Interactionanalysisofcolumn-andplate-like ......Contents 4.1.4 Interpolationbetweenplateandcolumnbehaviour. . . . . . . . 39 4.2 Eurocode’sapproachbasedonreducedstressmethod.

Interaction analysis of column- and plate-likebehaviour of stiffened plate structures

Zsombor Illés

Thesis to obtain the Master of Science Degree in civil engineering

Supervisors: Professor Balázs Géza Kövesdi

Professor Carlos Manuel Tiago Tavares Fernandes

Examination Committee

Chairperson: Professor António Manuel Figueiredo Pinto da Costa

Supervisor: Professor Balázs Géza Kövesdi

Members of the Committee: Professor Luís Manuel Calado de Oliveira Martins

July 2018

Declaration

I declare that this document is an original work of my own authorship and that it fulfillsall the requirements of the Code of Conduct and Good Practices of the Universidadede Lisboa.

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Abstract

At the Budapest University of Technology and Economics, Department of Bridges andStructures and at the Instituto Superior Técnico, Departamento de Engenharia Civil,Arquitectura e Georrecursos during the 2nd semester of the year 2017/2018 the title ofthe master thesis written was: “Interaction analysis of Column and Plate like behaviourof stiffened plate structures”, registration number MSc-SZ-C-013-17/18/2. The super-visors were: Dr. Balázs Kövesdi, Associate Professor of BME and Dr. Carlos Tiago,Assistant Professor of IST.The subject of the thesis is the analysis of the physically and geometrically non-linearbehaviour of stiffened plated structures, determining the buckling resistance and ex-amining the interpolation function between the column-like and plate-like behaviour.In the first part of the dissertation simply supported columns and plate structures wereexamined. The numerical solutions were compared against the analytical ones. Finiteelements with different interpolation functions were used to model beams and plates.Convergence analysis was made and the theoretical and numerical rates of convergenceof the elements was assessed.At the beginning of the section on design approach of plated structure an overview ofthe relevant parts of Eurocode was provided, Eurocode 3: Design of steel structures,Part 1-5: Plated structural elements, focusing on the interpolation function of platelike and column like behaviour.A new imperfection is proposed and compared against the ones in Eurocode. A para-metric study of the orthotropic plates was conducted. The geometric configuration ofthe plates were the input parameters, while the material properties were kept the samefor all models. Three different types of evaluation methods were chosen: (i) Eurocodebased, (ii) Eurocode-FEM based and (iii) mainly FEM based. For each evaluationmethod the reduction factors obtained from the finite element models have been con-sidered.Finally, from the evaluations conclusions have been drawn and future research prospec-tive have been construe.Finite element models through out the thesis were created in ANSYS Mechanical APDLfinite element program. To analyse the results, MATLAB R2017b was applied.

keywords: Stability of structures, buckling analysis, column-like and plate-likebehaviour, convergence analysis, parametric study, imperfection, arc-length method,eurocode

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Összefoglaló

A Budapesti Műszaki és Gazdaságtudományi Egyetem Hidak és Szerkezetek Tanszé-kén, illetve a Lisszaboni Instituto Superior Técnico Építőmérnöki Karán 2017/2018-astanév II. félévében elkészített diplomamunka címe: „Oszlopszerű és Lemezszerű stabi-litásvesztési mód interakciójának vizsgálata merevített lemezek esetén”, nyilvántartásiszáma MSc-SZ-C-013-17/18/2. Konzulensek: Dr. Kövesdi Balázs a BME egyetemi do-cense és Dr. Carlos Tiago az IST egyetemi adjunktusa voltak. A diplomamunka témájaa hosszbordákkal merevített lemezes szerkezetek horpadási viselkedésének elemzése, ahorpadási ellenállás meghatározása, az oszlopszerű és a lemezszerű viselkedés közti in-terpolációs függvény vizsgálata.A diplomamunkám első részében csuklósan megtámasztott rúd és lemez szerkezeteketvizsgálata kerül bemutatásra, a numerikus megoldások az analitikus megoldásokkaltörténõ összehasonlítása. Lemezek esetén a különböző interpolációs függvénnyel ren-delkező végeselemek kerültek megvizsgálásra. Konvergencia vizsgálat készült, illetve azelemek elméleti és gyakorlati konvergencia sebessége is meghatározásra került.A lemezes szerkezetek tervezésével foglalkozó rész elején az Eurocode szabvány csomagtémához tartozó, Eurocode 3: Design of steel structures, Part 1-5 : Plated structuralelements része kerül áttekintésre, kitérve a vizsgálni kívánt interpolációs függvényre.Az Eurocode által javasolt és a diplomamunkában használt, a témavezetõim által ja-vasolt imperfekciókat egy példa keretében bemutatom és összehasonlítom. A diplo-mamunkám keretében paraméter vizsgálatot végeztem, a bemenõ változók a szerkezetgeometriáj volt, míg a numerikus kísérletekhez használt anyagmodell változatlan voltvégig, S355 acél került használatra. Konzulenseimmel három különböző fajta kiértéke-lési módszer mellett döntöttünk, Eurocode alapú, Eurocode-FEM alapú és FEM alapú.Mindegyik kiértékelésnél természetesen felhasználásra kerültek a végeselemes futtatá-sok eredményei.A kiértékelések eredményeiből végezetül pedig konklúziót vontam, illetve az esetlegestovábbi kutatási lehetőségeket elemeztem.A diplomamunkám során ANSYS Mechanical APDL végeselemes programot használ-tam, az összes modellt ebben készûlt. Az eredmények kiértékeléséhez a MATLABR2017b verziójában történt.

kulcsszavak: stabilitás vesztés, kihajlás, horpadás, oszlopszerű és lemezszerű vi-selkedés, konvergencia vizsgálat, paraméter vizsgálat, imperfekció, arc-length módzser,eurocode

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Resumo

A presente dissertação foi efectuada no Department of Bridges and Structures da Bud-apest University of Technology and Economics e no Departamento de Engenharia Civil,Arquitectura e Georrecursos do Instituto Superior Técnico da Universidade de Lisboadurante o segundo semestre do ano lectivo 2017/2018 e tem o título: “Interaction ana-lysis of Column and Plate like behaviour of stiffened plate structures” e o número deregisto MSc-SZ-C-013-17/18/2. Os orientadores foram: Dr. Balázs Kövesdi, ProfessorAssociado da BME e o Dr. Carlos Tiago, Professor Auxiliar do IST.

O tema da dissertação é a análise do comportamento de placas de aço reforçadas.Para tal foi determinada a sua resistência considerando os efeitos geometricamente efisicamente não lineares e examinando a função de interpolação entre o comportamentode coluna e de placa.

Na primeira parte da dissertação apenas colunas e placas simplesmente apoiadasforam analisadas. As soluções numéricas obtidas foram comparadas com as analíticas.Elementos finitos com diferentes funções de aproximação foram usadas para modelarcolunas e placas. Foi efectuada uma análise de convergência com o refinamento dasmalhas e foram obtidas e comparadas as taxas de convergência teóricas e numéricasdos elementos.

No início da secção sobre abordagem ao dimensionamento de placas reforçadas éapresentada uma visão geral das partes relevantes do Eurocódigo 3: Design of steelstructures, Part 1-5: Plated structural elements, destacando o papel da função deinterpolação entre os comportamentos de placa e coluna.

Uma nova imperfeição é proposta e comparada com as apresentadas no Eurocódigo.Foi realizado um estudo paramétrico de placas ortotrópicas. Os parâmetros em análisesão constituídos pela configuração geométrica das placas, enquanto as propriedades domaterial são constantes para todos os modelos. Três diferentes tipos de métodos deavaliação foram escolhidos: (i) baseado no Eurocódigo, (ii) baseado no Eurocódigo eMétodo dos Elementos Finitos (MEF) e (iii) principalmente com base no MEF. Paracada método de avaliação foram considerados os factores de redução obtidos a partirdos resultados fornecidos pelo MEF.

Finalmente, a partir do estudo efectuado, foram apresentadas as conclusões e pro-postos desenvolvimentos futuros.

Os modelos de elementos finitos ao longo da tese foram criados no programa ANSYSMechanical APDL. A análise dos resultados foi efectuada através do MATLAB R2017b.

palavras-chave: Estabilidade de estruturas, análise de encurvadura, comporta-mento de coluna e de placa, análise de convergência, estudo paramétrico, imperfeição,método do comprimento do arco, eurocódigo.

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Acknowledgements

First of all, I would like to thank my supervisors, Dr. Balázs Kövesdi from BME andDr. Carlos Tiago from IST. During the semester I had consulations with Dr. CarlosTiago week-bye-week in Lisbon, while I had skype meetings with Dr. Balázs Kövesdi.During the summer this set-up was reversed and I had skype meeting with Prof. CarlosTiago and personal meetings with Prof. Balázs Kövesdi.I would also like to thank Dr. Ádány Sándor and Dr. Joó Attila, who were the reviewerof my defence in Budapest. During my last year, I had received the Campus MundiScholarship from the Tempus Public Foundation to study in Instituto Superior Técnico.Here I meet new students and new teachers, whom influenced my attitude and pointof view towards studying, examination and what is knowledge. It was a smashingexperience to study for a whole year in Lisbon after completing five long years in theTechnical University of Budapest.Last but not least, I would like to express my gratitude to my family for supportingme all the way through my studies.

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Contents

Contents xiii

List of Figures xv

List of Tables xvii

List of symbols xix

I Introduction to analytical and numerical approach 1

1 Introduction 3

2 Column buckling 72.1 Analytical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Finite Element solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Example: Pin ended column’s, buckling load . . . . . . . . . . . 132.3 Finite Element Solution (Software) . . . . . . . . . . . . . . . . . . . . . 16

2.3.1 Beam model (Example) . . . . . . . . . . . . . . . . . . . . . . . 16

3 Plate buckling 213.1 Analytical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.1 Simply Supported Rectangular Plates under Forces in the MiddlePlane of the Plate . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.2 Example: Rectangular, simply supported plate . . . . . . . . . . 233.2 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.2 Type of elements used . . . . . . . . . . . . . . . . . . . . . . . . 253.2.3 Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.4 Rate of convergence . . . . . . . . . . . . . . . . . . . . . . . . . 28

II Design approach 31

4 Design and production technology 334.1 Eurocode’s approach based on the reduced cross-section concept . . . . 33

4.1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.1.2 Plate behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.1.3 Column behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . 37

xiii

Contents

4.1.4 Interpolation between plate and column behaviour . . . . . . . . 394.2 Eurocode’s approach based on reduced stress method . . . . . . . . . . 424.3 Production technology . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 Model description and numerical research strategy 455.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.2 Geometric properties of the examined stiffened plate . . . . . . . . . . . 465.3 Material model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.4 Buckling modes of stiffened plate . . . . . . . . . . . . . . . . . . . . . . 485.5 Description of Imperfections . . . . . . . . . . . . . . . . . . . . . . . . 515.6 The imperfections’ effect on ultimate force . . . . . . . . . . . . . . . . 54

5.6.1 Results of displacement governed experiments . . . . . . . . . . . 545.6.2 Results obtained by arc-length method . . . . . . . . . . . . . . . 55

6 Numerical Experiments 596.1 Parametric study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.2 Layout of experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.2.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 626.2.2 Imperfections used during the experiments . . . . . . . . . . . . . 63

6.3 Evaluation of reduction factors . . . . . . . . . . . . . . . . . . . . . . . 646.3.1 Column-like reduction factors . . . . . . . . . . . . . . . . . . . . 646.3.2 Plate-like reduction factors . . . . . . . . . . . . . . . . . . . . . 65

6.4 Examination of a single geometry . . . . . . . . . . . . . . . . . . . . . 676.4.1 Force-displacement curves . . . . . . . . . . . . . . . . . . . . . . 676.4.2 Evaluation of reduction factors for a single geometry . . . . . . . 68

6.5 Eurocode based evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 706.6 Eurocode-FEM based evaluation . . . . . . . . . . . . . . . . . . . . . . 746.7 FEM based evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

7 Conclusions and furher studies 797.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797.2 Further studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Bibliography 81

A Plate buckling results 83

B Force-displacement curves of plated structures 87

xiv

List of Figures

1.1 Different stiffeners used for steel plates (Stiffeners, 2018) . . . . . . . . . . 4

2.1 Column effective length factors for Euler’s critical load (Ádány et al., 2007) 82.2 Pin ended column under the effect of Buckling load adapted from Galambos

and Surovel (1998) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 First three modes of buckling load adapted from Timoshenko and Gere (1961) 102.4 Unknown displacement for the eth element . . . . . . . . . . . . . . . . . . 112.5 Cross section of a trapezoid stiffener . . . . . . . . . . . . . . . . . . . . . . 142.6 Convention for discretisation with a single finite element . . . . . . . . . . 142.7 Convention for discretisation with two finite elements . . . . . . . . . . . . 152.8 Boundary conditions of simply supported, pin ended beam . . . . . . . . . 172.9 The first three buckling modes of simply supported, pin ended beam . . . . 172.10 Rate of convergence, first three eigenvalues . . . . . . . . . . . . . . . . . . 19

3.1 Plate in tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Factor k according to the ratio of the sides . . . . . . . . . . . . . . . . . . 233.3 Deflected shape of a compressed plate . . . . . . . . . . . . . . . . . . . . . 233.4 Boundary conditions of a simply supported plate (hard support) . . . . . . 253.5 Convergence analysis first seven buckling modes . . . . . . . . . . . . . . . 273.6 Convergence analysis first four buckling modes . . . . . . . . . . . . . . . . 293.7 Rate of convergence for linear shell elements . . . . . . . . . . . . . . . . . 303.8 Rate of convergence for quadratic shell elements . . . . . . . . . . . . . . . 30

4.1 Ac and Ac.eff.loc for a stiffened plate element (European Committee for Stan-dardisation, 2006) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2 Excentricities of the stiffeners (Johansson et al., 2007) . . . . . . . . . . . . 394.3 Interpolation formula between column and plate behaviour . . . . . . . . . 404.4 Interaction of plate and column like behaviour, adapted from Johansson

et al. (2007) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.5 Temperature and thermal stress distribution during welding . . . . . . . . 444.6 Common types of weld distortion . . . . . . . . . . . . . . . . . . . . . . . 44

5.1 Boundary conditions of examined plates . . . . . . . . . . . . . . . . . . . 465.2 Dimensions of the stiffened plate . . . . . . . . . . . . . . . . . . . . . . . . 475.3 Material model used during the numerical experiments . . . . . . . . . . . 485.4 1st, 2nd, 3rd eigenmodes of stiffened plate . . . . . . . . . . . . . . . . . . . 495.5 4th, 5th, 6th eigenmodes of stiffened plate . . . . . . . . . . . . . . . . . . . 505.6 IMP 3+, in a magnification of ×50 . . . . . . . . . . . . . . . . . . . . . . 51

xv

List of Figures

5.7 IMP 3−, in a magnification of ×50 . . . . . . . . . . . . . . . . . . . . . . 525.8 IMP 4+, in a magnification of ×50 . . . . . . . . . . . . . . . . . . . . . . 525.9 IMP 4− in a magnification of ×50 . . . . . . . . . . . . . . . . . . . . . . . 535.10 IMP proposed in a magnification of ×20 and zoom in of the elements . . . 545.11 Force displacement curves, (disp.-gov.) . . . . . . . . . . . . . . . . . . . . 565.12 Force displacement curves, (arc-length) . . . . . . . . . . . . . . . . . . . . 57

6.1 Boundary conditions in case of plate-like behaviour . . . . . . . . . . . . . 626.2 Boundary conditions in case of column-like behaviour . . . . . . . . . . . . 636.3 IMP proposed in a magnification of×50 . . . . . . . . . . . . . . . . . . . . 636.4 Von Mises stress at 0.01 time step . . . . . . . . . . . . . . . . . . . . . . . 646.5 Column reduction factor, χc . . . . . . . . . . . . . . . . . . . . . . . . . . 656.6 Reduction factors, relative slenderness calculated from EC results . . . . . 666.7 Reduction factors, relative slenderness calculated from FEM results . . . . 666.8 FEM based evaluation, maximums of (imp− /imp+) . . . . . . . . . . . . 686.9 Reduction factors, relative slenderness calculated from EC results . . . . . 696.10 Reduction factors, relative slenderness calculated from FEM results . . . . 706.11 Interpolation in case of EC based evaluation . . . . . . . . . . . . . . . . . 716.12 EC based evaluation, all results . . . . . . . . . . . . . . . . . . . . . . . . 716.13 EC based evaluation, no. of stiffener (Ns = 2) . . . . . . . . . . . . . . . . 726.14 EC based evaluation, number of stiffeners (Ns = 2) . . . . . . . . . . . . . 736.15 EC based evaluation, without inappropriate estimations . . . . . . . . . . . 736.16 Interpolation in case of EC-FEM based evaluation . . . . . . . . . . . . . . 746.17 EC-FEM based evaluation, (imp− /imp+) . . . . . . . . . . . . . . . . . . 756.18 EC-FEM based evaluation, maximums of (imp− /imp+) . . . . . . . . . . 756.19 EC-FEM based evaluation, (imp− /imp+) . . . . . . . . . . . . . . . . . . 766.20 EC-FEM based evaluation, maximums of (imp− /imp+) . . . . . . . . . . 766.21 Interpolation in case of FEM based evaluation . . . . . . . . . . . . . . . . 776.22 FEM based evaluation, (imp− /imp+) . . . . . . . . . . . . . . . . . . . . 786.23 FEM based evaluation, maximums of (imp− /imp+) . . . . . . . . . . . . 78

A.1 1st, 2nd, 3rd, 4th buckling modes of a simply supported plate . . . . . . . . . 85A.2 5th, 6th, 7th buckling modes of a simply supported plate . . . . . . . . . . . 86

B.1 Force displacement curve, with displaced geometries . . . . . . . . . . . . . 88B.2 Force displacement curve, with displaced geometries and von Mises stresses 89B.3 Columns, tp = 12 mm, b1 = 600 mm, tsf = 10 mm . . . . . . . . . . . . . . . 90B.4 Plates, tp = 12 mm, b1 = 600 mm, tsf = 10 mm . . . . . . . . . . . . . . . . 90B.5 Columns tp = 16 mm, b1 = 600 mm, tsf = 10 mm . . . . . . . . . . . . . . . 91B.6 Plates tp = 16 mm, b1 = 600 mm, tsf = 10 mm . . . . . . . . . . . . . . . . 91B.7 Columns tp = 20 mm, b1 = 400 mm, tsf = 12 mm . . . . . . . . . . . . . . . 92B.8 Plates tp = 20 mm, b1 = 400 mm, tsf = 12 mm . . . . . . . . . . . . . . . . 92B.9 Columns tp = 20 mm, b1 = 600 mm, tsf = 12 mm . . . . . . . . . . . . . . . 93B.10Plates tp = 20 mm, b1 = 600 mm, tsf = 12 mm . . . . . . . . . . . . . . . . 93B.11Columns tp = 25 mm, b1 = 400 mm, tsf = 10 mm . . . . . . . . . . . . . . . 94B.12Plates tp = 25 mm, b1 = 400 mm, tsf = 10 mm . . . . . . . . . . . . . . . . 94B.13Columns tp = 25 mm, b1 = 400 mm, tsf = 12 mm . . . . . . . . . . . . . . . 95B.14Plates tp = 25 mm, b1 = 400 mm, tsf = 12 mm . . . . . . . . . . . . . . . . 95

xvi

List of Tables

2.1 Convergence analysis of pin ended beam . . . . . . . . . . . . . . . . . . . 162.2 Convergence analysis of simply supported beam . . . . . . . . . . . . . . . 18

3.1 Number of semi-waves and buckling loads . . . . . . . . . . . . . . . . . . . 243.2 Hard supports and soft supports boundary conditions . . . . . . . . . . . . 25

5.1 Considered imperfections, corresponding ultimate forces and deviations inrespect to EC3, (disp.-gov.) . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.2 Considered imperfections, corresponding ultimate forces and deviations inrespect to EC3, (arc-length) . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.1 Considered parameters in the parametric study . . . . . . . . . . . . . . . . 596.2 Analysed stiffened plates, b1 = 400 mm . . . . . . . . . . . . . . . . . . . . 606.3 Analysed stiffened plates, b1 = 600 mm . . . . . . . . . . . . . . . . . . . . 616.4 Analysed stiffened plates, b1 = 800 mm . . . . . . . . . . . . . . . . . . . . 616.5 Total number of geometries analysed and number of runs . . . . . . . . . . 61

A.1 Convergence analysis with SHELL181 . . . . . . . . . . . . . . . . . . . . . 84A.2 Convergence analysis with SHELL281 . . . . . . . . . . . . . . . . . . . . . 84

xvii

List of Tables

xviii

List of symbols

Capital letters

A areaE modulus of elasticityF forceG gravity forceI inertiaM bending momentN normal forceR resistanceS stiffnessV shear forceW cross sectional elastic modulus

Small letters

a unloaded side of a plate, lengthb loaded side of a plate, breadthe exentricityf strengthh heighti radius of gyrationk buckling coefficientl lengthn integer valuet thicknessw widthIn case of plates:x parallel with the breadth of the platey parallel with the outward normal of the plate on the stiffened sidez parallel with the length of the plate

xix

List of symbols

Latin indexes

b bucklingc cross-sectionc columncr criticaleff effectiveel elasticf flangeloc localp platepl plasticsf stiffening, stiffenert thicknessw weby yield

Greek indexes

α imperfection parameter (EC)α ratio of a/b, length/breadthβ ratio of cross sectionsγ flexural stiffnessδ axial stiffnessδ displacement∆ differenceε strainΘ torsional stiffnessλ slendernessν Poisson-ratioξ “distance” between the elastic critical plate and column bucklingρ reduction factor, (ρ, global or ρp, plate like)σ normal stressτ shear stressφ auxiliary coefficient for the calculation of χcχ reduction factor (χc, column like reduction factor)ψ ration of extreme fiber stresses

xx

Part I

Introduction to analytical andnumerical approach

1

2

Chapter 1

Introduction

The analytical solutions for simple cases of pure plate buckling and pure column buck-ling are known. For special cases of stiffened plates analytical solution does exist. InTheory of Elastic stability by Timoshenko and Gere (1961) analytical solution for platereinforced by symmetrical ribs relatively to the middle plane is presented. In Timo-shenko and Woinowsky-Krieger (1959) a whole chapter is dedicated to the bendingof anisotropic plates. The behaviour of a stiffened plate can be described by partialdifferential equation, because of that fact advanced numerical tools can be used for ap-proximately solving them. Overall, the resistance of an orthotropic plate with complexstiffener geometry can be obtained by experiments or numerical simulations.

The first part of the thesis is an introduction to numerical simulations. A short inves-tigation was carried out to see what is the most suitable element for the modelling ofplates and carrying out linearised buckling analysis. Divided along this idea, in thefirst part numerical simulation results are compared with analytical solutions. Theconvergence rate was calculated for all elements and compared with the theoreticalresults (Hughes, 2000). As analytical solutions are not, in general, available, in secondpart plates are calculated according to the reduced cross sectional method presented inEurocode 3 (European Committee for Standardisation, 2006) and are compared againstthe results of the numerical experiments carried out by APDL.

The invention of welding marked a new era for steel structures and especially forplated structures. Before that, steel structures were mainly riveted and plates couldonly be connected by overlapping steel plates and bolting them together. These plateswere less thin and slender than the ones which nowadays are employed.The sweep of welding technologies also accounted for new problems such as fatigueunder cyclic load, improper connections, inclusions and, last but not least, these slen-der plates were more vulnerable to buckling. The stability of the plate always canbe increased by increasing its thickness, but such a design will not be economical inrespect to the weight of material used. A more economical solution is obtained bykeeping the thickness of the plate as small as possible and increasing its stability byintroducing reinforcing ribs (Timoshenko and Gere, 1961). However the mechanicalbehaviour of stiffened plates is very complex. Orthotropic plates or stiffened plates,according to the profiles of their ribs, can be specified as open sectioned (L and Tstiffeners) or close sectioned, which are usually trapezoidal cross sections. The crosssections of the stiffeners are presented in Fig. 1.1 For unstiffened plates, the plate-like

3

Chapter 1 Introduction

Figure 1.1: Different stiffeners used for steel plates (Stiffeners, 2018)

buckling is the common way of stability loss, while for orthotropic plates the plate-likeand column-like buckling behaviour is typical. These behaviours can appear separatelyor in a combination, common in case of stiffened plates.The ultimate strength of unstiffened plates has been studied by von Kármán et al.(1932) and Timoshenko and Woinowsky-Krieger (1959). The elastic buckling of stiff-ened plates under uniaxial compression has been well studied. Theoretical results forcompression panels stiffened by one or two stiffeners were carried obtained by Timo-shenko and Gere (1961). Klöppel and Scheer (1960) calculated various load cases andpractical configurations. An extensive summary of the available theories can be foundin Galambos and Surovel (1998). In the past 50 years numerous real scale and numer-ical investigations were carried out to examine the effects of stiffener/plate assemblygeometries on buckling and collapse strength. One of the first to compare numericaland experimental results were Yamada and Watanabe (1976). Later in the 20th cen-tury Mikami and Niwa (1996) carried out experiments and derive formulas to estimatethe ultimate strength of orthotropic plates. In recent years, at the Budapest Universityof Technology and Economics, research aiming to determine the optimal geometry ofI stiffeners Simon et al. (2014) was undertaken. This master thesis is meant to helpthe better understanding of orthotropic stiffened plates and throughout influence thedesign approach of Eurocode.

Design approach

Since the 1930s the progress in welding technology has facilitated the increased ap-plication of steel plated structures. The significant knowledge gained since then hasclearly influenced the design as well as the development of the design standards. Withthe Eurocodes, harmonised European rules have been established in the standard EN1993-1-5 “Design of steel structures - Plated structural elements”.Based on Eurocode the designer can choose between two different types of designmethod: (i) effective cross section method, described in section 4.1, which is very effi-cient for standard geometries because it accounts not only for the post-critical reservein a single plate element but also for load shedding between cross sectional elementsand (ii) reduced stress method, reported in section 4.2. The latter abstains from loadshedding between cross sectional elements, but it fully accounts for the post-critical

4

Chapter 1 Introduction

reserve in a single plate element. Beyond that, its general format facilitates its use forserviceability verifications and for the design of non-uniform members.

Scope of numerical experiments

A parametric study was conducted in order to assess the influence of several geometricparameters, allowing to draw conclusions on the plate geometrically and physicallynon-linear behaviour. More than 150 combinations of geometrical parameters wereanalysed, which resulted in more than 900 numerical experiments, as every geometryis analysed with plate- and column-like boundary conditions. In each case of bucklinganalysis, two geometrically materially nonlinear analysis with different imperfections(GMNI analysis) were performed. These numerical simulations were carried out in AN-SYS Parametric Design Language and analysed in MATLAB, which stands for MATrixLABoratory. The boundary conditions, type of elements and imperfections applied areall described in chapter 5. The results are evaluated according to different evaluationmethods. As an out come of the discussions with the supervisors, three main typesof evaluation methods were considered: (i) Eurocode based evaluation, presented insection 6.5, (ii) Eurocode-FEM based evaluation discussed in 6.6 and finally (iii) fullyFEM based evaluation method reviewed in section 6.7. Results of Eurocode’s reducedcross section method and the outcome of geometrically and materially nonlinear anal-ysis with imperfections (GMNIA) is also taken into account.

Organisation of the document

The first chapter was an introduction to the fabrication and design of stiffened platesand to the stiffener geometries, which were used. In chapter 2 the column bucklingproblem is outlined and the analytical solution of a particular problem was comparedwith solutions obtained by MATLAB and ANSYS. Although this chapter does notcontribute final conclusions of the work, it allowed to validate the implementation in theMechanical APDL environment of a simple example with analytical solution. Chapter3 deals with the plate buckling: convergence analysis and comparison of elements withdifferent shape functions were carried out. Chapter 4 focuses on the design methodspresented in Eurocode and also mentions the design related effects of the productiontechnology. Chapter 5 compares the different imperfections proposed by the Eurocodeand the author. After the decision of which imperfection to use for the GMNI analyses,the results of the Numerical Experiments are presented in Chapter 6. In chapter 7conclusions have been drawn and the possibilities of further studies were mentioned.

5

6

Chapter 2

Column buckling

Buckling phenomena of a column occurs, when the so called critical load is reached. Forloads less than the critical load the column will remain straight. In general, the criticalload of a real structure cannot be obtained analytically. Hence, numerical methodshave to be used. In the forthcoming sections both possibilities will be presented. Thefinite element solution will be demonstrated for particular cases, first implementedin MATLAB (2017) and then by the use of finite element software, ANSYS (2013b)Mechanical APDL.

2.1 Analytical solution

The formula for the analytical solution was derived in 1757, by the Swiss mathematicianLeonhard Euler. The column will remain straight for loads less than the critical load.The “critical load” is the greatest load that will not cause lateral deflection (buckling)in a real column. For loads greater than the critical load, a real column will deflectlaterally. After reaching the critical load a bifurcation point is found and two situationsare possible: (i) continue along the (unstable) fundamental path or (ii) follow one ofthe (stable) post-buckling paths. As the load is increased beyond the critical load itmay fail in other modes such as yielding of the material. The critical load is given by

Ncr =π2E I(ν L)2 (2.1)

where:

Ncr: Euler’s critical load (longitudinal compression load on column)E: modulus of elasticity of column materialI: area moment of inertia of the cross section of the columnL: length of columnν: column effective length factor (figure 2.1)

Note that for loads greater than the critical load, a real column will always deflectlaterally. Assumptions of the model

1. The material of the column is homogeneous and isotropic.2. The column is initially straight (no eccentricity of the axial load).

7

Chapter 2 Column buckling

ν = 1 ν = 0.7 ν = 0.5 ν = 2 ν = 1 0.5 6 ν 6 1 ν > 1

Figure 2.1: Column effective length factors for Euler’s critical load (Ádány et al., 2007)

3. The column is free from initial stresses.4. The weight of the column is neglected.5. Pin joints are friction-less (no moment constraint) and fixed ends are clamped

(no rotation deflection).6. The cross-section of the column is uniform throughout its length.7. The length of the column is very large as compared to the cross-sectional dimen-

sions of the column.8. The column fails only by buckling. This is true if the compressive stress in the

column does not exceed the yield strength of the material, fy

σ =Ncr

A=

π2E

(Lcr/i)2< fy (2.2)

where:Lcr = ν · L: buckling length regarding the axis of buckling (around whichbuckling occurs),i: is the radius of inertia, around the axis of buckling,

i =

√I

A

Consider the slenderness, λ, given by

λ =Lcri

If the compressive stress in the column equals to the yield strength of thematerial, the slenderness calculated in that case is called the Euler slender-ness :

λ1 = π

√E

fy

8


P PA B

y

x

M(x)

PP

A

w

L

Figure 2.2: Pin ended column under the effect of Buckling load adapted from Galambos andSurovel (1998)

Mathematical derivation of the Euler load for a pin ended column

Using the free body diagram in the right side of Fig. 2.2 and making a summation ofmoments about point A: ∑

M = 0 ⇒ M(x)

+ Pw = 0 (2.3)

where w is the lateral deflection.According to Euler–Bernoulli beam theory, the deflection of a beam is related with

its bending moment by

M = −EI d2w

dx2. (2.4)

By replacing (2.4) in (2.3) it is obtained the following partial differential equation

d2w

dx2+ λ2w = 0 (2.5)

whereλ2 =

P

EI

A classical homogeneous second-order ordinary differential equation is thus obtained.The various buckling loads are:

Pn =n2 π2EI

Lcr2 for n = 1, 2, 3... (2.6)

and depending upon the value of n, different buckling modes are produced as shownin Fig. 2.3. The load and mode for n = 0 is the nonbuckled mode. Theoretically, anybuckling mode is possible, but in the case of slowly applied load only the first modalshape is likely to appear Timoshenko and Goodier (1970).

9


P3 = 9 π2EIL2

P1 P1

P1 = π2EIL2

P2

P2 = 4 π2EIL2

P3P3

P2

L3

L2

L

Figure 2.3: First three modes of buckling load adapted from Timoshenko and Gere (1961)

2.2 Finite Element solution

Approximation by Trial Function

To find the beam element stiffness matrix, we assume that the deflection of a beamelement can be approximated by a trial function

w = w0 + w1 x+ w2 x2 + w3 x

3 + . . .+ wn xn (2.7)

where the coefficients of the polynomial are unknown constants. Retain only the firstfour terms of this series:

w =[1 x x2 x3

]w0

w1

w2

w3

= Nuw = wTNTu (2.8)

whereNu =

[1 x x2 x3

](2.9)

The series is written in terms of the unknown displacements at the ends of the beamelement. Express these unknown displacements for the eth element as

v =[wa θa wb θb

]T(2.10)

The convention for a single element is presented in Fig. 2.4. To construct a relationshipbetween the coefficients w and the unknown end displacements v, the following systemof equations is formed, where ` represents the length of the element,

w(0)

− dw(x)dx

∣∣∣x=0

w(`)

− dw(x)dx

∣∣∣x=`

=

wa

θa

wb

θb

⇔

1 0 0 0

0 −1 0 0

1 ` `2 `3

0 −1 −2 ` −3 `2

w0

w1

w2

w3

=

wa

θa

wb

θb

(2.11)

10


θa θb

wa wb

Figure 2.4: Unknown displacement for the eth element

It follows that the coefficients w can be expressed in terms of the discrete, unknownnodal displacements ve as

w = Gv (2.12)

where

G =

1 0 0 0

0 −1 0 0

−3/`2 2/` 3/`2 1/`

2/`3 −1/`2 −2/`3 −1/`2

(2.13)

The desired relationship between w and the unknown nodal displacement v is

w = Nu w = NuGv = Nv (2.14)

where N = NuG is called shape function matrix.

Principle of virtual work

The principle of virtual work can be used to formulate an equation that accounts,approximately, for the bending effects of the axial force N . The introduction of acompressive axial force of value N = −P in the principle of virtual work equation forthe eth element gives

−(δ Wint + δ Wext

)= 0⇔∫ `

0

(δ

d2w

dx2

)EI

d2w

dx2dx−

∫ `

0

(δ

dw

dx

)P

dw

dxdx−

∫ `

0

δ wpz dx = 0 (2.15)

The variation of (2.14) delivers the following equation:

δ w = δ(Nv)

= Nδv = δvTNT (2.16)

where δvT gathers the virtual displacements, and

w′′ = N′′uGv = BuGv = N′′v = Bv

δ w′′ = Bδv = δ vTBT(2.17)

where:

Bu(x) = N′′u(x) =d2 Nu(x)

dx2=[0 0 2 6x

](2.18)

11


and

BTu (x) =

0

0

2

6x

(2.19)

Substitute these expressions (δw′′, w′′, δw) in the principle of virtual work (2.15) it isobtained:

δvT((k− P kσ)v− p

)= 0 (2.20)

In the previous equation the stiffness matrix k for the element eth can be obtained asfollows:

k = GT

∫ `

0

{BTu (x)EIBu(x) dx

}G (2.21)

Inserting the expressions of Bu and G into the integral expression of the stiffnessmatrix, the following explicit form can be gained:

k =EI

`3

12 −6 ` −12 −6 `

−6 ` 4 `2 6 ` `2

−12 6 ` 12 6 `

−6 ` `2 6 ` 4 `2

(2.22)

Matrix kσ present in equation (2.20) stands for the geometrical stiffness matrix. Bysubstituting the first derivative of equations (2.14) and (2.16) into the term whichdepends upon the axial force, N , of eq. (2.15) we obtain the following expression:

kσ =

∫ `

0

N′T N′ dx = GT

∫ `

0

N′Tu N′u dxG (2.23)

where N′ = dN/dx. This matrix is referred to as the geometric, differential, or stressstiffness matrix. Matrix kσ is used in the study of structural stability or to performincremental nonlinear analysis of frame structures. The phenomenon of a tensile axialforce N increasing the bending stiffness is referred to as stress stiffening: hence theterminology stress stiffness matrix. The use of the same displacement trial functionto form kσ as in deriving k leads to a geometric stiffness matrix that is said to beconsistent. Substitution of the cubic polynomial Nu and matrix G into eq. (2.23)yields

kσ =1

30`

36 −3 ` −36 −3 `

−3 ` 4 `2 3 ` −`2

−36 3 ` 36 3 `

−3 ` −`2 3 ` 4 `2

(2.24)

The vector p is a column vector that gathers the equivalent nodal forces appliedperpendicularly on the eth element,

p =

∫ `

0

NTpz dx (2.25)

12


In the case of absence of perpendicular loads the expression p = 0.The solution of Eq. (2.20) is given by

δv = 0 or (k− P kσ)v = 0 (2.26)

As the solution must hold for arbitrary variations δv, then only the second solutionis acceptable. By assembling the elemental contributions and imposing the kinematicboundary conditions, it is obtained the global system of equations, which can be writtenas follows

(K− P Kσ)v = 0 (2.27)

As v means the solution is given by the undeformed configuration, then it is necessaryto impose that (K − P Kσ) is singular. Hence, the buckling analysis is given by ageneralised eigenvalue problem, where critical loads are the eigenvalues, formulatedaccording to the equation:

|K− P Kσ| = 0 (2.28)

where:

P : eigenvalue which represents a buckling loadK: global stiffness matrixKσ: global geometrical stiffness matrix

2.2.1 Example: Pin ended column’s, buckling load

The cross section of the column, which is made of a trapezoidal stiffener and the ad-herent part of the plate, is presented in Fig. 2.5. The cross sectional variables are alsomarked on the figure and their values are listed in the following. The buckling loadof the column is calculated by (i) the analytical solution presented in section 2.1, (ii)by MATLAB solving the generalised eigenvalue problem and (iii) with the use of finiteelement software in the subsection 2.3.1. The three solutions are then compared andconclusions are drawn.

Material properies

The only material property which is needed for the linear buckling analysis is the mod-ulus of elasticity.

Young’s Modulus: E = 210 000 MPa

Geometrical properies

The dimensions are referred to the midline of the walls of the cross section in Fig. 2.5.

13


h

b1/2

bsf

bspb1/2tp

tsf

y

z

Figure 2.5: Cross section of a trapezoid stiffener

d1 d2

Figure 2.6: Convention for discretisation with a single finite element

Plate thickness: tp = 10 mm

Distance between the stiff.: b1 = 400 mm

Length of the column: a = 6000 mm

Hight of the stiff.: hs = 160 mm

Base of the stiff.: bsp = 200 mm

Upper flange of the stiff.: bsf = 158 mm

Thickness of the stiff.: tsf = 6 mm

Cross sectional properies

The inertia moments are evaluated relatively to the center of gravity.

Area of the cross section: Atotal = 89.083 cm2

Inertia around y axis: Iyy = 19 768.025 cm4

Inertia around z axis: Izz = 3120.016 cm4

Elemental discretisation of the problem

With the stiffness and geometrical stiffness matrices presented in the previous sec-tion 2.2 and in Pilkey (2002), the critical buckling load of the beam example is cal-culated, by using one and two elements. The calculations were done by MATLABsymbolic package.

14


d1 d3 d4

d2

Figure 2.7: Convention for discretisation with two finite elements

With a single element the model has two degrees of freedom (dof ), so two eigen-values can be calculated, which are also the first two critical load values’. The dofconvention is presented in Fig. 2.6, where L is the length of the column and ` is thelength of the element used for the descritisation, which is the same in the present case.

After imposing the kinematic boundary conditions, the generalised eigenvalue prob-lem 2.28 renders: ∣∣∣∣∣EIL

[4 2

2 4

]− λ

[4 −1

−1 4

]L30

∣∣∣∣∣ = 0 (2.29)

Hence, the approximations of the buckling loads are:

λ1 = 12EI

L2λ2 = 60

EI

L2

With two identical elements, the number of dof is four, so with a four-by-four matrix,four eigenvalues can be obtained, which represents the first four buckling loads’ values.The convention used in case of two elements is indicated in Fig. 2.7, where the lengthof the column is divided into two elements, ` = L/2. Thus, the generalised eigenvalueproblem delivers∣∣∣∣∣∣∣∣∣

EI`3

4`2 6` 2`2 0

6` 24 0 −6`

2`2 0 8`2 2`2

0 −6` 2`2 4`2

− λ

4`2 3` −`2 0

3` 72 0 −3`

−`2 0 8`2 −`2

0 −3` −`2 4`2

130`

∣∣∣∣∣∣∣∣∣ = 0 (2.30)

The buckling loads approximations are, in this case,

λ1 =4(13− 2 ·

√31)

3

EI

`2≈ 9.94385

EI

L2

λ2 = 12EI

`2≈ 48

EI

L2

λ3 =4(13 + 2 ·

√31)

3

EI

`2≈ 128.722820

EI

L2

λ4 = 60EI

`2≈ 240

EI

L2

The values of the buckling loads obtained with one- and two-element discretisationsare summarised in table 2.1, as well as the analytical solutions for the first four modes.

15


mesh modes

size 1/h 1st 2nd 3rd 4th[mm

] [1/mm

]6000 0.000167 2184.00 10920.06 — —3000 0.000333 1809.79 8736.04 23427.67 43680.22

analytical solution 1796.28 7185.11 16166.50 28740.44

Table 2.1: Convergence analysis of pin ended beam

With the two elements, the first two modes can be captured more accurately than inthe one element discretization. Also, as four dofs are employed, the first four modescan be estimated.

As the approximations are strictly compatible, the obtained values of the bucklingloads bound the exact ones from above.

2.3 Finite Element Solution (Software)

The buckling of the same pin ended beam is also modeled in ANSYS MechanicalAPDL finite element software package. The modeling of the beam is carried out withBEAM4 element, implemented in ANSYS. By comparing the Ansys results and theresults obtained in the previous section, it will be possible to verify the correctness ofthe current implementation in the ANSYS Mechanical APDL.

2.3.1 Beam model (Example)

Element

For the beam model BEAM4 element of ANSYS Mechanical APDL is used. It isa uniaxial element with tension/compression, torsion and bending capabilities. Theelement has six degrees of freedom at each node: translations in the nodal x, y, and zdirections and rotations about the nodal x, y, and z axes.The shear deformation capabilities of the element are restricted, to make it behave asa Euler–Bernoulli beam element.

Boundary conditions

The boundary conditions of the simply supported beam are represented in Fig. 2.8. Thebeam is loaded in one end, while on the other end longitudinal movement is restrained.Rotations are restricted in plan X − Y and free around Z axis.

Buckling modes

The first three buckling modes are presented in Fig. 2.9, while convergence resultsare summarised in table 2.2. The finite element solution is converging very quickly.The results for the buckling loads are determined by the finite element software are

16


X

Y

ZELEMENTS

U

ROTF

Figure 2.8: Boundary conditions of simply supported, pin ended beam

Figure 2.9: The first three buckling modes of simply supported, pin ended beam

equal to the ones that were obtained by MATLAB. It means that the beam modeledwith BEAM4 elements in ANSYS Mechanical APDL was truly behaving as an Euler–Bernoulli beam.

Rate of convergence – Standard error estimates

The optimal theoretical convergence rate of the Galerkin finite element method for thestandard classes of elliptic eigenvalue problems is given by Hughes (2000)

λl ≤ λhl ≤ λl + c h2(k+1−m) λ(k+1)/ml (2.32)

where:

h: element size,λl: lth eigenvalue,c: is a constant independent of h and λl,k: maximum order of the complete polynomial used to construct the shape func-tions,m: maximum order of the derivatives in the equilibrium/compatibility differentialoperator.

17


mesh modes

size 1/h 1st 2nd 3rd[mm

] [1/mm

]6000 0.000167 2184.02 10920.09 —3000 0.000333 1809.80 8736.08 23427.771500 0.000667 1797.20 7239.19 16704.96750 0.001333 1796.34 7188.82 16206.95375 0.002667 1796.29 7185.37 16169.21187.5 0.005333 1796.28 7185.15 16166.7393.75 0.010667 1796.28 7185.14 16166.5746.875 0.021333 1796.28 7185.14 16166.56

analytical solution 1796.28 7185.11 16166.50

Table 2.2: Convergence analysis of simply supported beam

As seen in equation (2.14), the Euler–Bernoulli beam element uses cubic shape func-tions, hence (k = 3). As the problem demands for C1 approximations, then (m = 2).Hence, the expected rate of convergence for the eigenvalues is

2(k + 1−m

)= 2

(3 + 1− 2

)= 4

By plotting the log of element size, (h), against the log of relative error (e), the nu-merical rate of convergence is given by the slope of the obtained line. In this case theslope of the error of the buckling load should be around 4. In Fig. 2.10 the gradientsare indicated under the legend of every line. The calculated rates of convergence arein a good agreement with the theoretical value.

18


0.0001

0.0001

0.001

0.01

0.1

1100

log(e)

log(h)

1st

2nd

3rd

3.795

3.629

3.788

1000 10000

0.00001

Figure 2.10: Rate of convergence, first three eigenvalues

19

20

Chapter 3

Plate buckling


3.1.1 Simply Supported Rectangular Plates under Forces in the MiddlePlane of the Plate

Assume that a plate is under uniform tension in the x direction, as shown in Fig.3.1. The deduction presented in this section follows Timoshenko and Woinowsky-Krieger (1959). The uniform lateral load (or self weight) q0 can be represented by thetrigonometric series:

q =16 q0π2

∞∑m=1,3,5,...

∞∑n=1,3,5,...

1

mnsin

mπ x

asin

nπ y

b(3.1)

The following equation was derived from the compatibility, constitutive relationsand equilibrium equations of a laterally loaded plate:

∂4w

∂ x4+ 2

∂4w

∂ x2∂ y2+∂4w

∂ y4=

1

D

(q +Nxx

∂2w

∂ x2+Nyy

∂2w

∂ y2+ 2Nxy

∂2w

∂ x∂ y

)(3.2)

In this example we only have load in one direction, so Nyy and Nxy will fall out, onlyleaving Nxx in the right-hand-side of equation (3.2). By substituting the trigonometric

a

O

b

x

y

Nxx Nxx

Figure 3.1: Plate in tension

21

Chapter 3 Plate buckling

representation of the q load in the previous equation, the following will be obtained:

∂4w

∂ x4+2

∂4w

∂x2∂y2+∂4w

∂y4−Nxx

D

∂2w

∂ x2=

16 q

Dπ2

∞∑m=1,3,5,...

∞∑n=1,3,5,...

1

mnsin

mπ x

asin

nπ y

b(3.3)

The boundary conditions at the simply supported edges will be satisfied if the deflec-tion, w, is taken in the form of the series

w =∑∑

amn sinmπ x

asin

nπ y

b(3.4)

If we substitute this series into equation (3.3), after some symbolic manipulations, itis possible to obtain the following formula for amn:

amn =16 q

Dπ6mn

[(m2

a2+ n2

b2

)2

+ Nxxm2

π2Da2

] (3.5)

where m and n are odd numbers 1, 3, 5, . . . , and amn = 0 if m or n or both are evennumbers. Hence the deflection surface of the plate is

w =16 q

π6D

∞∑m=1,3,5,...

∞∑n=1,3,5,...

1

mn

[(m2

a2+ n2

b2

)2

+ Nxxm2

π2Da2

] sinmπ x

asin

nπ y

b(3.6)

If, instead of the tensile forces Nxx, there are compressive forces the same magnitude,the deflection of the plate is obtained by substituting −Nxx in the place of Nxx inexpression (3.6). The smallest value of Nxx at which the denominator of one of theterms in expression (3.6) becomes equal to zero is the critical value of the compressiveforce Nxx. It is evident that this critical value is obtained by taking n = 1.

(Nxx

)cr

=π2a2D

m2

(m2

a2+

1

b2

)2

=π2D

b2

(mb

a+

a

m b

)2

(3.7)

Plotting the buckling coefficient, kσ,

kσ = k =

(mb

a+

a

m b

)2

(3.8)

against the ratio α = a/b for various values of m, which is an integer representing thenumber of half sine waves in the direction of compression to be associated to the aspectratio α, it is obtained the set of curves shown in Fig. 3.2. The portions of the curvesthat must be used in determining kσ are indicated by heavy lines.

In practice, the above expression of kσ is relevant only when m = 1, which corre-sponds to α ≤

√2; indeed, larger values of n result in values of kσ which are only slightly

larger than 4. The value of 4 is a good approximation also in the range 1 ≤ α ≤√

2.Accordingly, for the so-called long plates (1 ≥ α), it is usual to adopt conservativelykσ = 4. The Fig. 3.3 shows the buckling of a rectangular plate, due to compressionfrom one direction. It represents the buckled shape described by w in equation (3.4)and the corresponding representation of the plate by beams.

22


√6

√21 5 64

√12320

0

2

4

6

8

10

kσ

α = a/b

m = 1 2 3 4

Figure 3.2: Factor k according to the ratio of the sides

Figure 3.3: Deflected shape of a compressed plate

3.1.2 Example: Rectangular, simply supported plate

A rectangular plate, with the following dimensions was analysed: a = 6000 mm,b = 3000 mm and h = 10.0 mm (thickness of the plate). The layout of the plate isthe same as in Fig. 3.3.Material properties (steel plate):


Poisson’s Ratio: ν = 0.3

Yield strength: fy = 355 MPa (not necessary)

Flexural rigidity: D =E h3

12(1− ν2

) = 1.9 · 107N/mm

23


m n Ncr

[N/mm2

]2 1 84.35563 1 99.00071 1 131.8064 1 131.8065 1 177.3586 1 234.3217 1 302.238

Table 3.1: Number of semi-waves and buckling loads

The Table. 3.1 summarises the number of semi-waves in each direction of the plateand the buckling load associated with the first seven mode shapes obtained with thesolution presented in subsection 3.1.1. The number of longitudinal waves increase from1 to 7. The mode shapes can then be arranged according to their ascending bucklingload value. It is important to point out that two of the mode shapes have the samebuckling load.

The buckling loads and the corresponding mode shapes can be determined by finiteelement analysis, as showed in the next section.

3.2 Numerical solution

The plate mentioned in section. 3.1.2 was modelled using the finite element softwareANSYS Mechanical APDL. The same geometrical and material properties where ap-plied as for the analytical calculation so the results could be compared.

3.2.1 Boundary conditions

Although in section 3.1.1 thin plate or classic plate solution was introduced, as thepresent plate is considered thin (a/h = 300), modelling with thick shell elementswouldn’t produce much difference compared to modelling it with thin shell or so calledKirchhoff–Love shell elements (as long as the thick based elements do not suffer fromshear-locking).

Consider a side parallel to the y axis, as in Fig. 3.1. In this case, the simply sup-ported sides of a thick plate can be modelled with the hard or soft support conditionssummarised in Table 3.2. Hard support should be used in case of simply supportedsides of Kirchhoff–Love shell elements. In case of modelling very thin plates, whereshear deformation tends to be zero, and it is desired to recover the Kirchhoff–Lovesolution, then hard supports should be used regardless the type of element used formodelling the plate.

In case of the finite element model, hard supports were used as it is meant to bevalidated against the analytical solution, which assumes thin plate. As the plate isvery thin, shear deformation would not introduce a significative effect in the solution.Consider now the reference system introduced in Fig. 3.4. The displacements along thethree axis x, y and z will be from now on denoted by ux, uy and uz, respectively. On

24


hard support soft supportw = 0 displacement w = 0

mxx = 0 moment mxx = 0

θy = 0 rotation/moment mxy = 0

Table 3.2: Hard supports and soft supports boundary conditions

X

Y

Z

UROT

Figure 3.4: Boundary conditions of a simply supported plate (hard support)

the sides of the model the generalized displacements uy, θx and θy were restricted. Theplate is loaded along the two shorter sides. At the middle of the longer sides the uz dofwas also restricted. At the middle of the shorter sides the ux dof was also restricted.

3.2.2 Type of elements used

The plate is modelled with the finite elements SHELL181 and SHELL281. Both of theelements are suitable for analysing thin to moderately-thick shell structures as theirformulation is based on the thick shell theory. They possess six degrees of freedom ateach node: translations in the x, y, and z directions, and rotations about the x, y, andz-axes. SHELL181 is a four-node element (including complete linear approximationfunctions), while SHELL281 is an eight-node element (including complete quadraticapproximation functions) (Bathe and Dvorkin, 1986). Elements based with higherorder shape functions are not available in ANSYS.

Both of the shell elements are well-suited for linear, large rotation, and/or largestrain nonlinear applications. Change in shell thickness is accounted for in nonlinearanalyses. The elements accounts for follower (load stiffness) effects of distributed pres-sures. The degenerate triangular option should only be used as filler elements in meshgeneration. It was sometimes used when the plates had geometrical imperfections (AN-

25


SYS, 2013b,a).

As mentioned in Bathe and Dvorkin (1986)“The requirements we have set upon our development of shell elements aregoverned by our desire to render the elements widely applicable in routineapplications.”

The authors summarise the requirements towards shell elements as:

1. The element should be formulated without the use of a specific shell theory sothat it is applicable to any plate/shell situation.

2. The theoretical formulation of the element should be strongly continuum mechan-ics based with assumptions in the finite element discretisation that are mechani-cally clear and well-founded.

3. The element should be ‘numerically sound’: it must not–ever–contain any spuriouszero energy modes, it must not–ever–lock and must not be based on numericallyadjusted factors.

4. The element should be simple and inexpensive to use with, for shell analysis, fiveor six engineering degrees of freedom per node and for plate analysis the threeengineering degrees of freedom per node.

5. The predictive capability of the element should be high and be relatively insensi-tive to element distortions.

3.2.3 Convergence analysis

Convergence analysis considering successive mesh discretisations is called h-refinement.This convergence analysis was performed both with SELL181 and SHELL281 elements.

During the convergence analysis the buckling loads were examined for the first sevenmodes. These results are presented in Tables A.1 and A.2, which are available in theappendix A. In these, it is visible how the finite element solution is converging to theanalytical solution. In Figure 3.5 the results of the convergence analysis are presented.In order to facilitate the reading of the graph, Figure 3.6 presents the results justfor the first four bucking loads. The dashed lines are the analytical solutions, so itmeans that the same dashed line represents Ncr.an.3 and Ncr.an.4. Letters l and q in thelegends stand for, linear approximation in case of SHELL181 element and quadraticapproximation in case of SHELL281, respectively. The buckled shape modes of theplate are also represented.

Although the analytical bucking load associated with modes 3 and 4 is the same,the FEM solution is obviously not equal. However, of course, the FEM solutions tendto the exact value.

When SHELL281 element was used to model the plate, a numerical error occurredin case of the finest refinement (h = 46.875 mm) of the convergence analysis. Thisnumerical error is also visible in Fig. 3.5, where the last point of each quadratic conver-gence line diverges. The cause of this divergence was not identified, but it was verifiedthat, at least, until h = 65.00 mm the results are still converging.

On the mentioned figure it is clear that the analytical solution is a lower boundfor the numerical solutions. For design consideration it means that numerical simu-lation results is always on the unsafe side. As it was mentioned before, the first fourmode shapes convergence are plotted in Fig. 3.6. In case of the loosest refinement

26


2nd

mod

eq

0

100

200

300

400

500

600

0,00

02

Ncr[kN/mm]

1/h

[1/m

m]

Ncr

.an.

2

Ncr

.an.

3

Ncr

.an.

5

Ncr

.an.

6

Ncr

.an.

7

1st

mod

el

2nd

mod

el

3rd

mod

el

4th

mod

el

5th

mod

el

6th

mod

el

7th

mod

el

1st

mod

eq

3rd

mod

eq

4th

mod

eq

5th

mod

eq

6th

mod

eq

7th

mod

eq

Ncr

.an.

1

700

0,00

20,

02

Figure3.5:

Con

vergen

cean

alysis

first

sevenbu

cklin

gmod

es

27


the element with a linear approximation SHELL181 usually produces better results,the only exception being the second mode shape, where the quadratic elements resultswere superior from the beginning. From the second refinement onwards the SHELL281models clearly produced better results.

3.2.4 Rate of convergence

Convergence rates can be again calculated according to equation (2.32), which is pre-sented in Hughes (2000). In the case of SHELL181 element, shape functions with linearcompleteness have been used, while the elements are formulated as thick shell elements.So the values (k = 1) and (m = 1) apply and

2(k + 1−m

)= 2(1 + 1− 1

)= 2 (3.9)

In the case of SHELL281 element, shape functions with quadratic completenesshave been used, while the elements are formulated as thick shell elements. So thevalues (k = 2) and (m = 1) apply and

2(k + 1−m

)= 2(2 + 1− 1

)= 4 (3.10)

In Figures 3.7 and 3.8 the log of element size, (h), is plotted against the log of relativeerror on the bucking load, (e). The slope of these lines provide an estimate of thenumerical rate of convergence, which is indicated under the legend of every line. Thetheoretical rate of convergence can be calculated according to Equation (2.32).

In case of SHELL181 element, the rate of convergence should be around 2, as seenin Equation (3.9). As it is visible in Fig. 3.7, all the calculated numerical slopes areslightly higher than the theoretical one. Nevertheless, they are in a good agreement.

In case of SHELL281 element the rate of convergence should be around 4, as seenin Equation (3.10). It can be observed in Fig. 3.8 that the slopes differ in both direc-tion compared to the theoretical rate of convergence but, again, they are close to thetheoretical estimate.

Although when elements with quadratic approximation were used (SHELL281), thethird mode’s estimation was slightly better then the lower modes, in general, the qualityof the buckling modes’ estimation deteriorates as higher modes are approached. Thisfact is theoretically explained by the presence of the term λ

(k+1)/ml in Equation (2.32).

Note that the third buckling mode (in Figure A.1) is the “simplest” in term of geometry.To conclude, the implementation of buckling analysis of simply supported plates

in the ANSYS mechanical APDL using SHELL181 and SHELL281 elements can beconsidered validated. In the next chapters this implementation is extended to thebuckling and incremental physically and geometrically nonlinear analysis of plates withstiffeners.

28


50100

200

400 0,

0002

Ncr[kN/mm]

1/h

[1/m

m]

Ncr

.an.

1

Ncr

.an.

2

Ncr

.an.

3

1st

mod

el

1st

mod

eq

2nd

mod

el

2nd

mod

eq

3rd

mod

el

3rd

mod

eq

4th

mod

el

4th

mod

eq

0,00

20,

02

Figure3.6:

Con

vergen

cean

alysis

first

four

bucklin

gmod

es

29


0.0001

0.001

0.01

0.1

1

10

1

log(h)

1st mode l

2nd mode l

3rd mode l

4th mode l

5th mode l

6th mode l

7th mode l

10 100 1000 10000

log(e)

2.118

2.430

2.186

2.170

2.263

2.250

2.044

Figure 3.7: Rate of convergence for linear shell elements

0.0001

0.001

0.01

0.1

1

10

1

log(h)

1st mode q

2nd mode q

3rd mode q

4th mode q

5th mode q

6th mode q

7th mode q

10 100 1000 10000

log(e)

3.967

3.971

3.484

4.897

4.200

4.310

3.8340.00001

Figure 3.8: Rate of convergence for quadratic shell elements

30

Part II

Design approach

31

32

Chapter 4

Design and production technology

4.1 Eurocode’s approach based on the reduced cross-sectionconcept

In this secion a short description of the reduced cross-sectional method (EuropeanCommittee for Standardisation, 2006) is provided. The section is divided into foursubsections: 4.1.1, where an overview of the procedure is provided, 4.1.2, where thecalculation is carried out according to plate supports, 4.1.3, where supports along thelongitudinal sides are not taken into account, and finally 4.1.4, where the interpolationbetween the two behaviours is made. Overviews of same style are provided in Eurocodecommentaries and design manuals, see Johansson et al. (2007) and Beg et al. (2010).

4.1.1 General

1. The buckling verification of a longitudinally stiffened girder has several steps.These should be carried out according to the approach based on the reducedcross-section method. The following exposition is taken from Johansson et al.(2007).a) The stress distribution determined based on the assumption of a fully effec-

tive cross-section.b) From the obtained stress distribution, the reduced cross-section of each in-

dividual plate element, composing the whole cross section, is calculated.c) The determination of the stress distribution is based on the properties of

the reduced cross-section of the stiffened plate member, being the lattercomposed of the reduced sections of all the plate elements forming this cross-section.

d) In case of cross sections of Class 4, when the stress distributions obtained inStep a) and Step c) are undoubtedly different, a refinement of the reducedcross-section is made for each of the individual plate element based on thestress distribution obtained in Step c).

e) The above process is repeated till the stress distribution is consistent withthe properties of the reduced cross-section.

2. There is no limitation in the stress due to local plate buckling as it is takeninto account by the “concept of effectivep width”. The concept is applied to anyunstiffened plate element of the orthotropic plate, including both plating andstiffeners.

33

Chapter 4 Design and production technology

3. “For a given loading, the amount of post-buckling strength reserve is highly de-pendent of the aspect ratio of the plate element under consideration; it dependsmoreover on the orthotropy degree when this plate element is longitudinally stiff-ened.” (Johansson et al., 2007). Due to this fact, attention should be paid tothe two extreme situations—the so-called plate type behaviour and column typebehaviour (presented in sections 4.1.2 and 4.1.3)—and then interpolating betweenthe behaviours according to the interpolation formula 4.1.4, in European Com-mittee for Standardisation (2006).

4.1.2 Plate behaviour

1. The elastic critical plate buckling stress could be determined by appropriate soft-wares such as: EBPlate (CTICM, 2018) and ANSYS (2013a). Alternatively, forthat purpose, the Eurocode provides two simple approaches regarding to the num-ber of longitudinal stiffeners located in the compression zone of the girder :

– At least 3 longitudinal stiffeners, in which case it is referred to so-calledmultiple stiffeners ;

– One or two longitudinal stiffeners.

2. In the case of multiple stiffeners, the stiffened plate element may be treated asan orthotropic plate, the term refering to this type of calculation menthod issmeared stiffeners. The total rigidity of all the stiffeners is distributed acrossthe plate width. In this way a “fictitious” plate is created, where the concept ofsubpanels is irrelevant. This concept is described, e.g., by Johansson et al. (2007).

3. When the plate is longitudinally stiffened by one or two stiffeners, then a sim-plified procedure can be used. The elastic critical plate buckling stress, σcr,p, isdeduced from the elastic critical column stress, σcr,sl.

4. The effectivep width bc.eff of the compression zone of an unstiffened plate elementis a proportion, ρ, of the actual geometric width bc of the compression zone of thisplate element. This proportion is a reduction factor. It depends on the directstress distribution of ψ, which is the ratio of extreme fiber stresses, across thegeometric width, b, of the plate element and on the boundary conditions alongthe longitudinal edges.

– For internal compression in plate elements with two longitudinal edges sup-ported (ECCS Technical Committee 8, 1986), the reduction factor for effec-tive area follows:

ρ =1

λp− 0.055(3 + ψ)

λ2

p

≤ 1 (4.1)

– For outstand compression plate elements with one longitudinal edge sup-ported and the other free (Johansson and Veljkovic, 2001), the reductionfactor for effective area follows:

ρ =1

λp− 0.188

λ2

p

≤ 1 (4.2)

34

4.1 Eurocode’s approach based on the reduced cross-section concept

where λp is the relative plate slenderness. It is defined latter, similarly tocolumn slenderness. The relative plate slenderness is the square root of theratio between the “squash load” and the elastic critical load of the wholecompression zone of the plated structure. in consideration:

λp =

√Ac fyAc σcr.p

=

√fyσcr.p

(4.3)

Taking into account that the elastic critical plate buckling stress, σcr.p, isgiven as:

σcr.p = kσ σE = kσπ2E

12(1− ν2)

(t

b

)2

(4.4)

where kσ is the buckling coefficient presented in Equation 3.8, the relativeplate slenderness λp can be written more explicitly (with E = 210 GPa,ν = 0.3 and the yield factor ε =

√235/fy) as:

λp =

(tb

)28.4ε

√kσ

(4.5)

In both above expressions (4.1) and (4.2) of ρ, the first term is the well-known von Kármán contribution, which accounts for post-buckling strengthreserve. It is supposed to provide the same behaviour as an ideally elastic per-fectly undeformed plate. The second term is a penalty, which was calibratedagainst test results so as to account for the adverse effects of out-of-planeimperfections of the plate element, residual stresses and interaction betweenmaterial yielding and plate buckling, according to Johansson et al. (2007).During the numerical modelling these effects were modelled by placing ge-ometrical imperfections and providing the elements with residual stresses.The reduction factor, ρ, depends on the stress ratio, ψ, in such a way that,with some approximations, a full effectivness (ρ = 1) is consistent with theb/t limits related to Class 3 plate elements.

5. Similarly, the effectivep width bc.eff of the compression zone of a longitudinallystiffened plate element is a proportion ρloc of the actual width bc of this zone. Theexpression of the relevant reduction factor ρloc is the same as for the unstiffenedplate element:

ρloc =1

λp− 0.055(3 + ψ)

λ2

p

≤ 1 (4.6)

The relative plate slenderness λp should be modified to take the effects of localplate buckling into account. The “squash load” then results from the yield strengthapplied on a reduced cross-sectional area, Ac.eff.loc, to take into account the localplate buckling effects. This slenderness, λp, can be formulated as the following:

λp =

√Ac.eff.loc fyAc σcr.p

=

√βA.c fyσcr.p

(4.7)

35


b1 b2 b3

0.5b30.5b1

AcAc.eff.loc

ρ1 b12

ρ2 b22

ρ3 b32

b1 b2 b3

b1.bord.eff b1.bord.eff

ρ2 b22

Figure 4.1: Ac and Ac.eff.loc for a stiffened plate element (European Committee for Standard-isation, 2006)

where:

βA.c =Ac.eff.locAc

(4.8)

The elastic critical plate buckling stress is computed based on an equivalent or-thotropic plate, i.e. a plate with smeared stiffeners, so that local plate bucklingis irrelevant here.

6. When computing βA,c, the cross-sectional areas, Ac and Ac.eff.loc, should be cal-culated from the same compresion area and the same compression width, respec-tively. This width differs from the actual width by the part of the stiffened platesubpanels, which are supported by the adjacent structures. As Ac and Ac.eff.locare calculated from the same compression width, they are comparable. The cross-sectional area Ac shall not include the mentioned parts of subpanels. Fig. 4.1 isdescribing how the areas Ac and Ac.eff.loc are calculated. This figure, and therespective explanation, are avalaibele in Johansson et al. (2007) and in Beg et al.(2010).

7. The critical plate buckling stress, σcr.p, for both stiffened or unstiffened plateelement is given by:

σcr.p = kσ.pπ2E

12(1− ν2

)( tb

)2

= 190000 kσ

(t

b

)2

(4.9)

8. The buckling coefficient, kσ, for simply supported unstiffened compression plateelements—including wall elements of longitudinal stiffeners—subjected to uni-form compression is evaluated according to Eq. (3.8).For stiffened plates with at least three equally spaced longitudinal stiffeners, theplate buckling coefficient, kσ,p, (global buckling of the stiffened panel) may beapproximated by the following formulas according to Eurocode European Com-mittee for Standardisation (2006):

36


kσ.p =

2

((1 + α2

)2+ γ − 1

)α2(ψ + 1

)(1 + δ

) if α ≤ 4√γ

kσ.p =

4

(1 +√γ

)(ψ + 1

)(1 + δ

) if α > 4√γ

(4.10)

where:

α = a/b aspect ratioψ direct stress distribution

δ =Acb tp

axial stiffness

γ =ItotalIp

stiffness ratio

Ip the second moment of area for bending of plates

4.1.3 Column behaviour

1. The expressions of ρ (plate buckling reduction factor) in Equations (4.1) and (4.2)account for a post-buckling strength reserve. However, a column type behaviourwith no such post-buckling reserve at all may be exhibited when: (i) the aspectratio a/b < 1 is small (for a non stiffened plating) and/or (ii) the plate orthotropyis large (if longitudinally stiffened plate element). Then, a reduction factor, χc,relative to column buckling is required, which is more severe than, ρ applicableto typical plate buckling.

2. Modelling the column behaviour is simply achieved by removing the longitudinalsupports of the plate element.

3. The elastic critical column buckling stress σcr.c is computed as follows:a) For unstiffened plate element, the Equation (4.4) is used.b) For a stiffened plate element, it is first identified the buckling stress (buckling

load divided by the cross sectional area) σcr,sl of a hinged, axially loadedcolumn composed of: (i) the stiffener that is located closest to the paneledge with the highest compressive stress, and (ii) an adjacent contributivepart of plating, given by:

σcr.sl =π2EIsl,1Asl,1a2

(4.11)

where:

Isl,1: is the second moment of area for the gross cross section, relative tothe out-of-plane bending of the stiffened plate element,

37


Asl,1: is the gross cross-sectional area of the stiffener and the adjacentparts of the plate,a: in this case, is the buckling of a stiffener, which is usually the panellength of the orthotropic plate distance between rigid transverse girders.

4. The relative column slenderness λc is given as the square root of the ratio betweenthe “squash load” and the elastic critical column buckling load of:a) A plating strip of unit width when unstiffened plate element is concerned:

λc =

√fyσcr.c

(4.12)

b) A column composed of the stiffener and the adjacent part of plating:

λc =

√Asl.1.eff.loc fyAsl.1 σcr.c

=

√βA.c fyσcr.c

(4.13)

where:βA.c =

Asl.1.eff.locAsl.1

(4.14)

The cross sectional area Asl,1,eff is the reduced section, when local plate bucklingin the plating and/or possibly in the wall elements of the stiffener are taken intoaccount (for the plating, see column “effective area” in Figure 4.1).

5. Following European Committee for Standardisation (2006), the expression for therelevant reduction factor χc is the same as for usual column buckling

χc =1

φ+

√φ2 − λ2c

(4.15)

where:φ = 0.5

[1 + αe

(λc − 0.2

)+ λ

2

c

](4.16)

and αe is a modified imperfection parameter, which accounts for larger initialgeometric imperfection.

6. It is common to stiffen a plate with one-sided longitudinal stiffeners. It resultsa shift in the neutral axis, from the mid line of the plate to the centroid of theorthotropic plate. The eccentricity of the stiffeners with respect to the plating inFig. 4.2 is regarded for determining the increase of the value of the generalisedimperfection parameter α governing the analytical expressions of the bucklingcurves:

αe = α +0.09

i/e(4.17)

where:

i =

√Isl.1Asl.1

(4.18)

is the radius of inertia.e = max(e1, e2) is the largest distance from the respective centroids of the platingand the one-sided stiffener (or of the centroids of either set of stiffeners when

38


e1

e2

G1: centroid of the stiffener

G: centroid of the stiffenerincluding the adherent plating

G2: centroid of the plate

Figure 4.2: Excentricities of the stiffeners (Johansson et al., 2007)

present on both sides of the plating) to the neutral axis of the stiffener includingthe contributive plating represented in Fig. 4.2.The use of closed section stiffeners results in a greater stability and in less residualstresses (because of thin walls and one-sided fillet welds) justifies α = 0.34; a largervalue α = 0.49 is required for open section stiffeners.

4.1.4 Interpolation between plate and column behaviour

1. First, the reduction factors are computed based respectively on a plate behaviour(reduction factor ρ) and on a column behaviour (reduction factor χc), as indicatedin Sections 4.1.2 and 4.1.3.

2. The actual behaviour is usually somewhere between the mentioned two extremesituations. In case of longitudinally stiffened plate the resistance in the ultimatelimit state (ULS) has to reflect the in-between nature of plate- and column-likebehaviour. For the resulting reduction factor, ρc, the following relation holds:

χc ≤ ρc ≤ ρ (4.19)

3. The reduction factor, ρc, is obtained by the following interpolation formula, pro-posed by Eurocode:

ρc = ξ(2− ξ

)(ρ− χc) + χc (4.20)

where the parameter ξ is a kind of measure of the “distance” between the elasticcritical plate and column buckling stresses according to:

ξ =σcr.pσcr.c

− 1 but 0 ≤ ξ ≤ 1 (4.21)

The limits designated to the parameter ξ are physically justified. The criticalstress related to plate behaviour may never be exceeded by the critical stress re-lated to column behaviour, so the σcr.p ≥ σcr.c relation always holds, resulting apositive value of ξ.According to Eurocode column behaviour, for the evaluation of ξ, it is irrelevantwhen σcr.p is significantly larger than σcr.c. When σcr.p ≥ 2σcr.c, from Eq. (4.20),it can be derived that the whole behaviour of the orthotropic plate is equal toρc = ρ, ξ ≤ 1.

39


ρc = (ρp − χc)ξ(2− ξ) + χc

ξ = 0 ξ = 1 ξ > 1

Plate-like buckling

Column-like buckling

ρc = ρ

ρc = χc

Interaction

Figure 4.3: Interpolation formula between column and plate behaviour

4. This interpolation formula between column and plate behaviour is plotted in Fig-ure 4.3.

The whole procedure of the reduced cross sectional method is summarised onFig. 4.4, where the column and plate buckling branches are clearly separated and,at the end, they are combined with the interpolation function according to Eq. (4.20).The process of interpolation between plate- and column-like behaviour, according tothe reduced cross-section method, is similarly summarised in Johansson et al. (2007)and in Beg et al. (2010). Resemblant interpolation plots to Fig. 4.3 are also providedin those references.

40


σ

ψ · σa

b

for webs b = hw

σ

ψ · σ

Column buckling Plate buckling

λc =√βA.c

fyσcr.c

λp =√βA.c

fyσcr.p

Account for local plate buckling βA.c =Ac.eff.loc

Ac(column) Asl.eff.loc

Asl

αe = 0.34(0.49) + 0.09i/e

ϕ = 0.5[1 + αe(λc − 0.2) + λ2c ]

χc =1

ϕ+√ϕ2−λ2c

ρp =λp−0.055(3+ψ)

λ2p

Interaction between columnlike and platelikebehaviour

ξ =σcr.pσcr.c− 1

ρc = (ρp − χc)ξ(2− ξ) + χc

Simplifiedprocedure:

disregard platelikebehaviour:ξ = 0

(plate)

Figure 4.4: Interaction of plate and column like behaviour, adapted from Johansson et al.(2007) 41


4.2 Eurocode’s approach based on reduced stress method

Besides the effective width method, EN 1993-1-5 European Committee for Standard-isation (2006) also provides the reduced stress method. The reduced stress methodapplies not only for standard steel plated cross sections, such as I- and box-girders, butalso for members with non-parallel flanges and webs with regular or irregular openingsand non-orthogonal stiffeners.In contrast to the effective width method, the reduced stress method assumes a linearstress distribution up to a stress limit of the plate element which buckles first. Untilthis stress limit has been reached, the cross section is fully effectivep.For a single plate element the reduced stress method fully corresponds to the effec-tive width method. However, for steel plated cross sections the reduced stress methoddoes not take into account load shedding from highly stressed to less stressed plateelements. As a result, the weakest plate element in a steel plated cross section governsthe resistance of the entire cross section (Beg et al., 2010).The reduced stress method uses the von Mises criterion to take into account the interac-tion between different stress types, followed by a combination of these load types. Dueto the fact that von Mises stress are analysed, an interaction equation is unnecessary.The verification of the plate subject to the complete stress field is:

√√√√( σx,Edρx fy/γM1

)2

+

(σz,Ed

ρz fy/γM1

)2

−

(σx,Ed

ρx fy/γM1

)2(σz,Ed

ρz fy/γM1

)2

+ 3

(τEd

χw fy/γM1

)2

≤ 1

(4.22)where:

Design stresses (loads): σx,Ed; σz,Ed; τEd

Yield strength (resistance): fy

Reduction factors: ρx; ρz; χw

Partial safety factor: γM1

Conservatively, if only the minimum reduction factor ρ is taken into account,

ρ = min(ρx; ρz; χw)

and Eq. (4.22) simplifies to:√√√√( σx,Edfy/γM1

)2

+

(σz,Edfy/γM1

)2

−

(σx,Edfy/γM1

)2

·

(σz,Edfy/γM1

)2

+ 3 ·

(τEd

fy/γM1

)2

≤ ρ

(4.23)The reduction factors are determined only with one plate slenderness λp follows:

λp =

√αult,kαcr

(4.24)

where:

42

4.3 Production technology

αult,k minimum load amplifier for the design loads to reach the characteristic valueof the resistance, presented later on.αcr minimum load amplifier for the design loads to reach the elastic critical valueof the plate, described at the end of the present section.

For simple cases the load amplifiers αult,k and αcr can be determined by hand calcu-lations or based on adequate literature, see Klöppel and Scheer (1960) and Klöppeland Möller (1968). The equivalent stress σeq,Ed is used to illustrate the procedure.Assuming a plane stress field, σeq,Ed is defined as follows:

σeq,Ed =√σ2x,Ed + σ2

z,Ed − σx,Ed · σz,Ed + 3 · τ 2Ed (4.25)

The load multiplier αult,k is the smallest factor for which the design equivalent stressσeq,Ed has to be increased to reach the characteristic yield strength fy. It can be writtento the following:

αult,k =fy

σeq,Ed(4.26)

The load amplifier αcr is the smallest factor for which the design equivalent stressσeq,Ed has to be increased to reach the elastic critical equivalent stress σeq,cr. It can beformulated as the following:

αcr =σeq,crσeq,Ed

(4.27)

4.3 Production technology

Orthotropic plates are manufactured by plates welded together in a factory. During thewelding the steel plates are subjected to very high temperatures, which causes changein the microstructure of the weld and the heat affected zones of the surrounding plates.The heat of the welding can be modeled with different heat source models. The twomost common ones are Gaussian heat source model and Goldak’s double ellipsoidalheat source model (Kollár, 2016). As a result of intensive research, plates and boxsections can be virtually fabricated (Kollár et al., 2017). The highly concentratedheat input creates an uneven temperature distribution of stresses in the welding. Thisresults in the development of residual stresses and deformations. Figure 4.5 showsa schematic representation of the temperature and thermal stress distribution duringwelding (Masubuchi, 1980). The high tensile stress near the welding might reach ashigh as the yield limit, it may reduce the strength of the structure and increases thechances of fatigue crack development. The compressive stresses further away from theweld, maintain equilibrium with the tensile stress in the unloaded structure, reducethe buckling strength of the structure (Néző, 2011). The residual stresses used for theimperfection are presented in Fig. 5.10.Figure 4.6 shows the most common modes of distortion observed in welded components.These deformations can be very significant and beside their unfavourable effects on thestrength of the structure they also make the assembly of the structure more difficult.

43


Figure 4.5: Temperature and thermal stress distribution during welding

Figure 4.6: Common types of weld distortion

44

Chapter 5

Model description and numericalresearch strategy

First of all, imperfections were compared with an example presented in “Design ofPlated Structures, ECCS Eurocode Design Manuals” (Beg et al., 2010) in section2.11.5. The example with different imperfections, combining geometrical irregulari-ties and residual stress, was analysed. After the comparison a decision was taken onwhich imperfection to use for all the numerical experiments, whose results are describedin chapter 6. In the present chapter, boundary conditions for the plate like behaviourand the material model used are described. These model properties are used through-out the thesis, so they are only presented here. The plate- and column-like boundaryconditions are compared in chapter 6.

During the numerical analysis geometric and material non-linearity with initial imper-fections was used to determine the resistance of a stiffened plate in pure compression.In this particular case, which was thoroughly examined, the panel is stiffened with twoparallel stiffeners, which are actually single plates. These stiffeners divide the plateinto three equal sub-panels. The examined plate can be a part of a more complex crosssection, a side flange of a steel girder bridge or any other similar engineering structure.The imperfection modelling will be based both on eigenmode shapes and equivalentgeometric imperfection, as proposed in EN 1993-1-5 (European Committee for Stan-dardisation, 2006). Both procedures are compared in the section 5.6.

5.1 Boundary conditions

The whole layout of the considered plate is presented in Fig. 5.1. The sides of the platein the longitudinal direction (axis Z) are supported against out of plane and sidewaysmovement. On the lateral sides, at the center of gravity of the cross-section, masternodes were placed on the upper and on the lower face of the stiffened plate. Thesemaster nodes were connected to all of the nodes on that face, which became slavenodes. The mentioned master nodes were supported against displacement in Y andX direction and rotation around the Z axis. The load or the displacement (in case ofimposed displacement experiment), was applied on the top node (Z = 0), while thebottom node (Z = length) was supported against longitudinal movement.

45

Chapter 5 Model description and numerical research strategy

X

Y

Z

UROTCE

Figure 5.1: Boundary conditions of examined plates

5.2 Geometric properties of the examined stiffened plate

The dimension parameters of the analysed stiffened plate are presented in Fig. 5.2,while their numerical values are listed in the following. The two stiffeners are dividingthe plate longitudinally to three equal subpanels. The height of the stiffeners are tentimes its thickness.

Plate thickness: tp = 12 mm

Distance between the stiff.: b1 = 600 mm

Length of the stiffened plate: a = 1800 mm

Hight of the stiff.: h = 100 mm

Thickness of the stiff.: tsf = 10 mm

46

5.3 Material model

X

Y

Z

a

b1

b1

b1

b

h

h

Figure 5.2: Dimensions of the stiffened plate

5.3 Material model

The material is elastoplastic with strain-hardening, as presented in Fig. 5.3. The ma-terial is linear elastic in the strain range of 0− fy/E. The yield plateau is modelled inthe strain range fy/E−0.02 during which a maximum stress of fy+5 MPa is reached.From ε = 0.02 until ε = 0.15 the hardening is taken into consideration. The ultimatestress is reached at the strain level of 0.15. In some cases the yield plateau steepnesswas not enough and the analysis terminated before the prescribed displacement wasreached, due to lack of convergence. In these occasions, to overcome this problem, thesteepness of the plateau was slightly raised.


Poisson’s Ratio: ν = 0.3

Yield strength: fy = 355 MPa

Ultimate strength: fu = 510 MPa

47


Figure 5.3: Material model used during the numerical experiments

5.4 Buckling modes of stiffened plate

A linear bifurcation analysis was performed to get the buckling shapes which will beused as initial imperfections in the nonlinear analysis. The material used in the linearbifurcation analysis is linear with the same elastic modulus as in case of the hardeningmodel. In Fig. 5.4 and in Fig. 5.5 the first six buckling modes are plotted. The 1st

buckling mode is a global one, while the rest of them are local buckling modes.In the remaining analysis, the 1st and the 2nd buckling modes were used as initialimperfections to assess the elastoplastic resistance at ULS.

48

5.4 Buckling modes of stiffened plate

X

Y

Z

X

Y

Z

X

Y

Z

Figure 5.4: 1st, 2nd, 3rd eigenmodes of stiffened plate

49


X

Y

Z

X

Y

Z

X

Y

Z

Figure 5.5: 4th, 5th, 6th eigenmodes of stiffened plate

50

5.5 Description of Imperfections


In the present section 12 different imperfections were compared for the example pre-sented in section 5.6. The majority of them were applied according to EN 1993−1−5,Annex C), (European Committee for Standardisation, 2006). In the description of theexperiment (Beg et al., 2010) the sign of the imperfection is unclear. In the presentwork, both positive and negative imperfections were considered on the model. Eu-rocode also proposes an imperfection, for local stiffeners or flanges, subjected to twist.This imperfection was not modeled for two reasons: (i) alone it would not contributeto the critical force according to (ECCS Technical Committee 8, 1986) and (ii) thetwist of the stiffeners are included in the magnification of the first eigenmode.

A short overview of the 12 imperfections is the following: IMP1 (+/−) is an im-perfection obtained from the amplification of the first eigenmode, IMP2 (+/−) is animperfection gained from the amplification of the seconde eigenmode, IMP3 (+/−) isalso a bow shaped imperfection alike IMP1. IMP4 (+/−) is also a local imperfectionsimilar to IMP2 eigenmode imperfection. IMP6 (+/−), numbered according to ECCS,Eurocode design manual (ECCS Technical Committee 8, 1986), is a combination ofglobal and local imperfections. Another type of imperfection was proposed by the su-pervisors and the author of the master thesis. This imperfection is referred as IMPpro.(+/−) or IMPown, briefly described further, in this section. The magnified imperfectshapes and the equations describing the surfaces are presented below.

• IMP 1+ is the 1st buckling mode with amplitude:w1 = min(a/400, b/400) = min(1800/400, 1800/400) = 4.5 mm

• IMP 1− is the 1st buckling mode with negative amplitude, w1, with imperfectionapplied opposite to the stiffeners side, in the negative y direction.

• IMP 2+ is the 2nd buckling mode with amplitude:w2 = min(a/200, b1/200) = min(1800/200, 600/200) = 3.0 mm

• IMP 2− is the 2nd buckling mode with negative amplitude, w2.

• IMP 3+ is a global panel imperfection, presented in Fig. 5.6 and w3 is describingthe surface.

XY

Z

Figure 5.6: IMP 3+, in a magnification of ×50

51


w3 = min(a/400, b/400) · sin

(π · xb

)· sin

(π · za

)

• IMP 3− is a global panel imperfection with negative amplitude, w3, shown inFig. 5.7.

XY

Z

Figure 5.7: IMP 3−, in a magnification of ×50

• IMP 4+ is local sub-panel imperfection, appears in Figure 5.8 and the surfaceevoked by w4.

XY

Z

Figure 5.8: IMP 4+, in a magnification of ×50

w4 = min(a/200, b1/200) · sin

(π · xb1

)· sin

(π · za

)

• IMP 4− is also a local sub-panel imperfection with negative amplitude, w4, indi-cated in Fig. 5.9.

52


XY

Z

Figure 5.9: IMP 4− in a magnification of ×50

• IMP 6, is a combined imperfection of global plate and local sub-panel imperfec-tion.

IMP6(+) = IMP3(+) + 0.7 IMP4(+)

IMP6(−) = IMP3(−) + 0.7 IMP4(+)

• IMP proposed - it is a combination of global and local imperfections and residualstresses. global-IMP = length/1000 local-IMP = b1/500

The residual stresses are in plane σzz stress due to the welding of the plate. Theirestimated values are fy tension stress and 0.25 fy compression stress. These stressesare equilibrated in a way that one element receives σzz = fy as a tension stress and fourother elements σzz = −0.25 fy as a compression stress. The resultant of these initialstresses in the cross-section is zero. The elements with residual stress are coloureddifferently on Fig. 5.10. Compression stress are represented with teal and red elementsin the plate and in the stiffeners, respectively. Tension stress are represented withpurple and light blue in the plate and in the stiffeners, respectively.

53


XY

Z

compression zone of stiffener (−0.25 fy)tension zone of stiffener (fy)

tension zone of plate (fy)

compression zone of plate (−0.25 fy)

Figure 5.10: IMP proposed in a magnification of ×20 and zoom in of the elements

5.6 The imperfections’ effect on ultimate force

For the geometric and material non-linear analysis two different methods were used toobtain the ultimate resistance force and the force-displacement curve. The two tech-niques to solve the resulting non-linear algebraic system of equations were (i) imposeddisplacement experiment and (ii) arc-length method.

The ultimate force related to the introduced imperfections are summarised in Tables 5.1and 5.2, while the force displacement curves were plotted in Figures 5.11 and 5.12. Inthe first row of the mentioned tables the resistance of the plate is calculated accordingto the procedure presented in section 4.1. The second row is a geometrically and phys-ically nonlinear analysis (without imperfection) and the remaining rows are GMNIA.At first the results obtained using imposed displacement are analysed, followed by theones of the arc-length method.

5.6.1 Results of displacement governed experiments

Table 5.1 shows that global imperfection (IMP 1− and IMP 3−) has the highest in-fluence on the capacity of the plate from the imperfections prescribed in Annex C5 ofEN 1993− 1− 5.Also the imperfections proposed by the supervisors and the author result in an even

54


Imperfection Ultimate Force [kN] Deviation[%]

Section 4.1 2525.461 0.00

No imperfection 6540.220 158.97IMP 1+ 5389.464 113.41IMP 1− 3650.063 44.53IMP 2+ 5844.864 131.44IMP 2− 6019.914 138.37IMP 3+ 5264.814 108.47IMP 3− 3517.932 39.30IMP 4+ 5793.596 129.41IMP 4− 6112.649 142.04IMP 6+ 5131.899 103.21IMP 6− 3658.181 44.85

IMP pro. + 3787.534 49.97IMP pro. − 3408.525 34.97

Table 5.1: Considered imperfections, corresponding ultimate forces and deviations in respectto EC3, (disp.-gov.)

lower ultimate force.In Fig. 5.11 the force-displacement curves obtained by displacement-governed experi-ments are presented. A displacement of 5 mm is applied in 100 steps on the top masternode. The resulting reaction force, Fz, is measured and plotted against the masternode’s displacement along z, denoted by δ.The force-displacement curves of IMP 1−, IMP 3− and IMP 6− are basically on top ofeach other, due to the fact that, they all have the same global imperfection, which is thenegative bow shape. The curves of IMP 1+, IMP 3+ and IMP 6+ are also close to eachother, although their peaks are a bit shifted compared to each other. They are based onthe positive bow shaped global imperfection. In case of IMP (4 + /−) and the perfectmodel, there is a “jump” in the force-displacement curve. When displacement-governedexperiment are used the displacement have to increase constantly, in these cases thedisplacements would probably decrease, but this behaviour cannot be captured by theexperiments.

5.6.2 Results obtained by arc-length method

In case of arc-length method the ultimate forces are summarised in Table 5.2. Similar tothe displacement-governed experiment, IMP 3− yields the lowest ultimate force amongEurocode imperfections. When the analysis is carried out by arc-length method theproposed imperfections seem to have less reserve over Eurocode (EC). The force dis-placement curves are plotted in Fig. 5.12. Similar to the Figure 5.11, the global imper-fection seem to have the largest influence on the value of ultimate force. In case of thesub-panel imperfections, such as IMP 2 and 4, the arc-length method sometimes failedto converge. One of the reasons could be that the steepness of the material model’s

55


Figure 5.11: Force displacement curves, (disp.-gov.)

hardening branch is not adequate. However, when the method properly converged, itproduced clear snap-back curves, which means that force and displacement are bothbeing reduced along the equilibrium path. As expected, displacement-governed exper-iments where unable to capture this behaviour as in their case displacements have toincrease throughout the whole experiment.

56


Imperfection Ultimate Force [kN] Deviation[%]

Section 4.1 2525.461 0.00

No imperfection 6549.603 159.34IMP 1+ 5389.446 113.40IMP 1− 3668.930 45.28IMP 2+ 6000.662 137.61IMP 2− 6020.821 138.40IMP 3+ 5275.613 108.90IMP 3− 3541.850 40,25IMP 4+ 5706.525 125.96IMP 4− 6113.595 142.08IMP 6+ 5089.517 101.53IMP 6− 4627.550 83.24

IMP pro. + 3001.763 18.86IMP pro. − 3073.146 21.69

Table 5.2: Considered imperfections, corresponding ultimate forces and deviations in respectto EC3, (arc-length)

Figure 5.12: Force displacement curves, (arc-length)

57

58

Chapter 6

Numerical Experiments

As it was mentioned in the introduction, 900 runs were completed with ANSYS me-chanical APDL. To be able to complete the sequence of multiple runs necessary for theparametric study, an ANSYS macro was written. The results of the force-displacementcurves were written to txt files. To organise and visualise these txt files a MATLABscript was developed, which opens the ANSYS macro, reads the txt files and storestheir values in arrays. MATLAB is used further on to sort, evaluate and plot the resultsof the completed runs. The flow chart presented in Fig. 4.4 was also implemented inthe same environment.

6.1 Parametric study

The five key variables included in the parametric study are listed in Table. 6.1. Inthe ANSYS macro other three arguments were specified, which were the following: (i)boundary conditions, whether to behave as a plate or as a column, (ii) sign of imper-fection, which was discussed in section 5.5 and (iii) the hight of the stiffener, which wastaken as (10 · tsf ). The name of the file reflects all these parameters. Every geomet-rical configuration was analysed as column and plate with both positive and negativeimperfection. The height of the stiffener is not an independent variable as it is linkedto its thickness.

The research matrix, or more precisely, the research matrices for the parametric

Designation Parameter Range of variables

Ns Number of stiffeners {2, 4, 6, 8}tp Plate thickness {12, 16, 20, 25, 40}tsf Thickness of the stiffener {8, 10, 12, 16, 20}b1 Distance between the stiffeners {400, 600, 800}α Ratio of stiffened plate {0.5, 2/3, 1.0, 2.0}

Table 6.1: Considered parameters in the parametric study

59

Chapter 6 Numerical Experiments

b1 Thickness of the plate(tp)[mm

]400 12 14 16 18 20 22 25 30 35 40

Thickn

essof

the

stiffen

er( t sf)

[ mm] 8 0.356 — 0.308 — — — — — — —

10 0.578 — 0.517 — 0.473 — 0.437 — — 0.42012 — — 0.690 — 0.650 — 0.609 — — —14 — — — — — — — — — —16 — — — — 0.845 — 0.822 — — 0.55018 — — — — — — — — — —20 — — — — — — — — — 0.822

Table 6.2: Analysed stiffened plates, b1 = 400mm

study, are presented in this section, see Tables 6.2 to 6.5. To understand these matriceswe have to look at the sorting principles. At each table the thickness of the plate ishorizontally, while the thickness of the stiffener is listed vertically. In their intersectionthe number indicates that the plate with that specific geometry was analysed. Thenumbers in intersections of the columns and rows are the column-like reduction factors.The distance between the stiffeners are marked at the upper left corner under “b1”. Thelength of the stiffened plate was kept constant regardless of the number of stiffeners.The length is determined in a way that a plate with four stiffeners would yield theaspect ratio, α = 1. In case of b1 is equal to 400 mm, 600 mm and 800 mm, the lengthof the plate is 1600 mm, 2400 mm and 3200 mm, respectively.As the distance between the stiffeners and the length of the plate were increased, thecolumn-like reduction factor decreases. In some cases, especially with b1 = 800 mm,this factor falls below the threshold value of χc < 0.2. However, these experimentswere still carried out and, during the evaluation, special care was taken.In Table 6.5 the 144 geometries analysed are summarised. In case of each analysedgeometries 6 runs were completed. Two of the runs are to gain the critical bucklingforce of the column and the critical buckling force of the plate-like behaviour. From thiscritical buckling forces (Fcr), stresses were calculated dividing the forces by the effectivecross sectional area of the plate. Special care was taken to class 4 cross sectional areasas in these cases effective cross sectional area is different from the cross section.

σFEMcr,c =Fcr,cAc,eff

; σFEMcr,p =Fcr,pAc,eff

(6.1)

The analysed stiffened plates are include global and local imperfections. Accordingto the global imperfections the structures are classified as imp− and imp+. In the caseof column and plate behaviour, the models were run with both positive and negativeimperfections. With these four geometrical and material nonlinear analysis, the previ-ously mentioned two buckling analysis adds up to six runs in case of each geometry.The data gained from the closely 900 runs was sorted, stored and analysed by MAT-LAB.

60

6.1 Parametric study


]600 12 14 16 18 20 22 25 30 35 40

Thickn

essof

the

stiffen

er( t sf)

[ mm] 8 — — — — — — — — — —

10 0.335 — 0.249 — 0.205 — 0.188 — — —12 — — 0.404 — 0.337 — 0.302 — — —14 — — — — — — — — — —16 — — — — 0.619 — 0.574 — — 0.48818 — — — — — — — — — —20 — — — — — — — — — 0.691



]800 12 14 16 18 20 22 25 30 35 40

Thickn

essof

the

stiffen

er( t sf)

[ mm] 8 — — — — — — — — — —

10 0.206 — 0.146 — 0.115 — 0.194 — — —12 — — 0.251 — 0.197 — 0.156 — — —14 — — — — — — — — — —16 — — — — 0.427 — 0.349 — — 0.28218 — — — — — — — — — —20 — — — — — — — — — 0.473


Distance between stiff.(b1)[mm

]No. stiff. 400 600 800

2 14 11 114 14 11 116 14 11 118 14 11 11

Geom. 144

No. of runs 864

Table 6.5: Total number of geometries analysed and number of runs

61


6.2 Layout of experiments

In this section the set up of the experiments are described and boundary conditionsin case of plate-like and column-like behaviour are compared. Installation of the loadand the measuring of displacements and forces are also described. Last but not least,a brief overview of the used imperfections are provided in addition to section 5.5.

6.2.1 Boundary conditions

Boundary conditions of the plate-like and column-like experiments are provided inFig. 6.1 and in Fig. 6.2, where the elemental mesh, the boundary conditions and theapplied loads are shown. The teal pyramids are the restrictions against movement.The orange double pyramids are rotational restrictions. The magenta pyramids andcords are related to the CERIG command connecting the master and the slave nodes.In both plate-like and column-like experiments, the top and bottom sides are supportedagainst side way (ux = 0), out of plain (uy = 0) movements and rotation around the Zaxis. These support are applied at the master nodes, which are in the center of gravity.In both cases the load is applied at the top surface’s master node, marked by a redarrow. Due to the CERIG command the load is distributed equally to all of the nodes.On the bottom surface the structures are supported against longitudinal movements(uz = 0).In case of a plate like behaviour the longitudinal side of the plates are supportedagainst side way (ux = 0) and out of plain (uy = 0) movements. These side supportsare missing in case of columns. The plate like supports, in general provide a higherresistance compared to column like supports.

X

Y

ZUROTFCE

Figure 6.1: Boundary conditions in case of plate-like behaviour

62

6.2 Layout of experiments

X

Y

ZUROTFCE

Figure 6.2: Boundary conditions in case of column-like behaviour

XY

Z

Figure 6.3: IMP proposed in a magnification of×50

6.2.2 Imperfections used during the experiments

The imperfection used during the experiments is the same as the one described insection 5.5 under IMP proposed. The only modification to that imperfection is that onthe side panels, as their breadth is only b1/2, the amplitude of the local-imperfectionis also halved. In Figure 6.3 a magnified version of IMP proposed + is visible. Thevon Mises stresses of a plate at 0.01 pseudo time is presented in Fig. 6.4. Stresses closeto the yield stress are still visible at places, where the welding would have occurredduring the fabrication of the orthotropic plate.

63


XY

Z

45.836780.0492

114.262148.474

182.686216.899

251.111285.324

319.536353.749

Figure 6.4: Von Mises stress at 0.01 time step

6.3 Evaluation of reduction factors

First, the results of the numerical experiments, were treated separately. Column be-haviour and plate behaviour reduction factors were compared to the reduction factorsand curves provided in the Eurocode. After that, fractions of the difference of plate-likereduction factors and column-like reduction factors were created and compared to theinteraction curve of Eurocode. Three different type of fractions were examined, mainlyEurocode based, Eurocode and FEM based and FEM based. These cases are presentedin the following three subsections.

Moreover, in this section all those experiments, which force-displacement curve didnot reach the desired displacement of 2.5 mm were not taken into account. If theexperiment reached a displacement between 2.5 mm and 5.0 mm it can still mean thatthe arc-length method could not find convergence and terminated before the desireddisplacement was attained or the capacity of the stiffened plate is reached before theend of the experiment.

6.3.1 Column-like reduction factors

For the analysis of column behaviour according to Eurocode, α parameter of bucklingcurve b is used, see European Committee for Standardisation (2006, page 20). The αconstant is modified, according to Eq. (4.17), to determine the increased values of theimperfection parameters. For each geometry αe is calculated. To plot the bucklingcurve, the average value of the imperfection parameters was used, which in case ofthese experiments is 0.60377. The column behaviour reduction factors in case of bothpositive and negative imperfections, the reduction factors according to EC and theestimated buckling curve were plotted in Fig. 6.5. The relative column slenderness,λc, was calculated according to Eurocode. When the column-like reduction factor,

64

6.3 Evaluation of reduction factors

Figure 6.5: Column reduction factor, χc

χc, falls below 0.2 the results of the geometrically and materially nonlinear analysiswith imperfections start to scatter. The reason for this phenomena is the fact that inthese cases only a small fraction, 0.1− 0.2, of the total load was enough for the failureoccurs. While the specimens fail in the lower range, the error of the numerical analysiscompared to the ultimate resistance is relatively higher than in the upper range.

6.3.2 Plate-like reduction factors

Similarly to the column-like reduction factors, in case of plate-like reduction factors,they can be obtained by running the model with both positive and negative imperfec-tions and calculating the factor according to the EC. The factors obtained by GMNIAand by the procedure according to the EC were then compared and presented in Fig. 6.6.Most of the reduction factors obtained by the numerical simulation are smaller thenthe ones calculated by Eurocode (European Committee for Standardisation, 2006).The plate buckling curve, the so called Winter-curve and the buckling-curve acts as anenvelope curve for the numerical results.The relative plate slenderness can be calculated from FEM results by dividing the crit-ical buckling force by the effective area. Hence, the critical plate stress is found, aspresented in Eq. (6.1). This critical plate stress is not the same as the one calculatedaccording to Eurocode Eq. (4.9). From the FEM based critical plate stress, the relativeplate slenderness is calculated according to Eq. (4.3). In Figure 6.7 the FEM basedrelative slenderness and the reduction factors obtained from the the FEM analysis arepresented. These results overtop the Winter-curve in many cases. In both cases thecolumn buckling curve is also presented.

65


Figure 6.6: Reduction factors, relative slenderness calculated from EC results

Figure 6.7: Reduction factors, relative slenderness calculated from FEM results

66

6.4 Examination of a single geometry


To inspect the post-critical behaviour of the plates, the arc-length method was used tosolve the resulting nonlinear system of equations. This method can trace the equilib-rium path and provides proper treatment of the limit and bifurcation points. Ordinarysolution techniques, such as Newton–Raphson and displacement governed experimentslead to instability near the limit points and also are unable to follow paths including asnap-back.

6.4.1 Force-displacement curves

The force-displacement curve was obtained for all of the analysed geometries. Approx-imately, 600 curves were traced out. Some of these were selected and are presented inAppendix B.

In this section the load-displacement response is described only for one specificgeometry, which was ‘20− 4− 600− 12’ (plate thickness of 20 mm, with four stiffeners,which were 600 mm apart and had a thickness of 12 mm). For both column- andplate-like behaviour the response curves were plotted in Fig. 6.8. This geometry has aresponse which well describes the responses of the numerical models in general. Thedescending phase of the load-displacement response is properly traced due to the use ofarc-length method. The column behaviour with negative imperfection has a snap-backresponse as both the force and the displacement of the loaded node dropped duringthe numerical experiment. The response is detailed in Fig. B.1. The column behaviourwith a positive imperfection has no snap-back. In case of the plate behaviour withnegative imperfection, hardening is visible in the analysed displacement. The platewith positive imperfection has an even higher residual strength than the plate withnegative imperfection.

In case of Eurocode-FEM based evaluation, in this particular example, the differ-ence between the positive imperfection’s reduction factors has been used, because thesmallest ultimate force belongs to the column behaviour with positive imperfection.

The force-displacement curve of the column with negative imperfection is presented inFigure B.1. The displacement geometry is plotted at three discrete points.The force-displacement curve increases linearly until a peak. During this period thephenomena of local buckling also occurs, besides the global one. However, during thedescending phase the buckled geometry only shows global buckling.

The other case which was analysed in detail is the plate-like support with appliednegative imperfection. This case is presented in Fig. B.2. As in case of Fig. B.1, theforce-displacement curve can be divided into two parts: (i) a linear relation in the force-displacement curve until a peak and (ii) a descending path, which now happens to hashardening. Until the peak local and global buckling modes appear in combination and,after that, global buckling is the leading failure mode. To support that assumption,the displaced geometries are presented under the line. The von Mises stresses of theplate are also visualised at the two examined points. At the peak of the curve, the twomiddle stiffeners and the corners of the plate are yielding. In the hardening phase a

67


Figure 6.8: FEM based evaluation, maximums of (imp− /imp+)

typical yield pattern is visible. The described figures are presented in the Appendix B,due to the limitations on the extent of the main body.

6.4.2 Evaluation of reduction factors for a single geometry

In case of a single geometry, which was ‘20−4−600−12’ (plate thickness of 20 mm, withfour stiffeners, which were 600 mm apart and had a thickness of 12 mm), the column-like reduction factors (χc and χFEMc ) and the plate like reduction factors (ρp and ρFEM)are plotted, to present how the interaction formula works.In Figure 6.9 the plate (λp) and column (λc) relative slenderness was calculated ac-cording to the EC standard. It is visible what is the distance between the reductionfactors assumed by the standard and what is, in reality, the one obtained from thefinite element analysis.The formula (4.20) interpolates between the two behaviours. The result of the inter-polation, ρc, is also plotted. Asterisks mark the results calculated by EC, while circlesmark the ones of finite element analysis.

68


FEM based resultsEC based results

Figure 6.9: Reduction factors, relative slenderness calculated from EC results

In case of Figure 6.10, the relative slenderness for the plate reduction factor iscalculated from critical buckling stress.

69


EC based resultsFEM based results

Figure 6.10: Reduction factors, relative slenderness calculated from FEM results

6.5 Eurocode based evaluation

The evaluation method is based on a ratio (6.2), where the numerator contains the dif-ference of global reduction factor obtained from the finite element analysis (ρFEM) andcolumn-like reduction factor according to Eurocode (χECc ). The denominator containsthe difference of Eurocode base plate- and column-like reduction factor. Hence, theevaluation method is

fEC1 =ρFEM − χECcρECp − χECc

(6.2)

The numerator and the denominator of the fraction is described in Fig. 6.11. Thedenominator of the fraction is the maximal possible “distance” between the column-and plate-like reduction factor according to the Eurocode standard. The numeratorrepresents the current state of interpolation. It is assumed that the value of the columnreduction factor in case of EC and FEM calculation is similar or equal.In case of Eurocode based evaluation method, the results are first presented in

Fig. 6.12. The fraction of fEC1 was plotted against ξ, which is the fraction of thecritical stresses according to Eq. (4.21).It is visible that some of the experiment results fall under the interaction curve pro-posed by Eurocode. The reason for that could lay in the way that the critical stiffenedplate stress is calculated. The buckling coefficient was calculated according to equa-tion (4.10), which can be applied in case of “three equally spaced longitudinal stiffeners”.The criteria was not satisfied in case of one-quarter of the experiments, when only twoequally spaced longitudinal stiffeners were applied to stiffen the plate.

70


1 σECcr,p/σECcr,c

ρECp

χECc ' χFEMc

2

ρFEM − χECcρECp − χECc

Figure 6.11: Interpolation in case of EC based evaluation

Figure 6.12: EC based evaluation, all results

The same graph is presented in Fig. 6.13, but in those cases, when the stiffened platehad only two stiffeners, the asterisks are coloured in light blue. It is clear that mostof the cases have fallen under the interaction line. This is due to the restriction in theuse of kσ,p’s formula (4.10).The same fEC1 fraction is plotted against the ratio of σENcr.p/σENcr.c. The Ns = 2 cases aredistinguished in Fig. 6.14 with light blue circles. In the final figure of this section, 6.15,experiments with only two stiffeners were not presented as their σcr.p cannot be esti-mated according to the formula (4.10). From that figure it is clear that only two, outof 111, specimens have fallen below the interaction line (as the number of experimentswere reduced by 37, which is the number of examined stiffened plates with only two

71


Figure 6.13: EC based evaluation, no. of stiffener (Ns = 2)

stiffeners), so it is less than 2.0% of the cases, which satisfies our expectations.

72


Figure 6.14: EC based evaluation, number of stiffeners (Ns = 2)

Figure 6.15: EC based evaluation, without inappropriate estimations

73


6.6 Eurocode-FEM based evaluation

In the Eurocode-FEM based evaluation method, the numerator is the difference be-tween the global reduction factor (ρFEM) and the column-like reduction factor (χFEMc ),while the denominator is the same as in case of Eurocode based evaluation. In this case,the stiffened plates with negative and positive imperfections are evaluated separately.Hence, in the Eurocode-FEM based evaluation method:

fEC/FEM =ρFEM − χFEMc

ρECp − χECc(6.3)

The parts of the fraction are revealed in Fig. 6.16. The evaluation method is basedon the assumption the that the column-like reduction factor approximated by theEurocode is similar or equal to the factor obtained by the numerical experiments.The column-like reduction factor in the nominator changed from the one calculatedaccording to the EC to one obtained by the numerical results.The results presented in Figure 6.15 are already filtered according to the mentionedcriteria of stiffeners (Ns ≤ 3) are not plotted. The fill of the circles represents theway of the imperfection (imp− /imp+). The perimeter of the circle indicates weatherthe difference between the plate- and column-like behaviour is the maximum or theminimum from the two imperfections applied.Figure 6.17 is further filtered. The pair of the results with the lowest column bucklingreduction factor and maximum difference between the plate- and column-like reductionfactors was selected and plotted in Fig. 6.18. The inside of the circle indicates, weatherthat difference is coming from the negative or positive imperfection. With only one ofthe imperfections plotted in case of each geometries it is more clear that only a few ofthe analysed cross sections have fallen under the interaction curve.Along the same governing idea Fig. 6.19 is also filtered to obtain Fig. 6.20. Out of 111plotted experiments only 5 have fallen under the interaction line, which is more than inthe previous evaluation case, but still less than 5%, so the 5% quartile of the Eurocodeis still satisfied.

1 σECcr,p/σECcr,c

ρECp

χECc ' χFEMc

2

ρFEM − χFEMcρECp − χECc

Figure 6.16: Interpolation in case of EC-FEM based evaluation

74

6.6 Eurocode-FEM based evaluation

Figure 6.17: EC-FEM based evaluation, (imp− /imp+)

Figure 6.18: EC-FEM based evaluation, maximums of (imp− /imp+)

75


Figure 6.19: EC-FEM based evaluation, (imp− /imp+)

Figure 6.20: EC-FEM based evaluation, maximums of (imp− /imp+)

76

6.7 FEM based evaluation

6.7 FEM based evaluation

In case of FEM based evaluation all the parts of the function, fFEM have been derivedfrom the finite element simulations. The parts in the numerator ρFEM and χFEMc are theplate- and column-like reduction factors introduced earlier. The ρFEMp was calculatedfrom the critical stress σFEMcr,p linked to the plate like buckling with Eq. (6.1). Fromthat value, the relative plate slenderness was calculated based on Eq. (4.7). Finally,the plate-like reduction factor ρFEMp was obtained according to Eq. (4.1), i.e.,

fFEM =ρFEM − χFEMc

ρFEMp − χFEMc

(6.4)

In case of the purely FEM based evaluation method no stress was calculated accordingto Eurocode European Committee for Standardisation (2006), so the formula (4.10)of the buckling coefficient was not used. It means that all of the geometries can beevaluated regardless the number of stiffeners they have. Similar to the previous twosections, the meaning of the fraction is briefly explained in Fig. 6.21.The results were plotted in Fig. 6.22. As in the previous evaluation method, thesame geometries with different imperfections were distinguished. The interaction curveproposed by Eurocode European Committee for Standardisation (2005) cannot be usedin this case of evaluation. The supervisors and the author of the master thesis proposesa linear line from (1, 0) to (3, 1) as a line to help interpolate between the plate- andthe column-like behaviour.

1 σFEMcr,p /σFEMcr,c

ρFEMp

χFEMc

2

ρFEM − χFEMcρFEMp − χFEMc

Figure 6.21: Interpolation in case of FEM based evaluation

77


Figure 6.22: FEM based evaluation, (imp− /imp+)

Figure 6.23: FEM based evaluation, maximums of (imp− /imp+)

78

Chapter 7

Conclusions and furher studies

7.1 Conclusions

• Stiffened plates load-displacement response includes snap-back behaviour. Newton–Raphson and imposed displacement techniques fail to provide the whole force-displacement path, as presented in Figures 5.11 and 5.12. During the test pro-gram arc-length method had to be used to trace out the whole load-displacementresponse.

• The imperfection proposed by the supervisors and the author provides a lowerresistance for stiffened plates than imperfections advised by Eurocode EuropeanCommittee for Standardisation (2006). According to the example in section 5.5,the imperfection proposed, most probably, always stays on the safe side.

• During the test program plates had different number of stiffeners, Ns = 2; 4; 6; 8.According to equation (4.10), the critical stress of plates can be calculated if theyhave at least three stiffeners. As this method was used for the first two evaluationmethodologies, those experiments with only two stiffeners were excluded from theassessment.

• The Eurocode based evaluation method was presented in section 6.5. In this casethe global resistance factor, ρFEM , was used. The evaluation method agrees withthe current Eurocode interpolation function (4.20). Only a few results remainlower than the interpolation line.

• The Eurocode-FEM based evaluation method was discussed in section 6.6. In thismethod both the global resistance factor, ρFEM , and the column reduction factor,χFEMc , had been taken over from the numerical experiments. Among the modelswith positive and negative imperfection, the most probabilistic failure mode isthe one with the lowest column-like reduction factor. In most cases this is themodel with the positive imperfection.

• The FEM based evaluation method was described in section 6.7. The interpola-tion formula provided by Eurocode is not sufficient in this case. The supervisorsand the author proposed a new linear interpolation line between the column- and

79

Chapter 7 Conclusions and furher studies

plate-like behaviours for such an evaluation case.

7.2 Further studies

To carry out the analyses and evaluation of the numerical experiments the author hadlimited time. This limitation was the submission data of the master thesis. To de-scribe and show the results the author had a limitation in size of the master thesis.The master thesis was written with these boundaries in mind. The further analysis ofthe results and data obtained by the geometrically and materially nonlinear analysis isinevitable. The runs created around 500 GB of data. With the use of ANSYS macroand the MATLAB script numerous other configuration and stiffener geometry can beanalysed.

• The parametric study should be extended to different plate ratios and to a highernumber of stiffeners. Stiffeners with considerable torsional rigidity should be alsoanalysed.

• Different stiffener geometries should be also considered, like �, � and trapezoidstiffeners. To analyse stiffened plates with different stiffeners is necessary forunderstanding the behaviour of stiffened plates.

The work flow for the mass analysis of stiffened plate is ready. Analysing and evalu-ating further geometries would be less time consuming than it was to set up the worksequence.

80

Bibliography

Ádány, S., E. Dulácska, L. Dunai, S. Fernezelyi and L. Horváth (2007). AcélszerkezetekTervezése az Eurocode alapján. Buisness Media Magyarország Kft., second edition.(in hungarian).

ANSYS, Inc. (2013a). ANSYS Mechanical APDL Basic Analysis Guide. ANSYS, Inc.

ANSYS, Inc. (2013b). ANSYS Mechanical APDL Modeling and Meshing Guide. AN-SYS, Inc.

Bathe, K.-J. and E. N. Dvorkin (1986). A formulation of general shell elements –the use of mixed interpolation of tensorial components. International Journal forNumerical Methods in Engineering, 22 (3), 697–722.

Beg, D., U. Kuhlmann, L. Davaine and B. Braun (2010). Design of Plated Structures.Ernst & Sohn, first edition.

CTICM (2018). Ebplate software: version 2.01.

ECCS Technical Committee 8 (1986). Structural Stability, Behaviour and Design ofsteel Plated Structures, volume 44. Applied Statics and Steel Structures, ETH Zurich.

European Committee for Standardisation (2005). Eurocode 3: Design of steel structures- Part 1-1: General rules and rules for buildings. third edition.

European Committee for Standardisation (2006). Eurocode3: Design of steel structures,Part 1-5: Plated structural elements.

Galambos, T. V. and A. E. Surovel (1998). Structural stability of steel: Concepts andapplications for structural engineers. John Wiley & Sons.

Hughes, T. J. R. (2000). The Finite Element Method: Linear Static and DynamicFinite Element Analysis. Dover Publications, Inc, Mineola, New York, USA.

Johansson, B., R. Maquoi, G. Sedlacek, C. Müller and D. Beg (2007). Commentraryand Worked Examples to EN 1993-1-5 ”Plated Structrual Elements”. JRC Scientificand Technical Reports, first edition.

Johansson, B. and M. Veljkovic (2001). Steel Plated Structures, Progress in StructuralEngineering and Materials, volume 31. John Wiley & Sons, Inc.

Klöppel, K. and KH. Möller (1968). Beulwerte augestefter Rechteckplatten (Band II).Ernst & Sohn Verlag, Berlin.

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Klöppel, K. and J. Scheer (1960). Beulwerte ausgesteifter Rechteckplatten (Band I).Ernst & Sohn Verlag, Berlin.

Kollár, D. (2016). Numerical simulation of a welding process, determination of residualstresses. MAGÉSZ–Journal of the Hungarian Steel Structure Association, 13 (3), 86–99.

Kollár, D., B. Kövesdi and J. Néző (2017). Numerical simulation of welding process− application in buckling analysis. Periodica Polytechnica Civil Engineering, 61,98–109.

Masubuchi, K. (1980). Analysis of Welded Structures. Pergamon Press.

MATLAB (2017). MATLAB, The Language of Technical Computing. The MathWorksInc. Version R2017b.

Mikami, I. and K. Niwa (1996). Ultimate compressive strength of orthogonally stiffenedsteel plates. Journal of Structural Engineering, (122), 674–682.

Néző, J. (2011). Virtual Fabrication of Full Size Welded Steel Plate Girder Specimens.Ph.D. thesis, Heriot−Watt University School of Engineerig and Physical Sciences.

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Stiffeners (2018). Stiffeners — Steelconstruction, the free encyclopedia for uk steelconstruction information. Online, accessed 21-may-2018.

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82

Appendix A

Plate buckling results

A convergence analysis was carried out, during which the first seven buckling modes ofa simply supported plate were estimated, by modeling the plate in ANSYS mechanicalAPDL.The Tables A.1 and A.2 are summarising the convergence analysis, in case of elementswith four nodes (SHELL181) and in case of elements with eight nodes (SHELL281),respectively.The buckling modes are visualised and collected in Figures A.1 and in A.2.

83

Appendix A Plate buckling results

mesh modes

ratio size 1/h 1st 2nd 3rd 4th 5th 6th 7th

[mm] [1/mm]

2:1 3000 0.00033 280.668 — — — — — —4:2 1500 0.00067 145.131 278.403 302.067 — — — —8:4 750 0.00133 94.9341 171.894 154.744 191.377 362.504 583.802 667.43716:8 375 0.00267 86.804 103.312 137.011 142.739 203.488 290.769 379.77432:16 187.5 0.00533 84.953 100.036 133.074 134.351 183.189 246.249 324.44564:32 93.75 0.01067 84.500 99.249 132.117 132.416 178.748 237.138 307.414128:64 46.875 0.02133 84.388 99.0553 131.879 131.942 177.673 234.964 303.422

analytical solution 84.356 99.001 131.806 131.806 177.358 234.321 302.238

Table A.1: Convergence analysis with SHELL181

mesh modes

ratio size 1/h 1st 2nd 3rd 4th 5th 6th 7th

[mm] [1/mm]

2:1 3000 0.00033 — — — — — — —4:2 1500 0.00067 192.097 248.456 356.745 — — — —8:4 750 0.00133 94.387 133.491 135.287 221.289 332.857 422.091 462.53716:8 375 0.00267 84.393 99.145 131.820 132.249 178.500 236.910 307.56332:16 187.5 0.00533 84.351 98.992 131.792 131.801 177.341 234.316 302.27964:32 93.75 0.01067 84.350 98.9906 131.785 131.801 177.319 234.252 302.123128:641 46.875 0.02133 89.387 108.995 131.793 159.244 207.154 268.714 341.777

analytical solution 84.356 99.001 131.806 131.806 177.358 234.321 302.238

Table A.2: Convergence analysis with SHELL281

1These line contains the numerical error pointed out in section 3.2.3.

84


X

Y

Z

X

Y

Z

XZ

X

Y

Z

Figure A.1: 1st, 2nd, 3rd, 4th buckling modes of a simply supported plate

85


1

X

Y

Z

1

1

X

Y

Z

1

X

Y

Z

Figure A.2: 5th, 6th, 7th buckling modes of a simply supported plate

86

Appendix B

Force-displacement curves of platedstructures

Without striving for completeness, the initial intention of this appendix was to presentone example of each plate thickness and stiffener distance. For sake of conciseness, thisaim is not possible, and only a small set of the obtained force-displacement curves ofplated structures is presented.

The two force-displacement curves (column20 − 4 − 600 − 12imp− and plate20 −4 − 600 − 12imp−), described in subsection 6.4.1, are presented in the following. Incase of Fig. B.1, the displaced geometries are presented at three points of the force-displacement curve. In case of Fig. B.2, the displaced geometries and, under them,the von Mises stresses of the stiffened plate are presented at two points of the force-displacement curve.

In the remaining of the present appendix the force-displacement curves of the fol-lowing geometries are also depicted:

• column/plate 12−Ns − 600− 10






Ns, after the value of plate thickness, indicates that the force-displacement curves forgeometries with the different number of stiffeners (Ns = 2; 4; 6; 8) were plotted.

87

Appendix B Force-displacement curves of plated structures

0.464889 .9297781.39467 1.859562.32444 2.789333.25422 3.719114.184

X YZ

MX

MN

01.78996 3.579915.36987 7.159828.94978 10.739712.5297 14.319616.1096

YX

Z

MX

MN

05.54626 11.092516.6388 22.18527.7313 33.277538.8238 44.37

49.9163

XY

Z

MX

MN

Figure

B.1:

Forcedisplacem

entcurve,w

ithdisplaced

geometries

88


XY Z

XY Z

XY Z

XY Z

X

Y

Z

7.53

655 48

.225

588.9

145 12

9.60

3170.

292 21

0.98

1251.

6729

2.35

9333.

048 37

3.73

7

X

Y

Z

1.94

965 47

.715

93.4

804 13

9.24

6185.

011 23

0.77

7276.

542 32

2.30

7368.

073 41

3.83

8

FigureB.2:Fo

rcedisplacementcu

rve,

withdisplacedgeom

etries

andvo

nMises

stresses

89


Figure B.3: Columns, tp = 12mm, b1 = 600mm, tsf = 10mm

Figure B.4: Plates, tp = 12mm, b1 = 600mm, tsf = 10mm

90


Figure B.5: Columns tp = 16mm, b1 = 600mm, tsf = 10mm

Figure B.6: Plates tp = 16mm, b1 = 600mm, tsf = 10mm

91




92




93




94




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Documents

Interactionanalysisofcolumn-andplate-like ......Contents 4.1.4 Interpolationbetweenplateandcolumnbehaviour. . . . . . . . 39 4.2 Eurocode’sapproachbasedonreducedstressmethod.