Preface xi
Acknowledgements xv
1 Measures of Structural Safety 1
1.1 Introduction 1
1.2 Uncertainties in Structural Design 1
1.2.1 Uncertainties in the Properties of Structures and Their Environment 1
1.2.2 Sources and Types of Uncertainties 4
1.2.3 Treatment of Uncertainties 5
1.2.4 Design and Decision Making with Uncertainties 9
1.3 Deterministic Measures of Safety 9
1.4 Probabilistic Measure of Safety 10
1.5 Summary 10
2 Fundamentals of Structural Reliability Theory 11
2.1 The Fundamental Case 11
2.2 Performance Function and Probability of Failure 16
2.2.1 Performance Function 16
2.2.2 Probability of Failure 17
2.2.3 Reliability Index 19
2.3 Monte Carlo Simulation 24
2.3.1 Formulation of the Probability of Failure 24
2.3.2 Generation of Random Numbers 25
2.3.3 Direct Sampling for Structural Reliability Evaluation 28
2.4 A Brief Review on Structural Reliability Theory 33
2.5 Summary 35
3 Moment Evaluation for Performance Functions 37
3.1 Introduction 37
3.2 Moment Computation for some Simple Functions 39
3.2.1 Moment Computation for Linear Sum of Random Variables 39
3.2.2 Moment Computation for Products of Random Variables 42
3.2.3 Moment Computation for Power of a Lognormally Distributed Random Variable 45
3.2.4 Moment Computation for Power of an Arbitrarily Distributed Random Variable 50
3.2.5 Moment Computation for Reciprocal of an Arbitrary Distributed Random Variable 52
3.3 Point Estimates for a Function with One Random Variable 53
3.3.1 Rosenblueth’s Two-Point Estimate 53
3.3.2 Gorman’s Three-Point Estimate 55
3.4 Point Estimates in Standardised Normal Space 60
3.4.1 Formulae of Moment of Functions with Single Random Variable 60
3.4.2 Two- and Three-Point Estimates in the Standard Normal Space 63
3.4.3 Five-Point Estimate in Standard Normal Space 64
3.4.4 Seven-Point Estimate in Standard Normal Space 65
3.4.5 General Expression of Estimating Points and Their Corresponding Weights 68
3.4.6 Accuracy of the Point Estimate 70
3.5 Point Estimates for a Function of Multiple Variables 76
3.5.1 General Expression of Point Estimates for a Function of n Variables 76
3.5.2 Approximate Point Estimates for a Function of n Variables 78
3.5.3 Dimension Reduction Integration 83
3.6 Point Estimates for a Function of Correlated Random Variables 89
3.7 Hybrid Dimension-Reduction Based Point Estimate Method 94
3.8 Summary 96
4 Direct Methods of Moment 99
4.1 Basic Concept of Methods of Moment 99
4.1.1 Integral Expression of Probability of Failure 99
4.1.2 The Second-Moment Method 100
4.1.3 General Expressions for Methods of Moment 102
4.2 Third-Moment Reliability Method 104
4.2.1 General Formulation of the Third-Moment Reliability Index 104
4.2.2 Third-Moment Reliability Indices 106
4.2.3 Empirical Applicable Range of Third-Moment Method 110
4.2.4 Simplification of Third-Moment Reliability Index 113
4.2.5 Applicable Range of the Second-Moment Method 117
4.3 Fourth-Moment Reliability Method 122
4.3.1 General Formulation of the Fourth-Moment Reliability Index 122
4.3.2 Fourth-Moment Reliability Index on the Basis of the Pearson System 124
4.3.3 Fourth-Moment Reliability Index Based on Third-Order Polynomial Transformation 127
4.3.4 Applicable Range of Fourth-Moment Method 130
4.3.5 Simplification of Fourth-Moment Reliability Index 136
4.4 Summary 138
5 Methods of Moment Based on First- and Second-Order Transformation 139
5.1 Introduction 139
5.2 First-Order Reliability Method 139
5.2.1 The Hasofer-Lind Reliability Index 139
5.2.2 First-Order Reliability Method 141
5.2.3 Numerical Solution for FORM 147
5.2.4 The Weakness of FORM 153
5.3 Second-Order Reliability Method 156
5.3.1 Necessity of Second-Order Reliability Method 156
5.3.2 Second-Order Approximation of the Performance Function 157
5.3.3 Failure Probability for Second-Order Performance Function 170
5.3.4 Methods of Moment for Second-Order Approximation 175
5.3.5 Applicable Range of FORM 187
5.4 Summary 191
6 Structural Reliability Assessment Based on the Information of Moments of Random Variables 193
6.1 Introduction 193
6.2 Direct Methods of Moment without Using Probability Distribution 195
6.2.1 Second-Moment Formulation 195
6.2.2 Third-Moment Formulation 196
6.2.3 Fourth-Moment Formulation 196
6.3 First-Order Second-Moment Method 197
6.4 First-Order Third-Moment Method 203
6.4.1 First-Order Third-Moment Method in Reduced Space 203
6.4.2 First-Order Third-Moment Method in Third-Moment Pseudo Standard Normal Space 204
6.5 First-Order Fourth-Moment Method 219
6.5.1 First-Order Fourth-Moment Method in Reduced Space 219
6.5.2 First-Order Fourth-Moment Method in Fourth-Moment Pseudo Standard Normal Space 220
6.6 Monte Carlo Simulation Using Statistical Moments of Random Variables 233
6.7 Subset Simulation Using Statistical Moments of Random Variables 246
6.8 Summary 252
7 Transformation of Non-Normal Variables to Independent Normal Variables 253
7.1 Introduction 253
7.2 The Normal Transformation for a Single Random Variable 253
7.3 The Normal Transformation for Correlated Random Variables 254
7.3.1 Rosenblatt Transformation 254
7.3.2 Nataf Transformation 255
7.4 Pseudo Normal Transformations for a Single Random Variable 265
7.4.1 Concept of Pseudo Normal Transformation 265
7.4.2 Third-Moment Pseudo Normal Transformation 267
7.4.3 Fourth-Moment Pseudo Normal Transformation 274
7.5 Pseudo Normal Transformations of Correlated Random Variables 293
7.5.1 Introduction 293
7.5.2 Third-Moment Pseudo Normal Transformation for Correlated Random Variables 295
7.5.3 Fourth-Moment Pseudo Normal Transformation for Correlated Random Variables 298
7.6 Summary 306
8 System Reliability Assessment by the Methods of Moment 307
8.1 Introduction 307
8.2 Basic Concepts of System Reliability 307
8.2.1 Multiple Failure Modes 307
8.2.2 Series Systems 308
8.2.3 Parallel Systems 311
8.3 System Reliability Bounds 318
8.3.1 Uni-Modal Bounds 318
8.3.2 Bi-Modal Bounds 320
8.3.3 Correlation Between a Pair of Failure Modes 322
8.3.4 Bound Estimation of the Joint Failure Probability of a Pair of Failure Modes 324
8.3.5 Point Estimation of the Joint Failure Probability of a Pair of Failure Modes 327
8.4 Moment Approach for System Reliability 338
8.4.1 Performance Function for a System 339
8.4.2 Methods of Moment for System Reliability 342
8.5 System Reliability Assessment of Ductile Frame Structures Using Methods of Moment 352
8.5.1 Challenges on System Reliability of Ductile Frames 352
8.5.2 Performance Function Independent of Failure Modes 353
8.5.3 Limit Analysis 355
8.5.4 Methods of Moment for System Reliability of Ductile Frames 356
8.6 Summary 364
9 Determination of Load and Resistance Factors by Methods of Moment 365
9.1 Introduction 365
9.2 Load and Resistance Factors 366
9.2.1 Basic Concept 366
9.2.2 Determination of LRFs by Second-Moment Method 366
9.2.3 Determination of LRFs Under Lognormal Assumption 369
9.2.4 Determination of LRFs Using FORM 370
9.2.5 An Approximate Method for the Determination of LRFs 378
9.3 Load and Resistance Factors by Third-Moment Method 380
9.3.1 Determination of LRFs Using Third-Moment Method 380
9.3.2 Estimation of the Mean Value of Resistance 382
9.4 General Expressions of Load and Resistance Factors Using Methods of Moment 388
9.5 Determination of Load and Resistance Factors Using Fourth-Moment Method 390
9.5.1 Basic Formulas 390
9.5.2 Determination of the Mean Value of Resistance 391
9.6 Summary 396
10 Methods of Moment for Time-Variant Reliability 397
10.1 Introduction 397
10.2 Simulating Stationary Non-Gaussian Process Using the Fourth-Moment Transformation 397
10.2.1 Brief Review on Simulating Stationary Non-Gaussian Process 397
10.2.2 Transformation for Marginal Probability Distributions 398
10.2.3 Transformation for Correlation Functions 399
10.2.4 Methods to Deal with the Incompatibility 403
10.2.5 Scheme of Simulating Stationary Non-Gaussian Random Processes 404
10.3 First Passage Probability Assessment of Stationary Non-Gaussian Processes Using the Fourth-Moment Transformation 415
10.3.1 Brief Review on First Passage Probability 415
10.3.2 Formulation of the First Passage Probability of Stationary Non-Gaussian Structural Responses 415
10.3.3 Computational Procedure for the First Passage Probability of Stationary Non-Gaussian Structural Responses 417
10.4 Time-Dependent Structural Reliability Analysis Considering Correlated Random Variables 419
10.4.1 Brief Review on Time-Dependent Structural Reliability Methods 419
10.4.2 Formulation of Time-Dependent Failure Probability 420
10.4.3 Fast Integration Algorithms for the Time-Dependent Failure Probability 422
10.5 Summary 434
11 Methods of Moment for Structural Reliability with Hierarchical Modelling of Uncertainty 435
11.1 Introduction 435
11.2 Formulation Description of the Structural Reliability with Hierarchical Modelling of Uncertainty 436
11.3 Overall Probability of Failure Due to Hierarchical Modelling of Uncertainty 438
11.3.1 Evaluating Overall Probability of Failure Based on FORM 438
11.3.2 Evaluating Overall Probability of Failure Based on Methods of Moment 441
11.3.3 Evaluating Overall Probability of Failure Based on Direct Point Estimates 442
11.4 The Quantile of the Conditional Failure Probability 447
11.5 Application to Structural Dynamic Reliability Considering Parameters Uncertainties 456
11.6 Summary 462
12 Structural Reliability Analysis Based on Linear Moments 463
12.1 Introduction 463
12.2 Definition of L-Moments 463
12.3 Structural Reliability Analysis Based on the First Three L-Moments 465
12.3.1 Transformation for Independent Random Variables 465
12.3.2 Transformation for Correlated Random Variables 468
12.3.3 Reliability Analysis Using the First Three L-Moments and Correlation Matrix 471
12.4 Structural Reliability Analysis Based on the First Four L-Moments 479
12.4.1 Transformation for Independent Random Variables 479
12.4.2 Transformation for Correlated Random Variables 486
12.4.3 Reliability Analysis Using the First Four L-Moments and Correlation Matrix 491
12.5 Summary 495
13 Methods of Moment with Box-Cox Transformation 497
13.1 Introduction 497
13.2 Methods of Moment with Box-Cox Transformation 498
13.2.1 Criterion for Determining the Box-Cox Transformation Parameter 498
13.2.2 Procedure of the Methods of Moment with Box-Cox Transformation for Structural Reliability 499
13.3 Summary 512
Appendix A Basic Probability Theory 513
Appendix B Three-Parameter Distributions 561
Appendix C Four-Parameter Distributions 579
Appendix D Basic Theory of Stochastic Process 605
References 615
Index 631
