Palle Thoft-Christensen Michael 1. Baker

Structural Reliability Theory and Its Applications

With 107 Figures

Springer-Verlag Berlin Heidelberg New York 1982

PALLE THOFT-CHRISTENSEN, Professor, Ph. D. Institute of Building Technology and Structural Engineering Aalborg University Centre Aalborg, Denmark

MICHAEL J. BAKER, B.Sc. (Eng) Department of Civil Engineering Imperial College of Science and Technology London, England

Library of Congr~S5 ('ataloging In Publication DatH

Thoft-Christensen, Falle. Structural relialxillty theory and its applica­

tions.

Includes lxi bliograIilies and index. 1. Structural stability. 2. Reliability

(Engineering) 1. Baker, Michael J. (Michael John), 1940- . II. Title. TA656.5.T47 1982 624.1 1 71 82-10277 ISBN-13:978-3-642-68699-3(U.S.)

ISBN -13:978-3-642-68699-3 e-ISBN -13 :978-3-642-68697-9 DOl: 10.1007/978-3-642-68697-9

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© Springer-Verlag Berlin, Heidelberg 1982 Softcover reprint of the hardcover 1st edition 1982

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206113020 - 543210

PREFACE

Structural reliability theory is concerned with the rational treatment of uncertainties in struc­

tural engineering and with the methods for assessing the safety and serviceability of civil en­

gineering and other structures. It is a subject which has grown rapidly during the last decade

and has evolved from being a topic for academic research to a set of well-developed or develop­

ing methodologies with a wide range of practical applications.

Uncertainties exist in most areas of civil and structural engineeri'1.g and rational design decisions

cannot be made without modelling them and taking them into account. Many structural en­

gineers are shielded from having to think about such problems, at least when designing simple

structures, because of the prescriptive and essentially deterministic nature of most codes of

practice. This is an undesirable situation. Most loads and other structural design parameters are

rarely known with certainty and should be regarded as random variables or stochastic processes,

even if in design calculations they are eventually treated as deterministic. Some problems such

as the analysis of load combinations cannot even be formulated without recourse to probabilistic

reasoning.

There is a need for all structural engineers to develop an understanding of structural reliability

theory and for this to be applied in design and construction, either indirectly through codes or

by direct application in the case of special structures having large failure consequences, the aim

in both cases being to achieve economy together with an appropriate degree of safety. The sub­

ject is now sufficiently well developed for it to be included as a formal part of the training of

all civil and structural engineers, both at undergraduate and post-graduate levels. Courses on

structural safety have been given at some universities for a number of years.

In writing this book we have tried to bring together under one cover the major components

of structural reliability theory with the aim of making it possible for a newcomer to see and

VI

study the subject as a whole. The book should be of value to those with no prior knowledge

of reliability theory, but it should also be of interest to those engineers involved in the de­

velopment of structural and loading codes and to those concerned with the safety assessment

of complex structures. The book does not try to cover all aspects of structural safety and no

attempt is made, for example, to discuss structural failures except in general statistical terms.

It was the intention to make this book moderately self-contained and for this reason chapter

2 is devoted to the essential fundamentals of probability theory. However, readers who have

had no training in this branch of mathematics would be well advised to study a more general

text in addition. Topics such as the statistical theory of extremes, methods of parameter esti­

mation and stochastic process theory are introduced in later chapters as and when they are re­

quired. The main core of the book is devoted to the so-called level 2 methods of analysis which

have provided the key to fast computational procedures for structural reliability calculations.

Other chapters cover the reliability of structural systems, load combinations, gross errors and

some major areas of application.

The work is set out in the form of a textbook with a number of examples and simple exer­

cises. The purpose of these is to illustrate the important principles and methods and to extend

the scope of the main text with economy of space. The reader is warned against a too literal in­

terpretation of some of the simple examples as these were not included to provide insight into

particular practical problems. In some examples, the parameters of the probability distributions

used in the calculations have been chosen quite arbitrarily or in such a way as to demonstrate

the calculation procedure with maximum effect. This does not mean that the practical aspects

of structural reliability theory have been overlooked - indeed, the theory would be of little

value if it could not be applied. Chapter 11, on the application of reliability theory to the de­

velopment of level 1 codes, attempts to address many of the practical problems faced by code

writers in the selection of partial coefficients (partial factors); and in chapter 3, the modelling

of load and resistance variables has been approached with applications strongly in mind. How­

ever, a complete book would be required to cover this subject in depth. Chapter 12 on off­

shore structures should be of interest to those working in this field.

In compiling the bibliography our approach has been to list only a selection of the more im­

portant works in each subject area, along with other works to which specific reference is made.

Whilst many important contributions to the literature are thus omitted, it is considered that

this selective approach will be of more help to the new reader.

We should like to acknowledge the major contributions to the field of structural reliability

theory that have been made by a relatively small number of people, mainly during the last 10

to 15 years, and without which this book would not have been possible. The subject has bene­

fitted from a large degree of international co-operation which has been stimulated by various

bodies - in particular, the Joint Committee on Structural Safety under the chairmanship of

J. Ferry Borges. The responsibility for this book must, however, rest with the authors and we

should be pleased to receive notification of corrections or omissions of any nature.

VII

Thanks are due to our respective colleagues in Aalborg and London for their helpful comments

and contributions and in particular to Mrs. Kirsten Aakj<er and Mrs. Norma Hornung who

have undertaken the type-setting and drawing of figures, respectively, with such skill and

efficiency.

We conclude with some words of caution. Structural reliability theory should not be thought

of as the solution to all safety problems or as a procedure which can be applied in a mechanical

fashion. In the right hands it is a powerful tool to aid decision making in matters of structural

safety, but like other tools it can be misused. It should not be thought of as an alternative to

more traditional methods of safety analysis, because all the information that is currently used

in other approaches can and should be incorporated within a reliability analysis. On occasions

the theory may give results which seem to contradict »experience». In this case, either »experi­

ence» will be found to have been incorrectly interpreted or some part of the reliability analysis

will be at fault, generally the modelling. The resolution of these real or apparent contradictions

will often provide considerable insight into the nature of the problem being examined, which can

only be of benefit.

March, 1982

Palle Thoft-Christensen

Institute of Building Technology

and Structural Engineering

Aalborg University Centre

Aalborg, Denmark

Michael J. Baker

Department of Civil Engineering

Imperial College of Science and Technology

London, England

IX

CONTENTS

1. THE TREATMENT OF UNCERTAINTIES IN STRUCTURAL ENGINEERING.. .. 1

1.1 INTRODUCTION................................................ 1 1.1.1 Current risk levels, 2 1.1.2 Structural codes, 3

1.2 UNCERTAINTy................................................. 4 1.2.1 General, 4 1.2.2 Basic variables, 5 1.2.3 Types of uncertainty, 6

1.3 STRUCTURAL RELIABILITY ANALYSIS AND SAFETY CHECKING...... 7 1.3.1 Structural reliability, 8 1.3.2 Methods of safety checking, 10

BIBLIOGRAPHY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11

2. FUNDAMENTALS OF PROBABILITY THEORY. . . . . . . . . . . . . . . . . . . . . . . . . .. 13

2.1 INTRODUCTION................................................ 13

2.2 SAMPLE SPACE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13

2.3 AXIOMS AND THEOREMS OF PROBABILITY THEORY. . . . . . . . . . . . . . .. 15

2.4 RANDOM VARIABLES. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19

2.5 MOMENTS..................................................... 22

2.6 UNIVARIATE DISTRIBUTIONS. . . . . . . . .. .. . . . . . . .. . . . . . . . . . . . . . . .. 25

2.7 RANDOM VECTORS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 28

2.8 CONDITIONAL DISTRIBUTIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31

2.9 FUNCTIONS OF RANDOM VARIABLES. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 32

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35

3. PROBABILISTIC MODELS FOR LOADS AND RESISTANCE VARIABLES. . . . .. 37

3.1 INTRODUCTION................................................ 37

3.2 STATISTICAL THEORY OF EXTREMES. . . . . . . . . . . .. . . . . . . . . . . . . . . .. 37 3.2.1 Derivation of the cumulative distribution of the ith smallest value of

n identically distributed independent random variables X j , 38 3.2.2 Normal extremes, 39

3.3 ASYMPTOTIC EXTREME-VALUE DISTRIBUTIONS. . . . . . . . . . . . . . . . . . .. 40 3.3.1 Type I extreme-value distributions (Gumbel distributions), 40 3.3.2 Type II extreme-value distributions, 42 3.3.3 Type III extreme-value distributions, 42

x

3.4 MODELLING OF RESISTANCE VARIABLES - MODEL SELECTION ...... , 44 3.4.1 General remarks, 44 3.4.2 Choice of distributions for resistance variables, 52

3.5 MODELLING OF LOAD VARIABLES - MODEL SELECTION. . . . . . . . . . . .. 54 3.5.1 General remarks, 54 3.5.2 Choice of distributions of loads and other actions, 58

3.6 ESTIMATION OF DISTRIBUTION PARAMETERS. .. . . . . . . .. . . . . . . .. .. 59 3.6.1 Techniques for parameter estimation, 59 3.6.2 Model verification, 63

3.7 INCLUSION OF STATISTICAL UNCERTAINTY .......... , . . . . . . .. .. .. 63

BIBLIOG RAPHY ................................................... , 64

4. FUNDAMENTALS OF STRUCTURAL RELIABILITY THEORY ... _ .......... , 67

4.1 INTRODUCTION................................................ 67

4.2 ELEMENTS OF CLASSICAL RELIABILITY THEORY.. . . . . . . .. . . . . .. .. 67

4.3 STRUCTURAL RELIABILITY ANALYSIS. . . . . . . . . . . . . . . . . . . . . . . . . . .. 70 4.3.1 General,70 4.3.2 The fundamental case, 71 4.3.3 Problems reducing to the fundamental case, 75 4.3.4 Treatment of a single time-varying load, 77 4.3.5 The general case, 77 4.3.6 Monte-Carlo methods, 79

BIBLIOGRAPHY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 80

5. LEVEL 2 METHODS _ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 81

5.1 INTRODUCTION................................................ 81

5.2 BASIC VARIABLES AND F AlLURE SURFACES ...................... , 81

5.3 RELIABILITY INDEX FOR LINEAR FAILURE FUNCTIONS AND NOR-MAL BASIC VARIABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 83

5.4 HASOFER AND LIND'S RELIABILITY INDEX.. . . .. . . . . . . . . . . . . .. .. .. 88

BIBLIOGRAPHY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 93

6. EXTENDED LEVEL 2 METHODS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95

6.1 INTRODUCTION................................................ 95

6.2 CONCEPT OF CORRELATION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 96

6.3 CORRELATED BASIC VARIABLES ................................. 101

6.4 NON-NORMAL BASIC VARIABLES ................................. 108

BIBLIOGRAPHY .................................................... 110

XI

7. RELIABILITY OF STRUCTURAL SYSTEMS ............................. 113

7.1 INTRODUCTION ............................................... 113

7.2 PERFECTLY BRITTLE AND PERFECTLY DUCTILE ELEMENTS ........ 114

7.3 FUNDAMENTAL SySTEMS ...................................... 115

7.4 SYSTEMS WITH EQUALLY CORRELATED ELEMENTS ............... 122

BIBLIOGRAPHY .................................................. 127

8. RELIABILITY BOUNDS FOR STRUCTURAL SYSTEMS ................... 129

8.1 INTRODUCTION ............................................... 129

8.2 SIMPLE BOUNDS ............................................... 130

8.3 DITLEVSEN BOUNDS ........................................... 133

8.4 PARALLEL SYSTEMS WITH UNEQUALLY CORRELATED ELEMENTS .. 134

8.5 SERIES SYSTEMS WITH UNEQUALLY CORRELATED ELEMENTS ...... 136

BIBLIOGRAPHY .................................................. 143

9. INTRODUCTION TO STOCHASTIC PROCESS THEORY AND ITS USES ....... 145

9.1 INTRODUCTION ............................................... 145

9.2 STOCHASTIC PROCESSES ....................................... 145

9.3 GAUSSIAN PROCESSES ......................................... 148

9.4 BARRIER CROSSING PROBLEM .................................. 150

9.5 PEAK DISTRIBUTION ........................................... 156

BIBLIOGRAPHY .................................................. 159

10. LOAD COMBINATIONS ............................................. 161

10.1 INTRODUCTION ............................................... 161

10.2 THE LOAD COMBINATION PROBLEM ................. ' ............ 162

10.3 THE FERRY BORGES·CASTANHETA LOAD MODEL ................. 166

10.4 COMBINATION RULES .......................................... 168

BIBLIOGRAPHY ................................................... 175

11. APPLICATIONS TO STRUCTURAL CODES .............................. 177

11.1 INTRODUCTION ............................................... 177

11.2 STRUCTURAL SAFETY AND LEVEL 1 CODES ...................... 178

XII

11.3 RECOMMENDED SAFETY FORMATS FOR LEVEL 1 CODES. . . . . . . . . .. 180 11.3.1 Limit state functions and checking equations, 180 11.3.2 Characteristic values of basic variables, 182 11.3.3 Treatment of geometrical variables, 183 11.3.4 Treatment of material properties, 185 11.3.5 Treatment of loads and other actions, 185

11.4 METHODS FOR THE EVALUATION OF PARTIAL COEFFICIENTS ...... 188 11.4.1 Relationship of partial coefficients to level 2 design point, 188 11.4.2 Approximate direct method for the evaluation of partial coeffi-

cients,190 11.4.3 General method for the evaluation of partial coefficients, 194

11.5 AN EXAMPLE OF PROBABILISTIC CODE CALIBRATION .......... , .. 196 11.5.1 Aims of calibration, 196 11.5.2 Results of calibration, 198

BIBLIOGRAPHY ................................................... 201

12. APPLICATIONS TO FIXED OFFSHORE STRUCTURES .................... 203

12.1 INTRODUCTION ............................................... 203

12.2 MODELLING THE RESPONSE OF JACKET STRUCTURES FOR RELIA­BILITY ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 203 12.2.1 Sea-state model, 207 12.2.2 Wave model, 215 12.2.3 Loading model, 217 12.2.4 Natural frequency model, 219 12.2.5 Evaluation of structural response, 219 12.2.6 Evaluation of peak response, 220 12.2.7 Other models, 222

12.3 PROBABILITY DISTRIBUTIONS FOR IMPORTANT LOADING VARI-ABLES ....................................................... 223 12.3.1 Wind speed, 223 12.3.2 Morison's coefficients, 225

12.4 METHODS OF RELIABILITY ANALYSIS ........................... 226 12.4.1 General, 226 12.4.2 Level 2 method, 227

12.5 SOME RESULTS FROM THE STUDY OF A JACKET STRUCTURE ....... 232

BIBLIOGRAPHY ................................................... 234

13. RELIABILITY THEORY AND QUALITY ASSURANCE .................... 239

13.1 INTRODUCTION ............................................... 239

13.2 GROSS ERRORS ............................................... 239 13.2.1 General, 239 13.2.2 Classification of gross errors, 241

XIII

13.3 INTERACTION OF RELIABILITY AND QUALITY ASSURANCE ........ 243 13.3.1 General,243 13.3.2 The effect of gross errors on the choice of partial coefficients, 244 ...... .

13.4 QUALITY ASSURANCE ......................................... 247

BIBLIOGRAPHY ................................................... 247

APPENDIX A. RANDOM NUMBER GENERATORS ........................... 249

1. GENERAL ................................................... , 249

2. UNIFORM RANDOM NUMBER GENERATORS ...................... 249

3. MULTIPLICATIVE CONGRUENCE METHOD ........................ 250

4. GENERATION OF RANDOM DEVIATES WITH A SPECIFIED PROB-ABILITY DISTRIBUTION FUNCTION Fx ........................... 251

5. SPECIAL CASES: GENERATION OF RANDOM DEVIATES HAVING NORMAL AND LOG-NORMAL DISTRIBUTIONS ..................... 252

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 253

APPENDIX B. SPECTRAL ANALYSIS OF WAVE FORCES .................... 255

1. INTRODUCTION ............................................... 255

2. GENERAL EQUATIONS OF MOTION .............................. 255

3. MODAL ANALYSIS .......................... , .................. 257

4. SOLUTION STRATEGY ......................................... 258

5. MULTIPLE PILES .............................................. 261

6. COMPUTATIONAL PROCEDURE .................................. 261

BIBLIOGRAPHY ................................................... 261

INDEX .............................................................. 263