Stanility and safety of ships  risk of capsizing

At the same time, in the Intact Stability Casualty Records, there were found several vessels which perished though their stability was considered sufficient according to the IMO draft..

STABILITY AND

SAFETY OF SHIPS

Risk of Capsizing VADIM BELENKY NIKITA B SEVASTIANOV

www.sname.org

Editors: R Bhattacharyya and M E McCormic

Cover design: Susan Evans Grove

© The Society of Naval Architects and Marine Engineers

Printed in the United States of America

by Automated Graphics Systems, Inc (AGS)

Bibliographical Note

This SNAME edition, first published in 2007 is republication of the work originally published by Elsevier, Amsterdam; Boston in 2003 A new preface, Appendix 2, and a list of additional references have been prepared especially for this edition

Library of Congress Cataloging-in-Publication Data

A catalog record from the Library of Congress has been applied for

ISBN 0-939773-61-9

Preface to the Second Edition

The first edition of this book was published by Elsevier as Volume 10 of Elsevier Ocean Engineering Book series edited by R Bhattacharyya and M E McCormick

Originally, this book was the second part of the two-volume monograph united under the

common title Stability and Safety of Ships The first part was published as Volume 9 of

Elsevier Ocean Engineering Book series with the subtitle “Regulation and Operation” authored by Kobylinski and Kastner It described the state of the art and historic perspective of intact stability regulations as well as covered the operational aspect of ship stability Volume 10, subtitled “Risk of Capsizing,” contained descriptions of contemporary approaches and solutions for evaluation of dynamic stability as well as a detailed review of the research results in the field and was meant to serve as an extended reference source for the development of future intact stability regulations Both parts were written with the same philosophy but could be read separately

The appearance of new types of naval and commercial vessels with unconventional dynamics in waves made conventional methods of evaluation of dynamic stability unreliable for the most part, as these methods are based on previous experience and statistics It is well known that the best approach is to use the physically sound solution for ship motion in waves employing Nonlinear Dynamics and theory of stochastic processes This allows developing new views on different types of stability failures including capsizing in dead ship conditions, surf-riding and broaching, parametric resonance and pure loss of stability on the wave crest The above approach has defined the increased interest of maritime industry to the problems of ship dynamics Understanding the importance of these problems motivated IMO to resume discussion on new approaches to intact stability regulations in 2002

Among the naval architects whose research results and organizational efforts determined these new views in recent years, I would like to mention: P R Alman, H P Cojeen,

J O de Kat, A Francescutto, Y Ikeda, L Kobylinski, M A S Neves, J R Paulling,

L Peres Rojas, P Purtell, A M Reed, R Sheinberg, K Spyrou, A W Troesch,

N Umeda and D Vassalos

This list, of course, is far from being complete, so I would like to ask those colleagues who were not mentioned in this list to accept my sincere apology

To assist this development the Society of Naval Architects and Marine Engineers (SNAME) decided to publish a second edition of Volume 10 since the first edition is out

of print Volume 9 remains available I am very grateful to R Bhattacharyya, W France,

S Evans Grove and R Tagg for their help with organization of the second edition

My special thanks are due for William Belknap, Michael Hughes and Arthur Reed for their detailed review of Chapter 3, for Marcello Neves for his thorough review of Chapter

4 and for Yury Nechaev for additional corrections to regression coefficients in Appendix I

The second edition is almost an exact reproduction of the first edition with the exception

of corrected typographical errors, updated text for some chapters to account for the most recent development in parametric roll and numerical simulation of irregular roll motions Corresponding updates were made in the list of references

I am grateful to all my colleagues, discussions with whom were very helpful in updating the book, in particular: G Bulian, A Degtyarev, P Handler, B Hutchison, B Johnson,

W M Lin, L McCue, K Metselaar, W Peters, and K Weems

The author considers it as a pleasant duty to thank management and employees of the American Bureau of Shipping and first of all: G Ashe, R I Basu, A J Breuer,

C J Dorchak, T Gruber, T Ingram, B Menon, D Novak and H Yu – all of whom shared the author’s interest to the problems of ship dynamics and made possible for the author to continue working in this direction, including publication of the second edition

XV

Table of Contents

Preface to the Second Edition v

Series Preface vii

Foreword ix

Preface xi

Part 1 Probabilistic Approach to Stability and Risk Assessment Chapter 1 Philosophy of Probabilistic Evaluation of Stability and Safety 3

1.1 General Concepts of Probabilistic Evaluation of Stability, Safety and Risk at Sea 3

1.2 Vectors of Assumed Situations and Loading Conditions Risk Function 12

1.3 The Probability of Survival and Its Interpretation in the Task of Stability Estimation 16

1.4 The Problems of Criteria and Norms in the Probabilistic Approach to Stability Standards 22

1.5 Algorithm of Averaging of Risk Function 25

Chapter 2 Probabilistic Evaluation of Environmental and Loading Conditions 31

2.1 Lightweight Loading Conditions 31

2.2 Time Varying Components of Loading Conditions 33

2.3 Meteorological Components of Assumed Situation 40

2.4 Operational Components of an Assumed Situation 52

Part 2 Dynamics of Capsizing Chapter 3 Equations for Nonlinear Motions 57

3.1 General Equations of Fluid Motions 57

3.1.1 Forces and Stresses in Fluid 57

3.1.2 Relationship of Volume and Surface Integrals Transport Theorem 60

3.1.3 Conservation of Mass and Momentum 61

3.1.4 Continuity Equation Euler’s Equations 61

3.1.5 Navier-Stokes Equations 62

3.1.6 Boundary Conditions 63

3.2 Motions of Ideal Fluid 64

3.2.1 Model of Ideal Fluid 64

3.2.2 Potential Laplace and Bernoulli Equations Green’s Theorem 66

3.2.3 Hydrodynamic Pressure Forces 67

3.2.4 Forces on Moving Body in Unbounded Fluid Added Masses 68

3.3 Waves 71

3.3.1 Free Surface Boundary Conditions 71

3.3.2 Linearized Free Surface Boundary Conditions Theory of Small Waves 72

3.3.3 Plane Progressive Small Waves 73

3.4 Ship Response in Regular Small Waves 75

3.4.1 System of Coordinates 75

3.4.2 Formulation of the Problem 76

3.4.3 Hydrostatic Forces 78

3.4.4 Added Mass and Wave Damping 81

3.4.5 Wave Forces: Formulation of the Problem 83

3.4.6 Froude-Krylov Forces 83

3.4.7 Hydrodynamic or Diffraction Wave Forces 86

3.4.8 Body Mass Forces 88

3.4.9 Linear Equation of Motions 91

3.5 Linear Equation of Roll Motions 94

3.5.1 Adequacy of Linear Equation of Motions 94

3.5.2 Calculation of Forces and Motions 95

3.5.3 Isolated Linear Equation of Roll Motions 96

3.5.4 Other Forms of Linear Equation of Roll Motions 97

3.5.5 Solution of Linear Equation of Roll Motions 98

3.5.6 Linear Roll Motions in Calm Water 99

3.5.7 Linear Roll in Waves 101

3.5.8 Steady State Roll Motions Memory Effect 102

3.6 Nonlinear Roll Equation 103

3.6.1 Classification of Forces 103

3.6.2 Inertial Hydrodynamic Forces and Moments 104

3.6.3 Hydrodynamic Wave Damping Forces 104

3.6.4 Viscous Damping Forces 104

3.6.5 Other Forces 105

3.6.6 Wave Excitation Forces 106

3.6.7 Hydrostatic Forces: Structure of Nonlinear Roll Equation 106

Chapter 4 Nonlinear Roll Motion in Regular Beam Seas 109

4.1 Free Roll Motion 109

4.1.1 Free Oscillations of Nonlinear System 109

4.1.2 Free Motions of Piecewise Linear System 111

4.2 Steady State of Forced Roll Motions 114

4.2.1 Equivalent Linearization 114

4.2.2 Harmonic Balance Method 116

4.2.3 Perturbation Method 119

4.2.4 Method of Multiple Scales 121

4.2.5 Numerical Method 125

4.2.6 Steady State Solution of Piecewise Linear System 128

4.3 Stability of Equilibrium 131

4.3.1 Identification of Equilibria 131

4.3.2 Original or “Normal” Equilibrium 132

XVII

4.3.3 Equilibrium at Angle of Vanishing Stability 134

4.3.4 Equilibrium at Capsized Position 136

4.3.5 Phase Plane in Vicinity of Equilibria 137

4.4 Stability of Roll Motion 140

4.4.1 Lyapunov Direct Method 140

4.4.2 Floquett Theory 141

4.4.3 Poincare Map and Numerical Method for Motion Stability 145

4.4.4 Motion Stability of Piecewise Linear System 149

4.5 Bifurcation Analysis 151

4.5.1 General 151

4.5.2 Fold Bifurcation 152

4.5.3 Period Doubling and Deterministic Chaos 154

4.5.4 Bifurcations of Piecewise Linear System 156

4.6 High Order Resonances 158

4.6.1 General 158

4.6.2 Ultra-harmonic Resonance 159

4.6.3 Sub-harmonic Resonance 161

Chapter 5 Capsizing in Regular Beam Seas 165

5.1 Classical Definition of Stability 165

5.1.1 Concept of Separatrix 165

5.1.2 Calculation of Separatrix 168

5.1.3 Separatrix, Eigenvalues and Eigenvectors 170

5.1.4 Numerical Validation of Classical Definition of Stability 173

5.2 Piecewise Linear Model of Capsizing 174

5.2.1 General 174

5.2.2 Capsizing in Piecewise Linear System 175

5.2.3 Piecewise linear System and Classical Definition of Stability 177

5.2.4 Shapes of Capsizing Trajectories 178

5.3 Nonlinear Dynamics and Capsizing 182

5.3.1 General 182

5.3.2 Sensitivity to Initial Conditions: Safe Basin 182

5.3.3 Concept of Invariant Manifold 185

5.3.4 Invariant Manifold and Erosion of Safe Basin Melnikov Function 188

5.3.5 Loss of Motion Stability and Capsizing 191

Chapter 6 Capsizing in Regular Following and Quartering Seas 195

6.1 Variation of the GZ Curve in Longitudinal Waves Pure Loss of Stability 195

6.1.1 Description of Phenomenon 195

6.1.2 Methods of Calculations 196

6.1.3 Pure Loss of Stability 198

6.1.4 Equation of Roll Motions 198

6.2 Parametric Resonance 199

6.2.1 Description of Phenomenon 199

6.2.2 Parametric Resonance in Linear System Mathieu equation 200

6.2.3 Parametric Resonance in Nonlinear System 202

6.3 Surf-Riding in Following Seas 207

6.3.1 General 207

6.3.2 Forces and Equation of Motions 207

6.3.3 Equilibria 208

6.3.4 Stability of Equilibria 210

6.3.5 Bifurcation Analysis 212

6.4 Model of Ship Motion in Quartering Seas 214

6.4.1 General 214

6.4.2 Equations of Horizontal Ship Motions 215

6.4.3 Surging and Surge Wave Force 219

6.4.4 Swaying and Sway Wave Force 219

6.4.5 Yaw Motions and Yaw Wave Moment 221

6.4.6 Roll Equation for Broaching Study 222

6.4.7 Equation of Autopilot 224

6.4.8 Model for Broaching 224

6.5 Ship Behavior in Quartering Seas 225

6.5.1 Equilibria of Unsteered Vessel 226

6.5.2 Stability of Equilibria of Unsteered Ship 227

6.5.3 Stability of Equilibria of Steered Ship 232

6.5.4 Large Ship Motions in Quartering Seas 234

6.5.5 Global Analysis 236

6.5.6 Broaching as the Manifestation of Bifurcation of Periodic Motions 237

6.6 Broaching and Capsizing 240

6.6.1 Analysis of Equilibria 240

6.6.2 Invariant Manifold 241

6.6.3 Capsizing 242

Chpater 7 Other Factors Affecting Capsizing 245

7.1 Aerodynamic Forces and Drift 245

7.1.1 Steady Drift 245

7.1.2 Aerodynamic Forces 247

7.1.3 Hydrodynamic Drift Forces 249

7.1.4 Sudden Squall of Wind 253

7.1.5 Method of Energy Balance 255

7.2 Influence of Freeboard Height and Water on Deck 261

7.2.1 General 261

7.2.2 Experimental Observations Pseudo-static Heel 262

7.2.3 Behavior of Water on Deck 264

7.2.4 Influence of Deck in Water 270

7.2.5 Model of Ship Motions 275

7.2.6 Behavior of Ship with Water on Deck 277

7.3 Stability in Breaking Waves 279

7.3.1 General 279

7.3.2 Geometry and Classification of Breaking Waves 280

7.3.3 Impact of Breaking Wave: Experiment and Theory 282

7.3.4 Probabilistic Approach to Capsizing in Breaking Waves 285

XIX

Chapter 8 Nonlinear Roll Motions in Irregular Seas 289

8.1 Fundamentals of Stochastic Processes 289

8.1.1 General 289

8.1.2 Moments of Stochastic Process Autocorrelation 290

8.1.3 Stationary and Non-stationary Processes 292

8.1.4 Ergodicity 292

8.1.5 Spectrum and Autocorrelation Function 293

8.1.6 Envelope of Stochastic Process 295

8.2 Probabilistic Models of Wind and Waves 299

8.2.1 Gusty Wind 299

8.2.2 Squalls 301

8.2.3 Spectral Model of Irregular Waves 302

8.2.4 Method of Envelope 302

8.2.5 Autoregression Model 304

8.2.6 Non-Canonical Presentation 305

8.3 Irregular Roll in Beam Seas 306

8.3.1 Linear System Weiner–Khinchin Theorem 306

8.3.2 Correlation of Irregular Roll 308

8.3.3 Statistical Linearization 311

8.3.4 Energy-Statistical Linearization 313

8.3.5 Method of Multiple Scales 316

8.3.6 Monte-Carlo Method 324

8.3.7 Non-Canonical Presentation and Monte-Carlo Method 327

8.3.8 Parametric Resonance in Irregular Beam Seas 328

8.4 Roll in Irregular Longitudinal Seas 331

8.4.1 Probabilistic Model of Irregular Longitudinal Seas 331

8.4.2 Surging in Irregular Seas 331

8.4.3 Changing Stability in Longitudinal Irregular Seas 332

8.4.4 Parametric Resonance in Irregular Longitudinal Seas 333

8.5 Influence of Gusty Wind 335

8.5.1 Distribution of Aerodynamic Pressures 335

8.5.2 Fourier Presentation for Aerodynamic Forces 339

8.5.3 Swaying and Drift in Beam Irregular Seas 339

8.5.4 Roll Under Action of Beam Irregular Seas and Gusty Wind 342

8.6 Probabilistic Qualities of Nonlinear Irregular Roll 343

8.6.1 Ergodicity of Nonlinear Irregular Roll 343

8.6.2 Distribution of Nonlinear Irregular Roll 346

8.6.3 Group Structure of Irregular Roll 350

8.6.4 Application of Markov Processes 352

Chapter 9 Probability of Capsizing 357

9.1 Application of Upcrossing Theory 357

9.1.1 General 357

9.1.2 Averaged Number of Crossings 358

9.1.3 Crossings as Poisson Flow 360

9.1.4 Time before Crossing 363

9.2 Probability of Capsizing in Beam Seas 364

9.2.1 Mathematical and Physical Modeling 364

9.2.2 Classical Definition of Stability 369

9.2.3 Method of Energy Balance 373

9.2.4 Piecewise Linear Method 377

9.2.5 Combined Piecewise-Linear-Numerical Method 382

9.2.6 Methods Based on Motion Stability 385

9.2.7 Methods Based on Nonlinear Dynamics 387

9.2.8 Markov Processes Application 391

9.3 Probability of Capsizing in Following Seas and Risk Caused by Breaking Waves 392

9.3.1 Classical Definition of Stability and Pure Loss of Stability 392

9.3.2 Piecewise Linear Method 394

9.3.3 Probability of Surf-Riding 399

9.3.4 Risk of Capsizing Caused by Breaking Waves 400

Appendix I Nechaev Method 403

Appendix II Basic Statistics and Ergodicity of Stochastic Process 413

A2.1 Statistical Estimates of Stochastic Process as Random Numbers 413

A2.2 Confidence Interval of Statistical Estimates 414

A2.3 Measure of Ergodicity 415

References 419

References for the Second Edition 443

Subject Index 445

Part 1

Probabilistic Approach to Stability and Risk Assessment

Chapter 1

Philosophy of Probabilistic Evaluation of Stability and Safety

1.1 General Concepts of Probabilistic Evaluation of Stability, Safety and Risk at Sea

It is clear from the review [Blagoveshchensky, 1932, 1951; Lugovsky, 1971, Rahola,

1935, 1939] of the existing national and international practices of stability standardization that the development of stability standards was initiated by experts in the theory of naval architecture Therefore, this problem is usually considered by naval architects and seamen as an aspect of the theory of ships However, it is not quite so

The final goal of setting ships' stability standards is to ensure their safe operation without fatal capsizing casualties during their service lives Similar tasks are considered practically concerning any other products of technology, see [Sevastianov, 1982] Indeed,

if we substitute in the former sentence the word "stability" for the word "strength" and the word "capsizing" for the word "failure" then the aim of standardization will be changed

in a specific technical aspect However, the very essence of setting standards will remain the same: to ensure operation of some object (product, system, etc.) without failure during a given time Only the nature, the causes and the form of the failure are specific Failures in a complicated system may occur both in the system as a whole and in some of its elements This should be especially borne in mind when we are developing the standards of sea-keeping qualities Such failures as capsizing, foundering, loss of longitudinal strength practically mean the total loss of the system called a vessel Damage

to the propeller, steering engine or hatch cover does not mean the immediate total loss of

a vessel, as a rule, although sometimes they may happen to be an important link in the chain of events which result in the total loss of the vessel In our further consideration of stability we shall deal only with such failures as capsizing or catastrophic heeling

First of all, it is necessary to consider the general plan of any standardization It can be represented as a set of the following four sub-problems:

1 The definition of the aims of standardization,

2 The choice of criteria,

3 The setting of the norm (standard) for each criterion,

4 The evaluation of the likelihood of achieving the goal and the technical and economical consequences of implementing standards

The goal of any standard is preventing certain types of casualties with certain objects during a certain time interval Such an interval may be determined as the duration of a missile’s flight to the target or the navigation period up and down the rivers which become frozen in winter, or lifetime of a vessel from its launch until it is sold for scrap metal

The choice of criteria is a specific task for each type of casualty The word "criterion” in Greek means "an instrument for judgment" We shall examine and use the criteria for estimating how great the risk of a casualty is It has become a custom in modern practice

to judge the possibility of casualties not by a single criterion but by a set of some criteria Apparently, the characteristics chosen as criteria should be functions of properties of the object itself Besides the arguments determining the criterion, its value should take into consideration the external condition parameters under which the object operates As far

as this concerns stability such conditions should include the forces affecting the vessel, their orientation, dynamic or static application as well as the ability of the crew to maintain the necessary safety level Simple and effective stability control methods are very important in this respect Therefore, it is clear that the sub-task of criteria choice necessitates participation of such experts who know in detail the properties of the corresponding objects and their operational conditions

The setting of the norms is the indication of some conventional boundary between the permissible and impermissible values of the criteria, that is between the points "good" and "bad" for providing stability

It is obvious that the risk of capsizing will change continuously with continuous alteration of the usual stability criteria Unfortunately, the actual interdependence between the risk and practical stability criteria cannot be expressed in explicit form by a simple formula It exists as a rule in a latent, implicit form This problem will be discussed in detail below

We will consider a simple example explaining the nature of usual stability norms

Let us imagine an ordinary deckless vessel or a boat affected by a static heeling moment,

Would it be reasonable to adopt as a norm the angle of inclination, If, corresponding to the immersion of the lowest point of the vessel's freeboard? A “pure" theoretician would probably agree with such a suggestion Really, at the angle of heel I < If the boat has some residual freeboard and water cannot enter into the vessel Nevertheless it is not necessary to be a seaman to have some doubts about the practicality of such a norm We shall not criticize this norm for the inaccuracy of the metacentric formula at large angles

of heel Indeed, it is possible to find the angle I more precisely on the basis of a stability diagram if the heeling moment, M h is known But who can guarantee proper accuracy of the assigned moment M h or of the initial value of the actual freeboard? Are we sure of the fact that all the mistakes will always increase safety? And what is the practical validity of the implicit assumption about heeling under the condition of absolutely still water? That's why even in the simplest example it is impossible to accept the value If as a practical norm though it is theoretically an indisputable boundary between zero risk and the situation when the boat is doomed to perish It would be reasonable to introduce some reserve into the norms foreseeing the unpredictable random external conditions (sea state, ship motion, etc.) and inaccuracy of the given initial data In other words, we have to insure ourselves against our own ignorance of all the actual circumstances of the situation being considered That is another matter that we can introduce such a reserve in an explicit or in an implicit form, and it is clear that the magnitude of this reserve depends

on the factors and circumstances which lie beyond the framework of ordinary theory with its deterministic approach

The last sub-task of setting the safety standard concerns the evaluation of the guarantee of safety, which should be ensured by the standards being suggested It also concerns foreseeing technical and economic consequences of introducing these standards

It should be noted that practical implementation of various seakeeping standards has not yet raised the problem of guarantee in its explicit form

Probably the experts faced this problem for the first time while developing the IMO Stability Recommendations This problem is rather new and practically important Therefore, we shall consider it and its history in detail

The draft of the IMO Stability Recommendations was developed in 1967 The interested countries tried to evaluate the acceptability of this draft on the basis of the numerous stability calculations made for their ships These were in accordance with the criteria and norms suggested by IMO [1967a], resolutions A-167 and A-168 Maritime Authorities of countries, which had already used their own national stability regulations, compared the results of such calculations with the stability estimations made in compliance with existing regulations They found out that the stability of some vessels, which were considered safe, was deemed insufficient according to the IMO draft At the same time,

in the Intact Stability Casualty Records, there were found several vessels which perished though their stability was considered sufficient according to the IMO draft The comparison of the results obtained in different countries did not give any definite answer even to a rather trivial question: which of the existing stability standards are more strict and which ones are less so? The answers obtained by different countries did not coincide

in some cases

Chapter 1

6

No doubt the clear understanding of the contradiction in the balance of merits and demerits caused by introducing new stability standards was a significant achievement of IMO The economic cost and other difficulties were evident First, they were connected with the ballasting of many vessels (decreasing their dead-weight); recalculations of stability according to new requirements; a greater number of inclining tests; the issue of the new information on stability properties for many vessels All these consequences require rather significant expenses For many developing countries these expenses would increase still more due to the necessity to invite foreign experts to carry out the corresponding calculations and tests However, there has never existed any method for evaluating how far the risk of capsizing can be reduced by introducing new stability standards

The deadlock, which arose in further IMO activities for developing Stability Recommendations, was overcome due to the coincidence of three circumstances First, some dramatic casualties took place late in the 1960’s with Russian, English, Japanese, Canadian and other countries fishing vessels, particularly under the conditions of severe icing The 19th of January 1965, was a dark day in the history of the Russian fishing fleet Four medium-sized fishing trawlers were lost during one night in the Bering Sea, where more than one hundred vessels were catching herring The catastrophe was caused

by heavy icing

These and similar accidents were a powerful incentive for the Maritime Administrations

to concentrate their efforts on stability problems and on the development of reliable stability standards

The second important reason for further IMO activities was the fact that by 1968 IMO had completed an unprecedented five-year collection of Intact Stability Casualty Records The majority of IMO members were engaged in this work No separate country would be able to collect such a representative set of statistical data concerning the stability characteristics of lost vessels and the circumstances of their loss The stability casualty is generally fatal not only for the ship itself, but also for the crew and the passengers As a rule, there are no surviving witnesses In the case of the four Russian trawlers, only one man was taken alive from the bottom of the capsized trawler "Boxitogorsk" which was floating with its keel up The other 96 crew members of these vessels perished with their ships

These Casualty Records gave the basic data for collective investigations by German and Polish experts and for their suggestions on the stability criteria and norms These suggestions were then submitted in 1967 as the first draft of the IMO Stability Recommendations for cargo, passenger and fishing vessels, [IMO, 1967], IMO resolutions A-167 and A-168

In 1968, the Russian delegation in the IMO Working Group on Safety of Fishing Vessels suggested its method for comparison of rigidity and effectiveness of various stability criteria and norms [Sevastianov, 1968a; IMO, 1967a] The method uses the concept of the so-called "critical" height of the vessel's centre of gravity above a baseline This concept is very convenient for the comparison with the results achieved in accordance with different sets of stability criteria and norms The same unified measure of stability was applied to compare the actual stability of capsized vessels with that stability level

which would be a sufficient minimum for the same vessels in accordance with various

stability standards The detailed description and foundations of these vessels were

published in Russian journals and books [Sevastianov, 1968a, 1970] As far as we know,

some concepts were introduced there for the first time One of these concepts was an

average reserve provided by the given stability standard

Where:i=1,2,3, n is an index of a certain vessel in the list of the lost vessels; n is the

total number of the vessels in the list;

cri

cri i

i

KG

KG KG

(1.2) The sign of the f i -value is to be determined by the numerator of this fraction

It is obvious that, if x i> 0 the lost vessel's stability would be recognized insufficient in

accordance with the given standard (KG i > KG cri) and such a conclusion would be

justified by the fact of the loss But the inequality x i < 0 means an error of the given

standards because the stability of this capsized vessel would be evaluated as quite

sufficient

It is evident that those standards which give less scatter (variance) of their values xi are

more relevant to the real circumstances of vessel loss That's why it was possible to

suggest a special measure of inadequacy for any stability standards This measure is the

n

i

i

The introduction of the concepts of a mean stability reserve x and V xlead to one more

concept, namely a concept of the ideal stability standard Such an imaginary standard

would be able to predict for all registered stability casualties the true "critical" KG -

values, that is just those values which actually took place at the moment of loss of the

perished vessels These predicted KG cri -values would exactly correspond to the

circumstances of each casualty In other words, such ideal standards applied to the lost

vessels would give the values:

0{



cri

cri i

i

KG

KG KG

Chapter 1

8

01

n

i

Of course, such ideal standards should take into account the smallest variations of any

arguments and parameters influencing stability in any imaginable situation But we must

admit that mankind will hardly be able to compile such precise equations of ship motion,

such detailed descriptions of external forces and in addition to have at its disposal a

sufficient number of absolutely precise and detailed stability casualty records This is

why it is impossible to achieve such an ideal standard in practice However, it is equally

true for any other ideal concepts; for example, for the ideal fluid, ideal propeller or ideal

thermodynamic cycle in combustion engines Nevertheless, this consideration does not

make such ideal concepts useless because they indicate those limits, which cannot be

exceeded by efforts of human wisdom and inventiveness In any case, the ideal models

permit comparison of the suitability of different approximate standards, mathematical

models of a fluid, construction of machines and propulsive devices

It was found that, in particular, that the set of x i- values taken from a sufficient number

of capsized vessels may give the evaluation of some conventional guarantee of safety,

which is provided by the given set of stability criteria and norms

Such a conventional guarantee is the average probability of safe navigation of a vessel

belonging to a fleet This fleet consists of vessels of the same type, which are included in

the list of casualty records It is also supposed that the vessels of this fleet keep -

permanently - the position of their centers of gravity at the height which is critical for

each of them in accordance with the given stability standard and its criteria and norms

Such a conventional guarantee can be calculated if one knows the law of distribution of

thex i values besides the x and V x values In the terms of the theory of probability and

mathematical statistics it means that we consider the total number of casualties as a set of

test results in which the random continuous variable f was realized in the values of x i A

special checking procedure was carried out in accordance with so called F2-criterion of

compliance It confirmed that normal (Gaussian) distribution law might be used with an

acceptable degree of accuracy to express the distribution of x The density of distribution

in accordance with the normal law may be written in the form

2 2

2 ) (

2

1)

x

x

e x

The curves in Fig 1.1 show the normal distribution obtained on the basis of IMO Intact

Stability Casualty Records [Sevastianov, 1970] for fishing vessels Curve N 1

corresponds to IMO Stability Recommendations for Fishing Vessels, [IMO, 1968] Curve

N 3 is drawn for a simplified stability criterion (see formula 1.19 below), which was

discussed at an early stage of IMO activity Curve N 2 is calculated according to the

Rules for Classification and Construction of Seagoing Steel Vessels issued by the

Russian Register in 1967

Fig 1.1 Normal distribution of accidental value x i:

1.- IMO Stability recommendation, 1968 2.- Draft Stability Standards of the Russian Register, 1967

3.- Simplified criterion, formula (1.19)

It is useful to recall that the law of distribution of random variables may be written in an

integral form:

dx x f x

If the probability density fx corresponds to the normal law (1.7) then:

dt e x

x t

M

f

 V





SV

2 2

2 ) (

2

1)

The function P(x) may be interpreted as the probability of such an accidental event,

which is to be determined by the double inequality:

)

(x P x x

The integral (1.9) may be calculated by numerical methods or by application of so called

Laplace function or by the integral of probability tabulated in the mathematical manuals

dz e y

2

2

2)

Chapter 1

10

x x at y

x at x

y

x at y

x

V



rfrf

0

Thus the value of Px 0 expresses the probability of an event that the value of x will be

within the interval fdx id0 , that is x i chosen at random will be negative

The value of Px  is given by the formula:

)



f)



x x

x x

x x

x

2

1)

(2

1

)

0

We have already come to the conclusion that a value x i means an error for the given

stability standard Such a standard would consider the corresponding capsized vessel

quite stable since its actual height of the centre of gravity was below the critical value at

the time of loss Therefore, the probability of Px 0 is a measure of unreliability for

such a standard Then the probability of correct sign of xi

)0(

1

It may be called a conventional guarantee of safety provided by the given stability

standard By using the word "conventional" we must bear in mind some special meaning

We assume that the master of a ship thoroughly controls the height of the actual centre of

gravity of his vessel to prevent its shifting above the critical level KG cr determined by the

given stability standard It means that at any moment the following inequality takes place:

cri

KG

If we assume that the real guarantee would be greater than the conventional one, then the

conventional guarantee may be considered as an estimation of the minimal value of a real

guarantee which is provided by the given stability standard Fig 1.2 shows the curves

P x for the stability requirements of the Russian Register (curve 1), for the IMO

Recommendations, 1968 (curve 2, fishing vessels) and for two rather simple but primitive

standards N1 and N2 which were considered at the very beginning of the discussion on

Stability Recommendations These two standards may be reduced to the setting of the

critical initial metacentric height values:

m,035.020.0

2

f

B

Here: GM cr1 is the critical (the least permissible) initial metacentric height in accordance

with the first variant of the standard (1.18); GM cr2 is the same for the second variant

(1.19); B is breadth of a vessel amidships; f is the least actual freeboard value in the given

loading conditions Other particulars of size and form of a vessel are not taken into

account by these variants of suggested standards Table 1.1 contains calculated numerical characteristics x,Vx , Px=0 and *

Fig 1.2 clearly shows that a high value of guarantee, * may be achieved by two alternative methods

The first method suggests improving the stability standard by the minimization of its measure of inadequacy Vx, that is by reducing VM as close to zero as is possible (see

formula 1.15) For a constant value x ,Vx decreases, the slope of the curves Px increase

and the conventional safety guarantee*is tending to the ideal value * 1 (Fig 1.2)

Fig 1.2 Integral distribution law of the accidental value x

1.- Draft Stability Standards of the Russian Register 2 - IMO Stability recommendation, 1968.,

1967 3.- Simplified criterion, formula (1.18) 3.- Simplified criterion, formula (1.19) 5.-”Ideal”

Standards

The second advantage of this method is that the average reserve x may be decreased

with decreasing Vx-value at a constant or even increased value *

These advantages make the first method of improving stability standards the most rational

The second method is much simpler, but not so rational It may be implemented by assigning more conservative norms for each criterion Let us make this point clear by the example of standards N1 and N2 They differ from each other only in the critical values

of GM cr which for standard N2 is 0.3 m larger than N1 It means that the critical height of

the centre of gravity KG of a certain vessel would be smaller than KG by the same

20 10 0 10 20 30 40

0.2 0.4 0.6 0.8

Chapter 1

12

value of 0.3 m Consequently the average stability reserve x2 should be larger than x1

Therefore, the curve P2x will be shifted along the x -axis towards the greater x -values

But the Vx-value remains almost constant That's why the slope of the curve P2x