Trends in the analysis and design of marine structures : proceedings of the 7th International Conference on Marine Structures (MARSTRUCT 2019, Dubrovnik, Croatia, 6-8 May 2019) 9780367278090, 036727809X, 9780429298875, 0429298870, 9781000024364, 1000024369, 9781000024449, 100002444X, 9781000024524, 1000024520

Table of contents :
Content: Cover
Half Title
Title Page
Copyright Page
Table of contents
Preface
Organisation
Methods and tools for loads and load effects Probabilistic models for extreme wave loads
Extreme loads estimation using genetic algorithm approach
Offshore responses using an efficient time simulation regression procedure
Evaluation of extreme design wave loads for hull girder strength assessment
Wave scatter diagrams
Impact of weather source selection on time-and-place specific vessel response predictions Study on design loads for fatigue strength assessments using Automatic Identification System (AIS) dataIce modelling
Peridynamic approach for modelling ice-structure interactions
Development of compressive failure test and analysis method for ice
An ice material model addressing the influence of strain rate, temperature, confining pressure and porosity
Non-Gaussian and non-linear effects
A frequency domain approach for wide-band non-Gaussian process
Estimation of Volterra kernel coefficients of a nonlinear dynamic system Efficient methods for the prediction of non-Gaussian stochastic response of offshore structureMethods and tools for strength assessment Buckling and ultimate strength of plates and stiffened panels
Residual ultimate strength analysis of stiffened panels exposed to fire
Residual strength of dented stiffened cylinders under combined loads
Buckling behaviour of thin plated structures
Ultimate strength of hull girders
Examination of the dynamic effects on the hull girder ultimate strength of ultra large container ships A cyclic progressive collapse method to predict the bending response of a ship hull girderProgressive collapse analyses of a stiffened box-girder under pure bending
Ultimate strength of pressure hulls
Simulation and evaluation of ultimate strength based safety factor for titanium alloy spherical pressure hull
Residual strength analysis of spherical pressure hull with partial damage
Fatigue
Optimum weight of the torsion box, in terms of fatigue life, of an ultra large container ship
The fatigue assessment of offshore monopile structure considering corrosion Effects of proof loading test on fatigue life of mooring chain linksCrashworthiness
Structural impact of circular buckling fenders on tanker hulls
Numerical simulation for collision performance of subsea cable protector
A simplified method for assessing safety of ship collision event
A new technique to consider hydrostatic and hydrodynamic forces in case of ship collision
Effect of weld modelling on crashworthiness optimization
Ship collision analysis with a floating offshore fish farm
Structural response to impulsive loads

Citation preview

TRENDS IN THE ANALYSIS AND DESIGN OF MARINE STRUCTURES

Proceedings in Marine Technology and Ocean Engineering

BOOK SERIES EDITOR Carlos Guedes Soares EDITORIAL BOARD MEMBERS R. Ajit Shenoi, Fenando Lopez-Peña, Jani Romanov, Joško Parunov, Enrico Rizzuto ABOUT THE SERIES The ‘Proceedings in Marine Technology and Ocean Engineering’ series is devoted to the publication of proceedings of peer-reviewed international conferences dealing with various aspects of ‘Marine Technology and Ocean Engineering’. The Series includes the proceedings of the following conferences: the International Maritime Association of the Mediterranean (IMAM) Conferences, the Marine Structures (MARSTRUCT) Conferences, the Renewable Energies Offshore (RENEW) Conferences and the Maritime Technology (MARTECH) Conferences. The ‘Marine Technology and Ocean Engineering’ series is also open to new conferences that cover topics on the sustainable exploration and exploitation of marine resources in various fields, such as maritime transport and ports, usage of the ocean including coastal areas, nautical activities, the exploration and exploitation of mineral resources, the protection of the marine environment and its resources, and risk analysis, safety and reliability. The aim of the series is to stimulate advanced education and training through the wide dissemination of the results of scientific research. BOOKS IN THE SERIES Volume 1: Advances in Renewable Energies Offshore, 2019, C. Guedes Soares (Ed.). Volume 2: Trends in the Analysis and Design of Marine Structures, 2019, J. Parunov and C. Guedes Soares (Eds.).

Proceedings in Marine Technology and Ocean Engineering (Print): ISSN: 2638-647X Proceedings in Marine Technology and Ocean Engineering (Online): eISSN: 2638-6461

PROCEEDINGS OF THE 7TH INTERNATIONAL CONFERENCE ON MARINE STRUCTURES (MARSTRUCT 2019), DUBROVNIK, CROATIA, 6–8 MAY 2019

Trends in the Analysis and Design of Marine Structures

Editors

J. Parunov Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Croatia

C. Guedes Soares Instituto Superior Técnico, Universidade de Lisboa, Portugal

Front cover image is provided by Shipbuilding Industry Split Inc. Back cover image is provided by Marin Šperanda (Centre for Advanced Academic Studies (CAAS) Dubrovnik, University of Zagreb)

CRC Press/Balkema is an imprint of the Taylor & Francis Group, an informa business © 2019 Taylor & Francis Group, London, UK Typeset by V Publishing Solutions Pvt Ltd., Chennai, India All rights reserved. No part of this publication or the information contained herein may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, by photocopying, recording or otherwise, without written prior permission from the publisher. Although all care is taken to ensure integrity and the quality of this publication and the information herein, no responsibility is assumed by the publishers nor the author for any damage to the property or persons as a result of operation or use of this publication and/or the information contained herein. Library of Congress Cataloging-in-Publication Data Applied for Published by: CRC Press/Balkema Schipholweg 107C, 2316 XC Leiden, The Netherlands e-mail: [email protected] www.crcpress.com – www.taylorandfrancis.com ISBN: 978-0-367-27809-0 (Hbk + CD-ROM) ISBN: 978-0-429-29887-5 (eBook)

Trends in the Analysis and Design of Marine Structures – Parunov & Guedes Soares (Eds) © 2019 Taylor & Francis Group, London, ISBN 978-0-367-27809-0

Table of contents

Preface

xi

Organisation

xiii

Methods and tools for loads and load effects Probabilistic models for extreme wave loads Extreme loads estimation using genetic algorithm approach F. Mauro, L. Braidotti & J. Prpić-Oršić

3

Offshore responses using an efficient time simulation regression procedure S.Z.A. Syed Ahmad, M.K. Abu Husain, N.I. Mohd Zaki, M.H. Mohd & G. Najafian

12

Evaluation of extreme design wave loads for hull girder strength assessment S. Ravinthrakumar, B.J. Leira, S. Haver, S. Ehlers & M. Klein

23

Wave scatter diagrams Impact of weather source selection on time-and-place specific vessel response predictions M.L. Schirmann, M.D. Collette & J.W. Gose Study on design loads for fatigue strength assessments using Automatic Identification System (AIS) data N. Yamamoto, T. Sugimoto & K. Ishibashi

33

42

Ice modelling Peridynamic approach for modelling ice-structure interactions B. Vazic, E. Oterkus & S. Oterkus

55

Development of compressive failure test and analysis method for ice S.R. Cho, J. Jang, H.S. Kang & Y.H. Ko

61

An ice material model addressing the influence of strain rate, temperature, confining pressure and porosity Y. Xu, Z.Q. Hu, J.W. Ringsberg, G. Chen & P. Kujala

66

Non-Gaussian and non-linear effects A frequency domain approach for wide-band non-Gaussian process H.-J. Kim & B.-S. Jang

79

Estimation of Volterra kernel coefficients of a nonlinear dynamic system J.-H. Son & Y. Kim

87

Efficient methods for the prediction of non-Gaussian stochastic response of offshore structure N.A. Mukhlas, N.I. Mohd Zaki, M.K. Abu Husain & G. Najafian

v

93

Methods and tools for strength assessment Buckling and ultimate strength of plates and stiffened panels Residual ultimate strength analysis of stiffened panels exposed to fire L.M. Zhang, H.X. Xue & W.Y. Tang

107

Residual strength of dented stiffened cylinders under combined loads Q. Thang Do, T. Muttaqie, S.-H. Park, H. Kyoung Shin & S.-R. Cho

116

Buckling behaviour of thin plated structures S. Zhang, L. Jiang, J. Tong, L. Zhu & H. Zhou

126

Ultimate strength of hull girders Examination of the dynamic effects on the hull girder ultimate strength of ultra large container ships G. Jagite, H. Le Sourne, P. Cartraud, F. Bigot, Q. Derbanne & Š. Malenica

137

A cyclic progressive collapse method to predict the bending response of a ship hull girder S. Li, Z.Q. Hu & S.D. Benson

149

Progressive collapse analyses of a stiffened box-girder under pure bending Z. Lin, L. Zhu, J.H. Liu & C. Guedes Soares

158

Ultimate strength of pressure hulls Simulation and evaluation of ultimate strength based safety factor for titanium alloy spherical pressure hull S. Yuan, Q. Chen & K. Ye Residual strength analysis of spherical pressure hull with partial damage X.B. Li, D.Y. Sun & H.R. Wu

167 175

Fatigue Optimum weight of the torsion box, in terms of fatigue life, of an ultra large container ship A. Silva-Campillo, M.A. Herreros-Sierra & J.C. Suárez-Bermejo

183

The fatigue assessment of offshore monopile structure considering corrosion M.S. Azad, W. Punurai, C. Sinsabvarodom & P. Asavadorndeja

188

Effects of proof loading test on fatigue life of mooring chain links X. Xue, N.-Z. Chen & Y.C. Pu

195

Crashworthiness Structural impact of circular buckling fenders on tanker hulls A. Grm & M. Perkovič Numerical simulation for collision performance of subsea cable protector using fluid-structure interaction analysis C.Y. Song, C.S. Shim & H.C. Song A simplified method for assessing safety of ship collision event H.-B. Ju & B.-S. Jang A new technique to consider hydrostatic and hydrodynamic forces in case of ship collision J. Choung, I. Kvan & K. Lee

203

210 218

225

Effect of weld modelling on crashworthiness optimization M. Kõrgesaar, J. Romanoff, L. St-Pierre & P. Varsta

231

Ship collision analysis with a floating offshore fish farm Z.L. Yu, J. Amdahl & D. Kristiansen

238

vi

Structural response to impulsive loads Study on anti-explosion performance of double-layer structures subjected to far-field underwater explosion W.B. Sun, X.B. Li & Z.Z. Ye

249

Application of Lagrangian Energy Approach to determine the response of isotropic circular plates subjected to air and underwater blasts Y.P. Sone Oo, H. Le Sourne & O. Dorival

255

Hydro-plastic slamming Z.L. Yu & J. Amdahl

265

Experimental analysis of structures Structural health monitoring Displacement and stress monitoring of a curved stiffened panel based on inverse finite element method A. Kefal

277

Estimation of the deflection field over a ship structure model based on pointwise measurements F. Saltari, D. Dessi, E. Faiella & F. Mastroddi

285

Structural health monitoring of submarine pressure hull using inverse finite element method M.Y. Li, A. Kefal, B. Cerik & E. Oterkus

293

Fatigue experiments and crack length measurements Crack length measurement in the surface of vessel structures based on computer vision K. Zhang & M.D. Collette Validation and model updating of weld in finite element model of K-node structure using experimental data for fatigue analysis M.L. Larsen, V. Arora, M. Lützen, R.R. Pedersen & E. Putnam

303

310

Model and full-scale measurements Full scale measurements of the hydro-elastic behavior of a 13000 TEUs container ship A. Andoniu, J. de Lauzon & V. Lamaison

319

Bending response of steel-concrete-steel sandwich beam with angled flat-plate shear connectors M. Leekitwattana

329

Experimental testing of spray rails for the resistance reduction of planing crafts M. Lakatos, T. Sahk, R. Kaarma, K. Tabri, M. Kõrgesaar & H. Andreasson Numerical prediction and experimental investigation on hydroelastic loads of a very large container ship in waves Y. Lin, N. Ma, D. Wang & X. Gu

334

344

Materials and fabrication of structures Composites and hybrid materials Adhesive free GFRP-Steel connection for maritime applications J. Jahnke, L. Molter & R. Luterbacher Mus Structural design of helicopter deck with hybrid materials and its joints with conventional steel M. de Vicente, R. Pérez Fernández & M. Toman Fernández

vii

353

359

Flexural behaviour prediction of a conventional composite-to-steel butt-joint of ships using layerwise HSDT N. Kharghani & C. Guedes Soares

369

Experimental and numerical damping behavior analyses of carbon/epoxy laminated plates including viscoelastic material R. Mateu Pastor, H. Le Sourne, P. Cartraud & E. Le Gal La Salle

376

Fibreglass repair behaviour as a function of the scarf angle G.V.W. Rothbarth & R.A.Q. Pinto

385

Dynamic buckling of composite mast panels of sail ships M. Gaiotti, S. Ghelardi & C.M. Rizzo

391

Corrosion A comparison of some multi-parameter distributions related to estimation of corrosion rate of aging bulk carriers Š. Ivošević, R. Meštrović & N. Kovač Implementation of in situ corrosion measurements in structural analysis Y. Wang, E.C. Ilman, N. Sohan Roy, A. Sobey, J. Wharton & R.A. Shenoi

403 411

Weld-induced imperfections Influence of weld-induced distortions on the stress magnification factor of a thin laser-hybrid welded ship deck panel A. Niraula, M. Rautiainen, A. Niemelä, I. Lillemäe-Avi & H. Remes Numerical study on weld residual stress reduction of a stiffened steel plate using vibration R. Pradhan, M.R. Sunny & A. Sarkar

423 433

Methods and tools for structural design and optimisation Advanced structural analysis Tanker structural design with improved hull structural safety P. Prebeg, J. Andric & S. Rudan

441

Research on tank impact resistance mechanism and structure optimization L. Zhang & P.D. Zhao

455

An intelligent design system for lift points arrangement and structural analysis L. He, J. Cui & D. Wang

462

Structural integrity of fixed offshore platforms by incorporating wave-in-deck load N.U. Azman, M.K. Abu Husain, N.I. Mohd Zaki & E. Mat Soom

468

Fatigue design Fatigue life as a factor in assessing warship design flexibility to support batch-building programs D.M. Dwyer, T. Magoga, B.A. Morris & G. Condon

479

Models for design and fatigue analysis of dynamic power cables for wave energy converters J.W. Ringsberg, A. Kuznecovs, S.H. Yang & E. Johnson

488

Structural optimisation methods Weight optimization of an enclosed stressed skin derrick designed for Arctic regions J.W. Ringsberg & Z.Q. Hu

501

Automated structural optimisation of ships’ midship section in concept design phase A. Bayatfar, R. Warnotte & Ph. Rigo

510

A review of artificial intelligence applications in ship structures A. Mikulić & J. Parunov

515

viii

Improving multi-objective structural optimization with a novel constraint-handling method F. Samanipour & J. Jelovica

524

Vibration analysis Vibration analysis in the design stage of the coastal patrol vessel J. Parunov, M. Ćorak, Z. Šperanda, M. Bezovnik & M. Brlić

537

Study on compound vibration isolation technology in ship foundation structure S. Wang, J.X. Yue, X.B. Li, W.J. Tu & L. Chen

544

LNG and FLNG Thermal analysis for LNG tanks of floating storage regasification units K.M. Doshi

553

Impact of cryogenic spills on FLNG hulls T. de Beer, B. Schrier & M. van den Berg

558

Structural reliability, safety and environmental protection Jack-ups Managing risk for damaged offshore structures E. Mat Soom, M.K. Abu Husain, N.I. Mohd Zaki & N.U. Azman System reliability calculation of jacket platforms including fatigue and extreme wave loading H. Khalili, S. Oterkus, N. Barltrop, U. Bharadwaj & M. Tipping Marine growth inspection for jacket structures by behaviour and sensitivity analysis S.M.Y. Wan Alwi, M.K. Abu Husain & N.I. Mohd Zaki

569

576 586

Environmental protection Coupled 3D wave and 2D bottom deposit transportation models for the prediction of harmful phenomena in coastal zone A.I. Sukhinov, A.E. Chistyakov, S.V. Protsenko & V.V. Sidoryakina

597

Suspension and deposit transport models for bottom relief prediction A.I. Sukhinov, A.E. Chistyakov, V.V. Sidoryakina & E.A. Protsenko

604

Vibration study due to controlled trial blasting for rock dredging and berth dismantling J.M. John, N. Saha, R. Sundaravadivelu, S.K. Modi & V.M.S.R. Murthy

610

Coastal structures Foundation for a caisson structure near an island in Indian Ocean R. Sundaravadivelu, N. Saha, S. Sakthivel, V. Rajesh, S.M. Madhumathy & K. Anbazhagan

619

Retrofitting of berthing structure in a major port in East Coast of India R. Sundaravadivelu, S. Sakthivel, R. Kalpana & V. Sasikumar

625

Stability of iron ore berth to handle deep draft vessels R. Sundaravadivelu, S. Sakthivel & S.M. Madhumathy

631

Impact on dry dock extension due to deepening of floor slab R. Sundaravadivelu, S. Sakthivel & K.S. Roja

636

A study on the displacement measurement of fire-structure experiment for verification of thermal-structural analysis M. Suk Ki, B.J. Park, B. Park, K. Lee & S.-Y. Lee

644

Author index

649

Book series page

651

ix

Trends in the Analysis and Design of Marine Structures – Parunov & Guedes Soares (Eds) © 2019 Taylor & Francis Group, London, ISBN 978-0-367-27809-0

Preface

This book contains the papers presented at the 7th International Conference on Marine Structures, (MARSTRUCT 2019), held in Dubrovnik, Croatia between 6 and 8 May. This is the seventh in the MARSTRUCT Conference series and follows on from previous events held in Glasgow – Scotland, Lisbon – Portugal, Hamburg – Germany, Espoo – Finland, Southampton – UK and Lisbon – Portugal, in 2007, 2009, 2011, 2013, 2015 and 2017 respectively. The main objective of the MARSTRUCT Conferences is to provide a specialised forum for academics, researchers and industrial participants to discuss progress in their research directly related with structural analysis and design of marine structures. It was the intention that the MARSTRUCT Conferences be specifically dedicated to marine structures, which complements other conferences on general aspects of ships and offshore structures already available. This series of Conferences is one of the main activities of the MARSTRUCT Virtual Institute, an association of research groups interested in cooperating in the field of marine structures, which was created in 2010 after the end of the Network of Excellence on Marine Structures (MARSTRUCT), which was funded by the European Union. The MARSTRUCT Virtual Institute was created with the same members as the EU project, but with the aim to extend that membership to other interested groups in the future. The 2019 Conference reflects the work conducted in the current trends in the analysis and design of marine structures, including the full range of methods, modelling procedures and experimental results. The aim is to promote knowledge that enables marine structures to be more efficient, environmentally friendly, reliable and safe using the latest methods and procedures for their design and optimisation. This book also deals with the fabrication and new materials of marine structures. The 75 papers are categorized in the following themes and areas of research: • • • • • •

Methods and tools for loads and load effects Methods and tools for strength assessment Experimental analysis of structures Materials and fabrication of structures Methods and tools for structural design and optimisation Structural reliability, safety and environmental protection

The MARSTRUCT 2019 Conference also includes 3 keynote lectures given by experts from academic, administrative and industrial sectors covering ships and offshore structures: • An overview of ship hydroelasticity, Ivo Senjanović & Nikola Vladimir, Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb • Decarbonisation of short sea shipping, Darko Bandula, Independent Expert of European Commission • Structural design of expedition passenger vessel according to Polar Code requirements – case study NB484, Oliver Stanić & Stipe Lambaša, Shipbuilding Industry Split Inc. We hope that the overview coming from experts of different sectors will be an important contribution to the audience. The articles in this book were accepted after a peer-review process, based on the full text of the papers. Thanks are due to the Technical Programme and Advisory Committees who had most of the responsibility for reviewing the papers. We are also grateful to the additional anonymous reviewers who helped the authors deliver better papers by providing them with constructive comments, and hope that this process contributed to a consistently good level of the papers included in the book. J. Parunov & C. Guedes Soares

xi

Trends in the Analysis and Design of Marine Structures – Parunov & Guedes Soares (Eds) © 2019 Taylor & Francis Group, London, ISBN 978-0-367-27809-0

Organisation

CONFERENCE CHAIRMAN J. Parunov, University of Zagreb, Croatia TECHNICAL PROGRAMME COMMITTEE C. Guedes Soares, IST, University of Lisbon, Portugal (Chair) J. Andrić, University of Zagreb, Croatia E. Begović, University of Naples – Frederico II, Italy D. Dessi, INSEAN, Italy L. Domnisoru, University “Dunarea de Jos” at Galati, Romania S. Ehlers, Hamburg University of Technology, Germany Y. Garbatov, Instituto Superior Técnico, Portugal A.M. Horn, DNVGL, Norway Z.Q. Hu, Newcastle University, UK H. Le Sourne, ICAM, School of Engineering, France E. Oterkus, University of Strathclyde, Glasgow, UK J. Parunov, University of Zagreb, Croatia J. Peschmann, DNV GL, Germany J. Prpić-Oršić, University of Rijeka, Croatia P. Rigo, University of Liège, Belgium J.W. Ringsberg, Chalmers University Technology, Sweden C.M. Rizzo, University of Genova, Italy J. Romanoff, Aalto University, Finland M. Samuelides, National Technical University of Athens, Greece S. Sriramula, University of Aberdeen, UK K. Tabri, Tallinn University of Technology, Estonia M. Taczala, West Pomeranian University of Technology, Poland ADVISORY COMMITTEE S. Aksu, Defence Science & Techn. Org., Australia S. Chandrasekaran, Indian Institute of Techn., Madras, India S.R. Cho, University of Ulsan, Korea M. Collette, University of Michigan, USA W.C. Cui, Shanghai Ocean University, China S.F. Estefen, Federal University of Rio de Janeiro, Brazil B.S. Jang, Seoul National University, Korea J. Jelovica, University of British Columbia, Canada H.W. Leheta, Alexandria University, Egypt Y.M. Low, National University Singapore, Singapore N.R. Mandal, Indian Institute of Techn., Kharagpur, India A. Morandi, Gusto MSC, USA L. Moro, Memorial University, Canada P.T. Pedersen, DTU, Denmark

xiii

C. Tian, CSSRC, China D.Y. Wang, Shanghai Jiao Tong University, China G. Wang, Jiangsu Univ. Techn., China X. Wang, ABS, USA Y. Yamada, National Maritime Research Institute, Japan N. Yamamoto, ClassNK, Japan L. Zhu, Wuhan University of Technology, China LOCAL ORGANIZING COMMITTEE J. Parunov, University of Zagreb, Croatia (Chair) J. Andrić, University of Zagreb, Croatia M. Ćorak, University of Dubrovnik, Croatia P. Prebeg, University of Zagreb, Croatia S. Rudan, University of Zagreb, Croatia TECHNICAL PROGRAMME SECRETARIAT Maria de Fátima Pina, IST, University of Lisbon, Portugal Sandra Ponce, IST, University of Lisbon, Portugal

xiv

Methods and tools for loads and load effects Probabilistic models for extreme wave loads

Trends in the Analysis and Design of Marine Structures – Parunov & Guedes Soares (Eds) © 2019 Taylor & Francis Group, London, ISBN 978-0-367-27809-0

Extreme loads estimation using genetic algorithm approach F. Mauro & L. Braidotti University of Trieste, Trieste, Italy

J. Prpić-Oršić Faculty of Engineering, University of Rijeka, Rijeka, Croatia

ABSTRACT: The analysis of time records, coming from seakeeping experiments in irregular waves, is used to determine the occurrence of extreme events. The common procedure used for data analysis is to assume that the statistics of record’s peaks is following two or three parameters Weibull distribution. For particularly severe sea states, it can happen that the peaks assume a multi-modal distribution and the use of a Weibull distribution may lead to errors in the extreme value estimate. It is than possible to use multimodal distributions, or to change the peaks extraction technique, adopting a certain threshold. Here, the determination of extreme values probability distribution parameters by genetic algorithm is applied to improve the methodology of extreme sea state prediction. A data analysis procedure is here proposed and tested on a time record coming from seakeeping model-scale experiments and on a set of wave heights record, in comparison with standard Weibull approach. 1

INTRODUCTION

It can be observed that the peaks distributions extracted from experiments on complex structures or from wave data recording may highlight a multimodal behavior. In such cases, standard analysis is not suitable to reproduce the peaks distribution (Islam et al. 2016). A possible solution is given by the adoption of mixed distributions (Mauro & Nabergoj 2016a), otherwise the GPD approach can be followed (Mauro & Nabergoj 2016b, 2017). In case the first approach is used, then it is necessary to determine a larger amount of regression parameters compared to standard or GPD analysis. For such a reason a fitting procedure based on genetic algorithms (GA) has been used and tested against standard regression methods in order to fit mixed distribution. The same procedure is also applied to GPD analysis and standard Weibull analysis, in such a way to properly compare the resulting extreme values. The procedure will be applied on the modelling of a sea environment (Prpić-Oršić et al. 2007) and on a record coming out forces measurements on a complex structure in severe sea state. The results highlight the differences in the final estimation of extreme values, resulting in different design waves and extreme loads.

During the design process of an offshore vessel/ structure, the estimation and prediction of extreme loads and motions in harsh condition is extremely important. This problem is strongly coupled with the estimation of an appropriate extreme design wave on the bases of hindcast or recorded data (Soares et al. 1996). In both cases, the analysis is based on a series of sampled data relative to a relatively short time compared to vessel/structure life cycle. Moreover, the extreme values are extrapolated in a region lying above the maximum value of the extreme data. That means extremes are scarce of definition and the analysis of extreme values becomes hard due to the lack of available observations. The approach that can be used to analyze the data records, depend on the peaks extraction technique. Theory of the extreme (Hahn & Samuel 1967) gives an indication on the distributions that should be used according to the extraction technique selected for the analysis (Berliant et al. 1996). Once all the peaks are considered, then it is advisable to use a Generalized Extreme Value Distribution (GED) (Gumbel 1958), i.e. according to the Weibull formulation. This is also accepted by the standard ITTC procedures (ITTC 2011). However also the general Gamma distribution is adopted (Ochi 1973) in case of severe sea state modelling. Once the analysis is carried out considering only the peaks above a given threshold, then Generalized Pareto Distribution (GPD) should be used (Haan & Ferreira 2006).

2

EXTREME VALUE THEORY

To model in a correct way the extreme values of a time record analyzing the maxima, Extreme Value

3

The cases not covered by the two above mentioned distributions, are covered by the Weibull distribution. Weibull distribution is also defined with a positive shape parameter. This distribution, especially in the 2 and 3 parameters form is widely used in engineering as, for example, to perform defect data analysis, weather forecasting and, as already mentioned, for the prediction of extreme loads in seakeeping and offshore experiments. For such a reason, in this study, the Weibull analysis is always considered as a reference for the comparison between different approaches.

Theory plays a fundamental role, defining the limiting distributions of the observed phenomena. The first statement in the analysis of a record of observations is the extraction of peaks. Typically, two kind of extraction techniques can be adopted, and depending to the method, the limiting distributions will change. The first method is the so-called Block-Maxima, which is considering the maxima on several periods inside the analyzed record. Once the period is equal to the sample time, then all the maxima of the time series are extracted. Another possibility is given by the extraction of all the peaks above a certain threshold value, adopting the Peaks Over Threshold (POT) method. The limiting distributions that have to be used in according to the selected peaks extraction technique are the GED and the GPD distributions.

2.1.1 2 and 3 parameters Weibull distribution Weibull distribution is usually adopted in two forms, the 2 and the 3 parameter ones. By adopting the cumulative distribution of Equations 1, 2 and 3, a 3 parameters distribution will be described. However, the 2 parameters distribution is still used for preliminary calculations and extreme value evaluation, since it is easier to determine the parameters, and, for some distributions, the location parameter γ can be omitted. When this occurs, the resulting cdf can be expressed as follows:

2.1 GED distribution According to extreme value theory, once the BlockMaxima is selected for peaks extraction, then the limiting law to identify the maxima distribution is given by the Fisher-Tippet-Gnedenko theorem. In such a case the GED distribution should be used to describe the phenomenon. GED can be expressed in a compact form according to the following cumulative density function (cdf): F ( x ) = e −t (x )

F ( x ) = 1 − e − (x

(1)

z=

x −γ η

)−1 / β

if β ≠ 0 if iif β = 0

(4)

where β and η are the same as in Equations 2 and 3. Weibull 2 parameters distribution has a particular property that makes it suitable for easy extreme value determination. In fact, on a log-log plot, the distribution can be represented with a straight line. In case the location parameter γ should be taken into account, then the Weibull distribution should be considered in the 3 parameters form. In such a case, the cdf is exactly equal to the GED case for a non-negative β parameter. The main difference between these two kinds of Weibull distributions is that 2 parameters one is defined for x > 0 while 3 parameters one for x > γ. In case of a 3 parameters distribution, the representation on a log-log plot will no more results as a straight line but will present a concavity or convexity according to the sing of γ. A straight line can be obtained only by considering in the abscissae ln(x–γ) instead of ln(x).

where t(x) is defined as: ⎧( + z t (x) = ⎨ e ⎩

)β

(2) (3)

The cdf results expressed as function of three real constants: β (shape parameter), η (scale parameter) and γ (location parameter). The shape and the scale parameter are defined in (0, +∞), while the location parameter is defined in (−∞, +∞). The shape parameter is defining three particular sub-cases of the GEV distribution. Once the shape parameter is equal to zero (β = 0), then the Gumbel distribution is identified. This distribution is used for data following an exponential distribution. For a positive value of β, then two possibilities can be identified. By adopting a reverse formulation of Equation 1 (changing sign to x axis), the Frechet distribution is obtained. This particular distribution is suitable for populations having a significant amount of data in the tale, being also defined as fat-tale distribution.

2.1.2 Mixed Weibull distribution By analyzing records coming from experimental measurements or calculations (Mauro & Monacolli 2018), sometimes, especially for severe sea state conditions, it is possible to observe that the peaks distribution is showing more than a single convexity/concavity due to the non-linear nature of the phenomenon. This is the case of multi-modal distributions, where the sample data contains more than one significant population. Once this kind of situation is present, a Weibull

4

such a way to ensure a sufficiently accurate estimation of the other unknown distribution parameters. A suitable method to perform the threshold selection is given by the sample mean excess function, which, despite its simplicity, is still considered one of the most appropriate method for threshold selection (Hogg & Klugmann 1984). The function is expressed in the following form:

distribution, also in 3 parameters form, is no more suitable to reproduce the peaks behavior in the extreme region. To describe this model use can be made of mixed Weibull distributions. This particular limiting law is given by the combination of two or more 3 parameters Weibull distributions, resulting in the cdf: N

F ( x ) = 1 − ∑ qi e

β ⎛ x −γ i ⎞ i −⎜ ⎟ ⎝ ηi ⎠

en (u ) =

iif β ≠ 0 iiff β = 0

+

(7)

Xi u

(

e (u) = E X − u X

u

)

(8)

The mean excess function is representative of the expected overshoot of a threshold, given that the exceedance occurs. The sample mean excess function can be evaluated according to a dedicated plot (Embrechts et al. 1997). Once the empirical plot seems to follow a straight line above a certain value of u, then the excesses will follow a GPD distribution. Being the threshold determined according to the sample mean excess plot, then for the GPD only the other 2 parameters should be determined to evaluate the extremes of the analyzed records.

The adoption of a POT method for peaks extraction imply the necessity to estimate a distribution function Fu of x values above a specified threshold u. Function Fu is called conditioned excess distribution function and, according to PickandsBalkema-de Haan theorem (Balkema & de Haan 1974), can be identified with a GPD distribution (Pickands 1975), having the following cdf: −1 β −1

u)

i

means the sum of the excesses over the threshold u divided by the number of data point exceeding u. According to this definition, the mean excess function can be interpreted as an empirical estimate of the mean excess function, which is defined as:

GPD distribution

⎧1 − ( + z ) F (x) = ⎨ ⎩ 1− e

i =1 n

i =1

where N is the number of sub-populations considered in the analysis, while qi are the percentile of the subpopulations on the total sample. That means the sum of the qi should be equal to 1. The other parameters remain the same of a Weibull distribution. There are no limits to the number of subpopulations that can be considered. In any case, by increasing the number of subpopulations, the number of parameters will consequently increase. As example, considering two sub-populations, 7 parameters should be estimate (3 parameter per distribution and a single percentile q), once 3 subpopulation are considered, the parameters number increases to 12 and so on, since the last percentile is obtained by considering that the total sum of the qi should be 1. In the present study, the case of 2 subpopulations has been investigated, leading to the determination of 7 parameters per each analysis. 2.2

∑ (X ∑ 1 n

(5)

i =1

3

GA APPLICATION

Since the application of determined distribution will led to the determination of multiple parameters on non-linear functions, it would not be easy to determine these parameters with conventional fitting methods. For data fitting, a possible solution is the adoption of genetic algorithms (Coit & Smith 2002, Strelen 2003), considering as objective the minimization of the discrepancies between sample data and distribution data generated with randomly generated parameters. GA are evolutionary problem solving techniques based on the Darwinian principle of the “survival of the fittest”. In this way, an efficient solution is obtained. Moreover, being the method starting from multiple distinct points, local optimum solutions can be avoided, increasing the flexibility of GA as global optimization tool. In this way nonlinearities and constraints can be easily applied to the objective function providing an efficient search inside large spaces, using a single suitable representation of the problem and a “black-box” evaluation system.

(6)

Shape parameter β, scale parameter η, location parameter γ and z are defined as per Equations 2 and 3. The limitations to take into account are different, β is defined in (−∞, +∞), η in (0, +∞) and γ in (−∞, +∞). In the case of GPD distribution, γ is representing the threshold value; therefore, it can be identified as u. Once GPD distribution is selected for extremes modelling, one of the most relevant issues is related to the determination of the threshold u. Threshold selection process should ensure that a sufficient number of events lie above the selected value, in

5

Table 1.

Comparison between different parameter estimation techniques on a 2 parameters Weibull distribution. GA

Record 1 Record 2 Record 3 Record 4 Record 5 Record 6 Record 7 Record 8

MLE

MOM

β

η

R2

β

η

R2

β

η

R2

Best

1.426 1.775 1.597 1.633 1.569 1.600 1.735 1.523

23.86 25.63 25.85 25.39 24.77 24.76 24.45 24.60

0.996 0.986 0.997 0.987 0.996 0.981 0.995 0.989

1.401 1.755 1.590 1.621 1.541 1.566 1.708 1.505

23.68 25.65 25.86 25.39 24.59 24.66 26.44 24.57

0.996 0.987 0.997 0.987 0.995 0.980 0.994 0.990

1.451 1.782 1.603 1.642 1.574 1.614 1.738 1.531

23.61 25.67 25.88 25.36 24.35 24.72 25.84 24.77

0.997 0.985 0.998 0.986 0.994 0.980 0.994 0.988

MOM MLE MOM GA GA GA GA MLE

The mechanism on the basis of a simple GA is as follows (Goldberg 1989). An initial population with a fixed number of individuals is randomly generated. Each population individual is characterized by a string of genes, representing a possible solution to the analyzed problem. The strings can be evaluated according to a fitness function, stating how good the different solutions are. Then strings with a higher fitness value have more chance to survive during the natural selection process with respect to weaker ones. Survived individuals are than paired up randomly ad subjected to a crossover procedure, forming new offspring strings according to a probability value pc. This process generates new individuals by the combinations of two parents, combining the good properties of the old generation to form an even better offspring. After crossover, a mutation process, with probability pm, starts, where entire parts of the offspring are changed to ensure diversity between generated individuals. New population is then reevaluated according to the fitness function. The process continues iteratively until it is stopped, after a certain number of fixed iteration or once fitness function reaches a determined threshold. Best strings can be therefore considered as nearoptimal solutions to the analyzed problem. For regression purposes, it has been selected to choose as fitness function the determination coefficient R2, defined as: R2 = 1 −

SSE SS Stot

y=

n

i =1

3.1

n

SS Stot = ∑ ( yi i =1

(10)

y)

(11)

2

2

Parameters determination

The proposed method based on GA, is different compared to traditional parameter estimation techniques used to fit simple extremes distributions. To evaluate whether GA approach is giving a reliable solution for the fitting of extremes, then a comparison is made with Method of Moments (MOM) and Maximum Likelihood Estimation method (MLE) on a standard 2 parameters Weibull distribution. In fact, those methods are usually adopted for the extreme fitting of extremes according to Weibull approach (Oliver & Ochi 1981, Ochi 1998). On this purpose, 8 random generated data records have been created according to a predetermined Weibull distribution with β  = 1.50 and η = 24.00. The parameters obtained per each fitting technique have been compared according to the R2 value. Results are reported in Table 1, highlighting that the selected methods are almost equivalent for the selected case. Then, it is the opinion that adopting a GA approach to fit an extreme distribution may ensure a sufficient accuracy for the obtained extreme values.

(9)

fi )

(12)

Being yi the observations values and fi the data values coming from the distributions cdf evaluated with a string of parameters. In such a way, the problem is automatically solved as a maximization, which is advisable for GA fitness function. The variables can be expressed in two forms: binary or real number. From previous studies (Dejhalla et al. 2002) it has been highlighted that by using a real coding approach, the mutation probability pm must have a higher value compared to binary coding. From past experience with GA (Mauro & Nabergoj, 2015, 2016c) it has been selected to proceed with real coding of variables.

where: SSE = ∑ ( yi

1 n ∑ yi n i =1

6

present, where the arguments are elevated at different exponents. For such a reason, an iterative process, converging at the desired 1-p level, determines the quantiles for the mixed Weibull case. Adopting the above-mentioned equations and methods it is possible to determine the desired extremes values according to the selected distribution.

In Table 2 an overview is given on the GA settings used in this study for the different distributions that have been analyzed. It can be noted that the parameter estimation of the Mixed Weibull distribution is more complicated, leading to the selection of different parameter settings to reach the convergence of the algorithm. In fact, the mixed Weibull approach with 2 subpopulations needs the estimation of 7 regression parameters, requiring then dedicated settings with respect to less complicated regression models. 3.2

4

The above described extreme value analysis based on GA parameter determination will be here applied on two different kinds of problem, significant for offshore industry. At first the analysis will be applied on a data record of a seakeeping experiment, where the loads on a tubular structure have been measured. Then, the same methodology will be applied to predict extreme wave height values in the North Atlantic Sea.

Extreme value determination

Once the parameters are evaluated, then the extreme values of the population can be calculated from the fitted distributions. Usually the values of interest are the events with probability p equal to 3%, 1% and 0.1%. To determine these values, the inverse cumulative distributions (quantiles) of the limiting laws should be used. For the distributions described in the present work, the quantiles have the following forms: Q ( p,η, β ) = η ( − ln ( − p))

1β

Q ( p,γ ,η, β )

Q ( p, u,η, β ) = u +

4.1

1β

(14)

β ⎞ ⎞ β⎛ ⎛ n ( − p)⎟ ⎟ ⎜1 − ⎜ η ⎝ ⎝ Nu ⎠ ⎠

(15)

which are representative of the 2, 3 parameters Weibull distribution and of the GPD respectively. In particular, for the GPD, n represents the total number of observations and Nu is the number of samples above the selected thresholds. For the mixed Weibull distribution, a general formulation for the quantile is not easy to determine, since in the cdf a sum of logarithms is Table 2.

GA settings for the different distributions. Weibull

Chromosome length Population size Number of generations Number of evaluations Crossover probability pc Mutation probability pm

2 par.

3 par.

Mixed

GPD

16

24

56

16

120 120

120 120

200 200

120 120

14400

14400

40000

14400

0.90

0.90

0.90

0.90

0.30

0.30

0.50

0.30

Extreme load estimation on a structure

As first case, a time record coming out a seakeeping test has been analyzed. The data refer to the total force measured on a vessel tubular appendage, and the time series is reported in Figure 1. On the selected record, peaks have been extracted both with block maxima method and POT one. Therefore, it was possible to analyze the extremes with standard Weibull, mixed Weibull and GPD approaches. In Figure 2 the result of the analysis is reported on the so-called Weibull plane. The data points clearly follow a multimodal distribution, since they present more than one concavity/convexity. This is a signal that probably a Mixed Weibull approach is a good solution. In fact, a 3 parameters distribution Weibull is not the best choice to reproduce the extremes, being strongly influenced by one of the two subpopulations. In such a case, a 2 parameters distribution is less influenced by the lower subpopulation, giving results comparable with more complex approaches. In the case of the GPD distribution, having to consider a POT peaks extraction technique, a threshold u has been selected at a value of 29.7 kN.

(13)

λ η ( ln ( − p))

TEST CASES

Figure 1.

7

Force time series.

Table 3. Extreme values of FW according to the different distributions. Probability p Regression type

1.0% 3.0% (kN) 0.1%

2 parameters Weibull 3 parameters Weibull Mixed Weibull Generalized Pareto

39.92 48.21 42.01 42.04

R2 (−)

SSE (−)

51.50 88.22 0.990 0.011 69.13 119.48 0.995 0.006 49.43 70.01 0.999 1.7E-4 49.78 71.23 0.991 1.4E-4

higher than the other distributions for all the three p levels extracted for the analysis, reaching a maximum overestimation above 65% at p = 0.1%. This is clearly an index that the 3-parameter distribution is absolutely undesirable for the fitting of the selected data sample. By adopting this kind of distribution, a designer is supposed that the structure under analysis will face a possible extreme load of about 120  kN. Adopting the other distributions, the extreme load is reduced to about 70  kN by using a Mixed Weibull or a GPD approach. Using a 2 parameters Weibull, then the load results in about 88 kN, which is still 25% higher than the two more complex approaches. A wrong selection of the distribution will for sure lead to a wrong dimensioning of the structure, then it is mandatory to properly analyze the records to avoid a possible mistake in the design process of the structure. In this particular case, the adoption of more complex distributions as the Mixed Weibull, lead to a better fitting of the whole population considered for the analysis. It is important to underline that, in case of the selection of a block maxima method for peaks extraction, all the peaks should be considered. For practical reason, once a multimodal behavior of the maxima is observed as per Figure 2, a designer may consider to cut all the data above a certain value and refit the remaining data according to another 3 parameter Weibull distribution. This is a mistake that has to be avoided. In fact, once a threshold is selected a POT technique is automatically chosen and then a GPD should be used according to extreme value theory. That means, once a situation as per Figure 2 is faced, Mixed Weibull or GPD distribution should be used in accordance with the selected peaks extraction technique.

Figure  2. Weibull plot of the extreme value of force measurement data according to 2 par. Weibull, 3 par. Weibull, Mixed Weibull and GPD distributions.

An overview of the obtained extreme values is given in Table  3. Here, besides the extreme values, also the statistic of the obtained regression is reported in terms of R2 and SSE, evaluated as per Equations 9 and 10 respectively. It can be observed that, for all the adopted distributions, the realization coefficient is quite high, being above 0.99 in all the tested cases. However, as it is represented in Figure  2, the 2 and 3 parameters distributions are not fitting really well the effective shape of the data. That means, traditional statistics coefficients are not suitable to clearly rank the distributions. Moreover, the regression of the GPD is made on a lover number of samples, since the threshold u is automatically excluding more than a half of the data sample. It is than of utmost importance the analysis of the extreme values obtained for each distribution, analyzing the differences between them. In the reported case, the 3 parameters Weibull distribution is giving results not comparable with the other methods, having estimates that are always

4.2

Extreme wave height estimation

To tackle the issue related to extreme wave height prediction, use has been made of the SHIPREL (Reliability Methods for Ship Structural Design) wave scatter diagram, reported in Table 4, describing the North Atlantic sea environment (Hogben

8

Table 4.

Significant wave height distribution according to SHIPREL data.

Total

0.87

12.40

62.60

166.1

254.3

242.7

155.7

71.03

25.4

7.86

0.96

1000

11–12 10–11 9–10 8–9 7–8 6–7 5–6 4–5 3–4 2–3 1–2 0–1

– – – – – – – – – 0.09 0.17 0.62

– – – – – –

– – – –

– –

– –

– –

–

0.17 0.35 0.68 1.00 4.50 10.00 20.00 25.00

0.17 0.22 0.43 1.40 3.00 6.90 17.00 37.00 63.00 37.00

0.17 0.76 1.50 3.50 8.40 20.00 42.00 74.00 82.00 22.00

0.76 1.68 3.40 7.00 15.00 31.00 54.00 72.00 51.00 6.90

0.77 1.08 1.95 4.40 8.30 16.00 27.00 39.00 38.00 18.00 1.20

0.5 0.63 1.02 1.88 3.40 5.80 9.80 14.00 17.00 13.00 4.00 –

– 0.43 0.67 1.20 1.80 2.70 4.00 5.30 4.90 3.30 1.10 –

– – 0.30 0.44 0.72 1.10 1.50 1.50 1.40 0.90 – –

– – – – 0.16 0.32 0.32 0.16 – – – –

0.50 1.83 4.17 8.13 15.98 30.47 58.87 108.05 180.52 249.79 242.37 99.32

Hs, (m)