structural capacity of the hull girder · departmen tofna v al arc hitecture and o shore...

Lars Peter NielsenOctober 1998

Structural Capacityof the Hull Girder

Department of

Naval Architecture

And Offshore Engineering

Technical University of Denmark

Structural Capacity of the Hull Girder

Lars Peter NielsenM.Sc. in Engineering

DEPARTMENT OF NAVAL ARCHITECTURE AND OFFSHORE ENGINEERING

TECHNICAL UNIVERSITY OF DENMARK � LYNGBYOCTOBER 1998

Department of Naval Architecture and O�shore EngineeringTechnical University of Denmark

Building 101E, DK-2800 Lyngby, DenmarkPhone +45 4525 1360, Telefax +45 4588 4325

email [email protected], Internet http://www.ish.dtu.dk

Published in Denmark byDepartment of Naval Architecture and O�shore Engineering


c L. P. Nielsen 1998

All right reserved. Copying of the full extent of this publication(including this notice) in any form and by any means, is herebygrated. However, no single part of this publication may be repro-duced, stored in a retrieval system, or transmitted, in any form orby any means, electronic, mechanical, photocopying, recording, orotherwise, without the prior permission of the Department of NavalArchitecture and O�shore Engineering.

Publication Reference Data

Nielsen, L. P.

Structural Capacity of the Hull Girder.Ph.D. Thesis | 211 pages.Department of Naval Architecture and O�shore Engineering,Technical University of Denmark, October, 1998.ISBN 87-89502-03-5Keywords: Ultimate Capacity, Strength, Combined Loading,

Damaged Condition, Beam-Column.

Typeset in Times Roman using LATEX and printed by LTT Tryk, Lyngby, Denmark.The cover illustration shows a cross section in a typical very large crude carrier (VLCC)double hull structure | c Hugo Heinicke 1998.

Preface

This thesis is submitted as partial ful�llment of the requirements for the Danish Ph.D. degree.The research documented has been performed at the Department of Naval Architecture andO�shore Engineering (ISH), Technical University of Denmark (DTU), and at the Departmentof Naval Architecture and O�shore Engineering (NAOE), University of California at Berkeley(UCB). The work was initiated on October 1st 1994 and concluded on June 30th 1998.

The supervisory committee counts Professor Preben Terndrup Pedersen, Associate ProfessorJ�rgen Juncher Jensen, and Associate Professor Peter Friis Hansen { all faculty membersat ISH. Supervision was further received from Professor Alaa Mansour, faculty member ofNAOE1 during my stay at UCB.

Financial support has been received from:

� The Danish Technical Research Council (STVF),

� The Department of Naval Architecture and O�shore EngineeringTechnical University of Denmark,

� Civilingeni�r Kristian Rasmussen og hustru Gunild Katrine Rasmussens Fond,

� Knud H�jgaards Fond, and

� The Danish Society for Naval Architecture and Marine Engineering

and is greatly acknowledged.

Further, a special thanks to all of my colleagues at ISH, friends and family, and especiallyto my supervisors for invaluable help and support.

Lars Peter Nielsen

October, 1998

1At present Professor Alaa Mansour is a faculty member of the Mechanical Engineering (ME) departmentat University of California at Berkeley (UCB)

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ii Preface

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Executive Summary

This thesis addresses the structural performance of the hull structure of a vessel in theextreme response state of a combined loading consisting of moment, shear, torque, and hy-drostatic pressure. The research is part of a larger study onMarine Structural Design aimingat the establishment of a rational, computer-based analysis system for marine structures.

The motivation for conducting this research task is the importance of the ultimate and post-ultimate strength in relation to the reliability of intact as well as damaged vessels. Moreover,in an emergency situation, knowledge of the structural capacity of the vessel in whateverdamaged or intact condition it may be in, is crucial to allow for a rational decision to bemade regarding possible salvage, or at worst disembarkation of the crew, in the interest ofsafety for human life, environmental protection, and capital investment.

The common denominator for both these spheres of interest, i.e. reliability and emergencyresponse, is the absolute requirement of rapid calculations. In the case of reliability assess-ment, simulations requiring vast numbers of ultimate capacity evaluations are frequently thetool employed to establish the reliability index of the vessel. Thus, a speedy procedure forstructural analysis is a prerequisite for fast reliability assessment. Concerning emergencyresponse, then huge investments and human safety are at stake. Time becomes an obstacle,and an ill informed decision may prove to have dire consequences. Quick formation of arational basis for decision making thus becomes of paramount importance.

To facilitate these requirements, the overall objective of the present research therefore be-comes to build a rapid computer based analysis tool for calculation of the ultimate andpost-ultimate capacity of the hull girder in the general state of combined loading consistingof moment, shear, torque, and hydrostatic pressure. Then, to verify the accuracy of theprocedure against available experimental results and by comparison with other theoreticalresults. To meet this objective essentially requires three sub-tasks to be completed. Hence,the scope of the work becomes

1. A study of literature for the available methods usable to evaluate the ultimate andpost-ultimate capacity. This study will reveal three possible candidates: The �niteelement method (FEM), the idealized structural unit method (ISUM), and the beam-column method. To select among these three methods, available comparative studiesare reviewed. Based on the �ndings in those, the beam-column method is chosen forfurther use in the present research project.

iii

iv Executive Summary

2. Development of a scheme for the evaluation of the ultimate and post-ultimate ca-pacity of the hull girder, utilizing the beam-column approach. Assuming that theload-displacement response of any beam-column is known beforehand, further prereq-uisites and assumptions must be identi�ed and decided on. Based on this, a schemesuitable for implementation in a computer code must be established.

3. Establishment of a framework designed to evaluate the load-displacement response of abeam-column. Key e�ects must be identi�ed and if possible, included in the procedure.

Through this work, a procedure relying on the beam-column approach and the asymmetricalforced curvature principle has been established. The procedure is capable of accounting forthe following key aspects:

� Overall System Modeling:

{ General combined loading consisting of moment, shear, and torque.

{ Asymmetrical cross section description.

{ Direct calculated instantaneous beam-column load-displacement response.

� Beam-Column Modeling:

{ Di�erent yield stress for the plating and the sti�ener.

{ Beam-column speci�c modulus of elasticity and shear modulus.

{ Initial de ection of the beam-column.

{ Pressure loading on the beam-column.

{ Beam-columns without a sti�ener, i.e. plating alone.

The following e�ects are however ignored in the present formulation:

� Load Modeling:

{ The warping part of the torsional moment.

{ Interaction between the bending moment and the shear forces.

{ Redistribution of shear stresses after ultimate capacity.

� Beam-Column Modeling:

{ Necking of the beam-column.

{ Tripping of the sti�ener.

{ Residual welding stresses.

{ Initial de ection of the plating.

Executive Summary v

The present procedure has then successfully been applied to two test-cases. The �rst ofthese was one of the so called Nishihara box girders investigated by Nishihara [31] bothby experiment and an analytic approach. The second test case was a double hull tankerstructure analyzed by Melchior Hansen [17] for combined vertical shear force and bendingloading. From the �ndings in these two test-cases, it has been concluded that the presentprocedure is indeed capable of accurately predicting the ultimate and post-ultimate strengthof the intact hull girder.

A new test-case has then been selected for a more detailed investigation. The vessel chosenfor this part of the work is a design study of an ultra large crude carrier (ULCC). Thus,the vessel has newer actually been build. However, it has been scantled in accordance withthe classi�cation rules for steel ships issued by Det Norske Veritas [7]. The purpose of theinvestigations performed on this ultra large crude carrier has been twofold: The �rst hasbeen to illustrate the present procedures ability to handle asymmetrical bending of an intactvessel and hydrostatic pressure loading. The second has been to demonstrate the presentprocedures applicability to the analysis of a vessel in a damaged condition. Towards thisend, the following four scenarios have been analyzed:

1. Intact, as-build condition.

2. Ballast condition.

3. Grounding damaged condition.

4. Fire and explosion damaged condition.

The response hereby obtained has been found to be in accordance with the expected behaviorof the midship section. Thus, based on the �ndings in these four test-cases, it has beenconcluded that the present procedure is capable of predicting the ultimate and post-ultimatestrength of the damaged hull girder.

Finally, the possible substitution of the present procedure by a simple moment interactionformula has been investigated for the intact, as-build condition. This e�ort has howeverproven disappointing, as the highly nonlinear sagging response of the ultra large crude carrierpredicted by the present procedure, eliminates any such simple formulation. It has thereforebeen concluded that substituting the present procedure by a simple interaction formula,although being an unarguably much faster approach, is a not recommendable practice in thequest for rapid evaluation of the ultimate hull girder strength.

vi Executive Summary


Synopsis

Denne afhandling omhandler den strukturelle opf�relse af et skibs skrogstruktur i den ek-streme design tilstand af en kombineret last best�aende af moment, forskydning, torsion oghydrostatisk tryk. Forskningsarbejdet er del af et st�rre studie over maritime konstruktioner,som sigter mod etableringen af et rationelt, computerbaseret analyse system for maritimekonstruktioner.

Motivationen for at udf�re dette forskningsarbejde er den ultimative og post-ultimativestyrkes vigtighed for p�alideligheden af s�avel intakte som skadede skibe. Ydermere, i enn�dsituation, vil kendskab til den strukturelle kapacitet af skibet i dets enten skadede ellerintakte tilstand, v�re altafg�rende for at kunne tr��e en beslutning om mulig bj�rgn-ing eller, i v�rste fald, redning af bes�tningen, med henblik p�a menneskelig sikkerhed,milj�beskyttelse og �konomi.

F�llesn�vneren for begge disse interesseomr�ader, p�alidelighed og n�dberedskab, er det ab-solutte krav om hurtige beregninger. I p�alidelighedsanalyser er simulationer, der kr�ver etstort antal af ultimative kapacitets beregninger ofte det v�rkt�j der bliver anvendt til atetablere p�alidelighedsindekset for et skib. Derfor er en hurtig procedure for den strukturelleanalyse en foruds�tning for en hurtig p�alidelighedsanalyse. Ang�aende n�dberedskab, s�a erder her store investeringer og menneskelig sikkerhed p�a spil. Tid bliver en forhindring og enfejlinformeret beslutning kan vise sig at have frygtelige konsekvenser. Hurtig etablering afen rationel basis, hvorfra beslutninger kan tr��es, bliver derfor af yderste vigtighed.

For at kunne im�dekomme disse krav bliver m�alet for det n�rv�rende forskningsarbejdederfor at bygge et computerbaseret analysev�rkt�j, der kan beregne den ultimative og post-ultimative styrke af skibsskrogbj�lken i den generelle kombinerede lastkondition best�aendeaf moment, forskydning, torsion og hydrostatisk tryk. Derefter, at veri�cere n�jagtighedenaf proceduren mod tilg�ngelige eksperimentelle resultater og ved sammenligning med andreteoretiske resultater. At opn�a dette m�al kr�ver basalt set l�sning af tre underopgaver:

1. Et litteraturstudie for de tilg�ngelige metoder der kan anvendes til at bestemme denultimative og post-ultimative kapacitet. Dette studie vil fremkomme med tre muligekandidater: De endelige elementers metode (FEM), den idealiserede strukturelle delsmetode (ISUM) og bj�lkes�jlemetoden. For at kunne v�lge mellem disse tre metoderer tilg�ngelige sammenlignings-studier blevet gennemg�aet. Baseret p�a disse fundneresultater, er bj�lkes�jlemetoden blive valgt til det n�rv�rende forskningsarbejde.

vii

viii Synopsis

2. Udvikling af en plan for beregning af den ultimative og post-ultimative kapacitet afskrogbj�lken under anvendelse af bj�lkes�jlemetoden. Under antagelse af, at arbe-jdskurven for en hvilken som helst bj�lkes�jle er kendt p�a forh�and, m�a yderligereforuds�tninger og antagelser identi�ceres og bestemmes. Baseret p�a dette m�a en bereg-ningsstrategi, der er egnet for implementering i et computerprogram, blive etableret.

3. Etablering af en strategi for beregning af arbejdskurven for en bj�lkes�jle. Vigtigee�ekter p�a denne m�a identi�ceres og om muligt inkluderes i beregningsstrategien.

Gennem dette arbejde er en procedure der anvender bj�lkes�jlemetoden og det asymmetrisketvungne krumnings princip blevet etableret. Proceduren er i stand til at medtage f�lgenden�glee�ekter:

� Overordnet systemmodellering:

{ Generel kombineret last best�aende af moment, forskydning og torsion.

{ Asymmetrisk tv�rsnitsbeskrivelse.

{ Direkte beregnet �jeblikkelig bj�lkes�jle-arbejdskurve.

� Bj�lkes�jlemodellering:

{ Forskellig ydesp�nding for plade og stiver.

{ Bj�lkes�jlespeci�k elasticitetsmodul og forskydningsmodul.

{ Initiel udb�jning af bj�lkes�jlen.

{ Tryklast p�a bj�lkes�jlen.

{ Bj�lkes�jler uden stiver, dvs. plader alene.

F�lgende e�ekter er imidlertid ignoreret i den n�rv�rende formulering:

� Lastmodellering:

{ Hv�lvings delen af torsionsmomentet.

{ Sammenvirket mellem det b�jende moment og forskydningskr�fterne.

{ Refordelingen af forskydningssp�ndingerne.

� Bj�lkes�jlemodellering:

{ Indsn�vring af bj�lkes�jlen i tr�k.

{ Kipning af stiveren.

{ Svejsesp�ndinger.

{ Initiel udb�jning af pladefeltet.

Synopsis ix

N�rv�rende procedure har herefter v�ret anvendt p�a to testmodeller med succes. Denf�rste af disse var en af de s�akaldte Nishihara kassedragere unders�gt af Nishihara [31]b�ade eksperimentelt og analytisk. Den anden model var en dobbeltskroget tanker som eranalyseret af Melchior Hansen [17] for kombineret vertikal forskydning og b�jende moment.P�a baggrund af resultaterne opn�aet i disse to testtilf�lde er det blevet konkluderet, atn�rv�rende procedure er i stand til n�jagtigt at forudsige den ultimative og post-ultimativestyrke af den intakte skrogbj�lke.

En ny testmodel er derefter blevet udvalgt for en mere grundig unders�gelse. Skibet valgttil denne del af forskningsarbejdet er et designstudie af en ultra stor olie tanker (ULCC).F�lgelig er skibet aldrig blevet bygget. Imidlertid er det blevet dimensioneret i henholdtil klassi�kationsreglerne for st�alskibe udsendt af Det Norske Veritas [7]. Form�alet medunders�gelserne udf�rt for denne olietanker har v�ret dobbelt. Den f�rste har v�ret at viseat n�rv�rende procedures evne til at h�andtere asymmetrisk b�jning af et intakt skib tilligemed hydrostatisk tryklast. Den anden har v�ret at demonstrere den n�rv�rende proceduresanvendelighed p�a skadede skibe. I den retning er f�lgende �re scenarier blevet analyseret:

1. Intakt kondition.

2. Ballast kondition.

3. Grundst�dningsskadet kondition.

4. Brand- og eksplosionsskadet kondition.

Det herved opn�aede gensvar er blevet fundet i overensstemmelse med den forventede opf�relseaf midtskibssektionen. F�lgelig er det, baseret p�a resultatet af de �re testmodeller, blevetkonkluderet at den n�rv�rende procedure ogs�a er istand til at forudsige den ultimative ogpost-ultimative styrke af den skadede skrogbj�lke.

Endelig er muligheden for en substituering af den n�rv�rende procedure med en simpelmoment interaktionsformel blevet unders�gt for den intakte kondition. Dette har imidlertidvist sig sku�ende, idet den meget ikke-line�re sagging opf�relse af olie tankeren, forudsagtaf den n�rv�rende procedure, eliminerer muligheden for en s�adan simpel formulering. Deter derfor blevet konkluderet, at erstatning af den n�rv�rende procedure med en simpelinteraktionsformel, selvom det uomtvisteligt er en meget hurtigere metode, ikke er tilr�adeligpraksis i bestr�belsen for hurtig evaluering af den ultimative styrke af skrogbj�lken.

x Synopsis


Contents

Preface i

Executive Summary iii

Synopsis (in Danish) vii

Contents x

Symbols and Nomenclature xvii

1 Introduction 1

1.1 Overview and Project A�liation . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Objective and Scope of Work . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.5 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Ultimate and Post-Ultimate Capacity of the Hull Girder 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Longitudinal Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Ultimate Longitudinal Strength . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Ultimate and Post-Ultimate Strength . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Pre-Selection of Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

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xii Contents

3 Ultimate and Post-Ultimate Capacity of a Laser Welded Panel 15

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 De�nition of Laser Welded Panel . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 Summary of Performed Calculations and Methods . . . . . . . . . . . . . . . 18

3.4 Basis for Comparison of Results . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5 Discussion of the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.6.1 The Finite Element Methods . . . . . . . . . . . . . . . . . . . . . . . 27

3.6.2 The Beam-Column and Idealized Structural Unit Methods . . . . . . 28

4 The Beam-Column Method 31

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 Load Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2.1 Load Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2.2 Load Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2.3 Accidental Loads & Probabilistic Reliability Analysis . . . . . . . . . 38

4.3 System Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3.1 Pure Horizontal Bending . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3.2 Beam-Column End Displacement and Rotation . . . . . . . . . . . . 42

4.3.3 General Load Condition . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.4 Ultimate Capacity Criterions . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.4.1 Ultimate Shear Capacity . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.4.2 Ultimate Moment Capacity . . . . . . . . . . . . . . . . . . . . . . . 49

4.5 Implementation Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Contents xiii

5 Beam-Columns in Combined Loading 53

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.2 Initial Imperfections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.3 Idealized Beam-Column Behavior . . . . . . . . . . . . . . . . . . . . . . . . 55

5.4 Plastic Tension Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.5 Elastic Tension Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.5.1 Stress Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.6 Elastic Compression Region . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.6.1 Marguerre's Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.6.2 Application { The von Karman Equations . . . . . . . . . . . . . . . 61

5.6.3 Post-Buckling Behavior { The Perturbation Technique . . . . . . . . 62

5.6.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.6.5 Collapse Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.6.6 Plate Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.6.7 Plate Collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.6.8 Stress Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.7 Plastic Compression Region (Unloading) . . . . . . . . . . . . . . . . . . . . 76

5.8 Limitations of Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.9 Implementation Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.10 Test Application of the Procedure . . . . . . . . . . . . . . . . . . . . . . . . 83

5.11 Beam-Columns without Sti�eners . . . . . . . . . . . . . . . . . . . . . . . . 85

5.11.1 Overall Buckling Mechanism . . . . . . . . . . . . . . . . . . . . . . . 85

5.11.2 Local Folding Mechanism . . . . . . . . . . . . . . . . . . . . . . . . 87

5.11.3 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

xiv Contents

6 Veri�cation of the Procedure 93

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.2 Nishihara Box Girders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.2.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.2.2 Collapse Analyzes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.2.3 Discussion of the Results . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.3 Double Hull Tanker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.3.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.3.2 Combined Loading Analyzes . . . . . . . . . . . . . . . . . . . . . . . 103

6.3.3 Discussion of the Results . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7 Application of the Procedure 109

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.2 Ultra Large Crude Carrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7.3 As-Build Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.3.1 Moment Interaction Formulas . . . . . . . . . . . . . . . . . . . . . . 117

7.4 Ballast Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7.5 Grounding Damage Condition . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7.6 Fire and Explosion Damage Condition . . . . . . . . . . . . . . . . . . . . . 129

7.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7.8 Final Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

8 Conclusion and Recommendations 137

8.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

8.2 Recommendation for Future Work . . . . . . . . . . . . . . . . . . . . . . . . 139

Contents xv

A The Asymmetrical Forced Curvature Principle 141

A.1 Purpose & Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

A.2 Plane Geometric Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

A.2.1 Beam-Column End De ection and Rotation . . . . . . . . . . . . . . 143

A.3 Local Beam-Column Description . . . . . . . . . . . . . . . . . . . . . . . . . 144

A.3.1 E�ective Beam-Column End Rotation . . . . . . . . . . . . . . . . . 144

A.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

B Solution to the Classical Beam-Column Problem 149

B.1 Purpose & Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

B.2 Beam Di�erential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

B.3 De ection Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

B.4 Sectional Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

B.5 Midspan Stress Distribution in Beam-Column . . . . . . . . . . . . . . . . . 154

B.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

C Solution of the von Karman Equations 159

C.1 Purpose & Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

C.2 The Finite Di�erence Method . . . . . . . . . . . . . . . . . . . . . . . . . . 160

C.3 Buckling Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

C.3.1 Buckling Solution Scheme . . . . . . . . . . . . . . . . . . . . . . . . 162

C.3.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

C.3.3 Buckling Coe�cients . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

C.3.4 Buckling Interaction in Combined Loading . . . . . . . . . . . . . . . 166

C.4 Post-Buckling Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

C.4.1 Post-Buckling Solution Scheme . . . . . . . . . . . . . . . . . . . . . 174

C.4.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

C.5 Veri�cation of Collapse Criterion . . . . . . . . . . . . . . . . . . . . . . . . 181

C.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

xvi Contents

D Alternative Modeling of the Idealized Beam-Column Behavior 187

D.1 Purpose & Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

D.2 Basic Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

D.3 The Beam Di�erential Equation . . . . . . . . . . . . . . . . . . . . . . . . . 189

D.4 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

D.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

Bibliography 193

List of Figures 199

List of Tables 209

List of Ph.D. Theses Available from the Department 213

Symbols and Nomenclature

The symbols used in this thesis are explained when they are �rst introduced. The followingis a list of the main symbols used. The list is divided into a general section followed by topicspeci�c sections. Finally, a section with notation notes is included.

General

A area

E Young's modulus { modulus of elasticity

I moment of inertia { second moment of area

M moment

N normal force

P direct load

Q shear force

x coordinate

y coordinate

z coordinate

" strain

"y yield strain

� curvature

� Poisson's ratio

� stress

�cr critical stress

�v =q�2x + �2z � �x�z + 3�2xz (von Mises stress)

�y yield stress

� shear stress

�cr critical shear stress

�y yield shear stress

xvii

xviii Symbols and Nomenclature

Hull De�nition

B breadth of vessel

D depth of vessel

L length of vessel

Mhog hogging moment

Msag sagging moment

Mp plastic moment

My �rst yield moment

My horizontal moment

Mz vertical moment

Mx torsional moment { torque

N normal force

Qy horizontal shear force

Qz vertical shear force

x coordinate { length direction

y coordinate { width direction

z coordinate { height direction

r displacement

Plate De�nition

b width of plate

D =Et3

12(1� �2)(plate sti�ness)

K =Et

(1� �2)

KT =Et

2(1 + �)

kz direct buckling coe�cient

kxz shear buckling coe�cient

` length of plate

Nx direct edge line load { length-wise

Nz direct edge line load { width-wise

Nxz shear edge line load

t thickness of plate

Symbols and Nomenclature xix

u in-plane displacement { width-wise

~u shape of in-plane displacement { width-wise

v in-plane displacement { length-wise

~v shape of in-plane displacement { length-wise

w out-of-plane de ection

~w shape of out-of-plane de ection

x coordinate { width direction

Y in-plane normal load

y coordinate { thickness direction

y? initial out-of-plane de ection

z coordinate { length direction

" perturbation parameter or strain

"c perturbation parameter at collapse

arbitrary scaling parameter

� stress function (�;xx = Nz ; �;zz = Nx ; �;zx = �Nxz)

' tan(') = Nz=Nxz (direct/shear load ratio)

�ec mean axial edge stress at collapse

�ac mean axial stress at collapse

�x direct stress { width-wise

�z direct stress { length-wise

�xz shear stress

�z;cr critical direct stress

�xz;cr critical shear stress

�sc mean shear stress at collapse

Finite Di�erence

h size of cell

i node { width-wise

j node { length-wise

M dimension of grid { width-wise

N dimension of grid { length-wise

ui;j in-plane displacement at node i; j

vi;j in-plane displacement at node i; j

wi;j out-of-plane de ection at node i; j

� eigenvector

� eigenvalue

xx Symbols and Nomenclature

Beam-Column De�nition

A total cross sectional area

As cross sectional area of sti�ener

Ap cross sectional area of plate

H total height of beam-column

` length of beam-column

Pcr critical direct load

PE = EI��

`

�2(Euler load)

Py = �yA (squash load)

q uniformly distributed line-load

w de ection

w0 initial de ection

� =

sjP jEI

� instantaneous rotation

� end displacement

' end rotation

"t strain at tension collapse

"u strain at compression collapse

�xx direct stress

Global System Analysis

M moment response of beam-column

P direct response of beam-column (force)

x global coordinate { length-wise

y global coordinate { breadth-wise

y0 principal coordinate { breadth-wise

z global coordinate { depth-wise

z0 principal coordinate { depth-wise

� angle of instantaneous neutral axis

� rotation of cross section plane

�i end displacement of ith beam-column

�i arm to ith beam-column

�INA perpendicular distance from instanta-neous neutral axis to center of baseline

Symbols and Nomenclature xxi

Ultimate Capacity Description

MINA total moment reponse about the instantaneous neutral axis of the cross section

MINA,u ultimate moment about the instantaneous neutral axis of the cross section

Mx,u ultimate torsional moment, i.e. about the global x-axis

My,u ultimate horizontal moment, i.e. about the global y-axis

Mz,u ultimate vertical moment, i.e. about the global z-axis

Qy,u ultimate horizontal shear force, i.e. in the global y-direction

Qz,u ultimate vertical shear force, i.e. in the global z-direction

�u ultimate shear stress distribution

Notes

(a) For derivatives of single variable functions the prime notation is used. That is,

f 0(x) =df(x)

dx

For derivatives of multi variable functions the comma notation, e.g.

f;x =@f(x)

@xor f;xz =

@f 2(x)

@x@z

(b) Functions/expressions evaluated at a speci�c value is indicated by a vertical barfollowed by the value of the speci�c variable, e.g.

Zf(x)dx

��x = a

(c) Vectors are in the text written in bold normal, e.g. X = (X1; X2; : : : ; Xn). Indrawings and sketches an arrow over the letter is used to indicate vectors.

(d) Matrixes are written like vectors, but with a bar below, e.g. A.

(e) Inverse matrixes are denoted by `-1', e.g. A�1.

(f) Zero vectors are denoted by a bold zero, i.e. 0 = (0; 0; : : : ; 0).

xxii Symbols and Nomenclature


Chapter 1

Introduction

: : : to encourage and facilitate the general adoption of the highest practicable

standards in matters concerning maritime safety, e�ciency of navigation and

prevention and control of marine pollution from ships.

Extract of Article 1(a)The IMO Convention

March 1948

1.1 Overview and Project A�liation

This thesis addresses the structural performance of the hull structure of a vessel in theextreme response state of a combined loading. The research is part of a larger study onMarine Structural Design aiming at the establishment of a rational, computer-based analysissystem for marine structures. Within this project, the thesis is closely related to the work ofMelchior Hansen [17] on \Reliability Methods for the Longitudinal Strength of Ships" and maybe seen as a continuation of this research. Further, the present work may o�er contributionsto the works of Friis Hansen [18] on \Reliability Analysis of a Midship Section" and of CerupSimonsen [43] on \Mechanics of Ship Grounding".

1.2 Background

The structural design of any vessel is strongly regulated both by class rules and regulationsfrom both national and international authorities. All this goes to insure safe operation withinan acceptable safety level of the vessel. While it is mainly the task of the authorities to setthe standards for what constitutes an acceptable safety level, it falls upon the classi�cationsocieties to formulate the appropriate design rules insuring this level of safety.

1

2 Chapter 1. Introduction

Focusing on the design rules, these are chie y based on rational design principles developedover the years and are continually revised and updated according to the latest achievementsin knowledge. Hence, for the structural aspects of a vessel, the ful�llment of the class rulesinsures safe operation given a speci�c design load level also set by the class rules.

Thus, it can reasonably be expected that a vessel will not undergo buckling/plastic collapsewhile the working load is below the design load. However, there is a risk that the vesselwill experience extreme loading conditions far beyond the design load in an unintended,accidental, or emergency situation.

On July 21st 1980, the VLCC Energy Concentration broke into two during discharge of oilat Rotterdam, The Netherlands. This is an excellent illustration of how irresponsible cargohandling caused transgressing of the design load and is well documented by Rutherfordand Caldwell [42]. Collision and grounding also constitutes a dangerous situation for anyvessel, as it did on the morning of March 24th 1989, where the VLCC Exxon Valdez wentaground onto a reef and spilled some 10.1 million barrels of oil into Prince William Sound,Alaska, USA. Although she su�ered extensive damage to her hull, overall structural integrityremained uncompromised by the accident. Less fortunate was the oil tanker Braer, whichon January 5th 1993 ran aground o� Sumburgh Head in Shetland, UK. Seven days later, onJanuary 12th she broke up into three sections after having been continually thrown againstthe rocks of the island and the entire cargo of some 620 thousand barrels of oil was spilledinto the sea around the southern end of the main Shetland Island.

These are just a few examples of real accidents arising from cargo handling and grounding.Other probable causes could be e.g. severe weather conditions, �re, explosions, etc. Also,the aspects of corrosion damage and fatigue cracking may solely be responsible for { or anattributing factor in { exceeding the design load of any vessel. Freak occurrences like thebreaking into two of the container vessel MSC Carla on the November 24th 1997 falls intothis last category. Although the exact cause of the loss has not been ascertained as of yet, itis known that the twenty-�ve year old vessel experienced heavy weather in the north Atlanticon its �nal voyage from Le Havre, France to Boston, USA. The vessel had twelve years intoits service life been re-�tted with a new container hold. Further, she was �tted with anexceptional power plant of some 75; 000 bhp, which made her capable of sustained servicespeeds in excess of 30 knots. Now, the correlation between high age and corrosion, highspeed and fatigue, together with the changing of the initial design loads caused by the addednew container hold, all puts the loss of the MSC Carla in lot with the above mentionedprobable causes. Thus, until the exact cause of failure is identi�ed, she serves as a perfectexample of all but �re and explosion.

1.3 Motivation

In an emergency situation at sea, the course of best action to save crew, cargo, and ship,while protecting the marine environment, may not immediately be obvious. Time wasted

1.4. Objective and Scope of Work 3

or an ill-informed decision made, could cause irretrievable damage. Hence, precise technicalinformation about the ship and its damage condition will be of paramount importance inrestraining the crisis.

Grounding and collision, �re and explosion, all presents extreme crisis situations for vessels atsea. Time becomes an obstacle, as each passing minute several factors may worsen. Possibleoil out ow, water ingress and the ships damage stability may all be worsening, particularlyif exacerbated by heavy weather or strong tides.

To lessen the consequences of casualty at sea, a full appreciation of the vessels damagedstability and damaged strength is essential before decisions are taken about transferringcargo or initiating other remedial actions to salvage the vessel. Thus, using a computermodel of the vessel which will allow rapid calculation of damage stability, damage strength,and damage oatability, will render the possibility of determining how the vessel will respondunder various rescue scenarios, and hereby enable the crew to make the best possible decisionin selecting a rescue option. Further, pre-analyzes of likely emergency scenarios are advisableto keep on the vessel for ready reference during a crisis when human stress factors andtime limitations might impair crew actions. This preparation is part of standard crisismanagement planning required by MARPOL1 73/78 and OPA2 '90.

Hence, it is of interest to develop an e�cient computer code suitable for repeated structuralanalyzes of intact as well as damaged ship structures with di�erent stages of corrosion.

Further, such a tool will also be of great value in the evaluation of the reliability of shipstructures. Here it is necessary to have a tool which can calculate the strength of the hull.Together with a probabilistic method for evaluation of the probability that the loading on thehull exceeds the strength the reliability of the vessel can be established. Such probabilisticanalyzes demand much computational work. Therefore, it is important that the underlyingultimate strength analyzes can be performed fast and e�ective. Just as it is the requirementin the case of emergency scenarios. This observation is substantiated by current e�orts bye.g. Paik et al. [35], Paik and Terndrup Pedersen [33] and Paik and Mansour [32] wherea simple formulation for the evaluation of the ultimate strength is applied to predict theresidual strength of e.g. grounded ship hulls.

1.4 Objective and Scope of Work

The overall objective is therefore to build a rapid computer based analysis tool for calcu-lation of the ultimate and post-ultimate capacity of the hull girder. Then, to verify theaccuracy of the procedure against available experimental results and by comparison withother theoretical results.

1International Convention for the Prevention of Pollution from Ships, IMO, 1973/1978.2Oil Pollution Act of 1990, USA, 1990.


To meet this objective essentially requires three sub-tasks to be completed. Hence, the scopeof the work to be performed becomes

1. A study of literature for the available methods usable to evaluate the ultimate andpost-ultimate capacity. This study will reveal three possible candidates: The �niteelement method (FEM), the idealized structural unit method (ISUM), and the beam-column method. To select among these three methods, available comparative studiesare reviewed. Based on the �ndings in those, the beam-column method will be chosenfor further use in the present research project.

2. Development of a scheme for the evaluation of the ultimate and post-ultimate capacity,utilizing the beam-column approach. Assuming that the load-displacement responseof any beam-column is known beforehand, further prerequisites and assumptions mustbe identi�ed and decided on. Based on this, a scheme suitable for implementation ina computer code must be established.

3. Establishment of a framework designed to evaluate the load-displacement response of abeam-column. Key e�ects must be identi�ed and if possible, included in the procedure.

When these three points have been addressed, and the objectives solved, all that is neededfor the development of a computer code for the ultimate strength is readily available. Theremaining �nal task will thus be to benchmark the code against available experimental resultsand other theoretical results.

1.5 Organization of the Thesis

The thesis is compiled as follows: First, a general introduction to the ultimate and post-ultimate capacity of vessels is given in Chapter 2. The historical development of the methodsfor evaluating the ultimate strength of ships is reviewed, and state-of-the-art is identi�edalong with the pros and cons for each method. Based on the �ndings in this chapter, thebeam-column method is pre-selected for further investigation.

Chapter 3 therefore continues to investigate the accuracy of di�erent methods for the pre-diction of the ultimate and post-ultimate capacity of a sti�ened panel, to rea�rm the choiceof the beam-column method. This is based on the works of the Technical Committee III.1of ISSC'97 [12] to which the author has contributed.

In Chapter 4 a scheme for evaluating the ultimate and post-ultimate capacity of the midshipcross section based on the beam-column method is formulated. The formulation for thisscheme is continued in Chapter 5 which is devoted to the analysis of beam-columns in com-bined loading. A framework for the evaluation of the load-displacement response, suitable

1.5. Organization of the Thesis 5

for implementation in a computer code, is developed. Thus, the total model has upon theconclusion of this chapter been derived.

The established procedure is then veri�ed in Chapter 6 against available third party resultsin the form of experimental data from Nishihara [31] and theoretical results from MelchiorHansen [17]. Chapter 7 concludes the documentation by presenting the results obtainedwhen the procedure is applied to a typical double hull, ultra large crude carrier (ULCC)structure.

Finally, conclusions from the present study and recommendations for future work are o�eredin Chapter 8.

To further facilitate the reading of this thesis, a selected few topics are presented in appen-dices. Of these, Appendix A which explains the derivation of the forced curvature principlefor a general asymmetrical loaded cross section will be of interest for general reference. Theremaining appendices are intended primarily for readers unfamiliar with the current topic ofultimate strength, or readers who which to implement their own procedure along the samelines used in this research.

Towards this end, the solution to the classical beam-column problem, including imperfec-tion and lateral loading, is given in Appendix B. In Appendix C the numerical solutionscheme to the von Karman equations is outlined in detail and veri�ed against experimentalresults. Further, Appendix D presents an alternative method for establishing the idealizedbeam-column response, involving the direct evaluation of the full set of ordinary di�erentialequations (ODE's) describing the beam behavior.

Chapter 2

Ultimate and Post-Ultimate Capacity

of the Hull Girder

2.1 Introduction

When assessing the ultimate capacity of a cross section made up of numerous individualstructural components in a typically complex way, the need for simpli�cations is evident. Ofcourse, one could go all the way: Forget about simpli�cation and just apply state-of-the-artstructural analysis tools, e.g. �nite elements { Build a highly complex model { Solve it, andarrive at an answer. But what would be the credibility of that answer?

First of all, the more complex the model gets, the bigger the room for error and hence,the more dire need for veri�cation. An obvious solution to this would be to test the realthing. Tab. 2.1, taken from Yao [47], list the tests conducted since the turn of the century.However, full scale experimental results for modern ships are hard to come by as it wouldrequire vast �nancial resources to perform these test, especially when the scope of interest ison the leviathans of the sea like bulk carriers and very large crude carriers (VLCC's). Thus,benchmarking against experiments is not an viable option. Realizing this enforces the notionof need for simpli�cation. Abandoning experiments leads to a simpli�ed model as the onlypossible cause for veri�cation of the answer obtained from the complex model.

Another aspect is the resources required to perform an analysis. A highly complex �niteelement analysis will allocate huge resources both in man hours going into creating the modeland in computer resources for solving of the model. In an academic/research environmentthis may be acceptable but for application in e.g. design and emergency response, fastcalculation of the ultimate capacity is of utmost importance.

Altogether, this lead to the inevitable conclusion that if evaluation of the ultimate capacityof a vessel is to enter the practical design and operation of any ship, the procedure must

7

8 Chapter 2. Ultimate and Post-Ultimate Capacity of the Hull Girder

Table 2.1: Characteristics of full scale tested ships (Adopted from Yao [47]).

Name of vessel Type Length Breadth Depth Disp. Year Test(m) (m) (m) (Tonf) tested group

Wolf Destroyer 68.10 6.25 4.11 381 1902 Adm.Preston Destroyer 95.81 9.44 6.30 1,190 1930 BCRBruce Destroyer 95.81 9.44 6.30 1,190 1931 BCRFrank Purnell Ore/Bulk 181.36 18.29 10.67 20,960 1943 USMCCadillac Ore/Bulk 181.36 18.29 10.67 20,960 1943 USMCJohn Hutchinson Ore/Bulk 184.22 18.29 10.67 21,058 1943 USMCChamplain Ore/Bulk 184.22 18.29 10.67 21,058 1943 USMCShilon Tanker 159.56 20.73 11.96 21,880 1943 USMCPhilip Schuyler Cargo 131.52 17.34 11.38 14,230 1944 USMCNeverita Tanker 147.95 17.98 10.36 16,793 1944 ASWCAntelope Tanker 159.56 20.73 11.96 21,888 1944 OSRDElk Hills Tanker 159.56 20.73 11.96 21,888 1944 OSRDVentura Hills Tanker 159.52 20.73 11.96 21,888 1945 USMCNewcombia Tanker 147.95 17.98 10.36 16,790 1945 OSRDFullerton Hills Tanker 159.56 20.73 11.96 21,888 1945 OSRDFort Mi�in Tanker 159.56 20.73 11.96 21,888 1945 USMCClan Alpine Cargo 134.57 17.34 11.38 13,764 1947 ASWCPresident Wilson Passenger 185.60 23.01 18.76 23,500 1947 USMCOcean Vulcan Cargo 134.57 17.34 11.38 13,752 1947/48 ASWCAlbuera Destroyer 115.52 12.27 6.71 2,315 1949/50 NCREAbbreviations: Adm. : Admiralty

BCR : Bureau of Construction and RepairUSMC : United States Maritime CommissionASWC : Admiralty Ship Welding CommitteeOSRD : O�ce of Scienti�c Research and DevelopmentNCRE : Naval Construction Research Establishment

be rapid. This will require simpli�cations to be made. Hence, it is necessary to gain anunderstanding of the inner working of the ship structure from which possible simpli�cationschemes can be established and evaluated.

2.2 Longitudinal Strength

A ship's hull is basically a box girder structure composed of sti�ened plating. The globalforces acting on it arises from the distributed hull weight, cargo weight, buoyancy, and waveenvironment. Together, these forces will create the traditional sectional loading consisting

2.2. Longitudinal Strength 9

of moment, shear, and torque all resulting in stresses and strains in the structure. Anyocean going vessel will experience these loadings. However, for an intact vessel the bendingmoment about the horizontal neutral axis will normally be the predominant sectional force.Because of the very high stress levels this moment may create in the deck and bottom of thestructure, these regions may su�er failure due to buckling or plastic collapse. Consequently,the structural response to this load component has received enough attention during timeto get its own name { Longitudinal strength { by which is understood the ship's strengthagainst the horizontal bending moment.

As already mentioned, the ship hull is basically a box girder, and this was indeed used in the�rst approach to calculate the longitudinal strength. What was done was simply to applystandard beam theory to the hull girder and then calculate the shear force and bendingmoment assuming a distribution of weight and a calculated buoyancy distribution along thehull based on some assumed wave model. One of the �rst to do this on steel vessels wasWilliam John [22] who in 1874 proposed that the maximum bending moment at midshipsapproximately could be expressed as

Mmax = rL� (2.1)

where r is the displacement, L is the length, and � is the fraction of the length thatapproximately represent the leverage of the moment. He established the relation based onthe observation that the maximum moment occurred for a wave of length equal to the lengthof the vessel. � was then taken in the region of thirty-�ve through �fty depending on theloading condition of the vessel. When the moment was known from Eq. (2.1), applyingNavier's equation for beam cross section, he determined the maximum stress as

�max =zMmax

I(2.2)

where I is the moment of inertia of the midship section and z is the distance from the elasticneutral axis to the stringer deck1. Based on this, John discussed the possibility of tensionfailure of the deck in the hogging condition.

In retrospect, with the bene�t of hindsight, it may seem negligent to discard the problemof compressive buckling as a failure mode. However, in those days naval architects collec-tively thought the main criterion to be failure/breaking of plating and rivets in the tensionside of bending. It was not until the loss of the torpedo-boat/destroyer HMS Cobra onSeptember 18th 1901, that the compressive buckling was moved up on the list of failuremodes (cf. Faulkner et al. [14]). Nevertheless, John's suggested method still covers almostall important items for the longitudinal strength assessment.

After John, many improvements has been achieved. Both on the method of wave loadcalculation and on the method of stress analysis, as well as on failure criterions where fatigue,yielding, buckling now all come into play. Still, John's fundamental idea remains unchanged.

1In ships of those days there was no point in including the distance z as the maximum distance from theneutral axis to either deck or bottom { It was always maximum to the deck.


2.3 Ultimate Longitudinal Strength

As already stated, the loss of the Cobra in 1901 lead to the recognition of the key in uenceof compressive buckling collapse. One of the �ndings of the investigation committee was,that the measured de ection of Cobra's sister ship, Wolf, during a static bending test waslarger than those calculated by application of beam theory. This lead to the introduction ofthe e�ective width concept where the e�ectiveness of some members is considered reduced bythe occurrence of buckling in compression. This �nding was further substantiated in Vasta's1958 paper [46], where he summarizes on the lessons learned from full scale test. Speci�cally,the tests of the two destroyers Preston and Bruce showed that the cross section attained itsultimate strength when the deck and/or bottom undergoes buckling/plastic collapse whileexposed to compressive longitudinal bending. Further, Vasta found that the stress at thebuckled plates calculated using � = zM=I (i.e. simple beam theory as in Eq. (2.2)) with Mequal to the collapse bending moment, had good correlation with the experimental �ndingsfor isolated plates.

In essence, this discovery lead to the very important realization that the structural behaviorof each individual member of the cross section must be considered separately. One cannotjust look at the whole section as one. One of the �rst to use this observation was Caldwell[5] who in 1965 suggested a new method of calculating the ultimate longitudinal strengthof a midship section. In his method the individual structural members are lumped togetherinto panels. The assumption is, that the collapse load for each of these panels is knownbeforehand, either by experiments or calculation. Thus, with this assumption ful�lled, theultimate longitudinal strength { or collapse moment { can be estimated by simple summation.

2.4 Ultimate and Post-Ultimate Strength

The method proposed by Caldwell fail, however, to account for the post-collapse strength ofthe individual structural members in the cross section. This shortcoming was later addressedby Smith [44]. He proposed an approach where each sti�ened panel is considered to be anumber of beam-columns, each comprised of one sti�ener and a certain width of the plateit is attached to. Then, by dividing the cross section of the beam-column into a number ofhorizontal �bers and by straining (loading) the structure incrementally, the method allowsfor the elastic-plastic behavior of the sti�ener and for the buckling of the plate. For eachincrement in the straining, the corresponding stress increment in each �ber is derived usingthe tangent modulus of elasticity. That is, for the sti�ener the slope of the stress-strain curveat the current level of straining is used, and for the plate the slope of the average stress-straincurve. It should also be noted that the occurrence of brittle fracture is not included in thediscussion. That is, it is assumed that the material has su�cient ductility to eliminate thepossibility of brittle fracture.

The work of Smith has been used by several other researchers in this �eld with some dif-ferences and improvements added on. The main di�erence in these improved methods, is

2.5. Pre-Selection of Method 11

the way the average stress-strain relations for the beam-columns are formulated. Rutherfordand Caldwell [42] did a very thorough analysis of the VLCC Energy Concentration whichcollapsed during discharge in Rotterdam on July 21st 1980. Their work focuses on the sen-sitivity of the ultimate capacity caused by material properties, corrosion, lateral pressure,methods of manufacture, and modeling uncertainties. Yao and Nikolov [48] used the sameconcept as Smith to formulate a method that includes residual stresses in the plating andinitial de ection of both plate and sti�ener. They later improved the formulation (Yao andNikolov [49]) by further including the coupled exural behavior of the sti�ener which en-abled their method to account for the possible tripping failure of the sti�ener. In the laststudy they further investigated the e�ect of the torsional sti�ness of the sti�eners on theultimate capacity. The �nding was, as expected, that the inclusion of the tripping mode forthe sti�ener caused a reduction in the ultimate capacity.

All these methods have common heritage with Smith's method, and can collectively bedescribed as beam-column methods. In contrast to these, a di�erent approach would beto apply the �nite element method (FEM). In 1983 Chen et al. [6] described a method forthe analysis of the ultimate strength based on the �nite element method. The method isexcellent in the way that it accounts for all major e�ects i.e. elasto-plastic properties ofthe material, geometrically nonlinear behavior of the elements, and their buckling and post-buckling strength. However, the drawback of the method is, that the solution procedure isextremely time consuming. This problem was addressed by Ueda and Rashed [45] in 1984,by the introduction of the Idealized Structural Unit Method (ISUM). This method adoptsthe �nite element terminology relating to nodal forces and displacements, but di�ers fromthe standard �nite element method approach suggested by Chen in the formulation and sizeof the used elements. In the idealized structural unit method approach, the conventionalelements used in the �nite element method are replaced with the Idealized Structural Units(ISU). These are large elements capable of describing the structural behavior of an entiremember of the structure. Thus, by application of the idealized structural unit method, thenumber of degrees of freedom is considerably reduced and a large reduction in computertime for the solve is thereby achieved. The idealized structural unit method approach holdsa lot of promise and is still under continuous development by researchers such as Paik et

al. [34] and Bai et al. [1], to cover structural members like both sti�ened and unsti�enedplate elements, and specialized beam-column elements.

2.5 Pre-Selection of Method

As re ected in the previous section, it becomes evident by studying the literature and publica-tion on the topic of ultimate and post-ultimate capacity of ship hulls, that state-of-the-art atpresent either relies on a beam-column approach, or on the idealized structural unit method.It may, however, be argued that the �nite element method also constitutes state-of-the-artas it assures inclusion of all major e�ects in a consistent rational way. Nevertheless, due the(at the moment) very high consumption of computing time associated with the �nite element


Figure 2.1: Calculated and measured moment-curvature relations for the 1/3-scalefrigate. (Adopted from ISSC'94, Technical Committee III.1 [11]).

method, it must be discarded as a candidate for the project at hand, where rapid calculationsis of the utmost importance. Hence, the choice is limited to either the beam-column methodor idealized structural unit method.

In the author's opinion, this choice is �rst of all a matter of taste. Both methods rely on someanalytical solution to whatever is considered a structural member, whether it is a solution tothe beam-column or a set of advanced elements describing also a beam-column or perhaps asti�ened panel, and both method has their advantages and drawbacks. For some purposesone may be preferable to the other, and vice versa. Nevertheless, it is essentially a choicebetween a model formulated in framework of either beam theory, or �nite elements.

This conclusion is substantiated by the benchmark investigation of di�erent computer codesfor the evaluation of the ultimate and post-ultimate capacity of ship hulls performed by theTechnical Committee III.1 of ISSC'94 [11]. The benchmarking was based on the experimen-tal test of a 1/3-scale model of a steel frigate representing a typical warship hull structure

2.5. Pre-Selection of Method 13

subjected to bending loads (cf. Dow [8]). The ISSC contributers each performed a calculationof the ultimate and post-ultimate capacity based on their individual selection of method andthe obtained results were compared in the committee report with the experimental �ndingsof Dow (see Fig. 2.1). Thus, all three of the above mentioned methods (i.e. FEM, ISUM,and beam-column) where represented in the comparison.

Studying the comparison of the moment-curvature relations shown in Fig. 2.1, two apparentobservations can be made. Firstly, it is evident that there is a pronounced discrepancy be-tween the di�erent solutions. Secondly, the best result appears to be the idealized structuralunit method based solution presented as number eight. However, considering the variousdi�erences in how the boundary conditions, residual stresses, initial de ections, etc. wereapplied in the di�erent solutions, defuses a conclusion, as also pointed out in [11], and al-though the idealized structural unit method yields a very potent solution, beam-columnapproaches such as number three, seven, and nine also quali�es as good approximations tothe experimental result.

Also interesting in this respect are the �ndings of the Technical Committee V.1 of ISSC'94[13] on the ultimate strength on ten di�erent sti�ened panels. Here, a benchmark testfocusing on mainly model uncertainties and human error was conducted. The methods usedwere all beam-column approaches, and it it was found that on e.g. the e�ective width used,a variation in the range of zero to �fteen percent arose with one odd forty-�ve percentlower value reported. The study was later extensively reported by Rigo et al. [41] with anumber of elaborations, e.g. a comparison with experimental data. The conclusion remainsthe same though: The prediction of the ultimate strength is very sensitive to the modelinguncertainties and the possibility of human error.

Hence, it seems that the accurate modeling of the boundary and initial conditions is moresigni�cant than the choice of method { leading back to the previously stated opinion, namelythat it is a matter of personal preference. But, there is one last objective measure to basea choice on { speed. Even though the idealized structural unit method is much faster thanthe standard �nite element method, the beam-column method will still, in general, be thefastest of the three, and is therefore the strongest candidate for the method to base thepresent research on. Especially if the software is to be used in reliability analyzes or as apart of an emergency response system.

Chapter 3

Ultimate and Post-Ultimate Capacity

of a Laser Welded Panel

3.1 Introduction

The previous chapter presented a study of literature for the available methods to evaluate ofthe ultimate and post-ultimate strength of the hull girder. Three possible candidates werefound, namely

� Finite Elements Method (FEM)

� Idealized Structural Unit Method (ISUM)

� Beam-column Method

Further, a qualitative comparison of the three methods [11] was reviewed, and based onthis a pre-selection of the beam-column method was made. Still, before making the �nalselection, it would be preferable to gain more insight into the performance of especially thebeam-column approach. Towards this end, it is of interest to investigate the behavior ofsti�ened panels as predicted by the three methods.

Such investigations have been performed by the Technical Committee III.1 of ISSC'97 [12] towhich the author has contributed. Here, the ultimate and post-ultimate capacity of both alaser welded panel and a glass-�ber reinforced plastic (GRP) panel was investigated, and thenumerical predictions was compared with the experimental results. Hence, the investigationmay be seen as a continuation of the work reported in [13, 41].

In the following, the scope will be limited to the laser welded panel, as this specimen repre-sents the most common design in shipbuilding. The laser welded panel is described in detailby Dow [9]. However, to facilitate the reading a short description is presented.

15

16 Chapter 3. Ultimate and Post-Ultimate Capacity of a Laser Welded Panel

Table 3.1: Measured welding residual stresses (MPa).

Panel A B C D E F

Longitudinal -73 -50 -65 -22 -49 -58Transverse -79 -34 -70 -35 -59 -51

Longitudinal S1 S2 S3 S4 S5 S6 S7 S8

Web -43 30 21 32 -115 8 12 -47Flange -50 -47 -73 lost -191 42 20 -48

3.2 De�nition of Laser Welded Panel

The geometry of the panel is shown in Fig. 3.1. It is a laser welded, orthogonally sti�enedgrillage approximately 3300 mm in total length by 1250 mm in total width. TransverseT sti�eners of 46 � 127 mm dimensions (labeled T1 through T3) divide the length of thepanel equally into bays of 1000 mm of which the two center bays are complete, with theend bays only being approximately half lengths of 650 mm. Four longitudinal T sti�ener of25 � 75 mm dimensions (labeled S1 through S8) divide the width in �ve sections of whichthe three central sections are of 300 mm width, with the two outside sections being only175 mm wide. Thus, this arrangement provides six complete panels of 1000� 300 mm and6 mm thickness (labeled A through F), plus fourteen half panels surrounding these to provideboundary conditions.

Further, extensive information on material properties, initial geometric imperfection, andresidual stresses was reported by Dow, and are summarized in the following.

The material is steel with a modulus of elasticity E = 207 GPa. For the plates the 0.2%tensile stress was between 437 and 447 MPa, whereas the yield stress for the sti�eners waseither 350 or 359 MPa, according to measurements. Based on these, the average yield stresseswere selected as �y,p = 440 MPa and �y,s = 359 MPa for the plate and sti�ener respectively.

The shape of the initial geometrical imperfections is shown in Fig. 3.2. The imperfections arebased on values measured in steps of 50 mm and 200 mm in the transverse and longitudinaldirections respectively, taken from the specimen description [9]. In the plot the displacementshave been magni�ed by a factor of approximately twenty-�ve to make the de ection patternmore pronounced. From the plot, it is evident that there exists a large overall imperfectionof the panel, superimposed with smaller inter-frame imperfection patterns.

The welding residual stresses was established based on measured strains. Thus, the residualstress presented in Tab. 3.1 are actually calculated values derived from the measured strains.By observing the values, it seen that a large variation of the residual stress exist throughoutthe grillage.

3.2. De�nition of Laser Welded Panel 17

Figure 3.1: Geometry of laser welded panel. All dimensions shown are in mm.

Figure 3.2: Initial geometric imperfections of laser welded panel, Magni�cation factor 25.


In the experiments, the grillage was clamped at both ends and at the transverse framelocations along each side. However, with respect to the transverse frames, translation in thelongitudinal direction was allowed due to the bolted end plates and the guide system used.The grillage was exposed to an in-plane load, applied in the longitudinal direction and greatcare was taken in the experiment to avoid any moment resultant of the load.

Hence, in the calculations, a pure axial load can be assumed. Further, as a result of the thedoubler plates at the bays closest to the longitudinal ends of the panel, the ultimate strengthof the grillage is expected to be governed by the behavior of the six plate �elds A-F with theattached sti�eners S1-S8.

3.3 Summary of Performed Calculations and Methods

Calculations has been performed by the following people:

R. Damonte, Cetena, ItalyJ.M. Gordo, Instituto Superior T�ecnico, PortugalT. Hu and N.G. Pegg, Defense Research Establishment Atlantic, UKT. Yao, Hiroshima University, JapanU. R�ohr and B. Jackstell, Universit�at Rostock, GermanyS.-R. Cho, University of Ulsan, KoreaM.L. Kaminski, Nevesbu, HollandR.S. Dow, DRA Dunfermline, UKL.P. Nielsen and J. Juncher Jensen, Technical University of Denmark.

In the following a short description of each calculation is presented. The description is basedon the documentation received from the individual participants in the ISSC'97 committeeinvestigation. This material is more or less extensive which also is re ected in the following.The focus of the description is on the method used. Further, a graphical presentation of theobtained load-displacement curve(s) is presented. Finally, each calculation has been assigneda reference number for use in the overall comparison of the results given in Fig. 3.3.

To facilitate this comparison between the di�erent results the average longitudinal stress �and strain " have been non-dimensionalized by the average yield stress �y and yield strain"y respectively. These are found as

�y =Ap�y,p + 4As�y,s

Ap + 4As= 423:9 MPa and "y =

�yE

= 2048�

where the cross sectional areas for the plate Ap and the longitudinal sti�eners As are givenin Fig. 3.1 and an average yield stresses �y,p = 440 MPa and �y,s = 359 MPa for the plateand sti�ener respectively is used.

3.3. Summary of Performed Calculations and Methods 19

Damonte: (#1, 2)

FEM analyzes using MARC, release K6.2,utilizing both an updated Lagrangian fullNewton-Raphson, and an automatic load in-crementation procedure to solve the materialand geometric nonlinear response.The grillage within the two transverse framesT1 and T3 was modeled using a four-node,thick shell element utilizing bilinear interpo-lation of coordinates, displacements and ro-tations, and with three global displacements

and three global rotations as degrees of freedom at each node. In total the �niteelement model consisted of 1644 elements, 1682 nodal points, and 10092 active degreesof freedom. Clamped boundary condition of the transverse frame T1 and T3 wasassumed. The middle transverse frames T2 was supported for vertical de ection, butallowed to rotate. Further, a rigid I bar, positioned at the neutral axis of the section,was used to transfer the in-plane load to one edge of the model. The geometricalimperfections were approximated by a sinusoidal shape with the same maximum valueas measured in the experiment.Finally, the residual stresses were applied to the model in two di�erent ways. Inthe �rst (#1), the measured values from Tab. 3.1 was included directly as initialstresses. However, as these stresses are not self-equilibrating another calculation (#2)was carried out using a temperature distribution to create a self-equilibrating initialstress pattern closely matching the measured stresses.

Gordo: (#3, 4)

FEM analysis using PANFEM, version 2.PANFEM is a university developed nonlin-ear �nite element code for at plates and atsti�ened panels with initial imperfection ex-posed to lateral and/or in-plane loads (seee.g. Kmiecik [24]).The span of a single longitudinal sti�enerwith attached plating supported at the mid-dle by a transverse frame was modeled (#3).Thus, the model was two times the frame

spacing in length and one longitudinal sti�ener spacing in width. Symmetry conditionswere assumed at the top and edges of the model. The geometrical imperfections weremodeled by a sti�ener out-of-plane imperfection of 1 mm and a plate imperfection of2 mm. The residual stresses were not included in the model.In addition a simpli�ed beam-column approach by Gordo and Guedes Soares [16]including residual stresses (#4), was also used.


Hu and Pegg: (#5)

FEM analysis using ADINA and automaticload-displacement control to solve the geo-metric and material nonlinearities.The grillage within the transverse framesT1 and T3 was modeled using a four-node,quadrilateral, isoparametric shell elementwith a 2� 2� 2 integration order. The kine-matic assumption was large displacementand rotation, but small strain. The materialwas assumed to be bilinear elastic-perfectly-

plastic with a von Mises yield condition. Simply supported boundary conditionswere assumed at frame T1 and T3. Also, the ends of these sti�eners, along with thecenter T2 sti�ener, were allowed to rotate in the longitudinal plane. The geometricalimperfections were approximated by half sine curves between the transverse sti�enerswith an amplitude of 5 mm. The residual stresses were not included in the model.

Yao: (#6)

FEM analysis using ULSAS, including bothmaterial and geometrical nonlinearities.Half the panel was modeled using a four-nodeshell element for the plating and the webs ofthe longitudinals and beam-column elementsfor the anges of the longitudinal sti�enersand for the entire transverse frames. In to-tal the �nite element model had 1836 nodalpoints for shell elements and 180 nodal pointsfor the beam-column elements. The materialwas model by plastic ow theory considering the von Mises yield function as a plas-tic potential. Further, the material was assumed to follow the combined hardeninglaw, although elastic-perfectly-plastic material was assumed in the analysis. Lateralde ections along the loading edges and the end points of the transverse frames wereconstrained, and symmetry conditions were applied along the longitudinal center line.The geometrical imperfections were approximated by

w = A0 sin�x

`+

11Xm=1

Am sinm�x

`sin

�y

b

where ` = 2000 mm, b = 300 mm, and A0 = 0:1, A1 = 1:1165, A2 = 0:2233,A3 = 0:3304, A4 = 0:061, A5 = 0:1541, A6 = 0:0308, A7 = 0:0725, A8 = 0:0145,A9 = 0:0826, A10 = 0:0057 and A11 = 0:0303 in mm. Finally, the residual weldingstresses were included by increasing the compressive inherent strain incrementally inthe measured heat a�ected zone in the plates and sti�ener webs until these zonesreached yield in tension.


R�ohr and Jackstell: (#7)

FEM analysis using MARC, release K6.1,utilizing an updated Lagrangian approachwhere the Lagrangian frame of reference isrede�ned at the beginning of each load in-crement.The complete grillage was modeled using afour-node, isoparametric, bilinear shell ele-ment. In total the �nite element model con-sisted of 3660 elements, 3705 nodal points,and 18346 active degrees of freedom. Lateral

de ections along the loading edges and the end points of the transverse frames wereconstrained together with rotations of the end points of the transverse frames. Thematerial was assumed to follow the isotropic hardening law and the von Mises yieldcriteria. The geometrical imperfections were modeled as forced displacements in eachnodal point using interpolation in the measured imperfection pattern. Finally, theresidual stresses were applied as a set of self-equilibrating stresses, constant throughthe thickness and over the panel.

Cho: (#8)

FEM analysis using NISA.The whole grillage was modeled in the com-putation. However, to mimic the e�ects ofdoubler plates at end bays, the �rst 600 mmfrom the edge and inward of both end bayswere assumed to be rigid. Therefore, only atotal length of 2200 mm of the grillage wasassumed to be deformable. This part in-cludes two middle bays within the transverseframes T1 and T3. The full width of the gril-

lage was included in the model. The four-node, Mindlin-Reissner plate element wasadopted for the deformable part of the model. The structure was modeled with a totalof 1681 nodal points and 1640 elements. To include the initial geometric imperfections,care was taken while meshing the model, such that nodes which coincide with thelocations where the initial shape imperfections were measured were generated in theprocess. The initial geometric imperfections where then introduced to the model asthe initial vertical coordinates of the corresponding nodes. The residual stresses wereconsidered in the calculations by introducing the e�ective yield stress for each element.That is, if the residual stress at a speci�c location was e.g. compressive, then the yieldstress for the the speci�c element at that location was reduced by the amount of theresidual stress.


Kaminski: (#9, 10, 11)

Three di�erent FEM models have been used.The code CSA/GENSA, a companion toCSA/NASTRAN, was applied in two di�er-ent modes: Explicit (#9) and implicit (#10).The former is a time integration procedure,which can cope with hugely dynamic prob-lems and the latter is the usual, nonlinear,quasi-static analysis. The complete grillagewas modeled by 950 elements yielding 5400degrees of freedom. The loaded edges were

restrained from rotation and lateral translations, and further speci�ed to remainstraight during the loading. The geometrical imperfections were divided into globaland inter-frame imperfections, both of which with an assumed sinusoidal shape andwith amplitude of 5 mm and 1 mm, respectively. Finally, utilizing that the pre-stressload f�0

and thermal load f�T in the FEM formulation are introduced as

f�0=ZBT�0dV and f�T =

ZBTE��T dV

respectively (B is the strain-nodal displacement matrix), the welding residual stresseswere included through a self-equilibrating thermal loads given through �0 = E��T .The third analysis (#11) was done in ANSYS release 5.0, using a plastic, quadrupleshell element with six degrees of freedom at each node. The modeling of boundaryconditions, geometrical imperfections, and residual stresses was essentially the sameas used in the GENSA analyzes. The complete model has 2270 elements, 2011 nodalpoints, and 11576 degrees of freedom.

Dow: (#12, 13)

A beam-column approach (#12) (Dow et

al. [10]) was used before performing the ac-tual experiments using average imperfectionlevels not based on the measured imperfec-tions. Later, a detailed FEM analysis (#13)was made using the code ASAS-NL. The fullgrillage was modeled using a eight-node, de-generate, isoparametric shell elements with a2� 2 Gauss integration order in the plane ofthe shell and a �ve-point Newton Coates

integration through the thickness. Plating, longitudinal girders, and transverse frameswere all modeled using shell elements. The �nite element model consisted of 920elements, 2905 nodal points, and 17430 degrees of freedom. Material properties,boundary conditions and both inter-frame and overall components of the initial, asmeasured, distortions were modeled accurately. Finally, the welding induced residualstresses were included as a self equilibrating set of initial stresses.


Nielsen and Juncher Jensen: (#14, 15)

FEM analysis using ANSYS release 5.3, uni-versity edition, utilizing the full Newton-Raphson equilibrium iteration scheme andbisection to solve the geometric and materialnonlinearities (#14).The complete grillage was modeled using afour-node, bilinear, isoparametric, thin tomoderately thick shell element with six de-grees of freedom at each node. The elementincorporates von Mises isotropic hardening

plasticity and has full through-the-thickness integration. In total the �nite elementmodel consisted of 8400 elements and 8537 nodes with an estimated 50088 activedegrees of freedom. The three transverse frames T1, T2, and T3 were clamped atthe boundaries except for free motion in the loading direction. The doubler plateswere assumed rigid and thus replaced with clamped boundary conditions yielding atotal length equal to 2700 mm of the model. Initial geometrical imperfections wereincluded at each nodal point using interpolation in the measured imperfection pattern,but residual stresses were neglected.Later, also a beam-column approach (#15) was applied to one longitudinal sti�ener inbetween two transverse frames, thus having a total length of 1000 mm. The boundaryconditions were assumed to be clamped at both ends and symmetry at the plate edges.The residual stresses were ignored and the geometrical imperfections where modeledby a prescribed end-rotation.

To summarize, a total of �fteen distinct calculation have been performed on the laser weldedpanel by nine contributors. Of these, most are nonlinear �nite element method calculations,but also beam-column approaches and to some extent idealized structural unit method cal-culations have been submitted. In numbers, this totals to

Method Total Calculation reference number(s)Finite element 11 1, 2, 3, 5, 7, 8, 9, 10, 11, 13, 14Idealized structural unit (1) 6Beam-column 3 4, 12, 15

Whether calculation number six actually can be characterized as being based on the idealizedstructural unit method is somewhat debatable. The fact that it uses special beam-columnelements to model the sti�eners in combination with standard shell elements for the platingof the grillage, de�nitely distinguish it from the other traditional �nite element modelspresented. In the author's opinion, this feature makes it an idealized structural unit methodbased solution, but it is recognized that this conclusion may be contested. Nevertheless, forcomparative purposes this will be the interpretation in the following sections.


3.4 Basis for Comparison of Results

Obviously, for design purposes of a single sti�ened panel, the ability to accurately predictthe ultimate load level of the panel is very interesting, whereas the post-ultimate behavioris of less importance when only one panel is considered. However, when the panel becomespart of a larger structure, the post-ultimate behavior becomes equally as important as theultimate load level.

In a large structure it must be expected that some members will exceed their ultimatecapacity before the entire structure reaches the ultimate load level. Hence, when evaluatingthe ultimate capacity of the entire structure, these members will contribute with their post-ultimate response. Consequently, to accurately predict the ultimate strength of e.g. a hullgirder made up of numerous such sti�ened panels, it is equally important to know both theultimate and post-ultimate strength of each panel.

Therefore, in the following discussion of the presented results, the focus will be on these twokey features of the solutions, i.e. the ability to match the experimental ultimate load level,and predict of the collapse behavior. The collapse behavior of the laser welded panel can besummarized in four tempi:

1. At a relative low load level (�=�y � 0:25) yielding initiates in the longitudinals. Thisbehavior is caused by a combination of two factors: One, the yield stress for thelongitudinals is lower than for the plate and two, the shape of the initial imperfectionpattern.

2. Upon continued up-loading, buckling �rst occurs in the plate �elds with unsupportedboundaries. Then, at a higher load level, buckling of the interior plate �elds A-F takesplace.

3. At the ultimate load carrying capacity, signi�cant plastic deformations have takenplace in both the plates and the longitudinals. Up to this point the behavior is mainlygoverned by inter-frame buckling, but in the post-ultimate region overall bucklingbecomes the dominant failure mechanism with a signi�cant torsion of the sti�ener T2.

4. Just after the ultimate load, a rapid decrease in sti�ness appears caused by the tippingfailure of the longitudinals. Thereafter, a slower rate of decrease in sti�ness is observed,mainly related to an increased yielding in the panel.

Regarding the ultimate capacity of the panel, the experiments yielded a collapse load equalto sixty percent of the squash load Py (average yield stress). This information together withthe described collapse behavior constitutes the basis for the following discussion. It should,however, be mentioned that the input data, as given in Fig. 3.1 and Tab. 3.1, may notnecessary re ect all pertinent data for the panel tested which may cause some methods tobeen judged too harshly in the comparison.

3.5. Discussion of the Results 25

Figure 3.3: Calculated and measured non-dimensionalized, axial stress-strainrelationships for the laser welded panel.

3.5 Discussion of the Results

In Fig. 3.3 the calculated load-displacement curves for all �fteen solutions is presented to-gether with the experimental result. Further, Tab. 3.2 lists the ultimate capacity �c predictedby the same �fteen calculations. Also listed in the table is the relative deviation from theexperimental data of both the non-dimensionalized axial stress �c=�y and strain "c="y atcollapse.

From the graphs in Fig. 3.3 it is seen that nearly all of the results re ect the prescribedcollapse behavior. Calculation eight stops short after the ultimate load is reached, and thuscannot describe the post-ultimate behavior. Other calculations, such as two, �ve, eleven,twelve, and �fteen displays a less signi�cant drop in sti�ness just after the ultimate capacity,but still captures the overall collapse behavior reasonably well. Also, both calculation twoand eleven seems to allow for a quite signi�cant straining at the ultimate load level beforeany drop in sti�ness is observed, which is in contrast with the experimental �ndings.


Table 3.2: Ultimate capacity predictions.

Calculation Collapse Normalized Relative Beam- Finite Idealizedreference load �c stress-strain deviation column element structuralnumber [MPa] "c="y �c=�y �" �� method method unit method

Experiment 254 0.65 0.60 0% 0%#1 346 0.93 0.82 42% 35%

p#2 311 1.18 0.73 81% 23%

p#3 300 0.87 0.71 33% 18%

p#4 273 0.96 0.64 47% 8%

p#5 234 0.71 0.55 9% -8%

p#6 233 0.71 0.55 8% -8%

p#7 311 0.76 0.73 17% 23%

p#8 309 0.82 0.73 26% 22%

p#9 272 0.69 0.64 6% 7%

p#10 256 0.73 0.60 12% 1%

p#11 344 0.96 0.81 46% 35%

p#12 259 0.77 0.61 18% 2%

p#13 240 0.68 0.57 5% -5%

p#14 278 0.72 0.66 11% 9%

p#15 250 0.85 0.59 30% -2%

p

Whereas the load-displacement behavior is predicted well in most of the calculations, theultimate load levels obtained di�er rather signi�cantly, being between �fty-�ve and eightypercent of the average yield stress (the squash load). This can also be seen from the relativedeviations in the non-dimensionalized stress at collapse �� as listed in Tab. 3.2, which isbetween eight and thirty-�ve percent below and above the experimental result respectively.

Most of the �nite element method calculations (one, two, three, seven, eight, and eleven)predict an ultimate load �fteen to thirty-�ve percent larger than the measured. Only the�nite element method calculations six, nine, ten, thirteen and fourteen yield predictionswithin plus or minus ten percent of the measured value (cf. Tab. 3.2). Of these, calculationnumber �ve allows for rotation of the transverse sti�eners T1, T2, and T3 and should thereforebe expected to give a low prediction.

These di�erences cannot be explained by gross errors in the modeling as all of the calculationshave been immaculately done within the chosen assumptions for each method. However, thediscrepancies may be due to inaccurate modeling of the tripping failure mode in the �niteelement models. On the other hand, the three calculations (four, twelve, and �fteen) basedon a beam-column approach, are all observed to give very reasonable results within a fewpercent of the measured ultimate load. This even though number �fteen does not accountsfor the tripping failure mode in its formulation.

3.6. Summary 27

Also, some of the �nite element method calculations have been performed both with andwithout geometrical imperfections and/or welding residual stresses. In comparison, it seemsthat the residual stresses only marginally in uences the load-displacement curve, illustratedby �nite element method calculations three, �ve, and fourteen which do not include theresidual stresses, but still predicts both ultimate and post-ultimate behavior just as good asthe other �nite element method calculations.

3.6 Summary

Except for solution thirteen by Dow who also performed the experiment on the panel, and�fteen which was added later on, all other calculations were made without prior knowledgeof the experimental result. Thus, it can be assumed that the presented solutions representsa fair description of the di�erent methods accuracy as a design tool, which is perfect for thetask at hand.

To restate the object of this investigation in relation to the current project, the interest ison selecting a method for rapid and accurate prediction of the ultimate strength of a hullgirder. A pre-selection of the beam-column approach was made in previous chapter, andthe purpose of the present chapter was to seek re-con�rmation of this choice. However,in choosing the beam-column method, the two other possibilities, being the �nite elementmethod and the idealized structural unit method were discarded, both mainly on the groundof poor performance in speed. It is therefore interesting to see if this conclusion still holdsin light of the new evidence provided by the investigation of the laser welded panel.

3.6.1 The Finite Element Methods

Based on the �ndings in this test of the laser welded panel, it can be concluded that it iswithin the framework of the �nite element method, that the most signi�cant discrepanciesfrom the experimental results arises. Most of these calculation predict an ultimate load �fteento thirty percent larger than what was measured in the experiment. They do, however, allpredict the collapse behavior reasonably well, although there is a tendency for some to missthe rapid loss of sti�ness occurring just after collapse. A plausible explanation for this couldbe that the e�ect of tripping failure of the longitudinal sti�eners is underestimated in thosemodels, must likely due to modeling of those members.

Thus, it seems that if this method is chosen, great care has to be taken in modeling to accountfor tripping failure. This could possibly be done by introducing a �ner mesh (i.e. addingmore elements) to describe the longitudinal sti�eners, as there seem to be a tendency ofhigher accuracy of the models with the largest number of elements { or possibly by takinggreat care in selecting suitable elements with a preference toward higher order elements(i.e. using more nodes) to model the sti�eners.


Figure 3.4: Calculated and measured non-dimensionalized, axial stress-strainrelationships for the laser welded panel.Beam-column and idealized structural unit methods only.

However, this will inevitable slow down the method even further, as either option will addmore degrees of freedom to solve for, disqualifying it even more for application in the currentstudy. Thus, with respect to the �nite element method the conclusion remains unchanged:It is not suitable for the current project.

3.6.2 The Beam-Column and Idealized Structural Unit Methods

With respect to the beam-column and idealized structural unit methods, then a plot ofthe load-displacement curves for only those methods is presented in Fig. 3.4 to clarify theconclusions. Even though only four calculations �ts in this category, and even worse onlyone of those is an idealized structural unit method, they will for lack of other informationbe considered representative in the following.

3.6. Summary 29

From the graphs it is observed that all four calculations predicts the ultimate capacity ofthe panel very well, as the discrepancies is for all of the calculation less than ten percentof the measured ultimate load. Concerning the path, then the idealized structural unitmethod slightly underestimates the ultimate capacity but captures the post-collapse behaviormuch better than all of the beam-column solutions. On the other hand, the beam-columnsolutions seems to estimate the ultimate loading capacity of the panel better, but have apoor description of the following unloading behavior in the post-collapse region.

The reason for this is most likely to be found in the mechanism used to model the post-ultimate structural behavior. Calculation four and twelve take tripping failure of the sti�enerinto account, whereas calculation �fteen rely on a simple plastic hinge mechanism with noinclusion of possible tripping, i.e. the cross section remains in its initial prismatic form.Obviously, this makes a fundamental di�erence which also is clearly visible in the plot inFig. 3.4 where calculations four and twelve shows a much more rapid loss of sti�ness inthe post-collapse region of the load-displacement response than is predicted by calculation�fteen.

Why four and twelve still do not come as close the the experiments as six does, even thoughthey include tripping may be caused by inherent assumptions in the beam-column approachwhere the overall de ection pattern of the sti�ened plate �eld is neglected. That is, the beam-column calculations only model one sti�ener in between two transverse sti�eners, i.e. onebay is modeled only, whereas the idealized structural unit method models the entire grillage.Still, the overall performance of both the idealized structural unit method and the beam-column approach appears quite comparable.

In conclusion therefore, there do not seem to be any signi�cant reason, based on the twocriterions of ability to predict collapse load and post-collapse behavior, that really distinguishone method form the other. Selecting between the two di�erent methods is therefore hardto do even with the extra evidence from the laser welded panel test. It still falls back to thequestion of speed.

Regarding the speed of the remaining two di�erent methods, it is the author's opinion, thatthe beam-column can be made { and in general will be { much faster than the idealizedstructural unit method. This mainly due to the very simple way in which the global systemanalysis of e.g. an entire hull cross section, can be made based on the beam-column approach.

Therefore, having exhausted the available information on which to make a choice of method,it is the �nal subjective conclusion to base the current project on a beam-column approach.

Chapter 4

The Beam-Column Method

4.1 Introduction

Based on the previously reviewed comparative studies of the three di�erent methods forthe evaluation of the ultimate capacity of the hull girder, the beam-column approach wasselected. The aim is then to make these beam-columns mimic the behavior of the real struc-ture as closely as possible. However, to apply the beam-column method, a few assumptionsmust be made.

First of all, it is assumed that collapse occurs locally between two adjacent frames. Other-wise, the beam-columns would not behave like columns, since there would be what couldbe considered as supports along the length of the beam-column at each intermediate frameposition. It is therefore a necessary assumption. The implications of this assumption is thatthe overall deformation of the hull girder is neglected. For most ship structures this is anexcellent simpli�cation, as failure of the midship section is the predominant event { obviouslydue to the occurrence of the maximum bending moment at this location. In the case of adamaged vessel, it may be some other location along the hull girder that will be the weakestdue to e.g. extensive fracture damage from a collision and/or grounding. Nevertheless, theassumption of failure in between two adjacent frames will still be an acceptable simpli�cationto most real scenarios.

The method of overall system analysis is then to divide the total hull cross section intoa number of beam-columns each consisting of a sti�ener and a part of the hull plating asillustrated in Fig. 4.1. This approach leads to the second assumption: Each beam-column isassumed to react independently of the adjacent beam-columns.

The downside to this assumption is that the overall structural behavior of e.g. a large sti�enedpanel between two stringers or girders, will be neglected. That is, by subdividing this�ctitious panel into a number of beam-columns, all reacting independently of each other,

31

32 Chapter 4. The Beam-Column Method

Figure 4.1: De�nition sketch for the beam-column approach.

e.g. the overall buckling collapse of the panel will be missed by the method. On the otherhand, the gain of the assumption is that the structural behavior of the beam-column becomesonly a function of how the ends of the beam-column are displaced and/or rotated.

The bene�t of this fact is quite immense when the system analysis of the entire cross sectionis to be performed. A tremendous gain in speed is achieved and the iteration for the ultimatecapacity can be performed simply and very rapidly. As overall performance of the procedureis one of the key concerns of the project, the bene�ts of this assumption are too good tomiss. Moreover, from the benchmarks previously reviewed, it does not seem to be the overallcollapse mode of a sti�ened panel that de�nes its ultimate capacity. Of course, by neglectingit even though it apparently wont in uence the prediction of the ultimate capacity, an errormay still be introduced in the post-ultimate response. However, in light of the bene�tsobtainable, it seems fair to neglect this slight error and proceed with the assumption as partof the scheme presently being formulated.

Thus, to recap: At this point two fundamental assumptions has been made, namely

1. Collapse occurs locally between two adjacent frames.

2. Each beam-column is assumed to react independently of the adjacent beam-columns.

Based on theses two assumptions the beam-column approach can easily be applied to a hullcross section, given that the load-displacement response of any beam-column is known asa function of how its ends are displaced and/or rotated, and of course its geometric andmaterial properties as well as the initial state of the members making up the beam-column.

4.2. Load Description 33

Figure 4.2: Load de�nition sketch for the midship cross section.

At this point it would be premature to consider how this load-displacement response can beestablished. The methods for doing this will be highly dependent on what the actual loadingcondition is and how this is selected to be transfered to the individual beam-column. Thus,at present it will be prudent to press on with a description of the loading.

4.2 Load Description

The object of investigation is a general loading condition consisting of moment, shear, andtorque as show in Fig. 4.2. That is, in the present contents, loading is understood to be theglobal sectional forces acting on the cross section being investigated. Further, it is noted thatwith the present sign convention shown in Fig. 4.2, the hogging condition will be caused bypositive horizontal moments and thus, the sagging condition by negative horizontal moments.

However, before continuing this discussion of the loading, it is necessary �rst to considerwhat type of scalable quantities the ultimate capacity are to be expressed in terms of. Thatis, which of the �ve load components, if not all, are to be varied until the ultimate capacityis reached?

First of all, it is important to recognize that there are in fact two di�erent types of ques-tions being asked of the procedure depending on the use. When the research objective wasformulated, it was simply stated that a rapid procedure for the evaluation of the ultimatecapacity was to be developed for use in both emergency response e�orts, and for input in aprobabilistic reliability analysis of the vessel. At that point, no concern was paid to whetherthese two di�erent usages would actually be in contradiction or if it posed the exact same


requirements to the procedure. A clari�cation of this question will therefore be sought inthe following.

Considering �rst the probabilistic reliability analysis, then in this framework what is basicallywanted, is a full limit state description of the cross section. Since there are a total of �vedi�erent load components, this is actually a limit state surface in a �ve-dimensional space.To establish this would require that the ultimate capacity for each load component alonewas to be established �rst, one-by-one. Then, at whatever discreteness level chosen, a fullvariation of all �ve component within there extremes as determined in the �rst step, wouldhave to be performed. Needless to say, this will be extremely laborious to do and would ofcourse, require an immense amount of computational work.

On the other hand, if the scope is the emergency response then the situation is totallydi�erent. First of all, in this case there will be an actual vessel in an actual conditionwhich will result in a base set of loads acting on the cross section. In the reliability analysisframework, this was not interesting as there in essence is no initial load condition in thisframework. What is being done, to put it very simply, is to match a huge number of possibleload conditions with the total limit state surface and then determining whether it is a safecondition or not. Thus, there is no single set of initial loads where the analysis is startedfrom. This is however, the exact case in the emergency response scenario. The problem hereis, given this set of initial loads, how much more (hopefully) can the cross section cope withbefore the ultimate capacity is reached.

How these initial loads are determined is beyond the scope of the present research. Theyare of course induced by the exterior forces acting on the vessel arising from the distributedhull weight, cargo weight, buoyancy, and wave environment in the case of an intact vessela oat. If the scenario involves e.g. grounding and/or collision, then some outer mechanicsand/or dynamics will further add to these loads. Nevertheless, standard textbooks1 on shipscantling covers this subject very well. Thus, for the present investigation the initial globalsectional loads acting on the hull cross section being investigated, are assumed to be knownquantities.

What is interesting in the emergency response case is, that it will probably be of greaterbene�t to have a fast method to evaluate the ultimate capacity based on a scaling of some orall of the initial load components, than a somewhat slower scheme that evaluates the totallimit state surface. The reasoning behind this statement would be, that given a vessel inan emergency scenario being e.g. a grounding, it may be fair to say that to investigate thee�ects of e.g. falling tide given the same sea state, the in uence on the sectional forces wouldbe comparable with a simple scaling of the initial condition.

1Excellent books on this subject are e.g. Terndrup Pedersen and Juncher Jensen [38, 39, 40] (in Danish)or Hughes [20]


4.2.1 Load Interpretation

Entertaining this idea of scaling the initial loading condition makes it interesting if the loadson the cross section perhaps may be arranged in a way that will allow for a simple scalingand thus make the computational work easier. Looking at the �ve load components it isobserved that they can basically be replaced with two equivalent loads in the form of adirect and a shear stress distribution. The two bending moments will result in one directstress distribution or alternatively one asymmetrical bending moment. Further, the twoshear forces will together result in one shear stress distribution. Regarding the torque, thenthis can be split into two parts: The St. Venant and the warping torsional moment. Both ofthese two moments will produce shear stresses in the structure (see Fig. 4.3), but only thewarping torsional moment will cause direct stress in the structure. However, for closed-boxcross sections like tankers, the warping part of the total torsional moment is small comparedwith the St. Venant part. Thus, the warping torsional moment can reasonably be considerednegligible for those types of structures2. Hence, only St. Venant torsion will be include inthe present formulation. This then means, that only shear stresses will be produced by thetorque together with the two shear forces.

Drawing on general knowledge of structural analysis of typical ship cross sections, it is knownthat regarding the shear stress distribution, by far the major part of these stresses are actingon the plating of the hull. That is, the shear stresses in the sti�eners are small compared tothe shear stresses in the plating as illustrated in Fig. 4.3. Hence, a fair assumption wouldbe that the shear stress in the sti�eners, as a �rst approximation, may be neglected. If itis further assumed that the shear forces and the torque, all are constant along the part ofthe vessel that is to be investigated (i.e. in between two frames), and that the shear stress isconstant between two adjacent sti�eners, then the shear stress distribution along an isolatedplate is also constant.

Such a situation will be ideal to have when the load-displacement response for a beam-columnis to be established. The implication will be that all the e�ects of the shear forces and thetorque can be transfered to the plate part of the beam-column only. Thus, if a solution to thein-plane compressive loading of the plate given a constant shear level can be established, thenthe compressive part of the entire beam-column response can be approximated by simplebeam theory. This because the beam-column thus in essence just is a regular beam madeup of two di�erent \materials" each having di�erent load-displacement behavior.

There are however problems with this assumption. A constant shear stress distribution alongthe length of the vessel implies a linear distribution of the bending moment. Nevertheless,the advantages gained by adopting this assumption are so great that it will be worthwhileto neglect this variation, and thus take the ultimate bending moment as the mean value ofthe bending moment along the particular length of the vessel being investigated.

2It is noted that this assumption is inaccurate in the case of open cross sections like those found incontainer vessels, etc. Here the warping torsional moment can be the dominant part of the torque.


Figure 4.3: Shear stress distribution in a typical tanker structure exposed to a unit of:Vertical and horizontal shear force, and St. Venant and warping torsionalmoment.


4.2.2 Load Modeling

Considering the originally stated objective of overall speed of the procedure to be developed,the �ndings in the previous section strongly advocate for the simple approach to the loadmodeling. That is, an approach where the �ve individual load components are transformedinto a shear stress distribution accounting for the shear forces and the torque, and a singlemoment acting about an asymmetrical axis. For a �xed level of shear stress this asymmetricalaxis could then be rotated full circle in appropriate steps, yielding the ultimate capacity ofany moment combination at that �xed level of shear forces and torque.

However, before committing to this scheme it needs to be considered what e�ects this ap-proach will have on the applicability of the procedure in the framework of probabilisticreliability analysis. Earlier considerations on this topic lead to the conclusion that what wasneeded here was really a full limit state surface spanning over all load components. If theabove mentioned scheme of load transformation is adopted, this limit state surface can stillbe produced by the procedure, although less gracefully than it otherwise could be done.

Basically, it would entail the establishment of a separate three-dimensional limit state surfacein the three shear stress creating forces, i.e. the horizontal and vertical shear force andthe torsional moment. For any point within the safe domain de�ned by this limit statesurface, a corresponding constant shear stress distribution will exist which is safe, i.e. thestructure can still withstand more loading which only can be in the form of a combination ofhorizontal and vertical bending moments, as these are the only remaining load components.Thus, at this point the proposed procedure can then be applied to establish exactly howmuch more load can be applied until the ultimate capacity is reached for a given numberof possible combinations of the two bending moments. This approach can then be appliedto a given number of the possible safe shear load conditions, and thereby allow for the full,�ve-dimensional limit state surface to be established for the cross section.

On the other hand, if the shear stress response is assumed linear and if the correlationbetween moment and shear force is neglected, then the following approach could also beapplied. Given the shear stress distributions �Qy , �Qz , and �Mx for a horizontal shear force,a vertical shear force, and a torsional moment respectively, all of unit magnitude, then theprinciple of superposition could be used to represent any combination of these shear stresscausing loads in the form of the resulting shear stress distribution � . Actually, even a linearelastic, ideal plastic behavior of the the shear stress could easily be incorporated into thisscheme.

This would de�nitely be the most open way to select for representation of the shear stresscausing loads, as this will allow for any type of scaling. That is, one could e.g. choose to letthe scaling be uniform for all three components, or keep one constant and scale the remainingtwo components by di�erent rations, etc. The possibilities are numerous and can thus mostlikely be made to �t the requirements of any end user.

Hence, as this approach �ts perfectly with the initially proposed procedure for �nding theultimate capacity based on a �xed level of shear stress and a single moment acting about an


asymmetrical axis, then this is the idea that will be adopted for implementation. This willthen be done in a way where the user will de�ne three relational scaling factors for each ofthe three shear stress distributions, i.e. Kx, Ky, Kz, and the code will then vary the resultingsuperposed shear stress distribution between zero and yield by a scaling factor K. That is,the shear stress distribution is scaled as

� = K�Ky�Qy +Kz�Qz +Kx�Mx

�(4.1)

Thus, the method for the overall system analysis is to apply a single moment acting aboutan asymmetrical axis and a �xed level of shear stress � de�ned as given in Eq. (4.1). Theasymmetrical axis is then assumed rotated full circle in appropriate steps, thus yielding theultimate capacity of any moment combination at that �xed level of shear forces and torque.It is noted that by selecting this formulation for shear inclusion, the dependency betweenbending moments and shear forces are e�ectively neglected as there is no redistribution ofthe shear stresses in the present formulation. That is, the relation between e.g. the bendingmoment about the horizontal axis My and the vertical shear force Qz is not included in thepresent load model. Further, the applied shear stress distribution � given by Eq. (4.1) isassumed linear elastic, ideal plastic, such that the shear stress in any individual plate of thehull cross section cannot exceed the yield shear stress �y de�ned as the mean shear stress atplate collapse �sc (see Eq. (5.54), Section 5.6.7) for each plate.

4.2.3 Accidental Loads & Probabilistic Reliability Analysis

Before leaving this section on the load modeling and continuing to the overall system analysisof the entire hull cross section, a few comments on the aspects of accidental loads would beappropriate to conclude the loading topic. Further, a few observations about the probabilisticreliability analysis will be presented.

As stated earlier the task of determining both the initial and the accidental loads is consideredbeyond the scope of the present research. However, two methods of obtaining informationfor the collision scenario can be suggested. First of all, there is the very obvious method ofconsulting actual collision damage reports. Here information about the location and extentof the structural damages caused by the collision are available for consideration. Based onthis information the appropriate members (in the form of beam-columns) can be removedfrom the hull cross section to simulate the damage and thus, an evaluation of the ultimatecapacity of the damaged vessel can be performed.

An alternative approach would be to try to estimate the potential damages arising from a�ctitious collision. A very elaborate scheme for doing exactly this, has been published byTerndrup Pedersen et al. [37, 36]. Here a probabilistic analysis of the risk of collision is takenas basis for an overlying evaluation of the consequences of such a collision in the form ofprobable damage extent and location on the struck vessel. The method is applied to Roll-on-Roll-o� (RoRo) vessels traveling on a given route with crossing tra�c of general merchant

4.3. System Analysis 39

vessels etc. Such a framework would obviously also be applicable to obtain information oncollision damages.

Regarding the grounding scenario, then again there is the obvious means of getting informa-tion by consulting actual damage reports. Alternatively, analytical methods for evaluatingthe extent of grounding damages, e.g. as those suggested by Cerup Simonsen [43], may beemployed to obtain the damage condition of the structure.

Finally, on the topic of probabilistic reliability analysis two possible usages of the presentmodel can be thought of: One, direct implementation for the evaluation of the hull capacityor two, basis for calibration of even more simpli�ed models. Along these lines the directimplementation scheme was used by e.g. Melchior Hansen [17] when he performed a meanout-crossing rate analysis to evaluate the reliability of a vessel, in the combined loadingof a horizontal bending moment and a vertical shear force. Here, the ultimate capacitywas determined by a beam-column approach. Further, Friis Hansen [18] applied MelchiorHansen's beam-column approach to, in a probabilistic sense, calibrate a simple plastic hingemodel for the ultimate strength based on the model correction factor method.

4.3 System Analysis

As described in the introduction the basic idea is to consider each sti�ened panel in the crosssection as a beam-column. Moreover, it was assumed that each beam-column reacts inde-pendently of the adjacent beam-columns. The beauty of this scheme is that the structuralbehavior of the beam-column then only is a function of how the ends of the beam-columnare displaced and/or rotated. Thus, when the load-displacement pattern is known for eachbeam-column in the whole hull cross section, i.e. Pi(�i; �) is known, the system analysis caneasily be performed.

Following the assumption that collapse occurs between two adjacent frames and furtherassuming that the frames do not deform (i.e. plane sections remain plane), the systemanalysis is performed by application of the forced curvature principle. Since plane sectionsremain plane, the curvature � will be proportional to the angle � the strain-plane is rotatedabout the instantaneous neutral axis of the cross section de�ned as being where the strainis zero. This angle � will again be equivalent with a bending moment about the same axis,which is obtained by summation of the end forces Pi(�i; �) times the leverage �i. Thus, byvarying the rotation angle of the strain-plane, the cross section is e�ectively loaded with aset of di�erent bending moments. That is, by forcing a rotation of the strain-plane, i.e. aspeci�c curvature, any bending moment can be reproduced, explaining the name { forcedcurvature principle.


Figure 4.4: De�nition sketch for the forced curvature principlein the case of pure horizontal bending.

4.3.1 Pure Horizontal Bending

Application of the principle is easiest explained for the simple case of pure horizontal bendingwith a constant shear stress level. Assuming an intact hull, the cross section will be sym-metric about the vertical axis, and consequently the instantaneous neutral axis about whichthe strain-plane rotates, will be horizontal. In this case, the position of the strain-plane canbe described by only two parameters (See Fig. 4.4):

1. The translation of the origin of the strain-plane, i.e. the position �INA of the instanta-neous neutral axis, and

2. the angle of rotation � of the strain-plane.

It is observed that since the instantaneous neutral axis is horizontal, the position of eachbeam-column can be described by just one coordinate, namely zi. Thus, following thenotation on Fig. 4.4 the distance for the instantaneous neutral axis to the i th beam-columnbecomes �i = zi � �INA and the total end displacement for the same beam-column �i = 2��iwhere the times two is because both ends of the beam-column are being displaced.


Further, as the translation �INA of the origin of the strain-plane is identical to the positionof the instantaneous neutral axis it can be found by demanding that the summation of theend forces Pi(�i; �) of all the, say k, beam-columns equals zero, i.e.

N(�) =kXi=1

Pi (�i; �) = 0 (4.2)

However, due to the nonlinear response of the beam-columns, the position of the instanta-neous neutral axis will be dependent on the prescribed rotation � of the strain-plane. Hence,the force equilibrium becomes

N(�) =kXi=1

Pi (2�[zi � �INA(�)]; �) = 0 (4.3)

from which the position �INA(�) of the instantaneous neutral axis can be determined itera-tively for any prescribed curvature.

When the position of this axis is known, the leverage �i to each of the beam-columns is alsoknown. Thus, a relation between the total moment response MINA of the entire cross sectionabout the instantaneous neutral axis and the rotation angle � can be established as

MINA(�) =kXi=1

fPi (�i; �) �i +Mi(�)g (4.4)

i.e. as the sum over all beam-columns of the axial response times the leverage plus the localclamped end moment. With respect to the later, i.e. the local clamped end moment, thenthis will amount to an insigni�cant contribution compared with the moment produced bythe force times leverage for all the beam-columns. Of course neglecting this contribution willcause the �nal total moment to be slightly less than otherwise, but in view of the overalllevel of simpli�cation inherent to the beam-column approach, this is a quite acceptableapproximation. Hence, the total moment response about the instantaneous neutral axis istaken to be

MINA(�) =kXi=1

Pi (�i; �) �i (4.5)

for a given rotation � of the strain-plane.


Figure 4.5: De�nition of the local beam-column reference system.

4.3.2 Beam-Column End Displacement and Rotation

Up until this point, the end rotation ' of the beam-column has simply been taken as equalto the rotation � of the strain-plane. This is of course true, but only if the load-displacementresponse of the beam-column is known based on a reference system which is coinciding withthe global one shown in Fig. 4.4. However, with respect to the structural description ofa each beam-column, it would be much more convenient if this was established in a localreference system, natural to the beam-column, as shown in Fig. 4.5. This local referencesystem would however, only in special cases be coinciding with the global reference system forthe entire hull cross section, due the di�erent orientation of the beam-columns. Therefore, adescription is needed of how the end rotation ' of the beam-column in local reference systemrelates to the rotation � of the stress plane.

Obtaining this relation will �rstly require de�nition of the beam-columns orientation withrespect to the global reference system. Fig. 4.6 shows an arbitrary oriented beam-columnlocated in the global reference system. From the �gure, it is noted that the local referencesystem is rotated an angle �/2 about the local y-axis, compared to the global reference system.This is for reasons pertinent to conveniently describing the beam-column behavior, whichwill be dealt with in the next chapter. This does not have any in uence on the end rotation' of the beam-column, as the orientation of both the global and the local y-axis is una�ectedby this, i.e. are identical.

Therefore, introducing the angle between the global y-axis and the local y-axis, as shownin the �gure, will be su�cient to uniquely de�ne the orientation of the beam-column. If thisangle is di�erent form zero, then the beam-column will be exposed to an end rotation aboutboth the local y- and z-axis. However, to simplify the description of the beam-columnsstructural behavior, it is assumed that only an end rotation 'e� about the local y-axis isacting on the beam-column.


Figure 4.6: Sketch of the transformation between the local beam-column referencesystem, the global coordinate system, and the current strain-plane.

Thus, given the angle de�ning the orientation of the beam-column, the relation betweenthis e�ective end rotation and the rotation � of the strain-plane becomes

'e� = � cos( ) (4.6)

which easily can be seen from the de�nition sketch in Fig. 4.6. Hence, as the angle isa constant, then the axial response of the beam-column, in relation to the overall systemanalysis, still remains only a function of the end displacement �i and the rotation � of thestrain-plane. Moreover, as the end displacement depends only on the �xed geometry of thehull cross section and the rotation of the strain-plane, then in the context of the global systemanalysis, the beam-column response is essentially just a function of one single parameter,namely the rotation � of the strain-plane.

4.3.3 General Load Condition

The expansion of the forced curvature principle to a asymmetrical instantaneous neutral axiswhich then inherently will be able to handle damaged cross sections, i.e. no requirement onany symmetry about the vertical axis, is now a straight forward process.

First of all, it is now necessary to use both coordinates to describe the position of eachindividual beam-column, namely (yi; zi). Moreover, by still requiring that plane sectionsremain plane, the asymmetrical instantaneous neutral axis can be described by only oneextra parameter being the angle � between it and the horizontal global y-axis as shown in


Figure 4.7: De�nition sketch for the forced curvature principleextended to asymmetrical bending.

Figure 4.8: Plane sketch of the the forced curvature principlewith asymmetrical bending.


Fig. 4.7. Combined with the original parameter used, i.e. the distance �INA from the baselineto the instantaneous neutral axis, the position is fully described. However, a rede�nitionof �INA is required to avoid problems when the instantaneous neutral axis is parallel to thevertical z-axis, i.e. for � = �/2.

If the previously used de�nition of �INA in the case of pure horizontal bending was used,it would be impossible to determine what �INA would be for � = �/2. Either, it would bean in�nity of possible solutions in the case where the global vertical z-axis would coincidewith the instantaneous neutral axis, or there would be no solution if the two axis whereparallel at a distance. To overcome this, the de�nition of the distance �INA is changed tobe the perpendicular distance from the center of the baseline (i.e. the origin of the globalcoordinate system) to the instantaneous neutral axis as shown in Fig. 4.8. By use of planegeometry the distance from the instantaneous neutral axis to the i th beam-column, i.e. theleverage, then becomes3

�i = ��INA � y sin� + z cos� (4.7)

where the formulation used to obtain this expression is such that it insures that the leverage�i will be positive in the tension zone, and negative in the compression zone of the hull crosssection. Thus, given a rotation � of the strain-plane about the instantaneous neutral axis,the resulting total end displacement �i for the i

th beam-column can be obtained from theleverage in exactly the same way as in the simple case of pure horizontal bending, i.e. as�i = 2��i where, again, the times two is because both ends of the beam-column are beingdisplaced.

With respect to the e�ective end rotation 'e� of the beam-column, then in this case it simplybecomes

'e� = � cos( � �) (4.8)

Thus, with the total end displacement �i and the e�ective end rotation 'e� established, theresponse of the beam-column is obtainable, as this is a function of only those two parameters.The rest of the system analysis therefore follows the exact same procedure as in the case ofpure bending. That is, the position of the instantaneous neutral axis �INA is found as

Xi

Pi (�i; 'e�;i)

��;�

= 0; �INA (4.9)

and the corresponding bending moment about the instantaneous neutral axis becomes

MINA =Xi

Pi (�i; 'e�;i) �i

��;�;�INA

(4.10)

for a given orientation � of the instantaneous neutral axis and rotation � of the strain-plane.

3See further Appendix A which presents the entire derivation of the forced curvature principle for anasymmetrical instantaneous neutral axis.


Figure 4.9: Typical moment-curvature response curve.

4.4 Ultimate Capacity Criterions

Having established the means to perform a system analysis of the entire hull cross section,based on the beam-column method, the resulting moment-curvature response is obtainable.In Fig. 4.9 a typical such moment-curvature curve is shown for a �ctitious hull cross sec-tion. Also, the de�nitions of the hogging and sagging condition along with the curvature, isgraphically presented in the sketch.

Regarding the curvature �, then there is a simple geometric relation between it and therotation � of strain-plane, as shown in the �gure. The relation can easily be seen to be� = 2�=`, where ` is the length of the cross section being investigated. Thus, even thoughthe derived system analysis has been expressed in terms of the rotation � of strain-plane,then this could actually just as well have been done with respect to the curvature �. Thereason for the shift to the curvature when presenting the results is purely historical { thisis the way it is normally done { hence it eases comparison of the results obtained by thepresent method with results published by other authors.

4.4. Ultimate Capacity Criterions 47

4.4.1 Ultimate Shear Capacity

The quantity of actual interest in this study is the ultimate moment capacity of the crosssection in combination with any given, allowable shear stress distribution. By allowable isunderstood a non-failure causing shear stress distribution. Hence, to evaluate if a shearstress distribution is safe, a de�nition of the ultimate shear capacity of the cross section isneeded.

Shear stresses are, as previous mentioned, assumed to act only on the plating of the crosssection. Consequently, failure is in this context only referring to the state of the plating { notthe entire hull cross section. Moreover, the shear response behavior is assumed linear elastic,ideal plastic as described in Section 4.2.2. Thus, a reasonable measure for the ultimate shearcapacity will be the yield shear stress distribution �y equal to total shear yielding/collapseof all the plating in the entire cross section.

Hence, given the scaling de�ned in Eq. (4.1) of the shear causing loads, i.e. horizontal andvertical shear forces and torque, and under the assumptions described in Section 4.2.2, therewill be a maximum global scaling factor Kmax equivalent to total yielding. That is,

�y = Kmax

�Ky�Qy +Kz�Qz +Kx�Mx

�(4.11)

where Kx, Ky, and Kz are the relational scaling factors between the three load components,which are assumed to be given constants. It is noted that the shear behavior is assumedlinear elastic, ideal plastic. Thus, in the discrete formulation of the hull cross section wherethe plating is divided into, say N , individual plates, the shear stress � for the ith plate,globally scaled by K, is taken as

� =

(K�Ky�Qy +Kz�Qz +Kx�Mx

�if K

�Ky�Qy +Kz�Qz +Kx�Mx

�< �y;i

�y;i else(4.12)

Hence, by applying the cut-o� scheme in Eq. (4.12), the maximum global scaling factor Kmax

can be determined iteratively using Eq. (4.11). Thereafter, any scaling K between zero andthis value (Kmax) will then represent a safe, non-failure causing shear stress distribution.

Relating a speci�c scaling factor K to a set of equivalent loads in the form of Mx, Qy, andQz poses however, quite bigger problems because of the assumed linear elastic, ideal plasticbehavior. As long as all the individual plates are in the linear elastic range of the shear, thenas the shear stress distributions �Qy , �Qz , and �Mx are assumed equivalent to a unit loading,the shear stress distribution � resulting from the scaling by K can be decomposed into thethree shear causing loads as

Qy = KKy (Horizontal shear) (4.13)

Qz = KKz (Vertical shear) (4.14)

Mx = KKx (Torque) (4.15)


However, this scheme cannot be applied in the plastic region. Alternatively, in this part ofthe response, the two shear loads could be approximated as

Qy =NXi

KKy�Qy;iAi (4.16)

Qz =NXi

KKz�Qz;iAi (4.17)

where Ai is the e�ective shear area of the ith plate, and the the cut-o� scheme in Eq. (4.12)

is applied to the KKy�Qy;i and KKz�Qz;i terms respectively. With respect to the torque,the knowledge of the center of torsion is needed to perform a similar approximation. Thiscenter will however, move as plasticity caused by the shear stress becomes more and moredominant. Thus, to establish the shear loads equivalent to a scaling factor K a completegeometric cross section analysis will have to be performed.

Including this in the present procedure would on the other hand undermine the selectedmethod of simply scaling the shear stress distributions caused by a unit loading to obtainthe net shear loading on the hull cross section. If the complete geometric cross sectionanalysis was to be performed, then one could more rationally use it the other way around,namely to establish the net shear stress distribution for any selected set of shear loads andtorque. Moreover, the primary reason for initially selecting the method of simply scalingthe shear stress distributions caused by a unit loading was to avoid performing a completegeometric cross section analysis, mainly in the interest of overall speed of the procedure.

The establishment of the exact shear loads corresponding to any global scaling factor K, hastherefore been deemed beyond the scope of the present research and consequently neglectedin the following. However, in the cases of zero torsional loading on the hull cross section,an exception is made and the scheme in Eqs. (4.16) and (4.17) is applied to approximaterespectively the vertical and horizontal shear load corresponding to the current global scal-ing factor. This approximation is then used later, both in the veri�cation of the presentprocedure in Chapter 6, and in Chapter 7 where the present procedure is applied to an ultralarge crude carrier in intact as well as damaged condition.

Finally on the topic of ultimate shear capacity of the hull cross section, it is noted, thatbecause the shear stresses are assumed to act exclusively on the plating of the hull crosssection, there will be a moment capacity of the section provided by the sti�eners even at theultimate shear capacity. Therefore, given the assumptions imposed on the system analysisscheme, total failure of the entire hull cross section cannot be caused solely by the twoshear forces and/or the torque, in any combination. This since the applied modeling of theshear stress, always will allow for a residual moment capacity of the sti�eners to carry anadditional bending load. Hence, the present procedure can only reveal an upper limit forthe magnitude of the shear causing forces.

4.5. Implementation Notes 49

4.4.2 Ultimate Moment Capacity

Given a shear stress distribution � the moment-curvature response of the hull cross sectionwill be as sketched in Fig. 4.9. Thus, it is obvious that the ultimate moment capacity easilycan be determined, simply as the maximum and minimum moment giving the ultimatecapacity in hogging and sagging condition respectively.

The ultimate moment capacity for a given rotation � of the instantaneous neutral axis, canthen be expressed as

MINA,u =

(M(�)

�� dMINA(�)

d�= 0 ^N(�) = 0

)(4.18)

given that the moment MINA(�) is at least C1. This moment can then easily be decomposed

into the equivalent horizontal and vertical moments as

My,u = MINA,u cos� (Horizontal moment) (4.19)

Mz,u = MINA,u sin� (Vertical moment) (4.20)

However, it should be noted that in the present formulation, the moment-curvature responseis only C0, i.e. continuous. This is due to the idealized, piecewise linear beam-columnresponse explained in detail in Chapter 5. Nevertheless, this poses no real problem, asthe method is intended for implementation in a numerical framework, and thus numerousmethods are available for �nding (local) extrema without knowledge of any gradient.

4.5 Implementation Notes

Implementation of the system analysis in a computer code is generally a simple and straightforward procedure. Thus, only a very general outline of the implementation scheme will bepresented.

The �rst step will be to establish the maximum shear loading the cross section can withstand.That is, �nd Kmax from Eq. (4.11). With this value known, the shear loading can then bevaried at what ever discreteness level wanted between K 2 [0;Kmax], i.e. from zero to totalyield shear loading. For each level of shear loading, i.e. a given K, the next step would thenbe to vary the angle �, de�ning the orientation of the instantaneous neutral axis, full circle.That is, at a chosen level of subdivision let � 2 [0; 2�]. For each angle � the task will thenbe to determine the ultimate moment capacity about the instantaneous neutral axis in bothhogging and sagging condition, i.e. �nd the maximum and minimum response momentMINA.


When this has been completed the response of the cross section is known, and the objectiveof the code obtained.

Hence, �nding the maximum and minimum response momentMINA is then the core facility ofthe code. First of all, looking on the problem from a top-down perspective, then the momentabout the instantaneous neutral axis is essentially known as a function of the rotation � of thestrain-plane through Eq. (4.10). Thus, obtaining the ultimate moment is a simple questionof applying an iterative procedure hunting for either a minimum or a maximum. However,for each di�erent rotation � there is actually the following three steps to complete to obtainthe equivalent response moment.

Step one is to build the load-displacement curves for each beam-column in the entire hull crosssection. This is necessary because the end rotation ' of the individual beam-columns { whichis a part of the loading on the beam-column, and thus will change the load-displacementresponse { is a function of the rotation � of the strain-plane as shown in Eq. (4.8). Howthe load-displacement curves can be established is explained in Chapter 5 and will thereforenot be addressed presently. The response will simply be taken as known information. Thus,with the load-displacement curves available for the current �, step two is then to iteratefor the position �INA of the instantaneous neutral axis, the location of which is found byrequiring force equilibrium, as expressed in Eq. (4.9). Upon completion of this task theresponse moment is then obtainable by application of Eq. (4.10), which then is the third anlast step to complete.

The procedure is then completed, and the result can if wanted be decomposed into equivalentloadings in the global reference system by application of Eqs. (4.13) through (4.15) for theshear loading part, i.e. horizontal and vertical shear forces and torque, and Eqs. (4.19)and (4.20) for the moment part.

4.6 Summary

To summarize, a procedure for the establishment of the ultimate moment capacity about anarbitrary oriented instantaneous neutral axis has been established. The structural responseof the hull cross section is based on Navier's hypothesis, with the position of the instantaneousneutral axis de�ned as the zero strain line. The procedure can handle the description of bothintact and damaged conditions of the hull cross section. Moreover, the derived method ofsystem analysis can, for a given relational scaling between the three shear stress causingloadings, solve for the ultimate net shear loading, de�ned as total shear yielding of theplating part of the hull cross section.

In relation to this, the obtained knowledge of the ultimate shear capacity of the hull crosssection, has been incorporated in a scheme capable of establishing the ultimate momentresponse for any shear loading between zero and yield shear loading. Further, this schemehas been presented in a form suitable for implementation in a computer code.

4.6. Summary 51

Finally, for the purpose of clarity, the question of how the beam-column load-displacementresponse can be obtained has, in the present formulation, been left out for later concern.However, necessary assumptions pertinent to the beam-column description has been madewhen needed throughout the derivation of the system analysis.

Chapter 5

Beam-Columns in Combined Loading

5.1 Introduction

In the previous chapter a scheme for the evaluation of the ultimate and post-ultimate strengthof a hull cross section was established based on a beam-column approach. That is, theidea is to consider the entire cross section to be made up of a number of beam-columnseach consisting of one sti�ener and the attached plating. The assumption of the schemewas then, that the load-displacement response of these beam-columns exposed to a generalloading consisting of moment, shear and torque was known.

It was further shown that the shear and torque part of the general load condition of thebeam-column, essentially could be broken down to just a shear force, or more accurately ashear stress level in the plate part of the beam-column. Therefore, to meet the assumptionsmade so far, a description of the structural behavior of beam-columns in a combined loadingconsisting of a bending moment and a shear force is required. Once such a descriptionhas been established, the needed load-displacement curves for each beam-column can becalculated and thereby, the circle is closed and the previously stated beam-column schemecan work. The rest of this chapter will therefore be devoted to establishing such a frameworkfor describing and calculating the structural behavior of a beam-column exposed to thisloading, i.e. a bending moment and a shear force, from collapse in tension to collapse incompression.

When previously formulating the the beam-column approach, it was necessary to make twoassumptions: One, collapse occurs locally between two adjacent frames and two, each beam-column react independently of the adjacent beam-columns. Based on theses two assumptionsthe beam-column approach can easily be applied to mimic the behavior of the entire hull crosssection. However, to take the formulation of the procedure one step further in detailedness,there are a few more points that need to be considered before the derivation of the beam-column analysis can be completed.

53

54 Chapter 5. Beam-Columns in Combined Loading

5.2 Initial Imperfections

So far, nothing has been stated about the state of the cross section. Implicitly then, it mustbe understood to be in perfect initial state. Knowledge of the real world however, tells thatthis is never the case. There will always be imperfections in any real structure. Consequently,if these imperfections have any in uence on the ultimate capacity of a structure they shouldpreferably be included in any model designed to calculate that quantity.

Numerous such imperfection can be listed for any given structure. Some related to themethod of manufacture, some related to properties of the materials used, etc. For shipstructures three key imperfection can readily be listed:

� Initial de ection of the sti�eners.

� Initial de ection of the plating between sti�eners.

� Residual stress caused by welding.

All of these must be expected to have some in uence on the ultimate capacity of the crosssection. However, the level of di�culty each poses with respect to possible implementationis vastly di�erent. In the previous chapter, considerations made on the load modeling aspectof the ultimate capacity evaluation scheme, lead to the conclusion that the beam-columnwould be considered as a two-component beam made up of a sti�ener part and a plate part.This approach was proven very bene�cial when the e�ects of the shear forces and the torquewas to be applied to each beam-column. In this scheme an initial de ection of the sti�enerin the plane perpendicular to the plate is easily accounted for. However, initial de ection ofthe plate, and especially residual stresses in the plate, poses far bigger problems.

Initial de ection of the plate and residual stresses could both be accounted for by choosing theMarguerre equations to describe the plate behavior. The solution scheme would then haveto be an incremental procedure, where the collapse load was found by slowly incrementingthe applied load. Needless to say, this would be very time consuming to solve for all platesin the entire cross section. On the other hand, if the the initial de ection was ignored,then the Marguerre equations reduces to the von Karman equations. The solution of theseequations would nevertheless still have to be incremental due to the presence of residualstresses. If however, also these where ignored, then the von Karman equations becomessolvable in the post-buckling region by application of the perturbation technique which isa much faster solution scheme than the incremental method otherwise needed. Moreover,seen in light of the �ndings in Chapter 3, where the ultimate strength of a laser weldedpanel was investigated, one of the conclusions was that the inclusion of residual stressesonly marginally in uences the load-displacement response { then the exclusion of this e�ectseems justi�able. Thus, to keep the analysis reasonably simple in light of the requirementfor rapidness of the procedure, the choice has been made to only include the imperfectionof the sti�ener and to disregard the e�ect of both an initial de ection of the plate, and anyresidual stresses in the plate.

5.3. Idealized Beam-Column Behavior 55

Figure 5.1: Idealized stress-strain curve for a single beam-column.

5.3 Idealized Beam-Column Behavior

Having decided on the inclusion/exclusion of initial imperfection an idealized model of thestructural behavior of a beam-column can be established. In Fig. 5.1 an idealized stress-strain curve for a beam-column is shown. The behavior can be separated into four di�erentregions:

� Plastic tension region.

� Elastic tension region.

{ Plate yielding.

� Elastic compression region.

{ Plate buckling.

{ Plate collapse.

� Plastic compression region (unloading).

Together, these make up the entire idealized load-displacement behavior of the beam-column.In the following sections each of these four regions will be addressed. First however, a shortgeneral outline of the procedure is presented to facilitate the further reading.

The requirement of the procedure will be to produce the load-displacement response of abeam-column exposed to the loading previous de�ned in Chapter 4. Hence, the variable


loading applied will be an axial force P and an end rotation ' of the beam-column resultingfrom the bending moment. Further, two constant load components in the form of a uniformlydistributed load q mimicking e.g. the hydrostatic pressure, and a shear stress � arising fromthe global shear and torsional loading, was to be accounted for.

Moreover, it was decided to model the beam-column as a two-component beam made up ofa sti�ener part and a plate part. The shear stress was assumed to act solely on the platepart of the beam-column, and regarding the uniformly distributed load then this load willbe assumed to act solely on the sti�ener part of the beam-column. Otherwise, the solutionof the von Karman equations cannot be obtained by the perturbation technique.

With this in mind, and looking at the idealized load-displacement behavior in Fig. 5.1, it isobserved that in essence three di�erent types of models are needed to accurately describethe entire response range. Starting in the middle, the behavior in the elastic tension andcompression region can be handled by an elastic solution to the classical beam-column prob-lem { at least up until the �rst nonlinearities are introduced at plate buckling and yielding,respectively. Moreover, the range in which the elastic solution is valid can be extended fur-ther by application of the e�ective width concept making it possible to trace the responsebehavior all the way up to the �nal elastic compressive collapse de�ned as �rst yield basedon the von Mises stress. With respect to tensile collapse, then the e�ective width can alsobe applied to model the assumed yielding of the plate prior to the sti�ener.

The compressive post-collapse response is described by a simple plastic three-hinge mecha-nism, allowing for the modeling of the decreasing load-carrying capacity the beam-columnwill exhibit after collapse. Further, to account for the buckling/collapse behavior of the platepart of the beam-column, the e�ective width principle is applied allowing the plastic hingesolution to re ect the lower than yield strength compressive load capacity of the plate.

Finally, the tensile plastic behavior is assumed ideal in the sense that the response of thebeam-column will remain constant for any further straining beyond tensile collapse. Thatis, the plastic response in tension will remain equal to the tensile collapse load.

This is then the foundation upon which the load-displacement response for the beam-columnwill have to be established. Four di�erent regions of response describable by three di�erentmodels, which all will be elaborated on in the following.

5.4 Plastic Tension Region

In this region the whole cross section of the beam-column is stressed beyond yield capacityand thus totally plasticized. The behavior is assumed perfectly plastic implying that theslope of the stress-strain curve equals zero as shown in Fig. 5.1. Thus, the post-ultimateresponse of the beam-column will be constant and equal to the ultimate tensile responsewhich marks the end of the elastic tension range of the response.

5.5. Elastic Tension Region 57

The ultimate load in tension will in the case of zero shear stress and zero uniformly dis-tributed load, be equal to the the squash load Py. However, in the general case the ultimateload will be less than the squash load, due to the presence of shear stress in the plate partof the beam-column, and the bending moment caused by the uniformly distributed load onthe sti�ener part. Nevertheless, as this is all accounted for in the modeling of the elastictension region, the plastic tension region is modeled as constant and equal to the maximumload obtained from the elastic region.

Finally, there is the necking phenomena pertaining to the plastic tension region that needs tobe considered to fully describe the response. When the axial load reaches the point of neckingthe load rapidly drops to zero, and the beam cross section is torn apart. This is normallyrelated to a very extensive straining of the beam-column. One method to include the neckingphenomena would be to e�ectively remove beam-columns from the global system analysisonce transgression of a certain strain limit e.g. �ve percent, used to identify the initiation ofnecking occurs. However, in the present idealized formulation it is assumed that collapse ofthe entire hull cross section occurs before necking initiates. That is, the necking phenomenais e�ectively neglected.

5.5 Elastic Tension Region

In the elastic tension region the behavior is assumed linear elastic. Further, it is assumedthat the plate will reach yielding before the entire beam-column is plasticized. This assump-tion is valid for typical ship structures mainly due to design consideration pertinent to thecompressive performance of a sti�ened panel. Here, general practice dictates that compres-sive failure of the sti�ener should be insured to occur after plate buckling, which in generalwill require the yield stress of the sti�ener at least to equal the yield stress of the plate.

In the case of zero shear stress and zero uniformly distributed load, plate yielding prior tototal beam-column failure implies that �y,p < �y,s is assumed { at least if the e�ect of theforced end rotation of the beam-column is also ignored { where �y,p and �y,s is the yieldstress of the plate and the sti�ener respectively. As this, for reasons just explained, normallywill be valid for ship structures, it seems for all practical purposes a fair assumption to makethat the plate will reach yielding before the sti�ener in relation to ship structures.

Considering the implications of this assumption in the presence of shear stress in the plate,then theses shear stresses are accounted for by considering tensile yielding of the plate astransgressing of �y,p by the von Mises stress �v. Thus, shear stresses will reduce the axialload-carrying capacity of the plate, but will not have any e�ect on the sti�ener, which ingeneral will help to ful�ll the assumption of the plate yielding before the sti�ener.

On the other hand, the presence of a uniformly distributed load on the sti�ener actingperpendicular to the plate, clutters the validity of the assumption. Together with the forcedend rotation of the beam-column this line load on the sti�ener will cause a bending moment to


be present in the beam-column. While the resulting moment arising from the end rotationfor all practical reasons will be insigni�cant due to the comparatively very small anglesactually occurring before total collapse of the entire hull cross section, there is no ignoringthe contribution of the line load. Depending on the sign of this moment, the e�ect on thevalidity of the assumption can go either way.

If the moment results in tensile stresses in the plate part of the beam-column, then obviouslyit will only further enforce the assumption of the plate yielding before the sti�ener. Onthe other hand, if compressive stresses are produced in the plate, then there is really noway of assessing the net result on the assumed behavior of the entire beam-column. Thee�ect with respect to axial response of the beam-column would be that part of the sti�enerwill be plasticized before yielding initiates in the plate, a situation that could possibly beaccounted for by application of a transformed cross section of the sti�ener much like theplanned e�ective width concept to extend the validity of the linear elastic behavior of thebeam-column even after plate yielding.

Meanwhile, the introduction of the transformed sti�ener cross section seems, in the author'sopinion, somewhat dubious with respect to its ability to model the growing fully plasticzone of the sti�ener. Moreover, choosing to consider sti�ener yielding as the ultimate tensileresponse of the entire beam-column in this situation, will at worst be a somewhat conservativeassumption. This last approach has therefore been decided on as a further needed assumptionto conveniently describe the tensile response of the beam-column.

Consequently, under these assumptions the response will either be linear up until the extreme�ber of the sti�ener yields, or until the average von Mises stress in the plate reaches yieldcapacity. In the �rst case this response will be interpreted as the ultimate tensile capacityof the entire beam-column. In the second case, reaching the yield capacity of the plate willmanifest it self in form of a drop in sti�ness of the beam-column at continued uploadingas indicated in Fig. 5.1. This loss of sti�ness is modeled by means of the e�ective widthprinciple for the plate. That is, the e�ective width of the plate is reduced such that whenstressed to the tensile yield strength of the sti�ener, the equivalent stress in the real, fulle�ective plate will be the actual tensile yield strength of the plate.

Increasing the tension loading further will eventually lead to the beam-column becomingtotally plasticized. At this point the average von Mises stress in the entire beam-columnequals the yield stress and the behavior changes to the plastic tension region describedearlier.

5.5.1 Stress Assessment

As the assumed behavior is linear elastic, the beam-column can adequately be described bythe standard di�erential equation for beams, i.e.

EI@4w(x)

@x4+ P

@2

@x2(w(x) + w0(x)) = q(x) (5.1)

5.6. Elastic Compression Region 59

Applying the actual boundary conditions and then solving this equation (Eq. (5.1)) will give�rst the out-of-plane de ection w(x), and then derivable from this, the momentM(x). Then,applying the standard Navier relation between direct stress �xx and the sectional forces, thenormal stress distribution at the middle of the beam-column (x = /2) becomes1

�xx(z) = �PA+

�Ez

sinh(� /2)

�Pw0

`(P � PE)+q`

2P+ '

!� �2EPw0z

`2(P � PE)� Eqz

P(5.2)

for tensile loading (P < 0), where

� =

s�PEI

PE = EI��

`

�2(The Euler load)

Thus, using Eq. (5.2) the stress in the plate and the extreme �ber of the sti�ener can befound for any tensile axial loading P . This allows for an iteratively scheme to be establishedby which the two characteristic loads describing �rst plate yielding, and then total tensilefailure of the entire beam-column can be found.

When the axial response load is known for these two characteristic points in the tensileload-displacement behavior of the beam-column, then the corresponding displacement � atthe neutral axis of the beam-column cross section can be approximated by application of thelinear relation

� =P`

EA(5.3)

in which the contribution by the out-of-plane de ection of the beam-column has been ignored.Actually, there will be a nonlinear contribution to the displacement by the de ection, butthis has been found to be insigni�cant in comparison with the contribution in Eq. (5.3) andis therefore ignored in the present analysis. Further, seen in light of the other assumptionsmade so far in the idealization of the beam-column response, it is quite reasonable to neglectthis higher order contribution to the displacement.

5.6 Elastic Compression Region

The elastic compression region covers the behavior of the beam-column in compression fromthe unloaded state to the point where the behavior is totally plastic. Included in this

1See Appendix B, Eq. (B.8)


transition from elastic to plastic behavior are two distinct phenomena related to the plateas shown in Fig. 5.1. That is, one: Buckling of the plate �eld and two: Final collapse of theplate �eld.

Therefore, considering the plate part of the beam-column �rst, it is for an initially perfectplate (i.e. without any imperfections) the case that when loaded in compression, the platewill remain perfect until the buckling load is exceeded. That is, there will only be in-planedisplacements but no out-of-plane de ection up until the buckling load is reached.

Continued uploading beyond the critical buckling load will still be characterized as elasticcompression but will very rapidly lead to the collapse load of the plate being reached. Atthis point the load-carrying capacity of the plate is exhausted. Assuming that the sti�enerwhich together with the plate makes up the beam-column still is below yielding2, then thebeam-column will still be in the elastic region because it still has load-carrying capacity left.This capacity will of course eventually be depleted and the collapse load of the beam-columnwill be reached marking the end of the elastic compression region and thus also the transitionto the plastic compression region.

In the elastic compression region it is therefore of special concern to establish how to modelthe behavior of the plate part of the beam-column to adequately describe the critical bucklingand collapse behavior of the plate alone. In the previous two sections concerning the tensionside of the beam-column response, the plate did not act di�erently from the sti�ener, exceptfor the possibility of di�erent tensile yield strength of the two. Further, in the plasticcompression region of the response, the reaction of the plate will again be similar to that ofthe sti�ener, as they both will be fully plasticized.

Thus, the elastic compression region is in fact the only part of the response of the beam-column, where the plate part needs to be considered as a plate. That is, this is the only regionwhere a plate theory is required to accurately describe the response of the beam-column.

Based on previous considerations made in reference to the possible inclusion of initial imper-fections in the beam-column model, where the emphasis was on the acceptable complexity ofsuch a plate theory, is was concluded to ignore any imperfections and/or out-of-plane loadingin the plate description. That decision led to the selection of the von Karman equations asan ideal theory for the plate description.

A description of these von Karman equations will therefore be presented in the following.However, as the von Karman equations are a sub-set of the more general Marguerre equa-tions, which allow for combined in-plane loading, out-of-plane loading, and initial de ectionof the plate, these equations will be presented �rst and then reduced to the von Karmanequations.

2This assumption is generally ful�lled in ship structures, as argued previously while discussing the elastictensile behavior of the beam-column, and thus seems fair to make. Further, if the yield stress of the plateand the sti�ener are identical, then this assumption will be even more valid.


5.6.1 Marguerre's Equations

Assuming the following notation:

� The plate carry in-plane load and a normal load Y per unit area.

� The plate has an initial shape given by y? = y?(x; z) assumed small enough to makethe usual approximation for small slopes.

� The de ection w of a point in the middle surface of the plate is de�ned so as afterdeformation the total de ection is w + y? in the x� z plane.

And by introducing a stress function � which satisfy

�;xx = Nz ; �;zz = Nx ; �;zx = �Nxz (5.4)

Then, following Murray [30] the usual stress-strain relationships given as

Nz =Et

1� �2("z + �"x) ; Nx =

Et

1� �2("x + �"z) ; Nxz =

Et

2(1 + �)"xz (5.5)

where the strains are "z = v;z + y?;zw;z +w2;z

2, "x = u;x + y?;xw;x +

w2;x

2, and ";xz = v;x + u;z +

y?;zw;x + y?;xw;z + w;zw;x yields the following two equations

r4� + Et�y?;zzw;xx � 2y?;zxw;zx + y?;xxw;zz + w;zzw;xx � w2

;zx

�= 0 (5.6)

Dr4w � [�;xx(y? + w);zz � 2�;zx(y

? + w);zx + �;zz(y? + w);xx]� Y = 0 (5.7)

which are the compatibility and equilibrium equation respectively. These two equations(Eqs. (5.6) and (5.7)) are Marguerre's simultaneous nonlinear partial di�erential equations.

5.6.2 Application { The von Karman Equations

The governing equations of an initially perfectly at plate (i.e. y? = 0), carrying uniformlydistributed in-plane loading (i.e. � is quadratic in x and z, cf. Fig. 5.2) and no transverseload (i.e. Y = 0) are the force equilibrium in the x- and z-direction (Eq. (5.8)) and momentequilibrium (Eq. (5.9)) { together the von Karman equations without transverse loading.

Nz;z +Nxz;x = 0 ; Nx;x +Nxz;z = 0 (5.8)

Dr4w �Nzw;zz�2Nxzw;zx �Nxw;xx = 0 (5.9)

It is seen that the moment equilibrium reduces to a linear homogeneous form which de�nesan eigenvalue problem. Thus, with Nx and Nz being the tensile loads per unit width orlength of the plate, the critical value of Nz will have a negative sign i.e. be in compression.


Figure 5.2: Square plate with in-plane load �z and �x.

5.6.3 Post-Buckling Behavior { The Perturbation Technique

To obtain the post-buckling behavior the perturbation technique is applied (cf. Murray [30]).The perturbation technique assumes that the displacement components u, v, and w can beexpanded in terms of an arbitrary perturbation parameter " about the point of buckling.At this point the perturbation parameter is by de�nition zero (" = 0) which conforms withthe singularity existing at the point of buckling, where up until buckling occurs the plateremains perfectly at (w = 0), and then when the critical stress is reached snaps out-of-planeand assumes its lowest buckling mode. The perturbation parameter " is a function of theapplied load and the central de ection. It is independent of the coordinates x and z. Thus,the expansion of the displacement components becomes

w =1X

n=1;3;:::

w(x; z)(n)"n ; u =1X

n=0;2;:::

u(x; z)(n)"n ; v =1X

n=0;2;:::

v(x; z)(n)"n (5.10)

Inserting these power series (Eq. (5.10)) into the strain expressions neglecting second orderterms of the in-plane displacements u and v, but keeping the second order terms of thede ection w yields

"z =1X

n=0;2

v(n);z "n +

1

2

1Xm=1;3

1Xn=1;3

w(m);z w(n)

;z "m+n (5.11)

"x =1X

n=0;2

u(n);x "n +

1

2

1Xm=1;3

1Xn=1;3

w(m);x w(n)

;x "m+n (5.12)

"xz =1X

n=0;2

�v(n);x + u(n);z

�"n +

1Xm=1;3

1Xn=1;3

w(m);z w(n)

;x "m+n (5.13)

where { for writing convenience { the dependency of x and z in the functions u, v, and w hasbeen left out and the in�nite continuation of the power series in either even or odd integersis self implied.


Substituting this into the stress-strain relationships (Eq. (5.5)) gives the following equationsfor the in-plane forces

Nz =Et

1� �2("z + �"x)

=1X

n=0;2

Et

1��2�v(n);z + �u(n);x

�| {z }

:=N

(n)z

"n +1X

m=1;3

1Xn=1;3

1

2

Et

1��2�w(m);z w(n)

;z + �w(m);x w(n)

;x

�| {z }

:=N

(mn)z

"m+n

Nx =Et

1� �2("x + �"z)

=1X

n=0;2

Et

1��2�u(n);x + �v(n);z

�| {z }

:=N

(n)x

"n +1X

m=1;3

1Xn=1;3

1

2

Et

1��2�w(m);x w(n)

;x + �w(m);z w(n)

;z

�| {z }

:=N

(mn)x

"m+n

Nxz =Et

2(1 + �)"xz

=1X

n=0;2

Et

2(1 + �)

�v(n);x + u(n);z

�| {z }

:=N

(n)xz

"n +1X

m=1;3

1Xn=1;3

Et

2(1 + �)w(m);z w(n)

;x| {z }:=N

(mn)xz

"m+n

which { with the introduced notation { can be written in a short form as

Nz =1X

n=0;2

N (n)z "n +

1Xm=1;3

1Xn=1;3

N (mn)z "m+n (5.14)

Nx =1X

n=0;2

N (n)x "n +

1Xm=1;3

1Xn=1;3

N (mn)x "m+n (5.15)

Nxz =1X

n=0;2

N (n)xz "

n +1X

m=1;3

1Xn=1;3

N (mn)xz "m+n (5.16)

where

N (n)z =

Et

1��2�v(n);z + �u(n);x

�(5.17)

N (mn)z =

Et

2(1��2)�w(m);z w(n)

;z + �w(m);x w(n)

;x

�(5.18)

N (n)x =

Et

1��2�u(n);x + �v(n);z

�(5.19)

N (mn)x =

Et

2(1��2)�w(m);x w(n)

;x + �w(m);z w(n)

;z

�(5.20)


N (n)xz =

Et

2(1 + �)

�v(n);x + u(n);z

�(5.21)

N (mn)xz =

Et

2(1 + �)w(m);z w(n)

;x (5.22)

Written out, the �rst terms in the power series Eqs. (5.14) through (5.16) the followingexpressions, truncated to n = 0; 1; 2 and m = 1 is obtained

Nz = N (0)z "0 +N (2)

z "2 +N (11)z "1+1 (5.23)

Nx = N (0)x "0 +N (2)

x "2 +N (11)x "1+1 (5.24)

Nxz = N (0)xz "

0 +N (2)xz "

2 +N (11)xz "1+1 (5.25)

which inserted into the force and moment equilibrium equations (Eqs. (5.8) and (5.9)) yieldsafter collecting the terms having the same order of "

Nx;x +Nxz;z = 0;�N (0)x;x +N (0)

xz;z

�"0 +

�N (2)x;x +N (2)

xz;z +N (11)x;x +N (11)

xz;z

�"2 = 0

Nz;z +Nxz;x = 0;�N (0)z;z +N (0)

xz;x

�"0 +

�N (2)z;z +N (2)

xz;x +N (11)z;z +N (11)

xz;x

�"2 = 0

Dr4w �Nzw;zz�2Nxzw;zx �Nxw;xx = 0;nDr4w(1) �N (0)

z w(1);zz � 2N (0)

xz w(1);xz �N (0)

x w(1);xx

o"1+n

Dr4w(3) ��N (2)z +N (11)

z

�w(1);zz �N (0)

z w(3);zz � 2

�N (2)xz +N (11)

xz

�w(1);xz � 2N (0)

xz w(3);xz

��N (2)x +N (11)

x

�w(1);xx �N (0)

x w(3);xx

o"3+n

��N (2)z +N (11)

z

�w(3);zz � 2

�N (2)xz +N (11)

xz

�w(3);xz �

�N (2)x +N (11)

x

�w(3);xx

o"5 = 0

Since the perturbation parameter " is an arbitrary parameter, each coe�cient of the powerseries must vanish. This leads to the in�nite set of successive linear partial di�erentialequations

N (0)x;x +N (0)

xz;z = 0 (5.26)

N (0)z;z +N (0)

xz;x = 0 (5.27)

Dr4w(1) �N (0)z w(1)

;zz � 2N (0)xz w

(1);xz �N (0)

x w(1);xx = 0 (5.28)

N (2)x;x +N (2)

xz;z = ��N (11)x;x +N (11)

xz;z

�(5.29)

N (2)z;z +N (2)

xz;x = ��N (11)z;z +N (11)

xz;x

�(5.30)

Dr4w(3) �N (0)z w(3)

;zz � 2N (0)xz w

(3);xz �N (0)

x w(3);xx =�

N (2)z +N (11)

z

�w(1);zz + 2

�N (2)xz +N (11)

xz

�w(1);xz +

�N (2)x +N (11)

x

�w(1);xx (5.31)


and so on for higher powers of " (i.e. greater than 3). Together, these six equations thenconstitutes the third order perturbation approximation to the post-buckling behavior of theplate �eld.

Let a given set of adequate boundary conditions for the plate be given. Then, the informationobtainable by solving these six equation can be investigated in general terms. Doing this,it is �rst of all seen that the �rst two equations (Eqs. (5.26) and (5.27)) express the pre-buckling in-plane force equilibrium. Thus, the solution to these two equations will be thepre-buckling in-plane displacements u(0) and v(0).

Next, the moment equilibrium in Eq. (5.28) is recognized as an eigenvalue problem of theform (A � �B)� = 0. Its solution is the buckling load and the out-of-plane de ectionpattern immediately after buckling occurs. However, no information on the magnitude of thede ection can be derived from this equation. That is, the de ection solution obtained fromthe eigenvalue problem, say ~w(1), relates to the real out-of-plane de ection as ~w(1) = w(1),where is an arbitrary scaling factor.

Finally, the force equilibrium in Eqs. (5.29) and (5.30) gives the in-plane displacement patternafter buckling, i.e. the post-buckling displacements. However, as the right-hand side ofthese two equations is a function of the out-of-plane de ection w(1) squared, which is knownonly in shape not in magnitude, then these two equation only gives a scaling between thepost-buckling in-plane displacements (u(2) and v(2)) and the out-of-plane de ection w(1) atbuckling. In other words, solving the force equilibrium based on the solution to the eigenvalueproblem, ~w(1) = w(1), yields the in-plane displacements ~u(2) = 2u(2) and ~v(2) = 2v(2).

In summary then, solving the �rst �ve of the equations will give the pre-buckling in-planedisplacements and the buckling load in true quantities, given the boundary conditions. Re-garding the out-of-plane de ection at buckling, and the post-buckling in-plane displacements,then these are only determined in shape not magnitude. They are in other words found aspatterns which are true in size but for the arbitrary scaling factor . The question thenarises whether knowing the post-buckling solution only in form is su�cient information forthe present case of investigation?

At this point it will be appropriate to recap the objective of the plate investigation. Referringback to the initial discussion of the elastic compression region, the questions that need to beanswered by the analysis are:

� What is the buckling load of the plate? And

� what is the post-buckling collapse load of the plate?

The �rst question can readily be answered by solving the eigenvalue problem in Eq. (5.28) fora given set of adequate boundary conditions. However, to answer the second question moreinformation is needed. A collapse criterion is a pre-requisite for determining the collapse load.


Hence, a choice of collapse criterion need to be made to continue this investigation. To do thisin a consistent rational way, it is not su�cient just to state the loading condition to be anyset of adequate boundary conditions. Prior knowledge of the exact actual loading conditionis needed. That is, up until now, all that has been stated about the information obtainableby solving perturbation equations has been general in the sense that it would apply to anyset of adequate boundary conditions. Unfortunately, this generalized interpretation cannotbe stretched any further, and thus the speci�c set of boundary conditions for the presentproblem must be de�ned to continue the interpretation of the perturbation equations.

5.6.4 Boundary Conditions

Recalling the assumption made when the beam-column approach was introduced in theprevious chapter { each beam-column reacts independently of adjacent ones, and shear forcescauses constant shear stress in the plate only { reasonable boundary conditions will be thatthe plate edges are simply supported and remain straight, and that the in-plane loading isuni-axial in the longitudinal direction only, given a constant shear stress level in the plate.Thus, recapping the boundary condition for the plate, they are:

� The shear stress is constant along the edges of the plate, i.e. the �rst derivative of theshear strain equals zero, yielding

"xz;z��x = � b/2

= 0 and "xz;x��z = � /2

= 0

� The plate is simply supported, i.e. both the de ection it self, and the curvature of thede ection (the moment) along the edges of the plate equals zero which requires

w��x = � b/2 ; z = � /2

= 0 ; w;xx

��x = � b/2

= 0 , and w;zz

��z = � /2

= 0

� The plate edges remain straight, i.e. the curvature of the in-plane displacement per-pendicular to the edge equals zero, giving

u;xx��x = � b/2

= 0 and v;zz��z = � /2

= 0

� In-plane uni-axial direct loading in longitudinal direction only, i.e.

Z b/2

� b/2Nzdx

��z = � /2

= P andZ /2

� /2Nxdz

��x = � b/2

= 0

combined with a constant shear force proportional to the direct in-plane load by afactor #, i.e.

Z b/2

� b/2Nxzdx

��z = � /2

= #P andZ /2

� /2Nxzdz

��x = � b/2

= #`

bP


At a �rst glance, these boundary conditions all seems to be applicable in their direct form,and thus the further solution seems to be a straight forward process. However, there arehidden problems in the above listed boundary conditions, as the uni-axial in-plane loadingP is in fact the quantity sought by the post-buckling analysis.

This becomes evident from expanding e.g. the in-plane, direct load boundary condition bysubstitution with Eq. (5.23), yielding

P =Z b/2

� b/2N (0)z dx+ "2

Z b/2

� b/2

�N (2)z +N (11)

z

�dx

� P (0) + P (2)"2 (5.32)

where " is the perturbation parameter.

Similar expansions can be made also for the two shear force boundary conditions. This posesa problem as the added in-plane load in the post-buckling region P (2) now is coupled withthe perturbation parameter ". Thus, to obtain the post-buckling solution to the in-planedisplacements u(2) and v(2) given only as a function of the perturbation parameter ", onemore relation which can e�ectively eliminate the P (2) term is needed. This relation can onlycome through Eq. (5.31) as this is the only equation not used so far.

Looking at this equation, it is seen that the left-hand-side is formally equal to the left-hand-side of moment equilibrium in Eq. (5.28). Hence, the left-hand-side di�erential operator,say L(�), of Eqs. (5.28) and (5.31) is the same, and will with suitable boundary condition(present in this analysis) be a self-adjoined di�erential operator. That is, it has the property

ZA

hw(1)L(w(3))� w(3)L(w(1))

idA = 0 (5.33)

Further, as the right-hand-side of Eq. (5.28) is zero, it then follows from the self-adjoinedproperty that

ZA

h�N (2)z +N (11)

z

�w(1);zz + 2

�N (2)xz +N (11)

xz

�w(1);xz +

�N (2)x +N (11)

x

�w(1);xx

iw(1)dA = 0 (5.34)

i.e. the right-hand-side of Eq. (5.31) times the de ection at buckling w(1) integrated overthe entire area of the plate equals zero. Unfortunately, this expression does not immediatelyyield a solution to the problem.


However, by observing the structure of the two equations expressing the force equilibriumin the post-buckling state (Eqs. (5.29) and (5.30)) two observations about the equations canbe made:

1. If applying only an out-of-plane de ection load, then the post-buckling solution to thein-plane forces will vary linearly with the de ection squared, and

2. if applying only an in-plane loading, i.e. keeping the plate at, then the post-bucklingsolution to the in-plane forces will vary linearly with the load.

The �rst point can easily be seen by substituting Eqs. (5.17) through (5.22) into the right-hand-side of the force equilibrium. The second point arises from the boundary conditionswhere a constant ratio # between the direct in-plane load Nz and the shear load Nxz isprescribed along with the linear behavior of the load expansion found in Eq. (5.32) for agiven ". Thus, it can be concluded that the post-buckling in-plane forces N (2) resulting froma combined in-plane loading and out-of-plane de ection at a given " can be decomposed into

N (2) = �N (2) + N (2)P (2) (5.35)

i.e. as the sum of two parts: One proportional to the buckling de ection w(1) squared andindependent of the in-plane loading, and one proportional to the added in-plane post-bucklingload P (2) and independent of the de ection.

In other words, it can be concluded that the ratio between the in-plane forces resulting fromthe out-of-plane de ection and in-plane loading respectively, will remain constant equal toexactly P (2) for any given perturbation (loading) parameter ".

This observation yields the solution to the problem of P (2) being coupled with the perturba-tion parameter ", as substituting Eq. (5.35) into the equation resulting from the self-adjointproperty (Eq. (5.34)), i.e.

ZA

h��N (2)z +N (11)

z

�w(1);zz + 2

��N (2)xz +N (11)

xz

�w(1);xz +

��N (2)x +N (11)

x

�w(1);xx

iw(1)dA

+P (2)ZA

hN (2)z w(1)

;zz + 2N (2)xz w

(1);xz + N (2)

x w(1);xx

iw(1)dA = 0

leads to a solution to the P (2) load term given as

P (2) = �

ZA

h��N (2)z +N (11)

z

�w(1);zz + 2

��N (2)xz +N (11)

xz

�w(1);xz +

��N (2)x +N (11)

x

�w(1);xx

iw(1)dAZ

A

hN (2)z w(1)

;zz + 2N (2)xz w

(1);xz + N (2)

x w(1);xx

iw(1)dA

(5.36)


Thus, the post-buckling in-plane displacements ~u(2) and ~v(2) for a given given direct/shearload ratio #, can be obtained by running through the following four steps:

1. Solve the eigenvalue problem in Eq. (5.28) obtaining the in-plane buckling load N (0)

and the buckling de ection ~w(1) = w(1).

2. Solve in the post-buckling force equilibrium in Eqs. (5.29) and (5.30) for the followingtwo load conditions:

� Zero in-plane load combined with the buckling de ection ~w(1) yielding the in-plane

displacements �~u(2)

= 2�u(2) and �~v(2)

= 2�v(2).

� Unit in-plane load combined with zero de ection yielding the in-plane displace-ments u(2) and v(2).

3. Obtain the constant ratio 2P (2) between the two in-plane displacement componentsfrom Eq. (5.36) based on the buckling de ection and the two in-plane displacementsolutions.

4. The post-buckling in-plane displacements then becomes

~u(2) = �~u(2)

+ 2P (2)u(2)

~v(2) = �~v(2)

+ 2P (2)v(2)

It is noted that the obtained post-buckling in-plane displacements still is determined as~u(2) = 2u(2) and ~v(2) = 2v(2). Hence, the question of whether or not it is su�cient forthe present case of investigation to know the in-plane displacements only in shape not inmagnitude, still remains. The �nal answer to this question is to be found in the formulationof the collapse criterion.

5.6.5 Collapse Criterion

The purpose of the collapse criterion is to set a relation between the perturbation (loading)parameter and the occurrence of plate collapse, thereby enabling the determination of theultimate collapse load of the plate �eld. To gain such a relation from which the perturbationparameter " can be found, the following collapse criterion is introduced:

Collapse of the plate occurs when the mean von Mises stress �v at the mid-planeof the plate along one of the unloaded edges reaches the yield stress �y.

which is the natural extension of the already introduced tension failure criterion for theplate, used in the elastic tensile region of the beam-column response.


The von Mises stress, �v is given as

�2v = �2x + �2z � �x�z + 3�2xz (5.37)

for the case of plane stress. Substituting the stress-strain relationships (Eq. (5.5)) into thisexpression (Eq. (5.37)) yields

�2yt2 � 1

`

Z /2

� /2

�N2x +N2

z �NxNz + 3N2xz

�dz

��x = b/2

= 0 (5.38)

where, for the �nite set of terms in the power series expanded in Eqs. (5.26) through (5.27)

Nx = N (0)x +

�N (2)x +N (11)

x

�"2 (5.39)

Nz = N (0)z +

�N (2)z +N (11)

z

�"2 (5.40)

Nxz = N (0)xz +

�N (2)xz +N (11)

xz

�"2 (5.41)

Thus, expanding the collapse criterion (Eq. (5.38)) even further by substituting the aboveexpressions for the in-plane loads, leads to the following second-order equation in "2

Ac

�"2�2

+Bc

�"2�+ Cc = 0 (5.42)

where

Ac =1

`

Z /2

� /2

��N (2)x +N (11)

x

�2+�N (2)z +N (11)

z

�2 � �N (2)x +N (11)

x

� �N (2)z +N (11)

z

�+

3�N (2)xz +N (11)

xz

�2�dz (5.43)

Bc =1

`

Z /2

� /2

n2N (0)

x

�N (2)x +N (11)

x

�+ 2N (0)

z

�N (2)z +N (11)

z

��N (0)

x

�N (2)z +N (11)

z

��

N (0)z

�N (2)x +N (11)

x

�+ 6N (0)

xz

�N (2)xz +N (11)

xz

�odz (5.44)

Cc =1

`

Z /2

� /2

nN (0)x

2+N (0)

z

2 �N (0)x N (0)

z + 3N (0)xz

2odz � �2yt

2 (5.45)


So, if the perturbation expansion is limited to second order, then the post-buckling collapseload can be determined by solving Eq. (5.42) as

"2c =�Bc �

pBc � 4AcCc

2Ac(5.46)

and then inserting the collapse perturbation parameter "c in Eqs. (5.39) through (5.41). But,what happens when the de ection and displacements are known but for a scaling factor?

The previous discussion of the successive linear partial di�erential equations, which theperturbation expansion ended up in (Eqs. (5.26) through (5.31)), lead to the conclusionthat the information gained from solving these equations with respect to the post-bucklingde ection and displacements was limited to ~w(1) = w(1), ~u(2) = 2u(2), and ~v(2) = 2v(2)

where is an undetermined scaling factor. With respect to the pre-buckling displacementsthen these are known exact, i.e. the true u(0) and v(0) are known.

Referring back to the stress-strain relations expressed in terms of the series expanded dis-placement components u, v, and w in Eqs. (5.17) through (5.22) it is observed that basedon the present solution the in-plane forces in Eqs. (5.39) through (5.41) becomes

Nx = N (0)x +

�N (2)x +N (11)

x

� 2"2 (5.47)

Nz = N (0)z +

�N (2)z +N (11)

z

� 2"2 (5.48)

Nxz = N (0)xz +

�N (2)xz +N (11)

xz

� 2"2 (5.49)

From this it is evident that the collapse criterion then changes to

Ac

� 2"2

�2+Bc

� 2"2

�+ Cc = 0 (5.50)

i.e. a second-order equation in 2"2 which can be solved yielding the same result as inEq. (5.46) only the collapse parameter will be in the form of ( ")2c. Hence, the parameter is just an arbitrary fractional part of the perturbation parameter " and therefore of nosigni�cance for the result.

Obtaining the critical buckling load and further the ultimate collapse load of the plate �eldin the form of stresses, is thus possible by solving the von Karman equations under theconditions assumed so far.


5.6.6 Plate Buckling

To determine the buckling load the eigenvalue problem in Eq. (5.28), i.e.

Dr4w(1) �N (0)z w(1)

;zz � 2N (0)xz w

(1);xz �N (0)

x w(1);xx = 0

needs to be solved. As mentioned earlier (page 64) the eigenvalue problem has the form(A � �B)� = 0 where � is the eigenvalue and � the corresponding eigenvector. This isthe form of the generalized eigenvalue problem and there are numerous references on itssolution (e.g. Bathe and Wilson [2]). In this case the solution is done most conveniently ina numerical framework consisting of a �nite di�erence description and the inverse iterationmethod (cf. Appendix C).

Introducing the direct/shear ratio Nz=Nxz = tan(') and applying a base load of

N =�2D

b2; Nz = cos(')N ; and Nxz = sin(')N

then from the eigenvalue � the classical buckling coe�cients becomes kz = cos(')� andkxz = sin(')�, relating the critical stresses as

�z;cr = kz�2E

12 (1� �2)

�t

b

�2(5.51)

�xz;cr = kxz�2E

12 (1� �2)

�t

b

�2(5.52)

Referring back to when the boundary condition was de�ned for the plate problem, it wasassumed that the in-plane loading consisted of a uni-axial direct loading in the longitudinaldirection combined with a constant shear force proportional to the direct in-plane loadingby a factor #. Thus, the above de�nition then replaces this factor as # = tan(').

In Fig. 5.3 the critical buckling coe�cients kz and kxz for direct and shear loading respectivelyare shown as buckling interaction curves, i.e. kxz versus kz. The results are obtained byapplying the present procedure to three plates with di�erent aspect ratios of 1.0, 1.5, and3.0. The buckling coe�cient for pure direct loading is well known to be kz = 4 for a squareplate and also for integer aspect ratios. This is clearly also obtained by the present solution.The results have further been compared with the buckling solutions in standard text books3

and with results by Murray [30], which all showed excellent agreement.

It can therefore with con�dence be conclude that the present model predicts the bucklingbehavior of the plate �eld su�ciently accurately for the purpose at hand.

3See further Appendix C, Figs. C.8 and C.9, and adjoining text.


Figure 5.3: Buckling interaction curves for three di�erent length over width ratios.

5.6.7 Plate Collapse

Determining the ultimate load for the plate �eld entails �rst the solution of the post-bucklingin-plane displacements which is done by running through the four steps previously established(see page 69) and then applying the collapse criterion in Eq. (5.46) to determine the collapseperturbation (loading) parameter "c.

Once the collapse loading parameter is known the corresponding collapse load can be estab-lished in the form of stresses. More precise, the mean axial stress at collapse can be foundas

�ac = �z;cr +1

t

1

b

Z b/2

� b/2

�N (2)z +N (11)

z

�dx

��z = /2

"2c (5.53)

along with the mean shear stress at collapse found as

�sc = �xz;cr +1

t

1

b

Z b/2

� b/2

�N (2)xz +N (11)

xz

�dx

��z = /2

"2c (5.54)


Figure 5.4: Collapse interaction curves for a square plate with threedi�erent plate slenderness ratios.(Experimental results adopted from Harding [19, Fig. 8.21]).

Applying this procedure to a square plate �eld with three di�erent slenderness ratios,b=t = 60, 120, and 180 produces the interaction curves for the collapse stress shown inFig. 5.4 in comparison with experimental results from Harding [19]4. Actually, only the low-est slenderness ratio of b=t = 60 is really of interest for typical ship structures. Signi�cantlyhigher ratios, which indicates thinner plates, are uncommon in merchant ship structures,mainly to avoid the possibility of elastic buckling of the plate �elds, which is in essence theconsequence of selecting that low a slenderness of the plate, i.e. a relatively thick plate.

From the plot it seems fair to conclude that the present collapse criterion, based on thetransgression of the von Mises stress at one of the unloaded edges, yields a reasonablyaccurate approximation to the experimental �nding by Harding. Especially when consideringthat the present collapse formulation is build on a second order perturbation expansion ofthe plate behavior beyond the singular perturbation point at the critical buckling load.

Of further interest among the results derivable at plate collapse is the mean axial edge stressat collapse obtainable as

�ec = �z;cr +1

t

1

`

Z /2

� /2

�N (2)z +N (11)

z

�dz

��x = b/2

"2c (5.55)

4See further Appendix C, Fig. C.10, and adjoining text.


This because it together with the mean axial stress at collapse �ac expresses the e�ectivewidth of the plate in the post-buckling region up until collapse occurs.

The importance of this is due to the e�ective width principle being used to model the loss insti�ness the plate will experience after buckling. Similarly, the mean axial stress at collapse�ac will also, together with the yield stress of the sti�ener �y,s, de�ne the e�ective widthapplicable in the post-collapse region. Moreover, the mean shear stress at collapse �sc is usedto de�ne the shear yield stress �y of the plate (see Section 4.2.2). This is used to enforce theassumed linear elastic, ideal plastic shear behavior of the total shear stress distribution ofthe entire hull cross section, when scaled by Eq. (4.11) as described in Section 4.4.1.

5.6.8 Stress Assessment

The only task remaining before the elastic compressive load-displacement response can bedetermined, is to �nd the axial load corresponding to the occurrence of the di�erent charac-teristic stresses in the plate and in the extreme �ber of the sti�ener.

This is done in exactly the same way as for the elastic tensile region, as the behavior still isassumed linear elastic. Thus, for compressive loading (P > 0) the normal stress distributionat the middle of the beam-column can be found as5

�xx(z) = �PA+

�Ez

sin(� /2)

�Pw0

`(P � PE)+

q`

2P+ '

!� �2EPw0z

`2(P � PE)� Eqz

P(5.56)

where

� =

sP

EI

PE = EI��

`


An elastic pre-buckling, buckling, and post-buckling analysis of the beam-column can thenbe performed for which the assumed load-displacement response can be described by thefollowing three characteristic points:

1. Plate buckling: The fully e�ective beam-column will �rst reach the limiting axial loadfor buckling of the plate �eld. That is, it is assumed that when the compressive stressin the plate part of the beam-column reaches the critical level �cr, then the stress inthe extreme �ber of the sti�ener is still below compressive yielding.

5See further Appendix B, Eq. (B.8).


2. Plate collapse: Beyond the critical load, the beam will be modeled with a reducede�ective width of the plate part, to describe the loss of sti�ness after buckling. Thewidth is reduced by the ratio between the mean edge and axial stress at collapse, i.e. by�ec=�ac. Collapse of the plate �eld is then taken to occur when the stress in the platereaches the mean edge stress at collapse �ec for the e�ective beam-column. Thus, forthe real full e�ective beam-column the stress in the plate at collapse will actually bethe mean axial stress at collapse, i.e. �ac. It is still assumed that the collapse stress willbe reached in the plate before the extreme �ber of the sti�ener reaches yield capacity.

3. Total collapse: After the plate collapses, the �nal point will be reaching the yieldcapacity of the sti�ener. The e�ective width of the plate is reduced even further bythe ratio between the yield stress of the sti�ener and the mean axial stress at collapse,i.e. by �ac=�y,s. Thus, a further loss of plate sti�ness is modeled such that when thestress in the extreme �ber of the sti�ener reaches yield capacity, the stress in the platefor the full e�ective beam-column will remain equal to the mean axial stress at collapse.

The behavior of the beam-column for loading beyond this point will be assumed totallyplastic, and thus point three will mark the ending of the elastic compression range of thebeam-column response.

However, the concerns about the in uence of an initial out-of-plane de ection w0 of thebeam-column, and of the presence of a uniformly distributed load q, still remains as outlinedin the previous description of the elastic tensile behavior of the beam-column6. That is,the possibility exist, that for certain combinations of w0 and q, the stress in the extreme�ber of the sti�ener may happen to reach its compressive yield capacity prior to any of thecharacteristic loads for the plate part.

In this situation, it will be assumed that the yielding of the sti�ener will be the ultimateresponse, based on the same argumentation as was the case in the tensile region. Namely,that while probably being a somewhat conservative assumption, it is nevertheless the mostreasonable to make within the limitations of the present solution scheme.

5.7 Plastic Compression Region (Unloading)

In the plastic compression region the axial load of the beam-column decreases with increasingdisplacement, leading to a negative slope of the stress-strain curve as shown in Fig. 5.1.Hence, the beam-column is exhibiting its post-collapse behavior in this region of the load-displacement response.

To describe the post-collapse behavior of structures, models based on plastic mechanismsare commonly used. The fundamental assumption of the plastic mechanism concept is that

6See Section 5.5, page 57.

5.7. Plastic Compression Region (Unloading) 77

Figure 5.5: Plastic hinge con�guration.

the material behavior is rigid-plastic. That is, it is assumed that the material strain is zerofor stresses less than the yield stress. Further straining of the material is assumed to happenat a constant stress level equal to the yield stress.

Moreover, the idea of using plastic mechanisms to describe the post-collapse behavior ofbeam-columns has successfully been applied by e.g. Rutherford and Caldwell [42]. Based onthis, the concept of plastic mechanisms is chosen for application in the present procedure, byassuming that three plastic hinges forms in the beam-column as shown in Fig. 5.5, i.e. onein each end of the beam-column and one at the middle. Further, it is assumed that the crosssection of the beam-column initially is { and remains { prismatic.

The type of plastic mechanism used is then the probably most simple thinkable, namely justthe plastic moment of the total cross section about a horizontal axis in each hinge. Moreelaborate mechanisms can be constructed as shown by e.g. Kierkegaard [23] and Murray[30]. These take into account the actually observed folding mechanism occurring when thebeam-column is crushed in compression and thereby yields a much more accurate descriptionof the post-collapse behavior.

Consequently, these more sophisticated models will in general predict a more rapid unloadingof the beam-column compared to the currently proposed simple three-hinge model whichmust be expected to be somewhat more rigid than the more advanced models. This mightbe essential in obtaining a good description of the unloading response of the beam-column.However, the simple model has the advantage of being very rapid to evaluate in a numericalframework. Thus, even though the use of this model is an non-conservative approximationto the real behavior, this simple approach will be used because of its better computationalperformance.

Looking at a half model of the beam-column as shown in Fig. 5.6, and assuming uniformyield stress �y for the cross section, then the force equilibrium of the beam cross section iseasily found to be

Z a1

0�y(�)b(�)d� �

Z H

a1

�y(�)b(�)d� = P (5.57)

Z a2

0�y(�)b(�)d� �

Z H

a2

�y(�)b(�)d� = �P (5.58)


Figure 5.6: Plastic hinge force and moment equilibrium.

which may be rewritten in a more convenient form using thatR a0 =

R b0 +

R ab yielding

2Z a1

0�y(�)b(�)d� �

Z H

0�y(�)b(�)d� = P (5.59)

2Z a2

0�y(�)b(�)d� �

Z H

0�y(�)b(�)d� = �P (5.60)

Thus, for a given axial force P the extent of both the compression and the tension zonesin the cross sections of the beam at each of the three plastic hinges, can be found fromEqs. (5.59) and (5.60). That is, a1 and a2 are known parameters. Hence, the de ection ofthe w at the middle of the beam can now be determined using moment equilibrium

M1 +M2 + 1/8 q`2 � wP = 0) w =

M1 +M2 + 1/8 q`2

P

M1 =Z a1

0�y(�)b(�)�d� �

Z H

a1

�y(�)b(�)�d�

M2 =Z a2

0�y(�)b(�)�d� �

Z H

a2

�y(�)b(�)�d�

P =Z H

a2

�y(�)b(�)d� �Z a2

0�y(�)b(�)d� (Eq. (5.58))

9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;

;

5.7. Plastic Compression Region (Unloading) 79

Figure 5.7: Plastic displacement de�nition.

w =

Z a1

0�y(�)b(�)�d� �

Z H

a1

�y(�)b(�)�d� +Z a2

0�y(�)b(�)�d� �

Z H

a2

�y(�)b(�)�d� + 1/8 q`2

Z H

0�y(�)b(�)d� � 2

Z a2

0�y(�)b(�)d�

which can be rewritten as

w =

2

Z a1

0�y(�)b(�)�d� +

Z a2

0�y(�)b(�)�d� �

Z H

0�y(�)b(�)�d�

!+ 1/8 q`2Z H

0�y(�)b(�)d� � 2

Z a2

0�y(�)b(�)d�

(5.61)

This is then the predicted de ection of the initially perfect beam-column. Included in thissolution is the e�ect of a uniformly distributed load q. Further, by reducing the e�ectivewidth of the plate part of the beam-column by the ratio between the mean edge stress atcollapse and the yield stress of the sti�ener, i.e. �ec=�y,s, then both the presence of shearstress in the plate, and the collapse behavior of the same, is accounted for in the model.That leaves only the initial out-of-plane de ection w0 to be accounted for.

In this context, the initial de ection is in fact the total de ection of the beam-column atultimate collapse, not to be confused with the assumed sti�ener imperfection w0. Therefore,associated with the initial elastic de ection w0;e there will be a corresponding initial elasticend displacement �0;e. This give rise to the following problem: If the collapse load foundin the elastic collapse analysis is inserted �rst in to Eqs. (5.59) and (5.60) and then intoEq. (5.61), then an out-of-plane de ection wp,c is found which is general will be di�erentfrom the elastic de ection w0;e. However, the di�erence will be small. Thus, to overcomethis problem the elastic de ection is regarded as an initial small out-of-plane de ection thatexists just before the unloading of the beam-column is initiated. Any further straining isthen considered to be totally plastic.


Under these assumptions the situation is as shown in Fig. 5.7 and the plastic contributionto the axial displacement of the beam-column �p can then be found from the total plasticdisplacement wp (obtained by Eq. (5.61)) by applying the Pythagorean theorem yielding

�p(P ) = 2

0@s`2

4� w2

0;e �s`2

4� w2

p(P )

1A (5.62)

The total axial displacement in the plastic unloading region is then found by adding theelastic displacement at collapse �0;e based on Eq. (5.3) with the plastic contribution justfound as �p for a given post-collapse axial load P .

Thus, with the plastic response now established, the �nal piece of the total load-displacementresponse of the beam-column in combined loading has been established ranging from plastictensile response to post-ultimate, plastic compressive response.

5.8 Limitations of Method

In the course of establishing the idealized load-displacement behavior of a beam-column incombined loading, a number of assumptions have proven necessary to make, to facilitate thefurther development of the procedure and obtaining the �nal model. Thus, having completedthis work as outlined in the previous sections of this chapter, it would be appropriate to recapthese assumptions for added clarity. The most important of these will be presented in thefollowing.

The presence of shear stresses in the plate part of the beam-column lead to the introductionof a collapse criterion based on transgression of the plate yield stress by the von Mises stress.In the tensile range of the response this was simply done based on the average shear stress� and the prescribed yield stress �y,p such that the tensile yielding stress became

�tc =q�2y,p � 3� 2 (5.63)

In the compressive range, a more elaborate scheme was deployed to describe collapse for theplate �eld. Here it was assumed the collapse happened when the mean von Mises stress �vat the mid-plane of the plate along one of the unloaded edges reached the prescribed yieldstress �y,p. This criterion was veri�ed against experimental results and concluded usable forthe present research.

In the post-ultimate compressive region is was further assumed that the behavior could bemodeled su�ciently accurate by a simple three-hinge plastic mechanism. And, in combina-tion here with, the possibility of sti�ener tripping was ignored. Similarly in the post-ultimate

5.9. Implementation Notes 81

tensile region, the possibility of necking of the sti�ener was ignored, as it was assumed thattotal collapse of the entire hull cross section would occur before necking initiated in any ofthe beam-columns.

Moreover, in the linear part of the response is was assumed that the ultimate loading capacityin both tensile and compressive loading, was either when the extreme �ber of the sti�ener,or when the plate, reach its maximum capacity, which ever occurs at the lowest load level.This, no matter what the remaining load carrying capacity of the non-failed component was.This assumption was chie y made necessary by the possibility of bending moment in thebeam-column, caused by the presence of a uniformly distributed loading. It is noted, thatthis is a slightly conservative assumption.

Finally, it was made necessary to assume linear behavior in the immediate vicinity of zeroaxial load, i.e. around P = 0. More accurately, it was assumed that there must exist anegative (tensile) load for which nonlinear e�ects in neither the plate, nor the sti�ener, hasonset. Likewise, there must exist a positive (compressive) load ful�lling the same require-ment, i.e. linear behavior of both plate and sti�ener. If this is not the case, the developedprocedure cannot describe the behavior7.

5.9 Implementation Notes

The implementation of this procedure in a computer code is a straight forward process.However, a few points concerning how this was done, will be presented in the following.

First of all, the load-displacement curves are to be used by the overlying code performingthe overall system analysis of the entire hull cross section. Thus, the task we actually wantthe code to perform can be speci�ed as:

Given a forced rotation � of the strain-plane, and further for each i th beam-column, a constant shear stress �i, an initial uniformly distributed load qi, andan initial de ection w0;i { Then build the corresponding load-displacement curvefor each of the beam-columns.

The solution for the critical and collapse behavior of the plate is independent of all but oneof these parameters, namely the shear stress �i. Thus, it will be possible to solve the plateproblem only for a discrete set of shear stress and axial stress combinations, and the useand interpolation scheme based on those discrete values to obtain the plate results for anyload combination. The actual number of discrete values needed for a su�ciently accuratedescription will of course depend highly on how smooth the function will be. This is alldiscussed in detail in the Summary section of Appendix C. As the plate problem is solved

7See further the discussion of the beam-column stress solution at the end of Appendix B.


using the �nite di�erence approach, it is by far the most time consuming part of the code.Thus, it will be highly bene�cial for the performance on the code, if this scheme is adopted.

With this done, the remaining work to be carried out can be divided into two tasks. One,establish the linear elastic part of the response, and two build the post-ultimate compressiveresponse by means of the plastic hinge mechanism. Regarding the later, then this is quitesimple and thus mandates no further attention. However, with respect to the linear elasticpart a few comments should be made.

The key interest in this region of the response is the establishment of the following �veload-displacements (�,P ) points:

1. Tensile yielding of the entirebeam-column.

2. Tensile yielding of the sti�ener.

3. Buckling of the plate.

4. Compressive collapse of the plate.

5. Compressive collapse of the entirebeam-column.

as illustrated graphically to the right. In this order these points will make up the responsefrom tensile failure to compressive collapse, where the plastic hinge solution will take overthe description. Each of these points are in fact described by two parameters, namely thestress in the mid-plane of the plate and in the extreme �ber of the sti�ener. As it is veryunlikely, that both these stresses will be produced by the same axial load, the assumptionis that it is the lowest of the two forces that describes the response. Hence, one approachwould be to solve for both these forces one-by-one, and then select the lowest.

This will however not be the most e�cient approach, as one may end up in the situationwhere the �rst found force turns out to be the smallest, and thus the resources used to �ndthe second force was kind of wasted. An alternative scheme has therefore been successfullyimplemented in the present code. It involves the generation of a sort of safety index function.Assuming that the target stress in the plate is within the range [�lowp ; �highp ] and in the sti�ener[�lows ; �highs ], then this safety index function (P ) is de�ned as

(P ) = min [ p(P ); s(P )] where

p(P ) =

8<: �highp � �p(P ) for �p(P ) > 0

�p(P )� �lowp for �p(P ) � 0

s(P ) =

8<: �highs � �s(P ) for �s(P ) > 0

�s(P )� �lows for �s(P ) � 0

5.10. Test Application of the Procedure 83

This function will have the property, that a positive value will imply that the stress inthe plate and in the sti�ener, both are within the ranges speci�ed. A negative value willindicated that one of these stresses is out of bounds, and �nally a zero value will be the sameas one of the stresses exactly is equal to one of the four bounds.

This can be utilized in the following way. Say the object is to �nd the axial load at bucklingof the plate (i.e. the third point). Then the limit for the stresses will be de�ned by thebuckling stress �cr for the plate and the yield stress �y,s for the sti�ener. The axial responsesought is then de�ned as the load where the stress in the plate within the range [�tc; �cr],i.e. higher than the stress causing tensile yielding and lower than the buckling stress. For thesti�ener the stress range simply becomes [��y,s; �y,s], i.e. within the yielding stress. Then,by solving for a positive root (positive because of compression) in the safety index function,the axial force obtained will inherently be the lowest giving either �cr in the plate, or �y,s inthe sti�ener.

This approach has proven not only very fast, but also very robust. Thus, this is the procedureimplemented and used to identify the �ve points in the elastic part of the response in thepresent code. An alternative procedure based on the full numerical solution to the di�erentialequation governing the beam-column has been tested as outlined in Appendix D. However,this approach was deemed in vain due to numerical problems and overall slow performance.This procedure has therefore been abandoned as unusable within the context of the presentresearch.

In relation to the elastic part of the load-de ection response, the possible problem of nonlin-ear behavior in the immediate vicinity of zero axial load which was explained previously, is inthe present code tackled by simply ignoring the beam-column. That is, if the beam-columnis in an unsafe condition when it is unloaded with respect to axial force, it is in essenceconsidered to be ine�ective in the total hull cross section.

The �nal comment to be made is pertaining to the �nite di�erence solution used to obtainthe descriptive stresses for the plate part of the beam-column8. The procedure requiresthe repeated solution of large matrix system. These are however very sparse in the presentformulation, thus a solver utilizing this property will be advantageous to use. Further, asa lot of the beam-columns in a typical hull cross section will be identical, checking for thisand only solving once for each distinct plate also improves the performance of the codesigni�cantly.

5.10 Test Application of the Procedure

To test the implementation of the procedure a test beam-column has been analyzed. Thedimensions of the beam-column is shown in Fig. 5.8, along with the results for di�erent levels

8A detailed description of all the aspect of implementation of this procedure is presented in Appendix C.


Figure 5.8: Load-displacement curves demonstrating the e�ect of shear stress.

of shear stress � . The shear stresses range from zero to total yield shear stress �y in steps often percent, giving a total of eleven investigated levels. Of these the zero shear stress, the�fty percent of yield, and the total yield shear stress has been highlighted.

The material properties for the beam-column is a Young's modulus E of 207 GPa and aPoisson's ratio � of 0:3. The yield stress for the plate and the sti�ener is identical being359 MPa. Further, the beam-column is assumed without any initial imperfection, i.e. w0 = 0,and the end rotation ' of the beam-column was set to zero. Finally, no uniformly distributedloading q was applied to the beam-column.

Hence, the result presented in Fig. 5.8 is for an ideal beam-column. From the graphs it isobserved how the increasing presence of shear stresses in the plate reduces the axial loadcarrying capacity of the beam-column, just as expected (see e.g. similar results published byMelchior Hansen [17]). Further, it is noted that the behavior when the plate is exposed to

5.11. Beam-Columns without Sti�eners 85

yield shear stress �y is that of the sti�ener alone, which is the expected behavior as shearstresses only a�ect the plate part of the beam-column.

The e�ect of end rotation and initial de ection of the beam-column has also been investigatedfor this test case. It was found that the behavior also here was as expected. That is, theintroduction of an initial out-of-plane de ection whether caused by an initial imperfectionor an end rotation of the beam-column, causes a reduction in the load-carrying capacity ofthe beam-column, though comparatively small with respect to the ideal behavior. Moreover,the presence of a uniformly distributed load on the beam-column has been tested. Here,the behavior again was in compliance with the expectation, being a reduction in the loadcarrying capacity of the beam-column. However, a graphical presentation of the changes inthe load-displacement curves due to these deviation from the ideal behavior has been leftout, as the comparatively small changes made distinction of the di�erent curves at best veryhard when keeping the loads and the sti�ener imperfection within reasonable ranges.

5.11 Beam-Columns without Sti�eners

The overall analysis method has so far been derived under the assumption that the entirehull cross section could be represented as a number of discrete beam-columns, each consistingof a sti�ener and the attach plating. However, the presence of plate �elds in the form ofstringers in the side and girders in the bottom, which are unsti�ened in the longitudinaldirection or with light secondary sti�ening, are not uncommon in typical hull structures.Although these members constitutes only a small fraction (a few percent or so) of the totallongitudinal strength, it will nevertheless be preferable if the present method could accountfor this.

For a plate alone, the elastic response from tensile yielding to compressive collapse can easilybe established by application of the previously derived results. Further, continuing alongthe same path as used previously, the tensile post-ultimate behavior can be assumed idealplastic. Thus, only the post-ultimate compressive response needs to be reconsidered whenthe beam-column consists only of one plate. Here, the problem is that without a sti�ener,the collapse mechanism of the plate cannot reasonably be represented by the three-hingemechanism used previously.

5.11.1 Overall Buckling Mechanism

To address this problem a new plastic mechanism based on the beam-column acting as a platerather than a beam has been introduced. The mechanism is based on an assumed overallbuckling mode of the plate as illustrated in Fig. 5.9. That is, an adoption of the classicalplastic yield-line model for a laterally loaded plate �eld, with the e�ect of an in-plane axialloading incorporated.


Figure 5.9: Overall buckling mechanism.

For this mechanism the internal work by the plastic moment Mp = �yt2=4, corresponding to

an end displacement � is

V i = 2Mpb�1 + 2Mp(`� � � b tan )�2 + 4(Mp`DE sin �2 +Mp`DE cos �1) +ZV"�ydV

= 4Mpw

2

tan +`� �

b

!+ 1/2tb�y� (5.64)

where the geometrical relations yields the two angle as �1 = 2w/b tan and �2 = 2w/b. Further,the length between point D and E is `DE = b/2 cos and the de ection is w =

p�2 + �b tan .

The external work by the axial load P and the lateral load q becomes

V e = q�(`� � � b tan )b

w

2+ b2

w

3tan

�+ �P

=qwb

6(3(`� �)� b tan ) + �P (5.65)

Requiring equilibrium between the internal and external work then leads to the followingrelation between the axial load P and the end displacement � for a given angel

P (�; ) =

s1� b

�tan

(4Mp

2

tan +`� �

b

!+1

2tb�y� � qb

6(3(`� �)� b tan )

)(5.66)


Figure 5.10: Straight edge folding mechanism.

As this is an upper bound solution, the response force P corresponding to an end displace-ment � is obtained by minimizing the expression with respect to the angle . Doing thisanalytically resolves in a third order equation in tan in which one of the roots will be theright solution. However, solving for the roots analytically in this third order equation is quitetedious, why in the present formulation, the minimization of Eq. (5.66) will be done numer-ically. Thus, the post-ultimate compressive load-displacement behavior can be estimated bythe use of Eq. (5.66).

This post-collapse model of overall buckling is justi�able for plates which are heavily laterallyloaded. However, if the lateral loading is moderate or non-existing, then overall buckling willnot be the expected response of the plate �eld in the post-ultimate region. Especially not,as the in-plane straining of the plate is ignored in the present formulation. A local straightedge (multiple) folding mechanism like the one illustrated in Fig. 5.10 will be more in linewith the expected behavior. Thus, to assess the validity of the overall buckling model, alsothis local folding mechanism will be investigated.

5.11.2 Local Folding Mechanism

The geometry of the straight edge folding mechanism is shown in Fig. 5.10 for an axiallyloaded plate. Following Kierkegaard [23] the rate of internal energy dissipation for half thewidth c = b/2 of this mechanism is,

_Ei = �ytkH2 ( _"eI + _"eII) (5.67)


where _"eI and _"eII are the e�ective strain rates in the the two triangular regions at the endof the fold (see Fig. 5.10), given as

_"eI =sin 2�

kp3

s1� cos2 2�

4k2 + 1

_� and _"eII =2p3sin� _�

The external energy dissipation caused by the end displacement � alone then simply becomes_Ee = P _� where the end displacement relates to the folding angle � as � = 2H(1 � cos�)giving

_Ee = 2PH sin� _� (5.68)

The height H of half the fold is dependent on the plastic moment Mp = �yt2=4 and the

squash load Ny = �yt. Further, the factor k de�ning the length kH is found by minimizingthe rate of internal energy dissipation at an early stage of collapse, i.e. for � = 0, leading tok = 0:5733. Thus the height H becomes

H =

vuutp3�Mpc

Nyk=

sp3�tb

8k

Requiring equilibrium between the rate of internal and external energy dissipation ( _Ei = _Ee)then leads to the following relation between the axial load P and the folding angle � for thefull width b of the plate

P (�) =�ytkH ( _"eI + _"eII)

sin�(5.69)

from which the load-displacement behavior can be obtained using the relation between theend displacement and the folding angle, i.e. � = arccos (1� �/2H). Moreover, the meancrushing force Pm can be found by integration over the entire folding process giving

Pm =2p3NyH

0@sk2 +

1

4arcsin

1p4k2 + 1

+ k

1A (5.70)

These last two expressions (Eqs. (5.69) and (5.70)) can thus be used in comparison with theoverall buckling model (Eq. (5.66)) to test the performance of the later with respect to pureaxial loading on the unsti�ened plate �eld.


Figure 5.11: Compressive load-displacement behavior of a stringerwithout lateral loading.

5.11.3 Comparison

This comparison has been done for a test plate of typical dimensions, and the results areshown in Fig. 5.11 in the form of load-displacement curves. The length and width of the testplate are ` = 1:0 m and b = 0:3 m respectively, and the thickness of the plate is t = 6 mm.The material parameters for the plate are a yield stress of �y = 359 MPa, a Young's modulusE = 207 GPa, and Poisson's ratio � = 0:3.

With these scantlings, the theoretical compressive length of one fold becomes 2H = 0:092 m,which also is observed in Fig. 5.11 to be the length of each of the repeated patterns ofunloading indicating multiple folds. Of course, this length is higher than what will be thereal length, as the thickness of the plate is not accounted for in this approximation. In theliterature this discrepancy is normally handled by introducing an e�ective folding length in


the range of sixty to seventy percent of the theoretical value, and then redistributing theenergy spend in one fold over that length.

Nevertheless, for the purpose of comparison, this hardly makes no di�erence, as the interesthere is more on the mean crushing force Pm, than on the repeated pattern of unloading causedby the multiple fold collapse of the plate. Also, it is noted that the transition from one to twofolds in the post-collapse response occurs at a signi�cant level of straining (approximatelynine percent), at which collapse of the entire hull cross section will at least be eminent if notalready surpassed, for typical hull structures.

Therefore, observing in Fig. 5.11 the load-displacement curve resulting from the overallbuckling approach in comparison with the mean crushing force as predicted by Eq. (5.70),with special emphasis on the response within the moderate straining range, a good correlationis found. Granted, the unloading of the overall buckling model is not as rapid as that of thestraight edge folding mechanism. However, the response predicted by the overall bucklingmodel has the, for the present scope, advantage over the straight edge folding mechanism inbeing at least one time di�erentiable, i.e. C1, and further yields a smooth unloading fromthe collapse load of the plate �led. These two features are important for the interpolationscheme applied in the overall system analysis of the entire hull. Therefore, as the correlationbetween the two solution methods is reasonably, it has been decided to use the overallbuckling approach in the present procedure. The implementation then follows the exactsame path as for the full beam-column, i.e. one having both a plate and a sti�ener part,with the elastic end displacement taken as an initial small displacement.

Finally, it is noted that because of the ignored contribution from the in-plane straining inthe overall buckling mechanism, the load-displacement response predicted by this mecha-nism falls below the mean crushing force Pm given by the straight edge folding mechanism.However, this occurs after excessive straining of the plate and is therefore insigni�cant inthe scope of the present investigation for reasons already described.

5.12 Summary

A procedure for establishing the load-displacement response of a beam-column exposed toand axial force P , an end rotation ', a constant shear stress level � in the plate part only,and a uniformly distributed load q acting only on the sti�ener perpendicular to the plate�eld, has been established. The procedure further allows for an initial imperfection w0 ofthe sti�ener only. Out-of-plane loading of the plate part is ignored as is initial out-of-planede ection of the plate. The beam-column can either be made up of both a plate and asti�ener, or it can be only a plate without any sti�ener.

The plate part of the beam-column has been model by a �nite di�erence solution to the vonKarman equations, and the result obtain hereby has been compared with known result forclassic plate �elds showing perfect accordance. Moreover, collapse of the plate part of the

5.12. Summary 91

beam-column has been based on the von Mises stress as the limiting capacity of the plate.The use of this approach has also been successfully veri�ed against available experiment.

Finally, the response from tensile yielding to compressive collapse of the entire beam-column,has been modeled by a linear elastic solution to the classical beam-column problem, modi�edby application of the e�ective width concept for the plate part. The response beyond tensileyielding has been assumed ideal plastic, whereas in the compressive post-collapse rangea simple three-hinge plastic mechanism has been applied in the case of a beam-columnconsisting of both a plate and a sti�ener. If the beam-column is made up only of a plate,i.e. no sti�ener, then the compressive post-collapse range is modeled by a plastic yield-linemechanism based on an assumed overall buckling mode of the plate �eld. The validity ofthis last approach has been veri�ed against a true compressive folding mechanism, wheregood agreement was obtained.

The procedure has then been applied to a test beam-column, the results of which has instilledcon�dence in its accuracy and performance.

Chapter 6

Veri�cation of the Procedure

6.1 Introduction

In the previous two chapters a procedure for the evaluation of the ultimate and post-ultimatecapacity of a hull cross section has been established. During this derivation, the di�erentparts, which together makes up the entire procedure, was tested and veri�ed. Thus, withthe complete analysis procedure readily available, it is possible to continue the veri�cationprocess by applying the procedure to test cases for which results are known. Either in formof experiments or as theoretical solutions by other methods.

Two test cases has been selected for use in this veri�cation process. The �rst is one of the socalled Nishihara box girders investigated by Nishihara [31]. The second test case is a doublehull tanker structure analyzed by Melchior Hansen [17]. In the following these two caseswill both be analyzed by the present procedure and the hereby obtained results will then becompared with results published by the two authors mentioned above.

6.2 Nishihara Box Girders

Nishihara [31] performed a number of experiments to obtain the ultimate moment capacityfor four di�erent ship-like cross sections. Each of these was build to simulate the typicalconventional ship types, being tankers, bulk carriers, and container carriers as shown inFig. 6.1. Of these, especially the �rst one simulating a single skin tanker is interesting inthe present study because it is a double symmetric cross section. Hence, bending about thehorizontal axis will be identical to bending about the vertical axis. Further, there should beno di�erence between the ultimate hogging and sagging capacity of the cross section. Theseproperties rarely exist in real ship structures, but for the purpose of testing the asymmetricalforced curvature principle used in the present formulation, they are very convenient to have.

93

94 Chapter 6. Veri�cation of the Procedure

Figure 6.1: Nishihara test cross sections.

Moreover, nonlinear �nite element calculations on this test section have been published byMelchior Hansen [17] along with results from a beam-column analysis somewhat similarto the present procedure. All this, makes the Nishihara box girder an excellent and welldocumented test case to benchmark the present procedure against.

6.2.1 Model description

The geometry of the cross section for the box girder simulating a single skin tanker is shown inFig. 6.2. Further, Tab. 6.1 lists the geometric and material parameters for the cross section.The loading on the cross section was, in the experiments performed by Nishihara, purebending. No information about the initial imperfections in the cross section was reported in[31]. Nevertheless, this information was needed by Melchior Hansen [17] to verify a beam-column method developed speci�cally for the longitudinal strength of hull girders. Thus, toside-step this problem, Melchior Hansen performed a total of four nonlinear �nite elementcalculations. Each of these with a well de�ned initial imperfection being:

1. Only plate imperfection in the shape of two half sine waves between the frames, andone half sine wave between the sti�eners. The magnitude of the plate de ection washalf the thickness of the plate, i.e. 1:5 mm.

2. Plate and sti�ener imperfection, but only in the compressed deck panel. The shapewas one half sine wave both between the frames, and between the sides of the crosssection. The magnitude of the imperfection at the middle of the deck panel was set to2`=300 = 6 mm outwards from the center of the cross section.

3. Same overall plate and sti�ener imperfection as above, only changed in direction toinwards to the center of the cross section.

4. Same as the �rst model (i.e. only plate imperfection, but in the entire structure), butwith a residual stress �R of magnitude �R=�y = 0:50 imposed on the plating.

6.2. Nishihara Box Girders 95

Table 6.1: Sectional parameters for the single skin tanker box girder.

Plate thickness (overall) : 0.003 mYoung's modulus : 207 GPaYield stress : 288 MPaSectional area : 9.960�10�3 m2

Sectional modulus : 2.379�10�3 m3

Moment of inertia : 0.856�10�3 m4

Position of elastic neutral axis : 0.360 mElastic moment (�rst yield) : 685 kNmPosition of plastic neutral axis : 0.360 mPlastic moment : 787 kNm

Figure 6.2: Cross section (top) and beam-column model (bottom)of the single skin tanker box girder.


Table 6.2: Ultimate moment capacity for the Nishihara box girder.(Adopted from Melchior Hansen [17]).

ID Imperfection Moment capacity [kNm]# identi�cation Finite element Beam-column1 Plate imperfections 525 5602 Sti�ener imperfections inwards 583 5303 Sti�ener imperfections outwards 525 5304 Residual stresses 520 535

The moment capacities hereby obtained for pure bending, both from the �nite elementanalyzes and the corresponding beam-column solution are listed in Tab. 6.2. Of these, caseone and four are of no interest for the current project, as the procedure developed cannothandle plate imperfections, neither geometrical nor residual stresses. However, as sti�enerimperfections are accounted for in the current procedure, it can approximate the imperfectionimposed to the box structure in analysis cases two and three.

6.2.2 Collapse Analyzes

Two di�erent types of investigations have been performed for the Nishihara box girder. First,a total of three solutions for the ultimate moment capacity in pure bending about a horizontalaxis have been obtained using the present procedure. One of these solutions were obtainedwithout any initial imperfection in the structure, i.e. for a initially perfect condition of thebox girder. The remaining two solutions were derived with initial sti�ener imperfectionsidentical to the modes imposed in �nite element model two and three by Melchior Hansen[17]. The motivation for performing these three investigations has been to verify the presentprocedures ability to accurately predict the ultimate moment capacity. Hence, the e�ects ofshear stresses and asymmetrical bending has been ignored, and the focus has been set solelyon the moment capacity in pure bending.

The second type of investigations addresses one of these e�ects previously ignored, namelythe asymmetrical bending of the box girder. Here, the focus is on veri�cation of the procedureto handle this asymmetrical bending, more than on the hereby obtained moment capacities.This, because there is no other data for asymmetrical bending found to compare with.Nevertheless, the present box girder has the, for the present investigations very appropriate,property of being symmetrical about not only a horizontal axis, but also about a verticalaxis. Hence, the moment-curvature response will be identical for any two axes which areorthogonal, e.g. the vertical and horizontal global axes. This will be used to verify theprocedure.


Figure 6.3: Moment-curvature relations for the Nishihara box girder.(Comparative results adopted from Nishihara [31] and Melchior Hansen [17]).

Pure Bending Analyzes

The results obtained from the pure bending investigations are shown in Fig. 6.3 in the formof moment-curvature response curves. The curves are based on very small increments in thecurvature to insure a smooth and accurate representation. Further, the ultimate capacitiesfound by the present procedure are listed in the �gure for both hogging and sagging. Also,the corresponding result from both Nishihara [31] and Melchior Hansen [17] are listed.

Regarding these results, then neither author distinguish between hogging and sagging inthere listed results. For Nishihara this is understandable as the model used was symmetricabout the horizontal axis, and thus there is no di�erence between the hogging and saggingcapacity. However, for the two models used by Melchior Hansen the imposed imperfectionswere only present in the deck part of the structure. Consequently, there will be a di�erencebetween hogging and sagging moments, as the result obtained by the present procedure alsoclearly shows. Nevertheless, no distinction was made in [17].


Table 6.3: Calculated ultimate moment capacity for the Nishihara box girder.

Calculation Moment capacity [kNm]Identi�cation Mhog Msag FEM [17] Beam [17]Experimental data [31] 576 -576 { {Present: Initially perfect structure 572 -572 { {Present: Deck sti�eners inwards 573 -608 583 530Present: Deck sti�eners outwards 534 -529 525 530

Observing the response curves in Fig. 6.3 the �rst observation to be made is how the post-ultimate response clearly shows the trace of the one-by-one collapse of the three beam-columns making up each side of the box girder cross section. The collapse behavior ofthe entire cross section is initiated when the deck or bottom reaches their ultimate capacity.Unloading then sets in, with a signi�cant drop in response each time one of the three verticalbeam-columns reaches its ultimate capacity. This behavior is in accordance with the expectedresponse of the box girder.

Regarding the ultimate moment capacities as predicted by the present procedure, then theseare listed in Tab. 6.3 together with comparative results from [31] and [17]. To recap, thethree models investigated are: One, initially ideal, i.e. with no imperfections. Two, sti�enerimperfection in the deck structure only with the shape on one half sine wave and magnitudeequal to 2`=300 = 6 mm inwards to the center of the cross section. Three, identical to two,but with the sti�ener imperfection being outwards from the center of the cross section.

From the previous Tab. 6.1 the elastic (�rst yield) and plastic moments for the box girder wasdetermined to be 685 kNm and 787 kNm respectively. Thus, the �rst observation which canbe made is that all of the predicted moment capacities is below not only the plastic moment,but also below the elastic moment. This is atypical of ship structure where the ultimatemoment capacity normally is found to be above the �rst yield moment, and below theplastic moment. Nevertheless, comparing the capacities obtained by the present procedurewith both the experimental data from [31] and with the nonlinear �nite element resultsfrom [17], it is seen that a very satisfactory correlation is achieved. Thus, the less than �rstyield moment capacity of the box girder is attributed to the thin plates in the scale modelcompared to the cross sections dimensions.

Finally, it is noted that the present procedure not only yields a very good correlation withthe nonlinear �nite element results from [17] (less than �ve percent deviation is achieved),but also produces a comparatively better result than the beam-column procedure reportedin [17]. This even though the present procedure in its formulation is closely related tothe procedure used by Melchior Hansen in [17]. This is attributed to a more stringentand thorough inclusion primary of the �nite di�erence description for the plating, but alsoimprovements in the procedure used to establish the entire load-displacement response ofthe individual beam-columns compared with the approach used in [17].


Figure 6.4: Moment-curvature relations for the Nishihara box girder atdi�erent orientations � of the instantaneous neutral axis.

Asymmetrical Bending Investigation

The second investigation aims at verifying the the present procedures ability to describeasymmetrical bending of the hull cross section. The results hereby obtained are shownin Fig. 6.4 as moment-curvature response curves for four di�erent orientations � of theinstantaneous neutral axis (INA). These being 0o (horizontal), 45o, 90o (vertical), and 135o,as also shown in the �gure. As stated previously, the emphasis in this investigation is not onthe the moment capacity, but rather on the moment-curvature response, as this will show ifthe symmetry of the box girder is re ected also in the predictions of the present procedure.


Following the double symmetric property of the box girder cross section, the moment-curvature response will be identical for any two axes which are orthogonal. This explainsthe selected orientations � of the instantaneous neutral axis previously listed. Of these, 0o

and 90o should yield identical moment-curvature response, and likewise 45o and 135o, mustbe expected to predict the same response.

Regarding the later, looking at the response curves in Fig. 6.4 is it seen that the presentmethod indeed yields the same result for � equal to 45o and 135o. However, with respect tothe horizontal and vertical orientations of the instantaneous neutral axis (� equal to 0o and90o) a small discrepancy is observed in the two predicted moment-curvature curves. Theexplanation for this di�erence is to be found in the way the real box girder cross section isidealized as beam-columns.

One of the limitations of the present procedure is that it can only handle beam-columnsmade up of a plate part and a sti�ener part in a T-pro�le shape with the sti�ener assumedto be positioned at the middle of the plate part. Thus, the four corners of the box girder,which really are sort of V-pro�le beam-columns, are in the idealized representation given aswhat becomes a T-pro�le beam-column (see sketch in Fig. 6.4 or Fig. 6.2). Consequently,the idealized cross section of the box girder is not the same with respect to bending abouta horizontal and vertical axis respectively. It is, however, the same for bending about axesat 45o and 135o which explains why these results are the same as shown in Fig. 6.4.

A �ctitious cross section without the four corner beam-columns has therefore been analyzedby the present procedure to verify the explanation made. For this cross section bendingabout a horizontal and vertical axis yields the same result. The conclusion is therefore, thatthe present procedure is capable of accurately handling and describing asymmetrical bendingof a hull cross section.

6.2.3 Discussion of the Results

To summarize on the results obtained by analyzing the Nishihara box girder, two conclusionscan be made. First, the ability of the present procedure to accurately predict the ultimatemoment capacity is convincing. Secondly, the present procedure can handle asymmetricalbending of the cross section. Moreover, the predicted moment-curvature response �ts accu-rately with the expected post-ultimate behavior of structure, which promotes con�dence inthe idealized beam-column description implemented in the present procedure.

Thus, the veri�cation of the asymmetrical bending prediction by the present procedure isconcluded, leaving only the e�ects of shear stresses in the structure to be veri�ed. This willbe addressed in the following.

6.3. Double Hull Tanker 101

6.3 Double Hull Tanker

The e�ect of shear stresses on the ultimate moment capacity was investigated by MelchiorHansen [17] for a typical double hull tanker of some 30.000 dead-weight-tons. The loadingapplied was a combination of bending moment about a horizontal axis, and a vertical shearforce. For this loading, results are presented in [17] both as moment-curvature responsesat di�erent levels of shear, and as ultimate capacity interactions charts, i.e. the ultimatehogging and sagging moment as a function of the shear force. Further, two conditions ofthe structure was studied. One, where the scantling of the cross section was equal to theas-build tanker and one where a severe state of corrosion was imposed on the plating throughreduced thicknesses by two-thirds at di�erent locations such that the section modulus of thecorroded cross section equals ninety percent of the minimum requirement prescribed by theInternational Association of Classi�cation Societies [21].

For the purpose of veri�cation, these results will be established by application of the presentprocedure, and then compared with the �ndings published in [17]. The scope of the in-vestigation will therefore be limited to only those conditions reported in [17] to allow for acomparison. That is, only shear caused by a vertical shear force and bending about a hori-zontal axis will be investigated. Moreover, only the original, as-build condition of the tankerwill be investigated. This because the purpose at hand is veri�cation of the present proce-dure only { not a parameter study as reported in [17]. However, on the topic of corrosion itshould be mentioned that including this simply as a uniform reduction of the plate thicknessas done in [17] may lead to an overestimated ultimate capacity. Resent �ndings publishedby Mateus and Witz [28] indicates that using a quasi-random thickness surface model, asopposed to the traditional uniform thickness reduction, shows a signi�cant discrepancy (ashigh as eight percent) between the ultimate capacity predicted by the two methods. Theconclusion in [28] is that the uniform thickness reduction produces optimistic results andis therefore arguably inadequate for design purposes. Although the present procedure aimsmore at being a emergency response tool than a design tool, it may still be important tothoroughly consider the modeling of corrosion, to obtain realistic ultimate capacities.

6.3.1 Model Description

The geometry of the midship section for the double hull tanker is shown in Fig. 6.5. Further,a list of the principal dimensions for the vessel along with the sectional parameters for themidship section is given is Tab. 6.4. Regarding the existence of imperfections in the crosssection, then [17] o�ers no description of any geometric imperfection. However, detailedinspection of the background material for the tanker model, made available to the authorby Melchior Hansen, revealed the imperfections applied to the model in [17] to be an initialsti�ener de ection of /200 = 0:02 m where ` is the frame spacing, i.e. the length of theindividual beam-columns in the idealized hull cross section.


Table 6.4: Principal dimensions and sectional parameters for the double hull tanker.

Principal Dimensions:Length between perpendiculars Lpp : 168.56 mLength overall Loa : 172.56 mBreadth B : 28.00 mDepth D : 14.90 mDraught T : 10.90 mDisplacement r : 42500 m3

Frame spacing amid ship ` : 3.925 mSectional area A : 2.80 m2

Sectional Parameters:Moment of inertia Iyy : 101.18 m4

Elastic neutral axis �ENA : 6.41 mPlastic neutral axis �PNA : 4.13 mFirst yield moment My : 2.92 GNmPlastic moment Mp : 4.64 GNmSection modulus (deck) Wdeck : 110.13 m3

Section modulus (keel) Wkeel : 157.79 m3

Figure 6.5: Midship section of the double hull tanker.


The tanker is made with the use of high tensile steel (HTS) for the longitudinals and theinner skin of the double hull, whereas almost all of the outer plating is normal mild steel.The yield stress for the two types of steels are 353 MPa and 265 MPa respectively. Thus, asthe yield stress of the sti�ener is higher than, or equal to, the yield stress of the plate, thetanker ful�lls the assumption made by the present procedure, allowing application withoutany further concerns.

6.3.2 Combined Loading Analyzes

For this tanker, Melchior Hansen [17] performed an analysis with combined loading consistingof bending moment about the horizontal axis, and vertical shear force. This was done for onlyhalf of the midship section as the method used in [17] relies on symmetry of the cross section.The e�ect of the vertical shear force was in the analysis approximated by an assumed shearstresses distribution were the shear stress along the height of the cross section was constantand along the width varying linearly from zero at the center to the constant magnitude atthe side. The magnitude of this approximate shear stresses distribution at the side of thecross section was then found by equating it to the resulting vertical shear force. Regardingthe accuracy of this approximation, then referring back to the previously shown shear stressdistributions caused by a unit loading on a typical hull cross section (Fig. 4.3), it can beconcluded that this assumed shear stress distribution is not a bad approximation to the realbehavior for the present vessel. The results hereby obtained, i.e. for half of the cross section,are in [17] listed as an ultimate shear force capacity of 81:1 MN, and the ultimate momentcapacities 2:03 GNm in hogging and 1:63 GNm in sagging equivalent to approximately 87%and 70% of the plastic moment respectively.

The same vessel has been analyzed by application of the present procedure which appliesthe true shear stress distribution and the ultimate capacities hereby obtained are listed inTab. 6.5 for shear force and moment. Also listed in the table are the longitudinal strengthparameters, i.e. the �rst yield moment and the plastic moment of the cross section withrespect to bending about a horizontal axis. Moreover, a graphical presentation of the pre-dicted moment-curvature response for eleven di�erent levels of shear, ranging from non to

Table 6.5: Predicted ultimate capacities for the double hull tanker.

Longitudinal Strength:First yield moment : 2.92 GNmPlastic moment : 4.64 GNm

Predicted Ultimate Capacities:Vertical shear force capacity : 169 MNBending moment capacity { hogging : 3.94 GNm ' 85% of plastic momentBending moment capacity { sagging : -3.11 GNm ' 67% of plastic moment


Figure 6.6: Moment-curvature response for the double hull tanker.

ultimate shear capacity, is shown in Fig. 6.6, and �nally the predicted interaction betweenthe vertical shear force and the ultimate bending moment capacity in both hogging andsagging condition is shown in Fig. 6.7.

6.3.3 Discussion of the Results

Observing the moment-curvature responses shown in Fig. 6.6 it is noted how the predictedunloading of the cross section becomes less pronounced with higher levels of shear loading.The explanation for this behavior lies within the assumed three-hinge plastic mechanismused to model the post-ultimate response of the individual beam-columns (see Section 5.7).As the level of shear loading increases the e�ectiveness of the plate with respect to directloading rapidly decrease and hereby the beam-column becomes more and more just a sti�enerwithout any attached plating which makes the response predicted by the assumed plasticmechanism somewhat dubious for these high levels of shear. In general, the three-hinge


Figure 6.7: Moment-shear interaction chart for the double hull tanker.

plastic mechanism will for high levels of shear predict a very slow unloading as shown inFig. 5.8. However, for beam-columns with very high sti�eners the unloading predicted will bealmost ideal-plastic in the sense that after the ultimate capacity of the beam-column has beenreached, excessive straining is required before any signi�cant unloading is achieved. This,not to mention the ignored e�ect of tripping which also becomes increasingly signi�cant forhigh sti�eners, is an expected short-coming of the assumed method, and therefore must beexpected to manifest it self in the moment-curvature response as shown in Fig. 6.6. Althoughhard to see from the graphs, a maximum for the hogging bending moment is actually achievedat a curvature roughly equal to 0:0006 m�1 for the 90% and full shear capacity response.Hence, the present procedure manages to capture the ultimate capacity of the entire loadingrange from zero to ultimate shear capacity as also shown in the moment-shear interactionchart presented in Fig. 6.7. Here, the change in the unloading response at high levels ofshear cannot explicitly be deducted although a somewhat abrupt change in the moment-shear interaction is observed around �fty percent shear capacity. Nevertheless, a continuous,generally smooth reduction of the moment capacity is observed all the way to ultimate shearcapacity. It is further noted that the present procedure predicts a moment capacity of thecross section even at ultimate shear loading. This is to be expected, as the the shear stresses


caused by the vertical shear force are assumed to act only on the plating. Thus, the sti�enersare una�ected by the presence of shear and will therefore produce a moment capacity of thecross section at any shear loading.

Comparing these results with the curves presented in [17] a good overall agreement is found.Moreover, the present procedure produces smooth moment-curvature response predictionsin comparison with the results shown in [17] where sharp drops in moment response occursafter the ultimate capacity is reached. Similarly, when comparing the ultimate momentcapacities an excellent agreement is achieved as shown below:

Ultimate capacity By [17] Present �%Hogging moment 2� 2:03 GNm 3:94 GNm 2.9%Sagging moment 2� 1:63 GNm 3:11 GNm 4.6%Shear force 2� 81:1 MN 169 MN 4.2%

The di�erence is three percent for the hogging capacity and �ve percent in the saggingcondition. Also, with respect to the ultimate shear capacity of the cross section an excellentagreement with the result presented in [17] is achieved with a discrepancy of some fourpercent.

Finally, as it was one of the initial objectives of the present research to develop a rapidprocedure for the evaluation of the ultimate capacity of the hull girder, it is relevant to discussthe time consumption of the present procedure when applied to a typical, real structure. Forthe double hull tanker, the time used for calculation of one-thousand ultimate capacities atdi�erent shear stress levels, evenly spread from zero to full shear capacity, was 1 hour andforty-eight minutes when performed in-core on a Hewlett Packard C200 C-class workstation1,giving an average computation time per ultimate capacity of 6.48 seconds. This excludesthe time used to solve the initial �nite di�erence description of the plate part for each of152 beam-columns that makes up the entire cross section. This task alone required one hourand twenty-�ve minutes when utilizing the identical properties of some of the plates makingit necessary only to solve for about one-third of the total 152 plates. The reason for leavingthis part of the calculation out of the per ultimate moment performance rating, is that itcan most reasonably be considered as a post-processing task. Once this has been solvedfor a given cross section in can be stored in a �le along with the other data making up thegeometry de�nition and thereafter be reused for any subsequent calculations.

6.4 Summary

To summarize, two di�erent test cases has been investigated with the purpose of veri�cationof the present procedure. The results hereby obtained has been compared with other pub-

1The Hewlett Packard Company lists the performance of this model as: 14.3 SPECint95 / 21.4 SPECfp95for integer and oating point operations respectively. The performance in million oating point operationsper second is at best listed to 550 MFLOPS.

6.4. Summary 107

lished �ndings and in general the agreement can be concluded to be acceptable. In the caseof the Nishihara box girder the present procedure predicted the ultimate moment capacityof the initially perfect structure within one percent of the experimental result by Nishihara[31]. Compared with nonlinear �nite element calculations by Melchior Hansen [17] for thestructure with assumed imperfections, the present procedure predicted results within two to�ve percent accuracy. Moreover, asymmetrical bending investigations of the Nishihara boxgirder proved successful in capturing the symmetry conditions existing in the cross sectionand thus lead to the conclusion that the present procedure indeed is capable of handlingasymmetrical bending.

Further, the midship section of a double hull tanker was analyzed in the combined loadingof bending about a horizontal axis and vertical shear force. The moment-curvature responsealong with the moment-shear interaction as predicted by the present procedure was comparedwith results published by Melchior Hansen [17] showing an excellent agreement both withrespect to the ultimate moment capacities, and with the predicted ultimate shear forcecapacity. Hence, it is concluded that the capacities predicted by the present procedure areaccurate. Moreover, the comparison with the moment-curvature response in [17] showedthat the present procedure yields a more smooth unloading description which contradictsthe explanation o�ered in [17] where the very abrupt drops in response during unloading isattributed to the collective collapse of large parts of the cross section. The response obtainedby the present procedure cannot substantiate this conclusion, as the post-ultimate responseof the structure for all levels of shear loading are indeed smooth.

Finally, the performance of the present procedure was evaluated and an average calculationtime of approximately seven seconds per ultimate moment capacity was achieved. Conse-quently, the objective of developing a rapid procedure for the evaluation of the ultimate hullgirder strength can be concluded to have been meet by the present procedure.

Chapter 7

Application of the Procedure

7.1 Introduction

When originally formulating the objective of the present research, the goal was set to bethe development of a rapid procedure that, apart for being able to describe the structuralbehavior of an intact vessel, also was applicable to damage conditions arising from grounding,collision, �re, and explosions. So far, the veri�cation of the present procedure presented inthe previous chapter has been based on two test-cases of which one was a real, full scaleship structure the other an experimental scale model. The objective of these two test-caseswas to demonstrate the present procedures capabilities to accurately assess the ultimate andpost-ultimate strength of the intact hull girder by comparison with known results. Havingsuccessfully done that, the remaining task will be to demonstrate that the present procedurecan handle damaged conditions of the hull girder.

To do this, a vessel is selected for which the following three scenarios are analyzed:




For the intact condition, the aim is to analyze the structure both with and without the e�ectof the hydrostatic pressure included as initial loading on the beam-columns. The pressureloading will be taken as the ballast condition. Regarding the analyzes of the two damagedconditions, then these will be based on �ctitious scenarios made up from reference dataof real accidents when possible. Otherwise, best judgment will be used to create realisticdamage scenarios.

109

110 Chapter 7. Application of the Procedure

Table 7.1: Principal dimensions for the ultra large crude carrier.

Length between perpendiculars Lpp : 353.17 mLength overall Loa : 370.40 mBreadth moulded B : 60.00 mFrame spacing amidship ` : 7.23 mDepth moulded to upper deck D : 35.00 mDraught in cargo condition Tc : 23.00 mDraught in ballast condition Tb : 11.30 mDisplacement r : 414000 tDeadweight DWT : 359500 t

Figure 7.1: Pro�le of the ultra large crude carrier.

7.2 Ultra Large Crude Carrier

The vessel selected for use in these investigations is a design study of a 360,000 deadweighttons ultra large crude carrier (ULCC) with double hull. The pro�le of this vessel is shown inFig. 7.1 and the principal dimensions are listed in Tab. 7.1. Being a design study the vesselhas never actually been build. However, the midship section scantlings are in accordance withthe classi�cation rules for steel ships by Det Norske Veritas [7]. Following these guidelines,the maximum allowable still-water bending moment Ms and wave-induced bending momentMw for the vessel are:

Ms =

(8:83 GNm in hogging

�8:07 GNm in saggingand Mw =

(12:90 GNm in hogging

�13:66 GNm in sagging

Hence, the midship section has been designed to withstand a maximum bending momentof 21:73 GNm. These design loads are relevant for evaluation of damaged conditions of thevessel, as they form a basis { although somewhat crude { to evaluate whether the damagevessel is in a structural safe condition or not. The entire midship section is made from hightensile steel with a yield stress equal to 390 MPa as listed in Tab. 7.2. The table further liststhe sectional parameter for the midship section. Finally, the structural layout of the midshipsection is shown in Fig. 7.2 with the transverse web-frames to the left and the longitudinal

7.2. Ultra Large Crude Carrier 111

Table 7.2: Sectional and longitudinal strength parameters for the ULCC.

Sectional Parameters:Frame spacing amidship ` : 7.23 mYield stress �y : 390 MPaYoung's modulus E : 210 GPaSectional area A : 14.05 m2

Longitudinal Strength:Moment of inertia Iyy : 1883.98 m4

Elastic neutral axis �ENA : 13.41 mPlastic neutral axis �PNA : 10.57 mFirst yield moment My : 34.03 GNmPlastic moment Mp : 52.56 GNmSection modulus (deck) Wdeck : 87.25 m3

Section modulus (keel) Wkeel : 140.52 m3

Figure 7.2: Structural layout of the ULCC midship section.


elements to the right. Regarding the imperfections, then an initial sti�ener de ection equalto /200 = 36 mm is assumed, where ` is the frame spacing. This initial de ection will beapplied to all of the following analyzes.

From the longitudinal strength parameters listed in Tab. 7.2 it is noted that compared withthe maximum design bending moment of 21:73 GNm, the �rst yield moment My equal to34:03 GNm is signi�cantly larger. At a �rst glance, this seems to indicate a somewhatover-scantled midship section. However, this is to be expected for a double hull structurethis large. Here, the scantlings will in general be dictated by the pressure loading on thesti�ened panels, where the class rules will require quite large dimensions of both the platingand the sti�eners, to ensure the avoidance of local buckling. Thus, this will normally causethe overall longitudinal strength of the midship section to be more than adequate comparedwith the design bending moment.

7.3 As-Build Condition

The ultra large crude carrier has been analyzed in the as-build condition with the assumedinitial de ection of 36 mm imposed on the sti�eners. The loading was chosen to be a verticalshear force in combination with an asymmetrical bending moment. The remaining loadcomponents, i.e. horizontal shear force, torsional moment, and pressure loading, were setto zero. Thus, this initial investigation of the ultra large crude carrier is, except for theasymmetrical bending, identical to the one previously performed on the double hull tankerstructure as part of the veri�cation of the present procedure.

Looking �rst at the longitudinal strength of the midship section in this loading condition,then the ultimate capacities predicted by the present procedure are listed in Tab. 7.3. Fur-ther, a graphical illustration of the predicted moment-curvature responses for eleven di�erentlevels of vertical shear force, ranging from zero to ultimate shear capacity, is shown in Fig. 7.3.

Table 7.3: Predicted ultimate capacities for the ultra large crude carrier.

Longitudinal Strength:First yield moment : 34.03 GNmPlastic moment : 52.56 GNm ' 154% of 1st yield moment

Predicted Ultimate Capacities:Vertical shear force capacity : 1.60 GN

Hogging moment capacity : 45.01 GNm '(

86% of plastic moment132% of 1st yield moment

Sagging moment capacity : -30.46 GNm '(


7.3. As-Build Condition 113

Figure 7.3: Moment-curvature response for the ultra large crude carrier.

To validate the predicted longitudinal strength a comparison with the reserve strength fac-tors for the ultimate longitudinal strength of various ship types published by the TechnicalCommittee III.1 of ISSC'94 [11] is made. For a similar vessel, being a double hull very largecrude carrier of same general cross section layout, [11] lists the following factors:

� Plastic / �rst yield moment Mp=My = 1:42

� Ultimate hogging / �rst yield moment Mhogy;u =My = 1:23

� Ultimate sagging / �rst yield moment M sagy;u =My = 0:95

Compared with the corresponding fractions in Tab. 7.3 an acceptable correlation is observed,although it is seen that the present procedure predicts a little higher plastic moment and hog-ging capacity, but also a little less sagging capacity than reported in [11]. This is attributed


Figure 7.4: Interaction chart for the vertical shear force and the ultimate moment aboutthe instantaneous neutral axis oriented between 0 and 180 degrees.

to the current vessel being a design study rather than a real tanker which has actually beenbuild. Consequently, it must be expected that the scantlings of the current ultra large crudecarrier will be somewhat conservative, explaining the larger plastic moment and hogging ca-pacity. This was also observed previously when comparing the design bending moment withthe �rst yield moment of the midship section. Here the present procedure found a �rst yieldmoment roughly �fty percent larger than the required design moment which also indicatesa somewhat conservative design.

Moving on to consider the e�ect of asymmetrical bending of the cross section, then Fig. 7.4shows the predicted interaction between the vertical shear force and the ultimate bendingmoment about the instantaneous neutral axis oriented between 0 and 180 degrees. Thehorizontal plane in the �gure is a polar representation of the vertical shear force for di�erentorientations (angles) � of the instantaneous neutral axis relative to horizontal. The uppersurface then shows the ultimate hogging moment and similarly the lower surface shows theultimate sagging moment for di�erent levels of the vertical shear force at di�erent angles�. Thus, any vertical plane through the origin of the reference system depicts the moment-shear interaction at a speci�c orientation of the instantaneous neutral axis. It is noted thatthe terms hogging and sagging in this context refers to the positive and negative part ofthe moment response with respect to the current orientation of instantaneous neutral axis.Hence, at � = 90o the physical meaning of the two terms swaps with respect to how thesection actually de ects. Further, due to the symmetry of the cross section the sagging


Figure 7.5: Interaction chart for the vertical shear force and the ultimate moment aboutthe instantaneous neutral axis oriented at 0, 45, and 90 degrees.

response shown in the �gure will be identical to the hogging response for orientations � ofthe instantaneous neutral axis in the range 180o to 360o, only with reversed sign. Likewise,will the shown hogging response also be identical to the sagging response except for signin the � range 180o to 360o. This can also be observed from Fig. 7.4 where the sagging(negative) response for � = 0o is identical to the hogging (positive) response for � = 180o

and vice versa for hogging and sagging also at these two angles.

To obtain a more detailed picture of the moment-shear interaction, Fig. 7.5 shows the ul-timate moment capacity versus the vertical shear force for the instantaneous neutral axisoriented at zero degrees (horizontal), forty-�ve degrees, and ninety degrees (vertical). Theplot shows an increasing moment capacity as the instantaneous neutral axis rotates fromhorizontal to vertical. This is to be expected as the cross section is nearly twice as wide asit is high (B = 60 m, D = 35 m). Thus, the leverage of the axial response forces from eachbeam-column increases with the rotation of the instantaneous neutral axis yielding a highermoment at collapse.


Figure 7.6: Ultimate moment interaction chart for the intact condition. The showninteraction formula is proposed by Mansour et al. [26].

The same observation can be made from Fig. 7.6 where the the interaction between the ulti-mate bending moment about the vertical and horizontal axis is shown for �ve di�erent levelsof vertical shear force. Here, it is consistently seen that the ultimate bending capacity aboutthe vertical axis is larger than about the horizontal axis. Another interesting observationcan be made from this �gure. Looking �rst at the ultimate moment about the horizontalaxis in the case of pure bending, it is observed that in the sagging part of the response,the moment actually increases when the instantaneous neutral axis in rotated just a smallangle away from horizontal orientation. This behavior is persistent for all levels of shear.Moreover, at higher levels of shear the same behavior is observed �rst for the hogging partof the response, and then also around the ultimate moment about the vertical axis. Theexplanation for this behavior is the highly nonlinear change in the leverage to the individualbeam-columns, and thereby also in the corresponding end-displacements for a speci�c curva-ture. Thus, just a small rotation angle away from either a horizontal or a vertical orientation


of the instantaneous neutral axis will have a large e�ect on the moment capacity of the crosssection. The reason why this behavior is �rst observed for the sagging part of the responselies in the double bottom structure being comparatively much sti�er than the deck structure.Consequently in the sagging condition where the deck is in compression and the bottom is intension, the assumed linear elastic, ideal plastic response of the individual beam-columns intension will yield practically the same axial response force for each beam-column at ultimatecapacity, but with a signi�cantly larger leverage, whereas the beam-columns in compressionyields a comparatively smaller axial response force due the the implemented unloading. Thenet e�ect must therefore be expected to be an increase in the moment capacity just as it isobserved in Fig. 7.6. The same argument can be applied to explain the moment interactionaround all four of the horizontal and vertical extremes.

Finally, a comment pertinent to the predicted moment interaction at full shear capacityshould be made. From the plot in Fig. 7.6 it is obvious that the response predicted by thepresent procedure is somewhat non-smooth and further lacks a little the expected symmetry.This behavior is directly related to the assumed idealized load-displacement response for eachbeam-column. At full shear capacity the plate cannot withstand any direct loading. Thus,all the beam-columns are in essence reduced to being just a sti�ener. As previously describedin the discussion of the results obtained for the double hull tanker during the veri�cationprocess (see Section 6.3.3) the predicted load-displacement response at full shear capacity issomewhat dubious with respect to the compressive unloading of the beam-column. Too muchcon�dence should therefore not be put in the response predicted by the present procedureat the ultimate shear capacity. However, as a loading condition consisting almost entirelyof shear is unrealistic for any real vessel, this is an acceptable limitation of the presentprocedure. This because it, for the same reason, poses no real limitation with respect toemergency response application of the present procedure. With respect to probabilisticanalyzes, it should not matter either as even if the response is requested, the probability ofoccurrence will be extremely low yielding practically no net e�ect on the analysis result.

7.3.1 Moment Interaction Formulas

The interaction between the ultimate horizontal and vertical moments has by many authorsbeen sought expressed through simple interaction formulas. The use of an interaction formulawill of course be a much faster procedure for obtaining the moment response of the crosssection. Thus, as one of the main objectives of the present research is to develop a rapidprocedure for the evaluation of the ultimate capacity of the hull girder, it is interesting toinvestigate if the response obtained by application of the present procedure can be describedjust as good by a simple interaction formula.

In Fig. 7.6 the predicted interaction by one such interaction formula, published originally byMansour and Thayamballi [27], is shown �tted to mimic the pure bending response, i.e. forzero vertical shear force. The same formula was applied in a later study by Mansour et

al. [26], where four di�erent vessels were investigated. One of these vessels was a tanker and


the results obtained in [26] showed an excellent agreement between the numerical data andthe interaction formula given in Eqs. (7.1) and (7.2), taken from [26].

My

My,u+ k

Mz

Mz,u

!2

= 1 if

�� Mz

Mz,u

�� <�� My

My,u

�� (7.1)

Mz

Mz,u+ k

My

My,u

!2

= 1 if

�� My

My,u

�� <�� Mz

Mz,u

�� (7.2)

Assuming a fully plastizied cross section, without any imperfections or buckling, the �tparameter k was in [27] derived to be

k =(A+ 2As)

2

16As(A� As)� 4(Ad � Ab)2(7.3)

in which Ad, As, and Ab is the cross sectional area of the deck, the side, and the bottomrespectively, all including the sti�eners. Thus, the total cross sectional area is A = Ad +2As + Ab. It is noted that because of the inherent assumption of a fully plastizied crosssection in Eq. (7.3), the interaction formula in Eqs. (7.1) and (7.2) cannot { based on thek parameter in Eq. (7.3) { reasonably be expected to mimic the behavior predicted by thepresent procedure in the sagging part of the response. This because, in the present procedure,the ultimate capacity always will be reached before the cross section is fully plastizied due tothe inclusion of imperfection, buckling, etc. Thus, in the sagging part of the response, wherethe deck of the tanker is in compression, the behavior will be far from plastic as evidentlyshown in Fig. 7.6. However, the interaction formula is applicable in the hogging part ofthe response. Here, the deck will be in tension and the bottom will be in compression. Asthe tensile behavior of the deck will be almost plastic, and because of the very sti� doublebottom, the compressive behavior of this part of the cross section will also be close to plastic,it can be expected that the response predicted by the present procedure can be related tothe interaction formula in Eqs. (7.1) and (7.2).

By insertion into Eq. (7.3) the �t parameter k is, for the present case, found to be equal to0.88. In comparison a numerically �tted value of 0.80 was in [26] found to yield the bestapproximation to the numerical data for the tanker investigated in that study. However,observing the interaction curve in Fig. 7.6 based on k = 0:88, the correlation between theinteraction formula and the numerical data predicted by the present procedure for pure bend-ing of the cross section, is somewhat disappointing. The interaction predicted by Eqs. (7.1)and (7.2) is obviously very low compared to the present pure bending results. This may bebecause of the very sti� double bottom and double side structure for the present ultra largecrude carrier. When evaluating the longitudinal strength of the vessel, it was found that thestrength of the tanker was very high compared to the design requirements by Det NorskeVeritas [7]. In correlation with the assumed fully plastic response in the derivation of the


k parameter in Eq. (7.3), this is most likely the major contributing factor explaining whythe present case yields such poor comparison with the interaction formula. Especially whenconsidering that excellent correlation was found by Mansour et al. [26] for another tanker.

However, as in [26], a numerical �t of the k parameter has been performed also, yieldinga value of 0.38. The �t was performed in the hogging part of the moment response only,as the interaction formula cannot be expected to give good correlation with the saggingresponse due to the highly nonlinear behavior of the structure in this region. Looking atthe interaction curve in Fig. 7.6 based on this new value k = 0:38 the correlation with thenumerical data is seen to be quite good. Of course, the discontinuous response where theformula changes from Eq. (7.1) to Eq. (7.2) becomes more pronounced with the lower kparameter, as also seen in Fig. 7.6. The large di�erence in the k parameter obtained byEq. (7.3) and the numerical further substantiates the previous conclusion, i.e. that for thepresent tanker, the behavior of the cross section is far from ideal plastic in the ultimate statebecause of the very sti� double bottom and side structure. Thus, as the expression for thek parameter given in Eq. (7.3) is based on the assumption of a fully plastic cross section, itis obvious that the interaction formula should be judged on the numerically �tted k ratherthan on the k given by Eq. (7.3). Doing this, the agreement is, as already stated, quite good.

Nevertheless, the problem with the non-smooth response caused by the division of the for-mula into two parts still exists, even more pronounced in the numerically �tted response.To address this, a classical interaction formula is suggested de�ned as

My

My,u

!�1+

Mz

Mz,u

!�2+

Qz

Qz,u

!�3= 1 (7.4)

i.e. formulated in the both the combined bending moments and the vertical shear force.However, when trying to �t this formula (Eq. (7.4)) to the response predicted by the presentprocedure, it became evident that the e�ect of the vertical shear force Qz already entered theformula through the ultimate bending capacities My,u and Mz,u which obviously are highlycorrelated to the vertical shear force. Thus, the formulation of the suggested interactionformula is changed to

My

My,u

!�1+

Mz

Mz,u

!�2= 1 (7.5)

This interaction formula (Eq. (7.5)) has been numerically �tted to the data obtained by thepresent procedure yielding the two powers �1 and �2 equal to 1.95 and 3.66 respectively. Aswith the previous �t of the interaction formula proposed by Mansour et al. [26], only thehogging part of the response was �tted. This because also the present interaction formula(Eq. (7.5)) cannot be expected to mimic the highly nonlinear behavior in the sagging partof the response as it simply is impossible for the expression in Eq. (7.5) to assumed theshape of the sagging response. Thus, focusing on the hogging response, the obtained �t is


Figure 7.7: Ultimate moment interaction for the intact condition as predictedby the interaction formula in Eq. (7.5).

shown in Fig. 7.7 for �ve di�erent levels of vertical shear force ranging from zero to ultimateshear capacity. Observing the �tted curves in comparison with the response predicted bythe present procedure (shown as points in the plot) a smooth set of curves with an overallgood correlation is seen to be achieved by the �tted interaction formula { especially whenconsidering that the �tted powers �1 and �2 are valid for all levels of vertical shear. Thus,it is the same interaction formula used to produce all �ve curves shown in Fig. 7.7.

Of course, the correlation in the sagging part of the response is as expected, less favorableas it was for the formula proposed by Mansour et al. [26], because of the nonlinear behaviorfound here. Neither of these two interaction formulas can be made to mimic the shape ofthe sagging response as predicted by the present procedure. Therefore, even though thesecond formula (Eq. (7.5)) �ts very nicely in the hogging region of the moment response,substituting the present procedure with an interaction formula is not recommendable.

7.4. Ballast Condition 121

Figure 7.8: Sketch of the pressure distribution in ballast condition.

7.4 Ballast Condition

To investigate the e�ect of pressure loading on the hull, the ballast condition of the ultralarge crude carrier has been modeled and analyzed by the present procedure. In the ballastcondition the entire double hull is �lled with sea water and the tanker has a draught of11:3 m. The resulting hydrostatic pressure distribution is sketched in Fig. 7.8. The ballastcondition thus represents the most severe pressure loading of the inner plating of the doublehull. Of course, in the cargo condition the draught is larger (23:0 m) yielding a higher

Table 7.4: Predicted ultimate capacities for the ultra large crude carrierin its ballast condition.

Longitudinal Strength:First yield moment : 34.03 GNmPlastic moment : 52.56 GNm

Predicted Ultimate Capacities:Vertical shear force capacity : 1.60 GN

Hogging moment capacity : 42.67 GNm '(


Sagging moment capacity : -30.95 GNm '(



Figure 7.9: Moment-curvature response for the ultra large crude carrierin its ballast condition.

hydrostatic pressure on the outer skin of the midship section. However, the cargo carriedby the tanker will typically have a lower speci�c density than sea water. Thus, the pressureon the inner hull will in the cargo condition be lower than that caused by the ballast water.Moreover, as the plate thickness of the outer skin is larger than the inner skin, then thehigher pressure on the outer skin in the cargo condition is not as sever a loading of the entirecross section, as a the ballast condition is with respect to the inner skin.

The obtained prediction of the longitudinal strength of the tanker in this ballast conditionare listed in Tab. 7.4. Further, the corresponding moment-curvature response for bendingabout a horizontal axis is shown in Fig. 7.9 for eleven di�erent levels of shear ranging fromzero to ultimate shear capacity. The shear loading is the same as for the intact, as-buildcondition previously investigated. That is, only a vertical shear force is assumed to act onthe cross section together with the bending moment. Horizontal shear and torque is thusignored in the present analysis.

7.4. Ballast Condition 123

Figure 7.10: Ultimate moment interaction chart for the ultra large crude carrierin its ballast condition.

Comparing the predicted capacities in Tab. 7.4 with the corresponding capacities for theintact, as-build condition of the tanker listed in Tab. 7.3, it is observed that the presence ofhydrostatic pressure on parts of the cross section, equivalent to the ballast condition, causesa �ve percent reduction of the ultimate moment capacity in hogging, relative to the plasticmoment. The ultimate moment capacity in sagging is however only reduced by one percentagain relative to the plastic moment. Regarding the e�ect of the vertical shear force, thenby comparing the moment-curvature response for the ballast condition shown in Fig. 7.9with the previously obtained response for the intact, as-build condition in Fig. 7.3, twoobservations are made: First, the presence of hydrostatic pressure tends to atten the post-ultimate moment response and second, the moment capacity of the cross section at ultimateshear capacity is signi�cantly increased in the hogging condition, whereas the response isonly slightly decreased in the sagging condition. This even though the ultimate momentcapacity at zero shear loading, i.e. in pure bending, is decreased by �ve percent.

The explanation for this behavior is to be found in the sign of the pressure and initial


de ection. Being in ballast condition, the hydrostatic pressure will be acting on the platingfrom the same side the longitudinal sti�ening is attached to the plating. The initial de ectionon the other hand is assumed to be in the opposite direction. Thus, with the decreasing axialstrength of the plate at increased levels of shear loading, the pressure loading on the sti�enerwill tend to make the beam-column more sti� in its compressive response, than it was inthe intact, as-build condition without any pressure loading. On the other hand, in the post-ultimate tensile response, the relative small initial imperfection will almost be non-existingbecause of the excessive staining (elongation) associated with the response. Hence, the e�ectof pressure on the beam-column in tension will be a slight decrease in axial load capacity.Therefore, in the sagging condition where the deck is in compression and the bottom intension, only a small e�ect of the hydrostatic pressure is to be expected. This because thedeck is una�ected by the pressure and the bottom, being in tension, only will exhibit a slightdecrease in axial load capacity. In the hogging condition on the other hand, the deck will bein tension and the bottom in compression. Hence, as a major part the the structure beingin compression will also be a�ected by the hydrostatic pressure, it is to be expected that adecrease of the moment capacity will occur which also is observed in Fig. 7.9.

Another e�ect caused by the presence of hydrostatic pressure is observed from the momentinteraction shown in Fig. 7.10. Compared with the result obtained for the intact, as-buildcondition shown in Fig. 7.6 it is seen that the presence of hydrostatic pressure on parts ofthe structure, apart from having a smoothening e�ect on the moment interaction predictedby the present procedure, also tends to make the nonlinear response in the sagging conditionless pronounced. As the hydrostatic pressure is highest in the bottom which in the saggingcondition is in compression and as the hydrostatic pressure has the greatest impact on theload capacity in compression, then in light of the previous �ndings concerning the moment-curvature response, this reduction of the nonlinear response in the sagging condition is tobe expected.

The smoothening of the moment interaction is attributed to the reduced compressive load-displacement response of the individual beam-columns. This together with the almost unaf-fected tensile load-displacement response causes a general reduction of the nonlinear behaviorof the cross section resulting in a more smooth moment interaction.

7.5 Grounding Damage Condition

To investigate the e�ect of grounding damage on the ultimate capacity of the hull girder,knowledge of the extent of the damage is naturally a prerequisite. However, as the tankerpresently being analyzed has never been build, information on a real grounding conditionis non-existing. In lieu, a �ctitious bottom damage has been de�ned based on a real lifegrounding accident of a single skin, very large crude carrier (VLCC) reported by Kuroiwa[25]. The accident occurred on January 6th 1975, as a 240,000 deadweight tons tanker ranonto the Bu�alo Reef o� the coast of Singapore, spilling more than 10,000 tons of oil. A

7.5. Grounding Damage Condition 125

Figure 7.11: Bottom damage on single skin VLCC after grounding.(Adopted from Kuroiwa [25]).

sketch of the reported damage to the hull structure is shown in Fig. 7.11. During the initialgrounding, the bottom of the vessel was torn open for about 180 m from the bow to themiddle of center tank number three. Moreover, after the initial grounding, the vessel wasstuck for ten days on the reef, explaining the severe expansion of the damage around centertank number three.

Considering the tearing part of this grounding near amidship, the �ctitious bottom damagehas been assumed to have the horizontal and vertical extent as shown in Fig. 7.12. Thatis, the outer skin of the double bottom from the centerline to the hopper tank plus thegirder in the double bottom under the longitudinal bulkhead, is assumed torn away duringthe grounding. The longitudinal extent of the damage is assumed far beyond the length ofthe section being investigated, i.e. greater than ` = 7:23 m. All the structural members inthe damaged zone are thus ignored in the model of the damaged cross section. The shearloading in this damaged condition is assumed to be a uniform scaling of all the three shearstress distributions arising from vertical and horizontal shear forces, and from the St. Venanttorsion. That is, it is assumed that the two shear forces and the torsional moment all scaleequally.


Figure 7.12: Sketch of the �ctitious bottom damage.

Looking �rst at the longitudinal strength of the ultra large crude carrier in this groundingcondition, the moment capacities of the midship section, as predicted by the present proce-dure, are listed in Tab. 7.5 along with the predicted �rst yield moment and the fully plasticmoment. Granted, in the current damage condition the cross section is asymmetrical, andthe concept of longitudinal strength is therefore somewhat dubious. The results presented inthe table as longitudinal strength are in fact the results obtained for a horizontal orientationof the instantaneous neutral axis. Thus, the cross section is forced to bend about a horizontalaxis, which for the present case is considered representable for longitudinal strength. Thelongitudinal strength data is further presented as moment-curvature responses in Fig. 7.13for eleven di�erent levels of shear loading ranging from zero to full shear capacity.

Table 7.5: Predicted ultimate capacities for the ultra large crude carrierafter a bottom damage as shown in Fig. 7.12.

Longitudinal Strength:First yield moment : 32.24 GNm ' 95% of intact conditionPlastic moment : 48.38 GNm ' 92% of intact condition

Predicted Ultimate Capacities:

Hogging moment capacity : 40.70 GNm '(84% of plastic moment90% of intact condition

Sagging moment capacity : -28.99 GNm '(60% of plastic moment95% of intact condition

7.5. Grounding Damage Condition 127

Figure 7.13: Moment-curvature response for the ultra large crude carrier in the assumedgrounding condition with uniformly scaled, fully combined shear loading.

From Tab. 7.5 it is seen that the presence of the assumed bottom damage causes a �ve toeight percent reduction in the �rst yield moment and plastic moment respectively comparedwith the intact, as-build condition listed in Tab. 7.3. The predicted ultimate moment ca-pacity in hogging is seen to have been reduced by only two percent relative to the plasticmoment, whereas the ultimate sagging moment has been increased by a similar two percentagain relative to the plastic moment. Of course the absolute magnitude of the two momentcapacities has both been decreased compared to the as-build condition. However, it is inter-esting to look at the change in hogging and sagging moment capacities relative to the currentplastic moment, as this gives an impression of the e�ect of the asymmetric cross section onthe moment capacities. Thus, judging by these numbers, the e�ect of asymmetry seemsto be quite modest. This is further substantiated by comparison with the intact capacitieswhich also are listed in Tab. 7.3. Here the present procedure predicts an ultimate hoggingcapacity which is ten percent less than the intact capacity. Regarding the ultimate sagging


Figure 7.14: Ultimate moment interaction chart for the assumed grounding conditionwith uniformly scaled, fully combined shear loading.

capacity it is seen to be reduced by only �ve percent compared to the intact capacity. Thisdi�erence in reduction is to be expected as the extend of the damage is localized to thebottom part of the structure. Thus, because of the di�erent handling of the post-ultimateload-displacement response of the individual beam-columns in compressive and tensile load-ing, where unloading only occurs for the compressive part, the e�ect of the bottom damagemust be expected to be bigger when the bottom is in compression (hogging) than when it isin tension (sagging).

To get a better impression of the e�ect of the asymmetric cross section, the interactionbetween the ultimate bending moment about the vertical and horizontal axis is shown inFig. 7.14. The �rst observation which can be made from this �gure is that the momentinteraction exhibits the same overall behavior as in the intact condition with respect toincreased moment capacity for orientations of the instantaneous neutral axis just slightly o�horizontal and vertical. This behavior has already been explained in relation to the intactcondition. However, compared with the response behavior for the intact condition shown

7.6. Fire and Explosion Damage Condition 129

Figure 7.15: Assumed temperature distribution by a �re in the right wing cargo tank.

in Fig. 7.6, it is seen that the somewhat elliptic shape of the response is slightly tilted tothe right. This is to be expected because of the change in the principal axes caused by theasymmetry of the cross section. The cross sectional analysis yielded a principal coordinatesystem rotated 2:29o as shown with dashed lines in Fig. 7.14. Obviously, this is also re ectedin the slightly tilted moment response predicted by the present procedure. Moreover, theultimate moment response depicted in the �rst and third quadrant are equivalent to thebottom damage zone being in compression, whereas the second and fourth quadrant representthe damage zone in tension. This explains why the predicted moment capacity in higher inthe second and fourth quadrant than in the �rst and third quadrant.

7.6 Fire and Explosion Damage Condition

To further demonstrate the present procedures applicability to vessels in damage conditions,a �ctitious damage scenario involving �re and explosion has been de�ned to illustrate theability of the present procedures to analyze this kind of damage to the structure. A �rein the right wing cargo tank is supposed to have been started due to an explosion in thatcargo hold following a collision. The situation being investigated is during the �re where theresulting temperature distribution in the immediate vicinity of the �re is assumed to be assketched in Fig. 7.15. The consequence of this �re is included in the model of the cross sectionthrough reduced sti�ness and strength of the steel in the heat a�ected zone. That is, theYoung's modulus and the yield stress are reduced according to the assumed temperatures.The reduction is based on the behavior of carbon steel at elevated temperatures given in theGuidance Notes on explosion on �re by the Steel Construction Institute [3] listed in Tab. 7.6.


Table 7.6: Sti�ness and strength of carbon steel at elevated temperatures.

Steel Fraction Fraction Steel Fraction Fractiontemperature of Young's of yield temperature of Young's of yield

in oC modulus stress in oC modulus stress20 1.00 1.00 700 0.13 0.075100 1.00 1.00 800 0.09 0.050200 0.90 0.807 900 0.0675 0.0375300 0.80 0.613 1000 0.0450 0.0250400 0.70 0.420 1100 0.0225 0.0125500 0.60 0.360 1200 0.0000 0.0000600 0.31 0.180 1350 0.0 0.0

The shear loading in this damaged condition is identical to the shear loading applied in thegrounding condition. Thus, it is assumed that the shear loading in the damaged conditionis representable by a uniform scaling of all the three shear stress distributions arising fromvertical and horizontal shear forces, and from the St. Venant torsion. That is, it is assumedthat the two shear forces and the torsional moment all scale equally.

The predicted ultimate capacities relating to the longitudinal strength of the ultra largecrude carrier in this �re damage condition is listed in Tab. 7.7. As for the grounding damagescenario, bending about the horizontal axis is considered to be representable as longitudinalstrength even though the cross section is asymmetric. From the table it is observed that thereduction in strength, caused by the elevated temperatures in the heat a�ected zone, is quitesigni�cant. The predicted full plastic moment of the cross section about the horizontal axisis reduced to only 76% of the intact, as-build condition. Regarding the ultimate momentcapacities, then a reduction of 23% is observed both for the hogging and sagging capacity.The relative utilization of the cross section, hereby understood the relative to the currentfull plastic moment, is however quite high even in this severely damaged condition, being

Table 7.7: Predicted ultimate capacities for the ultra large crude carrier in the assumed�re damage condition with uniformly scaled, fully combined shear loading.

Longitudinal Strength:Plastic moment : 40.05 GNm ' 76% of intact condition

Predicted Ultimate Capacities:

Hogging moment capacity : 34.84 GNm '(87% of plastic moment77% of intact condition

Sagging moment capacity : -23.55 GNm '(59% of plastic moment77% of intact condition

7.6. Fire and Explosion Damage Condition 131

Figure 7.16: Moment-curvature response for the ultra large crude carrier in the assumed�re damage condition with uniformly scaled, fully combined shear loading.

87% and 59% of the full plastic moment for the ultimate hogging and sagging momentrespectively.

The same data for the longitudinal strength is presented also as moment-curvature responsesin Fig. 7.161 for eleven di�erent levels of shear loading ranging from zero to full shear capacity.It is noted that the comparison with the �rst yield moment has been left out in this case.This is due to the variation in the material parameters for the cross section as a consequenceof the elevated temperatures in the heat a�ected zone. With the yield stress in the zoneexposed to 900 oC being less than four percent of the nominal value, the concept of �rstyield losses its practical meaning for comparison with the ultimate capacities of the cross

1The ranges on the two axes, i.e. curvature and moment, as well as the scale of the plot, has deliberatelybeen made identical to the two previous moment-curvature plots (Figs. 7.3 and 7.13) to allow for easycomparison.


Figure 7.17: Ultimate moment interaction chart for the assumed �re damage conditionwith uniformly scaled, fully combined shear loading.

section. Thus, it has been left out, both in Fig. 7.16 and also in Tab. 7.7.

As for the previous cases, the interaction between the ultimate bending moment aboutthe vertical and horizontal axis is interesting to investigate as it helps to provide a betterunderstanding of how the heat-induced asymmetry of the cross section a�ects the ultimatecapacities. The predicted ultimate moment interaction is therefore shown in Fig. 7.17. Fromthe plot it is observed that the curves in the present case are tilted to the left. This isto be expected as the location of the �re damage is in the upper right-hand corner of thecross section as illustrated in Fig. 7.15. The non-smooth behavior at full shear capacity isalready explained. However, for the present case it is observed also to begin a�ecting thepredicted response at three-quarters of the full shear capacity. However, a quite signi�cantpart of the structure has almost no residual load carrying capacity because of the �re. Theload-displacement behavior of these members is thus very di�cult to accurately obtain forall levels of shear loading. Hence, it must be expected that non-smooth behavior of thepredicted moment capacities will arise from this. Moreover, with the �re damage located in

7.7. Summary 133

Table 7.8: Summary of the longitudinal strength of the ultra large crude carrierin pure bending as predicted by the present procedure.

Longitudinal strength Predicted ultimate capacityFirst yield Plastic Hogging Sagging

Investigated condition moment moment moment momentMy Mp Mhog

u;y M sagu;y

[GNm] [GNm] [GNm] [GNm]Intact, as-build 34.03 52.56 45.01 (132%) -30.46 (90%)Ballast 34.03 52.56 42.67 (125%) -30.95 (91%)Grounding damage 32.24 48.38 40.70 (120%) -28.99 (85%)Fire/explosion damage { 40.05 34.84 (102%) -23.55 (69%)NOTE: Parenthesized numbers indicates percent of the �rst yield moment for the intact,

as-build condition, i.e. of My = 34:03 GNm.

the upper right-hand corner of the midship section, the ultimate moment response depictedin the �rst and third quadrant are equivalent to the damage zone being in tension, whereasthe response in the second and fourth quadrant is equivalent to the damage zone being incompression. This is why the ultimate capacities in the �rst and third quadrant are higherthan those in the second and fourth quadrant.

7.7 Summary

In the previous sections the results of the four investigations performed on the midshipsection of an ultra large crude carrier has been reported. However, in this process very littlee�ort has been made to cross reference the hereby obtained ultimate capacities and momentinteraction responses. Therefore, to complete the reporting this will be addressed in thefollowing.

Considering �rst the longitudinal strength of the ultra large crude carrier, then Tab. 7.8lists the capacities of the midship section as predicted by the present procedure in thefour di�erent conditions of the vessel which has been analyzed. For reference, the midshipsection of the ultra large crude carrier being investigated, has been designed to withstand amaximum bending moment of 21:73 GNm. Observing the listed capacities in both hoggingand sagging condition, it can be concluded that in none of the presently investigated casesdoes a transgression below the maximum withstandable bending moment occur. Thus, forthe present ultra large crude carrier, none of the investigated damage conditions { andcertainly not any of the intact conditions { poses any real threat to the survival of the vessel,judging by the design criterions set by Det Norske Veritas [7] on the loading. However, aspreviously stated, the midship section of the ultra large crude carrier seems to be somewhatover-scantled, which also can be deducted from the percentages of the �rst yield moment


Figure 7.18: Comparative plot of the moment-curvature response for the ultra largecrude carrier as predicted by the present procedure.

in the intact, as-build condition shown in parentheses in Tab. 7.8. From these percentagesit is further noted that the ultimate hogging capacity is consistently greater than the �rstyield moment in the intact, as-build condition, whereas the opposite is true for the saggingcapacity, being consistently lower than the �rst yield moment.

This is of course to be expected due to the double bottom structure being very sti� comparedwith the sti�ness of the deck structure. Thus, the compressive strength of the deck will besmall compared with the bottom. Therefore, in the hogging condition, where the deck isin tension and the bottom in compression, the moment capacity of the cross section willbe quite formidable, whereas in the sagging condition, the deck structure will collapse priorto �rst yield capacity as indicated in Tab. 7.8, and clearly shown in Fig. 7.18 where thepredicted moment-curvature response for all four conditions is plotted for the case of purebending and for the case of ultimate shear capacity.

7.8. Final Assessment 135

Figure 7.19: Comparative plot of the moment interaction for the ultra largecrude carrier as predicted by the present procedure.

Regarding the interaction between horizontal and vertical bending moment, then Fig. 7.19shows the predicted response for all four analyzed condition both in the pure bending caseand ultimate shear loading case. From the plot it is observed that the intact, as-buildcondition clearly constitutes the envelope for the interaction of the ultimate moment. Thisis to be expected as the three other condition each represent either a more severely loadedcondition or a damaged condition with resulting less structural capacity.

7.8 Final Assessment

It has been the objective of this chapter to demonstrate that the present procedure wascapable of accurately predicting the ultimate capacity of vessels in both intact and damaged


conditions. Towards this end, an ultra large crude carrier was selected for investigation andfor this vessel the following conditions has been analyzed:

1. Intact, as-build condition and ballast condition.



The results hereby obtained has been presented throughout this chapter and has been sum-marized in the previous section. Based on these results, the overall assessment of the presentprocedure for the ultimate capacity of the hull girder is, that it has achieved the two primaryrequirements. This �rstly by being capable of accurately describing the ultimate responseof both intact and damaged vessels and secondly, by being a rapid procedure.

Chapter 8

Conclusion and Recommendations

8.1 Conclusion

It has been the objective of the present research to develop and build a rapid computer basedanalysis tool for calculation of the ultimate and post-ultimate capacity of the hull girder inthe general combined loading condition where the hull is exposed to moment, shear, torque,and hydrostatic pressure. Then, to verify the validity of the procedure by comparison withother methods and experimental results.

A performance evaluation of the available theoretical formulations { being the �nite elementmethod, the idealized structural unit method, and the classical beam-column method { hasinitially been conducted to form a basis for selection within the current context. Focusing notonly on overall speed of the three di�erent methods, but also on accuracy, the beam-columnapproach has been selected to form the foundation for the current research.

An idealized procedure based on the beam-column approach has then been developed. Theprocedure relies on the asymmetrical forced curvature principle and is thus formulated di-rectly in the bending moment about the instantaneous neutral axis, with the remaining shearcausing sectional forces as secondary parameters. Hereby, the present procedure allows foranalysis of asymmetric cross sections not only geometrically, but also in the form of materialproperties such as the modulus of elasticity and the yield stress. The present procedureaccounts for all of the loading components speci�ed in the objective, and further includesthe e�ect of an initial sti�ener de ection.

The present procedure has then been veri�ed against both experimental �ndings reported byNishihara [31] and, analytical results published by Melchior Hansen [17]. Although these tworeferences only includes bending about the horizontal axis together with a vertical shear force,the overall evaluation of the performance of the present procedure proved very satisfactory.Not only was the correlation between the response predicted by the present procedure and

137

138 Chapter 8. Conclusion and Recommendations

the �ndings published in [31] and [17] very convincing by achieving a less than �ve percentdeviation in all cases, the present procedure further excelled in being very fast indeed,requiring less than seven seconds per ultimate moment evaluation for a full-sized, doublehull tanker cross section.

To further test the validity of the present procedure a series of four scenarios, involving bothintact and damaged conditions of the midship section of an ultra large crude carrier, was setup to be analyzed. The conditions de�ned were:


2. Intact, ballast condition.



The aim of these four investigations has been to demonstrate the ability of the presentprocedure to handle not only geometric and material asymmetry modeled by condition 3and 4 respectively, but also the hydrostatic pressure loading on the hull structure as modelby condition 2. The initial, as-build condition (1) has been analyzed to provide both areference for comparison and to investigate the use of a simple moment interaction formulaas an alternative to the present procedure.

The response predicted by application of the present procedure to these four scenarios hasproven equally as satisfactory as the previously obtained results relating to the veri�cationprocess. The results conforms in every detail with the expected response behavior of theultra large crude carrier in each of the four analyzed conditions of the midship section. Withrespect to the possible use of an interaction formula instead of the present procedure, thenthe highly nonlinear moment response predicted most dominantly in the sagging condition,proved this not to be a recommendable approach.

Finally, the developed computer code has, apart from being highly e�cient, also proven to bevery robust. This has been achieved through great care taken not only in the formulation ofthe numerical algorithms incorporated in the solution scheme, but also through a consistentattention paid to the achievement of the best preconditioned analytical formulation of thederived solution scheme.

In conclusion, the present procedure has proven capable of accurately predicting the ulti-mate hull girder strength for the general combined loading condition in a robust and highlypro�cient manner. This, both for intact and damaged condition of the hull girder. There-fore, it can be stated that the overall objective of the present research has been met by thederivation and implementation of the present procedure.

8.2. Recommendation for Future Work 139

8.2 Recommendation for Future Work

The present research is concluded here. However, the current topic of structural capacity ofthe hull girder is far from exhausted. Even limiting the scope to the beam-column approach,as done in the present procedure, a number of sub-topics remains prone for further investi-gation which might lead to some improvement of the present procedure. Should the researchbe continued, the following subjects would be obvious candidates for further investigation:

� The loss of sti�ness caused by tripping of the sti�ener should be incorporated into theload-displacement description of the beam-columns.

� The post-collapse behavior of the beam-columns can be investigated further to possiblyobtain a more accurate description of the unloading response.

� The possible inclusion of also the warping part of the torque should be looked into. Thiswould greatly expand the type of structures which can be analyzed by the procedure.

\I've seen the �rst and the last ... I've seen the beginning,

and now I see the ending."

William FaulknerThe Sound & The Fury

April 8, 1928

140 Chapter 8. Conclusion and Recommendations


Appendix A

The Asymmetrical Forced Curvature

Principle

A.1 Purpose & Objective

When accessing the global behavior of the entire hull cross section the forced curvatureprinciple is evoked as described previously in Chapter 4. Therefore, it is the purpose of thefollowing to explain the exact derivation of the system analysis in combined loading.

The idea behind the asymmetrical forced curvature principle is simple. First of all it buildson the key assumption of Navier's hypothesis, i.e. that plane sections remain plane. Thisis a fair description of the hull girder de ection when exposed to an asymmetrical bendingmoment based on the assumed collapse mode being between two adjacent frames. Therefore,de�ning the instantaneous neutral axis as the line where the strains equals zero, bendingwill occur about this axis. Further, any bending moment about the instantaneous neutralaxis will be equivalent to a rotation � of the strain-plane about the same axis.

Finally, assuming the the individual beam-column reacts independently of the adjacentbeam-columns, the response of each individual beam-column in the cross section then be-comes only a function of how the ends of the beam-column is displaced and rotated, i.e. theaxial response force for the i th beam-column can be written as Pi(�i; �). Hence, it is of keyinterest in the overall system analysis to determine what the displacement of an individualbeam-column becomes given a pre-de�ned rotation � of the strain-plane.

To do this requires knowledge of the perpendicular distance � from the instantaneous neutralaxis to the reference point of the beam-column. Obtaining this distance is an exercise inbasic plane geometry which will be described in the following section.

141

142 Appendix A. The Asymmetrical Forced Curvature Principle

Figure A.1: De�nition sketch of the asymmetrical forced curvature principle.

A.2 Plane Geometric Description

Based on all the assumptions made so far, the position of the strain-plane can be describedby three parameters (see Fig. A.1):

1. The rotation � of the plane about the instantaneous neutral axis,

2. the angle � between the global horizontal axis and the instantaneous neutral axis, and

3. the perpendicular distance �INA from the instantaneous neutral axis to the origin of theglobal coordinate system, i.e. the center of the baseline.

Let the two unit vectors e and n be de�ned by the angle � between the global horizontalaxis and the instantaneous neutral axis as

e =

cos�sin�

!; n = e =

� sin�cos�

!

which are oriented parallel and perpendicular to the instantaneous neutral axis respectively,then with the given orientation of the rotation � of the strain-plane, the normal vector nobtains the property of always being positively oriented into the tension zone.

A.2. Plane Geometric Description 143

Referring to Fig. A.1 the origin of the global coordinate system is located in point O. PointS=(y; z) is the reference point of a single beam-column and point P is the intersection of theline from the origin O perpendicular to the instantaneous neutral axis. Thus, the referencepoint of the beam-column can be described by the vector OS, and similarly the point P bythe vector OP. These two vectors then becomes

OS =

yz

!and OP = �INAn = �INA

� sin�cos�

!

The aim is then to �nd the distance � from the beam-column reference point S perpendicularto the instantaneous neutral axis. From Fig. A.1 it is observed that this distance can befound by projecting the vector from point P to point S onto the normal vector n. Thus, thePS vector needs to be determined. This is easily done as

PS = OS�OP =

yz

!� ��INA sin�

�INA cos�

!=

y � �INA sin�z + �INA cos�

!

and thus the distance � becomes

� = n �PS = �y sin�� INA sin2 � + z cos�� INA cos

2 �

= ��INA � y sin� + z cos� (A.1)

A.2.1 Beam-Column End De ection and Rotation

The total end displacement � of a beam-column at reference position S can now be foundsimply as � = 2� tan � ' 2�� if the rotation of the strain-plane is small. The times two isbecause both of the ends of the entire hull cross section is exposed to the forced curvatureangle �. The end rotation of the beam-column is simply � as Navier's hypothesis is assumedand thus, given a rotation angle � of the instantaneous neutral axis (zero strain line) and arotation � of the strain-plane about this axis, the response of any beam-column can now bedetermined. This, because the the �nal parameter �INA (i.e. the location of the instantaneousneutral axis) can be determined by requiring force equilibrium about that axis, i.e.

Xi

Pi(�i; �)

��;�

= 0; �INA (A.2)

It should be noted that since the de�nition of the normal vector n was made such that it wasalway positively oriented into the tension zone of the cross section, and since the projectionto obtain the distance was made such that the the \length" is positively oriented in the samedirection as n, the total end displacement �i = 2�� is inherently oriented such that positivedisplacements results in the tension zone and similarly negative displacements results in thecompression zone of the cross section.


Figure A.2: De�nition of the local beam-column reference system.

A.3 Local Beam-Column Description

In the previous section it was stated that the end rotation ' of the beam-column simply wasequal to the rotation � of the strain-plane. This is of course true given that the local coordi-nate system for the beam-column coincides with the current orientation of the instantaneousneutral axis de�ned through the angle �. However, a description of the beam-column re-sponse in an arbitrary oriented local coordinate system would be a very ine�cient approachas it would require a complete solution of the beam-column problem for each di�erent ori-entation, i.e. for each angel �.

From an ease of calculation point of view, it would be preferable to have a �xed local referencesystem for the beam-column in which the beam solution is performed. This has the advantagethat the solution only has to be performed once for each beam-column. Entertaining thisidea would require an analysis of how the end displacement � and rotation ' transformfrom the current strain-plane de�ned by the three parameters �INA, �, and � into the localreference system for the beam-column shown in Fig. A.2.

A.3.1 E�ective Beam-Column End Rotation

The idea is to perform all evaluations of the beam-column response in the local referencesystem shown in Fig. A.2. The �rst thing to note is that the local reference system is rotatedby ��/2 compared to the global reference system shown in Fig. A.1. Thus, there will be aswap of sign on the end displacement � of the column and the corresponding axial load Pwhen moving from the global to the local reference system. Angles about the y-axis willhowever remain unchanged.

A.3. Local Beam-Column Description 145

Figure A.3: Sketch of the transformation between the local beam-column referencesystem, the global coordinate system, and the current strain-plane.

Thus, with respect to the magnitude of the end displacement � of the column, this di�erencein orientation of the local and global reference system, really makes no di�erence (exceptfor the sign) provided that the right local end rotation ' is used, and as long as the globalreference to the beam-column is equal to the position of the local origin of the beam-column.

Pertaining to the �rst requirement, the local end rotation is given with respect to the horizon-tal y-axis and bending of the beam-column is, from the local point of view, also perceived tobe about the horizontal axis. Thus, as the real axis about which the beam-column is exposedto an end rotation, and subsequently also is bending about, most likely will be di�erent formthe local horizontal axis, a description of the relation between these two axes is needed.

Addressing the handling of di�erent initial orientation of each individual beam-column withrespect to the entire hull cross section �rst, the angle between the global and local horizon-tal axis is introduced as shown in Fig. A.3. The angle is de�ned as being between the globalhorizontal y-axis and the positively oriented part of the local z-axis in the beam-columnreference system. The angle is thus positively oriented in the counter-clockwise direction asshown in the sketch. With this de�nition, any orientation of the cross section of a beam-column with respect to the total hull cross section is uniquely determinable. Further, themaintained sign of the angle will ensure that the end rotation of the beam-column in thelocal reference system will be perceived correctly. Hence, given a rotation of the strain-plane� about an instantaneous neutral axis rotated an angle � with respect to the global horizon-tal y-axis, then the corresponding e�ective end rotation of the beam-column about the localy-axis de�ned through the angle , becomes

'e� = � cos( � �) (A.3)

directly derivable from Fig. A.3.


Thus, this is the angle for which the load-displacement response has to be established in thecase of and rotation � of the strain-plane. Hence, the axial response force of the beam-columnthen becomes

P (�; 'e�) = P (2��; � cos( � �)) (A.4)

by means of which the position of the instantaneous neutral axis �INA is determinable thesame way as in Eq. (A.2), i.e.

Xi

Pi (�i; 'e�;i)

��;�

= 0; �INA (A.5)

and the corresponding bending moment about the instantaneous neutral axis becomes

MINA =Xi

Pi (�i; 'e�;i) �i

��;�;�INA

(A.6)

It should be noticed that with respect to the axial response force hereby obtained, then thiswill also have to have the sign swapped just as the end displacement of the beam-column,due to the opposite orientation of the local reference system in comparison with the globalsystem.

A.4 Summary

In conclusion it has been proven durable to model asymmetrical bending of the entire hullcross section by means of the asymmetrical forced curvature principle. Further, the in-troduction of a �xed local reference system for each beam-column has been successfullyincorporated in the formulation.

With respect to this local reference system for the beam-columns, the introduced, oppositeoriented system may not seem the most logical choice, as this causes some signs to beswapped. When �rst introduced there really was made no argumentation for exactly sucha choice. Moreover, a system oriented the same way as the global reference system wouldhave eliminated such problems, so why not use that kind of system? The reason for this lieswithin the formulation of the beam-column load-displacement response.

Here, the traditional way to orientate the coordinate system for a beam-column is such thatcompressive loading yields positive axial loads and corresponding positive end displacements.This because, for a column, the most interesting part of the response would be compression

A.4. Summary 147

due to the inherent stability problems in this region, and thereof following special solutionscheme used.

Therefore, to ease the derivation of the beam-column response, and the veri�cation by com-parison, the same orientation has been used in this study. However, to make the codingof the global system analysis, the load-displacement curves are simply reversed to matchthe global system, i.e. such that compressive forces are negative for negative displacementsand similarly tensile forces are positive for positive displacements. This is easily done, justby swapping the sign of both forces and displacement once the load-displacement responsehas been established consistently in the local reference system. As the orientation of theend rotation of the beam-column is una�ected by this, the transformed load-displacement isdirectly usable in the overall system analysis of the entire hull cross section. This way, a lotof possible confusion and possible coding errors is e�ectively removed yielding a much morereliable procedure.

Appendix B

Solution to the Classical

Beam-Column Problem

B.1 Purpose & Objective

Figure B.1: Sign convention for the classical beam-column problem.

The classical beam-column problem, including imperfection, is shown in Fig. B.1. Thesolution to this problem is well known, but for the purpose of consistency and thoroughnessthe derivation will be repeated here. The interest is on the normal stress distribution �xx(z)at the middle of the beam-column under the following conditions:

� The beam-column is loaded at the ends with the normal force P and laterally with theuniformly distributed line load of magnitude q.

� The rotations at the ends of the beam-column are prescribed equal to '.

� The initial de ection of the beam-column is assumed to have the shape of one half sinewave with an amplitude equal to w0.

� The sectional properties (i.e. EI and A) are assumed constant and the length of thebeam-column is `.

149

150 Appendix B. Solution to the Classical Beam-Column Problem

B.2 Beam Di�erential Equation

The initial de ection may be written as w0 sin��x`

�which is seen to ful�ll the assumed bound-

ary conditions. Thus, as known form any standard textbook (See e.g. Gere and Timoshenko[15]) the di�erential equation of the beam-column may be formulated as

EI@4w(x)

@x4+ P

@2

@x2

�w(x) + w0 sin

��x

`

��= q ;

EI@4w(x)

@x4+ P

@2w(x)

@x2� w0

��

`

�2sin

��x

`

�!= q (B.1)

B.3 De ection Solution

The solution scheme for the di�erential equation in Eq. (B.1) is simply to integrate twiceand apply the boundary conditions to determine the constants arising from the integration.These boundary conditions are:

� On the left hand side (x = 0)

{ Zero de ection, i.e. w(0) = 0

{ Prescribed end rotation, i.e. w0(0) = '

� On the right hand side (x = /2)

{ Zero end rotation, i.e. w0(/2) = 0

{ Zero shear force, i.e. EIw000(/2) = 0

By integration of Eq. (B.1) once, and applying the boundary condition at the right handside x = /2 (the middle) of the beam-column, the following is obtained

EI@3w(x)

@x3+ P

@w(x)

@x+ w0

�

`cos

��x

`

��

x = /2 ; 0!= qx + C1

Zero end rotation :@w(/2)

@x= 0

Zero shear force : EI@3w(/2)

@x3= 0

9>>>>>>>>>=>>>>>>>>>;; C1 = �q /2

B.3. De ection Solution 151

Integrating once more yields

EI@2w(x)

@x2+ P

�w(x) + w0 sin

��x

`

��=

1

2qx2 � q

`

2x+ C2

+EI

@2w(x)

@x2+ Pw(x) = �Pw0 sin

��x

`

�+qx2

2� q`x

2+ C2

The solution to this di�erential equation is well known to be

P > 0 : w(x) = C3 sin(�x) + C4 cos(�x) + wp(x) ; � =

sP

EI

P < 0 : w(x) = C3 sinh(�x) + C4 cosh(�x) + wp(x) ; � =

s�PEI

which is the solution to the homogeneous equation plus a particular solution wp(x). For thepresent case this particular solution is found to be

wp(x) = C5 sin��x

`

�+ C6x

2 + C7x+ C8

which by insertion into the di�erential equation yields

EI@2wp(x)

@x2+ Pwp(x) = �Pw0 sin

��x

`

�+qx2

2� q`x

2+ C2

+�C5EI

��

`

�2sin

��x

`

�+ 2C6EI + C5P sin

��x

`

�+ C6Px

2 + C7Px+ C8P

= �Pw0 sin��x

`

�+qx2

2� q`x

2+ C2 ; C6 =

q

2P; C7 = � q`

2P

Introducing the Euler load PE =��`

�2EI, this further reduces to

C5(P � PE) sin��x

`

�+qEI

P+ C8P = �Pw0 sin

��x

`

�+ C2

; C5 =Pwo

PE � P; C8 =

1

P

�C2 � qEI

P

�

Thus the particular solution rewrites to

wp(x) =Pwo

PE � Psin

��x

`

�+qx2

2P� q`x

2P+C2

P� qEI

P 2


Considering �rst the compressive part, i.e. P > 0 and applying the zero de ection at theleft hand side boundary condition yields

w(x) = C3 sin(�x) + C4 cos(�x) +Pw0

PE � Psin

��x

`

�+

qx2

2P� q`x

2P+C2

P� EIq

P 2

��x=0

= 0 ; C4 +C2

P� EIq

P 2= 0

Further, the prescribed end rotation at the left hand side, gives

w0(x) = C3� cos(�x)� C4� sin(�x) +Pw0

PE � P

�

`cos

��x

`

�+ 2

qx

2P� q`

2P

��x=0

= '

; C3 =1

�

Pw0

P � PE

�

`+

q`

2P+ '

!� A

Finally, zero end rotation at the right hand side, gives

w0(/2) = C3� cos(� /2)� C4� sin(� /2) +Pw0

PE � P

�

`cos

�`

2`

!��

��1

0

+ 2q`

4P� q`

2P��*

0

= 0

; C3� cos(� /2)� C4� sin(� /2) = 0+

C4 = C3 cot(� /2) =EIq

P 2� C2

P+C4 = A cot(� /2) ; C2 =

EIq

P� A cot(� /2)P

With all the integration constants now determined by application of the boundary conditions,the �nal de ection then becomes

w(x) = A sin(�x) + A cot(� /2) cos(�x) +Pw0

PE � Psin

��x

`

�+

qx2

2P� q`x

2P+EIq

P 2� A cot(� /2)� EIq

P 2

= A (sin(�x) + cot(� /2) cos(�x)) +Pw0

PE � Psin

��x

`

�+

qx2

2P� q`x

2P� A cot(� /2) (B.2)

B.4. Sectional Moment 153

Performing the exact same derivation for the tensile part, i.e. P < 0 yields the de ection

w(x) = A (sinh(�x) + coth(� /2) cosh(�x)) +Pw0

PE � Psin

��x

`

�+

qx2

2P� q`x

2P+ A coth(� /2) (B.3)

B.4 Sectional Moment

From the de ection just found, the sectional moment can be established from

M(x)

EI=@2w(x)

@x2(B.4)

By insertion of Eqs. (B.2) and (B.3) into this di�erential equation (Eq. (B.4)), and afterperforming some rearranging, the moment, or rather M=EI, becomes

�M(x)

EI=

8>>>><>>>>:A�2 (sin(�x) + cot(� /2) cos(�x)) +

Pw0

PE � P

��

`

�2sin

��x

`

�� q

PP > 0

A�2 (coth(� /2) cosh(�x)�sin(�x)) + Pw0

PE � P

��

`

�2sin

��x

`

�� q

PP < 0

Of special interest for the present investigation is the moment at the middle of the fullbeam-column. This is now easily found as

M (/2) = �EI"A�2

sin(� /2) +

cos2(� /2)

sin(� /2)

!+

Pw0

PE � P

��

`

�2sin

�`

2`

!��

1

� q

P

#

= �EI"

A�2

sin(� /2)+

Pw0

PE � P

��

`

�2� q

P

#

= � EI�2

sin(� /2)A+

PEPw0

P � PE

+EIq

P

= � �EI

sin(� /2)

Pw0

P � PE

�

`+

q`

2P+ '

!+PEPw0

P � PE

+EIq

P(B.5)

for the compressive part of the loading (P > 0), and similarly as

M (/2) = � �EI

sinh(� /2)

Pw0

P � PE

�

`+

q`

2P+ '

!+PEPw0

P � PE

+EIq

P(B.6)

for the tensile part of the loading (P < 0).


B.5 Midspan Stress Distribution in Beam-Column

With the sectional moment known, the quantity sought to be established by this investiga-tion, i.e. the normal stress distribution �xx over the beam-column cross section, is obtainableby application of Navier's equation1

�xx(x; z) = �PA� M(x)

Iz (B.7)

By inserting the moment given in Eqs. (B.5) and (B.6) into Navier's equation Eq. (B.7), thenormal stress distribution at the middle of the beam-column (i.e. at x = /2) is found to be

�xx(z)

��x = /2

= �PA+�Ez

�

�Pw0

`(P � PE)+q`

2P+ '

!� �2EPw0z

`2(P � PE)� Eqz

P(B.8)

where

� =

8>>>>><>>>>>:

sin(� /2) ; � =

sP

EI; P > 0 (Compressive loading)

sinh(� /2) ; � =

s�PEI

; P < 0 (Tensile loading)

PE = EI��

`


Looking at this expression, the �rst observation to be made is that it is unde�ned for anaxial load P equal to zero. Actually no e�ort has been made to produce a solution for thespecial case of zero axial loading. This is however not a real problem, since within in scopeof the current study, zero axial loading is really of no interest.

The use of the result in Eq. (B.8) is for the establishment of the load-displacement responseof a beam-column as outlined in Chapter 5. In this context, zero axial load is implicitlyunderstood to an initial safe situation of the beam-column in which the response of theentire beam-column is linear elastic. That is, it is assumed that there will be a tensileloading less than zero at which the the �rst deviation from the linear elastic behavior willoccur. Similarly, there is assumed to be an compressive axial loading grater than zero upuntil which the behavior also is linear elastic.

If this is not the case, the implication would be that the plate part of the beam-column willbe either yielding in tension or buckling in compression already in its initial state without any

1Formulated such that compressive loads (P > 0) yields negative stresses. This is done for reasonspertinent to the way the plate part of the beam-column considered in this research later on will be treated.

B.5. Midspan Stress Distribution in Beam-Column 155

axial load applied. That would be in contradiction to the assumed initial safe condition ofthe beam-column for an axial load equal to zero. Consequently, any beam-column exhibitingthis behavior would be discarded in the overall system analysis performed on the entire hullcross section. Thus, it can be concluded, that the unde�ned nature of Eq. (B.8) for an axialload P equal to zero poses no limitation for the purpose at hand and can thus be ignored2.

The second observation to be made is that Eq. (B.8) also is unde�ned for an axial load Pequal to the Euler load PE. Thus, at �rst glance, it would seem that the buckling load wouldbe equal to the Euler load, indicating that the beam-column response is that of a simplysupported beam-column.

Nevertheless, even though the right-hand-side of Eq. (B.8) is unde�ned for P = PE, theexpression has a limiting value for P ! PE. This can easily be seen by expanding the twotroublesome terms at P = PE, i.e.

�Ez

sin(� /2)

�Pw0

`(P � PE)� �2EPw0z

`2(P � PE)

��P=PE

(B.9)

Now looking �rst at the � expression

� =

sP

EI

��P=PE

;

vuutEI��`

�2EI

=�

`(B.10)

leading to

�Ez

sin(� /2)

��P=PE

=�Ez

` sin��2

� =�Ez

`(B.11)

Thus, at P = PE the two terms in Eq. (B.9) expands to

�Ez

`

�Pw0

`(P � PE)� �2EPw0z

`2(P � PE)= 0 (B.12)

which are seen to cancel out each other and thereby eliminating the apparent problem at aload equal to the Euler load.

2A solution is of course obtainable either by solving the di�erential equation EI @4w@x4

= q(x) or simply by�nding the limiting value of the axial stress �xx in Eq. (B.8) for P ! 0. The later can easily be done usinga successive Taylor expansion of the �= sin(� /2) term giving

limP!0

�xx(z) =Ez

/2'+

q`2z

24I


Figure B.2: Axial stress �xx versus axial load P at the middle of a typical beam-column.

Thus, the real buckling load then becomes four times the Euler load, i.e. Pcr = 4PE, as thedenominator sin(� /2) then equals zero and the stress �xx consequently escapes to in�nity.This value is recognized as the classic value for a clamped beam-column, which correspondsexactly to the assumed boundary conditions.

These conclusions can also be reached by observing the plot in Fig. B.2 where the axial stressat the extreme �ber of the ange in a typical ideal3 beam-column �xx given by Eq. (B.8) isshown as a function of the applied compressive axial load P . The plot clearly shows thatthere is C0 continuity at the Euler load (P = PE), and that the stress escapes to in�nitywhen approaching the critical compressive load of four times the Euler load (Pcr = 4PE).

Another obvious observation which can be made from the plot, is that the stress level for thepresent beam-column is much higher than the yield stress (�y = 359 MPa) long before theEuler load is reached. This is however to be expected as the presented solution is based ona linear-elastic model. Nevertheless, within the range of reasonable loads, the linear elasticbehavior will produce realistic results usable in the present study.

3By ideal is understood no initial imperfection (de ection) of the sti�ener (w0 = 0), no line load (q = 0),and zero end rotation of the beam-column (' = 0).

B.6. Summary 157

B.6 Summary

In conclusion, a expression for the normal stress distribution �xx over the beam-columncross section have been established in Eq. (B.8). The behavior of the expression has furtherbeen investigated and found to be in compliance with the assumed initial conditions for thebeam-column.

It can therefore be concluded that the obtained expression is indeed usable for applicationwithin the linear elastic response range of the load-displacement behavior of the beam-columns, both in tension and in compression.

Appendix C

Solution of the von Karman Equations

C.1 Purpose & Objective

When previously discussing the elastic compression region of the idealized beam-columnresponse in Chapter 5, the solution to the eigenvalue problem (cf. Eq. (5.28))

Dr4w �Nzw;zz � 2Nxzw;xz �Nxw;xx = 0 (C.1)

was requested. Moreover, the in-plane displacement pattern after buckling is given by thefollowing two equation (cf. Eqs. (5.29) and (5.30))

N (2)x;x +N (2)

xz;z = ��N (11)x;x +N (11)

xz;z

�(C.2)

N (2)z;z +N (2)

xz;x = ��N (11)z;z +N (11)

xz;x

�(C.3)

which also needs to be solved.

So far, all that has been stated about the solution of these two problems, is that it is donein a numerical framework consisting of a �nite di�erence description and, for the eigenvaluepart, the inverse iteration method. The purpose of this appendix is to elaborate on howthis is done in the �nite di�erence framework, and further to present some �ndings andcomparison of these with known results. The later being not only theoretical results, butalso experimental �ndings.

159

160 Appendix C. Solution of the von Karman Equations

Figure C.1: Finite di�erence approximation to the de ection w = f(x; z).

C.2 The Finite Di�erence Method

In the �nite di�erence approach, the continuous function w = f(x; z) is discretized inM�Npoints as shown in Fig. C.1, which then constitutes the �nite di�erence approximation to thefunction w. The idea is then to substitute the derivatives of the function w with di�erencein the discrete points (m;n) and thus, the method allows an ordinary di�erential equation tobe replaced by an equivalent set of simultaneous linear algebraic equations with the functionvalues w1;1: : :wM;N as the unknowns1.

Looking �rst at the simplest case of dependency in only one variable, the derivatives statedas central di�erences at point k becomes,

1st derivative = (wk+1 � wk�1) =2h

2nd derivative = (wk+1 � 2wk + wk�1) =h2

3rd derivative = (wk+2 � 2wk+1 + 2wk�1 � wk�2) =2h3

4th derivative = (wk+1 � 4wk+1 + 6wk � 4wk�1 + wk�2) =h4

The expansion from this to two dimensions is obvious. However, for the problem at hand, be-ing the plate equation, there is only two extra mixed derivatives that needs to be formulated.Thus, for the purpose of consistency at point (i; k) these two are

r4w = (wi+2;k + 2wi+1;k+1 + 2wi+1;k�1 � 8wi+1;k � 8wi�1;k + wi�2;k + 20wi;k + wi;k+2

+2wi�1;k+1 + 2wi�1;k�1 � 8wi;k+1 � 8wi;k�1 + wi;k�2) =h4

@2w

@x@z= (wi+1;k+1 + wi�1;k�1 � wi+1;k�1 � wi�1;k+1) =4h

2

1For further general reference on the �nite di�erence method see e.g. Mitchell and Gri�ths [29], orspeci�cally applied to structural mechanics, Murray [30] or Brush and Almroth [4]

C.3. Buckling Behavior 161

C.3 Buckling Behavior

The �rst step in the analysis is to �nd the critical buckling load. Utilizing the �nite di�erencemethod, the eigenvalue problem in Eq. (C.1) can, applying central di�erences, be written as

wi+2;j +2wi+1;j+1 + 2wi+1;j�1 � 8wi+1;j � 8wi�1;j

+wi�2;j + 20wi;j + wi;j+2 + 2wi�1;j+1

+2wi�1;j�1 � 8wi;j+1 � 8wi;j�1

+wi;j�2 = �h2

D[�Nx (wi+1;j � 2wi;j + wi�1;j)

�1

2Nxz (wi+1;j+1 + wi�1;j�1 � wi+1;j�1 � wi�1;j+1)

�Nz (wi;j+1 � 2wi;j + wi;j�1)]

at point (i; j) in the �nite di�erence grid. Alternatively, the same expression and be writtenin the form of stencils as

� � +1 � �� +2 �8 +2 �+1 �8 +20 �8 +1� +2 �8 +2 �� +1 � �

�w = �h2

D

8>><>>:�Nz

� � �+1 �2 +1� � �

� w

-z

6x

�Nxz

2

�1 � +1� � �+1 � �1

� w �Nx

� +1 �� 2 �� +1 �

�w

9>>=>>; (C.4)

As stated previously, the generalized eigenvalue problem can in matrix form be written as(A � �B)� = 0 with � being the eigenvector complemented by � as the correspondingeigenvalue. Thus, re-arranging Eq. (C.4) yields

0BBBBBBB@

� � +1 � �� +2 �8 +2 �+1 �8 +20 �8 +1� +2 �8 +2 �� +1 � �

� h2

D

12Nxz �Nx �1

2Nxz

�Nz 2(Nx +Nz) �Nz

�12Nxz �Nx

12Nxz

1CCCCCCCAw = 0 (C.5)

with the same x-z-axis orientation as in Eq. (C.4). In this expression (Eq. (C.5)) the twomatrices A and B is easily recognized as the result of the two stencils in frame boxes set inthe entire calculation domain. Thus, all that is need now to solve the buckling problem is anumerical solution scheme to the generalized eigenvalue problem.


C.3.1 Buckling Solution Scheme

The solution scheme is to apply the inverse iteration procedure which ensures that thebuckling load found is the lowest load (see e.g. Bathe and Wilson [2]). Following the previousderived �nite di�erence description, the eigenvalue problem is formulated in matrix form as

(A� �B)� = 0 ) B� =1

�A� ) A�1B� =

1

�� (C.6)

The algorithm for the inverse iteration procedure for the eigenvector � then becomes

A�?k = B�k�1 ) �?

k = A�1B�k�1 (C.7)

�k = �?k

1

�k(C.8)

where �k is the numerically largest component in the �?k vector, i.e. in the kth approxima-

tion to the real eigenvector �. Thus, the last step (Eq. (C.8)) in the algorithm is just anormalization of the current approximation to the eigenvector. This is required for stabilitywhen the inverse iteration procedure is used in a numerical framework.

Since the, say N , eigenvectors �i constitutes an orthogonal basis in the N dimensionalspace, it is possible to let the initial guess on the eigenvector w0 be given as a combinationof all these eigenvectors. That is,

w0 =NXi=1

Ci�i (C.9)

which multiplied with the B matrix yields

Bw0 =NXi=1

CiB�i = A

NXi=1

Ci

�i�i

!(C.10)

Thus, applying the algorithm in Eqs. (C.7) and (C.8) the �rst approximation becomes

Aw?1 = Bw0 = A

NXi=1

Ci

�i�i

!) 1

�1w1 =

NXi=1

Ci

1

�i�i (C.11)

Similarly, the second approximation becomes

1

�1

1

�2w2 =

NXi=1

Ci

�1

�i

�2�i (C.12)


Continuing this scheme to the kth approximation yields

1

�1

1

�2: : :

1

�kwk =

NXi=1

Ci

�1

�i

�k�i (C.13)

which re-writes to

kYi=1

1

�i

!wk =

�1

�1

�k 0@C1�1 + C2

�1�2

!k�2 + : : :+ CN

�1�N

!k�N

1A (C.14)

From this expression (Eq. (C.14)) is is observed that if �1 is the the numerically lowestof all N eigenvalues, i.e. j�1j < j�2j < : : : < j�N j, then the successive use of the inverseiteration algorithm will cause wk to converge towards being proportional to �1 with �kconverging towards �1. Alternatively the corresponding eigenvalue can be found by dividingeach component of wk�1 by the corresponding component of wk. All these fractions willyield the same eigenvalue �1, and thus if this approach is used selecting the fraction wherethe wk component is numerically greatest is advisable as this will be the most accuratenumerical approximation to the eigenvalue.

The requirement that �1 is the the numerically lowest of allN eigenvalues, is ful�lled for all ofthe present load cases, except for the pure shear load. In this case two numerically identicalsolutions exists, namely �Nxz;cr, which is terms of eigenvalues is equivalent to �1 = ��2.However, observing the regularity of the successive approximations, it is seen that in everyeven iteration, i.e. k = 2; 4; : : :, the corresponding eigenvalue approximation is squared, thuseliminating the problem of sign of the eigenvalue. Hence, in the case of pure shear loading,the inverse iteration scheme can still be applied and the corresponding lowest eigenvaluesquared is found by dividing each component of the wk�2 by the corresponding componentof wk.

C.3.2 Implementation

The prerequisites for establishing a �nite di�erence solution to the buckling problem of aplate �eld is now available. However, one thing remains to be addressed, namely the bound-ary conditions. For the out-of-plane de ection part, only one of the boundary conditionsdescribed in detail previously in section 5.6.4 applies. That is, that the edges all are simplysupported, i.e.

w��x = � b/2 ; z = � /2

= 0 ; w;xx

��x = � b/2

= 0 , and w;zz

��z = � /2

= 0

The general requirement of the �nite di�erence method for this fourth order problem wouldbe to introduce two �ctitious nodes outside each of the edges of the calculation domain.


Figure C.2: De�nition sketch of the plate �eld as used in the �nite di�erence solution.

However, in the present case only one �ctitious node is required, as the simply supportedboundary condition is a requirement to the second derivative of the de ection at the edge,and this derivative requires only one node on each side of the edge.

Thus, limiting the �nite di�erence description to handle only square cells, the numericalmodel of the plate �eld becomes as illustrated in Fig. C.2. As shown, only one set of�ctitious nodes exist outside the calculation domain, i.e. the plate �eld. The cell size ish � h, i.e. square, and the total size of the �nite di�erence grid is M � N , width- andlength-wise respectively.

Therefore, following the nomenclature of the previously discussed eigenvalue solution scheme,the stencils for the two matrices in (A� �B)� = 0 becomes

A =

� � +1 � �� +2 �8 +2 �+1 �8 +20 �8 +1� +2 �8 +2 �� +1 � �

(C.15)

B =

h2

2DNxz �h

2

DNx � h2

2DNxz

�h2

DNz

2h2

D(Nx +Nz) �h

2

DNz

� h2

2DNxz �h

2

DNx

h2

2DNxz

(C.16)


These two stencils are then set for all nodes except for the ones on the plate edges and the�ctitious nodes. These other nodes are used to apply the boundary conditions. In the �gurebelow to the right, the upper left-hand corner of the plate �eld shown in Fig. C.2 is depictedwith the �nite di�erence nodes indicated. The nodes marked with � are the ones where thetwo stencils in Eqs. (C.15) and (C.16) are set. The remaining nodes marked with and �are the ones used to apply the boundary conditions.

The boundary condition is simply supported. To enforce this, theequations for the nodes on the edges (marked with ) are usedto set the zero de ection condition, giving for e.g. node n theequation wn = 0 and so forth. The remaining equations for eachof the �ctitious nodes (marked with�) are used to set the rest ofthe simply supported condition, i.e. that the second derivativeequals zero at the edge. Hence, the equation for e.g. node d

becomes wd = �wl ; wd + wl = 0 and so forth along the edgesof the plate �eld. For the �nal four �ctitious nodes in the corners

the requirement that the second derivative equals zero leads to the equation wa = wk ;

wa � wk = 0 in the case of the upper left-hand corner and like-wise for the other threecorners.

C.3.3 Buckling Coe�cients

The only remaining topic to address regarding the buckling analysis is then the in-plane loaddescription, i.e. Nx and Nz width- and length-wise respectively, and the shear load Nxz. Inthe present analysis it is assumed that there is no transverse loading. That is, it is assumedthat Nx equals zero. That leaves only one direct and the shear component to be considered.Therefore, introducing the direct/shear ratio Nz=Nxz = tan(') and applying a compressive(i.e. negative) base load of

N = ��2D

b2; Nz = cos(')N ; and Nxz = sin(')N

then from the eigenvalue � the classical buckling coe�cients becomes kz = cos(')� andkxz = sin(')�, relating the critical stresses as

�z;cr = kz�2E

12 (1� �2)

�t

b

�2(C.17)

�xz;cr = kxz�2E

12 (1� �2)

�t

b

�2(C.18)

Hereby, all that is needed to perform an evaluation of the critical loading (i.e. �z;cr and �xz;cr)of an arbitrary plate �eld exposed to any direct/shear ratio ' is readily available.


C.3.4 Buckling Interaction in Combined Loading

To verify the numerical method three test cases have been solved. The geometric andmaterial properties for each of these three plate �elds is given in Tab. C.1. The calculationshave been performed with a �nite di�erence cell size h = b=23 and twenty-one di�erentdirect/shear ratios ' ranging from pure direct stress, over intermediate combined stresses,to pure shear stress, have been investigated for each plate.

The resulting buckling modes (or de ection patterns) for these three plate �elds is shown inFig. C.3, C.4, and C.5 for length over width ratios of 1.0, 1.5, and 3.0 respectively. However,only half of the intermediate loading conditions have been used to limit the extensiveness ofthe plots. Further, as the magnitude of the de ection is undetermined, a normalization hasbeen performed so that the maximum de ection numerically equals one.

To evaluate the obtained buckling modes, a prior description of the expected result would beappropriate. By observing the buckling coe�cients for the two classical cases of pure directstress loading and pure shear stress loading shown in Figs. C.8 and C.9 it is noted that theshift in the number of half-waves in the de ection pattern at buckling occurs at di�erentlength over width ratios for the two load cases.

Generally, the shear loading has a sti�ening e�ect on the response of the plate �eld andthus the shift from e.g. one to two half-waves �rst occurs at length over width ratios greaterthan approximately 2.2, whereas for the pure direct stress load case the shift happens atapproximately 1.4. Consequently, it must be expected that a change in the number of half-waves in the de ection pattern will change to a lower number for a given plate when theload is varied from pure direct to pure shear loading.

From the de ection plots (Fig. C.3, C.4, and C.5) it is observed that the solution capturesthe shift from the direct stress dominated buckling mode to a shear dominated mode withone less half-wave in the de ection pattern. It it further seen that the buckling modes eachcomplies with the respective number of half-waves in the extreme load cases { being puredirect and pure shear load { as can be read o� Figs. C.8 and C.9 for the corresponding lengthover width ratios (i.e. 1.0, 1.5, and 3.0 respectively).

Table C.1: Dimensions and material properties for the three test plates.

`=b = 1:0 `=b = 1:5 `=b = 3:0Length (`) 1.0 m 1.5 m 3.0 mWidth (b) 1.0 m 1.0 m 1.0 mThickness (t) 5.5 mm 5.5 mm 5.5 mmYoung's modulus (E) 205 GPa 205 GPa 205 GPaPoisson's ratio (�) 0.3 0.3 0.3Yield stress (�y) 245 MPa 245 MPa 245 MPa


Figure C.3: Buckling modes for a length over width ratio `=b=1.0. (Nz=Nxz = tan(')).


Figure C.6: Critical buckling load (eigenvalue) for three di�erent length over width ratios.

Figure C.7: Buckling interaction curves for three di�erent length over width ratios.


Buckling coe�cient for simply supported plates (Adopted from Brush and Almroth [4])

Figure C.8: In-plane compressive loading.(Adopted from [4, Fig. 3.8])

Figure C.9: In-plane shear loading.(Adopted from [4, Fig. 3.11])

Moving on to the buckling coe�cients obtained from the numerical solution, Fig. C.6 showsthe critical buckling load coe�cient (eigenvalue) for the three di�erent plates as a functionof the direct/shear ratio de�ned as tan(') = Nz=Nxz. The same information is shown inthe form of the direct buckling coe�cient kz = cos(')� versus the shear buckling coe�cientkxz = sin(')� in Fig. C.7. Note that in these plots the positive orientation of the referencesystem has been swapped, such that compressive stresses now are positive. This is done toobtain positive values for the eigenvalue and thereby also for the two buckling coe�cients,which is the classical way to present these quantities.

Looking at the resulting eigenvalue plot for the plate with a length over width ratio of 1.5,it is observed how the shift from a direct stress dominated buckling mode with two half-waves to a shear stress dominated mode with only one half-wave results in a slightly reducedcontinuation of the eigenvalue curve. It is also observed (easiest from Fig. C.7) that thebuckling coe�cients for the cases of both pure direct and pure shear stress, are in excellentcompliance with the result shown in Figs. C.8 and C.9 for all of the length over width ratios.

Consequently, based on all of these observation, it is concluded that the proposed methodadequately describes the buckling behavior. However, before leaving this topic and con-tinuing with the post-buckling behavior, a �nal interesting conclusion must be presented.From the the classic buckling coe�cients shown in Figs. C.8 and C.9, the conclusion can bedrawn that for length over width ratios above six, the continuing behavior of the bucklingcoe�cients becomes asymptotic constant.

Thus, for plate �elds with very high length over width ratios the geometric extent of theplate can be truncated to a length equal to six times the width of the plate. This is, for the


general case, an important observation, as it reduces the number of nodes needed for the�nite di�erence approximation and thereby also the time resources needed for the solving ofthe associated system of equations. However, for typical ship structures aspect ratios abovesix for the sti�ened plate �elds are rare, and the observation is therefore more of a generalsimpli�cation without real e�ect for the current project. Nevertheless, the truncation hasbeen implemented in the developed computer code.

C.4 Post-Buckling Behavior

To determine the collapse load of the plate �eld, the next step in the solution procedure is�rst of all to determine whether or not the plate has residual load carrying capabilities afterbuckling occurred. To evaluate this the collapse criterion is employed. The criterion wasde�ned as transgression of the yield stress by the mean von Mises stress along one of theunloaded edges (i.e. along the length of the plate). Thus, the following in-equality

�2z;cr + 3�2xz;cr > �2y (C.19)

based on the critical buckling stresses can be formulated. If true, then the load carryingcapacity of the plate �eld has been exhausted and the plate will have collapsed even beforebuckling occurred. In this case the mean axial stress at collapse �ac and mean shear stressat collapse �sc can be determined simply by scaling the yield stress as

�ac =�yq

1 + 3 tan(')2(C.20)

�sc =�y tan(')q1 + 3 tan(')2

= �ac tan(') (C.21)

where tan(') = Nz=Nxz is the direct/shear ratio of the given in-plane loading. That is,the collapse stresses are simply set to the right fractions of the yield stress such that thecorresponding von Mises stress will equal the yield stress i.e. collapse per de�nition.

On the other hand if the in-equality in Eq. (C.19) is false, then the plate will still have loadcarrying capacity left after buckling occurred. In this case the post-buckling behavior of theplate �eld will then have to established to assess the collapse load of the plate.

The post-buckling behavior is governed by the force equilibrium expressed in the followingtwo equations

N (2)x;x +N (2)

xz;z = ��N (11)x;x +N (11)

xz;z

�(C.22)

N (2)z;z +N (2)

xz;x = ��N (11)z;z +N (11)

xz;x

�(C.23)

C.4. Post-Buckling Behavior 173

The right-hand-side of these two equations is given in terms of the buckling de ection w(1).Thus, after having solved the buckling problem the post-buckling in-plane displacements u(2)

and v(2) can be found by solving these two equations. With these in-plane displacementsknown, the corresponding stresses in the post-buckling region can be determined, and thecollapse criterion can then be applied to obtain the collapse load of the plate. Looking atthe left-hand-side of Eqs. (C.22) and (C.23) the four terms expands to2

N (2)x;x =

Et

(1� �2)

d

dx

nu(2);x + �v(2);z

o= K

�u(2);xx + �v(2);zx

�(C.24)

N (2)z;z =

Et

(1� �2)

d

dz

nv(2);z + �u(2);x

o= K

�v(2);zz + �u(2);xz

�(C.25)

N (2)xz;x =

Et

2(1 + �)

d

dx

nv(2);x + u(2);z

o= KT

�v(2);xx + u(2);zx

�(C.26)

N (2)xz;z =

Et

2(1 + �)

d

dz

nv(2);x + u(2);z

o= KT

�v(2);xz + u(2);zz

�(C.27)

and thus, the left-hand-side of Eqs. (C.22) and (C.23) re-writes to

N (2)x;x +N (2)

xz;z = K�u(2);xx + �v(2);zx

�+KT

�v(2);xz + u(2);zz

�(C.28)

N (2)z;z +N (2)

xz;x = K�v(2);zz + �u(2);xz

�+KT

�v(2);xx + u(2);zx

�(C.29)

where

K =Et

(1� �2)and KT =

Et

2(1 + �)

Expressed in the form of stencils Eq. (C.28) becomes

N (2)x;x +N (2)

xz;z =1

h2

8>><>>:K

� +1 �� 2 �� +1 �

u(2) +K� +KT

4

�1 � +1� � �+1 � �1

v(2)

-z

6x

+KT

� � �+1 �2 +1� � �

u(2)

9>>=>>; (C.30)

2See further Eqs. (5.17) through (5.22) formulated when the perturbation expansion was previouslyperformed in Chapter 5.


and similarly for Eq. (C.29)

N (2)z;z +N (2)

xz;x =1

h2

8>><>>:K

� � �+1 �2 +1� � �

v(2) +K� +KT

4

�1 � +1� � �+1 � �1

u(2)

-z

6x

+KT

� +1 �� 2 �� +1 �

v(2)

9>>=>>; (C.31)

Looking at the right-hand-side of Eqs. (C.22) and (C.23) the four terms expand to

N (11)x;x =

Et

2(1� �2)

d

dx

��w(1);x

�2+ �

�w(1);z

�2�= K

�w(1);xxw

(1);x + �w(1)

;zxw(1);z

�(C.32)

N (11)z;z =

Et

2(1� �2)

d

dz

��w(1);z

�2+ �

�w(1);x

�2�= K

�w(1);zzw

(1);z + �w(1)

;xzw(1);x

�(C.33)

N (11)xz;x =

Et

2(1 + �)

d

dx

nw(1);z w

(1);x

o= KT

�w(1);zxw

(1);x + w(1)

;z w(1);xx

�(C.34)

N (11)xz;z =

Et

2(1 + �)

d

dz

nw(1);z w

(1);x

o= KT

�w(1);zzw

(1);x + w(1)

;z w(1);xz

�(C.35)

These derivatives of the de ection can all readily be determined in the �nite di�erenceformulation from the previously obtained buckling solution. Hence, the post-buckling equa-tions for the in-plane displacements is now ready for implementation in the �nite di�erenceframework.

C.4.1 Post-Buckling Solution Scheme

Solution of the post-buckling problem is complicated by the fact that the collapse loadsought by the application of the perturbation technique enters the equation system throughthe boundary conditions. This obstacle is overcome by utilizing the self-adjoint property ofthe di�erential operator in the previously solved eigenvalue problem3.

The solution to the problem presents it self, when realizing that the post-buckling in-planebehavior of the plate as described by Eqs. (C.22) and (C.23) is linear with both the out-of-plane de ection and the externally applied in-plane loading. Based on this observation, it

3See further the initial discussion of the boundary conditions in Section 5.6.4, page 66 in Chapter 5 onbeam-columns in combined loading.


can be concluded that the post-buckling in-plane forces resulting from a combined in-planeloading and out-of-plane de ection at a given perturbation (loading) parameter ", can bedecomposed into a sum of two parts: One proportional to the buckling de ection squaredand independent of the in-plane loading, and one proportional to the added in-plane post-buckling load and independent of the de ection.

In other words, it can be concluded that the ratio between the in-plane forces resulting fromthe out-of-plane de ection and in-plane loading respectively, will remain constant for anygiven perturbation (loading) parameter ". This is where the self-adjoint property of thedi�erential operator come into play, as the constant ratio between the in-plane forces arisingfrom the two load cases proves to be determinable by application of exactly this property.Hence, the solution scheme then becomes a three step process:

First, the post-buckling in-plane behavior must be determined for two distinctively di�erentload cases: One with zero external in-plane loading, but with the previously obtained out-of-plane de ection at buckling, normalized such that the absolute value of the maximumde ection equals one, applied. The other case is where an external in-plane unit loading isapplied, but where the out-of-plane de ection equals zero.

This will result in two sets of in-plane displacements u and v. Based on these two solutions,the constant ratio between the in-plane forces resulting from the out-of-plane de ection andin-plane loading respectively can be determined by application of the self-adjoint property.This task then becomes the second step from which the true in-plane displacements corre-sponding to a unit loading can be determined.

The third and last step will then be to scale the unit load solution by means of the per-turbation parameter " until collapse occurs as de�ned through the collapse criterion. Thiswill give the �nal in-plane displacements at collapse. These can then be integrated alongthe loaded edges yielding the mean stress level at collapse. The stress obtained this way isequivalent with the collapse load and thus, the solution to the post-buckling behavior of theplate is found and the process can be concluded.

C.4.2 Implementation

The implementation in the �nite di�erence framework is a straight forward process. It di�ersslightly from the previously solved eigenvalue problem in being a set of two coupled equationsthat needs to be solved simultaneously. The problem can be expressed in matrix form as

24 [Au] [Av]

[Bu] [Bv]

3524 [u]

[v]

35 =

264h��N (11)x;x +N (11)

xz;z

�ih��N (11)z;z +N (11)

xz;x

�i375 (C.36)


where the left-hand-side coe�cient matrix, expressed in the form of stencils, is made up ofthe following four sub-matrices

Au =1

h2

� K �KT �2(K +KT) KT

� K �; Av =

1

4h2

��K �KT � �K +KT

� � ��K +KT � ��K �KT

Bu =1

4h2

��K �KT � �K +KT

� � ��K +KT � ��K �KT

; Bv =1

h2

� KT �K �2(K +KT) K� KT �

Thus, the post-buckling problem has the size 2M � 2N in the �nite di�erence formulation,i.e. four times the size of the previous eigenvalue problem. The right-hand-side of Eq. (C.36)will be dependent on the applied loading. At a �rst glance, it seems only to depend on theout-of-plane de ection w(1) at buckling. This is true in the sense that equation system aswritten in Eq. (C.36) is for the plate behavior only. However, when the boundary conditionsare introduced in the system through the �ctitious nodes, the right-hand-side becomes afunction also of the applied in-plane loading.

Boundary Conditions

Looking at the prescribed boundary conditions, one has already been used when the eigen-value problem was formulated, namely the simply supported condition along the plate edges.The remaining condition needed for the solution of the post-buckling problem can be dividedinto two types: Those depending only on the displacements, and those which couples withthe applied loading.

Regarding the �rst type, there are two boundary conditions which can be formulated solelyas functions of the two in-plane displacements, i.e. u and v. These are the required constantshear stress along the plate edges and that the plate edges are required to remain straight.The constant shear stress along the plate edges is equivalent to requiring that the �rstderivative of the shear strain equals zero at the edges of the plate. Thus, utilizing thestress-strain relations, the requirement at the four edges becomes

"xz;z = (v;xz + u;zz + w;zz��

0

w;xz)��x = � b/2

= (v;xz + u;zz)��x = � b/2

= 0

"xz;x = (v;xx + u;zx + w;zxw;xx��

0

)��z = � /2

= (v;xx + u;zx)��z = � /2

= 0

where the derivatives of the out-of-plane de ection cancels out because of the assumed simplesupport along the plate edges. Similarly, the required straight edges can be enforced by


requiring that the curvature of the in-plane displacement perpendicular to the edge equalszero, i.e.

u;xx��x = � b/2

= 0 and v;zz��z = � /2

= 0

Thus, there are eight requirements to set on four edges. This can be done as there are twosets of equations for each node in the entire system { one set for the u-displacement andone set for the v-displacement. The resulting equations are then set in each of the �ctitiousnodes along the edges of the plate �eld just as it was done for the previous eigenvalueproblem. Having done that, leaves two times four equations (one for each of the �ctitiouscorner nodes, cf. Fig. C.2) by which the boundary conditions whit coupled dependency onthe displacements and the applied loading can be enforced.

Looking �rst at the direct in-plane load description, then the requirement is a given constantuni-axial loading in the longitudinal direction of magnitude P and zero transverse loading.The corresponding boundary conditions to these two requirements then becomes integralequations which, when the stress-strain relations4 are inserted, takes the form

Z b/2

� b/2Nzdx

��z = � /2

= P ;

KZ b/2

� b/2(v;z + �u;x) dx

��z = � /2

+K

2

Z b/2

� b/2

�w2;z + w2

;x��

0�dx

��z = � /2

= P

for the uni-axial loading in the longitudinal direction and

Z /2

� /2Nxdz

��x = � b/2

= 0 ;

KZ /2

� /2(u;x + �v;z) dz

��x = � b/2

+K

2

Z /2

� /2

�w2;x + w2

;z��

0�dz

��x = � b/2

= 0

for the transverse loading. Here again, the simply supported boundary condition previouslyenforced in the eigenvalue problem cancels out two of the four derivatives of the out-of-planede ection w.

Similarly for the shear loading, a constant shear force proportional to the direct in-plane load-ing in the longitudinal direction by a factor of # is required. That is, after the direct/shear

4These are the standard relations previously de�ned in Chapter 5 where the strains are "z = v;z +w2

;z

2,

"x = u;x +w2

;x

2, and ";xz = v;x + u;z + w;zw;x.


ratio ' was introduced in the solution of the eigenvalue problem, the applied shear loadingis required to by proportional by # = tan('). The equivalent boundary condition to thisrequirement is also an integral equation. However, as the aspect ratio (`=b) of the plate isnot required to be equal to one, the boundary condition expands to two integral equations,which when the stress-strain relations are inserted, takes the form

Z b/2

� b/2Nxzdx

��z = � /2

= #P ;

KT

Z b/2

� b/2(v;x + u;z) dx

��z = � /2

+ KT

Z b/2

� b/2w;zw;x�

��0

dx

��z = � /2

= tan(')P

for the transverse direction and

Z /2

� /2Nxzdz

��x = � b/2

= #`

bP ;

KT

Z /2

� /2(v;x + u;z) dz

��x = � b/2

+ KT

Z /2

� /2w;xw;z�

��0

dz

��x = � b/2

= tan(')`

bP

for the longitudinal direction. Here the simply supported boundary condition completelycancels out any dependency on the out-of-plane de ection.

It is through these four integral equations that the coupling between the in-plane displace-ments and the applied loading arises. Referring back to the previously discussed solutionscheme for the post-buckling problem, the procedure was to solve the system for two di�erenttypes of loading: One with zero externally applied in-plane forces (i.e. P = 0) but with aloading in the form of a normalized out-of-plane de ection in the shape of w(1) at buckling,and one with zero out-of-plane de ection (i.e. w = 0) and an externally applied in-plane unitforce (i.e. P = 1). Thus, the right-hand-side of the post-buckling equation system will inthe two loading conditions depend only on either the out-of-plane de ection, or the in-planeload P and corresponding direct/shear ratio '.

Implementation of these four integral equations can easily be done by application of standardtrapezoidal integration approximation. This will then give the remaining eight equations toset in the �ctitious corner nodes. Eight equations as each of the four integral equation is tobe evaluated at two edges. However, due to the symmetry of the problem, there is a choiceof substituting one of each of the equations with a kinematic boundary condition. That is,because of the the symmetry each of the equations actually only needs to be set at one of theedges it applies to. The restrain thereby freed, can the be used to set a kinematic restraininstead which of course will have to be done for the system to remain solvable.


This has be used in the present implementation such that only the shear load condition atx = � b/2 is set at both the edges where it applies. The remaining three equations are onlyset for one edge and instead the three kinematic boundary conditions

u = v = 0 at x = � b/2 and z = /2u = 0 at x = � b/2 and z = /2

are enforced. This way, only compression in the longitudinal direction will occur which isthe desired behavior of the plate �eld derived from the assumption of no interaction betweenadjacent beam-columns.

Thus, with all the necessary boundary condition de�ned, the two right-hand-sides corre-sponding to the two loading scenarios can be established and the system can be solved forthe two in-plane displacement �elds. Denoting the displacements corresponding to the zeroexternal force P = 0 and normalized unit de ection w(1) loading by a bar, i.e. �u and �v, andthe displacements corresponding to unit external force P = 1 and zero de ection loadingby a hat, i.e. u and v, the real in-plane displacements corresponding to the combined trueloading can be found as

~u(2) = �u(2) + P (2)u(2) (C.37)

~v(2) = �v(2) + P (2)v(2) (C.38)

where the constant ratio P (2) between the two components is given as

P (2) = �

ZA

h��N (2)z +N (11)

z

�w(1);zz + 2

��N (2)xz +N (11)

xz

�w(1);xz +

��N (2)x +N (11)

x

�w(1);xx

iw(1)dAZ

A

hN (2)z w(1)

;zz + 2N (2)xz w

(1);xz + N (2)

x w(1);xx

iw(1)dA

which is the enforcement of the self-adjoint property that must be ful�lled by the solutiongiven the speci�ed boundary conditions.

Collapse Identi�cation

Having determined the in-plane displacement �eld, the post-buckling collapse load can thenbe found by application of the assumed collapse criterion i.e. that collapse occurs when themean von Mises stress �v at the mid-plane of the plate along one of the unloaded edgesreaches the yield stress �y. This leads to the following second order equation which can besolved for the collapse perturbation (loading) parameter squared "2c

Ac

�"2c�2

+Bc

�"2c�+ Cc = 0 (C.39)


where

Ac =1

`

Z /2

� /2

��N (2)x +N (11)

x

�2+�N (2)z +N (11)

z

�2 � �N (2)x +N (11)

x

� �N (2)z +N (11)

z

�+

3�N (2)xz +N (11)

xz

�2�dz (C.40)

Bc =1

`

Z /2

� /2

n2N (0)

z

�N (2)z +N (11)

z

��N (0)

z

�N (2)x +N (11)

x

�+

6N (0)xz

�N (2)xz +N (11)

xz

�odz (C.41)

Cc =1

`

Z /2

� /2

nN (0)z

2+ 3N (0)

xz

2odz � �2yt

2 (C.42)

in which the boundary condition stating zero transverse loading, i.e. N (0)x = 0 has been

utilized and where N (0)z = t�z;cr and N (0)

xz = t�xz;cr, i.e. the buckling solution from theeigenvalue problem.

Collapse Loading

When the collapse perturbation parameter "c has been determined, the post-buckling so-lution is completed and the resulting collapse loading can easily be found. In the presentinvestigation three measures of stress at collapse are of interest. These are:

� Mean axial edge stress at collapse

�ec = �z;cr +1

t

1

`

Z /2

� /2

�N (2)z +N (11)

z

�dz

��x = b/2

"2c (C.43)

� Mean axial stress at collapse

�ac = �z;cr +1

t

1

b

Z b/2

� b/2

�N (2)z +N (11)

z

�dx

��z = /2

"2c (C.44)

� Mean shear stress at collapse

�sc = �xz;cr +1

t

1

b

Z b/2

� b/2

�Nxz

(2) +Nxz(11)�dx

��z = /2

"2c (C.45)

C.5. Veri�cation of Collapse Criterion 181

Table C.2: Dimensions and material properties for the three test plates.

b=t = 60 b=t = 120 b=t = 180Length (`) 1.0 m 1.0 m 1.0 mWidth (b) 1.0 m 1.0 m 1.0 mThickness (t) 16.67 mm 8.33 mm 5.56 mmYoung's modulus (E) 205 GPa 205 GPa 205 GPaPoisson's ratio (�) 0.3 0.3 0.3Yield stress (�y) 245 MPa 245 MPa 245 MPa

With these stress known, along with the critical stresses at buckling, all the informationabout the plate behavior that is needed for the further establishment of the idealized load-displacement behavior of the entire beam-column in the elastic compression region is avail-able.

The object of the �nite di�erence description of the plate �eld has thus been achieved.However, it still remains to veri�ed that the chosen collapse criterion is an adequate meansof identifying the ultimate collapse load of the plate �eld.

C.5 Veri�cation of Collapse Criterion

To verify the implemented collapse criterion a test case has been performed. The object ofthe test was to see if the current implementation could reproduce the experimental �nding re-ported by Harding [19, Fig. 8.21]. From Harding, results in the form of interaction curves forthe collapse stress are available for a square plate �eld with three di�erent slenderness ratiosexposed to compression, tension and shear. However, as the present investigation is solelyconcerned with the plate response in the elastic compression region, only the compressionand shear part is considered for comparison.

The dimensions and material properties for the plate is given in Tab. C.2 for each of thethree di�erent slenderness ratios, b=t = 60, 120, and 180. Of these three, only b=t = 60is really interesting for typical ship structures. Signi�cantly higher ratios, which indicatesthinner plates, are e�ectively ruled out by the need for the vessel to be in compliance withthe design codes set down by the classi�cation societies. The reason for this requirement bythe class, is mainly to eliminate the possibility of elastic buckling of the plate �elds, whichis in essence the consequence of selecting that low a slenderness of the plate, i.e. a relativelythick plate.

Nevertheless, interaction curves for the collapse stress for all three slenderness ratios havebeen calculated using the present implementation to allow for the most thorough assessment


Figure C.10: Collapse interaction curves for a square plate with threedi�erent plate slenderness ratios.(Experimental results adopted from Harding [19, Fig. 8.21]).

of the proposed collapse criterion. The hereby obtained interaction curves are shown togetherwith the experimental result in Fig. C.10.

The computed interaction curves are based on a �nite di�erence cell size h = b=23 and�fty-one di�erent direct/shear ratios ' ranging from pure direct stress, over intermediatecombined stresses, to pure shear stress, have been investigated for each plate.

From the plot in Fig. C.10 it seems fair to conclude that the present collapse criterion,based on the transgression of the von Mises stress at one of the unloaded edges, yieldsa reasonably accurate approximation to the experimental �ndings by Harding. Especiallywhen considering that the present collapse formulation in essence is build on a second orderperturbation expansion of the plate behavior beyond the singular perturbation point at thecritical buckling load.

With this in mind, it is quiet impressive that the parabolic post-critical approximationmanages to move the critical solution that close to the real behavior as found in the Hardingexperiments. Just how much the critical load level is actually lifted is shown in Fig. C.11where the normalized critical stress is plotted along with the normalized collapse stress foundby the present formulation.

C.5. Veri�cation of Collapse Criterion 183

Figure C.11: Buckling and collapse interaction for a square plate �eld withthree di�erent slenderness ratios.Note: Buckling interaction for b=t = 60 has been truncated.

An added post-critical load capacity in the order of approximately two times the critical load,is predicted by the post-buckling solution for the two relatively slender plates (b=t = 120,and 180). That the second order approximation then manages to describe the experimental�ndings is in the author's opinion very impressive and seriously promotes the present collapsecriterion as being indeed usable.

Even better compliance between the numerical solution and the experimental results mightperhaps be obtainable by the inclusion of higher order terms in the perturbation expansionof the post-buckling behavior. However, as the computational work required to solve thepresent second order approximation is already quiet extensive, the conclusion is that theimplemented solution procedure is su�ciently accurate for the purpose at hand.

Another observation that can be made from Fig. C.11 is the explanation of the discontinuityat �=�y ' 0:85 for the plate with lowest slenderness ratio (b=t = 60). Evidently, this iswhere the critical buckling load exactly equals the collapse load. Thus, for higher levels ofshear load the plate with this slenderness ratio will collapse even before the critical bucklingload is achieved. That is, the plate will not buckle before its load capacity is depleted.


This is in excellent accordance with the previously mentioned reasons for the classi�cationsocieties to require slenderness ratios in the range of b=t = 60 in their design codes to avoidthe possibility of elastic buckling of the plate �elds in the hull construction.

C.6 Summary

Having established and veri�ed the present solution to the von Karman equations, the fol-lowing can be concluded: The application of the perturbation technique in combination witha collapse criterion based on transgression of the von Mises stress at one of the unloadedplate edges is indeed usable. Further, a numerical scheme based on the �nite di�erencemethod has been outlined and proven su�ciently accurate in its ability to predict both thebuckling load, and the post-buckling collapse load, for the purpose at hand.

There are however two free parameters left to be decided on. These are the cell size h and thenumber of di�erent direct/shear ratios '. Both of these parameters has a strong in uenceboth on the accuracy and on the rapidness of the procedure. A compromise between therequirement of a speedy procedure and a very accurate solution thus has to be made.

In the results presented so far the cell size has been set to h = b=23. The number ofdirect/shear ratios ' has been twenty-one in the buckling investigation and �fty-one in thecollapse load analyzes.

Looking �rst at the cell size, then numerical experimentation with di�erent numbers of cellsalong the width of the plate has been carried out. First of all, the number has to be at least�ve because of the need for two times two �ctitious nodes outside the boundaries of the realphysical plate. Thus, �ve would mean that only one cell is used describe the actual plate�eld. This is of course the theoretical minimum cell size and is not applicable at all.

On the other hand, the upper limit for the number of cells is really dictated by the memoryrequirements for holding the di�erent matrices in the computer code. In the present imple-mentation with 23 cells along the width of the plate a total of some 170 MB of memory isallocated for the entire plate solution procedure of which by far the majority is used to holdthe matrices. This even though the sparseness of the matrices has been used to minimizememory requirements.

Regarding the accuracy associated with the di�erent cell sizes. As expected the accuracyincreases with higher and higher number of cells, all the way until around twenty. Abovethis level of plate partitioning, there is no longer any signi�cant increase in accuracy. Con-sequently, it has been decided to maintain the current level of cells by setting the cell sizeto h = b=23. It is noted however, that this choice is based more on accuracy than on speed.If speed is really of the utmost concern, the a number as low at twelve to �fteen yieldsacceptable accuracy at a much better performance rate of the code.

C.6. Summary 185

With respect to the number of direct/shear ratios ', then looking �rst at the collapse in-teraction curves in Fig. C.10, which has been produced for a total of �fty-one direct/shearratios, it can be seen that this high a number of ratios is almost overkill. The producedcurve is practically totally smooth which is a far better representation than actually needed.Looking then at the buckling coe�cients presented in Fig. C.7, where the level of discretiza-tion is set to twenty-one direct/shear ratios, these also produce very smooth curves. Stillmuch better that actually needed.

A number of di�erent settings for the direct/shear ratios has therefore been tried, and theconclusion on these numerical experiments has been that eleven direct/shear ratios repre-sents an acceptable level of accuracy. This number has then been chosen as the number ofdirect/shear ratios which will be used in the following investigations performed in this study.

Appendix D

Alternative Modeling of the Idealized

Beam-Column Behavior

D.1 Purpose & Objective

An alternative approach to obtaining the the idealized beam-column load-de ection responsewould be to perform a full numerical solution to the di�erential equation governing the beam-column. That is, to simply solve for the axial response for any requested end displacementof a beam-column, instead of the present solution scheme developed in Chapter 5, where theidealized load-displacement response is �rst build for each beam-column, and then by meansof simple interpolation in these load-displacement curves, the axial response to a speci�edend displacement is obtained.

The bene�t of the direct calculation approach would be that a large number of specialcases, each with its on specialized solution, would be replaced by one rational and consistentsolution valid in the entire response range from tensile plastic behavior to post-ultimatecompressive behavior. Obviously, this would make the procedure much more reliable, as theneed to consider each special case, with the possibility of missing one, would be avoided.

D.2 Basic Idea

The idea is to assume an idealized stress-strain relation for the sti�ener and plate part of thebeam-column as shown in Fig. D.1. For the sti�ener part, the assumption is linear elastic,ideal plastic behavior. Thus, the stress will vary linearly up until the yield stress �y,s for thesti�ener is reached, both in tension and in compression. Thereafter, the stress will, for anyfurther straining, remain constant equal to the yield stress, i.e. the ideal plastic behaviorassumed.

187

188 Appendix D. Alternative Modeling of the Idealized Beam-Column Behavior

Figure D.1: Assumed stress-strain relation for the sti�ener and plate partof the beam-column respectively.

For the plate part the assumed stress-strain relation is a bit di�erent, as the response iscomplicated by the buckling behavior of the plate prior to compressive collapse. Further,the possibility of shear stresses present in the plate, also changes the behavior. The overallassumption is however still linear elastic, ideal plastic as for the sti�ener. For the tensilepart of the response, linear behavior is assumed up until the tensile yield capacity �tc forthe plate is reached. This will if shear stresses are present, be at a stress level which is lowerthan the direct yield stress for the plate �y,p.

In the compressive range, the behavior is assumed linear up until the critical buckling stress�cr is reached. Thereafter, the sti�ness is reduce to half the initial sti�ness, i.e. the tangentmodulus is assumed equal to half Young's modulus (Et = 1/2E). This linear response thencontinues until �nal collapse of the plate is reached at �ac. Beyond this, the compressivebehavior is assumed ideal plastic. Thus, further straining of the plate will happen at theconstant stress level �ac.

The governing equation for the beam-column is traditionally expressed in terms of bend-ing moments and axial forces. If this was changed to be in stresses and strains, then thesolution would be straight forward as this behavior is known. However, the re-writing tostresses and strains is not possible. Nevertheless, it is possible to obtain a governing equa-tion expressed in a combination of moment/forces and curvature/strains, within a numericalsolution procedure.

The scheme is then, that with the stress-strain relation known for both the sti�ener andplate part of a beam-column, then a given combination of bending moment M and axialforce P at a given position along the length of the beam-column, can be made equivalentwith a corresponding curvature � and axial strain ".

D.3. The Beam Di�erential Equation 189

Figure D.2: Simple beam model of one beam-column.

Thus, by using this scheme to solve the beam equation, the solution would be valid in theentire range of the response. Because, by doing it this way around, then the momentsM andaxial forces P would automatically re ect the assumed stress-strain behavior of the beam-column, i.e. e�ects such as the reduced sti�ness of the plate after buckling and plasticitywould automatically be included in the modeling.

D.3 The Beam Di�erential Equation

Let the sign convention be as shown in Fig. D.2, then the standard beam di�erential equationwhich can be found in any textbook on the subject of beams (See e.g. Gere and Timoshenko[15]) may be re-written as the following set of �ve ordinary di�erential equations (ODE's)formulated in the curvature � and strain " of the beam

dQ(x)

dx= �q(x)

dM(x)

dx= Q� P�

d�(x)

dx= �� d2u0(x)

dx2

du(x)

dx= �

dv(x)

dx= d`+ "

;

8>><>>:u0(x) = w0 sin

��x

`

�

d` = 1�q�2 + 1

where an initial sinusoidal de ection u0(x) has been assumed, d` is the end-shortening, and� is the instantaneous rotation of the beam.


Table D.1: Geometric and material properties for the test beam-column.

Geometric properties:Plate width : 300 mmPlate thickness : 6 mmWeb height : 69.85 mmWeb thickness : 4.445mmFlange width : 25.4 mmFlange thickness : 6.35 mm

Material properties:Young's modulus : 207 GPaYield stress : 359 MPa

D.4 Solution Method

In a numerical framework, this system of equations can be solved by the standard Runge-Kutta method, given that a relation between the current combination of moment M andaxial force P , and the corresponding curvature � and strain " combination, is known. Sucha relation can easily be established numerically by application of a simple iterative schemewhen the stress-strain relations are known. Thus, all the requirements for solving the beamequation is now present.

To test if this will work in practice, a computer code has been written implementing thisscheme, and the procedure has been applied to a test beam. The properties for the testbeam are listed in Tab. D.1. The result hereby obtained is shown in Fig. D.3 in comparisonwith the result obtained by the existing procedure described in Chapter 5. Only resultsfor the compressive region of the response is presented in the �gure, as this is the mostinteresting part of the response to base the evaluation of the new procedure on. However,before commenting on the obtained results, a short description of the problems encounteredwhile performing the new solution procedure will be presented.

The application of the Runge-Kutta method, means that the system of equations is formu-lated as an initial value problem, which is sought solved by a shooting method. That is, aguess on the condition in one end of the beam is set, and the system of equations is the usedto calculated the resulting condition arising in the other end of the beam. This result is thencompared with what the condition is known to be (the boundary conditions) and iterationis performed until convergence is obtained. However, when the behavior of the system ofequations becomes nonlinear, obtaining a new guess proved to be very troublesome. A widearray of error corrective measures was therefore include in the solution procedure, eventuallyallowing for the solution of the system of equations to be obtained. Further, the assumedideal plastic behavior had to be changed to a tangent sti�ness of one-thousand of the Young'smodulus for numerical reasons.

D.5. Summary 191

Figure D.3: Load-displacement curves for test beam-column.

D.5 Summary

Observing the results presented in Fig. D.3 it is seen that the comparison between the twosolution procedures is very good right up until just after the ultimate compressive loadis reached. At this point the new solution starts to deviate, predicting a slower rate ofunloading. Also at this point, the accuracy of the new solution starts to decrease (numericallyincrease). That is, convergence starts to prove impossible to obtain, and �nally when thenew solutions stops at approximately 2:5 mm of end displacement the new procedure totallyfails.

This problem of convergence may have been solvable by �ne-tuning the iteration procedure.However, the computational time spend by the new solution procedure was at this pointin the order of �ve hours, whereas the existing procedure was performed in less than halfa second for the comparatively same task. Consequently, as speed is one of the primaryconcerns in the current study, any further development on the new solution procedure wasdeemed in vain, and thus the procedure has been abandoned as unusable within the contextof the present research.

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List of Figures

2.1 Plot of the calculated and measured moment-curvature relations for the 1/3-scale frigate test specimen investigated for ultimate and post-ultimate capac-ity of the hull by Technical Committee III.1 of ISSC'94 [11]. A total of tencalculations are shown in comparison with the experimental result. . . . . . . 12

3.1 Geometry of the laser welded panel test specimen as reported by Dow [9]. Thepanel is an orthogonally sti�ened grillage approximately 3300 mm in totallength by 1250 mm in total width, which was investigated by the TechnicalCommittee III.1 of ISSC'97 [12], with the purpose of benchmarking numericalpredictions of the ultimate and post-ultimate capacity against experimentalresults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Graphical representation of the initial geometric imperfections present in thesame laser welded panel as in Fig. 3.1. The presentation is based on valuestaken from Dow [9]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Plot of the calculated and measured non-dimensionalized, axial stress-strainrelationships for the laser welded panel investigated by the Technical Com-mittee III.1 of ISSC'97 [12]. A total of �fteen calculations are shown in com-parison with the experimental result and the linear elastic slope. . . . . . . . 25

3.4 Plot of the same data as in Fig. 3.3. That is, a plot of the calculated andmeasured non-dimensionalized, axial stress-strain relationships for the laserwelded panel { But only for results obtained using either the beam-columnapproach or the idealized structural unit method. A total of three beam-column results and one idealized structural unit method result is shown incomparison with the experimental result. . . . . . . . . . . . . . . . . . . . . 28

4.1 Sketch illustrating how the hull cross section is idealized in the beam-columnapproach by decomposition into a number of discrete members, all of whichwith an assumed beam-column response. . . . . . . . . . . . . . . . . . . . . 32

199

200 List of Figures

4.2 Sketch illustration the sign de�nition for the sectional forces action on the hullcross section. The sectional forces considered are: Moment and shear { bothhorizontal and vertical, and further also torsional with respect to the momentload. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3 Graphical representation of the calculated shear stress distribution in a typ-ical double hull tanker structure subjected to a unit load of: Vertical shearforce (top left), horizontal shear force (top right), St. Venant torsional mo-ment (bottom left), and warping torsional moment (bottom right). The �gureillustrates that when exposed to a shear causing loading, by far the major partof the resulting shear stresses are acting on the plating of the hull. . . . . . . 36

4.4 Sketch illustrating the forced curvature principle used for the overall systemanalysis of the entire hull cross section. The loading is restricted to the thesimple case of pure horizontal bending given a constant shear stress level. . . 40

4.5 Sketch illustrating the orientation of the local beam-column reference coordi-nate system. The coordinate system has its origin located at the intersectionpoint between the plate and the web of the sti�ener. . . . . . . . . . . . . . . 42

4.6 Sketch illustrating the de�nition of the local beam-column reference systemin relation to both the global coordinate system of the hull cross section, andthe orientation of the instantaneous neutral axis and the strain-plane in thecase of pure horizontal bending. . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.7 Sketch illustrating the forced curvature principle extended to asymmetricalbending. The loading is the full �ve component load condition as illustratedin Fig. 4.2 where the the three shear stress causing forces are equated bya constant shear stress and the asymmetrical bending moment equates theremaining two moments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.8 Plane sketch of the the forced curvature principle with asymmetrical bending,i.e. for the same conditions as in Fig. 4.7. . . . . . . . . . . . . . . . . . . . . 44

4.9 Sketch of a typical moment-curvature response curve. The sketch furthershows the de�nition of the two response condition, i.e. sagging and hogging,graphically along with the de�nition of the curvature �. . . . . . . . . . . . . 46

5.1 Graphical illustration of the assumed idealized stress-strain curve for a singlebeam-column. The behavior of the beam-column is subdivided into four char-acteristic regions being: Plastic tension, elastic tension, elastic compression,and plastic compression (unloading). . . . . . . . . . . . . . . . . . . . . . . 55

5.2 Graphical illustration of an initially perfectly at, square plate subjected toin-plane loading. De�nition of sign convention is further shown. . . . . . . . 62

List of Figures 201

5.3 Graphical presentation of the buckling interaction for three similar plates, butwith di�erent length over width ratios (`=b), these being 1.0 (i.e. square), 1.5,and 3.0. See further Fig. C.7. . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.4 Graphical presentation of the collapse interaction at di�erent direct/shear loadratios for a square plate �eld. Curves are shown for three plate slendernessratios (b=t = 60, 120, and 180) along with experimental results adopted fromHarding [19]. See further Fig. C.10. . . . . . . . . . . . . . . . . . . . . . . . 74

5.5 Sketch showing the assumed con�guration of the three plastic hinge mecha-nism used to describe the post-ultimate compressive behavior of one beam-column. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.6 Sketch of the plastic hinge force equilibrium with the used sign conventionshown. Further shown is the two cross sections in the beam-column at theplastic hinge locations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.7 Sketch showing the assumed displacement behavior of the plastic hinge mech-anism with the elastic displacement and de ection at collapse interpreted asinitial conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.8 Graphical presentation of eleven load-displacement curves which demonstratethe e�ect of di�erent shear stress levels in the plate part of the beam-column.Curves are shown for zero through full yield shear stress �y in steps of tenpercent. Further, the curves for zero, �fty percent, and full yield shear stresshave been highlighted in the plot. Finally, also the dimensions for the beam-column are shown in the �gure. . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.9 Sketch showing the assumed overall buckling mechanism describing the post-ultimate region of the unsti�ened plate response. The model includes thein-plane axial load P , and the lateral pressure q. . . . . . . . . . . . . . . . . 86

5.10 Sketch of the local straight edge (multiple) folding mechanism for the post-ultimate behavior of an in-plane loaded, unsti�ened plate �eld. The mecha-nism is adopted from Kierkegaard [23]. . . . . . . . . . . . . . . . . . . . . . 87

5.11 Graphical presentation of the compressive post-ultimate (unloading) load-displacement curves for an unsti�ened plate �eld. Results are shown for twodi�erent plastic mechanisms: One based on an assumed overall buckling mode,and one based on a local straight edge (multiple) folding mechanism adoptedfrom Kierkegaard [23]. The linear elastic - ideal plastic behavior is furthershown along with the mean crushing force as predicted by the straight edgelocal folding mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

202 List of Figures

6.1 Sketch of the four cross section tested by Nishihara [31]. The cross sectionsare all rectangular of some 720�720 mm and are made to simulate the typicalconventional ship types, being tankers (single skin and double bottom), bulkcarriers, and container carriers. . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.2 Sketch of the Nishihara box girder simulating a single skin tanker structure.Principal dimensions are shown (top) along with the idealized beam-columnmodel (bottom) of the same structure. Further, The material properties arefurther listed in Tab. 6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.3 Graphical presentation of the calculated moment-curvature relations for theNishihara box girder simulating a single skin tanker structure. Three di�erentlevels of imperfection are presented: Perfect, sti�ener imperfection inwards,and outwards from the center of the cross section. Whit respect to the sti�-ener imperfection then this is only present in the deck part of the structure.Further, experimental data from [31] and by Melchior Hansen [17] are shown. 97

6.4 Graphical presentation of the calculated moment-curvature relations for theNishihara box girder simulating a single skin tanker structure. Results arepresented for four di�erent orientations � of the instantaneous neutral axis(INA), being 0o (horizontal), 45o, 90o (vertical), and 135o. . . . . . . . . . . 99

6.5 Sketch of the midship section of the double hull tanker structure analyzed incombined loading of moment and shear by Melchior Hansen [17]. Principaldimensions for the tanker vessel are listed in Tab. 6.4. . . . . . . . . . . . . . 102

6.6 Graphical presentation of the moment-curvature response for bending of themidship section of the double hull tanker shown in Fig. 6.5 about the hori-zontal axis. The response is shown for eleven cases of vertical shear loadingranging from non to ultimate shear capacity. Further, the �rst yield momentalong with the plastic moment of the cross section is indicated in the plot. . 104

6.7 Plot of the moment-shear interaction at ultimate capacity of the midshipsection of the double hull tanker shown in Fig. 6.5. That is, the ultimatemoment capacity in both hogging and sagging condition versus the verticalshear force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.1 Pro�le sketch of the 360,000 deadweight tons, double hull Ultra Large CrudeCarrier (ULCC) used as test case to demonstrate the present procedure. Theprincipal dimensions are given in Tab. 7.1 and Fig. 7.2 shows the structurallayout of the midship section. . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7.2 Graphical presentation of the midship section for the 360,000 deadweight tons,double hull Ultra Large Crude Carrier (ULCC). Further, dimensions of thelongitudinal sti�ening is shown. . . . . . . . . . . . . . . . . . . . . . . . . . 111

List of Figures 203

7.3 Graphical illustration of the predicted moment-curvature responses for theultra large crude carrier in its intact, as-build condition. The response isshown for eleven di�erent levels of vertical shear force, ranging from zero toultimate shear capacity. Further shown are the �rst yield moment and theplastic moment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.4 Graphical presentation of the interaction between the vertical shear force andthe ultimate moment about the instantaneous neutral axis for the ultra largecrude carrier in its intact, as-build condition. The interaction is shown fordi�erent orientations of the instantaneous neutral axis between 0 and 180degrees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.5 Plot of the interaction between the vertical shear force and the ultimate mo-ment about the instantaneous neutral oriented at zero degrees (horizontal),forty-�ve degrees, and ninety degrees (vertical). Further, the ultimate verticalshear force capacity is indicated in the plot. The plot is for the ultra largecrude carrier in its intact, as-build condition. . . . . . . . . . . . . . . . . . . 115

7.6 Plot of the interaction between the ultimate bending moment about the verti-cal and horizontal axis, i.e. My;u versus Mz;u, for the ultra large crude carrierin its intact, as-build condition. Interactions curves are shown for �ve di�er-ent levels of vertical shear force loading, being zero, one-quarter, half, three-quarters, and full ultimate shear force Qz;u. The two remaining load compo-nents, i.e. horizontal shear force Qy and torsional moment Mx are both zeroin the presented case. Further shown in the plot is the response as predicted bythe interaction formula

Mz

Mz,u

+k

My

My,u

!2

= 1

i.e. a parabolic �t, proposed by Mansour et al. [26]. . . . . . . . . . . . . . . 116

7.7 Plot of the interaction between the ultimate bending moment about the verti-cal and horizontal axis, i.e. My;u versus Mz;u, for the ultra large crude carrierin its intact, as-build condition. The data plotted are the same as in Fig. 7.6.However, in the present plot the the response as predicted by the interactionformula

My

My,u

!�1+

Mz

Mz,u

!�2= 1

is shown. The powers �1 and �2 entering the formula have been numerically�tted to �1 = 1:95 and �2 = 3:66. The present formula is presented as analternative to the one proposed by Mansour et al. [26]. . . . . . . . . . . . . 120

204 List of Figures

7.8 Sketch of the pressure distribution on the ULCC midship section in ballastcondition. The draught in this condition is 11:3 m. . . . . . . . . . . . . . . 121

7.9 Graphical illustration of the predicted moment-curvature responses for theultra large crude carrier in its ballast condition, i.e. at a draught of 11:30 m.The response is shown for eleven di�erent levels of vertical shear force, rangingfrom zero to ultimate shear capacity. Further shown are the �rst yield momentand the plastic moment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.10 Plot of the interaction between the ultimate bending moment about the verti-cal and horizontal axis, i.e. My;u versus Mz;u, for the ultra large crude carrierin the ballast condition. Interactions curves are shown for �ve di�erent levelsof vertical shear loading, being zero, one-quarter, half, three-quarters, andfull ultimate shear capacity �u. The shear loading is assumed to be a verticalshear force only. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.11 Graphical presentation of the bottom damage su�ered by a single skin VLCCafter grounding at Bu�alo Reef o� Singapore on January 6th 1975. The �gureis adopted from Kuroiwa [25]. . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7.12 Sketch of the �ctitious bottom damage to the ULCC midship section. Thevertical extent of the damage is assumed below the inner bottom of the 3 mhigh double bottom. The horizontal extent has been judged based on a realgrounding scenario of a single skin VLCC reported by Kuroiwa [25] (see sketchin Fig. 7.11). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.13 Graphical illustration of the predicted moment-curvature responses for theultra large crude carrier in the �ctitious grounding condition. The response isshown for eleven di�erent levels of total shear, ranging from zero to ultimateshear capacity. The total shear loading is assumed to be a uniform scalingof all the three shear stress distributions arising from vertical and horizontalshear forces, and form the St. Venant torsion. Further shown are the �rstyield moment and the plastic moment. . . . . . . . . . . . . . . . . . . . . . 127

7.14 Plot of the interaction between the ultimate bending moment about the verti-cal and horizontal axis, i.e. My;u versus Mz;u, for the ultra large crude carrierin the assumed grounding condition. Interactions curves are shown for �vedi�erent levels of vertical shear loading, being zero, one-quarter, half, three-quarters, and full ultimate shear capacity �u. The shear loading is assumedto be a uniform scaling of all the three shear stress distributions arising fromvertical and horizontal shear forces, and form the St. Venant torsion. . . . . 128

7.15 Sketch of the assumed temperature distribution in the midship section of theultra large crude carrier during a �ctitious �re in the right wing cargo tank.In the three indicated zones, the modulus of elasticity and the yield stress areassumed reduced according to the fractions listed in table Tab. 7.6. . . . . . 129

List of Figures 205

7.16 Graphical illustration of the predicted moment-curvature responses for the ul-tra large crude carrier in the �ctitious �re / explosion condition. The responseis shown for eleven di�erent levels of total shear, ranging from zero to ultimateshear capacity. The total shear loading is assumed to be a uniform scalingof all the three shear stress distributions arising from vertical and horizontalshear forces, and form the St. Venant torsion. Further shown are the �rstyield moment and the plastic moment. . . . . . . . . . . . . . . . . . . . . . 131

7.17 Plot of the interaction between the ultimate bending moment about the verti-cal and horizontal axis, i.e. My;u versus Mz;u, for the ultra large crude carrierin the assumed �re / explosion condition. Interactions curves are shown for�ve di�erent levels of vertical shear loading, being zero, one-quarter, half,three-quarters, and full ultimate shear capacity �u. The shear loading is as-sumed to be a uniform scaling of all the three shear stress distributions arisingfrom vertical and horizontal shear forces, and form the St. Venant torsion. . 132

7.18 Comparative plot of the moment-curvature response for the ultra large crudecarrier as predicted by the present procedure in the four conditions investi-gated. These four conditions are:

� Intact, as-build condition� Ballast condition� Grounding damage condition� Fire/explosion damage

for which the moment-curvature response in pure bending and at ultimateshear capacity is shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

7.19 Comparative plot of the moment interaction for the ultra large crude carrieras predicted by the present procedure in the four conditions investigated (seeFig. 7.18 above). The interaction is shown for the case of pure bending andfor the ultimate shear capacity. . . . . . . . . . . . . . . . . . . . . . . . . . 135

A.1 Plane geometric de�nition sketch of the asymmetrical forced curvature princi-ple used to perform the global system analysis of the entire hull cross section.For the purpose of clarity, the sketch is made for a simpli�ed box shaped hullcross section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

A.2 Sketch illustrating the orientation of the local beam-column reference coordi-nate system. The coordinate system has its origin located at the intersectionpoint between the plate and the web of the sti�ener. (Same as Fig. 4.5.) . . 144

A.3 Sketch illustrating the de�nition of the local beam-column reference systemin relation to both the global coordinate system of the hull cross section, andthe orientation of the instantaneous neutral axis and the strain-plane. . . . . 145

206 List of Figures

B.1 Sketch of the classical beam-column problem including initial imperfection,with the de�ned sign convention indicated. The loading is de�ned by a normalforce at the ends and a uniformly distributed line load along the length of thebeam-column. The initial imperfection is taken as one half sine wave, and theends of the beam-column are subjected to a prescribed rotation. . . . . . . . 149

B.2 Plot of the the cross sectional axial stress at the middle of a typical beam-column as a function of the applied compressive load. The stress is based ona linear-elastic solution to the standard beam di�erential equation. . . . . . . 156

C.1 Graphical illustration of the �nite di�erence approximation to the de ectionof a plate �eld showing the used coordinate system along with the conventionfor subdivision into cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

C.2 De�nition sketch of the plate �eld showing the used coordinate system alongwith the convention for subdivision into cells. Also, �ctitious nodes as opposedto \real" nodes are indicated along with the grid size. . . . . . . . . . . . . . 164

C.3 Graphical presentation of the buckling modes (de ection pattern) at di�erentloadings ranging from pure direct stress, over intermediate combined loadingsde�ned through Nz=Nxz = tan('), to pure shear stress. All for a plate �eldwith a length over width ratio (`=b) equal to 1.0 (i.e. square). The dimensionsand material properties for plate is given in Tab. C.1. . . . . . . . . . . . . . 167

C.4 Graphical presentation of the buckling modes at the same loadings as inFig. C.3, but here for a plate �eld with a length over width ratio (`=b) equalto 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

C.5 Graphical presentation of the buckling modes at the same loadings as inFig. C.3, but here for a plate �eld with a length over width ratio (`=b) equalto 3.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

C.6 Graphical presentation of the critical buckling load (or rather, the eigenvalue)for three similar plates, but with di�erent length over width ratios (`=b), thesebeing 1.0 (i.e. square), 1.5, and 3.0. The dimensions and material propertiesfor the three plates are given in Tab. C.1. The eigenvalues are plotted for dif-ferent ratios of direct versus shear loading, de�ned through Nz=Nxz = tan(').Hence, a direct/shear ratio ' of zero equals pure direct loading, and like wisea ratio of ninety degrees equals pure shear loading. . . . . . . . . . . . . . . 170

List of Figures 207

C.7 Graphical presentation of the buckling interaction for the same three plateswith di�erent length over width ratios as in Fig. C.6. The critical load is de-�ned as

�cr = k�2E

12 (1� �2)

�t

b

�2

where k is one of the classical buckling coe�cients i.e. kz for direct load andkxz for shear load. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

C.8 Plot of the classic buckling coe�cient kc for simply supported plates subjectedto in-plane compressive loading. Values are shown for length over width ratios(`=b) in the range one-half to �ve. . . . . . . . . . . . . . . . . . . . . . . . . 171

C.9 Plot of the classic buckling coe�cient ks for simply supported plates subjectedto in-plane shear loading. Values are shown for length over width ratios (`=b)in the range one to �ve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

C.10 Graphical presentation of the collapse interaction at di�erent direct/shear loadratios for a square plate �eld. Curves are shown for three plate slendernessratios (b=t = 60, 120, and 180) along with experimental results adopted fromHarding [19]. The dimensions and material properties for the three plates aregiven in Tab. C.2. Further, the stresses are non-dimensionalized by the yieldstress �y and �y for pure direct and pure shear stress respectively. . . . . . . 182

C.11 Graphical comparison of the buckling and collapse interaction at di�erentdirect/shear load ratios for a square plate �eld with three di�erent slendernessratios (b=t = 60, 120, and 180). That is, for the same three plates as inFig. C.10. Note that the buckling interaction curve for the slenderness ratiob=t = 60 has been truncated at �=�y = 1:2. . . . . . . . . . . . . . . . . . . . 183

D.1 Sketch of the assumed stress-strain relation for the sti�ener and plate partof the beam-column respectively. These relations are used to establish thecurvature of the beam-column at any given strain in association with thesolution of the beam di�erential equation as a set of �ve ordinary di�erentialequations (ODE's). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

D.2 Sketch of the assumed beam-column model used for the solution of the beamdi�erential equation as a set of �ve ordinary di�erential equations (ODE's).The chosen sign convention is shown, along with the loading and boundaryconditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

D.3 Plot of the load-displacement of the test beam-column as predicted by theidealized beam-column approach and the direct solution of the beam di�er-ential equation. The geometric and material properties for the beam-columnis given in Tab. D.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

208 List of Figures


List of Tables

2.1 Characteristics of full scale experiments conducted since the turn of the cen-tury, investigating the ultimate longitudinal strength of the vessel. Year oftest, principal dimensions, and test group are listed. The list is originallycomplied by Yao [47]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1 Measured welding residual stresses in the laser welded panel as reportedby Dow [9]. The panel is an orthogonally sti�ened grillage approximately3300 mm in total length by 1250 mm in total width, which was investigated bythe Technical Committee III.1 of ISSC'97 [12], with the purpose of benchmark-ing numerical predictions of the ultimate and post-ultimate capacity againstexperimental results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Summary of the ultimate capacity as predicted by the �fteen calculationsof the laser welded panel investigated by the Technical Committee III.1 ofISSC'97 [12]. The average axial stress �c at collapse is listed along with thenon-dimensionalized stresses and strains also at collapse (cf. Fig. 3.3). Further,the relative deviation from the experimental data is shown. . . . . . . . . . . 26

6.1 List of material properties and sectional parameters for the Nishihara [31]box girder simulating a single skin tanker (Principal dimensions are shown inFig. 6.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.2 Ultimate moment capacity for the Nishihara box girder simulating a singleskin tanker structure. The results were obtained by Melchior Hansen [17]using both a nonlinear �nite element analysis (Table 2.5 [17]), and a beam-column analysis (Figure 2.17 [17]) somewhat similar in its formulation to thepresent procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.3 Calculated ultimate moment capacity for the Nishihara box girder simulatinga single skin tanker structure. The results are listed in comparison with boththe experimental data from [31], and the capacities reported in [17] (see furtherTab. 6.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

209

210 List of Tables

6.4 List of principal dimensions and sectional parameters for the double hulltanker structure analyzed in the combined loading of bending moment andvertical shear by Melchior Hansen [17]. A sketch of the midship section forthis vessel is shown in Fig. 6.5. . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.5 List of the predicted ultimate capacities for vertical shear force, hogging,and sagging moment for the midship section of the double hull tanker shownin Fig. 6.5. Also listed are the longitudinal strength parameters, i.e. the�rst yield moment and the plastic moment. The listed results are presentedgraphically as moment-curvature responses in Fig. 6.6, and as moment-shearinteraction in Fig. 6.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.1 List of the principal dimensions for the 360,000 deadweight tons, double hullUltra Large Crude Carrier (ULCC) used as test case to demonstrate thepresent procedure. The pro�le of the tanker is shown in Fig. 7.1, and thestructural layout of the midship section in Fig. 7.2. . . . . . . . . . . . . . . 110

7.2 List of the sectional and longitudinal strength parameters for the midshipsection of the 360,000 deadweight tons, double hull Ultra Large Crude Carrier(ULCC) used as test case to demonstrate the present procedure. The midshipsection is shown in Fig. 7.2. Further, the pro�le of the tanker is shown inFig. 7.1 and Tab. 7.1 lists the principal dimensions of the vessel. . . . . . . . 111

7.3 List of the predicted ultimate capacities relating to the longitudinal strengthfor the midship section of the ultra large crude carrier in its intact, as-buildcondition. That is, the ultimate capacities for vertical shear force, hogging,and sagging moment about the horizontal axis. Also listed in the table arethe �rst yield moment and the plastic moment. . . . . . . . . . . . . . . . . 112

7.4 List of the predicted ultimate capacities relating to the longitudinal strengthfor the midship section of the ultra large crude carrier in its ballast condition,i.e. at a draught of 11:30 m. That is, the ultimate capacities for vertical shearforce, hogging, and sagging moment about the horizontal axis. Also listed inthe table are the �rst yield moment and the plastic moment. . . . . . . . . . 121

7.5 List of the predicted ultimate capacities relating to the longitudinal strengthfor the midship section of the ultra large crude carrier in the �ctitious ground-ing condition. That is, the ultimate capacities for hogging and sagging mo-ment about the horizontal axis. Also listed in the table are the �rst yieldmoment and the plastic moment. . . . . . . . . . . . . . . . . . . . . . . . . 126

7.6 Table of the sti�ness and strength of carbon steel at elevated temperatures.The Young's modulus and the yield stress are given as fraction of their ini-tial values at twenty degrees Celsius. The table is adopted from the SteelConstruction Institute's Guidance Notes [3]. . . . . . . . . . . . . . . . . . . 130

List of Tables 211

7.7 List of the predicted ultimate capacities relating to the longitudinal strengthfor the midship section of the ultra large crude carrier in the �ctitious �re /explosion condition. That is, the ultimate capacities for hogging and saggingmoment about the horizontal axis. Also listed in the table are the �rst yieldmoment and the plastic moment. . . . . . . . . . . . . . . . . . . . . . . . . 130

7.8 Summary listing of the longitudinal strength for the ultra large crude carrier aspredicted by the present procedure in the four conditions investigated. Thesefour conditions are:

� Intact, as-build condition� Ballast condition� Grounding damage condition� Fire/explosion damage

for which the �rst yield and plastic moment are listed along with the ultimatecapacities in both hogging and sagging. . . . . . . . . . . . . . . . . . . . . . 133

C.1 List of dimensions and material properties for the three plates with di�erentlength over with ratios used to test the developed �nite di�erent methodsability to predict the buckling behavior of an in-plane loaded plate. . . . . . 166

C.2 List of dimensions and material properties for the three plates with di�erentslenderness ratios used to test the developed �nite di�erent methods abilityto predict the collapse behavior of an in-plane loaded plate. . . . . . . . . . . 181

D.1 Geometric and material properties for the beam-column used to test the al-ternative solution scheme to the idealized behavior. This alternative beingthe direct solution of the �ve ordinary di�erential equations describing thestandard beam equation (see Fig. D.3). . . . . . . . . . . . . . . . . . . . . . 190

212 List of Tables


Ph.D. ThesesDepartment of Naval Architecture and O�shore Engineering

Technical University of Denmark � Lyngby

1961 Str�m-Tejsen, J.Damage Stability Calculations on the Computer DASK.

1963 Silovic, V.A Five Hole Spherical Pilot Tube for three Dimensional Wake Measurements.

1964 Chomchuenchit, V.Determination of the Weight Distribution of Ship Models.

1965 Chislett, M.S.A Planar Motion Mechanism.

1965 Nicordhanon, P.A Phase Changer in the HyA Planar Motion Mechanism and Calculation of PhaseAngle.

1966 Jensen, B.Anvendelse af statistiske metoder til kontrol af forskellige eksisterende tiln�rmelses-formler og udarbejdelse af nye til bestemmelse af skibes tonnage og stabilitet.

1968 Aage, C.Eksperimentel og beregningsm�ssig bestemmelse af vindkr�fter p�a skibe.

1972 Prytz, K.Datamatorienterede studier af planende b�ades fremdrivningsforhold.

1977 Hee, J.M.Store sideportes ind ydelse p�a langskibs styrke.

1977 Madsen, N.F.Vibrations in Ships.

1978 Andersen, P.B�lgeinducerede bev�gelser og belastninger for skib p�a l�gt vand.

1978 R�omeling, J.U.Buling af afstivede pladepaneler.

1978 S�rensen, H.H.Sammenkobling af rotations-symmetriske og generelle tre-dimensionale konstruk-tioner i elementmetode-beregninger.

1980 Fabian, O.Elastic-Plastic Collapse of Long Tubes under Combined Bending and Pressure Load.

213

214 List of Ph.D. Theses Available from the Department

1980 Petersen, M.J.Ship Collisions.

1981 Gong, J.A Rational Approach to Automatic Design of Ship Sections.

1982 Nielsen, K.B�lgeenergimaskiner.

1984 Rish�j Nielsen, N.J.Structural Optimization of Ship Structures.

1984 Liebst, J.Torsion of Container Ships.

1985 Gjers�e-Fog, N.Mathematical De�nition of Ship Hull Surfaces using B-splines.

1985 Jensen, P.S.Station�re skibsb�lger.

1986 Nedergaard, H.Collapse of O�shore Platforms.

1986 Junqui, Y.3-D Analysis of Pipelines during Laying.

1987 Holt-Madsen, A.A Quadratic Theory for the Fatigue Life Estimation of O�shore Structures.

1989 Vogt Andersen, S.Numerical Treatment of the Design-Analysis Problem of Ship Propellers using VortexLatttice Methods.

1989 Rasmussen, J.Structural Design of Sandwich Structures.

1990 Baatrup, J.Structural Analysis of Marine Structures.

1990 Wedel-Heinen, J.Vibration Analysis of Imperfect Elements in Marine Structures.

1991 Almlund, J.Life Cycle Model for O�shore Installations for Use in Prospect Evaluation.

1991 Back-Pedersen, A.Analysis of Slender Marine Structures.

List of Ph.D. Theses Available from the Department 215

1992 Bendiksen, E.Hull Girder Collapse.

1992 Buus Petersen, J.Non-Linear Strip Theories for Ship Response in Waves.

1992 Schalck, S.Ship Design Using B-spline Patches.

1993 Kierkegaard, H.Ship Collisions with Icebergs.

1994 Pedersen, B.A Free-Surface Analysis of a Two-Dimensional Moving Surface-Piercing Body.

1994 Friis Hansen, P.Reliability Analysis of a Midship Section.

1994 Michelsen, J.A Free-Form Geometric Modelling Approach with Ship Design Applications.

1995 Melchior Hansen, A.Reliability Methods for the Longitudinal Strength of Ships.

1995 Branner, K.Capacity and Lifetime of Foam Core Sandwich Structures.

1995 Schack, C.Skrogudvikling af hurtigg�aende f�rger med henblik p�a s�dygtighed og lav modstand.

1997 Cerup Simonsen, B.Mechanics of Ship Grounding.

1997 Olesen, N.A.Turbulent Flow past Ship Hulls.

1997 Riber, H.J.Response Analysis of Dynamically Loaded Composite Panels.

1998 Andersen, M.R.Fatigue Crack Initiation and Growth in Ship Structures.

216 List of Ph.D. Theses Available from the Department


Department of Naval Architecture

And Offshore Engineering


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structural capacity of the hull girder · departmen tofna v al arc hitecture and o shore...

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