Reliability Analysis of a Midship Section - DTU · midship section with the desired reliabilit y...

Peter Friis HansenJanuary 1994

Reliability Analysis of aMidship Section

Department ofNaval ArchitectureAnd Offshore Engineering

Technical University of Denmark

Reliability Analysis of aMidship Section

by

Peter Friis Hansen

Department of Naval Architectureand Offshore Engineering

Technical University of Denmark

January 1994

Preface

This thesis is submitted as a partial ful�llment of the requirements for the Danish Ph.D.degree. The work was carried out at the Department of Ocean Engineering, The Techni-cal University of Denmark, during the period of February 1991 to January 1993. ProfessorPreben Terndrup Pedersen and Regional Manager Henrik O. Madsen, DNV, Japan, super-vised the study.

The study was �nancially supported by The Danish Technical Research Council (STVF),and the support is greatfully acknowledged.

While I made this study Professor A. E. Mansour and Professor R. G. Bea, U.C. Berke-ley, gave me the opportunity to study at the Institute of Naval Architecture and O�shoreEngineering at U.C. Berkeley, California, USA, in the spring of 1992 { a very inspiring andfruitful period, and I am certainly indebted to them. The six-month stay at U.C. Berkeleywas �nancially supported by the Danish Research Academy, and the support is greatfullyacknowledged.

When sitting here { almost done { I honestly feel I am the most fortunate Ph.D. studentever been. All those people I had the opportunity to meet, to cooporate with, or just to con-sult frequently when theory or practice were obstacles to basic demands. They are: Henrik O.Madsen (DNV), Preben Terndrup Pedersen (TUD), J�rgen Juncher Jensen (TUD), StevenWinterstein (UC-Stanford), Espen Cramer (DNV), Ove Ditlevsen (TUD), Alaa Mansour(UCB), Roger Bea (UCB), Anil Thayamballi (UCB), Peter Bjerager (DNV), Robert L�seth(DNV), and Anders M. Hansen (TUD). Frankly, can anyone imagine a stronger group? Iam grateful to all these persons.

Especially, I would like to thank cordially Henrik O. Madsen for persuading me to returnto University and to join his group while he was Professor at the Civil Engineering Academy.He is one of the most inspiring, dynamic, and intelligent persons I have ever met, and I amvery proud of being one of his students.

Also very special thanks go to Steven Winterstein. During my stay in San Francisco,Steve and I started a hectic e-mail correspondence discussing all sorts of technical matters.Within the con�nes of a single cocktail napkin three or more alternative approaches to thecurrent subject would already be laid out, before we had visualized (or better, experienced)

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ii Preface

the �rst Guiness at our getting together at the BBC .... Ah .... those were the days! Theideas on the \Asymptotic expected values" and the \Quasi-stationary narrow-band model"originated during these \getting together" meetings.

Special thanks to J�rgen Juncher Jensen and Preben Terndrup Pedersen for invaluablediscussions, suggestions, interest, and patience with me when I { the civil engineer { did notunderstand the fundamentals of ocean engineering. Special thanks also to Espen Cramer forteaching me so much during our many late night discussions. To P. Ananthakrishnan forpatiently trying to teach me fundamentals of the art of technical writing { in return I taughthim to use fork and knife. To everybody here at the Department of Ocean Engineering forthe inspiring and diligent atmosphere. Our secretary Marianne Bonde for teaching me thedi�erence between the singular and the plural.

Finally, special thanks go to my family and my �anc�ee, Elena. Elena for her patiencewith me all those nights when I woke up at four o'clock realizing that everything done waswrong. For making co�ee to me and comforting me with, \Maybe you can talk to Juncheror Ditlevsen tomorrow morning { they'll help you".

Maybe the results of my sojourn will be but like rings from a stone thrown into the water.... but then again .... who cares .... it has been great fun doing this.

Peter Friis HansenLyngby, January 21, 1994

Executive Summary

A reliability analysis of a midship section is illustrated for the purpose of establishing utility-based design principles. The long-term goal is the speci�cation of an optimal set of partialsafety factors for a traditional deterministic design. To arrive at such partial safety factors,models for loading and strength are formulated. These models contain random elements,allowing that a reliability analysis with an associated sensitivity analysis can be carried outfor the relevant failure modes.

The desired reliability level is computed by minimizing a total expected cost expression,including construction cost, expected failure cost, cost of fatigue crack repair, and costof steel replacement due to corrosion. Three design parameters are considered: the platethickness of deck, side shell, and bottom. The partial safety factors are derived for a selecteddeterministic checking equation with respect to giving the same structural dimensions as forthe midship section with the desired reliability level.

The objective of this treatise is rather to remedy the establishment of a consistent proba-bilistic model universe (with due respect to model realism) within which a code format maybe calibrated, than to establish a model code itself.

The main aspects covered in this treatise are:

� Establishment of an analytical continuous two-dimensional distribution model for de-scribing the joint distribution of the wave scatter diagram experienced during thelifetime of the vessel.

� De�nition of short-term statistics for the linear case both for fatigue analysis and forextreme-value analysis. Clustering e�ects are taken into account in the extreme-valueanalysis.

� Formulation of \quasi-stationary narrow-band model" for the analysis of non-lineare�ects. Based on an approximate non-linear strip theory, motions and forces (in headsea) are calculated for a container ship in large-amplitude sinusoidal waves. Proceduresfor obtaining the response statistics are also given. The model indicates the existenceof an upper limit on the maximum wave-induced sagging moment.

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iv Executive Summary

� Extension of the \quasi-stationary narrow-band model" to the prediction of the fatiguedamage in the side shell of ship structures. The model is compared to registered failurestatistics and good agreement is found.

� Extension of the \quasi-stationary narrow-band model" to the calculation of the statis-tics of the combined low-frequency wave-induced bending moment and high-frequencyslamming-induced bending moment. The method accounts for the clustering e�ect ofthe slamming impacts, and the combination of low-frequency wave-induced bendingmoment with the high-frequency slamming-induced bending is \exact". The methodrequires that heave and pitch motions are su�ciently accurately described by linearwave theory. This is usually so.

� Formulation of a model for obtaining the long-term statistics both for fatigue analysisand extreme-value analysis. The model accounts for the operational philosophy inrough sea states, and avoidance of bad weather.

� Formulation of a procedure for establishing the combined wave-induced and still-water-induced response on a voyage. The procedure was extended to take into account theuncertainty in voyage duration. The uncertainty in voyage duration has no signi�cantimpact on the long-term distribution.

� Formulation of a probabilistic model for corrosion which takes into account the corro-sion initiation period, the ageing e�ect, and the location-dependent corrosion rates.

� Establishment of a probabilistic model for calculating the ultimate bending capacityof the hull section. A \model correction factor" formulation was used for the fullyplastic bending moment capacity, and surprisingly good agreement was obtained tothe elaborate model at a fraction of computing time. It was found by use of theelaborate model that the uncertainty in sti�ener imperfection has no impact on thefailure probability { only the mean values are appreciable.

� Formulation of a fatigue crack growth model of a semi-elliptical two-dimensional surfacecrack by application of a fracture mechanics approach, including a crack initiationperiod and threshold on the stress intensity factor.

Synopsis

En sikkerhedsanalyse af en midtskibssektion er illustreret med det form�al at etablere nytte-baserede design principper. Det langsigtede m�al er at speci�cere et optimalt s�t af par-tielle sikkerhedsfaktorer anvendelige for et traditionelt deterministisk design. For at etableres�adanne partielle sikkerhedsfaktorer er modeller for belastning og styrke formuleret. Dissemodeller indeholder usikre elementer, hvorved udf�relsen af en p�alidelighedsanalyse med endertil h�rende f�lsomhedsanalyse muligg�res for de relevante svigt m�ader.

Det �nskede sikkerhedsniveau er beregnet ved at minimere et udtryk for de samlede for-ventede omkostninger, der inkluderer konstruktions omkostninger, forventede svigtomkost-ninger, omkostninger ved reparation af udmattelsesrevner, og omkostninger ved st�aludskift-ninger p�a grund af korrosion. Der er anvendt tre design parametre: pladetykkelsen af d�k,af sideskallen og af bunden. De partielle sikkerhedsfaktorer er udledt for en valgt determin-istisk kontrol ligning s�aledes at denne resulterer i de samme konstruktionsdimensioner somfor midtskibs-sektionen med det �nskede sikkerhedsniveau.

Form�alet med denne afhandling er snarere at afhj�lpe etableringen af et konsistentsandsynlighedsbaseret model univers (under n�je krav til realistiske modeller), indenforhvilket et kodeformat kan kalibreres, end at etablere kodeformatet i sig selv.

De v�sentligste aspekter som er behandlet i denne afhandling er:

� Etableringen af en kontinuert analytisk to-dimensional fordelingsmodel til at beskriveden joint fordeling af b�lge-scatter diagram som opleves gennem fart�jets levetid.

� De�nition af kort-tids statistik for det line�re tilf�lde, b�ade for udmattelsesanalyse ogfor ekstremv�rdianalyse. Der er taget hensyn til klumpningse�ekter i ekstremv�rdianalysen.

� Formulering af en \kvasi-station�r smalb�andet model" til analyse af ikke-line�re ef-fekter. Baseret p�a en approksimativ ikke-line�r strip teori, er bev�gelser og kr�fterberegnet (i mod-s�) for et container-skib i stor-amplitudede sinus-formede b�lger. Derer ogs�a givet procedurer til at �nde responsstatistikken. Modellen indikerer at dereksisterer en �vre gr�nse for det b�lge-inducerede sagging moment.

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vi Synopsis

� Udvidelse af den \kvasi-station�re smalb�andede model" til forudsigelse af udmat-telsesskaden i side-skallen af skibskonstruktioner. Modellen er sammenlignet med re-gistrerede skadesstatistikker, og der er fundet god overensstemmelse.

� Udvidelse af den \kvasi-station�re smalb�andede model" til beregning af statistikkenfor det kombinerede lav-frekvente b�lge-inducerede b�jningsmoment med den h�j-fre-kvente slamming-indicerede b�jningsmoment. Metoden tager hensyn til klumpningsef-fekten af slammingslagene, og kombinationen af det lav-frekvente b�lge-inducerede b�j-ningsmoment med det h�j-frekvente slamminginducerede b�jning er \eksakt". Meto-den kr�ver, at heave og pitch bev�gelserne er tilstr�kkeligt akkurat beskrevet vedline�r b�lge-teori. Dette er s�dvanligvis tilf�ldet.

� Formulering af modeller til at �nde langstids statistikken b�ade for udmattelsesanalyseog ekstremv�rdianalyse. Modellen tager hensyn til operations�loso�en i h�ard s� ogundg�aelse af d�arligt vejr.

� Formulering af en procedure til at etablere den kombinerede b�lge-inducerede og stille-vands inducerede respons i en rejse. Proceduren blev udvidet til at inkludere usikker-heden i rejsevarigheden. Usikkerheden i rejsevarigheden har ingen signi�kant e�ekt p�alangtidsfordelingen.

� Formulering af en sandsynlighedsbaseret model for korrosion som tager hensyn til kor-rosionsinitieringsperioden og de lokalitetsafh�ngige korrosionsrater.

� Etablering af en sandsynlighedsbaseret model til beregning af det ultimale b�jnings-moment for skrog tv�rsnittet. En \model korrektionsfaktor" formulering blev anvendtp�a den fuldt plastiske b�jningskapacitet, og der blev fundet en overraskende god over-ensstemmelse med den komplicerede model til en fraktion af beregninstid. Ved brug afden komplicerede model blev det fundet, at usikkerheden i stiver imperfektioner ikkehar nogen ind ydelse p�a svigtsandsynligheden { kun middelv�rdierne er af betydning.

� Formulering af en revnev�kstmodel for en to-dimensional semi-elliptisk over aderevneved brug af brud-mekanik. Modellen inkluderer en revneinitieringsperiode og thresholdp�a sp�ndingsintensitets faktoren.

Contents

Preface i

Executive Summary iii

Synopsis(in Danish) v

Contents vii

1 Introduction 1

1.1 Overview and background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objectives and scope of the work . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Probabilistic Tools 7

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

2.2 Combination of stochastic processes . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Turkstra's rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.2 The Ferry Borges-Castanheta load model . . . . . . . . . . . . . . . . 9

2.2.3 General outcrossing models . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Asymptotic expected values . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 System analysis of correlated identical events . . . . . . . . . . . . . . . . . . 17

2.4.1 General theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4.2 Order statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4.3 System modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

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viii Contents

3 Environmental Modeling 23

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Sea state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 Wave spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4 Wave energy spreading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.5 Joint models of wave amplitude and frequencies . . . . . . . . . . . . . . . . 32

3.5.1 The Sveshnikov model . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.5.2 Longuet-Higgins model . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.5.3 Other models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Short-Term Response Statistics 37

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Still-water response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3 Wave action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3.1 Structural dynamic response model . . . . . . . . . . . . . . . . . . . 40

4.3.2 Hydrodynamic forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.4 Linear wave responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.4.1 Transfer function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.4.2 Response spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.4.3 Short-term response statistics . . . . . . . . . . . . . . . . . . . . . . 44

4.5 Non-linear wave responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.5.1 Second-order strip theory . . . . . . . . . . . . . . . . . . . . . . . . 49

4.5.2 Time-domain simulation . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.5.3 Quasi-stationary narrow-band non-linear model . . . . . . . . . . . . 58

4.6 Quasi-stationary fatigue analysis . . . . . . . . . . . . . . . . . . . . . . . . . 64

Contents ix

4.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.6.2 Stress range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.6.3 Veri�cation of model . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.6.4 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.7 Quasi-stationary model for impact slamming . . . . . . . . . . . . . . . . . . 80

4.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.7.2 Basic idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.7.3 Basic considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.7.4 Conditional mean excursion path . . . . . . . . . . . . . . . . . . . . 85

4.7.5 The Slepian model process . . . . . . . . . . . . . . . . . . . . . . . . 90

4.7.6 Joint distribution of wave amplitude and frequency . . . . . . . . . . 91

4.7.7 Extreme-value distribution . . . . . . . . . . . . . . . . . . . . . . . . 93

4.7.8 Slamming response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.7.9 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5 Long-Term Response Statistics 103

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.2 Operational philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.2.1 Avoidance of bad weather . . . . . . . . . . . . . . . . . . . . . . . . 105

5.2.2 Maneuvering philosophy . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.3 Stress range distribution for fatigue . . . . . . . . . . . . . . . . . . . . . . . 109

5.3.1 Long-term stress range distribution . . . . . . . . . . . . . . . . . . . 110

5.4 Extreme-value analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.4.1 Extreme wave-induced response on a voyage . . . . . . . . . . . . . . 113

5.4.2 Combination of wave-induced and still-water-induced response on avoyage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.4.3 Simpli�ed model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.4.4 Uncertainty in the duration of a voyage . . . . . . . . . . . . . . . . . 119

x Contents

6 Limit State Formulation 125

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.2 Uncertainty sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.2.1 Physical uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.2.2 Statistical uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.2.3 Model uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.2.4 Gross errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.3 Corrosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.3.1 Probabilistic model for corrosion . . . . . . . . . . . . . . . . . . . . 131

6.4 Hull strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.4.1 Ultimate bending capacity of hull section . . . . . . . . . . . . . . . . 135

6.4.2 Model correction factor method . . . . . . . . . . . . . . . . . . . . . 136

6.5 Fatigue models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.5.1 Palmgren-Miner fatigue damage model . . . . . . . . . . . . . . . . . 141

6.5.2 Fracture mechanics model . . . . . . . . . . . . . . . . . . . . . . . . 143

6.5.3 Safety margin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.5.4 Threshold on stress intensity factor . . . . . . . . . . . . . . . . . . . 149

6.5.5 Numerical values in analysis . . . . . . . . . . . . . . . . . . . . . . . 150

6.5.6 Introduction of a crack initiation period . . . . . . . . . . . . . . . . 155

7 Partial Safety Factors 157

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

7.2 Tail sensitivity problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

7.3 Calibration of partial safety factors . . . . . . . . . . . . . . . . . . . . . . . 160

7.4 Assessment of target reliability . . . . . . . . . . . . . . . . . . . . . . . . . 162

7.4.1 Optimization problem . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7.4.2 Cost modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

7.5 Safety factors for ultimate failure . . . . . . . . . . . . . . . . . . . . . . . . 170

Contents xi

8 Conclusion and Recommendations 173

8.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

8.2 Recommendations for future work . . . . . . . . . . . . . . . . . . . . . . . . 177

8.2.1 On the asymptotic expected values . . . . . . . . . . . . . . . . . . . 177

8.2.2 On the probabilistic model universe . . . . . . . . . . . . . . . . . . . 177

8.2.3 On the quasi-stationary narrow-band model . . . . . . . . . . . . . . 178

Bibliography 180

A Stresses in L-Sti�eners 192

B Simulation of the Process 195

C Uncertainty in Long-Term Distribution 198

C.0.4 Bias factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

C.0.5 Stochastic modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

C.0.6 Bootstrapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

List of Figures

2.1 Outcome of the 3-combination FBC-process. . . . . . . . . . . . . . . . . . . 10

3.1 Weighed global scatter diagram for Marsden zones 15, 16, 24, and 25. . . . . 24

3.2 Fitting of marginal 3-parameter Weibull distribution in HS. . . . . . . . . . 25

3.3 Fitting of conditional 3-parameter Weibull distribution in Tz. . . . . . . . . . 26

3.4 Progress of Weibull parameters in the conditioned distribution of TZ . . . . . 27

3.5 Dimensionless gamma spectrum for selected values of �. . . . . . . . . . . . . 30

4.1 Midship bending moment of a container vessel under sinusoidal waves, ! = 0:75. 59

4.2 Sagging moment variation as a function of wave amplitude and frequency {contour and topology plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3 Hogging moment variation as a function of wave amplitude and frequency {contour and topology plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.4 Comparison of upcrossing rates obtained by means of a direct approach withthe Hermite transformation model. . . . . . . . . . . . . . . . . . . . . . . . 62

4.5 Crack in longitudinal at connections to transverse web frames. . . . . . . . . 64

4.6 Contour plot of registered fatigue damage, frame vs. sti�ener. . . . . . . . . 65

4.7 Fatigue damage along side shell. . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.8 Registered number of cracks at frames 52 and 54. . . . . . . . . . . . . . . . 66

4.9 Stress variation for selected signi�cant wave heightsHs and mean zero crossingperiod Tz. Left: sti�ener no. 45, right: sti�ener no. 46. . . . . . . . . . . . . 71

4.10 Mean fatigue damage vs. number of simulations. . . . . . . . . . . . . . . . . 72

xii

List of Figures xiii

4.11 Comparison of damage obtained by the proposed model and frequency domainanalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.12 General outline of the analyzed vessel. . . . . . . . . . . . . . . . . . . . . . 75

4.13 Midship section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.14 Calculated fatigue damage rate for laden and ballast condition for I-shapedsti�eners. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.15 Calculated fatigue damage rate for laden and ballast condition for L-shapedsti�eners. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.16 Comparison of calculated fatigue damage and registered at frame 52 for I-shaped sti�eners. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.17 Comparison of calculated fatigue damage and registered at frame 52 for L-shaped sti�eners. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.18 Narrow-band Gaussian process with associated envelope. . . . . . . . . . . . 84

4.19 Comparison of analytical and simulated marginal distribution functions. . . . 92

4.20 Series system for extreme bending moment during the lifetime. . . . . . . . . 93

4.21 Slamming-induced bending moment combined with wave-induced bendingmoment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.1 Illustration of the load process. . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.2 Heading angle domain as a function of the sea state Hs. . . . . . . . . . . . . 108

5.3 Domain of ship speed change as a function of wave angle � and sea state Hs. 109

5.4 Dependency of expected fatigue damage on speed and direction. . . . . . . . 112

5.5 Comparison of calculated long-term distribution and di�erent �tted distribu-tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.6 Comparison of di�erent calculated long-term distributions and �tted Gammadistribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.7 Dependency of 98% fractile extreme value on speed and direction. . . . . . . 116

5.8 Combined maximum moment (hogging) during the lifetime. . . . . . . . . . 119

5.9 Combined maximum moment (sagging) during the lifetime. . . . . . . . . . . 120

xiv List of Figures

5.10 Comparison of simulated and theoretical density function for the number ofvoyages. Ten voyages expected. . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.11 Comparison of simulated and theoretical density function for the number ofvoyages. Thirty voyages expected. . . . . . . . . . . . . . . . . . . . . . . . . 122

5.12 E�ect of uncertainty in voyage duration. . . . . . . . . . . . . . . . . . . . . 123

5.13 E�ect of uncertainty in voyage duration. . . . . . . . . . . . . . . . . . . . . 124

6.1 Causes of failure in ship structures { from Akita [2]. . . . . . . . . . . . . . . 129

6.2 Illustration of the time function for corrosion. . . . . . . . . . . . . . . . . . 132

6.3 Corrosion rates in a double-hull tanker. L�seth et al. [82]. . . . . . . . . . . 133

6.4 Schematic load-end shortening curves for sti�ener in compression. . . . . . . 134

6.5 Structural outline of the examined vessel { from Paik et al. [122]. . . . . . . 137

6.6 Comparison of reliability index calculated by use of the exact model and themodel correction factor approach to the fully plastic model. . . . . . . . . . . 139

6.7 Comparison of lumped importance factors obtained by use of the exact modeland the model correction factor approach to the fully plastic model. . . . . . 139

6.8 The three modes of loading, Broek [15]. . . . . . . . . . . . . . . . . . . . . . 141

6.9 Crack in an arbitrary body, Broek [15]. . . . . . . . . . . . . . . . . . . . . . 143

6.10 Semi-elliptical surface crack in a plate under tension or bending fatigue loads. 144

6.11 E�ect of applying the threshold model to stress intensity factor. . . . . . . . 150

6.12 Calibration of S-N curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.13 Reliability - Lifetime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

6.14 Reliability { Thickness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

6.15 Calibration against the DNV X-curve. . . . . . . . . . . . . . . . . . . . . . 154

6.16 Reliability - Lifetime (DNV X-curve). . . . . . . . . . . . . . . . . . . . . . . 154

A.1 Geometry of sti�ener. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

A.2 Section for calculation of secondary forces. . . . . . . . . . . . . . . . . . . . 193

Chapter 1

Introduction

1.1 Overview and background

E�ects of commercial trading (that is, operation, loading and discharging, repair and mainte-nance practices, etc.) on hull life and potential structural failure of large vessels have duringthe last decade been the subject of considerable interest. In the past year, there has alsobeen a major concern about the loss of 25 large bulk carrier vessels worldwide during 1991and 1992. A major factor to the cause of these losses is considered to be catastrophic struc-tural failure. In addition, there was a number of structural failures resulting in signi�cantdamage to other large vessels. The majority of these vessels carried iron ore.

The industrial cost of disasters which involve large bulk carriers and oil tankers are notonly counted in terms of loss of human life, the ship and its cargo, and of environmentaldamages, but also in terms of on-going increases in insurance premiums, and loss of businesscaused by bad publicity.

To some extent the structural arrangements of modern large bulk carriers and tankershave been extrapolated from the (codi�ed) structural performance of much smaller dead-weight vessels built in the 1960s. The requirement for optimized cargo handling, the use ofhigher-yield steels, and the introduction of unconventional ship designs have further reducedthe relevance of these earlier ships as compared to their more sophisticated replacements.

Commercial pressures to reduce turn-round time have led to the use of conveyor loadingsystems, and large grabs used for discharge. Bulldozers are also used to dislodge residualore from bulk carriers' cargo hold. The stresses to the vessel's structure imposed by thesepractices are exacerbated by the use of higher-strength, thinner-section steel members tomaximize these loading/discharge practices. Other aspects which also contribute to thestructural degradation include corrosive cargoes, such as coal with a high sulphur content,which can cause sweating of the steel and concentrated corrosion of bulkheads and hull.

1

2 Chapter 1. Introduction

Currently, the design of a vessel is based mainly on rules developed by the classi�ca-tion societies. These rules are based on semi-analytical models for response and strength,empirically modi�ed to obtain agreement with observations. The safety level of existingship structures is thus, to a large extent, de�ned by registered failure statistics and provenby successful operational experience. Consequently, it is not straightforward to comparealternative designs, if these designs are not closely related to traditional design.

Recent trends in ship design have been towards increased use of high-tensile steel platesand development of unconventional ship types, such as double-bottom and double-hull struc-tures. Despite past success, the semi-empirical design concept can hardly with con�dencebe extrapolated to cover this new technology and new design. Therefore, the need for prob-abilistic models to estimate the reliability of the ship section is becoming increasingly clear.Applying probabilistic models will empower an objective comparison of traditional and newdesigns, in the sense that the designs may be ordered according to their reliability withrespect to any well-de�ned adverse event.

The semi-empirical design rules have not been calibrated against a uniform reliabilitylevel. On the contrary, several example analyses have shown very large variations in theimplied reliability level. To be fair, semi-empirical design rules are, above all, simple in formatand familiar to the shipbuilding industry. Moreover, the rules provide valuable guidelineson how to treat �rst-order design issues such as primary loads and structural strength. Therequirements of semi-empirical rules must, however, be interpreted rather as guiding criteriathan in legalistic terms. It is not feasible to perform a probabilistic design for everydayengineering practice, and there is consequently an indispensable need to establish a modelcode for ship structures for design by partial safety factors which are calibrated on the basisof a probabilistic reliability analysis.

The preparation of a design code implies evaluation of characteristic values, design values(partial safety factors ( -values)), combination factors ( -values), and formulation of loadcon�gurations etc. by calibration against the results of a set of representative examples ofprobabilistic reliability analyses. This means that a set of probability distribution models forthe material parameters and the actions must be established with respect to the requirementsof model realism. Moreover, the uncertainties as regards the representative accuracy ofthe applied mathematical models (geometry, mechanics, actions, strength, etc.) must bequanti�ed in terms of probability distributions. When these models are formulated, theprobabilities of di�erent relevant adverse events occurring within a pre-speci�ed time canbe calculated by use of existing computer programs. The relevant adverse events concernmalfunctioning with respect to serviceability or they concern regular collapse situations.The adverse events are represented in the reliability analysis as di�erent categories of limitstates, and special reliability requirements are assigned to each category. The usual limitstate categories are serviceability limit state (SLS), ultimate limit state (ULS), and accidentallimit state (ALS).

1.2. Objectives and scope of the work 3

1.2 Objectives and scope of the work

The objective of this treatise is to address some important aspects of the establishment ofa consistent model universe for the probabilistic description of the response and strength ofa midship section. Within this model universe probabilistic reliability analyses of a midshipsection can be performed. The long-term objective of the research work from which thistreatise emanates is to develop and calibrate a reliability-based code format for ship struc-tures. This treatise is thus a precursor in the achievement of the long-term objective. Thestudy deals with:

� Development of methods and procedures for modeling of the linear wave-induced re-sponse and still-water response. The model includes uncertainties in e.g. wave climateand calculation methods. The output of the model is probability density functions forthe maximum response and long-term distribution for calculation of the fatigue life ofthe structure.

� Development of methods to describe the non-linear wave-induced response.

� Development of methods and procedures for the combination of the stochastic processesduring the lifetime of the ship structure.

� Development of probabilistic methods for calculation and analysis of the structuralstrength of the sti�ened panels and the strength of the hull girder. An essential task isinclusion of corrosion. Further, the fatigue life of structural connections is considered.

� Assessment of optimal reliability requirements through a reliability-based optimizationof a midship section, in which both tangible and intangible costs are considered.

It is not the intention to give an overview of relevant references on the present status ofseakeeping analysis. Instead reference is made to the encyclopedic paper by Hutchison [60]on the status of seakeeping analysis and application. This paper contains 179 references.Wherever it is relevant in this treatise proper reference is given to relevant literature.

When the reliability of a midship section is evaluated, several di�erent adverse eventsmust be considered. The reliability requirements of these adverse events should in general bespeci�ed by the maritime administrations and classi�cation societies. To re ect the di�erentconsequences of the events, the reliability requirements will be di�erent for the di�erentadverse events. At least the following adverse events should be considered:

1. ultimate collapse of hull section including buckling of sti�ened plate elements,

2. excessive fatigue crack growth and corrosion,

3. excessive deformation of plate structures,


4. accidental events (collisions, grounding, etc.).

Only adverse events 1 and 2 (ULS) will be dealt with in this treatise.

The probabilistic reliability analysis of the hull section involves two basic problems: (1)Analysis of the bending capacity of the hull section, and (2) combination of the still-water-and wave-induced responses for the di�erent loading conditions the vessel is assumed tooperate under. Both of these two basic problems will be dealt with. The e�ects of wind-induced loads and thermal loads have not been treated in this study.

Based on a speci�c trade, a vessel will travel through given geographical areas in each ofwhich the wave conditions are assumed to be fairly uniform in a statistical sense. In each ofthese geographical areas information of the wave process is given in terms of the probabilityof occurrence of di�erent pairs of signi�cant wave height and zero crossing period. The wavedata are obtained by visual observation or by measurements on the assumptions of station-arity and ergodicity (in a short-term period { usually up to three hours). These assumptionsare crucial in the analysis of the probabilistic nature of the wave-induced responses. Other-wise complications prohibits the estimation of the probabilistic nature of the response. Ineach short-term period response statistics can be calculated and from the mission pro�le ofthe vessel be extended to long-term statistics.

Traditionally, the short-term response statistics are obtained through a linear frequencydomain analysis. The linear theory, however, falls short of capturing the measured di�er-ences between the wave-induced hogging and sagging moments. Jensen and Pedersen [65]formulated a quadratic theory which was successfully used to analyze the measured di�er-ences between the hogging and sagging moments. The quadratic theory uses a second-orderStokes wave and thus includes a non-Gaussian wave elevation. Jensen and Pedersen con-cluded that the primary reason for the non-linerities in wave-induced bending was causedby the non-verticality of the sides of the vessel and not by the non-Gaussian wave elevation.

A direct consequence of the quadratic model departing from a perturbational theoryis that the wave-induced responses are overestimated for severe sea states due to bottomemergence or deck emergence. As an alternative to frequency domain formulations, one mayresort to time-domain simulation and solve the equations of motions for irregular sea. Butdespite the growing speed of computers, time-domain simulation methods are not used in thisstudy. Several factors militate against their use: (1) The number of simulations required toadequately estimate rare failure events, and (2) the need for response analysis over multiplesea states.

In this treatise a \quasi-stationary narrow-band model" is suggested instead. The \quasi-stationary narrow-band model" carries less details, yet it is capable of re ecting larger non-linearities than the quadratic model. The idea is to impose a regular sinusoidal wave andthen �nd the corresponding steady-state non-linear force/response. The regular, sinusoidal

1.2. Objectives and scope of the work 5

wave is an important concept in the study of ocean waves. It o�ers a simple solution to thelinearized wave equations, the dynamic variables are easy to calculate, and regular waves areeasy to generate in hydraulic laboratories. Moreover, the sinusoidal wave approach allowsfor a simple approximation of the joint density function of wave amplitude and frequency ina given sea state.

The \quasi-stationary narrow-band model" is used to estimate the fatigue damage in theside shell of ship structures. The fatigue damage in the side shell is caused partly by outsidewater pressure and partly by wave-induced bending. Furthermore, the \quasi-stationarynarrow-band model" is applied to the combination problem of impact slamming and waveinduced bending.

An ocean-going cargo vessel will always operate under at least two di�erent loadingconditions, namely a laden and a ballast condition. In the prediction of the long-termdistribution it is therefore important to take into account both the loading condition inwhich the vessel operates and the duration of the voyage. The duration of a voyage in aspeci�c loading condition is uncertain, and the formulated probabilistic model takes thisuncertainty into account in the calculation of the long-term distribution.

The factors determining the ultimate collapse capacity of the hull section are the initialscantlings, the material characteristics, the level of initial imperfections such as distortionand residual stresses, and the amount of corrosion. The formulated probabilistic model forthe ultimate collapse capacity takes these uncertainties into account.

This treatise is composed as follows. Chapter 2 introduces some basic probabilistic con-cepts and reviews di�erent load combination models for linear and non-linear combinationsof stochastic processes. A model for handling an ordered set of correlated identical events ina probabilistic framework is also formulated. This model is used in the probabilistic modelfor the ultimate collapse capacity. Finally, the Chapter presents a new asymptotic methodfor calculating expected values of random variables.

In Chapter 3 the environmental modeling is de�ned. A scatter diagram is obtainedfrom the world-wide mission pro�le during the lifetime of the vessel and is �tted in withan analytical model. At the end of the Chapter, some well established models for the jointdensity function of wave amplitude and frequency in a sea state are reviewed.

Chapter 4 describes how the short-term statistics are obtained. Both linear and non-linear models are considered. The theory behind the Volterra models and the fundamentalsof the second-order strip theory are reviewed. Based on the sinusoidal wave approach, aquasi-stationary narrow-band transfer function analysis is formulated. The use of the quasi-stationary narrow-band model is exempli�ed by the analysis of fatigue damage in the sideshell and by impact slamming response analysis.


In Chapter 5 the short-term statistics of Chapter 4 are extended to long-term analysis.The long-term distribution is presented both for fatigue and for extreme-value analyses. Thelong-term prediction accounts for the operational philosophy in severe sea states, avoidanceof bad weather, and uncertainty in voyage duration under the di�erent loading conditionsthe vessel is assumed to operate under.

Chapter 6 formulates a probabilistic model for corrosion including a corrosion initiationperiod and de�nes the limit state formulation both for ultimate collapse of the midshipsection and for fatigue analysis.

In Chapter 7 it is discussed how the partial safety factors are calibrated. The targetreliability level to be used in the calibration procedure is assessed by a reliability-based costoptimization of a midship section. The cost formulation includes initial cost, cost of failure,cost of crack repair, and cost of steel replacement due to corrosion. A set of partial safetyfactors is calibrated on the basis of the optimal target reliability.

Finally, in Chapter 8 conclusions and recommendations for future work are given.

Chapter 2

Probabilistic Tools

2.1 Introduction

The design and operation of engineering systems involve decision-making under conditions ofuncertainty. This uncertainty may relate to both necessary capacity and the actual capacityof the system. The uncertainty is due to random uctuations of signi�cant physical quantitiesand due to limited information on these physical quantities. The uncertainty is also due tomodel idealization of unknown credibility introduced because of lack of knowledge as well asthe need for operability of the engineering model.

Within the �eld of civil and ocean engineering the solution of such decision-makingproblems has been facilitated by the use of codes of practice. In the simplest form thesecodes simply record common practice developed in the course of long-term experience ofsuccessful design. However, shortcomings of such codi�ed experience are evident when aproblem arises outside the �eld on which the experience is based. Moreover, to establishconsistent codes, a theoretical quanti�cation of the uncertainties present is imperative. Suchtheoretical considerations may lead to engineering systems which are optimal in a well-de�nedsense.

Structural reliability theory is a discipline for dealing with uncertainties in a consistentand rational way. The theory has developed rapidly in the last two decades, up to the presentstage of providing a conceptually and operationally satisfactory reliability methodology. Itallows of a rational comparison between alternative structural designs, maintenance plans,etc., and it is based on simple principles of including both physical, statistical, and modeluncertainty. The representation of uncertainty may be based on recorded physical observa-tions as well as subjective, professional judgements. As more information becomes availableit can be included in the reliability model by using Bayesian updating methods.

An essential result of a structural reliability analysis is the reliability measure. The mea-sure may depend on a number of assumptions adopted by the analyst. The reliabilitymeasure

7

8 Chapter 2. Probabilistic Tools

shall be interpreted as an engineering factor expressing the current knowledge/informationabout the structure and its environment as regards ability of the structure to meet theconsidered requirements { under the assumptions regarding probability distribution typesand the structural model adopted. The reliability measure introduces thus a system of theconsidered structures with respect to reliability which can be used in comparative studies.

Structural reliability concepts may be used directly in the design and operation phasesof a structure. Another important application so far is its use in the formulation and ratio-nalization of structural codes of practice. This has resulted in signi�cant changes in codeformats and has led to consistent calibration and optimization of the adjoined safety factors.Practical use of structural reliability methods is now possible, due to the availability of anumber of automated reliability computational methods, [25, 128, 153].

The two broad classes of complementary probability computational methods are theMonte Carlo Simulation (MCS) and the First-Order ReliabilityMethods or the Second-OrderReliability Methods (FORM/SORM). Since FORM/SORM are analytical methods, they donot apply when the problem concerned does not ful�ll the necessary analytical requirements,such as continuity and di�erentiability. On the other hand, for the large class of engineeringproblems to which FORM/SORM do apply, and for which the failure probability is small,FORM/SORM are generally preferable to MCS.

The methods and the application of structural reliability theory have been documentedin an increasing number of textbooks, and the fundamentals of the theory will not be givenhere. Instead reference is given to textbooks such as Ditlevsen [30], Thoft-Christensen andBaker [147], Ang and Tang [5], Madsen et al. [84], Melchers [102], and Ditlevsen and Madsen[33].

The purpose of this Chapter is to introduce and formulate probabilistic tools which { tosome extent { are used in this treatise. Three aspects will be addressed: (1) A review ofsome existing models for the combination of stochastic processes, (2) formulation of a newasymptotic theory for the calculation of expected values, and �nally (3) a system analysis ofcorrelated identical events is formulated to support the reliability analysis of the hull section,see Section 6.4.

2.2 Combination of stochastic processes

An important application of reliability methods to advanced structural design is in the treat-ment of combined loading. Let Y = g(X(t); t) de�ne a general non-linear, time-dependentcombination of the individual load processes X(t). When two or more of the loads act-ing on the structure are time-varying, the probability distribution of the lifetime maximumFmaxY;T (y), in the reference period [0; T ], cannot in general be derived solely from the prob-ability distribution of the lifetime maxima of the individual loads.

2.2. Combination of stochastic processes 9

A general evaluation of the probability distribution of the combined lifetime maximumis a complicated task. However, some approximate models exist for the case of a linearcombination of the individual load processes. These are:

� Turkstra's rule.

� The Ferry Borges-Castanheta load model.

and for general combination problems:

� General outcrossing models.

2.2.1 Turkstra's rule

Turkstra's rule [150] often provides a good estimation of the maximal e�ect of a linearcombination of independent processes. According to this rule, only the points in time atwhich one of the processes is at its maximum value are considered. The extreme-valuedistribution for the single load can often be approximately obtained, and the distribution ofeach of the accompanying load values is simply their marginal distribution. For uncorrelatedprocesses the corresponding value in time is thus easily obtained, whereas it is more involvedto apply the rule to correlated processes.

The k-combination problem de�ned by Turkstra's rule is thus

max[0;T ]

Y � max

8>><>>:

max[0;T ]X1(t) +X2(t) + � � �+Xk(t)X1(t) + max[0;T ]X2(t) + � � �+Xk(t)...X1(t) +X2(t) + � � �+max[0;T ]Xk(t)

(2.1)

The combined extreme-value distribution function is underestimated by Turkstra's rule, butthe error turns out to be small in most practical cases.

2.2.2 The Ferry Borges-Castanheta load model

The Ferry Borges-Castanheta (FBC) load model, cf. [33], is a simpli�ed load model for in-dependent loads which facilitates the mathematical problems associated with the estimationof the extreme of a sum of load processes.

For each load process it is assumed that the load changes at equal elementary timeintervals �i. The reference period T is thus divided into ni = T=�i intervals (that is the


Figure 2.1: Outcome of the 3-combination FBC-process.

number of repetitions), see Figure 2.1. The loads are further assumed to be constant in eachinterval and statistically independent from interval to interval. For a load process havingthe marginal distribution FXi

(xi) the extreme-value distribution in a time period T is thendetermined as

FmaxT Xi(xi) = FXi

(xi)ni (2.2)

The k-combination in the FBC-model is a vector X1(t); X2(t); � � � ; Xk(t) of scalar FBC-processes. It is assumed that the FBC-processes are sorted so that their repetition numbersare n1 � n2 � � � � � nk, and that they possess the unique property of ni being an integermultiplication factor of nj for any j � i. This property may be illustrated for the k = 3with repetition numbers n1 = 2, n2 = 3n1 = 6, and n3 = 2n2 = 6n1 = 12, see Figure 2.1.The extreme value for the two-combination y = x1 + x2 is the convolution

FmaxT Y (y) =�Z

Fmax�2 X1(y � x)fX2

(x)dx��2=�1

(2.3)

This equation is then recursively applied for each additional load process.

2.2.3 General outcrossing models

The distribution function for the maximum of Y is approximately given as follows, cf.Ditlevsen [29]:

FmaxY;T (y) � P0 exp

(� 1

P0

Z T

0�(t)dt

)(2.4)

2.2. Combination of stochastic processes 11

in which P0 = P [Y (0) < y] and �(t) are the mean upcrossing rate of the process Y (t) oflevel y at time t. The upcrossing rate of the process can in some cases be calculated by thegeneralized Rice formula. It is, however, highly di�cult to evaluate the integration involvedwhen the load processes are combined non-linearly or the processes are correlated.

Equation 4.4.2 in Ditlevsen [29] applies to the evaluation of the �rst crossing of stochasticprocesses, stationary or non-stationary, of a di�erentiable curve. In a somewhat similarapproach Madsen [87] formulated the upcrossing rate of scalar stochastic processes as aparticular sensitivity measure of a suitably modeled parallel system. The formulation is byHagen and Tvedt [51, 52] named Madsens's formula.

Hagen and Tvedt [51, 52] applied Madsen's formula to the calculation of Gaussian andnon-Gaussian vector process outcrossing from safe domains and derived closed-form expres-sions for the mean outcrossing rate of Gaussian processes into convex polyhedral sets.

In the following, the basic theory will be reviewed. For a complete treatment see [50, 51,52, 87].

Let X(t) be a stochastic vector process and G(X(t); t) the system failure function

G(x; t) =

8><>:< 0 unsafe domain= 0 failure surface> 0 safe domain

(2.5)

X(t) may in general be stationary or non-stationary, Gaussian or non-Gaussian.

The probability that a continuously di�erentiable time-dependent vector process X(t) =[X1(t); � � � ; Xk(t)]

0 leaves a safe domain during a time interval [0; T ] is of interest. The jointprobability density function (PDF) ofX(t) and its time derivative _X(t) is f

X _X(x(t); _x(t)).

It is assumed that G(x; t) is continuously di�erentiable in x and t.

The probability PF (T ) that failure occurs in the time interval [0; T ] is

PF (T ) = P [ mint2[0;T ]

G(X(t); t) < 0] (2.6)

General analytical methods for the calculation of Eq. 2.6 do not exist. However, if thezero downcrossings of G(X(t)) are independent, then PF (T ) can be approximated by use ofEq. 2.4 as

PF (T ) � (1� PF (0)) exp � C[0; T ]

1� PF (0)

!(2.7)


where PF (0) is the instantaneous failure rate, and C[0; T ] is the expected number of zerodown-crossings of the function G(X(t); t). This rate can be expressed as

C[0; T ] =Z T

0�(t)dt (2.8)

where �(t) is the instantaneous mean crossing rate. This rate is usually determined from thegeneralized Rice formula

�(t) =Z@G

E�[ _xn � @ _Gn j x]fX (x)d(@G) (2.9)

in which @G is the surface G(X; t) = 0, d(@G) is the surface element, E�[�] denotes theexpectation of negative values, and _xn and _Gn are the velocity components of the processand surface in the direction of the normal of the surface pointing into the safe domain.

Rice's formula gives the rate in terms of an integral expression which, however, may bedi�cult to solve analytically. The sensitivity factor formulation only requires a di�erentiationto be carried out. In the following, the method of determination of C[0; T ] proposed by Hagenand Tvedt [51, 52] will be reviewed.

C[0; T ] is derived from a formula for �(t) developed for a continuously di�erentiablescalar process upcrossing a �xed level �, [87]. The formula applied to a continuously di�er-entiable vector process X down-crossing the failure surface @G, the function G(x; t) beingcontinuously di�erentiable in x and in t, yields (Madsen's formula)

�(t) =d

d�P ( _G(X; _X; t) < 0 \G(X; t) + � _G(X; _X; t) < 0) j�=0 (2.10)

in which

_G(X; _X; t) = rxG(X; T ) _X +@G(X ; t)

@t(2.11)

Eq. 2.10 is Madsen's formula, and it can be derived as follows:

�(t) = lim�t!0+

1

�tP (G > 0 \G+ _G�t < 0)

= lim�t!0+

1

�tfP ( _G < 0 \G+ _G�t < 0)� P ( _G < 0 \G < 0)g

=d

d�P ( _G < 0 \G+ � _G < 0) j�=0 (2.12)

2.3. Asymptotic expected values 13

The equal sign from the �rst to the second step is easily seen by sketching the sets involvedin the equation, by use of G and _G as coordinate axes.

The importance of this result is that methods for the calculation of parametric sensitivityfactors for parallel systems become applicable for the calculation of the mean crossing rate.Provided the random variablesX and _X can be mapped jointly into a set of independent andstandardized normal variables, the FORM/SORM approach can be applied. It is noteworthythat a change of � from 0, Eq. 2.10, implies a rotation of the surface G + _G� = 0. Thesensitivity factor formulation by Madsen [87] accounts for this rotation and is moreoverconsistent with the FORM approximation.

For non-stationary processes or time-dependent surfaces, the parallel system formulationcan be applied to the calculation of C[0; T ] with the following result, Hagen and Tvedt [52]

C[0; T ] =Z T

0�(t)dt =

Z T

0

d

d�P (t; �)dt = T

d

d�

Z T

0P (t; �)

1

Tdt

= Td

d�

Zs2R

P (s; �)fS(s)ds

= Td

d�

ZGp(x; _x;s;�)<0

f(x; _x j s)fs(s)dxd _xds (2.13)

in which fS(s) is the uniform pdf in the interval [0; T ], f(x; _x j s) is the joint density functionof x and _x conditional on s, and Gp(x; _x; s; �) is the failure function for the associatedparallel system. The time parameter t is here replaced by a random variable, S, uniformlydistributed on the interval [0; T ]. Alternatively, the formulation in the subsequent Sectionmay be used.

2.3 Asymptotic expected values

The only way of calculating moments or expected values of complicated distributions hasso far been to use simulation techniques. The advantage of using simulation techniques isthat these methods are well established, easy to use, and readily available in most reliabilityanalysis programs. Moreover, no special consideration has to be given to the shape of thedensity function of the joint distribution. The disadvantage is, however, that a relativelylarge number of function calls is needed, and that the expected value to be estimated maybe slowly drifting as a function of the number of simulations. Furthermore, the sensitivitymeasures with respect to the parameters de�ning the distribution function are only availablein terms of a numerical di�erentiation (this is generally not the case for the sensitivitymeasure of a failure probability using simulation techniques).

The Hagen-Tvedt formulation presented in the previous Section is in essence identicalto the nested probability method formulated by Madsen and Moghtaderi-Zadeh [89] or Wen


and Chen [158]. In the calculation of the expected value of the failure probability E [PF (Z)],Wen and Chen interpreted the integrand PF (Z) as the density function of a normal variableU , and they formulated the expected value as

E [PF (Z)] =Zall z

PF (z)fZ(z)dz

=Zall z

Zu<��1(PF (z))

�(u)fZ(z)dudz (2.14)

that is a FORM/SORM problem in n + 1 variables: the original n variables in Z, and anauxiliary U variable. It is nested because PF (Z) appears in the g-function for the outerFORM/SORM.

For bounded problems, in which the integrand h(Z) is less than some value hmax, the ratioh(Z)=hmax may be interpreted as a \failure probability" and the expected value E [h(Z)]may be calculated by the basic Wen-Chen approach. However, there seems to be no generalway of selecting the maximum value hmax for unbounded problems. Of course, hmax could beassumed to be some high, relatively unlikely value of h(Z), but the calculated result wouldseem to depend importantly on what hmax is assumed to be.

Here, a new asymptotic method of calculating expected values is presented. The methodmaps the logarithm of the function into an independent super Rayleigh distributed space(that is a Weibull distributed space with shape parameter larger than 2). In this space, thepoint of maximum contribution to the expected value is obtained by use of a standard opti-mization routine. Around the point of maximum contribution the integrand is approximatedby a second-order Taylor expansion, and an analytical integration is carried out.

2.3.1 Theory

The expected value of the function h(z), with respect to the joint density fZ(z) of z, is

E [h(z)] =Zall z

h(z)fZ (z)dz (2.15)

It is assumed that the integrand h(z)fZ(z) is piecewise regular in ]0;1). This identity isensured by the mapping to the super Rayleigh distributed space. The integral in Eq. 2.15 isthen written as

E [h(z)] =Zall z

exp[L(z)]dz (2.16)

where L(z) = ln[h(z)fZ(z)] is the log-likelihood function. The idea is to �nd the designpoint z�, which maximizes the log-likelihood function, and to approximate the log-likelihood

2.3. Asymptotic expected values 15

function to its second-order Taylor expansion around this point. The integration is thenperformed analytically. The optimization may be performed by use of standard optimizationroutines for unconstrained optimization, e.g. the MINCF routine [92] or the SQP routine[135].

In the usual FORM formulation, the integration region is fg(z) < 0g, and the pointwhere L(z) is maximum (the most likely failure point) is on the boundary. In this case L(z)has a non-zero gradient at the boundary, so L(z) is replaced by, cf. Breitung [12, 13]

L(z) = L(z�) +@L(z�)@z

(z � z�) + � � � (2.17)

The integration is then performed, and an expression for the integral (Pf in this case) interms of f(z) and its gradient at z� is obtained.

In the present case, the integration domain is in�nite and the point z�, which maximizesthe log-likelihood function L(z), occurs at an interior point within the domain. Consequently,the �rst-order gradients vanish (due to optimality), and L(z), must be replaced by its second-order Taylor expansion

L(z) � L(z�) +1

2(z � z�)0H(z�)(z � z�) (2.18)

where H(z�) is the Hessian matrix at the design point

H(z�) =

@2L(z�)@zi@zj

!i;j=1;��;n

(2.19)

It is noted that @L(z�)@z = 0 has been inserted in Eq. 2.18.

An asymptotic approximation to the expected value then becomes

E [h(z)] �Zall z

exp[L(z�)] exp[1

2(z � z�)0H(z�)(z � z)�]dz (2.20)

=exp[L(z�)](2�)n=2qj det(H(z�)) j

(2.21)

The advantages of the formulation are that

� The expected value may be found by considerably fewer function evaluations.


� The sensitivity factors are { as in standard FORM analysis { almost a direct by-productof the method.

� Beyond giving E [h(z)], the approach gives a critical \design" point z� and importancefactors (related to the curvature of L(z) at z�).

� The \design" point re ects the relative increase of h(z) versus decrease of fZ (z) in�nding maxfh(z)fZ(z)g.

The disadvantages of the formulation are that

� The method is only directly applicable when the considered function is continuous anddi�erentiable.

� The method is not very useful if L(z) has multiple optima (of course, FORM is not ofmuch help in that case).

� The accuracy hinges on how peaked L(z) is near z�. It is not immediately clear how\asymptotic" the expected value is { its contribution may be less locally concentratedthan those of rare PF .

2.3.2 Example

Consider a 1-D example which is simply the Rayleigh damage rate, for which the exactanswer is known:

E [D] =Z 1

0

sm+1

�2exp

"�s22�2

#ds (2.22)

=Z 1

0exp[L(s)]ds (2.23)

where the log-likelihood function is

L(s) = (m + 1) ln s� 2 ln� � s2

2�2(2.24)

Optimization of the log-likelihood function gives s� = �pm+ 1. Hence the expected damage

asymptotically becomes

E [D] � exp[L(s�)]

s2�

�L00(s�) (2.25)

= �mq�(m+ 1)(m+1) exp[�0:5(m + 1)] (2.26)

2.4. System analysis of correlated identical events 17

For m = 3 the asymptotic result is within 2% of the exact result:

E [D] = �m2m=2�(1 + 0:5m) (2.27)

Comparing Eqs. 2.26 and 2.27 reveals but a \reinvention" of Stirlings approximate formulafor large factorials (replace m = 2n).

The proposed model possesses the opportunity of becoming an attractive alternative tosimulation techniques in the calculation of expected values. Future studies will elucidatewhat happens when the integrand is transformed to a space in which it looks more peaked,and hence a local asymptotic approximation may do better. Moreover, future studies willquantify limitations of the model.

2.4 System analysis of correlated identical events

E�cient methods for the evaluation of the reliability of a single event based on full distri-bution reliability methods have been developed, [10, 102, 151, 154], and it is therefore veryattractive to make use of these methods in the evaluation of the failure probability of asystem consisting of correlated identical events.

For parallel (small intersection) and series systems (large intersection), the probabilityof failure is usually calculated by use of multi-point FORM/SORM procedures. For seriessystems the failure probability may be bounded by the well-known second-order Ditlevsenbounds. Although these procedures are well developed, e�cient and reliable for a systemwhich consists of a relatively small number of events, their e�ciency may be questionable incases where a relatively large number of events make up the system.

Assume that n failure functions of a series system are linearized in relevant joint designpoints, and that the partial �rst-order reliability indices �i and the correlation coe�cient�ij = �

Ti �j for the set of linear approximating safety margins, Mi, are computed. The �rst

order approximation to the failure probability is (e.g. Madsen et al. [84])

PF = �n(�;R) =Z 1

�1�(v)�n

i=1�

�i �p�vp

1� �

!dv (2.28)

where the last equality only applies to the case of all the safety margins having the samecorrelation coe�cient �. Reference to [84] is given for results in the case of general correlationstructures. When all the failure events have the same reliability index �e and the samecorrelation coe�cient �, then Eq. 2.28 condenses to

PF =Z 1

�1�(v)

"�

�e �p�vp

1� �

!#ndv (2.29)


This failure probability is easily evaluated by numerical integration techniques. In caseswhere the smallest event (in a series system) or a subset of the smallest events (subsidiarilylargest) enters an ensuing reliability analysis, this procedure is not feasible.

The evaluation of the system reliability of correlated identical events is of interest inmany areas of structural and mechanical engineering. From a point of view of design thismay represent areas as:

� Evaluation of the reliability requirement for a basic event based on (codi�ed) require-ments for the reliability of a system.

� Study of the correlation e�ects between the events on the total reliability level of thesystem.

These areas have been the focus of interest by Ximenes and Mansour [165] in the study ofTLP-tether reliability, and in Cramer [20] in the study of the fatigue reliability of continuous,welded structures. In those studies the system reliability was calculated on the basis ofstricter system requirements than are presented here. In this treatise special interest is givento obtain an ordered set of the correlated identical events. The proposed model is used inthe reliability analysis of the ultimate bending capacity of the hull, see Section 6.4.

The following presents a procedure which utilizes the simplicity of ordinary componentevents reliability evaluation in the evaluation of the reliability of a system consisting ofcorrelated identical events. The procedure is based on the use of order statistics, and anytype of correlated identical components can therefore be applied in the modeling of thesystem. The proposed procedure hinges on the nested FORM/SORM method, and is thusin this regard similar to the procedure proposed by Wen and Chen [158]. Apart from thisthe two procedures should not be related.

2.4.1 General theory

The basic assumption is that the performance of all the basic events is described in terms ofan identical limit state function, G(Z):

G(Z) =

8><>:< 0 unsafe domain= 0 failure surface> 0 safe domain

(2.30)

where Z is a vector of basic variables including loading variables, material properties, geo-metrical variables, statistical estimates, and model uncertainty parameters. The value of


some basic variables is known, while the value of others is uncertain. Some of the basic vari-ables may be independent from event to event, whereas others may be identical and commonfor all the events making up the system. The limit state function is therefore written as

G(Z) = G(ZI ;ZC) (2.31)

where ZI is modeled as the vector of the basic but independent variables, whereas ZC ismodeled as the vector of the basic dependent variables.

The single-event failure probability may then be written as

PF = P [G(ZI ;ZC) < 0] =ZG(zI ;zC)<0

fZ(zI ; zC)dzIdzC (2.32)

=ZzC

Zu<��1(F (�jzC))

�(u)fzC (zC)dzCdu (2.33)

where fzCis the joint density function of the common variables, and F�jzC (� j zC) the

conditioned distribution function of the limit state function in the independent variables:

F�jzC (� j zC) = P [G(ZI ; zC) < �]

=ZG(zI ;zC)<�)

fZ(zI ; zC)dzI (2.34)

fZ(zI ; zC) is the joint density function of the basic variables. With the given formulation,F�(�) gives the conditioned failure probability of the independent variables.

In cases where the dimension of the vector of independent variables ZI is equal to one,and G(ZI ; zC) is a monotonic increasing or decreasing function of this variable, then thedistribution function of F�(� j zC) is uniquely determined in terms of this variable. Ingeneral, the distribution function, Eq. 2.34, has to be obtained by application of e.g. theFORM/SORM method. Given the distribution function F�(� j zC), it is straightforward toapply any type of order statistics to this distribution.

Failure of the system event is given as a function of the extreme values of the independentevent. For a pure series system, only the lowest out of n is of interest, whereas it is the highestout of n for the parallel system. In general, the system failure will be described in terms ofa subset of the ordered basic events. The failure probability of the system is then written as

PFsys = P [Gsys(fGi(ZI ;ZC))gn) < 0] (2.35)

=ZGsys(f�ign)<0

ff�jzC (�i j zC)gnfzC (zC)dzCdf�ign (2.36)


where Gsys is the limit state function for the system, and �i is the i'th lowest out of nconditioned variables. fF�jzC (� j zC)gn is the distribution function for the sorted n-set.

It follows that the failure probability of the system can be obtained by evaluating anequivalent component described in terms of the joint set of ordered basic events. The fol-lowing Section describes how this set may be obtained.

2.4.2 Order statistics

The variables �i are independent, but when these variables are ordered as

�1 < �2 < � � � < �N�1 < �N (2.37)

they are no longer mutually independent. In order to apply the distribution function toEq. 2.34 to the probabilistic analysis, it is useful to demonstrate the �rst step in the reliabilitycalculation, namely the Rosenblatt transformation [84] of the basic variables into a set ofmutually independent and standardized normal variables.

Let the basic variables be collected in the basic variable vector �

� = (�1;�2; � � � ;�N+1�k; � � � ;�N) (2.38)

The Rosenblatt transformation of interest is de�ned on the basis of the conditional distribu-tion function

Fi(�i j �i+1; �i+2; � � � ; �N) (2.39)

For �N = �N the distribution function follows directly

F�N(�N) = F�(�)

N = �(uN) (2.40)

giving

�N = F�1� [�(uN)

1=N ] (2.41)

where uN is a standard normal distributed variable. The distribution of any �i other than�N , given that �N = �N , is the original distribution function truncated from above at �N :

P (�i � � j �N = �N) =F�(�)

F�(�N); � < �N (2.42)


The Rosenblatt transformation from the largest of N � 1 random variables with the distri-bution function in Eq. 2.42 is then

F�N�1(�N�1 j �N ) =

F�(�)

F�(�N)

!N�1= �(uN�1) (2.43)

giving

�N�1 = F�1� [F�(�N )�(uN�1)1=(N�1)] (2.44)

Continuation of the procedure leads to the distribution function of the N � k highest valueof the variable:

F�N�k(�N�k j �N ; �N�1; � � � ; �N+1�k) =

F�(�)

F�(�N+1�k)

!N�k= �(uN�k) (2.45)

giving

�N�k = F�1� [F�(�N+1�k)�(uN�k)

1=(N�k)] (2.46)

Similarly, the joint distribution of the k lowest values may be obtained, in which case,however, the distribution has to be truncated from below. By de�nition of n auxiliary stan-dard normal variables in the probabilistic analysis of Eq. 2.36, it is thus possible successivelyto obtain the needed values of �i from Eq. 2.46.

2.4.3 System modeling

The problem left is to invert Eq. 2.34 in the general case, where the dimension of the vectorZI is larger than one. When the dimension of the vector ZI is equal to one, the inversion ofmost distributions is readily available. For a larger dimension of the vector ZI the inversionmay, however, be performed when the distribution function in Eq. 2.34 is obtained by use ofFORM/SORM. The sensitivity with respect to the threshold level � is a direct by-productof the FORM/SORM analysis. The sought threshold level � giving the requested targetprobability p (or preferably target ��1(p), as this function is much more linear) is then infew iterations obtained by means of a standard Newton-Raphson scheme.

To review the procedure of the system modeling, two limit state functions have to bede�ned. One describing the system failure, and another describing the basic event failurein terms of a threshold level �. The set of basic variables is divided into two separate sets.One consists of common basic variables ZC , which { including a set of k auxiliary standardnormal variables { is used in the call to the limit state function describing the system failure.Within this limit state function, the k requested values of the ordered set are obtained fromEq. 2.46 by yet another target FORM/SORM evaluation of the basic event, which is de�nedin terms of the other set of the independent set of basic variables ZI .


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Chapter 3

Environmental Modeling

3.1 Introduction

The physical and statistical characteristics of the sea surface elevation are described math-ematically by considering the sea surface elevation as a sum of random waves of variousheights and frequencies, grouped into sea states according to severity.

The wave climate experienced by a vessel during its lifetime is described by models oftwo di�erent time scales: (1) A short-term model, in which a sea state is described by aset of parameters and is approximately de�ned as a stationary condition, and (2) a long-term model which de�nes the variation of the describing set of parameters. The describingparameters are the wave data which include the signi�cant wave height HS, the mean wavezero crossing period TZ , and the wave direction �0. Precise de�nitions of the signi�cant waveheight HS and the mean wave zero period TZ are given in connection with Section 3.3.

In each short-term sea state the sea elevation �(t) is described at a speci�c position asa stationary (over a short period of time { 1�3 hours), relatively narrow-banded, Gaussianrandom process. However, the Gaussian assumption is only acceptable for deep-water waves.The wave pro�le of wind-generated waves in shallow waters shows a de�nite excess of highcrests and shallow troughs and is de�nitely non-Gaussian, Ochi [117]. Within each sea statethe distribution of wave energy upon di�erent frequencies and directions is expressed by awave spectrum.

The sea surface of the earth is divided into squares, known as Marsden zones [14]. Eachof these zones covers a geographic area over which the wave conditions are assumed to befairly uniform. The wave data for each Marsden zone are obtained by visual observations orby measurements, on the assumption of the ergodic property. From the worldwide missionpro�le experienced by the vessel, the frequency of occurrence of di�erent sea states duringthe lifetime is found. Each sea state is, as mentioned, described through signi�cant waveheight HS and mean zero crossing period TZ .

23

24 Chapter 3. Environmental Modeling

Figure 3.1: Weighed global scatter diagram for Marsden zones 15, 16, 24, and 25.

3.2 Sea state

Based on the speci�ed trade of the vessel, the relative time period within each Marsdenzone is estimated, and the frequency of occurrence of di�erent sea conditions is found as theweighed average of the available wave statistics in the di�erent zones:

fHS; TZglifetime =NXi=1

�i fHS; TZgi (3.1)

where fHS; TZgi is the scatter diagram for the i'th Marsden zone, �i the fraction of thelifetime during which the ship is in Marsden zone i, and N the total number of zonespassed by the ship during its lifetime. In Figure 3.1 the weighed global scatter diagramfor a transatlantic route through Marsden zones 15, 16, 24, and 25, see [14], is shown. Itis obtained by application of annual values and all directions added. It is of interest to�nd a continuous analytical formulation of the joint distribution of the obtained weighedglobal discrete scatter diagram. Unfortunately, there is no known theoretical reason whyany particular joint distribution function should �t the data best. The evaluation of thedi�erent joint distribution functions must rely solely on an empirical basis (or codi�ed), andmany di�erent formulations have been suggested.

Ochi [116] applied a bivariate log-normal distribution to the wave data, but he pointedout that rather poor �tting is obtained for cumulative probabilities exceeding 0.9. All dis-tribution parameters are, however, given directly from the second moment statistics of HS

and TZ .

3.2. Sea state 25

Figure 3.2: Fitting of marginal 3-parameter Weibull distribution in HS.

Gran [48] studied a number of bivariate distributions extensively, and he proposed tointroduce a statistical dependency between two marginal Gamma distributions of which boththe bivariate log-normal and the bivariate Weibull distribution are special cases. Mathisenand Bitner-Gregersen [100] used the Gamma distribution suggested by Gran [48], but in allcases analyzed, the Gamma distribution reduced to the Weibull distributions.

Haver [57] proposed (for North Sea conditions) to express the joint distribution througha marginal two-parameter Weibull distribution in the signi�cant wave height HS, and aconditional 2-parameter log-normal distribution or Weibull distribution in the mean zerocrossing period TZ. This distribution was also used in the study by Mathisen and Bitner-Gregersen [100] and in the study by Schall et al. [134]. In all these approaches, the lowerlimit of the mean zero crossing period TZ was considered to be constant. By means of the2-parameter approach, the distribution parameters are uniquely described by the mean valueand standard deviation of the wave height or the conditional wave period.

In this treatise it is suggested, Cramer and Friis Hansen [22], to use a 3-parametermarginal Weibull distribution of HS and a 3-parameter conditional Weibull distribution ofTZ given HS where all three Weibull parameters are determined by non-linear least-square�tting. The least-square �tting is obtained by application of a modi�ed version of theMarguard-Levenberg method, Friis Hansen [42]. This method implies that the lower limitof the conditional distribution of the wave period TZ is estimated as a function of HS (seeFigure 3.1):

FHs(hs) = 1� exp(��h(hs � h)�h) ; hs > h (3.2)

FTz(tz j hs) = 1� exp(��tjh(tz � tjh)�tjh) ; tz > tjh (3.3)


Figure 3.3: Fitting of conditional 3-parameter Weibull distribution in Tz.

3.3. Wave spectrum 27

Figure 3.4: Progress of Weibull parameters in the conditioned distribution of TZ .

Figure 3.2 shows the �tting of the marginal distribution of HS for the weighed global scatterdiagram in Figure 3.1. In Figure 3.3 plots of the conditioned distribution of TZ are presented.Good �ttings were obtained in all cases. The �tting presented in Figure 3.3d with HS = 8:5represents the case with the largest residual variance of all �ttings performed. The residualvariance obtained was of the order 0:68 � 10�5 � 0:11 � 10�4 for all cases (HS = [0.5 { 14.5]m) except for HS = 8:5 m which has residual variance of 0:30 � 10�3. Figure 3.4 shows theprogress of the Weibull parameters in the conditioned distribution of TZ as a function ofHS. As it is seen from the �gure, one could argue that the parameters �tjh and �tjh could be�xed values whereas a linear variation of the parameter tjh, which models the lower limitof the distribution, seems appropriate for the case analyzed. A more thorough study of theparameter variations for di�erent Marsden zones is nevertheless necessary to state generalconclusions with respect to such a parameter variation.

3.3 Wave spectrum

The wave spectrum is an important environmental input to the reliability evaluation ofmarine structures, as it expresses the distribution of wave energy upon the wave frequencyrange in a given sea state. For a speci�ed sea state, that is a speci�ed combination of HS

and TZ , the wave spectrum is estimated on the assumption of stationarity.

The signi�cant wave height HS is de�ned as the expected value of the largest one-thirdof the waves, and the mean zero crossing period { the time scale { TZ is de�ned as theupcrossings of the mean still-water level. If the wave elevation process �(t) is assumed to be


Gaussian, the mean zero crossing wave period is

TZ = 2�

s�0�2

(3.4)

where �n is the moment of the one-sided wave spectrum S�(!)

�n =Z 1

0!nS�(!)d! (3.5)

If the wave elevation process is furthermore assumed to be narrow-banded (that is, thewave peaks are Rayleigh distributed), then it follows that the expected value of the highestone-third of the waves is

HS

��= 3

Z 1

r2=3

2rfR(r)dr = 4 (3.6)

that is, HS = 4��, where �� =p�0 is the standard deviation of �(t). Factor 3 in the above

equation is the normalization factor, and factor 2 is from the de�nition of the wave heightbeing measured from wave trough to wave crest.

In reality, the shape of wave spectra observed in the ocean waves varies considerably(even though the signi�cant wave height and mean wave period are the same) depending ongeographical location, duration and fetch of wind, stage of growth and decay of storm, andexistence of swell. Since an ocean-going vessel encounters a large variety of wave conditions,and since the magnitude of the responses is signi�cantly in uenced by the shape of the wavespectrum, the uncertainty of the predicted responses must be expected to be high unless thevariability of wave conditions is re ected in the prediction of the wave spectra.

Several simple spectral formulations have been developed for some ideal conditions.These include wave spectra such as the Pierson-Moskowitz spectrum, Bretschneider's two-parameter spectrum, and those from the International Towing Tank Conference (ITTC),and the International Ship Structure Congress (ISSC), [84]. The JONSWAP spectrum wasestablished partly to deal with the phenomenon called peak enhancement. This phenomenonis caused by the limited development of waves due to the presence of coastlines; the wavepattern is said to be fetch limited.

The Pierson-Moskowitz wave spectrum is presently the most applied wave spectrum, butit is only a special case of the more general Gamma spectrum [48]

S�(! j hs; tz) = A!�� exp(�B!��); ! > 0 (3.7)

3.3. Wave spectrum 29

The parameter � represents the power of the high-frequency tail, and the parameter � de-scribes the steepness of the low-frequency part. A and B are uniquely related to HS and TZ ,which leads to a simple description of the wave spectrum for di�erent sea states:

A =1

16H2S��2�

TZ

��1 � ��1�

� ��32

��3�

� ��12

(3.8)

B =�2�

TZ

�� 1�

� �2

��3�

� �2

(3.9)

For � = 4 and � = 5, the Gamma spectrum is equivalent to the Pierson-Moskowitz spectrum.In a structural response analysis both of the parameters � and � may be modeled as stochasticvariables. It is noteworthy that although the parameter choice � = 5 has been justi�ed byasymptotic results, cf. Phillips [125], the exponent � can be used to adjust the width of thespectrum.

The n'th spectral moment �n of the Gamma spectrum is

�n =AB�(��1�n)=�

��

� � 1� n

�

!(3.10)

For a Gamma spectrum satisfying the asymptotic behavior � = 5, the bandwidth measure

� =

s1� �21

�0�2(3.11)

becomes

� =

vuut1� �(3=�)2

�(2=�)�(4=�)! 0:333 for � !1 (3.12)

(for � = 4 then � = 0:391). The frequency !p at the spectral density maximum is

!p = (B�=5)1=� (3.13)

with the corresponding spectral density maxima S� max

S� max = A!�5p exp(�5=�) (3.14)


Figure 3.5: Dimensionless gamma spectrum for selected values of �.

Figure 3.5 shows a dimensionless representation of the Gamma spectrum:

S�2�

H2STZ

=1

16�!�5

�(4=�)

�(2=�)2exp

24�

�(4=�)

�(2=�)2

!�=2!��

35 (3.15)

in which ! = !TZ2�

. It is seen from Figure 3.5 that it is possible to enhance the peakby increasing the value of � beyond 4. For � ! 1 the peak enhancement of the Gammaspectrum will at most be a factor of 3.5 compared to that of the Pierson-Moskowitz spectrum.Unfortunately, the low frequency tail will be lost by this procedure, and the spectral densitywill always lie below the Phillips asymptote. None of these properties are of general validity,Gran [48].

To model the more peek-enhanced spectra, Gran [48] proposed to modify the Gammaspectrum by adding a narrow-banded wave with a spectral density closely centered aroundthe peak frequency !p. The spectrum developed by Gran will not be given here, instead theJOint North Sea WAve Project, JONSWAP, spectrum is given (Hasselmann et al. [55]):

S�(!) = �g2!�5 exp

24�5

4

!

!p

!�435 a (3.16)

where

� =5�4H2

Sf4m

g26

5 + (3.17)

3.4. Wave energy spreading 31

= exp

"5:75� 0:367TZ

sg

HS

#; 1 < < 5 (3.18)

fm =1

TZ

s5 +

10:89 + (3.19)

a = exp

24� 1

2

! � !p�!p

!235 ; � =

(0:07 ! < !p0:09 ! > !p

(3.20)

The above described spectra are mainly intended to describe wind-driven seas which aregenerated locally. For this reason the spectra are limited to uni-modal, or single-peakedfunctions. Empirical spectra have often an additional low-frequency peak or ridge due tothe swells from previous storms in remote sea. Such features will not be discussed in thistreatise, instead see Torsethaugen [149] and Soares [140, 143].

The assumption of a pure wind-generated sea seems reasonable for most severe sea states.Concerning low and moderate sea states, the actual sea will, however, often be of a combinednature and the double-peakedness may therefore a�ect the results, Bitner-Gregersen andHaver [9].

3.4 Wave energy spreading

The wave �eld de�ned by a wave spectrum can be interpreted either as a 2-D (long-crested)or 3-D (short-crested) wave �eld. Information about the wave amplitude a is provided bythe wave spectrum as 1

2a2 = S�(!)d!, whereas the directionality of the waves is de�ned by a

distribution of the main wave direction (typically the wind direction). In the 2-D wave �eld,all waves are propagating in the same direction, namely the main wave direction, whereasthe 3-D wave �eld consists of contributions of wavelets from all directions. The rate ofwavelet contribution from the di�erent directions to the resultant wave, is de�ned by thewave spreading function f�, and the weighed sum of all wavelets then de�nes the resultant3-D wave with amplitude a, which, however, still propagates in the main wave direction.The directional spectrum is generally simpli�ed as

S�;�(!; �) = S�(!)f�(�; !) (3.21)

The wave spreading function f� depends on geographical location, changes in wind di-rection, duration and fetch of wind, stage of growth and decay of storm, and existence ofswell. The spreading function is often expressed in the form of a cosine power function (cf.[47, 132])

f�(�; !) =22s�1

�

�2(s+ 1)

�(2s+ 1)cos2s

� � �02

!(3.22)


in which �0 is the main wave direction, and s a parameter dependent on frequency and windspeed. For the purpose of engineering applications Goda [47] proposed the following form:

s(!) =

8><>:smax

�!!p

�5! � !p

smax

�!!p

��2:5! � !p

(3.23)

with smax = 20. The range of the energy spreading is � 2 [��=2; �=2]. Alternative forms ofthe wave spreading function may be found in Sand [132].

The spreading of waves in the 3-D wave-�eld tends to result in forces (on �xed structures)that are somewhat less than those predicted on the basis of a 2-D wave-�eld, Sarpkayaand Isaacson [133]. This may be illustrated simply by considering two regular waveletspropagating at an angle � 45 degrees to the main wave direction: in the resultant wave,horizontal velocities and accelerations (which add vectorially), and thus the resultant dragforce and inertia force will be 1=2 and 1=

p2, respectively, of that owing to the corresponding

2-D wave. The vertical velocities are the same in 2-D and 3-D waves. Generally, it is notpossible to conclude, whether or not a 3-D wave-�eld will increase or decrease the wave-induced responses to oating structures, as the 3-D shape may signi�cantly increase roll andpitch motions. In a linear analysis, the e�ect of 3-D waves can be obtained by calculatingthe spectral moments of the response based on the directional wave spectrum, and thenintegrating over the entire wave spreading. In a non-linear analysis, however, it is notdirectly possible to include the e�ect of 3-D waves as the superposition principle is not valid.

3.5 Joint models of wave amplitude and frequencies

The regular, sinusoidal wave is an important concept in the study of ocean waves. It o�ersa simple solution to the wave equations, dynamic variables are easy to calculate and regularwaves are simple to generate in hydraulic laboratories.

Fortunately, the wave spectra are typically uni-modal with rather limited bandwidth,and within a narrow-band assumption the wave elevation �(t) is formulated as

�(t) = a cos� ; � = !t+ � (3.24)

that is, a function of the instantaneous wave amplitude a, frequency !, and phase �. Ap-proximations for the joint probability density function of wave amplitude a and frequency! can be made on the assumption of a normal process of �(t). The wave process �(t) inEq. 3.24 is required to be stationary. This implies that, whatever be the choice of bivariatedensity f(a; !), the phase � should be uniformly distributed and independent of a and !:

f(� j a; !) = 1

2�; 0 � � � 2� (3.25)

3.5. Joint models of wave amplitude and frequencies 33

Traditionally, the wave amplitude is commonly de�ned in terms of the Cramer-Leadbetter[24] envelope of the wave elevation as

a = [�2 + �2]1=2 (3.26)

Here � = a sin� is the Hilbert transform of �, see Cramer and Leadbetter [24] p.142. Asboth � and � are assumed to be Gaussian variables, it follows directly that a is Rayleighdistributed:

f(a) =a

�2�exp

� a2

2�2�

!; a � 0 (3.27)

This equivalent envelope amplitude is useful, as it essentially �ts equivalent sinusoids to anirregular wave history.

Various choices of instantaneous frequency ! can be adopted. The simplest choice is toassign a constant frequency to all wave cycles:

! = �! =�1�0

for all a (3.28)

where the spectral moments �n, cf. Eq. 3.5, are de�ned in terms of the one-sided wave spec-trum S�(!). Unless the wave process is extremely narrow-banded, this should be regardedonly as a �rst-order approximation of real wave behavior.

3.5.1 The Sveshnikov model

The simple model of Eq. 3.28 neglects the systematic association of large periods with largewave amplitudes. This trend is shown both by E [! ja] and Var [! ja], the conditional meanfrequency and its variance. A simple model proposed by Sveshnikov [146] corrects for thistrend, although the mean frequency is left unchanged. The alternate Longuet-Higgins modeldescribed in the next subsection accounts for trends in both mean and variance.

The underlying idea is again based on the envelope interpretation of Eq. 3.24. Theinstantaneous frequency ! is de�ned as the rate of change of the phase process �(t) (! =_� = d�=dt, see Eq. 3.24). The joint density is then found (see Madsen et al. [84], equation10.69):

fS(a; !) =a2

�3� �!�p2�

exp

"� a2

2�2�

1 +

�! � �!

�!�

�2!#

=a

�2�exp

� a2

2�2�

!a=��

�!�p2�

exp

0B@�1

2

0@! � �!

�!�a=��

1A21CA (3.29)


in which

� =

s�0�2�21� 1 =

�p1� �2 (3.30)

in which � is the bandwidth of Eq. 3.11. Thus the parameter � re ects the bandwidth. Thesubscript S indicates the Sveshnikov model.

It is seen that Eq. 3.29 can be factorized into the form f(a)f(! j a), in which f(a) isagain the Rayleigh distribution. The conditional distribution f(! j a) is Gaussian with

E [! j a] = �! ; Var [! j a] = �2!ja =

�!�

a=��

!2

(3.31)

3.5.2 Longuet-Higgins model

To avoid the possibility of having negative frequencies, Eq. 3.29 may be truncated to includeonly positive frequencies (see Longuet-Higgins [79, 80]). The resultant joint density needsto be rescaled only by a factor L(�) to ensure the unit area after truncation.

fLH = fS(a; !)L(�) ; a; ! � 0 (3.32)

L(�) =2

1 +�

11+�2

�1=2 (3.33)

Similarly, the conditional normal distribution of !, given a, from the Sveshnikov modelmust be renormalized in the Longuet-Higgins model over positive frequencies:

F(! j ! > 0) =�((! � �!)=�!ja)� �(��!=�!ja)

�(�!=�!ja)

=�(a(! � �!)=�� !�)� �(�a=��)

�(a=��)= �(u) (3.34)

For a given a, the frequency ! in the Longuet-Higgins model may be expressed in terms ofan auxiliary standard normal variable u.

3.5. Joint models of wave amplitude and frequencies 35

3.5.3 Other models

A more involved model of the joint density of amplitude and frequency has been formulatedby Lindgren and Rychlik [77]. Their model is determined by means of the local extremaof the wave process and therefore requires the spectral moment �4. However, as a directconsequence of the asymptotic behavior of the ocean wave spectrum, �4 will only rarely beavailable.

The above joint density functions are all derived on the basis of a Gaussian wave pro�le.Huang et al. [59] derived the probability density function of � for a third-order Stokes wave,both for deep-water waves, and waves in �nite depth. It is noteworthy that the analysisby Bitner-Gregersen [8] concluded that the statistical model of deep-water waves is notvery sensitive to non-linearities (that is discrepancies between the linear and the non-linearformulation).

Chapter 4

Short-Term Response Statistics

4.1 Introduction

Responses induced in an operating vessel have �ve primary sources: Static load (still-waterloading), dynamic load (wave-induced loading), wind-induced load, loading/unloading atharbor, and thermal load. Although many interesting and important aspects are raised withrespect to the responses induced by the three latter load sources, only the responses inducedby still-water- and wave-induced loading will be dealt with in this treatise.

The stochastic modeling of the response is { as in the environmental description { dividedinto a short-term and a long-term period. A short-term period means a period in the servicelife of the vessel, during which the loading condition is practically unchanged and speci�edin more or less detail. A long-term condition of the response means a long period of theservice life, during which the vessel experiences a set of speci�ed loading conditions withgiven statistical weights.

It is evident that the wave-induced response is highly correlated to the still-water loading(that is the loading condition) and the analysis of the wave-induced response should thereforebe analyzed conditional on a speci�ed loading condition. Thus, the uncertainties may bedivided into three main contributions:

1. The uncertainties connected with each individual short-term condition.

2. The uncertainties connected with the long-term description.

3. The statistical variations from one loading condition to the next (say, from a ballastcondition to a laden condition).

The third contribution is usually more weighty than the �rst two ones.

37

38 Chapter 4. Short-Term Response Statistics

This Chapter treats stochastic modeling of the responses in a short-term period and for aspeci�ed loading condition. Both linear and non-linear response analyses are considered. Thesubsequent Chapter addresses the long-term combination problem of the short-term periods,and the statistical variations from one loading condition to the next. For convenience theconditioning on the loading condition is omitted in all equations in this Chapter.

4.2 Still-water response

The calculation of the still-water induced response consists of three parts:

1. Calculation of the mass and buoyancy distribution.

2. Calculation of the equilibrium position.

3. Calculation of sectional forces.

The calculation procedure is well established, o�ers no complications, and will not be re-viewed here.

When the joint distribution of all the individual masses and buoyancy forces on the vesselis known, the joint distribution of the still-water responses may in principle be calculated.Such a general calculation is, however, a complicated and time-consuming procedure, whichmust be simpli�ed.

There is not much literature on statistical still-water load data. The statistical still-waterbending moment data reported by Soares [141, 142, 144] are mixtures from very di�erentvessels in di�erent trades, and may at best be used to check the statistical relevance ofdata derived from the loading booklet. This loading booklet is exclusively prepared for aparticular vessel with attention to the intended trade.

Gran [48] suggested to obtain the second-moment statistics of the still-water responseby means of a modal expansion approach. The calculation method is based on a spectralapproach to random �elds where the responses are expanded in orthogonal modes. In thisway a set of joint random variables, i.e. draft, trim, and moment expansion coe�cients, isobtained. The procedure proposed by Gran has several advantages such as the direct waythe joint second-moment statistics of the sectional forces is obtained at any location alongthe hull. Unfortunately, the procedure demands a high number of expansion modes to obtainaccurate second-moment statistics.

Instead it is suggested to establish the stochastic variation of the responses in terms ofa �rst-order Taylor expansion in the individual stochastic masses (cargo loads). A Taylorexpansion should be available both for the sectional forces, and for draught and trim. The

4.3. Wave action 39

values of draught and trim are the two most important parameters in the description ofthe wave-induced responses, as they determine the immersion of the hull. For each of theindividual uncertain massesmi on the vessel, derivatives should be calculated for the responseR, and for the draught T and trim �. Ideally, the Taylor expansion should be performedaround the design point obtained from the probabilistic analysis, but for practical reasonsthe mean value of the individual masses may be used instead:

@R

@mi

@T

@mi

@�

@mi

(4.1)

These derivatives are recognized as the in uence-coe�cients, and they are in principle aby-product of the still-water analysis. Alternatively, these coe�cients might be establishednumerically. It should be noted that constraints would typically be imposed on the totalmass as regards leight weight and cargo loading.

4.3 Wave action

Responses induced by wave action include both low- and high-frequency responses. Thelow-frequency responses are primarily due to the slow motions of the instantaneous wavesurface. The high-frequency responses are caused by entirely di�erent mechanisms than theslowly varying wave responses and are partly induced by slamming or green water on deck,and partly by ship hull vibrations.

In the analysis of the high-frequency responses due to ship hull vibrations, only thevertical modes of vibration have usually been the focus of interest. This is partly becausethe vertical wave loads are the largest, partly because the two-node vertical hull mode isnormally associated with the lowest eigenfrequency, Kaplan and Sargent [69].

These high-frequency vibratory structural responses may give rise to signi�cant stressesin the hull, [61], and are usually classi�ed as either whipping or springing, depending onwhether the vibration mode is transient or steady-state. Whipping is often associated withthe occurrence of large ship motions which may be accompanied by emersion of the bowregion, leading to impact forces when the vessel re-enters the water { that is the ordinaryimpact slamming phenomena (alternatively it might be green water on deck). Anothercase is vessels with large bow are where the forces developed due to the bow are shapevariations lead to whipping. Finally, whipping may also be caused by green water on deck.The force mechanisms associated with the motions are thus highly dependent on non-lineare�ects, although the rigid body motions themselves are usually su�ciently well representedas linear responses, [65].

The phenomenon of springing occurs when the vessel has small (or insigni�cant) motionsin relatively short waves so that the encounter frequency with the waves is close to that of


the lower structural eigenfrequencies of the vessel. Since these eigenfrequencies are belowthe range of notable wave energy, there is an indication that such responses are caused bynon-linear wave forces. In the springing analysis by Jensen and Dogliani [63] it was alsoconcluded that the non-linear contribution to springing is at least as important as the linearcontribution, and that the e�ect of springing may result in some increase in the extreme-valueprediction.

However, great care must be taken in the analysis of springing. It must be recognizedthat the springing phenomenon involves considerations of short waves relative to the shiplength, and that the theoretical basis for evaluating the wave excitation forces by any striptheory is at its limit. Moreover, little attention has been paid to the establishment of thetail of the wave spectrum. Only a small change in the wave spectrum parameters may thuschange the e�ect of springing by several orders of magnitude. Therefore, some questionsremain to be answered before full con�dence can be put in such analyses.

Whether the analysis is linear or non-linear, the reliability analysis of the wave-inducedresponse consists of the following general steps:

1. Structural dynamic response analysis { via a structural model, displacementsand rotations of the vessel are described in terms of the wave surface elevation. Thecorresponding wave-induced forces and moments in the vessel are calculated.

2. Short-term statistics { rates of extreme responses and fatigue accumulation areestimated. This also includes non-Gaussian e�ects due to non-linear mechanisms.

3. Long-term statistics { results for various stationary conditions (sea states, forwardspeed, heading, and weight distribution (i.e. loading condition)) are weighed by theirprobability of occurrence in order to estimate extreme response levels as well as averagefatigue damage rates.

Only the short-term analysis in step 2 di�ers in a linear and a non-linear analysis. Thestructural dynamic response model and the evaluation of the long-term statistics will inprinciple be identical in a linear and a non-linear analysis (except, of course, in the derivationof the forcing function).

4.3.1 Structural dynamic response model

The lower modes of the hull vibrations can generally be determined quite accurately bymodeling the hull as a non-prismatic Timoshenko beam. The equations of motion for theship beam are then written as

@

@x

"EI

1 + �d

@

@t

!@�

@x

#+ �GA

1 + �d

@

@t

! @w

@x� �

!= ms�

2@2�

@t2(4.2)

4.3. Wave action 41

@

@x

"�GA

1 + �d

@

@t

! @w

@x� �

!#= ms

@2w

@t2� F (x; t) (4.3)

in which I(x) is the sectional moment of inertia for vertical bending, A(x) sectional sheararea, ms(x) mass per unit length, ms(x)�

2 equatorial mass moment of inertia, and �d thedamping factor. � is a cross-section-dependent constant. �(t; x) is the slope due to bending,w(t; x) the total de ection, and F (x; t) is the external forcing function per unit length whichis non-linear in w.

A solution procedure for Eq. 4.2 and Eq. 4.3 is outlined in Section 4.7.

4.3.2 Hydrodynamic forces

The calculation of the hydrodynamic forces is based on the time derivative of the momentumof the added mass of water surrounding the hull. Furthermore, a damping term and arestoring term dependent on the relative motion must be included. The force per unitlength of the hull at x is taken in the form:

F (x; t) = �"D

Dt

�mH(�z; x)

D�z

Dt

�+NH(�z; x)

D�z

Dt+Z ��z

�TB(z; x)

@p

@z

��z+w

dz

#(4.4)

in which �z denotes the di�erence between the absolute displacement of the vessel in thevertical direction, w(x; t), and the surface of the ocean, h(x; t) (corrected with respect to theso-called Smith e�ect). mH is the added mass per unit length and NH the damping. Theoperator D

Dtrepresents the total derivative in a water strip with respect to time t:

D

Dt=

@

@t+dx

dt

@

@x=

@

@t� V @

@x(4.5)

where V is the forward speed of the vessel. The breadth of the vessel is B(z; x), T (x)represents the draft, and �nally p the Froude-Krylov uid pressure. It should be notedthat the �rst term in Eq. 4.4 is a momentum slamming term. This term describes the timederivative of the momentum of the added mass. Measurements have shown that momentumslamming is important for severe ship motions.

The slamming sequence is often split into two parts: (1) Bottom slamming (or impactslamming), which occurs at the instant when the hull strikes the water surface, and (2)momentum slamming (or bow are slamming), which is due to the bow entering the wave.These are two phenomena but they are actually both included in the momentum term.However, the time derivative of the added mass coe�cient becomes a problem because ofexcursions in and out of the water.


In the linear strip theory the dependence of added mass per unit length, damping term,and breadth on the relative motion of the vessel is neglected. Eq. 4.4 thus becomes linear inthe wave height.

The second-order strip theory [65] evaluates the dependence of the above-mentionedfactors on the forcing function by a perturbational method, taking linear and quadraticterms in the relative displacement into account. The perturbation is performed around themean still-water line.

Finally, a fully non-linear analysis will evaluate the forcing function at the instantaneousposition of the vessel.

4.4 Linear wave responses

By assuming that the vessel responds linearly to wave action, the total response in a seawaymay be described by a superposition of the response to all regular wave components thatconstitute the irregular sea. This leads to a frequency domain analysis. Given the linear-ity, the response is described by a stationary, ergodic, but not necessarily narrow-bandedGaussian process.

The linear assumption is generally adequate for the description of the motions. For severesea states, however, other responses may be highly non-linear for certain ship types, and anon-linear analysis should be conducted, [63].

4.4.1 Transfer function

The transfer function H(!), which for di�erent frequencies models the response due to asinusoidal wave with a unit amplitude, is usually obtained either from towing tank experi-ments or from calculations based on the equations of motion of the vessel by application of alinear potential wave theory. The transfer function depends on the speci�ed ship velocity Vand wave heading angle �. However, a continuous description of the transfer function in theV � � plane is required in the analysis. In the present analysis, a continuous description isachieved by applying a two-dimensional bicubic, semi-cyclic spline in the modulus squaredof the transfer function, cyclic in the � direction. A natural spline is used in the frequencyplane.

It is noteworthy that the transfer function for any linear combination of the sectionalforces is easily obtained by combination of the complex valued transfer functions. Thismeans that the transfer function for the in-plane stress is readily available at any location(z; y) in the cross-section by use of Navier's formula:

H�(!) =HMyy(!)

Iyyz � HMzz(!)

Izzy (4.6)

4.4. Linear wave responses 43

where HMyy and HMzz are the complex transfer functions for vertical and lateral bending,respectively. Also in-plane stresses owing to torsion may be included in the above formulationin terms of the transfer function for the bimoment distribution.

4.4.2 Response spectrum

Based on the linear model the response spectrum of the vessel is directly given in terms ofthe wave spectrum:

Se�(!e j hs; tz; v; �) = j H�(!e j v; �) j2 Se�(!e j hs; tz; v; �) (4.7)

where !e is the encountered wave frequency and j H�(!e) j is the modulus of the transferfunction. The wave spectrum experienced by the vessel, Se�(!e), is di�erent from the wavespectrum estimated from the speci�ed sea state, S�(!), since the latter wave spectrum isdescribed with respect to a non-moving coordinate system.

The modi�cation of the wave spectrum due to encounter frequency !e is based on fre-quency mapping (see e.g. Price and Bishop [127]). The relative velocity between the wavevelocity and the ship velocity is given by

Vrel = Vwave � Vship cos� (4.8)

The encountered wave frequency is therefore

!e = j Vwave � Vship cos� j k = j ! � kVship cos � j (4.9)

where the wave velocity is expressed as !=k, k = 2�=�w is the wave number, and �w is thewave length. For deep-water gravity waves !2 = kg, and hence, the encounter frequency is

!e = j ! � !2

gVship cos � j (4.10)

Based on energy conservation, the response spectrum expressed in wave frequency is

S�(!)@! = Se�(!e)@!e (4.11)


The n'th spectral moment, �n, of the encountered response spectrum experienced by thevessel is

�n =Z 1

0!e

nSe�(!e j hs; tz; v; �)d!e

=Z 1

0j ! � !2

gVship cos� jn S�(! j hs; tz; v; �)d! (4.12)

This formulation avoids the complication of the { in some cases multiple (up to three) {encounter frequencies in following sea, see e.g. [63, 127]. The response spectrum in anabsolute frequency domain is given by

S�(! j hs; tz; v; �) =j H�(! j v; �) j2 S�(! j hs; tz) (4.13)

and H�(!) is the transfer function in the wave frequency domain.

The spectral moments may be e�ciently obtained by using integration by parts andthereby performing an analytical integration with respect to the wave spectrum. This pro-cedure is possible since the modulus squared of the transfer function is conveniently expressedin terms of a spline. The integral in Eq. 4.12 is then rewritten as a sum of the spline coe�-cients multiplied by an incomplete Gamma function expression.

Note that calculation of the n'th order spectral moment of the response spectrum in theencounter wave frequency domain requires evaluation of the 2n'th order spectral moment inthe absolute wave frequency domain. Therefore caution must be used in calculations of thehigher-order spectral moments due to the possibility of a divergent integral { the spectralmoments of the Pierson-Moskowitz spectrum are divergent for n � 4.

4.4.3 Short-term response statistics

From the estimated response spectrum, the peak distribution of the response in each sta-tionary short-term period is determined by means of the response spectral moments.

Peak distribution

On the assumption that the wave elevation within each short-term period is a stationaryzero mean Gaussian process, the response process for the linear system is also a stationaryzero mean Gaussian process. For a narrow-band response process, the peaks are Rayleighdistributed:

Fp(a) = 1� exp

� a2

2�0

!(4.14)


where �0 is the spectral moment of order zero, equal to the mean square of the responseprocess. Depending on the response transfer function, the narrow-band assumption of theresponse process might not be adequate. It is shown by Rice (see [24]) that the peakdistribution of a general stationary zero mean Gaussian process has the form

Fp(a) = �

a

�p�0

!�p1� �2 exp

� a2

2�0

!�

p1� �2�

ap�0

!(4.15)

where �2 is the bandwidth parameter, de�ned as

�2 = 1� �22

�0�4(4.16)

and �n is the spectral moment of order n. The distribution is usually referred to as theRice distribution. It is seen that the Rice distribution is an interpolation between thenormal distribution for broad-band processes (� = 1) and the Rayleigh distribution fornarrow-band processes (� = 0). Both these distributions are directly given as functions ofthe spectral moments of the response spectrum. It should be emphasized that the abovedistributions are conditional on HS; TZ; V;�. The e�ect of the narrow-band assumption onthe estimated extreme wave load distribution on a ship structure within a short-term periodwas investigated by Mansour [95].

The number of peaks within each period of time is estimated from the rate of peaks:

�p =1

2�

s�4�2

(4.17)

For a narrow-band process, the rate of peaks is approximated by the rate of zero crossings:

�p � �0 =1

2�

s�2�0

(4.18)

Stress range distribution for fatigue analysis

In the fatigue analysis, the stress range distribution is interesting. For a zero mean narrow-band process, the stress range �� is equal to two times the amplitude, which leads to thefollowing stress range distribution for a narrow-band process:

F��(��) = 1� exp

��

2

8�0

!(4.19)


With increasing bandwidth, the process starts to include both negative and positivemaxima. A fatigue analysis based on the narrow-band model ignores the e�ect of an increas-ing number of small-amplitude, high-frequency oscillations. On average this leads to actual,smaller peak and stress range values than the narrow-band model predicts, and consequently,the narrow-band assumption will generally lead to conservative results.

Veers et al. [156] suggested a procedure for evaluation of the fatigue damage of a wide-band process based on an empirical modi�cation of the estimated Rayleigh distributed stressrange and the rate of peaks. The modi�cation is based on racetrack �ltering of simulatedtime series from di�erent power spectra, leading to nearly equivalent fatigue damage as theoriginal data. This takes the form

F��(��) = 1� exp

� ��2

8(1� �2)�0

!(4.20)

with the modi�ed rate of peaks, �p

�p = �p

exp(�R

2th

8�0)� 3z1

2

5z1 + 1(1� exp(

3Rthp�0

))

!(4.21)

where

�2 =(Mz1 +B)2

1 + (Mz1 +B)2(4.22)

and

z1 =

s�0�2�1

2 � 1 ; M = 1:7� 0:8Rthp�0

; B = 0:18Rth

2

�0� 0:36

Rthp�0

(4.23)

As suggested by Veers et al. [156], the threshold level for the racetrack �ltering Rth isregarded as 1:25

p�0 for crack growth material parameter m = 3.

Wirsching and Light [164] obtained a wide-band correction factor for the narrow-bandnumber of peaks. This factor was obtained by computing the fatigue damage from a rain owanalysis by digital simulation. They produced the empirical formula

�p = �p[a(m) + (1� a(m))(1� �)b(m)] (4.24)

where

a(m) = 0:926� 0:033m ; b(m) = 1:587m� 2:323 (4.25)


� is the bandwidth parameter, and m the crack growth material parameter.

It is not clear whether the peak correction factor by Veers et al. is preferable to that ofWirsching and Light. It may be argued, however, that the Veers et al. procedure containstoo many empirical constants to be superior to the simple one of Wirsching and Light. Bothprocedures are based on simulations by use of di�erent spectra and the peak correction israther modest (approximately 20%). Moreover, it is obvious that the Wirsching and Lightpeak rate indeed coalesces into the Rayleigh peak rate for � ! 0. This is not immediatelyclear to be the case of the procedure of Veers et al.

Extreme-value distribution

In the design phase, it is of great interest to determine the �rst passage of the process X(t)of a critical response level a within a given time interval [0; T ]. For a stationary Gaussianprocess, the cumulative density function for crossing a speci�ed level a is bounded by thewell-known formula

Fmax(a) = exp f��X(a)Tg (4.26)

where �X(a) is the mean crossing rate of the level a

�X(a) = �0 exp

� a2

2�0

!(4.27)

However, the bound can be quite conservative because the narrow-band character of theprocess tends to concentrate the upcrossings in clumps and it thus violates the assumptionof independence of the individual outcrossings of the process X(t) embedded in Eq. 4.26.Instead, an upper-bound envelope process for the given process is usually constructed, andthe upper bound of the �rst-passage probability of the envelope process is then considered.

The upcrossing rate of the envelope process is given by Cramer and Leadbetter [24]

�R(a) =p2��

ap�0�0 exp

� a2

2�0

!(4.28)

where �2 = 1 � �21

�0�2is the bandwidth parameter given in Eq. 3.11. Use of this upcrossing

rate may not lead to a decreased bound, because the envelope may have excursions abovethe critical level without any upcrossings of the original process during the time of theseexcursions. This is easily seen by comparing Eq. 4.27 and Eq. 4.28, as the ratio �R=�X !1for a!1. Such envelope excursions are said to be \empty" (Ditlevsen and Lindgren [32])


otherwise they are said to be \quali�ed" (see Vanmarcke [155]). The problem at hand isto estimate the long-run fraction of quali�ed envelope excursions for ergodic narrow-bandprocesses.

Vanmarcke [155] made a crude estimate of the long-run fraction of quali�ed excursions:

rv(a) =�X(a)

�R(a)

(1� exp

� �R(a)�X(a)

!)=

1

1(1� exp(� 1)) (4.29)

where

1 =p2�

ap�0� (4.30)

It is seen that Eq. 4.29 is a kind of interpolation between �R(a) for small values of a and�X(a) for large values.

Ditlevsen and Lindgren [32] observed relatively large deviations from the Vanmarckeprediction of the long-run fraction (1 � rv) of empty envelope excursions. They concludedthough that the Vanmarcke model gives the right order of magnitude, so that the practicalconsequence of the deviations is not serious, in particular for applications to highly reliablestructures.

By use of the Slepian model process method, Ditlevsen and Lindgren [32] improved thelong-run fraction of empty envelope excursions. The main conclusion of their study wasthat the long-run fraction of empty excursions of the envelope above the level a for zero ormoderate spectral skewness can be approximated by the formula

1� rv(a) = 2Z u

0�(�)

8<:1�

p2�

�� u2��2

u

�� 1

2

�� u2��2u

9=;d� (4.31)

where

� =

s�0�2�1

2 � 1 =�p

1� �2 ; u =ap�0

(4.32)

The formula applies to narrow-band processes, but seems applicable to wide-band pro-cesses also. It should be noted, however, that Eq. 4.31 for larger bandwidth � > 0:4 andu 2 [2:5; 3:5] may result in upcrossing rates which are larger than that of the narrow-bandapproximation.

4.5. Non-linear wave responses 49

The cumulative density function of the maximum value of a stationary ergodic Gaussianprocess then becomes, [29]

Fmax(a) =n1� exp[�a2=(2�0)]

oexp

(� rv(a)�R(a)T

1� exp[�a2=(2�0)]

)(4.33)

where the �rst factor gives the approximate probability of having no excursions ofX(t) abovea in the interval 0 � � < 1=�0. The above expression for the extreme-value distributionduring the lifetime applies to each short-term period. In Eq. 4.33 the considered period oftime T should be taken as the duration of a sea state, i.e. the duration of the short-termcondition.

4.5 Non-linear wave responses

Ship motions generally tend to behave linearly even for large wave amplitudes. The onlyship motion that tends to exhibit non-linear behavior, even in small waves, is roll. This isbecause roll is a lightly damped response and such damping is usually dominated by viscousprocesses which are non-linear with roll amplitude. Contrary to motions, accelerations { andthus sectional forces { will generally show marked non-linear e�ects for large wave amplitudes.This is typically seen in measured di�erences between the wave-induced hogging and saggingbending moments in ships.

For a sinusoidal wave elevation, the hydrodynamic force F (x; t) in Eq. 4.4 is no longersinusoidal, and the superposition principle used in linear strip theory is no longer justi�able.For an irregular sea, two possible solution methods have so far been used. The �rst is aperturbational expansion of Eq. 4.4 in which it is assumed that products of the componentwave amplitudes involving terms which are higher than quadratic are small and can beignored. The second method is time simulation of Eq. 4.4.

In Jensen et al. [67] various non-linear strip theories for the prediction of ship motionsand wave-induced loads in moderate, stochastic seaways were examined. These non-linearformulations are based on rather empirical generalizations of the more concise linear striptheory.

4.5.1 Second-order strip theory

The most well-known non-linear problems in oating structures are the added resistanceand slow drift forces in waves. These problems are more or less quadratic in the waveamplitude. Non-linear quadratic problems are conveniently solved in the frequency domain


by use of the so-called second-order Volterra model, see e.g. Neal [109]. Basically, the second-order Volterra model is a statistical method for handling quadratic relations in stationary,stochastic processes. The quadratic relations are established by perturbational methods.Some fundamentals of the second-order Volterra model are outlined in the subsequent section.

From the perturbational model, [78], as point of departure, Jensen and Pedersen [65] for-mulated a quadratic strip theory for calculation of the wave-induced loads on ships sailingin uni-directional random seas. The second-order theory was derived by a perturbationalprocedure in which the linear part is identical to the usual linear strip theory. The perturba-tion is performed around the mean still-water line. The quadratic terms are determined bytaking into account: (1) The non-linearities of the exciting waves, (2) the non-verticality ofthe ship sides, and (3) the variations of the hydrodynamic forces during the vertical motionof the vessel. The responses of the vessel are calculated on the basis of a Timoshenko beamformulation of the ship hull and by application of the modal superposition method.

The second-order strip theory has successfully been used to analyze measured di�erencesbetween the hogging and the sagging moments, [66]. Jensen and Dogliani [63] address boththe long-term fatigue and the extreme-value analysis using the second-order strip theory.

The advantages of the second-order strip theory formulation are that:

� The formulation is in the frequency domain. This means that the correct frequencydependency on added mass and damping is used.

� The formulation is a generalization of the linear strip theory approach, thus the linearsolution is identical to that obtained by the linear theory.

� Hull vibrations are easily included.

� Some e�ect of momentum slamming is included in the formulation.

� The method is computationally fast.

The disadvantages are that:

� Asymmetric non-linearities are of a cubic nature and cannot be taken into account.The procedure is limited to the symmetric vertical responses (heave, pitch, verticalbending moments and shear force). Generally, tankers traveling in oblique seas mayencounter horizontal bending moments of the same order of magnitude as of verticalbending, Mansour [94].

� For severer sea states a quadratic expansion might perform rather poorly due to tran-sient e�ects like slamming and green water on deck. Moreover, the immersed sectionalarea is overestimated when the vessel is experiencing large bow motions. This is adirect consequence of the perturbational expansion of the breadth of the vessel.


Second-order Volterra model

The second-order Volterra model is particularly convenient to model systems with a singleinput: here, the wave elevation �(t) at midship. The corresponding output X(t), whichmay be any structural response (e.g. the midship bending moment), is modeled in the timedomain as

X(t) = X1(t) +X2(t) (4.34)

where

X1(t) =Z 1

�1h1(�)�(t� �)d� (4.35)

X2(t) =Z 1

�1

Z 1

�1h2(�1; �2)�(t� �1)�(t� �2)d�1d�2 (4.36)

Thus, in addition to the �rst-order impulse response function h1(�) that de�nes a linearsystem, quadratic non-linear e�ects are included through the second-order impulse responseh2(�1; �2). The Fourier transforms of h1(�) and h2(�1; �2) are the �rst- and second-ordertransfer functions H1(!) and H2(!1; !2).

One of the �rst attempts to describe the probability distribution of the response inEq. 4.34 was presented by Neal [109]. Following this contribution, several contributions havesupplemented Neal's �rst results, notably Vinje [157], Stansberg [145], Naess [104, 105],Langley [73], Kato et al. [70]. These papers, however, focus attention on the statistics of theslow-drift problem of o�shore platforms. The high-frequency springing response was treatedby Winterstein and Marthinsen [162] and by Naess [107]. This section will review somefundamentals behind the second-order Volterra models:

The surface elevation, �(t), of a stationary irregular sea can be expressed as a Fouriersum

�(t) = <NXn=1

anei�n ; �n = !nt + �n (4.37)

where the wave amplitudes an are related to the (one-sided) wave spectrum S�(!):

an =q2S�(!n)�!; (4.38)

�n is the random phase angles, uniformly distributed on the interval [0; 2�], and �! =!n � !n�1. < is the real part.


The hydrodynamic force F (x; t), Eq. 4.4, is evaluated by a perturbational method, whichtakes linear and quadratic terms in the surface elevation into account, see [65]. The �rst-orderexciting force is then written as the sum

F1(t) = <NXn=1

anF1(!n)ei�n (4.39)

where F1(!n) may be referred to as the �rst-order (force) transfer function. The second-orderforce can also be expressed in terms of a transfer function. However, as the force dependson terms which are products of sums as in Eq. 4.37, the transfer function will depend ontwo frequencies !n and !m. The force expression will therefore include terms which oscillatewith frequencies which are sums and di�erences of the frequencies in the �rst-order incomingwave:

F2(t) = <NXn=1

NXm=1

anamnF+2 (!n; !m)e

i(�n+�m) + F�2 (!n; !m)e

i(�n��m)o

(4.40)

where the second-order force transfer functions (F+2 and F�

2 ) satisfy (see e.g. Langley [74])

F+2 (!n; !m) = F+�

2 (!m; !n) (4.41)

F�2 (!n; !m) = F��

2 (!m; !n) (4.42)

where the star denotes the complex conjugate.

The quadratic expression for the force

F (t) = F1(t) + F2(t) (4.43)

may now be inserted in the equations of motion, Eqs. 4.2 and 4.3. As each term in theforcing function is harmonic with time, the equation of motion can be solved by means ofstandard frequency domain techniques which yield the response quantity X:

X(t) = <NXn=1

anR1(!n)ei�n

+ <NXn=1

NXm=1

anamnR+2 (!n; !m)e

i(�n+�m) + R�2 (!n; !m)e

i(�n��m)o

(4.44)


where R1 is the �rst-order response transfer functions, and R+2 ; and R�

2 the second-orderresponse transfer functions. Eq. 4.44 forms the basis on which the statistics of the second-order response can be determined. It is noteworthy that also the second-order responsetransfer functions (R+

2 and R�2 ) satisfy the relations

R+2 (!n; !m) = R+�

2 (!m; !n) (4.45)

R�2 (!n; !m) = R��

2 (!m; !n) (4.46)

By collecting real parts, Eq. 4.44 can be written in matrix notation as

X(t) = l0�+ �0Q� (4.47)

where a prime indicates vector transpose, and the vector � contains the standard independentGaussian variables

� = [<fei�ngN ;=fei�ngN ]0 (4.48)

and

l = [<fanR1gN ;=fanR1gN ]0 (4.49)

Q =

" <fanam[R�2 + R+

2 ]gN�N =f�anam[R�2 � R+

2 ]gN�N=fanam[R�

2 + R+2 ]gN�N <fanam[R�

2 � R+2 ]gN�N

#(4.50)

in which f�gN indicates a vector with N elements, and = the imaginary part.

Short-term statistics

As mentioned earlier, Eq. 4.44 forms the basis on which the statistics of the second-order re-sponse can be determined. The �rst four moments of the response may be directly calculatedfrom Eq. 4.47, [63]:

� = ��2NXn=1

�nn (4.51)

�2 = �2(1 + 2�2) (4.52)

�3�3 = �3

6�

2NXn=1

2NXm=1

�n�nm�m + 8�32NXn=1

2NXm=1

2NXl=1

�nm�ml�ln

!(4.53)

(�4 � 3)�4 = �448�2

2NXn=1

2NXm=1

2NXl=1

�n�nm�ml�l

+�22NXn=1

2NXm=1

2NXl=1

2NXk=1

�nm�ml�lk�kn

!(4.54)


where � = l0l, �n = ln=� the n'th normalized element of l. Moreover � =P2N

m;n=1QnmQmn=�,and �nm = Qnm=(��), the normalized elements of Q. These four moments provide the inputto the Hermite transformation model, Winterstein [159], from which the e�ects of non-linearity on extreme value and fatigue damage can be analytically estimated.

In the Hermite transformation model [159], the �rst four statistical moments of a responsequantity X are used to calibrate the four deterministic coe�cients c0, c1, c2, and c3 in theexpression

X(t) = c0 + c1U(t) + c2U2(t) + c3U

3(t) (4.55)

in terms of a standard Gaussian variable U . The coe�cient ci may be determined numericallyby a Newton-Raphson scheme from, [62]:

E [X] = �x = c0 + c2 (4.56)

Eh(X � �x)2

i= �2x = c21 + 6c1c3 + 2c22 + 15c23 (4.57)

Eh(X � �x)3

i= �3�

3x = c2(6c

21 + 8c22 + 72c1c3 + 270c23) (4.58)

Eh(X � �x)4

i= �4�

4x = 60c42 + 3c41 + 10395c43 + 60c22c

21 + 4500c22c

23

+ 630c21c23 + 936c1c

22c3 + 3780c1c

33 + 60c31c3 (4.59)

It should be noted that the transformation is only monotonic for 3c1c3 > c22.

The crossing rate of level a is, [159]:

�x(a) = �0 exp

�u

2(a)

2

!(4.60)

where u(a) is the real solution to a =P3

i=0 ciui(a), which can be analytically found. From

the conventional Poisson upcrossing model, the extreme-value distribution is

Fmax(a) = 1� exp (��x(a)T ) (4.61)

Alternatively to the Hermite series the statistics of the second-order response can bedetermined by Gram-Charlier series [78, 117], or by the Kac-Siegert [68] analysis, see Naess


[104] for a detailed description of the procedure. The basis of the Kac-Siegert approach isto decompose X(t) of Eq. 4.47 into

X(t) = l0�+ �0S0AS� (4.62)

= (Sl)0 S�+ �0S 0AS� (4.63)

= B0U +U 0AU (4.64)

=2NXn=1

[�nUn(t) + �nU2n(t)] (4.65)

where �n are the eigenvalues of Q, and S represents a matrix whose rows are the corre-sponding eigenvectors. Further, the matrix S has the property: S0S = I.

Generally, the algebraic formulation should be performed in the complex variables. Thecomplex matrix corresponding to Q is a complex Hermitian kernel matrix (�� = �0), whichimplies that the eigenvalues are real, and that the eigenvectors (R) satisfy the relationR0R� = I. Thus, the above real domain derivation implies no restriction, [74].

It has been shown by Naess [105] that only when the slowly varying part of the second-order process is considered, that is when the Hermitian kernel �(!1; !2) is set equal to zerofor !1 � !2 < 0, then the eigenvalues �n will occur in pairs, say �2n�1 = �2n. Similarly,it can be shown that in the opposite case, in which only the sum-frequency is considered,the eigenvalues occur in pairs with opposite signs, �2n�1 = ��2n. The reason is that thekernel function will then be zero in the �rst and third quadrant, and the kernel function willconsequently generate a set of eigenvalues with opposite signs, see [105] for a general signconvention of the Hermitian kernel.

The fact that the random variables Un are Gaussian and statistically independent impliesthat Eq. 4.65 represents the response as a sum of statistically independent variables xn, say,where

xn = �nUn + �nU2n (4.66)

The characteristic function Mn(�) of xn is de�ned as the expected value of ei�xn

Mn(�) = E[ei�xn] =Z 1

�1ei�(�nU+�nU

2)fU(u)du (4.67)

where fU(u) is the standard Gaussian probability density function. Eq. 4.67 can thus beevaluated to

Mn(�) = (1� 2i�n�)�1=2 exp

(� �2n�

2

2(1� 2i�n�)

)(4.68)


It follows that the characteristic function of the response X is

M(�) = E[ei�X ] =2NYn=1

Mn(�) (4.69)

An expression for the probability density function of X can now be obtained by a Fouriertransform of the characteristic function:

fX(x) =1

2�

Z 1

�1ei�xM(�)d� (4.70)

In general it is not possible to evaluate analytically the integral in Eq. 4.70, although avery e�cient result can be obtained by means of a Fast Fourier Transform (FFT). However,by making the variable substitution w = i� it becomes even more attractive to evaluatethe integral by the saddle point integration technique, Naess and Johnsen [108]. Recently,Langley and McWilliam [75] provided an attractive closed-form series solution for the prob-ability distribution of the combined �rst- and second-order response, which in McWilliamand Langley [101] was extended to extreme-value prediction.

The fact that the Volterra model only provides forces and responses up to second orderactually implies that the leading-order contribution to the kurtosis cannot be computedconsistently. Longuet-Higgins [78] performed a consistent derivation of all contributionsto the �rst four moments of the stochastic process that is obtained by a perturbationalmethod. These derivations reveal that the kurtosis has contributions to the leading orderfrom interactions of the �rst- and third-order terms as well as the �rst- and second-orderterms. Similarly, the �rst non-linear correction to the variance includes both the square ofthe second-order terms and the product of the �rst- and third-order terms.

Although it is possible to establish very accurate estimates of the probability densityfunction for the quadratic process, [108], it seems to be more attractive to estimate theprobability density by the Hermite transformation model. This simpli�cation seems to bemore consistent with the level of physical modeling, which only retains second-order non-linear e�ects (i.e. skewness-driven). However, more detailed investigation of the third-ordere�ect is required before �rm conclusions can be drawn.

Langley [72, 73] reported that the number N of frequency components should be in theregion of 200{300 before results of the slow-drift forces on a moored vessel converge. Onthe other hand, Jensen and Dogliani [63] found that only 30 frequency components wererequired in their analysis of the second-order e�ects in ship structures.

Finally, an extension of the second-order Volterra model to short-crested random seas isdescribed by Naess [106]. It follows from this analysis that the theory for short-crested seasis a natural extension of existing theory with the long-crested waves as a special case.


4.5.2 Time-domain simulation

For severer sea states it is necessary to resort to time-domain simulations and solve theequations of motion for the irregular sea, as a solution including third- or higher-order termsin the Volterra series expansion seems to be infeasible.

The time-domain formulation is able to take slamming e�ects, and the instantaneousimmersion-dependency on added mass, damping, and restoring coe�cients into account. Asummary of the historical development of the time-domain formulation may be found inBuus Petersen [123].

Following Buus Petersen [123] the equation of motion may be written as

" RL f1(x)dx

RL xf1(x)dx

� RL xf1(x)dx RL x

2f1(x)dx

# "�w��

#=

" RL f2(x)dxRL xf2(x)dx

#(4.71)

in which w is the heave, � the pitch, L the ship length, and

f1(x) = mH(x; �z) +ms(x) + �d(x; �z)Bd(x) (4.72)

f2(x) =

@mH(x; �z)

@�z

n� _w + x _� + �(x) _�(x; t)

o� V @mH(x; �z)

@x+NH(x; �z)

!�

n� _w + x _� + �(x) _�(x; t)� V �

o+mH(x; �z)

n�(x)��(x; t)� 2V _�

o+

g f�A(x; �z)�ms(x)g (4.73)

Here ms is the sectional mass per unit length, d(x; �z) height of water on deck, Bd(x) breadthof deck, � the Smith correction factor, and A(x; �z) the immersed sectional area. Other vari-ables are de�ned in connection with Eq. 4.4.

The advantages of time-domain simulation are that:

� The e�ect of momentum and bottom slamming is included in the formulation.

� Structural deterioration or other \evolving" non-linearities may (in principle) be in-cluded.

� Almost any type of non-linearity can be taken into account.

The disadvantages are that:

� The method is very time-consuming (cpu-time >> real time);


� Added mass and damping are frequency-dependent as a consequence of memory e�ects,but there is no consistent way of choosing this frequency in irregular seas.

� It is di�cult to obtain reliable response statistics as the process may be slowly drifting,Winterstein and Torhaug [163].

� It is di�cult to obtain su�cient reliable extreme-value statistics due to the limitedlength of the simulated time series.

Jensen et al. [67] made a comparison of the quadratic theory with the time-domainsimulation. They concluded that the e�ects of non-linearities are smaller in the time-domainsimulation than in the quadratic theory. For the sagging bending moment the di�erencesbetween the two theories were found to be signi�cant, whereas the hogging bending momentswere found to be more comparable. It is argued that the observed di�erences to some extentmay be due to the calculation procedure of the hydrodynamic coe�cients.

4.5.3 Quasi-stationary narrow-band non-linear model

While the second-order Volterra models are a common way of representing non-linear oceanmechanisms, they have several disadvantages. One is their relative complexity, as theyrequire the response amplitudes and phases generated at sums and di�erences of all possiblewave frequencies. In practice, a separate analysis of non-linear e�ects is needed for all possiblefrequency pairs (!1; !2) of incoming waves. Due to the resultant computational expense, the2-D frequency meshes (!1; !2) at which the quadratic transfer H2 is estimated may be quitecoarse. This complexity also complicates the ensuing stochastic analysis, which requires aneigenvalue analysis, [109]. Finally, as mentioned earlier, the second-order Volterra model islimited to quadratic non-linearities. Thus it re ects only a limited, smoothly varying subsetof possible non-linear mechanisms.

In this treatise an alternative non-linear analysis which is simpler yet capable of capturingless regular non-linearities is considered. The idea is to impose a sinusoidal wave

�(t) = A cos� � = !t+ � (4.74)

and �nd the corresponding steady-state non-linear force/response

X(t) = X(A; !; �) (4.75)

Although the response is no longer sinusoidal, it is assumed that X(t) will display a periodicsteady-state behaviour. As re ected from Eq. 4.75, X may be regarded as a function ofrelative wave phase � rather than absolute time t. Time changes of length �t are equivalentto phase changes ��=!�t.


Figure 4.1: Midship bending moment of a container vessel under sinusoidal waves, ! = 0:75.

As an example, Figure 4.1 shows the midship bending moment of a container vesselunder sinusoidal waves. After a transient period which is not shown, results indeed becomeperiodic with the wave frequency. One cycle of this steady-state response represents thetransfer function x(A; !; �) at the imposed wave frequency !. Here, the response is found bysolution of the non-linear equation of motion, Eq. 4.71, by a time-step analysis. Note thatthe ship's are shape induces considerable non-linearity and asymmetry between maximumand minimum moments. Note also that this non-linearity is considerably less smooth than asecond-order Volterra model, which would predict a combination of two pure sinusoids, oneat the wave frequency ! and the other at 2!. Main particulars of the analyzed vessel aregiven in Table 4.1. Figures 4.2 and 4.3 show the variation of the peak sagging moment andhogging moment, respectively, with wave amplitude and wave frequency.

What is proposed here is a \quasi-stationary narrow-band non-linear transfer functionanalysis," which proceeds as follows:

� Apply regular sinusoidal waves at a selected set of wave amplitudes ai and frequencies!j. Calculate the (cyclic but non-sinusoidal) non-linear transfer function, x(ai; !j; �),over the entire phase � = [0; 2�] of the wave cycle.

� Obtain the force statistics by weighing the calculated result by the joint probability ofthe various pairs of wave amplitude and frequency (ai; !j) in actual random seas. The


Table 4.1: Main Particulars.

Length between perpendiculars (Lpp) 163.000 mBreadth molded 26.000 mDraught even keel 8.340 mLCG aft of midship 7.360 mSigni�cant wave height (Hs) 5.00 mZero crossing period (Tz) 7.94 sShip speed v 0.0 m/sStill-water moment (hogging) -1076.6 MNm

joint probability distributions are based solely on random vibration theory, see Section3.5, and require only the wave elevation power spectrum.

The net result is an estimate of response statistics { moments, extremes, and/or fatiguedamage { under irregular, random waves, which requires only a limited number of regularwave analyses. A somewhat similar procedure has been applied by Jensen et al. [64] for �xedo�shore platforms. In that study the wave kinematics and wave forces were determined onthe basis of Stokes' �fth-order wave theory. However, a one-to-one relationship between waveamplitude and wave frequency was assumed, and only the wave amplitude was randomizedin the ensuing reliability analysis.

The simpli�cation of this analysis is not without expenses. The basic assumption isthat the structural response to a single wave cycle is well approximated by the steady-stateresponse to an in�nite number of such cycles. This becomes true in two marginal cases:

1. Arbitrary system: The bandwidth of the wave spectrum becomes increasingly narrow(and the wave becomes in essence a single sinusoid).

2. Arbitrary wave response: The structure behaves quasi-statically (memoryless response),so that its instantaneous response depends only on the corresponding instantaneouswave elevation.

This is in general only an approximation and must be veri�ed on a case-by-case basis.

Short-term statistics

Given the response X(a; !; �) due to one wave cycle, force statistics in random sea can beexpressed as the integrals

E [Xn] =Zall a

Zall !

Zall �

[X(a; !; �)]nf(a; !; �)dad!d� (4.76)

=Zall a

Zall !

Zall �

[X(a; !; �)]nf(� j a; !)f(a; !)dad!d� (4.77)


Figure 4.2: Sagging moment variation as a function of wave amplitude and frequency {contour and topology plot.

Figure 4.3: Hogging moment variation as a function of wave amplitude and frequency {contour and topology plot.


Figure 4.4: Comparison of upcrossing rates obtained by means of a direct approach with theHermite transformation model.

where the uniform distribution should be used for f(� j a; !). Either the Sveshnikov Eq. 3.29or the Longuet-Higgins Eq. 3.32 model may be used for f(a; !).

From the �rst four moments of the response process, upcrossing rates can be obtained byapplication of the Hermite transformation model, Eq. 4.60. Alternatively, upcrossing ratesmay be obtained directly by examining the response due to one wave cycle

�X(a) =Zall a

Zall !

!

2�1lX(a;!;�=[0;2�])>af(a; !)dadw (4.78)

in which !2�

is the zero upcrossing rate of the local sinusoid and 1lX(a;!;�=[0;2�])>a is a zero-oneindicator for the response being larger than the level a.

The upcrossing rates obtained by the two models are compared in Figure 4.4. Note thatthe upcrossing rate obtained by the Hermite model is larger than the upcrossing rate pre-dicted by direct examination of the response. This is because the response process deviatesconsiderably from from the Gaussian by having more narrow throughs and wider crests.Consequently, the �rst four moments, as used in the Hermite transformation model, do notdescribe the response accurately. To better model the response, the Hermite transformationcould be established for the hogging and sagging moment separately. This, however, wouldrequire a logarithmic transformation (for instance) of the response in order to preserve therequired in�nite de�nition domain of the Hermite transformation model.

Note also (see Figures 4.2 and 4.3) that the present example indicates the existence of anupper limit on the maximum wave-induced sagging moment. This e�ect is captured by thedirect approach but not by the Hermite model. A similar tendency of to an upper limit was


not observed for the hogging moment. These observations may partly explain the di�erencesobserved by Jensen et al. [67] in their comparison of the quadratic theory with the time-domain simulation. Further studies need, however, to be conducted before �rm conclusionscan be drawn about the existence of a de�nite upper limit of the wave induced response.

Remark: The peak statistics obtained in the general linear case by application of atraditional response spectrum analysis will in general be di�erent from the peak statisticsobtained by the proposed narrow-band transfer function model. The two models are basedon di�erent approximating assumptions: The narrow-band transfer function model assumesa narrow-band model of the waves, and the conventional response spectrum model assumes anarrow-band model of the response. Recall, of course, that the narrow-band transfer functionmodel is introduced as an alternative simplifying approximation that better generalizes tonon-linear systems, and not to better model linear systems.


Figure 4.5: Crack in longitudinal at connections to transverse web frames.

4.6 Quasi-stationary fatigue analysis

The �rst example of application of the quasi-stationary narrow-band model concerns fatiguedamage in the side shells of ship structures, �rst presented in Friis Hansen and Winterstein[45]. Fatigue damage in the side shell is primarily observed in the connections of longitudinalsto transverse bulckheads, see Figure 4.5. The fatigue damage is caused partly by verticaland horizontal wave-induced hull bending and partly by outside water pressure on the sideshell. Due to the non-linear nature of the outside water pressure the fatigue damage of thecombined stress cannot be calculated via a traditional frequency domain analysis. However,the combination problem may be solved by application of the quasi-stationary narrow-bandmodel.

4.6.1 Introduction

In Schulte-Strathaus [136], survey reports of nine vessels are organized into a database. Forfour of the vessels of the same class (single hull, 165000 dwt), about 1990 fatigue cracks werefound and reported. The remaining �ve vessels, which were double-bottomed or double-hulled, ranged from 35000 dwt to 188000 dwt and represented about 1660 registered cracks.Reportedly, about 42% of all fatigue damage in the vessels was observed to occur in the sideshell longitudinals near the connection with the transverse web frames. Figures 4.6 and 4.7show the observed number of fatigue cracks as a function of the location along the side shell.In order not to mix data from di�erent populations, only the data for the four vessels of thesame class are presented in this study.

The stress induced in the sti�ener at a certain location along the side shell is partly causedby wave-induced vertical and horizontal hull bending, and partly by outside water pressure.

4.6. Quasi-stationary fatigue analysis 65

Figure 4.6: Contour plot of registered fatigue damage, frame vs. sti�ener.

For large ship structures like tankers, the stresses induced by vertical and horizontal hullbending are in most cases su�ciently described by linear wave theory, at least for fatiguedamage. However, the stresses induced by the outside water pressure are non-linear withrespect to wave height (dry and wet areas). This implies that the expected fatigue damagein the side shell due to the combined stress cannot be calculated by a traditional frequencydomain analysis.

The purpose of the present study is to apply the quasi-stationary narrow-band modelto the prediction of the fatigue damage along the side shell. The model is well suited tocombine stresses from wave-induced vertical and horizontal hull bending with stresses fromoutside water pressure, taking wet and dry areas into account.

In Cramer and Friis Hansen [22] a general procedure for obtaining the long-term stressrange distribution or extreme-value distribution was presented. The procedure outlined builton a speci�ed travel route of the vessel and took into account di�erent loading conditionsand the e�ect of maneuvering. The method was based on linear frequency domain analysiswhich, as stated, is inapplicable to the present study. Here, it is attempted to bridge thegap by combining the ideas with the application of the quasi-stationary narrow-band modelin the analysis of the fatigue damage in the side shells.

It is assumed that procedures are available for the determination, either analytical ornumerical, of all ship motions and forces due to a regular sinusoidal wave system with givenamplitude and frequency.

Further assumptions embedded in the model are that:


Figure 4.7: Fatigue damage along side shell.

Figure 4.8: Registered number of cracks at frames 52 and 54.


1. The wave spectrum is narrow-banded.

2. The dynamic transients are small and evolve slowly, so that the structure respondsdirectly to the local wave sinusoid without signi�cant e�ects of transients from previouswaves.

For ship structures these assumptions are justi�able, whereas for o�shore structures,these assumptions may be rather crude, as o�shore structures can generally show markeddynamic e�ects.

Environmental modeling and the joint distribution of wave amplitude and frequencyfollow the outlines in Chapter 3.

4.6.2 Stress range

Motions and forces will be induced in the vessel when it passes through a regular sinusoidalwave of amplitude a and frequency !. These motions and forces may, for example, becalculated by linear strip theory, [46, 131], or by second-order strip theory, [65]. In thisstudy the variation of the total resultant axial stress in a sti�ener at a speci�ed location(x; z) along the side shell is of particular interest. The resultant axial stress consists ofcontributions by wave-induced

� vertical and horizontal hull bending,

� internal pressure exerted by the oil in the tank due to vertical acceleration,

� outside water pressure.

It is convenient to combine these stress contributions in the complex plane, and thus\automatically" account for the relative phase information between the stress components.In the following section all complex variables will be denoted by a capital Z.

The following paragraphs describe how the individual stress contributions may be ob-tained by use of the transfer functions from the linear ship-motion program. A coordinatesystem is introduced with its vertical axis through the center of gravity of the ship. Forsimplicity the longitudinal axis is selected through the neutral axis of the midship section.The cause of this origin selection is that the linear ship-motion program used relates allforces to a unit sinusoidal wave with a wave crest at the center of gravity of the vessel.

Vertical and horizontal hull bending. The bending-induced stresses Z�m at a speci�clocation z along the side shell are obtained by means of Navier's theory for the complexresponses:

Z�m(z; a; !) = a

ZMy(!)

Iyyz � ZMz(!)

IzzB=2

!(4.79)


where ZMy and ZMz are the complex frequency-dependent vertical and horizontal hull bend-ing moments obtained from the ship-motion program. Iyy and Izz are the vertical andhorizontal moments of inertia, and B the width of the vessel.

Internal pressure in tank. The wave-induced vertical accelerations Z�z in the vessel causea variation in the internal hydrostatic pressure Zip in the tank. At the point z the internalpressure is:

Zip(a; !) = a�oilZ�z(!)(zu � z) (4.80)

where �oil is the density of the oil, and zu the distance from the elastic neutral axis to thedeck.

Outside water pressure. In calculations based on linear strip theory, the local hydrody-namic pressure can be decomposed into:

1. The incident wave pressure (the \Froude-Krylov" pressure), which is the pressure �eldof the incident wave in the absence of the ship, evaluated at the ship's surface.

2. The e�ect of changes in hydrostatic pressure due to vessel oscillations in calm water.

3. The pressure due to di�racted waves on the hull, assuming the vessel is �xed.

4. The pressure due to radiated waves generated by ship oscillations in calm water.

Only items 1 and 2 are considered here, as these are believed to cover the major part of thesubject. Furthermore, the contribution by hydrostatic pressure is assumed to be controlledby instantaneous wave elevation, even though the motion calculations are based on lineartheory. This means that the dynamic pressure caused by ship motions is neglected in thecalculation of stresses, but the non-linearities in hydrostatic pressure are accounted for. Atmidship this assumption is justi�able because of wall-sidedness, whereas for bow and aftsections these assumptions might be rather crude because of the presence of are.

To calculate the wave-induced pressure in sti�eners, the instantaneous or apparent waveelevation Zaw at location xl from the center of gravity is calculated on the basis of the transferfunctions for heave and pitch:

Zaw(xl; a; !) = afZw(xl; !)� [Zh(!) + Zp(!)xl]g (4.81)

where Zw is the incoming wave, Zh the transfer function for heave, and Zp the transferfunction for pitch. Heave and pitch are again relative to the center of gravity, and smallpitch angles are assumed. When gravity waves are assumed, the incoming wave must be


shifted in phase by sign(!e) � !2exl=g radian. Here !e = ! � !2v=g cos � is the frequency of

encounter, where v and � are ship speed and heading direction, respectively.

The water pressure Zwp at the point of consideration (xl; z) at the side shell then becomes:

Zwp(a; !) = �swgfzmsw � z +Kp(zmsw � z)Zaw(a; !)g1l<fzmsw�z+Zawg>0 (4.82)

where �sw is the density of the salt water, zmsw the mean still-water level, Kp(�) is a pressureresponse factor, 1l<f�g is a zero-one indicator, and <f�g represents the real part of the complexvariable. Thus, the zero-one indicator triggers when the wave elevation is above the pointof consideration.

The pressure in Eq. 4.82 is seen to consist of two terms. The �rst term is the hydrostaticterm, which would exist without the presence of the waves. The second term is called thedynamic pressure. The dynamic pressure is a result of two contributions. The �rst andmost obvious contributor is the surcharge pressure due to the water surface displacement.However, the wave motion is associated with a vertical acceleration, which is 180� out ofphase with the free-surface displacement. This second contribution counteracts the directhydrostatic pressure arising from the �rst contribution. The pressure response factor has amaximum of unity at mean still-water level, and below it is always less than unity. Belowthe mean still-water level, the pressure response factor approaches asymptotically ek(zmsw�z)

for deep water, where k = !2=g is the wave number. Above the mean still-water level thepressure response factor may for simplicity be regarded as unity.

Combined stress. As a simpli�cation, it is assumed that the calculated water pressureZwp is constant between two consecutive web frames. When it is further assumed that thesti�ener is clamped onto the web frames, the stress Z�p induced by the outside water pressureand internal oil pressure is

Z�p(z; a; !) = � bsl2w

12Wstiff

(Zwp(a; !)� Zip(a; !)) (4.83)

where bs is the spacing between the sti�eners, lw the spacing between the transverse webframes, and Wstiff is the section modulus of the sti�ener with respect to the outer ange(hence the negative sign). is a factor which accounts for the increase in nominal stress inthe ange due to an asymmetric sti�ener. For a symmetric sti�ener = 1 whereas AppendixA presents an approximate method of calculating the factor for asymmetric sti�eners.

The total combined stress in the sti�ener near the transverse web frames then simplybecomes:

�(a; !) = <fZ�m(a; !) + Z�p(a; !)g (4.84)


The stress de�ned in Eq. 4.84 is calculated on the basis a sinusoidal wave of amplitude a andfrequency ! having a wave crest at the center of gravity of the vessel. The stress trajectoryis obtained by letting the phase � run through an entire cycle [0; 2�]. Figure 4.9 shows thevariation of the combined stress in the sti�eners near the mean still-water level as a functionof the phase shift of the wave. The stress variation is calculated for the mean value of thewave amplitude and the frequency in the di�erent sea states. The stress variation is shownfor sti�eners 45 and 46 in laden condition (see also Figure 4.13).

Fatigue damage calculation

Within the narrow-band assumption, it is possible to de�ne directly the stress amplitude ��for a given wave amplitude a and frequency ! in a stress response cycle. The stress rangeis given as the di�erence between the maximum and minimum stresses within a stress cycleobtained from Eq. 4.84:

�� = maxf�(a; !)g �minf�(a; !)g (4.85)

Then there is solved, once and for all, for the mean fatigue damage rate by unconditioningwith respect to the variables:

E[D] =T

KE[�(��)m]

=T

K

ZHs

ZTz

ZA

Z

ZV

Z��(��(a; !; v; �))m

f(a; ! j Hs; Tz)f(v; � j Hs; Tz))f(hs; tz)dad!dhsdtzdvd� (4.86)

where T is the time, K a material fatigue parameter, f(a; ! j Hs; Tz) is the joint distributionof wave amplitude and frequency within each sea state, f(v; � j Hs; Tz) accounts for thee�ect of maneuvering in severe sea states. Eq. 4.86 is conveniently solved by simulation. InFigure 4.10 the variation of the mean fatigue damage is shown as a function of the numberof simulations.

4.6.3 Veri�cation of model

In order to verify the proposed damage model, the results obtained by Eq. 4.86 are comparedto results obtained from a traditionally linear frequency domain analysis. The veri�cationprocedure is performed for two marginal cases: (1) Due to wave-induced hull bending, and(2) due to outside water pressure.


Figure 4.9: Stress variation for selected signi�cant wave heights Hs and mean zero crossingperiod Tz. Left: sti�ener no. 45, right: sti�ener no. 46.


Figure 4.10: Mean fatigue damage vs. number of simulations.

To facilitate the reading, only the fatigue damage in a short-term sea state is given inthe following. This means that

RHs

RTz

RV

R� � � �f(v; � j Hs; Tz))f(hs; tz)dhsdtzdvd� is omitted

in the equations.

Wave-induced hull bending. If all non-linear pressure e�ects are neglected, the distribu-tion of the peaks, and thus the stress range ��, follows a Rayleigh distribution

F��(��) = 1� exp

�(��)

2

8�0

!(4.87)

where �n is the n'th spectral moment of the response spectrum experienced by the vessel,cf. Eq. 4.12.

Given the Rayleigh distribution assumption, the fatigue damage caused by wave-bendingis in linear statistics written as:

E[D] =T

KE[�(��)m]

=T

K

Z 1

��=0�(��)m+1

4�0exp

�(��)

2

8�0

!d��

=T

K�(8�0)

m=2�(1 +m=2) (4.88)


The peak rate � is

� =1

2�

s�2�0

(4.89)

T is the time, and K is a material fatigue parameter.

Outside water pressure. The fatigue damage from the outside water pressure is calculatedby use of the linear dependency between the stress range and the Rayleigh distributed wavepeaks. However, it is necessary to distinguish between sti�eners above and below the meanwater level. Let y0 be the location of the sti�ener relative to the mean still-water level.

y0 � 0 : E[D] =T

KE[�(��)m j a > y0]

=T

K

Z 1

a=0�(�(a� y0))ma

�0exp

� a2

2�0

!1la>y0da

= �m�T

K

Z 1

a=y0

((a� y0))ma�0

exp

� a2

2�0

!da (4.90)

y0 < 0 : E[D] =T

KE[�(��)m j a � �y0] + E[�(��)m j a > �y0]

=T

K

Z 1

a=0� [(�2a)m1la��y0 + (�(a� y0))m1la>�y0 ]

a

�0exp

� a2

2�0

!da

= �m�T

K

"Z �y0

a=0

2mam+1

�0exp

� a2

2�0

!da+

Z 1

a=�y0

a(a� y0)m�0

exp

� a2

2�0

!da

#(4.91)

in which

� =�swgbsl

2w

12Wstiff

(4.92)

For any real value of m the integrals in Eqs. 4.90 and 4.91 involving the term (a � z)mmust be solved numerically. For integer values of m the integrals may be evaluated in termsof the incomplete and complementary incomplete Gamma function, see e.g. Abramowitzand Stegun [1]. An analytical evaluation of the integrals may be found in [23].

It should be noted that the spectral moments �n are conditional on wave height, waveperiod, velocity, and direction. Eq. 4.88, Eq. 4.90, and Eq. 4.91 are solved by simulation andcompared to the results obtained by the regular sinusoidal wave approach, Eq. 4.86. Figure4.11 shows the comparison of the estimated fatigue damage obtained by the two di�erentmethods. It is seen that the results obtained by the two di�erent models coincide almostperfectly, with a deviation of less than 5%. This slight di�erence is expected as only a limitednumber of simulations (10000) are performed in the evaluation of the integrals.


Figure 4.11: Comparison of damage obtained by the proposed model and frequency domainanalysis.

Table 4.2: Principal dimensions.

Length overall 274.170 mLength between perpendiculars 262.130 mLength at 17.450 m waterline 269.600 mScantling length 261.520 mBreadth molded 52.410 mDepth molded 22.810 mScantling draft 17.450 mDisplacement molded (17.450 m WL) 196500 TBlock coe�cient 0.809Draught laden 16.765 mDraught ballast 8.270 m


Figure 4.12: General outline of the analyzed vessel.

Table 4.3: Sectional properties of midship section.

Distance from bottom to c.g. 11:158mArea 5:494m2

Moment of inertia Izz 1688:3m4

Moment of inertia Iyy 544:78m4

4.6.4 Numerical example

To compare a theoretically calculated damage rate along the side shell, a midship with theregistered number of fatigue cracks, a 164000 dwt segregated ballast tanker was analyzed.Figure 4.12 shows the general outline of the segregated ballast tanker and Table 4.2 givesthe principal dimensions.

Figure 4.13 presents a drawing of the midship section including the numbering of thelongitudinal sti�eners. The geometry of the sti�eners is given in Table 4.4, and the locationalong the side shell relative to the elastic neutral axis in Table 4.5.

The vessel is assumed to operate under two di�erent loading conditions only, namely aladen and a ballast condition. Further, it is assumed that the vessel operates 50% of thetime in the ballast condition. The intended trade of the vessel during the lifetime is a NorthAtlantic route with equal occupation in Marsden zones 15, 16, 24, and 25 (see [14]).

The model was implemented into the reliability program PROBAN [49, 120], and for eachof the sti�eners along the side shell of a midship, the mean fatigue damage was calculated.The mean fatigue damage was calculated for two di�erent types of sti�eners: (1) A symmetricI sti�ener, and (2) an asymmetric L sti�ener. The expected value of E[�(��)m] from Eq. 4.86


Detl. 2

L1 L23L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 L13 L14 L15 L16 L17 L19 L20 L21 L22 L24 L25L18 L26 L27 L28

L29

L30

L31

L32

L33

L34

L35

L36

L37

L38

L39

L40

L41

L42

L43

L44

L45

L46

L47

L48

L49

L50

L51

L52

L29

L30

L31

L32

L33

L34

L35

L36

L37

L38

L39

L40

L41

L42

L43

L44

L45

L46

L47

L48

L49

L50

L51

L52

Detl. 3

Detl. 4

Detl. 6

Detl. 4

Detl. 6

Detl. 6

Detl. 4

Detl. 6

Detl. 6

Detl. 4

Detl. 6

Detl. 6

Detl. 4

Detl. 6

Detl. 4

Detl. 6

Detl. 6

Figure 4.13: Midship section.

Table 4.4: Section properties of longitudinal sti�eners.

Sec. # Web [mm] Flange [mm] W [106mm3] 1 585� 12 180� 20 3.031 1.89952 585� 12 165� 20 2.883 1.82413 530� 12 165� 20 2.531 1.83634 510� 12 150� 20 2.276 1.74415 455� 12 140� 20 1.880 1.65386 430� 12 110� 20 1.520 1.41137 380� 12 100� 20 1.128 1.30008 305� 12 100� 20 0.906 1.23569 180� 10 100� 10 0.282 1.0981


Table 4.5: Location of longitudinal sti�eners.

Sti�. # Loc. Sec. Sti�. # Loc. Sec. Sti�. # Loc. Sec.52 11.012 9 45 4.712 7 36 -3.388 451 10.112 9 44 3.812 6 35 -4.288 350 9.212 8 43 2.912 6 34 -5.188 349 8.312 8 42 2.012 6 33 -6.088 348 7.412 8 41 1.112 5 32 -6.988 247 6.512 7 40 0.212 5 31 -7.888 246 5.612 7 39 -0.688 5 30 -8.788 2

38 -1.588 4 29 -9.688 137 -2.488 4

Figure 4.14: Calculated fatigue damage rate for laden and ballast condition for I-shapedsti�eners.

with m = 3 was estimated on the basis of 10000 simulations. The obtained results for theladen and the ballast conditions are presented in Figure 4.14 for I-shaped sti�eners, and inFigure 4.15 for L-shaped sti�eners. Although the calculated fatigue damage at each sti�eneris connected with a solid line, an interpolation between the sti�eners is absurd. The pointsare connected only to support the visualization. It is seen from the �gures that the fatiguedamage induced by the outside water pressure accounts for the majority of the expectedfatigue damage. This trend is especially clear in the laden condition. It is seen that themajority of the fatigue damage is expected to occur just below the mean waterline, wherethe e�ect of wet and dry areas is most likely to be seen. The mean waterline is in theladen condition located between sti�eners no. 45 and 46 (5.606 m above the centroid of thesection), whereas between sti�eners no. 36 and 37 for the ballast condition (2.888 m belowthe centroid of the section).


Figure 4.15: Calculated fatigue damage rate for laden and ballast condition for L-shapedsti�eners.

Note that the asymmetric shape of an L-shaped sti�ener results in a remarkable increasein expected fatigue damage. Note also that the e�ect of stress combination results in a rathersigni�cant increase in fatigue damage. For the I-shaped sti�ener no. 45 the fatigue damagerate E[��m] is 5170 for the combined case whereas, due to vertical hull bending alone, itis 75, due to horizontal hull bending alone, 56, and due to pressure alone, 2958. The peakrate � is approximately 0.1 and m = 3. Thus, the combination of the stresses results in anincrease in \mean", in some sense, stress range due to pressures from 26 MPa to 31 MPa.Note also that the damage resulting from horizontal hull bending alone is approximately60% larger for the ballast condition than for the laden condition. This is expected since thehorizontal hull bending in the ballast condition is typically 15-20% higher than in the laden.

Figures 4.16 and 4.17 show the combined damage rate in the laden and in the ballastconditions for I- and L-shaped sti�eners. The calculated expected damage rate is comparedto the relative number of registered fatigue cracks. It is seen that the model predicts theobserved trend. However, the model overestimates somewhat the registered fatigue damagein the upper part of the side shell. No fatigue cracks were reported in the upper part ofthe side shell. This might indicate that either the inspection of the upper part of the sideshell was performed under di�cult circumstances, or that the upper part was subjected toexcessive corrosion, so that these areas have to be replaced in a major overhaul. In somesurveys little e�ort would be made to look for or identify cracks in highly corroded areas.Indeed, the upper part of the tank is classi�ed among the most critical corrosion areas, butno information was provided about corrosion wastage for the inspected tankers.

The location of the calculated peak in Figures 4.16 and 4.17 is highly in uenced by thesectional draft in the laden condition. A more accurate modeling of the load distributionmight change the sectional draft.

Table 4.6 presents sensitivity factors obtained for sti�ener no. 45. Not surprisingly, it


Figure 4.16: Comparison of calculated fatigue damage and registered at frame 52 for I-shaped sti�eners.

Figure 4.17: Comparison of calculated fatigue damage and registered at frame 52 for L-shaped sti�eners.


Table 4.6: Sensitivity factors.

Laden Ballast CombinedParameter Value d�

dPard�

dPard�dPar

Cruising speed [m/s] 7.21 -141.0 14.2 -63.4� - wave spectrum 5.0 -40.3 7.54 -16.4� - wave spectrum 4.0 3.95 4.88 4.41m - material parameter 3.0 1:95 � 104 5:18 � 102 9:99 � 103Ratio laden, ballast 0.5 4566 -144 4422

is seen that the expected fatigue damage is most sensitive to the material parameter m.However, the analysis shows a negative sensitivity factor for the cruising speed in the ladencondition. No straightforward explanation was found of this, as both the stress range andthe peak rate relate to the velocity in a complicated manner. Also, the sensitivity withrespect to the parameters in the wave spectrum is rather insigni�cant.

4.7 Quasi-stationary model for impact slamming

The second example of an application of the quasi-stationary narrow-band model is on thecombination of impact slamming- and wave-induced responses, Friis Hansen [43].

4.7.1 Introduction

The term slamming refers to the impact generated when the ship bottom hits the water sur-face after a series of large heave and pitch motions have forced a part of the ship's bottomto emerge and thereafter to re-enter the water. Thus, (impact) slamming requires \bottomemergence" contrary to bow are slamming. In this study, only impact slamming is con-sidered. Full-scale measurements have shown that the slamming-induced stresses amidshipcan be of the same order of magnitude as the bending-induced stresses, see Ochi and Motter[119]. Therefore, slamming stresses must be carefully evaluated in the design phase, andbe suitably combined with the low-frequency wave-induced bending stresses. According toOchi and Motter [119], the necessary and su�cient conditions leading to slamming impactare:

� relative motion must exceed sectional draft (bottom emergence), and

� relative velocity at the instant of re-entry must exceed a certain magnitude, called thethreshold velocity.

4.7. Quasi-stationary model for impact slamming 81

Ochi and Motter [119] suggested a truncated exponential probability distribution of theimpact pressure and proposed a Poisson process model for slamming interarrival time by�tting experimental data. Further, they derived statistical properties for the slammingpressure. Conservatively, they proposed to calculate the slamming moment from the dynamicanalysis of a hull girder under the impact forces using the extreme slamming pressure ateach point. Moreover, they suggested combining this extreme upper bound on the slammingmoment with the extreme wave moment by use of a suitable (unspeci�ed) phase angle.

A more rigorous mathematical approach to obtain the statistical properties has beenpresented by Mansour and Lozow [96]. They assumed that slamming impacts follow aPoisson pulse process with independent amplitudes and interarrival times.

Nikolaidis and Kaplan [113] performed a Monte-Carlo simulation study to estimate theuncertainty in the combination of low-frequency wave bending moments and slamming bend-ing moments in ships. They concluded that Turkstra's rule [150] underestimates the meanvalue of the combined moment, and that the design load, estimated from Turkstra's rule,had a larger variability than the actual load. However, Nikolaidis and Kaplan did not ap-ply Turkstra's rule correctly as Turkstra's rule requires the maximum value of one of theprocesses to be combined with the corresponding value in time of the other process. Foruncorrelated processes the corresponding value in time is easily obtained, whereas it is moreinvolved for correlated processes.

Nikolaidis and Kaplan [113] stated the following arguments for the incorrectness of thePoisson assumption:

1. The times of occurrence of slamming impacts are not independent because of theperiodic nature of the ship motions and waves.

2. The times of occurrence of the slamming and the wave-induced stress peaks are highlycorrelated. As it was reported by Ochi and Motter, a slamming impact usually gen-erates the �rst peak of a compressive (sagging) slamming stress in the deck as thewave-induced stress passes from hogging to sagging.

3. When the wave-induced stress is high, it is very likely that the slamming-induced stressis also high. Therefore, from a probabilistic point of view, the slamming and the wavestress intensities are positively correlated.

So far no general mathematical method has been developed to calculate the probabilisticstructure of the slamming and low-frequency stresses, although Ferro and Mansour [40]proposed to apply Tukstra's rule. Ferro and Mansour based their work on the earlier workby Mansour and Lozov, which as mentioned assumed the slamming impact to follow aPoisson pulse process. However, the slamming process is not exactly a Poisson process butapproaches a Poisson process as the sectional draft approaches in�nity. The reason is thatthe narrow-band character of the process of the relative motion tends to concentrate the


slamming impacts in clusters and thus violates the assumption of mutual independence ofthe individual slamming impacts. When slamming plays a signi�cant role in design, thenon-Poisson character of the slamming process may therefore a�ect the results signi�cantly.

The purpose of this study is to formulate a probabilistic model which takes the non-Poissonian character of slamming impact into account { more speci�cally the clusteringe�ect. Further, the probabilistic model should give a combination rule for the low-frequencywave-induced bending moment, and the high-frequency slamming-induced bending moment.

It is assumed that procedures are available to determine, either analytically or numeri-cally, all internal ship forces (wave-induced bending and slamming-induced bending) due toa passing regular sinusoidal wave with given amplitude and frequency. Further assumptionsembedded in the model are:

1. The ship motions are su�ciently described by linear wave theory, and not in uencedby the slamming impacts.

2. The spectrum of the relative motion is narrow-banded.

3. The dynamic transients are small and evolve slowly, so that the structure respondsdirectly to the local wave sinusoid without signi�cant e�ects of transients from previouswaves.

These assumptions are justi�able for ship structures.

The basic idea is to model the joint density function of the wave amplitude and the fre-quency for those waves resulting in local maximum wave-induced slamming response withina cluster of slamming impacts. The procedure to be followed is to consider an envelopeprocess for the process of relative motion at the bow section in order to take the cluster-ing e�ect into account. For a regular sinusoidal wave with �xed values of amplitude andfrequency, the maximum/minimum value of the combined moment response is calculated,Figure 4.21. Given the joint density function for the wave amplitude and the frequency,this density can be used to weigh the calculated combined response, so that the responsestatistics (say, the �rst four moments) are obtained. Thus the analysis is quasi-stationary.Finally, the extreme-value distribution is found on the basis of the theory for �rst-passagetime distributions in Poisson pulse processes. The mean interarrival times of the pulses areapproximated by use of the upcrossing rate of the envelope process, modi�ed for so-called\empty" envelope excursions.

To facilitate the reading the following overview is given:

1. Construct an envelope process for the process of relative motion at the bow section.

2. Formulate the so-called Slepian model process for the envelope process when the en-velope process exceeds the sectional draft at the bow section (i.e. when slamming willoccur).


3. Construct a second-order Taylor expansion to the Slepian model process, and �ndthe point of maximum envelope excursion. This point corresponds to the maximumslamming response within a cluster of slamming impacts.

4. Obtain the joint probability distribution of the wave amplitude and the frequency forthe waves that give the maximum slamming response.

5. Make a deterministic analysis of the slamming- and wave-induced response due toregular sinusoidal waves of selected amplitude and frequency.

6. Obtain the force statistic by weighing the calculated response by the probability den-sities of the various pairs of wave amplitude and frequency.

4.7.2 Basic idea

For each short-term sea state, the distribution of the wave energy upon the frequency is givenby the wave spectrum. When the response is a linear function of the wave height, the responsespectrum is obtained by combining the transfer function (response amplitude operator) withthe wave spectrum. Di�erent statistics of the response are then easily obtainable from theresponse spectrum. However, when the response is a non-linear function of the wave heightit is necessary to replace the wave spectrum by a wave model that allows a formulation of ajoint distribution of the wave amplitude and the wave frequency. This distribution is thenused to weigh the calculated response for a wave with a given amplitude and frequency toobtain the force statistics.

In the present study the combination problem of wave- and slamming-induced bendingis of particular interest. Thus the modeling is concentrated on the joint density functionof wave amplitude and frequency for the waves which cause slamming. Given that the seastate is Gaussian, the process of relative motion at the bow section is also Gaussian. Ifit is assumed that all motions are su�ciently described by linear wave theory, the complextransfer function for the relative motion ZRM at the bow section may be obtained from linearstrip theory programs. Consequently the process of relative motion X(t) at the bow sectionmay be written in terms of the process of wave amplitude and frequency as

X(t) = a(t)<fZRM [!(t)]g (4.93)

or

a(t) =X(t)

<fZRM [!(t)]g (4.94)

where <f�g indicates the real part of the complex variable. Here, the process of relativemotion X(t) is assumed to be positive downwards. If the de�nition of slamming is that


Figure 4.18: Narrow-band Gaussian process with associated envelope.

it occurs when the relative motion X(t) exceeds the sectional draft u at the bow section,slamming will occur whenever X(t) upcrosses the level u. Strictly speaking, this means thatthe latter of Ochi's and Motter's two conditions leading to slamming (the de�nition of athreshold velocity) is disregarded. The slamming pressure is assumed to be proportional tothe square of the relative velocity at re-entry, and there seems to be no physical reasons forthis relationship not to apply to small velocities.

It is well known that the narrow-band character of the process X(t) tends to concen-trate the upcrossings (the slamming impacts) in clusters and thus violates the assumptionof independence of the individual upcrossings of the process X(t). Therefore it is convenientto construct an envelope process for X(t) and then examine the excursions of the envelopeprocess. This envelope process will automatically account for the clustering e�ect of theunderlying process. Given the envelope process, it is in particular of interest to describe thevalue of the envelope process for which the slamming-induced response reaches its maximumvalue within a cluster, see Figure 4.18. This point in time is the point of maximum down-crossing velocity at level u of the process X(t). The phase process is slowly varying, whichleads to the occurrence of maximum slamming-induced response at approximately the samepoint in time as when the envelope excursion reaches its maximum value after an upcrossingof level u.

Ditlevsen and Lindgren [32] modeled the Slepian model process for the envelope in theirstudy of empty envelope excursions. The following work is based on, and inspired by theirwork. By formulating the Slepian model process of the Cramer-Leadbetter [24] envelopeprocess and conditionally on an upcrossing of level u, it is possible to �nd approximations forboth the maximum value of the envelope process and the duration of the excursion. Further,at the maximum point of the envelope the corresponding value of the wave frequency can beobtained. The joint density function of relative amplitude is then found by unconditioningwith respect to those variables on which basis the Slepian model process was formulated.The corresponding value of the wave amplitude at the occurrence of maximum slammingresponse is uniquely determined from Eq. 4.94.


4.7.3 Basic considerations

Let the Gaussian response process of the relative motion at the bow section be X(t). Thespectral moments of the response process are given as

�n =Z 1

0!ndG(!) <1 ; at least for n � 3 (4.95)

dG(!) is the spectral increment of the response process at frequency ! given as

dG(!) =j ZRM(!) j2 S�(!)d! (4.96)

where S�(!) is the wave spectrum.

The envelope process R(t) is de�ned (Cramer and Leadbetter [24]) as the norm

R(t) =qX2(t) + X2(t) (4.97)

which satis�es the conditions for being an envelope process to X(t) since R(t) � X(t) forall t. X(t) is the Hilbert transform of X(t).

Since X(t) is assumed to be ergodic it implies that dG(0) = 0, and that the commonvariance of X(t) and X(t) is the spectral distribution function G(!). The covariance matrixof (X(t); X(t)) is

"Cov[X(0); X(t)] Cov[X(0); X(t)]

Cov[X(0); X(t)] Cov[X(0); X(t)]

#=Z 1

0

"cos!t sin!t� sin!t cos!t

#dG(!) (4.98)

Note that since the matrix in the integrand is orthogonal and equal to the unit matrix fort = 0, it follows that the probabilistic structure of the vector process (X(t); X(t)) is invariantto a rotation of the coordinate system.

4.7.4 Conditional mean excursion path

Following Ditlevsen and Lindgren [32] this section derives an approximate second-order Tay-lor expansion of the envelope process conditional on the occurrence of an upcrossing of agiven level at time t = 0. In the following a prime indicates vector transpose.


The vector process (X(t); X(t)) is written as X(t), and for convenience (X(0); _X(0)) as(X; _X). Since X(t) is a Gaussian process, the conditional mean is determined as the linearregression E[X(t) jX; _X] for (X; _X) = (x; _x).

Standard calculations by means of Eq. 4.98 at t = 0 give

Cov

""X_X

#; [X 0 _X

0]

#�1=

"I�0 J�1�J�1 I�2

#�1

=1

�0�2 � �21

"I�2 �J�1J�1 I�0

#(4.99)

in which

I =

"1 00 1

#J =

"0 1�1 0

#(4.100)

Furthermore

Cov[X(t); [X 0 _X0]] =

Z 1

0

"cos!t � sin!t ! sin!t ! cos!tsin!t cos!t �! cos!t ! sin!t

#dG(!) (4.101)

Thus, the linear regression of X on (X; _X) is

E[X(t) jX; _X] = E[X(t)]

+ Cov[X(t); [X 0 _X0]]Cov

""X_X

#; [X 0 _X

0]

#�1 "X_X

#� E

"X_X

#!

=1

�0�2 � �21Z 1

0[I cos!t� J sin!t !(I sin!t+ J cos!t)]�

"I�2 �J�1J�1 I�0

# "X_X

#dG(!)

=1

�0�2 � �21Z 1

0f[(�2 � !�1)IX � (�1 � !�0)J _X] cos!t

� [(�2 � !�1)JX + (�1 � !�0)I _X] sin!tgdG(!)(4.102)

From this it follows that

E[X(t) jX; _X]t=0 =1

�0�2 � �21Z 1

0[(�2 � !�1)X � (�1 � !�0)J _X]dG(!)


= X (4.103)

d

dtE[X(t) jX; _X]t=0 =

�1�0�2 � �21

Z 1

0![(�2 � !�1)JX + (�1 � !�0) _X]dG(!)

= _X (4.104)

d2

dt2E[X(t) jX; _X]t=0 =

�1�0�2 � �21

Z 1

0!2[(�2 � !�1)X � (�1 � !�0)J _X]dG(!)

=�1�3 � �22�0�2 � �21

X +�2�1 � �0�3�0�2 � �21

_X (4.105)

The existence of the k'th derivative requires the existence of the (k+1)'th spectral moment.

Owing to the stationary characteristic and the rotational symmetry of the probabilisticstructure of the vector process, it will be su�cient to study excursions starting at (x; x) =(u; 0) for t = 0. In a polar coordinate system, the representation of the vector processbecomes

X(t) = R(t)

"cos�(t)sin�(t)

#(4.106)

with R(0) = u and �(0) = 0. R(t) is then the envelope process, while �(t) is the phaseprocess. The conditional mean is correspondingly represented by

E[X(t) jX; _X] = r(t)

"cos �(t)sin �(t)

#(4.107)

The �rst few time derivatives of r and � follow from Eqs. 4.104 and 4.105:

d

dtE[X(t) jX ; _X] = _r(t)


#+ r(t) _�(t)

" � sin �(t)cos �(t)

#(4.108)

giving

_r(0) = _x (4.109)

_�(0) = _x=u (4.110)


and

d2

dt2E[X(t) jX; _X] = �r(t)


#+ 2 _r(t) _�(t)

"� sin �(t)cos �(t)

#

+ r(t) _�2(t)

" � cos �(t)� sin �(t)

#+ r(t)��(t)

" � sin �(t)cos �(t)

#(4.111)

giving

�r(0) =�1�3 � �22�0�2 � �21

x+�1�2 � �0�3�0�2 � �21

_x +_x2

u(4.112)

��(0) =�0�3 � �1�2�0�2 � �21

_x

u� _x _x

u2(4.113)

For convenience introduce �n as the n'th normalized central spectral moment, that is thespectral moment with respect to �1=�0:

�2�0

= �2 +

�1�0

!2

(4.114)

�3�0

= �3 + 3

�1�0

!�2 +

�1�0

!3

(4.115)

Further introduce �; �, and � as

_x =p�2� ; _x =

p�2� +

�1�0u ; t =

�p�2

(4.116)

Then the second-order Taylor expansion of �(t) and r(t) from t = 0 become, by use ofEqs. 4.109 and 4.110 and Eqs. 4.112 and 4.113

�(�) =

�

u+1

!� �

�

u� �

2

!�

u� 2 + o(� 2) (4.117)

r(�) = u+ �� 1

2u(u2 � �2 + �u�)� 2 + o(� 2) (4.118)

in which

=�0p�2

�1; � =

�3�2

p�2 (4.119)


and o(�)=� ! 0 for � ! 0. The parameter re ects the bandwidth of the process. Fornarrow-band processes the value of is small. For such processes the term �= will be thedominating term in Eq. 4.117. The excursion of r(t) above the level u terminates for � = �2.From Eq. 4.118 it follows that

�2 =2u�

u2 � �2 + �u�; (��

p4 + �2)

u

2< � < (�+

p4 + �2)

u

2(4.120)

The truncation interval on � is applied in order to preserve the positivity properties of �2.The corresponding value of the phase is

�2 =

�

u+1

!�2 �

�

u� �

2

!�

u� 22 (4.121)

For narrow-band processes ( ! 0) the dominating term of the phase is

�2 � �2 =

1

2u�

u2 � �2 + �u�(4.122)

Note that the neglected terms are independent of .

The maximum value of the excursion of the envelope process r(t) occurs approximatelyat �max = �2=2. The corresponding value of the Taylor expansion of the envelope process is

rmax = u+�2u

2(u2 � �2 + �u�)(4.123)

The instantaneous frequency ! is de�ned as the rate of change of the process �(t) (i.e.! = _�(t) ). After di�erentation of Eq. 4.117 with respect to time and substitution of thevalue of �max, the wave frequency at the maximum value of the excursion is obtained as

!max =

�

u+1

!p�2 �

�

u� �

2

!2�2�2

u2 � �2 + �u�(4.124)

When using only the dominating term of the frequency, we obtain,

!(�max) =

p�2

(4.125)


4.7.5 The Slepian model process

The set of those sample paths of the vector X(t) that in the time interval [0; dt] passesthrough the vertical line element dx at (u; 0) with velocity vector ( _x; _x) has the probability

fX;X; _X;

_X(u; 0; _x; _x) _xdtdxd _xd _x (4.126)

in which fX;X; _X;

_Xis the joint Gaussian density of (X; X; _X;

_X) = (X(0); X(0); _X(0);

_X(0)).

Ditlevsen and Lindgren [32] showed that _X and_X given (X; X) are uncorrelated, and that

the density of ( _X;_X) is proportional to

f _X;_X( _x; _x) / _x�

_xp�2

!�

_x� �1u=�0p

�2

!; _x > 0 (4.127)

Thus the corresponding random variables

� =_xp�2

; Z =_x� �1u=�0p

�2(4.128)

are mutually independent with densities

f�(�) = � exp[��2=2] ; (Rayleigh) (4.129)

fZ(�) = �(�) ; (Standard normal) (4.130)

By substituting the random variables _X(0) =p�2 � and

_X =

p�2Z + �1u=�0 into

Eq. 4.107 and setting X(0) = u, X(0) = 0, the following non-stationary random Slepianmodel process

X(t)ju�upcross = E[X(t) jX=(u; 0); _X=(p�2 �;

p�2Z +

�1�0u)] + S(t) (4.131)

is obtained for the probabilistic structure of the random process X(t) at an arbitrary u-upcrossing. The residual process S(t) is a zero mean non-stationary Gaussian vector processwith common variance function of the components. The variance and its time derivativesassume the value zero for t = 0. It is seen that the Slepian model process, Eq. 4.131, is nota Gaussian process.


The covariance function of the Gaussian residual vector process S(t) is

Cov[S(t);S(s)0] = Cov[X(s)� E[X(s) jX; _X];X(t)]� E[X(t) jX; _X]]

= Ifr(t� s)� 1

�0�2 � �21[�2(r(t)r(s) + c(t)c(s))

+ �1(c(t) _r(s)� r(t) _c(s) + _r(t)c(s)� _c(t)r(s))

+ �0( _r(t) _r(s) + _c(t) _c(s))]g+ Jfc(t� s) + 1

�0�2 � �21[�2(r(t)c(s)� c(t)r(s))

+ �1(c(t) _c(s) + r(t) _r(s)� _c(t)c(s)� _r(t)r(s))

+ �0( _r(t) _c(s)� _c(t) _r(s))]g (4.132)

in which

r(t) =Z 1

0cos(!t)dG(!) ; c(t) =

Z 1

0sin(!t)dG(!) (4.133)

are the covariance function of X(t) and X(t) and the cross-covariance function between X(t)and X(t), respectively. The residual variance at time t is

Var[S(t)] = If�0 � 1

�0�2 � �21[�2(r(t)

2 + c(t)2)

+ 2�1(c(t) _r(t)� r(t) _c(t)) + �0( _r(t)2 + _c(t)2)]g (4.134)

4.7.6 Joint distribution of wave amplitude and frequency

The main purpose of the present analysis is to establish the joint distribution of wave am-plitude and frequency at the occurrence of maximum slamming response within a cluster ofslamming impacts. This joint distribution may be determined by the Slepian model process,Eq. 4.131. However, the presence of the residual vector process complicates and prohibitsexact calculations. The residual vector process has the e�ect that the maximum envelopeexcursion rmax for given _X uctuates in the vicinity of a speci�c value of rmax determineduniquely as the maximum of the linear regression. Since the interest here is only in an aver-age value, in some sense, of rmax, the error committed by neglecting the residual process isjudged not to be serious in the weighed average over � and �. Further, the maximum enve-lope excursion is located so closely in time to t = 0 that it is almost completely dominatedby the conditional value of _X.


Figure 4.19: Comparison of analytical and simulated marginal distribution functions.

In terms of the two independent random variables given in Eq. 4.129 and Eq. 4.130,outcomes of the maximum envelope excursion and the corresponding wave frequency areobtained from Eq. 4.123 and Eq. 4.124, respectively. In turn these values of the maximumrelative motions and frequency determine the wave amplitude of interest by use of Eq. 4.94.

The marginal distribution function of the maximum envelope excursion may, from Eq. 4.123upon integration with respect to the random variables, be found to be

FR(r) =1

c

Z H

L

"1� exp

�(r � u)(u

2 � �2 + �u�

u

!#�(�)d� (4.135)

where L = (��p4 + �2)u=2, H = (�+p4 + �2)u=2, and c = �(H)��(L). The distribution

function FR(r), de�ned by Eq. 4.135, can be calculated by numerical integration.

Figure 4.19 shows the marginal distribution function of FR(r) given by Eq. 4.135 incomparison with the empirical distribution function obtained by simulation. Appendix Bdescribes how the envelope process is simulated. It is seen from the �gure that the analyticaldistribution function of the maximum envelope excursion as given by Eq. 4.135 agrees verywell with the simulated results. This justi�es the neglection of the residual process.


maxMSlam

+MWavejSlam

maxMWavejNo Slam

Figure 4.20: Series system for extreme bending moment during the lifetime.

4.7.7 Extreme-value distribution

The distribution function of the maximum bending moment M during the lifetime T isexpressed as:

FmaxM(m) = P [maxT

M � m] = P [maxTslam

MSW � m \ maxT\Tslam

MW � m]

= 1� P [maxTslam

MSW � m [ maxT\Tslam

MW � m]

� 1� P [maxTslam

MSW � m]� P [maxT

MW � m] (4.136)

in whichMSW is the combined slamming- and wave-induced moment,MW the wave-inducedmoment, Tslam the random part of the lifetime T of which the ship is in a slamming condition,and T\Tslam the remaining time. The error in the last approximation in Eq. 4.136 is less thanthe smallest of the two probabilities. If the maximum value of the wave-induced responsecannot occur within a slamming period, then the equal sign is valid.

Turkstra's rule [150] often provides a good estimation of the maximal e�ect of a linearcombination of independent processes. According to this rule, only the points in time whereone of the processes is at its maximum value are considered. The extreme-value distributionfor the single load can often be approximately obtained, and the distribution of each of theaccompanying load values is simply its marginal distribution. The combined extreme-valuedistribution function is underestimated by Turkstra's rule, but the error turns out to besmall in most practical cases.

When applying Turkstra's rule to the present extreme-value analysis in which the pro-cesses are dependent, a series system with two components must be considered, see Figure4.20. The two components represent the maximum bending moment when slamming ispresent and the maximum bending moment when slamming is not present, respectively.The second term in Eq. 4.136, the extreme-value distribution of the bending moment whenslamming is not present, is easily found by use of standard techniques as given in e.g. Cramerand Friis Hansen [22] equation (35). The �rst term gives the extreme-value distribution of


the bending moment when slamming is present. This term is, however, more involved, butcan be obtained as follows.

As mentioned earlier the basic idea of the proposed model is to consider regular sinusoidalwaves for selected pairs of wave amplitudes ai and frequencies !i, and to calculate thecorresponding maximum bending moment amidship over the entire phase interval [0; 2�].The calculated bending moment should include both the low-frequency wave-induced, andthe high-frequency slamming-induced bending moment.

Given the largest (or smallest) moment M(a; !) due to the regular sinusoidal wave, theforce statistics (n'th order statistical moments) can be expressed as the integral

E[Mn] =Zall a

Zall !

[M(a; !)]nf(a; !)dad!

=Zall �

Zall �

[M(a; !)]nfZ(�)f�(�)d�d� (4.137)

In practice, this integral might conveniently be evaluated by either simulation or numericalquadrature techniques. From the �rst four moments an approximation to the distributionfunction FM(m) of the bending moment can be obtained by a Hermite transformation model,Winterstein [159, 160].

Since the Gaussian process is stationary and ergodic, the sequence of consecutive dis-tances between clusters are identically distributed, [24]. As an assumption the point processof maximum slamming impacts may be regarded as a Poisson pulse process with intensityequal to the upcrossing rate of level u of the envelope process, in which the distributionfunction of the pulses is given by FM(m).

From the theory of Poisson pulse processes, [84], the extreme-value distribution of themaximum combined response during the time period T is

F slammaxM(m) = exp[��T ]

1Xn=0

1

n![�TFM(m)]

n

= exp[�(1� FM(m))�T ] (4.138)

where � is the mean rate of \quali�ed" envelope excursions, that is excursions which are notempty, see below. The mean outcrossing rate of level u of the envelope process is given inCramer and Leadbetter [24] p. 254 as

�e(u) =�p2�

up�0

exp

� u2

2�0

!(4.139)

in which � =q(�0�2 � �21)=�20 is the so-called spectral bandwidth parameter.


Envelope excursions above level u are characterized by being empty in the sense thatduring the entire excursion the process itself stays below the level u. The long-term fractionof quali�ed envelope excursions pu above level u is approximated by Ditlevsen and Lindgrenas

pu(u) = 1� 2Z �

0�(�)

8<:1�

p2�

�� 2�

�2��2�

�� 1

2

2��2��2�

9=;d� (4.140)

where

=

s�0�2�21

; � =up�0

(4.141)

The fraction pu of quali�ed envelope excursions acts as a thinning probability on theoutcrossing rate of the envelope process. Therefore, the intensity � of the approximatingPoisson pulse process becomes

� = pu(u)�e(u) (4.142)

By application of Eq. 4.138 and the traditional extreme-value distribution in the right-hand side of Eq. 4.136, a lower bound on the extreme-value distribution of the responseduring the lifetime is obtained.

4.7.8 Slamming response

This section outlines a method for calculation of the slamming-induced response. The calcu-lation of the slamming pressure follows the theory by Ochi and Motter [119]. The dynamicresponse due to the slamming impacts is calculated by solution of the equations of motionfor the ship beam.

Prerequisites

Two prerequisites, both of which can be achieved from the ship lines, have to be preparedin advance.

1. Ship motions in a given sea state.Information on acceleration of the ship at the bow, and motions and velocities relativeto the wave at several forward locations are required. These quantities are obtainablefrom standard ship motion computer programs and are generally given in terms oftransfer functions.


2. Evaluation of the form coe�cient associated with slamming impact.The hull form of interest generally refers to the section shape below the low waterlineand has for convenience been categorized as U -form or V -form, depending on thedegree of bluntness of the section. It has been observed that the pressure magnitudefor the U -form is much larger than for the V -form in the same environment, evenif the ship motions are of the same order of magnitude for the two shapes. Theform coe�cient gives the slamming characteristics for the intermediate sections. Thecalculation procedure is described in Ochi and Motter [118].

Slamming pressure

From these two prerequisites, the slamming pressure at a certain location x can be estab-lished. The following functional relationship between the pressure and the velocity is given:

p = k _r2 =1

2�k1 _r

2 (4.143)

where _r is the relative velocity between wave and ship at the instant of impact, k is thedimensional constant depending upon the section shape, and k1 is the corresponding non-dimensional constant. � is the density of water.

Several examples of time histories of slamming impact pressures observed during full-scaletrials as well as during model experiments have shown that the impact is of a triangular type,and that the time duration varies considerably from one impact to another. For example,the duration of the impact has been observed to be within the range of 0.025 to 0.25 sec,but the time duration for most impacts falls within the range of 0.08 to 0.12 sec (convertedto a 160 m vessel). For other ship lengths, the time duration may be considered as, [119]:

Ts = T160

sLpp160

(4.144)

where T160 is the time duration of the impact for a 160 m vessel and Lpp the ship length.

Slamming impact force on the hull

In order to estimate the main hull girder response to slamming impact, impact pressures act-ing on the forward bottom of the ship must be estimated. Since slamming impact pressuresusually travel forward or aft with changing magnitude, the spatial distribution of pressureon the bottom as a function of time is necessary for prediction of the load.


It is assumed that the impact pressure is of a triangular shape in time. The impactpressure travels along the ship length with changing magnitude with a certain velocity.

In this work, the slamming-induced response due to a regular sinusoidal wave with givenwave amplitude and frequency is calculated. Consequently, both the velocity with whichthe impact pressure travels and the length direction for the slamming impact are uniquelyde�ned in this case. With respect to the determination of the distribution of pressure aroundthe ship hull, see Ochi and Motter [119] p.158.

Hull response to slamming impact

A slamming impact force on the forward bottom of the ship produces a shudder throughoutthe entire hull. This results in a vibratory stress called whipping stress and a sudden deceler-ation which are superimposed to the steady-state wave-induced hull stress and acceleration,respectively.

It has been observed in many full-scale trials that although high-frequency accelerationsand whipping stresses are excited by slamming impact, only the fundamental modes aregenerally appreciable, since the higher-mode vibrations die out quickly because of strongstructural damping characteristics.

Let the slamming impact load on the ship be given as the time-varying load p(x; t). Thenthe di�erential equations of motion for the ship beam are written as

@

@x

"EI(x)

1 + �d

@

@t

!@�

@x

#+ �GA(x)

1 + �d

@

@t

! @w

@x� �

!= ms(x)�

2@2�

@t2(4.145)

@

@x

"�GA(x)

1 + �d

@

@t

! @w

@x� �

!#= ms(x)

@2w

@t2� p(x; t) (4.146)

in which I(x) is the sectional moment of inertia for vertical bending, A(x) sectional sheararea, ms(x) mass per unit length, ms�

2 equatorial mass moment of inertia, and �d thedamping factor. � is a cross-section-dependent constant.

Following the procedure outlined e.g. in Price and Bishop [127], the solution (w(x; t); �(x; t))to Eqs. 4.145 and 4.146 may be expressed in terms of the eigenfunctions as the series

�(x; t) =NXi=0

ui(t)�i(x)

w(x; t) =NXi=0

ui(t)vi(x) (4.147)


where ui represents coe�cients to be determined, and �i and vi are the orthonormalizedeigenfunctions, ordered so that fv0; �0g is the heave mode, fv1; �1g the pitch mode, fv2; �2gthe two-node mode, and so on. If we substitute Eq. 4.147 into Eqs. 4.145 and 4.146, multiplyEq. 4.145 by �j, and Eq. 4.146 by vj, add the equations, and �nally integrate from 0 to Lppusing the orthogonality relations, the following set of equations

�uj + �d2j _uj + 2

juj =Z Lpp

0vjp(x; t)dx ; j = 0; 1; 2; :::: (4.148)

is obtained with j as the j0th eigenfrequency. Because of linearity it can without any loss

of generality be assumed that the normal wave-induced loads are known from the traditionalrigid-body analysis. The equations for j = 2; 3; ::: in Eq. 4.148 thus represent additive terms,in which p(x; t) is the impact slamming de�ned in the previous section. Consequently, thesolution to Eq. 4.148 may be expressed so that the vertical de ection w(x; t) is written as

w(x; t) = w0(t) + x�1(t)

+NXj=2

vj(x)[Aje��j t cos(�jt) +Bje

��jt sin(�jt)]

+NXj=2

vj(x)

j

Z t

0e��j(t��) sin j(t� �)

Z Lpp

0p(x; �)vj(x)dxd� (4.149)

where

�j =1

2�d

2j ; j =

q2j � �2j (4.150)

The �rst in�nite summation in Eq. 4.149 represents the free vibrations. These vibrationsare assumed to be negligible. The second in�nite summation represents the forced vibrationdue to the slamming load p(x; t).

The moment distribution is determined from

M(x; t) = Mrigid body(x; t)

+NXj=2

�uj

Z x

0[msr

2(�)�j(�)� (x� �)msvj(�)]d� (4.151)

in which

uj =1

j

Z t

0e��j(t��) sin j(t� �)

Z Lpp

0p(x; �)vj(x)dxd� (4.152)


Figure 4.21: Slamming-induced bending moment combined with wave-induced bending mo-ment.

is the so-called Duhamel integral, recognized from Eq. 4.149.

Figure 4.21 shows the slamming-induced bending moment and the wave-induced bendingmoment, calculated from Eq. 4.151 by use of the �rst �ve eigenmodes. The moment is due toa regular sinusoidal wave of amplitude 6.75 m and frequency 0.56 rad/sec. For this particularwave the �gure indicates that dynamic transients are rather small.

4.7.9 Numerical example

The analysis outlined above was performed for a container vessel in head sea. The selectedcontainer vessel is that described by Flokstra [41]. The main particulars of the vessel aregiven in Table 4.7. The presented results are conditional on a certain sailing direction �,a certain ship velocity v, and a certain short-term sea state Hs; Tz. To obtain a long-termprediction, it is necessary later to uncondition with respect to these variables.

The signi�cant wave height was Hs = 6:0 m, and the mean zero crossing period Tz = 8:5s. The wave energy was assumed to follow the description of the Pierson-Moskovitz wavespectrum. The velocity of the ship was 6 m/s in head sea. The combined slamming- andwave-induced bending moment was calculated for regular sinusoidal waves with di�erentamplitudes and frequencies. The maximum and minimum bending responses within a wavecycle were found and weighed according to Eq. 4.137. The �rst four moments of the combinedpeak response are shown in Table 4.8. In Table 4.9 the corresponding \marginal" peakmoments of the wave-induced and slamming-induced response are shown. The correlation



Length between perpendiculars (Lpp) 270.000 mBreadth molded 32.200 mDraught even keel 10.850 mBlock Coe�cient 0.598LCG aft of midship 10.120 mLongitudinal radius of gyration 0.248 LSigni�cant wave height (Hs) 6.00 mZero crossing period (Tz) 8.5 sShip speed v 6.0m/s

between the slamming- and wave-induced peaks in hogging was found to be -0.050 and insagging to be -0.032. Note that the correlation coe�cients of the wave-induced and slamminginduced peaks are calculated conditioned on the occurence of slamming. It is also noted thatthe wave induced peak moment is not a Rayleigh distributed variable (skewness 6= 0:631 andkurtosis 6= 3:245). This is a consequence of the non-Gaussian character of the Slepian modelprocess.

The analysis indicates that although the slamming-induced response does not in uencethe mean value of the combined response signi�cantly, it results in a much larger variability.A consequence of this will be larger extreme values.

Even though both the skewness and the kurtosis of the combined and the marginal peakresponses are large, the values are not unrealistic compared to known distribution functions.The cumulative density function of the Generalized Gamma distribution is [152]:

F (x) =

8>><>>:

(a;( x�dc )b)

�(a)for b > 0

�(a;( x�dc )b)

�(a)for b < 0

; x � d; a > 0; b 6= 0; c > 0 (4.153)

where (�; �) is the incomplete gamma function, and �(��) the complementary incompletegamma function. a; b; c; d are distribution parameters. The n'th order statistical momentsof the Generalized Gamma distribution are

E[(X � d)n] = cn��a+ n

b

��(a)

(4.154)

The parameter choise (a; b; c; d) = (1.710, 0.360, 8.195, 131.79) gives (E[X], D[X], skewness,kurtosis) = (197, 278, 10.1, 238), and the parameter choice (a; b; c; d) = (0.933, 0.334, 8.165,0.026) gives (E[X], D[X], skewness, kurtosis) = (43.2, 196, 20.7, 1126). These moments are


Table 4.8: Moments of combined peak response.

Condition E[M] D[M] Skewness Kurtosis[MNm] [MNm]

Hogging 195 312 10.6 226Sagging -202 276 -59.7 79.3

Table 4.9: Moments of marginal peak response.

Type Condition E[M] D[M] Skewness Kurtosis[MNm] [MNm]

Wave Hog/Sag � 169 227 29.5 11.9Slamming Hogging 40.3 218 29.5 1016

Sagging -53.0 156 -26.2 823

satisfactory in a comparison to the moments obtained in hogging for the combined and themarginal peak responses, respectively.

In this analysis the wave-induced bending moments were calculated by a �rst-order striptheory, but they might as well have been obtained by a more re�ned second-order theory,e.g. as in Jensen and Pedersen [65]. One of the advantages by using a second order striptheory would be that the e�ect of springing and bow are slamming could thus be included.

The e�ect of clustering may for the present example be approximately quanti�ed bycomparing the rate of quali�ed envelope excursions � in Eq. 4.142 with the rate of processupcrossings �p above level u, i.e. the sectional draft at the bow section. The upcrossingrates were found to be � = 0:0420 and �p = 0:0452. Hence, the clustering e�ect results in anupcrossing rate which approximately only is 7% less than that of the peak rate alone. Forother ship types or sea states where the occurence of slamming may be more pronouncedthe e�ect of clustering will be di�erent.

Chapter 5

Long-Term Response Statistics

5.1 Introduction

An ocean-going cargo vessel will always operate under at least two di�erent loading condi-tions, namely a laden and a ballast condition. Often some partial loading conditions shouldbe considered as well. In general, however, it is expected that the intermediate loadingconditions are rare, although such loading conditions may be expected on short-distancevoyages.

For tanker structures the still-water bending moment is generally a hogging moment inthe ballast condition and a sagging moment in the laden condition. Furthermore, the wave-induced vertical bending moment is typically 15% to 20% larger for the ballast condition thanfor the laden condition. Consequently, it is an imperative that the long-term distributionis evaluated with due consideration to di�erent loading conditions, and to the time spentin each loading condition. The ballast condition will thus control the extreme hoggingmoment, whereas the laden condition will control the extreme sagging moment. For generalcargo vessels the above �gures may di�er.

Each of the di�erent loading conditions will be described by their probability of occur-rence. The probability { or the relative fraction of time { of the di�erent loading conditionsmay depend on several factors. For tanker structures the probability of being in the ballastor in the laden condition is expected to be in the interval of 40% to 50% of the lifetime,whereas all intermediate loading conditions may be expected in the range from 5% to 20%of the lifetime of the vessel.

There will always be uncertainties in connection with the speci�cations of the loadingconditions, for example

� inaccurate �llings of tanks,

103

104 Chapter 5. Long-Term Response Statistics

� containers with unknown content,

� movable units on board, such as trucks and cranes (live loads),

� re-ballasting during the voyage,

� consumption of bunkers and stores (state between the departure and the arrival con-ditions).

Inaccuracies in the weights and weight distribution will lead to random errors in both thestill-water-induced response and the wave-induced response.

Uncertainties will also be connected with the prediction of the extreme wave-inducedloads on a voyage. These uncertainties will be a result of

� the uncertainty in the speci�cation of the wave climate the vessel experiences on avoyage,

� the e�ect of operational philosophy in severe sea states,

� the uncertainty in the duration of a voyage.

The di�erent loading conditions under which the vessel is assumed to operate may bemodeled in macrotime as distinct, independent events separated by periods of no activity.A useful event type model which guarantees that events in the same process do not overlapin time is the stationary renewal pulse process, [76]. Figure 5.1 illustrates the variation ofthe load process.

In the previous Chapter di�erent procedures for obtaining the short-term distribution ofa speci�ed response quantity were formulated. The short-term distribution could be a resultof a solution to a linear or non-linear combination problem. The short-term distributionwas obtained conditional on loading condition, forward speed, and heading direction toa random sea of speci�ed characteristics. In this Chapter the long-term distribution willbe established by unconditioning with respect to the uncertain parameters considering theoperational philosophy in rough sea states. Moreover, the uncertainty in voyage durationand the uncertainty in each loading condition will be dealt with in the prediction of the long-term distribution. The long-term distribution will be established both for fatigue analysisand for extreme-value analysis.

5.2 Operational philosophy

An important aspect of the long-term prediction of the wave-induced response is the oper-ational philosophy. The operational philosophy is almost completely controlled by human

5.2. Operational philosophy 105

Figure 5.1: Illustration of the load process.

decisions and aims at describing how the captain decides to operate his vessel. The opera-tional philosophy is of major interest in bad weather, that is rough sea states.

Two aspects are considered in the operational philosophy: (1) On the basis of the weatherforecast, the captain may decide to avoid bad weather, either by staying longer in port or byselecting a di�erent route. (2) While the vessel is in rough sea states it may su�er from greenwater on deck, slamming, large roll motions, or large vertical or transverse accelerations. Inorder to avoid such heavy responses the ship speed and/or the heading must occasionally bereduced by the captain's command.

5.2.1 Avoidance of bad weather

Avoidance of bad weather may a�ect signi�cantly the long-term wave-induced response, andthe probabilistic modeling should therefore account for avoidance of bad weather. Avoidanceof bad weather is included in the present formulation by de�nition of a joint probabilitydensity of avoiding di�erent signi�cant wave heights, HS, and zero crossing periods, TZ .One minus this probability then acts as a thinning probability of the occurrence of thedi�erent HS; TZ values in the combined scatter diagram. Avoidance of bad weather maye�ortlessly be included in the establishment of the combined scatter diagram. Here, thescatter diagram is anyhow renormalized before the actual non-linear least square �tting inwith the Weibull distribution is performed.

Table 5.1 presents a relation between Beaufort number and signi�cant wave height HS

proposed by Lloyd's Register for the North Atlantic zone. On the basis of these results


Table 5.1: Relation between Beaufort number and signi�cant wave height HS.

Beaufort number HS

BN [m]1 1.632 1.883 2.264 2.745 3.326 4.007 4.768 5.619 6.5510 7.5711 8.6712 9.85

a bad-weather avoidance philosophy de�ned by the signi�cant wave height can be relatedto the more amenable Beaufort numbers (BN). An example of a bad-weather avoidancephilosophy could be that BN = 10 is avoided 25% of the time, BN = 11 50%, and BN = 1275% of the time. Such a philosophy results in the following avoidance probabilities

favoid(av j hs) =

8>>><>>>:

0 hs < 6:55m0:25 hs = 7:57m0:50 hs = 8:67m0:75 hs = 9:85m

(5.1)

where the probability for intermediate signi�cant wave heights, Hs, is obtained by linearinterpolation.

5.2.2 Maneuvering philosophy

Criteria for voluntary speed reduction are all related to human decision and have no precisephysical level. According to NordForsk [114] the ship speed would voluntary be reduced ifit is judged that

� green water on deck occurs more than 7% of the time,

� slamming occurs more than 3% of the time,

� roll amplitudes larger than 25 degrees occur more than 0.1% of the time,

5.2. Operational philosophy 107

� signi�cant roll amplitude is larger than 7 degrees,

� ship bow acceleration is larger than 0.4g more than 7% of the time,

� ship bow acceleration larger than 0.5g more than 3% of the time,

� signi�cant midship acceleration larger than 0.2g.

Not only a voluntary speed reduction will be observed, but also an involuntary speedloss. Added resistance is an involuntary speed loss, see Faltinsen [39]. The added resistanceis mainly connected with the ability of the vessel to create waves. There will be no addedresistance if the vessel does not cause waves either due to ship motions or due to di�ractionof incident waves around the ship hull. In practice, however, all ships create waves whensailing.

Instead of reducing the speed, the captain could decide to change course. The criteriafor decision-making may be based on shortest time, lowest fuel consumption, or lowest levelof accelerations and discomfort.

The joint probability density function fV�jHs for the combined e�ect of course change(relative to the main wave heading direction) and speed reduction as a function of thesigni�cant wave height is modeled as

fV�jHs(v; �0 j hs) = fV j�Hs(v j �0; hs)f�jHs(�0 j hs) (5.2)

that is a density function f�jHs for course selection as a function of signi�cant wave height,and a conditional density function fV j�Hs de�ning the associated speed.

The long-term response distribution is, as mentioned, sensitive to high sea states, andthe e�ect of maneuvering should therefore be included in the response analysis. However,a complete model which describes the maneuvering philosophy on the basis of the abovestrategy would be very involved. Instead the simpli�ed model proposed by Cramer and FriisHansen [22] is used. This model is described in the following.

Heading angle - f�jHs

Under normal wave conditions the vessel generally travels independently of the wave headingangle �. For larger wave heights the captain tries to reduce the wave-induced response onthe vessel by changing the heading direction. Soares [139] has shown results which describehow the change of course in high sea states is conducted in practice, they indicate a reductionin the occurrence of beam sea.

The proposed procedure for modeling the distribution of ship heading angles relative tothe wave direction in di�erent sea states is to apply a directional distribution function within


Figure 5.2: Heading angle domain as a function of the sea state Hs.

speci�ed feasible domains of the heading angle. The feasible domains are given as a functionof HS, where the feasible domain for the ship heading angle is [0; 2�] in low sea states. Forsevere seas, the feasible interval decreases continuously as a function of the signi�cant waveheight, in the sense that the possible occurence of beam waves is reduced. In extreme seastates, it is assumed that all the waves are encountered as head waves. The possible areas ofwave heading direction as a function of the signi�cant wave height are shown in Figure 5.2,where the density function for possible wave directions for reasons of simplicity is assumedto be uniform within each area. This implies that a possible long-term e�ect of directionalityin the wave heading direction is ignored.

Ship speed - fV j�;Hs

The vessel is assumed to travel at a speci�ed cruising speed VC under normal sea conditions.At a certain signi�cant wave height H1, depending on the wave heading angle, the captaindecreases the speed (or changes the heading direction) in order to reduce the wave response.At another, higher signi�cant wave height H2, it is assumed that the wave-induced responseis so drastic that the captain is forced to reduce the speed to steering speed VS. In theintermediate phase between H1 and H2, a linear reduction of the ship speed with HS isassumed. The signi�cant wave heights H1 and H2 are functions of the wave heading angle.Figure 5.3 shows the above description of modeling the mean ship speed as a function of �and Hs.

5.3. Stress range distribution for fatigue 109

Figure 5.3: Domain of ship speed change as a function of wave angle � and sea state Hs.

5.3 Stress range distribution for fatigue

The long-term peak distribution of the response e�ect in a given loading condition is obtainedby unconditioning the short-term peak distribution in Eq. 4.19:

F��(�� j L = l) =ZHs

ZTz

Z�

ZV

�Hs;Tz;�;VF��(�� j hs; tz; v; �; l)fV�(v; � j hs; tz)�fHsTz(hs; tz) dv d� dtz dhs (5.3)

where L is the loading condition, and �Hs;Tz;V;� is a weighing factor which expresses therelative rate of response peaks within each sea state. fV�(v; � j hs; tz) accounts for the e�ectof maneuvering in heavy weather with respect to sailing speed and relative heading angle,and fHsTz is the two-dimensional description of the sea state experienced by the vessel duringits lifetime { modi�ed as regards the avoidance of bad weather.

It is not possible to perform a closed-form integration of Eq. 5.3. Therefore the value ofthe integral is obtained by Monte Carlo simulation (MCS). The MCS is generally preferableto complex integral evaluations in comparison with other numerical integration techniques, asthere are less strict requirements for the analytical properties of the function to be integrated,and functions of non-structured, \black-box" type can be used. The basic concepts of theMCS method is described in numerous papers and textbooks, e.g. Rubinstein [129], andonly the basic philosophy will be reviewed here.

Consider an integral

p =Zx2Rn

fx(x)

hx(x)hx(x) dx (5.4)


where hx(x) is the non-negative sampling density. By performing N simulations of thevector x with respect to hx(x), p is estimated as the average of the sampling values:

p � 1

N

NXi=1

pi =1

N

NXi=1

fx(x)

hx(x)(5.5)

For evaluation of the integral in Eq. 5.3, hx(x) is conveniently considered as

hx(x) = fV�(v; � j hs; tz)fHsTz(hs; tz) (5.6)

thus approximating Eq. 5.3 as

F��(�� j L = l) � 1

N

NXi=1

�Hs;Tz;�;V;iF��(�� j hsi; tzi; vi; �i; l) (5.7)

As an alternative to the MCS technique, the integral in Eq. 5.3 might have been solved bythe asymptotic theory outlined in Section 2.3.

5.3.1 Long-term stress range distribution

It is assumed that the slowly cyclic variation of the still-water response does not a�ect thelong-term stress range distribution for fatigue. If it is further assumed that uncertainty incargo load distribution in a negligible way a�ects the stress range distribution, then thelong-term distribution is obtained by weighing the stress range distribution in the individualloading conditions by their probability of occurrence

F��(�� j L = l) =NLXi=1

F��(�� j L = li)P (li) (5.8)

Even with the use of MCS technique, Eq. 5.8 is too complex to be directly applied toa structural reliability analysis. Therefore, an equivalent long-term Weibull distribution iscalibrated to the outcome of the MCS result:

Flong ��(��) = 1� exp

"��

A

�B#(5.9)

5.3. Stress range distribution for fatigue 111

The �tting of the Weibull parameters is based on the 0.95 and 0.99 fractile values, �0:95and �0:99. These fractile values divide the contribution to the fatigue damage (E[�m]) intothree areas of equal magnitude (for B = 1 and m = 3):

lnA =k ln�0:99 � ln�0:95

k � 1; B =

ln(� ln 0:05)ln�0:95 � lnA

(5.10)

where

k =ln(� ln 0:05)ln(� ln 0:01)

For other values of B and m, optimal fractile values are given by Cramer [20].

The average rate of stress cycles during the lifetime is found in the simulation procedurefor the evaluation of the long-term response distribution:

�0 =1

N

NXi=1

�Hs;Tz;�;V;L;i (5.11)

where �Hs;Tz;�;V;L;i is the rate of load cycles for the speci�ed short-term condition i. Thenumber of load cycles the vessel is exposed to during the lifetime TL is then:

Npeak = �0rLTL (5.12)

where rL is the fraction of the lifetime the vessel is expected to be at sea.

Figure 5.4 shows the variation of the expected fatigue damage (E [�(��)m]) as a functionof ship speed and heading direction for the vessel with main particulars given in Table 6.1.The �gure is obtained by unconditioning with respect to the sea state parameters. It isseen that the expected fatigue damage in head sea increases almost linearly with ship speed,whereas the opposite tendency is observed for the following sea { although not as pronouncedas for head sea. This is mostly because of the change in the rate of load cycles. From the�gure it may be concluded that for fatigue damage both the heading direction and the shipspeed are of importance in the long-term prediction.


Figure 5.4: Dependency of expected fatigue damage on speed and direction.

5.4 Extreme-value analysis

The total extreme response in an ocean-going vessel consists of both the still-water responseand the wave-induced response. The extreme-value analysis should be carried out for thecombined response under due consideration to the dependency of the wave-induced responseon the still-water response. Moreover, the duration of the voyages must also be carefullyconsidered as the maximum wave-induced response is dependent on the duration of a voyage.In a comparison of the uncertainty associated with the still-water response and the wave-induced response (in a speci�ed loading condition), uncertainty in wave-induced response is,however, more dominant in nature and is therefore of special importance.

Mansour and Sinha [97] established the long-term distribution for the vertical bendingmoment by calibrating a Gumbel distribution (extreme-value type I) to the extreme wave-induced response during the lifetime of the vessel, and they assumed the still-water responseto be normally distributed. The long-term distribution was then obtained by convolution ofthe densities.

Shi [138] proposed to obtain the long-term distribution by a log-likelihood maximiza-tion. Except for the log-likelihood maximization (performed in the real variable space) theprocedure proposed by Shi is somewhat similar to the procedure proposed by L�seth andBjerager [81]. The procedure by Shi depends on to the calculation of the upcrossing rateof the combined response. This upcrossing rate is calculated by an upper-bound procedure

5.4. Extreme-value analysis 113

and must in general be calculated by nested reliability methods. It is judged, however, thatthe outcrossing rate is more easily calculated using the parallel system approach by Hagenand Tvedt [52] (Madsen's formula).

None of the above-mentioned studies addressed either the uncertainty in the voyageduration, or the correlation between the still-water response and the wave-induced response.In this treatise the extreme-value analysis is performed as follows. (1) Conditional on aloading condition, the extreme-value distribution for the combined wave-induced and still-water-induced response quantity of interest is established for a voyage of duration � . (2)The combined wave-induced and still-water-induced response on a voyage is then established(3) The e�ect of uncertainty in the duration of a voyage, and thus the number of voyagesduring the lifetime, is addressed next. (4) Finally, the long-term distribution function forthe response is established by combining the di�erent loading conditions under which thevessel is expected to operate.

5.4.1 Extreme wave-induced response on a voyage

Within each (stationary) short-term sea state the distribution function for the maximumvalue, Fmax(a j hS; tZ), is given by Eq. 4.33 (or the corresponding Hermite distribution fornon-linear responses, Eq. 4.61). The extreme-value distribution on a voyage is identical tothe �rst crossing by the stochastic process of a constant level curve. This is easily seen asthe probability of the maximum value remaining below the level is equal to the probabilityof being below the level initially and not crossing the level within the interval. For any typeof process, it is a mathematically complicated task to describe how a passage is indeed the�rst passage. The complication disappears for Poisson processes in the limit of spikes, thatis the duration of the event is equal to zero.

It is assumed that the sea states are independent, and that they arrive as points in aPoisson process with intensity �Hs;Tz = fHs;Tz(hS; tZ)=tsea, where tsea is the duration ofthe sea state. If the spikes (that is the maximum value within each sea state) are furtherassumed to be independent, then the problem consists of determining the probability thatall the maximum values within the sea states are below the given level a. This probabilityfollows as a special result from the observation that the spikes exceeding level a may beconsidered as events in a new Poisson process with intensity

�sea(a j hS; tZ) = fHs;Tz(hS; tZ)

tsea(1� Fmax(a j hS; tZ)) (5.13)

The proposed modeling of the operational philosophy is conditional on the sea state, andEq. 5.13 can therefore without loss of generality be extended to cover the operational par-ameters as well. The duration of a sea state tsea is a parameter which can be adjusted toapproximately accounting for a correlation between two consecutive sea states.


Figure 5.5: Comparison of calculated long-term distribution and di�erent �tted distributions.

From this it follows that the distribution function for the maximum wave-induced re-sponse within the time interval [0; � ] is

Fext(a) = exp f��sea(a)g (5.14)

in which the upcrossing rate is

�sea(a) =ZHs

ZTz

Z�

ZV

1

tsea(1� Fmax(a j hs; tz; v; �))fV�(v; � j hs; tz)�

fHsTz(hs; tz) dv d� dtz dhs (5.15)

It is noteworthy that given the upcrossing rate �sea by Eq. 5.15, the distribution for themaximum wave-induced response for any voyage duration � is readily available from Eq. 5.14.

Note that the extreme-value distribution formulated above is closely linked to the du-ration of the sea state as a deterministic value. An extension of the above result to anuncertain duration of the sea states does not seem possible.

Figure 5.5 shows a comparison of a calculated long-term distribution for the wave-inducedvertical moment with di�erent known distributions. The distributions have been �tted bymeans of non-linear least squares. It is seen that both the Gumbel distribution and the


Figure 5.6: Comparison of di�erent calculated long-term distributions and �tted Gammadistribution.

Weibull distribution perform rather poorly, and that the Gamma distribution and the gen-eralized Gamma distribution both �t the calculated extreme-value distribution well. TheGamma distribution is a two-parameter distribution (the lower limit is �xed) and is thussomewhat simpler than the three-parameter (initially four) generalized Gamma distribution.In what follows, all extreme-value distributions will be �tted by non-linear least square tothe Gamma distribution with the density function

fX(x) =�

�(�)(�(x� �))��1 exp (��(x� �)) (5.16)

and distribution function

FX(x) = (�; �(x� �))

�(�)(5.17)

where � = 0 is the lower limit, and � and � are the distribution parameters to be �tted. �(�)is the Gamma function, and (�; �) the incomplete Gamma function.

Figure 5.6 shows a comparison of di�erent calculated long-term distributions and the�tted Gamma distribution. It is seen that the Gamma distribution for all loading conditions�ts the calculated data well.


Figure 5.7: Dependency of 98% fractile extreme value on speed and direction.

Figure 5.7 shows the variation of the 98% fractile values in the extreme-value distributionof the vertical bending moment as a function of ship speed and heading direction. The �gureis obtained by unconditioning with respect to the sea state parameters. It is seen that the98% fractile values are almost independent of the ship speed and only in uenced by theheading direction. It may be concluded from the �gure that only the heading direction is ofimportance in the long-term prediction of the vertical bending moment. The maximum shipspeed in the analysis corresponds to the Froude number Fn =

VpgL

= 0:17. The importance

of the ship speed to the 98% fractile values may, however, increase for larger Froude numbers.

5.4.2 Combination of wave-induced and still-water-induced response

on a voyage

For any linear or non-linear combination of wave-induced and still-water-induced response,the response may be established directly within each short-term sea state by the distributionfunction for the maximum value, Fmax(a j hS; tZ), Eq. 4.33. The outcrossing rate shouldthen be calculated by Madsen's formula. The uncertainty in still-water response would beaccounted for by a nested FORM/SORM.

A di�erent procedure may be formulated for a linear combination of the responses. In


this case the total response at time t on a voyage is:

Rtot(t) = Rsw(t) +Rwi(t) (5.18)

in which Rsw(t) is the still-water response, and Rwi(t) the wave-induced response. Let thethe envelope process to the sum process be de�ned as

RV OY (t) = Rsw(t) + max[ti;ti+1]

Rwi(t)

= Rsw(t) +R� wi (5.19)

where R� wi represents the maximum value of the wave-induced response (from the previoussub-section) on a given voyage. Thus the envelope process may be viewed as a rectangularpulse process, with pulse duration �i = ti+1� ti equal to the duration of a voyage. It shouldbe noted that the peak values and the pulse durations are not independent, and that themaximum value of wave-induced response is dependent on the still-water load response.

As mentioned previously, the dependence of the still-water response and the wave-inducedresponse �nds its expression in the uncertain cargo loads. This uncertainty may be includedin the analysis by approximating the maximum response in Eq. 5.19 to its �rst-order Taylorexpansion:

RV OY (t) � Rsw +MXi

dRsw

dmi�mi

+ R� wi +dR� wi

dT

MXi

dT

dmi

�mi +dR� wi

d�

MXi

d�

dmi

�mi (5.20)

where the summation takes place over the M stochastic masses in the particular load con-dition. dT

dmiis change in draught due to load mi, and

d�dmi

is the corresponding change intrim.

The distribution function for the rectangular pulse of duration � in a given loadingcondition is

FR�voy(�) = P [Rsw +R� wi < �] (5.21)

This distribution function for the maximum response on a voyage of a speci�ed duration �may be obtained using FORM/SORM.


5.4.3 Simpli�ed model

It is possible to formulate a simpli�ed model of the combined response by assuming thatthe uncertainty in the cargo load distribution a�ects insigni�cantly the distribution of thewave-induced response. This is not judged to be a crude assumption. Mansour and Sinha[97] examined the e�ect of a correlation between the wave-induced load and the still-waterload. The result of their analysis was that the correlation e�ect on the failure probabilitywas insigni�cant. However, the study by Mansour and Sinha did not account for the e�ectof change in the load level on the wave-induced load, only the correlation in the marginaldistributions was considered.

In this simpli�ed model, it is assumed that the still-water bending moment in the consid-ered loading condition is normally distributed with mean value �sw and standard deviation�sw. The extreme-value distribution of the wave-induced load for each loading condition isobtained as previously described and is �tted to the Gamma distribution:

FMwi(mwi) =

(�; �mwi)

�(�)(5.22)

where mwi is the extreme wave-induced moment on a voyage of duration � . � and � are thedistribution parameters.

The distribution function for the maximum hogging moment on a voyage is given as

FMvoy(mvoy) = P (msw +mwi < mvoy) (5.23)

in which mvoy is the maximum moment on a voyage of duration � , and msw the still-watermoment. The random variablesmsw andmwi are assumed to be independent, and the densityfunction of their sum mvoy = msw + mwi thus equals the convolution of their respectivedensities:

FmaxMvoy(mvoy) =Z 1

0

�

�(�)(�x)��1 exp (��x) �

�mvoy � x� �sw

�sw

�dx (5.24)

�(�) is the standard normal distribution function, �sw and �sw are the mean value and thestandard deviation of the still-water bending moment. FMvoy in Eq. 5.24 is de�ned for themaximum moment (hogging condition). The corresponding distribution for the minimal

moment (sagging condition) is obtained by substituting ��mvoy�x��sw

�sw

�for �

�mvoy+x��sw

�sw

�.

It follows that the extreme-value distribution for the maximum value (hogging) upon Nvessel voyages of equal duration is:

Fmax(M)(m) = fFmaxMvoy(m)gN (5.25)


Figure 5.8: Combined maximum moment (hogging) during the lifetime.

fFmaxMvoy(m)g is the extreme-value distribution for loading condition i. The extreme-valuedistribution of the minimum value (sagging) is obtained similarly:

Fmin(M)(m) = 1� f1� FminMvoy(m)gN (5.26)

Figures 5.8 and 5.9 show the obtained extreme-value distribution for hogging and sagging,respectively.

5.4.4 Uncertainty in the duration of a voyage

The distribution function for the maximum value of a voyage of duration � has been estab-lished. It has been discussed how the obtained distribution function is conditional on theactual load distribution. Moreover, it has been shown how the unconditioned distributionfunction may be established for a voyage of duration � . This means that the distributionfunction for the maximum value is conditional on the duration � .

If independence between the individual voyages is assumed, the probability that themaximum value A during the lifetime of the vessel stays below a given level a is

F (a) = P [A � a] =NLadenYi=1

FLadenext (a j �i)�

NBallastYj=1

FBallastext (a j �j)� � � � (5.27)


Figure 5.9: Combined maximum moment (sagging) during the lifetime.

in which

NLadenXi=1

�i +NBallastXj=1

�j + � � � = TL(pLaden + pBallast + � � �) (5.28)

TL is the considered lifetime of the vessel. In the above equations, both the duration of avoyage �i in a speci�c loading condition and the number of voyages N are uncertain.

When the duration of a voyage � is a deterministic quantity, the distribution functionin Eq. 5.27 follows directly. The duration of a voyage is, however, an uncertain quantity,and under a certain loading condition the voyages are independent, identical distributedvariables with density function fT (�), having mean value E[� ] and standard deviation D[� ].The dependence of � on the extreme-value distribution of a voyage complicates and prohibitsexact calculation of the distribution function in Eq. 5.27.

However, it is possible to establish an asymptotic expression for the extreme-value dis-tribution, if it is assumed that the extreme value on a voyage with su�cient accuracy canbe conditioned on the mean duration of a voyage �i = Tc=n, where Tc is the expected timespent under a loading condition and n the uncertain number of voyages. Eq. 5.27 is thenreduced to

F (a j nLaden; nBallast; � � �) = FLadenext (a j �i)nLaden � FBallast

ext (a j �j)nBallast � � � � (5.29)


in which nc is the uncertain number of voyages under each loading condition. The time spentunder a speci�c loading condition may be considered as the sum

Tc(n) = �1 + �2 + � � �+ �n (5.30)

The problem is to �nd the �rst-passage time distribution function for the random numberof voyages n for which

T (n) � Tc (5.31)

in which Tc = TLpc is the total time spent under the speci�c loading condition.

From the central limit theorem it follows that the process T (n) asymptotically for n!1is a Gaussian process with independent and stationary increments. This means that the T (n)asymptotically for n!1 becomes a Wiener process { although restricted to integers { withdrift E[� ]� (the mean function) and dispersion D[� ]

p� (the standard deviation function) {

here � should be viewed as a continuous parameter of n.

The density function of the �rst-passage \time" � from 0 to Tc { i.e. of a Wiener processwith one absorbing barrier { may be found to be, cf. Cox and Miller [19] equation (5.74):

fN(�) =Tc

D[� ]�p��

Tc � E[� ]�

D[� ]p�

!(5.32)

This density function belongs to the family of so-called inverse Gaussian densities.

The corresponding density function for the discrete variable n may then be approximatedas

fN(n) =Z n

n�1Tc

D[� ]�p��

Tc � E[� ]�

D[� ]p�

!d� (5.33)

The integration is performed \backwards" (that is, from n� 1 to n) as Eq. 5.32 is obtainedby solution of the backward Fokker-Planck equations, cf. Cox and Miller [19]. Figures5.10 and 5.11 show the asymptotic theoretical density function for ten and thirty expectednumbers of voyages compared to simulated results. The simulated results are presented forthe duration of a voyage which is normally, log-normally, and exponentially distributed. Itis seen that theoretical density function agrees well with the simulated results, when thevoyage duration follows a normal distribution (the circles in the �gures). The e�ect of thevoyage distribution is negligible for larger values of the expected number of voyages.


Figure 5.10: Comparison of simulated and theoretical density function for the number ofvoyages. Ten voyages expected.

Figure 5.11: Comparison of simulated and theoretical density function for the number ofvoyages. Thirty voyages expected.


Figure 5.12: E�ect of uncertainty in voyage duration.

Note that an expression for the density function of the �rst-passage time of the integervalued process (the generalized random walk) may in principle be obtained by use of Wald'sidentity, see [19]. Using Wald's identity will, however, not result in a simple expression forthe density function.

From this it follows that the extreme-value distribution for the considered time periodunder each loading condition becomes

F cext(a) =

1Xn=1

F cext(a j �i = Tc=n)

nfN(n) (5.34)

which may be inserted in Eq. 5.29. Figures 5.12 and 5.13 show the extreme-value distri-bution after one and �fteen years. It is seen that the uncertainty in voyage duration hasalmost no in uence on the extreme-value distribution.


Figure 5.13: E�ect of uncertainty in voyage duration.

Chapter 6

Limit State Formulation

6.1 Introduction

A limit state means a condition under which the structure is at its limit to ful�ll a certainrequired demand. A given limit state de�nition divides the de�nition domain of the structureinto two sets, a safe set, and a failure set, in which the demand is satis�ed and not satis�ed,respectively. The boundary between the safe set and the failure set is called the limit state(or failure surface). Traditionally, the limit state function is denoted g(z) and is largerthan zero in the safe set, and less than zero in the failure set. For computational reasonsa di�erentiable g-function is generally chosen whenever possible. The g-function usuallyresults from a mechanical analysis of the structure. Many di�erent formulations are possiblefor the probabilistic model which is used for evaluation of the safety of a given structure.These models can be more or less detailed and thus more or less suited to handle additionalinformation.

Although available programs for reliability calculations allow a direct formulation of theconsidered limit state function great care must be taken to ful�ll the necessary probabilisticrequirements. These requirements relate both to treatment of processes and to system e�ects.Other aspects to be considered are the treatment of time-dependent e�ects such as crackinitiation periods and degrading e�ects due to corrosion.

Marley [99] studied time variant reliability analysis under fatigue degradation. Thebehavior of the time variant model was examined, and on that basis an approximate methodwas formulated. The main conclusion of Marley was that the time variant problem could besolved as a time invariant problem with a resistance strength equal to the strength at theend of the considered reference life. The approximate method is conservative, but was foundto yield satisfactory, accurate results for practical problems.

The �rst section in this Chapter is devoted to the di�erent uncertainty sources. In thefollowing section, aspects of corrosion of ship structures are discussed and a probabilistic

125

126 Chapter 6. Limit State Formulation

model for corrosion is formulated. The model takes into account the corrosion initiationperiod and the ageing e�ect. A model for the ultimate hull bending strength which includessti�ener imperfections and residual stresses is also formulated. Finally, two fatigue crackgrowth models are formulated. These include the Palmgren-Miner fatigue model and atwo-dimensional fracture mechanics model.

6.2 Uncertainty sources

The uncertainty sources may be divided into four groups: physical uncertainty, statisticaluncertainty, model uncertainty, and gross errors, Madsen et al. [84]. In the following thevarious sources of uncertainty are described.

6.2.1 Physical uncertainty

The most obvious type of uncertainty which a�ects the safety of structures, is the uncer-tainty of the material property itself showing up as more or less random uctuations ofthe physical properties from sample to sample. This type of uncertainty is called physicaluncertainty. It may be measured in terms of relative frequencies of observed values of thephysical characteristics in speci�ed intervals or other relevant sets.

6.2.2 Statistical uncertainty

Information on physical uncertainty is obtained through a sample of observations of therelevant physical quantity. This information may be speci�ed without essential uncertaintyonly if the sample size is large. Frequently, only a small sample of observations is available.The uncertainty of the speci�cations of the measures of the physical uncertainty is calledstatistical uncertainty. Just as the physical uncertainty itself, the statistical uncertaintyshould be taken into account in the reliability analysis.

Within the framework of reliability theory the Bayesian statistical model for inferenceseems particularly attractive. Assume that information about a random variable Z, withknown distribution type but unknown distribution parameters, is given through a samplez1; :::; zk of k independent observations of Z. Let the mean value � and the standard deviation� be unknown. According to the Bayesian model, the mean value and standard deviationof Z are modeled as random variables M and �. On the basis of prior information about(M;�), a prior density fM;�(�; �) is assigned. According to Bayes' theorem, the posteriordensity is

f 0M;�(�; �) / fM;�(�; �)kYi=1

fz(zi j �; �) (6.1)

6.2. Uncertainty sources 127

where / means \proportional to". The product is called the likelihood function.

This Bayesian method of dealing with statistical uncertainty requires a choice of a priordistribution of unknown parameters. Even though there are prior distributions which maye�ectively be considered to be non-informative, general professional knowledge may very welljustify the choice of prior distribution that carries some information. Mathematical conve-nience points to the family of natural conjugate distributions associated with the distributionfamily of the input variables. By canonization of this distribution family, professional judge-ments concentrate on the assessment of location and scale parameters. This means that theprior distributions become conceptional vehicles for the transformation of information onthe basis of professional judgement into reliability models.

6.2.3 Model uncertainty

Decisions based on structural reliability analysis depend, naturally, on the mathematicalmodel which is set up for the analysis by the engineer. However, if careful real-life decisionsare made, it is necessary that considerations about the uncertainty of the model itself arequanti�ed within the model. Model uncertainty can only be quanti�ed either by comparisonswith other more involved models which exhibit a closer representation of nature, or bycomparisons with collected data from the �eld or laboratory. These so-called real data are,however, also representatives of model outputs, because there is some model behind anyperformance of data collection and data processing which is never a faultless and { muchless { a complete model of reality. Consistently with this view, uncertainty caused by lessperfect measuring procedures is classi�ed as model uncertainty.

Model selection is guided by a balance between the ability to represent reality and thepragmatic need for such simple mathematical properties of the model that a large variety ofproblems can be analyzed by the model. Thus it is obvious that it is not particularly helpfulto deal with the uncertainty of the simple model by actually calculating the di�erences ofthe results from the simple and the more complicated model. Model uncertainty shouldbe introduced in such a way that the pragmatic level of simplicity is not a�ected seriously.Further, it is convenient if model uncertainty can be represented in a form in which it isinvariant to the mathematical transformations of the equations of the model. This is thecase if it can be directly connected with the set of basic input variables of the model.

Model uncertainty in a structural reliability problem has two sources. First, the numberof basic physical variables has been limited to a �nite number n, leaving out possibly anin�nite set of parameters which in the model idealization process have been judged to beof secondary or negligible importance for the problem at hand. By the realization of thestructure in its environments of lifetime duration, the set of neglected parameters takes onsome set of values. For this set of values, there is a subset of the formulation space, inwhich outcomes of the n explicitly considered variables will not cause failure. This is thesafe set conditional on the outcome of the neglected variables. For another outcome of these


variables, another slightly di�erent safe set will result. Obviously the neglected variables actas generators of a background noise, the mechanism of which is usually only partly known.Thus the safe set may be modeled as a random set in accordance with a suitable probabilitylaw.

Secondly, model uncertainty is caused by the idealization down to operational mathe-matical expressions. Besides this cause of pragmatic simpli�cation, it may be due to lack ofknowledge about the detailed interplay between the considered variables. For a given set ofvalues of the neglected parameters, the lack of knowledge beyond the actual modeling of thelimit state surface invites to consider the \true" failure surface as some perturbation of theidealized limit state surface. If this perturbation is considered to be an unknown elementfrom a set of possible perturbations, an evaluation of the uncertainty may be given as somedeviations from the idealized surface in terms of the entirety of perturbations. In this view,the second source of uncertainty can also be modeled probabilistically, even if the adjoinedprobability measure should not be interpreted in the relative frequency sense.

Irrespective of the source of uncertainty, the above discussion points out that model un-certainty may be modeled as a deformation of the space by which the idealized limit statesurface deforms randomly into \a possible true" limit state surface. Professional judgementsare hardly suited to point out speci�c distribution types for this purpose on the basis ofobjective evidence. Mathematical convenience is e�ectively the sole guidance for the choice.However, it may be possible to give judgemental assessments of location and scale parameterssuch as the mean and standard deviation. The basis for this is general engineering experiencefrom work with relevant idealized models and from comparisons of results with observed dataor other predictions calculated by use of more detailed models. Obviously, such formal modeluncertainty distributions play the same role as the prior distribution in the Bayesian sta-tistical methods. Therefore judgemental random variables are occasionally called Bayesianrandom variables. Der Kiureghian [26, 27] formulated a Bayesian approach to assessingmodel uncertainty and including its e�ect in structural reliability analysis. The work of DerKiureghian was applied by Maes [93] in connection with structural code development.

6.2.4 Gross errors

Uncertainty related to potential possibilities of occurrence of non-imaginable gross errors isof a quite di�erent nature than the three types of uncertainty discussed above. It seemsnot to be obvious how to include this source of uncertainty in structural reliability analy-sis. Fortunately, there are reasonable arguments for keeping gross error analysis separatedfrom the structural reliability analysis that takes account of physical uncertainty, statisticaluncertainty, and model uncertainty.

6.3. Corrosion 129

Figure 6.1: Causes of failure in ship structures { from Akita [2].

6.3 Corrosion

The single most important cause of failure of ship structures is possibly corrosion. From sur-vey reports of ships registered with Nippon Kaiji Kyokai (reported in the period from 1976to 1981), Akita [2, 3] observed that almost 80% of all failures in ships were due to corrosion.Figure 6.1 taken from [2] shows the reported failure causes. Although corrosion is very im-portant it is, however, complicated to deal with in a structured probabilistic framework, ascorrosion wastage is highly in uenced by initial design, building supervision, local environ-mental conditions (PH, temperature, etc.), inspection, and planned maintenance. Moreover,the exposure to corrosion and the type of corrosion vary signi�cantly around the hull.

The common type of corrosion on steel ships is electrolytic. Generally, pitting and groov-ing corrosion are variants of principally the same electrolytic type. The smaller the anodic(corroding) surface is in relation to the cathodic (non-corroding) surface, the more localized,concentrated, and rapid is the corrosion process. In Pollard and Bea [126] average pittingand grooving corrosion rates were found to be in the order of 1.0{2.0 mm per year, comparedto the bulk of average general corrosion rates from 0.05 to 0.3 mm per year. Grooving cor-rosion typically occurs at the connection of the horizontal longitudinal sti�eners to verticalplating when the coating is broken down in a region just above the sti�ener. Corrosionfatigue and stress corrosion cracking are also electrolytic, though combined with mechanicalaction. Dry, high-temperature corrosion may occasionally occur under special conditionssuch as in machinery [36].


For steel submerged in sea water, the accessibility of oxygen to the surface controls thecorrosion rate. This implies that the corrosion rate for di�erent steel grades in submerged,static conditions is approximately the same, independent of minor alloying elements.

High-tensile steels may, more often than common steels, be subjected to high stress levelsor cyclic stresses. This may lead to increased corrosion rates because of the stresses as such(stress corrosion cracking or corrosion fatigue), and also because a protective layer of rustcannot be formed at spots which receive high stress levels or fatigue loads, [36]. The side shellarea between the laden and ballast waterlines is subjected to the very high cyclic loadingthroughout the lifetime of the ship because of the passage of waves along the side, [45].

The areas where a double-hull tanker may be particularly exposed to corrosion includethe cargo tanks, the double-hull vertical wing spaces, and the double bottom. The cargotanks will most likely only carry cargo oil throughout the service life of the vessel, althoughdesignated cargo tanks may be used for heavy weather ballast in emergency situations. Thecorrosion risk within these tanks is therefore normally very low except in the following areas:

� The deckheap vapour area may be subjected to atmospheric corrosion by water vapour.

� The water settling from the cargo oil corrodes the upper surface of horizontal surfacesin the cargo tanks such as bottom plating and horizontal web stringer plating.

� For the coated cargo tanks, the horizontal surface wastage, particularly in the innerbottom, takes the form of local pitting corrosion and of grooving corrosion in wayof coating failure { for the unprotected cargo tanks this wastage is more uniform ingeneral.

The International Association of Classi�cation Societies' (IACS) uni�ed requirements forcoating of the ballast tanks preserve the structural integrity from corrosion wastage, exceptin those areas where the coatings have broken down. The corrosion rate in areas wherecoatings have broken down is likely to accelerate and needs to be dealt with.

Experience has shown that coal tar epoxy coatings with a good standard of applicationhave given successful protection for well over ten years of service life, L�seth et al. [82].However, the coating is susceptible to local breaking down in way of sharp edges and surfacedefects which can lead to accelerated corrosion of the steel structure. The DNV Guidelinesfor Corrosion [36] specify requirements for corrosion protection systems of two durabilitylevels for ballast tanks, oil cargo tanks, and holds in bulk carriers/OBO:

� Speci�cation I: Durability 5 years (� 3 years).

� Speci�cation II: Durability 10 years (� 3 years).

6.3. Corrosion 131

In these speci�cations the life of the coating system is considered to be until 10 to 20%breakthrough of rust on the coated surface.

Experience has shown that the corrosion rate is initially lower than a certain averagecorrosion rate, and that the corrosion rate increases with time. There are two quoted causesfor accelerated corrosion rates:

1. The ageing e�ect is a manifestation of the weakening of the plate because of corrosion.In areas where wastage has reduced the thickness of the plate, the strength of the panelis reduced so that the thinner section de ects more, which causes the corrosion productwhich previously protected the base metal to crack to some extent, and una�ected steelis thus exposed to the corrosive environment. The corrosion rates for such an \old"plate is higher than for a \young" plate, [126].

2. The necking e�ect, whose mechanism is similar to the ageing e�ect, is caused by stresson the original scantling plate so that corrosion is accelerated.

6.3.1 Probabilistic model for corrosion

A general probabilistic model for corrosion should cover the following items:

� Coating protection time (corrosion initiation period).

� Ageing e�ect.

� Location-dependent corrosion rates when coating no longer o�ers successful protection.

� E�ect of pitting/grooving corrosion.

� Replacement of highly corroded members in connection with an annual survey if theloss of thickness exceeds a certain limit (and if the corroded area is detected and it isdecided to repair the corroded area).

In this treatise the following generalization of the often used model for the corrosionwastage W as a function of time is applied:

W (t) = Af(t)B ; t > t0 (6.2)

in which A is the location-dependent corrosion rate, f(t) represents the corrosion time func-tion, and B the ageing parameter. Basically, the corrosion time function is the real timet minus the corrosion initiation time t0. However, modeling the corrosion time function assuch may pose problems to available reliability programs as a spike at zero is thus imposed


Figure 6.2: Illustration of the time function for corrosion.

on the density function of the corrosion wastage, which implies that FORM/SORM is notapplicable. Instead the corrosion time function is modeled as a continuous di�erentiablefunction, see Figure 6.2:

f(t) =fln cosh(a(t� c))� ln cosh(�ac)g=a+ t tanh ac

tanh ac+ 1(6.3)

in which a is a curvature parameter, and c a location parameter. The relation between c, a,and t0 is determined from

limt!1ff(t)� (t� t0)g = f� ln 2=a� c� ln cosh(�ac)=ag

tanh(ac) + 1:0+ t0 = 0 (6.4)

For values of a larger than 1.5, c may with su�cient accuracy be set equal to t0.

The corrosion initiation period t0 is regarded as lognormally distributed with a meanvalue of 5 years and a standard deviation of 3 years. This is in accordance with the DNVGuidelines for Corrosion [36].

Pollard and Bea [126] performed an extensive study of corrosion rates in ship hulls. Morethan 7000 registered corrosion rates were grouped into 106 populations dependent on tanktype, detail type, and location, and average corrosion rates in each population were �tted inwith a Weibull distribution. Application of the results of Pollard's and Bea's analysis to the

6.4. Hull strength 133

Figure 6.3: Corrosion rates in a double-hull tanker. L�seth et al. [82].

present study would, however, make the analysis much to complicated. Instead the averagecorrosion rates reported by L�seth et al. [82] are applied. The corrosion rates around thehull section for a double hull are shown in Figure 6.3. The mean corrosion rates reportedby L�seth et al. are comparable to the values found by Pollard and Bea. However, Pollardand Bea report coe�cients of variation which tend to be almost twice as large comparedto those reported by L�seth et al. The ageing e�ect of the structure is a�ected by the seatemperature in which the vessel operates, the loading and discharging procedures, repair andmaintenance procedures, etc. The ageing parameter B is therefore modeled as a commonparameter for the entire hull. It is assumed that average corrosion rates are estimated onthe basis of measurements of wastage after ten years, and that the corrosion initiation timeis considered to be �ve years. The average corrosion rate after twenty years is approximately1.5 times the average corrosion rate after ten years [82]. This results in a mean value of theparameter B = 1:0, with a subjectively chosen COV of 10%. The corrosion rate A is assumedto be lognormally distributed with a mean value of twice the average mean rate given inFigure 6.3 and a standard deviation equal to the reported average standard deviation.

6.4 Hull strength

The hull section consists of large plate structures sti�ened by a relatively large number of lon-gitudinal sti�eners, see e.g. Figure 4.13. Traditionally, the load capacity of the hull section iscalculated by considering the capacity of the individual sub-elements consisting of a sti�enerand a part of the plating. The ultimate load capacity of these individual sub-elements is


Figure 6.4: Schematic load-end shortening curves for sti�ener in compression.

in uenced by the level of initial imperfections such as distortion and residual stresses { atleast as far as compression is concerned. Figure 6.4 depicts schematically load-end shorten-ing curves for a sti�ener in compression for di�erent levels of initial imperfections. It shouldbe noted that the sti�ener has a signi�cant post-ultimate carrying capacity Consequently, itis necessary to take this post-ultimate load capacity into account, when the ultimate loadcapacity of the hull section is assessed. Disregarding the post-ultimate load capacity of theindividual sti�eners might lead to a too conservative design.

In the probabilistic analysis presented here, it is assumed that the initial imperfectionsof the individual sti�eners (distortions and residual stresses) are independent but identicallydistributed, whereas the material and the corrosion parameters are identical for large areasof the hull section.

Before proceeding, it might be illustrative to examine how the independent initial im-perfections a�ect the ultimate load capacity of the hull section in the case of a perfectlyhorizontal deck. When a vertical bending moment is applied to the hull section, then allidentical sti�ened panel elements in the compression zone initially carry the same load. Thisis a consequence of the identical, initial sti�ness of the sti�ened panel elements. Whenthe applied bending moment is continuously increased, the ultimate load capacity of theweakest sti�ened panel element will be reached at a certain level of the moment. Afterthis panel has reached its ultimate capacity, it will, however, still carry a part of the load,although decreasing in magnitude. Successively, when the applied bending moment is in-creased, the ultimate load capacity of the second- and third-weakest sti�ened panel elementwill be reached. This process will continue until �nally the ultimate load capacity of, say: ksti�ened plate elements, is exceeded, and the ultimate bending capacity of the hull sectionis reached.


It is recognized that the FORM/SORM method transforms all basic stochastic variablesinto a standard normal distributed space and in this space solves an appropriately formulatedoptimization problem. The optimization aims at �nding the point of maximum failurelikelihood. However, special consideration has to be given to the load capacity of the sti�enedplate panels as their load capacity is correlated through identical material parameters andindependent, identically distributed initial imperfections. The presence of these correlatedidentical events unfortunately jeopardizes the optimization performed by FORM/SORM asthey introduce multiple design points. In most cases the optimization routine will be unableto converge because the location of the weakest, second-weakest, third-weakest, etc. sti�enerat a certain stage is but interchanged. Popularly said, the surface on which the optimizationis performed may be visualized as \egg tray"-shaped.

Application of the procedure outlined in Section 2.4 on correlated identical events to thesti�ened panel elements conquers the above problem.

6.4.1 Ultimate bending capacity of hull section

In this treatise, the method given by Hansen [53] is used to estimate the load-end shorteningcurve of a sti�ened plate panel with initial imperfection. This procedure is more completethan the somewhat similar procedure proposed by Chatterjee and Dowling [18] in the sensethat the Hansen method [53] predicts both the ultimate load and the associated strain. TheChatterjee and Dowling method [18] was used by Rutherford and Caldwell [130] to calculatethe ultimate moment capacity of a hull section. Rutherford and Caldwell obtained goodagreement with the registered maximum load capacity.

The approximate method given in [53] is based on the original work of Caldwell [17],but extended in various ways. The method consists of a simple beam-column analogy ofthe actual behavior of the structural system. The method takes the e�ects of (1) lateralpressure, (2) initial de ection of the sti�eners, (3) initial de ection of the plating betweenthe sti�eners, and (4) residual stresses caused by welding into account in the prediction ofthe load-end shortening behavior of the sti�ened plate panel.

The method divides the stress-strain curve of each beam into four regions: (1) A plas-tic tension region in which the material behavior is assumed to be perfectly plastic, (2) aperfectly elastic tension region, (3) an elastic compression region, where the stress-straincurve is derived from a Bernoulli-Euler beam theory in connection with an e�ective-widthapproach, and �nally (4) a plastic compression unloading region, where the stress-straincurve is modeled by assuming that a plastic hinge is formed at mid-span and the ends of thepanel element.

Given the stress-strain relationship for all individual sti�ened plate panels in the hullsection, it is a straightforward task to calculate the resultant bending moment M and theaxial force N as functions of the vertical location of the neutral axis � and the curvature �.


The ultimate bending moment of the hull section is then obtained by solving the optimizationproblem

max M(�; �) (6.5)

s.t. N(�; �) = 0 (6.6)

The optimization problem is solved by use of the MINCF [92] routine. In [53] the methodwas veri�ed against numerical results obtained by the �nite-element method, and the agree-ment was good. Application of this model in the reliability analysis in connection with theprocedure outlined in Section 2.4 is in the following termed the complete model.

6.4.2 Model correction factor method

A direct application of the complete model outlined in the previous section within a reliabilityanalysis is very time-consuming, and not feasible for practical purposes. Themodel correctionfactor method proposed by Arnbjerg-Nielsen [6] and Ditlevsen and Arnbjerg-Nielsen [31]seems to be an attractive alternative.

The idea behind the model correction factor method is to formulate a simpli�ed structuralmodel { e.g. an ideal rigid-plastic yield hinge structural model { and then, in a probabilisticsense, perform a calibration of the simpli�ed model to the complicated, but more realistic,model. A simpli�ed model such as the ideal rigid-plastic yield hinge structural model isvery attractive due to the clear way of functioning. However, such a model may not besu�ciently realistic with respect to the physical conditions at the yield hinge, or with respectto capturing all second-order bending e�ects. The probabilistic calibration procedure assuresthat the simpli�ed model is made \realistic" { at least around the design point.

The method is as follows: Let the random plastic moment Mideal(Z) of the simpli�edmodel everywhere be corrected by a random model correction factor �. � is a function ofthe vector Z of the basic random load and strength variables. The model correction factorfunction �(Z) ensures that the relation

Mreal(Z) = �(Z)Mideal(Z) (6.7)

holds everywhere. Mreal(Z) is the ultimate moment from the advanced model. It is evidentthat the reliability analysis is una�ected by whether the left-hand side or the right-handside of Eq. 6.7 is used in the analysis. However, the evaluation of Mreal is much moretime-consuming than the evaluation of Mideal.

A complete evaluation of the model correction factor �(Z) would be even more compli-cated and time-consuming than solving the original realistic problem. Therefore Ditlevsen


Figure 6.5: Structural outline of the examined vessel { from Paik et al. [122].

and Arnbjerg-Nielsen [31] suggested approximating �(Z) by a zero- or �rst-order Taylorexpansion with expansion point around the design point Z� obtained by means of the idealmodel. Initially, the Taylor expansion may be performed around the mean-value point of Z.

Within a few iterations the series fZ�g of design points, obtained by consecutive relia-bility calculations by use of the ideal model in connection with the model correction factor,will converge. Of course, the accuracy improving iteration procedure is at the expense ofthe necessity of solving Eq. 6.7 an increasing number of times.

The model correction factor method is preferable to traditional response surface tech-niques, as it departures from a simpli�ed analogue model which is able to capture some ofthe e�ects of the realistic model. The simpli�ed model is, to zero- or �rst-order, made equalto the realistic model at a speci�c chosen point, that is the design point of the simpli�edmodel. The evaluation of the realistic model is therefore performed in carefully selected do-mains, contrary to traditional response surfaces whose domain is selected in a more arbitrarymanner. Moreover, the model correction factor approach is preferable in the sense that thesimpli�ed model advances the understanding of the most important features with respectto the load carrying capacity. Such properties decrease the risk of committing gross errorsin the design due to a misconception of the way the structure carries its loads close to thefailure situation.

As an exempli�cation of the model correction factor method, the reliability of the ves-sel shown in Figure 6.5 has been evaluated by application of both the complete model andthe model correction factor method. The examined vessel is a new double-hull tanker withtransverseless system, see Paik et. al [122]. The principal dimensions of the vessel are givenin Table 6.1. The vessel has nine cargo oil tanks. The ship's bottom, sides, and deck are sur-



Length between perpendiculars (Lpp) 234.000 mBreadth molded 42.600 mDepth molded 19.200 m

rounded by a double-hull. The width of the double-sides and the depth of the double-bottomare 2 m, which satis�es the OPA 90 and IMO requirements for oil pollution prevention. Wingand bottom tanks will be used for ballast, while the deck space will be void space. The hullstructure has only longitudinal girders and no transverse members. Horizontal and verticalsti�eners of the at bar type are attached to bottom, side, and deck girders, while inner andouter shell platings have no sti�eners.

The vessel is assumed to operate under the �ve di�erent loading conditions for which theyearly long-term extreme-value distribution of the combined wave and still-water-inducedresponse was calculated in the previous Chapter, see Figures 5.8 and 5.9.

The uncertainty modeling of the parameters which describe the ultimate bending capacityhas the following appearance:

Variable Name DistributionYield stress Log-normal (314.0, Cov=7% )Model uncertainty strength Normal ( 1.0, Cov=7% )Plate imperfections Normal ( t=2, Cov=20% )Sti�ener imperfections Normal ( L=100, Cov=20% )� (in residual stress) Normal (5.25, Cov = 12% )

The residual stress is de�ned by the width �t of the heat-a�ected zone:

�r =2�y�t

b� 2�t(6.8)

in which �r is the residual stress, �y the yield stress, t the thickness of the plate, and b thebreadth of the plating.

Figure 6.6 shows a comparison of the reliability index calculated by means of the ex-act model and the simpli�ed fully plastic model multiplied by the model correction factor.The model correction factor �(Z) was calculated only on the basis of a zero-order Taylorexpansion. Only three to four iterations led to the �nal design point. Although the fullyplastic model only in a mean sense describes the sti�ener uncertainty, it is seen that thereliability indicies of the two models coincide almost perfectly. The reason is that the mostdominant uncertainty in the reliability calculation is due to the loading. The uncertainty of


Figure 6.6: Comparison of reliability index calculated by use of the exact model and themodel correction factor approach to the fully plastic model.

Figure 6.7: Comparison of lumped importance factors obtained by use of the exact modeland the model correction factor approach to the fully plastic model.


the independent, identically distributed sti�eners has a mutually cancelling e�ect, and onlythe mean value of the sti�ener uncertainty is thus appreciable.

Figure 6.7 shows a comparison of the corresponding importance factors obtained by thetwo models. For simplicity the importance factors are lumped into variable groups of (1) aloading group, which covers uncertainty in wave loading and still-water loading, and modeluncertainty, (2) a strength group of yield strength, imperfections, and model uncertainty ofstrength calculation, and (3) a corrosion group, containing corrosion rates, ageing parameter,and corrosion initiation time. It is seen that also the importance factors obtained by the twomethods coincide satisfactorily.

The cpu-time in obtaining the reliability index due to the realistic model was aroundseven days compared to approximately one hour for the simpli�ed model.

6.5 Fatigue models

Flaws are inherent in most metallic structures due to notches, welding defects etc. Fromthese aws, cracks may initiate and propagate under time-varying loading and possibly growto a critical size causing failure. Since fatigue crack growth in most steel structures undertime-varying loading can hardly be prevented in the long run, it is important to be able topredict the propagation of cracks in order to control the reliability.

The fatigue crack growth mechanism will not be reviewed here. Instead reference is madeto e.g. Madsen [84] or Broek [15]. It is only stated that crack growth is the geometricalconsequence of slip and blunting at the crack tip. As these events are controlled by thelocal stresses at the crack tip, these stresses must be described as a function of the appliedstress and the crack geometry. Such a relationship can be established by applying elasticitytheory to the cracked element. In order to describe the problem, the stress �eld at the cracktip is classi�ed as one of three di�erent basic types, each associated with a local mode ofdeformation as shown in Figure 6.8. The three di�erent modes are referred to as

I opening mode or tension modeII in-plane shear modeIII out-of-plane shear mode or tearing mode

Only mode I is considered to be relevant for this analysis.

The conditions governing the fatigue crack growth are the geometry of the structure,the crack initiation site, the material characteristics, the environmental conditions, and theloading. These conditions are of a random nature, and consequently, analysis and designbased on probabilistic methods are most relevant. The design criterion is then generally alimiting value of the probability that the crack depth exceeds the critical value during thedesign lifetime.

6.5. Fatigue models 141

Figure 6.8: The three modes of loading, Broek [15].

Inspections are undertaken to control the possible development of cracks in structuresduring service. An inspection will either result in detection of a crack or not, but whether ornot a crack is detected, the inspection provides additional information, which can be usedto update the reliability for the structure and/or the distributions of the basic variables.

The fatigue crack growth may be described by two basically di�erent approaches. One isthe simple S �N fatigue approach which relates a constant stress range �S to the numberof cycles N , which leads to failure. The Palmgren-Miner model extends this approach tovariable amplitude loading. The second approach is the fracture mechanics approach whichdescribes the crack growth under due consideration of the stress �eld at the crack tip. Bothof the two fatigue models will be described in the following.

6.5.1 Palmgren-Miner fatigue damage model

As stated above the fatigue strength in the S � N fatigue approach is de�ned by �S �N ,which relates a constant stress range �S to the number of stress cycles N leading to failure:

N�Sm = K (6.9)

where K and m are fatigue material parameters, ASCE [7]. The model is often used with apositive lower threshold for �S, below which no damage can occur.

Out in the real world, however, the stress range amplitude is not constant but varies dur-ing the lifetime of the structure. The Palmgren-Miner fatigue damage model [103] accountsfor varying stress range amplitudes by assuming that the damage imposed on the structureper load cycle is constant at a given stress level �Si. By ignoring any sequence e�ects of


the stress cycles, the total damage is expressed as the accumulated damage from each loadcycle at di�erent stress range levels:

�NL=

NLXi=1

1

n(�Si)(6.10)

where �NLis the accumulated damage over the period of time with NL load cycles, and

n(�Si) is the number of load cycles of range �Si. A combination of Eqs. 6.9 and 6.10 leadsto the following expression for the accumulated damage:

�NL=

1

K

NLXi=1

�Smi (6.11)

If the number of load cycles is su�ciently large, Eq. 6.11 can be simpli�ed to the sum of theexpected value of the stress range process:

�NL=

1

K

NLXi=1

�Smi =1

KNL

NLXi=1

1

NL�Smi �

1

KNLE[�S

m] (6.12)

The failure criterion is regarded as the accumulated damage exceeding the critical Palm-gren-Miner damage index �c, which for instance may de�ne through-thickness crack. Nor-mally, this damage index is assumed to be one. The design life of the structure, TD, maythen be de�ned as

TD =�cK

E [�0�Sm](6.13)

in which �0 is the zero crossing period of the load process.

The safety margin M against fatigue failure during the lifetime TL of the structure isthen given by

M = TD � TL = �cK

E [�0�Sm]� TL (6.14)


Figure 6.9: Crack in an arbitrary body, Broek [15].

6.5.2 Fracture mechanics model

The stress �eld solution for mode I can be expressed as:

�ij(r; �) =KIp2�r

fij(�) (6.15)

where �ij represents the stresses at a distance r from the crack tip (Figure 6.9) and at anangle � from the crack plane { fij stands for dimensionless functions of �.

Physically, the stress intensity factor { KI for mode I { can be interpreted as a parameterwhich re ects the redistribution of stresses in a body due to the emergence of a crack. Thestress-intensity factor is normally written in the form

K = Y �p�a (6.16)

where the geometry function Y accounts for the e�ect of all the boundaries, � is a referencestress at the analyzed point, and a represents the crack depth.

Fatigue crack growth model

In order to predict the fatigue crack growth of a surface crack, it is assumed that the crackgrowth per stress cycle at any point along the crack front obeys the Paris and Erdoganequation. This equation states that the increment in crack size dr(�) during a load cycle


Figure 6.10: Semi-elliptical surface crack in a plate under tension or bending fatigue loads.

dN , at a speci�c point along the crack front, see Figure 6.10, is related to the range of thestress intensity factor �Kr(�) for that speci�c load cycle by means of

dr(�)

dN= Cr(�)(�Kr(�))

m ; �Kr(�) > 0 (6.17)

where Cr(�) and m are material parameters for that speci�c point along the crack front.The di�erential equation, Eq. 6.17, must be satis�ed at all points along the crack front.

To simplify the problem it is assumed that the fatigue crack has initially a semi-ellipticalshape, and that the shape remains semi-elliptical as the crack propagates. That is the crackdepth (a) and the crack length (2c) are su�cient parameters for the description of the crackfront, see Figure 6.10.

As a consequence of this assumption, the general di�erential equation, Eq. 6.17, can bereplaced by a pair of coupled di�erential equations, Madsen [86]:

da

dN= CA(�KA)

m ; a(N0) = a0 (6.18)

dc

dN= CC(�KC)

m ; c(N0) = c0 (6.19)

The subscripts A and C refer to the deepest point and an endpoint of the crack at thesurface, respectively. Due to the general triaxial stress �eld, experimental results and stress


intensity factors, as calculated by e.g. Newman and Raju [110], di�er slightly. To accountfor this, the material parameters CA and CC may be regarded as di�erent values. It isproposed by Shang-Xian [137] and Newman and Raju [110] that CA and CC are related bythe formula:

CA = 1:1mCC (6.20)

The material property m depends mainly on the fatigue crack propagation, Shang-Xian[137]. It is therefore reasonable to assumed that it is independent of the crack size, both inthe depth and surface directions.

Normally, the failure criterion refers to a critical value of the crack depth a or the cracklength c, therefore the equations are conveniently rewritten as:

dc

da=

CC

CA

��KC

�KA

�m; c(a0) = c0 (6.21)

dN

da=

1

CA(�KA)m; N(a0) = N0 (6.22)

Alternatively, Eqs. 6.21 and 6.22 may be written in terms of c as the independent par-ameter.

The general expression for the stress-intensity factor is K = Y �p�a, where the ge-

ometry factor Y accounts for the e�ect of all the boundaries, i.e. the relevant dimensions ofthe structure (width, thickness, crack front curvature, etc.). The individual e�ects of theseboundaries can be found in handbooks, and their composite e�ect is obtained by multiplica-tion of all the individual e�ects. As an example of compounding, Broek [16] lets the variousboundary e�ects be due to: back free surface (BFS), front free surface (FFS), width (w),and crack front curvature (CFC). The stress intensity factor is in this case written as:

K = YBFSYFFSYwYCFC�p�a = Y �

p�a (6.23)

Y = YBFSYFFSYwYCFC (6.24)

Stress intensity factor equation for the surface crack in a plane plate

An empirical equation for the stress intensity factor KI(�) for a surface crack in a �niteplate subjected to tension and bending is given by Newman and Raju [110]. The equationhas been �tted from �nite-element results for two di�erent types of loads applied to thesurface-cracked plate: remote uniform tension and remote bending, see Figure 6.10. The


remote uniform tension stress is designated as �t and the remote outer-�ber bending stressas �b. The stress intensity factor equation for combined tension and bending loads is [110]:

KI(�) = (�t +H�b)

s�a

QF (a=t; a=c;

c

b; �) (6.25)

for 0 < a=c � 1:0; 0 � a=t < 1:0; cb< 0:5 and 0 � � � �. � is the angle that de�nes the

position of the point considered, a is the crack depth (semi-minor axis), c is half the cracklength (semi-major axis), t is the thickness of the plate, and b is the half width of the panel,see Figure 6.10. Q is the shape factor, F and H de�ne the boundary-correction factors. TheNewman-Raju formula in Eq. 6.25 has passed several independent, published investigationsas the best available parametric formula.

By de�nition of the stress intensity factor as in Eq. 6.25 it follows that the solution tothe �rst di�erential equation, Eq. 6.21, depends only on the exponent m, the geometry, theratio between bending and membrane stresses and the initial condition a0 and c0, but it isindependent of the loading magnitude.

A useful approximation for the shape factor Q is given by [110]:

Q = 1:0 + 1:464 (a=c)1:65 ; a=c � 1 (6.26)

The boundary-correction factor functions F and H are de�ned so that the boundary-correc-tion factor for tension is equal to F and the boundary-correction factor for bending is equalto the product of H and F . The following expression for F is recommended by Newmanand Raju [110]:

F = [M1 +M2 (a=t)2 +M3 (a=t)

4]f(�)g(�)fw (6.27)

where

M1 = 1:13� 0:09 (a=c) (6.28)

M2 = �0:54 + 0:89

0:2 + (a=c)(6.29)

M3 = 0:5 + 14 (1� a=c)24 � 1

0:65 + a=c(6.30)

g(�) = 1 + [0:1 + 0:35 (a=t)2](1� sin�)2 (6.31)


The function f(�), an angular function from the embedded elliptical crack solution, is:

f(�) = [(a=c)2 cos2 �+ sin2 �]1=4 (6.32)

The function fw, a �nite-width correction, is:

fw = [sec(�c

2b

qa=t)]1=2 ; c=b � 0:5 (6.33)

The function H taking bending into account has the form:

H = H1 + (H2 �H1) sinp � (6.34)

where

p = 0:2 + (a=c) + 0:6 (a=t) (6.35)

H1 = 1� 0:34 (a=t)� 0:11 (a=c) (a=t) (6.36)

H2 = 1 +G1 (a=t) +G2 (a=t)2 (6.37)

G1 = �1:22� 0:12 (a=c) (6.38)

G2 = 0:55� 1:05 (a=c)0:75 + 0:47 (a=c)1:5 (6.39)

Newman and Raju investigated several combinations of parameters for which a=t � 0:8,and in all cases the stress intensity factor obtained by Eq. 6.25 was found to be within �5%of the �nite-element result.

In a later study [111], Newman and Raju determined stress intensity factors for pipesand rods. The di�erence in stress intensity factor between a semi-elliptical crack in a plateand in a pipe with small thickness/diameter ratio was found to be inconsiderable.

In the study by Kirkemo and Madsen [71], a one-dimensional crack description was usedby de�ning a=c = 0:15. In that study �b=�t = 4 and KI(�=2) is

KI(�=2) = (�t + �t)p�aY (a) (6.40)


with Y (a):

a=t 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Y (a) 1.08 1.00 0.94 0.91 0.89 0.88 0.86 0.83 0.80 0.75 0.69

This function was �tted by Kirkemo and Madsen [71] as

Y (a) � 1:08� 0:7a=t (6.41)

which leads to very close �tting for 0 < a=t < 0:2 and therefore { since this interval accountsfor more than 95% of the crack growth time { also for the calculated lifetimes.

In the probabilistic analysis the stress intensity factor is multiplied by a model uncertaintyfactor Y1.

Solution strategy

Since the stress intensity factors KA = KI(�=2) and KC = KI(0) depend on the crack size ina very complicated manner, it is not possible to obtain a closed-form analytical solution tothe coupled di�erential equations, Eqs. 6.21 and 6.22. Numerical integration techniques havetherefore to be applied in the solution procedure [38]. In the present study the integrationroutine LSODA is used. LSODA is based on Adam's predictor corrector method and solves aset of coupled ordinary �rst-order di�erential equations by means of the Livermore strategy.A description of LSODA is given in [58, 124] and the routine has been extensively tested formany di�erent problems.

6.5.3 Safety margin

The failure criterion is that the crack depth exceeds a critical crack depth aC equal to thethickness t during the design lifetime period T with NT stress cycles

aC � aT � 0 (6.42)

This failure criterion can be de�ned by a safety margin M given by

M = NC �NT (6.43)

where NC is the number of cycles to reach a crack size aC . Failure takes place when M � 0and the failure probability PF is

PF = P (M � 0) (6.44)


6.5.4 Threshold on stress intensity factor

The crack growth, Eq. 6.17, was formulated without a threshold value. Two formulationsincluding a threshold are

dr(�)

dN= Cr(�)(�Kr(�))

m ; �Kr(�) > �Kthr (6.45)

dr(�)

dN= Cr(�)[(�Kr(�))

m � (�Kthr)m] ; �Kr(�) > �Kthr (6.46)

The �rst formulation is the most conservative while the second is proposed in the FatigueHandbook [4]. The �rst approach is used in the following.

The stress intensity factor KI(�=2) is

KI(�

2) = (�t + �b)

1 +H�b=�t1 + �b=�t

s�a

QF (a=c; a=t; 0;

�

2)

= �sp�aYa(a=t; a=c) (6.47)

(6.48)

where the stress at the surface is �s = �b+�t. The bending/membrane ratio �b=�t is regardedas constant and may be determined from a FEM analysis. With �Kthr = 0 Eq. 6.18 is

da

dN= Ca(

p�a)mYa(a=t; a=c)

m(��s)m;S > 0 (6.49)

The long-term stress range distribution for ��s = S is the Weibull type with distributionscale parameter A and shape parameter B determined from the global response analysis:

FS(s) = 1� exp(��s

A

�B) ; S > 0 (6.50)

The factor Sm varies from cycle to cycle, and the expected value is used in the di�erentialequation which becomes

da

dN= Ca(

p�a)mYa(a=t; a=c)

mAm�(1 +m

B) (6.51)

where �(�) is the gamma function.


Figure 6.11: E�ect of applying the threshold model to stress intensity factor.

Based on the same procedure Eqs. 6.45 and 6.46 become for � = �2:

da

dN= Ca(

p�a)mYa(a=t; a=c)

mAm�

0@1 + m

B;

�Kthr

AYa(a=t; a=c)p�a

!B1A (6.52)

da

dN= Ca

0@(p�a)mYa(a=t; a=c)mAm�

0@1 + m

B;

�Kthr

AYa(a=t; a=c)p�a

!B1A

� (�Kthr)m exp

0@�

�Kthr

AYa(a=t; a=c)p�a

!B1A1A (6.53)

where �(�; �) is the incomplete gamma function.

Figure 6.11 shows the e�ect of using the two di�erent threshold models given by Eqs. 6.45and 6.46.

6.5.5 Numerical values in analysis

The numerical values in Table 6.2 are suggested on the basis of a review of literature. A verysmall initial a=c ratio may indicate that in the early phase several small cracks are formed


Table 6.2: Suggested distributions for basic variables.Variable Distribution CommentInitial crack depth a0 exponential Bokalrud and Karlsen [11]

E [a0] = 0:11 mmInitial aspect ratio a=c lognormal Distribution type and COV

E [a=c] = 0:3 ; COV=0.50 subjective estimates,results not very sensitive to these

Crack growth parameter m �xed DNV [28]m = 3:1

Crack growth parameter Ca lognormal DNV [28]E [lnCa] = �29:84 ; D [lnCa] = 0:55

Threshold �Kthr �xed Fatigue Handbook [4] p. 167�Kthr = 3:0MPa

pm = 95Nmm�3=2

Model uncertainty SIF Y 1 lognormal subjective assessmentE [Y 1] = 1:0; COV=0.10

Weibull parameters A;B from global response analysis

Figure 6.12: Calibration of S-N curve.


and coalesce into a long, shallow crack. The e�ect of crack coalescence has been investigatedin Friis Hansen et al. [44]. With the parameters in Table 6.2 an equivalent S-N curvemay be determined. A mean-fracture mechanics (FM) curve is determined (� = 0) togetherwith a characteristic curve (� = 2, i.e. mean minus two standard deviations), see Figure6.12. These curves can be compared to the S-N curves DeN T-curve and DNV X-curve. Thecharacteristic DeN T-curve is

lnN = 28:0� 0:75 ln�t

32

�� 3 lnS ; S > 19 MPa (6.54)

whereas the characteristic DNV X-curve is

lnN = 33:55� 4:1 lnS (6.55)

The characteristic FM-curve is closer to the DeN T-curve than the DNV X-curve butgives somewhat shorter lifetimes { approximately a factor of 2 in di�erence. In the FM-studyby Kirkemo and Madsen [71] the characteristic FM and the DeN T-curves coincided almostcompletely when a crack initiation period of 10% of the crack growth time was added to theFM-analysis. Kirkemo and Madsen [71] used

E[lnCa] = �29:75; D[lnCa] = 0:5; m = 3:0;�Kthr = 0 (6.56)

It has been con�rmed in this analysis that with these values for m;Ca and �Kthr theFM-approach reproduces the DeN T-curve almost completely. The relatively small di�erencebetween the material parameters applied here and those applied by Kirkemo and Madsen[71] has a rather signi�cant e�ect { a factor of 2 { on the FM-based S-N curve. The changein material parameter has been made to obtain agreement with DNV-rules, appendix C.Values in this appendix cover a large class of steel types, but may be slightly conservativefor this case.

A comparison between the FM-curve and the DNV X-curve shows a large di�erence { afactor 5 of lifetime { for the relevant stress levels.

Figure 6.13 shows a comparison of reliability indices computed by FM and an S-N ana-lysis applying the DeN T-curve. The factor of 2 in di�erence between times correspondingto a reliability level is again observed.

The DeN T-curve has a reduction in fatigue strength when the thickness increases fortubular joints. The thickness e�ect is also predicted by the FM-analysis as shown in Figure6.14. The DNV X-curve has no such thickness reduction.


Figure 6.13: Reliability - Lifetime.

Figure 6.14: Reliability { Thickness.


Figure 6.15: Calibration against the DNV X-curve.

Figure 6.16: Reliability - Lifetime (DNV X-curve).


6.5.6 Introduction of a crack initiation period

In experiments with fatigue failure at approximately 106 cycles, an initiation period is indeedobserved experimentally at typically 10% of the lifetime. For smaller stress ranges (longerlifetimes) the initiation period is larger, but no experimental results exist.

The initial condition of Eq. 6.18 is de�ned as N(a0) = N0, where N0 is the crack initiationperiod. In the analysis, the crack initiation period is related to the stress range as:

N0 = K1S�m1 (6.57)

In the uncertainty modeling lnK1 is assumed to be normally distributed with mean value�lnK1

and standard deviation �lnK1, and m1 is assumed to be �xed.

The parameters �lnK1, �lnK1

, and m1 are calibrated so that the fracture mechanics curve,with initiation and crack growth, and the DNV X-curve give approximately identical resultsat � = 0 (mean curve) and � = 2 (design curve) for 107 and 109 cycles of variable amplitudeloading.

The calibration has been performed for t = 20 mm, as the experimental results behindthe X-curve relate to joints of thickness less than or equal to this value. The result of thecalibration is shown in Figures 6.15 and 6.16.

Chapter 7

Partial Safety Factors

7.1 Introduction

The rules developed by the classi�cation societies have not been calibrated against a uniformreliability level. On the contrary, several analyses concerning speci�c structures have shownvery large variations in reliability level.

Mansour et al. [98] formulated a pioneering rationale for selecting and calibrating aformat of reliability-based strength standards for use in ship design. Based on a set ofsample calculations for typical values of target reliability, recommendations were made forchecking format and for the associated partial safety factors. However, the partial safetyfactors were related to mean values of still-water and wave-induced bending moments, and adirect application to code values was not possible. Moreover, the partial safety factors weregiven as a linear regression formula of the vessel length only.

Recently, �Ostergaard [121] extended the work towards direct application of code valuesby presenting partial safety factors as { highly { non-linear functions of length and beam ofthe vessel. In the study by �Ostergaard, however, the stochastic models of hull strength andstill-water bending moment were but randomized models of the present code format. Thestochastic model for the wave-induced loads was established by a traditional linear frequencydomain analysis. Obviously, as a consequence of the study by �Ostergaard almost entirelyrelying on present empirical code formulas, it veri�es the dubiousness of using the presentcode format within a partial safety factor format (at least as far as ultimate failure concerns).To arrive at a con�dent set of partial safety factors, the structure must be examined in itsentirety.

The preparation of a design code implies evaluation of characteristic values, design values(partial safety factors ( -values)), combination factors ( -values), and formulation of loadcon�gurations etc. by calibration against the results of a set of representative examples of

157

158 Chapter 7. Partial Safety Factors

probabilistic reliability analyses. This means that a set of probability distribution modelsfor the material parameters and the actions (the loads) must be established with respectto the requirements of model realism. Moreover, the uncertainties as regards the accuracyof the applied mathematical models (geometry, mechanics, actions, strength, etc.) mustbe quanti�ed in terms of probability distributions. When these models are formulated, theprobabilities of di�erent relevant adverse events occurring within a pre-speci�ed time can becalculated by use of existing computer programs, e.g. PROBAN [153].

An example of optimal code calibration was given by Hauge et al. [56] who calibratedtwo simple design rules for a jack-up structure. The paper discusses important aspects to beconsidered in code calibration and formulates both a suitable penalty measure for deviationsfrom target reliability and the associated optimization problem.

In this Chapter the tail sensitivity problem and the indispensable need for a standardizeddistribution model universe as a common reference are discussed. Moreover, the Chapterreviews the calibration procedure of the partial safety factor format. The optimal targetreliability level is in the following section determined from a reliability based cost optimiza-tion. Finally, the Chapter deals with an example of a calibration of a set of partial safetyfactors applicable to a particular double-hull tanker.

7.2 Tail sensitivity problem

For models of real { highly reliable { structures the theoretical failure probability is very small{ of a smaller order of magnitude than 10�3 per year, say. Hence the failure probability isvery sensitive to di�erent choices of probability distributions of the stochastic variables usedin the limit state formulation. In a design situation with a given target failure probabilityof e.g. 10�5, the consequence of this sensitivity is that quite varying structural dimensionsoccur, dependent on the choice of type of probability distribution. The engineer mightideally rely on available empirical data set to guide his choice. However, due to the smallprobabilities such data sets must be very large, in order to distinguish between di�erentdistribution tail hypotheses with reasonable con�dence. Real data sets are usually not muchlarger than to allow for an estimation of mean values, variances, and covariances. Even thisestimation possesses a considerable statistical uncertainty due to modest sample size { anuncertainty which should be included in the structural reliability analysis, of course. Thedata may allow for a con�dent choice between a limited number of distribution types, asregards the central part of the distribution. However, the data do not justify the use of thetails of the selected mathematical distribution in the structural reliability analysis. Onlyoccasionally a kind of mechanistic model is available from which the distribution type maybe predicted. Examples of the most common mechanistic arguments are references to thecentral-limit theorem and to the asymptotic theory of extremes. There are reasons to bevery critical to such arguments, which are often too easily stated without proper justi�cationof their basic premises. Asymptotic theories are laws of large numbers. In particular for the

7.2. Tail sensitivity problem 159

extreme-value asymptotic theory, these large numbers must be extremely large in order toensure that the asymptotic distribution is well approximated in the tail region.

These di�erences are all summarized in the phrase \tail sensitivity problem". This prob-lem causes the computed failure probability to be of limited informative value except forreliability comparisons made in the same model universe of probability distributions. There-fore, for the advancement of the use of modern reliability analysis to support structuralengineering decisions in practice, there is an indispensable need for agreement among (com-peting) engineers and the general public on using a standardized distribution model universeas a common reference. In other words, there is an indispensable need for a code of practicefor structural reliability analysis.

An attempt to formulate code-like guidelines has recently been published by the JointCommittee on Structural Safety (JCSS) in the form of a working document with the title:\Proposal for a Code for the Direct Use of Reliability Methods in Structural Design"Ditlevsen and Madsen [34]. The Nordic Committee for Building Structures (NKB) [115]earlier presented a model code for design by use of the partial safety factor method withpartial safety factors being calibrated on the basis of a reliability analysis. This work hasbecome the basis for the partial safety factors in the Danish system of civil engineering codes.

An important point in the code proposal for the direct use of reliability methods is thatthe distribution types to be used in the reliability analysis are standardized. The way inwhich these standardizations are introduced is best illustrated by direct quotation from theproposal. In section 7 on action modeling the code type text is as follows:

Standardized distribution and process types to be used in action models for speci�c reli-ability investigations can be given in an action code to be used in parallel with this code onreliability methods. In such cases the action load model standardization given in this codeare secondary to the standardizations of the action code.

and similarly in section 8 on structural resistance modeling:

Standardized distributions of material properties to be used in structural resistance mod-els can be given in material oriented codes to be used in parallel with this code on reliabilitymethods. Standardized distributions given in such material codes are superior to the stan-dardizations given in this code. It is required that a standardized distribution of a materialproperty assigns zero probability to any set in which no value is possible due to the physicalde�nition of the considered material property.

In section 9 on reliability models the code type text is as follows:

If no speci�c distribution type is given as standard in the action and material codes thiscode for the purpose of reliability evaluations standardizes the clipped (or, alternatively, thezero-truncated) normal distribution type for basic load pulse amplitudes. Furthermore, thelogarithmic normal distribution type is standardized for the basic strength variables.


Deviations from speci�c geometrical measures of physical dimensions as length are stan-dardized to have normal distributions if they act at the adverse state in the same way as loadvariables (increase of value implies decrease of reliability) and to have logarithmic normaldistribution if they contribute to the adverse state in the same way as resistance variables(decrease of value implies decrease of reliability).

and further

In special situations other than the code standardized distribution types can be relevant forthe reliability evaluation. Such code deviating assumptions must be well documented on thebasis of a plausible model that by its elements generates the claimed probability distributiontype. Asymptotic distributions generated from the model are allowed to be applied only if itcan be shown that they by application on a suitable representative example structure lead toapproximately the same generalized reliability indices as obtained by application of the exactdistribution generated by the model.

Experimental veri�cation without any other type of veri�cation of a distributional as-sumption that deviates strongly from the standard is only su�cient if very large representativesamples of data are available.

Distributional assumptions that deviate from those of the code must in any case be testedon a suitable representative example structure. By calibration against results obtained on thebasis of the standardizations of the code it must be guaranteed that the real (the absolute)safety level is not changed signi�cantly relative to the requirements of the code.

When arguing within a speci�c anticipatory model universe it is important to ensurethat near-zero probability value results are used for comparisons only within the model itself.Transferring the results to the outside world and associating them with the usual probabilityinterpretation of relative frequency of occurrence in the real world of the considered event,will generally be highly misleading even though the model has been carefully calibratedagainst real-world data. This insight is not new. It has, however, now become urgentto focus attention on this subject because of the recent maturing of practicable reliabilityanalysis methods.

7.3 Calibration of partial safety factors

The two main objectives of any code development and calibration of partial safety factorsare the determination of a code which is simple to use and yet achieves a uniform safety levelfor any design which is based on the code. Unfortunately, simplicity and safety are mostlytwo con icting objectives.

In the calibration procedure a class of structures must be de�ned for which the code isto be adopted, and also a class of relevant adverse events for all structures encompassed by

7.3. Calibration of partial safety factors 161

the class. Further, a (trial) code format must be selected for each considered adverse event.A key in the calibration procedure is that any given structure or structural element underspeci�ed stochastic action conditions can be designed to have a pre-speci�ed probability ofa considered adverse event. The pre-speci�ed probability is obtained by variation of themean material strength properties or the geometrical dimensions. When the design result isknown the relevant partial safety factors in the selected code format can be chosen from aset of values so that exactly the same design is obtained by design-to-limit according to thecode format using these partial safety factor values. Unfortunately, each considered examplestructure (within a class of structures designed to a common adverse event probability)will in this way generally de�ne a set of partial safety factor values deviating from the setsobtained for other example structures of the class. Thus the calibration of the partial safetyfactors in a code format must have the reverse direction of this procedure.

First of all several of the partial safety factor values of the code format for the consideredclass of structures might be �xed in advance. This could conveniently be the case for allthe partial safety factor values related to the material property. The partial safety factorvalues to be applied to the characteristic action values are then free to be calibrated againstcommon values for the considered structural class.

The calibration is in principle made as follows:

1. For speci�ed values of the partial safety factors, a representative set of example struc-tures is �rst designed according to the code format.

2. All the structures within the class are then subjected to a probabilistic analysis, inwhich the probability of the adverse event is calculated.

3. A suitable penalty measure of the deviations from the desired common value of theprobability (the target reliability) is then calculated.

4. An optimization procedure that aims at minimizing the penalty measure as a functionof the partial safety factor values is set up.

The load combination factors ( -values) are obtained by the same kind of calibration proce-dure. The details of the calibration procedure of the partial safety factors and combinationfactors are described in Ditlevsen and Madsen [33] and in Madsen et al. [84].

For some load types it may be more convenient to calibrate the characteristic value ofthe load and keep the partial safety factor at a value which is common to the value for otherload types. Typically in civil engineering, the characteristic values are de�ned as the 98%fractile value of the yearly extreme. Keeping this de�nition of the characteristic value may insome cases imply an extraordinarily large value of the partial safety factor. Alternatively, alarger characteristic value could be speci�ed without reference to a pre-speci�ed �xed fractilevalue.


It should be noted that traditional Hessian-based (Gauss-Newton) optimization algo-rithms are not well suited for optimizing the sum function of penalty function measures.This is because the Hessian matrix of the system becomes almost singular, and if conver-gence is obtained at all, it will be extremely slow. Instead it is suggested using specialnorm optimization algorithms, like e.g. the modi�ed version of the Marguardt-Levenbergalgorithm [42].

7.4 Assessment of target reliability

It is not obvious for what target reliability level a given adverse event should be designed. Itcould be argued, however, that the target reliability level should be selected in accordancewith present code formulations. Unfortunately, as previously mentioned, the present codeformats have not been the subject of a calibration against a uniform reliability level, and atarget reliability level can consequently not be uniquely selected from the code formats. Inthis treatise it is suggested performing a reliability-based cost optimization and thus arrivingat reliability levels which are optimal in some well-de�ned sense.

Methods for structural optimization and methods for structural reliability analysis havein recent years been developed considerably and also been implemented to some extent indaily design work. Less work has been carried out to combine the two subjects and thus arriveat procedures for reliability-based optimal design. It is therefore of considerable interest tocombine these subjects and to demonstrate the practical applicability and advantages foran actual design. The main application of such a methodology is expected to be to unique,expensive structures and to structures which are to be produced in large numbers.

Procedures for probability-based, cost-optimal inspection planning have been presentedfor fatigue-sensitive structures, cf. Madsen [85], Madsen et al. [90] Madsen et al. [91], andThoft-Christensen and S�rensen [148]. These procedures establish excellent approaches tosolution of the probabilistic optimization problem of a single fatigue-sensitive detail. Theearlier works on reliability-based optimization were extended by Cramer and Friis Hansen[21] to the analysis of multi-component welded structures.

This section introduces di�erent formulations of the optimization problem, and formu-lates a total cost expression for a midship section. The proposed formulation includes theinitial cost of steel and design, and cost of failure in hogging and in sagging. Moreover,the formulation includes cost of fatigue cracking repair and cost of steel replacement due tocorrosion. The result of the reliability-based optimization is presented in terms of reliabil-ity index requirements of ultimate failure in hogging and sagging for two di�erent coatingprotection systems.

7.4. Assessment of target reliability 163

7.4.1 Optimization problem

Traditional structural optimization based on a deterministic analysis is typically formulatedas

min: C(p; q) (7.1)

s:t: simple constraints on design parameters

s:t: code requirements

The optimization aims at minimizing the objective function C(�), which is a function ofthe cost parameters p and the design parameters (optimization variables) q. The objectivefunction is often the cost of design and construction, while in recent years attempts havebeen made to also include the cost of maintenance as well. The optimization must obeyconstraints. Simple constraints on the design parameters and constraints imposed by thecode requirements are typically formulated. The code requirements can refer to both ulti-mate and serviceability limit states. This formulation does not explicitly include reliabilityrequirements, but it is implicitly assumed that the reliability level obtained by use of thecodes is optimal (in terms of the general public acceptance of experienced failure rates ver-sus cost of the structure). It is sometimes argued that otherwise the codes would have beenchanged.

In connection with a code for the direct use of probabilistic methods (Ditlevsen andMadsen [34]) an optimization problem is often formulated as

min: C(p; q) (7.2)


s:t: PF � PmaxF

The constraints related to code requirements in Eq. 7.1 are then replaced by a constrainton the failure probability, PF . The code de�nes this limit value for the failure probabilityand speci�es the basis for computing PF by speci�cation of distribution types, inclusion ofmodel uncertainty, inclusion of statistical uncertainty, load and load combination modeling,resistance modeling and system reliability modeling. Optimization formulations of the typein Eq. 7.2 have been the subject of considerable research.

In this treatise the optimization is formulated as

min: CT (p; q) (7.3)



The objective function CT is the total expected cost including the expected cost of failure.It is expressed as

CT (p; q) = CI(p; q) + CF (p; q)PF (q) (7.4)

CI(�) is the cost of design, construction, and expected cost of maintenance, while CF (�) isthe cost of failure. The cost of failure should include both tangible and intangible cost (lossof life, environmental damages, bad publicity, etc.). Hence, the reliability level obtained bythe optimization in Eq. 7.4 is optimal in a well-de�ned sense.

It is recognized that the optimization aims at minimizing the expected total cost. It istherefore only the expected values of the cost items which enter the analysis. These costsare in practice only assessed with uncertainty and it is recommended to perform a sensitivityanalysis to determine the change in the solution of the optimization to a change in the costinput parameters.

The failure probability is computed by a reliability method. The failure probability isoften expressed in terms of the reliability index � de�ned as

� = ��1(PF ) (7.5)

Here the reliability index and failure probability are computed by �rst-order reliabilitymethods. The limit state function is denoted by g(�) and is a function of the determin-istic design parameters q and uncertain parameters u (transformed into a standard normalspace). The uncertain parameters describe the loading, resistance and geometry of thestructure as well as model and statistical uncertainty. The �rst-order reliability methoddetermines the reliability index by solution of an optimization problem, Madsen et al. [84]:

min: j u j (7.6)

s:t: g(u; q) = 0

The solution is

� =j u� j (7.7)

where u� is the solution point { the design point.

It is recognized that the optimization in Eq. 7.3, when applied in connection with a�rst-order reliability method, has an objective function with a value determined by solvinga second optimization.

The optimization can be formulated as a nested optimization, and it has been imple-mented into a development version of the reliability analysis program PROBAN [153] at DetNorske Veritas Research. The advantages of formulating the analysis as a nested optimiza-tion are:


� An existing reliability analysis program can be used.

� The gradient of the objective function with respect to the design parameters is easy todetermine. The only di�culty in this gradient calculation is to determine the gradientof the failure probability with respect to the design parameters q, but this gradient isa by-product of the �rst-order reliability analysis.

� The gradient of the constraints involves the �rst, but not the second derivatives of thelimit state function.

The disadvantages of the nested optimization are:

� Convergence problems may occur due to a di�culty in performing proper scaling forthe outer optimization.

� The computational time may be unnecessarily long as the reliability analysis calculationmust be performed in each step of the outer optimization. Intuitively, it does not seemnecessary for the �rst steps in the outer optimization to carry the inner optimizationto an end.

� No standard optimization program can be used for the complete optimization.

Instead of using the nested approach the two optimizations can be combined into one,Madsen and Friis Hansen [88]. The formulation becomes

min: CT (p; q) = CI(p; q) + CF (p; q)�(� j u j) (7.8)


s:t: g(u; q) = 0

s:t:u

j u j = �rug(u; q)

jrug(u; q) j

where rug(�) is the gradient vector of the g-function with respect to the u-variables. Thelast constraint { the Kuhn-Tucker condition for the reliability index optimization { can berewritten as

u0rug(u; q)+ j u jjrug(u; q) j= 0 (7.9)

where u0 is the transpose of u.The advantages of the combined formulation are:

� A standard optimization program can be used.


� Scaling problems can be handled as in standard optimization programs.

� Sensitivity factors are almost a by-product of the optimization analysis.

The disadvantages of the formulation are:

� Optimization routines which utilize the gradient of the constraints require the calcu-lation of the second derivative of the limit state function.

� The transformation from the physical variables to the standard normal u-variablesmust be included explicitly. This can be done by using modules from a reliabilityanalysis program but it requires some additional work.

� The size of the optimization problem and the number of constraints increases, whichmay lead to convergence problems.

For two complicated structural analysis problems, Madsen and Friis Hansen [88] com-pared the e�ciency of the two approaches by comparing the number of calls to the limitstate function, since this calculation is the main time consuming operation. The conclusionof the study was that the combined approach required up to 50% more computational workfor the considered examples. Thus, through further improvements the combined approachhas the potential of becoming an attractive alternative to the nested analysis. Optimizationalgorithms proposed by Han and Powell [92] and by Schittkowski [135] were applied in thecombined analysis.

7.4.2 Cost modeling

The objective is to determine the midship section design leading to the minimum totalexpected cost and thus arrive at reliability indices for ultimate failure which are optimal ina well-de�ned sense. Costs to be considered in the analysis are those related to:

� steel and e�ect of steel weight on fuel consumption,

� initial analysis,

� steel replacement as a consequence of corrosion,

� repair of fatigue cracks,

� time out of service,

� ultimate failure.


A factor not speci�cally considered in the assessment of the failure cost is the impact ofchange in fatigue cracking on pollution risk. It can be argued, however, that signi�cantcases of pollution are related to accidents and not to the number of fatigue cracks.

The total expected cost CT during the period of time t is regarded as

CT = CI(q) + CF (PFhog(q; t) + PFsag(q; t)) +

CRP [rep(q; t)] + CreplE [Corr: Area(q; t)] + CfatE [# cracks(q; t)] (7.10)

in which CI is the initial cost of design (including fuel consumption, loss of payload, etc.),CF the failure cost, CR the initial cost of having a repair, Crepl the cost of steel replacement,and Cfat the cost of fatigue crack repair. The design parameters q are the thickness of thedeck plating, the side shell plating, and the bottom plating. The design parameters q aremodeled as multiplication factors of the thicknesses of the conceptual design.

The total failure probability is approximately regarded as the sum of failure in hogging(PFhog) and failure in sagging (PFsag). This is not a crude assumption { although thetwo failure modes are correlated { since hogging and sagging failures represent two distinctloading conditions, and since the loading accounts for approximately 90% of the uncertainty.The expected corroded area (E [Corr: Area]) to be replaced in a major overhaul in deck, side,and bottom of the vessel is regarded as the probability of corrosion wastage in each zone beinglarger than 25% of the plate thickness times the considered area. Similarly, the expectednumber of fatigue cracks (E [# cracks]) in the sti�eners is regarded as the probability of athrough-thickness crack times the number of sti�eners in each of the considered zones. Themodel thus assumes one representative sti�ener to be identi�ed in each of the consideredzones.

The proposed model for estimating the total expected cost clearly leaves room for im-provement since all system e�ects with respect to both corrosion and fatigue cracking havebeen neglected. Although crudely capturing major involved cost items, the model aims pri-marily at verifying the feasibility of a reliability-based design. Note that the model assumesno (or little) maintenance in the considered period of time!

Cost data related to initial design and repair are taken from L�seth et al. [82], whoidenti�ed the cost data in cooperation with two yards, a newbuilding and a repair yard, andtwo shipowners. All the cost data are given in Table 7.1. It is assumed that the initial costof steel and initial design in Table 7.1 also include the loss of payload and the e�ect of steelweight on fuel consumption.

The reliability-based cost optimization has been performed for the vessel shown in Figure6.5 and with main particulars given in Table 6.1. The midship section data for the conceptualdesign are given in Table 7.2.

The vessel is assumed to have one major overhaul after a ten years' period of service.Therefore, the cost optimization is performed for a time period t of 10 years. The failure


Table 7.1: Cost data.

Cost of steel and initial design 2000 US$/tonSteel replacement 5000 US$/tonCrack repair 2500 US$/crackInitial cost of repair 30000 US$/dayDocking cost 7500 US$/dayYard repair capacity 30 ton/day

Table 7.2: Midship section data { conceptual design.

Zone Deck Side BottomWidth of zone 17.0 m 11.64 m 17.0 mInitial thickness of outer plating 10.0 mm 15.0 mm 14.5 mmInitial thickness of inner plating 12.0 mm 15.0 mm 16.0 mmCorrosion rate LN(0.1,0.03) LN(0.3,0.04) LN(0.2,0.04)Weibull parameter lnA for fatigue 2.425 3.076 2.000Number of sti�eners 20 11 20

probability in sagging and hogging at the time t is calculated by use of the model correctionfactor approach of Chapter 6 in connection with resistance strength equal to the strength atthe considered reference time, and the ten years' extreme-value distribution of the combinedwave-induced and still-water loading. The time-dependent reliability problem is thus {according to Marley [99] { solved as a corresponding time-invariant problem.

The fatigue crack growth parameters, except the Weibull parameters, are given in Table6.2. As concerns the side shell zone, a sti�ener around the laden waterline has conservativelybeen selected as representative of the fatigue damage assessment. Note that in Table 7.2the Weibull parameter lnA (stress range) due to the e�ect of the outside water pressure issigni�cantly higher for the side shell zone than the parameters for the deck and bottom zones.The uncertainty in the Weibull parameters lnA and 1=B has been calculated by means ofthe bootstrapping technique outlined in Appendix C, and the results are D [lnA] = 0:240,E [1=B] = 1:113, D [1=B] = 0:15, and �(lnA; 1=B) = �0:79, which holds for all zones.

The optimization is performed for two estimated failure costs (US$ 200,000 and US$1,000,000). The failure cost is estimated for a section, that is the total failure cost dividedwith the length of the vessel. Moreover, to illustrate the e�ect of coating quality, the op-timization has been carried out for two coating durability systems with a mean corrosioninitiation period of 5 and 10 years, respectively.

The optimization problem was formulated both as a nested optimization in the develop-ment version of PROBAN and as the combined optimization. However, the combined for-


Table 7.3: Result of optimization.

Corrosion protection: 5 yearsCF Factor tdeck Factor tside Factor tbott CT �H �S Weight [ton/m]

200,000 1.306 1.670 1.192 30563 2.45 2.80 12.671,000,000 1.624 1.979 1.402 34719 2.97 3.21 15.19

Corrosion protection: 10 yearsCF Factor tdeck Factor tside Factor tbott CT �H �S Weight [ton/m]

200000 1.003 1.372 0.809 22477 2.40 2.76 9.561000000 1.404 1.601 0.971 27304 2.94 3.20 11.90

Table 7.4: Optimal annual reliability index.

Corrosion protection: 5 yearsCF �H �S

200,000 3.741 3.9331,000,000 4.120 4.251

Corrosion protection: 10 yearsCF �H �S

200000 3.246 3.5241000000 3.692 3.892

mulation did not succeed in convergence. The result of the optimization is shown in Table7.3. The reliability index for hogging and sagging is given in terms of the ten years' (timevariant) reliability index. Table 7.4 gives the corresponding annual (�rst-year) reliability in-dex. For plate panels with T-pro�les designed in accordance with current ship design rules,L�seth et al. [83] obtained annual reliability indices in the range of 3.53 to 4.72. Thus, theannual (�rst-year) reliability indices obtained by the proposed cost optimization procedureagree with those obtained from current design practice.

Note that in Table 7.3 an improvement of the corrosion protection system from a meanprotection time of 5 years to 10 years has a notable impact on the optimal plate thicknesses.Note also that the noteable increase in the thickness of the side shell primarily is a resultof fatigue sensitivity. For real structures, however, the fatigue sensitivity would rather beimproved by redesign than just by increasing the plate thickness. Note �nally that dueto the impact of the failure cost on the total cost, the ten years' (time-variant) reliabilityindex is almost insigni�cantly dependent on the coating protection system { compared withrespect to the same failure cost, of course. The signi�cant impact of improving the coatingprotection system on the annual (�rst-year) reliability index is obvious in Table 7.4.


In comparison, the total expected cost of the conceptual design for the two protectionsystems were US$ 42,210 (�ve years' protection) and US$ 24,858 (ten years' protection) forthe US$ 200,000 failure cost assessment. The corresponding ten years' reliability indices were�H = 1:69 and �S = 2:18 for the �ve years' protection system and �H = 2:36 and �S = 2:72for the ten years' protection system.

The sensitivity of the result to the cost of failure is observed to be rather moderate interms of the weight of the section. The sensitivity is approximately 3.1 ton/mill US$ and2.9 ton/mill US$ for coating protection times of 5 and 10 years, respectively. Such moderatesensitivity is desirable as the failure cost assessment can easily be disputed.

From the above analysis it is suggested that the following annual (�rst-year) target reli-ability index requirements can be adopted:

Failure mode Moderate Goodprotection protection

Ultimate hogging bending failure � =3.70 � =3.25Ultimate sagging bending failure � =3.95 � =3.50

It should be noted, however, that the above reliability index requirements are based ononly one analyzed vessel, and care should be taken in generalizing the requirements directlyto other vessel types.

7.5 Safety factors for ultimate failure

As an illustration of the calibration procedure, sets of partial safety factors have been ob-tained. The sets of partial safety factors are calibrated to meet the optimal vessel designobtained in the previous section by design-to-limit according to di�erent proposed designmodels. Using a particular design model with the associated calibrated set of partial safetyfactors results in a design which satis�es the pre-speci�ed reliability requirements.

Four di�erent design models have been considered in the calibration procedure of thepartial safety factors:

Model 1 :WHS�d m

� (E [Mstill] + HSMwave;98%) (7.11)

Model 2 : MHS;ult(�d= m)� (E [Mstill] + HSMwave;98%) (7.12)

Model 3 :WHS�d m

� HS(Mstill +Mwave)98% (7.13)

Model 4 : MHS;ult(�d= m)� HS(Mstill +Mwave)98% (7.14)

7.5. Safety factors for ultimate failure 171

Table 7.5: Partial safety factor HS.

Moderate protection Good protectionModel Hogging Sagging Hogging Sagging1 2.52 2.41 2.04 2.062 1.73 1.95 1.39 1.673 1.54 1.69 1.32 1.474 1.17 1.39 1.02 1.21

in which index HS indicates hogging or sagging and index 98% the 98% fractile value.WHS is the moment of resistance calculated either for the bottom (hogging) or for the deck(sagging). MHS;ult is the ultimate bending moment for hogging or sagging calculated underdue consideration to all buckling and plasticity e�ects. m is the partial safety factor of thecharacteristic material strength �d, and HS the partial safety factor of the characteristicaction value.

In design models 1 and 3 it is assumed that the ultimate failure capacity of bending canbe represented by the elastic yield limit in the compression zone. Hence, buckling e�ectsin the compression zone are re ected in the partial safety factor of the characteristic actionvalue. On the contrary, design models 2 and 4 require an exact calculation of the ultimatebending capacity of the hull section.

A comparison of design models 1 and 2 with design models 3 and 4 re ects two di�erentapproaches of the action modeling. In models 1 and 2 all uncertainty in the still-water loadis lumped into the partial safety factor of the characteristic value (the 98% fractile) of theyearly extreme wave-induced loading. Design models 3 and 4, on the other hand, de�ne thecharacteristic value of the action as the 98% fractile value of the yearly extreme combinedstill-water and wave-induced response.

The characteristic value of the yield stress is here chosen to be the 2% fractile value.The yield stress is log-normally distributed with a mean of 314 MPa and COV of 7% andthe characteristic value is thus �d = 271 MPa. The partial safety factor of the material ischosen to be m = 1:28. This particular choice of characteristic material value and associatedpartial safety factor is in accordance with the Danish Code on Steel Structures, DS-412 [35].

The calibrated partial safety factor HS is given in Table 7.5 for the four di�erent designmodels and for the two di�erent protection systems. It follows from an examination of thepartial safety factors that the partial safety factor of the loading in sagging is larger thanthe corresponding safety factor in hogging { apart from model 1 in moderate protection.This might indicate that model 1 is too simple for application to a general code format. Forgeneral design purposes it may, however, be desirable to have the same partial safety factorsof the characteristic action value in both hogging and sagging.


Finally, it must be realised that the partial safety factors are calculated for one vesselonly and thus have not been subject to an optimization. Therefore, great care should betaken in applying these factors to other vessel designs as their validity is limited to a speci�cmechanical formulation and uncertainty modeling.

Chapter 8

Conclusion and Recommendations

From these stochastic models many interesting conclusions can bedrawn about the sea { some of which might be true.

T. Francis Ogilvie

8.1 Conclusion

Although probabilistic models are always superior to deterministic models of an equal levelof complexity (in the sense that the former have a considerably higher threshold of realismwhen dealing with phenomena which take place in uncertain environments) simple deter-ministic models in everyday design work are needed. The reason is the large number ofdecisions involved in engineering work, and the decision models should therefore preferablybe pushed in the direction of simplicity. The simple deterministic models should be producedby authorities in terms of a model code for design by partial safety factors with partial safetyfactors being calibrated on the basis of probabilistic models. In such a model code, all rulesof probabilistic calculus are neglected. Therefore the code is only able to reproduce theresults of a truly probabilistic model when applied within narrow classes of standard typestructures for which the safety factors have been calibrated. Due to its simplicity the deter-ministic format of the partial safety factor format has obvious advantages in everyday routinework. Therefore it is not considered wise to enforce a replacement of the deterministic rulesby a set of probabilistic rules. However, a synergetic e�ect of performing a probabilisticdesign suggests that if a designer handles the loading well and subsequently pursues crediblestructural response analyses, then the likelihood of ending up with a reliable ship structureis greatly enhanced.

The aim of this treatise has not been on the establishment of a model code for shipstructures, but merely a remedy to ameliorate the establishment of a consistent probabilistic

173

174 Chapter 8. Conclusion and Recommendations

model universe (with due respects to model realism) within which the code format may becalibrated. This objective has been achieved through:

� Establishment of an analytical continuous two-dimensional distribution model for de-scribing the joint distribution of the wave scatter diagram experienced during thelifetime of the vessel. The analytical distribution was based on a three-parametermarginal Weibull distribution and a three-parameter conditional Weibull distribution.A model for modi�cation of the wave scatter diagram according to avoidance of badweather was also given.

� Short-term statistics were given for the linear case both with respect to fatigue analysisand extreme-value analysis. Clustering e�ects were taken into account in the extreme-value analysis.

� Formulation of a \quasi-stationary narrow-band model" for the analysis of non-lineare�ects. Based on an approximate non-linear strip theory, motions and forces (in headsea) were calculated for a container ship in large-amplitude sinusoidal waves. Proce-dures for obtaining the response statistics were also given.

� Extension of the \quasi-stationary narrow-band model" to the prediction of fatiguedamage in the side shell of ship structures. The model was based on motions andforces from linear strip theory, with a straightforward extension to inclusion of non-linearities arising from integration of the wave pressure over the instantaneous positionof the hull relative to the waves.

� Extension of the \quasi-stationary narrow-band model" to the calculation of the statis-tics of the combined low-frequency wave-induced bending moment and the high-frequen-cy slamming-induced bending moment. The method accounted for the clustering e�ectof the slamming impacts, and the combination of low-frequency wave-induced bendingmoment with high-frequency slamming-induced bending was \exact". The model onlyrequires deterministic calculations of the response at regular sinusoidal waves for se-lected pairs of amplitude and frequency. This implies that more elaborate calculationsof the wave-induced bending moment can be applied as for example calculations basedon the second-order strip theory or integration of the water pressure to the instanta-neous water surface. The only restriction on a more elaborate model for wave-inducedbending is that the method requires that heave and pitch motions are su�ciently ac-curately described by linear wave theory. This is usually so.

� Formulation of a model for obtaining the long-term statistics both for fatigue analysisand extreme-value analysis. The model accounted for the operational philosophy inrough sea states.

� Formulation of a procedure for establishing the combined wave-induced and still-water-induced response on a voyage. The procedure was extended to taking into account theuncertainty in voyage duration.

8.1. Conclusion 175

� Formulation of a probabilistic model for corrosion which took into account the corrosioninitiation period, the ageing e�ect, and the location-dependent corrosion rates.

� Establishment of a probabilistic model for calculating the ultimate bending capacityof the hull section. By formulating the model called \system analysis of correlatedidentical events" it became possible to include the uncertainties in initial imperfectionssuch as distortion and residual stresses, and localized corrosion of the sti�eners in theprobabilistic analysis of the ultimate load capacity of the hull section.

� Formulation of a fatigue crack growth model of a semi-elliptical two-dimensional surfacecrack using a fracture mechanics approach. A crack initiation period was introduced tomake conventional S �N curve and the present analysis correspond. The duration ofthe crack initiation period is well documented for life times in the order of 106 cycles.For longer lives, as it is relevant in the present study, the applied initiation periodrepresents an extrapolation from documented results. Moreover, the model includedtwo di�erent formulations of a threshold on the stress intensity factor.

After this model universe was established, some important aspects of the calibration of amodel code based on the partial safety factor approach were discussed. In passing, it isworth recalling that the objectivity in the process of inference from data is a �ction { simplybecause any data set has to be passed through a statistical model. The choice of this modelis not unique, and there is no objective principle by which this ambiguity may be removed.Some subjective judgements must always be made.

It is not obvious to what target reliability level a given adverse event should be designed.It could be argued, however, that the target reliability level should be selected in accordancewith present code formulations. Unfortunately, the present semi-empirical rules for shipdesign exhibit a large variation, and no unique target reliability level can be deduced fromthe code formats. In this treatise the required target reliability level was decided to bededuced from a reliability-based cost optimization. Initial cost of steel and design, cost offailure in hogging and sagging, cost of fatigue cracking repair, and cost of steel replacementdue to corrosion were included in the cost formulation. The reliability level resulting fromthe reliability-based optimization is claimed to be optimal in a well-de�ned sense. By useof the thus obtained target reliability level requirements a set of partial safety factors wascalibrated against a single double-hull tanker structure.

The following main conclusions may be drawn from this treatise:

� The use of the \quasi-stationary narrow-band model" in connection with the non-linearstrip theory indicates the existence of an upper limit on the maximum wave-inducedsagging moment. A similar tendency was not observed for the hogging moment.


� The e�ect of the maneuvering philosophy in severe sea states has an impact on boththe stress range distribution for fatigue damage and the extreme-value distribution.

� The uncertainty in voyage duration has no signi�cant impact on the long term extremevalue distribution { only the mean voyage duration is important.

� The uncertainty in the level of initial imperfections such as distortion and residualstresses has no signi�cant impact on the ultimate failure probability of the hull section{ only their mean values are appreciable.

� The model correction factor method applied in connection with the fully plastic bendingmoment was found to be a surprisingly accurate tool in the probabilistic analysis ofthe midship section { and then at a fraction of computer time.

� The reliability-based cost optimization (for one particular vessel) suggests that thefollowing annual (�rst-year) reliability index requirements can be adopted:

Failure mode Moderate Goodprotection protection

Ultimate hogging bending failure � =3.70 � =3.25Ultimate sagging bending failure � =3.95 � =3.50

in which \Moderate protection" assumes a 5 years' mean corrosion initiation time, and\Good protection" a 10 years'.

� By application of the design formula: MHS;ult(�d= m) � HS(Mstill + Mwave)98% thefollowing partial safety factors may be adopted for the loading:

Moderate protection Good protection Hogging Sagging Hogging Sagging1.17 1.39 1.02 1.21

and m = 1:28 for the material.

It must be realised that both the target reliability requirements and the partial safetyfactors are calculated for one vessel only. Therefore, great care should be taken in generalizingthese factors to other vessel designs as their validity is limited to a speci�c mechanicalformulation and uncertainty modeling.

8.2. Recommendations for future work 177

8.2 Recommendations for future work

8.2.1 On the asymptotic expected values

The proposed model for asymptotic expected values has the opportunity of becoming anattractive alternative to simulation techniques in the calculation of expected values. Itis recommended that future studies focus attention on what happens when the integrandis transformed to a space in which it looks more peaked and hence a local asymptoticapproximation may do better. One possible technique could be to select the power of the\super Rayleigh" distributed space so that the third-order derivative of the log-likelihoodfunction equals zero. The third-order derivative may, however, be di�cult to establishaccurately. Moreover, future studies should quantify limitations of the model.

8.2.2 On the probabilistic model universe

Several additional limit states need to be formulated in the establishment of the probabilisticmodel universe for ship structures. These count primarily limit states taking into account thecombined e�ect of shear stresses and horizontal and vertical bending stresses. Such modelsmust deal with the combination of stochastic processes and may therefore conveniently beformulated by use of Madsens's formula. The failure probability would then be estimated asthe zero downcrossing rate.

In the case of linear response analysis, the variables and their time derivatives are zeromean joint Gaussian, and the covariance structure is readily available:

Cov [Zi(t); Zj(t)] =Z 1

�1Sij(!)d! = 2

Z 1

0<[Sij(!)]d!

CovhZi(t); _Zj(t)

i=

Z 1

�1(i!)Sij(!)d! = �2

Z 1

0!=[Sij(!)]d!

Covh_Zi(t); _Zj(t)

i=

Z 1

�1!2Sij(!)d! = 2

Z 1

0!2<[Sij(!)]d!

in which Zi is the i'th considered response, <[�] denotes the real part, =[�] denotes theimaginary part, and Sij the cross-spectral matrix:

Sij(!) = S�(!)Hi(!)Hj(!)

S�(!) is the spectral density function of the excitation process. Overbar denotes the complexconjugate.


Given the joint statistics of the sectional forces, the failure probability during the lifetimeis calculated for each short-term period by use of Madsens's formula. Depending on the an-alyzed response, di�erent models must be applied for the entering series system in Madsen'sformula. The calculated failure probability (downcrossing rate) is uniquely related to theresponse spectral moments, conditional on the short-term sea state, the loading condition,the wave heading angle, and the velocity of the vessel. The long-term failure probabilitymay �nally be found as it is outlined in this treatise.

8.2.3 On the quasi-stationary narrow-band model

Veri�cation through simulation

The simpli�cation of quasi-stationary narrow-band analysis does not come without expenses.The basic assumption is that the structural response to a single wave cycle is well approx-imated by the steady-state response to an in�nite number of such cycles. This becomestrue in two marginal cases: (1) The bandwidth of the wave spectrum becomes increasinglynarrow (and the wave becomes in essence a single sinusoid), or (2) the structure behavesquasi-statically, so that its instantaneous response depends only on the corresponding in-stantaneous wave elevation. This is in general only an approximation and must be veri�edon a case-by-case basis.

The purpose of future studies will therefore be to compare predicted response statisticsfrom this analysis with the results of simulation. The objective will be to ascertain (1) limitson the physical problem { type of non-linearity, wave spectrum, etc. { for which the methodapplies and (2) what response statistics it predicts most accurately. One may, for example,directly estimate the extreme response by combining a corresponding extreme wave withthis narrow-band transfer function. Alternatively, one may use the narrow-band model justto estimate response moments and �t an alternate random process model (e.g. Hermite) toestimate corresponding extremes. The strategy preferred here depends on which statisticsthe narrow-band model is best at estimating.

Optimal choices of wave amplitudes and frequencies

As noted earlier, the joint probability density of the various wave amplitudes and frequenciesis estimated theoretically, from random vibration models. A number of such models exist.Some are not applicable to wave applications, however, because they require higher spectralmoments

�n =Z 1

!=0!nS�(!)d!

8.2. Recommendations for future work 179

which do not exist when n � 4 for common wave spectra. Various models using only low-order moments (e.g., �0, �1, �2) have been proposed by Sveshnikov and Longuet-Higgins.Some of these models have been compared with actual wave amplitude-frequency data.None, however, have been studied with regard to applicability for the purpose of non-linearresponse analysis.

By comparison with \exact" results from the previous simulation task, the theoreticalmodel which gives most accurate response statistics should be determined. As a separatebut related task, e�cient discretization of these continuous amplitude-frequency distributionsshould be established. The objective is here to establish an optimal (minimal) set of (ai; !i)pairs at which the non-linear transfer function must be evaluated. For the non-linear system,this step is the most costly analysis step. The minimal set may be selected by means of themodel for asymptotic expected values.

Comparison with quadratic theory

The results of the narrow-band non-linear model should be compared with those based onthe quadratic theory. While the quadratic theory carries greater detail, as previously noted,it has its own disadvantages (e.g. assumed quadratic non-linear form, not fully capturing lesssmooth non-linearities). Conditions should be considered { e.g. range of non-linear models{ for which this narrow-band analysis is su�cient, or perhaps even superior to the quadratictheory.

Dynamic response models

Finally, it should be noted that the narrow-band model is most applicable to quasi-staticresponses. For lightly damped resonant response, alternate models should be developed.One possible technique could be to establish the Slepian model process conditional on amaximum and a wave frequency. The most probable wave pro�le during a period of timemight then be established, and response statistics may be obtained similarly to the foregoing.More general veri�cations should pursue here and alternative models should be formulated,if necessary.

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Appendix A

Stresses in L-Sti�eners

The major problem of using L-shaped sti�eners is their asymmetry, which causes a sidewaysdeformation of the ange when the sti�ener is laterally loaded (e.g. due to outside waterpressure). These secondary stresses induced by the sideways deformation are of the sameorder of magnitude as those induced by pure bending and are thus non-negligible. In thisappendix a simpli�ed model for calculating the secondary stresses in an L-shaped sti�eneris suggested.

When the sti�ener is laterally loaded, the unsymmetric shear ow in the ange causes asideways deformation in the ange. The ange, however, is somewhat restrained from thesideways deformation because of the sti�ness of the web and plating. It is proposed here toload the ange and part of the web with the shear stresses from pure bending of the sti�enerand then calculate the stresses in this sub-section as a beam on an elastic foundation. Thesestresses are then superimposed to the stresses arising from pure bending.

The sideways deformation of the ange due to a unit horizontal load attacking the ange,see Figure A.1, is

�h =h2sE�

4hst3s

+Bp

t3p

!(A.1)

in which hs and ts are the height and thickness of the sti�ener, Bp and tp the width andthickness of the plate, and E� = E

1��2 is the plate sti�ness.

The fraction kh { the integrated shear stresses in the ange { of the lateral load Q = qBp

is

kh =h2f tf (hs � y0)

2Is(A.2)

192

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......................................

�� Bp ��!

� hf �!"jjjhsjjj#

......................................................... ........ ......................................................... ........ ......................................................... ........ ......................................................... ........ ......................................................... ........ ......................................................... ........ ......................................................... ........ ......................................................... ........ ......................................................... ........ ......................

y0

tf

ts

tp

Figure A.1: Geometry of sti�ener.

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.................................................................................................................................................................................................................

......................................khQ

k = 1=�h �� hf ��!

"j0.25hsj# .

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y0f

Figure A.2: Section for calculation of secondary forces.

194 Appendix A. Stresses in L-Sti�eners

in which hf and tf are the height and thickness of the ange, y0 the location of the neutralaxis for the plating and sti�ener, and Is the corresponding moment of inertia.

The di�erential equation for the beam on the elastic foundation is

(EIw00)00 + kw � p = 0 (A.3)

with the boundary conditions w(�l=2) = w(l=2) = 0 and w0(�l=2) = w0(l=2) = 0, as theelastically supported beam is assumed to be clamped in both ends. The applied load isp = khQ, and the sti�ness of the elastic support k = 1=�h. By use of the symmetry, thesolution to Eq. A.3 may be given as

w(x) = C1 cosh(�x) cos(�x) + C2 sinh(�x) sin(�x) +p

k(A.4)

where � =�

k4EI

�1=4.

The constants C1 and C2 are determined from the boundary conditions as

C1 = �pk

cosh(�a) sin(�a) + sinh(�a) cos(�a)

cosh(�a) sinh(�a) + cos(�a) sin(�a)(A.5)

C2 =p

k

sinh(�a) cos(�a)� cosh(�a) sin(�a)

cosh(�a) sinh(�a) + cos(�a) sin(�a)(A.6)

where a = l=2.

The moment at the clamped end is

Me(a) = w00(a)EI (A.7)

=2p�2EI

k

sinh(2�a)� sin(2�a)

sinh(2�a) + sin(2�a)(A.8)

If the factor de�nes the ratio between the nominal, resultant stress and the bending-induced stress, hence, by use of the proposed model, may be approximated as

=�e + �b�b

(A.9)

in which �b is the traditional Navier stress in the sti�ener, and �e =Mey0f=If is the stress inthe beam on the elastic foundation, y0f is the location of the neutral axis, and If the momentof inertia of the ange section. Table 4.3 shows the obtained values of for the sti�enersin the analyzed vessel. For sti�ener sections no. 1 and no. 6 the model has been comparedto results obtained from a non-linear �nite-element analysis. The resultant nominal stressobtained by the present method was 4% and 0.5%, respectively, lower than the �nite-elementresults.

Appendix B

Simulation of the Process

Simulation methods for a random process X(t) fall broadly into two classes. Sequentialtechniques, such as ARMA models, use simple algorithms to generate the current value froma limited number of past observations. By their nature, these can re ect the stochasticdynamic characteristics of X(t) in only a limited way. In contrast, simultaneous techniquessuch as FFT (Fast Fourier Transform) require simultaneous generation and storage of theentire history but allow a complete discretized power spectrum to be matched. The moreaccurate simultaneous methods are applied here. An e�cient and attractive simulationtechnique is Hartley's transform method, Winterstein [161], as it reduces computationaltime and storage requirements. However, as the process after upcrossing of a speci�ed levelis the object of interest, it seems more attractive to simulate the stochastic stationary processX(t) following Hasofer [54]. Hasofer suggests approximating the stationary Gaussian processover a �nite time interval by a trigonometric polynomial with predetermined error.

The process is de�ned as

X(t) =NXn=1

an

�Vn cos

n�t

2T+Wn sin

n�t

2T

�(B.1)

and the Hilbert transform as

X(t) =NXn=1

an

�Vn sin

n�t

2T�Wn cos

n�t

2T

�(B.2)

The coe�cients an are de�ned as the square root of the non-negative Fourier coe�cients �n(a2n = max(0; �n)), given as

�0 =Z 1

0

sin 2!T

2!TdG(!) (B.3)

�n = (�1)nZ 1

0

4!T sin 2!T

4!2T 2 � n2�2dG(!) (n > 0) (B.4)

195

196 Appendix B. Simulation of the Process

where G(!) is the one-sided spectral increment of the process. The coe�cients Vn and Wn

are mutually independent standard normal random variables. The process is periodic withthe period 4�.

Here the outcomes of the Slepian model process are dealt with, de�ned after a particularoutcrossing of the underlying process. The initial conditions of the Slepian model processmay be written in terms of a random vector Y = (Y1; Y2) and Z = (Z1; Z2), Ditlevsen andLindgren [32], where

Y1 =NXn=1

anVn �X(0) (B.5)

Y2 =�

2T

NXn=1

nanVn � _X(0) (B.6)

Z1 =NXn=1

anWn � X(0) (B.7)

Z2 =�

2T

NXn=1

nanWn � _X(0) (B.8)

It is seen that the initial conditions Y = (0; 0) and Z = (0; 0) only have an in uence on thecoe�cients Vn and Wn, respectively. Since the residual variance of the conditional randomGaussian vector [V j Y = (0; 0)] is constant for any Y, the conditional vector may be writtenas

[V j Y=(0; 0)]= V�E[V j Y]+E[V j Y=(0; 0)] = V�Cov[V;Y0]Cov[Y;Y0]�1Y(B.9)

The i'th component of the vector V becomes

[Vi j Y = (0; 0)] = Vi � ai�0�2 � �21

24 NXj=0

aj(�2 � �1 �2T

(i+ j) + ij��

2T

�2�0)Vj

�X(0)(�2 � �1i �2T

)� _X(0)(�0i

�

2T� �1)

�(B.10)

and similarly

[Wi j Z = (0; 0)] = Wi � ai�0�2 � �21

24 NXj=0

aj(�2 � �1 �2T

(i + j) + ij��

2T

�2�0)Wj

+X(0)(�2 � �1i �2T

)� _X(0)(�0i�

2T� �1)

�(B.11)

197

In order to obtain numerical stability in the simulated conditioned process, the spectralmoments must be calculated from the trigonometric polynomial as

�m =NXi=0

��

2Ti�m

a2i (B.12)

Appendix C

Uncertainty in Long-Term

Distribution

This appendix gives a brief description of the uncertainties in the proposed model of the longterm response distribution. A description of uncertainties involved in the stress analysis onship structures is given in Nikolaidis and Kaplan [112].

C.0.4 Bias factors

Soares [139] has conducted an extensive study of the various bias terms e�ecting the transferfunction calculation. Including the bias factors, the transfer function may be rewritten as

H(!) = L SHHL(!) (C.1)

where L is a bias factor based on comparison between experiments and the mathematicallyestimated transfer function and SH is a non-linear bias factor. When the calculation ofthe transfer functions is based on the theory of Salvesen, Tuck, and Faltinsen [131], the biasfactor L is given as [139]

L =

( �0:005� + 0:42Fn + 0:70CB + 1:25 ; 90 < � � 1800:0063� + 1:22Fn + 0:66CB + 0:06 ; 0 � � < 90

(C.2)

where Fn is the Froude number and CB is the block coe�cient.

The non-linearity bias factor SH accounts for the di�erence in sagging and hoggingmoments, and it is dependent on the accuracy of the assumption of the verticality of theship sides.

S = 1:74� 0:93CB (sagging) (C.3)

198

199

H = 0:26 + 0:93CB (hogging) (C.4)

Note that, when applying these non-linearity factors to fatigue analysis, one should use( S + H)=2 = 1, which implies that the non-linear sagging/hogging e�ect on the estimatedfatigue damage has no in uence. However, several studies show, e.g. Winterstein [160] andJensen [61], that the non-linearities in the longitudinal stress component will lead to anincrease in the fatigue damage of the order of magnitude of 50�100 percent, depending onthe forward speed of the ship.

C.0.5 Stochastic modeling

Uncertainty in wave scatter diagram

From the available wave scatter diagram, estimates of the Weibull distribution parameters�h, �h, h and �t(h), �t(h), t(h) are obtained. In order to include the uncertainties in thewave scatter diagram, these estimates are modeled as stochastic variables with a correlationmatrix equal to the one obtained from the non-linear least squares �tting. The coe�cientof variation of the variables should be chosen according to the quality of the data. Thequality of the data is classi�ed according to the quality of the observation method, i.e.visual observation, instrumental measurements or hind-cast simulation. The available waveheights and wave periods are often obtained from measurements of short duration, whichmay not adequately account for seasonal and climatological variations.

Uncertainty in wave spectrum

The parameter � might be set equal to 5 (see Phillips [125]), and the bandwidth parameter �be regarded as a random variable in order to account for uncertainties in the wave spectrumand the bandwidth variation. The standard deviation of the � parameter �� is selected inaccordance with the con�dence in the spectrum.

Uncertainty in transfer function

The uncertainty in the transfer function consists of two terms, namely model uncertaintyand interpolation uncertainty. The model uncertainty is the dominant uncertainty and isdescribed by the uncertainty in L. Concerning the interpolation uncertainty, it is rec-ommended to multiply the interpolated modulus squared of the transfer function by a log-normally distributed random variable with mean value equal to 1 and a coe�cient of variationdetermined on the basis of the accuracy of the interpolation.

200 Appendix C. Uncertainty in Long-Term Distribution

Uncertainty in operational philosophy

The operational philosophy has an important in uence on the long-term peak distribution. Itis therefore of great importance to include uncertainties in the modeling of the wave heightsfor which heading direction and sailing speed are changed, see Figures 5.2 and 5.3. Theoperational philosophy is subjectively judged by the captain/shipowners and may vary fromship to ship. Consequently, it is recommended to assign a rather high standard deviation tothe signi�cant wave heights, for which changes are conducted. It is suggested to multiplyeach of the wave heights by an independent lognormally distributed random variable.

Uncertainty in ship speed

It is assumed that the ship operates at a fairly constant speed in a particular sea stateaccording to the operational philosophy. Consequently, the ship speed could be taken as anormally distributed value with a rather small coe�cient of variation of, say, 5 percent.

Uncertainties in bias factors

According to Soares [139], the standard deviation of L was found to be equal to 0:38 andthe standard deviation of SH to be equal to 0:12.

C.0.6 Bootstrapping

To include the above-mentioned uncertainties in the calibration of the long term Weibulldistribution or the long-term Gamma distribution, the bootstrapping technique is applied,see Efron [37]. For consecutive outcomes of the uncertain parameters, values of lnA and Bin Eq. 5.9 or the values of � and � in Eq. 5.17 are calculated. The mean values, the standarddeviations, and the correlation of the parameters are then obtained. The simpli�ed estimatedlong-term Weibull or long-term Gamma response distribution with correlated stochasticdistribution parameters may then be directly applied in an ensuing structural reliabilityanalysis.

In all cases analyzed, the parameter estimation showed stable results after approximately200 simulations of the uncertain parameters.

Ph.d. ThesesDepartment of Naval Architecture and O�shore Engineering, DTU

1961 Str�m-Tejsen, J.: \Damage Stability Calculations on the ComputerDASK".

1963 Silovic, V.: \A Five Hole Spherical Pilot Tube for three DimensionalWake Measurements".

1964 Chomchuenchit, V.: \Determination of the Weight Distribution ofShip Models".

1965 Chislett, M.S.: \A Planar Motion Mechanism".

1965 Nicordhanon, P.: \A Phase Changer in the HyA Planar Motion Mecha-nism and Calculation of Phase Angle".

1966 Jensen, B.: \Anvendelse af statistiske metoder til kontrol af forskelligeeksisterende tiln�rmelsesformler og udarbejdelse af nye til bestemmelseaf skibes tonnage og stabilitet".

1968 Aage, C.: \Eksperimentel og beregningsm�ssig bestemmelse af vind-kr�fter p�a skibe".

1972 Prytz, K.: \Datamatorienterede studier af planende b�ades fremdriv-ningsforhold".

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�al�gt vand".

1978 R�omeling, J.U.: \Buling af afstivede pladepaneler".

1978 S�rensen, H.H.: \Sammenkobling af rotations-symmetriske og generelletre-dimensionale konstruktioner i elementmetode-beregninger".

1980 Fabian, O.: \Elastic-Plastic Collapse of Long Tubes under CombinedBending and Pressure Load".

1980 Petersen, M.J.: \Ship Collisions".

1981 Gong, J.: \A Rational Approach to Automatic Design of Ship Sec-tions".

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 usingB-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 Estima-tion of O�shore Structures".

1989 Vogt Andersen, S.: \Numerical Treatment of the Design-AnalysisProblem of Ship Propellers using Vortex Latttice 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 Ma-rine Structures".

1991 Almlund, J.: \Life Cycle Model for O�shore Installations for Use inProspect Evaluation".

1991 Back-Pedersen, A.: \Analysis of Slender Marine Structures".

1992 Bendiksen, E.: \Hull Girder Collapse".

1992 Buus Petersen, J.: \Non-Linear Strip Theories for Ship Response inWaves".

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 MovingSurface-Piercing Body".

1994 Friis Hansen, P.: \Reliability Analysis of a Midship Section".

1994 Michelsen, J.: \A Free-Form Geometric Modelling Approach with ShipDesign Applications".

1995 Melchior Hansen, A.: \Reliability Methods for the Longitu-dinalStrength of Ships".

1995 Branner, K.: \Capacity and Lifetime of Foam Core Sandwich Struc-tures".

1995 Schack, C.: \Skrogudvikling af hurtigg�aende f�rger med henblik p�as�dygtighed og lav modstand".

1997 Cerup Simonsen, B.: \Mechanics of Ship Grounding".

1997 Riber, H.J.: \Response Analysis of Dynamically Loaded CompositePanels".

Department of Naval Architecture

And Offshore Engineering

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