basic scantling was then assigned the previously discussed set of loading scenarios and run through Brokkr (with the iterative design functionality re-enabled) twice, once for each set of safety factors. This resulted in the generation of two new structural designs. The first used only the overall safety factor, and the second used the series of partial safety factors for each structural type and failure mode. Both of these Brokkr-generated designs were then analyzed and compared with the 5415 hull that they were trying to approximate. Exact geometries and the highest ultimate limit state fractions of each deck section were compared, as well as overall features like volume, frame spacing, mass, and maximum ultimate limit state fraction generated under loading. As the latter two – mass and maximum generated ultimate limit state fraction – represent the objective function and the major constraint, respectively, they were examined closely as the major comparison points [Table 4]. Table 4 Major design comparisons. Design

Mass ratio of new design to baseline 5415

Ratio to 5415 maximum ultimate limit state fraction+

Overall safety factor constraint design

0.759

0.998

Partial safety factor constraint design

0.991

0.999

+ Accounting for safety factors

5 Analysis The new design generated using the overall safety factor [Fig. 2] differed significantly from the 5415 in mass, though it shared approximately the same maximum ultimate limit state fraction. The purpose of redesigning a structure using a single overall safety factor was to test whether the design is being driven principally by the failure mode that is closest to its ultimate limit state. In other words, is it sufficient for early-stage design to only consider the limit state closest to failure? If so, this would reduce the number of safety factors needed to specify prior a new design effort. However, we see that the mass for this overall safety factor design was lower by almost a quarter compared with the baseline 5415. Given the breadth and depth of experience that informed the baseline 5415 design, we conclude that a single overall safety factor is insufficient to obtain a safe and efficient structural design using the ultimate limit state approach of Brokkr. Upon deeper analysis, the reason for the significant difference in mass is because more

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severe failure modes, such as collapse of a stiffener panel (mode 1) were allowed to be more highly loaded. This incorporates more risk into the structural design, despite the fact that the most highly loaded mode is unchanged.

Fig. 2. Scantling bottom section comparison of overall safety factor design (bottom) and original 5415 (top)

The design generated using the set of partial safety factors, on the other hand, [Fig. 3.] matched the 5415 quite closely. Not only was its highest ultimate limit state fraction nearly identical to that of the 5415, but also its mass matched almost perfectly as well. While the partial safety factor design was certainly more constrained than its overall factor counterpart, it still had a large solution space in which to vary and yet converged to a design that is similar in many regards to the baseline 5415 model. This is an exciting result. It shows that with only a basic scantling, a loading parameter set, and a small set of safety factors, an ultimate limit state design procedure – such as the one used by Brokkr – can quickly and effectively match the previous efforts of human designers exercising long-established procedures such as an allowable stress approach. This experiment verifies the design behavior of the ultimate limit state-based design procedure for a conventional Navy hull form.

6 Conclusions These iterative design runs took only a matter of hours with minimal human input and achieved the reinvention of a test hull that was originally developed over years. This result shows the capability that ultimate limit state-based methods have to dramatically speed and standardize early-stage ship design efforts, while also incorporating more first-principles analysis. While further verification in the form of additional varied test

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Fig. 3. Scantling top section comparison of partial safety factor design (bottom) and original 5415 (top)

cases is prudent to more fully examine Brokkr’s capabilities, the fact that a tuned, automated, ultimate limit state-based procedure has matched the results of long-established historically-informed design processes is both exciting and promising. Future work will examine the transferability of this partial safety factor and ultimate limit state-based procedure to novel designs that do not share the benefit of historical design precedent (e.g., multihulled vessels). The verification procedure outlined in this paper will also be repeated for other ship classes to determine if the partial safety factors varying significantly between classes, or if a common set of partial safety factors can be deduced that translate to between classes, and hence to completely novel hull forms. Acknowledgements. The authors would like to gratefully acknowledge the support from the U.S. Office of Naval Research under the technical direction of Ms. Kelly Cooper through contract N00014-18-C-2007.

References 1. Ashe, G., et al.: Committee V.5 Naval Ship Design, Seoul, Korea (2009) 2. Benson, S., Downes, J., Dow, R.S.: Overall buckling of lightweight stiffened panels using an adapted orthotropic plate method. Eng. Struct. 85, 107–117 (2015) 3. Collette, M.D.: Rapid Analysis Techniques for Ultimate Strength Predictions of Aluminum Structures. Advances in Marine Structures, Hamburg, Germany, pp. 109–117 (2011) 4. Faulkner, D.: Toward a better understanding of compression induced tripping. In: Steel and Aluminum Structures. Applied Science, pp. 159–175 (1987) 5. Faulkner, D., Adamchak, J., Snyder, G., Vetter, M.: Synthesis of welded grillages to withstand compression and normal loads. Comput. Struct. 3(2), 221–246 (1973)

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6. Hughes, O.F., Paik, J.K.: Ship Structural Analysis and Design. The Society of Naval Architects and Marine Engineers, Jersey City (2010) 7. Knight, J.T.: Rapid, early-stage ultimate limit state structural design for multihulls. In: 6th International Conference on Marine Structures, Lisbon, Portugal (2017) 8. Paik, J.K., Duran, A.: Ultimate strength of aluminum plates and stiffened panels for marine applications. Mar. Technol. 41(3), 108–121 (2004) 9. Paik, J.K., Thayamballi, A.K., Kim, B.J.: Advanced ultimate strength formulations for ship plating under combined biaxial compression/tension, edge shear, and lateral pressure loads. Mar. Technol. 38(1), 9–25 (2001) 10. Smith, M., Szlatenyi, C., Field, C., Knight, J.: Improved ultimate strength prediction for plating under lateral pressure. In: International Marine Design Conference (IMDC), Helsinki (2018)

Application of Automatic FE-Modeling Using 3D CAD Hyokwon Son(B)

, Junghyuk Chun , and Ohseok Kwon

Hyundai Heavy Industries Co., Ltd., 1000, Bangeojinsunhwan-doro, Don-gu, Ulsan 44032, South Korea {HKSON,chunjh,kwonosca}@hhi.co.kr

Abstract. Case analyses were conducted to examine the efficiency of automatic FE-model generation for the purpose of reducing the time required to create FEmodels for structural analyses. A total of 3 practical case studies were implemented with using commercial softwares. In each case, a 3D CAD model was created first, and then a FE-model was generated based on the 3D CAD model. After the FE-model was generated, a set of tasks were conducted for finalization using a commercial pre-processor. On average, approximately 30% of overall man-hour for FE-modeling was saved compared to the conventional manual FE-modeling method. Further and wider application of automation may streamline the FE-modeling procedure. Keywords: Automation · 3D CAD · FE-modeling

1 Introduction With developments in structural analysis procedure and computer performance, wider range of Finite-Element Analysis (FEA) is required in basic hull design stage. Complex FE-models are needed so as to provide reliability to the FEA. However, the conventional modelling was mostly manual, which inevitably results in significant time for meshing and inconsistent mesh quality depending on an engineer’s experience. Therefore, HHI developed a new FE-modeling automation procedure using 3D CAD model, for the purpose of generating quality FE-model in a timely manner. This paper describes 3 case studies of FE-modeling automation, in order to find out the practical effect of the automation. In each case, 3D CAD model was created and then corresponding FE-model was automatically generated. Juan Manuel Nunes Prieto1 implemented a research on utilization of 3D CAD model in global FEA with the purpose of developing automatic method to create FE-model and loads. Meanwhile, in car manufacturing industry, many studies were carried out to figure out the effect of automation of CAE modeling. In this context, this study was intended to contribute to development in application of automation of CAE to hull structure design. 1 Juan Manuel Nunes Prieto [3].

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2 Description of the Automation Tools 2.1 Basic Understanding of 3D CAD Both 2D and 3D CAD tools basically have same function as representation of design ship, however the working methods are quite different. While all geometric information is represented with lines in conventional 2D CAD, geometries are represented with surface defined by limitations in 3D CAD. One remarkable benefit of 3D CAD is the fact that interactive modeling is possible. Interactive modeling means when one object is modified, referenced objects will be updated automatically. This feature may provide flexibility to design procedure, in that it could reduce time for modifying CAD model if design changes. In addition, 3D CAD model contains not only geometric information but also properties such as thickness of plate, material, beam profile, etc. Thus, all information included in the model can be reproduced in FE-model. Furthermore additional properties, such as corrosion reduction derived from the compartment model, and grouping can also be included in the FE-model.2 Standard workflow of 3D CAD modeling is shown as following figure. Starting from main structures, modeling is carried out downstream into all necessary structural details such as stiffeners, seams, openings, brackets, etc. (Fig. 1).

Set Reference System

Create Reference Surfaces

Create Surface Objects

Create Structural Details

Set Properties

Fig. 1. Standard workflow of 3D CAD modeling

There are several design tools that provide 3D CAD modeling and automatic generation of FE-model from its 3D CAD model, such as AVEVA, TTM, and NAPA. Among them NAPA was used in this research. 2.2 AVEVA AVEVA Hull Structural Design is used for the preliminary definition and arrangement of the ship’s structures using 3D CAD components.3 It provides parametric modelling of hull steel structures and outputs such as 2D drawing, section modulus, and FE-model can be exported from its 3D CAD model. It has a strong point of compatibility with other AVEVA marine products for detailed design and the preparation of production information (Fig. 2). AVEVA Hull Finite Element Modeller is a tool for creating FE-model from AVEVA 3D CAD model. It has its own rule-based idealization techniques driven by user-configurable options. Such options can be grouped into 3 subgroups; filtering components, simplifying geometry and idealization (Fig. 3).4 2 NAPA [8]. 3 AVEVA [6]. 4 AVEVA [7].

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Fig. 2. Interface of AVEVA Hull Structural Design

Fig. 3. Interface of AVEVA Hull Finite Element Modeller

AVEVA Hull Finite Element Modeller can generate FE-model in the form of ANSYS APDL format and XML. 2.3 TTM TTM is a 3D CAD design package developed by Korean developer, TimeTech. TTM consists of several 7 sub-systems; Initial Structure, Initial Outfitting, Hull Production, Outfitting Production, Plant structure, Plant Outfitting and Retrofit. TTM Initial Structure Design (hereafter termed TTM ISD) is a 3D CAD tool for basic hull design. It also features full 3D modeling and automatic creation of outputs for example 2D drawing and FE-model (Figs. 4 and 5). TTM ISD 3D CAD information can also be exported to other tools, such as TTM Ship Production Design (hereafter termed TTM SPD), KR Hullscan and AVEVA Marine.

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Fig. 4. 3D CAD model of TTM ISD [11]

Fig. 5. FE-model generated through TTM ISD

2.4 NAPA NAPA (Naval Architecture Package) is a program for design of ships and offshore platforms. It covers various design disciplines with solutions such as stability calculation, computational fluid dynamics (CFD), hydrostatics and so forth. The core of NAPA system is the topological 3D CAD Model which can be exported to other software for further analysis e.g. FEA. NAPA Designer is a new interface of NAPA. Graphical User Interface (GUI) of NAPA Designer has been developed to be more intuitionally understandable than that of NAPA. In this study, 3D CAD model was created using NAPA Designer (Fig. 6). NAPA FEM is one of NAPA’s subsystems. It converts a 3D CAD model into a CAE model. FE-models are generated based on the geometry of the 3D CAD model. Meshing is controlled based on some sets of parameters. Generated FE-models can be exported in several formats for FEM softwares such as Patran, Nastran and Ansys.

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Fig. 6. User interface of NAPA Designer [10]

The workflow of FE-model automation is divided into 3 parts; setting mesh parameters for limited model area, extending model to all area of interest and finalizing. Each step can be described as below (Fig. 7). Set Model Extent (One Section)

Set Model Extent (All Area)

FEM EXPORT

Mesh Parameter

Mesh Parameter

Finalization (FEGate)

OK

NOT OK

Plot FEM

Plot FEM

Model Check

NOT OK OK

Check Mesh Quality

Check Mesh Quality

Set Boundary& Load Condition

Finish

Fig. 7. Standard workflow of FE-model generation

The first step is ‘setting mesh parameters’ for limited model area often taken as one section. Mesh parameters are predefined in the software and they cover options for surfaces, stiffeners and openings. The reason why setting parameters for limited area needs to be done first is that it is more efficient to process iterations with small model area.

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‘Setting mesh parameters’ herein includes 3 tasks as indicated below. (1) Select a meshing policy: Global, Local, Fine (2) Select structures and structural details to be included in FE-model (3) Select idealization and meshing parameters As a consequence, NAPA continues automatically with the following FE-model creation process5 (Fig. 8): • • • • •

Finding topological connection of a 3D CAD model Idealization of geometry 1D meshing Surface meshing Boundary meshing and smoothing (optional)

Fig. 8. Steps in the FE-model creation process (Ibid.)

Once mesh parameters are defined for selected small area, next step is to extend the model limit to all area of interest. It could be either global or local model. This step is often less of a hassle, because mesh parameters defined in the first step tend to be suitable for other areas. Normally a little amount of modification is required for those options which become inappropriate. The second step is completed by exporting FE-model. In this study, the model was exported as Nastran bulk data format (bdf). The last step is finalization of generated FE-model. Exported model should be finalized with manual tools through Pre-processors. The finalization includes model check, 5 NAPA [8].

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partial modification for mesh quality and setting load/boundary conditions. In this study, SVD FEGate was used for these tasks.

3 Case Study 3 practical case studies were implemented to examine the efficiency of the automatic FE-modeling method as described above. In each case, FE-model was created as a result. Following steps were taken for the process (Fig. 9).

3D CAD Modeling

NAPA FEM Parameter

Finalization

Fig. 9. Procedure for a case study

3.1 LNGC D/H Model The first case study is on LNG Carrier’s (hereafter termed ‘LNGC’) deckhouse (hereafter termed ‘D/H’) model. LNGC and LNG Floating Storage & Regasification Unit (hereafter termed ‘LNG FSRU’) have distinctive structural feature in that D/H structure is connected to main longitudinal members such as trunk deck and inner deck. For this structural arrangement, LNGC’s D/H structure is subjected to hull-girder bending moment, which brings about reinforcement for the area. In addition, classification societies like Lloyd’s Register (hereafter termed ‘LR’) require shipbuilders to implement FEA on this area. In this case a FE-model for LNGC’s D/H area was generated for the purpose of fulfilling LR’s ‘Analysis of Trunk Deck Scarphing Arrangements with the Aft End’.6 In this procedure, the model is required to extend from the transom to the middle of the aft cargo hold and include the D/H and all internal structure. At the point when carrying out this analysis, whole ship FE-model existed but as for the D/H area internal structures were not included and its mesh quality was less than that of the hold area. In this circumstance, the target was set to create partial model of D/H so that it could be combined to existing global model. One remarkable advantage in 3D CAD modeling stage was the fact that 2D drawing was utilizable. This feature dramatically reduced modeling time by skipping reproducing geometry. The 3D CAD model was created in the following steps (Fig. 10). (1) (2) (3) (4) (5)

Importing 2D curves into NAPA Designer Creating surface objects by extruding curves Repeating step (1) & (2) at each deck level Modeling structural details Setting properties

6 Lloyd’s Register [9].

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Fig. 10. Steps for 3D CAD modeling of D/H structure

Next step is setting parameters in NAPA FEM. For this model, few parameters were required to generate quality FE-model, since D/H structures consist of decks and walls which are connected in perpendicular joints, in other words, it has simple structural arrangement. Original settings other than the default ones were set and used, which includes options such as ‘Coordinate planes’, ‘Tolerance’ and ‘Reduction shape’. Following figure shows the result of generated FE-model (Fig. 11).

Fig. 11. Result of FE-model generation for case1

Lastly finalization was implemented in FEGate. The software has a ‘model diagnosis’ function, which combines more than 20 model check criteria. After model diagnosis was implemented, some meshes were modified to improve quality. For instance, ‘collapsed elements’, ‘isolated entities’ and ‘duplicated elements’ were fixed. Finally, boundary and load condition were set for NASTRAN calculation.

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3.2 LNG Liquid Dome Opening LNG liquid dome opening structure is one of a critical point in terms of structural engineering, since this area is subjected to stress concentration due to geometrical discontinuity. Thus, most classification societies require fine mesh analysis on this area. For this reason, fine mesh model of LNG liquid dome area was selected for case2. The target was to create sub-model of LNG liquid dome structure including fine meshed area adjacent to the dome opening. The mesh size in fine mesh zone was about 50 × 50 mm. For the accuracy of analysis, fine mesh model should reflect the actual geometry as accurately as possible. In this respect, 3D CAD modeling has an advantage in that it is easier to describe complex geometry than FE pre-processors. The figure below shows the 3D CAD model of LNG liquid dome Opening (Fig. 12).

Fig. 12. NAPA 3D model of LNG liquid dome opening structure

For a fine mesh model, additional setting is required to define fine mesh zone other than regular parameter set in NAPA FEM. Two areas are required to be defined; a fine area and a transition area. In the left figure below, the green box and the red box represent the fine area and the transition area, respectively. On the right side of the figure shows the generated model (Fig. 13).

Fig. 13. Setting fine mesh zone and result

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It was found that mesh quality of generated fine mesh area depends on how the fine and transition area were defined. Undergoing some trials and errors, setting the edge of the box in the middle of coarse mesh nodes seemed to give the best results. Finalization was carried out in FEGate mainly for improving mesh quality in way of the transition zone. Generated sub-model was combined to global model, to set boundary and load condition for calculation. 3.3 LNG FSRU Re-Gas Unit In structural point of view, Regasification unit (hereafter termed Re-gas unit) structure distinguishes LNG FSRU from LNGC. Re-gas unit and its supporting structures are distinctive feature of LNG FSRU and they should be reinforced properly. Recently HHI has made a major design modification to its FSRU by which re-gas unit was rearranged from hold part to fore part of ship. This brought out significant change in structural arrangement, and strength of the new structure and its supporting structure had to be examined by FEA. The target was to create local FE-model of fore end structure. Since there existed a global model, local model was supposed to be combined to it afterward (Fig. 14).

Fig. 14. 3D CAD model and generated FE-model of case3

4 Result In all 3 cases, time required for pre-processing was measured, pre-processing here means creating FE-model with operable quality for NASTRAN calculation. As a result, however the effect varied from case to case, time for pre-processing was decreased in general. In the chart below, there are 6 bar graphs which represent time needed for FEmodeling for 3 cases in 2 methods; the conventional manual method (AS-IS) and the new automation method (TO-BE). Each bar consists of sub tasks such as 3D CAD modeling, parameter setting, meshing, model check and fix, boundary and load setting. In the Conventional manual methods, there is no subtask as 3D CAD modeling and Parameter setting. Meshing is a major task in this case. In the new automation method, subtasks as 3D CAD modeling and parameter setting were added and account for over 50% of the total. On the other hand, time for meshing turned out to be significantly reduced.

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In sum, as mentioned above, total time for FE-modeling was reduced in every case. On average, time saving was about 30% (Fig. 15).

Fig. 15. Result comparison; time required for FE-modeling

With regard to the quality of FE models, the new automation method gave more or less same quality with the models made by manual method. Owing to the ‘model check and fix’ task, qualities of automatically generated models were improved.

5 Discussion 5.1 Benefits The main expected effects of applying automation in FE-modeling are time saving and standardization. As for the time saving, 3 case studies were implemented and time needed to create FE-model was measured in each case. Compared to conventional manual modeling, time for FE-modeling was reduced about 30%. The time saving effect mentioned above, however, is not enough for generalization, mainly because the data was not plenty enough to result in statistically meaningful number. Besides, in each case, the target model was quite different from each other. Nevertheless this result remains worthwhile in that the effect of FE-modeling automation was measured with quantitative approach.

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Meanwhile the benefit of the standardization is harder to quantify. However this beneficial effect exists and leads to consistent quality of FE-model, since most of meshes are created automatically. 5.2 Challenges On the other hand there are some challenges to be resolved. Here are some suggestions to enhance more efficiency of the FE-modeling automation. Firstly, better quality of generated FE-model is required. The generated models were not operable quality, thus some meshes needed to be fixed. If quality of generated model is straight operable, there will be no additional touch-up. Secondly, setting boundary and load condition should be included in NAPA FEM. In the same context of the first suggestion, if generated models are straight operable, there will be no need for the finalization step, which will lead to a more streamlined procedure. Finally, the user interface of NAPA FEM should be improved to be more intuitive and provide instant feedback. Users find it difficult to catch which parameter set has which effect. In addition, iteration of modifying a parameter and checking the result is not smooth, i.e. user can’t easily get the feedback.

6 Conclusion For the market’s growing demands for wider range of engineering scope and developments in CAD and CAE tools, it is an inevitable challenge and also a chance for ship builders to streamline their FEA procedure to seize competitiveness. And this may be achieved by automation of the process. In this paper, efforts were made to examine the effect of FE-modeling automation. 3 case studies were implemented, and as a result about 30% of time for pre-processing was reduced on average. Moreover standardization of FE-model was also beneficial. Meanwhile, there was still room for improvements. Although NAPA was selected as a main tool in this research, other tools mentioned earlier in this paper such as AVEVA and TTM should also be examined to compare their performances. Further application of this automation method is planned to be addressed. It will result in more efficient pre-processing which is a bottleneck of FEA.

References 1. Kujala, A.: Measuring initial mesh quality of ship finite element models. Dissertation, Aalto University (2010) 2. Anserwadekar, A., Patel, S., Bhattacharya, I.: Automation of FE-modeling and analysis using CAE driven product development. Int. J. Latest Technol. Eng. Manage. (IJLTEM) (2016) 3. Prieto, J.J.N.: Utilization of modern 3D product model in global finite element strength assessment. Dissertation, Aalto University (2013)

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4. Helenius, L.: Utilization of weight calculation data in finite element vibration analysis of a passenger vessel. Dissertation, Aalto University (2011) 5. Yanagisawa, M., Kato, M.: Automatic generation of High-quality CAE model using ANSA. In: 7th BETA CAE International Conference (2017) 6. AVEVA: AVEVA Hull Structural Design (2018) 7. AVEVA: AVEVA Hull Finite Element Modeller (2018) 8. NAPA: NAPA for design manuals 2018.3, p. 31 (2018) 9. Lloyd’s Register: Structural design assessment: procedure for membrane tank LNG ships, p. 46 (2018) 10. NAPA Homepage. https://www.napa.fi/Design-Solutions/NAPA-Steel/Modelling. Accessed 19 March 2019 11. TimeTech Homepage. https://www.timetec-ttm.com/eng/basics/isd.html. Accessed 22 March 2019

Design Contours for Complex Marine Systems Harleigh C. Seyffert1(B) , Austin A. Kana1 , and ArminW. Troesch2 1

Delft University of Technology, 2628 Delft, The Netherlands [email protected] 2 The University of Michigan, Ann Arbor, MI 48109, USA

Abstract. This paper examines the performance of 6 stiffened ship panel designs in different operational profiles. The main question of interest is: which sea states will lead to the worst panel performances in terms of reliability? As stiffened panel collapse is governed by combined lateral and in-plane loading effects (non-linear functions of the wave environment) this is not a simple problem and does not easily fit into the confines of traditional analyses. Interesting sea states for stiffened panel collapse are identified by a low-order design contour method which uses order statistics and extreme value theory. The resulting multimodal design contours pinpoint areas of interest and the panel performances are confirmed using a higher-order reliability analysis: the non-linear Design Loads Generator process. Such results have impact for creating and interpreting environmental and design contours, as well as assumptions about which operational profiles will lead to the worst system responses. Keywords: Design contours · Environmental contours of extreme responses · Combined non-Gaussian loading

1

· Return period · Reliability

Introduction

The reliability of marine systems is an important design consideration, but one that is not easily analyzed in the early stages of the design process. Despite continuing advances in computational efficiency, it is generally not feasible to use brute-force simulations in a cell-based approach of each possible operational profile for a long-term probabilistic reliability analysis. Therefore, a common approach is to use a short-term probabilistic analysis in which the presumed “worst-case-scenario” system response is examined due to a few operational profiles, generally known as the equivalent or design sea state method [1]. 1.1

Review of Environmental Contour Methods

Sea states which lead to the most extreme loading may be identified by the environmental contour method paired with some coefficient of contribution determination, as in [2]. For marine systems, it is often assumed that a load on a system c Springer Nature Singapore Pte Ltd. 2021  T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 168–183, 2021. https://doi.org/10.1007/978-981-15-4672-3_10

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with a given probability of exceedance is excited by a sea state with that same probability of exceedance. A design sea state is chosen based on the sea states aligning with the probability contour associated with some allowable risk level along with information about a representative spectral period that maximizes the load variance [3]. Sea states corresponding to rare load responses with very low probabilities of exceedance will have correspondingly short exposures. The exposure to such harsh excitation is a small fraction of the total expected lifetime, which allows for a short-term probabilistic analysis of the desired load. But in general, failure of marine systems may be due to combined non-linear loading effects. In such cases, the connection between extreme environments, extreme loads, and resulting failure occurrences (or limit surface exceedances) is less clear. To determine the reliability of a system over all possible operational profiles requires information about the ocean excitation, how that environment excites the relevant load effects, and how those load effects interact with some limit surface which determines the occurrence of failure. This paper examines such a system: the collapse of stiffened ship panels due to combined lateral and in-plane loading effects. Given a set exposure and excitation profile, the reliability of different panel designs can be efficiently examined and compared, as in [4]. For individual linear loads, it is easily determined which excitation regime (short exposure to harsh excitation vs. long exposure to mild excitation) leads to larger load values using extreme value theory (see, e.g., [5]). But stiffened panel failure is due to the combined interaction of lateral and in-plane loading effects with a limit surface which is a non-linear function of panel properties. For such a system, it is reasonable to question whether a short exposure to the harshest expected environment actually leads to the worst-case system response. Similar risk levels may be possible given a longer exposure to a milder sea state, especially if simultaneous moderate values of combined loading can excite failure. To examine whether this may be possible, this paper expands on the example in [4]. In that paper, the performances of six different stiffened panel designs at a specific location on the David Taylor Model Basin (DTMB) vessel 5415 were compared by the non-linear Design Loads Generator (NL-DLG) process [6]. This paper examines the potential of defining design contours for stiffened ship panels. A low-order design contour method utilizing linear surrogate processes and extreme value theory is presented which identifies sea states which lead to interesting panel performances. As opposed to environmental contour methods which de-couple environmental parameters from structural responses, this loworder model examines the structural response in each possible operational profile in a cell-based approach. The resulting design contour identifies areas of interest which are examined more in-depth by the NL-DLG process. The resulting group of chosen design sea states present a surprising group of sea states which lead to extreme panel reliability responses.

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Problem Definition

2.1

David Taylor Model Basin 5415 Stiffened Panel Designs

The stiffened panels considered in this paper are potential design options for the inner bottom external shell strake of the DTMB 5415, a modern destroyerlike hull with parameters given in Table 1. The 17 th International Ship and Offshore Structures Congress (committee V.5 Naval Ship Design) used existing naval structural rules from 6 classification societies to design an optimal stiffened panel (in terms of minimal longitudinal structural weight) for the DTMB 5415 [7], as described in Table 2. Table 1. DTMB 5415 particulars. Parameter

Value

Length between perpendiculars (Lpp)

142 m

Length on water line (Lwl)

142.18 m

Beam on water line (Bwl)

19.06 m

Draft (T)

6.15 m

Displacement (∇)

8424.4 m3

Block Coefficient (CB)

0.507

Longitudinal Center of Buoyancy (LCB) (% Lpp fwd+) −0.683 Panel Location (fwd of midships +)

13.96 m

Web frame spacing

1905 mm

Stiffener & Plate Yield Stress, σY

355 MPa

Steel Young’s modulus, E

200 GPa

Table 2. Panel and stiffener designs for the DTMB 5415 from 2009 ISSC report [7]. Panel

1

Design pressure [kPa]

pstiffener = 60.6 103.6

2

Plate thickness [mm]

9

11

Hweb × Tweb [mm]

160 × 6.2

150 × 9 154.4 × 6

113.64 × 6.35 246.9 × 5.8 220 × 6

Hflange × Tflange [mm]

120 × 9.8

90 × 14 101.8 × 8.9

63 × 13.36

101.6 × 6.9 200 × 6

Stiffener spacing [mm] 672

700

500

364

600

400

Bottom cross-section modulus [m3 ]

4.60

4.14

3.64

3.34

4.93

21,121

19,276

15,844

18,329

19,733

pweb = 33.6

3.77

Weight of longitudinal 16,520 structure [kg]

3

4

pstiffener = 59.75 86.6

5

6

127.45

174.55

8

10

pweb = 33.89 8.1

7

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Operational Profile

The reliabilities of these panels are examined for a 30-year lifetime with a probability of non-exceedance P N E = 0.990, resulting in a total 300-year exposure. As directed by many classification societies the possible operational profiles are based on the North Atlantic environment described in [8]. The possible significant wave height-zero crossing period (Hs −Tz ) pairs that define the 2-parameter ITTC spectrum for the operational profile are shown in Fig. 1, along with the probability of experiencing those sea states over the 300-year exposure. These probability occurrences are defined by [8]. For the sake of brevity in this analysis and to follow [4], only the head seas condition is considered.

Fig. 1. Probability of occurrence for different sea states in the North Atlantic [8].

3 3.1

Methodology Stiffened Panel Failure Mechanism & Reliability Estimation

Stiffened ship panels fail due to combined lateral and in-plane loading effects with a limit surface described in the form of Fig. 2. A detailed description of the process to define a panel limit surface is given in [9]. In [4], the reliability of the panels described in Table 2 for a 1000-hour exposure in head-seas Hurricane Camille-type conditions for failure modes 2–3 was determined using the NL-DLG process [6]. Lateral loading due to slam events (based on the relative velocity between the panel location and the water surface) and in-plane loading effects due to global ship bending are considered as the main loading drivers, restricting the limit surface to just modes 2 and 3, or the first quadrant of Fig. 2. The NLDLG process uses the Design Loads Generator [10,11] to construct an ensemble of irregular wave profiles which lead to a distribution of extreme responses of a specified linear function corresponding to the given operational profile and exposure period. In the NL-DLG process, linear surrogate processes act as indicators of extreme behavior for the associated non-linear load effects and estimate which DLG wave profiles excite extreme responses of a system governed by combined non-Gaussian loading.

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Fig. 2. Limit surface of a stiffened panel due to lateral and in-plane loading effects [9].

3.2

Surrogate Processes of Extreme Loading Effects

As described in [4], extreme relative velocity (RV) at the panel location is an indicator for extreme lateral loading effects on the panel. In the same way, extreme global bending moment (BM) at the panel location is an indicator for extreme in-plane loading effects on the panel. The estimation of the failure probability for the panels in [4] was quite accurate and efficient compared to Monte Carlo Simulations (MCS), indicating that the choice of these surrogates is a good one for the non-linear loading models used. Using linear surrogate processes maintains a clear connection between wave excitation (sea spectrum defined by Hs − Tz ), extreme vessel “load” responses (RV and BM at the panel location), and characteristics which impact panel reliability (interaction of lateral and inplane loading effects with the limit surface definition). As another dimension, a criterion is added for a change in vessel speed as a function of the service speed, Vs = 20 knots, given the significant wave height. This change in speed based on sea state is often prescribed when assessing lifetime design loads, see, e.g. [12]. Hs ≥ 10.5 m → 25% of Vs = 5 knots 10.5 m > Hs ≥ 7.5 m → 50% of Vs = 10 knots 7.5 m > Hs ≥ 4.5 m → 75% of Vs = 15 knots Hs < 4.5 m → 100% Vs = 20 knots

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3.3

173

Most-Likely Extreme Responses of Surrogate Processes

Using the probability of sea state occurrences from Fig. 1 and extreme value theory, contours of the most-likely extreme RV and BM value at the panel location, given the sea state and associated exposure, are constructed and shown in Figs. 3 and 4. The left insets of Figs. 3 and 4 give the most-likely extreme RV and BM value normalized by the respective standard deviation given the sea state and exposure. The right insets give the most-likely extreme RV and BM value for each sea state in physical dimensions (m/s and Nm, respectively).

Fig. 3. Left inset: Most-likely extreme RV value (normalized by σRV associated with the given sea state) at panel location, right inset: most-likely extreme RV value in m/s in the given sea state.

Fig. 4. Left inset: Most-likely extreme BM value (normalized by σBM associated with the given sea state) at panel location, right inset: most-likely extreme BM value in Nm in the given sea state.

The left insets of Figs. 3 and 4, which give the most-likely extreme RV and BM value as a function of the respective standard deviation describe how rare this extreme value is in the sea state (e.g. a 5σ event) based on the exposure. As

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the cycle period of RV is much lower than BM there are more RV cycles than BM cycles over a given exposure, meaning the relative extrema of RV are larger than the relative extrema of BM given an operational profile. This is reflected in the left insets of Figs. 3 and 4, in which all of the contours of most likely extreme RV values are at higher σ events than the most-likely BM values at the same sea state. The contours in the left insets indicate how rare an event is, but this does not necessarily mean the magnitude of the most-likely extreme RV or BM value in that sea state will be high, since the process σ value in that particular sea state may be relatively low. The right insets of Figs. 3 and 4 show that there are specific ranges of wave Tz that lead to large extreme RV and BM values in physical dimensions. However, the range of sea states that lead to the largest most-likely extreme BM values (the brightest yellow contour, right inset of Fig. 4) includes many more sea states than the equivalent contour for RV. Note also that the contour of most-likely extreme RV values (in physical dimensions, right inset of Fig. 3) is multimodal. This may be due to the fact that the most-likely extreme value is a function of both the process variance and the exposure. The transfer function for the panel RV has a sizable high-frequency content, meaning that sea states with low zero-crossing periods will excite larger RV values. The exposure associated with a given sea state will also affect the mostlikely extreme value, though this most-likely  extreme scales with the number of process cycles over the exposure, m, as ln(m). The variance of RV and BM scales with the variance of the exciting sea state and has a more appreciable impact on the dimensional most-likely extreme value than does the effect of the exposure. However, there are some cases where a lower process variance paired with a long exposure may still lead to an appreciable most-likely extreme RV or BM value. Such a sea state could also lead to poor panel performance. 3.4

Constructing Design Contours Using Surrogate Processes

For stiffened panel collapse due to combined lateral and in-plane loading effects, an approximate design contour can be constructed using the surrogate models identified in [4]. In [4], extreme RV and BM at the panel location were identified as good indicators of extreme lateral and in-plane loading effects on the panel, respectively. Therefore, some model reductions can be made to assemble approximate design contours. In this case, the aim of the design contours is not to give an exact estimation of the stiffened panel failure probability given the sea state but to efficiently compare the stiffened panel performance over all possible sea states to identify sea states of interest. These sea states may then be evaluated by the NL-DLG process, which gives an efficient estimation of the failure probability as compared to brute-force MCS. Stiffened panel collapse is governed by the panel limit state, which follows the form in Fig. 2 and is a non-linear function of the lateral and in-plane loading effects on the panel. The lateral load effect is a non-linear function of the RV at the panel location, and the in-plane loading effect a non-linear function of the global BM at the panel. Even though the lateral and in-plane loading effects

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are non-linear functions of the RV and BM, respectively, these functions are still one-to-one, meaning the limit surface can equivalently be written as a function of the RV and BM at the panel location. The limit surface described in the BM−RV space is a piece-wise non-linear limit surface as a function of linear effects (BM and RV). The limit surface for each panel in a sea state can further be expressed by the relative weighting of RV and BM by normalizing both axes by the respective standard deviation in the given sea state. Therefore, the limit surface for a panel in the normalized BM−RV space describes the rareness of a BM or RV event required in the given sea state to lead to failure. A surrogate process can be formulated which is a weighted sum of RV and BM normalized by their respective standard deviations, as in [4], which is described by Eq. 1. Each point on the limit surface in the normalized BM−RV space in a given sea state can then be expressed as an individual surrogate process, SPi , as illustrated in Fig. 5 for panel 4. RV BM + βi σRV σBM RVi for g = i, · · · , n αi = max(RVg ) BMi βi = for g = i, · · · , n max(BMg )

SPi = αi

(1)

where SPi = surrogate process i (BMi , RVi ) = discrete point on the limit surface for a panel in a given sea state max(BMg ) = maximum BM value on the limit surface max(RVg ) = maximum RV value on the limit surface

Fig. 5. Limit surface for panel 4 described in the in-plane load−lateral load space (left) and in the normalized BM−RV space given a Hs = 9.5 m Tz = 6.5 s sea state (right).

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Visualizing the limit surface for panel 4 in the normalized BM−RV space for the sea state Hs = 9.5 m Tz = 6.5 s already gives a hint about how panel 4 will perform in this sea state. For panel 4 to fail due to extreme lateral loading effects, or extreme RV, (failures on the limit surface near the y-axis), the relative velocity acting at the panel location must be at least a 2.8σRV event. Consider that the most-likely extreme RV value in this sea state is 4.2σRV (see Fig. 3, left inset), and that there is a 63.2% chance that the extreme RV value will be larger than 4.2σRV . It is expected that panel 4 will often fail when excited by an exposure-period-RV-maximum value in the Hs = 9.5 m Tz = 6.5 s sea state. In the same way, for panel 4 to fail due to extreme in-plane loading effects, or extreme BM, (failures on the limit surface near the x-axis) in the Hs = 9.5 m Tz = 6.5 s sea state, the BM must achieve at least a 7.4σBM event. According to the left inset of Fig. 4, the most-likely extreme BM value in the Hs = 9.5 m Tz = 6.5 s sea state is 3.98σBM . It is quite unlikely that this sea state will produce so large a BM event to cause failures in this part of the limit surface. Based on the information about the most-likely extreme RV and BM values in the Hs = 9.5 m Tz = 6.5 s sea state, along with the limit surface in the normalized BM−RV space in this sea state, it is expected that panel 4 will have a high failure probability in the Hs = 9.5 m Tz = 6.5 s sea state, and that any failures will likely be clustered on the limit surface intersect near the y-axis. As in Fig. 5, individual points on the limit surface for each panel in a given sea state can be written as surrogate processes in the form of Eq. 1. Since Eq. 1 is a linear function of Gaussian inputs (BM and RV) with known transfer functions, it has a known energy spectrum and extreme value distribution. Therefore, for each sea state included in Fig. 1, it is easy to calculate the extreme value distribution for each surrogate process SPi which corresponds to a specific point on the limit surface for a given panel in the normalized BM−RV space for that specific sea state. A low-order estimation for the failure probability in that sea state given waves which excite exposure-period maxima of SPi is the probability that the most-likely extreme value of SPi over the exposure exceeds the limit surface value at the location i on the limit surface in the normalized BM−RV space. That is:  p(failure for panel j|SPi ) = p(failj |SPi ) = p(SPi,m > sp i,m )  s pi,m g(spi,m ) =1−

(2)

0

where g(spm ) = extreme value distribution of surrogate process SPi , given m process cycles over the exposure in the given sea state i αi RVi + βi σBM SPi BM = σRV at point i on the limit surface in the normalized σRVi σRVi BM − RV space describing panel j in the given sea state

sp  i,m =

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In [6], there is a discussion on how to relate failure probabilities conditioned on the system being excited by waves which lead to exposure-period-maxima of a given surrogate process, such as p(failj |SPi ) in Eq. 2, to an overall failure probability estimate for panel j. This formulation requires determining how the different surrogate processes SPi may be related, and the probabilities p(failj |SPi ) are estimated via directed DLG simulations, as opposed to Eq. 2. But this overall failure probability estimate for panel j must be at least p(failj |SPi ), given each possible surrogate SPi . A low-order estimate of the failure probability for panel j in a given sea state is then: p(failj ) = maximum(p(failj |SP1 ), p(failj |SP2 ), · · · , p(failj |SPn ))

4

(3)

Results

Using the process described in Sect. 3, a low-order estimate of the panel failure probabilities given the range of possible operational profiles from Fig. 1 can be assembled. Contours of failure probability for each panel design given the operational profile for each sea state are shown in Fig. 6. The failure probabilities estimated from Eqs. 2 and 3 are a lower bound on the probabilities estimated from the NL-DLG process and what would be expected from brute-force MCS.

Fig. 6. Contours of failure probability for stiffened panels 1–6 given sea state and exposure. Some potential sea states of interest are highlighted with the different markers.

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Choice of Sea State Given Design Contours

The design contours in Fig. 6 bring up some interesting observations about the different panel designs from Table 2. First, clearly these panels do not all have the same general performance characteristics. Panels 3 and 4 both have large ranges of Hs and Tz where a high failure probability is expected. In contrast, the failure probability for panels 5 and 6 across all possible sea states seems to be bounded at about 40%. In addition, the design contours for panels 1, 2, 3, 5, and 6 all exhibit multimodal behavior. Figure 6 also highlights some sea states which are expected lead to interesting panel performances; these are noted as the different markers. The chosen sea states are: – Hs = 9.5 m − Tz = 6.5 s: This sea state appears to lead to the maximum failure probability possible for all panels. This sea state also happens to be one of the multimodal maxima for panels 1–3 and 5–6 (panel 4 does not exhibit any multimodal behavior based on Fig. 6). – Hs = 12.5 m − Tz = 7.5 s: This sea state appears to lead to another instance of maximum failure probability for all panels and represents a different mode for panels 1–2 and 5–6. For panels 3 and 4, this sea state is in the same probability contour as for the Hs = 9.5 m − Tz = 6.5 s sea state. – Hs = 10.5 m − Tz = 7.5 s: This sea state represents the saddle point between the two peaks in the failure probability contour for panels 2, 5, and 6. For panel 1, this sea state is within the same failure probability contour as for the Hs = 12.5 m − Tz = 7.5 s sea state. – Hs = 9.5 m − Tz = 7.5 s: This sea state represents the saddle point between the two major peaks in the failure probability contour for panel 1. – Hs = 6.5 m − Tz = 7.5 s & Hs = 5.5 m − Tz = 7.5 s: These sea states represent operational profiles which are expected to lead to an appreciable failure probability for a single panel (panel 4) but for no other panel. The panel failure probabilities based on the contours in Fig. 6, estimated using Eq. 2 and 3, along with the sea state and exposure, are given in Table 3. Figure 6 identified a few sea states that may lead to interesting panel responses, with a low-order estimate of the failure probability for that sea state given in Table 3. This reduction of the entire operating space to just a few cases allows a more in-depth analysis of the reliability of the stiffened panels, versus the loworder failure probability estimation offered by Eqs. 2 and 3. However, some of the identified sea states have exposure lengths which may be too long for bruteforce simulation to be a feasible option. As in [4], the NL-DLG process is used to estimate these failure probabilities, which are given in Table 4. As expected, the failure probabilities from the low-order estimate of Eqs. 2 and 3, presented in Table 3, are a low bound on the failure probabilities estimated by the NL-DLG process in Table 4. Additionally, the sea states identified in Fig. 6 for the specific characteristics (maximum failure probability in sea states from disjoint probability contours, saddle point between those disjoint contours, and a sea state which leads to poor performance for a single panel) do produce the expected behavior. The point of the design contours presented in Fig. 6 is not to

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Table 3. Failure probabilities from low-order design contours in Fig. 6. Contour failure probability for panel j

Sea state

Hs [m] Tz [sec] Exposure [hours] 1

2

3

4

5

6

9.5

6.5

5.26

0.88

1.0

1.0

0.41

0.41

12.5

7.5

2.63

0.99

0.77

1.0

1.0

0.33

0.33

10.5

7.5

31.55

0.98

0.51

1.0

1.0

0.11

0.11

9.5

7.5

113.03

0.88

0.25

1.0

1.0

0.031

0.032

6.5

7.5

4390

0.0002 0.0001 0.011

1.0

0.0001 0.0001

5.5

7.5

13,101

0.0004 0.0004 0.0004 0.13 0.0004 0.0004

0.99

Table 4. Failure probabilities from the NL-DLG process. Failure probability for panel j

Sea state

Hs [m] Tz [sec] Exposure [hours] 1

2

3

4

5

6

9.5

6.5

5.26

0.99

1.0

1.0

0.57

0.57

12.5

7.5

2.63

0.99

0.95

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1.0

0.43

0.42

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31.55

0.99

0.85

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113.03

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0.002 0.0005 0.018 1.0

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0

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0

0

0.058 0.063 0

0

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0

give an exact estimate on the failure probability of the different panels but to give a global comparison between the different panel performances and highlight areas of specific interest for further investigation. The panel performances in these specific sea states are discussed below. 4.2

Sea States Leading to Worst Panel Responses

Figure 6 indicated sea states which lead to the worst responses for all panels in terms of reliability, corresponding to the two major disjoint peaks in the probability contours for panels 1, 2, 5, and 6: Hs = 9.5 m − Tz = 6.5 s and Hs = 12.5 m − Tz = 7.5 s. The NL-DLG process confirmed that both sea states lead to the highest failure probabilities for all examined sea states. 4.3

Sea States Corresponding to Design Contour Saddle Points

Figure 6 indicated that some panels have multimodal contours of failure probability. Two sea states were identified as saddle points between the multiple peaks of failure probability. The sea state Hs = 10.5 m − Tz = 7.5 s represents a saddle point for the failure probability contour for panels 2, 5, and 6. Table 4 confirms that the failure probability for these panels is indeed lower at this saddle point

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sea state than for the two sea states that maximize failure probability for those panels (Hs = 9.5 m − Tz = 6.5 s and Hs = 12.5 m − Tz = 7.5 s). The sea state Hs = 9.5 m − Tz = 7.5 s represents a saddle point for the failure probability contour for panel 1. Again, Table 4 confirms that this is the case. 4.4

Sea States at Design Contour Boundaries

Two sea states were chosen (Hs = 6.5 m − Tz = 7.5 s & Hs = 5.5 m − Tz = 7.5 s) because they are expected to lead to poor performance for a single panel (panel 4) but negligible failure probabilities for the other panels. Table 4 confirms what may be suspected from Fig. 6. These two sea states are only interesting when considering the performance of panel 4. Specifically, for the sea state Hs = 6.5 m − Tz = 7.5 s, where the failure probability for panel 4 is 100% while the other panels have negligible failure probabilities, this result raises an interesting question about the nature of environmental contours. Environmental contours de-couple extreme environmental conditions from structural responses, meaning that the contours have a wide application. The fact that design sea states can be chosen solely from environmental characteristics, without including response characteristics, is noted as an advantage of the method, e.g. [1,13,14]. However, the different panel failure probabilities excited by these two sea states illustrate that the structural design strongly impacts whether a sea state leads to interesting design performances. That is, a design sea state may not prove equally interesting for all designs. Picking a sea state simply by the environmental characteristics may not reliably indicate which sea states lead to interesting structural performances. Of course, present design contour methods like the inverse First-Order Reliability Method (IFORM) aim to expand environmental contours to include the limit state of structural response [1]. However, IFORM cannot produce multimodal probability contours. Clearly for some marine systems, traditional reliability and environmental contour methods may not suffice to indicate which sea states may lead to the worst system performances. 4.5

Sea State Harshness of Excitation vs. Exposure Length

A key assumption of environmental contour methods is that the probability of exceeding a load value is associated with the probability of exceeding the harshest sea state which could excite that load, based on the process cycle period. This implies that rare load values are excited by rare sea states, meaning that the exposure to these sea states is likely to be short. It might be natural to say, then, that rare sea states lead to the worst possible system performance. Consider, though, the reliability assessment for panel 4 from Table 4. Clearly panel 4 does not perform well in almost all of the examined sea states. But it is interesting that panel 4 has a 100% failure probability over sea states with many different spectral properties, and crucially, exposure lengths. Panel 4 has a 100% chance of failure in the Hs = 12.5 m − Tz = 7.5 s sea state, with only a 2.63-h exposure out of the full 300-year lifetime. But panel 4 also has a 100% chance of failure in the sea state Hs = 6.5 m − Tz = 7.5 s, with a 4390-h exposure.

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Clearly a longer exposure to milder loading can be just as damaging to a system as a much shorter exposure to harsh loading. These sea states have the same Tz , Hs different by a factor of 2, and exposures different by a factor of about 1669. Yet they lead to the same reliability estimate for panel 4. From Fig. 6 it might be expected that these two sea states lead to a similar reliability estimate for panel 4. But traditional assumptions about which sea states are expected to lead to the worst-case system responses might not identify the Hs = 6.5 m − Tz = 7.5 s sea state as important. That this sea state can also lead to extreme response for the stiffened ship panel 4 implies that only examining design sea states as identified by traditional environmental contour methods may prove unwise. Extreme responses of complex marine systems may be excited by unexpected sea states given the right combination of Tz , Hs , and exposure.

5

Conclusions

This paper examined the identification of design sea states for complex marine problems like the collapse of stiffened ship panels, which is governed by combined lateral and in-plane loading. Environmental contour methods have been developed because cell-based reliability analysis, in which all possible operational profiles and sea states are individually examined for extreme loads and responses, is generally not a feasible option. Therefore, most environmental contour methods work as an up-down approach. That is, exceedance probability contours of environmental conditions, like Hs and Tz , are first constructed, and design sea states are chosen based on these contours. A key point of such methods is that the environmental conditions and resulting structural response are de-coupled. In this paper, a bottom-up cell-based approach was instead taken. Using linear surrogate processes along with order statistics and extreme value theory, the structural response to each potential operational profile was examined. With a low-order estimate of the stiffened panel failure probabilities given each cell, design contours over all possible operational profiles were constructed. These contours identified potential design sea states, which were then examined using the higher-order NL-DLG process, which was shown in [4,6] to estimate failure probabilities accurately compared to brute-force MCS. The low-order bottom-up approach to design contours presented in this paper avoided some key assumptions of traditional contour methods, and showed how these assumptions may indeed be limitations rather than helpful simplifications. The low-order design contour method used in this paper identified disjoint regions of sea states that lead to similar failure probabilities. Evaluating these sea states (Hs = 9.5 m − Tz = 6.5 s and Hs = 12.5 m − Tz = 7.5 s) along with the saddle points (Hs = 10.5 m − Tz = 7.5 s and Hs = 9.5 m − Tz = 7.5 s) via the NL-DLG process confirmed the results. By performing a low-order reliability estimate cell-by-cell, this multimodal reliability behavior was discovered, whereas in top-down approaches, this multimodal behavior might not be identified. Additionally, unexpected sea states may also be candidates for design sea states. This was clear for panel 4, where the same failure probability was excited

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by sea states ranging from a 2.63-hour exposure (Hs = 12.5 m − Tz = 7.5 s) to a 4,390-hour exposure (Hs = 6.5 m − Tz = 7.5 s). In general, this low-order design contour approach illustrated that a design sea state for one design may not be a design sea state for other designs. This was clearly the case for the Hs = 6.5 m − Tz = 7.5 s & Hs = 5.5 m − Tz = 7.5 s sea states, which resulted in poor performance for panel 4, but negligible failure probabilities for the rest of the panels. In this case, it may not be beneficial to determine design sea states without including some notion of the design limit state. Overall, such results indicate that complex marine systems may not easily be analyzed by traditional methods. By employing some specific system simplifications via linear surrogate processes that are indicators of extreme non-linear behavior, interesting design quirks were discovered using order statistics and extreme value theory. The low-order cell-based design contour approach is a potential way to examine complex marine systems and identify sea states which may be good candidates to evaluate reliability and performance under different operational conditions. This has major consequences for designing operational profiles for in-depth numerical or physical models. Design contours as presented in this paper could help direct which sea states are most worth investigating. Acknowledgements. The authors would like to thank Ms. Kelly Cooper and the Office of Naval Research for their support for this research which is funded under the Naval International Cooperative Opportunities in Science and Technology Program (NICOP) contract number N00014-15-1-2752.

References 1. Winterstein, S., Ude, T., Cornell, C., Bjerager, P., Haver, S.: Environmental parameters for extreme response: inverse form with omission factors. In: Proceedings of International Conference on Structural Safety and Reliability (ICOSSAR 1993), vol. 01 (1993) 2. Baarholm, G.S., Moan, T.: Estimation of nonlinear long-term extremes of hull girder loads in ships. Mar. Struct. 13(6), 495–516 (2000) 3. Fukasawa, T., Kawabe, H., Moan, T.: On extreme ship responses in severe shortterm sea state. In: Advancements in Marine Structures (2007) 4. Seyffert, H.C., Troesch, A.W., Collette, M.D.: Combined stochastic lateral and inplane loading of a stiffened ship panel leading to collapse. Accepted: Mar. Struct. (2019) 5. Ochi, M.K.: Applied Probability & Stochastic Processes in Engineering & Physical Sciences. Wiley Series in Probability and Mathematical Sciences (1990) 6. Seyffert, H.C.: Extreme Design Events due to Combined, Non-Gaussian Loading. Ph.D. thesis, The University of Michigan (2018) 7. Ashe, G., Cheng, F., Kaeding, P., Kaneko, H., Dow, R., Broekhuijsen, J., Pegg, N., Fredriksen, A., de Francisco, J.F., Leguen, F.V., Hess, P., Gruenitz, L., Jeon, W., Kaneko, H., Silva, S., Sheinberg, R.: 17th International Ship and Offshore Structures Congress, Committee V.5 Naval Ship Design (2009) 8. IACS Rec. No. 34. Standard Wave Data (2001) 9. Hughes, O.F.: Ship Structural Design: A Rationally-Based, Computer-Aided Optimization Approach. The Society of Naval Architects and Marine Engineers (1988)

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10. Alford, L.K.: Estimating Extreme Responses using a Non-Uniform Phase Distribution. Ph.D. thesis, The University of Michigan (2008) 11. Kim, D.-H.: Design Loads Generator: Estimation of Extreme Environmental Loadings for Ship and Offshore Applications. Ph.D. thesis, The University of Michigan (2012) 12. Lloyd’s Register. ShipRight Design and Construction, Structural Design Assessment: Global Design Loads of Container Ships and Other Ships Prone to Whipping and Springing. Technical report, January 2018 13. Baarholm, G.S., Haver, S., Økland, O.D.: Combining contours of significant wave height and peak period with platform response distributions for predicting design response. Mar. Struct. 23(2), 147–163 (2010) 14. Vanem, E.: A comparison study on the estimation of extreme structural response from different environmental contour methods. Mar. Struct. 56, 137–162 (2017)

Potential of Homogenized and Non-local Beam and Plate Theories in Ship Structural Design Jani Romanoff1(B) , Bruno Reinaldo Goncalves1 , Anssi Karttunen1,2 , and J. N. Reddy2 1 Marine Technology, Department of Mechanical Engineering, School of Engineering, Aalto University, Puumiehenkuja 5 A, 00076 Aalto, Finland [email protected] 2 Advanced Computational Mechanics Laboratory, Texas A&M University, 180 Spence Street, College Station, TX 77843, USA

Abstract. The paper gives an overview of the recent development on application of homogenized, non-local beam and plate theories in design of marine structures. The homogenization considers all steps of the derivation of the prevailing differential equations from kinematics to equilibrium with external loading. This enables an accurate localization process that recovers the microstructural effects from the homogenized solution of the prevailing differential equations. Presently, the method is handle limit states of serviceability and ultimate strength. The theory is validated by full 3D-FE simulations on periodic beams and plates. The theory converges to the physically correct solutions in case of infinite and zero shear stiffness; especially the limit of zero shear stiffness is important as there the traditional Timoshenko beam theory (TBT) fails to predict the response correctly. We also show the case of plate under bi-axial loading where the bifurcation buckling solutions fail to predict correctly the physics, but the non-local framework is able to handle this case. Keywords: Beam theory · Plate theory · Structural design

1 Introduction In terms of transport efficiency, ships operating with relatively slow speeds, are the most efficient form of transport in terms of sustainability and costs [1]. Thin-walled structures form the basis of ship structures and they contribute significantly to the overall lightweight and to the payload. Structural performance is measured by the ratio between the payload and the structural weight and therefore we aim to build increasingly lighter vehicles, to diminish the environmental impact of transportation. This can be done by using wide range of engineering materials and structural topologies and geometries; see Fig. 1. However, the limits to which we can push the structural performance ratio are defined engineering approaches used for lightweight design. When utilizing new materials and structures, the level of confidence to simulation methods one has to rely on, becomes easily a limiting design issue. Currently in materials engineering, the ultra-lightweight features are obtained by applying to ever smaller scales the ideas of civil engineers, bridge-builders, aeronautical © Springer Nature Singapore Pte Ltd. 2021 T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 184–196, 2021. https://doi.org/10.1007/978-981-15-4672-3_11

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Fig. 1. Structural design and optimization of ship structures by use of equivalent beam, plate and shell elements and evolutionary optimization algorithms. Different classical length-scales at primary, secondary, tertiary levels and new ones introduced by lattice type materials seen as new regions in the Ashby diagram [2].

and marine engineers who have over long time found feasible lightweight solutions in format of ship hull girders, double bottoms and stiffened/sandwich structures [2].

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This work is realized in the so-called Ashby diagram of materials as new regions of structured, lattice materials; see Fig. 1. These ultra-lightweight materials and structures have relative densities, ρ mat /ρ bulk , often as low as 0.05–0.2, but stiffness and strength values are not reduced as much; this low ratio is caused by significant volume of voids. Typically, the material reserve in terms of strength capacity with respect to the elastic limit (design point) is significantly reduced especially when the material contains flaws or perforations, what is typical for ship structures. This forces designers to impose larger safety factors for these ultra-lightweight builds or to use more advanced modeling strategy. In essence, an additional length scale is introduced to the ship structural design (see e.g. Ref [3]. for the classification) in addition to the classical primary (σ1 -level), secondary (σ2 -level) and tertiary (σ3 -level). This emerging level could be denoted as quaternary (σ4 -level) and in theory, as the technology develops, we could introduce length-scales even beyond this. The problem in introduction of these additional lengthscales is the fact that computational models based on classical continuum mechanics are pushed to their limits due to violation of the continuum assumption. These homogenized continuum models become at this situation infeasible. However, being computationally cheap and relatively accurate, they possess a lot of potential in use of early design stages (e.g. concept design) of ship design. These models are often based on equivalent single layer (ESL) assumption. This means that if deformations are known at this layer, they can be explicitly described anywhere in thin-walled structure. In many cases, the First Order Shear Deformation Theory (FSDT) is used in this context. The equivalent single layer element mesh allows modeling of any beam or plate type of structure that follows the ESL-FSDT kinematics to be modelled accurately by a single FE-model. As the stiffness properties are equivalent, the changes in material and structural topology can be changed directly in the stiffness matrix, without changing the FE-mesh [3–6]. In the structural design where large design space must be explored, this is beneficial as the large and geometrically complex FE-model of the ship needs to be created only once. In practical optimization the scantlings of stiffened panels, lattice type sandwich panels and laminates are discrete and we need to exploit global evolutionary optimization algorithms. On one hand this allows parallelization of the computations, but on the other hand the structural analysis is performed in a “black box”. The approach has been used to optimize the structures on panel [4], deck [5] and hull girder levels [6]. When optimizing the structure with these methods, we may explore the design space with inaccuracy in modeling, which results in false conclusions about feasibility of various designs [7, 8]. This is seen in a simple example of an orthotopic plate under biaxial stress, simply supported from four edges, and by investigating the corresponding bifurcation buckling load often used as strength constraint in optimization. The buckling load for this case is given as [9]:   c11 2 + β 2 DQ2 α c + 55 DQ2 DQ1 1   (1) N0 = 2 2 D Q2 c c 1 α + kβ 1 + 11 + 22 + c33 DQ2

DQ1

c11 DQ1

c11

Potential of Homogenized and Non-local Beam and Plate Theories

α= c44 =

mπ a ,β

187

=

nπ 2 2 2 b , c11 = c22 c33 − c44 , c22 = D11 α + D33 β , 2 2 c33 = D33 α + D22 β (D12 + D33 )αβ, c55 = D11 α 4 + 2(D12 + 2D33 )α 2 β 2 + D22 β 4 ,

k = Ny /Nx

where DQ1 and DQ2 are the shear stiffnesses in principal directions 1 and 2 respectively. In similar fashion, D11 , D22 , D12 , D33 correspond the bending and twisting stiffnesses about directions 1, 2 and 3 with first subscript denoting the direction of edge normal and second the direction of rotation. Symbols m and n are used to denote the number of half-waves in directions 1 and 2 respectively. In case of high orthotropy in shear is large, i.e. DQ2 /DQ1 -ratio is very small (order of magnitude 1% or less) while the axial and bending stiffnesses do not suffer from such orthotropy. Due to this, for the smallest bucking load, the amount of buckling half waves in y-direction (n) must be very large to reach minimum. In practice this leads to situation in finite element calculations where the buckling load becomes mesh size dependent and the finer is the mesh, the lower is predicted buckling load. This situation occurs for example in lattice structures that are currently emerging to the Ashby diagram as the future most effective structural materials. The utilization of these materials is however limited due to model properties as shown by this example. We could use direct modeling of the material microstructure, but in these cases the computational task becomes enormous and too costly for design processes where competitiveness must be maintained [10–12]. This paper gives an overview of the recent developments on the application of homogenized beam theories based on non-classical continuum mechanics used to predict the micro- and macrostructural stresses in the design of marine structures. These theories are shown to be able to push the continuum limit assumption to the range of lsmall /l large = 0.1…0.25 which is often seen in ship structures. We first present the main assumptions introduced by homogenization and localization and discuss their relevancy in terms of marine structures. Then we move to the resulting differential equations and discuss their differences and numerical solutions with Finite Element Method. In order to demonstrate the gains and remaining challenges we present case studies where comparison to high-fidelity finite element simulations are performed.

2 Method Description 2.1 Scale Separation – Homogenization and Localization In the assessment of structural performance of ships, the response is needed to ensure that the load-carrying mechanism is correct. Thus, homogenized models that predict average responses are enough. The strength is needed to ensure capacity with respect to environmental load. This in turn requires localization of stresses for example for fatigue analysis. The challenge between the two approaches is the scale transition. In order it to be fully consistent, the energy, stresses and strains must be all be equal between the continuum model and the sub-model where localization and strength is defined. In practice we are forced to satisfy these equalities only partially. This in turn means that many methods are feasible only for either stiffness or strength prediction. In Romanoff

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et al. [13] this issue is discussed through example of a web-core sandwich beam in bending; see Fig. 2.

Fig. 2. Web-core sandwich beam in bending with local response due to warping deformation of unit cells and global deformation due to bending deformation of unit cells; discontinuity functions as solid line and homogenized solutions as dashed line.

The beam deflection by using discontinuity functions (i.e. Heaviside, Dirac’s delta and unit doublet) is given as [13, 14]: wface (x) = wl,bend (x) + wg,bend (x) wl,bend (x) = w0l + θ0l x  I   H x−aiM Mi x−aiM 2 + D1i 2! +

w

g,bend

(x) =

g w0

g + θ0 x

i=1    3 H x−ajF Fj x−ajF

3! j=1    K H x−aq q x−aq 3  k k k + 4! k=1 L    1 Mlt + Mlb H x

−

(2)

J 

Dg

l=1

 2  − alMi x − alMi 2!

where wl,bend is the local deflection of the face due to point forces and moments and distributed external and internal loads applied directly to the face and wg,bend is the

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global deflection caused by elongation of the faces to opposite directions (w0 and θ 0 are the displacement and rotation boundary conditions). This needs to be solved for each location of flexible core members [14] and boundary conditions. The solution can be simplified by neglecting distributed loads and moving all loads to the locations of webs. The result is then L  g g H (x−al )(x−al )2 wface (x) = w0l + θ0l x + w0 + θ0 x + 2! l=1  

  F (x−a ) 1 l l − D1g Mlt + Mlb Di Ml + 3

(3)

which is periodic with respect to stiffener spacing. In localization, the peak response at the location of high gradients must be assessed, and we can use Taylor-series expansion around point a as w(x) = w0 (a) +

w1 (a) w2 (a) (x − a) + (x − a)2 + . . . 1! 2!

(4)

However, this is accurate only at the neighborhood of point a and if we want to extend the solution over entire beam, the solution becomes lengthy and dependent on the number of webs in the beam. To reduce the size of the problem to smooth and periodic fields far away from peak response, the homogenization theory can be used. There, the two length-scale asymptotic expansion is used instead to give wk (x) = w0 (x, y) + k 1 w1 (x, y) + k 2 w2 (x, y) + . . . micro k = llmacro ν1, wk (x, y) = wk (x, y + lmicro )

(5)

where the microlevel coordinate y = x/k and k is the length scale ratio -parameter. Comparison of Eq. (2) to Eqs. (3) and (4), reveals that the Heaviside operator is dropped in both localization and homogenized solution. In addition, the homogenized solution approximates the (x−a)n -terms xn -terms. Thus, the homogenized models change the reality, by 1) moving all loads to the hard points of the structure; 2) do not consider the actual positioning of the unit cell along the beam; 3) the continuity conditions on deflection and its derivatives at the edges of unit cell is not considered and 4) by assuming that the length-scale ratio is infinitely small. Thus, the assumptions are very severe as we impose loads directly on the surfaces of structures, the discrete supports are not always evenly spaced and the distance between discrete supports are not infinitesimal (e.g. hull girder of the ship, l1 ~300 m; bulkheads and decks, l2 ~30 m; stiffened or corrugated sandwich panels, l 3 ~3 m). The approximation can be improved when more terms are included in the asymptotic expansion and continuum description. Here we present the influence of adding microrotation to ESL-FSDT kinematics of beams and plates. This model is called micropolar, and it has been discussed broadly in solid mechanics community. The main advantage of using it comes from the fact that we obtain additional deformation mode to the FSDT-formulation which classically consists of in-plane stretch, linear bending- and constant shear-strain(s) over the thickness. The added microrotation allows bending of the microstructure to have piecewise linear strains over the thickness.

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2.2 Beam and Plate Models The displacements for a micropolar plate can be written as [8] ux (x, y, z) = ux (x, y) + zφx (x, y), uy (x, y, z) = uy (x, y) + zφy (x, y) uz (x, y, z) = uz (x, y) x (x, y, z) = ϕx (x, y) y (x, y, z) = ϕy (x, y) z (x, y, z) = 0

(6)

where the classical displacements u and rotation φ are complemented by a non-classical microrotation ϕ and assumption of incompressibility is made (ϕ and uz are independent of z-coordinate). The strains are [8] εxx = ux,x + zφx,x , εyy = uy,y + zφy,y , εxy = uy,x + φy,x , εyx = ux,y + φx,y , εxz = uz,x + ϕy , εzx = φx − ϕy , εyz = uz,y − ϕx , εzy = φy + ϕx , χxx = ϕx,x , χyy = ϕy,y , χxy = ϕy,x , χyx = ϕx,y

(7)

and the symmetric and antisymmetric shear strains read γxs = uz,x + φx , γys = uz,y + φy γxa = uz,x − φx + 2ϕy , γya = uz,y − φy − 2ϕx

(8)

Thus, shear strain is composed from symmetric antisymmetric parts, with symmetric part having the same form as conventional first order shear deformation theory. The plate equations are obtained by employing a constitutive model and variational principles and they result in micropolar moments in addition to classical stress resultants of membrane and shear forces and bending and twisting moments. Thus, the relation between strains and stress resultants is needed. This relation is done by use of micromechanics; see Fig. 3. There, the unit cells of the periodic structures are exposed to displacement fields corresponding in-plane stretch, bending and shear, but also to microrotation. The relation between these micro strains and curvatures and stress resultants give the equivalent stiffness properties of the beams and plates [8, 10, 16]. From the homogenized smooth deformation field, the periodic oscillating fields can be recovered by use of localization [5, 8, 14, 16], see Fig. 3. Beam and plate models based on modified couple stress theory are obtained, instead of formulating shear into antisymmetric and symmetric parts, by  2 considering the average shear strain and taking curvature χxy = 1/4 φx,x − dd xw2 of the deformations. The beam models are obtained from plate models by considering the terms with x-variation only and uniaxial stress. 2.3 Finite Elements The fact that the strains include microrotations or gradients of local rotations increases the total order of the governing differential equations, but do not affect the discretization

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Fig. 3. Derivation of equivalent, homogenized stiffness properties by unit cell analysis, localization of periodic stresses from, smooth and homogenous solution and coupling between shell and offset beam elements oriented along x-axis. Reproduced from [5, 16].

of the FE-model; see Fig. 3. The analytical solutions will include higher-order terms that appear as exponential or hyperbolic functions and these terms are significant in the vicinity of strain gradients and in structures where internal stiffness of the microstructure is significant in comparison to macroscale strain gradients. In finite element approximations these high-order terms create a need to include higher-order polynomials to shape function approximations. Karttunen et al. [15, 17] derived exact shape functions for beams which do not suffer from poor convergence. For plates such elements do not exist, but are currently being developed. Thus, in practical structural design the effect

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of gradients is modelled by adding a stabilizing layer of elements on top of the classical ESL-FSDT mesh. This stabilizing layer has the stiffness related to microrotation. Due to it, the FE-mesh can deform only to finite curvature. In the examples the importance of this stabilization is demonstrated.

3 Case Studies 3.1 Size Effect and Classical Continuum Assumption The model has been validated to numerous cases and limit states on beams and plates including linear and geometrically non-linear bending, bifurcation buckling, vibrations [15–18]. Figure 4 collects some of the most important ones to demonstrate in which cases the enhanced formulations become important. In Fig. 4A bending of a web-core beam with 12 unit cells along its length is demonstrated [18]. The rotation stiffness of the interface between webs and faces is varied; this in case of sandwich panels is related to laser-stake-welding, but in double bottom or ship scale would mean influence of brackets to beam responses. As the rotation stiffness is reduced, the classical Timoshenko beam theory (TBT, i.e. FSDT) fails to predict the response correctly while the micropolar model is in perfect agreement with high fidelity 3-D FEA. Figure 4B presents a similar comparison on bifurcation buckling load based on a couple stress beam model. The scale parameter based on ratio of local unit cell resistance to curvature (DL ) and to shear flexibility of the beam (DQ L 2 ) increases for decreasing joint rotation stiffness, which causes the deviation of classical beam theory from accurate solution. Figures 4C and D extend the investigation for other core types in bending. From these figures we can see that core topology has significant effect to the observed behavior. The stretch-dominated X-core is less sensitive for the local stiffness effect than the bending dominated web- or Y-cores. Thus, in order to have full confidence of the computational model, the non-local extension is needed as it is able to handle wider variety of design alternatives accurately. 3.2 Influence of Micro-Rotation to Bifurcation Buckling of Orthotropic Plate In the plate formulation the phenomena due to strain gradients become more complex due to the fact that the load-carrying mechanism is 2-dimensional, see Fig. 5. On the other hand, finite elements are merging and the examples created are therefore based on either engineering computations based on FEA or analytical solutions (Navier solution). Figure 5A shows an example of numerical model where a stabilizing element layer is added on top of classical ESL layer to account finite curvature of the plate bending; this way we can omit the size dependency observed in Eq. (1) [8, 19]. The analytical Navier solution for micropolar plate has been derived in Ref. [8]. Figure 5B shows the bi-axial buckling case of a simply supported web-core sandwich plate (analogous to double side or bottom). The uniaxial compression in weak direction is shown in Fig. 4C and biaxial buckling with load ratio, N xx /N yy = 1/4, in Fig. 4D. While the improved non-classical plate formulation is able to handle the problem with perfect agreement to 3D-FEA, the classical solution fails completely and has significant size effect, as depicted in Eq. (1).

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Fig. 4. A) micropolar model: bending of a web-core beam with 12 unit cells along its length and B) couple stress model: bifurcation buckling load comparison. Bending of clamped C) X-core and D) Y-core beams under a centered point load in bending (couple stress model)

4 Conclusions The paper gives an overview of the recent development on application of homogenized, non-local beam and plate theories in design of marine structures. The main addition to classical beam and plate theories is introduction of a micro-rotation or strain gradient effect into the continuum description, which allows handling of sharp gradients in shear forces. The following derivation of differential equations follow the normal process exploited in solid mechanics. As an outcome additional stress resultant are obtained. The homogenization considers all steps of the derivation of the prevailing differential equations from kinematics to equilibrium with external loading. This enables accurate localization process that recovers the micro-structural effects from the homogenized solution of the prevailing differential equations. Paper presented examples from different examples on limit states of serviceability and bifurcation buckling. The theory was validated by full high-fidelity 3D-FEsimulations on periodic beams and plates. The theory converges to the physically correct solutions in case of infinite and zero shear stiffness and was able to handle significant

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Fig. 5. A) introduction of stabilizing mesh to classical ESL. B) load case of bi-axial bifurcation buckling load for simply supported web-core sandwich panel with results in C) weak direction and D) and N xx /N yy = 1/4.

orthotropy in plates where classical continuum mechanics-based plates produce mesh size-dependent results. This is significant problem in structural optimization where structural analysis is performed hundreds or thousands time to explore the design space with different materials and structural alternatives. As the developed method can better handle the size effect and orthotropy, it increases the confidence to structural optimization from viewpoint of structural model. The future work should be extended towards development of finite element for plates and shells. This development should account for von Karman kinematics, so that geometrical non-linearity can be handled as part of structural design. In our present model we used first gradient of strain to improve our response predictions. Due to this, significant improvements were obtained, but still there is a difference between validation models and theory. Increasing the number of gradients would improve the estimates even further, but would at the same time lead to more complex analytical and numerical solutions. Thus, the balance between accuracy and complexity of these solutions is interesting question related to practical engineering work. The fact that strain-gradients are used in

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plasticity give possibilities to include these recent developments to the structural models. Although examples presented in this paper focus on beam and plate scale, similar phenomena are present in levels of secondary and primary structures. Thus, research should be extended to these scales. However, this is left for future work.

References 1. Trancossi, M.: What price of speed? a critical revision through constructal optimization of transport modes. Int. J. Energy Environ. Eng. 7(4), 425–448 (2016) 2. Fleck, N.A., Deshpande, V.S., Ashby, M.F.: Micro-architectured materials: past, present and future. Proc. Roy. Soc. A 466, 2495–2516 (2010) 3. Hughes, O.F., Paik, J.K., Beghin, D., Caldwell, J.B., Payer, H.G., Schellin, T.E.: Ship Structural Analysis and Design. Society of Naval Architects and Marine Engineers (SNAME), New Jersey (2010) 4. Romanoff, J., Klanac, A.: Design optimization of steel sandwich hoistable car decks applying homogenized plate theory. J. Ship Prod. 24(2), 108–115 (2008) 5. Romanoff, J.: Optimization of web-core steel sandwich decks at concept design stage using envelope surface for stress assessment. Eng. Struct. 66, 1–9 (2014) 6. Raikunen, J., Avi, E., Remes, H., Romanoff, J., Lillemäe-Avi, I., Niemelä, A.: Optimisation of passenger ship structures in concept design stage. Ships Offshore Struct. 14, 320–334 (2019) 7. Jelovica, J., Romanoff, J.: Buckling of sandwich panels with transversely flexible core: correction to the equivalent single-layer model using thick-faces effect. J. Sandwich Struct. Mater. 22, 1612–1634 (2019) 8. Karttunen, A., Reddy, J.N., Romanoff, J.: Two-scale micropolar plate model for web-core sandwich panels. Solids Struct. 170, 82–94 (2019) 9. Reddy, J.N.: Mechanics of Laminated Composite Plates and Shells – Theory and Analysis, 2nd edn. CRC Press, Boca Raton (2004) 10. Matous, K., Geers, M.G.D., Kouznetsova, V.G., Gillman, A.: A review of predictive non-linear theories for multiscale modeling of heterogenous materials. J. Comput. Phys. 330, 192–220 (2017) 11. Hulkkonen, T., Shin, S.H., Yi, N.H., Jang, D.H., Jang, T.G.: Enabling a paradigm shift in ship structural design with a 3D approach. In: Proceedings on International Conference on Computer Applications in Shipbuilding 2017, Royal Institution of Naval Architects, Singapore , 26–28 September 2017 (2017) 12. Son, M.J., Woo, J.J., Park, H.G., Lee, J.Y.: Auto-fine mesh generation for local analysis based on the consistent finite element model. In: Proceedings on International Conference on Computer Applications in Shipbuilding 2017, Royal Institution of Naval Architects, Singapore, 26–28 September 2017 (2017) 13. Romanoff, J., Karttunen, A., Reinaldo Goncalves, B., Reddy, J.N.: Homogenized and nonclassical beam theories in ship structural design – challenges and opportunities. In: Bensow, R., Ringsberg, J. (eds.) Proceedings of the VIII International Conference on Computational Methods in Marine Engineering - MARINE 2019, Gothenburg (2019) 14. Romanoff, J., Varsta, P.: Bending response of web-core sandwich beams. Compos. Struct. 73, 478–487 (2006) 15. Karttunen, A., Romanoff, J., Reddy, J.N.: Exact microstructure-dependent Timoshenko beam element. Int. J. Mech. Sci. 111–112, 35–42 (2016) 16. Reinaldo Goncalves, B., Karttunen, A., Romanoff, J.: A nonlinear couple stress model for periodic sandwich beams. Compos. Struct. 212, 586–597 (2019)

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17. Reinaldo Goncalves, B., Karttunen, A., Romanoff, J., Reddy, J.N.: Buckling and free vibration of shear-flexible sandwich beams using a couple-stress-based finite element. Compos. Struct. 165, 233–241 (2017) 18. Karttunen, A., Reddy, J.N., Romanoff, J.: Two-scale constitutive modeling of a lattice core sandwich beam. Compos. Part B Eng. 160, 66–75 (2019) 19. Romanoff, J., Varsta, P., Remes, H.: Laser-welded web-core sandwich plates under patchloading. Mar. Struct. 20, 25–48 (2007)

Structural Design of Hinge Connector for Very Large Floating Structures Ye Lu1(&), Bei Teng2, Yikun Wang3, Ye Zhou1, Xiaoming Cheng1, and Enrong Qi1 1

2

Department of Offshore Structure, China Ship Scientific Research Center, Wuxi, China [email protected] School of Ship Engineering, Wuxi Institute of Communications Technology, Wuxi, China 3 FSI Group, University of Southampton, Southampton, UK

Abstract. Very large floating structures (VLFS) are usually composed of a number of structural modules. These are a set of semi-submersible floating bodies with similar sizes, connected by several connectors with a specific rigidity. Due to the potential severe sea states that the VLFS may encounter, the connectors become the most critical devices and inevitably the weakest link, determining the strength capacity of the entire structure. To ensure the structural safety and the integrity of the VLFS, the strength of the connectors needs to be evaluated accurately. This study numerically explores the strength of the hinge type connectors. Considering the dynamic characteristics of the connectors, a strength model of flexible connector is proposed and analysed. The results indicate that the hinged type flexible connector with hollow circular variablecross-section pin shaft is capable of adjusting the stiffness of the whole structures and decreasing the shocks and abrasion by filling with flexible materials. Therefore, the safety and reliability of the module connections in a VLFS can be guaranteed. Keywords: Structural strength


Hinge connector
VLFS

1 Introduction The very large floating structure (VLFS) is a multi-module floating system composed of a set of semi-submersible floating bodies with the same sizes, which are connected by connectors with a specific rigidity. The VLFS has been intensively studied in the field of marine structural research. There are four international conferences on the very large floating structures organised in Japan (1991, 1999) and in USA (1996, 2001) separately. The box-type floating structures studied in Japan, due to the simple shapes and assumption, can be easily deemed two-dimensional (2D) flexible buoyant plate at sea surface. Owing to excellent seakeeping, the mobile offshore bases with the columns connected along the length could be the optimised choice at the marine structural design.

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However, due to the large scale of structures and long durations floating at sea, it is inevitable for the VLFS to encounter severe sea states. Among various structural components within a VLFS, the connectors are the most critical devices and nevertheless the weakest link members. When the environmental load is greater than the designed load, the connector will be partially damaged and in severe scenarios, the overall structure of the VLFS will fail. To ensure the safety and the structural integrity of the VLFS, the strength of the connectors needs to be evaluated. In general, connectors for VLFS use an articulated, flexible connection that adjusts the stiffness of the whole structures and decreases the loads by filling with flexible materials. The hinge and the flexible material result in nonlinearities in the mechanical analysis, and hence it could be difficult to assess the strength of such connectors. Furthermore, in order to reducing the total weight, the connectors can be designed for the hinge type with the hollow shaft. Therefore, before the structural assessment of the connectors, the loads prediction should be applied precisely at first. In order to achieve this, the three-dimensional (3D) linear hydroelastic theory was initially proposed by Wu [1]. He studied the fluid-structure interactions of multi-rigidbody modules elastically connected and obtained the motions and deformation of structures at different wave directions, as well as the dynamic response of the connectors. Based on the 3D potential flow theory and the Green’s function method, Ertekin et al. [2] and Riggs et al. [3] took into account the hydrodynamic interaction between rigid-body modules and calculated the motion and loads of five-module mobile offshore base. Comparison of unconnected, flexible connected, and rigid connected conditions was achieved. Yu [4] analysed the impact of the interaction forces between modules of mobile offshore bases, and other influences such as connector stiffness, wave angles, and sea conditions. Qi et al. [5] used rigid module flexible connector model to calculate the hydrodynamic performances of the floating body near the island reef based on the 3D potential flow theory and obtained the dynamic response of the connectors. In addition, Qi et al. [6] designed a flexible connector model. Through the static tensile and compression tests of connectors with different amplitudes and load combinations, the stiffness characteristics of super-large floating body connectors were studied. The influence of combined loads on connector stiffness was discussed. Zhang et al. [7] calculated the hydrodynamic motions and dynamic response of a mobile base system composed of different numbers of semisubmersible modules under regular and irregular waves. Liu et al. [8] compared the dynamic responses of three different models under the condition of sea state 7 and explored the effect on the dynamic characteristics of the connector in shallow water. Furthermore, Xu et al. [9] gave an adaptive optimal control method, named amplitude death, to suppress the oscillation of VLFS system consists of multiple modules connected by flexible connectors with strong nonlinearity in geometry. Although the loads prediction and dynamic responses of connectors are well known in terms of rigid and flexibility, the structural details have not been solved yet, especially for two different materials in contacted. Flexible connectors with uncertain stiffness need to be analysed considering nonlinearity. Zhu et al. [10] proposed a flexible connector structure to model by the finite element’s method for nonlinear analysis. Lu et al. designed a hinged type flexible connector with hollow circular variable-cross-section pin shaft for the load characteristics of super-large floating

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structures, and basically analysed the strength of the flexible connector with varying stiffnesses [11, 12]. Zhou et al. [13] compared the strength of connectors with different diameters of solid shaft considering contact pairs and gave the suggestion that the diameter of the pin and the thickness of the ears should be determined according to the design load and the preliminary design dimensions. In this paper, the VLFS with horizontal pontoons is taken as the research object. Considering the dynamic characteristics of the connectors, a strength model of flexible connector is proposed and analysed.

2 Methodology 2.1

Very Large Floating Structures

The very large floating body, which is design by China Ship Scientific Research Center (CSSRC), is composed of five modules and eight connectors. The modules are represented by M1-M5, and the connectors are represented by C1-C8 in the Fig. 1. The five modules have the same geometry. Each module i (i = 1, 2, 3, 4 and 5) has the corresponding Cartesian coordinate system oi-xiyizi. The xi-axis is from the aft perpendicular (A.P.) to the forward perpendicular (F.P.); yi-axis is from the starboard side to port side and zi-axis is pointing upward from the oi, which is denoted the centre of gravity of the module. With the same directions, the Cartesian coordinate system OXYZ is defined as follows, the origin of the system is the gravity of the VLFS; the Xaxis points out from the M1 to M5; Y-axis is from the starboard side to port side and Zaxis points upward from the origin. Due to the identical geometry of the modules, the origin of the system, which is the centre of gravity of the VLFS, is coincident with the centre of gravity of module M3. Table 1 shows the design parameters for a single module. The connector installation position between the modules of VLFS is shown in the Fig. 2. The centre of the connector is located at the centre of the column and at the centre of the upper platform.

Fig. 1. Sketch of very large floating structures systems

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Y. Lu et al. Table 1. Principle particulars of VLFS Location Items Upper platform Length (m) Breadth (m) Height (m) Columns Height (m) Diameter (m) Spans in horizon (m) Spans in longitudinal (m) Pontoons Length (m) Breadth (m) Height (m) Others Vertical of gravity (m) Design drought (m) Design displacement (kg) Ixx (kg.m2) Iyy (kg.m2) Izz (kg.m2)

Values 300 100 6 16 18 58 60 30 96 5 10.91 12 9.192  7.451  6.848  7.407 

107 1010 1011 1011

Fig. 2. Location of the connectors of VLFS (dimensions are in mm)

2.2

Design Requirements of the Connectors

The design load of the connector is closely related to its rigidity. If a rigid connector is used, the connector loads, especially vertical and horizontal bending moment, will be very large. In addition, there will be significant fatigue issues. Therefore, an ideal design is that the connector should be flexible with suitable rigidity. Natural rubber and synthetic rubber exhibit many unique physical and chemical properties. They have high mechanical strength and energy absorption characters. Some rubbers are also wearing and corrosion resistant in various environments.

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Therefore, the connectors are designed as a combination of high-strength steel and rubber. Rubbers provide flexibility and damping. In this article a simple flexible hinge connector was studied, consisting of single and double ears and bolts. The rubber mats in the double ears not only fix the displacements but also absorb impaction with a flexible connector, as shown in the Fig. 3, where the double ears are coloured with blue, while the single ear is in pink. The white area denotes the bolt surrounded by green rubber mats.

Fig. 3. Concept of flexible connector

The hinged flexible connector contains a base, an articulated ring, a latch and a rubber ring. Consistent with the coordinate system of the super large floating body, the right-handed rectangular coordinate system is defined at the connector. Considering the location of the connectors, the origin of the coordinates is at the centre of the rings, with the x-axis pointing from double ears to the single ear, the y-axis pointing to the left side, and the z-axis pointing vertically upwards. The six degrees of relative freedom of the very large floating body module are surge (DX), sway (DY), heave (DZ), roll (RX), pitch (RY), and yaw (RZ). Due to the reduction in the vertical bending moment, constrains of articulated flexible connectors are assumed that DX, DY, and DZ are not rigid, RX and RZ are determined through three axis directional constraints, and RY is released, as shown in Table 2. Table 2. Constrains of six degrees of relative freedom modules of VLFS Relative freedom Surge (DX) Sway (DY) Heave (DZ)

Constraints Flexible Flexible Flexible

Relative freedom Roll (RX) Pitch (RY) Yaw (RZ)

Constraints Flexible None Flexible

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3 Design of Connectors 3.1

Basic Properties of the Connectors

The 3D potential theory was employed to calculate the responses and loads between modules. The rigid module and flexible connectors (RMFC) models were used to obtain the loads of connectors. Figure 4 shows the hydrodynamic grids of a single module with five transverse pontoons.

Fig. 4. Hydrodynamic panels of single module with five transverse pontoons

The analysis on loads and responses of the hinged connector was carried out using the five-module VLFS. Each module has 5 horizontal pontoons beneath the water surface. Using RMFC method, the maximum loads of the connector C4 under different wave angles is shown in the Figs. 5 and 6 and listed in Table 3. When the wave angle is less than 60°, the connector loads, and module motion responses are small without much change with the wave angle. Also, the longitudinal load Fx is less than 40 MN, which can be used as the working load of the connectors. The connector loads and the module motion are changed sharply with the wave angle from 60° to 90°. The maximum longitudinal load can reach nearly three times of the working load, which is the extreme load to be avoided in the design. Consequently, the longitudinal loads were applied to design the details of the connectors due to lower values of horizontal and vertical loads. According to the loads and responses of connectors of VLFS, articulated connectors are designed in detail based on the conceptual picture as shown in the Fig. 7. It mainly consists of single ear, double ears, bases and bolt. The main parameters are shown in Table 4.

Structural Design of Hinge Connector

Fig. 5. Maximum forces of the connector under different wave angles

4.0M

Mx Mz

3.5M

Moment(N.m)

3.0M 2.5M 2.0M 1.5M 1.0M 500.0k 0.0 0

10

20

30

40 50 60 Wave angle(deg)

70

80

90

Fig. 6. Maximum moments of the connector under different wave angles Table 3. Design loads of connectors Maximum Fx (N) Fy (N) Fz (N) Mx (N.m) Mz (N.m) 0°–60° 4.03  107 5.01  106 2.75  107 6.61  105 1.26  106 0°–90° 1.14  108 1.60  107 6.06  107 1.84  106 3.80  106

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Fig. 7. 3D views of flexible connector of VLFS in full scale (dimensions are in mm) Table 4. The main parameters of the full scale of the flexible connector of VLFS Items External diameter of double ears Inner diameter of double ears External diameter of single ear Inner diameter of single ear Thickness of nylon 66 External diameter of hollow cylinder Inner diameter of hollow cylinder Height of ring rib Breadth of ring rib Spans between single ear and double ears

3.2

Full scale (mm) 2000.0 1200.0 2000.0 1230.0 15.0 1200.0 1000.0 200.0 250.0 5.0

Material

Mild steel properties (with Young’s modulus of 2.0  1011 N/m2, density of 7850 kg/m3, Poisson’s ratio of 0.3) and yield strength of 235 MPa were applied to bases, single ear, double ears and bolt. The inner surface of the hollow cylinder contained a circular rib between the single and the two ears. Such structure was adopted to reduce the weight and to guarantee that the connector would fail prior to the module and the base at the limit states.

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3.3

205

Modelling in Finite Element Analysis

The finite element analysis (FEA) software ANSYS was used to calculate the strength of the full-scale connector structures. The SOLID186 element was used to model the connector’s rings, varying sectional latch. The TARGE170 and CONTA174 elements were used to simulate the target surface and contact surface by surface-surface contact. In this situation, the inner surfaces of ears are deemed to be target surfaces and the outer plane of the bolt is the contact surface. The finite element model of the three-ring connector with hollow bolt is shown in the Fig. 8, while the connector with ringstiffened is shown in the Fig. 9. In terms of constrains, the two-ear base surface was rigidly fixed. The single-ear base surface used the MPC184 multi-point restraint. Two identical connectors were arranged at the upper platform of VLFS, with a 25% safety factor. Therefore, the x-axis direction applied load is 5.0  107 N.

Fig. 8. FEM of connector without ring-stiffened cylindrical shell

Fig. 9. FEM of connector with ring-stiffened cylindrical shell

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4 Strength Results The Von Mises stress distributions of longitudinal section of the full-scale connector with varying stiffeners heights such as 0 mm, 50 mm, 100 mm and 20 mm are shown in the Fig. 10. 1

1

NODAL SOLUTION

Y STEP=1 SUB =8 Z TIME=1 SEQV (AVG) DMX =.001949 SMN =630721 SMX =.204E+09

NODAL SOLUTION

Y STEP=1 SUB =8 Z X TIME=1 SEQV (AVG) DMX =.001836 SMN =.164E+07 SMX =.157E+09

MN

X

MX

MX

MN

630721

.232E+08

.458E+08

.684E+08

.910E+08

.114E+09

.136E+09

.159E+09

.181E+09

.164E+07

.204E+09

.189E+08

.361E+08

.533E+08

.706E+08

.878E+08

.105E+09

.122E+09

.139E+09

.157E+09

CONNECTOR4

CONNECTOR1

(a) 0 mm

(b) 50 mm

1

1 NODAL SOLUTION

Y STEP=1 SUB =8 Z TIME=1 SEQV (AVG) DMX =.001796 SMN =.325E+07 SMX =.136E+09

NODAL SOLUTION

Y STEP=1 SUB =8 Z TIME=1 SEQV (AVG) DMX =.001469 SMN =.388E+07 SMX =.132E+09

X

MX

MN

X

MX

MN

.325E+07

.180E+08

.328E+08

.475E+08

.623E+08

.770E+08

.918E+08

CONNECTOR2

.107E+09

.121E+09

.136E+09

.388E+07

.181E+08

.324E+08

.466E+08

.609E+08

.752E+08

.894E+08

.104E+09

.118E+09

.132E+09

CONNECTOR3

(c) 100 mm

(d) 200 mm

Fig. 10. Von Mises stress of the longitudinal section of connector with ring-stiffened cylindrical shell (a) 0 mm, (b) 50 mm, (c) 100 mm, (d) 200 mm

According to the FEA results, there are some phenomenon listed as follows. (1) With the ring-stiffened in hollow cylinder, the connector has smaller deformation and von Mises stresses. The maximum displacements of the connector system are 1.949 mm, 1.836 mm, 1.796 mm and 1.469 mm in Figs. 10(a)–(d) respectively, while the maximum von Mises stresses are 204 MPa, 157 MPa, 136 MPa and 132 MPa. Under the same tensile loads, in comparison with the connector bolt with strength structures, the distortion without circular rib is closely 33% larger and stresses are 55% bigger.

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(2) With the same pin diameter and the thickness, the minimum stresses of the three-ring connector without strengthen are smaller than those with ring-stiffened, as seen from Fig. 10. It indicates that the circulate ribs have the ability of load balance to eliminate the stress concentration. (3) The maximum deformation and stresses appear in the area of the gap between double ears and single ear due to the shear failure mode. (4) The strengths analysis of connectors of VLFS shows that the stresses meet the strength requirements if the connector attached with circulate ribs.

5 Conclusions and Discussion Based on the loads calculated by the rigid module flexible connector model, the connector structures of the VLFS are designed to be suitable for the application and service. The mechanical model of the flexible connector was modelled using FEA. Through the comparison of the numerical simulation results, the following conclusions are drawn: (1) According to the load prediction and the preliminary design dimensions, the diameter of the bolt and the thickness of the ears could be determined. A proper size can minimize the stress level. Even if a hollow connector is used, as long as the corresponding principal particulars of connectors and the material parameter are optimised, it will also have the effect of reducing the stresses but not significantly increasing the weight. (2) The thickness of the hollow bolt should be met with the thickness of the single and double ears. If the thickness of the pin is too small, the structures will fail easily under extreme deformation and stress. Of course, if the thickness of the bolt is sufficiently large, the system of the connector will have large masses. (3) Using a flexible connector with a rubber ring can reduce the stiffness without significantly increasing the stresses. Therefore, surrounded by the rubber, the hollow connectors not only satisfy the dynamic characteristics of very large floating body, but also achieve the safety and reliability of the connection between modules. Acknowledgements. The work was supported by the Ministry of Science and Technology with the research project (Grant No. 2013CB036100 and No.2017YFB0202702), the National Natural Science Foundation of China, PR China (51809241) and the Ministry of Industry and Information Technology with the research project in the fields of high-tech ships ([2016]22).

References 1. Wu, Y.S.: Hydroelasticity of Floating Bodies. Brunel University, UK (1984) 2. Ertekin, R.C., Riggs, H.R., Che, X.L., et al.: Efficient methods for hydroelastic analysis of very large floating structure. J. Ship Res. 37(1), 58–76 (1993) 3. Riggs, H.R., Ertekin, R.C., Mills, T.R.J.: Wave-induced response of a 5-module mobile offshore base. Mar. Struct. 13, 217–232 (2000) 4. Yu, L.: Study on dynamic responses of connectors of mobile offshore base. Shanghai Jiaotong University (2004)

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5. Qi, E.R., Song, H., Lu, Y.: Study on dynamic response of flexible connectors of very large floating structures. J. Ship Mech. s1, 209–217 (2017) 6. Qi, E.R., Liu, C., Xia, J.S., Lu, Y., et al.: Experimental study of functional simulation for flexible connectors of very large floating structures. J. Ship Mech. 19(10), 1245–1254 (2015) 7. Zhang, B., Qi, E.R., Lu, Y.: Influence of module number on dynamic responses of very large mobile offshore base. J. Ship Mech. 17, 49–55 (2013) 8. Liu, C., Qi, E.R., Lu, Y.: Dynamic response of connectors of very large floating structures under shallow draft. J. Ship Mech. 5, 581–590 (2014) 9. Xu, D.L., Xia, S.Y., Zhang, H.C., Wu, Y.S.: Adaptive optimal control of multi–modular floating platforms in random seas. Nonlinear Dynamic, 1–14 (2017) 10. Zhu, X., Liu, C., Qi, E.R., et al.: Design and finite element analysis for very large floating base connector. J. Ship Mech. 11, 1361–1366 (2014) 11. Lu, Y., Qi, E.R., Liu, C.: Strength assessment on the strength of flexible connectors with several stiffness of very large floating structures. In: Proceedings of the China Steel Construction Society Ocean Steel Structure Branch Association Conference, pp. 117–122 (2015) 12. Lu, Y., Qi, E.R., Liu, C.: Strength assessment on the keypoint of connectors of very large floating structures. In: Proceedings of the China Steel Construction Society Ocean Steel Structure Branch Association Conference, pp. 80–85 (2016) 13. Zhou, Y., Wang, Y., Lu, Y., Qi, E.R.: Strength of connectors with varied stiffness for very large floating structures. In: Third International Conference on Safety and Reliability of Ships, Offshore & Subsea Structures, Wuhan (2018)

Structural Design and Strength Estimation of Energy Saving Y-FIN by Using Finite Element Method Wen-Huai Tsou1(&), Pai-Chen Guan2, Wen-Hsuan Chang2, and Chao-Jieh Chen2

2

1 Department of Design, CSBC Corporation, 3 Jhonggang Road, Siaogang District, Kaohsiung, Taiwan [email protected] Department of Systems Engineering and Naval Architecture, National Taiwan Ocean University, Keelung, Taiwan

Abstract. In this paper, we study the strength design of energy saving Y-fin installed on the stern of 1800TEU container vessel (in front of the propeller). The main purpose of this device is to recover rotational kinetic energy and enhance propulsion efficiency. The structural design processes and the strength estimation of Y-fin are presented. The commercial Finite Element Method (FEM) software Abaqus is used as the analyzing tool for the structural analysis of ultimate strength and fatigue strength. During the analysis, we found that at the joint between the casting and plate, the inconsistency of degree-of-freedoms between shell and solid element may cause non-physical stress concentration. This is later prevented by introducing “shell to solid coupling” in Abaqus to link these two types of elements. This function is using the penalty method to constraint the translational and rotational degree-of-freedoms at the joints of shell and solid elements. To save analyzing time, we first perform the dynamic analysis to verify that the structural response is closer to quasi-static condition. Then the rest of the fatigue analysis can be simplified to several static analysis with peak hydraulic force as boundary conditions. In fatigue life estimation, the S-N curves for different welding methods from ABS (American Bureau of Shipping) rule are considered. We use the Goodman and Gerber fatigue rules to correct the stress magnitude, then we employ the Miner’s rule to estimate fatigue accumulation. The result shows that the fatigue life is from 840 days to 2471 days. Further structural optimization will be performed in the future to extend the life time of the energy saving device in the future. Keywords: Energy saving Y-fin Method


Ship structural design
Finite Element

1 Introduction As the fuel price rising and the Energy Efficiency Design Index (EEDI) regulations becoming stricter for the environmental protection, designing the high efficiency ships has become the major task of ship yard nowadays. One of the useful way to achieve the © Springer Nature Singapore Pte Ltd. 2021 T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 209–231, 2021. https://doi.org/10.1007/978-981-15-4672-3_13

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energy efficient ship is to develop proper energy saving devices. The mechanical type energy saving devices are mostly installed on the upstream of the propeller or on rudder to improve the propulsion efficiency. There are many types of energy saving devices, and each type causes different improvement on propulsion efficiency. H.J. Prins et al. [1] has compared the propulsion efficiency of bulk carrier equipped with pre-swirl stator and rudder bulb. The pre-swirl stator (PSS or fin) could improve the propulsion efficiency about 6.8% under 16 knots. However, most of the articles only discuss the energy efficiency performance about the PSS. The detail discussion regarding the structural safety assessment of PSS are usually absent. In 2016, DongBeom Lee et al. [2] brought up a procedure of structural analysis which contains the calculation of nonlinear lifting force and lifting moment on PSS. The nonlinear hydrodynamic loading is caused by the pitch motion of large ship under extreme sea condition. To save analyzing time, the neural network is adopted to learn the results from CFD (Computational Fluid Dynamics) to predict the hydrodynamic loading under extreme condition by statistical point of view. However, this method could only discuss under one ship speed and the same PSS geometry. The drastic increase of CFD analysis is still needed after change the ship speed or geometry of PSS. To solve the problem mentioned above, Han-Baek Ju et al. [3] brought up an explicit formula to calculate nonlinear lifting force and moment directly instead of neural network. In this paper, we focus on the practical structural design and analysis of PSS (or Yfin in this paper). The energy saving Y-fin is installed on the stern of 1800TEU container vessel. The structural design process and the strength estimation of Y-fin are as follows: 1. Initial Y-fin structural design: To understand the deformation and structural feature of Y-fin. The Y-fin is to be designed and analyzed alone in this stage. 2. Full model structural design: The Y-fin is attached to the stern. This model is to check if Y-fin installation affects the original ship hull structure. Some combinations of the Y-fin and stern structural design are considered to study the health of structure after the Y-fin installation. The main structural members of Y-fin and stern are plate, stiffener and casting. Shell, beam and solid elements are the main FE elements used to describe these structures. At the joint between the casting and plate, solid and shell elements cannot directly connect with each other due to the inconsistency of degree of freedoms defined in the elements. There are several methods to connect shell and solid element. Some of the methods are Penalty method, Nitche’s method and constraint degree of freedoms. Penalty method and Nitche’s method are also adopted to impose the essential boundary condition in meshfree methods [4]. These methods force the boundary value to fit the essential boundary conditions by numerical way. In this paper, we use the function “shell to solid coupling” in Abaqus to link these two types of elements. This function directly connects the common degree of freedoms between shell and solid element. Then, we also check the continuity of the displacement field to confirm if the connections are properly done by this function. The dynamic pressure distribution is calculated from Computational Fluid Dynamics (CFD) under forward motion and wave loading condition. To save analysis time, the dynamic structural analysis is simplified since natural frequency between structure and the surrounding fluid are greatly different from each other. The frequency

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of fluid loading is very low when comparing with structural vibration frequencies. Therefore, the structural response is closer to quasi-static condition. We performed one dynamic analysis to proof this point of view and the following analyses are done with the sense of quasi-static conditions to save computation time. In fatigue life estimation, the S-N curves for different wielding methods from ABS (American Bureau of Shipping) rule are considered. The eleven kinds of wave conditions are chosen from the North Atlantic scatter diagram. The stress range and average stress are also calculated under each wave loading conditions. The Goodman and Gerber fatigue rules are adopted to revise the stress and accumulate the fatigue damage by Miner’s rule. The optimized structural design is decided in the full model structural design stage and the fatigue life is estimated by the process mentioned above.

2 Model Description and Design Process The geometries of Y-fin and partial 1800TEU container vessel stern are shown in Fig. 1. After the optimization design by CFD, the Y-fins are installed on both sides: two on the portside and one on the starboard side. The position of Y-fins are fixed in the following structural design: By changing the internal structure of Y-fin and stern, several design cases are listed. Then compare the analysis results and find optimized structural design case. The structural design process is as follow. 1. Initial Y-fin structural design: In this design stage, the main purpose is to design the internal structure of Y-fin. Then understand feature of Y-fin structure during the design process and offer the design reference to the next design stage. 2. Full model structural design: In this design stage, the Y-fins are installed on the stern according to the original design position. We try to find the optimized design by changing the structural position and scantling to reduce the maximum von-Mises stress. Finally, the fatigue life of optimized design case is estimated.

Stern Y-fin 2

Y-fin 1

z x y Y-fin 3 Fig. 1. Geometry of Y-fin and 1800TEU container ship stern

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3 Initial Y-Fin Structural Design In this design stage, we want to study whether the Y-fin structure itself could sustain all type of wave loads. Therefore, only single Y-fin and the fix root are built. According to the CFD results, the maximum pressure occurs on the Y-fin 2 shown in Fig. 1. Therefore, in the following design processes, we will use the pressure distribution on Y-fin 2 as the critical loading example. There are three design cases in this section. They are “Original Y-fin structural design” (Type A), “Adjusting the position of longitudinal walls design” (Type B) and “Adjust the thickness of Y-fin shell design” (Type C). 3.1

Original Y-Fin Structural Design (Type A)

The original structural design of Y-fin is shown in Fig. 2. The transverse walls are spaced equally in the longitudinal direction to resist the torsion of Y-fin. The two longitudinal walls are located at 1/4L and 1/2L from leading edge (where L is the chord length). The thickness of the stiffened walls is 14 mm and the thickness of the other panels is 12 mm. The stress distribution under forward motion and forward motion with wave loading condition are shown in Fig. 3 and Fig. 4 respectively. According to the results, the high stress region is near the root of the Y-fin. The stress distribution of Y-fin is similar to cantilever beam. Therefore, enlarging the thickness of Y-fin shell could provide more bending strength. However, the longitudinal walls are closer to the leading edge of the Y-fin in x-direction, this causes the insufficient structural strength near the trailing edge. 3.2

Adjust the Position of Longitudinal Walls Design (Type B)

According to the results of Type A, the structural strength near the trailing edge is insufficient due to the arrangement of longitudinal walls. Therefore we shift their locations to 1/3L and 2/3L from the leading edge as shown in Fig. 5. The thickness is the same as in design type A. The stress distribution under forward motion and forward motion with wave loading condition are shown in Fig. 6 and Fig. 7 respectively. The results show that the high stress region changes to the middle part of the chord length and the maximum stress is smaller than the “Type A” results. We can summarize that the high stress region at the root of the Y-fin could be reduced by the equally spaced longitudinal walls. 3.3

Adjust the Thickness of Y-Fin Shell Design (Type C)

From the results above, we could observe that the high stress region is on the root of Yfin and the stress changes from high to low along the y-direction (shown in Fig. 2), which agrees with the cantilever beam behavior. To reduce the maximum stress, we change the thickness of Y-fin shell from the design “Type B” and the new thickness arrangement is shown in Fig. 8. The thickness of shell on the root increases to 14 mm and the stress distribution under forward motion is shown in Fig. 9. Compare the result

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with “Type B”, we can find that the maximum stress reduces about 14% and the stress distribution remains the same during the thickness change. 3.4

Summarize Initial Y-Fin Structural Design

We have finished the initial Y-fin structural design and summarize two points of view as follows: • The high stress region on the root of Y-fin could be reduced by adjusting the position of longitudinal walls in x-direction. • Increasing shell thickness near the root of Y-fin could reduce the maximum stress without changing the stress distribution.

Transverse walls Longitudinal walls

y x 1/2L

1/4L

1/4L

Fig. 2. Inner structural arrangement of Y-fin: Type A

MAX. 108.2MPa

Fig. 3. von-Mises stress distribution under forward motion: Type A

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MAX. 148.6MPa

Fig. 4. von-Mises stress distribution under forward motion with wave loading: Type A

y x

1/3L

1/3L

1/3L

Fig. 5. Inner structural arrangement of Y-fin: Type B

MAX. 85.5MPa

Fig. 6. Von-Mises stress distribution under forward motion: Type B

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MAX. 129.4MPa

Fig. 7. Von-Mises stress distribution under forward motion with wave loading: Type B

12mm 14mm

Fig. 8. Thickness of Y-fin: Type C

MAX. 73.2MPa

Fig. 9. Von-Mises stress distribution under forward motion: Type C

4 Full-Model Structural Design From the results of Y-fin structural design, the structural feature of Y-fin is close to cantilever beam. Adjusting the arrangement of longitudinal walls could change the high stress region and enlarge the shell thickness at the root of Y-fin could lower the maximum stress.

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In this design stage, we first consider the transient analysis to study the dynamic effect of the wave loading. The inconsistency of degree of freedoms between shell and solid element is also coupled by adopting the function in Abaqus. Y-fin is installed on the position of the original design by CFD. There are three design cases by changing the structural arrangement and scantling. One is “Original full model analysis” (Case A), another is “Keep structural continuity design” (Case B) and the other is “Enlarge structural scantling design” (Case C). Then analyze the maximum von-Mises stress of those design cases and the design criteria is shown in Table 1. Finally, choose optimum structural design case and estimate the fatigue life. Table 1. Criteria of steel (Yielding stress, MPa) Mesh size 800 mm 400 mm 266 mm 200 mm 160 mm T by T

4.1

M.S. 235 249 263 277 294 352

H32 298 315 333 351 372 441

H36 320 339 358 378 400 480

Shell and Solid Element Coupling

Y-fin is installed on the boundary of the hull shell and casting as shown in Fig. 10. The shell and solid elements are employed to describe the shell and casting parting respectively. However, these two type of elements cannot connect with each other directly due to the inconsistency of the degree of freedoms. In this paper, we use the function “shell to solid coupling” in Abaqus to link these two types of elements. The main concept of the coupling is to introduce the constraint between the nodal displacement of the solid elements with the nodal displacement (in-plane and out-of-plane translation) and rotation (along two in-plane axis and out-of-plane axis) of the shell/plate element by introducing the penalty method. (The penalty method uses a modified Galerkin weak form to force the solution to weakly satisfies the optimum solution that best satisfy both the minimum energy condition and the constraint conditions.). The results without and with this functional link are shown in Fig. 11 and Fig. 12, respectively. The maximum stress occurs on the connection of shell and solid element in Fig. 11 and occurs on the shell element in Fig. 12. The stress concentration in Fig. 11 is because of the inconsistency of the degree of freedoms. We will employ this link function to connect the casting to both side shell and the Y-fin in the following studies.

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Side shell (shell element)

Casting (solid element) z x

Y-fin boundary

Boundary of shell and solid element Fig. 10. Boundary of shell and solid element on stern model

MAX. 235.4MPa

z x Fig. 11. von-Mises stress: Uncouple shell and solid element

MAX. 205MPa

z x Fig. 12. von-Mises stress: Couple shell and solid element

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Study of Dynamic Effect

The structures under dynamic loading conditions should be studied with the dynamic analysis to understand the possible resonance and the fatigue life estimation. However the dynamic analysis usually takes a long analyzing time. The external loading of Y-fin is obtained by considering one of the general operating environment and modeled by the CFD simulation. The loading condition is shown in Fig. 13. For the static analysis, we tested the instance when the maximum loading happens. Then we compare the maximum stress calculated by the dynamic analysis and static analysis in Table 2. The dynamic and static results are very close and the maximum difference is about 4.1%. From this validation, we can know that the structural response of Y-fin is close to quasi static condition during the full loading history and we can simplify the dynamic analysis to static analysis in the following structural design.

Y-fin2 Y-fin3 Y-fin1

Fig. 13. Hydrodynamic force history under ship forward motion with extreme wave loading Table 2. Dynamic and static analysis result comparison

Dynamic Static Error

4.3

Y-fin maximum stress 148.6 MPa 148.7 MPa 0.6%

Y-fin minimum stress 117.4 MPa 112.6 MPa 4.1%

Hull maximum stress 258.1 MPa 258.1 MPa 0.0%

Hull minimum stress 193.8 MPa 193.8 MPa 0.0%

Full Model Analysis

In this section, the Y-fin is attached to the stern model and apply pressure under ship forward with wave loading condition. The material of stern and Y-fin are mild steel and the mesh size is T by T (about 15 mm). The corresponding criteria is in Table 1.There are three design cases in this section and the von-Mises stress of each cases is shown. The optimized case is chosen to estimate the fatigue life in next section.

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Case A: Original Full Model Design In design case A, we keep original design of Y-fin and hull, therefore the internal structure of Y-fin and hull is discontinuity (as shown in Fig. 14). The thickness of Y-fin shell is 12 mm, internal structure of Y-fin is 14 mm and the hull shell is 20 mm. By applying the pressure mentioned above, the result is shown in Fig. 15 and the maximum stress is about 462.2 MPa. The stress distribution is discontinuity at the connection between Y-fin and hull. The maximum stress concentrates on Y-fin due to the structural discontinuity and it is possible to cause Y-fin yielding even fracture. To reduce the stress concentration and make the stress distributes continuously. There are two methods might improve these conditions: • Keep structural continuity between Y-fin and hull structure. • Thicken the shell of Y-fin and hull to strength the structure. The following two design cases will follow the methods mentioned above.

Hull structure Y-fin structure

z x y

Casting Fig. 14. Structural arrangement of Y-fin and Hull: Case A

Y-fin: MAX. 404.6MPa

Hull: MAX. 462.2MPa

Fig. 15. Von-Mises stress on Y-fin and hull: Case A

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Case B: Keep Structural Continuity Design In this design case, we move the internal structure of Y-fin to keep the structural continuity. The structure arrangement is shown in Fig. 16 and the thickness of Y-fin shell is 12 mm, internal structure of Y-fin is 14 mm and the hull shell is 20 mm. By applying the pressure mentioned above, the result is shown in Fig. 17. According to the result, the stress becomes more continuous than “Case A” and the concentration point moves to the joint between internal structure of Y-fin and hull structure. However, the bending strength of hull shell is not enough to transfer the stress from Y-fin to hull and the maximum stress is about 568.4 MPa which is higher than “Case A”. The structural strength is obviously lower than the “Case A”. Therefore, the key point is to strengthen the hull shell in the next design.

Hull structure Y-fin structure

z x y

Casting Fig. 16. Structural arrangement of Y-fin and Hull: Case B

Y-fin: MAX. 502MPa

Hull: MAX. 568MPa

Fig. 17. Von-Mises stress on Y-fin and hull: Case B

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Case C: Enlarge Structural Scantling Design According to the result of “Case B”, keeping the structural continuity could not strengthen the Y-fin structure effectively. In this section, we thicken the Y-fin shell to 20 mm and part of hull shell to 40 mm. To prevent the strength discontinuity on hull shell, the thick plate region increases bigger than the Y-fin attached area and opens working hole on hull shell for the construction requirement. The structure arrangement is shown in Fig. 18, where the red region is thick plate area. The results is shown in Fig. 19. The maximum stress is about 258 MPa on hull shell and this value is lower than criteria 352 MPa. These results show that the maximum stress point transfer from Y-fin to hull shell. Then compare to the “Case A”, the maximum stress decreases about 35%.

Thicken plate area

z x Fig. 18. Structural arrangement: Case C

Y-fin: MAX. 258.1MPa

Hull: MAX. 148.7MPa

Fig. 19. Von-Mises stress on Y-fin and hull: Case C

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Fig. 20. Convergence plot of case C

In this case, the element sensitive analysis is also checked and the convergence plot is as Fig. 20. When the mesh size decreases from 80 mm to 15 mm (this element size is the smallest size refers from class rule), the slope reduces from 0.68 to 0.48. This result shows that the stress is approaching to converge. Summarizing above, the design of “Case C” could transfer the maximum stress point and reduce the maximum stress effectively. This design case is adopted to estimate the fatigue life in the next section.

5 Fatigue Life Estimation By the analysis mentioned above, we could confirm that the Y-fin and hull structure will not fracture at the extreme loading condition. However, the fatigue due to the cycling load might also cause the Y-fin fracture during the normal operating. The classification society usually only specifies rules to prevent the fatigue damage of hull structure. And the Y-fin belongs to appendage according to the class category. There is no clear rule to check fatigue life for appendage. In this section, we borrow the fatigue life estimation procedure based on the ABS (American Bureau of Shipping) rule for the container vessel [5]. The wave loading condition is decided from the north Atlantic wave scatter diagram and the fatigue damage caused by different wave loading condition is checked. 5.1

ABS Fatigue Rule

The cumulative damage equation in ABS fatigue rule is as follow. D¼

X J ni i N i

ð1Þ

where D is the cumulative damage, ni is the number of fatigue damage in each loading case and Ni is total fatigue life of each cycling load.

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In ABS rule, the S-N curve is decided in the following Eqs. (2) and (3), and each curve is consisted by two segments with different slope. The corresponding coefficients are shown in Table 3 where N is the number of loading cycles under stress range SB, K1, K2 and K3 are constants, r is the standard deviation of log N, m is the inverse slope of S-N curve and fq is the stress range at joint of two segments. The S-N curves plotted according to the following equations are shown in Fig. 21. Each S-N curve represents different type of structural geometry. The F2 type of S-N curve is chosen for the following fatigue analysis. The first segment of S-N curve is as follow logðN Þ ¼ logðK2 Þ  mlogðSB Þ

ð2Þ

Where logðK2 Þ ¼ logðK1 Þ  2r The second segment of S-N curve is as follow logðN Þ ¼ logðK3 Þ  ðm þ 2ÞlogðSB Þ

ð3Þ

  Where logðK3 Þ ¼ logðK2 Þ  2log fq Table 3. Coefficients for different curve class [5]

5.2

Wave Condition Refers from North Atlantic Scatter Diagram

In normal operating condition, the Y-fin is under static and dynamic pressure. The static pressure is caused by still water pressure, self-weight and lifting force in ship forward motion. The dynamic force is caused by wave which is a periodic loading and might cause fatigue damage during the whole ship life. In fatigue life estimation, the waves with different wave height and period should be considered. According to the requirement of trade routes, the North Atlantic scatter diagram is chosen to check the probability of different waves as shown in Fig. 22. The horizontal axis is wave period and the vertical axis is wave height, the happening times per one hundred thousand times for each wave condition is shown in the chart. However it is inefficiency to

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Fig. 21. S-N curve for different welding direction [5]

calculate all wave loading conditions. To increase the analysis effectiveness, the eleven kinds of representative waves are chosen (the red frame in Fig. 22). The fatigue life estimation is to be calculated under the eleven wave loading conditions in the next section.

Fig. 22. North Atlantic scatter diagram

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5.3

225

Fatigue Damage Accumulation

In this section, the fatigue damage and fatigue life is to be calculated. The pressure history under the representative wave loading conditions is calculated by CFD. The maximum and minimum stress during each wave loading history is calculated by dynamic analysis and computes the fatigue damage. Take wave with period 7.5 s and height 6.5 m (for abbreviating, T75H65) as an example. The stress distribution under maximum and minimum pressure load are shown in Fig. 23 and Fig. 24 respectively. The maximum and minimum stress are 187.6 MPa and 166.8 MPa. The stress range under T75H65 wave load is 10.4 MPa and the average stress is 177.2 MPa. The stress range and average stress for the remaining wave conditions are shown in Table 4. To apply S-N curve from ABS standard, the stress range ra with non-zero average stress rm should be revised to standard stress range r-1 with zero average stress. There are two common methods to revise the stress range in general metal fatigue analysis. One is Goodman relation as follow. r1 ¼ 

ra 1  rrmb



ð4Þ

Where rb is the ultimate tensile stress of the material and it is about 460 MPa in this paper. According to the equation, Goodman relation assumes that the relation between stress range and average stress is linear. The other is Gerber relation as follow. ra r1 ¼   2  1  rrmb

ð5Þ

Gerber relation assumes that the relation between stress range and average stress is quadratic. These two relations show that the standard stress range r-1 increases as the average stress rm increases. The stress range revised by Goodman and Gerber is shown in Table 5. Due to the finite analysis time, it is difficult to get all the pressure history under the wave conditions chosen by blue frame in Fig. 22. However the eleven wave conditions are extreme loading. To make the analysis result more reasonable, the other wave conditions should be considered and the standard stress range r-1 under these wave conditions are interpolated by eleven data points. The relation between the wave height and standard stress range in the wave period 8.5 s could be obtained by fitting the data point in Table 5. The curve fitting results are shown in Fig. 25 and Fig. 26. Then the standard stress range r-1 in the wave period 8.5 s could be interpolated by the curve and linear interpolation in wave height direction. The interpolated results are shown in Table 6, Table 7 and values in red color are data points from Table 5. By comparing with S-N curve Fig. 21, the total fatigue life Ni of each standard stress range could be obtained. Assume the total times of stress cycling load is X and the probability of each wave condition is Pi. By applying the coefficients to the cumulative damage equation mentioned above, the equation could be rewritten as follow.

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D¼

X X  Pi

ð6Þ

Ni

Then the total fatigue life (T) could be obtained by the following equation. X T¼ X  Pi  Ti

ð7Þ

The total fatigue life of Goodman and Gerber are 840 and 2471 days. The real fatigue life will locate between those two values in the most of papers and the Goodman relation is conservative way to estimate the fatigue life.

Table 4. Stress range under different wave loading condition (MPa)

T75H65

T85H05

T85H25

T85H45

T85H65

T85H85

σMAX

187.6

171.5

182.0

184.2

195.9

204.2

σmin σrange

166.8 20.8

157.8 13.7

163.5 18.5

155.8 28.4

158.2 37.7

179.9 24.3 12.15

10.4

6.85

9.25

14.2

18.85

T95H65

T105H65

T115H65

T125H65

T133H65

σMAX

231.5

191.6

242.1

245.7

251.1

σmin

133.0

141.3

137.7

157.2

162.3

σrange σa

98.5 49.25

50.3 25.15

104.4 52.2

88.5 44.25

88.8 44.4

σa

Table 5. Standard stress range under different wave loading condition (MPa)

σm σa Goodman Gerber σm σa Goodman Gerber

T75H65 177.2 10.4 16.91 12.21 T95H65 182.25 49.25 81.56 58.42

T85H05 164.65 6.85 10.66 7.85 T105H65 166.45 25.15 39.41 28.93

T85H25 172.75 9.25 14.81 10.76 T115H65 189.9 52.2 88.9 62.92

T85H45 170 14.2 22.52 16.44 T125H65 201.45 44.25 78.72 54.75

T85H65 177.05 18.85 30.64 22.12 T133H65 206.7 44.4 80.63 55.63

T85H85 192.05 12.15 20.85 14.71

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Table 6. Interpolated standard stress range r–1: Goodman

Tz(s) Hs(m) 0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5

7.5

8.5

9.5

10.5

11.5

12.5

13.5

5.89 7.00 8.18 9.97 12.43 15.07 16.89 16.37 11.45

10.67 12.69 14.81 18.06 22.51 27.30 30.60 29.65 20.74

28.40 33.78 39.42 48.07 59.93 72.67 81.46 78.93 55.20

13.72 16.32 19.05 23.23 28.95 35.11 39.36 38.14 26.67

30.95 36.81 42.97 52.39 65.31 79.20 88.78 86.02 60.16

27.41 32.60 38.05 46.40 57.84 70.14 78.62 76.18 53.28

28.07 33.39 38.97 47.52 59.24 71.83 80.52 78.02 54.56

Table 7. Interpolated standard stress range r–1: Gerber

Tz(s)

7.5

8.5

9.5

10.5

11.5

12.5

13.5

0.5

4.34

7.86

20.74

10.27

22.34

19.44

19.75

1.5

5.08

9.20

24.28

12.03

26.16

22.76

23.13

2.5

5.94

10.77

28.43

14.08

30.62

26.65

27.08

3.5

7.28

13.20

34.84

17.26

37.52

32.65

33.17

4.5

9.08

16.45

43.44

21.52

46.79

40.71

41.37

5.5

10.97

19.88

52.49

26.00

56.54

49.20

49.99

6.5

12.23

22.17

58.52

28.99

63.03

54.85

55.73

7.5

11.78

21.35

56.37

27.92

60.71

52.83

53.68

8.5

8.19

14.83

39.16

19.40

42.17

36.70

37.29

Hs(m)

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MAX. 187.6MPa

Fig. 23. Von-Mises stress under maximum wave loading: T75H65

MAX. 166.8MPa

Fig. 24. Von-Mises stress under minimum wave loading: T75H65

Fig. 25. Stress range and wave height: Goodman

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Fig. 26. Stress range and wave height: Gerber

6 Conclusion In this paper, the energy saving device Y-fin is installed on the stern of 1800TEU container vessel. There are two on the port side, one on the starboard side and these positions are the optimized design calculated by CFD. There are two main items during the design process. One is the single Y-fin structural design and the other is the full model structural design. In the single Y-fin structural design stage, there are three design cases. One is the “Original Y-fin structural design” (Type A), another is the “Adjust the position of longitudinal walls design” (Type B) and the other is the “Adjust the thickness of Y-fin shell design” (Type C). In design type A, the thickness of stiffened walls are 14 mm and the others are 12 mm. The position of each longitudinal wall is 1/4L and 1/2L (where L is the chord length of Y-fin). The analysis result shows that there is high stress region near the root of Y-fin and the stress distribution is close to cantilever beam. Therefore thicken the shell of Yfin could provide more bending strength. Due to the longitudinal walls arrange close to leading edge of Y-fin, the structural strength near the trailing edge is insufficient. The design type B changes the position of each longitudinal wall to 1/3L, 2/3L and remains the same thickness as design type A. The maximum stress is similar to the design type A but the high stress region transfers to the middle of the chord length. Therefore, the high stress region could be changed by adjusting the position of longitudinal walls. To lower the maximum stress, the design type C keeps the arrangement of stiffened walls and thickens the shell of Y-fin from 12 mm to 14 mm. The maximum stress reduces about 14% and the stress distribution is unchanged. In summary, the position of longitudinal walls concerns about high stress region and the thickness of Y-fin shell concerns about the value of maximum stress. In full model design stage, the Y-fin is installed to the stern at the original design position by CFD. To simply the analysis process, the dynamic result and the static result are compared and the maximum error is about 4.1%. This shows that the dynamic result is similar to static result and could simplify from dynamic analysis to static analysis in the following analysis. The element types used in the full model are beam, shell and solid element. Due to the inconsistency of degree of freedom between the shell and solid element. The “shell

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to solid” function in Abaqus is adopted to connect these two types of element. The results show that the unreasonable stress concentration could be eliminated by using this function. There are three design cases in full model design stage. One is “Original full model design” (Case A), another is “Keep structural continuity design” (Case B) and the other is “Enlarge structural scantly design” (Case C). In the design case A, the structural arrangement between Y-fin and stern keeps the original design. However, the structural discontinuity causes the stress concentration on shell of Y-fin and the maximum stress is over the criteria. To prevent the problem mentioned above, the structure keeps continue in design case B. However, keeping structure continuity could not transfer stress from Y-fin to stern effectively for the reason of insufficient strength at the joint of Y-fin and stern. In design case C, the structural arrangement follows design case A. Then thicken the shell of stern to 40 mm and open working hole according to construction requirement. This design case could transfer stress from Y-fin to stern and lower the maximum stress. The element sensitivity analysis shows that the stress is approaching to converge. In this stage, the design case C is the optimized design case. In fatigue life estimation stage, the ABS fatigue rules for container vessel is adopted. Choose eleven wave conditions from north Atlantic scatter diagram. Calculate the stress range by dynamic analysis and revise the stress range by Goodman and Gerber relation. Then interpolate the standard stress range under other wave conditions and find the maximum loading times under each wave condition by comparing with the S-N curve. Finally, accumulate the fatigue damage by Miner’s rule to estimate the fatigue life. The fatigue life is 840 days (about two years) by Goodman and 2471 days (about six years) by Gerber. In summary, the structure feature of Y-fin is similar to cantilever. Therefore, the structural arrangement and scantling near the root is the key point. After enlarge the scantling at the connection between Y-fin and stern. The stress could transfer from Yfin to stern and lower the maximum stress. The fatigue life is about two to six years and is much shorter than the vessel design life (about 25 years). The reasons might be the overestimation of wave conditions, inappropriate S-N curve and non-optimal structural design. Therefore more accurate choice of wave conditions, appropriate S-N curve and finding more optimized structural design is required in the future.

References 1. Prins, H.J., et al.: Green retrofitting through optimisation of hull-propulsion interaction-GRIP. J. Transp. Res. Procedia 14, 1591–1600 (2016). https://doi.org/10.1016/j.trpro.2016.05.124 2. Lee, D.B., Jang, B.-S., Kim, H.J.: Development of procedure for structural safety assessment of energy saving device subjected to nonlinear hydrodynamic load. J. Ocean Eng. 116, 165– 183 (2016). https://doi.org/10.1016/j.oceaneng.2016.02.038 3. Han-Beak, J., Jang, B.-S., Lee, D.B., Kim, H.-J., Park, C.-K.: A simplified structural safety assessment of a fin-typed energy saving devices subjected to nonlinear hydrodynamic load. J. Ocean Eng. 149, 245–259 (2018). https://doi.org/10.1016/j.oceaneng.2017.12.022

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4. Fernández-Méndez, S., Huerta, A.: Imposing essential boundary conditions in mesh-free methods. J. Comput. Methods Appl. Mech. Eng. 193(12–14), 1257–1275 (2003). https://doi. org/10.1016/j.cma.2003.12.019 5. ABS Rules for Building and Classing Steel Vessels 2007 Part 5C Specfic Vessel Types (Chapters 1–6)

Sustainable Sandwich Panels for Use in Ship Superstructures Jeanne Blanchard(B)

and Adam Sobey

University of Southampton, Burgess Road, Southampton SO16 7QF, UK [email protected]

Abstract. Ship superstructures are commonly manufactured from steel but composite sandwich structures could be an alternative leading to significant weight savings. In addition to these weight savings using sustainable materials could reduce the environmental impact of ship production, operation and recyclability in shipping. However, these materials must be capable of equal performance to those that are currently used. Comparing sandwich panels is complex as there are many objectives, stress to strength ratio, sustainability, cost and mass, and variables such as the skin and core materials and thicknesses. Due to this complexity Genetic Algorithms are used to compare potential designs, providing different material selections for different combinations of objectives. The comparison between different Genetic Algorithms demonstrates that HEIA is the most effective algorithm but with all of the algorithms having equivalent performance on these problems. The optimisation provides a set of 716 feasible designs, with balsa being the most popular core but with feasible solutions split between the flax, carbon and glass skins. Keywords: Sandwich structures · Sustainable materials · Optimisation

1 Introduction Large ships have historically been manufactured from steel or aluminum alloys. However, composite materials are becoming more attractive to the marine industry for a number of reasons including: high specific properties, the ability to tailor mechanical properties, no corrosion, low maintenance costs and excellent resistance to fatigue failure. This potential has led to an increase in their utilisation in ship applications and many companies wishing to explore the use of composite materials for the superstructures of new and refitted passenger ships, with the potential for a number of benefits for the industry. A typical ship superstructure is presented in Fig. 1. There are currently over 100,000 merchant ships operating globally and despite their efficiency the sheer volume of trade means that global shipping is responsible for 3.1% of anthropogenic CO2 emissions [1]. Weight saving is therefore an increasingly crucial aspect for the shipping industry as it leads to energy efficient vessels and reduced fuel costs. Superstructures of passenger ships and ferries are increasing in size and the possible weight savings if manufactured from composite sandwich materials is estimated to © Springer Nature Singapore Pte Ltd. 2021 T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 232–245, 2021. https://doi.org/10.1007/978-981-15-4672-3_14

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Fig. 1. Superstructure of a passenger ship made of steel with a possible composite sandwich materials replacement

be up to 70% [2]. In addition, reducing the superstructure weight has a positive effect on the stability performance. Composite sandwich panels are manufactured with stiff skins made of fibre reinforced laminates and a cheaper, low density, core material that provides thickness without much additional weight. The use of sandwich structures increases the available options for material use, including using natural fibres for the skin which will make the materials more sustainable and remove the dangerous fumes when burnt, opening up an increased range of applications for these materials. However, even if the good mechanical properties of natural fibres are highlighted in the literature, studies investigating their possible applications are limited. The cores used in ship superstructures are often made of synthetic materials, which have replaced the traditional balsa wood cores, but this goes against the general global trend for increasing sustainability. It is therefore important to investigate the feasibility of replacing synthetic cores with equivalent sustainable options, including reverting to balsa. 3D printed materials may also be an efficient substitute especially as they can benefit from an increased flexibility in internal geometries, where the cell size and type can be changed to give the same stiffness and strength but with reduced mass, which must also be balanced against the external properties, such as the thickness of the core and skins. This optimisation allows the full benefits of the sandwich configuration to be realised. However, with so many variables finding the optimum design can be difficult and an efficient optimisation tool is required. Genetic algorithms are widely used across many industries for this type of early investigation, where there is a large potential search space. Furthermore, genetic algorithms allow the search to be performed rapidly and most importantly without bias. This paper therefore presents an investigation into the potential for sustainable material use in ship superstructures. This will be through a topological optimisation of sandwich panels made of sustainable cores and natural fibre skins in comparison to conventional materials. The optimisation is performed with a range of leading Genetic Algorithms including MOEA/D-MSF, U-NSGA-III, HEIA, and cMLSGA. The stress to strength ratio of each structure together with the cost, sustainability and mass are optimised.

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2 Methodology 2.1 Materials Properties A range of materials available on the market for manufacturing composite sandwich structures are selected for this comparison. Composite sandwich structures are composed of stiff skins made of fibre reinforced laminates with a light core in the middle to increase the thickness and therefore rigidity of the structure for minimum additional weight. Therefore the replacement of conventional materials with sustainable options is investigated for both the fibres used in the skin and the core materials in order to determine if both the skin and the core materials can be replaced with more sustainable options or only one part of the sandwich structures. Sustainable materials such as flax fibres, cork, balsa and 3D printed PLA cores are compared with widely used conventional materials such as E-glass and carbon fibres with PVC foam core. For the skin of composite sandwich structures, E-glass fibres are the most commonly used reinforcement with carbon fibres used for more expensive applications with higher structural requirement but natural fibres are proposed as a sustainable replacement to E-glass in the literature. Among all the natural fibres available in the market, flax fibres are selected for this study for their high mechanical properties and wide availability. The mechanical properties of the fibre reinforced laminates selected for the skin of the sandwich structures are presented in Table 1. Table 1. Materials properties of the sandwich constituents Material

Flax/epoxy

E-glass/epoxy

Carbon/epoxy

Longitudinal Young’s modulus E1 MPa

22,800

43,000

135,000

Transverse Young’s modulus E2 MPa

4520

8000

10,000

Shear modulus G12 (MPa)

1960

4000

5000

Poisson’s ratio ν12

0.43

0.28

0.3

Tensile strength (MPa)

318

1100

1500

Compressive strength (MPa)

136

900

1200

Density (g/cm3 )

1.17

1.73

1.60

References

[3]

[4]

[5]

Conventional materials used for the core of composite sandwich structures such as PVC foam are also compared with different sustainable options: cork, balsa wood and 3D printed PLA honeycomb cores. Cork is one of the most sustainable materials available for the core and relatively cheap. Balsa wood has been used in the past by the marine industry and reverting to balsa can be an option for sustainable sandwich structures. 3D printed PLA honeycomb cores are recent materials with a number of advantages such as the ability to tailor the geometry or the density of the materials for specific load cases or applications. The mechanical properties of the different core materials selected for this study are presented in Table 2.

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Table 2. Materials properties of the core Material

Cork

Balsa

PLA

Airex PVC foam

Tensile modulus (MPa)

19

3921

40

28

Shear modulus (MPa)

44

157

29.55

13

Poisson’s ratio

0.3

0.3

0.3

0.3

Strength (MPa)

1.2

12.67

1.1

0.45

Density (kg/m3 )

120

151

80

40

References

[6]

[7]

[8]

[9]

In addition to the mechanical properties of the materials which is a key aspect for ship superstructures, a sustainability criteria is included into the optimisation to consider the environmental impact of each material. The sustainability of each material is assessed based on the embodied energy during primary production and the data are presented in Table 3. Table 3. Embodied energy, primary production and density of the different constituents of the sandwich structures Material

Embodied energy (MJ/kg)

References

Flax fibres

10

[10]

E-glass fibres

62.2

[10]

Carbon fibres

380

[10]

Cork core

4

[11]

Balsa core

13

[10]

PLA core

49

[10]

Airex PVC foam core

96

[10]

In addition to the environmental impact of the different materials, the economical aspect needs to be considered. The price of the different materials used to manufacture the sandwich structure are presented in Table 4. 2.2 Third Order Shear deformation theory The stresses on the sandwich structures are obtained with the Third Order Shear Deformation Theory from Reddy [17]. The Third Order Shear Deformation Theory can predicts more accurate stress and strain distributions and interlaminar stress distributions.

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J. Blanchard and A. Sobey Table 4. Price of the different constituents of the sandwich structure

Material

Price (Euros/m3 )

References

Flax fibre reinforced epoxy composites

49.8

[12]

E-glass fibre reinforced epoxy composites

34.84

[12]

Carbon fibre reinforced epoxy composites

75.5

[12] and [13]

Cork core

2420

[14]

Balsa core

2813

[15]

3D printed PLA core

22,375

–

AIREX foam PVC core

5690

[16]

The displacement is expanded to the cubic term in the thickness coordinates and therefore the shear strains of thick laminate or sandwich panels is more accurate than the widely used Classical Laminate plate theory [17]. The Third Order Shear Deformation Theory is based on the same assumptions than the Classical Laminate Plate Theory or the First Order Shear Deformation Theory except that the assumption on the straightness and normality of a transverse after deformation is relaxed by expanding the displacements (u,v,w) as cubic functions of the thickness coordinate. The strains are calculated with Eqs. (1) and (2) ⎧ ⎫ ⎧ ⎫ ⎫ ⎧ (0) ⎫ ⎧ (1) (3) ⎪ ⎪ ⎨ εxx ⎪ ⎬ ⎬ ⎨ εxx ⎪ ⎬ ⎨ εxx ⎪ ⎨ εxx ⎬ ⎪ (0) (1) (3) + z εyy + z 3 εyy (1) εyy = εyy ⎪ ⎪ ⎭ ⎪ ⎩ ⎩ (1) ⎪ ⎭ ⎭ ⎩ (3) ⎪ ⎭ ⎩ (0) ⎪ γxy γxy γxy γxy   (0) (2) γyz γyz 2 γyz = (2) (0) + z (2) γxz γxz γxz The stress resultants are related to the strains with Eq. (3) and (4) ⎧ ⎫ ⎡ ⎤⎧   ⎫ [A] [B] [E] ⎨ ε0  ⎬ ⎨ [N ] ⎬ [M ] = ⎣ [B] [D] [F] ⎦ ε1  ⎩ ⎭ ⎩ 3 ⎭ ε [P] [E] [F] [H ]      0 

Q [A] [D] γ  = γ2 [R] [D] [F]

(3)

(4)

With the matrixes, Ai,j , Bi,j, , Di,j , Ei,j, Fi,j , and Hi,j , calculated with Eqs. (5) to (10) Aij =

N k=1

Qijk (Zk+1 − Zk )

1 N Qk (Z 2 − Zk2 ) k=1 ij k+1 2 1 N Qk (Z 3 − Zk3 ) Dij = k=1 ij k+1 3 Bij =

(5) (6) (7)

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237

Eij =

1 N Qk (Z 4 − Zk4 ) k=1 ij k+1 4

(8)

Fij =

1 N Qk (Z 5 − Zk5 ) k=1 ij k+1 5

(9)

Hij =

1 N Qk (Z 7 − Zk7 ) k=1 ij k+1 7

(10)

A longitudinal bending moment equals to 30,551 Nm was applied on the sandwich panels as a representative load seen in ship superstructures. 2.3 Optimisation and Genetic Algorithms Comparison As stated by Sobey et al. [18], there is no free lunch and different Genetic Algorithms have different performance for a given problem. Specialist solvers are created to solve a specific problem more accurately but their accuracy can be reduced on different problems whereas general solvers are capable of solving a larger range of problems with reduced accuracy in comparison to specialist solvers, Sobey et al. [18]. Therefore, it is important to compare different Genetic Algorithms on this particular example to access their accuracy. 5 different Genetic Algorithms are used to optimise the sandwich structures. An extensive review of the different Genetic Algorithms is presented by Sobey et al. [18] and a summary of the different algorithms used in this study is presented hereafter. MOEA/D-MSF developed by Jiang et al. [19] as improved variants of MOEA/D for imbalanced and unconstrained problems. HEIA developed by Lin et al. [20] is an algorithm that demonstrates high proficiency across a diverse set of problems and is a general solver with a bias towards convergence. U-NSGA-III developed by Seada and Deb [21] “as the many-objective universal variant of NSGA-II. cMLSGA developed by Grudniewski and Sobey [22] “as the best general solver and an algorithm developed to be similar to the original Genetic Algorithm representing a solver which is still common in the marine literature.” The tests are performed over 15 separate runs, and the termination criterion is set at 300,000 function evaluations for each run on the 24 m and 50 m case studies and 50,000 for the 140 m simulations as some solvers perform the analysis slowly with the high number of objectives. The results are compared using the Hyper Volume (HV) and Inverted Generational Distance (IGD) indicators, as between them they provide comprehensive information on the convergence, accuracy, and diversity of the obtained solutions. HV is the measure of volume of the objective space between a predefined reference point and the obtained solutions which has a stronger focus on the diversity and edge points and can be calculated according to [23], where higher values indicates the better results. IGD is the measurement of the average Euclidean distance between the points in a true Pareto Optimal Fronts and the closest solution in the obtained set of solutions, and is to be minimised, where 0 indicates the perfect convergence. This metric has stronger emphasis on the convergence and uniformity of the points and for practical problems can be calculated according to [24]. Different population sizes have been evaluated and 1800 is selected as the best value for cMLSGA and 1000 for other algorithms. The crossover and mutation rates are set as 1 and 0.08 respectively, and the

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rest of algorithm-specific operation parameters are set as in the original publications. For all cases the objective normalisation strategy taken from [25] is used. The objectives of the optimisation are to minimise the maximum stress to strength ratio on the panel, to minimise the sustainability index, to minimise the mass and to minimise the cost of the final structure. The maximum stress obtained for each layer of the sandwich panel with the Third Order ShearDdeformation Theory is divided by the strength of the given material as presented in Table 1 and Table 2. The sustainability of the sandwich structure is calculated for a panel of 1 m2 based on the sustainability index presented in Table 3 for the different constituents of the sandwich structures with Eq. (11) Sustainability = Masscore ∗ Sustainabilitycore + Massskin ∗ Sustainabilityskin

(11)

The mass of the sandwich structures is calculated for a panel of 1m2 with Eq. (12) and the density of each material presented in Tables 1 and 2. Total Mass = thicknesscore ∗ densitycore + thicknessskin ∗ densityskin

(12)

The cost of the sandwich structures is calculated for a panel of 1m2 with Eq. (13) Cost = Masscore ∗ Costcore + Massskin ∗ Costskin

(13)

Table 5. Design parameters for the optimisation Parameters

Lower boundary

Upper Boundary

Fibre reinforced laminate skin properties

–

–

Core properties

–

–

Thickness of the skin (mm)

1

20

Thickness of the core (mm)

1

90

The design parameters for the optimisation are presented in Table 5. The skin properties are varying depending on the material selected between flax, E-glass or carbon fibre reinforced composites. The core properties are varying depending on the core selected between cork, balsa, 3D printed PLA or Airex PVC foam.

3 Results The solutions with stresses higher than the strength of the material are removed and the final pareto front provides 716 feasible designs across the different objectives. Of the final results 665 use a balsa core, 21 of cork, 5 of 3D printed PLA and 25 of the PVC foam. The superior mechanical properties for the balsa play a large role in the selection of this material in almost all of the scenarios. The results for the mass in comparison to

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239

the stress to strength ratio are shown in Fig. 2. The balsa core provides the solutions with the lowest stresses, though many of these are over designed with some having stress to strength ratios approaching 0. However, for the low mass solutions a number of different options become more important, these options would seem to be the most realistic. For the panels with a stress to strength ratio of between 0.1 and 0.4, which is assumed to be a more realistic value and similar to that used by classification societies, the selection is focused on cork and the PVA foam. Despite its prevalence in the literature the PLA core is rarely selected. 1.2 1

Mass

0.8 Cork

0.6

Balsa

0.4

PLA PVC Foam

0.2 0 0

0.2

0.4

0.6

0.8

1

1.2

Stress/Strength Fig. 2. Core material selection for stress in comparison to mass

For the skin 208 designs with flax are selected, 349 of glass and 159 of carbon. In this case the mechanical stresses are low and therefore any of the materials were able to provide solutions in combination with the core. In addition the carbon has both a high cost and low sustainability compared to the other materials. The carbon provides low specific properties and when it was selected it was at the expense of the core, forming a structure that was virtually monolithic and resulting in exceedingly high safety factors of up to 1000. The correlations between the mass, cost and sustainability are strong at large values, and it is only when these values are low that there is a noticeable difference between them. In the same manner as for the core, the carbon dominates at the low stresses, shown in Fig. 3, with a number of overengineered designs. When the stress to strength ratio is limited to 0.1 to 0.4 then half of the designs are found to be carbon, with high skin thicknesses and the lowest possible core. The range of thicknesses for the different skins and cores selected for the feasible designs are presented in Table 6. For the cork, PLA and PVC foam cores, the thickness is reduced by the optimisation process to the minimum value of 1 mm demonstrating that the addition of the core materials do not provide much benefits for the given objective compared to a monolithic composite; the cost is too high or the material properties are too low. However for the

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J. Blanchard and A. Sobey

1.2 1

Mass

0.8 Flax 0.6

E-glass

0.4

Carbon

0.2 0 0

0.2

0.4

0.6

0.8

1

1.2

Stress/Strength Fig. 3. Skin material selection for stress in comparison to mass

Table 6. Range of thicknesses for the different skins and cores selected for the feasible designs Materials

Minimum thickness (mm)

Maximum thickness (mm)

Mode thickness (mm)

Flax skin

6.1

20.0

20.0

E-glass skin

6.3

20.0

20.0

Carbon skin

2.9

20.0

20.0

Cork core

1

1

1

Balsa core

1

18

-

PLA core

1

1

1

Airex PVC foam core

1

1

1

balsa core, the thicknesses vary from the minimum thickness of 1 mm up to 18 mm with thicknesses in between equally selected for the different feasible designs. This study is conducted with a stress/strength criteria rather than a deflection criteria for which the results are likely to be different with thick cores adding stiffness for minimal additional mass. The results for the feasible designs with a stress to strength ratio ranging from 10 to 40%, to consider safety factors representative of the marine industry, are presented in Appendix A. Table A.1 shows that the minimisation of the mass of the structure also reduces the cost and environmental impact, as these factors are linked. Since the sustainability and cost in operation have not been investigated in this study and need further study, the mass is likely to be even more important to reduce the cost and environmental impact of the ship. The thickness of all of the panels is set to 1 mm in these cases, showing that the addition of a deflection criteria is vital with the stress criteria being easy to match. The skin made little difference to the final objectives, with all materials being capable of providing high or low values for each. The core makes the

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241

largest difference, with Airex panels being the least expensive and PLA the most. This is related to the mass, with Airex being the lightest, though Balsa provided the heaviest panels. Again sustainability was heavily related to mass but provides more of a mix of core types but with balsa providing the most environmentally sustainable core type. The different algorithms are compared in terms of their performance on the sandwich optimisation problem with results presented in Table 7. In this case HEIA is the top performing algorithm, providing the best convergence and uniformity while providing the second best diversity of the population. However, the separation between the algorithms is not significant and this indicates that this is a simple problem to solve, with any of the algorithms being equivalent. Table 7. Rankings for the different algorithms (green boxes indicate that algorithms with stronger convergence mechanisms)

Ranking 1

2

3

4

IGD

HEIA 0.2712 0.0056 U-NSGA-III 0.2889 0.0124 cMLSGA 0.2917 0.0253 MOEA/D-MSF 0.3401 0.0161

HV

MOEA/D-MSF 0.9565 0.0085 HEIA 0.9543 0.0043 U-NSGA-III 0.9389 0.0137 cMLSGA 0.9384 0.0241

4 Limitations Even if the utilisation of composite materials is rapidly growing in a number of sectors including for aerospace applications, steel is still the most commonly used material in the maritime industry. The majority of ship superstructures are manufactured with steel for a number of reasons including their well-defined mechanical properties even if sandwich structures might represent an interesting alternative in terms of weight saving. However, a number of obstacles are preventing the utilisation of sandwich structures for ships and some of these limitations are identified in this study. The load case used in this study is considered realistic for a ship superstructure but limited data are available on the loads encounter by superstructures during operation which are smaller than the hull. It is unrealistic and expensive to model with a Finite Element Analysis an entire ship to determine the load in the superstructure and determine the best material. More studies are required for more complex load cases to determine if sandwich structures can replace steel in ship superstructures.

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Environmental awareness is becoming more important in industry and the requirement for more sustainable options is increasing. However, it is difficult to determine how sustainable a structure is. Sustainability data are difficult to obtain for a large majority of the materials and the limited references available bring limited confidence in the values. Furthermore, the sustainability of a structure needs to be calculated during the production process but also needs to be assessed over its life time in terms of weight saving and therefore reduction in fuel consumption of the ship. The total life of the structure also needs to be considered and the recycling or end of life options available for the given materials. This study is a first attempt with the data available in the literature but more research is required. The material properties selected for this study and used in the Third Order Shear Deformation Theory represent values commonly seen in the literature but data are limited for less common properties or newer materials such as flax fibres, 3D printed PLA core and cork.

5 Conclusion This paper demonstrates the potential positive impact of an increased utilisation of composite panels for ship superstructures and determines if recently developed sustainable materials can be successfully used in ship manufacturing. The geometrically optimised panels will lead to further weight reductions of the superstructures, provide sustainable structural materials while improving the ship’s fuel efficiency and the stability. The analysis shows that balsa is the most popular core, but that there are a number of combinations of carbon, flax and glass skins that provide optimal results depending on the different objectives. The best optimisation algorithm in this case is HEIA but with limited separation in the performance of the different algorithms which indicates the relatively easy search space compared to many examples in composite structures.

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Appendix a Table A.1: Feasible design for a stress to strength ratio between 10 and 40%

Stress/ strength (%)

Cost (€)

Mass (Kg)

Sustaina bility (MJ)

Skin

0.10

0.011

0.039

0.173

E-glass

Skin thickness (mm)

Core

Core Thickness (mm)

17.9

Cork

1.08 1.17

0.13

0.015

0.042

0.152

E-glass

16.3

Cork

0.15

0.008

0.034

0.134

Carbon

15.1

Cork

1.00

0.19

0.018

0.044

0.115

Carbon

13.5

Cork

1.25

0.24

0.008

0.032

0.093

Carbon

12.1

Cork

1.00 1.42

0.27

0.024

0.050

0.089

E-glass

11.5

Cork

0.27

0.017

0.041

0.086

E-glass

11.5

Cork

1.22

0.33

0.021

0.045

0.074

Carbon

10.5

Cork

1.32

0.12

0.037

0.051

0.069

Flax

7.3

Balsa

1.19

5.9

Balsa

1.57 1.00

0.18

0.058

0.071

0.066

Carbon

0.14

0.206

0.019

0.212

Flax

15.2

PLA

0.16

0.233

0.022

0.211

E-glass

14.4

PLA

1.11

0.24

0.212

0.018

0.170

Carbon

12.0

PLA

1.02

11.3

PLA

1.00 1.05

0.27

0.206

0.017

0.159

Carbon

0.12

0.002

0.006

0.231

Flax

16.4

Airex

0.13

0.001

0.006

0.229

Carbon

16.1

Airex

1.04

0.14

0.002

0.006

0.221

Carbon

15.6

Airex

1.05

15.5

Airex

1.00 1.11

0.14

0.0001

0.005

0.215

E-glass

0.17

0.003

0.006

0.209

Carbon

14.4

Airex

0.18

0.0001

0.004

0.193

Carbon

13.9

Airex

1.00

0.21

9.0E-05

0.003

0.178

Carbon

12.8

Airex

1.00

12.5

Airex

1.00 1.00

0.22

8.7E-05

0.003

0.174

Carbon

0.25

7.5E-05

0.003

0.165

Carbon

11.9

Airex

0.29

6.3E-05

0.002

0.154

Flax

11.1

Airex

1.00

0.31

0.0005

0.002

0.150

Carbon

10.7

Airex

1.02

10.4

Airex

1.00 1.00 1.01

0.33

5.2E-05

0.002

0.145

Flax

0.35

4.8E-05

0.002

0.141

Flax

10.1

Airex

0.37

0.0002

0.002

0.138

Flax

9.9

Airex

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References 1. Smith, T.: Third IMO greenhouse gas study. In: Marine Environment Protection Committee 67th Session, Agenda item 6 (2014) 2. Karatzas, V., Hjornet, N.K., Kristensen, H.O., Berggreen, C., Jense, J.J.: The effects of the operating condition of a passenger ship retroffited with a composite superstructure. In: Proceedings of PRADS, Copenhagen, Denmark (2016) 3. Liang, S., Gning, P.B., Guillaumat, L.: Quasi-static behaviour and damage assessment of flax/epoxy composites. Mater. Des. 67, 344–353 (2015) 4. https://www.hexcel.com. https://www.hexcel.com/user_area/content_media/raw/Prepreg_T echnology.pdf?w=500&w=500. Accessed 29 Mar 2019 5. Performance composites. https://www.performance-composites.com/carbonfibre/mechanica lproperties_2.asp 6. https://amorimcorkcomposites.com/media/4201/mds-nl10.pdf 7. Le Duigou, A., Deux, J.M., Davies, P., Baley, C.: PLLA/Flax Mat/Balsa bio-sandwich manufacture and mechanical properties. Appl. Compos. Mater. 18(5), 421–438 (2011) 8. https://www.econcore.com/en/products-applications/bio-based-panels 9. https://www.3accorematerials.com/uploads/images/TDS-AIREX-T10-E-Rev_4_external_1 106.pdf 10. Ashby, M.F.: Materials and the Environment Eco-Informed Material Choice. Elsevier, Oxford (2013) 11. Bojic, M., Miletic, M., Bojic, L.: Optimization of thermal insulation to achieve energy savings in low energy house (refurbishment). Energy Convers. Manag. 84, 681–690 (2014) 12. Cihan, M., Sobey, A.J., Blake, J.I.R.: Mechanical and dynamic performance of woven flax/Eglass hybrid composites. Compos. Sci. Technol. 172, 36–42 (2019) 13. https://www.castrocompositesshop.com/gb/fibre-reinforcements/1220-tejido-de-carbonounidireccional-0-de-24-k-y-300-gm2.html 14. https://www.castrocompositesshop.com/gb/92-cork 15. https://www.castrocompositesshop.com/gb/91-balsa-wood 16. https://www.ecfibreglasssupplies.co.uk/product/airex-c70-75-structural-foam-10mm 17. Reddy, J.: Mechanics of Laminated Composite Plates and Shells Theory and analysis, 2nd edn. CRC Press, Boca Raton (2004) 18. Sobey, A., Blanchard, J., Grudniewski, P., Savasta, T.: There’s no free lunch: a study of genetic algortihm use in maritime applications. In: 18th Conference on Computer Applications and Information Technology in the Maritime Industries, Tullamore (2019) 19. Jiang, S., Yang, S., Wang, Y., Liu, X.: Scalarizing functions in decomposition-based multiobjective evolutionary algorithms. IEEE Trans. Evol. Comput. 22, 296–313 (2018) 20. Lin, Q., et al.: A hybrid evolutionary immune algorithm for multiobjective optimization problems. IEEE Trans. Evol. Comput. 20, 645–665 (2016) 21. Seada, H., Deb, K.: U-NSGA-III: a unified evolutionary optimization procedure for single, multiple, and many objectives, pp. 34–49. Springer, Heidelberg (2015) 22. Grudniewski, P.A., Sobey, A.J.: cMLSGA: co-evolutionary multi-level selection genetic algorithm. IEEE Evol. Comput. (under review) 23. While, L., Bradstreet, L., Barone, L.: A fast way of calculating exact hypervolumes. IEEE Trans. Evol. Comput. 16, 86–95 (2012) 24. Wang, Z., Bai, J., Sobey, A.J., Xiong, J., Shenoi, R.A.: Optimal design of triaxial weave fabric composites under tension. Compos. Struct. 201, 616–624 (2018) 25. Zhang, Q., Li, H.: MOEA/D: A multiobjective evolutionary aAlgorithm based on decomposition. IEEE Trans. Evol. Comput. 11(6), 712–731 (2007)

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26. Tseng, L.Y., Chen, C.: Multiple trajectory search for unconstrained/constrained multiobjective optimization. In: Trondheim (2009) 27. Li, M., Yang, S., Member, S., Liu, X.: Pareto or non-pareto: bi-criterion evolution in multiobjective optimization. IEEE Trans. Evol. Comput. 20, 645–665 (2016) 28. Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NLSGA-II. IEEE Trans. Evol. Comput. 6, 182–197 (2002) 29. Zitzler, E., Simon, K.: Indicator-based selection in multiobjective search. In: Birmingham (2004)

Design Principle of Wheel Patch Loaded Ship Plating Ling Zhu1,2 , Wei Cai1,2(B) , Yuansong He3 , and Youjun Wu3 1 Key Laboratory of High Performance Ship Technology of Ministry of Education, School of

Transportation, Wuhan University of Technology, Wuhan, People’s Republic of China [email protected] 2 Departments of Naval Architecture, Ocean and Structural Engineering, School of Transportation, Wuhan University of Technology, Wuhan, People’s Republic of China 3 Marine Design and Research Institute of China, Shanghai 200011, China

Abstract. Plastic deformation of plates in deck structures under wheel loads of heavy vehicles or helicopters is common in ships and offshore structures, which is a design aspect of significant interests to the designers of ro-ro/cargo ships, naval ships and offshore platforms. Important work on this topic is reviewed in association with recent development and the design principle of wheel patch loaded plating is studied together with the design criteria. The nonlinear response of ship plates is investigated by using nonlinear finite element simulations. A simple formula based on the relationship between the lateral load and permanent deflection of plates is proposed, which compared with the nonlinear FE analysis. The design formula to determine the plating thickness is derived based on the level of allowable permanent set. Comparisons are made with Hughes design formula and Jackson & Frieze design curves, and it shows that the proposed design formula is promising for practical design of ship deck plates. Keywords: Design principle · Plates · Wheel patch load · Plastic design · Permanent set

1 Introduction The deck structures of some ships or offshore platforms are common to be subjected to wheel loads of heavy vehicles or helicopters, as shown in Fig. 1. In some extreme wheel loading, such as the ‘ultimate’ or ‘emergency’ landing loads of helicopter, inertia loads of heavy vehicles from severe motion of the ship, aircraft/vehicles tie-down force, and wind force, some plastic deformations may occur in these deck plates as result of overloading. These permanent deformations will affect the stability and the ultimate strength of the hull deck structures, thus endangering the work performance and safety of ships and offshore platforms. In addition, such permanent deformations will cause trapped water in these areas and might become the source of corrosion. The design of deck structures under wheel patch loading is a significant point for designers since repairing is very complex and costly. The behavior of plates in deck structures under © Springer Nature Singapore Pte Ltd. 2021 T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 246–256, 2021. https://doi.org/10.1007/978-981-15-4672-3_15

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Fig. 1. (a) Ferry; (b) helicopter-carrying naval ship

wheel loads is a design aspect of significant interests to the designers of ro-ro/cargo ships, helicopter-carrying naval ships and offshore platforms. Many investigations on the inelastic deformations of deck plates under wheel patch loading have been conducted by using experimental, numerical and theoretical methods. Jackson and Frieze [1] carried out a series of tests to study the elastic-plastic behavior of a steel deck grillage model under patch loading cells representing wheel loading, and some nondimensional design curves were produced to calculate plating thickness of deck structures under wheel loading based on a given allowable permanent set. Hughes [2–4] presented a formula for the design of steel plates subjected to uniform lateral pressure, and investigated two types of concentrated loads, single location loads and multiple location loads. The design formula of laterally loaded plating under concentrated loads also derived. Konieczny and Bogdaniuk [5] studied the nonlinear response of ship plating subjected to load in form of pressure of vehicle wheels, and proposed a design formula used to determine the thickness of plates based on the serviceability criteria established as an allowable permanent deflection. Shi and Zhu et al. [6] performed a series of experiments to examine the response of clamped steel plates loaded quasistatically by a rigid rectangular indenter. The results of plastic deformations and loading force were obtained in loading and unloading tests, and compared to the numerical calculation results. The Reliability-based design methods applied to design deck structures subjected to wheel patch loading were studied by Omidali and Khedmati [7], and the case of stiffened plate structure subjected to the truck wheel loads was investigated and both probability of failure and reliability index were calculated for different axel load distribution functions. Based on these above works, the nonlinear response of deck plates under wheel patch loads is studied, and some experimental and theoretical works on this topic are reviewed in this paper. Simple formula based on the relationship between the lateral load and permanent deflection of plates is proposed by using numerical calculation results. The design formula to determine the plating thickness are derived based on two levels of allowable permanent set, which are compared with Hughes design formula and Jackson & Frieze design curves.

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2 Design Principle The design principles guiding the design of ship structures, and therefore the criteria used to develop the best designs, have evolved over time. The sustainability concept has been applied to establish the design principles and criteria of ship structures. There are many definitions of sustainability, however it is now widely accepted that the concept has three pillars: economic, societal, and environmental, which was first introduced in the 2009 Design Principles and Criteria Report (from ISSC Committee IV.1) [8]. In the structures design, the best economic, social and environmental principles and criteria are the final developing object, which also have been introduced in the 2012 Design Principles and Criteria Report (from ISSC Committee IV.1) [9]. ● Economic principles and criteria: design has always focused on minimizing the cost of constructing a vessel, and of operating it; Reduce design weight is an important point to minimize the cost. Besides, the design criteria are developed to prevent the costs due to accidental loss of the vessel and cargo. ● Social principles and criteria: the safety of those aboard the vessel, crew and passengers, grew as a concern in the 19th century, eventually being formalized in the first SOLAS convention in 1914. These concerns initially concentrated on the loss of life due to accidents, however during the 20th century an increasing emphasis has been placed on health as well as safety. In the ship lifecycle, some requirements of design principles are proposed to ensure the safety of vessel and human life. ● Environmental principles and criteria: environmental concerns are focused on the damage suffered by nature as a result of accidental pollution from events such as the loss of a tanker. It has been widely recognized that some design principles and criteria should be demanded to minimize or avoid the routinely damage to the environment. Therefore, the economic and social principles and criteria should be developed to direct the deck structures design under wheel patch loading. Initially, the deck structures design used the elastic design method, as shown in Fig. 2. In elastic design method, the allowable maximum stress of plates was the design criterion based on the material yield stress of deck plates with a defined safety factor. Using elastic design method, the design value of thickness of plate will be too large, thus this results in the heavy design weight and high cost of ship. The elastic theory applies in plate design only if the material of plate does not yield. However, the local loads applied to the deck plates which result in the onset of yield do not represent that the plates have withstood the maximum pressure that the plates can support. The plates may withstand a pressure several times greater than this before those fail in any significant way, or before the deformation becomes unacceptably large, which also proposed by Hughes [3]. Considered that the deck plates with small permanent deformations caused by extreme wheel loading still have high structural strength and carrying capacity, the elastic-plastic design method is gradually applied to the design of deck structures under wheel patch loading by defining a maximum allowable permanent set, as shown in Fig. 2. However, while the plastic theory is applied in the deck plating design, some plastic deformations are easy to occur in deck plates under some extreme wheel loading, such as the ‘ultimate’ or ‘emergency’ landing loads of helicopter, inertia loads of heavy vehicles from severe motion of the ship. Therefore, the design criteria established as allowable permanent set, the design loads and other design requirement are quite

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Fig. 2. Design method of deck structures under local patch loading

important in deck structures design, which will determine the structural reliability and safety. There are three key points to establish the design criteria of deck structures under wheel loads, using the elastic-plastic method. ● Feedback of service experience from operators, designers and constructors Some special and use requirements from service experiences from the operators, designers and constructors need to be followed. For example, the allowable permanent set need to follow the flight deck use requirements, such as flatness, which should not affect the helicopter’s landing performance; Such permanent deformations will increase the difficulty in manhandling vehicles or helicopters, besides, those will cause trapped water in these areas and might become the source of corrosion, hence it should be controlled and limited. ● Accidents analysis Collect some accident data and information, such as the design dimensions, accidental loads, accidental consequences, and analyze the causes of accidents. From these accidents, some design principles may be proposed to reducing the probability of accidents. ● Economic and social principles and criteria Focus on minimizing the cost of constructing a vessel, by reduced design dimensions and weight. Besides, the most important point, ensure the safety of vessel and human life, and establish the principles and criteria based on structures safety design.

3 Design Method 3.1 Design Parameters Generally, in the thickness design for deck plates under wheel patch loading, the stiffened plate model is always simplified as single panel with fixed support boundary, since the stiffeners in the side of the panel do not have plastic deformations in the elastic-plastic design method. Besides, compared to the stiffened plate model, the calculated plastic deformations for single panel with fixed support boundary are larger, thus this case will be more dangerous and tend to be conservative. In the deck plate design, there are some

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non-dimensional design parameters defined, such as aspect ratio α, plate slenderness C b , load parameter Qp and allowable permanent set C s , which are also proposed by Jackson and Frieze [1]. Aspect ratio is α = l/s, where l is the plate length, and s is the plate width. Plate slenderness:  s σy Cb = (1) t E where t is the plate thickness, σy is the yield stress, and E is the Young’s modulus. Load parameter: FE Qp =  2 sσy

(2)

where, F is the wheel force, F = pab, p is the wheel print pressure, a is the wheel print length, and b is the wheel print width. Allowable permanent set parameter:  ws E Cs = (3) s σy where the ws is the allowable permanent deformations of plate (Fig. 3).

Fig. 3. Single panel with fixed support boundary under wheel patch loading

3.2 Experimental and Numerical Investigation A series of experiments was performed by Shi and Zhu et al. [6] in order to gain further insight into the quasi-static behavior of ship plates loaded by a rigid rectangular mass which was similarly described to wheel print loads. The plate specimens used in the test were cut from high strength steel sheets with different dimensions. The indenter positioned at the mid-span of specimen had a rectangular face of a × b = 299 mm × 185 mm, whose dimensions were determined based on a typical wheel-on-deck interaction scenario. The quasi-static loading and unloading process, which was applied by the MTS 322 test-frame servo-hydraulic test machine (250 kN) under load control at

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a rate of about 25 kN/min, were carried out to investigate the plate behavior with different thickness and aspect ratios. The overall view of the test set-up was presented in Fig. 4. The numerical simulations also were conducted to study the behavior of plates in the loading and unloading process. The experimental results were compared well with numerical results, as shown in Fig. 5, which showed that the numerical results can predict accurately the permanent deflections and deformation modes (Table 1).

Fig. 4. Experimental set-up with a plate specimen (Shi and Zhu et al. [6])

3.3 Rigid-Plastic Theoretical Analysis Many works on the prediction of plastic capacity for plates subjected to lateral pressure based on yield line theory have been done for the case of fully clamped condition. The yield line theory and corresponding collapse mechanism have been proved to be practical and used for design of ship plating by Kmiecik [10]. The yield line model is used for plates subject to lateral patch loads, which also is applied to International Association of Classification Societies (IACS) rules for polar ship plating design. Many simplified and assumed patch loading mechanism models used for describing the plastic deformation mode of plate under lateral patch loads have been proposed. Shi and Zhu et al. [6] proposed some deformation modes from experimental and numerical observation. Based on the assumed deformation mode, the formula based on the relationship between the lateral load and permanent deflection of plates can be derived. Two general types of deformation model of plate under patch loading, “roof-top” model and “doublediamond” model, were introduced and investigated by Hong and Amdahl [11], and the formula of the plate resistance-deformation relationships was also proposed.

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B8

B6

A6

Experimental Numerical

A8 A4

Force (kN)

200

B4

150

100

50

0 0

4

8

12

16

20

24

28

32

36

40

Displacement (mm)

Fig. 5. Force-displacement curves of the experimental and numerical results with different tests Table 1. Experimental details for all tested plates Test No.

l × s × t (mm)

l/s

s/t

Cb

A4

700 × 350 × 4

2

87.5

3.43

A6

700 × 350 × 6

2

58.3

2.48

A8

700 × 350 × 8

2

43.75

1.93

B4

1200 × 350 × 4

3.428

87.5

3.43

B6

1200 × 350 × 6

3.428

58.3

2.48

B8

1200 × 350 × 8

3.428

43.75

1.93

Based on the assumed “Roof-top” deformation mode of full clamped plate under patch loading, the expression for load-carrying capacity of patch loaded plates was written as [11]: 16M0 Kp Km (4) s2 where, M 0 represents the plastic bending moment capacity of a strip, with unit width and thickness, K p is only a function of patch loading aspect ratio s/a, and the K m represents the membrane effect. Using the same method, based on the improved “Double-diamond” collapse model, combined with the numerical calculations, the formula of load-carrying capacity of patch loaded plates was written as [11]: p=

16M0 · KP · Km · fB · fL (5) s2 where, f B is the correction factor adjusting the resistance in pure bending, and the f L is a reasonable approximation for plate length restriction factor. p=

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In order to be compared with rigid-plastic method and numerical method, two cases are calculated according to Eq. (4), Eq. (5) and NLFEA for the resistance-deformation relationships of plates under lateral patch loading, as shown in Table 2. As shown in Fig. 6, in the small plastic deformations range, the rigid-plastic calculation results and FEM results have large difference. In fact, plastic hinges are gradually forming in the loading process which can be reflected in the FE method. However, in rigid-plastic method, all plastic hinges form in the same time. The principal limitation of the plastic yield line method is that it ignores the elastic-plastic behavior of plate in the loading process which foregoes edge hinge formation. Therefore, the proposed formula derived by using plastic yield line method to calculate small permanent deflections is not acceptable. The FE method is reasonable by using elastic-plastic material, which can reflect and calculate the process of real hinge formation. Table 2. The calculation cases of plates under patch loading l × s × t (mm)

a × b (mm)

Yield stress (MPa)

Young’s modulus (GPa)

➀

2400 × 700 × 14

700 × 700

355

206

➁

1050 × 700 × 14

700 × 700

355

206

7

Case1: Eq.(4) Case1: Eq.(5) Case1: NLFEA Case2: Eq.(4) Case2: Eq.(5) Case2: NLFEA

6

Qp

5 4 3 2 1 0 0.0

0.4

0.8

1.2

1.6

2.0

2.4

wp/t Fig. 6. Resistance predictions for plate according to Eq. (4), Eq. (5) and NLFEA

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4 Design Formula In the elastic-plastic design method, the allowable permanent set should be defined based on the work and safety requirement of deck structures, as shown in Fig. 2. Jackson and Frieze [1] conducted a preliminary survey of flight decks in existing British warships to establish the approximate ranges of the variable involved. In this survey, the allowable permanent set was found that was no more than the maximum design value of C s = 0.6 (corresponding to ws /s = 1/50 for mild steel). Some design curves were also derived by Jackson and Frieze [1] and Hughes [3], based on allowable permanent set C s = 0.1, 0.2, 0.4, 0.6. According to the Polish Register of shipping (PRS) experience collected from ship monitoring, it was established that the allowable permanent deflection C s caused by uniform pressure is 0.17 proposed by Konieczny and Bogdaniuk [5], which was the basic criterion in PRS rules [12] for the standard determination of plate thickness subjected to lateral loads. In this paper, the values of allowable permanent set are assumed as 0.1 and 0.2. Based on the proposed values of allowable permanent set (C s = 0.1, 0.2), the design formulas to determine the plating thickness are derived. According to some basic rigid-plastic formulas, such as Eq. (4), Eq. (5), some similar laws can be concluded, and the function based on the relationship between the lateral load and permanent deflection can be constructed. As the maximum allowable permanent deformation C s will be set, the design formula can be expressed as:   l a b Cs = constant Qp = f ( , , Cb ) · φα (6) s s s where the function φα (l/s) is used for a reasonable approximation for plate length restriction factor. Based on the proposed values of allowable permanent set (C s = 0.1, 0.2), a series of numerical calculations are conducted by using the commercially available FE-package, ABAQUS/standard version 6.14. The validation of numerical simulations has been conducted with experimental tests, introduced in Sect. 3.2. In the numerical simulation, the deck plate is constructed using S4R reduced integration shell elements with five integration points throughout the thickness, and the constraints limiting all degrees of freedom are assigned to the plate edges. The wheel patch load in form of uniformly distributed pressure is applied to the center of deck plates. Massive parameter analysis for nonlinear response of wheel patch loaded deck plating is carried out under different design parameters, such as aspect ratio l/s, wheel loading parameters b/s and a/s, plate slenderness C b and load parameter Qp . Based on these numerical calculations, the plate design thickness formula is fitting according to Eq. (6). The design formula to determine the plating thickness is

  (7) Qp = u + v · 1/Cb2 · φα where, u and v, are the functions of wheel print parameters,a/s and b/s, can be expressed as a 1.45 b u = x1 + x2 + x3 (8) s s

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v=

1 + x4 − x5 b/s





1 a 1.42 x6 − x7 b/s s

255

(9)

In the Eq. (8) and Eq. (9), the corresponding coefficients, x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , are the fitting constant, which are obtained from the fitting results of numerical calculations, as shown in Table 3. Besides, the function φα (l/s) in the Eq. (10), used for a reasonable approximation for plate length restriction factor, can be expressed as

  2 0.975 − 0.104 as + 3.72+0.261 s −2.22 a · sl l/s < 3.428 (10) φα = a s 1 l/s ≥ 3.428 Besides, as shown in Fig. 7, some results calculated by Jackson and Frieze (1980) method, Hughes method and proposed method are compared. The comparison of these results shows that, the results calculated by Hughes are larger than calculations from other methods, and the calculations of Qp ~ C b values according to proposed method are close to Jackson’s design curves and its experimental results, which can prove that the results calculated by proposed formula Eq. (7) give a good trend of variation and Table 3. The corresponding coefficients in the Eq. (8) and Eq. (9) x1

x2

x3

x4

x5

x6

x7

Cs = 0.1

0.0824

−0.0667

0.0589

0.441

−0.239

1.003

−0.463

Cs = 0.2

0.158

0.146

0.111

0.403

−0.229

1.030

−0.535

3.5

Cw=0.1, Jackson & Frieze curves Experimental results (Jackson & Frieze) Cw=0.1, Hughes design formula (1983)

3.0

Cw=0.1, proposed formula Eq.(7)

2.5

Qp

Cw=0.2, Jackson & Frieze curves Cw=0.2, Hughes design formula (1983)

2.0

Cw=0.2, proposed formula Eq.(7) 1.5

1.0

0.5 1.0

1.5

2.0

2.5

3.0

3.5

Cb Fig. 7. The comparisons of Qp ~ C b curves (l/s = 4, a/b = 2, b/s = 1/6)

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accuracy in comparison with experimental data. If the basic parameters of wheel loading pressure, plate dimensions and wheel print dimensions are given, according to Eq. (7), the thickness of deck plate under wheel loading can be calculated.

5 Concluding Remarks The design principle of wheel patch loaded plating is studied together with the design criteria in this paper. The elastic-plastic design method is introduced in this paper which is applied to the thickness design of wheel patch loaded plating. Some experimental, numerical and theoretical works based on the rigid plastic method on this topic are also reviewed in association with recent development. According to the relationship between the lateral load and slenderness parameter of plate, the design formula to determine the plating thickness is proposed based on the level of allowable permanent set, which compared with Hughes’s design formula and Jackson & Frieze design curves. The comparison shows that the proposed design formula is promising for practical design of ship deck plates.

References 1. Jackson, R.I., Frieze, P.A.: Design of deck structures under wheel loads. Trans. RINA 3, 119–144 (1980) 2. Hughes, O.F.: Design of laterally loaded plating-uniform pressure loads. J. Ship Res. 25, 77–89 (1981) 3. Hughes, O.F.: Design of plating under concentrated lateral loads. J. Ship Res. 27(4), 252–264 (1983) 4. Hughes, O.F.: Ship Structural Design (A Rationally-Based, Computer-Aided Optimization Approach). SNAME, Jersey City (1988) 5. Konieczny, L., Bogdaniuk, M.: Design of transversely loaded plating based on allowable permanent set. Mar. Struct. 12(7–8), 497–519 (1999) 6. Shi, S., Zhu, L., Karagiozova, D., Gao, J.: Experimental and numerical analysis of plates quasi-statically loaded by a rectangular indenter. Mar. Struct. 55, 62–77 (2017) 7. Omidali, M., Khedmati, M.R.: Reliability-based design of stiffened plates in ship structures subject to wheel patch loading. Thin-Walled Struct. 127, 416–424 (2018) 8. Moore, W.H., Arai, M., Besse, P., Birmingham, R., Bruenner, E., Chen, Y.Q., Dasgupta, J., FriisHansen, P., Boonstra, H., Hovem, L. et al.: Design principles and criteria. In: Report of Committee IV.1, The 17th International Ship and Offshore Structures Congress (ISSC), Seoul, Korea (2009) 9. Brunner, E., Birmingham, R.W., Byklum, E., Chen, Y., Cheng, Y., Dasgupta, J., Egorov, G., Juhl, J., Kang, B., Karr, D., Kawamura, Y. et al.: Design principles and criteria. In: Report of Committee IV.1, The 18th International Ship and Offshore Structures Congress (ISSC), Rostock, Germany (2012) 10. Kmiecik, M.: Usefulness of the yield line theory in design of ship plating. Mar. Struct. 8(1), 67–79 (1995) 11. Hong, L., Amdahl, J.: Plastic design of laterally patch loaded plates for ships. Mar. Struct. 20(3), 124–142 (2007) 12. Polish Register of Shipping, Rules for the Classification and Construction of Sea-Going Ships, Part II, Hull, Gdansk (1996)

Numerical Simulations of Grounding Scenarios–Benchmark Study on Key Parameters in FEM Modelling Lars Brubak1(B) , Zhiqiang Hu2 , Mihkel Kõrgesaar3 , Ingrid Schipperen4 , and Kristjan Tabri5 1 DNV GL, Maritime Technical Advisory, Structures, Høvik, Norway

[email protected] 2 School of Engineering, Newcastle University, Newcastle upon Tyne, UK 3 Estonian Maritime Academy, Tallinn University of Technology, Tallinn, Estonia 4 TNO, Delft, The Netherlands 5 Tallinn University of Technology, Tallinn, Estonia

Abstract. This paper presents a benchmark study on grounding simulations. The objective was to compare key parameters in modelling techniques and assumptions for finite element (FE) analysis. Five contributors participated in the benchmark study by performing numerical simulations, and results were compared with available results from experimental tests. Good agreement between experimental tests and numerical simulations was achieved which illustrates that non-linear finite element analysis can be used to estimate the damage extent in grounding scenarios. In addition, a sensitivity study was performed in order to investigate the effect of varying different parameters such as friction coefficient and failure strain values. For the numerical results following the guideline DNV GL-RP-C208 [5], a calibration case for mean failure strain was used in order to simulate real experimental tests (with measured yield and ultimate strength), and good agreement was achieved. Keywords: Grounding scenario · Benchmark study · Numerical FE simulations

1 Introduction The present paper presents a benchmark study with five contributors with numerical simulations of grounding scenarios. The numerical results for the grounding force and absorbed energy are compared with those from experimental tests performed in mid1990s at the Naval Surface Warfare Centre in Virginia, USA [3, 4]. In these experimental tests, rupture initiation and energy dissipation were compared for several different designs. The present paper presents an extended benchmark study presented in the ISSC committee for Accidental Limit States [1] where a conventional tanker design was studied. In the extended study, another more unconventional double bottom design is included. © Springer Nature Singapore Pte Ltd. 2021 T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 257–269, 2021. https://doi.org/10.1007/978-981-15-4672-3_16

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The models were analyzed under slightly different conditions based on the different contributors’ experience. In addition, for one of the contributors, the FE models are tested under the conditions as given in the guideline DNV GL-RP-C208 [5]. The main objective of the paper is to compare key parameters in modelling techniques and assumptions for finite element (FE) analysis, and in addition to demonstrate that nonlinear finite element analysis can be used to simulate grounding scenarios. A similar benchmark study was presented by Ringsberg et al. [2] where a collision scenario of an indenter impact with a ship side-shell structure was simulated with non-linear FE analysis. Another related study by Ehlers [11] investigated the influence of the material relation on the accuracy of collision simulations. The outcome from such studies and the present benchmark study can give useful recommendations on different parameters such as mesh size, failure criteria, material data, etc. to be used in numerical simulations with finite element analysis.

2 Model Description 2.1 Experiment and Model Setup In the benchmark study, two barges from the experimental tests were studied, one with a conventional tanker design and one more unconventional design as shown in Fig. 1 and Fig. 2, respectively. The grounding models are a 1:5 scale model of a ship double bottom and is meant to represent a ship in the range 30,000-40,000 displacement ton. Both models include a double bottom, two sides and transverse bulkheads in the front, middle and the aft as shown in the figures. The conventional design is built up with a double bottom with longitudinal stiffeners, transverse frames and one centerline longitudinal frame. In comparison, the unconventional design does not have transverse frames, and the double bottom is built up with many longitudinal frames with longitudinal web stiffeners.

Fig. 1. Model dimensions for the conventional design.

In the experimental tests, the barges were mounted to a rail cart and released down from an incline to build up velocity and accumulate kinetic energy. The total mass of the grounding models and the rail cart was about 220 tons. At the end of the slope was

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Fig. 2. Model dimensions for the unconventional design.

an artificial “rock” made of a concrete cone with steel cladding to simulate a grounding scenario. The shape of the artificial rock was a 90° steel cone with a rounded tip. The model setup is illustrated in Fig. 3 where the barge is mounted with a longitudinal inclination (trim) angle so that when it hits the tip of the rock, the top is in level with the inner bottom. As the barge moves forward, the rock tip will be forced farther up through the barge and if the barge eventually clears the rock completely, the rock tip will be at a penetration height equal to twice the double bottom height. This ensured that the inner bottom is ruptured at some point during the test. More details about the experiments and the geometries of the barges, etc. can be found in the literature [3, 4].

Fig. 3. Model setup where the barge is hitting with an angle.

2.2 Finite Element Models and Assumptions The contributors used both Abaqus [6] and Ls-Dyna [7] which are popular commercial software tools for advanced nonlinear finite element simulations. Software and analysis options for each contributor are given in Table 1. One advantage of using general, commercial software tools is that complex structure can be modelled relatively accurately with curved surfaces, stiffeners, etc. The numerical simulations in both Abaqus and Ls-Dyna were performed using an explicit dynamic solver. All FE models were discretized with shell elements with a relatively fine mesh in order to capture different failure modes such as folding/buckling, tearing, etc.

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Participant

Analysis software

Element type/analysis options

Brubak

Abaqus/Explicit

S4R elements, zero drilling stiffness, double precision calc.

Hu

Ls-dyna/Explicit

Belytschko-Tsay

Kõrgesaar

Abaqus/Explicit

S4R elements, zero drilling stiffness, double precision calc.

Schipperen

Ls-dyna/Explicit

Belytschko-Tsay, 3 integr. points, shear factor 5/6, Flanagan-Belytschko hourglass control

Tabri

Ls-dyna/Explicit

Belytschko-Tsay, 3 integr. points, shear factor 5/6, Flanagan-Belytschko hourglass control

The same geometry of the barge and the rock was used by all the contributors, with static and dynamic friction coefficient equal to 0.35 (except in a sensitivity study). The static and dynamic friction coefficients were kept equal throughout the study, even though the dynamic friction is somewhat lower compared to the static friction. However, as such displacement controlled analyses are often conducted at constant and not at realistic speeds, correct decomposition into static and dynamic friction is hard to achieve. Keeping the friction coefficients equal allows to remove uncertainty due to unphysical speed, but might lead to minor overestimation of resistance due to dynamic friction. The material was ASTM A569 steel with material characteristics from the experiment description as summarized in Table 2 Table 2. Material properties of ASTM A569 [3, 4]. Material parameter

Value

Young’s modulus, E

206 GPa

Poisson’s ratio, ν

0.3

Yield strength, σy

283 MPa

Ultimate strength, σu

345 MPa

Material flow stress, σ 0 = (σ y + σ u )/2

314 MPa

Strain hardening, n (assumed value by Simonsen [4]) 0.22

A common practice for the material modelling is to use a combination of a stepwise linear and a power law with a yield plateau, given in true stress and strain parameters. The part with a power can be described with the following equation  σ0 if ε¯ ≤ εL  σ¯ = n K ε¯ + ε0,eff ε¯ > εL σ0 − yield stress

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ε¯ - equivalent plastic strain εL - plateau strain ε0,eff = (σ0 /K)1/n − εL

(1)

where K is the hardening coefficient and n is the hardening exponent. More details on material modelling for nonlinear finite element calculations can be found in DNV GLRP-C208 [5]. Strain rate effects were neglected since the experiments were carried out at low speed. The material curve by three of the contributors was taken according to the guideline DNV GL-RP-C208 [5] with K = 600 and n = 0.22 (Brubak and Tabri), or a similar description using curve fitting (Kõrgesaar, σ0 = 250, K = 565, εL = 0) as shown in Fig. 4. The material curves by the two other contributors (Hu and Schipperen using a bi-linear material curve) are not shown here, and the corresponding results were carried out in an early phase of benchmark study for the ISSC committee for ALS [1]. Parameters relevant for the study (such as mesh size, failure strain, etc.) were assumed by each of the contributors as given in Table 3. Each of the contributors used different failure criteria formulations to model rupture. With respect to that, a clear distinction is made between damage softening (damage evolution in Abaqus terminology) and damage history. Damage softening refers to softening of the true stress-strain curve whereby element stiffness is gradually reduced once the equivalent plastic strain at the onset of damage is reached. Damage history refers to the way damage accumulates during deformation, see Kõrgesaar (2019). It is used in material fracture communities to model the fracture for any combination of proportional and non-proportional deformation path. Thereby, if the history of deformation is taken into account an integral expression is introduced: ε¯ p D= 0

d ε¯ p ε¯ f (η)

(2)

where ε¯ f (η) is a fracture strain that is a function of stress triaxiality. If damage history is not taken into account, damage is calculated as follows D=

ε¯ p ε¯ f (η)

(3)

In both cases, element is removed from simulation when D = 1. Note that both expressions lead to same failure point when failure strain is independent of stress triaxiality, i.e. a constant equivalent plastic strain.

3 Results for the Conventional Design The barge with a conventional design was presented in the ISSC report for committee V.1 for Accidental Limit States in 2018 [1], and for completeness the main results are presented here. An early phase of this study with nonlinear finite element calculations was also presented by Hareide, Brubak and Pettersen [10].

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L. Brubak et al. Table 3. Assumptions in FE modelling by each contributor

Contributor Failure criterion, damage models, mesh size, etc. Brubak

Conditions as given in the guideline DNV GL-RP-C208 [5] with fracture strain for mean values (triaxiality and element thickness/length depended), For uniaxial tension: necking strain εn = 0.22 (damage initiation in Abaqus) and failure strain εf = 0.266, Abaqus damage evolution, Mesh size = 10 mm (3.3tp )

Hu

Damage model: plastic_ kinematic, Failure strain εf = 0.17, No damage softening, Mesh size = 30 mm

Kõrgesaar

Kõrgesaar (2019), Failure criteria parameters: C1 = 0.25, C2 = 385, C3 = 0.972, KMMC = 680 MPa, nMMC = 0.205; damage history not considered; mesh size = 30 mm. These parameters, except C1 , were calibrated in Kõrgesaar (2019) based on the tensile test data of Karlsson et al. (2009). The effect of changing the parameter C1 in current study lead to 20% reduction in failure strain at uniaxial tension compared to data reported by Karlsson

Schipperen GL criterion, ADN-2015, Element thickness/length dependent failure strain, No damage softening, Mesh size = 30 mm Tabri

Lehmann, et al. (2001), damage history not considered, thru thickness failure strain εthru = 0.148-0.157, No damage softening, Mesh size = 30 mm (for conventional design), 10 mm (for unconventional design)

Fig. 4. Material description according to DNV GL RP C208 and simplified curve using curve fitting

In Fig. 5, the energy and force curves are plotted against the horizontal position (grounding distance) of the barge. These results are obtained for the base case with a friction coefficient equal to 0.35. Figure 6 presents the average curve for force and energy and the confidence interval corresponding to ± a standard deviation. The corrected standard deviation is used which gives an unbiased estimate of the variation and is more representative for small data sets. It can be seen that the agreement between the contributors as well as with the experimental test results is relatively good.

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Fig. 5. Results in terms of reaction force (left) and energy (right) versus grounding distance – base case for conventional design, comparison among contributors

Fig. 6. Average and standard deviation of results in terms of reaction force (left) and energy (right) versus grounding distance – base case for conversional design

In addition, a sensitivity study was presented in the ISSC report, but not repeated here, where the effect of varying different parameters such as friction coefficient, failure strain, mesh size, etc. was studied. A similar sensitivity study for the barge with an unconventional design is presented in Sect. 4.2.

4 Results for the Unconventional Design 4.1 Base Case The barge with an unconventional design was analysed as an extension of the benchmark study in the ISSC report for Accidental Limit States [1] by four of the five authors. Figure 7 presents the energy and force curves plotted against the horizontal position (grounding distance) of the barge. Agreement between experimental results and numerical simulations is relatively good. Figure 8 presents the average curve for force and energy and the confidence interval corresponding to ± a standard deviation. In the experimental tests for the unconventional design, the initial speed was not big enough so that the barge stopped moving just before 6 meters. This is the reason to the sudden drop in the force at the end for the experimental test. In the experimental test [3], it was concluded that with the same setup, a full-scale model will come to rests after 31 meters. Then, all initial kinetic energy is transformed into deformation and friction energy. In comparison, in the numerical simulations with finite element

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Fig. 7. Results in terms of reaction force (left) and energy (right) versus grounding distance – base case for unconventional design, comparison among contributors.

analysis, a constant speed was applied which allows the rock to go through the barge, and eventually the barge clears the rock completely. For the force curve, two large spikes are observed which corresponding to the structure resisting deformations as the rock passes through the two cargo hold bulkheads placed at mid-length and in the aft end. There is an overall increase in the force curves since the rock is moving further into the double hull due to the trim angle. The absorbed energy can be found by integrating the reaction force over the grounding distance. Following the pattern of the force curve, the energy plot shows a slightly higher absorption rate (energy absorption per meter) around the two bulkheads than in the rest of the cargo hold. The differences among the results reflect the different choices made by the analysts. If this set of analyses is considered as representative of the typical dispersion in prediction results by experts in the field, the average value of predictions can be adopted to evaluate the bias between numerical results and experiments, while the standard deviation of numerical results can be assumed as an indication of the uncertainty in predictions. The standard deviation among the predicted forces and energies, expressed as percentage of the corresponding average value, features a mean value (computed on the whole test i.e. averaged over the barge length) of 15% and 5%, respectively. The maximum value of the same quantity is 30% for the force (at 5.7 m displacement in Fig. 8 left) and 6% for the energy (at 7-m displacement in Fig. 8 right). This is summarized in Table 4. Table 4. Average and maximum standard deviation expressed as percentage of average value Average standard deviation

Maximum standard deviation

Force

15%

30%

Energy

5%

6%

In Fig. 9, the relative difference between the computed average value for the base case and the experimental result is plotted versus the displacements for energy and force. It can be seen that the average computed force deviates more from the corresponding

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experimental value (variations of the order of ±20%) than the average energy (±10% in respect to experiments). This is expected, since the energy is an integral value of force.

Fig. 8. Average and standard deviation of results in terms of reaction force (left) and energy (right) versus grounding distance – base case for unconversional design

Fig. 9. Ratio between average of computed values and experimental result for the energy

The damages due to the grounding are shown in Fig. 10 for a displacement around 6 m. The damages simulated by finite element analysis was very similar to what can be read from the pictures of the experimental test [3, 4]. This illustrates that non-linear finite element computations can be used to estimate the damage extent with reasonable accuracy. The total dissipated energy of the grounding scenario can be broken down into several components such as friction, plastic deformation, elastic strain and energy gone into tearing elements apart. In the analyses, most of the energy is dissipated by plastic deformation and friction while the other energy contributions are relatively small. The ratio between the energy going into friction and plastic deformation varies depending on the coefficient of friction. A sensitivity study of the influence of the friction coefficient is presented below.

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4.2 Sensitivity Study Key parameters in FE modelling. The results of the analyses are rather sensitive to different input parameters and a sensitivity study has been carried out by three of the authors (Brubak, Kõrgesaar and Tabri) to investigate to effect of friction coefficients and failure strain values. Typically, failure strain values are also related to the mesh size, plate thickness, etc. For instance, in the guideline DNV GL RP208 [5], there are calibration procedures to assess failure strain values dependent on different parameters. In order to isolate the effect of the different parameters, only one parameter per time is changed in respect to the assumptions for the base model assumptions in Table 3. Effect of friction coefficient. A sensitivity study to investigate the effect of the friction coefficient was carried out for three different values: the base case with 0.35, and two additional friction coefficients 0.3 and 0.4. The other parameters are the same as for the base case. The average of results between the contributors is plotted in terms of force (left) and energy (right) in Fig. 11.

Fig. 10. Damage extent of the unconventional barge (left) and close-up view of the damages (right) after about 6 meters penetration

The friction coefficient will have an effect on the results since the resistance is dependent on this. Normally, a higher friction coefficient means a higher resistance as shown in Fig. 11. It can be noted that the failure mode may change slightly for different values of friction, and actually this can result in a smaller resistance. Consequently, it is very useful to do a sensitivity study with respect to the friction coefficient in order to get an overview of the resistance of the hull. For the numerical computations with the different friction coefficients, the average of the total energy at the end (7 m displacement) and the relative difference to the base case are summarised in Table 5. As mentioned above, the total dissipated impact energy can be broken down into several components such as friction, plastic deformation, elastic strain, etc. In many cases, a target impact energy is required such as collision impacts in Norsok, and in such cases it is conservative to neglect the friction energy although friction is included in the analysis.

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Table 5. Average total energy for numerical computations with different friction coefficients Case

Total average energy [MJ]

Relative difference to base case

Average, Friction = 0.35 (base case)

6.56

1.00

Average, Friction = 0.3

6.08

0.93

Average, Friction = 0.4

6.79

1.04

Effect of failure strain values. A sensitivity study to investigate the effect of different failure strain values was studied in the same manner as for the friction, by varying failure strain values and keeping all other parameters at the same value as for the base case. Results are presented in Fig. 12 and in total three different values of failure strain are used: the one from Table 3 and two more, corresponding to variations of ±20%. In the same manner as for friction, the failure mode may change for different values of failure strain (Fig. 12).

Fig. 11. Sensitivity for friction coefficients (average values among contributors) Results in terms of force (left) and energy (right)

Fig. 12. Sensitivity to failure strain (average values among contributors) Results in terms of force (left) and energy (right)

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For the numerical computations with the different failure strain values, the average of the total energy at the end (7 m displacement) and the relative difference to the base case are summarised in Table 6. Table 6. Average total energy for numerical computations with different failure strain values Case

Total average energy [MJ]

Relative difference to base case

Average, Base case

6.56

1.00

Average, Base case minus 20%

6.19

0.94

Average, Base case plus 20%

7.20

1.10

5 Summary and Conclusion A benchmark study has been performed with nonlinear finite element analyses of two grounding scenarios and comparison with available results from experimental tests. Good agreement between the experimental tests and numerical simulations was achieved which illustrates that non-linear finite element analysis can be used to estimate the damage extent in grounding scenarios. Two different barges were studied, one conventional design and one unconventional deign. The same geometries of the barges and the rock was used by all the contributors, and the adopted friction coefficient was equal to 0.35. The other input for the construction of the FEM models (such as mesh size, failure strain, etc.) were assumed freely by each of the contributors. In addition, a sensitivity study was performed to investigate the effect of varying different parameters such as the friction coefficient and failure strain values. The total computed energy for friction coefficient 0.3 and 0.4 was respectively about 7% lower and 4% higher compared to the base case with friction coefficient 0.35. Similarly, the total energy for assumed failure strain values −20% and + 20% was about 6% lower and 10% higher relative to the base case. It can be noted that the benchmark was not ‘blind’ (i.e. participants knew the experimental results) and this is always considered to improve the quality of predictions. However, it can be concluded that the fairly good agreement between experimental values and numerical predictions coming from this exercise demonstrates that a complex grounding scenario can be effectively simulated with nonlinear finite element analysis. Moreover, numerical results following the guideline DNV GL-RP-C208 [5] was performed by one of the participants and good agreement was achieved. In this guideline, a relatively detailed description on how to perform nonlinear finite element analysis is given, such as material curves, failure strains, etc. Acknowledgement. A special thanks goes to Professor Enrico Rizzuto, the chairman of the ISSC committee V.1 for Accidental Limit States (Rizzuto et al. [1]) and all the members of the committee.

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References 1. Rizzuto, E., Brubak, L., Hu, Z., Kim, G.S., Kõrgesaar, M., Nahshon, K., Nilva, A., Schipperen, I., Stadie-Frohboes, G., Suzuki, K., Tabri, K., Wægter, J.: Accidental limit states - report of ISSC committee V. 1. In: Kaminski, M.L., Rigo, P. (eds.) Proceedings of 20th ISSC, Belgium & Amsterdam, The Netherlands (2018) 2. Ringsberg, J., Amdahl, J., Chen, B.Q., Cho, S-R., Ehlers, S., Hu, Z., Kubiczek, J.M., Kõrgesaar, M., Liu, B., Marinatos, J., Niklas, K., Parunov, J., Quinton, B.W.T., Rudan, S., Samuelides, M., Guedes Soares, C., Tabri, K., Villavicencio, R., Yamada, Y., Yu, Z., Zhang, Z.: MARSTRUCT benchmark study on nonlinear FE simulation of an experiment of an indenter impact with a ship side-shell structure. Mar. Struct. 59, 142–157 (2018) 3. Rodd, J.L.: Observations on conventional and advanced double hull grounding experiments. In: Proceedings of the International Conference on Design and Methodologies for Collision and Grounding Protection of Ships, San Francisco, California (1996) 4. Simonsen, B.C.: Mechanics of ship grounding. Ph.D. thesis, DTU, Technical University of Denmark (1997) 5. DNV GL Recommended Practice: DNVGL-RP-C208, Determination of Structural Capacity by Non-linear Finite Element Analysis Methods, September 2016 6. Abaqus: Dassault Systémes Simulia Corp., Rhode Island, USA (2018) 7. LS-Dyna: Livermore Software Technology Corporation, LS-DYNA software (2018) 8. Zhiqiang, H., Amdahl, J., Hong, L.: Verification of a simplified analytical method for predictions of ship groundings over large contact surfaces by numerical simulations. Mar. Struct. 24, 436–458 (2011) 9. Kõrgesaar, M.: The effect of low stress triaxialities and deformation paths on ductile fracture simulations of large shell structures. Mar. Struct. 63, 45–64 (2019). https://doi.org/10.1016/ j.marstruc.2018.08.004 10. Hareide, O.J., Brubak, L., Pettersen, T.: Modelling ship grounding with Finite Elements. In: Nordic Seminar on Computational Mechanics (NSCM-26), Oslo, Norway (2013) 11. Ehlers, S.: The influence of the material relation on the accuracy of collision simulations. Marine Struct. 23, 462–474 (2010)

Development of Analytical Formulae to Determine the Response of Submerged Composite Plates Subjected to Underwater Explosion Ye Pyae Sone Oo1(B)

, Hervé Le Sourne2 , and Olivier Dorival3

1 GeM Institute (UMR CNRS 6183) – Calcul-Meca, Nantes, France

[email protected] 2 GeM Institute (UMR CNRS 6183) – ICAM, Nantes, France 3 Clément Ader Institute (FRE CNRS 3687) – ICAM, Toulouse, France

Abstract. Closed-form analytical formulae are developed to analyze the bending response of submerged composite rectangular plates subjected to underwater explosions (UNDEX). These explosions are supposed to occur at a sufficiently large stand-off distance so that a uniformly distributed pressure pulse can be applied and the corresponding bubble effects can be ignored. The plate is considered in an air-backed condition. The derivation steps are divided into two main stages. In the first stage, the impulsive velocity due to the interaction of shock wave and structure is determined by using Taylor’s fluid-structure interaction (FSI) formulation while supposing a negligible structural deformation. Transmission of shock waves through the thickness of the plate is considered by assuming the material under uniaxial strain. At the end of the first stage, cavitation is supposed to occur all over the plate. In the second stage, deformation of the plate will commence which is followed by the collapse of the cavitation zone. The corresponding mechanical response of the plate is determined by imposing a simplysupported boundary condition and by applying Lagrangian Energy approach to derive the motion equation, taking into account the water inertial effects. The proposed method is then tested with isotropic (steel) and laminated composite (carbon-fiber/epoxy) plates to analyze for both impulsive velocity and UNDEX responses. The obtained analytical results are compared with those from non-linear finite element explicit code, LS-DYNA. Finally, the advantages and limitations of the present method are evaluated. Keywords: Underwater Explosion (UNDEX) · Fluid-Structure Interaction (FSI) · Analytical formulations · Composite rectangular plate · LS-DYNA

1 Introduction During recent years, composites have been widely used in the fields of civil and military naval structures due to their advantages over conventional materials such as steel. However, there is still a major concern about how these composite structures will respond © Springer Nature Singapore Pte Ltd. 2021 T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 270–290, 2021. https://doi.org/10.1007/978-981-15-4672-3_17

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when subjected to an intense dynamic loading such as underwater explosion or hydrodynamic impacts. These loadings are usually comprised of complicated physical phenomena such as shock wave propagation, fluid-structure interaction, cavitation, and so on. In order to capture all these phenomena accurately, one needs to use complex non-linear finite element codes such as LS-DYNA/USA. Nevertheless, this numerical approach is not only very complicated to set up but can also take a lot of computation time. It is, thus, not well-suited for the preliminary design phases especially when solutions with rapid and reasonable accuracy are only desired. In this context, this research is intended to solve the issue by introducing simplified analytical formulae to predict the response of submerged composite rectangular plates to a reasonable accuracy when subjected to underwater explosive loads. The primary objective is to propose various design solutions for the preliminary design of submarines, surface ships and fast composite boats. The application area will concern with the underwater shock loadings applied to the composite surface ship sonar domes, submarine acoustic windows as well as hydrodynamic impacts to the composite hulls. The analytical formulations will be developed by assuming that explosion occurs at a sufficiently large stand-off distance so that planar pressure pulse can be considered and the influence of bubble shock waves can be ignored. The calculation steps will involve two main stages. Stage I, which is the fluid-structure interaction phase, will include determination of the impulsive velocity due to the interaction of the shock waves with the structure by using Taylor FSI formulation [1]. The plate is considered to be in an air-backed condition. The deformation of the plate is assumed to be negligibly small during Stage I. Transmission of the shock waves throughout the thickness of the plate is taken into account by assuming the material under uniaxial strain. In Stage II, the mechanical response of the plate is determined by imposing a simply-supported boundary condition and by applying Lagrangian Energy approach to derive the equation of motion, accounting for the water-added inertial effects. Closed-form analytical solutions will be proposed at the end. The developed formulations are first tested with steel to check their validity. Only then, they are applied to a more complicated case of composite plates. The predicted results for both materials are compared with those from non-linear finite element explicit code, LS-DYNA. Finally, the accuracy as well as limitations of the proposed method will be evaluated.

2 Literature Review Underwater explosion has long been the focus of naval research since World War I and II. The majority of these extensive researches has been published through three volumes of ‘Underwater Explosion Research’ issued by the office of Naval Research in 1950 [2]. Cole [3] also systematically presented some of the useful summaries regarding the physical effects of UNDEX. A general literature review about the noticeable worldwide research efforts in the fields of blast loaded marine structures can be found in Porfiri and Gupta [4]. Experimental works dealing with underwater explosion can be divided into two: one testing with real explosive facilities and the other using laboratory environment. In the past, experiments of the former type were mostly performed. For example, in the

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1980s, experimental shock tests were conducted on a large number of glass-reinforced plastic (GRP) composite panels and a full-scaled midship section to determine suitable materials for the newly-built mine hunter [5]. During the 1990s, a series of underwater explosive tests were performed on GRP composite laminates by Mouritz [6–9] in order to analyze the damage response behavior of stitched or non-stitched GRP laminates, fatigue properties, flexural strength, and so on. Arora et al. [10] have conducted fullscale experimental studies on glass-fiber reinforced polymer (GFRP) and carbon-fiber reinforced polymer (CFRP) sandwich composite panels and laminate tubes when they are subjected to air and underwater blast loadings. A Dyno-Crusher test, an alternative way of studying a 1D response of multi-layered pyramidal core sandwich panel due to the water blast, can be found in Wadley [11, 12]. The experimental tests mentioned above are only concerned with the use of real explosives on the test samples. However, these tests are usually expensive, timeconsuming and involve extensive safety measures. Therefore, shock tests using labscaled environment are getting more popular in the recent years. In those experiments, a projectile impact-based shock tube is widely employed. It typically consists of a gas reservoir, a high-speed camera, a projectile, a long water-filled shock tube, a thin piston plate and a test specimen. This kind of apparatus was initially used by Deshpande et al. [13] who studied the effect of underwater shock loading on the structures and proved the finding of Taylor [1]. Later it was modified by Espinosa [14] by designing a divergent shock tube to overcome the dimensional limitation imposed by the apparatus size. Schiffer and Tagarielli [15] used a transparent shock tube in order to observe not only the dynamic response of circular composite plates but also the cavitation effect. LeBlanc and Shukla [16–18] studied UNDEX response of composite plates and curved composite panels by detonating an explosive charge at one end of a water-filled conical shock tube. From 2012 to 2017, a series of papers were published regarding the use of Underwater Shock Loading Simulator by Avachat and Zhou [19–23] to investigate the shock response behavior of laminated composite plates and sandwich structures. Qu et al. [24] also employed the same apparatus to analyze the dynamic response of thickand thin-walled composite cylinders. A recent paper of Huang [25] investigated the dynamic response and failure of composite circular laminates by employing a lab-scaled underwater explosive simulator and the 3D Digital Image Correlation (DIC) technique. With more development in computation power, numerical solutions have been widely employed in the fields of UNDEX and FSI problems. The most common approaches include the use of hydrocodes and Doubly Asymptotic Approximations (DAA)/Boundary Element Method (BEM) codes. Mair [26] gives a fairly comprehensive literature review about the use of various hydrocodes to predict the UNDEX responses. Analysis using DAA code uses boundary element method developed by Geers [27, 28]. Its application in the numerical analysis can be found in Underwater Shock Analysis (USA) program. The main advantage of using such method is that the governing equations are expressed in terms of the wet surface variables only and thus, there is no need to model explicitly the surrounding fluid. Applications of LS-DYNA/USA can be found in DeRuntz [29] and Le Sourne et al. [30]. Analytical works regarding the response of composite structures subjected to UNDEX are not very common. One of the earliest theoretical works includes Taylor

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[1] who proposed a 1D FSI solution to analyze the response of a free-standing rigid plate with an infinite length impacted by a plane shock wave. Librescu [31] proposed an analytical approach based on 3D elasticity theory in the Lagrangian description to study the dynamic response of anisotropic sandwich flat panels subjected to underwater and in-air explosions. In Liu and Young [32], Taylor’s air-backed FSI model was extended to a water-backed model by introducing a new FSI parameter and then solving the governing equations to give pressure, velocity and displacement. Wang et al. [33] has provided a novel solution method based on state space methodology, a numerical inversion of the Laplace transform, to yield the elastic dynamic response of the laminated composite plates subjected to UNDEX. A theoretical model that could take into account the stretching forces, transverse shear deformation effect, flexural wave propagation and cavitation induced non-linearity effect has been developed by Schiffer and Tagarielli [34]. Hoo Fatt and Sirivolu [35] presented an analytical method by coupling Taylor’s 1D FSI formulation with the Lagrange’s equation of motion and then by expanding the displacement terms into double Fourier Series to describe the in- and out-of-plane deflections of the sandwich facesheets. Recently, Sone Oo et al. [36], has proposed a simplified analytical solution to predict the elastic response of the isotropic circular plate subjected to air and underwater blasts. Equations of motions are derived based on Lagrangian Energy approach and the time-varying water-added mass term is predicted using Kirchhoff’s Retarded Potential Formulae (RPF) along with a constant averaging term.

3 Analytical Model 3.1 Problem Formulation Consider a simply-supported rectangular composite plate having the sides a, b and constant thickness h. A standard Cartesian coordinate (x, y, z) system is defined at the origin and mid-surface of the plate as shown in Fig. 1. The displacements in the x, y, z directions are denoted as u, v, and w respectively. Suppose that the plate has N orthotropic plies that are bonded together perfectly. The orthotropic axes of material symmetry of an individual ply do not necessarily need to coincide with the x, y, z axes of the plate. Kirchhoff’s thin plate theory is considered in this paper and hence, the thickness h of the plate is much smaller than the other physical dimensions. In-plane displacements u and v are assumed negligibly small as compared with the out-of-plane displacement w. The bending strains are assumed infinitesimally small so that each ply obeys linear stressstrain relationship. In the model presented in this paper, any transverse shear deformation effects are ignored. Also damage and failure is not addressed within the scope of this paper.

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Fig. 1. Panel geometry and coordinate system of the problem

3.2 Underwater Blast Loading The pressure submitted to the plate due to underwater explosion can be near-field or farfield depending on the stand-off distance. The treatment on the compressibility of water and of several other important parameters will be different depending on whether the explosion is near-field and far-field. In this paper, it is assumed that the plate is located at a sufficiently far distance from the explosive source point so that a uniformly distributed planar pressure pulse can be applied. The corresponding bubble effects are ignored in accordance with the far-field assumption. Then, the response due to the aforementioned pressure loading can be calculated in two stages. Stage I, which is the early-time FSI phase, is one in which the maximum impulsive velocity of the plate due to the shock wave is determined by employing the procedures described in [1] and [3] and by assuming a negligible plate deformation. At the end of the Stage I, cavitation is supposed to occur close to the plate. Then comes the Stage II in which the deformation of the plate commences. The corresponding mechanical response of the plate is determined by adapting Lagrangian Energy approach. Water added mass is derived by using the natural frequency for the rectangular plate with water on one side as provided by [37]. It should, however, be aware that during Stage II, no more incident pressure loading is considered. According to the long-time response, only the water added mass effect will be considered in this Stage. The calculation and derivation steps in this paper are indeed analogous to the analytical solutions presented in [36] except that [36] considered only fundamental mode shape of vibration. Stage I: Early-time Fluid-Structure Interaction Response. Suppose that a plane shock exponentially decaying pressure pulse PI is submitted onto the composite rectangular plate of density ρc . The plate is in contact with water of density ρw on one side and air ρa on the other side. Note that the incident pressure pulse PI is a 1D shock wave propagating in negative z direction at sonic speed in water cw , as depicted in Fig. 2. For any arbitrary time t and at any arbitrary distance z from the fluid-structure boundary OO , the incident pressure PI can be written as:

PI (z, t) = P0 e

  − t− czw /τ

(1)

Development of Analytical Formulae

275

where P0 is the peak pressure and τ is the decay time. Both quantities can be determined by applying the principle of similarity, which states that the pressure and other properties of the shock wave will be the same if the scales of length and time are varied by the same scale factor [3]. However, in this paper, both quantities P0 and τ will only be defined arbitrarily in order to test the accuracy of the proposed formulae. In Fig. 2, it can be seen that the problem is treated as a single degree of freedom (DOF) and hence, only transverse displacements are considered in the analysis. Any tangential components of velocity for both particles and plate are assumed to vanish. The particle velocities for incident, reflected and transmitted wave are denoted as z˙I , ˙ (t). Recall that during Stage I of the z˙R and z˙T respectively and the plate velocity as W response, the plate is assumed to behave like a free-standing rigid plate with negligible deformation. Upon arrival of the shock wave at the FSI boundary OO , the entire plate will respond with a high frequency, creating rarefaction waves during the process. Some portion of the incident pressure is reflected back into the fluid while the other part is transmitted through the plate depending on the mechanical impedance of the plate material. For composite, the mechanical impedance Zc can be written as: Zc = ρc c3

(2)

where c3 is the through-thickness wave speed in the transverse direction. According to Abrate [38], c3 can be calculated as:  E33 (1 − ν12 ν21 ) c3 = (3) ρc where  = 1 − ν12 ν21 − ν23 ν32 − ν31 ν13 − 2ν12 ν23 ν31 . At the FSI boundary OO (z = 0), the incident pressure is: PI (t) = P0 e−t/τ

(4)

Let us describe the reflected pressure as: PR (t) = P0 ϕ(t)

(5)

where ϕ(t) is a temporal function to be determined. Note that ϕ(t) will decay with time and is indeed a component of the reflection and radiated waves due to the transverse ˙ (t). Considering the incident wave, reflected wave, transmitted movement of the plate W wave and plate velocity shown in Fig. 2, the velocity continuity equation at the boundary OO can be written as follows: ˙ (t) + z˙T (t) z˙I (t) − z˙R (t) = W

(6)

Then assuming that the fluid density and propagation velocity do not change significantly from the undisturbed values and also supposing sufficiently small disturbances, the pressure and particle velocities in the waves can be related by the following equations, according to [3]: z˙I =

PI ρw cw

z˙R =

PR ρw cw

z˙T =

PT ρc c3

(7)

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where PI , PR and PT are incident, reflected and transmitted pressure respectively. The transmitted pressure PT can be given in terms of the incident pressure PI as: PT =

2ρc c3 PI ρc c3 + ρw cw

(8)

By substituting Eq. (7) and (8) into Eq. (6), the following equation for ϕ(t) can be obtained:  1 ˙ μP0 e−t/τ − ρw cw W ϕ(t) = (9) P0 w cw where μ = ρρcc cc33 −ρ +ρw cw is the reflection parameter. Now the total pressure P(t) applied to the plate can be expressed by summing Eq. (4) and (5) as follows:

˙ P(t) = (1 + μ)P0 e−t/τ − ρw cw W

(10)

With the use of Eq. (10), the following equation of motion for Stage I is obtained: ¨ (t) + ρw cw W ˙ (t) = (1 + μ)P0 e−t/τ ρc hW

(11)

˙ (t) and W ¨ (t) represent velocity and acceleration of the plate. Analytical where W solution of Eq. (11) has already been proposed by Taylor [1] in which the maximum impulsive velocity of the plate is given as: vimp =

(1 + μ)ξ P0 τ ρc h

(12)

Recall that at the end of the Stage I, cavitation is supposed to occur all over the plate. Assuming fluid cavitation pressure to be zero, the cavitation inception time or the peak response time can be determined as: tc = −

lnψ τ ψ −1

ψ

where ξ = ψ ψ−1 is the momentum reduction parameter and ψ = Taylor’s FSI parameter for the air-backed plate.

Fig. 2. Pressure and particle velocity at the fluid-structure boundary

(13) ρw cw τ ρc h

is the

Development of Analytical Formulae

277

Stage II: Mechanical Response of the Submerged Composite Plate. As discussed before, the plate deformation is supposed to commence in the second stage of the response. The maximum impulsive velocity obtained at the end of the first stage is applied as an initial condition for Stage II. The incident pressure field is assumed to have vanished completely and the associated plate deceleration causes collapse of the cavitation zone, promoting the water-added inertia. The equation of motion for the plate can be derived by adapting Lagrangian Energy approach and then by describing the out-of-plane displacement term w(x, y, t) in the form of double Fourier summation as: ∞  ∞ Wij (t)φij (x, y) (14) w(x, y, t) = i=1

j=1

where Wij (t) is the temporal term for the modal participation and φij (x, y) is the spatial term that accounts for the mode shape. Assuming that the plate is simply-supported on all four edges, the mode shapes that would satisfy the corresponding boundary conditions can be given as:   jπ y iπ x sin (15) φij (x, y) = sin a b where i and j represent the mode number in x- and y-directions respectively. It must be noticed that the mode shape equation given in Eq. (15) satisfies the following orthogonality relation: ¨ ρφij φmn d  = 0, for i, j = m, n (16) 

where  is the area of the plate considered. In accordance with the Eq. (16), all the equations of motion are uncoupled. Later this property will also be applied in the derivation of initial conditions for each uncoupled modal equation. According to the classical plate theory, only bending of the orthotropic plate is considered. Hence, the expression for the bending strain energy can be given as a form of modal summation using the following equation: U =

Kij =

π4 4a3 b3

1 2

∞ ∞

2 i=1 j=1 Kij Wij  D11 (bi)4 + 2(abij)2 (D21 + 2D66 ) + D22 (aj)4

where Kij is the stiffness term for each mode, Dmn =

1 3

N

k=1

(17)

  3 is the Qmn |k zk3 − zk−1

bending stiffness matrix for composite, N is the number of layers in the laminate, and Qmn |k is the reduced stiffness matrix for each k th layer. These matrices can be readily found in any classical composite literature, for example, see [39]. Similarly, the kinetic energy T can be derived as:

Mij =

ρc π 2 12ab

∞ ˙2 T = 21 ∞ i=1 j=1 Mij Wij 

N  3 2 2 3 k=1 zk − zk−1 (aj) + (bi) +

ρc hab 4

(18)

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where the first term in the expression of Mij corresponds to the rotatory inertia effect and the second term to mass inertia effect. Equations (17) and (18) are finally solved in general Lagrangian Equation:  

∂L d ∂L = Qij (19) − ˙ ij dt ∂ W ∂Wij where L = T − U and Qij is the non-conservative force function for each mode due to the external load. Since the current solution portrays the UNDEX response as a free vibration problem, any Qij term will be equal to zero. By solving Eq. (19), the following equation of motion can be derived: ¨ ij (t) + Kij Wij (t) = 0 Mij W

(20)

Note that Eq. (20) is valid only for AIRBLAST or impulsive velocity responses. In order to include the water-added inertia term, Eq. (20) must be modified into:   ¨ ij (t) + Kij Wij (t) = 0 Mij + Maij W (21) where Maij is the water-added inertia term and can be calculated by using wetted natural frequency of the submerged plate given by Greenspon [37], ρw l βαij Mij (22) 2ρc h   where l is the longer side of the plate, β = f ab is a correction term for various aspect ratios of the plate, αij is a correction term for boundary conditions and mode shapes. The term β is bounded between 0 and 1 for a/b = 0 and a/b = 1 respectively. For intermediate values of a/b, β can be expressed as a polynomial function:  a 3  a 2 a + 0.0098 (23) β = 1.5 − 3.12 + 2.6 b b b Maij =

When applying Eq. (23), it should be noted that b = l is the longer side of the plate so that 0 < a/b < 1. The modal term αij can be determined by using:



a

αij = 0



b

2 φij dxdy



a

/ ab

0

0



b 0

 φij2 dxdy

(24)

The initial conditions for each mode i and j can be derived by using the initial impulsive velocity calculated from Stage I and the orthogonality of the mode shapes. By differentiating Eq. (14), the transverse velocity of the plate can be written as: ∞ ∞ ˙ ij (0)φij (x, y) (25) W w(x, ˙ y, 0) = i=1

j=1

By multiplying Eq. (25) by φmn (x, y) on both sides and then integrating over the surface area, only one term will remain on right-hand side of the equation by virtue of

Development of Analytical Formulae

279

the orthogonality property. Knowing that w(x, ˙ y, 0) = vimp at the initial condition, the equations for initial conditions become: ˙ ij (0) = Aij vimp Wij (0) = 0 and W

(26)

      a b a b where Aij = 0 0 φij dxdy / 0 0 φij2 dxdy is the correction term for any odd number modes. Aij will be zero for any even number modes due to the fact that the loading and the deformation are axisymmetric. Finally, analytical solution can be given in terms of modal participation and natural frequency as: Wij (t) =

Aij vimp sin ωij t ωij

(27)

 Kij where ωij = Mij +M is the wetted natural frequency. Note that when Maij = 0, aij the response is simply analogous to that of AIRBLAST or impulsive velocity.

4 Numerical Models Numerical simulations are performed in LS-DYNA, a non-linear finite element explicit code. Two types of numerical simulations are studied. The first type concerns with the impulsive velocity response in which only 2D finite element plate was modelled. The second type includes both fluid and plate models. Detailed modelling steps are provided in the subsequent sections. 4.1 Materials Two types of material models, steel and CFRP/epoxy laminate, are considered. However, the same geometry of the plate is used for both models. A hypothetical square plate with the dimensions of 203.2 mm and an overall constant thickness of 6.12 mm is constructed. Therefore, the aspect ratio of the plate h/a is about 0.03. Material characteristics for isotropic material (steel) are given in Table 1. Table 1. Characteristics of material 1 (steel) Item

Values Units

Density ρ

7822.8 kgm−3

Young modulus E 207 Poisson ratio υ

0.3

Yield stress σ y

545

GPa MPa

280

Y. P. Sone Oo et al. Table 2. Characteristics of material 2 (carbon-fiber/epoxy laminate) Item

Values Units

Density ρ c

1548

Young modulus E 11

137.67 GPa

Young modulus E 22 = E 33 8.98 Shear modulus G12 3.66

kgm−3 GPa GPa

a

162

GPa

Shear modulus G31 b

183

GPa

Poisson ratio υ 12 = υ 13

0.281

Poisson ratio υ 23

0.385

Tensile strength X T

2214

MPa

Compressive strength X C

1030

MPa

Tensile strength Y T

47.5

MPa

Compressive strength Y C

80.7

MPa

Shear modulus G23

Shear strength S C 25.6 MPa a Out-of-plane shear stiffness G increased by 23 50 times (original value = 3.24 GPa)

b Out-of-plane shear stiffness G increased by 31

50 times (original value = 3.66 GPa)

As for the composite, a laminated carbon-fiber/epoxy plate with the layout [±45/0/0/0/ ± 45/0/0/0/90/90/0]S is considered. Each ply is unidirectional and has the thickness of 0.278 mm. For ±45◦ plies, each one will have about 0.139 mm thickness so that overall thickness of the laminate is about 6.12 mm. To make sure that the material shows only the bending response, the out-of-plane shear stiffness, that is, G23 and G31 , is artificially increased to 50 times that of the actual values. This has been done just to make sure that the analytical results are comparable to the numerical one, keeping in mind that the out-of-plane shear deformation will need to be coupled into the formulation in the near future. Material characteristics for composite laminate is given in Table 2.

Development of Analytical Formulae

281

4.2 Details of the Finite Element Models Models Using Only Impulsive Velocity. The finite element model for the plate is comprised of 2D shell elements. For steel, Belytschko-Tsay element formulation is applied. Material model for steel uses *MAT_PIECEWISE_LINEAR_PLASTICITY without taking into account the strain rate effect. Five through-thickness integration points are considered. A shear correction factor of 0.83 is applied to correct zero traction conditions at the top and bottom surfaces of the shell. Regarding the composite plate model, fully integrated shell elements are employed. With *PART_COMPOSITE card, the thickness and orientation of each ply can be defined. A total of 22 through-thickness integration points is used, each representing one laminate ply. The author has performed various sensitivity analyses regarding through-thickness integration points and concluded that there is not much influence. However, the simulation time could be affected a lot and thus, it is decided that one integration point for each layer is sufficient for the desired accuracy. A laminated shell theory is applied by setting LAMSHT = 1 to correct for the differences in the elastic constants from ply to ply. The material model for the composite uses *MAT_ENHANCED_COMPOSITE_DAMAGE. However, within the scope of this paper, the problem will be limited to only elastic response without having any damage. Simply-supported boundary conditions are imposed at edges of the plate for both material models. However, due to the symmetry of the problem, only a quarter of the plate has been modeled and the symmetric plane is defined through *BOUNDARY_SPC card in LS-DYNA. As explained above, modelling with the impulsive velocity requires neither pressure loading nor fluid elements. Only impulsive velocity is applied on the nodes of the plate model as an initial condition. The amplitude of the initial velocity is limited to small values so that materials remain in an elastic regime without suffering any damage. Models Including Fluid-Structure Interaction. The fluid is modeled using acoustic solid element formulation in LS-DYNA employing acoustic material (ρw = 1025 kgm−3 , cw = 1500 ms−1 ). These elements are similar to the Eulerian elements since only the nodes attached to the Lagrangian elements are allowed to move. The length of the fluid model needs to be selected very carefully in order not to have very long computation time as well as the returning wave effects that could come back from the far end of the fluid boundary. In this paper, the effect of varying the fluid column length is analyzed too. The mesh size of the fluid element is chosen as 1 mm in the thickness direction (negative z-direction). This fluid mesh size needs to be fine enough in order to accurately capture the shock wave propagation and cavitation behavior, satisfying the criterion 2ρw tw < 5ρs ts where tw and ts are the thickness of the fluid element and structure respectively. The nodes of the structure and the fluid are shared so that fluid-structure coupling is automatically treated in LS-DYNA. The lateral surfaces of the fluid elements are constrained in the x- and y-directions to ensure 1D wave propagation. Cavitation is considered in the analysis by activating the cavitation flag so that when the pressure becomes negative, it will be forced to zero. Numerical damping (BETA = 0.25) is applied on the acoustic fluid element for the stability issue. As a consequence, the peak pressure will become slightly less and the decay time slightly long. However,

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analyzing the impulse at the nearest element to the plate shows that the results are in good accordance with the transferred impulse value provided by Taylor’s formulations. A typical finite element quarter plate model with or without the fluid elements are shown in Fig. 3.

Fig. 3. Typical finite element model of the rectangular plate: (a) with impulsive velocity and (b) with acoustic fluid elements

4.3 Details of the Finite Element Analyses Performed In Table 3 and Table 4, various numerical simulations performed using steel and carbonfiber/epoxy laminate are shown respectively. The purpose is to analyze the effect of changing the mesh size. Here, it should be noticed that case 1 (steel) and case 4 (composite) contain only shell elements since these cases correspond to impulsive velocity simulations. On contrary, cases 2, 3, 5 and 6 are simulations with fluid models. As explained in the previous section, it is very important to use the correct length of the water column. It needs to be sufficiently long in order to avoid the returning pressure wave reflected from the free end of the fluid column. Therefore, in cases 2 and 3, the length of the fluid (water) column is varied from 0.25 m to 0.5 m respectively to be able to check if there is the returning wave effect. Note that the calculation time becomes almost doubled due to the increased number of solid elements. The size of the plate mesh is kept the same in those two cases. On the other hand, in cases 5 and 6, the effect of varying the plate mesh is studied by using 2 mm and 8 mm mesh size respectively. The tested plate mesh sizes are given in the last column of Table 3 and Table 4. It should also be aware that for composites, the impulsive velocity and applied loading is decreased by about 3 and 4 times respectively to guarantee that the response of the plate remains in elastic region.

Development of Analytical Formulae

283

Table 3. Numerical simulations performed in LS-DYNA (steel) Cases

Description

Water column length (m)

Peak pressure (MPa)

Decay time (ms)

Impulsive velocity (ms−1 )

Plate mesh size (mm)

1

Taylor impulsive modela

–

–

–

0.576 ms−1

≈3.4

2

FSI model

0.25

1

0.05

–

≈8

3

FSI model

0.5

1

0.05

–

≈8

a Modelling using Taylor impulsive velocity does not require fluid elements. Hence no information

regarding water column length, peak pressure or decay time is available. Only the initial impulsive velocity value is given.

Table 4. Numerical simulations performed in LS-DYNA (carbon-fiber/epoxy laminate) Cases

Description

Water column length (m)

Peak pressure (MPa)

Decay time (ms)

Impulsive velocity (ms−1 )

Plate mesh size (mm)

4

Taylor impulsive model

–

–

–

0.182 ms−1

≈8

5

FSI model

0.25

0.25

0.05

–

≈2

6

FSI model

0.25

0.25

0.05

–

≈8

5 Results and Discussion The current analytical formulations can be applied to steel by imposing E11 = E22 = E, E . The rest of the formulation will ν12 = ν23 = ν13 = ν and G12 = G23 = G13 = 2(1+ν) be the same except that the composite density ρc is replaced by steel density ρs . The purpose is to check the validity of the current formulation before directly solving for more complex cases of composite. Only after the result of isotropic material is verified, the formulations are applied to investigate the UNDEX response of composite plate. 5.1 Response of Isotropic Rectangular Plate The central-deflection time histories of the steel rectangular plate subjected to Taylor’s impulsive velocity as well as water blast loading are shown in Fig. 4 along with the LS-DYNA results. Based on these results, many important observations can be made. First of all, it can be observed that the present formulations work very well (with 1% discrepancy) for the AIRBLAST response in which only impulsive velocity is modelled. As for UNDEX, added mass inertia term becomes important especially for longer time step. As can be seen in Fig. 4, the response might be seriously underestimated (about

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30%) without the water-added mass. The two plots of LS-DYNA (Case 2 and 3) are overlapped. This means that either using water column length of 0.25 m or 0.5 m does not matter since the results are exactly the same. The analytical result of UNDEX is also similar to finite element results except that the response is slightly faster in the analytical. This is mainly due to the two stage approximations in the analytical calculations. Recall that added mass term could begin only after the end of Stage I. In numerical solutions, added mass inertia might have already evolved since Stage I because the consideration of negligibly small deformation during Stage I of analytical calculation is in fact an idealized assumption . As a consequence, there is slightly more added mass term, leading to slightly longer response (also slightly higher strain energy) in the numerical results. In Fig. 5, normalized Von-Mises stress spatial distribution is plotted with respect to normalized x-coordinates and then compared with the finite element solutions. It can be seen that present analytical results overestimate the peak by about 25%. But it is not surprising given the fact that the current analytical solution considers only bending. Other modes of deformation, for example, transverse shear or stretching or both are not negligible in the numerical results. 5.2 Response of Composite Rectangular Plate The results of the composite rectangular plate are shown in Fig. 6 and Fig. 7. It can be seen that the current solution predicts very well for the impulsive velocity response since the discrepancy does not exceed 6% and the profiles of the time history curves are very similar. However, there is an obvious difference in the two curves of UNDEX response between analytical and LS-DYNA although the peak amplitude shows only 4% difference. The peak response time in LS-DYNA is obviously faster than that of analytical one. This effect might come from the non-linearity of cavitation. Consideration of the through-thickness wave speed could also reduce the impulse transferred to the composite plate. Perhaps modelling using 2D shell elements in LS-DYNA may not consider this kind of effect, resulting higher response within a shorter period. This issue still needs to be investigated more in the future. The LS-DYNA results shown in Fig. 6 and Fig. 7 for case 5 (fine mesh) and case 6 (coarse mesh) are very similar, meaning that the convergence of the results has been reached. Evaluating the stress results in the material direction (σ11 ) for the lowest ply (45°) gives satisfactory results as can be seen in Fig. 7 although the numerical results clearly show more damped behavior. It is not surprising, however, because the present analytical solution describes the problem as free vibration response and therefore, there is no damping or compressibility of the fluid. 5.3 Advantages and Limitations Table 5(a) compares the time needed to finish one calculation and % discrepancies of the central-deflection calculated in all the analysis cases are shown in Table 5(b). The closed-form solutions are implemented in MATLAB (version R2015a). It can be seen in Table 5(a) that the present approach takes little or almost no time to finish the calculations as compared with LS-DYNA numerical approach. In fact, this much time is necessary in order to store the calculated data for each modal result at each time step. Anyway, it

Development of Analytical Formulae

285

is obvious that adapting analytical approach could improve the calculation time by as much as 1857 times as shown in the last column of Table 5(a). Of course, it depends on how much elements have been used in the numerical model. It is also worth noticing that the calculation time for both analytical and numerical approaches increases for cases 4, 5 and 6. This is normal since cases 4, 5 and 6 represent composite plate and so, requiring storage for the additional variables and of course, more calculation time. According to Table 5(b), the proposed analytical approach shows discrepancy less than 10% compared to LS-DYNA results. It also provides more physical insights into the complicated problem such as underwater explosion. Table 5. Comparison of the simulation time and the accuracy between LS-DYNA and present analytical method (a) Simulation time (sec) Case

LS-DYNA

Analytical

(b) Central-deflection (mm) Faster by

Case

LS-DYNA

Analytical

% −1%

1

24

0.2

120

1

0.212

0.209

2

199

0.25

796

2

0.292

0.3

3

363

0.25

1452

3

0.291

0.3

4

47

1.9

25

4

0.068

0.064

−6%

5

3547

1.91

1857

5

0.139

0.134

−4%

6

277

1.91

145

6

0.139

0.134

−4%

3% 3%

However, some drawbacks must be pointed out too. One of these is the ignorance of the transverse shear deformation. With the knowledge that transverse shear deformation could decrease the natural frequency of the plate, the actual response of the composite plate would be longer than what has been predicted by the current analytical method and the amplitude may be lower. Another drawback of the solution is the consideration of two calculation stages. As discussed before, the action of cavitation is in reality non-linear and depends on a lot of factors such as plate aspect ratio, load duration time as well as the material or orientation considered. That is why some numerical approach such as DAA considers early-time and long-time responses with a smooth transition between the two. The current analytical method simply does not consider this phenomenon and so there is no smooth transition between the two steps. This could be improved by coupling DAA method into the analytical model in which the pressure and structural equations are solved simultaneously for each time step.

Y. P. Sone Oo et al.

Central-deflection (mm)

286

0.30 0.25 0.20 0.15 0.10 0.05 0.00 -0.05 -0.10

UNDEX

Taylor’s impulsive velocity

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time (ms) Analytical - Impulsive

Analytical - UNDEX

LS-DYNA Case 1: Impulsive

LS-DYNA Case 2: UNDEX

LS-DYNA Case 3: UNDEX

Normalized Von-mises stress (σvm/σy)

Fig. 4. Response of isotropic rectangular plate (Central-deflection Vs time; a = b = 0.2032 m; h = 6.12 mm; First and third terms in Fourier Series i = j = 1, 3; Taylor’s impulsive velocity response: vimp = 0.576 ms−1 ; UNDEX: P0 = 1 MPa, τ = 0.05 ms)

0.2

Analytical - UNDEX

0.18 0.16

LS-DYNA Case 2: UNDEX (WC=0.25m)

0.14

LS-DYNA Case 3: UNDEX (WC=0.5m)

0.12 0.1 0.08 0.06 0.04 0.02 0 0

0.2

0.4

0.6

0.8

1

Normalized x-coordinates (x/a) Fig. 5. Spatial distribution of the normalized Von-mises stress (σVM /σY ) for the bottom plane with respect to normalized coordinates (x/a) at the time of the maximum UNDEX response (tmax analytical = 0.495 ms, tmax LS-DYNA = 0.514 ms; WC = water column length)

Development of Analytical Formulae

287

Central-deflection (mm)

0.15 0.10 0.05 0.00 0

0.2

0.4

0.6

0.8

1

1.2

-0.05 -0.10

Time (ms) Analytical - Impulsive

Analytical - UNDEX

LS-DYNA Case 4: Impulsive

LS-DYNA Case 5: UNDEX

LS-DYNA Case 6: UNDEX

Fig. 6. Response of composite rectangular plate (Central-deflection Vs Time; a = b = 0.2032 m; h = 6.12 mm; First and third terms in Fourier Series i = j = 1, 3; Taylor’s impulsive velocity response: vimp = 0.182 ms−1 ; UNDEX: P0 = 0.25 MPa, τ = 0.05 ms)

20 15

σ11 (MPa)

10 5 0 0.00 -5 -10

0.25

0.50

0.75

1.00

1.25

1.50

Analytical - UNDEX LS-DYNA Case 5: UNDEX LS-DYNA Case 6: UNDEX

-15

Time (ms)

Fig. 7. Response of composite rectangular plate (σ11 Vs Time; a = b = 0.2032 m; h = 6.12 mm; First and third terms in Fourier Series i = j = 1, 3; At bottom ply 45°; UNDEX: P0 = 0.25 MPa, τ = 0.05 ms)

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6 Conclusion and Future Works This paper presented simplified analytical formulae by adapting Lagrangian Energy approach and Taylor’s 1D FSI method. Results are analyzed for two different types of material models; isotropic rectangular plate and carbon-fiber/epoxy laminated plate. However, only the elastic perturbation of the problem has been focused since the main interest of this research is mainly to find an analytical solution for composite UNDEX response. Using steel material in this case should, in fact, be seen as a trial case before actually applying to more complicated composite cases. Comparisons of the results with LS-DYNA show that the current formulations could predict the maximum central deflection with a discrepancy less than 10%. The stresses are also captured quite well, only showing 13% discrepancy. However, it must be kept in mind that this solution considers only bending and so, as long as the plate aspect ratio remains in the correct range, it would be valid. Testing with different aspect ratios as well as varying loading levels still need to be performed. Also, it is of practical interest to couple the transverse shear deformation effect into the current approach. Moreover, the authors intend to develop analytical formulations for the water-backed plate. Extending the current formulae for the stiffened or curved plate cases would also be interesting. All of these mentioned above will be for future work and the corresponding results will be published elsewhere. Finally, degradation of the strength due to damage will be investigated by adapting some classical failure criteria and then by decreasing the elastic moduli of the plies. Acknowledgement. This research work has been conducted with the financial support of DGADGE. The authors would also like to thank Calcul-Meca and Multiplast companies for their technical support.

References 1. Taylor, G.I.: The pressure and impulse of submarine explosion waves on plates. In: The Scientific Papers of G. I. Taylor, vol. III, pp. 287–303. Cambridge University Press (1941) 2. Office of Naval Research: Underwater Explosion Research: a compendium of British and American reports, vol. I, II, III. Department of the Navy, Washington, D.C. (1950) 3. Cole, R.H.: Underwater Explosions. Princeton University Press, Princeton (1948) 4. Porfiri, M., Gupta, N.: A review of research on impulsive loading of marine composites. In: Major Accomplishments in Composite Materials and Sandwich Structures: An Anthology of ONR Sponsored Research, pp. 169–194 (2009). https://doi.org/10.1007/978-90-481-314 1-9_8 5. Hall, D.J.: Examination of the effects of underwater blasts on sandwich composite structures. Compos. Struct. 11, 101–120 (1989). https://doi.org/10.1016/0263-8223(89)90063-9 6. Mouritz, A.P., Saunders, D.S., Buckley, S.: The damage and failure of GRP laminates by underwater explosion shock loading. Composites 25, 431–437 (1994). https://doi.org/10. 1016/0010-4361(94)90099-X 7. Mouritz, A.P.: The damage to stitched GRP laminates by underwater explosion shock loading. Compos. Sci. Technol. 55, 365–374 (1995). https://doi.org/10.1016/0266-3538(95)00122-0

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25. Huang, W., Zhang, W., Chen, T., Jiang, X., Liu, J.: Dynamic response of circular composite laminates subjected to underwater impulsive loading. Compos. Part A 109, 63–74 (2018). https://doi.org/10.1016/j.compositesa.2018.02.043 26. Mair, H.U.: Review: hydrocodes for structural response to underwater explosions. Shock Vib. 6, 81–96 (1999). https://doi.org/10.1155/1999/587105 27. Geers, T.L.: Residual potential and approximate methods for three-dimensional fluid-structure interaction problems. J. Acoust. Soc. Am. 49, 1505–1510 (1971). https://doi.org/10.1121/1. 1912526 28. Geers, T.L.: Doubly asymptotic approximations for transient motions of submerged structures. J. Acoust. Soc. Am. 64, 1500–1508 (1978). https://doi.org/10.1121/1.382093 29. DeRuntz Jr., J.A.: The underwater shock analysis code and its applications. In: Proceedings of the 60th Shock and Vibration Symposium, pp. 89–107 (1989) 30. Le Sourne, H., County, N., Besnier, F., Kammerer, C., Legavre, H.: LS-DYNA applications in shipbuilding. In: 4th European LS-DYNA Users Conference, pp. 1–16 (2003) 31. Librescu, L.: Dynamic response of anisotropic sandwich flat panels to underwater and in-air explosions. Int. J. Solids Struct. 43, 3794–3816 (2006). https://doi.org/10.1016/j.ijsolstr.2005. 03.052 32. Liu, Z., Young, Y.L.: Transient response of submerged plates subject to underwater shock loading: an analytical perspective. J. Appl. Mech. Trans. ASME 75 (2008). https://doi.org/ 10.1115/1.2871129 33. Wang, Z., et al.: A novel efficient method to evaluate the dynamic response of laminated plates subjected to underwater shock. J. Sound Vib. 332, 5618–5634 (2013). https://doi.org/ 10.1016/j.jsv.2013.05.028 34. Schiffer, A., Tagarielli, V.L.: The dynamic response of composite plates to underwater blast: theoretical and numerical modelling. Int. J. Impact Eng. 70, 1–13 (2014). https://doi.org/10. 1016/j.ijimpeng.2014.03.002 35. Hoo Fatt, M.S., Sirivolu, D.: Marine composite sandwich plates under air and water blasts. Mar. Struct. 56, 163–185 (2017). https://doi.org/10.1016/j.marstruc.2017.08.004 36. Sone Oo, Y.P., Le Sourne, H., Dorival, O.: Application of Lagrangian Energy Approach to determine the response of isotropic circular plates subjected to air and underwater blasts. In: Proceedings of the 7th International Conference on Marine Structures, (MARSTRUCT 2019), Dubrovnik, Croatia, 6–8 May 2019 (2019) 37. Greenspon, J.E.: Vibrations of cross-stiffened and sandwich plates with application to underwater sound radiators. J. Acoust. Soc. Am. 33, 1485–1497 (1961). https://doi.org/10.1121/1. 1908480 38. Abrate, S.: Interaction of underwater blasts and submerged structures. In: Abrate, S., Castanié, B., Rajapakse, Y. (eds.) Dynamic Failure of Composite and Sandwich Structures. Solid Mechanics and Its Applications, vol. 192. Springer, Dordrecht (2013). https://doi.org/10. 1007/978-94-007-5329-7_3 39. Jones, R.M.: Mechanics of Composite Materials. Scripta Book Co., Washington, D.C. (1975)

Stress Concentration Factors of Damaged Ship Side Panels Cristian Lombardi, Bianca Pinheiro, and Ilson Paranhos Pasqualino(B) Subsea Technology Laboratory, Ocean Engineering Department, COPPE, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil [email protected]

Abstract. Floating production systems units (FPSOs or FSOs) converted from VLCCs are subject to damage in side panels due to overloading operations involving supply vessels. Although damages of small magnitude do not significantly affect the ultimate strength of the panel, the stress concentration in damaged regions may lead to the nucleation of cracks, in view of the long term operational life of these units. The proposed study evaluates experimentally and numerically stress concentration factors (SCFs) in damaged reinforced flat panels. A small scale model of a panel was manufactured and damaged to measure SCFs under cyclic in-plane compressive loading. The strain measurements where correlated with numerical results obtained from a finite element model which reproduces the laser mapped geometry of the panel. The calibrated numerical model can then be used to investigate SCFs of damaged panels under different conditions of damage position and depth. The study has shown that SCFs may significantly reduce the fatigue life of these panels when cyclic in-plane compressive loading is considered. Keywords: Fatigue · Stress concentration factor · FPSO · Mechanical damage

1 Introduction Due to the great market demand, the exploration of oil and natural gas has presented a systematic increase over the years. The volume of oil produced in the world in 2017 increased by 625.4 thousand barrels/day (0.7%) with respect to 2016, reaching a level of 92.6 million barrels/day [1]. Brazil was in 10th place, among the producing countries, after a 4.8% increase in the volume of oil produced. The increase in oil demand, coupled with technological advances, led to the exploration of new oil and gas fields at increasing water depths. In Brazil, these fields can reach water depths of more than 2,000 meters, in addition to being located at great distances from the coast [2], which can considerably increase the exploration costs involved. Thus, FPSO (Floating Production, Storage and Offloading) units are an interesting alternative. Such platforms operate with the aid of support vessels, called PSV (Platform Supply Vessel), which, due to adverse sea conditions or possible technical errors during operation may collide with the platform side panel. Although resulting damages of small magnitude may not lead to immediate failure of the panel, the stress concentration in the damaged region may compromise the fatigue © Springer Nature Singapore Pte Ltd. 2021 T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 291–304, 2021. https://doi.org/10.1007/978-981-15-4672-3_18

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life of the platform, which is a floating unit exposed to the action of cyclic loads over a period that may exceed 20 years [3]. This work addresses the accidental event where there is a collision between a Floating, Production, Storage and Offloading (FPSO) platform with a Platform Supply Vessel (PSV), with the objective of analyzing the stress concentration factors caused by the damaged geometry of the side panel, combined with its initial geometric imperfections.

2 Methodology For the development of the present work, the methodology comprised the development of a numerical model using the finite element software ABAQUS [4] for the simulation of denting (damage introduction) in an FPSO side panel model followed by in-plane cyclic compression. The geometric configuration of the panel was obtained after dimensional mapping of a small-scale model fabricated in the Subsea Technology Laboratory. For this mapping, a mechanical laser reading arm (FARO) was used, which outputs a cloud of points described by three-dimensional coordinates in a “.txt” file. The geometry obtained by the mechanical arm was then be subjected to a treatment comprising: post-processing of the file exported by FARO (1); cleaning reading “noises” (2); filling of lacking zones (mapping failures) (3), and generation of the input file for ABAQUS (“.inp”) (4). For the conduction of the steps (1), (2) and (4) three computational codes developed by FIGUEREDO [5] were used. After numerical simulations using the finite element method and experimental tests representative of the collision process and cyclic load action, the numerical-experimental correlation was carried out in order to verify the accuracy of the numerical model. 2.1 Small-Scale Model In order to perform the evaluation of stress concentration factors on damaged flat panels of FPSO sides, small scale panel models (1:20) were built at the Subsea Technology Laboratory. The panel consists of a 1 mm thick side plate, 240 mm length and 220 mm width, reinforced by six “T” shaped profiles, spaced by b = 44 mm, with web height and flange width of 21 mm and 7 mm, respectively. The web has a thickness of 0.7 mm and the flange 1 mm. The panel is fixed at two metallic bases, which are intended to be attached to the test equipment (Fig. 1). The mechanical properties of the steel used in the manufacturing of the panels were determined previously in [6] by tension tests whose mechanical properties are listed in Table 1, where E is Young modulus,  is the Poisson ratio, σ0 is the yield strength and σu is the ultimate tensile strength. 2.2 Panel Mapping The fatigue life assessment of a structure under cyclic loads should consider the evaluation of any stress concentrators that may be present which, in turn, have their effects

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Table 1. Mechanical properties of the steel used. E[GPa] ν 207,8

σ0 [MPa] σu [MPa]

0,3 230

582

Fig. 1. Small scale side panel (1:20).

combined with the initial geometrical imperfections of the structure. Thus, for a precise representation of the geometry of the small-scale panels, the FARO laser scanning equipment was used in a geometrical mapping of the model generating a cloud of points with three-dimensional coordinates. This cloud is acquired and exported to a file in the “.txt” format. 2.3 Mapping Post-processing The post-processing (treatment) of points generated during geometrical laser mapping by the FARO software consists of four steps: conversion of the “.txt” file into the appropriate format; cleaning the cloud of points with the removal of spurious points; filling of the mapping failures, and creation of the “.inp” file for finite element mesh generation in the ABAQUS program [4]. After mapping the panel geometry, a text file containing the coordinates of the cloud of points obtained is generated. However, to be directly imported in the ABAQUS modeling software, the text file should be rearranged. 2.3.1 Cleaning and Filling the Acquired Mapping As the mapping of the panel geometry is performed by a laser beam, some noises (spurious results) are picked up during the process, so cleaning is necessary. This cleaning process was conducted by manually removing sets of points that clearly resulted from spurious geometry acquisitions. In addition, the mapping process may sometimes not be able to fully represent a certain region of the panel, which results in setting faults in some locations. In order to fill the faults, the computational code developed by FIGUEREDO in [5] was used. The process of filling of fault regions adopted by the code considers the average of each surface area to be recovered and the orientation of its corresponding surface normal, estimated approximately according to the surrounding geometry.

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An example of the cleaning process, followed by the filling of faults, is shown in Fig. 2.

Fig. 2. a) Web before cleaning noises; b) Web after cleaning noises; c) Web after fault fills.

2.3.2 Generation of the ABAQUS Input File For formatting the ABAQUS input file, the code [5] uses the point clouds in the form of “.txt” files as input data, comprising the point (node) numbers and the web-plate and web-flange connection coordinates. The program also allows the choice of the desired mesh refinement level, setting as input data the approximated size of elements. The output files are a text file with the coordinates of the mesh nodes and the formatted “.inp” file. The type of element used was the S4R (thin shell, linear, with four nodes and reduced integration) in a mesh with a 1:1 aspect ratio.

3 Numerical Model The model geometry acquired by laser mapping (i.e. without the mechanical damage, but with initial geometric imperfections due to its fabrication), the numerical model was developed according to the finite element method using the software ABAQUS (2014) [4]. The numerical model comprises two analyses: the damage introduction caused by an indenter, followed by the application of cyclic compressive load in the longitudinal direction of the panel, in order to evaluate the stress concentration in the damaged region. 3.1 Finite Element Mesh The model is still composed of the panel and an indenter, which is a tool used for the introduction of the damage (dent). The indenter was modeled as a rigid body with a hemispherical shape, 17.5 mm in diameter (Fig. 3). In order to define the mesh refinement level to be used, aiming an accurate estimation of the stress concentration with a reasonable computational cost, a mesh sensitivity analysis was performed considering five different element sizes. As it can be seen in

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Fig. 3. Geometric representation of the model: panel/indenter.

Fig. 4, the coarser meshes result in lower stress (maximum stress in the damaged region) levels, due to higher model stiffness, but a stabilization of results is reached for a wide range of lower element sizes. Therefore, it was decided to use the 2.5 mm mesh, which can present stress results with enough precision (within the stabilization range) for a satisfactory computational time.

Fig. 4. Mesh sensitivity analysis.

3.2 Material The mechanical properties of the material were defined according to the characterization of the steel used in the model fabrication. The material behavior was set as linear in the elastic regime and assumed as an isotropic hardening model in the elastic-plastic regime, assuming the potential flow rule, based on the von Mises yield criterion. In Fig. 5 is presented the true stress versus logarithmic plastic strain curve of the steel used as input for the numerical model.

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Fig. 5. True stress versus logarithmic plastic strain of the used steel.

3.3 Boundary Conditions The definition of the boundary conditions seeks to reproduce the conditions undergone by the small-scale model during the conduction of experimental tests. The numerical simulation comprises the following load steps: indentation; indenter removal (spring back, or elastic return); longitudinal compression; load removal (stress relief). In the indentation and spring back steps, both panel edges are clamped. In the compression and stress relief steps, the longitudinal displacement (z-direction) at the upper edge are released to allow longitudinal compression and relief of the panel stresses. 3.4 Denting Since the aim of the numerical simulation is to allow a fatigue analysis based on the determination of stress concentration factors, it is important to ensure that the stresses attained during the compression of the panel remain in the linear elastic regime of the material behavior. Thus, after conducting preliminary analysis for definition of a penetration depth (d), and resulting remaining depth (d ), related to residual stresses bellow the yield strength, an indentation depth of 2.6 mm was defined for the test. 3.5 Cyclic Compression 0–2,5 KN After indenter removal, the deformed geometry (Fig. 6) was imported as the initial configuration of the second analysis, without the incorporation of the residual stress state, in order to simulate the experimental test. Strain gages were fixed on the panel after the indentation process, and the strain measurements were acquired during the cyclic compression. A range 0–2.5 kN was set for the cyclic compression load in the longitudinal direction, applied using the tool “Shell edge load”.

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Fig. 6. Damaged panel with a fixed lower base and the compression load application on the upper base.

3.5.1 Nominal Stress In order to determine the nominal stress to be used for the calculation of the stress concentration factor, an intact panel (without mechanical damage, but with initial geometric imperfections) was subjected to the same loading condition of the damaged panel. In order to eliminate edge effects and consider the stress distribution related to initial geometric imperfections, the nominal stress was defined as the mean the von Mises stress in a central region with area bxb, as indicated in Fig. 7.

Fig. 7. Intact panel – region bxb around the damaged region.

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The mean von Mises stress achieved in the central region of the intact panel was 7.4 MPa.

4 Numerical and Experimental Results 4.1 Indentation Graphs of the indentation force versus the penetration depth obtained from numerical and experimental results are shown in Fig. 8, where it is found a satisfactory numericalexperimental correlation of the panel response in terms of the force-displacement relation as well as the maximum penetration reached, corresponding to an error of only 4.2% (Table 2). A resulting dent of relative depth d/b of 4% was reached in the numerical model.

Fig. 8. Indentation force x d. Table 2. Numerical-experimental correlation of indentation. Max. Force (kN) Num.

2.19

Exp.

2.28

Error (%) −4.2

4.2 Cyclic Compression Strain gages allowed the acquisition of the strain behavior during the cyclic compression. Triaxial rosette gages were used, with axes oriented at 0°, 45° and 90°. Strain results were acquired by the five gages fixed on the panel, whose positions are shown in Fig. 9.

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Fig. 9. Damaged panel instrumentation.

The panel response showed a tendency to accommodate successively during compressive load cycles. Then, to assure that the results were acquired after stabilization of the panel, three compression sets of 50 cycles each were accomplished.

Fig. 10. Load x extension during compression load cycles (0–2.5 kN) in the experimental test.

Figure 10 shows clearly the occurrence of the panel accommodation along the application of the three load sets during the experimental test. The strain values used for numerical-experimental correlation were selected within the 3rd load set. The longitudinal and transverse strains acquired during the test are shown in Figs. 11 and 12, respectively.

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Fig. 11. Longitudinal strains during cyclic compression in the experimental test.

Fig. 12. Transverse strains during cyclic compression in the experimental test.

The results obtained for the longitudinal and transverse strain amplitudes in the numerical simulation and experimental test, as well as their respective discrepancies, are shown in Table 3. Table 3. Correlation: strain amplitudes during cyclic compression.  (εzz) 2 Gage Num.

Exp.

 (εyy) 2 Dif. (%) Num.

Exp.

Dif. (%)

1

2,5E-5 1,9E-5 31,6

3,0E-6 4,4E-6 −31,8

2

8,0E-6 5,6E-6 42,9

1,2E-6 2,6E-6 −53,8

3

2,0E-5 2,1E-5 −4,8

1,8E-6 1,2E-6 50,0

4

8,0E-6 6,4E-6 25,0

4,0E-6 4,3E-6 −7,0

5

1,5E-5 1,5E-5 0,0

2,3E-6 3,9E-6 −41,0

Assuming a plane stress state, the respective longitudinal stresses were determined for the same acquisition time (see Fig. 13).

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Fig. 13. Longitudinal stress during cyclic compression.

A comparison between the longitudinal stress amplitudes obtained during experimental test and numerical simulation is shown in Table 4. Table 4. Correlation between numerical and experimental results: longitudinal stress amplitude. Longitudinal stress amplitude σzz (MPa) Gage

Numeric

Experimental

Error (%)

1

10.9

8.2

32.9

2

3.5

3.0

16.7

3

9.5

9.0

5.6

4

3.1

2.9

6.9

5

6.6

6.1

8.2

The results presented in Table 4 show that the stress variations obtained by the numerical model are in the same order of magnitude of the experimental results, reaching a maximum discrepancy of 33%. These discrepancies may arise due to different causes. One cause may be related to the size of the strain gages (3 mm) used in the instrumentation of the panel model, since high stress gradients are found in the damaged region. Then, strains, and stresses, may significantly vary over each strain gage area, leading to discrepancies when values (strains or stresses) acquired in the experimental tests are correlated with corresponding numerical results, that are obtained at the prescribed positions for the strain gages. Despite of the mesh sensitivity analysis carried out to define a reasonable element size, the refinement of the finite element (FE) mesh may also be a cause of the observed discrepancies, since the element size of 2.5 mm actually seems not accurate enough and it should be reduced for further analysis. Still concerning the FE mesh, discrepancies

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may also arise due to the stiffness of the numerical model, which somewhat differs from that of the panel model. However, it is interesting to note that the discrepancies of the model are all positive, when compared with the experimental results, that is, the model is somewhat conservative, resulting in slightly higher stresses in the damaged region, which is beneficial from a safety point of view.

5 Stress Concentration Factors Once the numerical model has been calibrated and validated, the theoretical stress concentration factor resulting from the mechanical damage (dent) can be determined. The stress concentration factor is obtained, in terms of the von Mises equivalent stress, as the ratio between the maximum stress reached in the damaged panel and that reached in the intact panel, according to the following expression:   σdamaged kt = σintact Figure 14 shows the distribution of von Mises stresses (in MPa) in the damaged panel. The maximum von Mises stress reached in the damaged region was 12.6 MPa and, as shown in Sect. 3.5.1, the nominal von Mises stress was 7.4 MPa. The stress concentration factor, in terms of the von Mises stress, is then equal to:

Fig. 14. Distribution of von Mises stresses (MPa) in the damaged panel.

 kt =

12.6 7.4

 = 1.7

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The stress concentration factor was also determined in terms of the longitudinal stress, using as the nominal stress the ratio between the compressive load applied and the nominal cross-sectional area of the intact panel (without geometric imperfections). Thus, the nominal longitudinal stress is given by:   −2500 N = −7.1MPa σnominal = 350.2mm2 The maximum longitudinal stress reached in the damaged region was equal to 13.7 MPa, which leads to obtaining the following stress concentration factor:   −13.7 kt = = 1.9 −7.1 The stress concentration factors obtained in terms of the von Mises stress and longitudinal stress are close, since the last one is the preponderant stress component in this case. The stress concentration factor obtained in terms of the von Mises stress (equivalent stress) is recommended for fatigue life predictions of damaged steel panels, since it directly allows the use of uniaxial S-N curves.

6 Conclusions It can be concluded that the laser mapping performed with the help of the FARO facility and the post-processing of the mapped data using the codes developed in [5] are effective and of great importance for the determination of the geometric initial deformations of the panel. The numerical-experimental correlation using the data acquired by the strain gages during indentation and cyclic compression tests showed satisfactory results. The panel response with respect to the indentation force x penetration depth relation correlates well between numerical and experimental result, and an error of only 4.2% was attained between the maximum penetration reached. During compression load cycles, the correlation between experimental and numerical stresses showed satisfactory results, with the numerical model giving conservative results. Once the numerical model has been calibrated and validated, the theoretical stress concentration factor resulting from the mechanical damage was determined. According to the stress concentration factor obtained, it was found that the fatigue life of a damaged panel can be reduced to about 50%. The work is ongoing and more results will allow more discussion concerning this effect. Some recommendations can also be left for future works: – It has been found that the stresses reached in the panel remain far from the material yield strength, so the compressive load can be raised to ensure faster panel accommodation, avoiding the need of a high number of cycles. – Due to the large stress gradients around the damage, it would be interesting to use smaller strain gages, allowing the acquisition of more localized data and possibly improving the correlation results.

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As perspectives for this ongoing work, it is desired to fabricate and test four additional small-scale models to contribute to the numerical-experimental correlation results and numerical model validation. It is also possible to consider the variation of other model parameters, as dent depth, collision angle and position, to define the critical conditions in terms of stress concentration and obtain analytical expressions for stress concentration factors. The influence of the residual stresses on the stress concentration can also be investigated by the numerical model. Finally, it is also possible to perform analyses of typical sea state conditions of the operating region of an FPSO, in conjunction with a hull beam bending model, to estimate the average operating stresses, as well as the number of cycles associated with the operating time of the FPSO platform. Then, it is possible to obtain more representative stress magnitudes, which together with appropriate S-N curves, can determine stress concentration factors of existing dents and evaluate if they can lead to a failure fatigue.

References 1. Anuário estatístico brasileiro do petróleo, gás natural e biocombustíveis: 2018/ Agência Nacional do Petróleo, Gás Natural e Biocombustíveis. ANP, Rio de Janeiro (2018) 2. Pré-Sal. Disponível em. http://www.petrobras.com.br/pt/nossas-atividades/areas-de-atuacao/ exploracao-e-producao-de-petroleo-e-gas/pre-sal. Acessado em 27 Jan 2019 3. Pinheiro, B.D.C.: Notas de Aula: EEN424 Resistência estrutural II. Departamento de Engenharia Naval e Oceânica, UFRJ, Rio de Janeiro (2016) 4. Dassault Systèmes Simulia Corp.: Abaqus/CAE User’s Guide, Providence (2014) 5. Figueredo, R.B.: Mapeamento de danos e imperfeições em modelos de painéis de navios. Escola Politécnica UFRJ, Rio de Janeiro (2017) 6. Estefen, S.F., Chujutalli, H.J., Soares, C.G.: Influence of geometric imperfections on the ultimate strength of the double bottom of a Suezmax tanker. Eng. Struct. 127, 287–302 (2016)

A Procedure for the Identification of Hydrodynamic Damping Associated to Elastic Modes Daniele Dessi(B) , Edoardo Faiella, and Filippo Riccioli Institute of Marine Engineering, National Research Council, via di Vallerano 139, Rome, Italy [email protected]

Abstract. In this paper, the estimation of modal damping concerning the ship bending modes is addressed experimentally via physical model tests of a cruise ship carried out both in the towing-tank and in the vibration lab. In the perspective of applying the proposed procedure to full-scale ships, output-only modal identification techniques are needed, and consequently the Band-Pass Proper Orthogonal Decomposition (BP-POD) as described in [1] is employed for the mode identification. Thus, the Random Decrement Technique (RDT) and the Logarithmic Decrement Technique (LDT) are finally applied for extracting the modal damping. A key point of the present analysis is the application of BP-POD to both “wet” and “dry” response tests. This allows for both (i) the isolation of hydrodynamic damping and (ii) the comparison of the damping estimation with the input-output techniques based on the calculation of the frequency response functions (FRFs). This analysis points out that the correspondence between the procedures can be achieved only approximately in terms of modal damping, rising some concern about the values to be used for modal updating. Keywords: Modal damping · Output-only identification techniques · Elastic scaled model

1 Introduction There are several reasons to deal with experimental damping analysis for marine structures. Above all, ship and offshore structures are mechanical systems which require a damping matrix to be described properly; accounting for damping is crucial to get the correct response level close to resonance conditions. There are also other reasons to estimate damping; for instance, damping variation is considered also as an indicator of certain failure modes in composites. As a specific feature, damping for marine structures is composed of two contributions, a structural (or “dry”) damping and a hydrodynamic (or “wet”) damping; being the overall damping fundamental to describe correctly certain phenomena like vortex induced vibrations or springing. However, it is then rather difficult to provide reasonable damping values for the global modes of large structures like ships or offshore installations starting from the material properties and other hydroelastic principles. © Springer Nature Singapore Pte Ltd. 2021 T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 305–320, 2021. https://doi.org/10.1007/978-981-15-4672-3_19

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The “a prior” damping modeling on the basis of some simplified relations like proportional damping as well as the interpretation of damping itself seem still open problems; the availability of experimental results about damping identification is then fundamental also to validate damping models. On the experimental side, the different ways by which damping can be extracted give interesting options to the damping analyst but also pose some questions when the extracted values do not correspond to each other [2]. Furthermore, it is worth to underline that this variability appears also within the same techniques if some settings are varied [1, 3]. In the intriguing perspective outlined above, addressing the damping extraction from elastically scaled models is interesting for several reasons. First, this problem retains some features of the full-scale one which final efforts are aimed at [4]. For instance, it exhibits a relatively high modal density typical of complex structures without making the problem too much involved. Secondly, it offers the chance of identifying the hydrodynamic damping by comparing test results in dry and wet conditions. In turn, the hydrodynamic damping estimation can then be used to isolate structural damping at full-scale. Therefore, in this paper, the Band-Pass Proper Orthogonal Decomposition (BP-POD) is applied to the vibration mode identification of the vertical bending modes of an elastically scaled model. The segmented elastic model was initially designed to reproduce the full-scale bending response of a cruise ship, and then was tested in the towing-tank (Fig. 1) under different wave conditions to take into account also the whipping behavior; however, a complete structural identification had never been performed before for this segmented model. The experimental modal analysis is performed with “dry” and “wet” vibration tests; the latter are carried out in regular waves and irregular sea to provide an excitation distributed over all the modes of interests, while the “dry” vibration tests has required the elastic model to be suspended on springs in the laboratory. After the mode shape identification, the proper orthogonal coordinates (POCs) associated to each identified mode are processed with the random decrement technique and the linear decrement technique to achieve a damping estimation related to a particular mode. One of the important points in this work is the comparison of the “dry” damping estimations with those obtained with the input-output technique based on the evaluation of frequency response functions. In a general perspective, the identification results point out that the selection of the modes corresponding to the full-scale behavior has to be carried out carefully in order to avoid that the damping identification is carried out on spurious or unphysical modes due to the segmented model layout.

2 Damping Analysis for Elastic Physical Models 2.1 Damping Significance To obtain a dynamic response close to full-scale by using elastically scaled models, damping is a key parameter. Clearly, the overall damping has structural and hydrodynamic contributions. The latter depends on the hull form as well as on the inclusion of some details like appendages or other devices, especially for roll motion or torsional modes; thus, hydrodynamic damping can be correctly represented at model-scale if hull geometry is correctly reproduced. The structural damping, due to the different topology

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Fig. 1. Elastic model tested in the towing-tank.

and materials employed in the physical model layout, has in principle a weak relationship with that one experienced at full-scale. A poor correspondence with full-scale damping may affect the physical meaning of the recorded responses for some problems like springing and vortex induced vibrations. Nonetheless, this limitation does not preclude the possibility of testing damping identification techniques at model-scale. In principle, the separation of hydrodynamic damping from the overall damping through measurements in water can be done only at model scale; at full-scale exciting the whole ship in dry conditions is practically impossible, and also for smaller ships parked in a dry-dock the determination of point-to-point transfer functions from which damping characteristics can be identified is hard. At model-scale, experimental modal analysis on the dry structure, conveniently constrained, instead allows for the determination of the structural damping. Thus, the procedure for evaluating the structural damping at full-scale (S) can be done in principal according to the following relationship: (S)

(S)

(S)

(S)

(M )

(M )

(M )

ζstr = ζall − ζhyd with ζhyd = ζhyd = ζall − ζstr

which implies the equivalence of hydrodynamic damping at model and full-scale as well as the possibility to isolate the hydrodynamic damping by testing the physical scaled model. For the same ship, hydrodynamic damping is likely to vary with the forward speed due to modifications of the ship waterline. This variation was clearly recorded in [5] for the first vertical bending modes in the case of a fast monohulls. The abrupt change after Fr = 0.3 was probably due to the flat transom which causes a detachment of the flow from the hull stern at high speeds. Not much results are available in literature about the sensitivity of damping variation to forward speed. This is also due to the uncertainty in the damping estimation which depends on several reasons: uncertainty of the experimental data, dependence on the chosen technique, sensitivity to some parameters in the extraction procedure, nonlinear damping. Another factor that may affect modal damping on the flexible modes is the sea state. It is not well known to which extent this phenomenon occurs, and if short-period variations are filtered out by long-time averaging. Regarding techniques for modal damping extraction, there is apparently no need of identifying at the same time the vibration mode shape. Based on information provided

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by the Finite Element (FE) analysis, one can directly select the frequency range to filter the signal provided by just one accelerometer or strain-gage. Though this is true for the 2-nodes mode dominating the ship whipping response, for higher-order modes this straightforward procedure might lead to attribute the identified damping to modes different to those one has in mind. For this reason, the review of the damping identification procedures is limited here only to those including at the same time the vibration mode identification, which in general require multiple experimental degrees-of-freedom (dofs) as well as sensors to get the desired spatial resolution. As well known in vibration analysis, there are basically two approaches for the identification of the vibration modes: (i) input-output techniques, using single or multi-point (roving hammer) excitation with multiple or one sensor, respectively, and (ii) output-only techniques, employing multiple outputs but no excitation measurement. While, in the first case, damping can be obtained from the half-power points of the transfer function peaks, in the second case there are different approaches: Stochastic Subspace Identification [6] uses pole selection from a stabilization diagram, FDD [7–9] or POD [10] require the calculation of a free-decay function, which the linear decrement technique is applied to. In the following, we will focus on the use of the Proper Orthogonal Decomposition and, in particular, of the Band-Pass Orthogonal Decomposition [1, 5]. 2.2 Elastic Scaling As outlined in the previous section, the damping extraction techniques are here applied to a scaled flexible model. Testing elastically scaled models is motivated by two main technical objectives: first, scaled experiments give us information about the elastic response of the full-scale ship; second, the collected data can be used to validate FSI simulations under conditions closer to those that will be faced in real cases. Froude scaling is typically employed to transform the reference values from full to model scale, and a reduced-order representation of the real structure is typically implemented to focus on some features of the ship response. As in the case which will be presented here, the backbone technique is often adopted because it offers several advantages with respect to the hydro-structural model [11]. Thus, the elastic physical model used for comparing the damping extraction techniques uses a backbone beam to scale accurately the 2-node vertical bending mode provided by a finite element model of a cruise ship. The significant rotary inertia of the not-structural masses (hull segments and ballast masses) counteracts the slenderness effect of the backbone beam on the frequency spacing of the bending modes. The beam slenderness, according to the Euler-beam model, gives a too large frequency spacing with respect to real ship structures, whereas shear area or rotary inertia tends to group frequencies. Scaling 3-nodes or higher-order modes may appear a useless game because they are less excited in waves; however, since these modes ‘live’ in a frequency range with a high modal density, the identification issues are instructive about what one may expect in more complex structural cases.

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3 Operational Modal Analysis 3.1 Procedure Overview In this section, the focus is on extracting damping from the Proper Orthogonal Coordinates (POCs) provided by POD with pre-filtering. The POCs aj (t) are time functions which appear in the decomposition of the response measurement vector w(t): w(t) =

L 

aj (t)ϕ j

j=1

where w ∈ RN , and the number L of proper orthogonal modes ϕ j necessary to get most of the signal energy is usually less than N . This aspect is less faced in literature because SSI [6] and FDD [9] are more popular methods for the estimation of modal parameters. Being damping intrinsically linked to oscillations of decreasing amplitude, damping estimation techniques in time domain are mostly concerned with the extraction of a free decay from the examined signal. Random Decrement Technique (RDT), Impulse Response Decay Method (IRDM) and Reverberation Decay Method (RDM) differ essentially in the way the free decay is obtained, and this choice is dependent on the type of excitation the structure is exposed to. Being the ship subjected to stochastic excitation due to the encountered waves, the RDT will be used to extract the free response from the POCs together with the evaluation of the correlation. As recalled previously, the POCs contain the modal parameter information relative to a certain vibration mode (linear normal mode, LNM) only if pre-filtering is applied to processed measurements. This implies that in the BP-POD we need to repeat the POD for each selected frequency band where a vibration mode is supposed to be. To have a good isolation of the vibration mode, or in other words, to have a POM converging to a LNM, the POM must have a clear (energy) prevalence over the other modes. Therefore, its largest eigenvalue σ1 from the POD  eigenvalue problem Rϕ = σ ϕ, with R the correlation matrix, must be such that σj > γ , where the summation is extended to all eigenvalues and the threshold σ1 energy fraction γ is usually larger than 1/2. A simple filtering procedure (i.e., cancelling out the frequency amplitudes outside the range of interest) is sufficient for the scope of determining the mode shapes. To summarize the adopted procedure, in Fig. 2. a flow-chart accounting for the steps of the performed analysis is shown. Filtering appears at the beginning of the considered flow-chart. Then POD is applied and POC time histories relative to the isolated mode are provided. At this point, the free-decay is calculated with RDT and successively the LDT gives the time-dependent damping from the analysis of free-decay curve. To give a final value for damping, averaging is necessary, and this step can be performed in different ways. 3.2 Damping Extraction via RDT-LDT The RDT was introduced by Cole in the late 1960s. Cole (1968,1973) was trying to develop a method for on–line failure detection in structures by measuring the change in vibration characteristics, such as the decay rate of a structure. Cole faced the difficulty of

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Fig. 2. Signal analysis procedure for extracting modal damping.

measuring ambient loads acting on structures, inspiring him to develop a method based on the measurement of response only. Vandiver et al. (1982) provided a more rigorous mathematical basis for RDT. They proved that the randomdec signature, in general, is not equivalent to the free decay. In fact, the randomdec signature is proportional to the autocorrelation function. The randomdec signature is equal to free decay only for the SDOF system, excited by Gaussian white noise. As individuals and together, Ibrahim (2001), Brincker (1990) and Asmussen have published a substantial number of papers on RDT. Asmussen’s dissertation (1997), on the application of RDT for modal analysis, has the best mathematical description of RDT and the parameters associated with it. Brincker et al. have provided an excellent explanation of the mathematics and methodologies for estimation of autocorrelation and cross–correlation using RDT. Kijewski and Kareem (2000) published their work on the reliability of RDT for the estimation of structural damping when some of the trigger conditions, associated with RDT, are relaxed. The basic idea of RDT is to pick out time segments and average them whenever the time series fulfills a given so-called “trigger condition”: Nt  ˆ ww (τ ) = 1 w(ti + τ )|w(ti )∈[α, α + α] D N k=1

When making use of the above equation, the random part of the time series tends to ˆ ww (τ ) can be interpreted as free decay, average out. Thus, the obtained RDT-signature D which is the fundamental idea behind extracting damping ratios by means of RDT. The method is not very effective when two or more outstanding system frequencies coexist, but one tries to exploit band-pass filters to fulfill this requirement. Different triggering conditions can be conceived other the one indicating above. Assuming that the randomdec signal gives us a free decay, the damping ratio can be estimated by means of the logarithmic decrement δ: δ = ln

2π ζ wn = wn+1 1 − ζ2

where wn and wn+1 are amplitudes of two successive free vibrations of like sign. Thus, for small damping ratios, one finally gets ζ =

δ 2π

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The free response obtained with RDT may provide amplitude-dependent damping ratios. In many cases, damping seems indeed to increase significantly when the amplitude of oscillation gets to be small. In hydroelastic codes, using an amplitude-dependent damping poses several problems. Therefore, some kind of averaging is necessary to achieve a single, hopefully representative, value. Several types of averaging can be performed on these values. The choice depends also on the final analysis to which the response computation is devoted to.

4 Case of Application 4.1 Dry-Mode Identification via RHT The modal identification on the “dry” structure via the evaluation of frequency response functions is performed using the roving hammer technique (RHT), which employs the measurement of the impulsive force applied at different locations with an instrumented hammer and the measurement of the response with one or more accelerometers placed at given points. The RHT points to calculate at least one row of the FRF matrix by changing the excitation input position over the experimental dofs; in one of this point is also located the accelerometer. In this perspective, the RHT is a multiple-input, single-output technique which gives a mapping of the vibration modes in a sufficiently large range due to the broad band excitation provided by the hammer impulse and, consequently, it allows for a comparison with output-only techniques. The segmented hull is suspended with fifth spring pairs as shown in Fig. 3 at the Vibration and Structural Diagnostic Lab. of CNR-INM. This allows for rigid-body oscillation frequencies (heave and pitch) far less than one fifth of the lowest elastic vibration frequency, a ratio that is usually set as a minimum requirement.

Fig. 3. Segmented elastic model suspended with pairs of springs.

The experimental dofs, coincident with the points where the hammer sticks the structure, are distributed along the upper face of the beam. The list of identified vibration modes which have a relevant vertical component is reported in Table 1. The presence

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of multiple modes sharing the same number of nodes along the beam constitutes an obstacle in establishing a simple relationship with the reference mode shapes. This is of concern for mode shapes having 3-nodes, and to a less extent for modes with 4-nodes. Table 1. Dry vibration modes identified with the Roving Hammer Technique. Mode ID Frequency Nodes

Notes

1RHT

10.4 Hz

2-nodes Unique 2-nodes

2RHT

28.4 Hz

2-nodes Multiple 2-nodes

3RHT

33.2 Hz

3-nodes Multiple 2-nodes

4RHT

35.6 Hz

3-nodes Multiple 2-nodes

5RHT

38.8 Hz

3-nodes Multiple 2-nodes

6RHT

42.4 Hz

3-nodes 4-nodes shape

7RHT

47.1 Hz

3-nodes Stern-up shape

8RHT

54.0 Hz

3-nodes Bow-dw shape

9RHT

57.8 Hz

4-nodes Bow-flat shape

10RHT

60.7 Hz

2-nodes 3-nodes shape

11RHT

65.3 Hz

4-nodes Multiple 4-nodes

12RHT

74.3 Hz

4-nodes Bow-flat shape

To highlight the modes more related to the deformation of the backbone beam alone, the Modal Assurance Criterion index, or MAC index, is calculated. The MAC is defined as follows:  (i)T (j) 2 ϕ ϕ  MACij =  (i)T (j)  (j)T (j)  ∈ [0, 1] ϕ ϕ ϕ ϕ where i = j = 1, . . . , 12. Since in this case the vectors span over the same set of mode shapes, the index takes the name AutoMAC. The results of the AutoMAC evaluation are plotted in Fig. 4. The gray-scale of the cell belonging to the i-th rowand to the  j-th column is proportional to the AutoMAC value relative to the mode pair ϕ (i) , ϕ (j) ; close-to-zero values (white cells) indicate orthogonality between modes, close-to-one values (black cells) indicate non-orthogonal modes. Looking at the AutoMAC values in Fig. 4, a unique answer cannot still be obtained. Nonetheless, if one moves along the bottom row (or left column) several modes appear to be ‘near-orthogonal’ to the 2-node mode (10.4 Hz); (3-nodes) modes #3, #4, #5 and (4-nodes) modes #11 and #12. Next, taking into account the orthogonality between the selected 3-nodes and 4-nodes modes, the following ‘near-orthogonal’ combinations are admissible: – #1, #3, #11 and #1, #3,# 12 – #1, #4, #11 and #1, #4,# 12 – #1, #5, #11 and #1, #5,# 12

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Fig. 4. AutoMAC calculated between the modes identified with RHT. Greyscale ranging over the AutoMAC values (white = 0, orthogonal /black = 1, parallel). (Color figure online)

To investigate further this correspondence between ship beam “clean” modes and segmented hull modes, additional dry vibration tests have been performed by incrementally changing the physical model configuration from the bare one (only backbone) to the full one with all segments attached. The intermediate configurations had one or three segments attached. The MAC index reported in Table 2 correlates the bare backbone modes (i.e., without ballast) with those of the full physical model, with all the segments and minor components attached. The MAC values are reported inside each cell, which has again a greyscale color proportional to this value. It is clearly evident that the modes #1, #3 and #11 show the highest scores in terms of correlation with those of the “clean” backbone. This interpretation is confirmed also looking to the generation of the multiple-nodes modes as long as more structural components are linked to the backbone, not shown here for sake of conciseness. A simple finite element developed with MSC.NASTRAN using beam elements and all the segment masses lumped at the leg ends substantially confirms the interpretation of results. The results in terms of modal frequencies as reported in the Table 3 for the modes of interest alone. The approximation of considering the segment mass lumped at the leg ends tends to provide higher frequencies than expected with the exact distribution, but some differences may arise also in the use of beam elements for the legs. The numerical results suggest again that the most probable mode combination is the first one, that is, modes #1, #3 and #11. 4.2 Dry-Mode Identification via POD The same analysis has been carried out with the output-only analysis based on BP-POD. To perform this task, tip-tap excitation has been performed by hitting the top-face of the backbone beam with the hammer tip. The acceleration was recorded in the same ten

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Seg. Hull Mode ID 1RHT 2RHT 3RHT 6RHT 7RHT 8RHT 11RHT

Frequency 10.4 28.3 33.2 42.4 47.1 54.0 65.3

Bare backbone beam modes 2-nodes 3-nodes 4-nodes 23 Hz 71 Hz 124 Hz 93.4 15.7 1.6 64.8 1.4 1.3 20.6 77.5 0.1 0.1 0.1 61.1 69.8 45.2 4.7 0.1 59.0 15.2 9.4 0.1 84.7

Table 3. Segmented-hull vibration mode frequency obtained with a FE model. Mode ID

Frequency

Nodes

1FE-B

11.0 Hz

2

2FE-B

33.5 Hz

3

3FE-B

60.4 Hz

4

positions already assumed as experimental dofs for the input-output analysis. The FFT of the processed signals, i.e., averaged over all the accelerations, shows clearly several peaks in the same positions of the modes already identified with the RHT. This diagram allows for deciding the filter bands for the successive BP-POD analysis. The FFT in Fig. 5 shows some peaks that were not considered in the RHT analysis because the poles in the realization diagram were unstable or they were affected by mainly local modes.

Fig. 5. FFT of the sum of the acceleration signals acquired with tip-tap excitation for BP-POD analysis.

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A summary table (Table 4) gives a brief overview of all the identified the modes, which are also plotted in the figures below. It is worth noting that the modes #1, #3 and #11 are identified in frequency intervals where the energy associated to that mode is largely higher than the others, a feature which indicates the capability of separating the modes inside each frequency band. This property is based on the following index σˆ i which gives the fraction of the overall energy associated to the POM i: Table 4 Dry vibration modes identified with the BP-POD technique. Mode ID

Freq. band [Hz]

Freq. [Hz]

Nodes

σˆ 1 (POM prevalence)

1BP-POD

9.0–12.0

11.2 Hz

2-nodes

99.5

2BP-POD

27.5–29.5

28.7 Hz

2-nodes

77.8

3BP-POD

32.0–34.0

32.9 Hz

3-nodes

71.9

4BP-POD

35.5–37.5

36.5 Hz

3-nodes

63.2

5BP-POD

37.2–39.2

38.0 Hz

3-nodes

66.9

6BP-POD

41.0–43.0

41.9 Hz

3-nodes

46.5

7BP-POD

46.0–48.0

46.5 Hz

3-nodes

80.2

8BP-POD

53.0–55.0

54 Hz

3-nodes

60.4

9BP-POD

56.0–58.0

57 Hz

4-nodes

80.7

10BP-POD

58.0–61.0

59.0 Hz

2-nodes

79.9

11BP-POD

61.0–65.0

63.7 Hz

4-nodes

91.6

12BP-POD

69.0–75.0

71.3 Hz

4-nodes

84.3

σˆ i = σi



L j=1

σj

and is reported in the right column of Table 4 for the most excited mode (more details about the identification procedure can be ound in [1, 3]). In the following only the modes with a POM prevalence larger than 60.0% are plotted. It is worth noting that the filtering interval affects the mode prevalence. The 2-node bending mode is clearly identified Fig. 6, where the points represent the sensor measurements and a spline pass through them to give a better representation. In Fig. 7, while the “red” and “green” modes have similar frequencies as well as positions of nodal points, the “blue” one has a quite different shape near the bow, where the mode shape curvature changes sign. On the other hand the “green” mode exhibit a very slight crossing of the x axis. These remarks further support the considerations made in Sect. 4.1 which lead to identify mode #3 as the best representative of global ship 3-nodes mode. The two candidates for the 4-nodes mode are shown in Fig. 8. That one which was selected as 4-nodes mode has a worse shape (“red” curve) than the one that was excluded (“blue” curve). This condition is really opposite with respect to identification via RHT where the modal shapes are in the opposite order as shown in Fig. 9.

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Fig. 6. Dry 2-nodes vertical bending mode obtained via BP-POD.

Fig. 7. Dry 3-nodes vertical bending modes identified via BP-POD.

Fig. 8. Dry 4-nodes vertical bending modes identified via BP-POD.

4.3 Wet-Mode Identification The identification of the “wet” bending modes is carried out on the basis of the strain measurements, which are provided by ten strain-gages applied on the top-face of the backbone beam. The strain-gage outputs were transformed into vertical bending moments by determing experimentally the proper conversion coefficients. For this purpose, the backbone beam was loaded with known weights at given locations, and successively

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Fig. 9. Dry 4-nodes vertical bending modes identified via RHT: 65.3 Hz mode (left), 74.4 Hz mode (right).

the exact bending moment at the strain-gage locations was compared with the sensor voltage output to obtain these coefficients. The time-history of the beam displacements is obtained by a double spatial integration along the beam using the same procedure described in [1]. Two kind of tests have been analyzed: one in regular waves with a wave amplitude aw = 0.064 m and frequency fw = 0.622 Hz at model scale, and in other in irregular head waves (JONSWAP spectrum) at a forward speed of 15 kts (full-scale). In Fig. 10 the FFT of the sum of the strain signals is reported for the regular wave case; while it is easy to identify two frequencies around 7.3 Hz and 49.7 Hz, corresponding to near-symmetric modes with respect to midship with 2 and 4 nodes, respectively, no sharp frequency peak can be highlighted in-between. A more detailed analysis, based on the mode prevalence in narrow frequency intervals into which the frequency range of interest is divided [1], is also shown in Fig. 10. The mode prevalence (see also Table 5) leads to restrict the filtering interval for the BP-POD relatively to the 2-nodes and 4nodes modes, confirming their presence at the frequency peaks highlighted by FFT; however, also the POV prevalence curve provides only very weak suggestions about the 3-nodes mode. This seems to indicate that the energy associated to the 3-nodes mode is distributed over several modes sharing a similar shape; these modes probably differ for the different coupling with the legs flexible modes. 4.4 Damping Estimation Using the procedure described in Sect. 3, the modal damping is extracted. In particular, here we focus on the principal bending modes and on the variation of frequency between “dry” and “wet” conditions identified with BP-POD. The output-only technique (BPPOD) tends to overestimate the damping of dry modes with respect to the input-output (RHT) in some cases as reported in Table 6. It is worth to underline that the damping estimation is affected by three main factors: (i) the width of the filter pass-band around the mode natural frequency, (ii) the thresholds for the calculation of the free-decay response using the RDT and, finally, (iii) the way one obtains the average value by applying the LDT on the free-response. The results shown here refer to a frequency width of 10% of the mode natural frequency, a choice of the lower and upper signal RDT thresholds equal to the standard deviation of the signal and to 1.1 the standard deviation, respectively, and

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Fig. 10. FFT (green curve) and mode prevalence in the POD filter bands (red) from the strain gage signals relative to the regular wave tets.

Table 5. Wet vibration modes identified with the BP-POD technique. Mode ID Freq. band [Hz] Freq. [Hz] Nodes

Mode prevalence %

1WET

6.2–8.4

7.3 Hz

2-nodes 81%

3WET

49.0–52.0

49.7 Hz

4-nodes 92%

a damping average considering only the peak-to-peak decrement values greater than 0.1 the largest decrement in the free decay response. If we move to compare “dry” and “wet” cases, in the latter condition the modal damping is higher as expected. In the right-end column, in particular, the difference between the “dry” and the “wet” estimation using the output-only technique provides the hydrodynamic damping associated with that mode; thus, the hydrodynamic damping ratio is equal to 0.50% for the 2-nodes whipping mode (Table 7). Table 6. Comparison between dry damping estimations between RHT and BP-POD Test type Dry test Dry test

Dry test

Method

BP-POD

RHT

RHT

Var.

f [Hz]

Damping Ratio [%] Damping Ratio [%]

2-nodes

10.4

0.44

0.43 (0.65)

3-nodes

33.2

1.27

6.5 (8.1)

4-nodes

65.3

0.84

5.5 (6.2)

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Table 7. Comparison between dry and wet damping estimations with BP-POD technique. Test type

Dry test

Dry test BP-POD

BP-POD

BP-POD

BP-POD

Var.

f [Hz]

Damping Ratio [%]

f [Hz]

Damping Ratio [%]

Diff. [%]

2-nodes

10.4

0.43 (0.65)

7.3

0.93

0.50

Method

Wet test

Wet test

Hydrodynamic damping

5 Conclusions In this paper, the modal damping of an elastic ship model has been estimated using a procedure based on a sequential application of three techniques: (i) the BP-POD, which identifies the modes shapes and the associated time coordinate, (ii) the RDT which extracts a free decay curve from each band-limited application of the POD and, finally, (iii) the LDT which computes the damping ratio associated with the deacreasing oscillations of the free-decay curve. The comparison between the modal damping ratio identified in dry testing and in the towing-tank allows for isolating the hydrodynamic damping alone, provided that the mode is tracked correctly from the “dry” to the “wet” condition. The identification of hydrodynamic damping can be also considered as an intermediate result allowing for the identification of structural damping on the full-scale ship. Nonetheless, the present analysis points out some features as well as opens some questions concerning the application of this modal identification procedure. First, the correspondence between the full-scale and the model-scale modes needs attention when the modal density is high, as it occurs for the 3-nodes. Moreover, some modes may be less excited under certain conditions. Second, the BP-POD tends to provide higher values of the modal damping with respect to the RHT and, from a check performed with SSI on the same cases, this seems a common feature of output-only techniques in the time domain. Even if in the present paper the sensitivity of the damping estimation with respect to some settings has not been presented for sake of coinciseness, it is likely to occur that some of the identified values are affected by possible variations within a certain range and, correspondingly, some trends may need to be reconsidered.

References 1. Dessi, D., Faiella, E.: Modal parameter estimation for a wetted plate under flow excitation: a challenging case in using POD. J. Sound Vib. 449, 214–234 (2019) 2. Kim, Y., Park, S.G.: Wet damping estimation of the segmented hull model using the random decrement technique. J. Soc. Nav. Arch. Korea 50(4), 217–223 (2013) 3. Dessi, D.: Damping of ship global modes: techniques and analysis. In: PRADS 2013 - Proceedings of the 12th International Symposium on PRActical Design of Ships and Other Floating Structures (2016) 4. Kim, Y., Park, S.-G., Kim, B.-H., Ahn, I.-G.: Operational modal analysis on the hydroelastic response of a segmented container carrier model under oblique waves. Ocean Eng. 127, 357–367 (2016)

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5. Mariani, R., Dessi, D.: Analysis of the global bending modes of a floating structure using the proper orthogonal decomposition. J. Fluids Struct. 28, 115–134 (2012) 6. van Overschee, P., De Moor, B.: Subspace Identification for Linear Systems. Kluwer Academic Publishers (1996) 7. Asmussen, J.C.: Modal analysis based on the random decrement technique – application to civil engineering structures. Ph.D. thesis at the University of Aalborg (1997) 8. Brincker, R., Jensen, J.L., Krenk, S.: Spectral estimation by the random decrement technique. In: 9th International Conference on Experimental Mechanics, Copenhagen, Denmark (1990) 9. Brincker, R., Zhang, L., Andersen, P.: Modal identification from ambient response using frequency domain decomposition. In: Proceedings of the 18th International Modal Analysis Conference (IMAC), San Antonio (2000) 10. Feeny, B.F., Kappagantu, R.: On the physical interpretation of proper orthogonal modes in vibrations. J. Sound Vib. 211, 607–616 (1998) 11. Dessi, D., Brennan, F., Hoogeland, M., Li, X.B., Michailides, C., Pearson, D., Romanoff, J., Shi, X.H., Sugimura, T., Wang, G.: Experimental methods. In: Proceedings of the 19th International Ship and Offshore Structures Congress, ISSC 2019, vol. 1 (2019) 12. Cole, H.A.: On-the-line analysis of random vibrations. AIAA Paper No. 68-288 (1968) 13. Cole, H.A.: On-line failure detection and damping measurement of aerospace structures by random decrement signatures. In: NASA CR-2205 (1973) 14. Ibrahim, S.R.: Efficient Random Decrement Computation for Identification of Ambient Responses. IMAC XIX, Kissimmee, USA (2001) 15. Kijewski, T., Kareem, A.M.: Relibaility of random decrement technique for estimates of structural damping. In: 8th ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability (2000)

Vibration Damping of Large Containership in Operation Saeed Shakibfar1 , Ingrid M. V. Andersen2 , and Anders Brandt1(B) 1 Department of Technology and Innovation, University of Southern Denmark,

Odense, Denmark [email protected] 2 A.P. Møller-Mærsk, Copenhagen, Denmark

Abstract. Vibration damping is a paramount factor in understanding vibrations in structures because the response of a structure under stationary dynamic loads, at resonance, is directly (inversely) proportional to the damping. Also, transient response of a structure decays exponentially at a rate determined by the damping factor and eigenfrequency of each mode. Thus, to compute forced responses, the damping must be known. Despite damping thus being of large importance, there is limited knowledge of the damping of ships in general and of large container ships in particular. Some studies have suggested the damping is speed dependent, whereas others have not. In the last ISSC 2018 report, it was concluded that more data are needed from full-scale measurements. In the present paper the dynamics of a large container ship in operation is investigated, from full scale measurements in different operation conditions. State-of-the art, so-called operational modal analysis (OMA) is used for computing the eigenfrequencies, damping factors, and mode shapes of the ship. It is discussed how particularly the vibration damping varies with different operating parameters such as cargo, speed, and sea state. Keywords: Vibration damping · Operational modal analysis · Large container ship

1 Introduction Container ships have become larger over the past decades. This affects their structural response to wave-induced excitation particularly for global high-frequency vibrations. Understanding the structural damping of wave-induced vibration is required to correctly obtain the acceleration levels of the hull and the loads acting on the containers. However, assessing the actual damping of a ship in operation is cumbersome, and perhaps therefore much research has not been conducted in this area, and there is a large need for knowing target damping [1, 2]. Using large finite element (FE) models, this behavior can be assessed numerically, particularly natural frequencies can be accurately computed. However, FE models are dealing with uncertainties including effects of the water/hull interaction, wave impact location and duration, structural damping, the deformation modes, and the complex interaction between the ship structure and the vibration environment in different operating © Springer Nature Singapore Pte Ltd. 2021 T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 321–329, 2021. https://doi.org/10.1007/978-981-15-4672-3_20

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conditions (speed, sea state, etc.). Hydrodynamic mass has also been added to the FE analysis of the ship [3]. For methods to compute ship response, damping has to be input externally [4]. There is thus a need for more full-scale data under different operating conditions to correct and validate numerical models. Operating condition includes information on wind and wave conditions, ship speed, and cargo condition which may all affect natural frequencies and damping values. In order to estimate damping, modal analysis from full-scale measurements can be used. Operational modal analysis (OMA) is a special kind of experimental modal analysis (EMA), where the excitation is unmeasured (output-only) as opposed to classic EMA (input–output) where the excitation is artificial and measured, where the modal parameters: resonance frequency, damping, and mode shapes, are computed. OMA is a potential technique for validation of ship structure dynamics with some great advantages over EMA. Previous studies focused on the modal parameter estimation using accelerometer data on ship structures by operational modal analysis [5, 6]. Modal analysis provides frequency, damping ratio, and mode shape identification. The hydroelastic damping effects have been reported to depend on the operating condition, especially speed, in towing tank tests [7]. In a recent study on a full-scale container ship (a RO-LO ship) it was found that the damping was speed dependent [8]. It was therefore suggested that measurements of damping should be estimated under different operating conditions to establish dependence on such conditions. In addition, for full scale measurements, more data are required, as reported in the last ISSC report [9]. The present study aims to assess the effects of environmental and operational parameters, particularly speed, on the dynamic response of the ship. Using OMA for dynamic characterization based on full-scale measurements of a ship, modal parameters were calculated for the ship under different operating conditions. Measurements were obtained under three different operating conditions of the ship: 15.15-knots cruising speed, 16.35-knots cruising speed and 19.75-knots cruising speed.

2 Methods 2.1 Equipment and Measurement Setup The vessel is an 8,400 TEU post Panamax container ship with the main dimensions given below: Length overall: Length between perpendiculars: Breadth moulded: Depth moulded: Scantling draught:

353.0 m 336.4 m 42.8 m 24.1 m 15.0 m

The vessel is instrumented with strain gages and accelerometers as described in [10]. For the present investigation, only accelerometer signals were considered; two accelerometers were located at deck level amidships and in the aft of the ship, and one accelerometer was mounted in the bow of the ship.

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2.2 Operating Conditions The measurements contain three different data sets obtained under different operating conditions of the ship including: Speed (knots), Mean Draught (m), Sea Water Depth (m), Sea Relative Wind Speed (m/s) and Sea State (m). Table 1 shows three different operating conditions sorted on speed. The operating conditions are available from ship’s noon report and are reported manually by onboard crew. 2.3 Modal Parameter Estimation The modal parameters were estimated using the modified Ibrahim time domain method (MITD) described in [11]. The MITD method is one of the available methods for modal parameter extraction in the free ABRAVIBE MATLAB toolbox [12]. This method is a state-of-the-art parameter estimation algorithm, in principle equivalent with the covariance-driven stochastic subspace identification (Cov-SSI) method that is readily available in commercial software today [13]. Such methods are providing superior results, especially for damping, compared to, e.g., the logarithmic decrement technique. The MITD method uses correlation functions that are stacked in a block Hankel matrix, which is decomposed by a singular value decomposition. To account for the unknown model order, a so-called stabilization diagram is used, in which the modal parameters are presented for every model order from a low order to some upper limit. Stable poles, i.e. poles that are not changing between the model orders, are then selected from the stabilization diagram. In the present analysis, the full correlation function matrix was used, i.e. all channels were used as references. More information regarding the signal processing in OMA applications may be found in [14]. A total of 200 lines from the correlation functions were used for the block Hankel matrix, corresponding to 40 s. The first ten lines were discarded to avoid the influence of measurement noise. A typical autocorrelation function from the 15.15-knots condition is presented in Fig. 1. The poles of the compressed system matrix estimated at increasing system order (number of poles), up to 40 modes, were plotted in a stabilization diagram as shown in Fig. 2. From the stabilization diagram, stable poles were manually selected. The modal assurance criterion (MAC) [15] was used to detect similarity between two estimated modes. The MAC value between two mode shapes with indices r and s, respectively, is defined by 2  {ψ}H r {ψ}s   MAC(r, s) =  H {ψ}H r {ψ}r {ψ}s {ψ}s

(2.1)

where {ψ}r is mode shape r, and H denotes Hermitian transpose. It can be regarded as the normalized correlation coefficient between the two vectors. Thus, the MAC value is a number between 0 and 1, where a value of 1 means that the mode shapes are equal, except for a possible scaling factor, to which the MAC is insensitive. The MAC values are normally presented in a matrix with all pairs of mode shapes compared, either to the same set of mode shapes, or to another set of mode shapes.

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Fig. 1. Typical autocorrelation function for the 15.15 knot condition. Note that data are unscaled

Fig. 2. Stabilization diagram for the first condition (15.15-knots). Plus signs and circles indicate stable and unstable poles, respectively, and the tree modes for vertical, torsional and horizontal directions were found between 0.4 to 0.6 Hz (indicated by green plus signs)

3 Results and Discussion The modal parameters, natural frequencies, modal damping ratios, and mode shapes were estimated from the measurement data under the three operating conditions, described in Table 1. For all analyzes, stable poles were manually selected from the stabilization diagram, as shown by the example in Fig. 2.

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Table 1. Different operating conditions, sorted by speed variable. Condition nr. 1

Speed 15.2

MeanDraught 11.66

SeaWaterDepth 56

SeaRelativeWindSpeed 4.52

SeaState 0.94

2

16.4

12.29

95

18.40

1.64

3

19.8

13.23

101

8.21

0.32

The natural frequencies and modal damping ratios of the estimated modes for all mode shapes from the three operating conditions are presented in Table 2. The natural frequencies of all modes consistently decreased with increasing speed, from 0.522 to 0.515 Hz for the 2-node vertical bending. The changes for all modes are plotted in Fig. 3. This is consistent with the results in [8] and may be contributed by the added mass from the bow wave. Relative damping as a function of speed is plotted in Fig. 4. It can be seen that the damping increases with speed for all mode shapes except for the 2-node vertical bending, for which it is close to constant between the first two speeds. This is also reasonably consistent with the results in [8], where the damping did not change significantly between anchored and low speed (10 knots) conditions, whereas an increase in speed to rated speed of 18 knots resulted in a significant increase in damping for the vertical bending. The damping in the present study, however, increases significantly also for the horizontal 2-node mode and the torsion, whereas the study in [8] found insignificant change in damping for those modes. Table 2. Observations of the measurement modal parameters and the corresponding mode shapes for three different operating conditions. The modes are HB: horizontal 2-node bending, VB: vertical 2-node bending, and T: first torsional mode. fr [Hz] 0.49

Condition Nr. 0 1

0.52 2 0.60 2 0.47 0

2

0.51 5 0.57 2 0.45 5

3

0.48 8 0.52 2

ζr [-]

Mode

0.020

HB

0.010

VB

0.017

T

0.023

HB

0.010

VB

0.022

T

0.030

HB

0.016

VB

0.028

T

The auto-MAC matrix for the first condition is shown in Fig. 5. Here it can be seen that the off diagonal terms are very low, signifying that the three modes are well

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Fig. 3. Plot of natural frequencies versus speed

Fig. 4. Plot of damping ratio versus speed

separated by the chosen measurement locations. This is also a good indication that the modal parameter estimation has managed to separate all modes. All mode shapes are, furthermore, showing a very low complexity (normal modes, i.e. all points move in-phase, or out-of-phase). Next, the mode shapes between the modes obtained for the different operating conditions were compared. In Figs. 6 and 7. the cross-MAC matrices between the first and second, and the first and third operating conditions are shown, respectively. The figures

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Fig. 5. Auto-MAC for the first condition (15.15-knot)

show that there are some changes in the mode shapes between the mode shapes, but relatively small. The differences may be contributed to the change in boundary conditions between the conditions.

Fig. 6. Cross-MAC matrix between mode shapes for different conditions

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Fig. 7. Cross-MAC matrix between mode shapes for different conditions

4 Conclusions In the present study we have presented operational modal analysis results in the form of natural frequencies, damping factors, and mode shapes, of an 8,400-TEU container ship during three different operating conditions. Although many conditions are different between the three conditions, we have only compared the modal parameters versus speed. The speed was 15.2, 16.4, and 19.8 knots in the three conditions. Three modes were identified; the 2-node horizontal bending mode was found to have natural frequencies ranging from 0.49 Hz (first speed) to 0.46 knots for the third speed. The 2-node vertical bending mode changed from 0.52 to 0.49 knots, and the first torsion mode from 0.60 to 0.52 knots. The (relative viscous) damping factors were found to increase with increasing speed, from 2% to 3% for the horizontal mode, from 1% to 1.6% for the vertical bending mode, and 1.7% to 2.8% for the torsion mode. The mode shapes between the conditions were also compared. They were found to change, indicating some changed boundary conditions, but no particular pattern was found. Acknowledgments. The authors wish to gratefully acknowledge DNVGL for making the measured data available for research collaboration and the Danish Maritime Fund for supporting the research.

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References 1. Storhaug, G.: The measured contribution of whipping and springing on the fatigue and extreme loading of container vessels. Int. J. Naval Archit. Ocean Eng. 6, 1096–1110 (2014). https:// doi.org/10.2478/IJNAOE-2013-0233 2. Storhaug, G., Laanemets, K., Edin, I., Ringsberg, J.W.: Estimation of damping from wave induced vibrations in ships. In: Proceedings of the 6th International Conference on Marine Structures (MARSTRUCT), Lisbon, Portugal, 8–10 May 2107 3. Wilken, M.M.A., Voss, H., Cabos, C.: Efficient calculation of fluid structure interaction in ship vibration. In: Soares, G., Fricke, W. (eds.) Advances in Marine Structures. Taylor & Francis Group, London (2011) 4. El Moctar, O., Ley, J., Oberhagemann, J., Schellin, T.: Nonlinear computational methods for hydroelastic effects of ships in extreme seas. Ocean Eng. 130, 659–673 (2017). https://doi. org/10.1016/j.oceaneng.2016.11.037 5. Kim, Y., Kim, B.-H., Choi, B.-K., Park, S.-G., Malenica, S.: Analysis on the full scale measurement data of 9400TEU container carrier with hydroelastic response. Mar. Struct. 61, 25–45 (2018). https://doi.org/10.1016/j.marstruc.2018.04.009 6. Hageman, R.B., Drummen, I.: Modal analysis for the global flexural response of ships. Mar. Struct. 63, 318–332 (2019). https://doi.org/10.1016/j.marstruc.2018.09.012 7. Coppotelli, G., Dessi, D., Mariani, R., Rimondi, M.: output-only analysis for modal parameters estimation of an elastically scaled ship. J. Ship Res. 52, 45–56 (2008) 8. Orlowitz, E., Brandt, A.: Operational modal analysis for dynamic characterization of a Ro-Lo ship. J. Ship Res. 58, 216–224 (2014). https://doi.org/10.5957/JOSR.58.4.140015 9. Ergin, A., et al.: ISSC committee II. 2–dynamic response. In: Proceedings of the 20th International Ship and Offshore Structures Congress (ISSC 2018) (2018) 10. Storhaug, G., Kahl, A.: Full scale measurements of torsional vibrations on Post-Panamax container ships. In: Proceedings of the 7th International Conference on Hydroelasticity in Marine Technology, Split, Croatia (2015) 11. Allemang, R.J., Brown, D.: Experimental modal analysis and dynamic component synthesis. Volume 3. Modal Parameter Estimation (1987) 12. Brandt, A.: ABRAVIBE – A MATLAB toolbox for noise and vibration analysis and teaching. Department of Technology and Innovation, University of Southern Denmark (2018). http:// www.abravibe.com 13. Peeters, B., De Roeck, G.: Reference-based stochastic subspace identification for output-only modal analysis. Mech. Syst. Sig. Process. 13, 855–878 (1999). https://doi.org/10.1006/mssp. 1999.1249 14. Brandt, A.: A signal processing framework for operational modal analysis in time and frequency domain. Mech. Syst. Sig. Process. 115, 380–393 (2019). https://doi.org/10.1016/j. ymssp.2018.06.009 15. Allemang, R.J.: The modal assurance criterion - Twenty years of use and abuse. Sound Vib. 37(8), 14–23 (2003)

Study on the Effect of Liquid in Tanks on the Hull Girder Vibration Hiroyuki Takahashi1(B) and Yukitaka Yasuzawa2 1 Ship and Offshore Design Department, Design Division, Japan Marine United Corporation,

Tokyo, Japan [email protected] 2 Department of Marine Systems Engineering, Kyushu University, Fukuoka, Japan

Abstract. For the vibration response analysis of ship hull structure using whole ship model, it is important to consider the influence of liquid such as sea water around the hull, sea water in ballast tanks, crude oil in cargo holds of tanker. And it is known that when tank containing liquid vibrates in horizontally or rotationally, the effective mass of vibration does not coincide with the total mass of contained liquid due to the free surface effect of the liquid. In this paper, the influence of modeling method of liquid in tank for the whole ship vibration analysis has been investigated. First, the characteristics of effective mass of contained liquid when a tank is sinusoidally oscillated horizontally or rotationally respectively are reviewed qualitatively using series expansion method for liquid motion in the tank. Next, the influence of effective mass of liquid in tank on the vibration analysis of hull girder vibration modes has been investigated. Natural frequencies of hull girder vibration modes obtained by using accurate virtual mass method and simplified method for tanks are compared with the measurement results at the sea trial. And the reason of the discrepancy between the results of virtual mass method and simplified one is discussed and investigated quantitatively by using block models of tanks which are extracted from the whole ship model. Keywords: Vibration · Response analysis · Whole ship model · Virtual mass

1 Introduction These days, customer’s requirements for the vibration response level of the superstructure of large merchant ships such as tanker, bulk carrier and container ships become more severe than before. Therefore the accuracy of the estimation of vibration response of the superstructure at the design stage is very important in order to achieve customer’s satisfaction on the habitability in cabin area. It means that the accuracy of vibration response analysis using whole ship model is required. So far, the authors have introduced investigations to improve the accuracy of vibration response analysis using whole ship models [1–3]. For the vibration response analysis of ship hull structure, it is important to consider the influence of liquid such as sea water around the hull, sea water in ballast tanks, crude © Springer Nature Singapore Pte Ltd. 2021 T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 330–344, 2021. https://doi.org/10.1007/978-981-15-4672-3_21

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oil in cargo holds of tanker. In the finite element analysis of vibration using whole ship model, the virtual mass method such as MFLUID function of NASTRAN is commonly used to consider the added mass effect of outer liquid in contact with hull surface. But if virtual mass method is applied for inner liquid contained in tanks additionally, load on computer becomes remarkably higher because the mass matrix in the governing equation of FE(Finite Element) vibration analysis becomes denser compared with the mass matrix of ship structure and lumped mass only. Therefore simplified method such as application of additional density to shell elements or added masses at nodes are used for the inner liquid in tanks in the whole ship vibration analysis as approximated approach. But it is known that when tank containing liquid vibrates in horizontally or rotationally, the effective mass of the liquid due to the tank motion does not coincide with the total mass or rotational inertia of contained liquid due to the free surface effect of the liquid [4]. Therefore accuracy of the application of the simplified method to the tank liquid in the whole ship vibration analysis should be evaluated. In the present study, we focus on the influence of application of the simplified method for tank liquid on the results of whole ship vibration analysis. And the influence of different modeling methods of liquid in tanks for the whole ship vibration analysis has been investigated and discussed. First, the characteristics of effective mass of contained liquid when a tank is sinusoidally oscillated horizontally or rotationally respectively are reviewed qualitatively using series expansion method for liquid motion in the tank [5]. That is because horizontal tank motion appears in horizontal or longitudinal mode of hull girder vibration and the rotational motion of tank appears in torsional vibration and in the vicinity of the nodes of vertical modes of hull girder vibration. And effective masses of liquid in tanks for each vibration mode are presented as a function of ratio of height and width of tank. Next, the influence of effective mass of liquid in tank on the natural frequencies of hull girder vibration mode has been investigated. Natural frequencies of hull girder vibration modes obtained by using accurate virtual mass method and simplified method for tanks respectively are compared with the measurement results at the sea trial. And the reason of the difference between the results of virtual mass method and simplified one is discussed and investigated quantitatively by using block models of tanks which are extracted from the whole ship model.

2 Characteristics of Effective Mass of Liquid in Tank In order to improve the accuracy of the vibration analysis using whole ship model, it is necessary to model the mass distribution of the ship hull adequately. The mass of the ballast water at the ballast condition and the mass of the liquid cargo at the full load condition of the tanker are more than the half of the displacement of the ship. It is indispensable to adequately model the mass of the liquid in the tank for improving the accuracy of analysis. Then, the influence of the free surface of liquid in the tank to the effective mass of vibration is summarized in this section. The bulkheads and the bottoms of the tanks of the ship hull are made of steel and elastic bodies, but the following discussion assumes that the tank walls are treated as rigid bodies neglecting the elastic coupling with liquid to concentrate on the effect of free surface of liquid.

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In order to investigate the influence of the free surface of the tank, the vibration mode of the tank of horizontal motion and rotational motion are considered. The effective masses for the each vibration mode are summarized below. The tanks are assumed to be two-dimensional rectangular tanks, and the dimensions and the coordinate system is shown in Fig. 1. It is noted that the effective mass of the vertical vibration mode is the same as the total mass of the fluid. Therefore, the effective vibration mass in the vertical mode is considered to be equal to the total mass of the inner liquid. z H

x

0

L

Fig. 1. Model of the tank

2.1 Horizontal Vibration Mode The tank is assumed to vibrate parallel to the x axis of Fig. 1. The effective mass ratio of the liquid in the tank ηH is expressed by analytical solution based on the series expansion method as the following equation [5]. F = ma ηH

(1)

where, F: horizontal force acting on the wall of the tank from the liquid m: mass of liquid in the tank a: amplitude of horizontal acceleration of the tank The effective mass ratio ηH of Eq. (1) is presented as a function of ratio of height and width of tank as shown in Fig. 2. Figure 2 shows that the effective mass ratio of the liquid in the tank is less than 1.0 when the tank moves horizontally, and decreases as the H/L becomes smaller. It can be concluded that the effective mass of the liquid in the tank is smaller than the total mass of the liquid when the tank vibrates horizontally corresponding to hull girder longitudinal vibration or hull girder horizontal vibration. 2.2 Rotational Vibration Mode The center of rotation is selected at the central position of liquid volume where breadth and height of the liquid are L and H respectively as the origin of the coordinates in Fig. 1,

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Fig. 2. Effective mass ratio for horizontal vibration mode

and the rotational vibration in the xz plane around the origin is considered. The effective rotational inertia ratio of the liquid in the tank ηT is expressed by analytical solution based on the series expansion method as the following equation [5]. M = I ω˙ ηT

(2)

where, M: moment by the force acting on the wall and bottom of the tank from the liquid I: inertia when liquid is assumed as rigid body ω: ˙ amplitude of angular acceleration of the tank The effective rotational inertia ratio ηT of Eq. (2) is presented as a function of ratio of height and width of tank as shown in Fig. 3. When the tank rotates harmonically, the effective inertia ratio of the liquid in the tank is less than 1.0. And around H/L = 1.0, the effective rotational inertia ratio becomes minimum, and the effective rotational inertia tends to increase as H/L becomes smaller or larger than 1.0. That means when the tank rotates at the hull girder torsional vibration mode, the effective rotational inertia of the liquid in tank becomes smaller than that in the case the liquid in the tank is assumed as a solid.

3 Influence of Modeling Method of Liquid in Tank on Ship Vibration In the conventional method of the whole ship FE modeling for vibration analysis, the mass of the liquid in the tanks is modeled by adjusting the density of elements or by adding nodal masses at the tank structural elements contacted with inner liquid because modeling of liquid motion effect by considering the consistent virtual mass needs computing power and memories to solve. But as shown in the previous section, the equivalent mass and

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Fig. 3. Effective rotational inertia ratio for rotational vibration mode

rotational inertia of the liquid in the tank are smaller than the total mass of the liquid and change depending on the liquid shape when the tank vibrates horizontally or rotationally. Therefore, we have investigated the influence of modeling methods of liquid in tank for vibration of hull girder vibration mode. First, natural frequencies of hull girder vibration modes derived by the eigenvalue analysis using whole ship model has been compared between the results by using conventional modeling method and virtual mass one for modeling liquid in tanks. Next, modal masses estimated by those modeling methods have been compared quantitatively by using block models to ensure the effect of added mass of contained water with free surface discussed in the previous section. 3.1 Comparison of Natural Frequencies of Hull Girder Vibration Modes Regarding the modeling methods of liquid in the tanks, the following two cases of vibration analyses using whole ship model of a VLCC shown in Table 1 were carried out. Table 1. Principal dimensions L × B×D

322.0 m × 60.0 m × 28.5 m

Main Engine

7RT-flex84TD

Shaft Revolution (rpm)

70.1 (NOR), 74.0 (MCR)

Propeller Blade

5

Displacement (at Sea Trial) 344,800 ton

Case-1: Conventional Method Case-2: Virtual Mass Method

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In these analyses, added mass of seawater outside the shell plate of the ship hull is modeled by the virtual mass method. The cargo tanks consist of center tank and wing tanks as shown in Fig. 4.

NO.5CH NO.4CH NO.3CH

NO.2CH NO.1CH

Fig. 4. Cargo tank arrangement

In Case-1, the mass of the liquid in the cargo tank is modeled by density adjustment of structural elements contacting with liquid in the cargo tanks. This means that density which is the weight of liquid in the cargo tank divided by sum of multiplication of area and thickness of each element at the tank boundary of the model was added to the model. In general, the tank wall thickness is larger in the lower part. In this conventional method, density is uniformly added to wall plate elements of the cargo tank contacting with liquid so that the total added weight by this adjustment might be the same as that of liquid in the tanks. It means that effective mass of liquid is 100% of the liquid mass for the vibrations both in horizontal and vertical directions. As for the rotational vibration, all of the mass of the liquid is distributed on the tank boundary wall. Therefore the effective rotational inertia is considered to be larger than the actual rotational one. As for the conventional methods of mass adjustment, another technique can be selected by adding nodal masses which can consider different values in the each longitudinal, transverse, and vertical directions. However, when the liquid in the tank is modeled by the method instead of virtual mass method, it is inevitable to consider some assumptions. In Case-2 modeling the liquid in the cargo tank by virtual mass method, the added mass of water calculated based on BEM(Boundary Element Method) for ideal fluid with free surface. It is more consistent and accurate method than Case-1, but the load to the computer becomes higher because it needs large memory and CPU time to solve. The whole ship model used in these analyses is shown in Fig. 5. The numerical results of the natural frequencies of Case-1 and Case-2 for the vibration analyses using whole ship model are shown in Table 2. Measurement results were identified by the experimental modal analysis by LMS Test. Lab using data acquired during the sea trial [6]. The wireframe model connecting measurement points which is used for experimental modal analysis is shown in Fig. 6.

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Fig. 5. Whole ship model Table 2. Comparison for modeling method Mode

Natural frequency (Hz)

Conventional/virtual mass

Measurement Analysis Conventional method Virtual mass method V-2 nodes 0.458

0.432

0.453

0.954

V-3 nodes 0.987

0.962

1.017

0.946

V-4 nodes 1.499

1.494

1.576

0.948

V-5 nodes 1.936

1.981

2.059

0.962

Fig. 6. Wire flame model used in the experimental modal analysis

Even there are little differences between their mode shapes obtained by 2 types of modellings in each hull girder vibration mode, natural frequencies calculated by using the conventional method are lower by about 4 to 5% than those by using virtual mass method which difference is significant for the anti-vibration design. In the Table 2, it is observed that the deviation between the measurement results and the analysis results by virtual mass method increases, as the number of nodes of modal shapes increases. It is caused by the influence of the mesh size to the interaction of hold double bottom [7]. For reference, the hull girder vibration modes of virtual mass method are shown in Fig. 7. 3.2 Quantification of Modal Masses by Using Block Models Each tank vibrates vertically and rotationally due to the hull girder vertical vibration modes. As mentioned at the beginning of Sect. 2, it can be assumed that the effective

Study on the Effect of Liquid in Tanks

(V-2 nodes)

(V-3 nodes)

(V-4 nodes)

(V-5 nodes)

337

Fig. 7. Hull girder vibration mode

mass of contained liquid in the rigid tank for vertical motion is equal to 100% of the liquid mass. On the other hand, the effective mass for rotational motion is reduced due to the liquid motion induced by vibration by free surface effect. As the results, natural frequencies obtained using conventional modeling method for liquid in tanks is smaller than those obtained using virtual mass method as shown in Table 2. Then in order to make clear the reduced effective mass of contained liquid in rotational motion for vertical hull girder vibration of a real ship structure shown in Fig. 8, we divided the ship structure into 8 blocks as shown in Fig. 9 and investigated the effective mass of liquid in each blocks in rotational motion.

AP

ER NO.5CH NO.4CH NO.3CH NO.2CH NO.1CH

FP

Fig. 8. Extent of block model

The block approach is consists of 2 steps. First, effective rotational inertia of the contained liquid in each block structure was estimated by rotational vibration analysis. Next, modal masses were evaluated by the effective rotational inertia of each block model considering the displacement of the hull girder vibration mode. The detail is explained as follows. The whole ship model was divided into blocks which are supported at the center of rotation and also supported with springs at the fore and aft ends of the blocks. The modal masses were calculated for the block models’ vibration analysis by using virtual mass method or density adjustment one. And the influence from the effective rotational inertia of each block was evaluated. The evaluation using block approach was performed only for the two-node hull girder vibration mode because this is the simplest and the influence of elastic deformation of tank structure is considered to be small from the following reason. Actually the effective mass of the contained liquid in the previous section was discussed for rectangular rigid tank without elastic deformation though the tank wall and bottom are elastic in actual structure. However, the natural frequency of 2-node hull

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(AP)

(NO.2-4 CH)

(ER)

(NO.1 CH)

(NO.5 CH)

(FP)

Fig. 9. Block models

girder vibration mode of VLCC is around 0.5 Hz and sufficiently lower than the natural frequency of the tank wall structure, so the influence of the elastic deformation of the tank wall does not seem significant. Evaluation Method. Modal masses of whole ship using block models for conventional method and virtual mass method are compared based on the Rayleigh’s energy method as shown below. The model is shown in Fig. 10. M LWi is the sum of the mass of the light weight of ship hull, M Li is the sum of the mass of the liquid in the tank, and I i is the sum of rotational inertia of structure and the effective rotational inertia of liquid in the tank for each block model. I i is calculated by the FE Analysis. yi is the amplitude at the midpoint in longitudinal direction of each block, and θ i is the rotational angle at the same position of the 2-node hull girder vibration mode of the natural frequency analysis of the whole ship model which liquid in tank was modeled by virtual mass method. The kinetic energy of the each block model which structure is assumed as a rigid body is considered. The equation of kinetic energy of the block model is expressed by the following equation.  2  2   dyi dy d θi 2 1 1 1 M0i = (MLWi + MLi ) + Ii 2 dt 2 dt 2 dt

(3)

Where M 0i is modal mass of each block model which position is considered at aft end of the ship where the maximum amplitude of the mode is. And y is the amplitude of the displacement at the aft end of the ship. y0 , yi0 , θ i0 are expressed by the following expressions. y = y0 cos(ωt)

(4)

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y

339

MLWi, MLi, Ii θi

x yi

0

Block Model Fig. 10. Model for the estimation of modal masses

yi = yi0 cos(ωt)

(5)

θi = θi0 cos(ωt)

(6)

where, ω: angular frequency t: time Equation (7) is obtained by the rearrangement of Eq. (3) with considering Eqs. (4), (5), and (6).  2  2 yi0 θio + Ii (7) M0i = (MLWi + MLi ) y0 y0 In this study, yi0 and θ i0 when y0 = 1.0 are used. Then the modal mass M VLWi for vertical motion of the light weight of ship hull, the modal mass M VLi for vertical motion of the liquid in the tank and the modal rotational inertia M Ri for rotational motion which consider the deformation of 2-nodes hull girder vibration mode are defined as follows. 2 MVLWi = MLWi yi0

(8)

2 MVLi = MLi yi0

(9)

2 MRi = Ii θi0

(10)

It is noted that the application of this method for the higher vibration mode is inappropriate because the deviation increases from the assumption that each section vibrate as a rigid body. Evaluation of Modal Masses of the External Liquid of the Ship Hull The modal masses of the external liquid of the ship hull M Wi was calculated based on

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the Rayleigh’s energy method as shown below. The vertical component of work from the external liquid of the ship hull to the ship structure of the element j U j is shown as follows. Uj =

1 Pzj Aj δj 2

(11)

where, Pzj : amplitude of the vertical component of the pressure acting on the element of the outer shell plate by the external liquid Aj : area of the element δ j : amplitude of the displacement of vertical direction of the element The kinetic energy T j is expressed as follows. Tj =

1 mj vj2 2

(12)

where, mj : effective mass of the liquid of the element vj : amplitude of the velocity of vertical direction of the element As U j is equal to T j , the effective mass of the liquid mj is expressed as follows. mj =

Pzj Aj aj

(13)

where, aj : amplitude of the acceleration of vertical direction of the element The vibration mass of the liquid M W0i was calculated by taking the sum of mj for each block. And the modal mass M Wi is defined by the following equation. MWi = MW 0i yi2

(14)

Analysis Method for the Calculation of Rotational Inertia As for Nos. 1 to 5 CH which include cargo tanks, natural frequency analysis was carried out in order to calculate effective rotational inertia as shown below. The constraint points of the model for natural frequency analysis was assumed to be the center of the fore end and aft end of the model in the longitudinal direction, and the intermediate position between the tank bottom and the free surface of the liquid in the vertical direction. Longitudinal, transverse and vertical directions were constrained at the position of the outer shell plates, double side plates and longitudinal bulkhead plates. It is assumed that

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these restraint points are the midpoint of the tank liquid. In addition, the liquid filling ratio by volume of the cargo tank was the same as 87% except for 41% of the center tank of No. 2 CH. The influence of difference of the height of liquid for center of rotation of the center tank and the wing tank of No. 2 CH are assumed to be small. Then the rotation center of No. 2 CH model is assumed to be the same as the model of other cargo holds. The constraint points of the No. 3 CH model are shown by black circles in Fig. 11.

Fig. 11. Constrain and elastic support of the model

Elastic support of the vertical direction was applied to the model at the 4 corners at the same height as the node which applied the constraints. The positions applied the spring support of the No. 3 CH model is shown by white circles in Fig. 11. Since the adjacent tanks in longitudinal direction share the structure of the tank boundary walls, the density of the tank wall is reduced to 1/2 so as not to be considered duplicate in the calculation of effective rotational inertia. The vibration mode of the analysis result of No. 3 CH model is shown in Fig. 12.

Fig. 12. Vibration mode of block model

The effective rotational inertia I i was calculated by the following equation using the natural frequency obtained by the analysis. Ii =

Kθ (2π f )2

(15)

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where, K θ : rotational spring constant f : natural frequency The rotational spring constant K θ was set as the natural frequency of the rotational vibration of the tank was around 0.3 Hz which is almost same as the natural frequency of the 2-node hull girder vibration mode so that the influence of elastic deformation of the tank wall is not significant. For the block models which does not include the cargo tank (AP, ER, FP), the effective rotational inertia was calculated by using the tool according to the mass distribution of the model itself. Numerical Results of Modal Masses Table 3 shows summary of the modal masses of each block. Table 4 shows the evaluation results of the modal masses by using the block models as well as the modal masses by using whole ship model which are shown in Table 2. And ratios of modal mass of conventional method to virtual mass method are also shown. Table 3. Modal masses for 2 node hull girder vibration mode Modal masses by using block models (× 103 kg) M VLWi (Light Weight)

M VLi (Liquid in Tank)

M Wi (External Liquid)

M Ri Conventional method

Virtual mass method

AP

1,220

–

1,951

14

14

ER

3,631

–

6,819

167

167

No. 5 CH

501

4,028

2,928

1,818

680

No. 4 CH

178

1,656

1,581

1,234

408

No. 3 CH

804

7,490

8,921

34

12

No. 2 CH

31

230

257

1,472

517

No. 1 CH

933

8,742

7,367

2,459

693

FP

725

–

1,330

11

11

Total

8,023

22,146

31,154

7,209

2,502

The modal masses estimated by using block models are a little smaller than those by whole ship model for each model type. And the ratio of modal mass of conventional method to virtual mass method by block models is 1.074 and by whole ship model is 1.103 as shown in Table 4. The dominant causes of the difference are discussed that representative value of amplitude and rotational angle are used for the block model though actual amplitude are applied for the whole ship model.

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Table 4. Comparison of modal masses Model type

Modal masses of whole ship (× 103 kg) Block models

Whole ship model

Conventional method

68,532 (1.074)

71,000 (1.103)

Virtual mass method

63,825

64,399

By the assumption that the ratio of natural frequency is proportional to the square root of the sum of the modal masses ratio, the natural frequency ratio of the conventional method with respect to the virtual mass model is 0.965. Therefore, it can be explained that when the liquid in the cargo tank is modeled by the conventional method, the natural frequency is estimated to be a few percentage lower than when modeling by the virtual mass method. In this paper, structure of each tank assumed to be rigid bodies because the 2 node hull girder vibration mode was selected. Influence of deformation of tank structure for the higher vibration modes are to be investigated as future works.

4 Conclusion In this study, we have investigated the effective masses of tanks containing liquid for horizontal vibration and rotational vibration. And the influence of the free surface of the liquid in the tank on the natural frequencies of the hull girder vibration modes of a tanker has been investigated. As the results, the followings have been obtained. The conventional modeling method of liquid in the tanks of whole ship model evaluates the natural frequency of 2-node hull girder vibration mode lower than that of virtual mass method. It was verified that this discrepancy of natural frequency is resulted from the influence of the effective masses due to the rotational motion of the tank by considering the block model analysis. Therefore, it is necessary to model the liquid in the cargo tank by the virtual mass method in order to accurately simulate the vibration characteristics of hull girder vibration of tanker in the vibration analysis using whole ship model.

References 1. Takahashi, H., Kusumoto, H., Takeda, T., Deguchi, T.: Investigation on the Vibration Characteristics of the Superstructure of Mega Container Ships: Design & Operation of Container Ships, pp. 107–112. RINA, London (2006) 2. Takahashi, H., Investigation on the vibration response analysis using whole ship model. In: The 26th Asian-Pacific Technical Exchange and Advisory Meeting on Marine Structure, Fukuoka, pp. 257–261 (2012)

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3. Takahashi, H., Yasuzawa, Y.: Investigation on damping model for vibration response analysis using whole ship model. In: The 24th International Ocean and Polar Engineering Conference, Busan, pp. 890–895 (2014) 4. Kumai, T.: On the apparent mass of cargo oil in ship vibrations. J. Soc. Nav. Archit. Jpn. 117, 146–152 (1965) 5. Senda, K., Nakagawa, K.: On the Vibration of an Elevated Water-Tank 1. Technology Reports of the Osaka University, vol. 4, pp. 247–264 (1954) 6. Rocca, G., Peeters, B., Ota, R.: Experimental ship hull dynamic characterization using operational modal analysis. In: 4th International Operational Modal Analysis Conference (IOMAC), pp. 517–526 (2011) 7. Takahashi, H.: Investigation on the vibration response analysis using whole ship model. In: The 26th Asian-Pacific Technical Exchange and Advisory Meeting on Marine Structure, pp. 257– 261 (2012)

Calculation of Structural Damping of the Global Hull Structure from In-Service Measurements Remco Hageman(B) and Ingo Drummen MARIN, Wageningen, The Netherlands [email protected] Abstract. Ships operating in waves are continuously subjected to varying loads from waves. As a result of these loads the ship undergoes rigid body motions and deformations. The latter can be subdivided into two types, a quasi-static and a dynamic response. Two dynamic responses that are of interest for ships are springing and whipping. Springing is the steady-state resonant vibration of a flexural mode due to continuous wave loading. Whipping is the transient elastic vibration of the ship hull girder caused for example by slamming. Springing has an important contribution to increased lifetime consumption, whereas for assessment of ultimate stresses whipping is more important. One of the major uncertainties when examining global flexural response is the amount of damping to be incorporated. Because springing is a resonance phenomenon, damping is especially important. In this paper Operational Modal Analysis is used to retrieve damping from in-service measurements and is tested for a heavily instrumented frigate type vessel. The aim of this paper is both to explain and discuss the methodology and to provide typical damping parameters which are needed in the assessment of whipping and springing. The paper also shows sensitivity of frequency and damping of the flexural modes to selected operational parameters. The research in this paper is restricted to the two node vertical bending mode. Acceleration data is most commonly used to derive mode shapes. For the present application, however, accelerations and strains were combined. In order to obtain useful results stringent filtering on mode shape, frequency, damping and stationarity of operating conditions should be applied. Operating in confined waters also has a noticeable effect. Variations in the observed natural frequency can be as large as 10%. These variations must be related to mass variations. At average operating speeds in moderate environmental conditions, the damping of the two node bending mode is around 0.6 to 1%. However, analysis presented in this paper have indicated that damping is related to speed and wave height. Specifically, the damping in 26 knots increases to 2.5% from the 1% at 15 knots. In beam sea conditions, an increase in damping from 0.7% to 1.2% was found in wave heights ranging from 1 to 2 m. This indicates that when studying extreme conditions different damping ratios are applicable compared to intermediate conditions. Keywords: Operational modal analysis · Whipping Vibration frequency · Structural damping c Springer Nature Singapore Pte Ltd. 2021  T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 345–364, 2021. https://doi.org/10.1007/978-981-15-4672-3_22

· Springing ·

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Introduction

Ships operating in waves are continuously subjected to varying loads from waves. As a result of these loads the ship undergoes rigid body motions and deformations. The latter can be subdivided into two types, a quasi-static and a dynamic response. The quasi-static response depends only on the stiffness distribution of the structure. The dynamic response also depends on the mass distribution. Two dynamic responses that are of interest for ships are springing and whipping. Springing is the steady-state resonant vibration of a flexural mode due to continuous wave loading. Whipping is the transient elastic vibration of the ship hull girder caused for example by slamming. Whipping vibrations typically lead to higher accelerations, leading to potential loss of cargo, or higher stresses leading to increased fatigue accumulation [18] or exceedance of ultimate strength [16]. The dynamic response of a ship is characterised by its modal parameters. Design procedures need to adequately account for these dynamic effects. The whipping response of ships to impulsive slamming loads can be studied using either dedicated model tests or numerical simulations. Several tools and methods have been developed to assess whipping. These methods include purely numerical methods, experimental methods using flexible backbone models or hybrid methods combining slamming loads from model tests and finite element calculations. Springing has received less attention from the research community and current prediction methods for springing are less accurate compared to whipping. In the past, springing has been a rare phenomenon as typical encounter frequencies are much lower compared to the global hull frequencies. With increasing ship dimensions the global hull frequencies become lower and springing is seen more often. Springing has an important contribution to increased lifetime consumption, whereas for assessment of ultimate stresses whipping is more important. Modelling dynamic responses implies knowledge of the model parameters. In the design stage a rough estimate of the mass and stiffness distribution is usually available. Information about the damping ratio is much less well known and described in literature. Particularly when looking at fatigue the damping ratio is very important for both described dynamic effects. The damping ratio controls the decay of the whipping cycle and the amplitude of the springing, as the latter is a resonance phenomenon. Except for the design stage, knowledge of the modal parameters is also very interesting in the operational stage. The ability to determine the modal parameters from in service measurements allows to track small changes in these parameters over time. These changes may originate from changes in mass distribution, characteristics of the loads [11] or potentially damage to the structure [15,25]. Modal Analysis is most often executed to identify dominant deformation modes. These modes are most conveniently identified on flexible slender structures, such as beam and plate structures. The loading on these structures must also be such that the vibration modes of interest are sufficiently excited. Bridges are often successfully examined using Modal Analysis both through numerical analysis [25] as well as during their service life. Various types of bridges, such as cable-stayed bridges [3] and truss bridges [7] have been examined exten-

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sively. Modal Analysis has also been successfully applied to more rigid concrete bridges [22]. The examples discussed by these researchers generally feature a large number of sensors. Modal analysis has also been applied in offshore and ship structures. The offshore jacket presented by [24] is a typical example of a slender structure with random loading. A 9300 TEU container vessel has been examined using modal analysis by [8]. A third example can be found in the RO-LO vessel considered by [12,13]. The aforementioned vessel types are generally considered as more flexible structures. However, [17] shows the application of modal analysis on a smaller, more rigid polar research vessel. This example shows the potential for application of this method on a wide variety of ships. It should be noted that the examples mentioned here are based on a more compact instrumentation set-up featuring a smaller number of sensors compared to the analyses of bridges. Finding accurate modal parameters for modelling ships requires evaluation of in-service measurements. However, one important drawback of using in-service measurements is the general lack of control of the operations of the ship. Careful evaluation of the operating conditions is required to be able to understand the observed behaviour and gain insight into how parameters may change in relation to operating conditions. This paper will show such an analysis for a few sets of data obtained from a ship in service conditions. Correlations between modal parameters and operating conditions will be highlighted. Modal Analysis can also be executed on more rigid structures. An example of modal identification on a lantern housing is presented by [9]. The aim of these authors is to identify damage to the structure. To be able to do so on this structure, they executed FEA of the structure in intact conditions and featuring several types of damage. After executing the modal analysis, the authors compared their findings against these scenarios. This indicates the added complexity of executing Modal Analysis for non-slender structures. In the presented examples, dedicated analysis are executed on numerical analyses, model tests or in-service measurements. Modal identification is performed manually on small data sets through stabilization diagrams. These examples can therefore not be considered as Operational Modal Analysis (OMA). OMA is to apply a modal identification method to subsequent sets of measurements and thereby keeping track of changes in the modal characteristics. This requires careful definition of criteria to compare modes with a reference model and tolerances which should be applied. In order to perform model tests or numerical simulations for ship design, modal parameters need to be quantified. In the design stage the shape and frequency of the global flexural vibration modes are typically obtained from a finite element model. The damping ratio is however much more elusive. There is limited literature available on the damping ratios of actual vessels. This paper presents results on damping characteristics of a naval vessel using in-service measurements and OMA. Variability in modal parameters is examined and discussed. The article evaluates procedures to incorporate acceleration and strain measurements in a single analysis. Section 2 discusses the operational modal analysis procedure. The findings of OMA work on

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an in-service measurement campaign on the vessel are described in Sect. 3. In Sect. 4 a more in depth analysis of the damping ratios is done. A discussion of these results is provided in Sect. 5, before Sect. 6 provides some conclusions.

2 2.1

Operational Modal Analysis General

A data set containing multiple measurements of a structure inherently contains time and space dependent information. Consequently, such a set of information can be referred to as a State-Space model. The general scheme of the StateSpace model is shown in Fig. 1. The vector x(t) represents the true state of the system and is referred to as the state vector. y(t) are the actual observations or measurements of the system. The observations will in general not be a true observation of the system state due to systematic disturbances, represented by the matrix C and random measurement disturbances v(t). Vector w(t) represents the unknown input to the structure, while Δ denotes a time delay. Both inputs v(t) and w(t) are assumed to be white-noise processes. The relation between subsequent conditions is provided by matrix A. This matrix, called the system matrix, contains the inner system relation and will therefore yield the system dynamics we are looking for. It should be noted that the only parameter which can actually be observed from this system is y(t).

Fig. 1. Flow scheme of information in the state-space model used in Stochastic Subspace Identification

To analyze such systems, the Stochastic Subspace Identification (SSI) method is frequently used [21]. This method is a time-domain method which extracts the dominant components describing the inner workings of the system. For a mechanical system with significant internal vibrations, the obtained matrices can be further analysed to yield vibration characteristics [2]. Most commonly, acceleration measurements are used to derive these characteristics. The eigenvalues and eigenmodes of the system matrix are closely related to the physical vibrations. Let μi and ψ i represent the eigenvalue and eigenmode

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i of the matrix A. In that case, the vibration frequency ωi and damping ζi can be determined from: ln(μi ) dt ωi = |λi |

λi =

(1)

Re(λi ) ζi = − |λi | In which dt represents the time between subsequent samples or the inverse of the sampling frequency. The mode shape itself is also an important result of the analysis. As a result of measurement noise, the application of Modal Analysis will yield a number of spurious modes. In order to distinguish physical modes from spurious modes, two checks should be executed. First, physical modes exist in the Modal Analysis output as complex conjugate modes. The physical mode is the real part of any of these complex conjugate vectors. Secondly, when increasing the number of modes requested from the algorithm, the physical modes will consistently show up, while the non-physical modes change in shape and frequency. Mode shapes can be compared using the Modal Assurance Criterion (MAC) [1] which is defined as: M ACiR =

|ψ Ti ψ R |2 |ψ Ti ψ i ||ψ TR ψ R |

(2)

in which the modes ψ i and ψ R are compared and the subscript T denotes the vector transpose. ψ R specifically represents the reference mode shape against which mode i which results from the Modal Analysis is compared. By derivation of modal parameters at subsequent time instances, changes in these parameters, and therefore the structural behaviour, may be tracked over time. These changes can be the result of differences in mass distribution, characteristics of the loading [11] or significant damage to the structure which affects the global behaviour [15,25]. When executing OMA, specific criteria needs to be defined with respect to ensure that the comparison between subsequent conditions is justified [1]. 2.2

Strain Measurements

For the ship considered in this research both strain and acceleration measurements are available. A modal analysis can be executed using both types of data [23]. The analysis presented in this paper uses both types of data simultaneously. Mathematically, the modal frequencies and damping can be obtained in the exact same way for both strain and acceleration measurements. There are however two main challenges when combining both sets of data [6].

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First of all, the result obtained from the Modal Analysis have a different physical interpretation. The physical deformation modes are denoted with ψ i . Because strains are the spatial derivative of the deformation modes, the final result obtained from the Modal Analysis from strain measurements also yields i the spatial derivative, i.e. ∂ψ ∂x ¯ . When using accelerations, which are the double temporal derivatives of the deformation, the obtained modes are ωi2 ψ i . Note that frequencies associated with each mode, ωi , are also obtained from the procedure. This means that the acceleration modes are directly proportional to the deformation modes, while the modes from strains are proportional to the spatial derivative of the deformation modes. The mode shape ψ i is generally a convenient way to visualize the output. This can be done directly when using acceleration measurements, but when using strain measurements, this can only be achieved by fitting the output to a model. Both types of output can be compared directly to results from numerical analyses. Secondly, vibration modes on ships are much harder to identify from strain measurements. The reason is that the actual loading on marine structures is not a white-noise process as used in the theoretical derivation in Sect. 2. Rather wave loading features dominant frequencies ranging from 0.1 to 0.25 Hz. As a result, the modes obtained from the procedure will not feature pure structural resonance modes, but wave-induced components will also be represented strongly. Because the natural frequency of ships is higher compared to the wave loading and because the modes obtained from acceleration measurements are multiplied by ωi2 as discussed in the previous paragraph, the natural modes can still be determined well from acceleration measurements. The modes from strain measurements lack the frequency multiplication factor and therefore more wave driven components will be present in the modes obtained from strain measurements. Strains and acceleration measurements may also be used together in a single analysis. However, as the modal analysis is a purely mathematical operation, it is required to apply scaling of the measurements, which have different orders of magnitude. The scaling procedure used by the authors is defined in [10]. Besides scaling of the order of magnitude, additional scaling of the measurements can be applied. Considerations to do so, relate to the number of the different type of sensors or their respective accuracy. Because of simplicity and because the number of strain gauges and accelerometers used in this research are similar, the authors have not applied such additional scaling. Besides strain and accelerations, the deformation can also be measured directly through optical systems [14]. Although possible in laboratory conditions, this method is not suitable for in-service applications on ships, while in-service strain and acceleration measurements are very common.

Structural Damping of the Global Hull Structure

3 3.1

351

Continuous In-Service Measurements Measurement Set-Up

The vessel is equipped with 26 accelerometers, see also Fig. 2. In this research the vertical bending modes are the primary interest. Therefore only the 11 accelerometers which are oriented in the vertical direction will be used. These accelerometers are distributed over 7 ship sections. Note that no sensors are placed at the extreme forward and aft end of the ship. The vessel is also instrumented with 24 Long Base Strain Gauges (LBSG) which are oriented in the longitudinal direction of the ship. These sensors are evenly distributed over 6 ship sections, see also Fig. 3. Each LBSG is 1 m steel rod of which one side is fixed to the vessel hull. The displacement of the free end of the LBSG is measured using a ship mounted linear variable displacement transducer. This allows the sensor to capture global structural deformation without disturbance of local details. Both the strain and acceleration measurements are available at 20 Hz. Besides these devices, the ship has been instrumented with a variety of additional sensors [4]. Noteworthy are the motion sensor, GPS and wave radar. The GPS has been used to obtain speed and heading of the vessel at all times. The data obtained from the wave radar has been corrected using a wave data fusion method to obtain realistic wave heights [19]. This allowed for the determination of the operational profile of the vessel and scatter diagram it has seen.

Fig. 2. Accelerometers installed on the vessel used in this research. The green arrows indicate accelerometers oriented in the vertical direction which have been used in this research.

3.2

Analysis Setup and Criteria

The measurements have been analyzed in 30-min intervals. The time series of accelerations and strains measured in each interval have been processed using Modal Analysis to derive the characteristics of the two node bending mode.

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Fig. 3. Long Base Strain Gauges (LBSG) installed on the vessel. 6 cross sections have been installed with 4 LBSGs each, making for a total of 24 strain gauges available.

For each interval, the average speed, heading and wave spectral parameters were also determined. From numerical analyses the frequency and shape of the vertical bending modes is known. A mode shape is identified as the two node bending mode, if the frequency is close to the reference frequency, within 10%, and the mode shapes correspond. The MAC, which defines the agreement of the mode shape, is harder to define beforehand. Figure 4 has been used to specify a reasonable limit for the MAC for this ship and instrumentation set-up. In the figure on the left, the vessel is operating in beam seas and the excitation of the two node bending mode is low. As a result, frequency and damping are ill-defined and show large variations. In the right hand side figure, the excitation of this mode is higher, leading to a more consistent frequency, damping and mode shape. The figure indicates that using a MAC value of 0.99 will remove the majority of spurious results without compromising the data set too significantly. An example of the observed mode shape with a MAC exceeding 0.99 is compared against the numerical mode shape in Fig. 5. The frequency of the observed mode shape, 1.98 Hz, corresponds well with the numerically obtained value of 1.93 Hz. Note that the numerical value has been derived for a full load condition.

4 4.1

Damping Identification Data Selection

Mode shapes have been automatically obtained using OMA from the ship described in Sect. 3. In order to gain insight in the relation between modal parameters and operational or environmental conditions, it is important to isolate time frames of at least several hours during which the vessel is operating in constant conditions. Given the purpose of the vessel and varying external factors, only a limited set of periods can be identified which satisfy the requirement of stationary operations.

2

Frequency [Hz]

Frequency [Hz]

Structural Damping of the Global Hull Structure

1.95 1.9 1.85

2 1.95 1.9 1.85

1.8 1.75

1.8 0

12

24

36

1.75

48

8 6 4 2 0

12

24

36

24

36

48

0

12

24

36

48

0

12

24

36

48

6 4 2 0

48

1

1 0.95 MAC [−]

MAC [−]

12

8

0.95 0.9 0.85 0.8

0

10 Damping ratio [%]

Damping ratio [%]

10

0

353

0.9 0.85

0

12

24

Time [hours]

36

48

0.8

Time [hours]

Fig. 4. Frequency, damping and MAC during two-day periods used for selection of appropriate MAC values. The crosses indicate a MAC larger than 0.99; the diamonds indicate a MAC between 0.95 and 0.99; the circles indicate a MAC below 0.95. The difference in the observed natural frequency between both periods is the result of differences in operating conditions.

Fig. 5. The two node vertical bending mode obtained from modal analysis (♦) compared against the numerically determined modeshape (). The MAC obtained in this comparison is between 0.99 and 1.

Two different periods have been selected for detailed analyses. The frequency and damping of the two node bending mode in both periods is shown in Fig. 6. Both sets of data have been obtained from different deployments of the vessel. One can see some random scatter which implies randomness in the output of the

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procedure. In order to understand the variability which can be expected from the calculation itself, a detailed analysis of the results obtained during the second day in period 1 has been performed first. Then more global analyses of both period 1 and period 2 were performed. For ease of discussion, the frequency and damping of the two node bending mode will be referred to as simply frequency and damping throughout the remainder of this paper. 2.05

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4.2

Single Day Analysis

The frequency and damping recorded during the second day of the first period are shown in Fig. 7. This figure also shows speed and heading of the vessel. Overall, the random variation in determination of the vibration frequency appears to be minor. There are however a couple of notable events which will be examined further. The following observations have been made: – The initial frequency up to 7 h is 1.97 Hz. – The frequency between hour 9 and 18 is almost stable at 2.0 Hz. – Two exceptions of these stable results are found at hour 3 and 15 when the frequency momentarily reduces to 1.95 and 1.93 Hz, respectively. – After 18 h the frequency gradually reduces from 2.0 to 1.94 Hz. Vibration frequencies are related to structural stiffness and mass (distribution). The ship structure and stiffness remain unmodified during the deployment.

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The mass of the vessel itself will also not change considerably at once. However, the added mass of the vessel can change rather abruptly due to changes in operating conditions. Table 1 shows the estimated change in added mass associated with several flexural vibration frequencies for this ship. The change of frequency occurring around 8 hours occurs simultaneously with a reduction in speed from 25 to 20 knots. The change in frequency corresponds to an estimated change in mass of 280 tons. At the time of the speed reduction, the vessel is sailing in stern quartering seas with wave height 1.5 m and wave period around 9.5 s. Diffraction calculations have been executed for the 28 and 21 knots operating conditions. For these operating points, the difference in added mass is around 200 tons which is well in line with the estimated difference using the observed frequency. After around 15 h, the speed reduces further, but as the vessel is approaching confined waters which will also affect vibration characteristics. The change of heading around 6 h does not affect the vibration frequency. Due to this change of heading, the incoming wave direction changes from stern quartering waves over starboard to stern quartering waves over portside. Therefore, no significant change of the added mass is expected.

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Table 1. Estimated changes in mass associated with different frequencies compared to the reference frequency of 2.00 Hz. The frequencies presented here are frequencies found in the detailed analysis presented in Fig. 7 Frequency [Hz] Change in mass [%] Approximate change in mass [ton] 2.00

+0.0

+0

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+280

1.95

+5.2

+470

1.94

+6.3

+565

1.93

+7.4

+665

The temporary reduction in frequency at 15 h is probably shallow water effects. The vessel was found to be sailing over a shallow of 50 m in this period which can lead to a minor increase in added mass. A similar explanation does not hold for the observed reduction at 3 h. No reasonable explanation for this condition, including the increased damping, could be found and the situation is considered an analysis anomaly. At the end of this day, the vessel is sailing into the harbour. Confined water effects lead to an increase of added mass and therefore reduction of frequency which can be seen in the graphs [5]. When looking at the damping, a stepwise reduction over the day can be seen. The reduction steps correlate with speed changes. When sailing at 25 knots, the damping is around 2.6%. This reduces to 1.2% in the 21 knots condition and only 0.3% when the vessel is navigating at low speed inside the harbour. 4.3

Global Analysis First Period

In addition to the observations already made regarding the second day of the first period, a couple of additional observations can be made when looking at the 21 day period shown in Fig. 6: – – – –

Increase in damping around day 8. Step change in frequency from 2.0 Hz to 1.88 Hz during day 8. Gradual change in frequency from 1.88 to 1.90 Hz from day 8 up to day 20. Deviation in frequency of multiple observations between day 15 to 17.

The change in frequency presented around day 8 is the result of added mass effects. The frequency change from 2.00 to 1.88 Hz is equivalent to an increase of mass of 13% or 1200 tons. An overview of the operating and environmental conditions during day 8 is shown in Fig. 8. At an operating speed of 21 knots, the damping ratio is approximately 2.5%, while at 17 knots, the ratio reduces to about 1.2%. The correlation between speed and frequency is shown in Fig. 9. This relation does not explain all variations. Therefore, the environmental parameters have also been examined. Although there is little variation in wave height, a change in wave period from 9 to 8 s can be seen. Although the measurements

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show an abrupt change in period, it would be more likely that this shift happens more gradually. On top of that, the incoming wave direction varies between bow quartering (135◦ ) and beam seas (90◦ ). The added mass at these periods and speeds has been compared using the added mass obtained from diffraction calculations [20], see Table 2. The difference observed in this table amounts to approximately 500 tons. Because of the various changes in operating and environmental conditions, the estimated change in added mass is quite uncertain. The comparison shows that an increase in added mass such as the 1200 tons estimated from the change in frequency, may be possible in this period. Table 2. Changes in added mass for different operating conditions. The changes are relative to the lowest added mass observed at 8 second bow quartering waves in 21 knots. Wave direction Beam seas Bow quartering 18 kn Period [s] 8 9 21 kn 8 9

+305 +508 +305 +509

+18 +112 +0 +78

Between day 8 and 20 a gradual change in the frequency is observed from 1.88 to 1.90 Hz. This change is equivalent to a mass reduction of around 200 tons over the entire period. The vessel is consuming around 10 tons of fuel each day at this speed, around 130 tons in total, which corresponds well with the observed change in frequency. In the period between day 15 and 18, a number of outliers in the vibration frequency can be identified. The frequency, damping and speed in this period has been shown in Fig. 10. The changes show a correlation with speed variations. The difference in added mass between 0 knots and 15 knots is about 450 tons according to the numerical analyses. The change estimated from the natural frequency corresponds to 300 tons, so the temporary stops of the ship are likely responsible for the change in frequency.

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4.4

Global Analysis Second Period

A more detailed overview of the second period including environmental and operational parameters is shown in Fig. 11. This Figure shows speed and heading which were used to determine time frames with constant operations. The most consistent periods are during day 3 (albeit at two different speeds of 16 and 21 knots) and day 8 and 9 at 14 knots. During day 8, a gradual increase in damping from 0.5% to 1.5% can be seen. Day 9 sees a reduction of the frequency. The same period sees a change in wave height from 1 m to 2 m and back to 1 m. Figure 12 shows the amount of observed damping as function of the significant wave height. This figure indicates a correlation between both parameters although there are also other contributing factors. The speed effect which was observed earlier can also be detected in this graph with a higher damping found at higher speeds. The damping ratio on day 3 is somewhat higher than those on day 8 and 9. This can be related to the small difference in speed, but could also be related to different wave directions as the dominant wave direction on day 3 is bow quartering and on day 8 and 9 is beam seas. It is unknown if a similar relation can be observed in head sea conditions as well.

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Fig. 11. Frequency and damping on the vessel during the second period shown in Fig. 6 with associated navigational and environmental data. The wave direction as mentioned in this graph is relative to the vessel. 4 Day 3, V ≈ 16 kn Day 3, V ≈ 21 kn Day 8, V ≈ 14 kn Day 9, V ≈ 14 kn

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Fig. 12. Correlation between wave height and damping as observed during selected days of the second period. In day 3, the vessel is operating at bow quartering seas, while during day 8 and 9 the vessel is operating in beam seas.

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Discussion Filtering Results

A considerable amount of variability in the results of the OMA procedure can be found. For meaningful analyses of dependency between modal parameters and global loading effects, adequate filtering is required. The examples presented in this paper have shown the efforts undertaken to be able to understand the observed deviations in the data set. Parts of this information had to be obtained in hindsight from other public sources and was not monitored during the campaign. This explains the challenge to enable formal statistical analysis of such a dataset which was obtained from non-dedicated measurements. Based on the presented data, the authors recommend the following checks on the obtained results to ensure useful output: – Frequency A limited deviation between the observed frequency and reference frequency should be specified – Mode shape The comparison of mode shapes using the MAC should be used to eliminate deviating modes – Stationarity Verification of stationarity of operational parameters, speed and heading, is necessary to eliminate cases with large navigational changes which can lead to ambigious output of the analysis – Confined water In confined waters the added mass changes significantly and may obscure other correlations. Note that no threshold values on these criteria are specified. These depend on the type of ship, mode of operation, set-up of the measurements system etc. The parameters used by the authors have been mentioned throughout the article, but are no general applicable values. 5.2

Mass Sensitivity

The modal frequency is determined by stiffness and mass of the structure. As the structure itself does not change, the stiffness will not change. However, the mass or mass distribution can change significantly over time. This is related to both fuel consumption as well as changes in added mass which depends on both operational and environmental conditions. Frequency changes as a result of fuel consumption have been observed in a period of 12 days during which the vessel was sailing at constant speed. However, this effect is very subtle and fuel consumption has a negligible effect on overall vibration behaviour. Added mass of the vessel depends on speed, heading and sea states. Therefore, the added mass can change quickly when the operator changes speed or course. This effect has been noticed in the measurements and the results are in line with the observed changes in the measurements. This highlights the importance of using the correct added mass when studying whipping effects in different operational and environmental conditions.

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Speed Sensitivity

The vessel which was studied is deployed for homeland security which means it has a wide range of operating modes. As a result, the vessel was seen to be navigating significantly and often changes speeds. A small set of conditions have been isolated in which the vessel is sailing at similar speed and heading on open ocean for a significant amount of time. These examples have shown that the damping of the flexural vibrations are speed dependent. At higher speeds, the vessel will encounter a larger amount of water during each vibration cycle. Higher speed will therefore increase the added mass which was confirmed by the diffraction calculations. The increased speed will also give rise to increased radiation and viscous friction between the hull and the surrounding water, thereby leading to an increased damping. 5.4

Environmental Sensitivity

A correlation between damping and wave height was found for beam seas and bow quartering conditions. The damping changed from about 0.7% in a 1 m sea state to around 1.2% in a 2 m sea state. Although these are both mild conditions, they indicate that in extreme conditions, higher damping ratios may be appropriate. During this period a change in speed was also found which showed a much stronger change in damping than the contribution of the wave height.

6

Conclusions

OMA was successfully applied to in-service measurements on a fast displacement ship during normal operations. The ship was instrumented with accelerometers and strain gauges and OMA was applied to both types of measurements. In order to obtain useful results stringent filtering on mode shape, frequency, damping and stationarity of operating conditions should be applied. Operating in confined waters also has a noticeable effect. Note that only by applying this stringent filtering, one can mimic the stationary conditions which are also applied in model tests and simulations. Variations in the observed natural frequency can be as large as 10%. These variations must be related to mass variations. In specific cases, the effect of confined waters, heading and speed variations on added mass has been highlighted. Added mass in various operating conditions can be obtained from diffraction codes. The calculations are in line with the observed changes in vibration frequency. At average operating speeds in moderate environmental conditions, the damping of the two node bending mode is around 0.6 to 1%. However, analysis presented in this paper have indicated that damping is related to speed and wave height. Specifically, the damping in 26 knots increases to 2.5% from the 1% at 15 knots. In beam sea conditions, an increase in damping from 0.7% to 1.2% was found in wave heights ranging from 1 to 2 m. This indicates that when studying extreme conditions different damping ratios are applicable compared to intermediate conditions.

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Acknowledgements. This research is partly funded by the Dutch Ministry of Economic Affairs.

References 1. Allemang, R.J.: The modal assurance criterion - twenty years of use and abuse. Sound Vibr. 37(8), 14–21 (2003) 2. Andersen, P.: Identification of Civil Engineering Structures using Vector ARMA Models. Ph.D thesis, Aalborg University (1997) 3. Dascotte, E.: Vibration monitoring of the Hong Kong stonecutters bridge. In: Evaces (2011) 4. Drummen, I., Schiere, M., Dallinga, R., Stambaugh, K.A.: Full and model scale testing of a new class of US coast Guard cutter. In: Ship Structure Comittee (2014) 5. Faltinsen, O.M.: Sea Loads on Ships and Offshore Structures. Cambridge University Press, Cambridge (1990) 6. Hageman, R.B., Drummen, I.: Modal analysis for ships. Marine Struct. 63, 318–332 (2018) 7. Khoshnoudian, F., Esfandiari, A.: Structural damage diagnosis using modal data. Scientia Iranica 18(4), 853–860 (2011) 8. Koning, J., Kapsenberg, G.: Full scale container ship cross section load - first results. In: Conference on Hydroelasticity (2012) 9. Lauwagie, T., Dascotte, E.: A scenario-based damage identification framework. In: Topics in Modal Analysis II (2012) 10. Martens, H., Hoy, M., Wise, B.M., Bro, R., Brockhoff, P.B.: Pre-whitening of data by covariance-weighted pre-processing. J. Chemom. 17, 153–165 (2003) 11. Miao, S., Knobbe, A., Koenders, E.A.B., Bosma, C.: Analysis of traffic effects on a Dutch highway bridge. In: IABSE Symposium (2013) 12. Orlowitz, E., Andersen, P., Brandt, A.: Comparison of simultaneous and multisetup measurement strategies in operational modal analysis. In: Proceedings of the 6th International Operational Modal Analysis Conference (IOMAC) (2015) 13. Orlowitz, E., Brandt, A.: Operational modal analysis for dynamic characterization of a RO-LO ship. J. Ship Res. 58(4), 216–224 (2014) 14. Ozbek, M., Rixen, D.J.: Operational modal analysis of a 2.5 MW wind turbine using optical measurement techniques and strain gauges. Wind Energy 16(3), 367– 381 (2012) 15. Salawu, O.S.: Detection of structural damage through changes in frequency: a review. Eng. Struct. 19(9), 718–723 (1995) 16. Sheinberg, R., Cleary, C., Stambaugh, K.A., Storhaug, G.: Investigation of wave impact and whipping response on the fatigue life and ultimate strength of a semidisplacement patrol boat. In: International Conference on Fast Sea Transportation (2011) 17. Soal, K., Bienert, J., Bekker, A.: Operational modal analysis on the polar supply vessel the s.a. agulhas II. In: Proceedings of the 6th International Operational Modal Analysis Conference (IOMAC) (2015) 18. Stambaugh, K.A., Drummen, I., Cleary, C., Sheinberg, R., Kaminski, M.L.: Structural fatigue life assessment and sustainment implications for a new class of US Coast Guard Cutters. In: Ship Structure Comittee (2014) 19. Thornhill, E.M., Stredulinsky, D.C.: Real time local sea state measurement using wave radar and ship motions. SNAME Trans. 118, 248–259 (2010)

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20. Tuitman, J., Malenica, S.: Fully coupled seakeeping, slamming and whipping calculations. J. Eng. Marit. Environ. 223, 439–456 (2009) 21. van Overschee, P., De Moor, B.: Subspace Indentification for Linear Systems. Kluwer Academic Publishers, Dordrecht (1996) 22. Veerman, R.P., Koenders, E.A.B.:Structural monitoring and modal properties of a real time bridge and lab tests. In: Proceedings of the 4th International FIB Congress (2014) 23. Wang, T., Celik, O., Catbas, F.N., Zhang, L.M.: A frequency and spatial domain decomposition method for operational strain modal analysis and its application. Eng. Struct. 114, 104–112 (2016) 24. Xin, J., Hu, S.-L.J., Li, H.: Experimental modal analysis of jacket-type platforms using data-driven stochastic subspace identification method. In: Proceedings of the OMAE (2012) 25. Zhao, J., Zhang, L.: Structural damage identification based on the modal data change. Int. J. Eng. Manuf. 4, 59–66 (2012)

Vibration Characteristics of Separated Superstructure of a Ship Yukitaka Yasuzawa1(B) , Akihiro Morooka1 , and Kazuyuki Tanimoto2 1 Kyushu University, 744 Motooka, Nishi-ku, Fukuoka, Japan

[email protected] 2 Fukuoka Shipbuilding Co., Ltd., 3-3-14 Minato, Shuo-ku, Fukuoka, Japan

Abstract. The IMO noise code revised in 2012 is applied to newly built ships whose tonnage are equal or larger than 1600GT’s. Particularly for vessels of 10,000 GT’s or more, the noise regulation value of the living quarter became more severe than before. As a countermeasure to the noise code in the living quarter, it is recommended to separate accommodation area from funnel casing structure. However, as the result of the separation, natural frequencies of the superstructure is expected to decrease and possibly cause resonance with the excitation due to the engine or propeller excitation. Therefore, in the present research we aim to elucidate the vibration characteristics of a separated superstructure of a chemical tanker with L × B × D = abt.146 m × 24 m × 13 m. We have calculated principal natural frequencies and modes of the separated superstructure by using finite element vibration analysis for 3 types of models, as 1) superstructure (S.S.) only on the rigid foundation, 2) S.S. with the aft part of the ship including the engine room considering the effect of the elastic foundation, and 3) whole ship model considering the effect of hull girder vibration including the effect of added mass of outer sea water. Then the effect of shallow water and proximity related to added mass are investigated by using the whole ship FE model. Finally some structural modifications to increase the natural frequencies are investigated by using the numerical analysis, and the effects of reinforcement on the vibration characteristics have been discussed by comparing with those of the original structure. Keywords: Vibration · Superstructure · Whole ship FE analysis

1 Introduction In 2012, IMO Maritime Safety Committee adopted a draft amendment on the inboard noise code that recommends that noise generated from engine room and crew’s noise exposure be kept below a certain level in order to preserving health of crews. In the new noise code [1], noise regulation values, measurement methods, measurement devices, etc., have been revised. The revised noise code is applied to newly built ships whose tonnage are equal or larger than 1600GT’s. Particularly for vessels of 10,000GT’s or more, the noise regulation value of the living quarter was reduced by 5 dB. © Springer Nature Singapore Pte Ltd. 2021 T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 365–376, 2021. https://doi.org/10.1007/978-981-15-4672-3_23

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As a countermeasure to the noise code in the living quarter, it is recommended to separate accommodation area from funnel casing structure. However, as the result of the separation, natural frequencies of the superstructure are expected to decrease and possibly cause resonance with the excitation due to the engine or propeller excitation. Therefore, in the present research we aim to elucidate the vibration characteristics of a separated superstructure of a chemical tanker with L × B × D = abt.146 m × 24 m × 13 m. Firstly, the effect of modeled range of the ship structure are investigated by calculating natural frequencies and modes of the separated superstructure. FE analysis has been performed for 3 types of FE models, 1) superstructure alone model on the rigid foundation, 2) S.S. with the aft part of the ship including the engine room considering the effect of the elastic foundation, and 3) a whole ship model considering the effect of hull girder vibration including the effect of added mass of sea water. Secondly the effect of the shallow water or a quay on the natural frequencies and modes are investigated, because the excitation test is usually performed near the quay. To verify the FE analysis results, the numerical results were compared with the excitation test ones. Thirdly the effect of structural design change to raise natural frequencies are investigated to prepare for the possibility of resonance after construction. We have performed numerical analysis for some realistic cases by structural change and discussed by comparing with those of the original structure.

2 Effect of FE Modeling Range on Superstructure Vibration 2.1 Natural Vibration Analysis Using FEM In the present study, finite element analysis is used to derive the natural frequencies and their natural vibration mode by solving the following equation of an eigenvalue problem,   K − ω2 (M + M∗ u = 0 (1) where K, M, M*, and u are stiffness matrix of the structure, mass matrix, added mass matrix of sea water, and nodal displacement vector respectively. Solution of Eq. (1) are derived as natural angular frequencies, ωi and mode vector, ui . Mass matrix includes not only the structural mass but also all the other mass contained in the ship structure. Added mass matrix is obtained by the virtual mass method based on the potential fluid theory considering the free surface location. In the present study, only the surrounding sea water produces the added mass effect on the hull structure. Various inner liquid contained in the ship structure are replaced by equivalent nodal mass or distributed mass attached to the structural elements. All the matrices and solutions are calculated by using a commercial software, MSC Nastran. 2.2 FE Modellings for Superstructure Vibration Analysis The target ship to investigate is a chemical tanker. The principal particular is L × B × D = abt.146 m × 24 m × 13 m. Figure 1 shows a separated superstructure’s FE model

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above the upper deck level where accommodation area is separated from funnel-casing structure above upper deck. Deck plates, hull skins and side walls of the superstructures are reproduced with equivalent shell elements with 4 nodes, and stiffeners and pillars are created with beam elements. The mesh size of each element is 0.700 m or less which is the frame space of the present ship. Details of the method to derive the equivalent property is explained in Ref. [2]. For FE analysis, a commercial finite element analysis software, MSC Nastran was used.

Fig. 1. A FE model and constrained nodes of a separated superstructure alone model above the upper deck (Blue triangle markers are the restrained points)

Superstructure Alone Model The simplest FE model for the superstructure is a ‘superstructure alone model’ shown in Fig. 1. In this case, the range of the model is limited above the upper deck. In this analysis, comparatively rigid edges connected with the wall of the lower structure are constrained as the geometrical boundary condition as shown in the Fig. 1. Therefore, the elastic foundation effect of the lower structure is neglected in this model. Stern Structure Model of Superstructure with Engine Room The second FE model is the superstructure with the aft part of the ship including the engine room as shown in Fig. 2. For the vibration analysis, this FE model is constrained at the fore transverse bulkhead and the bottom narrow part. Added mass effect of sea water contacting the hull surface is considered. This FE model can take the elastic foundation effect for the superstructure into consideration, but the interaction with the hull girder vibration cannot be incorporated enough. Whole Ship FE Model The third FE model is a whole ship FE model shown in Fig. 3. There is no constraint all through the structure as kinematic boundary condition. Added mass effect of sea water contacting the hull surface is considered. Number of shell elements are 15,295 for the superstructure and 10,939 for hull girder where the maximum size of the shell

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Fig. 2. Stern structure FE model and constrained points (Light blue markers indicate constraints)

element in the hull girder is the same as one transverse frame space. Corrugate bulkhead in the tank holds are replaced by the equivalent orthotropic shell element referring to [3]. Number of shell elements in contact with outer seawater is about 2500. Numerical calculations are conducted for the loading conditions corresponding to empty, ballast, and full load conditions respectively. The empty condition is a special light weight condition where the ship is floating in the shallow water near a quay. Impact hammering test was conducted to investigate the vibration characteristics by using the technique of experimental modal analysis. Hull surface area contacting with seawater is dependent on the loading conditions. Loading conditions we considered are shown in Table 1.

Fig. 3. Whole ship FE model

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Table 1. Loading conditions and draft Loading condition

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Front draft df (m)

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abt. 5,000

4.7

0.3

Ballast

abt. 11,500

5.9

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Full load

abt. 25,000

10.0

9.4

2.3 Numerical Results of the Effect of Modelling Range We focus on the fore-aft vibration of separated superstructure in the present study. Representative fore-aft vibration modes of the present chemical tanker in full load condition by solving Eq. 1 for a whole ship FE model are shown in Figs. 4 and 5. Two modes in Fig. 4 are strongly affected by the hull girder vibration where the motion of superstructure is dependent on the global hull girder vibration with 6 nodes and 7 nodes respectively.

Fig. 4. Hull girder dominant modes (upper: 422 cpm (7.03 Hz), lower: 505 cpm (8.42 Hz))

On the other hand, two modes in Fig. 5 are not affected by global hull girder deformation. In this superstructure dominant modes, natural frequency of reverse mode is lower than that of in-phase mode. In the reverse mode, rocking motion of the structure of living quarter occurs because the front wall of the structure is rigidly supported by the transverse bulkhead while the aft wall of the structure is not supported generally due to the existence of the engine room. As for the in-phase mode, the superstructure of both living quarter and funnel casing structure moves in the same direction but the stern part above propeller moves vertically in reverse. Comparison of Natural Frequencies of Superstructure The superstructure dominant modes are observed both superstructure alone model and stern structure modes. Natural frequencies of fore-aft vibration modes of superstructure for empty, ballast, and full load condition have been calculated as shown in Fig. 6.

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Fig. 5. Superstructure dominant mode (upper: opposite phase mode, 568 cpm (8.47 Hz), lower: in-phase mode, 644 cpm (10.73 Hz))

In reverse phase mode, it is largely influenced by the elastic foundation effect below the upper deck level and natural frequency is reduced for stern structure model and whole ship model while the influence of the loading condition is small. In in-phase mode, natural frequency is largely influenced by the loading condition because this mode is coupled by the stern part. Loading condition changes the added mass of sea water largely on the stern part.

Fig. 6. Numerical results of natural frequencies of fore-aft vibration of superstructure in empty, ballast, and full load conditions (SS alone: Super structure alone model, Stern model with superstructure, Whole Ship: whole ship FE model)

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3 Effect of Shallow Water and Proximity of Quay 3.1 Modeling of Sea Bottom and Quay and Numerical Convergence Impact hammering test or exciter test is often conducted near quay because of the convenience of loading and unloading of the test machine. But it is located in shallow water and close to the quay wall. Generally the added mass effect may increase when the fluid motion is restricted. Also the loading condition is different from normal operating condition. Therefore it is difficult to predict the natural frequency or vibration response from the experimental test in quay area. In case of hull girder dominant vibration of superstructure, there might be the large effect of the shallowness and proximity on the natural frequencies. Therefore, a whole ship vibration analysis considering the effect of shallow water and quay wall has been performed. Those effect is realized by the function of MFLUID in MSC NASTRAN arranging the rigid large plate to restrict fluid motion around the whole ship FE model as shown in the right of Fig. 7. As the results of investigation of the numerical convergence, the length and breadth of the rigid plate is decided to be three times large as the ship length and 7 times large as the ship width respectively, and the mesh size of the square plate element is selected to be 2.8 m.

Fig. 7. Configuration of a ship, sea bottom, and quay and FE to restrict the fluid motion

3.2 Added Mass Effect of Shallow Water and Quay on the Natural Frequencies The effect of shallow water and the effect of proximity of quay wall on the natural frequencies of vertical hull girder vibrations are shown in Fig. 8 and Fig. 9 respectively. The parameters of shallowness to water depth and proximity to quay wall are defined respectively T/d and C/b referring to the left figure in Fig. 7. The effect of shallow water is negligible for T/d > 5, but cannot be negligible for T/d < 2. This result is consistent with refs. [4] and [5]. The effect of proximity of quay wall is negligible for C/b > 2, and much smaller for vertical hull girder vibration compared with the effect of shallowness.

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Fig. 8. Effect of shallow water on the natural frequencies of vertical hull girder vibration

Fig. 9. Effect of proximity of quay on the natural frequencies of vertical hull girder vibration

3.3 Comparison of Natural Frequencies Between Experimental Results and Whole Ship FE Analysis In Fig. 10, natural frequencies obtained by hammering test close to a quay is compared with those obtained by whole ship FE vibration analysis with and without considering the effect of shallow water and quay wall explained in the Sect. 3.2. Hammering test means the impact excitation method based on the experimental modal analysis. The impact excitation was performed using large hammering equipment fixed on the stern deck. And the structural responses are derived simultaneously from multiple accelerometers placed on the ship structure. Natural frequencies and corresponding vibration modes are identified from experimental modal analysis with FFT of the obtained data. In the FE analysis, water depth T = 7 m are used. Judging from the results in Fig. 10, shallow water effect can be observed. But some of the analysis results are too reduced by the effect compared with experimental ones. It may come from the difference of the flatness and inclination of the sea bottom between the numerical model and real experimental condition.

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Fig. 10 Comparison of natural frequencies between whole ship FEA and experimental test (Vert2: vertical hull girder vibration with 2 nodes, Tors-1: torsional hull girder vibration with 1 node, Trans-3: transverse hull girder vibration mode with 3 nodes, funnel: funnel casing structure, quarter: living quarter structure)

4 Effect of Design Modification on Natural Frequencies and Modes 4.1 Design Modification to Raise the Natural Frequencies Separated superstructure has lower natural frequencies than conventional integrated superstructure. Therefore possibility of resonance against the excitation source may be higher. It is important to know the ways to increase the natural frequencies of the superstructure effectively by structural modification to avoid resonance to excitation forces. The effect of three kinds of stiffening approach are considered and investigated by numerical analysis using whole ship FE model. 1. Large connecting structural members are added between the superstructure and hull girder. Large girder type or box girder support type are considered as shown in Fig. 11. The latter is introduced in this paper (Box girder support type). 2. Plate thickness of side walls of living quarter structure is increased to raise the vertical in-plane shear stiffness (Wall thickness increased type) as shown in Fig. 12 Breadth and depth of the cross section are 1.500 m and thickness is 10 mm. 3. Partial connections between living quarter and funnel-casing structures are built to avoid the settling with rocking motion (Partial bridge type) as shown in Fig. 13. Three girders are built on each floor of C deck and D-deck respectively. The cross section size of the girder is 600 mm × 300 mm each.

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Fig. 11. Box girder support

Fig. 12. Wall thickness increased to enhance the shear stiffness

D-deck C-deck

Fig. 13. Partial connection by 6 girder between superstructures

Increase of the natural frequency to the original structure is shown in Table 1. In case of box girder support type as shown in Fig. 11, natural frequencies of superstructure are remarkably increased for both reverse and in-phase modes. Especially this support is effective to in-phase mode in superstructure. But it should be noted that hull girder vibration modes are strongly influenced and natural frequencies are decreased as shown in Table 1. In case of increasing side wall thickness, effect is smaller than any other type though it is observed a small improvement in reverse mode of superstructure. In case of the partial bridge connection type, it is effective to both reverse and in-plane modes. This is especially effective to the reverse mode (Table 2).

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Table 2. Increasing rates of natural frequencies by structural modification (unit: %) Vibration Mode

Box girder support

Side wall thickness increased

Partial bridge connection

Vert-2

−10.3

−0.1

−0.0

Vert-3

−14.2

−0.0

−0.0

Vert-4

−16.5

0.0

−0.0

Vert-5

−12.2

0.1

−0.0

Vert-6

−6.0

0.1

0.0

Vert-7

4.7

0.1

0.1

SS-reverse

4.8

0.6

9.2

SS-in-phase

18.4

0.1

0.9

5 Conclusion Natural frequencies of separated superstructure are calculated for superstructure only model, superstructure with stern structure model, and whole ship mode. As the results, it is proved that the elastic foundation effect is large for reverse mode and that loading condition affects the in-phase mode related to the interaction with stern part vibration in contact with seawater. Superstructure dominant modes can approximately be derived by stern structure model. Shallow water effect and proximity effect to quay wall have been investigated. Hull girder dominant mode are strongly affected by the effect while superstructure dominant modes are hardly affected. Comparison between natural frequencies obtained from hammering test and those of whole ship FE model. It proved the reduction of natural frequencies due to the effect of shallow water and proximity of quay wall. The effects of some structural modifications are investigated using whole ship model to raise natural frequencies related to superstructure. As the results, connection between superstructure and hull girder with large support members may raise the natural frequencies effectively, but may largely reduce the natural frequencies of hull girder vibration. The method to increase the thickness of sidewall of the superstructure increases the natural frequency a little. Connection between the accommodation structure and funnel casing structure is very effective to reverse mode.

References 1. Solas regulation II-1/3–12: Code on noise levels on board ships, IMO Noise Code MSC 337(91) (2012) 2. Morooka, A., Yasuzawa, Y.: Vibration characteristics of separated superstructure of a ship. In: Proceedings of the 32th Asian-Pacific Technical Exchange and Advisory Meeting on Marine Structures (2018)

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3. Xia, Y., Friswell, M.I., Saavedra Flores, E.I.: Equivalent models of corrugated panels. Int. J. Solids Struct. 49, 1453–1462 (2012) 4. Japan Shipbuilding Research Association SR94: Report of experimental research on vibration prevention, No. 91 (1969) 5. Matsuura, Y., Kawakami, H.: Calculation of added virtual mass and added virtual mass moment of inertia of ship hull vibration by the finite element method. J. Soc. Naval Architects Jpn. 124, 281–291 (1968)

Ship Vibration and Noise Reduction with Metamaterial Structures Deqing Yang(B) State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China [email protected]

Abstract. Metamaterials and metamaterial structures show excellent performance in vibration reduction and energy absorption. The developments on modelling theories and additive manufacturing technology have enable the application of metamaterials to vibroacoustics design of ships. In this paper, metamaterial structures with auxetic effect and negative stiffness are designed to decrease vibration, underwater noise radiation and shock in naval structures. Functional element topology optimization (FETO) method to design lightweight metamaterials with Negative Poisson’s ratio(NPR), Negative Stiffness (NS) lattice metamaterals design by prefabricated buckling component method (PBCM) are introduced. A lightweight auxetic cellular vibration isolation base and a novel floating raft structures constituted by re-entrant hexagonal honeycombs showed the effect of vibration reduction in a geophysical vessel. A multilayer negative stiffness shock isolation base for marine structure show the application potentials of shock absorption of metamaterial structures. By replacing the conventional ribs with auxetic metamaterial rib in the submarine power equipment cabin between the double shell, the radiated sound power level and sound radiation directivity of a submarine are designed. Energy from power equipment can be blocked by using of auxetic metamaterial ribs. Keywords: Ship · Marine structure · Metamaterials · Vibration · Noise · Auxetic · Negative stiffness

Metamaterials are products of human ingenuity and their properties depend largely on the internal construction rather than on the parent materials [1]. Over the past few decades, programmed by suitable topology and configuration, these artificial materials have been studied and designed to obtain unconventional characteristics, such as 2D and 3Dbehaviors of negative or zero Poisson’s ratio, tunable magnitude and prescribed directionality of thermal expansion, negative effective mass density and negative effective elastic modulus and low-frequency sound absorption [2, 3]. Possessing the features of light weight, shock absorbing and vibration reduction, the study of metamaterials have attracted increasing attention of scholars for several decades. Negative Poisson’s ratio materials and structures are termed by Evans et al. as auxetics or auxetic cellular

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materials exhibiting the interesting property by virtue of the auxetic behavior [4]. In this paper, applications of metamaterials to vibroacoustics and lightweight design of ships are presented. Metamaterial structures with auxetic effect and negative stiffness are designed to decrease the underwater noise radiation, vibration and shock in naval structures.

1 Design Methods for Metamaterials with Negative Poisson’s Ratio and Negative Stiffness 1.1 FETO Method and Modeling Theory for NPR Metamaterials Design A functional element topology optimization (FETO) method to design metamaterials with arbitrary Poisson’s ratio is proposed by Yang [5] which is different from the existed methods [6–9]. Figure 1 illustrates the main idea of FETO method in which the shape of a functional element can be rectangular, triangular, circular or cubic solid.

Fig. 1. Schematic of FETO method

Two points are chosen to measure the macroscopic NPR effect, corresponding to points 1 and 2 in Fig. 2 in functional element.

Fig. 2. Functional element (rectangular and triangular)

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The macroscopic NPR of functional element is calculated as ν= −

ε2 u2 B =− · ε1 w1 H

(1-1)

where, the Z-direction displacement change of points 1 is denoted by w1 , and the X-direction displacement change of points 2 is denoted by u2 . Modeling theory to consider load bearing capacity, energy absorption, lightweight and vibration reduction were presented as the following formulations. ⎧ find X = {x1 , x2 , · · · , xN } ⎪ ⎪ ⎪ ⎪ ⎪ N ⎪  ⎪ ⎪ T ⎪ min C(X) = U KU = (xe )p uTe k0 ue ⎪ ⎪ ⎪ ⎪ e=1 ⎪ ⎨ V (X) (1-2) s.t. ≤ fvol ⎪ ⎪ V0 ⎪ ⎪ ⎪ ⎪ KU = F ⎪ ⎪ ⎪ ⎪ ⎪ ν = ν0 ⎪ ⎪ ⎪ ⎩ 0 < xmin ≤ xe ≤ xmax ≤ 1 (e = 1, . . . , N ) Model (1-2) is established with the objective function of minimal structural compliance, which emphasizes the structural load bearing capacity. X = {x1 , x2 , · · · , xN } is the vector of relative density values of design domain; xe is the design variable of e-th element relative density, e = 1, 2, · · · , N , N is the number of design variables; xmax , xmin are the upper and lower limits of the design variables, respectively (non-zero to avoid singularity); C(X ) is the structural compliance; F and U are the global loading and displacement vector; K is the global stiffness matrix; ue and k0 are the displacement vector and stiffness matrix of each mesh element corresponding to the design variables, respectively; V (X) is the total structural volume in the optimization progress; V0 is the initial total volume when the relative density of the design domain area is 1; fvol is the specified volume fraction of the design domain; ν = ν0 is the constraint of NPR; ν0 is the specified design value of NPR. ⎧ ⎪ ⎪ find X = {x1 , x2 , · · · , xN } ⎪ ⎪ ⎪ N ⎪  ⎪ ⎪ T ⎪ KU = max C(X) = U (xe )p uTe k0 ue ⎪ ⎪ ⎪ ⎪ e=1 ⎪ ⎨ V (X) (1-3) s.t. ≤ fvol ⎪ ⎪ V0 ⎪ ⎪ ⎪ ⎪ KU = F ⎪ ⎪ ⎪ ⎪ ⎪ ν = ν0 ⎪ ⎪ ⎪ ⎩ 0 < xmin ≤ xe ≤ xmax ≤ 1 (e = 1, . . . , N )

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Model (1-3) means that energy absorption performance of the configuration is the priority. The improvement of structural softness is advantageous to the impact resistance and energy absorption characteristics of the configuration, which can effectively filter dynamic vibration. ⎧ find X = {x1 , x2 , · · · , xN } ⎪ ⎪ ⎪ ⎪ ⎪ N ⎪  ⎪ ⎪ ⎪ min M = xe V0 (X) ⎪ ⎪ ⎪ ⎪ e=1 ⎪ ⎨ s.t. KU = F (1-4) ⎪ ⎪ ν = ν ⎪ 0 ⎪ ⎪ ⎪ ⎪ V (X) ⎪ ⎪ ⎪ ≤ fvol ⎪ ⎪ V0 ⎪ ⎪ ⎩ 0 < xmin ≤ xe ≤ xmax ≤ 1 (e = 1, . . . , N ) Model (1-4) established with the total minimal structural mass as the optimization objective function can realize the lightweight design. Unlike the optimization models described above, here the upper limit of the volume constraint is reduced to further control the amount of material volume. ⎧ find X = {x1 , x2 , · · · , xN } ⎪ ⎪ ⎪ ⎪ ⎪ max L¯ z0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ s.t. MU + KU=F (1-5) ν = ν0 ⎪ ⎪ ⎪ ⎪ V (X) ⎪ ⎪ ≤ fvol ⎪ ⎪ V0 ⎪ ⎪ ⎩ 0 < xmin ≤ xe ≤ xmax ≤ 1 (e = 1, . . . , N ) Model (1-5) established with maximization of the origin impedance level as objective function can realize the vibration reduction design. L¯ z0 denotes the synthesized origin p impedance level. M = N e=1 xe m0 is the total mass of the functional element. 1.2 Buckling Component Method for Negative Stiffness Metamaterials Design Negative Stiffness (NS) lattice metamaterial structures are designed by buckling component method [10]. The uni/bi/tridirectional NS metamaterials are illustrated in Figs. 3a– 3c. All the three designs are composed of relatively rigid supporting frames and flexible curved beams. The shape is given by w(x) = h/2·[1 – cos(2πx/l)] where l is the beam span and h is the initial apex height. Curved beams are assembled row by row toward one, two and three different directions respectively, to ensure that they are loaded laterally at the midpoints, thus negative stiffness effect can be utilized for multiaxial stress conditions. In both the bidirectional and tridirectional designs, adjacent curved beams are connected with short straight rods (See Figs. 3d). The basic lattice of the bidirectional design is a planar square frame with four oblique beams at its vertices and for the tridirectional

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Fig. 3. Designs of (a) unidirectional, (b) bidirectional and (c) tridirectional NS metamaterials and their unit lattices

design, it is a hexahedral frame with square cross-sections. The investigated bidirectional design contains 4 × 4 lattice units and the tridirectional one contains 3 × 3 × 3 units. It is worth highlighting that by adjusting their section sizes, the supporting frames can be designed to provide rigid or flexible boundary conditions for the curved beams, resulting in various overall properties of the metamaterials. Meanwhile, to avoid unwanted twisting and rotation at the midpoint and to increase the overall stiffness, two or more beams can be clamped together at the center for all the three designs (See Fig. 3a). In the current work, for the sake of simplification we set the frames as nearly rigid and have not added any clamped beams into the design. However, in further study on quantitative design of these metamaterials, these issues will be discussed in detail. To show the feasibility of these designs, three prototypes have been fabricated with a Raise3D® Fused Deposition Modeling (FDM) machine and the same polylactic acid (PLA) material, as shown in Fig. 4 below. Considering the manufacturing time, only several of the lattices are printed.

Fig. 4. Photographs of 3D printed prototypes: (a) unidirectional prototype, (b) bidirectional prototype and (c) tridirectional prototype

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2 Ship Vibration and Noise Reduction with Metamaterial Structures In this section, applications of metamaterials in vibroacoustics design of ships are presented. Metamaterial structures with auxetic effect and negative stiffness are designed to decrease the underwater noise radiation, vibration and shock in naval structures. 2.1 Vibration Reduction of a Geophysical Vessel with Novel Floating Raft Consist of NPR Metamaterial Figure 5 showed a novel floating raft in which raft frame is constituted by sandwich panel with auxetic cellular cores. Figure 6 and 7 demonstrate the FEM model of the floating raft and a geophysical vessel using the metamaterial floating raft.

Fig. 5. Metamaterial floating raft with auxetic cellular cores (Part C)

Fig. 6. FEM model of metamaterial floating raft

Figure 8 presented the vibration level difference using different metamaterial raft frame.

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Fig. 7. FEM model of a geophysical vessel with metamaterial floating raft

Fig. 8. Vibration level difference with metamaterial floating raft in a geophysical vessel

2.2 Multilayer Negative Stiffness Shock Isolation System for Marine Structure Unidirectional and bidirectional Buckling-based Negative Stiffness (BNS) lattice metamaterals design method by adding prefabricated curved beams into multidimensional rid frames are introduced by Yang and Ren [6]. A multilayer negative stiffness shock isolation base for marine structure show the application potentials of shock absorption of metamaterial structures (Fig. 9).

100

Kinetic energy Strain energy

energy (J)

80

60

40

20

0 0.00

0.01

0.02

0.03

0.04

0.05

time (s)

Fig. 9. Energy variations in 4 layer negative stiffness shock isolation system

0.06

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2.3 Vibration and Underwater Sound Radiation Performance Analysis of Submarine with Auxetic Metamaterial Ribs By replacing the conventional floor rib into auxetic metamaterial rib in the submarine power equipment cabin between the double shell, a submarine finite element model is established to analyze the vibration and radiation noise, the cabin vibration acceleration level of metamaterial rib and floor plate is compared under different quality constraint by adjusting the thickness of metamaterial rib, and the radiated sound power level and sound radiation directivity are calculated by using the coupled indirect boundary element method. It is indicated that auxetic metamaterial rib shows better noise reduction performance in synthetic acoustic power by comparing with submarine floor cabin structure, the energy from power equipment can be better blocked by using the auxetic metamaterial, which demonstrates the application value and prospect of the auxetic metamaterial rib. Figure 10, 11, and 12 presented the structures and the corresponding underwater sound pressure levels of using metamaterial ribs and conventional ribs (Fig. 13).

Fig. 10. FEM model of the submarine

Fig. 11. Cross section of auxetic metamaterial rib

a submarine with conventional ribs b

submarine with metamaterial ribs

Fig. 12. The half profile FEM model of the submarine

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Fig. 13. Acoustic power level of submarine with different ribs at 18 Hz

3 Conclusions In this paper, FETO method to design lightweight metamaterials with Negative Poisson’s ratio (NPR) and buckling component method to design Negative Stiffness (NS) lattice metamaterals are introduced. Modelling theories and additive manufacturing technology have enable the application of metamaterials to vibroacoustics design of ships. A lightweight auxetic floating raft constituted by re-entrant hexagonal honeycombs showed the effect of vibration reduction in a geophysical vessel. A multilayer negative stiffness shock isolation base for marine structure show the application potentials of shock absorption of metamaterial structures. By replacing the conventional ribs with auxetic metamaterial rib in the submarine power equipment cabin between the double shells, the radiated sound power level and sound radiation directivity of a submarine are designed. Energy from power equipment can be blocked by using of auxetic metamaterial ribs. Acknowledgment. This work was supported by the National Natural Science Foundation of China (51479115), High-tech Ship Research Projects by MIIT ([2014]148, [2016]548) and Opening Project by the State Key Laboratory of Ocean Engineering (GKZD010071).

References 1. Yu, X., Zhou, J., Liang, H., Jiang, Z., Wu, L.: Mechanical metamaterials associated with stiffness, rigidity and compressibility: a brief review. Prog. Mater. Sci. 94, 114–173 (2018) 2. Mizzi, L., Mahdi, E.M., Titov, K., et al.: Mechanical metamaterials with star-shaped pores exhibiting negative and zero Poisson’s ratio. Mater. Des. 14, 628–637 (2018) 3. Li, Y., Chen, Y., Li, T., Cao, S., Wang, L.: Hoberman-sphere-inspired lattice metamaterials with tunable negative thermal expansion. Compos. Struct. 189, 586–597 (2018) 4. Lakes, R.S., Lee, T., Bersie, A., Wang, Y.C.: Extreme damping in composite materials with negative-stiffness inclusions. Nature 410, 565–567 (2001) 5. Qin, H., Yang, D., Ren, C.: Modelling theory of functional element design for metamaterials with arbitrary negative Poisson’s ratio. Comput. Mater. Sci. 150, 121–133 (2018) 6. Sigmund, O.: Materials with prescribed constitutive parameters: an inverse homogenization problem. Int. J. Solids Struct. 31(17), 2313–2329 (1994) 7. Zhang, W., Sun, S.: Scale-related topology optimization of cellular materials and structures. Int. J. Numer. Meth. Eng. 68(9), 993–1011 (2006) 8. Yan, J., Cheng, G., Liu, S., et al.: Comparison of prediction on effective elastic property and shape optimization of truss material with periodic microstructure. Int. J. Mech. Sci. 48(4), 400–413 (2006)

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9. Xia, L., Breitkopf, P.: Recent advances on topology optimization of multiscale nonlinear structures. Arch. Comput. Meth. Eng. 24(2), 227–249 (2017) 10. Ren, C., Yang, D., Qin, H.: Mechanical performance of multidirectional buckling-based negative stiffness metamaterials: an analytical and numerical study. Materials 11, 1078 (2018). https://doi.org/10.3390/ma11071078

Experimental and Numerical Investigations About Hydraulic Top Bracings and Its Influence on Engine and Vessels Superstructure Vibrations Michael Holtmann1(B) and Martin H. Frandsen2 1 Department of Structural Engineering - Noise Vibration Shock, DNVGL, Hamburg, Germany

[email protected] 2 Department Engine Dynamics, Crankshaft & Bearings,

MAN Energy Solutions, Copenhagen, Denmark

Abstract. Top bracings are widely used to control vibration of main engines onboard ships. Currently two types of top bracings are used: hydraulic top bracings and mechanical top bracings with a friction connection. Main advantages of hydraulic top bracing are their improved ability to compensate relative deflections between the engine and the ship and that their dynamic characteristic can be controlled by different settings. Compared to conventional mechanical top bracings MAN-ES hydraulic top bracings have two basic settings that can be controlled by engine speed: the standard active setting and the passive setting. While in the active setting the hydraulic top bracing acts like a spring with a certain amount of stiffness, similar in its behavior to the mechanical type, in the passive setting it acts like a weak damper. To get more insight into the dynamic characteristics of hydraulic top bracings and their influence on the engine and structural vibrations, MAN-ES conducted several measurements on several ships. Based on the measurements, the working principle was, and their stiffness and damping properties were studied. Also, the condition with the top bracings being drained from oil and the correlation between the different top bracing settings and the vibrations response of the superstructure was investigated. In a joint project, DNVGL and MAN-ES aimed at correlating the measurement results by the simulation findings [1]. Based on a validated simulation model it was then even possible to investigate several more different nonstandard configurations and settings by additional simulations. Keywords: Ship structural vibration · Engine vibration · Top bracings

1 Controlling Transverse Engine Frame Vibration The engine produced guide force moments result in a rocking/rotational movement of the engine frame as seen in Fig. 4. This vibration type is normally controlled by a so-called top bracing system (Fig. 1). Currently two types of top bracings are used: hydraulic top bracings and mechanical top bracings with a friction connection (Fig. 2). © Springer Nature Singapore Pte Ltd. 2021 T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 387–407, 2021. https://doi.org/10.1007/978-981-15-4672-3_25

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Fig. 1. Principle installation of top bracing on engine and engine room structure

Fig. 2. Hydraulic type top bracing (left side) and Mechanical type top bracing (right picture)

1.1 Excitation/Guide Force Moments The engines guide force moments can be calculated for each engine type based on its gas and mass forces for each cylinder unit. The individual guide force moments produced from all cylinders are summarized by use of the engines firing angles creating resulting moments called the H and X type guide force moment. Each of these engine produced excitation types, can under adverse conditions, cause problematic resonances with well-known engine vibration mode shapes commonly known as the H-mode and the X-mode. 1.2 Vibration Mode Shapes A specific engine/hull combination will have a resulting set of natural frequencies and associated mode shapes. This set of vibration modes will to some extent be sensitive to H or X-type excitation and can cause high vibrations in case of resonance in the

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operational speed range of the engine. The resulting vibration is in most cases pronounced/dominant on the main engine structure but can also cause high vibration levels of vessels superstructures and local components/machinery in vessel’s main engine room. In the uncoupled vibration case, the mode shape is mainly composed by high motion of the main engine structure. An example can be seen left in Fig. 3.

Fig. 3. Uncoupled and coupled H-mode shape

Resonance can result in high local vibration of the engine frame but limited vibration of vessels superstructures. Uncoupled vibration is often seen in cases where top bracings have been omitted or installations where the top bracing system has low stiffness. This vibration type is also comparable to engines installed on testbed prior to vessel installation. The coupled vibration mode shapes are characterized by having both local engine frame movement in combination with high movement of global vessel structures such as the deckhouse and/or aft body. An example can be seen right in Fig. 3. The coupled vibration modes represent a large sub-group of vibration modes that can give different types of superstructure vibration such as transverse, torsion or longitudinal deckhouse vibration, but also more global modes such as aft body torsion or higher order transverse hull girder vibration. Often, the vibration type changes from uncoupled to coupled when top bracings are present. The top bracings shift the local engine mode to higher frequencies due to their contribution to the total installation stiffness. As many mode shapes of deck structures (transverse/torsional/longitudinal) are located at higher frequencies, the local engine mode can “shift up” and merge to global vessel modes and thereby become coupled (combined). Thereby the top bracing system can form an additional transmission path for engine vibrations to excite the more distant and global part of the vessel structure. It should however be noted that the effectiveness of top bracing is strongly depending on the design of its shipside foundation. It needs to be ensured that a sufficient rigidity in axial direction of the top bracings at the installation point is provided. Typically, a minimum stiffness is required by the top bracings and engine manufacturer.

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2 MAN-ES Hydraulic Top Bracing 2.1 Working Principle MAN-ES hydraulic top bracings have been designed to work in two operation modes: As a rigid connection between engine and hull side, or as a free connection with no stiffness. The two operation modes are referred to as active (TB-Active) or passive condition (TB-Passive). Pressurization. The top bracing consists of a single-acting hydraulic cylinder, an oil accumulator and a hydraulic control unit mounted on the cylinder housing. When the distance between the hull and engine increases, oil flows into the cylinder under pressure from the accumulator (7 Bar). When the distance between the hull and engine decreases, a non-return valve prevents the oil from flowing back to the accumulator and the pressure rises. TB-Active Condition. For steady state vibration and after pressurization has taken place, oil will not flow between cylinder and accumulator as the cylinder pressure is higher or equal to the accumulator pressure. In case vibrations increases and the minimum cylinder pressure drops below accumulator pressure during the positive displacement stroke, additional oil will flow into the cylinder. This results in an increase of the mean cylinder pressure (self-adjustment) so that the minimum pressure of 7 bar is restored. Thereby the top bracing is self-adjusting, by using the vibration energy emitted by the engine. If the pressure reaches a preset maximum value, a relief valve (Safety valve) allows the oil to flow back to the accumulator, maintaining the force on the engine and hull structures below the specified level. As the pressurized entrapped oil transmits forces between hull and engine side, the hydraulic top bracing will perform as a stiffness member only - as design-wise intended. So, when the engine is running, the unit builds up a pressure in proportion to the vibration level. Under normal working conditions and in TB-Active condition, the top bracing can therefore be idealized as a uniaxial spring element, having the same stiffness in both tension and compression, much like a pre-stressed bolted connection. The actual stiffness is mainly composed by the steel part of the top bracing and the stiffness of the enclosed oil volume. TB-Passive Condition. In the passive condition, the non-return valve is bypassed by the solenoid valve allowing oil flow between accumulator and cylinder during both positive and negative displacement strokes. Depending on top bracings internal oil flow restrictions, some level of energy dissipation will occur. This will in principle allow the hydraulic top bracing to perform as a vibration damper. To summarize, the top bracing system addresses a potential vibration problem by one of the following methods:

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Active Top Bracings/De-tuning Mode. Top Bracing working as de-tuner by adding installation stiffness in order to shift problematic resonances above main engine operational speed range or alternatively to engine speeds where hull is less sensitive. Shifting resonances above engine operational speed range is the most common approach and normally the most successful when possible. Passive Top Bracings/Damping Mode. In cases where active top bracings are not capable of shifting problematic resonances above operational speed range, the hydraulic top bracings can be set to “passive” condition. By this setting, the aim is to supply a sufficient amount of local vibration damping in order to reduce vibration responses to acceptable levels. Combination Active/Passive. The MAN-ES Hydraulic top bracing can be automatically controlled by engine speed signal (RPM), making it possible to select the most optimal condition (Active or Passive) in a pre-programmed speed range. No Top Bracings (Drained or Removed). Finally, a complete removal of the top bracings can be relevant. Removal of the top bracings will shift local X/H modes to lower frequencies where the hull can be less sensitive. This can in some cases be sufficient and thereby pose a cost down to the installation.

2.2 Calculated Stiffness of Top Bracings The length of the hydraulic top bracing can be adjusted within a certain working range in order to take up the distance between engine and hull side. This length can vary due to assembly tolerances of vessels steel structures, deformation of hull side from vessels loading condition and finally also heat expansions. Linear elastic finite element calculations have therefore been made in maximum, nominal and minimum length of the top bracing. The composition and resulting overall flexibility of the top bracings are shown for minimum distance to maximum distance in Fig. 4.

MDT Hydraulic Top Bracing

Flexibility [mm/kN] x 10-6

Flexibility composion.

Barrel shaping due to pressurizaon. Hydraulic oil volume. Adjustable steel part. Non adjustable steel part.

2000

1500

1000

500

0 Minimum distance.

Nominal distance.

Maximum distance.

Fig. 4. Flexibility composition of top bracing

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3 Onboard Vibration Measurements Detailed vibration measurements were conducted during sea-going voyage on two subject case vessels both installed with four (4) MAN-ES Hydraulic top bracings of the same type and size: Ship 1 VLCC: MAN B&W 6S90MC-C7, MK7, 29300 kW x 76 rpm. Ship 2 RoRo: MAN B&W 9L60MC-C, MK7, 20070 kW x 123 rpm. All signals were recorded continuously during slow increase of engine speed (Sweep measurement). Acceleration rate was aimed to be approx. 1 rpm/min. Thus, all signals are simultaneously measured and contained in one recording file per sweep. The low acceleration rate of 1 rpm/min was chosen to obtain a steady state condition of the quantities; “Engine load”, “Vessel speed” and “Vibration response”. The sampling frequency was set to 1.6 kHz in the measurement software program. Sweep TB Passive: All four top bracings are switched to passive condition (Damper mode). Sweep TB - Active: All four top bracings are switched to active condition (De-tuner mode) Sweep TB - Drained: All four top bracings are drained from oil so that no oil remains in the top bracing. Thereby the piston inside the top bracing can move freely and only transmit insignificant forces to hull side due to friction between piston and cylinder liner. Alternatively, a similar condition could be achieved by removing the top bracings completely from the engine. For practical reasons, draining oil from the top bracings was more convenient.

3.1 Measured Signals A subset of the many signals recorded are listed below and shown in Fig. 5 and Fig. 6: • P1: Main Engine Vibration – Fore end Measured on engine frame top level in transverse direction Transducer type: Accelerometer • P2: Main Engine Vibration – Aft end Measured on engine frame top level in transverse direction Transducer type: Accelerometer • P5: Navigation Deck Vibration Vessel center line. Measured in transverse direction Transducer type: Accelerometer

See Fig. 5

(continued)

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(continued) • TB_Press: internal Pressure Hydraulic oil pressure in top bracing cylinder chamber. Transducer type: Pressure Transmitter • TB_Dist: relative movement Relative movement between hull and engine side across top bracing Transducer type: Proximity sensor/probe • TB_SG_Fore: Stress in H-Beam web plate Fore side of web plate at center line Transducer type: Strain Gauge Signal • TB_SG_Aft: Stress in H-Beam web plate Aft side of web plate at center line Transducer type: Strain Gauge Signal

Fig. 5 Structure vibration measurements locations

Fig. 6 Local measurements on top bracings

See Fig. 6

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3.2 Measurements Results: 6S90MC-C – VLCC For the 6S90MC-C, the dominant forcing frequency is the ignition frequency (6th order). Optimally, the top bracing system should shift the H-mode resonance well above engines operational speed range (Top bracing active and working as “de-tuner”). However, such a sub-critical design was not obtained and 6th order H-type resonance is located within the running range with active top bracings. It is therefore of special interest if improved vibration performance could be achieved, by setting the top bracing system to passive and working as a discrete vibration damper. From Fig. 7 and Fig. 8, the results of the Order Track analysis is plotted for 6th order (0-peak) for increasing engine speed when vessel is in loaded condition.

Fig. 7. Engine frame (P1/P2) - transverse vibration

From Fig. 7 it is seen that the two signals, fore/aft (P1/P2), have practically the same 6th order vibration level in the three measurement sweeps (Active, Passive and Drained).

Vibraon Amplitude, [mm/s]

8

6th Order

TB Passive (P5) TB Acve (P5) TB Drained (P5)

6

4

2

0 45

50

55 60 Engine Speed [rpm]

65

70

Fig. 8. Navigation deck centre (P5) transverse vibration

75

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Furthermore, the signals P1/P2 were found to be in phase. The in-phase relationship indicates that the fore and aft end of the engine frame are moving in same direction indicating a clear H-type vibration. It is also clear that different conditions of the top bracing system have significant impact on both the engine frame and the navigation deck vibration responses. For the important 6th order, the highest vibration occurs for the TB-Drained condition (Fig. 8). In the important upper speed range (above 57 rpm), the lowest and optimal condition for both main engine and navigation deck, is seen for TB-Passive condition (Top Bracing working as damper). Furthermore, when taking the first 50 orders into consideration as seen in Fig. 9, the TB-Passive condition gives lower vibration levels in most speed ranges compared to “TB-Active” condition.

Synthesis (Sum)

TB Passive (P1) TB Acve (P1)

1/2 (Max-Min) of Time signal. Frequency Range : 50 Orders

Vibraon Velocity [mm/s]

30 25 20 15 10 5 45

50

55

60

65

70

75

Engine Speed [rpm]

Fig. 9. Signal P1: synthesis orders 1–50. Value: 1/2 (Max-Min)

Another important discovery was made. When comparing the drained condition with the passive condition, the resonance speed is practically unchanged (approx. 50–51 rpm in both cases). This observation suggests that the top bracing system is performing as a damper rather than a de-tuner when in passive condition (the top bracings contribution to the installation stiffness has practically no influence in passive condition). 3.3 Measurements RESULTS: 9L60MC-C/RoRo The MAN B&W 9L60MC-C has dominant guide force moment excitation for orders 4, 5 and 6th . In the drained condition (no top bracings) the 6th order X-type resonance falls within the engines operational speed range as seen in Fig. 10. Highest single order vibration is seen for 5th order (flank vibration) at MCR speed in drained condition (Fig. 11). When top bracings are active, resonance is shifted to higher engine speeds resulting in low and fully acceptable 5th and 6th order vibration levels. Thereby the top bracings work successfully applied as a de-tuner for 9L60MC-C/RoRo case.

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TB Passive (P1) TB Passive (P2) TB Acve (P1) TB Acve (P2) TB Drained (P1) TB Drained (P2)

Vibraon Amplitude, [mm/s]

25 20

6th Order X-Type Resonance (TB Drained)

15 10 5 0 65

75

85 95 Engine Speed [rpm]

105

115

125

Fig. 10. Engine frame (P1/P2) - transverse vibration 6th Order

40

Vibraon Amplitude, [mm/s]

5th Order

TB Passive (P1) TB Passive (P2) TB Acve (P1) TB Acve (P2) TB Drained (P1) TB Drained (P2)

35 30 25 20

X-Type Flank (TB Drained)

15 10 5 0 65

75

85 95 Engine Speed [rpm]

105

115

125

Fig. 11. Engine frame (P1/P2) - transverse vibration 5th Order

From Fig. 10 it is seen that the X-mode resonance speed for drained and passive condition are both located around 113 rpm. As was the case for the 6S90MC-C/VLCC, this indicates that the top bracings are performing as a damper where the stiffness properties of the top bracing have only marginal influence. 3.4 Damping Estimation by Measurements The physical approximation model used to represent the MDT Hydraulic top bracing is shown in Fig. 12. The model shows a series connection of a spring and a damper element.

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397

Fig. 12. Physical model of top bracing

F(t): Force in top bracing as a function of time (t) X(t): Displacement/deformation of spring element only. Y(t): Total displacement of spring and damper element. Forces are transmitted by in-series connection between entrapped oil and the steel part of the top bracing. Thus, two components (spring and damper) share the same force F(t), but have independent displacement and velocity. The spring compression is defined as X(t) and the damper right end displacement is defended as Y(t). For the in-series connection, the relationship between force, displacement and velocity is given below: Spring force is directly proportional to the displacement X(t). F(t) = KStiffTB · X (t)

(1)

F  (t) = KStiffTB · X  (t)

(2)

Differentiation of (1)

X  (t) =

F (t) KStiffTB

Damper force is directly proportional to the relative velocity of its two ends:   F(t) = CDamp_TB · Y  (t) − X  (t)

(3)

(4)

Substituting (3) into (4) and rearrangement of terms gives: F(t) = CDamp_TB · Y (t) −

CDampTB  · F (t) KStiffTB

(5)

F (t) KStiffTB

(6)

Rearrangement of (5) gives Y (t) =

F(t) CDampTB

+

The total velocities can be expressed as the sum of the damper and the spring velocities. F(t): Top Bracing Force. The top bracing force FTB_Press, was calculated by multiplying the pressure signal (TB_Press) with the area of the hydraulic piston (ATB).

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FTB_Press (t) = TBPress (t) · ATB [N]

(7)

For backup and verification, the top bracing force was also derived from the strain gauge signals by exploiting knowledge of the H-Beams cross section area. In order to compensate for longitudinal bending of the H-beam, the force (FTB_SG ) was calculated by averaging the two stress signals mounted on the fore/Aft side of the H-beams web plate. FTB_SG (t) =

TB_SG_Fore (t) + TB_SG_Aft (t) · AH _Beam [N] 2

(8)

It was found that after pressurization of the top bracing has taken place the two forces were found to be in good agreement. Right side of Eq. (6) forms the velocity based on the pressure signal (TB_Press):  YTB_Press (t) =

FTB_Press (t) F TB_Press (t) + [m/s] C DampTB K Stiff TB

(9)

The differentiation of top bracing force (Force impulse, F TB_Press ) was performed in the measurement software program:  FTB_Press (t) : Differentiation of FTB_Press (t) [N/s]

Left side of Eq. (6) forms the velocity based on the displacement signal (TB_Disp): Y(t) : Relative displacement [m] YTB_DISP (t) = TBDist (t) [m]  YTB_DISP (t) : Differentiation Y(t) [m/s]

(10)

Also the differentiation of the relative displacement was performed in the measurement software program. Solving by least squares method: Eq. (6) is the basis for the optimization function used to derive the stiffness and damping properties of the MDT Hydraulic top bracing:  YTB_Press (t) =

FTB_Press (t) F TB_Press (t)  + = YTB_DISP (t) C DampTB K Stiff TB

(11)

Equation (11) states that the velocity derived from the proximity probe signal (TB_DISP), equals the velocity derived from the pressure transmitter signal (TB_PRESS). The velocity error (residual) is then defined as: V Err (t) = Y  TB_Dist (t) − Y  TB_Press (t)

(12)

Least square summation function: V 2Min (N) =

t=t2 t=t1

V Err (t)2

(13)

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t1: Start time for crankshaft revolution N. t2: End time for crankshaft revolution N. The optimization Eq. (13), states the summation of the “squared velocity error” for one time block defined over the interval (t1–t2) for crankshaft revolution N. The optimization objective consists of adjusting the stiffness and damping parameters to minimize the sum of the squared velocity  error. This is done by optimal selection of the  two unknowns; (KStiff _TB ), CDamp_TB . The root mean square of the velocity errors is defined as:  1 n=SN · (14) RMSV (N) = [V Err (n)]2 n=1 SN SN : Number of samples contained in time block N n: Sample number. The number of samples SN is related to the sample frequency and engine speed by: SN =

60 · f Speed

(15)

Speed: Revolutions per min. [rpm] f: Sample frequency. [Hz] The differentiation of both the pressure and the displacement was performed on the full 1.6 kHz sampled signals. The signals were hereafter down-sampled to 200 Hz by the measurement software program, prior to export to the optimization software. TB-Active Condition. In the active condition, the many optimizations strongly converged to finite values of stiffness (KStiff _TB ) in range 400–850 kN/mm and damping values (CDamp_TB ) approaching very high values in excess of 250 kN/(mm/s). TB-Passive Condition. In the passive condition, the optimizations converged to finite damping values (C DampTB ) in range 1.0–2.0 kN/(mm/s). Summary Active Condition. In the active condition, there is no oil flow and the major part of the relative movement between engine/hull is due to elastic deformation of the top bracing. The damping can be considered infinite and thereby disregarded. The MDT top bracing can therefore be idealized as a uniaxial spring element only. Passive Condition. It was found that in passive condition, the stiffness property could practically be disregarded as the major part of the total displacement occurs in the damping part due to large piston movements as oil is oscillating between the accumulator and cylinder. The ratio between damper deflection and spring deflection was found to be a factor of 10–20. Due to the low levels of power dissipation, temperature increases will be small and therefore be of no practical importance for the operation and function of the top bracing working as a vibration damper.

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4 Shipboard Vibration Simulations The prediction of the vibration behavior has become a standard part of the early design stage for new ship designs using Finite Element (FE) models. Beside the propeller diesel engines are the main excitation source on board especially for cargo vessels. Because of the engine’s flexibility it is essential for a comprehensive vibration assessment to integrate a main engine model in the global finite element model. Besides studying the external and internal excitation this allows to assess the influence of top bracings on the vibration response of engine as well as on the ship structure. Other excitation like the dynamic axial forces introduced at the thrust bearing of the main engine into the hull girder as well as the influence of 2nd order mass moment compensators and external compensators can also be applied and studied by the simulation model. In order to validate DNVGL’s simulation and modelling procedure for hydraulic top bracings it was aimed for retracing the above described measurements by simulations. 4.1 Comparison of Measurements with Simulations Following the three settings during the above described measurements onboard the VLCC three simulation models were established. Only difference between the models was the idealization of the top bracing setting (Table 1). Table 1. Simulation models Model Setting Top bracing A

drained No elements (no stiffness, no damping)

B

active

C

passive Damper elements (damping only, no stiffness)

Truss elements (stiffness only, no damping)

Since no global finite element model of the vessel on board which the measurements were conducted was available, it was decided to use an existing model of a similar VLCC with same general dimensions. While the superstructure was not changed the finite element model was adjusted in way of the engine room and especially in the area of the top bracings’ shipside interface based on steel structural drawings. Consequently, only the engine vibration could be directly compared to the measurements and not the superstructure vibration. The engine model was merged into the global model as shown in Figs. 13 and 14. Subsequently the mass application was applied aiming to meet the loading condition of the vessel during the measurements. The simulations were conducted with DNV GL in-house vibration analysis software and for ballast condition only. At first the simulations with the top bracings drained from oil were conducted. In the simulations a quadratic dependency of the engine excitation forces on revolution rate was assumed. The comparison between the measurements and the simulations for drained from oil condition are presented in Fig. 15. Shown are the vibration velocities at the

Experimental and Numerical Investigations

Fig. 13. Model A

401

Fig. 14. Model with top bracings

engine’s top in transverse direction at the 6th order, representing the ignition frequency of the engine at the specific revolution rate.

Fig. 15. Comparison of engine top transverse vibration for drained condition

Good agreement between H-type natural frequency (50 rpm/53 rpm), the maximum resonant amplitude (70 mm/s/66 mm/s) and the general characteristic of the response curves was achieved. In a next step the top bracings in active condition were introduced into the simulation model. Since only for the foremost and aft most located top bracings TB1 and TB4 the stiffness could be derived from the measurements it was assumed that the stiffness of TB2 was equal to the one of TB1 and the stiffness of TB3 equal to the one of TB4. The same was assumed for the damping in passive setting. The comparison between the measurements and the simulations for active condition are shown in Fig. 16. Again, good agreement between H-type natural frequency (62 rpm/63 rpm) and the maximum resonant amplitude (18 mm/s/17 mm/s) was achieved. While the general characteristic of the response curve for revolution rates below the H-type natural frequency

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Fig. 16. Comparison of engine top transverse vibration for active condition

is matched very well it seems that it is suffering above the resonance. A possible explanation for this is the above-mentioned structural deviations between the two different ships which gains more relevance in the higher frequency range. The comparison between the measurements and the simulations for passive condition are shown in Fig. 17. The simulation was more on the conservative side with respect to the maximum resonant vibration velocity. The maximum velocity at engine top was calculated to be 15 mm/s whereas the measurements indicated only 10 mm/s. However, good agreement between H-type natural frequency (51 rpm/52 rpm) and the general characteristic of the response curves was achieved.

Fig. 17. Comparison of engine top transverse vibration for passive condition

Based on the comparisons between the measurements and the simulations it can concluded that the simulation models are in general suitable to satisfactorily predict the engine response for all three top bracing settings, the drained from oil, active and passive. Furthermore, the simulations can serve already in design stage to give decision

Experimental and Numerical Investigations

403

support for optimum setting and a possible engine revolution rate controlled switching point from passive to active setting. 4.2 Variant Investigations by Simulation Based on the validated simulation models it was then possible to investigate the influence of the different settings on the structural vibrations. Furthermore, the models allow studying different top bracing arrangements, combinations of different settings and to vary the stiffness and damping properties. Table 2 lists the different variants that were investigated. The shortcuts give information about the top bracing setting, where “S” indicates active setting and “D” passive setting. The first number indicates the number of top bracings and the number following “S” or “D” tells the variation of stiffness or damping compared to the nominal value. So for example the variant “2S_2D “is a variant with 2 active top bracings with nominal stiffness on one engine side and with 2 passive top bracings with nominal damping on the other engine side. Table 2. Variants Shortcut Variant

Shortcut Variant

4S05

4 top bracing active on one side; 0.5x nominal stiffness

4D

4 top bracing passive on one side; nominal damping

4S

4 top bracing active on one side; nominal stiffness

4D2

4 top bracing passive on one side; 2x nominal damping

4S2

4 top bracing active on one side; 2x nominal stiffness

4D10

4 top bracing passive on one side; 10x nominal damping

4S10

4 top bracing active on one side; 10x nominal stiffness

4D_4D

4 top bracing passive on both sides; nominal damping

4S_4S

4 top bracing active on both sides; 2S2D nominal stiffness

2 top bracing active and 2 passive on one side; nominal data

4D05

4 top bracing passive on one side; 2S_2D 0.5x nominal damping

2 top bracing active on one side and 2 passive on the other side

drained

No top bracing; drained from oil

In the following figures the different results of the variant investigation are shown. The vibration peak velocities at the 6th order, representing the ignition frequency of the engine, are given. The vertical line at 7.34 Hz is representing the nominal engine speed of 73.4 rpm. SUPERSTRUCTURE RESPONSE. The evaluation point for the superstructure vibration response was located at navigation deck level on starboard side (Fig. 18). In Fig. 19 the superstructure vibration in longitudinal direction for drained condition as well as for four top bracings on one side with nominal stiffness for active 4S and

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Fig. 18. Evaluation point at superstructure

nominal damping for passive setting 4D is shown. Compared to the drained and passive setting the response significantly increased for the active setting.

Fig. 19. Superstructure longitudinal response

So changing from active to passive setting, either revolution rate controlled at a predefined switching point or permanently, results in lower responses at nominal revolution rate for both the engine and the superstructure. Figure 20 show the superstructure vibration for the variants with varied stiffness and damping properties. As expected for increased stiffness of the top bracings the structural response increases at the superstructure as well as with increasing damping of the top bracings the superstructure vibration decreases. The combined variants with two active and two passive top bracings on one engine side 2S2D and with two active on one side and two passive top bracings on the other engine side 2S_2D lead to similar superstructure response. The vibration levels are between the pure active 4S and the pure damping 4D variants in the upper frequency range and on lower level at the lower frequency range (Fig. 21).

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Fig. 20. Superstructure longitudinal response S- and D-variants

Fig. 21. Superstructure longitudinal response combined variants

Engine Response. Figure 22 shows the calculated H-type natural frequencies of the engine for the different variants investigated. It can be concluded that not only the increase of stiffness but also the increase of damping increased to a certain extent the natural frequency.

Fig. 22. Change of engine H-type natural frequency

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Figure 23 show the engine vibration for the variants with varied stiffness and damping properties. As expected for increased stiffness of the top bracings the response decreased as well as with increasing damping of the top bracings the engine vibration decreases.

Fig. 23. Engine top transverse response S- and D-variants

The combined variants 2S2D and 2S_2D lead to similar engine response. The frequency response curves have similar characteristic as the pure active 4S slightly shifted towards lower frequencies (Fig. 24).

Fig. 24. Engine top transverse response combined

5 Conclusion The objective of the study was to get more insight into the dynamic characteristics of hydraulic top bracings and their influence on the engine and structural vibrations. While the stiffness and damping properties were successfully derived by experimental testing, based on these measurements and the subsequent simulations the following can be concluded: • The working principles of the three possible conditions of MAN-ES hydraulic top bracings were confirmed. In active setting the top bracing acts like a pure spring with

Experimental and Numerical Investigations

•

• • •

407

stiffness property only. In passive setting the top bracing acts like a pure damper with damping property only. In case the top bracings are drained from oil it is as if there would be no top bracing installed The passive condition reduced significantly the engine’s vibration amplitudes compared to the drained from oil and active condition at the engine’s H-type resonance condition. For nominal revolution rate the same tendency but less pronounced was observed The passive setting reduced the superstructure vibrations more effective than the active setting for both revolution rates, the engine’s resonance and nominal revolution rate Based on a careful selection of the top bracing’s setting, dynamic properties and arrangement it is possible to control the engine vibration and the structural vibration at the same time An optimum vibration response for both, the engine and the superstructure vibration, can be achieved either by a revolution rate controlled switching from active to passive or for some constellations also by a permanent passive setting over the entire revolution range

The top bracings and its dynamic characteristics were successfully implemented into simulation models and the measurement results were satisfactorily retraced by simulations. Based on the validated simulation model it was possible to investigate several nonstandard configurations and settings by additional simulations. Regarding the correlation between the different top bracing settings and the engine as well as the superstructure response the following was observed: • The change of stiffness had more influence on the vibration velocity of the engine vibration than the change in damping. • The change of stiffness had less influence on the H-type natural frequency compared to the change of damping. • Arranging four additional active top bracings on the other engine side reduced significantly the vibration amplitude and increased the H-type natural frequency. • Arranging four additional passive top bracings on the other engine side had less influence on vibration amplitude and the H-type natural frequency. • Replacing two active top bracings by two passive ones the vibration velocities are almost unchanged and the H-type natural frequency is only slightly decreased.

Reference 1. Holtmann, M., et al.: Experimental and numerical investigations about hydraulic top bracings and its influence on engine and vessels superstructure vibrations. In: Proceeding of 28th CIMAC World Congress, Helsinki, 6–10 June 2016 (2016)

Design Procedure to Estimate the Mechanical Behaviour of Resilient Mounting Elements for Marine Applications Jacopo Fragasso(B) and Lorenzo Moro Department of Ocean and Naval Architectural Engineering, Memorial University of Newfoundland, St. John’s, NL A1B3X7, Canada {jfragasso,lmoro}@mun.ca

Abstract. Transfer Path Analysis (TPA) allows the ship designer to predict how the vibrational energy is transmitted from marine diesel engines to ship structures. The most critical step of TPA is the characterization of the resilient mounting element placed between the engine and the ship foundation, due to its nonlinear static and dynamic behaviour. In this paper, the authors present an overview of a procedure developed to obtain the static and dynamic characterization of resilient mounting elements for marine applications. A nonlinear material model and frequency-dependent damping ratios are adopted to characterize accurately the isolator. The results of the numerical simulations are compared to the experimental static force-displacement curve and transmissibility frequency response function in order to estimate the optimal values of the hyperelastic coefficients and the damping ratio over the frequency range of analysis. Keywords: Transfer path analysis · Nonlinear material model · Ship design · Noise and vibration on ships · Resilient rubber mounts

1 1.1

Introduction General Frame of Reference

Onboard noise and vibrations are one of the main topics of investigation in the research of the growing maritime industry. The assessment of onboard level of noise and vibration and the implementation of efficient techniques to limit them is a crucial concern in the design of new vessels and in the retrofitting of existing ones. In particular, in the case of passenger ships, this is an issue of primary importance, as noise and vibrations are the main factor influencing the perceived comfort onboard. This is proven, for example, by the work [9], which, analyzing the results of a survey study conducted on cruise ships’ professionals, highlights the demand for lower levels of noise and vibrations onboard passenger c Springer Nature Singapore Pte Ltd. 2021  T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 408–423, 2021. https://doi.org/10.1007/978-981-15-4672-3_26

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ships. Furthermore, the report [21], considering the feedback provided by ferry users to a survey about comfort onboard, shows that the desirability of water transport is strongly related to dynamic factors, such as the vessel’s seakeeping, speed and transversal motions and rotation. Noise and low-frequency vibrations had primary relevance in the survey’s results. Unfortunately, it is difficult to achieve a global reduction of noise and vibrations levels onboard. Alongside the variety of operating conditions crew and passengers are exposed to, and the manufacturing constraints of the maritime industry, a source of complexity that cannot be neglected is the fact that onboard structure-borne noise is caused by several sources acting together. The work [6] highlights the most relevant ones: the prime movers, the shaft line, the propeller and the bearings, the air conditioning system and the auxiliary systems. A first way to simplify this problem is focusing the analysis on the effect of the ship’s propulsion plant, because it is generally the most relevant source of structure-borne noise and vibrations, as noted in [14]. In the case of existing vessels, it is possible to estimate the level of vibrations produced by the ship’s prime movers by following the guidelines provided by the ISO Standard 20283 [10]. The standard defines a procedure to characterize the level of vibrations exciting different areas of the vessel, in terms of frequencyweighted RMS vibration values. In the design of a new vessel, it is necessary to consider how vibrations are transmitted from noise sources to receiving points since the early stage of design. If the dominant paths of transmission of noise and vibration energy are known, and the transmitted vibration content is characterized in terms of frequency bands, it is possible to implement efficient vibration reduction techniques in the regions of the frequency spectrum presenting the highest vibrations’ intensity. This technique, called Transfer Path Analysis (TPA) [22] provides designers a useful framework to predict noise and vibration levels acting on the structure under analysis. In the case of marine structures, a limitation of general-purpose TPA, highlighted in [4], is the level of uncertainty in the predicted results, due to the complexity of the engine’s excitation, the number of transmission paths to be considered, and the effect of high-frequency vibrations. Therefore, the authors present a simplified approach, based on the representation of the resilient mounting elements, placed between the engine and the foundation, as point-sized components. The transmission of energy through the element is represented using the four-pole model, presented in [19], defining a relationship between the spectra of force ad velocity on the two sides of the element. This allows to characterize the transmission path using a lumped-parameter representation, by neglecting the reciprocal interactions between resilient mounting elements, and the effect of non-vertical stress components. In this way, the Transfer Path Analysis is split into the dynamic characterization of each component of the transfer path, and the four-pole model explains how vibrational energy is transferred between them.

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The mechanical characterization of resilient mounting elements presents some critical aspects. First, the dynamics of the element along all translational and rotational directions needs to be modeled. This can be done explicitly or simplifying the analysis to the vertical direction, as done in [4]. Furthermore, another layer of complexity is provided by the non-linear behaviour of the rubber material used in the resilient mount. The work [3] outlines a two-step procedure to obtain an accurate characterization of a simple resilient mounting element, that considers the non-linear behaviour of the rubber material. The first step of the procedure consists in the analysis of the response of the resilient mounting element to a static stress. An experimental force-displacement curve is acquired, for a set of displacement values. Then, the parameters of a hyperelastic model are calculated by fitting the behaviour of the finite element (FE) model of the resilient mounting element to the experimental forcedisplacement curve. The tests are performed on a simple cylindrical rubber element [11]. The model updating is performed by changing the hyperelastic coefficients of the numerical model one at a time, in order to match slope and curvature of the experimental curve [2]. Moro et al. [13] were able to extend the approach presented by Beijers et al. [3] to the case of resilient mounting elements having a more complex geometry: while the work by Beijers et al. analyzed a rubber cylinder constrained between two steel plates, Moro et al. apply a similar technique to the analysis of a marine resilient mounting element designed for medium-speed diesel engines. The difficulties to overcome in order to extend this approach are related to the complex representation of the contact between the rubber isolator and the steel cups, and the use of a multi-dimensional state of stress. The second step of the procedure presented in [3] yields the dynamic characterization of the resilient mounting element. After the hyperelastic coefficients have been determined, the dynamic characterization of the resilient mounting element is performed by finding the optimal value of the shear modulus of the rubber material. Similarly to the static characterization, this is done by comparing the numerical and experimental transfer stiffness curves. 1.2

Contributions of the Present Work

The purpose of this paper is to present a novel technique to perform the characterization of resilient mounting elements for marine applications. This technique builds up on previous studies conducted by Beijers et al. [3] and Moro et al. [13]. The first step of the technique, thoroughly described in the paper [8] consists in the numerical simulation of the quasi-static compression of the resilient mount, increasing the load applied on top. Compared to the work [13], the use of an implicit time-integration scheme increases the computational efficiency of the analysis, making it feasible with the demand of the maritime design industry. Moreover, the use of Response Surface Methodology (RSM) [15] is a step up from the previous state of the art: it allows to define a priori the number of simulations to be performed in the fitting procedure, to estimate the influence of

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each parameter of the hyperelastic model on the global response of the numerical model and it provides an accurate optimization technique to fit the coefficients of the hyperelastic model. The second step of the methodology is the dynamic characterization of the resilient mounting element. The application of this part of the procedure to two test cases will be accurately described in a forthcoming paper, but the preliminary results of the analysis are hereby presented. A static preload is applied to the resilient mount, constrained between two masses. The system is excited in the frequency range 50–1000 Hz with a shaker, that generates a random excitation signal. The test rig is built in such a way that the effect of the connection between the masses and the surrounding structure on the dynamic of the system is negligible. The vibration levels are measured at the top and bottom surfaces of the resilient mount. The transmittance (i.e. the frequency response function defined as the ratio between the acceleration levels at the opposite sides of the resilient mount) is then considered to assess the dynamic behaviour of the element under analysis. This procedure, presented in the ISO Standard 10846-3 [7], was applied by Beijers et al. [3] and Moro et al. [13] to perform the dynamic characterization of resilient mounting elements. In those works, an iterative procedure was applied to find the optimal values of the dynamic coefficients of the rubber material of the isolator. As an improvement over this, this paper will present a novel methodology to characterize the dynamic behaviour of the resilient mount under test. This methodology relies on the optimization of the frequency-dependent damping ratio of the system. Sets of values of damping ratio are generated using a Design of Experiments (DOE) technique and tested in the numerical model, then the optimal values are found using an optimization methodology, based on the fitting performance of the transmittance curves obtained with the FE analysis. The Least Squares Method is used to compare experimental and numerical data points [18]. Moreover, the effect of the mass attached to the resilient mount and the connection between the resilient mount and the frame are discussed accurately. The low number of FE simulations needed to estimate the optimal values of the coefficients, and the possibility to gauge the effect on each factor on the fitting performance of the system, provide the designer a fast and reliable way to assess the effect of the resilient mounting element on the TPA of the ship structures.

2 2.1

Methods Overview

The static and dynamic characterization of the vibration isolators follows the model-updating procedure shown in Fig. 1.

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The methodology presented in the block diagram applies to both the static and the dynamic characterization of the resilient mount under test, but the experimental procedure, and the type of coefficients used to generate the numerical model depend on the type of characterization—static or dynamic—that the designer needs to perform. For this reason, the main differences between static and dynamic characterization are denoted in the diagram using callout blocks. Firstly, the experimental data are acquired through laboratory tests. The experiments consist, respectively, of a mono-axial compression test for the static analysis, and a vibration measure for the dynamic analysis (Sect. 2.4). The results of the experimental tests need to be compared with the behaviour predicted by a finite element model—presented in Sect. 2.5. After the designer defines the range of each factor, sets of factors are generated according to a predefined DOE methodology and tested in the numerical model. For each numerical experiment, a response curve is calculated. The hyperelastic coefficients of the Yeoh model are tested performing a quasi-static compression of the resilient mounting element, solved with an implicit time-integration scheme. Then, the frequency-dependent values of the damping ratio are tested with a modal analysis of the resilient mounting element, attached to the blocking mass. The quality of the fit between the experimental data and each numerical simulation is expressed using a measure of performance. Different measures of performance are used in the two stages of the analysis. In the compression test, it is necessary to obtain a good matching between experimental and numerical data over the entire loading range. Therefore, the Nash-Sutcliffe efficiency (NSE), presented in [16], is calculated for each numerical force-displacement curve. In the modal analysis, it is necessary to represent accurately the behaviour of the resilient mount in proximity of the peak of resonance. Therefore, the measure of performance that was adopted is the sum of the squared differences of the transmittance values acquired with the experimental tests and calculated with the numerical modal analysis. The values of the measure of performance are used to produce a predictive model, which relates the values of the coefficients to the expected value of the measure of performance. The performance of the statistical model is evaluated using a set of test points, and, if the evaluation is successful, the best values of the factors are calculated with an optimization procedure. These values provide the best fit between the numerical and the experimental set of values, and therefore are used to characterize the rubber material. 2.2

The Yeoh Model

Rubber materials can experience large strain, non-linear elastic deformations [5]. The random distribution of length and orientation of the long-chain molecules of the rubber material requires the use of statistical models to obtain an accurate characterization of the stress-strain behaviour of the material. In these constitutive relations, derived from the statistical mechanical approach presented by

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Fig. 1. Graphical representation of the model-updating procedure followed in this research activity

Treloar in [20], the material deformation is expressed as a function of the elastic strain energy [1]. Treloar defines the elastic strain energy density of the material, W , as a function of the state of stress of the material [20]: W =

  1 N kθ λ21 + λ22 + λ23 − 3 2

(1)

where N is the number of network chains per unit volume, k is the Boltzmann’s constant, θ is the absolute temperature and λ1−3 are the components of a principal stretch state, defined as the ratio between the current and the original length. Yeoh [23] approximates Eq. 1 with a polynomial model. The rubber properties are expressed by the coefficients C10 , C20 and C30 , while the deformation state is represented exclusively using the first invariant of the stretch tensor (I1 ): 2

W = C10 (I1 − 3) + C20 (I1 − 3) + C30 (I1 − 3)

3

(2)

The hyperelastic coefficients C10 , C20 and C30 must be estimated in order to characterize the material under analysis. Since only the first invariant of the stretch tensor is needed, the coefficients of the Yeoh model can be obtained by performing a simple uniaxial tension test [17]. This feature, and the fact that the Yeoh model provides accurate results for moderate to large deformation states, make for the widespread use of this model for the characterization of rubber materials [5]. 2.3

The Four-Pole Model

The four-pole parameter model, presented in [12], defines a linear relation for the performance equations of an elastic system:  F1 = Z11 F2 + Z12 V2 (3) V1 = Z21 F2 + Z22 V2

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or, in matrix notation: F = [Z]V

(4)

In the system (3) force and velocity at the input point (denoted by the index 1) are expressed as a function of force and velocity at the output point (denoted by the index 2). The case under analysis can be reduced to the analysis of a massless spring, since the mass of the resilient mount is negligible when compared to the masses of the inertial elements composing the test rig (described in Sect. 2.4). The total response of the resilient mounting element is calculated by considering 6 degrees of freedom (3 translational directions and 3 rotational directions), and representing the dynamics of the system along each direction using Eq. (4). Generally, the energy transmitted along the vertical direction is considerably higher than the other degrees of freedom [13]. For this reason, in this analysis, only the vertical component of the dynamics of the system is considered. Following the procedure defined in the ISO Standard, the force at the receiving point (Fr ) can be estimated using the formula Fr ∼ = Z21 vs = k21 us

(5)

where k21 is the transfer stiffness of the resilient mounting element and us is the amplitude of the displacement at the source point. This equation is related to the transmittance function aa21 by the relation [13]:   a2 u2 F2 2 2 ∼ = − (2πf ) m2 = − (2πf ) m2 k21 = (6) u1 u1 a1 Therefore, the ratio aa21 , measured with the dynamic tests described in Sect. 2.4 provides an estimation of the isolator performance in a given frequency range. 2.4

Experimental Procedure

The experimental data were acquired at the Ship Noise and Vibration Laboratory (NVL) of the University of Trieste (Italy). Measurements of static compression and dynamic frequency response functions were performed. Two resilient mounts were tested: a simple cylindrical element, having a maximum static load of 36 kN, and a more complex mount, composed of a conical rubber ring constrained between two steel cups. The maximum static load for this element is 145 kN. The isolators are shown in Fig. 2. The test rig developed to characterize the resilient mounts is shown in Fig. 3. Details of the testing procedure are provided in the work [13]. Compression Test. The force-displacement curve was acquired by compressing the resilient mount using a hydraulic piston, applying an increasing force on the top surface of the element. For every load value, the corresponding displacement was measured 24 h after the load was applied.

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Fig. 2. Resilient mounts tested in the activity (figures not in scale)

Fig. 3. Test rig implemented to perform the characterization of the resilient mounting elements

Dynamic Test. The dynamic experiments were performed following the experimental procedure described in the EN ISO Standard 10846-3 [7]. The test rig is excited using an electrodynamic shaker, which creates a uniform acceleration spectra on a mass (excitation mass) placed on the vibration isolator under test. A second mass (blocking mass) supports the isolator. This group of masses is decoupled from the rest of the structure using soft isolators, whose dynamic properties are known. A static preload is applied to the structure using an hydraulic piston.

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The acceleration levels of the shaker are controlled using a closed-loop control system, in order to generate a uniform acceleration in the frequency range 50– 1000 Hz. Vertical acceleration levels are measured at the top and the bottom of the resilient mount under test, using piezoelectric accelerometers. The quantity analyzed in this research activity is the ratio between the amplitude of the acceleration signals (called transmittance). This quantity gives an estimation of the isolating properties of the resilient mount for each frequency of excitation. Other acceleration signals are acquired in different points of the test rig. Their levels are compared to the measurement signal, in order to guarantee that the assumptions of the ISO Standard are verified [13]. Then, the vibro-acoustic behaviour of the isolator was estimated with a measure of transmissibility, following the indirect method described in the ISO Standard 10846-3 [7]. 2.5

Numerical Model

The two resilient mounting elements analyzed in this research activity are modeled in LS-DYNA. The numerical models are composed of 8-noded hexaedral solid elements. A mesh convergence analysis was performed, in order to find the optimal edge size. In the case of the modal analysis, preliminary test runs showed that reducedintegration solid elements showed non-physical shear warping modes. For this reason, the element formulation was switched from reduced-integration to fullyintegrated solid elements, as they can avoid this problem, thanks to the higher number of integration points. As discussed before, in the experimental setup the model is excited by a shaker. A closed-loop control on the acceleration signal measured on the top surface of the isolator under test is designed to generate a uniform acceleration level over the frequency range 50–1000 Hz. In order to represent this condition in the finite-element model, the model is subjected to a base-excitation acceleration and constrained on the top surface of the resilient mount on all degrees of freedom. It is necessary to add to the model the geometry of the blocking mass, because, differently from the excitation mass, it cannot be assumed to be fixed in this analysis. The blocking mass (made of steel) is assumed to be considerably stiffer than the rubber part of the resilient mount. Therefore, a more coarse mesh is used to model it. The geometry of the model used in the modal analysis is shown in Fig. 5. In the case of the cylindrical resilient mount, the numerical and the experimental results presented a significant difference in the amplitude of the transmittance frequency response functions over the entire frequency range. The cause of this difference was assessed in the effect on the overall level of vibration of a resonance condition in the low-frequency range. The difference in the position of this peak (48 Hz in the experimental data vs. 12 Hz estimated by the numerical

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model) was attributed to the stiffness of the connection between the frame and the blocking mass. Contrarily to the hypothesis of the ISO Standard, in this analysis this quantity cannot be assumed negligible. Therefore, an additional one-dimensional spring element was created, connected to the bottom surface of the blocking mass. The stiffness of this spring was calculated by reducing the analysis of the model to a basic one-degree-of-freedom system, having a natural frequency of resonance at 48 Hz (Fig. 4).

Fig. 4. Geometry of the finite element models

2.6

Error Function

In order to compare Model and experimental FRFs, an error function is calculated. The work [18] suggests the use of a decibel-transformed sum of squared differences, defined as λdB =

M 

|AdB (ωm ) − XdB (ωm )|

2

(7)

m=1

where AdB and XdB are the measured and model FRFs in decibel, respectively, and m is the index of the M discrete frequency lines considered in the analysis. Lower values of λdB guarantee a better fit in the frequency range of interest.

3

Results

In this section, the results of the static and dynamic characterization of the two resilient mounting elements tested in this activity are presented.

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Fig. 5. Finite element models, blocking mass added to the geometry

Fig. 6. Experimental force-displacement curves

3.1

Experimental Tests

Figure 6 shows the static force-displacement curves of the elements tested in this activity, acquired with the compression test presented in Sect. 2.4. Both figures show a significant deviation from a linear force-displacement behaviour, proving the need for non-linear constitutive relations. In the case of the dynamic analysis, the experimental transmittance curves are plotted in Fig. 7. The conical resilient mount presents a resonance peak at

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Fig. 7. Experimental transmittance curves

540 Hz, while the cylindrical element shows no significant conditions of resonance in the frequency range under investigation. 3.2

Quasi-static Numerical Simulations

The numerical force-displacement curves obtained with the quasi-static analysis are presented in Fig. 8, compared with the experimental results. For each resilient element, 15 numerical simulations were performed. The values of the parameters C10 , C20 , and C30 were varied according to a Central Composite Design (CCD) methodology. In Fig. 8 the results of the numerical simulations are plotted alongside the experimental curves. The Nash-Sutcliffe Efficiency index is used as a measure of performance, in order to create a predictive model. Through the optimization of the equation of the predictive model, the optimal values of the Yeoh coefficients can be determined. Details of the optimization procedure are provided in [8]. The force-displacement curve of the numerical model having the optimal values of the Yeoh coefficients is shown in Fig. 9 for the two case studies. 3.3

Dynamic Simulations

Figure 10 shows a comparison between the experimental and numerical transmittance curves. 15 runs are performed. Modal damping ratios were defined for the frequency lines: 0, 200, 400, 800, 1000, 1200, and 1400 Hz. The values of these coefficients for each simulation are defined using a Latin Hypercube Design [15]. The transmittance curves, plotted in Fig. 10, show a good agreement between experimental and numerical values. In particular, in the case of the conical

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Fig. 8. Comparison of the experimental force-displacement curve with the results of the simulations of the runs of the design

Fig. 9. Comparison of the experimental force-displacement curve with the optimal numerical simulation

Fig. 10. Comparison of the experimental transmittance curve with the results of the simulations of the runs of the design

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resilient mount, it can be seen that the position and the amplitude of the resonance peak are correctly predicted by the numerical model. The fit of the numerical transmittance curves to the experimental results is calculated using the error function defined in Sect. 2.6. Equation (7) is applied to each numerical curve plotted in Fig. 10. The curves having the lowest values of the error parameter λdB are plotted in Fig. 11. It can be seen that, in the case of the conical isolator, the best run of the design is able to capture the position and the amplitude of the resonance peak. This is a critical point of this study, because it allows the designer to accurately represent in the simulation of the ship structures the resonance condition of the isolator. In the case of the cylindrical isolator, the curve having the minimum error value accurately matches the experimental data over the entire frequency range of interest.

Fig. 11. Representation of the transmittance curves having the minimum error value, compared with the experimental data

4

Conclusions

A novel methodology to estimate the effect of resilient mounting elements in the Transfer Path Analysis of marine structures was presented in the paper. The two steps of the procedure—static and dynamic characterization—were discussed, and two case studies were examined. The results show that the technique provides an accurate estimation of the static and dynamic properties of the resilient mounts. The use of Response Surfaces reduces the number of experiments needed to obtain accurate results, and provides the designer a predictive model to assess the effect of the parameters of the constitutive relations and dynamic models used in the Transfer Path Analysis on the overall fitting performance. Special attention is dedicated to the dynamic analysis, as it is shown that the element formulation, the geometry of the blocking mass, the definition of the boundary conditions, and the stiffness of the elements connecting the blocking mass to the frame all have a critical role in producing an accurate prediction of the transmittance curve.

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Acknowledgments. We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), [funding reference number 211146]. Cette recherche a ´et´e financ´ee par le Conseil de recherches en sciences naturelles et en g´enie du Canada (CRSNG), [num´ero de r´ef´erence 211146].

References 1. Beatty, M.F.: Topics in finite elasticity: hyperelasticity of rubber, elastomers, and biological tissues–with examples. Appl. Mech. Rev. 40(12), 1699–1734 (1987) 2. Beijers, C.: A modeling approach to hybrid isolation of structure-borne sound. Ph.d. thesis, University of Twente (2005) 3. Beijers, C., Noordman, B., de Boer, A.: Numerical modelling of rubber vibration isolators: identification of material parameters. In: Proceedings of the 11th International Congress on Sound and Vibration (2014), pp. 3193–3200 (2004) 4. Biot, M., Moro, L., Mendoza Vassallo, P.N.: Prediction of the structure-borne noise due to marine diesel engines on board cruise ships. In: Proceedings of the 21st International Congress on Sound and Vibration (ICSV21, 2014) (2014) 5. Boyce, M.C., Arruda, E.M.: Constitutive models of rubber elasticity: a review. Rubber Chem. Technol. 73(3), 504–523 (2000) 6. Carlton, J.S., Vlasic, D.: Ship vibration and noise: some topical aspects. In: Proceedings of the 1st International Ship Noise and Vibration Conference (2005), pp. 1–11 (2005) 7. EN ISO 10846-3. EN ISO 10846-3: Acoustics and vibration - Laboratory measurement of vibro-acoustic transfer properties of resilient elements - Part 3: Indirect method for determination of the dynamic stiffness of resilient supports for translatory motion. Standard, International Organization for Standardization (2008) 8. Fragasso, J., Moro, L., Lye, L., Quinton, B.W.T.: Characterization of resilient mounts for marine diesel engines: prediction of static response via nonlinear analysis and response surface methodology. Ocean Eng. 171, 14–24 (2019) 9. Goujard, B., Sakout, A., Valeau, V.: Acoustic comfort on board ships: an evaluation based on a questionnaire. Appl. Acoust. 66(9), 1063–1073 (2005) 10. ISO 20283-5:2016. Mechanical vibration – Measurement of vibration on ships – Part 5: Guidelines for measurement, evaluation and reporting of vibration with regard to habitability on passenger and merchant ships. Standard, International Organization for Standardization (2016) 11. Kari, L.: On the dynamic stiffness of preloaded vibration isolators in the audible frequency range: modeling and experiments. J. Acoust. Soc. Am. 113(4), 1909– 1921 (2003) 12. Molloy, C.T.: Use of four-pole parameters in vibration calculations. J. Acoust. Soc. Am. 29(7), 842–853 (1957) 13. Moro, L., Brocco, E., Mendoza Vassallo, P.N., Biot, M., Le Sourne, H.: Numerical simulation of the dynamic behaviour of resilient mounts for marine diesel engines. In: Guedes Soares, C., Shenoi, A.A. (eds.) Proceedings of the 5th International Conference on Marine Structures (MARSTRUCT 2015), pp. 149–157 (2015) 14. Moro, L., Biot, M.: Laboratory tests pave the way for the knowledge of dynamic response of resilient mountings on board ships. In: PRADS 2013, 12th International Symposium on Practical Design of Ships and Other Floating Structures, vol. 2, pp. 853–860 (2013)

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15. Myers, R.H., Montgomery, D.C., Anderson-Cook, C.M.: Response Surface Methodology: Process and Product Optimization Using Designed Experiments, 3rd edn. Wiley, New York (2009) 16. Eamonn Nash, J., Sutcliffe, J.V.: River flow forecasting through conceptual models: part I - a discussion of principles. J. Hydrol. 10(3), 282–290 (1970) 17. Renaud, C., Cros, J.-M., Feng, Z.-Q., Yang, B.: The Yeoh model applied to the modeling of large deformation contact/impact problems. Int. J. Impact Eng. 36(5), 659–666 (2009) 18. Suzuki, H., Yamaguchi, M.: Comparison of FRF curve fitting methods for the loss factor measurements. In: INTER-NOISE and NOISE-CON Congress and Conference Proceedings, vol. 2002, pp. 1060–1065. Institute of Noise Control Engineering (2002) 19. Ten Wolde, T., Gadefelt, G.R.: Development of standard measurement methods for structure-borne sound emission. Noise Control Eng. J. 28(1), 5–14 (1987) 20. Treloar, L.R.G.: The Physics of Rubber Elasticity, 2nd edn. Oxford University Press, London (1958) 21. Turan, O.: A rational approach for reduction of motion sickness & improvement of passenger comfort and safety in sea transportation. Technical report G3RD-CT2002-00809, COMPASS (2006) 22. van der Seijs, M.V., de Klerk, D., Rixen, D.J.: General framework for transfer path analysis: History, theory and classification of techniques. Mech. Syst. Signal Process. 68, 217–244 (2016) 23. Yeoh, O.H.: Some forms of the strain energy function for rubber. Rubber Chem. Technol. 66(5), 754–771 (1993)

Experimental and Simulate Research on Fatigue Crack Propagation Behavior Under Varying Loadings in High Strength Steel of Marine Structures Jingxia Yue(B) , Wenjie Tu, Hao Xie, Ke Yang, and Weiguo Tang School of Transportation, Wuhan University of Technology, Wuhan, China [email protected]

Abstract. Fatigue damage is one of the most serious problems marine structures facing with. Though the S-N assessment for structural fatigue life is a simulative method based on numerous experimental data, it ignores the complicacy of loads on structures. Nevertheless, the fatigue life assessment based on fracture mechanics can consider the effects of stress ratio, overload ratio and load sequence effects, etc., the assessment is reliable, still. In this paper, according to ASTM specification, designed a series of fatigue crack propagation tests by using CT specimens with AH36 high-strength steel for marine structures, and researched simultaneously how overload ratio influence Fatigue crack propagation. Then based on the extended finite element method XFEM, the article made a numerical research on the size of plastic zone at crack tip and calculated by Irwin Formula (a formula used to calculate the size of plastic zone at crack tip), and then make a contradistinction between the results of study and calculation. Ultimately, by using the amended Wheeler Model, the text predicted the crack propagation behavior of AH36 high-strength steel, and compared the applicability of the predictive model with the experimental results. The results show that: the greater the overload, the higher the delayed effect. When the overload reaches 2.5 times, growth of crack almost stops; the accuracy is much high when calculating the size of plastic zone at crack tip by XFEM; revised Wheeler Model made a great depiction of fatigue crack propagation behavior under single overload. Keywords: Fatigue crack propagation · Overload ratio · The size of plastic zone at crack tip · Modified wheeler model

1 Introduction Fatigue problem is imperative in the safety of Naval Architecture and Ocean Engineering, and plays an important role in the service life of marine engineering structures. Fatigue strength, generally, refers to when the materials and components received cyclic loads, a point or a part emerge partial permanent initial damage due to the existence of initial defects, further form crack nucleus. Then with addition of the number of cycles, the damage will expand and form macroscopic cracks visible to the naked eye, and © Springer Nature Singapore Pte Ltd. 2021 T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 424–441, 2021. https://doi.org/10.1007/978-981-15-4672-3_27

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keep expanding until the structure completely breaks. Because the existence of periodic cyclic loads, the components in such kind structure will frequently, suffer severe fatigue accidents, and cause huge loss. Ordinarily, we use S-N curve and Miner fatigue cumulative damage theory to assess the fatigue life of engineering structures, yet such a method is inclined to be a data fitting way which based on a large amount of experimental data and unable to consider the complex load problems during fatigue failure. While the fatigue life assessment method base on Fracture mechanics is to depict the fatigue damage process through observing fatigue crack propagation behavior. This method can ably conclude many factors, such like overload ration, load sequence, etc., which means more reliability. The structure is not only subjected to the constant amplitude loads, but also to the irregular low-load or overload during its service life. When one marine architecture sails in rough sea, for instance, it often gets one or more transient overloads irregularly. Thereby, for the marine structures’ fatigue cracks, they totally are under the activities of variable amplitude loads. For fatigue crack propagation under general constant amplitude loads, the Paris Formula [1] is a formula that only related to a single parameter–stress intensity factor K, and the formula can meet description of fatigue crack propagation behavior well. Through amounts of researches on the fatigue crack propagation, the such factors as overload ratio and load sequence, etc., have a great impact on the growth rate of fatigue crack [12–19]. Wheeler [3] believes, it is the overload peak that make crack tip develop to a large plastic zone, and this zone is one main barrier to fatigue crack propagation under invariable load and further will slow down the crack propagation. Meanwhile, Wheeler present an analysis device for promoting the prediction accuracy of metal’s crack propagation when under varying amplitude cyclic loads and amended linear cumulative growth method which considered both the yield zone of crack tip and the history of previous loads. Elber [2] gave a crack closure mechanism, and explained the overloaded crack propagation behavior with plastic induced crack closure, and proposed a concept named equivalent stress intensity factor: K eff , which be widely used in the prediction of overloaded crack propagation behavior. Based on above formulae, many scholars had done numerous researches on fatigue crack propagation behavior. Fitting the number of delayed cycles in a single overload experiment, Sheu [4] regulated the index in Wheeler model, and adopted an empirical parameter matched the plastic zone’s size. Thus, the value of plastic zone’s size can be corrected by specific experiment numbers. To reflect the effects of both load interaction and delayed retardation phase, Yuen and Taheri [5] made overload tests which conclude two coefficients: the coefficient of overload interaction and of delayed retardation, and got an amended Wheeler model. Huang [6] introduced the concept of equivalent stress intensity factor range K eq 0 , improved Wheeler model further, and make it better when describing the load sequence’s influence on crack propagation. Gu [7] analyzed the mechanism of fatigue crack growth, and evaluated the difference of two descriptions on three phases (threshold value, stable expansion phase, instability phase) of fatigue crack propagation. Those two descriptions respectively came from single-parameter model and bi-parameters model of structure’s fatigue strength. Based on the extended finite element method XFEM, Hayder [8] figured out the size of plastic zone of crack tip, and made a comparison with test results and improved Willenborg model [9] to

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analysis. They studied fatigue crack propagation behavior under overload to low-load, and verified the accuracy of the model and simulation results. In summary, this article researched the structure’s fatigue crack propagation behavior based on the material AH36 steel, which widely applied on ocean engineering. Through the test, we received the relevant crack growth rate curves and fitted those curves to the corresponding Fracture mechanics parameters. Then we studied how load effects act in the expansion of AH36 steel. With the FEM, this article numerically studied the size of plastic zone at crack tip, then compared and verified those numerical results with the calculation results that came from the Irwin Formula. By using the amended Wheeler Model in the end, we predicted the fatigue crack propagation behavior under varying amplitude loads, and make a comparison between the reliability of the prediction model and the laboratory results.

2 Fatigue Crack Propagation Model To study further on fatigue crack behavior, scholars had got a mathematical model that can depict fatigue crack propagation behavior. Generally, for fatigue crack propagation under constant amplitude loads, Paris, et al., had conducted a huge amount of experiments on fatigue crack propagation of A533 steel at room temperature. Based on the Fracture mechanics, they achieved the famous formula–Paris Formula: da = C(K)m dN

(1)

Where m and C present the material constants that related to the experimental conditions (surroundings, frequency, temperature, etc.), K represents the amplitude of the stress intensity factor, defined as K = K max − K min . Paris Formula can describe well on fatigue crack propagation behavior under constant amplitude loads. Considering the retardation acceleration effect in actual load expansion process, Wheeler, et al., added a load retardation coefficient fR to Paris Formula, then they introduced a primitive Wheeler Model:   da = φR C(K)n dN The retardation coefficient fR can expressed as: ⎧ morg rp,i ⎪ ⎨ when ai + rp,i < aOL + rp,OL aOL + rp,OL − ai φR = ⎪ ⎩ 1 when ai + rp,i ≥ aOL + rp,OL

(2)

(3)

Where aOL represents the length of crack when applying overload, ai represents the length of current crack, r p,OL represents the size of plastic zone that generated by applying overload, r p,I represents the size of plastic zone that emerged by constant amplitude load at the current length of crack, morg represents the shape parameter came from fitting experiment data.

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Generally, we can calculate the size of plastic zone by Irwin Formula, which expressed as:

 1 Ki 2 rp,i = 2π σys 

1 KOL 2 rp,OL = 2π σys

(4)

Where σ ys represents the flow stress of material, σ ys = σ s when plane stress, σ ys = σ s /(1−2ν) when plane strain state, K i represents the amplitude of current stress intensity factor, K OL represents the amplitude of stress intensity factor when applying overload. Both K i and K OL can be calculated by the formula provided in the ASTM E647-11. Subsequently, Sheu, et al., adopted an empirical parameter of the size of plastic zone to revised Irwin Formula, thus the size of plastic zone can be corrected by specific test values. The revised formula stated as follows:

2 Ki rp,i = α σy 

KOL 2 rp,OL = α (5) σy Where α represents the empirical parameter of the size of retardation plastic zone, can be obtained by calculating retardation phase data in unimodal overload tests, ar represents the size of retardation plastic zone acquired in tests, that is, the length of crack propagation from the start of retardation to the phase when the rate of crack propagation recover to the rate of constant amplitude load propagation, N r represents the corresponding load cycles. By Wheeler model, we can simulate the retardation effect of crack propagation. But the real retardation of crack propagation divided into three phases: instantaneous acceleration, delayed retardation and retardation [17]. This caused Wheeler Formula cannot reflect the influence of delayed retardation. To solve this defect and reflect the influences of load interaction and delayed retardation completely, Yuen and Taheri had revised Wheeler Model, added two coefficients, fI and fD . The amended Wheeler Model expressed as:   da = φR · φI · φD · C(Kac )n dN

(6)

Where fI represents the overload interaction, fD represents the delayed retardation coefficient, K ac represents the amplitude of stress intensity factor in acceleration. fI and fD can be represented as following:

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⎧ a + rd ,OL − ai m mod ⎪ ⎨ OL if ai + rd ,i < aOL + rd ,OL rd ,i φD = ⎪ ⎩ 1 if ai + rd ,i ≥ aOL + rd ,OL

m mod ⎧ rp,i aOL + rd ,OL − ai ⎪ ⎪ ⎪ · if ai + rd ,i < aOL + rd ,OL 1 − (1 − φmin,i ) 1 − ⎪ ⎪ a + r − a rd ,i ⎪ i OL p,OL ⎪ ⎪ ⎪

m mod ⎪ ⎨ rp,i φI = 1 − (1 − φmin,i ) 1 − if aOL + rd ,OL ≤ ai + rd ,i ⎪ aOL + rp,OL − ai ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ai + rp,i < aOL + rd ,i ⎪ ⎪ ⎪ ⎩ 1 if ai + rp,i ≥ aOL + rp,OL

(7) Where r d,OL represents the length of delayed retardation zone under overload, r d,i represents the current size of delayed retardation zone, mmod represents a shape index in the amended model, fmin,i represents minimum of the retardation coefficient fR , and can be obtained in the previous overload calculation. At the initial phase of propagation after overloading, it appeared an instantaneous acceleration phase that the velocity of propagation accelerated suddenly. Therefore, K ac which represents the amplitude of stress intensity factor in acceleration phase is added to reflect this phase as following:

Kac = Ki + (KOL − Ki ) · 1 −

m mod rd ,i if ai + rd ,i < aOL + rd ,OL aOL + rd ,OL − ai

(8)

Where K OL represents the amplitude of stress intensity factor when applying overload, K i represents the amplitude of current stress intensity factor. After obtaining corresponding data, we can get a constant α of the size of plastic zone according to connection between actual size of plastic zone and corresponding length of crack propagation. α can be calculated with the following formula expressed as:

 

KOL 2 Kr 2 − (9) ar = rp,OL − rp,r = α σyld σyld Commonly, the size of delayed retardation plastic zone is one part of the size of plastic zone. To determine the constant of the size of delayed retardation plastic zone β this article had analyzed the curves of propagation rate in unimodal overload tests and had obtained ad (the length of crack propagation in corresponding delayed retardation phase) and N d (the corresponding cycles).

 

KOL 2 Kd 2 − (10) ad = rd ,OL − rd ,d = β σyld σyld The two cycle times, N r (corresponding to ar ) and N d (corresponding to ad ), obtained in experiments, are used to fitting the shape parameter m in Wheeler model. Schematic diagrams of overload retardation plastic zone are shown in Fig. 1.

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Fig. 1. Illustration of the terms defining the retardation parameters. Figure on the left shows the retardation phase of the crack propagation. Figure on the right shows the delayed retardation phase of the crack propagation.

Due to significant differences of mechanical properties among various materials, the predicted size of plastic zone by Irwin Formula often has some errors. Thence many amended formulas had been proposed. Irwin Formula, amended by Sheu, et al., was obtained due to fitting the lab data. They got the empirical parameter α by deriving the size of actual plastic zone gained in tests in turn. Still, it is inconvenient to distinguish the start-stop of plastic zone through the size of actual plastic zone precisely. Therefore, the way Sheu obtained α couldn’t be identify with the real situation.

3 Experimental Data 3.1 CT Specimen The main body of the series of crack extension specimens is standard compact tensile specimens taken from one AH36 steel plate with 20 mm thickness. According to ASTM E647-11, the sampling directions of all samples are strictly controlled to the L-T direction–the rolling direction of the plate, to make sure each specimen has the same direction of grain flow and materials’ fatigue property keep in line. In order to eliminate the processing scratches on the samples and to easily observe a clear crack-expanded image under an optical microscope, all standard compact tensile specimens shall be polished to the coefficient of 0.8 or higher. Commonly, we estimate plastic zone of crack tip with Irwin Formula, thus it is necessary to obtain accurate material properties such like yield strength. To obtain detailed material parameters, two circular section tensile specimens were prepared in accordance with GB/T 228.1. The specific sizes of specimens are expressed at Fig. 2 and Fig. 3. According to the specification, the mechanical property parameters of AH36 steel are gained after steel standard tensile tests and shown in Table 1. the mechanical property parameters of AH36 steel.

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Fig. 2. Tensile specimens’ schematic diagram.

Fig. 3. Compact tensile specimens’ schematic diagram

Table 1. the mechanical property parameters of AH36 steel Mechanical property parameters

Value

Density (ρ)

kg/m3 7850

Elastic Modulus (E)

GPa

206

Passion ratio (ν)

–

0.3

Yield stress (σ y )

MPa

432

Tensile strength (σ u )

MPa

556

Elongation

%

21.0

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3.2 Test Fixture Tests were conducted by MTS fatigue testing machine, and the professional control systems MTS Model 793.10 Multi Purpose Test Ware and 793 Application Software were provided by MTS Company. The system can preset the required load form and the number of test cycles, and select real-time data to record the number of cycles and load values. When the collection is required, the program can adjust the load form and degraded the loading frequency promptly and make the collection of crack surface image data conveniently. The length of crack was monitored by CCD + microscope + s-eye measurement software. Rely on the CCD camera, the optical magnified images at the microscope eyepiece were transported to computer. Then the current length of crack is measured and photographed by s-eye software. That make sure all the surface crack images of specimens have been recorded during the whole process of crack propagation. For the crucial moments before and after overloading, video capture is conducted with sampling rate of 20FPS to ensure there are enough image data for analysis of plastic zone of crack. The layout of specific test tooling is shown at Fig. 4.

Fig. 4. The layout of specific test tooling

3.3 Analysis of Crack Growth Parameters Under Overload In this paper, the fatigue crack growth behavior of AH36 high-strength steel under different overload ratios is analyzed by the above-mentioned test tool, and the fatigue crack growth behavior of AH36 high-strength steel under overload ratios of 1.5, 2.0 and 2.5 is analyzed. The test results are as follows:

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Figure 5 and Fig. 6 are crack growth rate curves of AH36 steel under different overload ratios. The effect of 1.5 times overload on crack propagation is very small. The propagation curve is basically the same as that under constant amplitude. Some retardation occurs only in a small range after the overload point, but the fatigue life is almost the same throughout the crack propagation process. However, with the increase of overload ratio, the retardation effect of crack propagation is gradually obvious. Under the condition of 1.5 times overload, obvious retardation phase can already be seen from the a-N curve, and the propagation rate of this phase is much lower than that of the normal propagation phase. When the overload was 2.5 times, the crack growth almost stopped and only expanded about 1 mm in more than 600,000 cycles, even exceeding the life cycle under constant amplitude. The retardation response of crack growth is the decrease of crack growth rate on da/dN-K curve. In order to ensure the comparability of the test, all overloads are applied after the crack expands 4 mm in constant amplitude, and the amplitude of stress intensity factor at this time is 18.2 MPa·mˆ0.5. It can be clearly seen in Fig. 6 that the da/dN value decreases obviously near the overload position and assumes an inverted cone shape. It could be seen that the expansion rate has dropped to 10–7 levels under 2.5 times of overload, and the expansion is almost stopped. However, the influence of different overload ratios on the crack growth rate is also different. In the test, the retardation of crack growth increases with the increase of overload ratio.

Fig. 5. a-N curves under different overload ratios (The overload ratio represents the ratio of the load amplitude at the time of overload to the case of no overload)

Figure 7 shows crack growth data and piecewise fitting curves under 2 times overload. The extended data before, during and after overload are fitted piecewise linearly by least square method. As can be seen from the figure, the influence of expansion can be divided into three stages: the instantaneous acceleration stage, the delayed retardation stage and the retardation stage during expansion. According to experimental observation, the causes of instantaneous acceleration can be divided into two types. One is that excessive

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Fig. 6. Fatigue crack propagation rate da/dN versus K curves

force causes partial tearing of the crack tip. This is mainly due to a fact that the surface crack is often in a plane stress state during the crack propagation process. Compared with the thickness center crack, the stress intensity factor is smaller and the propagation speed is slower, resulting in the front edge of the crack taking on an arc shape. The larger force directly tears off the arc edge near the surface of the specimen when overload occurs, making the crack front approach a straight line, and the crack suddenly accelerates on the surface of the structure. Second, the larger force causes the closed part of the crack tip to open, and the larger overload force causes the crack tip to passivate, thus producing more obvious cracks.

Fig. 7. Piecewise fitting expansion curve under 2 times overload

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Through the experimental data above, data fitting can be carried out to obtain various parameters required by the fatigue crack growth prediction model, and the accuracy of the parameters in the model determines the reliability of the prediction results. Therefore, it is necessary to analyze and discuss the calculation methods of the parameters of the crack growth prediction model.

4 Prediction of Fatigue Crack Growth Rate Under Overload 4.1 XFEM Calculation of Crack Tip Plastic Zone Size Among the common crack propagation rate prediction models based on the plastic zone size at the crack tip, the most important work is to obtain the plastic zone size at the crack tip accurately. Therefore, researchers have proposed kinds of formulas to estimate the crack tip plastic zone size, such as the most classical Irwin formula mentioned in Sect. 2 and the improved formula obtained by subsequent researchers after various modifications based on Irwin formula. However, none of these formulas can describe the plastic size of the crack tip perfectly, which is the reason why the formulas need to be modified again and again. When using the basic FEM to calculate the crack growth parameters, it is necessary to ensure that the geometric boundary of the crack leading edge coincides with the boundary of the element, and it is inevitable to encrypt the crack leading edge mesh, which leads to the process to be cumbersome, and the calculation time increases with the number of girds increases. XFEM can handle these problems well. An augmented shape function is constructed for the cell node near the crack tip and the cell node penetrated by the crack based on the FEM, which solved the problem of discontinuity in crack position well. The XFEM displacement field formula for linear elastic fracture mechanics is as follows: u(x) =

 i∈I

Ni (x)ui +

 j∈J

Nj H (x)aj +

 k∈K

Nk

4 

φα (x)bk

(11)

α=1

Where i denotes a node in a general unit, j denotes a step expansion node, K denotes a crack tip propagation node, N i (x) is a standard node shape function of a conventional finite element, ui denotes a displacement vector of a general node, and aj denotes additional degrees of freedom for step expansion nodes, bk represents the additional degree of freedom of the crack tip expansion node. The extended node area is shown in Fig. 8. In Eq. (11), the first term of the function is a conventional finite element displacement field function. The first additional term is the displacement field function of the extended step node of the crack penetration unit, which contains a step function H(x) with +1 on one side of the crack and -1 on the other side. The second additional term is the displacement field function of the crack tip expansion node. fα (x) represents a crack tip propagation function:  √ √ θ √ θ √ θ θ (12) φα (x) = r sin , r cos , r sin sin θ, r cos sin θ 2 2 2 2

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Fig. 8. The node areas mentioned in the Eq. (11)

In which, r and θ is the position function at the crack tip. The model used in the XFEM analysis was the same CT specimen as the experiment, and the XFEM crack and enrichment unit was set to simulate the crack existing in the structure. And since the XFEM crack is on the axis of symmetry of the structure, the model will take the entire full-scale model for subsequent calculations. In order to ensure a smooth transition of the mesh, and to ensure the accuracy of calculation, the structural mesh was divided into four steps, and the mesh size from the vicinity of the crack tip to the periphery is controlled by setting seeds of different sizes in the crack growth region [10–12]. At the same time, the model is appropriately simplified in the analysis, the load and the constraint are directly applied to the loading hole node, the displacement of 5 nodes on the right side of the right-hand loading hole is fixed in the X direction, and the concentration is applied to 5 nodes on the left of the left loading hole. The plastic zone of the crack tip was calculated by setting the material stress threshold when check for the calculation results, and then, the plastic zone size can be measured by counting the number of grids. The FE model and the calculation results are shown in Fig. 9 and Fig. 10. The dark red portion in Fig. 10 is the region where the stress exceeds the yield stress by 448 MPa to produce plastic yield. 4.2 Comparison of the Calculation Methods of the Plastic Zone of the Crack Tip In this paper, the XFEM is used to calculate the plastic zone size, which is compared with the calculation result of Irwin formula. In order to obtain the difference of the plastic zone size calculated by different methods, the data under different load forms is listed in Table 2, including the plastic zone size r pX obtained by the XFEM method, the plane strain calculated by the Irwin formula, and the plastic zone size r pI-Stress under plane stress and r pI-Strain , the plastic zone size r pMI calculated by the modified Irwin formula, and the retardation propagation length ar obtained by analyzing the experimental expansion rate curve, the retardation loop number N r .

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Fig. 9. Model and meshing

Fig. 10. The plastic zone at the crack tip

Table 2. Parameters of the plastic zone Load forms

r pX (mm)

r pMI (mm)

r pI-Strain (mm)

r pI-Stress (mm)

ar (mm)

Nr

1.5OL

0.515

0.569

0.113

0.703

1.06

39479

2OL

0.638

0.714

0.201

1.251

1.99

121263

2.5OL

0.973

1.073

0.313

1.954

2.57

626377

Since the thickness of the test piece is 8 mm, the stress state of the structure is closer to the plane stress, so the plastic zone size of the crack tip is actually between the plastic zone size predicted by the Irwin formula under the plane stress and plane strain state. Compared with the calculation results of other methods, it is closer to the value of the plane stress state. It could be seen that the difference in the size of the plastic zone

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predicted by the Irwin formula is obvious when the stress state of the structure is hard to determine, and it is difficult to directly predict crack propagation. The plastic zone size r pX calculated by the XFEM method is not much different from the plastic zone size r pMI calculated by the modified Irwin formula with the trend to be the same as the load ratio changes. The reason for the difference between the two calculation results is mainly because the calculation result of the modified Iwrin formula is affected by the processing of the test data. There is a certain error in the selection of the data when the crack propagation length of the retardation phase is obtained through the test data. In the case of single peak overload, the calculation results of these two methods are better than those of the Irwin formula. However, the crack propagation length in the delayed retardation phase cannot be directly obtained by the XFEM method, and the crack propagation prediction cannot be performed by the modified Wheeler model. Therefore, the experimental data is used to obtain the delayed retardation crack propagation length and the corresponding delayed retardation crack tip plastic zone size by the modified Iwrin formula, and then the crack propagation prediction is performed by the modified Wheeler model. 4.3 Prediction Results and Analysis For the prediction of the fatigue crack growth rate of the marine high-strength steel, it is particularly important to fit the high-precision shape parameter m, the retardation plastic zone size constant α, and the delayed retardation plastic zone size β accurately through the experimental data. Referring to B.C. Sheu, Taheri et al. on the calculation of the parameters of modified Wheeler model, this paper will improve the fitting method based on it, making it suitable for the modified Wheeler model with more extended parameters. In this paper, the crack propagation parameters under 2 times overload conditions are used for prediction and analysis. The shape parameter m is calibrated by comparing the predicted number of crack propagation cycles with the number of crack propagation cycles obtained by the test. The specific steps for calibration are as follows: 1. Calculate the size of the overload plastic zone. 2. Calculate the coefficients fR and fD and fI for each cycle after overload according to Eq. (3) and Eq. (7). 3. Calculate the crack propagation amount corresponding to each load cycle by Eq. (6). 4. Summing the crack propagation amount after the overload to the crack propagation rate before returning to the corresponding constant amplitude load expansion rate. 5. Compare the number of retardation extended cycles with the test. If the data is consistent, use the coefficient. If not, correct the coefficient and repeat the above steps until the predicted data matches the test data. In order to obtain the crack propagation parameters of the high-strength steel, the data of the da/dN-K curve is processed by double logarithmic coordinates according to the crack propagation data under constant amplitude load firstly. The double logarithmic coordinates approximate a linear relationship, and the extended parameters C and n of the Paris formula are obtained by linear fitting with least squares (Fig. 11).

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Fig. 11. Linear fitting of crack propagation data under constant amplitude load

Then, according to the 2 times single peak overload data of AH36-5, α and β are fitted. In the delayed retardation phase of crack propagation, the point of the propagation rate at the constant amplitude load is selected as start, and the point where the propagation rate is reduced to the lowest point of the entire retardation phase is selected as the end, as the delayed retardation phase of crack propagation according to the experimental data. The region from the lowest point to the rate at which the crack propagation rate returns to the propagation rate under the corresponding constant amplitude load is used as the retardation propagation phase. The predicted results obtained by the above calculation steps are as Fig. 12.

Fig. 12. Comparison of test data and prediction results of crack growth rate under 2 times overload

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In Fig. 12, the solid red line is the result predicted according to Eq. 5. It could be seen that, based on the predictive retardation phase of the Wheeler model, the model predicts the transient acceleration and delayed retardation propagation phases after overload. The propagation rate prediction in the instantaneous acceleration phase after overload basically coincides with the test data, and there is a slight error at the lowest point of the delayed retardation propagation rate. The prediction of crack growth life is shown in Fig. 13.

Fig. 13. The prediction of crack growth life

The predicted crack propagation life in Fig. 13 agrees with the data in the actual test. The retardation propagation phase of the crack is corrected by the modified Wheeler formula and basically conforms to the experimental data. Only in the final stage of crack propagation, the constant amplitude propagation calculated by the Paris formula produces a slight difference. The total crack propagation life differs by 8904 cycles, which is equivalent to only 2.47% of the total structural lifetime of 360,642 from 1 mm to 20.26 mm. Therefore, it can be considered that the improved Wheeler model predicts the crack propagation of the high-strength steel under unimodal overload with relatively high precision. The extended data for each phase is shown in Table 3. Table 3. Data and error between test and predictive data at each stage ad (mm) ar (mm) af (mm) N d Test

Nr

N tot

0.64

1.99

20.26

41150 121263 360642

Prediction 0.62

2.01

20.26

39212 126820 369546

Error

1.00%

/

4.71% 4.58%

3.13%

2.47%

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ad and N d are the crack propagation length and the corresponding cycle number in the delayed retardation phase, ar and N r are the crack propagation length and the corresponding cycle number in the retardation phase, af is the final crack length, and N tot is the total number of cycles. In the case of single overload, the error of data in each stage does not exceed 5%, so the accuracy can be fully guaranteed.

5 Conclusion In this paper, aimed at AH36 steel which is widely used in marine and marine engineering, the fatigue crack growth rate of the AH36 steel under constant amplitude and variable amplitude loads is obtained. The crack propagation behavior under different overload ratios is studied. The effect of the plastic zone of the crack are analyzed by experimental data and finite element method. The numerical study on the plastic zone size at the crack tip was carried out. And the size of the crack tip plastic zone calculated by different methods was compared and analyzed. After that, the modified Wheeler model was used to predict the crack propagation behavior of AH36 high strength steel under variable amplitude load. The applicability of the prediction model was analyzed by comparing with the experimental results. The conclusions are drawn as follows: 1. Single overload will cause retardation in crack propagation. And the effect is related to overload ratio. With the increase of overload ratio, the retardation effect is strengthened. Under the action of 2.5 times overload, crack propagation is almost stopped, and the propagation rate in the retardation phase was once reduced to the order of 10–7. 2. The overload will cause obvious residual strain near the crack tip. In the subsequent constant amplitude load cycle, the crack tip structure is pressed to improve the shielding effect of the plastic zone at the crack tip, which causes retardation in crack propagation. 3. Predicting the plastic zone size of the crack tip by XFEM, the size of the crack tip plastic zone under the crack length can be predicted based on a single model, eliminating the need to re-mesh each crack length in the traditional FEM method, and reducing the workload effectively. However, this method can only obtain the crack retardation propagation length, and it is impossible to further use the modified Wheeler Model for crack propagation prediction. 4. The fatigue crack propagation under various loads can be predicted precisely based on the Wheeler model combined with the plastic zone of crack tip calculated by modified Iwrin formula, which also proves that the plastic zone theory of crack tip can better describe the fatigue crack growth behavior of AH36 steel.

References 1. Paris, P.: A critical analysis of crack propagation laws. J. Basic Eng. 85(4), 528–533 (1963) 2. Elber, W.: Fatigue crack closure under cyclic tension. Eng. Fract. Mech. 2(1), 44–45 (1970)

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3. Wheeler, O.: Spectrum loading and crack growth. J. Fluids Eng. 94(1), 181–186 (1972) 4. Sheu, B.C., Song, P.S., Hwang, S.: Shaping exponent in wheeler model under a single overload. Eng. Fract. Mech. 51, 135–143 (1995) 5. Yuen, B., Taheri, F.: Proposed modifications to the Wheeler retardation model for multiple overloading fatigue life prediction. Int. J. Fatigue 28(12), 1802–1819 (2006) 6. Huang, X.P., Torgeir, M., Cui, W.: An engineering model of fatigue crack growth under variable amplitude loading. Int. J. Fatigue 30(1), 2–10 (2008) 7. Yang, P., Gu, X.K.: Comparative analysis of some fatigue crack propagation models. J. Ship Mech. 17(3), 289–297 (2013) 8. Haydar, D., Yalcinkaya, T.: Fatigue crack growth under variable amplitude loading through XFEM. Procedia Struct. Integrity 2, 3073–3080 (2016) 9. Willenborg, J., Engle, R.M., Wood, H.A.: A crack growth retardation model using an effective stress concept (1971) 10. Newman, Jr., J.C., Crews, Jr., J.H., Bigelow, C.A.: Variations of a global constraint factor in cracked bodies under tension and bending loads. NASA-TM-lO9119 (1994) 11. Baptista, J.B., Antunes, F.V., Correia, L.: A numerical study of the effect of single overloads on plasticity induced crack closure. Theor. Appl. Fract. Mech. 88, 51–63 (2017) 12. Camas, D., Garcia-Manrique, J., Gonzalez-Herrera, A.: Numerical study of the thickness transition in bi-dimensional specimen cracks. Int. J. Fatigue 33, 921–928 (2011) 13. Tsukuda, H., Ogiyama, H., Shiraishi, T.: Transient fatigue crack growth behavior following single overloads at high stress ratios. Fatigue Fract. Eng. Mater. Struct. 19(7), 879–891 (2010) 14. Singh, K.D., Xu, Y.G., Sinclair. I.: Strip yield modelling of fatigue crack under variable amplitude loading. J. Mech. Sci. Technol. 25(12), 3025–3036 (2011) 15. Hammouda, M., Seleem, S.E.: Fatigue crack growth due to two successive single overload s. Fatigue Fract. Eng. Mater. Struct. 21(12), 1537–1547 (1998) 16. Otakar, J., Wei, R.P.: An exploratory study of delay in fatigue-crack growth. Int. J. Fract. 7(1), 116–118 (1971) 17. Mcevily, A.J., Ishihara, S., Mutoh, Y.: On the number of overload-induced delay cycles as a function of thickness. Int. J. Fatigue 26(12), 1311–1319 (2004) 18. Borrego, L.P., Ferreira, J.M., Cruz, J.M.: Evaluation of overload effects on fatigue crack growth and closure. Eng. Fract. Mech. 70(11), 1379–1397 (2003) 19. Datta, S., Chattopadhyay, A., Iyyer, N.: Fatigue crack propagation under biaxial fatigue loading with single overloads. Int. J. Fatigue 109, 103–113 (2018)

The Effect of Loading Sequence on Fatigue Crack Growth of a Ship Detail Under Different Load Spectra Xiaoping Huang1(B) , Honggan Yu2 , and Dimitrios G. Pavlou3 1 School of Naval Architecture, Ocean and Civil Engineering,

Shanghai Jiao Tong University, Shanghai, China [email protected] 2 School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, China 3 Department of Mechanical and Structural Engineering and Materials Science, Stavanger University, Stavanger, Norway

Abstract. In this paper, a method of constructing fatigue load spectrum sequences under multiple load and working conditions based on the simplified fatigue load calculation formula is proposed. For easy understanding the loading sequence on fatigue crack growth, three details in a large container ship are taken as examples for discussion. In the analysis, different generating fatigue load spectrum for lager container ship details combining with the unique crack growth rate curve model to calculate the crack growth under different fatigue loading spectra have been performed. For performing the calculation and guarantee the calculating accuracy, a program code, based on the proposed procedure, has been compiled in MATLAB. Using the codes, the effect of fatigue loading sequence has been discussed. The different order of fatigue loading stages has a big influence on the crack propagation when considering the SIF threshold. In addition, the method of constructing fatigue load spectrum sequence under multiple load and working conditions needs to be further examined. Keywords: Ship details · Fatigue loading · Crack growth · Loading sequence · Unique crack growth rate curve model

1 Introduction There are generally two methods for fatigue assessment of steel structural details: the Cumulative Fatigue Damage (CFD) method based on the S-N curves and the Fatigue Crack Propagation (FCP) method based on fracture mechanic. The former is still commonly used in engineering practice, mainly for crack initiation prediction. This method can utilize the Miner’s linear rule [1] or nonlinear ones [e.g. 2] in order to take into account the loading sequence effect. The CFD is generally very conservative method. The latter has developed rapidly in marine structural engineering practice. It can utilize the Paris linear FCG rule, or nonlinear ones [3–9] in order to take into account the load interaction effects. Furthermore the predicted results can be detected by NDT. The larger © Springer Nature Singapore Pte Ltd. 2021 T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 442–453, 2021. https://doi.org/10.1007/978-981-15-4672-3_28

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scale of ships has led to the application of higher strength thick steel plates in ship hulls [10]. The fatigue strength of materials is not linearly related to the yield strength or tensile strength, and the fatigue performance of some materials decreases with the strength of material increases [11]. These make the fatigue problems of large vessels become more and more prominent. Since the traditional method is limited by the strength and thickness of the steel plate, it is appropriate to adopt the FCP method to assess the fatigue life of high-strength steel plate. For the FCG calculations, the nonlinear crack propagation model [6–8] is adopted. The accuracy of the load spectrum is a key step to assess fatigue life using FCP method. In general, there are two ways to obtain the fatigue load spectrum; the spectral analysis method and the simplified method. Among them, the spectral analysis method has clear physical meaning and reasonable process, but the calculation is complex and needs to know the wave spectrum and transfer function. The simplified method is based on the equivalent design wave method. Although the physical meaning is not clear, the calculation process is simple and the result is conservative. Guideline for Fatigue Strength of Ship Structures introduces a simplified method for obtaining fatigue load. The fatigue assessment mainly adopts the traditional method, and the FCP method is briefly mentioned. The simplified fatigue load calculation method is adopted to calculate the fatigue loading and combined with fracture mechanics to calculate fatigue life of hull structures. The MATLAB is used for cycle-by-cycle calculation. There are many studies to calculate the fatigue life of hull structures by means of programs, such as the codes by Sumi [12], Lebaillif and Recho [13], and He and Liu [14]. There are some papers which studied the fatigue loading sequence generation techniques incorporate the fracture mechanics were summarized in ISSC III.2 report, there is still not a widely accepted procedure to generate the fatigue loading spectrum for a ship detail and the effect of the loading sequence is need to be studied further [15]. In present, the effect of loading sequence on fatigue crack growth is studied under the different fatigue loading spectra combined with the different loading blocks.

2 Fatigue Crack Growth Calculation Procedure The flow chart of FCP method is shown in Fig. 1, which divided into load spectrum module, crack propagation model module and post-processing module. In this paper, the simplified fatigue loading calculation method is adopted to calculate the fatigue load spectrum, and the calculated fatigue loading spectrum is input into the crack propagation model together with the initial crack information and material parameters. Then, the crack propagation is calculated cycle-by-cycle. Finally, the result is post-processed. 2.1 Simplified Fatigue Loading Calculation Method The nominal stress range of a structural spot under different load and working conditions is obtained by adopting the simplified fatigue load calculation method recommended in the Guidelines for Fatigue Strength of Ship Structures [16]. In fatigue analysis of ship structures, the Weibull distribution with two parameters is often used to express the long-term distribution of the stress range. The probability

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Crack Information

Crack Propagation Model

Load Spectrum (Simplified method)

(Huang Model)

Post-processing (Crack propagation curve etc.)

Material Parameters

Fig. 1. Fatigue assessment process of FCP

density function is:       σ h h σ h−1 exp − f (σ ) = q q q

(1)

where h is shape parameter which is generally determined by factors such as marine environment, structure type and response characteristics, position of the component in the hull. In HCSR, the shape parameter is 1.0 and the exceeding probability level 10−2 is selected to calculate the cumulative fatigue damage, so that the sensitivity to changes in shape parameter is smallest [17]. The scale parameter q can be represented by a stress range corresponding to a certain exceeding probability obtained by long-term analysis of the fatigue load recovery period n0 : q=

σ 1 (ln n0 ) / h

(2)

After getting the nominal stress ranges of all working conditions in each load condition, the working condition with the largest nominal stress range is selected in order to represent conservatively the corresponding loading condition. Then, according to Eq. 2, the scale parameter of each load condition is obtained, then generate load spectra of each different load conditions and finally process them to get the final fatigue load spectrum. There are many ways to process the load spectrum, e.g. the load spectrum of full load and ballast can be sorted or combined. 2.2 Crack Growth Rate Model The adopted crack propagation model is proposed Huang et al. [6–8] whose basic expressions are:  m da = C Keq0 − (Kth0 )m dN

(3)

Keq0 = MP MR K

(4)

The Effect of Loading Sequence on Fatigue Crack Growth of a Ship Detail

√ K = σ Y π a

445

(5)

where C and m are material constants corresponding to R = 0; Keq0 and Kth0 are the equivalent stress intensity factor (SIF) range and the equivalent SIF range threshold corresponding to R = 0; MP and MR are load sequence correction factor and stresses ratio correction factor; K, σ, and Y are SIF range, stress ranges, and geometric correction factors. For SIFs of through-thickness cracks, edge cracks and embedded cracks, the formulas given in BS7910 [18] are adopted. The SIF of a surface crack is calculated by formula proposed by Newman and Raju [19]. The formulas for calculating SIFs of butt joint and T-joint are given in BS7910. Luo and Huang [20] proved that the T-joint SIF calculation formula is applicable to cruciform joint. The empirical formula for calculating the SIF of a surface crack at longitudinal end proposed by Kong and Huang [21] are adopted when the detail is a longitudinal end. The nominal stress ranges σ are extracted from the load spectrum. Taking into account the geometry of the structural detail, the geometric parameter Y is calculated and the range K is derived by the Eq. 5. The Keq0 can be calculated from Eq. 4 and the amount of crack propagation per cycle can be obtained from Eq. 3. For a surface crack and an embedded crack, the increments of crack per loading cycle along depth and length direction can be calculated by the following equations.  m (6) da = Ca Keq0a − (Kth0 )m  m dc = Cc Keq0c − (Kth0 )m

(7)

In order to coincide with the experimentally observed crack shape, IIW [22] pointed out that the material constant C in Paris Law should have different values at the deepest point and the surface point of a surface crack, i.e. CC = 0.9m CA . To calculate fatigue life using FCP method, it is necessary to know the initial crack size and the critical crack size. The initial crack size is given by the specification or the actual inspection size, and the critical crack size is determined by the FAD (Level 2A) recommended by BS7910.

3 Outline of the Fatigue Strength Assessment Program In this paper, a program for fatigue assessment of ship structures is compiled with MATLAB, which can assess the fatigue strength of typical structural details on ship, such as a plate, butt-plate, T-joint/cruciform joint and longitudinal end. For some structural details, several crack types can also be considered (Fig. 2). In Fig. 3 the detailed configuration of the program is demonstrated. The dashed boxes from left to right show the load spectrum module, the crack propagation model module and post-processing module in turn. There are three ways to generate load spectrum in the load spectrum module. One way is to generate a load spectrum using the simplified method. The other two ways are

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Fig. 2. Function of the GUI program

importing load spectrum parameters of all load and working conditions in a specified format, the details can be found in references [20, 21, 23]. The fatigue crack propagation model is the core of the entire program and must ensure that the material parameters and crack information have been obtained before running this module. It’s noteworthy that j in Fig. 3 indicates the number of load cycles and i indicates the number of times the data read. A sufficient number of loads are randomly selected from the load spectrum, and the probability density of the entire load spectrum is approximately represented by a frequency histogram of the selected loads. The fatigue life is obtained from the intersection point of the failure assessment curve and failure assessment curve in FAD. The critical crack size can be derived from the number of load cycles corresponding to the fatigue life. Crack propagation curves are obtained by connecting the points represented by the crack sizes under different load cycles.

4 Effect of Loading Sequence on Fatigue Crack Growth Three structural details containing surface cracks (Fig. 4) on an ultra large container are considered as examples to calculate fatigue lives and critical crack sizes. The results are used to verify the reliability and practicality of the proposed program, and to further study the effect of loading sequence on crack propagation. The basic information of the three structural details is shown in Table 1, and the ship design life is 20 years. 4.1 Fatigue Loading Spectra and the Corresponding Crack Growth Before calculating the fatigue life of a detail, it is necessary to obtain the fatigue load spectrum mentioned in Sect. 2.1. The structural detail 1 is taken as an example to discuss

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Crack informaon (structure type, crack form etc.)

Material parameters

a(0),c(0) n(0)=1,i=0,j=0

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Crical crack size, fague life

Simplified method

Get da, dc from formula(6) and formula(7)

Import h and q of all load and working condions

a(j+1)=a(j)+da c(j+1)=c(j)+dc

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mod(n(j),1e4)==1?

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Fig. 3. Configuration of the program

the effect of different load spectra on the crack propagation. The length of the initial crack and the parameter co is a0 = 2 mm and c0 = 10 mm respectively. Assuming that h = 1, q = 16 under full load condition, and h = 1, q = 8 under ballast condition, the time distribution coefficients for full load and ballast are 0.65 and 0.2 respectively. These values are used to generate pseudo-random numbers representing the full load and ballast spectra through MATLAB. Taking into account the data of Table 1 and Fig. 4 the calculated crack propagation curves along crack depth direction are shown in Fig. 5. It can be seen that the three structural details, at given initial crack, are safe during service. The cracks at structural detail 1 and 2 are propagated to the critical crack depth and the critical aspect ratios (rc = ac /cc ) is approximately 0.6 which is consistent with experimental observations. In this subsection, only two load spectrum blocks are considered during the entire design life, namely, the previous block of the load spectrum is the full load and the next block is ballast (load 1) or opposite of this (load 2). Figure 6 shows the load fragment diagrams of the two load spectra. The crack propagation curves along depth direction under the two load spectra is shown in Fig. 7. It can be seen that the fatigue life under load 1 is shorter than load 2 and the structural detail failed before the load from full load block traversed to ballast block of the spectrum under load 1. Obviously, the processing of the load spectrum is unreasonable. The derivation of a reasonable fatigue loading

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Fig. 4. Structural details to be assessed on the container

Table 1. Basic information of structural details Item

Structural detail 1 Structural detail 2 Structural detail 3

Profiles

FB900 × 70

FB900 × 65

BP340 × 12

Material E47

E40

E32

x (m)

179.6

179.6

179.6

y (m)

23.565

24.84

25.43

z (m)

32.2

29.75

24.94

spectrum is based on the study results of the effect of loading sequence. The effect of different load spectra with different block lengths on fatigue life will be studied in the following subsection. 4.2 The Effect of Loading Sequence on Crack Growth We consider the effect of loading sequence under 4 different loading spectra which consists with the random loading (load1), the ascending (load2), the descending (load3) and the equivalent root mean square value (load4) of the same fatigue loading respectively. The load spectrum is equivalent to a constant amplitude spectrum via the root mean square value σrms and the effect of mean stress is not considered. The Model is

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structural detail1 structural detail2 structural detail3

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transformed into the Paris Law. The expression of σrms is as follows:

Nt σrms = (σi )2 /Nt i=1

(8)

where Nt represents the total number of load cycles; σi represents stress range of the ith cycle. Following the procedure in Sect. 4.1, and then ascending, descending, and transforming the load spectrum into a constant amplitude one (via σrms ), we generate a total of four different load spectra. The total number of load cycles is 6.5 × 107 for ship structures is selected here. The samples of the four load spectra segments are shown in Fig. 8. The initial crack size and the crack growth rate curve constants are the same

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Fig. 7. Crack propagation curves along depth direction

as that in Sect. 4.1. The crack propagation curves along depth direction under the four different load spectra are shown in Fig. 9. It can be seen that the crack propagation is extremely rapid and the fatigue life is very short when the load is in descending order. The longer fatigue lives were obtained for loads is in ascending order and the equivalent root mean stress range. The results is something opposite to our commonsense on the effect of loading sequence. It can be explained by the crack growth threshold was introduced in the fatigue crack growth law. For descending order, most of the SIF ranges are great than the threshold, but the opposite for the ascending order. This example shows the difficult to give a procedure to correct the effect of the fatigue loading sequence for fatigue loadings that marine structures will encounter during their service.

48

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The effect of loading sequence on fatigue life with different loading spectra which consists with different block lengths is studied. The time distribution coefficients for full load and ballast remain unchanged. The schematic diagram of the two load spectra

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Fig. 9. Crack propagation curves under different loads

(load 1 and load 2) are shown in Fig. 10, where Nt is the total number of load cycles corresponding to the design life, N0 is the sum of cycles of full load block and ballast block. The guideline stipulates that when the design life is 20 years, the total number of load cycles is 6.5 × 107 for ship structures, so the number of load cycles within one month after considering the operating coefficient is Nm = 2.3 × 106 . The corresponding time intervals for full loading and ballast loading conditions are different when the type of ships, route and marine environment are different. The fatigue lives of structural detail 1 are calculated under different loading spectra with N0 = 0.25 Nm , 0.5 Nm , Nm , 2 Nm , and 3 Nm in load 1 and load 2. Figure 11 shows the crack propagation curves along crack depth direction under different load spectra. It can be seen that the crack propagation curves are very close to each other, the smaller of the N0 , the longer of the calculated fatigue lives, but they are between the two curves in Fig. 6. The critical life is about 8 years. The difference between the calculated fatigue lives under load 1 and load 2 is not big, the load spectrum is processed by the first load spectrum sorting method and N0 = Nm are recommended. 129 86

load1(full load) load1(ballast)

Δσ/MPa

43 0 129 86

......

......

......

......

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Nt

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Fig. 10. Load spectra

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Fig. 11. Crack propagation curves of different N0

5 Conclusions In this paper, the fatigue load spectrum is generated based on the simplified method recommended in the Guidelines for Fatigue Strength of Ship Structures. An existing crack growth rate model developed in past by the first author is adopted to assess the fatigue strength of typical hull structures. The effect of loading sequence on fatigue life is discussed. Main conclusions: 1. A method of constructing fatigue load spectrum sequence under multiple load and working conditions based on the simplified method has been proposed. 2. The different order of loading spectrum has a great influence on the crack growth when considering the SIF range threshold. The load spectrum consisting of number of block loadings in one month or shorter is recommended. 3. The proposed method of constructing fatigue load spectrum sequence under multiple load and working conditions needs to be further examined.

References 1. Miner, M.A.: Cumulative damage in fatigue. J. Appl. Mech. 12(3), A159–A164 (1945) 2. Pavlou, D.G.: The theory of the S-N fatigue damage envelope: generalization of linear, doublelinear, and non-linear fatigue damage models. Int. J. Fatigue 110, 204–214 (2018) 3. Elber, W.: The significance of fatigue crack closure in fatigue. In: ASTM STP 1972, vol. 486, pp. 230–242 (1972) 4. Wheeler, O.E.: Spectrum loading and crack growth. J. Basic Eng. Trans. ASME D 94(1), 181–186 (1972)

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5. Willenborg, J.D., Engle Jr., R.M., Wood, H.A.: A crack growth retardation model using effective stress concept. AFDL-TM-71-1-FBR (1971) 6. Huang, X., Moan, T.: Improved modeling of the effect of R-ratio on crack growth rate. Int. J. Fatigue 29(4), 591–602 (2007) 7. Huang, X., Torgeir, M., et al.: An engineering model of fatigue crack growth under variable amplitude loading. Int. J. Fatigue 30(1), 2–10 (2008) 8. Huang, X., Moan, T., et al.: A unique crack growth rate curve method for fatigue life prediction of steel structures. Ships Offshore Struc. 4(2), 165–173 (2009) 9. Pavlou, D.G.: Prediction of fatigue crack growth under real stress histories. Eng. Struct. 22(12), 1707–1713 (2000) 10. Cui, W., Cai, X., et al.: Research status of ship structure fatigue strength assessment and progress in China. Ship Mech. (4), 63–81(1998) 11. Wang, J.: Crack Propagation Analysis and Fracture Assessment of High Strength Steel Plates for Container Ships. Shanghai Jiao Tong University, Shanghai (2014) 12. Sumi, Y., Mohri, M., et al.: Computational prediction of fatigue crack paths in ship structural details. Fatigue Fract. Eng. Mater. Struct. 28(1–2), 107–115 (2005) 13. Lebaillif, D., Recho, N.: Brittle and ductile crack propagation using automatic finite element crack box technique. Eng. Fract. Mech. 74(11), 1810–1824 (2007) 14. He, W., Liu, J., et al.: Numerical study on fatigue crack growth at a web-stiffener of ship structural details by an objected-oriented approach in conjunction with ABAQUS. Mar. Struct. 35(1), 45–69 (2014) 15. Kaminski, M.L., Rigo P.: COMMITTEE III.2 fatigue and fracture. In: Proceedings of the 20th International Ship and Offshore Structures Congress (ISSC 2018) (2018) 16. CCS: Guidelines for fatigue strength assessment of offshore engineering structures. China Classification Society (2013) 17. Feng, G., Ren, H., et al.: Study on fatigue dynamic load probability level and shape parameter of HCSR. Ship Ocean Eng. 42(6), 22–24 (2013). (in Chinese) 18. British Standards Institution: Guide to methods for assessing the acceptability of flaws in metallic structures. BS 7910 (2005) 19. Newman, J.C., Raju, I.S.: An empirical stress-intensity factor equation for the surface crack. Eng. Fract. Mech. 15(1–2), 185–192 (1981) 20. Luo, P., Huang, X.: A fatigue life prediction method for longitudinal deck stiffener nodes in large container vessels. Nav. Arch. Ocean Eng. 32(6), 1–10 (2016) 21. Kong, X., Hunag, X., et al.: Calculation of stress intensity factors of surface cracks at weld toe of longitudinal stiffener joints in a container. Shipbuild. China 2, 67–77 (2016) 22. IIW: Recommendations for Fatigue Design of Welded Joints and Components (2014) 23. Huang, X., et al.: Study on fatigue life prediction of details with a surface crack under spectrum loading. In: OMAE 2019, Madrid, Spain, 17–22 June 2018 (2018)

Study on Fatigue Strength of Welded Joints Subject to Intermittently Whipping Superimposed Wave Load Naoki Osawa1(B) , Luis De Gracia1 , Kazuhiro Iijima1 , Norio Yamamoto2 , and Kyosuke Matsumoto1 1 Osaka University, 2-1 Yamadaoka, Suita, Osaka 5650871, Japan

[email protected] 2 Nippon Kaiji Kyokai, 4-7 Kioichyo, Chiyoda-ku, Tokyo 102-8567, Japan

Abstract. The stochastic characteristics of hull vibration superimposed stress waveform experienced by 6500 TEU container ship’s deck longitudinal is examined by Non-Linear Hydro-Elasticity Analyses. It is found that the stress waveform’s characteristics can be simplified so that the slamming impact occurs once in every 4 to 5 waves and the impact stress range is comparable to the wave stress range. Based on these results, the whipping superimposed stress waveform which are applied in fatigue tests are chosen. An electric exciter-driven plate bending vibration fatigue testing machine, which can apply various hull vibration superimposed stress waveform in high speed, is newly developed. Performing the whipping superimposed fatigue tests of welded joints using this machine, the validity of rainflow cycle counting for whipping superimposed loadings is examined. It is found that rainflow counting led to conservative estimates of fatigue lives, and the fatigue damage up to the failure of intermittently superimposed cases is larger than that of constantly superimposed cases under conditions chosen. Keywords: Fatigue · Whipping · Hydro-elasticity analysis · Electric exciter

Nomenclature a BNF CA C, m C wh C wh,lim Drf DUF EE EM FE FEM

Threshold value parameter Bending natural frequency Constant amplitude Material parameters Coefficient of whipping interval C wh at limit ship forward speed Rainflow damage Rainflow damage Up to the Failure Electric Exciter Eccentric Mass Finite Element Finite Element Method

© Springer Nature Singapore Pte Ltd. 2021 T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 454–472, 2021. https://doi.org/10.1007/978-981-15-4672-3_29

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FFT f PW f whip HS HFE HV HVSSW IF IFFT MTR Nb NLHEA N PW N0 nCA ntotal nPW nSUP nslm PID PBV PW RFCC SP SUP S HV-PK S HV-VL S CREST S PK S PK,lim S PK_RNG,lim S PW,lim S PW S PW,lim T HRES TS VBM V lim (H S ) VS σ σCA σeq,rf σRng σSUP

455

Fast Fourier Transformation PW frequency Frequency of whipping vibration Significant wave height High-frequency Effect Hull Vibration Hull Vibration Superimposed Stress Waveform Intermittently superimposed with Fixed σeq Inverse Fast Fourier Transformation Minimum Time Route Number of load cycles at which the crack propagation path turns into the main plate Non-Linear Hydro-Elasticity Analyses Number of PW N b estimated by CA SN curve CA cycles Total number of cycles in one load set PW cycles in each sea state SUP cycles Number of slamming impact Proportional-Integral-Differential Plate Bending Vibration Primary Wave RainFlow Cycle Counting Set point Superimposed Peak stress of HV waveform Valley stress of HV waveform Crests of S PK sequence Difference between S HV-PK and S HV-VL Average values of S PK for V lim (H S ) Slamming peak stress range at the limit ship forward speed Primary stress range at the limit ship forward speed Stress range of PW stress waveform Average values of S PK for V lim (H S ) Threshold value Wave mean period Vertical Bending Moment Maximum V S achievable for a given H S Ship forward speed Nominal bending stress range σeq,rf in CA cycles Rainflow equivalent stress range Standard deviation of S PK σeq,rf in SUP cycles

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1 Introduction It is well known that ships at sea are not only subjected to wave-induced stresses. Ship’s hydro-elastic hull vibrations are referred to as ‘whipping’ and ‘springing’. Whipping is known as a transient vibratory phenomenon due to wave impacts, called slamming, and ‘springing’ as a resonance of hull vibration, consequence of the oscillating wave loads. Nowadays, these hydro-elastic vibratory phenomena are more likely to be more important for larger container ships due to large bow flare which induces higher slamming loads, high ship speed, the increase in the ship size reduce the natural frequencies, increase of the fatigue damage in fatigue sensitive detail, among others. Below, the hull vibration and its effect on fatigue strength are called HV (Hull Vibration) and HFE (High-Frequency Effect). Also, the HV superimposed stress waveform is called HVSSW (Hull Vibration Superimposed Stress Waveform). Let ‘rainflow damage’ Drf be the fatigue damage evaluated by RainFlow Cycle Counting (RFCC). Storhaug et al. [1–3] are the first to point out the importance of HFE. Moe et al. [4] performed one-year onboard measurement and reported that the increase in Drf due to HV was up to 56% of the total damage for a typical fatigue sensitive detail in a deck longitudinal member. Iijima et al. [5] reported that Drf increased 20%–40% by HV between different ship types. Fatigue tests of ship’s welded joints were not performed in these studies. The validity of RFCC for welded joints subject to HVSSW has not been fully verified yet. Fricke et al. [6, 7] examined the fatigue strength of cruciform welded joints subject to uniaxial springing superimposed wave loadings using a hydraulic testing machine. A simplified whipping waveform was superimposed constantly in their tests. They reported that the mean of Drf Up to the Failure (DUF) was about 0.75, that is, RFCC led to non-conservative estimates. The authors (Osawa et al. [8, 9]) measured fatigue lives of out-of-gusset welded joints subject to springing superimposed wave loadings by using Eccentric Mass (EM)-driven Plate Bending Vibration (PBV) fatigue testing machines. They reported that the fatigue life can be predicted with acceptable accuracy by RFCC when HV is superimposed constantly, while fatigue lives are prolonged substantially than RFCC’s estimation when the HV is superimposed intermittently. These results show that HFE strongly depends on test conditions such as HVSSW’s characteristics. Recently, for seakeeping analysis, various hydroelastic methods to consider nonlinearities have been proposed [10–13], and the effect/characteristics of HVSSW were examined by Non-Linear Hydro-Elasticity Analyses (NLHEA) for various wave conditions [14–16]. The validity of RFCC should be further examined by carrying out fatigue tests with various stress histories. In such study, HVSSWs applied should be determined so that they emulate the stochastic characteristics (amplitude ratio of HV to the Primary Wave (PW), the ratio of frequencies of HV and PW, slamming’s occurrence interval, HV’s decay time, etc.) of those experienced by structural members in ocean-going ships. Unacceptably long testing time is needed if those fatigue tests are performed by a hydraulic testing machine. The testing time can be reduced drastically by using EM-driven PBV machines (Osawa et al. [8, 9]), but it is difficult to control the waveform characteristics. In this study, the stochastic characteristics of HVSSW experienced by 6500 TEU container ship’s deck longitudinal are examined by performing a series of NLHEA for

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various conditions (short seas/heading angles/ship speeds). Based on these NLHEA results, HVSSWs which are applied in HFE fatigue tests are chosen. An electric exciter (EE)-driven PBV fatigue testing machine, which can apply various HVSSW in high speed, is newly developed. Performing fatigue tests of welded joints using this EE-driven PBV testing machine, the validity of RFCC for various intermittently HV superimposed loadings is examined.

2 NLHEA of 6500 TEU Container Ship 2.1 Theory The present numerical method combines a 3D panel method and FEA. The theoretical details are explained in Iijima et al. [13]. The 3D panel method based on linear potential theory is used to evaluate the hydrodynamic behavior. A zero-forward Green function, however, with so-called encounter frequency correction [17], is used for this purpose. The weakly nonlinear approach is adopted to account for the nonlinearity of the load. Slamming load is evaluated on 2D strips and integrated by using a classical von Karman’s momentum theory. For the structural deformation, a modal method is adopted. The modes are extracted based on the FE model. In the analysis, the whole structure is discretized into beam elements for structural modeling. The analysis procedure may be summarized as follows: 1. The structural model is established by using FEM to evaluate the Eigen (natural) frequencies and the associated modes. In the present model, only vertical bending modes are considered. 2. The 3D panel method is used to evaluate the hydrodynamic load properties in the frequency domain. Radiation for the rigid body motions and the flexible modes are considered. 3. A system of time-domain equations of motion is established and solved in modal space. IFFT (inverse FFT) is used to calculate the load in the time domain. The weakly nonlinear term is also considered while so-called hydrodynamic memory effects are not considered. 4. The motions and structural deformation are evaluated by using the modal superposition. In this study, the vertical bending moment is evaluated by using the modal superposition, and the stress is obtained by dividing by the sectional modulus. It is expected that the stress value converges even when we consider the first few modes of deformation if we refer to the stress at deck side where the effect of local pressure distribution is negligible. 2.2 Analysis Target, Numerical Model and Analysis Conditions A 6500TEU container ship is the target ship. The main particulars are L × B × D_d = 280 × 43 × 26_14 (m). It is assumed to be in the fully loaded condition. Cargoes are modeled by using mass elements distributed over the whole ship. It is assumed the

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target ship follows the container ship’s performance curve of the research committee P-27, of the Japan Society of Naval Architects and Ocean Engineers, shown in Fig. 1 and sails on a North Atlantic route following a weather routing algorithm which can decide the minimum time route (MTR) from a spatiotemporal distribution of sea states (significant wave height and wave direction). In this weather routing algorithm, the relationship between ship speed loss, relative heading angle and significant wave height is considered. Details can be found in [18]. The hydrodynamic mesh model of the subject ship contains 3004 nodes and 2820 elements on the hull. The whole ship is discretized into 24 beam elements. The sectional properties of the beam are given based on the mass and rigidity distribution data. The first natural frequency is found at 0.747 Hz. The associated mode is the 2-noded vertical bending. Ship’s structural damping is not clear. In our simulations, less than 2% critical damping is assumed for the respective flexible modes. The sea state is characterized by significant wave height (H S ) and wave mean period (T S ). All sea states are considered as fully developed (Pierson Moskowitz). In this study only long crested waves are considered. 180

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Fig. 1. The ship performance curve of a 6500 TEU container ship [18].

In real operational conditions, it is needed to consider voluntary ship speed reduction when sailing in heavy seas. In this study, the ship performance curve shown in Fig. 1 is adopted. The ship performance curve considers the added resistance (involuntary speed reduction) and operational limit (voluntary speed reduction) based on a model developed by Tamaru [18]. The operational limit is relevant when the ship speed is larger than 6 m, which is the reason why there is a sudden drop around 6 m. It is not purely due to added resistance. The short sea duration is 2.5 h. The forward speed is chosen to be constant in each short sea.

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3 Stochastic Characteristics of HVSSW Experienced by 6500 TEU Container Ship 3.1 Slamming Detection Methodology The details of the methodology are explained in Luis et al. [19]. The Vertical Bending Moments (VBM) is converted into deck stress on the amidships section. The wavefrequency response is obtained by low-pass filtering the original stress waveform. The upper cutoff frequency is 0.35 Hz. After PW stress is extracted, the HV stress component is obtained by high-pass filtering the original waveform. The upper cutoff frequency for PW stress filtering is the same as the lower cutoff frequency for HV stress filtering. Responses above 1.0 Hz are also removed. Figure 2 shows an example of the HVSSW experienced by the target ship in head sea condition (H S = 6.5 m, T S = 10.5 s). It is noted that hogging condition is positive, with slamming in sagging.

HVSSW (MPa)

250

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200 150 100 50 0 500

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600 Time(s)

650

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Fig. 2. An example of HVSSW experienced by 6500 TEU container ship (HS = 6.5 m, TS = 10.5 s, head waves).

Let ‘peak stress’ S HV-PK and ‘valley stress’ S HV-VL be the peak and valley of HV waveform, and ‘peak-valley range’ S PK be the difference between S HV-PK and S HV-VL (see Fig. 3). The slamming peaks are detected as follows (see Fig. 4): 1. HV stress waveform is generated by high-pass filtering of the original (HVSSW) stress waveform (the blue line in Fig. 4). 2. Create sequence lists of S HV-PK , S HV-VL and S PK (the orange line in Fig. 4), and calculate σRng = STDEV of S PK . 3. Let S CREST be the crests of S PK sequence’s skyline. Detect S CREST from the sequence of S PK by Savitzky-Golay filtering [21] (kernel k = 2 and half width n = 3). 4. Remove S HV-PK from the sequence when S PK-VL,Rng < T HRES = a × σRng (the green dashed line in Fig. 4). The remaining sequence of S PK (the red marks in Fig. 4) gives the sequence of the slamming stress ranges. 5. The sequence of the times at which S HV-PK occurs responsible for the S PK-VL,Rng in the remaining list gives the sequence of the slamming occurred times.

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20 0 -20 -40 2100

2200

2300 Time (s)

2400

2500

Fig. 4. An example of slamming impact detection procedure.

For T HRES in procedure 4, a = 1.5 was chosen so is equivalent to 1/5 of the period of the hull vibration to the period of the hull vibration superimposed stress waveform reported in [22]. 3.2 Occurrence Frequency of Whipping Vibration Let nslm and nPW be the numbers of slamming impact and PW cycles in each SS. nslm is determined once the S PK sequence is obtained. nPW can be approximated by the inverse of T S . Let the coefficient of whipping interval C wh be given by Eq. (1):  Cwh = nslam nPW (1) C wh becomes 1.0 when slamming occurs in every PW.

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Coefficient of whippping interval - Cwh

0.80 0.70 0.60 Cwh

0.50 0.40 0.30

0.20 0.10 0.00

0

5

10

15

20

25

30 35 40 Sea State

45

50

55

60

65

Fig. 5. The variation in the coefficient of whipping interval C wh for short sea with various significant wave heights and ship speeds– head seas–.

Figure 5 shows the calculated C wh in head sea conditions for various sea states. Results are plotted by different color/mark for each ship speed V S . In Fig. 5, presented data includes data for V S beyond the performance curve (Fig. 1), which is not experienced in actual practice. H S increases from left to right for each V S . It is shown that C wh ranges from 0.15 to 0.6, and it becomes larger as H S and/or V S becomes larger. Let ‘limit speed’ V lim (H S ) be the maximum V S achievable for a given H S , which can be determined by the performance curve (Fig. 1)”. Voluntary and involuntary speed reduction is considered. Let C wh,lim (H S ) be C wh for V lim (H S ). The relation between H S and C wh,lim (H S ) is shown in Fig. 6. In this figure, it is shown that C wh,lim (H S ) decreasing with the increase in Hs, and it ranges between 0.15 and 0.20 for cases with H S > 5.5 m. This means that it can be approximated that a slamming happens once in every 4 to 5 PW under conditions chosen. Coefficient of whippping interval - Cwh

0.80 0.70

0.60 Cwh

0.50 0.40

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0.10 0.00 2.5

3.5

4.5

5.5

6.5 7.5 Hs (m)

8.5

9.5

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Fig. 6. The relation between significant wave height H S and coefficient of whipping interval at the limit speed C wh,lim (H S ) −head seas−.

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Avg. ΔSPK per Hs - Head Waves Stress range (MPa)

200 150 100 50 0 2.5

3.5

4.5

5.5

6.5 7.5 Hs (m)

8.5

9.5

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Fig. 7. The relation between significant wave height H S and slamming peak stress range at the limit speed S PK_RNG,lim (H S ) –head seas-.

Avg. ΔSPW per Hs - Head Waves

400

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350 300 250 200 150 100 50 0 2.5

3.5

4.5

5.5

6.5 7.5 Hs (m)

8.5

9.5

10.5

11.5

Fig. 8. The relation between significant wave height H S and primary stress range at the limit speed S PW,lim (H S ) –head seas–.

3.3 Slamming Impact Stress Range Let S PW be the stress range of PW stress waveform, and S PK,lim and S PW,lim be the average values of S PK and S PW for V lim (H S ). S PK is analyzed in the slamming detection analysis of Sect. 3.1, and S PW is calculated as the range of the low-pass filtered stress. The relation between H S and S PK,lim is shown in Fig. 7, and that between H S and S PW,lim is shown in Fig. 8. These figures show that both S PK,lim and S PW,lim are almost linearly in proportion to H S , and the ratio of S PK,li to S PW,lim ranges from 0.4 to 0.7, while the ratio of S PK to S PW (not the average) can be up to 1.0 in some cases, as shown in Fig. 9.

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HVSSW time history

HVSSW (MPa)

200 150 100 50 0 -50 25

75

125 Time (s)

175

225

Fig. 9. An example of HVSSW/HV and PW comparison experienced by 6500 TEU container ship (HS = 6.5 m, TS = 10.5 s, head waves).

3.4 Approximated HVSSW for HFE Fatigue Tests From NLHEA results on slamming impact’s occurrence frequency and its peak stress range, presented in Sects. 3.2 and 2.3, the characteristics of whipping HVSSW experienced by the target ship (6500TEU container), can be simplified as follows when the ship follows weather routing which prevents the speed reduction: 1. The slamming impact occurs once in every 4 to 5 PW cycles; 2. The maximum stress range due to slamming impact can be comparable to the PW stress range; 3. The HVSSW show approximately similar waveform regardless of significant wave height.

4 Fatigue Testing Machine 4.1 Eccentric Mass (EM) Driven PBV Testing Machines Yamada [20] developed a fatigue testing machine for plate bending, which is driven by a motor with Eccentric Mass (EM) attached to the specimen. Hereafter, this machine is called ‘EM-driven Plate Bending Vibration (PBV) testing machine’. For this machine, specimen’s elastic vibration can be superimposed onto PW load by attaching an additional vibrator with higher rotation frequency or intermittent hammering. Osawa et al. [8, 9] developed an EM-driven PBV testing apparatus which could emulate the whipping superimposed waveform. The free end of the test specimen was hit by a rotary hammer so that whipping vibration was induced. It was difficult to perform HFE fatigue tests with realistic whipping stress histories by using this testing machine because of the following shortcomings: it is difficult to change the ratio of frequencies of whipping vibration and PW loading; the impact stress’s waveform differs each time because the positional relation between the hammer and the test specimen are not strictly identical due to the specimen’s elastic vibration; and the hammer’s impulse cannot be changed continuously, and the adjustment range of PW’s load amplitude is narrow.

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4.2 Electric Exciter (EE) Driven PBV Testing Machine The problems on EM-driven testing machine can be solved by using a digital controlled high-power electric exciter (EE) in place of EM. In this study, an EE-driven PBV testing machine is developed. The excitation unit of this machine is consist of Asahi-seisakusho R-3030 exciter, APD-5000 power amplifier, and K2-Sprint vibration control unit. The PBV welded joint specimen is excited by a rod which comments the specimen and the exciter’s driving shaft. The driving rod is attached to the specimen in the same position as the gravity center of the vibrator (motor with EM) for PW excitation in the EM-driven tests. A load cell is inserted between the exciter’s driving shaft and the rod to measure the exciting force.

(a) entire view

(b) load cell

Fig. 10. EE-driven PBV testing machine used in intermittently whipping superimposed loading tests: photos.

This apparatus possesses the following features: a) Excitation up to 3,920 N in a frequency range up to is 5 kHz can be applied; b) The excitation force is always proportional to the input current, and this makes the actuator load follow the input waveform precisely; c) User-defined arbitrary waveforms which include frequency element higher than 100 Hz with the nominal stress range larger than 200 MPa, which cause fatigue failure of the weld, can be applied; d) It is easy to apply stress waveform with arbitrary amplitude, which is analogous to the reference whipping superimposed waveform. Figure 10 shows the developed EE-driven PBV testing machine.

5 Constant Amplitude Fatigue Tests 5.1 Test Specimens Fatigue tests were carried out for out-of-plane gusset welded joint specimens made from 12 mm-thickness AH32 ship structural steel plates. Size and shape of specimens are shown in Fig. 11. The left end of a specimen is fixed onto the frame base while the

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free end was excited by the EE (exciter). Fatigue cracks initiate at the weld toe near the main plate’s centerline. The crack propagates along with the weld bead during a certain period, then turns and propagates into the main plate. Let σ be the nominal bending stress range.

Fig. 11. The out-of-plane gusset welded joint specimen (unit: mm).

Let N b be the number of load cycles at which the crack propagation path turns into the main plate. Plate surface strains were measured by 4 strain gages arranged at points Ch1 ~ 4 in Fig. 11. The responses of Ch1 and Ch2 include the local stress concentration due to the weld. Responses of Ch3 and Ch4 can be treated as the nominal bending strains. 5.2 Test Results Osawa et al. [8, 9] performed Constant Amplitude (CA) fatigue tests by using EM-driven PBV testing machines. Figure 12 shows the changes in strain amplitudes with loading cycles measured in the test with σ = 98.0 MPa. Strains affected by the local stress concentration (Ch1 and Ch2) decrease substantially with cycles after a certain period while nominal strains (Ch3 and Ch4) increase slightly. σ is determined from the initial strain amplitudes of Ch3 and Ch4. ch1

ch2

ch3

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ch4

65% drop

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1000.0 800.0 600.0 400.0

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Nb=168,862

0.0 0

50,000

100,000

150,000

200,000

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Fig. 12. Changes in strain amplitudes during a constant amplitude loading test (σ = 98.0 MPa, EM-driven) [9].

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As shown in this figure, Ch1 strain amplitude showed about 65% drop at Nb (determined by visual inspection) in all CA tests carried out in this study. This strain decline is caused by the crack propagation and the decrease in ligament width. Yamada [14] showed that the failure lives of welded joint specimens measured by conventional (hydraulic) fatigue testing apparatus almost agreed with the number of cycles to the 65% drop of host spot strain range measured in PBV fatigue tests. CA EM-driven PBV fatigue test results are presented as an SN plot in Fig. 13.

Nominal equiv. stress range Δσ (MPa)

300

30 1.00E+04

Const. amp. (EM-driven) Regress. Const. amp. (EM-driven) Const. amp. (EE-driven)

1.00E+05 1.00E+06 Number of cycles to failure Nb

Fig. 13. Comparison of CA fatigue test results performed by EM- and EE-driven testing machines.

The relationship between σ and Nb is approximated by Basquin’s law, σ = C Nbm ,

(2)

where, C = 3.443 × 103 MPa, and m = −0.293. CA EE-driven PBV test results performed in this study are also plotted in the same figure (Fig. 13). EE-driven test results are within the variability range of EM-driven CA test results. Therefore, C and m for EM-driven tests are adopted in following fatigue damage calculations for EE-driven results. Below, Drf is calculated by modified Miner rule, that is, the fatigue limit is not taken into account. Equivalent stress range σeq,rf is calculated by the equations below:   N m σeq, rf = C , (3) Drf where C and m are the coefficient and exponent of Eq. (2), and N is the number of PW cycles.

6 Whipping Superimposed Fatigue Tests 6.1 Whipping Superimposed Waveform The whipping superimposed loading tests are performed by using the EE-driven testing machine developed in this study. The load is applied by performing ProportionalIntegral-Differential (PID) control using the load cell’s output as the process variable.

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It is needed to keep high linearity between the load (load cell’s output) and the nominal strain (Ch. 3 or Ch. 4 strain gage’s output) because the nominal strain was treated as the loading signal in our previous tests performed by the EM-driven machine. Preliminary tests are carried out, and it is found that the high linearity can be achieved when the waveform does not include a frequency component higher than 20 Hz. In Sect. 3.4, it was found that the characteristics of whipping HVSSW experienced the target 6500 TEU container is simplified as: a) The slamming impact occurs once in every 4 to 5 PW cycles; b) The maximum stress range due to slamming impact can be comparable to the PW stress range. In order to emulate this waveform under the restriction condition for the frequency component, the drive force’s waveform parameters listed in Table 1 are adopted. The ‘damping time’ in Table 1 is the number of PW in which the whipping wave’s amplitude shows a 99% drop. The waveforms of setpoint (SP) signal (load) and that of measured nominal strain (Ch. 3 strain) are compared in Fig. 14. It is shown that good linearity between load and strain is established, and the waveform’s characteristics are almost the same as those of the above mentioned simplified HVSSW. Hereafter, this strain waveform is called ‘reference whipping superimposed waveform’. Table 1. Waveform parameters of the reference whipping superimposed waveform Primary wave

whipping

ΔσLF (Mpa) 2.5

Freq (Hz) 4

Hull vibration ΔσHV (Mpa) 2.5

Freq (Hz) 16

Phase diff (x π rad) 0.27

Slaming intvl (cycle)

Damping time (cycle)

Δσ eq,rf

5

12

116.8

6.2 Constantly Whipping Superimposed Loading Tests Constantly whipping superimposed loading fatigue tests are carried out by using stress waveforms proportional to the reference waveform (Fig. 14). The loading conditions are listed in Table 2. In these tests, the number of PW cycles, N PW , is treated as N. The measured failure lives (N b ), fatigue damages Drf and equivalent stress range σeq,rf are shown in Table 2. Table 2 shows that DUF (Drf up to N b ) for constantly whipping superimposed loadings ranges from 0.824 to 2.296, and the average is 1.391. These constantly whipping superimposed EE-driven PBV fatigue test results are presented as an SN plot of σeq,rf and N b by open triangular marks in Fig. 13. This figure also indicates the CA SN curve (solid line).

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'whip_lr2.5_lf4_N100_hr2.5_hfmag4_ho.27_decay12_intvl5.csv'

2

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0

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12

12.5

13

13.5

14

14.5

time (sec)

(b) Measured nominal strain waveform

Fig. 14. Load and nominal strain waveforms of reference whipping superimposed load.

Table 2. Loading conditions, fatigue lives and accumulated damage of constantly whipping superimposed loading tests. Primary wave

Hull vibration

test ID

ΔσLF (Mpa)

Freq (Hz)

ΔσHV (Mpa)

Freq (Hz)

CW-1 CW-2 CW-3 CW-4

85.9 51.54 68.72 77.31

4

85.9 51.54 68.72 77.31

16

Slaming Damping Δσ eq,rf for Phase intvl time Nb(cycl) Nc(cycl) diff SUP cycles (cycle) (cycle) (x π rad) 152280 28,180 116.8 70.08 440020 154,080 0.27 5 12 93.44 602660 205,700 105.12 159020 22,920

Drf

Δσeq,rf for whole life(Mpa)

1.523 0.824 2.296 0.921

118.0 72.2 88.9 100.5

This result is similar to those observed in the constantly HV superimposed loading tests performed by EM-driven PBV testing machine in Osawa et al. [8, 9], that is, the fatigue life could be predicted with fair accuracy by RFCC in which modified Miner rule and CA SN curve are adopted. It is also shown that RFCC leads to a little conservative estimate of failure lives under conditions chosen.

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300

30 1.00E+04

469

Const. amp. Regress. Const. Whipping Intermittently Whipping

1.00E+05 1.00E+06 Number of cycles to failure Nb

Fig. 15. The relation between σeq,rf and Nb for constantly and intermittently whipping superimposed fatigue tests.

6.3 Intermittently Whipping Superimposed Loading Tests

Fig. 16. Load set applied in intermittently superimposed loading tests.

Intermittently whipping superimposed loading fatigue tests are carried out by using the EE-driven PBV testing machine. A combination of nSUP superimposed (SUP) cycles and subsequent nCA constant amplitude (CA) cycles is called ‘load set’ (see Fig. 16). ntotal = nSUP + nCA is the total number of cycles in one load set. Let σeq,rf in SUP cycles be σSUP and that in CA cycles σCA . In this study, σCA is chosen so that σCA nearly equals σSUP . This case is called “IF (Intermittently superimposed with Fixed σeq )” series. Let N 0 be N b estimated by CA SN curve (Fig. 13) for the case with σ = σ SUP . Load sets are applied repeatedly until the Ch1 strain amplitude shows a 65% drop. (nSUP , nCA ) are chosen so that nSUP /ntotal is 0.5, and ntotal /N 0 is 1/4. The loading conditions are listed in Table 3. In the same manner as the constantly superimposed cases, the number of PW cycles is treated as N in both SUP and CA cycles. The measured failure lives (N b ), DUF (Drf up to N b ) and σeq,rf for SUP cycles and that for whole life are shown in Table 3.

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Table 3. Loading conditions, fatigue lives and accumulated damage of intermittently whipping superimposed loading tests. Primary wave test ID

ΔσLF (Mpa)

IF-1 IF-4 IF-5 IF-6 IF-7

85.9 85.9 51.5 44.7 38.7

Hull vibration

Freq (Hz)

ΔσHV (Mpa)

4

85.9 85.9 51.54 44.668 38.655

Freq (Hz)

16

Slaming Damping Phase Δσ eq,rf for intvl time Nb(cycl) Nc(cycl) diff SUP cycles (cycle) (cycle) (x π rad) 116.8 136420 30,940 116.8 166520 25,600 0.27 5 12 70.1 512,960 92,400 60.7 977980 190,300 52.6 2693800 749,600

Drf

Δσeq,rf for whole life(Mpa)

2.172 1.885 1.067 1.003 1.833

135.2 122.4 74.5 60.6 53.7

Table 3 shows that Drf for intermittently whipping superimposed loadings ranges from about 1.00 to 2.172, and the average is 1.592. Intermittently whipping superimposed EE-driven PBV fatigue test results are presented as an SN plot of σeq,rf, and N b in Fig. 15 by open rectangular marks. This result is similar to those of the constantly HV superimposed loading tests in Sect. 6.2, that is, the fatigue life could be predicted with fair accuracy by RFCC in which modified Miner rule and CA SN curve was adopted. It is shown that RFCC leads to conservative estimates of failure lives, and it is likely that the mean Drf of intermittently superimposed cases is larger than that of constantly superimposed cases under conditions chosen. The above results were obtained from a small number of test data, and it is unclear whether similar results can be found in the generality of cases. The validity of the RFCC for HVSSW should be further examined by carrying out fatigue tests with realistic stress histories which emulate intermittent occurrence of whipping in ship structure. In such study, the EE-driven PBV testing machine developed in this study is a great aid in performing high-speed, low-cost whipping superimposed loading tests with arbitrary complicated waveforms.

7 Conclusions The stochastic characteristics of hull vibration (HV) superimposed stress waveform experienced by 6500 TEU container ship’s deck longitudinal was examined by Non-Linear Hydro-Elasticity Analyses (NLHEA) for various heading angle/ship speed/sea conditions. Based on these NLHEA results, HV superimposed stress waveform (HVSSW) which are applied in high frequency effect (HFE) fatigue tests are chosen. An electric exciter (EE)-driven PBV fatigue testing machine, which can apply various HVSSW in high speed, is newly developed. Performing HFE fatigue tests of welded joints using this EE-driven PBV testing machine, the validity of rainflow cycle counting (RFCC) for various HV superimposed loading cases are examined. As a results, the following are found: (1) From NLHEA results on slamming impact’s occurrence frequency and its peak stress range, the characteristics of whipping HVSSW experienced by the target ship, which follows weather routing, can be simplified as a) the slamming impact occurs

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once in every 4 to 5 PW cycles; b) the maximum stress range due to slamming impact can be comparable to the PW stress range; c) the HVSSW show approximately similar waveform regardless of significant wave height. (2) The ‘reference whipping superimposed waveform’, which has almost the same waveform’s characteristics as the simplified HVSSW derived from NLHEA, has been proposed. The developed EE-driven PBV fatigue testing machine can apply this ‘reference waveform’ to out-of-gusset welded joint specimens with good linearity between load and strain at high speed. (3) The fatigue life of out-of-gusset welded joints subject to constantly and intermittently whipping superimposed loadings could be predicted with fair accuracy by RFCC in which modified Miner rule and constant amplitude SN curve were adopted. It was shown that RFCC led to conservative estimates of failure lives, and it was likely that the fatigue damage up to the failure of intermittently superimposed cases was larger than that of constantly superimposed cases under conditions chosen. (4) The validity of the RFCC for HVSSW should be further examined by carrying out fatigue tests with realistic stress histories which emulate intermittent occurrence of whipping in ship structure. In such study, the EE-driven PBV testing machine developed in this study is a great aid in performing high-speed, low-cost whipping superimposed loading tests with arbitrary complicated waveforms.

Acknowledgment. This research was partially supported by the Ministry of Education, Science, Sports and Culture of Japan, Grant-in-Aid for Scientific Research (A), 2016-2018, 16H02432. The authors would like to acknowledge Dr. Koji Terai (Nippon Kaiji Kyokai), Mr. Tetsuya Nakamura and Mr. Fukuhiko Kataoka (Japan Marine United) for their valuable advice and discussions. The authors would like to acknowledge Mr. Kohei Higaki and Mr. Kota Tsunedomi (Osaka University) for their contribution in the fatigue tests.

References 1. Storhaug, G.: The effect of heading on springing and whipping induced fatigue damage measured on container vessels. In: 6th International Conference on Hydroelasticity in Marine Technology, Tokyo, Japan, pp. 299–310 (2012) 2. Storhaug, G., Vidic-Perunovic, J., Rüdinger, F., Holtsmark, G., Helmers, J.B., Gu, X.: Springing/whipping response of a large ocean-going vessel - a comparison between numerical simulations and full-scale measurements. In: Proceedings of the 3rd International Conference on Hydroelasticity in Marine Technology, Oxford, UK, pp. 117–131 (2003) 3. Storhaug, G., Moe, E., Holtsmark, G.: Measurements of wave-induced hull girder vibrations of an ore carrier in different trades. J. Offshore Mech. Arct. Eng. 129(4), 279–289 (2007). https://doi.org/10.1115/1.2746398 4. Moe, E., Holtsmark, G., Storhaug, G.: Full scale measurements of the wave induced hull girder vibration of an ore carrier trading in the North Atlantic. In: Transactions of The Royal Inst. of Naval Architects Conference on Design & Operation of Bulk Carriers, London, UK (2005) 5. Iijima, K., Itamura, N., Fujikubo, M.: Comparison of long-term damage in bulk carrier, VLCC and container carrier subjected to wave-induced vibrations. In: Proceedings of the 11th International Conference on Fast Sea Transportation FAST2011, Hawaii, USA, pp. 547–554 (2011)

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6. Fricke, W., Paetzold, H.: Experimental investigation of the effect of whipping stresses on the fatigue life of ships. In: Proceedings of the International Marine Design Conference, IMDC 2012, vol. 3, pp. 3–10 (2012) 7. Fricke, W., Paetzold, H.: Effect of whipping stresses on the fatigue damage of ship structures. Weld. World 58(2), 261–268 (2014). https://doi.org/10.1007/s40194-014-0111-5 8. Osawa, N., Nakamura, T., Yamamoto, N., Sawamura, J.: Experimental study on highfrequency effect on fatigue strength of welded joint by using plate-bending-vibration type fatigue testing machine. In: ASME. International Conference on Offshore Mechanics and Arctic Engineering, Volume 4B: Structures, Safety and Reliability, V04BT02A003 (2014). https://doi.org/10.1115/omae2014-23856 9. Osawa, N., Nakamura, T., Yamamoto, N., Sawamura, J.: Experimental study on fatigue strength of ship’s welded joint under intermittently high frequency superimposed loads considering actual encountered condition. In: International Society of Offshore and Polar Engineers, Hawaii, USA. ISOPE-I-15-829 (2015) 10. Jensen, J.J., Dogliani, M.: Wave-induced ship hull vibrations in stochastic seaways. Mar. Struct. 9, 353–387 (1996) 11. Malenica, S., Tuitman, J.T.: 3D FEM-3D BEM model for springing and whipping analysis of ships. In: Proceedings of the International Conference on Design and operation of Containerships, London (2008) 12. Kim, Y., Kim, K.H., Kim, Y.: Analysis of hydroelasticity of floating ship-like structures in time domain using a fully coupled hybrid BEM-FEM. J. Ship Res. 53(1), 31–47 (2009) 13. Iijima, K., Yao, T., Moan, T.: Structural response of a ship in severe seas considering global hydro elastic vibrations. Mar. Struct. 21, 420–445 (2008). https://doi.org/10.1016/j.marstruc. 2008.03.003 14. Kim, J.H., Kim, Y., Yuck, R.H., Lee, D.Y.: Comparison of slamming and whipping loads by fully coupled hydroelastic analysis and experimental measurement. J. Fluids Struct. 52, 145–165 (2015). https://doi.org/10.1016/j.jfluidstructs.2014.10.011 15. Kim, J.H., Kim, Y.: Numerical analysis on springing and whipping using fully-coupled FSI models. Ocean Eng. 91, 28–50 (2014). https://doi.org/10.1016/j.oceaneng.2014.08.001 16. Kim, Y., Kim, J.H.: Benchmark study on motions and loads of a 6750-TEU containership. Ocean Eng. 119, 262–273 (2016). https://doi.org/10.1016/j.oceaneng.2016.04.015 17. Papanikolaou, A.D., Schellin, T.E.: A three-dimensional panel method for motions and loads of ships with forward speed. Ship Technol. Res. 39, 145–156 (1992) 18. Tamaru, H.: About optimum route by weather routing. In: Proceedings of the Japan Society of Naval Architects and Ocean Engineers Spring Meeting (JASNAOE), Okayama, Japan, 2016A-OS4-2 (2016). (in Japanese) 19. De Gracia, L., Osawa, N., Iijima, K., Fukasawa, T., Tamaru, H.: A study on the stochastic aspects of the whipping vibrations in a container ship. In: ASME. International Conference on Offshore Mechanics and Arctic Engineering, Volume 3: Structures, Safety, and Reliability, V003T02A027 (2018). https://doi.org/10.1115/omae2018-77888 20. Yamada, K.: Some new approaches to fatigue evaluation of steel bridges. Steel Struct. 6, 319–326 (2006) 21. Savitzky, A., Golay, M.J.E.: Smoothing and differentiation of data by simplified least squares procedures. Anal. Chem. 36(8), 1627–1639 (1964). https://doi.org/10.1021/ac60214a047 22. Toyoda, K., Matsumoto, T., Yamamoto, N., Terai, K.: Simplified fatigue assessment considering the occurrence probability of hydro-elastic response in actual sea state conditions. In: Proceedings of the 6th International Conference on Hydroelasticity in Marine Technology, Osaka, Japan, pp. 367–375 (2012)

Determination of a Methodology for the Fatigue Strength Evaluation of Transverse Hatch Coaming Stays on Container Ships Chi-Fang Lee1(B) , Yann Quéméner1 , Po-Kai Liao1 , Kuan-Chen Chen1 , and Ya-Jung Lee2 1 Research Department, CR Classification Society, Taipei, Taiwan (ROC)

[email protected] 2 Department of Engineering Science and Ocean Engineering,

National Taiwan University, Taipei, Taiwan (ROC)

Abstract. This study evaluated the fatigue strengthening of the foremost transverse hatch coaming stays of 6 sister container vessels of 2650 TEU capacity that suffered fatigue cracking after 1.0 year operation on a North Pacific route. For that, a simplified method was proposed to validate the fatigue strengthening by providing a target hot spot stress reduction factor as a function of the ship lifetime at the crack occurrence. Afterwards, further seakeeping and structural analyses enabled conducting the spectral fatigue analyses of the considered stays’ original and strengthened designs and, by then, to validate the fatigue strengthening of the stays and the simplified approach. The fatigue driving loads, as well as the effect of the long-term operational profiles uncertainties on the fatigue were also discussed on the basis of the spectral fatigue analyses. Finally, crack growth analyses confirmed the criticality of the examined stays with regards to rapid fatigue cracking when the fatigue damage is not properly considered at the design stage. Keywords: Container ship · Transverse hatch coaming · Fatigue

1 Introduction To fit the maximum of ‘boxes’ in their holds, the container ships include numerous structural discontinuities that result in as much hot spot areas prone to fatigue cracking. Therefore, for all structural details referenced as critical, Class rules provide fabrication standards and specific fatigue life assessment methodologies, so that the fatigue can be properly considered at the design stage. However, fatigue cracking in container ships is still an issue to be tackled that requires the continuous development of the design and fabrication methods. Recently, significant research efforts focusing on large container vessels are put on quantifying the influence of the springing and whipping phenomena on the fatigue. Besides, researchers are also addressing the issue of relieving the uncertainties with regards to long term operational profile, e.g. operation area, loading conditions and associated routing under weather and operation constraints. © Springer Nature Singapore Pte Ltd. 2021 T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 473–491, 2021. https://doi.org/10.1007/978-981-15-4672-3_30

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This study considered the case of 6 sister vessels of 2650 TEU capacity in which, despite an extensive fatigue life evaluation at the design stage, cracks appeared early in their operational life and propagated rapidly in the foremost bays’ transverse hatch coamings stays L2 to L11 at Fr.206 and L2 to L8 at Fr.210. Figure 1 shows an overview of the cracked stays locations in the vessel. The ship-owner reported that, for all of the 6 vessels, the cracking appeared 1.0 year after the beginning of their operations on the North Pacific route from Taipei to Los Angeles and resulted in crack length ranging between 50 mm to 350 mm when detected by the crew. Besides, waiting for the stays upgrade on the yet uncracked vessels, the crews kept a close watch on those critical areas and reported that some previously undetected cracks appeared and propagated extensively in the stays after a voyage with particularly adverse weather conditions. Based on those observations, this study proposed a method to validate the fatigue strengthening of new stays and examined the crack initiation by damage accumulation and the subsequent crack propagation through advanced numerical methods involving seakeeping and finite element analyses.

Fig. 1. Overview of the considered cracked transverse coamings stays in the vessel.

This study consists of three main sections. The first section presents a simplified methodology to validate the fatigue strengthening of the cracked stays. The second section presents the fatigue damage accumulation assessment conducted to validate the fatigue strengthening. The third section discusses on the crack driving loads, on the ship operational profile effect on the fatigue predictions and on the speed of the crack propagation.

2 Fatigue Strengthening 2.1 Hot Spot Stress Reduction Factor In practice, when a fatigue crack is detected in a structural detail, the most common corrective action consists in welding the crack [1] and thereby resetting the fatigue life

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of the structural hot spot. However, if the ship operational profile remains unchanged, the crack is expected to occur again after a period of time equivalent to the ship lifetime at the crack appearance (Tcrack ). For a crack occurring after half the ship design life (TD ), the welding repair might be deemed satisfactory since the cracking would not occur before the end of the design life. However, for a crack occurring before half the ship design life, the welding repair will not be sufficient to prevent the crack from reappearing within the design life. For that early cracking case, the redesign of the structural detail should be preferred in order to reduce the hot spot stress and, by then, increase its fatigue strength. This section proposed a simplified approach to quantify the required hot spot stress reduction of the new structural detail design, so that its remaining fatigue life would at least meet the ship design life. The fatigue damage for a single slope S-N curve can be calculated with Eq. (1) as provided in IACS [2]. Although most of the S-N curves are doubly sloped, this single slope assumption of the S-N curve is deemed satisfactory in the case of an early crack appearance for which most of the fatigue damage is consumed in the low cycle part of the S-N curve.   m σHS m ND (1) · ·  1 + D= K2 (lnNR )m/ξ ξ where σHS is the long term hot spot stress range at the reference probability level of exceedance of 1/NR that is set to 10−2 and a 2-parameter Weibul function () with a shape parameter ξ taken as 1.0. K2 and m are the S-N curve parameters, and ND is the number of wave cycles experienced by the ship during the design fatigue life that can be expressed as: ND = f0 × TD × 365.25 × 24 × 3600/4log(L)

(2)

where TD is the ship design life, f0 is the factor taking into account time in seagoing operations excluding time in loading and unloading, repairs, and L is the ship length. For validating the fatigue strengthening, this study employed a simplified approach that expressed the ratio of target fatigue damage to achieve the remaining design life of the vessel (Dtarget ) to the fatigue damage at the crack occurrence (Dcrack ) in the Eq. (3). mo  mn Dtarget ND,target × K2,o × (1 + mn ) σHS,target × lnNR,crack · = m  mo Dcrack ND,crack × K2,n × (1 + mo ) σHS,crack × lnNR,target n

(3)

where K2,o and mo are the S-N curve parameters applicable to the hot spot of the original stay design and K2,n and mn are the S-N curve parameters applicable to the hot spot of the new stay design, and where σHS,crack and σHS,target are the reference long-term hot spot stress range at the probability level of 1/NR,crack and 1/NR,target , respectively, that results in Dcrack and in Dtarget , respectively, as calculated by Eq. (1). And with Dtarget = TD /(TD − Tcrack )

(4)

Dcrack = TD /Tcrack

(5)

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By assuming that the ship operational profile would remain unchanged after the crack repair and, by then, the long-term distribution of load cycles and loading conditions, the required hot spot stress range reduction to ensure a remaining life (TD − Tcrack ) that meets the design life (TD ) can be calculated by Eq. (6). mn σHS,target mo σHS,crack

=

Tcrack K2,n × (1 + mo ) · (lnNR )(mn −mo ) · TD − Tcrack K2,o × (1 + mn )

(6)

In the case where the S-N curves applicable to the hot spots of the original and of the new design of the stays are the same, the Eq. (6) can be further simplified as expressed in Eq. (7): σHS,target = σHS,crack



Tcrack TD − Tcrack

1/m (7)

In the case where the S-N curves applicable to the hot spots of the original and of the new design of the stays are different, the reference hot spot stress range (σHS,crack ) for a probability of exceedance of 1/NR,crack , here set to 10−2 , that results in a given damage of the original design (i.e. Dcrack = TD /Tcrack ) is to be deduced first as per Eq. (1), so that the target hot spot stress range can be evaluated as expressed in Eq. (8).  σHS,target =

(m −m ) Tcrack K2,n · (1 + mo )  mo · lnNR,crack n o · σHS,crack · TD − Tcrack K2,o · (1 + mn )



1 mn

,

(8) and the hot spot stress range reduction factor formulation in Eq. (6) becomes Eq. (9). σHS,target = σHS,crack



Tcrack K2,n × (1 + mo ) · · TD − Tcrack K2,o × (1 + mn )



σHS,crack lnNR,crack

(mo −mn )  m1n (9)

For the original stay, the crack initiated from a hot spot located at the edge of a machine-cut soft toe. The FAT125 S-N curve with a slope set to m = 3.5 is appropriate for this type hot spot to evaluate the fatigue damage as provided by the IACS [2] based on IIW recommendation [3]. A first fatigue strengthening strategy has consisted in increasing the soft toe radius and consequently, the same FAT125 (m = 3.5) S-N curve was thus applicable. Another fatigue strengthening strategy was considered in line with IACS recommendation [4] that consisted in adding a face plate at the edge of the stay, so that the hot spot would now be located at the welded joint between the face plate and the deck. Consequently, the FAT90 (m = 3.0) S-N curve was selected because commonly used for the fatigue assessment of welded connections by using the stress interpolation technique [0.5t; 1.5t] as provided by IACS [2] based on Maddox [5] study. Figure 2 illustrates the S-N curve FAT125 (m = 3.5) and FAT90 (m = 3). Then, the target stress reduction factor for crack occurrence at various ship lifetimes was evaluated as per Eqs. (7) and (9) for the two fatigue strengthening strategies, as shown in Fig. 3. It can be observed that, for the ‘Enlarged soft toe radius’ strengthening, the target stress reduction factor was approximately 13% larger than that produced for

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Fig. 2. S-N curves applicable to the fatigue assessment of the original and the fatigue-strengthened stay.

the ‘Additional face plate’ strengthening, and as such, this first strategy looked more efficient to strengthen the stay against fatigue cracking. Over all the considered ships, the earliest crack occurred after 1.0 year operation. Therefore, to produce accordingly a satisfactory strengthening against fatigue cracking, the target hot spot stress reduction factor to be achieved by the new stay design should be approximately of 40% for a redesign involving a larger soft toe radius, or 28% for a redesign involving an additional welded face plate.

Fig. 3. Target hot spot stress reduction factor to be achieved by the fatigue strengthening.

2.2 Hot Spot Stress Reduction Evaluation This study employed finite element analyses (FEA) to evaluate the hot spot stress reduction between the original and the redesigned stays. For that, an identical nominal loading condition will be applied on both stays FE model, so that the produced hot spot stresses can be consistently compared. For those FE analyses, the orientation of the load must reflects the realistic load configuration that maximized the stress at the hot spot and, as such, contributed the most to the fatigue. For the hatch cover ultimate strength assessment, the IACS [6] requires to consider that the on-deck containers longitudinal component of the inertia loads transmitted to the hatch cover is carried by the stoppers, while the

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bearing pads, thereby assumed frictionless, would only transfer the vertical component of the inertia loads. However, in Fig. 1, the orientation of the crack path helped identifying the direction of the principal stresses driving the fatigue that, by definition for a mode I of crack opening, are perpendicular to the crack direction. This crack orientation was thus induced by an in-plane bending of the stay driven by the longitudinal inertia load fluctuations transmitted from the hatch cover to the top of the transverse coaming, so that non-negligible friction forces were transmitted through the bearing pads. Likewise, the IACS [4] provides several recommendations for the strengthening of transverse hatch coaming stays that suffered fatigue cracking, and identifies the source of the damage as being due to an “insufficient consideration of the horizontal friction forces in way of the bearing pads for hatch cover”. Finally, it worth being noted that, as located inside the line of hatch opening, the stays are not exposed to the global bending and torsion of the hull girder, and those effects were here disregarded accordingly. In addition, the IACS requirements [6] for the hatch covers ultimate strength evaluation provide design loads that, at the considered foremost container bays (see Fig. 1), set an extreme cyclic heave acceleration amplitude of 0.45 g’s and a longitudinal acceleration of 0.2 g’s. The combination of those vertical and longitudinal accelerations was thus used in this study as a first estimate to determine the orientation of the crack driving loads, here set to Fx = 44%Fz , to be applied for the hot spot stress reduction assessment. The stays local FE models were then made of shell elements with a very fine mesh size set to the plating thickness, so called ‘t × t’, and the steel material represented as linear elastic. Figure 4 shows the Finite Element models of the original stay and those of the strengthened stay designs. This model was deemed small enough to be rapidly built and sufficiently large to reduce the effect of the boundary conditions on the hot spot stress. Finally, a nominal load with an orientation of Fx = 44%Fz was applied at the top plate in way of the stay, where the bearing pads are located, and an amplitude arbitrarily set so that, for the original stay, a hot spot stress of 100 N/mm2 was obtained that enabled a clearer observation of the stress reduction factor (%) achieved by the strengthened stay designs. All the strengthened stay designs employed the same 11 mm mild steel plating as for the original design, so that new hot spots would not arise by mounting stays significantly stiffer than the adjacent hatch coaming structural members. Figure 4 shows, for the various designs of stay, the main modifications and the resulting hot spot stress produced by FEA. It can be observed that by enlarging the soft toe radius to 400 mm, the hot spot stress reduction factor (92%) did not meet the target value of 40% (see Fig. 3), whereas by adding a face plate, the hot spot stress reduction factor of ‘FP2’ (27%) did meet the target value of 28%. In addition, as recommended by the IACS [4] to improve the fatigue strength, a full penetration butt-weld was provided at the face plate connection to the deck.

3 Fatigue Life Assessment 3.1 Ship Operational Profile The 6 sister vessels were chartered on a North Pacific route between Taipei and Los Angeles with a time in seagoing condition corresponding to 80% of their operations.

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Fig. 4. Local FE model of the Stay-L2 at Fr.206 (left) and hot spot stress reduction assessment for the strengthened designs (right).

The average speed was set to 80% the vessel design speed of 22.7 knt and two loading conditions were considered by this study corresponding to 75% and 100% the total deadweight capacity. Table 1 lists the ship hydrostatic properties for those two loading conditions. The actual on-deck containers loading conditions of the foremost bays were not made available, but the loading manual indicated that, between 75% to 100% the total design capacity, the on-deck container mass distributions at those bays would be identical with an average mass of 24 t per 40’ containers. For this study, it was thus assumed that the bays were commonly fully loaded. Table 2 lists the details of the related on-deck container bay arrangement. Table 1. Ship hydrostatic properties. 75% DWT 100% DWT Displacement (t) 40516.1 LCG (m)

98.32

45880.8 97.54

VCG (m)

13.88

14.49

Draft - MS (m)

10.48

11.42

0.27

0.53

Trim (deg)

3.2 Ship Motions and Structure Analyses The structural model of the foremost hatches was made of shell elements with a mesh size set to the stiffener spacing (here ~600 mm) and the steel material was represented

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Bays (see Fig. 1)

14 15 13

10 11 09

06 07 05

02 03 01

Containers arrangement on hatch covers

4 Tiers × 11 Rows

4 Tiers × 11 Rows

4 Tiers × 7 Rows

2 Tiers × 5 Rows

Mass of bay (t)

1056

1056

672

240

XG (m)

140.6

154.7

170.5

184.6

ZG (m)

24.4

24.4

24.4

24.5

as linear elastic. In addition, at the hotspots, the transverse coaming stays on Fr.206 and Fr.210 were represented using the very fine mesh ‘t × t’ technique previously described in Sect. 2.2. Figure 5 shows the Finite Element model of the foremost hatches that extended vertically from the 2nd deck, with the container mass carried by the corresponding hatch covers represented through mass elements (red squares) located at the center of gravity provided in Table 2, and the interpolation elements (blue lines) that were employed to distribute the inertia loads from the mass elements to the corresponding hatch covers bearing pads. Finally, the aft and fore ends of the model, as well as the 2nd deck were set as simply supported. Though bigger than the local models presented in Fig. 4, this model was deemed small enough to be rapidly built and sufficiently large to reproduce realistically the inertia loads distribution on the hatch coaming and to reduce the effect of the boundary conditions on the hot spot stress. In addition, the sliding of the hatch cover was here disregarded, as discussed in Sect. 2.2, by the use of the interpolation elements that transmitted the whole of the longitudinal component of the inertia loads through the bearing pads. A first FE model was created based on the original design of the stays, and then a second version was made by replacing the critical stays-L2, -L5 and -L8 at Fr.206 by the strengthened stay design ‘FP2’ presented in Fig. 4. The structural response was evaluated by static analysis conducted with NX NASTRAN [7] and the loads in terms of ship motions and accelerations were transferred from seakeeping analyses carried out through Hydrostar [8]. First, the structural response in stillwater condition was evaluated by FEA. Table 3 lists the mean stress at the hot spots of each stay that appeared to be always in tension since this loading condition resulted in the inward deflection of the transverse hatch coamings at Fr.206 and Fr.210. The mean stress effect was thus disregarded in the next fatigue life assessment, since it is especially relevant for compressive mean stress. The seakeeping analyses were then carried out using the potential flow theory CFD software Hydrostar [8] for the two considered loading conditions (see Table 1). The ships motions were evaluated for 36 wave headings from 0° to 350° and for 40 wave frequencies from 0.05 rad/s to 2.0 rad/s. Figure 6 shows, for both loading conditions, the produced response amplitude operators (RAO) of ship heave acceleration and pitch motions. It appeared that the largest motions were obtained for the head sea (here, h = 180°) and quartering seas, and that the largest motions were induced for the loading condition set to 75%DWT. Therefore, since the foremost bays’ hatch covers carried the

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Fig. 5. Foremost hatches FE model.

Table 3. Hot spot mean stress at the transverse coaming stays. σmean (N/mm2 ) Fr.206 L2

Fr.210 L5

L8

L11 L14

L2

L5

L8

Original stays

34.5 20.3 25.5 8.8

−16.4 25.5 11.8 9.9

Strengthened stays

19.8 16.6 15.2 –

–

–

–

–

same on-deck container mass for both loading conditions, the inertia loads produced by the 75%DWT motions will be larger than for that of the 100%DWT, and, accordingly, the fatigue lives obtained for the 75%DWT motions were anticipated to be the lowest. After transferring the loads to both the original and the strengthened stays FE models, the finite element analyses were conducted and the RAO of hot spot stress were then extracted as needed for the spectral fatigue assessment. Figure 7 (left) shows the RAOs of hot spot stress of the stay-L2 at Fr.206 for a loading condition set to 75%DWT, and (right) the corresponding peak values of the original and the strengthened stays for loading conditions set to 75%DWT and 100%DWT. It can be observed that the largest stress RAOs occurred in head (here, h = 180°) and quartering sea, while rapidly dropping for beam sea and beyond until following sea (here, h = 0°). It appeared also that, for the strengthened stay, the peak values dropped significantly compare to those of the original stay, up to 3.6 times decrease for the head sea and both loading conditions.

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Fig. 6. RAOs of ship heave acceleration (top) and pitch motion (bottom) and corresponding peak values’ wave heading distribution.

Fig. 7. RAO of hot spot stress of the original stay-L2 at Fr.206 for 75%DWT (left), and corresponding peak values’ wave heading distribution for the original and the strengthened stays for loading conditions set to 75%DWT and 100%DWT (right).

3.3 Fatigue Damage Accumulation Spectral fatigue analyses (SFA) were conducted to evaluate the fatigue damage for 25 years of seagoing operations. The 3-h short-term stationary irregular sea states was

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described by a Pierson-Moskowitz wave spectrum combined to a cosine 2s wave spreading spectrum as recommended by IACS [9] to represent a fully-developed sea that corresponds to most of the sea-states encountered by the vessels. For the North Pacific route (i.e. ‘Taipei to Los-Angeles’) on which the ships were chartered, and the South China Sea route (i.e. ‘Taipei to Singapore’) on which the ships is presently chartered (see Fig. 8), the long-term wave environments were derived from the wave scatter diagrams provided by the Bmt [10] for various ocean areas. The equivalent wave scatter diagram was produced as the sum of the scatter diagrams corresponding to the ocean areas crossed by the ships along the shortest path and weighted by the fraction of time in each area which was taken as the ratio of the crossed distance in each area to the total distance. The encountered wave headings were taken as uniformly distributed. This approach disregarded thus the effect of the speeds and heading variations induced by the routing under weather and operations constraints. Finally, the S-N curves presented in Fig. 2 were used to evaluate the accumulated fatigue damage of the original and the ‘FP2’ fatigue-strengthened stays (see Fig. 4).

Fig. 8. Considered trading routes drawn on Bmt’s ‘Map of Area Subdivisions’ [10].

Table 4 lists the evaluated fatigue life of each stay for the two loading conditions and the North Pacific route long-term wave environments. It can be observed that the most critical fatigue life was obtained for the hot spot of the stay-L5 supporting the transverse hatch coaming at the Fr.206 with a fatigue life comprised between 0.2 and 0.3 years for loading conditions set to 75% and 100% the deadweight capacity, respectively. The evaluated fatigue lives were thus significantly lower than the 1.0 years actually observed in terms of cracking on board those ships. Those large deviations with the observations can be explained by the lack of operation data needed to produce a realistic fatigue assessment, especially the typical loading conditions and associated on-hatch-cover container mass distribution, and possibly the weather and operation routing constraints. Likewise,

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the evaluated fatigue of the strengthened stay-L5 only reached 4.7 and 9.0 years for loading conditions set to 75% and 100% the deadweight capacity, respectively. However, the fatigue strengthening can be validated based on the fatigue strengthening ratios of the evaluated fatigue life produced for the strengthened stays to that obtained for the original stays. In Sect. 2, the target fatigue strengthening ratio was of 24 (i.e. a fatigue life increase from 1 year at the observed cracking to 24 years to achieve the remaining design life). In Table 4, it appeared that, for both loading conditions, the minimum fatigue strengthening ratio was of 23.5 (L5 at 75%DWT) and reached 38.3 (L2 at 100%DWT). Therefore, the fatigue strengthening was satisfactory and the simplified approach presented in Sect. 2 to confirm the fatigue strengthening was thus validated by the predictions of the spectral fatigue analyses. Table 4. Evaluated fatigue life of the stays for the North Pacific route. Fr.206 75%DWT

100%DWT

Fr.210

L2

L5

L8

L11

L14

L2

L5

L8

Original stay (years)

0.4

0.2

0.9

14.2

>50 yrs

1.2

0.8

2.2

Strengthened stay (years)

14.6

4.7

27.3

–

–

–

–

–

Strengthened/Original (–)

36.5

23.5

30.3

–

–

–

–

–

Original stay (years)

0.8

0.3

1.8

20.4

>50 yrs

2.5

1.4

3.5

Strengthened stay (years)

30.6

9.0

53.4

–

–

–

–

–

Strengthened/Original (–)

38.3

30.0

29.7

–

–

–

–

–

4 Discussions 4.1 Fatigue Driving Loads Figure 9 shows the fatigue damage distribution of the original stay-L2 at Fr.206 as it relates to the wave heading (left) and to the probability level of loads (right). It can be observed that the maximum damage contribution was produced for the head sea (here, 0°) and for long-term loads with a probability level of approximately 10−2 . An equivalent design wave (EDW) was thus derived from the RAO of hot spot stress in head sea (see Fig. 7, h = 180°) with a long-term value at a probability level of 10−2 . Table 5 lists the EDW parameters. Figure 10 shows the corresponding evolution, over one EDW period, of the on-deck containers inertia loads applied on the transverse hatch coaming at Fr.206. It appeared that the longitudinal component of the applied inertia loads (Fx ), at its extremes, was 28% as large as the cyclic part of the vertical component of loads (Fz,cyclic ; here disregarding

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Fig. 9. Fatigue damage distribution vs. wave heading (left) and vs. probability level of loads (right) for the stay-L2 at Frame206.

Table 5. Equivalent design wave maximizing the hot spot stress. Loading conditions

Long-term hot Wave spot stress range heading at a P = 10-2 (RAO)

Peak value and associated wave frequency of RAO of hot spot stress

LT_σHS

ωmax

EDW amplitude = LT_σHS 2×RAO_σHS,max

h

RAO_σHS,max

N/mm2

deg

N/mm2

AEDW

rad/s

m

75%DWT

316

180

107

0.50

1.48

100%DWT

263

180

87

0.55

1.51

the static weight), that is significantly lower than the load ratio of 44% derived from the IACS recommendation in Sect. 2.2. In addition, it can be observed that Fx , at the extremes, fluctuated between 5% and 10% the whole of the vertical component of loads (Fz,total ; here considering the static weight) that is of the same order as the coefficient of friction of 0.1 for the steel-steel contact in dry condition as provided by the IMO [11]. It can be argued that the coefficient of friction of the bearing pads that are made of specific steel alloys (here, Hardox® [12]) to reduce the abrasion and thus the friction, should be lower than 0.1. Therefore, the foremost hatches FE model (see Fig. 5) would generate conservative results by transmitting the whole of the longitudinal component of the inertia loads acting on the hatch cover, whereas it should be limited by the sliding. 4.2 Wave Environments After the stays upgrade, the vessel continued operating on the North Pacific route for about 2 years before being redirected to a South China Sea route for the last 6 years, and yet no crack has been detected, thereby comforting the effectiveness of the strengthening. Figure 11 shows the evaluated fatigue of the strengthened stays for the two trading routes.

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Fig. 10. Evolution over one EDW period of the on-deck containers load applied on the transverse hatch coaming at Fr.206.

It can be observed that the evaluated fatigue life produced for the South China Sea route was approximately 2.5 times that obtained for the North Pacific route. Therefore, it is believed that the combined actions of strengthening and milder wave environment will prevent any crack from reoccurring before the end of the design life. However, it can also be observed that the evaluated fatigue lives produced for the North Atlantic wave environment, commonly considered by the Class rules for unrestricted operations, were approximately 0.6 times that obtained for the North Pacific route, thereby reducing the fatigue strengthening ratios listed in Table 4 to 14.1 (L5 at 75%DWT) and 23.0 (L2 at 100%DWT) that would be less than the target ratio of 24 (i.e. increase the 1.0 year fatigue life to the 24 years remaining design life). Therefore, for North Atlantic operations, the present strengthening would not have been sufficient. The definition of the trading route and the associated weather and operation routing can thus greatly influence the fatigue life and the assumption of unchanged operational profile made in Sect. 2.1 for assessing the fatigue strengthening must be carefully considered when using this approach.

Fig. 11. Evaluated fatigue life of the strengthened stays.

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4.3 Crack Propagation As mentioned in Introduction, the crew reported that, under severe conditions, ~50 mm long cracks appeared in the considered stays after a voyage lasting approximately 1 week. This study evaluated the crack propagation from an undetected level of crack length, here arbitrarily set to 5 mm, until the length of 350 mm observed in Fig. 1. For that, the crack was idealized as a sharp tipped crack that propagated at a crack growth rate calculated by Eq. (10). da = A(K)m dN

(10)

where K is the stress intensity factor range evaluated at the tip of the crack of length a, and A and m are constants that depends on the material and the applied conditions including environment and cyclic frequency. Figure 12 shows the crack growth law and the associated constants recommended by British Standard [13] for unwelded steels and operations in air as ensured by the protective coating good condition of the stays (see Fig. 1).

Fig. 12. Fatigue crack growth law for unwelded steel and in air environment.

The stress intensity factor was evaluated by finite element analyses conducted on a local finite element model of the stay including a crack as shown in Fig. 13. The initiation point and the direction of the crack were defined accordingly to that observed in Fig. 1. The cracked stay FE model was made of quadrilateral shell elements with centered nodes, referenced as S8R5 in Abaqus [14]. The meshing produced concentric rings of element centered on the crack tip. The elements in the innermost ring were then degenerated to triangles by merging the three nodes of one S8R5 element edge as provided by Abaqus to extract stress intensity factor as per the linear elastic fracture mechanic formulation. Besides, the crack propagation is a nonlinear process that highly depends on the loads history. A realistic approach would consist in simulating the crack propagation for various sequences of severe sea-states actually observed on the considered route. However, the

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Fig. 13. FE model of the original stay including a 150 mm long crack.

present study proposed to employ the statistical long-term load distribution produced by the spectral fatigue analysis previously described in Sect. 3.3. The method consisted then in scaling the 25-year long-term load distribution down to a 1-week load cycles block where the amplitude of the loads will increase during 3.5 days and decrease during the remaining 3.5 days, thereby reproducing in a statistical manner a ‘calm-severe-calm’ weather sequence met during a voyage. The 1-week load cycles block was then repeated until the crack met the target length of 350 mm. Therefore, for various crack lengths, the RAOs of stress intensity factor was obtained using a top-down scheme that applied the nodal displacements at the boundaries of the stay-L2 (see Fig. 13) previously produced by the FEA of the foremost hatches FE model (see Fig. 5) for each wave heading and frequency. A spectral analysis was then conducted to obtain the long-term distribution of stress intensity factor range. The load cycles block construction was then simplified by dividing the long-term distribution of the stress intensity factor range (K) for each crack length by that of the stay deflection range (Ux, see Fig. 13) that resulted in a stress intensity factor per mm of stay deflection (K/Ux) which the evolution as a function of the crack length was unified for any level of loads with a very good precision as shown in Fig. 14 (right). The 1-week load cycles block construction (see Fig. 14, middle) was thus derived from the long-term distribution of stay deflection range (see Fig. 14, left). The crack growth was then evaluated as per Eq. (10) at every load cycle illustrated in Fig. 14 (middle). Figure 15 shows the evaluated crack growth and the corresponding weekly crack growth rate for various wave environments, when assuming an initial undetected crack length of 5 mm. For the North Pacific route wave environment, it appeared that a 50 mm crack length was reached after simulating approximately 8 weeks of operations for both loading conditions. Those predictions can only be considered as the average crack propagation prediction over a sufficiently large period and not as the realistic status of the crack length at a specific time. Especially, during the first 75 mm of the crack propagation, the weekly crack growth rate varies rapidly. It can thus be argued that if the vessel had encountered successive or extensive severe weather conditions within a concentrate period that corresponds to the beginning of the crack propagation,

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Fig. 14. Long-term distribution of stay-L2 deflection (left) and the associated 1-week load cycles blocks sequence (middle), and the stress intensity factor relation to the crack length (right).

the crack growth would have been much accelerated, and a 50 mm to 100 mm crack could appear within 1 to 2 weeks as previously observed by the crews. The use of a more realistic storm model such as that recommended by DNVGL [15] calibrated on actual seasonal data would be necessary to further discuss on the crack propagation. However, it can already be confirmed that the integrity of this kind of structural detail can be dangerously and rapidly compromised if the fatigue damage is not correctly taken into account at the design stage.

Fig. 15. Evaluated crack growth (left) and the corresponding weekly crack growth rate (right) for various wave environments and ship loading conditions.

Besides, in Fig. 15, it can be observed that, contrary to the fatigue damage, the ship loading conditions had a limited effect on the fatigue crack growth, whereas the considered wave environment still had a significant influence on the predictions. Finally, with respect to the strengthened stay ‘FP2’, it is anticipated that the face plate will intrinsically greatly reduce the potential crack propagation rate since there would be three crack-tips to resist to the crack growth (i.e. one in the stay plus two in the face

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plate propagating on each side of the stay along the weld with the deck) instead of one crack-tip for the original design.

5 Conclusions This study evaluated the fatigue strengthening of the transverse hatch coaming stays of 6 sister container vessels that suffered cracking after 1.0 year operation on a North Pacific route. At first, this method proposed a simplified method to validate the fatigue strengthening by providing a target hot spot stress reduction factor relating to the ship lifetime at the crack occurrence. This method assumed that the operational profile of the ship remained unchanged after the stay upgrade, that consisted in adding a face plate at the edge of the stays, and considered in view of the crack orientation that a non-negligible part of the longitudinal component of the on-deck container inertia load was transferred through the hatch covers’ resting pads that are commonly considered as frictionless. Then, spectral fatigue assessment (SFA) were conducted by transferring the ship motions and accelerations on two FE models of the foremost hatches that included separately the original and the strengthened design of stays. The produced fatigue lives of the original stays were approximately of 0.2 years and as such did not reflect the observed fatigue life. However, the produced fatigue strengthening ratio of the fatigue life obtained for the strengthened stays to that produced for the original stay matched the target defined for the simplified method, thereby validating the accuracy of this approach. The detail examination of the SFA calculations showed that the assumptions made for the simplified method in terms of dominant fatigue driving loads were relevant. However, the SFA and crack propagation results showed also that the fatigue life assessment were very sensitive to the considered loading condition and the wave environment, so that more information would be needed to improve the accuracy of the fatigue life predictions. Besides, it appeared that the proper consideration of the coefficient of friction of the hatch covers’ bearing pads would also have a significant effect on the fatigue evaluation by limiting the inertia loads transmitted by friction from the hatch cover to the corresponding transverse coaming. Finally, crack growth assessments confirmed the criticality of the stays to the fatigue cracking when the fatigue damage is not appropriately considered at the design stage.

References 1. IACS: Shipbuilding and Repair Quality Standard - Part B. Repair Quality Standard for Existing Ships, Recommendation No. 47 (2017) 2. IACS: Common Structural Rules for Bulk Carriers and Oil Tankers - Pt.I Ch.9, CSR (2018) 3. Hobbacher, A.: Recommendations for Fatigue Design of Welded Joints and Components, IIW doc. 1823-07, Welding Research Council Bulletin 520, New York (2009) 4. IACS: Container Ships - Guidelines for Surveys, Assessment and Repair of Hull Structures - Example No. 12-a, Recommendation No. 84 (2017) 5. Maddox, S.J.: Recommended hot-spot stress design S-N curves for fatigue assessment of FPSOs. In: Proceedings of the Eleventh International Offshore and Polar Engineering Conference, Stavanger, Norway (2001)

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6. IACS: Evaluation of Scantlings of Hatch Covers and Hatch Coamings and Closing Arrangements of Cargo Holds of Ships, Unified Requirement concerning the strength of ships, UR-S21A (2011) 7. NX NASTRAN. www.plm.automation.siemens.com. Accessed 04 Mar 2019 8. Hydrostar. www.veristar.com. Accessed 04 Mar 2019 9. IACS: Standard Wave Data, Recommendation No. 34 (2001) 10. Hogben, H., Dacunha, N.M.C., Olliver, G.F.: Global Wave Statistics. British Maritime Technology Limited, Teddington (2000) 11. IMO: Code of safe practice for Cargo Stowage and Securing (CSS Code), London (2002) 12. SAAB. www.ssab.fr/products/brands/hardox. Accessed 04 Mar 2019 13. British Standard: Guide to methods for assessing the acceptability of flaws in metallic structures, BS 7910 (2015) 14. Abaqus. www.3ds.com. Accessed 04 Mar 2019 15. DNVGL: Environmental conditions and environmental loads, RP-C205 (2017)

An Investigation of Fatigue and Long-Term Stress Prediction for Container Ship Based on Full Scale Hull Monitoring System Chong Ma(B) , Masayoshi Oka, and Hiroshi Ochi National Maritime Research Institute, Mitaka-shi, Tokyo, Japan [email protected]

Abstract. To pursue higher transportation efficiency, the dimension of container ship be-comes larger and larger and it is usually concerned with huge economic loss and environmental pollution when structural failure occurs. It is important to quantitatively assess the structural health for operated container ship considering the encountered wave condition and corresponding ship operation. In this research, a full-scale hull monitoring system is introduced which has been installed on several operated container ships. Based on the monitoring stress for 1–2 years, long-term stress prediction and fatigue assessment according to zerocross method and rainflow method are conducted. According to the analysis with and without low-pass-filter, it is indicated that, the structural elastic vibration has significant influence on hull structure which may increase the maximum stress by 10%–60% correspondingly. The hindcast-based encountered wave condition which includes the wave spectrum, significant wave height, wave mean period and wave direction are utilized in this research. Relative numerical simulations based on panel method are performed for all the monitored container ships. The wave condition influence is investigated numerically by adopting global wave statistics (GWS) and actual encountered wave condition. It is proved that, the recommended wave tables (GWS) generally predict much higher stress comparing with the actualencountered wave-based simulation and monitoring results. By considering the actual encountered wave condition, the improvement of numerical prediction accuracy, with respect to the evaluation of long-term stress and fatigue damage, is clarified quantitatively. Keywords: Hull stress monitoring · Very large container ship · Long-term prediction · Fatigue damage · Elastic vibration effect

1 Introduction To pursue higher transportation efficiency, the dimension of container ship becomes larger and larger and it is usually concerned with huge economic loss and environmental pollution when structural failure occurs. It is important to quantitatively assess the structural health for operated container ship considering the encountered wave condition and corresponding ship operation. © Springer Nature Singapore Pte Ltd. 2021 T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 492–504, 2021. https://doi.org/10.1007/978-981-15-4672-3_31

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Okada et al. [1] analyzed the stress monitoring data for post-Panamax container ship and demonstrated that, the fatigue damage would be doubled due to the whipping effect. In Oka’s research [2], the predicted accuracy of wave load was investigated based on comparison between the stress monitoring data and numerical simulation result. It was concluded that, the simulation predicted value was usually about 100% larger than the monitored data and this difference can be mainly interpreted by the difference between predefined wave condition utilized in the simulation and the actual encountered sea condition. In addition, Storhaug et al. [3] studied the stress monitoring data with respect to the LNG ship and discussed the relationship between the influence of structural vibration, ship route and cargo condition. Hirdaris et al. [4] investigated the springing/whipping influence on the fatigue damage based on full-scale monitoring data. However, both have not been able to show the quantitative evaluation results as the number of objective ships is as small as 1–2 ships. Due to the recent enhancement of ship safety level and the development of measurement technology, utilization of monitoring system to verify the ship safety becomes more and more popular. One project involved with series ships is being carried out [5]. In this research, the stress data obtained by the monitoring system for series ships is analyzed and the influence of structure vibration such as whipping is evaluated. Besides, by the comparison between the monitored data and numerical simulation result, the effect of actual encountered sea condition is assessed, as one of the uncertain factors that cause the discrepancy between monitoring data and numerical simulation result.

2 Stress Monitoring System 2.1 Objective Ships Four 14, 000 TEU container ships which were built in 2016 were selected and the relevant stresses were monitored [5]. The main route of the objective ships is East Asia-Suez Canal-Europe. 2.2 Location of Stress Monitoring Points The stress monitoring points are shown in Table 1 and Fig. 1. All the five longitudinal bending stresses are measured by the strain measurement system based on optical fiber type sensor. The abbreviations and locations of sensors are presented in Table 1. The stress data is collected at a sampling period of 3 ms. While, for data analysis, data extracted by a sampling period of 30 ms is used. 2.3 Measured Stress Data Spectrum of Stress. An example of the power spectrum (single-side-spectrum) of timehistory data collected by stress monitoring system is shown in Fig. 2. It can be found by Fig. 2(a) that, the peak near frequency 0.1 Hz is the low frequency component (LF) concerning with the encounter wave period, and the peak near 0.72 Hz is the high frequency component (HF) due to the elastic vibration of the hull. While, in Fig. 2(b),

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Name

Location

1

BMC

Double bottom (center line)

Middle ship

2

GAP

Under deck passage (port)

1/4L from AP

3

GFP

Under deck passage (port)

1/4L from FP

4

GMP

Under deck passage (port)

Middle ship

5

GMS

Under deck passage (SB)

Middle ship

GAP

GMP,GMS

GFP

BMC Fig. 1. Arrangement of 5 stress monitoring points.

700

700

600

600

Power(MPa^2-s)

Power(MPa^2-s)

the encounter wave component is about 0.15 Hz and the elastic vibration component is about 0.53 Hz. It can be interpreted that, the frequency of elastic vibration varies depending on the measurement time because of the difference in the loading condition. In general, the natural frequency of the hull decreases with the increment of the loading capacity.

500 GMP

400 300 200

500 GMP

400 300 200 100

100

0

0 0

0.1

0.2

0.3

0.4

0.5 0.6 F(Hz)

(a)

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5 0.6 F(Hz)

(b)

Fig. 2. Power spectrum of the time-history stress data.

0.7

0.8

0.9

1

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Time History of Stress Data. The history of stress data corresponding to the above stress spectrum is given by Fig. 3. Figure 3 shows an example when whipping occurs, and it is confirmed that, transient damping oscillation (whipping) is induced by the occurrence of slamming at all measurement points. The stress vibration component with period about 10 s (marked by LF) is caused by the encountered wave. And the relatively high frequency stress vibration component with period about 2 s is caused by the elastic vibration of the hull. When the ship dimension increases, the elastic vibration of the hull tends to occur more easily, and the ratio of the high vibration component to the total stress becomes large. Therefore, it is important to evaluate the influence of the elastic vibration of the hull on the maximum value of stress and the fatigue damage. In this research, long-term prediction of stress and fatigue damage analysis are performed using raw data (RAW) including elastic vibration of hull and low pass filtered data (LF) in which, elastic vibration is removed. The influence of elastic vibration of hull is studied by the comparison between analysis results of RAW and LF. The threshold of the low-pass filter is set to 0.3 Hz according to the spectrum analysis results. In this research, a band pass filter (see Table 2) is applied to the extract raw data (RAW), low frequency component (LF) and high frequency component (HF), and the relevant analysis for the three series of data is carried out. The time histories of stress δ(t) for RAW, LF and HF satisfy the following equation: δRAW (t) = δLF (t) + δHF (t)

(1)

Table 2. Coefficients of band pass filter

Frequency range

RAW

LF

HF

0.01–2 Hz

0.01–0.3 Hz

0.3–2 Hz

2.4 The Long-Term Stress Prediction Zerocross Method. Measured stress data is analyzed by the zerocross method to obtain the frequency distribution of stress amplitude. After setting the zero level as the average value of one measurement period (1 h), stress amplitude (peak to peak) can be solved by taking the difference between the maximum and the minimum value within one zerocross period (zerocross period is calculated by the zero up cross points, see Fig. 4, circle symbol denotes the maximum value for each zerocross period). Then, frequency distribution of stress can be obtained in a short-term manner. All the short-term frequency distributions of stress are finally accumulated to derive the long-term frequency distribution. Two series of stress time-history data (RAW and LF, see Fig. 4) are utilized by zerocross method to investigate the effect of the elastic vibration of the hull.

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Fig. 3. Time history of stress data.

Results and Discussions. An example of the long-term distribution of stress amplitude obtained by zerocross method is shown in Fig. 5. The horizontal axis of Fig. 5 represents the excess probability with respect to the encounter wave number, and the excess probability Q = 10−8 (−Log(Q) = 8) corresponds to the possible maximum stress in 25 years. Here, maximum value of stress amplitudes during measured period are used. The zerocross analysis with and without considering the elastic vibration is respectively carried out and the results are plotted in Fig. 5, named by GMP/RAW and GMP/LF correspondingly. The definition of GMP can be found in Table 1 and Fig. 1. The effect coefficient W effect of the elastic vibration on maximum stress can be determined according to the Eq. 2.   σRAW Q = 10−8   −1 Weffect = (2) σLF Q = 10−8     where σRAW Q = 10−8 and σLF Q = 10−8 denote the maximum stress concerning excess probability Q = 10−8 based on RAW data and LF data correspondingly.

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Mw (Red line): Vertical bending moment in wave without considering elastic vibration effect (LF data) Mwh (Blue line): Vertical bending moment in wave with considering elastic vibration effect (RAW data) : Zerocross Period

Fig. 4. Time history of stress data analyzed by zerocross method [6]

Fig. 5. Long-term stress prediction results of measured stress for one ship

Figure 6 concludes all the elastic vibration effect coefficients W effect for five monitoring points of four container ships, according to which, the 10%–60% effect coefficients can be clarified. 2.5 Assessment of Fatigue Damage Rainflow Method. All the stress monitoring data is analyzed by the rainflow method to achieve the stress frequency distribution during the service period. Then, the fatigue

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Ship B BMC

GAP

Ship C

GFP

GMP

Ship D

GMS

Fig. 6. Elastic vibration effect coefficients W effect for long-term stress prediction.

damage D can be calculated by utilizing S-N curve based on the frequency distribution following Eq. 3. D=

k  i=1

Di =

n1 n2 nk + + ... + N1 N2 Nk

(3)

where ni denotes the repetition times of stress range Si . k corresponds the number of different stress ranges. Ni is the repetition times of stress range Si until which, structural failure occurs. Ni can be solved based on S-N curve according Eq. 4 Ni =

K Sim

(4)

where coefficients m and K are pre-defined according to the material property, wielding condition and structural information. In this research, S-N curve designed for butt welds (D-Curve) is applied to calculate the fatigue damage. The relevant coefficients of DCurve are shown in Table 3. Table 3. Coefficients of D-Curve [7] S

K

m

>53.4 MPa 1.519E12 3.0 53.4 MPa 4.239E15 5.0

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Results and Discussions. The measurement data obtained by stress monitoring system is analyzed to determine the fatigue damage at the measurement points. Raw data (RAW), wave component (LF) and elastic vibration component (HF) shown in Fig. 3 are analyzed separately. The growth curve of fatigue damage in the measurement period (horizontal axis: cumulative measurement time, vertical axis: fatigue damage D) is given in Fig. 7. The rapid increment of fatigue damage in the growth curve is observed which can be interpreted by the encountered rough sea conditions which can induce a large stress vibration.

Fig. 7. Growth curve of fatigue damage

The growth curve is extrapolated linearly to predict fatigue damage after 25 years. The results are shown in Fig. 8. It can be seen that, the fatigue damage of the middle hull (GMP, GMS) tends to be larger than that of the bow (GFP) and stern (GAP). Assuming that the hull is a free end beam, the longitudinal bending moment usually gets the maximum value at the hull middle section where, the highest stress occurs. Correspondingly, fatigue damage at the hull middle section is relatively large. The fatigue damage of the bottom (BMC) tends to be smaller than that of the deck (GMP, GMS) because the longitudinal bending neutral axis of the cross section of the container ship is close to the bottom so that, the stress on the bottom due to the longitudinal bending is relatively smaller than that of deck. To investigate the influence of elastic vibration on fatigue damage, the effect coefficient HVeffect is proposed as Eq. 5 where, DRAW and DLF represent the fatigue damage results based on RAW data (considering elastic vibration component) and LF data (ignoring elastic vibration component), relevantly. HVeffect =

DRAW −1 DLF

(5)

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BMC

D

GAP GFP GMP GMS

Ship A

Ship B

Ship C

Ship D

Fig. 8. Predicted fatigue damage after 25 years for five monitoring points of four container ships.

The effect coefficient HVeffect solved by Eq. 4 is shown in Fig. 9. As HVeffect is the relative ratio between DRAW and DLF , the reliability of HVeffect decreases when the absolute amount of fatigue damage is small. Therefore, GFP and GAP are excluded. Elastic vibration consists of whipping which is a transient vibration caused slamming and springing which is a steady state vibration. Both whipping and springing are mainly reflected by the modes of longitudinal bending deformation. As all the stresses are measured based on the longitudinal strength member, the measurement stresses follow the longitudinal direction. Except for BMC, all the other stresses are dominated by global longitudinal deformation of the hull. As the results, distinction caused by the different measurement locations is small. Besides, the difference of HVeffect among four ships is mainly caused by the different ship operation condition and different encountered weather condition. 400%

Hull vibra on Effect

350% 300% 250% BMC

200% 150%

GMP

100%

GMS

50% 0% Ship A

Ship B

Ship C

Ship D

Fig. 9. Elastic vibration effect coefficients HV effect for fatigue damage.

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3 Comparison of Stress Monitoring Data and Numerical Simulation Results 3.1 Numerical Simulation In order to confirm the reliability of stress monitoring system, validation works by numerical simulation is performed. By the comparison of stress monitoring data and numerical simulation results, the prediction accuracy of numerical simulation can be demonstrated as well. For numerical simulation, the stresses on the measured points are calculated based on panel method and FEM-loading coupling analysis. Then, long-term prediction [8] which is similar with that introduced in Sect. 2 is carried out for the stresses solved by the numerical model. Firstly, the numerical simulation is done according to the general pre-designed environment condition as shown by Table 4. In Fig. 10, besides the long-term prediction results based on measurement data (marked by blue circle, GMP/LF, in which the elastic vibration component is omitted), the numerical results based on general pre-designed environment condition (represented by gray line, GWS) is included as well. It is observed that, numerical results based on general pre-designed environment condition are always larger than that of measurement data which means that, the pre-designed environment condition may be too conservative than the real condition. This difference can be caused by kinds of uncertainty factors such as the difference between the pre-designed environment condition and actual condition, which includes the errors of waves, ship speed, ship loading condition, as well as, the errors existing in the monitoring data and accuracy of numerical calculation. By clarifying the influence of these uncertainty factors, the distinction between the stress monitoring data and numerical simulation results should be eliminated and then, the high-accuracy fatigue life estimation based on numerical simulation can be expected. Table 4. Assumed environment condition for numerical simulation Item

Model

Wave scatter diagram Global Wave Statistics (GWS [9]) corresponding to route of monitoring ship Wave spectrum

ISSC1964

Ship speed

V = 17.7knot (constant)

Wave direction

All headings

In this study, pre-designed environment condition is further improved based on actual encountered wave condition which is calculated based on a hindcast scheme [10]. Then, the influence of improved wave condition on long-term stress prediction and fatigue damage is checked.

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Fig. 10. Comparison of long-term stress prediction for ship A.

3.2 Results and Discussions In order to investigate the influence of the encountered wave condition, two kinds of wave conditions are considered which consist of the wave occurrence frequency table [11] modified by the ship route based on GWS [9] and the actual encountered wave condition of each objective ship according to a hindcast scheme based on wave database of Japan Weather Association [10]. The comparison for long-term stress prediction and fatigue damage are demonstrated below. Long-Term Stress Prediction. In addition to the long-term stress prediction results of measurement data (GMP/LF) and simulation results based on pre-designed environment condition (GWS), simulation results based on actual encountered wave condition is given by green line (JWA). The enhancement of the numerical prediction accuracy by the improvement of the wave condition is clearly confirmed. It implies that, the contribution of the utilization of actual encountered wave condition on long-term stress prediction is significant. Fatigue Damage. The comparison of fatigue damage among the measurement data (hull monitoring), the simulation results based on pre-designed condition (simulation using GWS) and the simulation results based on actual encountered wave condition (simulation using hindcast) is presented in Fig. 11. For an easier comparison, the fatigue damage value is converted by power of 1/4. Measurement fatigue damage is solved by the RAW data. Similar with the long-term stress prediction, it is known that, numerical models based on both GWS data and Hindcast data predict higher fatigue damage compared with the measurement value. However, the accuracy improvement for fatigue damage prediction by implementation of the actual encountered wave condition is obvious as well.

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Fig. 11. Comparison of fatigue damage for all four ships.

4 Conclusions In this research, the measurement stress data by hull health monitoring system is utilized and long-term stress prediction and fatigue damage evaluation for 4 objective ships are conducted. The following conclusions can be derived. By the band pass filter, the measurement stress data is separated into the wave component and structural elastic vibration component. The elastic vibration (like whipping) influence is discussed based on the separated components in terms of possible maximum stress and accumulated fatigue damage in 25 years. It is noteworthy that, the influence of elastic vibration on the possible maximum stress and is about 10%–60%, correspondingly. The comparison between the stress monitoring data and numerical simulation results is performed. It is known that, the long-term stress prediction and fatigue damage obtained by the monitoring data are usually smaller than the numerical simulation results which means that, the safety level of measured points is sufficient. By utilizing the actual encountered wave condition, the improvement of numerical prediction accuracy, with respect to the evaluation of long-term stress and fatigue damage, is demonstrated quantitatively. Based on the full-scale monitoring data, the presented results about the influence of elastic vibration and encountered wave condition can be utilized to improve the Classification Society Rules on the evaluation of hull structure health.

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Acknowledgment. This study was conducted as a part of the collaborative research project “Hull structure health monitoring of 14,000TEU large container ships” under the support of the Ministry of Land, Infrastructure, Transportation and Tourism of Japan for i-Shipping operation. The authors would like to thank the project members from Japan Marine United Corporation, ClassNK, Yokohama National University, the University of Tokyo, Japan Weather Association, NYK Line and MTI Co., Ltd. for their discussion and support.

References 1. Okada, T., Takeda, Y., Maeda, T.: On board measurement of stresses and deflections of a Post-Panamax containership and its feedback to rational design. Mar Struct. 19, 141–172 (2006) 2. Oka, M., Ogawa, Y., Kawano, H.: A study on fatigue strength of large container ship with taking the effect of hull girder vibration into account. PRADS (2010) 3. Storhaug, G., Pettersen, T.A., Oma, N., Blomberg, B.: The effect of wave induced vibration on fatigue loading and the safety margin against collapse on two LNG vessels. Hydorelastisity Marit. Technol. (2012) 4. Hirdaris, S.E., White, N.J., Angoshtari, N., Johnson, M.C., Lee, Y., Bakkers, N.: Wave loads and flexible fluid-structure interactions: current developments and future directions. Ships Offshore Struct. 5(4), 307–325 (2010) 5. Okada, T., Kawamura, Y., Kato, J., Ando, H., Yonezawa, T., Kimura, F., Toyoda, M., Yamanouchi, A., Arima, T., Oka, M., Matsumoto, T., Kakizaki, H.: Outline of the research project on hull structure health monitoring of 14,000TEU large container ships. Conf. Proc. Jpn. Soc. Naval Architects Ocean Eng. 24, 31–35 (2017) 6. Matsumoto, T., et al.: Activities of ClassNK for structural strength of large container ships, Class NK Technical seminar (2016). (in Japanese) 7. International Association of Classification Societies (IACS) Common Structural Rules for Bulk Carriers and Oil Tankers, January 2019 8. Fukuda, J.: Statistical prediction for ship response. In: 1st Symposium About Seaworthiness, The Society of Naval Architects of Japan, pp. 99–119 (1969) 9. Hogben, N., Dacunha, N.M.C., Olliver, G.F.: British Maritime Technology “Global Wave Statistics”. Unwin Brothers Limited, London (1986) 10. Matsuura, K., Maeda, M., Nakano, H., Kuroki, K., Koshita, S., Sato, Y.: Estimation of meteorological sea conditions and its accuracy. KANRIN Jpn. Soc. Naval Architects Ocean Eng. 77, 6–10 (2018) 11. Oka, M., Ogawa, Y., Takagi, K.: A fatigue design for large container ship taking long-term environmental condition into account OMAE (2011)

Evaluation of Long-Term Corrosion Fatigue Life of Ship and Offshore Structural Steel Won Beom Kim(B) Department of Naval Architecture and Ocean Engineering, Ulsan College, Ulsan, Korea [email protected]

Abstract. In this study, an evaluation of the long-term corrosion fatigue life of ships and offshore structural carbon steel was conducted, and the result is compared with data from other experimental studies. The evaluation was carried out as follows: 1) First, an evaluation of the increased stress, caused by general corrosion, was conducted to calculate the crack initiation life and propagation life. This stress increase, due to area reduction from the weight loss, was calculated through an evaluation of the corrosion rate obtained from 16 years of published weight loss data from continuous immersion in seawater. 2) Second, an evaluation of the moment of crack initiation (Ni), when a crack is initiated at the corrosion pit, was conducted. The depth of the corrosion pit was calculated from 16 years of continuous immersion test data of pit penetration. To determine the instant of crack initiation at the corrosion pit, the Komai threshold value for crack initiation at the corrosion pits was adopted. To calculate the stress intensity factor of a crack, the Newman-Raju formula was used. 3) Finally, the crack propagation life was evaluated from crack initiation to fracture. The results of the evaluation are compared with long-term S-N experimental corrosion fatigue test data. In this study, the long-term corrosion fatigue life estimation method, using weight loss and pit corrosion data by long-term immersion corrosion data, was investigated. The evaluation results were found to reflect the tendency of long-term S-N fatigue test results. Keywords: Long-term corrosion fatigue · Corrosion pit · General corrosion · Crack initiation threshold · Fracture mechanics

1 Introduction Fatigue strength is a key design factor contributing to the strength of ships and offshore structures that must withstand the constant application of loads in the ocean, such as waves, tides, and winds. Additionally, in marine structures such as ships and offshore © Springer Nature Singapore Pte Ltd. 2021 T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 505–516, 2021. https://doi.org/10.1007/978-981-15-4672-3_32

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structures, corrosion resistance as well as fatigue strength is required. As a countermeasure against corrosion in these structures, corrosion protection is generally obtained by, for example, coating, sacrificial anode, and impressed current cathodic protection. However, corrosion is not always completely prevented, for various reasons, and constantly occurs. The strength of a structure under such corrosive conditions should be fully examined. In this study, a long-term corrosion fatigue life estimation corresponding to the lifetime of a ship or offshore structure was performed. The purpose of this study is to propose a new concept that considers both general corrosion and pitting corrosion in the estimation of the corrosion fatigue life.

2 Background It is well-known that fatigue, a major structural failure mode, behaves differently in seawater than in the atmosphere. The phenomenon of corrosion fatigue has been recognized for many years and involves the synergistic effect of corrosion and fatigue, which is more prominent in seawater than in the atmosphere. In ships, corrosion has been dealt with by providing a corrosion margin on the plate. Previous studies have shown that the types of corrosion of certain alloy systems occur in specific forms. That is, general corrosion occurs in mild steel, while pitting corrosion and stress corrosion cracking occur in stainless steel [1]. Meanwhile, it has been established that corrosion pits play an important role in the occurrence of corrosion fatigue crack initiation in hull structural steel [2]. Many papers have also been published that focus on the mechanical characteristics of corrosion pits [3–6]. An important issue regarding corrosion pits is the possibility of a mechanical evaluation of the occurrence of cracks. In previous studies, crack initiation was determined to begin when the crack length reached 1 mm or when the fatigue crack reached the plate thickness size. This judgment is convenient but somewhat ambiguous to the mechanical or physical meaning. In addition, general corrosion was observed with the naked eye, and numerous corrosion pits were observed under an electron microscope in the data [7] of the plate taken from a ship that was in operation for 17 years, as shown in Fig. 1.

Fig. 1. Fatigue test results of steel plate taken from aged ship and corrosion pit of crack initiation site.

The fatigue test results under atmospheric and seawater conditions confirm that cracks begin from the corrosion pits, then coalesce and grow into longer cracks. Therefore, the main cause of the decrease in the corrosion fatigue strength of the structural

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steel is considered to be the appearance of the crack originating from the corrosion pit, while the increase in the load stress due to the decrease in the plate thickness caused by the general corrosion is another important factor.

3 Calculation of Long-Term Corrosion Fatigue Life 3.1 Corrosion Pit Data A flowchart of the process for estimating the long-term corrosion fatigue life is shown in Fig. 2. Here, the calculation process was performed in the order of the flowchart.

Fig. 2. Flowchart for the evaluation of long-term corrosion fatigue life.

Many studies have been conducted on the corrosion of carbon steel. Stainless steel has been known to corrode in the form of pitting corrosion. Furthermore, it is known that the corrosion of carbon steel occurs not only in the form of general corrosion, but also in the form of corrosion pits [2, 6–8]. These studies show that corrosion pits occur along with general corrosion in steel in a corrosive environment. Therefore, in this study, corrosion pits as well as general corrosion were included in the estimation of the corrosion fatigue life. This requires corrosion data for long-term general corrosion and pitting corrosion. In previous studies, corrosion data with a test duration of less than two months were found to follow a trend [6], but long-term corrosion data for long-term

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corrosion fatigue life evaluation are difficult to find. Therefore, in this study, we use the results of Southwell and Alexander [9] detailing 16 years of immersion on steel as the corrosion data for the evaluation of the long-term corrosion fatigue life. Figure 3 shows the results of the weight loss and pitting penetration trends for ferrous steel.

Fig. 3. Corrosion of carbon steel continuously immersed in seawater [9].

Fig. 4. Schematic of the concept of general corrosion and pitting corrosion in carbon steel.

According to Okazaki et al. [10], the regional dependence of seawater concentration is relatively small and its composition is almost the same. However, the gas components, such as dissolved oxygen, strongly depend on the temperature and water pressure, hence, caution must be paid to the evaluation of the corrosiveness. The concept of general corrosion and pitting corrosion in this study is shown in Fig. 4. The pitting penetration represents the depth of the pit and characterizes the corrosion

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pit. That is, the general corrosion by weight loss increases the applied stress, and the corrosion pit is involved in crack initiation. In this study, the corrosion values of steel were derived from the graph shown in Fig. 3. Here, the weight loss shows a nearly linear trend. Meanwhile, the pitting penetration shows a rapid increase up to approximately 2 years, but the weight loss gradient is gentle until the 8-year mark; then, a sharp weight loss tendency is observed again. This tendency affects the slope of the S-N curve of the corrosion fatigue. A computer program was developed to calculate the long-term corrosion fatigue life estimation according to the flowchart of Fig. 1. The relationship between the pit depth and time for each linear section, shown in Fig. 3, is expressed as a function in the calculation program, and the stress intensity factor (SIF) was calculated from the Newman-Raju equation by considering the pit as a crack. 3.2 General Corrosion Data Consideration of the general corrosion was conducted as follows. The weight loss data of Fig. 2 exhibits an almost linear increase over the entire periods of 1 year, 2 years, 4 years, 8 years, and 16 years. The effective stress range equation of Müller [11], displayed as Eq. (1), indicates the increase in the applied stress range according to the reduction in the cross-sectional shape. In this study, the long-term corrosion fatigue life of more than 20 years, corresponding to the life of the ship, was evaluated such that the phenomenon of the increased stress due to general corrosion could be expressed using Eq. (1). We calculated the corrosion rate from the slope of the weight loss for the steel and used it to calculate the effective stress range of Eq. (1) due to the increase in the applied stress range from the reduction in the cross-sectional area. In this study, this was considered to be the effect of general corrosion.   Rk · th −2 σ(th ) = σ0 1 − r0

(1)

where σ (t h ) is the effective stress range, σ 0 is the applied stress range, Rk is the corrosion rate, t h is time (= N/f , where f represents the frequency), and r 0 is the radius of the circular cross-section. Considering a wave period in the ocean, 10 cpm (cycles per minute) or 0.17 Hz was adopted for the frequency. In this equation, r 0 was set to 5 mm. 3.3 Calculation of Stress Intensity Factor of Corrosion Pit The stress intensity factor K CF at the time of crack initiation in the corrosion pit was calculated using Eq. (2), of the Newman-Raju equation [12].  πa · F(a/t, a/c, c/b, ∅) (2) K = (σt + H σb ) · Q where σ t is the remote uniform tension stress, σ b is the remote outer fiber bending stress, ø is the parametric angle defining the position of the point under consideration, a is the crack depth, 2c is the crack length, t is the plate thickness, 2b is the plate width for the tension and bending loads, Q is the elliptical crack shape factor, and F(a/t, a/c, c/b, ø) is

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the boundary correction factor. This equation is valid for 0 < a/c ≤ 1.0, 0 ≤ a/t ≤ 0.8, c/b < 0.5, 0 ≤ ø ≤ π. Figure 5 shows a surface crack in a finite plate. The stress intensity factor was calculated by considering the two-dimensional shape of the corrosion pit projected in the load direction as a crack. When calculating the stress intensity factor using Eq. (2), the effective stress range of σ (t) of Eq. (1) was used for the remote stress in Eq. (2). The pit shape was calculated as follows. The growth of the corrosion pits can be seen to continue when the corrosion pits grow or when the growth rate of the cracks in the corrosion pits is parallel to the x-axis in the corrosion fatigue crack growth curve. The growth of the corrosion pits is known to have a constant aspect ratio [3, 6]. Komai determined the maximum, average, and minimum aspect ratios in the corrosion pits where cracks occurred during the HT50-TMCP experiments. Here, the average aspect ratio of HT50-TMCP was 0.65. Therefore, the aspect ratio for the growth of the corrosion pits in this calculation was specified as 0.65. 3.4 Stress Intensity Factor of K CF at the Moment of Corrosion Fatigue Crack Initiation Komai [6] defined the moment of the occurrence of a crack from a corrosion pit as the crack initiation life of the corrosion fatigue by using high-tensile stress steel, a kind of carbon steel. The conversion of a corrosion pit to a crack is explained as follows [6]. For the growth law of the corrosion pit, Komai obtained the change over time of a corrosion pit under the conditions of f = 0.17 Hz and stress ratio R = −1 under cyclic stress. Next, the corrosion pits with cracks at the bottom were regarded as sharp semi-elliptical surface cracks and the K I values at the bottom of the corrosion pits were calculated using the Newman-Raju equation. Here, the aspect ratio of the corrosion pits was 0.65. From this, it was concluded that K at the bottom of the corrosion pit does not necessarily increase even when the number of cycles or pit depth is increased, and that the average value K CF is maintained at a substantially constant value, with K CF = 1.66 MPa (m)1/2 . From this, it was reported that the mechanical condition of the pit, with the crack obtained from the fracture test, reflects that at the time of cracking, and the occurrence of the crack from the bottom of the pit is well organized by the K CF obtained from the pit with the crack. Therefore, in this study, the aspect ratio of the growth of pits was a/c = 0.65 and the average K CF = 1.66 MPa (m)1/2 when considering the cracks generated from pits. In this study, this concept was used to estimate the corrosion fatigue life of carbon steel. Komai proposed K CF = 1.66 MPa (m)1/2 as the corrosion fatigue crack initiation life (Nc) of carbon steel, which was adopted in this study. Moreover, in this study, Nc was determined when the calculated stress intensity factor of the pit, considering a crack, reached K CF = 1.66 MPa (m)1/2 . Table 1 shows the crack size at the time of the crack initiation and the number of cycles of crack initiation life at each applied stress.

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Table 1. Calculation results of crack initiation life and crack depth. Stress amplitude (MPa)

Number of cycles (Ni)

Crack depth (mm)

10

83,195,892

2.11

20

4,872,132

0.74

30

2,280,924

0.35

40

1,307,844

0.2

50

845,172

0.13

60

589,968

0.09

70

434,520

0.07

80

333,540

0.05

90

263,772

0.04

100

214,200

0.03

200

53,856

0.01

300

23,868

0.004

3.5 Calculation of Crack Propagation Life The crack propagation life was calculated using the Newman-Raju equation with the fatigue crack initiation length as the initial crack length. The thickness t was set to infinity and the half-width b was set to 10 mm. When the thickness is thin, the length of the width has a great influence on the calculation result, but when the thickness is infinite, the calculation result is not changed when the half-width is compared between 10 mm and 20 mm. da/dN = C(K)m

Fig. 5. Surface crack in a finite plate.

(6)

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The aspect ratios in the fatigue crack growth calculations were determined using the fatigue crack growth patterns of surface cracks as presented in the appendix of the paper by Newman and Raju [12], as follows. At point A and B in the figure, the following equations can be given. da = CA KAn dN

(3)

dc = CB KBn dN

(4)

Because the stress intensity factor solution for the small semicircular crack shows that the stress intensity at point B is approximately 10% higher than that at point A, the coefficient C B was assumed to be: CB = 0.9n CA

(5)

Therefore, in this paper, additional crack growth calculations were carried out. The crack propagation life was calculated using the Paris equation, shown in Eq. (2). The relationship between the SIF and the crack propagation rate is shown in Fig. 6.

Fig. 6. Relationship between stress intensity factor K and da/dN [13].

The crack propagation characteristics used in the crack propagation calculation were adopted from the da/dN-K curve for the HT50-TMCP proposed by Komai et al. [13]. Komai et al. proposed the following for the crack propagation life evaluation. That is, in

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the case of small cracks, there is no crack closure phenomenon in corrosion fatigue crack growth. From this, it is possible to use the propagation characteristics of the long crack obtained by avoiding the occurrence of crack closure under the condition of constant K max when the propagation characteristics of a small crack cannot be obtained. Table 2. Calculation results of crack propagation life and fracture life. Number of cycles at K = 2.9 MPa:(m)1/2

Number of cycles (Nf) from K = 2.9 MPa:(m)1/2 until fracture (t = 9.99 mm)

10

99,719,892

117,785,609

20

12,851,388

31,179,436

30

5,601,636

17,130,798

40

3,292,560

11,123,643

50

2,153,016

7,750,287

60

1,476,756

5,620,265

70

1,119,960

4,315,199

80

861,696

3,406,142

Stress amplitude (MPa)

90

682,992

2,747,496

100

554,472

2,269,758

200

139,536

631,976

300

62,118

293,951

Table 3. Summary of the evaluation methods.

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Fig. 7. Detailed description of calculation result.

Fig. 8. Comparison between the calculation result from the current research and the experimental results from the literature [6].

For the crack closure, K eff was measured and considered as K eff = K, it is possible to evaluate the crack propagation life, including the case where the crack size is small, using long crack data. The crack propagation curves for HT50-TMCP thus obtained were used to calculate the crack propagation life of this study. This was then used for comparison with the corrosion fatigue test results by HT50-TMCP on the S-N curve. In this diagram, from the Paris equations at the slope line, c = 1.0339 × 10−8 and

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m = 2.7824. The results of the crack propagation life calculation are shown in Table 2. Table 3 shows summary of the evaluation methods. The weight loss effect in the integration calculation of the slope section of the Paris equation of Fig. 5 is not yet included in the calculation results. If the effect of weight loss in this interval is included in the results, the Nf (Life to Failure) will go further to the left. Figure 7 shows detailed description of calculation results. Figure 8 shows a comparison of the calculation results of the stress amplitude in the above calculation with the corrosion fatigue test results of carbon steel obtained in [6]. The results of the calculations are in good agreement with the experimental results in the low-stress region, and the consideration of the general corrosion and pitting corrosion reflects the experimental results well.

4 Conclusion In this study, the long-term corrosion fatigue life was estimated by considering the effect of general corrosion and pitting corrosion using the weight loss and pit corrosion information obtained from 16 years long-term corrosion data of carbon steel. The calculation of the corrosion fatigue life was performed under the same conditions as that of the corrosion fatigue strength for comparison of the calculated results with the results of the long-term S-N test conducted under the corrosive environment, with the stress ratio R = −1 and the load frequency f = 0.17 Hz. From the above calculation results, the applicability of the corrosion fatigue strength evaluation method, considering not only pit corrosion but also weight loss by using long-term corrosion data, was examined in the evaluation of the corrosion fatigue strength.

References 1. Jaske, C.E., Payer, J.H., Balint, V.S.: Corrosion Fatigue of Metals in Marine Environments. Springer-Verlag, New York Heidelberg Berlin, Battelle-Press, Columbus Ohio (1981) 2. Nagai, K., Mori, M., Yajima, H., Yamamoto, Y., Fujimoto, Y.: Studies on the evaluation of corrosion fatigue crack initiation life in notched mild steel plate. Soc. Naval Archit. Jpn. 142, 239–250 (1977) 3. Kondo, Y.: Prediction method of corrosion fatigue crack initiation life based on corrosion pit growth mechanism. Trans. Japanese Soc. Mech. Eng. (A) 53(495), 1983–1987 (1987) 4. Lindley, T.C., McIntyre, P., Trant, P.J.: Fatigue-crack initiation at corrosion pits. Metals Technol. 9(135) (1982) 5. Kim, W.B., Paik, J.K., Yajima, H.: Evaluation of corrosion fatigue crack initiation life of 13Cr steel. Key Eng. Mater. 326, 1007–1010 (2006) 6. Komai, K.: Environmental Strength Design of Structural Materials, 1st edn. Youkendou, Tokyo (1993). (in Japanese) 7. Kim, W.B., Paik, J.K., Iwata, M., Yajima, H.: Fatigue strength of rusting decayed hull steel plate in air and artificial seawater condition. J. Soc. Naval Archit. Korea 40(1), 63–68 (2006) 8. Funatsu, Y.: Building better oil tankers. ClassNK Magazine, pp. 11–14 (2009) 9. Southwell, C.R., Alexander, A.I.: Corrosion of metals in tropical environments, Part 9 – Structural ferrous metals – Sixteen years’ exposure to sea and fresh water. NRL Report 6862, Naval Research Laboratory, Washington, D.C. (1969)

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10. Ozaki, T., Ishikawa, Y., Akiyama, M.: Corrosion of seawater equipment – damage and countermeasures, 2nd edn. Kagakutoshoshuppan, Tokyo (2007). (in Japanese) 11. Müller, M.: Theoretical considerations on corrosion fatigue crack initiation. Metall. Trans. A 13A, 649–655 (1982) 12. Newman, J.C., Raju, I.S.: An empirical stress-intensity factor equation for the surface crack. Eng. Fract. Mech. 15(1–2), 185–192 (1981) 13. Komai, K., Minoshima, K., Kim, G.S.: An estimation method of long-term corrosion fatigue growth characteristics. Trans. Japanese Soc. Mech. Eng. (A) 54(499), 509–512 (1987)

Statistical Modelling and Comparison of Model-Based Fatigue Calculations and Hull Monitoring Data for Container Vessels Erik Vanem1,3(B) , Lars Holterud Aarsnes2 , Gaute Storhaug2 , and Ole Christian Astrup1 1

3

DNV GL Group Technology and Research, Høvik, Norway [email protected] 2 DNV GL Maritime Advisory, Høvik, Norway Department of Mathematics, University of Oslo, Oslo, Norway

Abstract. This paper presents the results of statistical analyses on the difference between measured and calculated fatigue rates for an oceangoing container vessel. Data from a hull monitoring system installed onboard this ship provides time-series of observed fatigue rates in actual operation, sea- and weather conditions. In addition, a hydrodynamic model has been used to calculate fatigue rates for selected locations in the mid-ship section due to vertical bending moment only, corresponding to experienced weather conditions over the same time period. The measured fatigue rates are then compared to the calculated fatigue rates and statistical regression models are established to explain the differences as a function of selected explanatory variables. This can then be used to correct for biases in the numerical models. Overall, the results indicate that the models perform reasonably well and are able to describe much of the variation in the difference. However, the models are found to not generalize very well and it may be challenging to find models for a whole fleet of ships.

Keywords: Hull monitoring analysis

1

· Fatigue · Sensor data · Statistical

Introduction and Background

Fatigue is a cumulative process that leads to gradually deteriorating hull strength of ocean going ships over time. It is a result of random cyclic loading mostly due to wave interaction with the hull and may appear in locations with high stress concentrations at locally highly loaded details. Over time, this may lead to the development of cracks which may grow and cause structural fracture, leakage and may ultimately lead to structural collapse [1]. In order to control the risk of fatigue damages, fatigue assessment of ship structures are carried c Springer Nature Singapore Pte Ltd. 2021  T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 517–536, 2021. https://doi.org/10.1007/978-981-15-4672-3_33

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out during the design phase [2], and structural monitoring may be performed during operations [3], see also [4]. By comparing fatigue rates calculated by numerical models with measured fatigue rates from hull monitoring systems, one may correct for possible biases and account for uncertainties in the numerical calculations. Fatigue is within the scope of ship classification rules [5,6] and are covered in the common structural rules for bulk carriers and oil tankers [7]. Fatigue is a serous risk to maritime transportation and fatigue damages to the hull may contribute to serious ship accidents. A number of historical ship accidents can be attributed to fatigue of the hull. Some notable examples are the break-up and sinking of the bulk carrier MS Flare [8] and the sinking and subsequent oil spills from the Erika and Prestige oil tankers in 1999 and 2002, respectively, both of which resulted in enormous economic losses and catastrophic environmental damages [9]. Fatigue life refers to the expected lifetime of a structural detail and is related to the number of stress cycles until failure. Typically, for an ocean going ship the design fatigue life will be at least 25 years [5]. Fatigue rate is the fatigue damage divided by the budget damage per unit time, for example, half hour. The capacity of welded steel joints with respect to the fatigue strength is characterized by S-N curves which give the relationship between the stress range applied to a given detail and the number of constant amplitude load cycles to failure, typically displayed on a log scale. Design S-N curves are based on results from experimental S-N curves with a survival probability of 97,5%. However, fatigue is a highly probabilistic process, and many factors influence fatigue, such as loading frequency, material characteristics, workmanship, mean loading and the environment to mention some. Moreover, in practice a ship will not experience constant cyclic loading but will be subject to variable amplitude loading. Variable amplitude loading is given by a load spectrum representative of the long-term cyclic loading. The load spectrum can be established by cycle counting using the rain-flow method and combined with the SN curve, the Miner’s rule is used to assess the fatigue life giving: k  ni = C, N i i=1

(1)

where k is the number of different stress magnitudes, Si (1 ≤ i ≤ k) and Ni is the number of stress cycles to failure for constant cyclic stresses Si and ni is the number of contributing cycles of stress level Si . For design purposes, C is typically set to 1, indicating that failure occurs when the cumulative fatigue rates sum to unity. The Miner’s rule is a simple model for cumulated fatigue rate with complex loading that assumes linear cumulative damage and that the fatigue rate is independent of the sequence of stress cycles. Hence, more elaborate models may be used in actual ship design [10–16]. In this paper, the fatigue rates induced by the midship bending moment as calculated by a numerical model will be compared to fatigue rates measured by a hull monitoring system for a container vessels. Moreover, the differences between the calculated and the measured fatigue rates will be modelled by statistical

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regression models to inform about the uncertainty of the calculated fatigue rates as a function of various input parameters. Assuming that the actual fatigue rate is F, which is unknown, the calculated fatigue rate from the numerical model, FModel will be an estimate of the true fatigue rate that will be associated with an error. Typically, this error will not be the same for different cases, and it is possible to model this as a function of selected input parameters. Similarly, the measured fatigue rate, FHmon , will be another estimate of the true fatigue rate (associated with the class notation HMON for hull monitoring). That is, one may write F(·) = FModel (·) + ε(·)

(2)

F(·) = FHmon (·) + ε¯(·),

(3)

where · denotes a set of relevant input parameters such as vessel speed, vessel heading, significant wave height (HS ), peak wave period (TP ), etc. For the purpose of this exercise, it will be tacitly assumed that the error made in the measured fatigue rate is negligible compared to the errors made by the numerical model. The mean stress effect is neglected in the measurement but is a good assumption for the deck details in hogging experiencing tension as mean stress. Hence, by studying the difference between the calculated and measured fatigue rates, one may get estimates of the errors associated with the numerical calculation model, FHmon (·) − FModel (·) = ε(·) − ε¯(·) ≈ ε(·).

(4)

Hence, in this paper, regression models will be established to describe this error as a function of the following input variables: – – – –

Significant wave height (HS ) Zero up-crossing wave period (TZ ) ship speed relative wave direction

Three different regression models will be investigated, i.e., generalized additive models (GAM), Gaussian processes regression (GP) and regression trees (RT). All these are flexible regression models that allow for non-linear relationships between the covariates and the dependent variable. The idea is that if statistical models with good predictive power can be established for the uncertainty in the numerical models for calculating the fatigue rates, this can be accounted for in order to improve the accuracy of fatigue calculations.

2

Data Description

Measured fatigue rates from a hull monitoring system are compared to calculated fatigue rates from a numerical model. One approximately 15 000 TEU container ship is selected for this study and sensor data for one year between 2016 and 2017 are available, covering different ocean areas and weather conditions. The dimensions of the ship is approximately 350 m in length, 50 m in breadth and 15 m in design draught.

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Hull Monitoring Data

The vessel included in this study are equipped with a hull monitoring system that collects measurements from various sensors. The data include raw sensor readings from different types of sensors as well as rainflow counting data, logs and summary statistics for 5- and 30-min intervals. For the purpose of this analysis, the data for the 30-min summary statistics are analysed. The 5-min data give operational input onboard to quickly see the effect of speed and course changes in a seaway. The half-hourly summary statistics data are time-series with a number of summary statistics collected at 30-min intervals for a large number of sensor signals. For all sensor signals, or channels, the following 9 summary statistics are collected 1. Number of observations 2. Mean 3. Standard deviation

4. 5. 6. 7.

Skewness Kurtosis Minimum value Maximum value

8. Mean up-crossing count 9. Max Peak-Peak value

In this study, only the mean values are analysed, but it is worth mentioning that additional information regarding the sensor signals are available, either in the form of other statistical parameters, or in the raw sensor signals. Only two of the sensor channels are considered in this analysis, i.e., corresponding to the fatigue rates induced by wave motion for the global midship bending moment for starboard (GMS) and port (GMP) sides, respectively. In these signals, the effects of vibration, which is important in practice, has been filtered out by a frequency filter so that what remains are effects from bending moment from wave actions. The measured stress contains contributions from vertical wave bending, horizontal wave bending, warping (torsional response) and axial forces. However, it is dominated by vertical wave bending. The fatigue rates are given in fatigue damage per second, and are normalized with respect to the design life. That is, a fatigue rate of 1 over half an hour means that half an hour of the overall fatigue budget will be spent. If this value is 1 throughout its life, the structure will fail at exactly the design life of the ship. If, for example, the value of the fatigue rate is 120 over a day in a storm, it means that it will spend about 1⁄3 year of fatigue budget, etc. The measured fatigue rates from the hull monitoring system are shown in Fig. 1 and clearly indicates the time periods when this ship has been exposed to bad weather and increased fatigue rates. It is also seen that there are high correlation between high fatigue rates for starboard and port side, which tend to appear at the same time, but that the port side appear to have experienced generally higher fatigue rates, suggesting off head sea directions. Some data points were removed from the analysis due to measurement noise. Most notably, a number of points with zero fatigue have been removed. This is not believed to influence the results much, as the interest is in relatively large responses. In addition, a few very high measured values have been removed as

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Fig. 1. Example of hull monitoring data: measured fatigue rates

they were regarded as non-physical. Hence, a threshold of 100 has been set, admittedly rather arbitrarily, and any measured fatigue rates above this value have been removed. This corresponds to half-hour intervals consuming more than 2 days of fatigue budget. In addition to the sensor signals for fatigue rate, information about the position of the vessels, the speed over ground and the course over ground were utilized together with the timestamps to identify where the vessels have sailed at all times. This information is then coupled to weather data for the time and location of the vessel in order to obtain information about prevailing sea states. Prevailing weather and sea state conditions were obtained by pairing vessel positions with hindcast data from NOAA (National Oceanic and Atmospheric Administration)1 , containing wind and wave parameters at 3-h temporal resolution. 30-min weather conditions are obtained from the 3-h data by interpolation to match the temporal resolution of the sensor data. 2.2

Numerical Model Data

In order to calculate the fatigue rates by the numerical model, a hydrodynamic model was run with actual weather conditions obtained from the hindcast data. In this manner, fatigue rates for mid-ship vertical bending moment corresponding to actual experienced weather conditions were calculated for each half hour interval where there are hull monitoring data available. Values of HS below 0.1 m have been set to 0.1 and values of TZ below 1.3 s have been set to 1.3 s. These values are associated with negligible fatigue rates so this should not have a 1

http://polar.ncep.noaa.gov/waves/hindcasts/.

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significant impact. The calculated fatigue rates are normalized in the same manner as the measured fatigue rates, that is relative to the design life of the vessel. The vertical wave bending moment has been estimated by direct calculation on a hydrodynamical model of the vessel in full loading condition. For a deck detail, the governing response will be from vertical bending but it is acknowledged that an improvement to the analysis would be to account for all responses. The resulting vertical bending moment is calculated at L/2. The structural response (nominal stress in deck) for the given vertical bending moment is calculated using beam theory and section modulus of the ship. The fatigue calculation has been carried out for all combinations of HS , TP , direction and speed using the following ranges: – – – –

Significant wave heights: 20 evenly spaced values between 0.1 and 20.0 m Wave periods: 20 evenly spaced values between 1.3 and 25.0 s Directions: 0, 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330◦ Speed: Corresponding to the hydrodynamic model, that is 0, 3, 6, 9, 12, 15, 18. 21 and 24 knots. If there are speeds above max speed (24 knots), they are set to be the maximum speed.

The fatigue result for a vessel of a given speed in a given sea state and with a given angle of encounter is estimated by linear interpolation of the nearby calculated results. The wave environment is described by the Pierson-Moskowitz spectrum and a spreading power of 2 [17] and the 2-parameter Weibull distribution is used to calculate the long-term stresses according to [6]. The stresses are found at the 5 × 10−3 and 1 × 10−4 probability levels and used to fit the Weibull parameters, ξ (shape) and q (scale). The fatigue damage, D, can then be estimated based on the two-slope (bi-linear) S-N curve: 

D = ND

qm Γ K2



1+

m ; ξ



Δσq q

ξ 

+

q m+Δm γ K3



1+

m + Δm ; ξ



Δσq q

ξ 

≤η

(5) where Δσq is the stress range where the change of slope occurs, K2 , m are S-N fatigue parameters for N < 107 cycles, K3 , m + Δm are S-N fatigue parameters for N > 107 cycles and γ(·; ·), Γ (·; ·) are the incomplete and complimentary incomplete Gamma functions, respectively, to be found in standard tables. The DNV GL S-N curve D (hot spot curve; FAT 90) has been used and this is further described in [18]. An additional stress concentration factor of 1.33 has been used as assumed in the measurements. The results from these calculations is a fatigue rate in terms of damage per second and this must be multiplied with the number of seconds in the sea state and summed up for each sea state to get the cumulative fatigue damage for the detail in question. The hull monitoring system supplies the damage for each 30 min interval as the ratio between actual measured damage and design damage. The same methodology has been used to estimate a ratio between the calculated fatigue and the design fatigue. The design life used for the calculation has been 20 years. Design life ratio = Damage rate [1/s] * 60 s * 60 min * 24 h * 365 days * 20 years

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Note that weather data have not been available for some data-points, and these points have been removed from the analysis. 2.3

Difference Between Measured and Calculated Fatigue Rates

The data that will be subject to the statistical analysis are simply the difference between the measured and the calculated fatigue rates. As all the data have been normalized with respect to target design life in the same way, the measured and the calculated fatigue rates should be directly comparable. Trace plots of measured fatigue rates (for both starboard and port sides) compared with calculated fatigue rates as well as the difference between the measured and the calculated values are shown in Fig. 2. Included in the plots are also the cumulative fatigue damage for both measured and calculated fatigue as well as the cumulative difference. The top plot shows results for starboard side, the middle plot shows for port side and the bottom plot shows the differences. Note that the calculated fatigue does not distinguish between starboard and port sides and are only induced by vertical wave bending. Hence, the calculated values in the top and middle plots are identical. The number of zero-values that has been removed from the dataset is also reported in the figure, and this turns out to be about 70% of the data. This is due to lack of weather data in some locations and also the fact that time-points with 0 measured or calculated fatigue rates have been removed. These are typically associated with port time, channel operations or transit in the Mediterranean (where wave data have not been available). Some interesting observations that can be made from these plots are that the calculated fatigue rates are consistently lower than the measured fatigue rates. However, this can be explained by the fact that the calculated rates only accounts for vertical bending moments, whereas other effects may be included in the measured rates, particularly in oblique seas. Another interesting observation is that the measured accumulated fatigue damage for this vessel is quite low, and well within the design life for the time period when data have been collected, indicating healthy operation of the vessel. Pairwise scatterplots showing how the response variables (fatigue rate difference) vary with the explanatory variables are shown in Fig. 3. The differences between measured fatigue rates on either side of the vessel and calculated fatigue rates are highly correlated, with a nearly linear relationship. However, the dependence between the response variables and the explanatory variables seems to be less clear. For example, for the wave period, TZ , the difference typically seems to become large for a particular value. Presumably, this can be explained by the resonance frequency of the ship structure. Similar relationships can also be seen for ship speed and relative wave direction.

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Fig. 2. Comparing measured and calculated fatigue

Fig. 3. Pairwise scatterplots of the variables

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525

Methodology: Statistical Models

Three types of statistical regression models are utilized in this study, that is, generalized additive models (GAM), Gaussian processes (GP) and regression trees (RT). The response variable will be denoted Y and the explanatory variables will be denoted X = (X1 , X2 , . . . , Xp ) and the basic problem is to construct a prediction rule for predicting Y conditioned on the explanatory variables; Yˆ = f (X). 3.1

Generalized Additive Models (GAM)

The generalized additive model has the form Y =α+

p 

fj (Xj ) + ε

(6)

j=1

where fj (·) are smooth functions of one covariate only specified by regression splines. Estimation of the smooth functions to fit the data as good as possible and to be as smooth as possible can be formulated as minimization of a penalized sum of squares, where a tuning parameter λ is introduced to control the degree of smoothing as follows: ⎛ ⎞ p  N N    ⎝ ⎠ yi − α − fj (xij ) + λj fj (tj )2 dtj (7) P RSS(α, fj ) = i=1

j=1

j=1

t

Interaction terms between two or more explanatory variables may be included in the models. The models have been fitted using R [19] and the mgcv -package. For the GAM-models, model evaluation is based on the adjusted R2 -value which describes how well the model fit to the data. Essentially, the R2 value, sometimes also referred to as the coefficient of determination, measure how large proportion of the variation in the data that is explained by the model. Hence, a larger value of R2 indicates a better fit to the data. There are several definitions of the coefficient of determination, but one general definition is R2 = 1 −

SSres SSmod = , SStot SStot

(8)

where SStot denotes the total variation in the data (total sum of squares), SSmod is the variation in the data that is explained by the model (explained sum of squares) and SSres is the variation in the data that is still unaccounted for by the model (residual sum of squares). The adjusted R2 is introduced to adjust for the fact that a model with additional explanatory variables will always get higher R2 than a simpler model, even if the added parameters are insignificant, and includes a penalty for the number of explanatory variables. In this way, models of varying complexity can be compared. The adjusted R2 is defined as n−1 2 Radj , (9) = 1 − (1 − R2 ) n−p−1 where n is the sample size and p is the total number of explanatory variables in the model.

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Gaussian Processes Models (GP)

A stochastic process is a collection of random variables typically associated with a set of continuous indices such as space, time or any other input variables. A Gaussian process is a stochastic process where every realization of the process, that is, every finite collection of the random variables, have a multivariate normal (Gaussian) distribution, and it can be regarded as an infinite-dimensional generalization of the multivariate normal distribution. The distribution of a set of multivariate normal random variables is fully specified by its mean vector μ, specifying the expected value of each random variable, and its covariance matrix Σ, specifying the variances of each random variable as well as the covariance between any pair of random variables. The density of a collection of random variables X T = (X1 , X2 , . . . , Xk ) which are T multivariate normally distributed with mean vector μ = (μ1 , μ2 , . . . , μk ) and covariance matrix ⎞ ⎛ 2 σ1 ρ1,2 ρ1,3 · · · ρ1,k ⎜ρ2,1 σ22 ρ2,3 · · · ρ2,k ⎟ ⎟ ⎜ ⎜ .. .. ⎟ .. ⎜ . ··· . ⎟ Σ = ⎜ . ··· ⎟, ⎜ . .. ⎟ . . . ⎝ . ··· ··· . . ⎠ ρk,k−1 σk2 ρk,1 · · · where σi2 is the variance of Xi and ρi,j = ρj,i is the covariance between Xi and Xj , is as follows: f (x) =

1 (2π)k |Σ|

1

e− 2 (x−μ)

T

Σ −1 (x−μ)

.

(10)

One important feature of the multivariate normal distribution is that if a set of random variables are jointly multivariate normally distributed, then the marginal distribution of any of the random variables will be Gaussian. That is, if X ∼ M V N(k) (μ, Σ), then Xi ∼ N (μi , σi2 )∀i ∈ {1, . . . , k}. Moreover, the conditional distribution of any of the random variables given the others will also be normal. In Gaussian processes, the mean vector is replaced by the mean function and the covariance matrix is replaced by a covariance function, which are typically continuous functions determining the process’ behaviour over the input space. Having specified these functions, the Gaussian process is fully specified over the input space. For a homogeneous Gaussian process, the covariance function, κ(x, x ) will only depend on the distance between points in the input space, d = |x − x |. In Gaussian processes regression, the properties of Gaussian processes are exploited, and if a random process can be modelled as a Gaussian process, it may be used to predict the distribution at unobserved points in the input space. Such predictions will not only be point predictions, but it will provide the full distributions at unobserved points and hence also provide an estimate of the uncertainty. Given a set of observations from the Gaussian process, predictions may be the conditional distributions given the observed values. Typically,

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in Gaussian processes regression, the dependent variables y are modelled as a Gaussian process over the space of the input variables, x, and the training data set is used to estimate the mean- and covariance functions as well as to provide predictive distributions for unobserved points in input space. In order to train a Gaussian process model, there is a need to specify the type of covariance function and in this study the radial basis kernel function is used, which is a function of the squared Euclidean distance between two points in input space. This kernel function has one hyper parameter, the inverse kernel width, σ, and takes the following form: 1

κ(x, x ) = e− 2σ2 ||x−x



||2

.

(11)

A nugget effect or variance may be included to account for noisy data, and this may typically be a constant value, for example on the form cδ(i, j) where δ(·) denote the delta function and c is the noise variance. Estimation can be done by maximum likelihood or by Bayesian methods [20]. The models in this study have been fitted using R and the kernlab-package. 3.3

Regression Trees (RT)

Regression trees represent a different approach to regress the dependent variable on the independent variables by dividing the input-space into regions and then estimate a simple predictor in each region. One advantage of tree-based methods is that they can be used to analyse the relative importance of each explanatory variable, Xj , in explaining the variation in the response variable, Y . However, since there are no functional relationship between the covariates and the response variable, direct interpretation of the influence of each input variable on the response is not straightforward. To grow a regression tree, first divide the X-space into M regions Rm , Rp = R1 ∪ R2 ∪ · · · ∪ RM

(12)

In each region, let the predictor for Y be flat by just taking the regional average fˆ(x) =

M 

cm I(x ∈ Rm )

(13)

m=1

where cˆm = ave(yi |xi ∈ Rm ). First, grow a preliminary tree, T0 with a large number of regions, M0 = |T0 |. Start with one region: fˆ(x) = ave(yi ) (one node) and find the variable xj and a split of it, s, such that the sum of square errors is minimized. Keep this pair (j, s) and continue with new split points until a relatively large number of nodes M0 is obtained. This tree will typically give over-fitting and needs pruning, i.e. removal of some of the nodes, to obtain a tree with M = |T | nodes or regions, where |T |≤ |T0 |. Random forest is an extension of regression trees that consist of fitting an ensemble of regression trees to the data and then let the predictions be the mean prediction of the individual trees. This will reduce the chance of overfitting a tree to the training data. However, in this study, only individual regression trees have been applied. The models have been fitted using R and the rpart-package.

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Statistical Analysis and Results

Three types of models have been applied to the difference between measured and calculated fatigue rates, as outlined above. That is, generalized additive models (GAM), Gaussian process models (GP) and regression trees (RT). In all cases, regression of the differences on the following four explanatory variables will be made for each individual ship: Significant wave height (HS ), zero up-crossing wave period (TZ ), ship speed and relative wave direction. Both the differences between port-side and starboard-side measured fatigue rates and the calculated fatigue rates will be analysed. 4.1

Generalized Additive Models

Three alternative GAM-models have been fitted to the data on the difference between the measured (port and starboard, respectively) and the calculated fatigue rates, as described above. The first model includes smooth terms for all the four explanatory variables, as well as a fixed term. The two alternative models include an interaction term between HS and TZ and interaction terms between HS and TZ and between ship speed and relative direction. Model fit is measured in terms of the adjusted R2 -value in Table 1. It is observed that the GAM-models are able to explain the variation in the data quite well, and somewhat better for the starboard side. Moreover, the addition of the first interaction term in the models improves the fit notably, indicating that such terms should be included in the model. However, the effect of the second interaction term appears to be negligible. GMS and GMP denote global midship starboard and port sides, respectively and refers to the measured fatigue rates from two sensors. 2 Table 1. Radj for model evaluation and comparison

GAM 1 GAM 2 GAM 3 No interaction term +ti (HS , TZ ) +ti (HS , TZ ) + ti (speed, dir) GMS 0.590

0.789

0.789

GMP 0.548

0.731

0.731

For all of the models the HS and TZ variables are significant, indicating that the environmental conditions are influential for the deviations between measured and calculated fatigue rates. Also the interaction term between HS and TZ are significant. However, the effect of ship speed and relative direction turns out to be not statistically significant for this particular case. Moreover, these smooth terms, including the interaction term, are highly non-linear. One advantage of GAM-models over more complicated models such as projection pursuit models and neural nets is that the results are easily interpretable. That is, due to its additive nature, it is possible to plot the estimated smooth functions to see how

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the response variable varies with the values of the various explanatory variables and this is shown in Fig. 4 (for port side only). The estimated functions illustrate how the response variable (difference in measured and calculated fatigue rates) varies with the different explanatory variables, if the value of all other explanatory variables remain unchanged. In this particular case, one may see that the effect of significant wave height is the most notable, and that this relationship is highly non-linear and with larger deviations for higher significant wave heights. In the rightmost plot, the effect of the interaction terms are also shown.

Fig. 4. Estimated smooth terms for the difference between calculated and measured (port side) fatigue rates; Without interaction terms (left) and with two interaction terms (right)

Some diagnostic tools are available for checking the adequacy of the GAM models, including the size of the basis and various residual plots such as qqplots and histograms as well as plots of observed values against fitted values and plots of residuals against the linear predictor. An example of a set of diagnostics plots are shown in Fig. 5 for the port side measurements. The first plots are quantile-quantile plots for the residuals. If the model assumption is correct then the qq-plot should follow a straight line. In the example shown here, the residuals appear to have somewhat fatter tails than the Gaussian distribution, indicating that the Gaussian assumption is not ideal. However, the third plot provides a histogram of the residuals and this shows that the residuals are symmetric around 0, and failure to capture the exact form of the error distribution might not be a serious problem if the model are to be used for prediction. The second plot shows the residuals against the linear predictor, and the plots reveal no particular patterns indicating that the model is reasonable. Finally, the last plot shows the actual response variable against the estimated response, and it is clear that increasing values of the response generally corresponds to increasing

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estimates. Hence, even though the model is not perfect, these simple diagnostic plots indicate that the model is reasonable.

Fig. 5. Diagnostic plots for model checking (port side only); Without interaction terms (left) and with two interaction terms (right)

4.2

Gaussian Processes Models

Gaussian processes regression is applied to the same dataset, regressing the difference between measured and calculated fatigue rates on the same explanatory variables. Gaussian processes regression assumes that the random variables being observed comes from a Gaussian process and estimates a mean function and the parameters of the covariance function. In this way, a response surface can be build up in the feature space that may be used for prediction at any point. Gaussian processes regression is performed in R with the package kernlab. A Gaussian kernel function is assumed, taking one parameter. Moreover, a mean function can be estimated over the input space. As outlined in [21] (eq. (2.27)), the mean function can be found from the α-vector and the kernel function, both of which are estimated in the Gaussian process model. Note however, that the nugget effect or initial variance is not estimated and is specified as input. In order to evaluate the model fit to the training data, the training error is reported in Table 2. It is observed that also the Gaussian Processes models perform better for the GMS data (starboard). Examples of the fitted response surface for different values of speed and direction in (HS , TZ ) space are shown in Fig. 6 for GMS; the fitted models emulate how the difference vary in the multi-dimensional space. The red stars on the surfaces represent data-points within (HS , TZ )-space in a bin around the speed and direction values. It can be observed that the difference is mostly small and centred around 0, but for some combinations of speed, relative direction, HS and TZ , the difference becomes quite notable.

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Table 2. Training error for model evaluation and comparison for the selected vessel GMS (Starboard) GMP (Port) 0.195

0.245

Fig. 6. Examples of response surfaces estimated by Gaussian process models in (HS , TZ )-space for some vessel speeds and relative wave directions

In Fig. 7, plots of the estimated function of the response together with confidence bounds based on the estimated variances are shown for each explanatory variable. This illustrates how the uncertainty varies for different values of the explanatory variables. However, simultaneous variation of several input parameters are not accounted for and the full model would give different curves for different combinations of the other input variables, corresponding to different slices of the input hypercube. However, some interesting observations can be made. For significant wave height, the model predicts increasing differences for increasing values of HS . For the wave period, there seem to be a peak with large differences around Tz = 10 s. Moreover, there seems to be certain periods with small uncertainty, presumably due to much data in this region. For speed, there seems to be small peaks between 4 and 8 knots and generally wider confidence bound for lower and higher speeds. For relative direction, there are notable peaks for directions around 145 and 225◦ , corresponding to ±45◦ from head sea. It should be realized, however, that the plots correspond to one slice of the input hypercube but they illustrate how uncertainty information can be achieved by estimating the standard deviation from the Gaussian process models.

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Fig. 7. Examples of response curves for the different input variables with confidence bounds. The black curve shows the predictive mean and the red lines are predictive mean ±2× predicted standard deviation (starboard side)

4.3

Regression Trees

Two regression trees is fitted to both GMS and GMP-differences. The first tree is an initially fitted tree and this is later pruned additionally to arrived at the final tree. The results of fitting such a model is contained in the information about the various splits as well as the predicted response within each region of the input space. The coefficient of determination is calculated based on the residuals between the predicted and observed values in the dataset and the adjusted R2 values for the initial and pruned regression trees are presented in Table 3. It is seen that these models perform slightly better for the port side data. The results of fitting a regression tree to the data can be illustrated by making a plot of the tree, and examples of this are shown in Fig. 8 for the port side.

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Table 3. Adjusted coefficient of determination for the pruned regression trees for the individual vessels GMS (Starboard) GMP (Port) Initial Pruned Initial Pruned 0.860

0.767

0.861

0.784

Fig. 8. Regression trees showing the splits in the input variables and the resulting nodes, initial tree (left) and pruned tree (right); port side

From the plotted trees, one may see the main splits in the input space, and the first split in both trees is for significant wave heights above or below 4.5 m. That is, for higher significant wave heights than this, the differences between measured and calculated fatigue rates are greatest. Moreover, the largest differences according to the pruned tree for GMP is for significant wave heights above 4.5 m, wave periods larger than 8.8 s and speed less than 5.8 m/s. One may also look at the relative importance of the input variables in explaining the variation in the response value. For all four trees, the most important variable is the significant wave height, followed by, in order of decreasing importance, wave period, relative direction and speed over ground.

5

Discussion

This paper has investigated three types of regression models to describe the variability of the differences between measured and calculated fatigue rates for ships as a function of various input variables. The results indicate that such models are able to describe the variations in the data quite well for the vessel considered in this study. Similar models have been fitted to data from other container ships, and results vary between vessels. Some models have also been

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tried fitted to combined data from several ships, where additional explanatory variables related to ship dimensions have been included. This is deemed important if the models are to be applied to other vessels than the ones being used to train the models. However, these models did not perform very well, indicating that they do not easily generalize. The reason for this is not clear but could be due to important information that are missing. For example, no information about loading conditions have been included in the models, and this could be important factors influencing the fatigue rates. Another issue is how good the wave description has been, both with respect to wave height and period and also relative to the wave energy spreading and bi-directional sea from wind sea and swell. Moreover, only the vertical bending moment has been calculated, but other structural responses may be included in the sensor measurements that influence the measured fatigue rates, especially in off head sea directions. Wave induced vibration has been neglected, but could possibly be included in future models. In this analysis, data of summary statistics for 30-min time intervals have been used. That means that the fatigue rates have been measured and calculated based on the average conditions over consecutive 30-min intervals. If there are large variability within these intervals, this will not be reflected in the explanatory variables, even if they are accounted for in the measured fatigue rates. Moreover, the data for the sea state conditions are initially given only every 3 h and these data have been translated to 30-min resolution by simple interpolation. Hence, if there are large variability in the wave conditions within each 3-hour period, this will not be reflected in the data, and is an additional source of uncertainty. Only three types of regression models have been explored. There are numerous other methods that could be investigated, and other alternatives that could be considered are projection pursuit models, neural networks and random forests (see e.g. [22–24]). The aim has been to explore whether statistical models can be fit for the expected error from numerical fatigue calculations and further study would be needed to establish the best model. This is out of scope of the current paper and it is merely suggested that improved models may exist for these types 2 -values for GAM and regression trees, it is clear of data. Comparing the Radj 2 -values, but that the regression trees appear to perform best, with higher Radj such models are also known to be prone to overfitting.

6

Summary and Conclusions

This study has investigated the difference between calculated and measured fatigue rates for a selected container vessel. The aim has been to develop statistical methods that can describe these differences as functions of various explanatory or input variables. Numerical analyses in combination with such statistical models could then achieve more accurate predictions of the responses. In this study, information about prevailing sea states as well as ship speed and relative direction have been utilized in the analysis of these differences.

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The results indicate that the models perform reasonably well in describing the 2 -values greater than 0.9 is obtained indicating data for the selected vessel. Radj a good fit to the data. Hence, it is demonstrated that it is possible to establish statistical models to describe and predict the errors in numerical fatigue calculations as a function of a few selected explanatory variables. Some models were tried out for other ships and for a group of ships. This showed that separate models for individual ships perform reasonably well, but that it is difficult to generalize to other ships. In practice, this means that training data need to be available for all ships one want to predict using the model and indicates that it is challenging to generalize these models to a whole fleet of ships. Possibly there are other parameters that distinguishes the error in the fatigue calculations than merely the ship dimensions and additional explanatory variables are needed. The reason for this has not been determined, but clearly demonstrates that there are large individual differences between the different ships. Notwithstanding these difficulties in generalizing to the whole fleet, if measured and calculated fatigue rates are collected for a particular vessel and used to fit a statistical model, it may be used in calibrating and accounting for errors in calculated fatigue rates. This could potentially be used in maintenance planning and inspection. Acknowledgments. The study presented in this paper is partly carried out within the centre for research-based innovation, BigInsight.

References 1. Fricke, W.: Fatigue and fracture of ship structures. In: Encyclopedia of Maritime and Offshore Engineering. Wiley (2017) 2. Cramer, E.H., Løseth, R., Olaisen, K.: Fatigue assessment of ship structures. Mar. Struct. 8, 359–383 (1995) 3. Storhaug, G.: Measurements and structural monitoring. In: Encyclopedia of Maritime and Offshore Engineering. Wiley (2017) 4. Hirdaris, S., White, N., Angoshtari, N., Johnson, M., Lee, Y., Bakkers, N.: Wave loads and flexible fluid-structure interactions: current developments and future directions. Ships Offshore Struct. 5, 307–325 (2010) 5. DNV GL: Rules for classification: Ships. part 3 hull. chapter 9 fatigue (2018). DNVGL-RU-SHIP Pt.3 Ch.9 6. DNV GL: Class Guideline: Fatigue assessment of ship structures. DNV GL (2018). DNVGL-CG-0129 7. IACS: Common structural rules for bulk carriers and oil tankers (2014) 8. The Transportation Safety Board of Canada: Break-up and Sinking. The Bulk Carrier “FLARE” Cabot Strait 16 January 1998 (1998). Marine Investigation Report M98N0001 9. Ask, M.: Ageing of ships, LPG tankers. University of Stavanger. Master thesis (2015) 10. Mao, W., Ringsberg, J.W., Rychlik, I., Storhaug, G.: Development of a fatigue model useful in ship routing design. J. Ship Res. 54, 281–293 (2010)

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11. Mao, W., Ringsberg, J., Rychlik, I., Li, Z.: Theoretical development and validation of a fatigue model for ship routing. Ships Offshore Struct. 7, 399–415 (2012) 12. Li, Z., Ringsberg, J.W., Storhaug, G.: Time-domain fatigue assessment of ship side-shell structures. Int. J. Fatigue 55, 276–290 (2013) 13. Li, Z., Mao, W., Ringsberg, J.W., Johnson, E., Storhaug, G.: A comparative study of fatigue assessments of container ship structures using various direct calculation approaches. Ocean Eng. 82, 65–74 (2014) 14. Storhaug, G.: The measured contribution of whipping and springing on the fatigue and extreme loading of container vessels. Int. J. Naval Archit. Ocean Eng. 6, 1096– 1110 (2014) 15. Li, Z., Ringsberg, J.W.: Fatigue routing of container ships - assessment of contributions to fatigue damage from wave-induced torsion and horizontal and vertical bending. Ships Offshore Struct. 7, 119–131 (2012) 16. Mao, W., Rychlik, I., Storhaug, G.: Safety index of fatigue failure for ship structure details. J. Ship Res. 54, 197–208 (2010) 17. DNVGL: Environmental conditions and environmental loads. DNV GL. DNVGLRP-C205 (2017) 18. DNVGL: Fatigue design of offshore steel structures. DNV GL. DNVGL-RP-C203 (2016) 19. R Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria (2017). https://www.R-project. org/ 20. Williams, C.K., Rasmussen, C.E.: Gaussian processes for regression. In: Touretzky, D.S., Mozer, M.C., Hasselmo, M.E. (eds.) Advances in Neural Information Processing Systems 8. MIT Press (1996) 21. Rasmussen, C.E., Williams, C.K.: Gaussian Processes for Machine Learning. MIT Press (2006) 22. Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning, 2nd edn. Springer (2009) 23. Vanem, E.: Statistical methods for condition monitoring systems. Int. J. Condition Monit. 8, 9–23 (2018) 24. Brandsæter, A., Vanem, E.: Ship speed prediction based on full scale sensor measurements of shaft thrust and environmental conditions. Ocean Eng. 162, 316–330 (2018)

Optimization of Superstructure Connection Design Base on Fatigue Strength Analysis Xianyin Chen1(B) , Wenyuan Zeng1 , and Shuo Li2 1 Department of Military Ship Design and Research, Marine Design and Research

Institute of China, Shanghai, China [email protected] 2 Department of Design, SAIC Motor Passenger Vehicle Company, Shanghai, China

Abstract. For warships with long superstructure and complex cabin structure, the obvious stress concentration at the superstructure end had generally led to serious fatigue problem. Taking a certain type of warship as an example, this paper had determined the serious fatigue problem of the joint of the superstructure end and the main hull by finite element calculation of the whole ship. Comparing the fatigue life of different structures, this paper provided an optimum structural design scheme for typical nodes of warship’s superstructure end. A real scale model fatigue experiment was designed in the research of this paper to obtain the fatigue life of nodes under typical load conditions, followed by the fitting of an S-N curve based on least square method and maximum likelihood method accordingly. The typical node’s fatigue strength was evaluated based on the fitted S-N curve. The fatigue life results were compared with those based on the existing S-N curves of CCS (China Classification Society) rules. The assessment results indicated that the fatigue life obtained by the experimental fitting S-N curve was larger. The experimental results and the fitted S-N curve have certain reference significance for the structural design based on fatigue strength of the ship’s superstructure. Keywords: Fatigue strength evaluation · Optimum structural design · Real scale model fatigue experiment · S-N curve

1 Introduction Fatigue damage is one of the main forms of structural damage of large and super large ships [1]. Fatigue problems are particularly serious in those large warships with complex hull structures and various sailing conditions [2]. These kinds of ships usually possess long superstructure, which to a large extent participate in the longitudinal bending of hull girders. The stress concentration is obvious because of the sudden decrease of the transverse section modulus of superstructure end, which is the key focus of ship structure design [3, 4]. In order to solve this problem, a certain type of warship with a long superstructure and complex cabin structure was taken as a study example. Through the finite element calculation of the whole ship, a serious fatigue problem was identified © Springer Nature Singapore Pte Ltd. 2021 T. Okada et al. (Eds.): PRADS 2019, LNCE 64, pp. 537–547, 2021. https://doi.org/10.1007/978-981-15-4672-3_34

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in the joint of the superstructure end and the main hull. The fatigue lives of different structural forms were compared to obtain the optimized design scheme of the typical joint at the superstructure end. A full-scale model fatigue experiment was designed to obtain the fatigue life of the joints under typical load conditions. Based on the maximum likelihood method, the S-N curve specialized for this kind of joint was obtained and compared with the existing CCS rule S-N curve.

2 Whole Ship Spectrum Analysis Fatigue Strength Calculation The spectral analysis direct calculation method of fatigue assessment is a widely accepted method which can truly reflect the fatigue performance of ship hull. The wave loads of fatigue strength assessment are computed by wave load calculation software and applied to the finite element structure to obtain the fatigue stress response and stress range, and then the fatigue life of the checked structure can be acquired through the calculation of cumulative damage degree. In this study, a certain type of warship was selected as the research object. The main dimensions and related parameters of the ship were shown in Table 1. The finite element model of the whole ship was established (as Fig. 1). A total of 2,788 nodes related to fatigue strength assessment were selected in the model according to structural strength rules, which covered all types of typical nodes of the whole ship, and all nodes were classified according to their structure type. The spectral analysis method was adopted to calculate the fatigue strength of the whole ship. The fatigue strength calculation of the ship was conducted using spectrum analysis method. 4 typical loading conditions of the example ship (normal displacement, full load displacement, displacement of returning to the port, maximum displacement) were selected as the working condition of this calculation study. Hypothesis was set that the probability of each loading condition is the same in the whole life of the ship, namely time distribution coefficient α = 0.25. The wave loads were calculated by Compass-Walcs-Basic, the wave load calculation software based on three dimensional potential flow theory, which is developed by Institute of Mechanics of Harbin Engineering University. The stress response transfer functions and stress response spectrums of the hull structure in each regular wave were calculated by applying wave load to the finite element model of the ship hull, and the cumulative fatigue damage degrees and fatigue lives in the 20-year fatigue recovery period were finally obtained. The nodes with the shortest fatigue life among all node types were sorted out to be the focused areas of fatigue assessment [5]. The calculation conditions and parameters of the wave load were shown in Table 2. The types of nodes with serious fatigue problems and the calculation results of fatigue life were shown in Table 3. It can be seen from the calculation results of spectrum analysis fatigue strength of the whole ship that the fatigue problem of corner of superstructure end was the most prominent. As the superstructure of the ship was built high and long, and end at the middle of the ship, where the total longitudinal bending moment of the ship girders reached its maximum, which led to a sudden decrease of the transverse section modulus at superstructure end. As a result, an obvious stress concentration existed at the connection of superstructure end and the main deck.

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Fig. 1. Whole ship Patran finite element model

Table 1. Hull main dimensions and related parameters Items

Unit

Value

Items

Unit

Value

Length

m

145

Diamond coefficient

–

Width

m

22.8

Waterplane coefficient

–

0.8046

Depth

m

10.8

Maximum section coefficient

–

0.9478

Draught

m

7.2

Design waterline radius angle

º

Block coefficient

–

0.5527

Buoyant center

m

0.5832

10.2 1.618

Table 2. Wave load response parameters Items

Unit

Value

Speed U

Knots

14

Heading angel θ

deg

0°–330° (Every 30°, Total: 12)

Probabilities of each heading P

–

1/12

Wave circular frequency ω

rad/s

0.1–1.8 (Total: 18)

Table 3. Results of fatigue life calculation Joints types

Fatigue degree (D) Fatigue life (years) Results evaluation

Corner of superstructure end

1.02

19.60

30 years

Intersection of girder & plate

0.61

32.78

>30 years

Intersection of side frame & deck 0.23

86.96

>30 years

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3 Optimum Structural Design The width of the superstructure of this ship was smaller than that of the main hull, and the superstructure ended on the main deck. In order to reduce the stress concentration in this area, the rear wall and the side wall of the superstructure were connected by circular arc overplate. A 16 mm thick cladding plate was added on the main deck. For the structure form of the optimized joint, three optimization schemes were given in this section: thickening scheme, vertical stiffener scheme and horizontal stiffener scheme. The lightweight principle was always the to-be-followed principle in ship structural design, which means to reduce the structural weight as much as possible on the premise of ensuring the structural strength. Stiffener layout is a common method to strengthen structural strength in structural design. In this study, the thickening scheme of plate thickness was added to compare the optimization effect of different structural optimization schemes. The strengthening scheme of vertical + horizontal stiffeners is usually used in the longitudinal bulkhead of large ships which mainly bear the total longitudinal bending moment. For the consideration of lightweight structure and avoiding excess strength reserve, the strengthening scheme of vertical + horizontal stiffeners was not considered in this study. 3.1 Thickening Scheme The thickness of the side wall panel at the optimized joint was modified continuously to obtain the appropriate thickness value that could meet the fatigue design requirements. The thickness of the side wall panel was eventually increased from original 10 mm to 20 mm. The fatigue calculation results of each structure type at the optimized joint before and after the optimization were shown in Table 4; the structure form of the optimized joint was shown in Fig. 2, the stress nephogram was shown in Fig. 3, and the RAO curves were shown in Fig. 4. Table 4. Optimization calculation results of thickening scheme (B-Before, A-After) Structure type

Fatigue degree (D)

Fatigue life (years)

Results evaluation

B

A

B

A

B

A

Intersection of side wall & cladding plate

1.85

0.58

10.80

33.92

30

Intersection of rear wall & stiffener (I)

4.35

0.36

4.58

54.32

30

Intersection of rear wall & stiffener (II)

6.91

0.87

2.89

22.95