Structural Reliability Analysis and Prediction
3. Edition December 2017
528 Pages, Softcover
Wiley & Sons Ltd
ISBN:
978-1-119-26599-3
John Wiley & Sons
Further versions
Structural Reliability Analysis and Prediction, Third Edition is a textbook which addresses the important issue of predicting the safety of structures at the design stage and also the safety of existing, perhaps deteriorating structures. Attention is focused on the development and definition of limit states such as serviceability and ultimate strength, the definition of failure and the various models which might be used to describe strength and loading. This book emphasises concepts and applications, built up from basic principles and avoids undue mathematical rigour. It presents an accessible and unified account of the theory and techniques for the analysis of the reliability of engineering structures using probability theory.
This new edition has been updated to cover new developments and applications and a new chapter is included which covers structural optimization in the context of reliability analysis. New examples and end of chapter problems are also now included.
Preface xv
Preface to the Second Edition xvii
Preface to the First Edition xviii
Acknowledgements xx
1 Measures of Structural Reliability 1
1.1 Introduction 1
1.2 DeterministicMeasures of Limit State Violation 2
1.2.1 Factor of Safety 2
1.2.2 Load Factor 3
1.2.3 Partial Factor ('Limit State Design') 4
1.2.4 A Deficiency in Some SafetyMeasures: Lack of Invariance 5
1.2.5 Invariant SafetyMeasures 8
1.3 A Partial Probabilistic SafetyMeasure of Limit State Violation--The Return
Period 8
1.4 Probabilistic Measure of Limit State Violation 12
1.4.1 Introduction 12
1.4.2 The Basic Reliability Problem 14
1.4.3 Special Case: Normal Random Variables 17
1.4.4 Safety Factors and Characteristic Values 19
1.4.5 Numerical Integration of the Convolution Integral 23
1.5 Generalized Reliability Problem 24
1.5.1 Basic Variables 24
1.5.2 Generalized Limit State Equations 25
1.5.3 Generalized Reliability Problem Formulation 26
1.5.4 Conditional Reliability Problems* 27
1.6 Conclusion 29
2 Structural Reliability Assessment 31
2.1 Introduction 31
2.2 Uncertainties in Reliability Assessment 33
2.2.1 Identification of Uncertainties 33
2.2.2 Phenomenological Uncertainty 34
2.2.3 Decision Uncertainty 34
2.2.4 Modelling Uncertainty 34
2.2.5 Prediction Uncertainty 35
2.2.6 Physical Uncertainty 36
2.2.7 Statistical Uncertainty 36
2.2.8 Uncertainties Due to Human Factors 37
2.2.8.1 Human Error 37
2.2.8.2 Human Intervention 40
2.2.8.3 Modelling of Human Error and Intervention 43
2.2.8.4 Quality Assurance 44
2.2.8.5 Hazard Management 45
2.3 Integrated Risk Assessment 45
2.3.1 Calculation of the Probability of Failure 45
2.3.2 Analysis and Prediction 47
2.3.3 Comparison to Failure Data 48
2.3.4 Validation--a Philosophical Issue 50
2.3.5 The Tail Sensitivity 'Problem' 50
2.4 Criteria for Risk Acceptability 51
2.4.1 Acceptable Risk Criterion 51
2.4.1.1 Risks in Society 51
2.4.1.2 Acceptable or Tolerable Risk Levels 53
2.4.2 Socio-economic Criterion 54
2.5 Nominal Probability of Failure 56
2.5.1 General 56
2.5.2 Axiomatic Definition 56
2.5.3 Influence of Gross and Other Errors 57
2.5.4 Practical Implications 58
2.5.5 Target Values for Nominal Failure Probability 59
2.6 Hierarchy of Structural ReliabilityMeasures 60
2.7 Conclusion 61
3 Integration and Simulation Methods 63
3.1 Introduction 63
3.2 Direct and Numerical Integration 63
3.3 Monte Carlo Simulation 65
3.3.1 Introduction 65
3.3.2 Generation of Uniformly Distributed Random Numbers 65
3.3.3 Generation of Random Variates 66
3.3.4 Direct Sampling ('Crude' Monte Carlo) 68
3.3.5 Number of Samples Required 69
3.3.6 Variance Reduction 72
3.3.7 Stratified and Latin Hypercube Sampling 73
3.4 Importance Sampling 73
3.4.1 Theory of Importance Sampling 73
3.4.2 Importance Sampling Functions 75
3.4.3 Observations About Importance Sampling Functions 76
3.4.4 Improved Sampling Functions 79
3.4.5 Search or Adaptive Techniques 80
3.4.6 Sensitivity 81
3.5 Directional Simulation* 82
3.5.1 Basic Notions 82
3.5.2 Directional Simulation with Importance Sampling 84
3.5.3 Generalized Directional Simulation 85
3.5.4 Directional Simulation in the Load Space 87
3.5.4.1 Basic Concept 87
3.5.4.2 Variation of Strength with Radial Direction 89
3.5.4.3 Line Sampling 90
3.6 Practical Aspects of Monte Carlo Simulation 90
3.6.1 Conditional Expectation 90
3.6.2 Generalized Limit State Function - Response Surfaces 91
3.6.3 Systematic Selection of Random Variables 92
3.6.4 Applications 92
3.7 Conclusion 93
4 Second-Moment and Transformation Methods 95
4.1 Introduction 95
4.2 Second-Moment Concepts 95
4.3 First-Order Second-Moment (FOSM) Theory 97
4.3.1 The Hasofer-Lind Transformation 97
4.3.2 Linear Limit State Function 98
4.3.3 Sensitivity Factors and Gradient Projection 101
4.3.4 Non-Linear Limit State Function--General Case 102
4.3.5 Non-Linear Limit State Function--Numerical Solution 106
4.3.6 Non-Linear Limit State Function--HLRF Algorithm 106
4.3.7 Geometric Interpretation of Iterative Solution Scheme 109
4.3.8 Interpretation of First-Order Second-Moment (FOSM) Theory 110
4.3.9 General Limit State Functions--Probability Bounds 112
4.4 The First-Order Reliability (FOR) Method 112
4.4.1 Simple Transformations 112
4.4.2 The Normal Tail Transformation 114
4.4.3 Transformations to Independent Normal Basic Variables 116
4.4.3.1 Rosenblatt Transformation 117
4.4.3.2 Nataf Transformation 118
4.4.4 Algorithm for First-Order Reliability (FOR) Method 121
4.4.5 Observations 124
4.4.6 Asymptotic Formulation 125
4.5 Second-Order Reliability (SOR) Methods 126
4.5.1 Basic Concept 126
4.5.2 EvaluationThrough Sampling 126
4.5.3 EvaluationThrough Asymptotic Approximation 127
4.6 Application of FOSM/FOR/SOR Methods 128
4.7 Mean Value Methods 129
4.8 Conclusion 130
5 Reliability of Structural Systems 131
5.1 Introduction 131
5.2 Systems Reliability Fundamentals 132
5.2.1 Structural System Modelling 132
5.2.1.1 Load Modelling 132
5.2.1.2 MaterialModelling 133
5.2.1.3 System Modelling 135
5.2.2 Solution Approaches 136
5.2.2.1 Failure Mode Approach 136
5.2.2.2 Survival Mode Approach 137
5.2.2.3 Upper and Lower Bounds--Plastic Theory 138
5.2.3 Idealizations of Structural Systems 139
5.2.3.1 Series Systems 139
5.2.3.2 Parallel Systems--General 141
5.2.3.3 Parallel Systems--Ideal Plastic 143
5.2.3.4 Combined and Conditional Systems 146
5.3 Monte Carlo Techniques for Systems 147
5.3.1 General Remarks 147
5.3.2 Importance Sampling 147
5.3.2.1 Series Systems 147
5.3.2.2 Parallel Systems 149
5.3.2.3 Search-Type Approaches in Importance Sampling 150
5.3.2.4 Failure Modes Identification in Importance Sampling 151
5.3.3 Directional Simulation 151
5.3.4 Directional Simulation in the Load Space 151
5.4 System Reliability Bounds 153
5.4.1 First-Order Series Bounds 153
5.4.2 Second-Order Series Bounds 154
5.4.3 Second-Order Series Bounds by Loading Sequences 157
5.4.4 Series Bounds by Modes and Loading Sequences 158
5.4.5 Improved Series Bounds and Parallel System Bounds 158
5.4.6 First-Order Second-Moment Method in Systems Reliability 159
5.4.7 Correlation Effects 164
5.4.8 Bounds by Matrix Operations and Linear Programming* 164
5.5 Implicit Limit States 168
5.5.1 Introduction 168
5.5.2 Response Surfaces 169
5.5.2.1 Basics of Response Surfaces 169
5.5.2.2 Fitting the Response Surface 170
5.5.3 Applications of Response Surfaces 172
5.5.4 Other Techniques for Obtaining Surrogate Limit States 173
5.6 Functionally Dependent Limit States 173
5.6.1 Effect of Order of Loading 173
5.6.2 Failure Mode Enumeration and Reduction 174
5.6.3 Reduction of Number of Limit States--Truncation 175
5.6.4 Applications 176
5.7 Conclusion 177
6 Time-Dependent Reliability 179
6.1 Introduction 179
6.2 Time-Integrated Approach 182
6.2.1 Basic Notions 182
6.2.2 Conversion to a Time-Independent Format* 184
6.3 Discretized Approach 185
6.3.1 Known Number of Discrete Events 185
6.3.2 Random Number of Discrete Events 187
6.3.3 Return Period 188
6.3.4 Hazard Function 189
6.4 Stochastic Process Theory 191
6.4.1 Stochastic Process 191
6.4.2 Stationary Processes 192
6.4.3 Derivative Process 193
6.4.4 Ergodic Processes 194
6.4.5 First-Passage Probability 194
6.4.6 Distribution of Local Maxima 196
6.5 Stochastic Processes and Outcrossings 196
6.5.1 Discrete Processes 196
6.5.1.1 Borges Processes 196
6.5.1.2 Poisson Counting Process 197
6.5.1.3 Filtered Poisson process 198
6.5.1.4 Poisson Spike Process 199
6.5.1.5 Poisson SquareWave Process 200
6.5.1.6 Renewal Processes 201
6.5.2 Continuous Processes 202
6.5.3 Barrier (or Level) Upcrossing Rate 202
6.5.4 Outcrossing Rate 205
6.5.4.1 Generalization from Barrier Crossing Rate 205
6.5.4.2 Outcrossings for Discrete Processes 207
6.5.4.3 Outcrossings for Continuous Gaussian Processes 209
6.5.4.4 General Regions and Processes 213
6.5.5 Numerical Evaluation of Outcrossing Rates 214
6.6 Time-Dependent Reliability 215
6.6.1 Introduction 215
6.6.2 SamplingMethods for Unconditional Failure Probability 216
6.6.2.1 Importance and Conditional Sampling 216
6.6.2.2 Directional Simulation in the Load Process Space 217
6.6.3 FOSM/FOR Methods for Unconditional Failure Probability 218
6.6.4 Summary for Time-Dependent Reliability Estimation 225
6.7 Load Combinations 226
6.7.1 Introduction 226
6.7.2 General Formulation 226
6.7.3 Discrete Processes 228
6.7.4 Simplifications 230
6.7.4.1 Load Coincidence Method 230
6.7.4.2 Borges Processes 231
6.7.4.3 Deterministic Load Combination--Turkstra's Rule 233
6.8 Ensemble Crossing Rate and Barrier Failure Dominance 234
6.8.1 Introduction 234
6.8.2 Ensemble Crossing Rate Approximation 234
6.8.3 Application to Turkstra's Rule and the Point Crossing Formula 235
6.8.4 Barrier Failure Dominance 236
6.8.5 Validity 237
6.9 Dynamic Analysis of Structures 237
6.9.1 Introduction 237
6.9.2 Frequency Domain Analysis 238
6.9.3 Reliability Analysis 240
6.10 Fatigue Analysis 241
6.10.1 General Formulation 241
6.10.2 The S-N Model 242
6.10.3 Fracture Mechanics Models 243
6.11 Conclusion 244
7 Load and Load EffectModelling 247
7.1 Introduction 247
7.2 Wind Loading 248
7.3 Wave Loading 252
7.4 Floor Loading 255
7.4.1 General 255
7.4.2 Sustained Load Representation 256
7.4.3 Equivalent Uniformly Distributed Load 260
7.4.4 Distribution of Equivalent Uniformly Distributed Load 263
7.4.5 Maximum (Lifetime) Sustained Load 265
7.4.6 Extraordinary Live Loads 267
7.4.7 Total Live Load 268
7.4.8 Permanent and Construction Loads 269
7.5 Conclusion 271
8 ResistanceModelling 273
8.1 Introduction 273
8.2 Basic Properties of Hot-Rolled Steel Members 273
8.2.1 Steel Material Properties 273
8.2.2 Yield Strength 274
8.2.3 Moduli of Elasticity 277
8.2.4 Strain-Hardening Properties 278
8.2.5 Size Variation 278
8.2.6 Properties for Reliability Assessment 279
8.3 Properties of Steel Reinforcing Bars 280
8.4 Concrete Statistical Properties 281
8.5 Statistical Properties of Structural Members 284
8.5.1 Introduction 284
8.5.2 Methods of Analysis 284
8.5.3 Second-moment Analysis 284
8.5.4 Simulation 287
8.6 Connections 290
8.7 Incorporation of Member Strength in Design 290
8.8 Conclusion 292
9 Codes and Structural Reliability 293
9.1 Introduction 293
9.2 Structural Design Codes 294
9.3 Safety-Checking Formats 296
9.3.1 Probability-Based Code Rules 296
9.3.2 Partial Factors Code Format 297
9.3.3 Simplified Partial Factors Code Format 299
9.3.4 Load and Resistance Factor Code Format 300
9.3.5 Some Observations 300
9.4 Relationship Between Level 1 and Level 2 SafetyMeasures 301
9.4.1 Derivation from FOSM / FORTheory 302
9.4.2 Special Case: Linear Limit State Function 303
9.5 Selection of Code Safety Levels 304
9.6 Code Calibration Procedure 305
9.7 Example of Code Calibration 310
9.8 Observations 315
9.8.1 Applications 315
9.8.2 SomeTheoretical Issues 316
9.9 Performance-Based Design 317
9.10 Conclusion 319
10 Probabilistic Evaluation of Existing Structures 321
10.1 Introduction 321
10.2 Assessment Procedures 323
10.2.1 Overall Procedure 323
10.2.2 Service-Proven Structures 325
10.2.3 Proof Loading 326
10.3 Updating Probabilistic Information 327
10.3.1 Bayes Theorem 327
10.3.2 Updating Failure Probabilities for Proof Loads 328
10.3.3 Updating Probability Density Functions 328
10.3.4 Pre-Posterior Analysis 332
10.4 Analytical Assessment 333
10.4.1 General 333
10.4.2 Models for Deterioration 334
10.5 Acceptance Criteria for Existing Structures 338
10.5.1 Nominal Probabilities 338
10.5.2 Semi-Probabilistic Safety Checking Formats 339
10.5.3 Probabilistic Criteria 340
10.5.4 Decision-Theory-Based Criteria 340
10.5.5 Life-Cycle Decision Approach 342
10.6 Conclusion 343
11 Structural Optimization and Reliability 345
11.1 Introduction 345
11.2 Types of Reliability-based Optimization Problems 346
11.2.1 Introduction 346
11.2.2 Deterministic Design Optimization (DDO) 347
11.2.2.1 Formulation 347
11.2.2.2 Example of DDO Using FOSM 348
11.2.3 Reliability-Based Design Optimization (RBDO) 349
11.2.3.1 Formulation 349
11.2.3.2 Example of RBDO using FOSM 350
11.2.4 Life-Cycle Cost and Risk Optimization (LCRO) 351
11.2.4.1 Formulation 351
11.2.4.2 Example of LCRO using FOSM 352
11.2.5 Comparison, Summary and Outlook 353
11.3 Reliability Based Design Optimization (RBDO) Using First Order Reliability (FOR) 354
11.3.1 Introduction 354
11.3.2 Alternative Robust Solutions Schemes 354
11.3.3 Comparison Between RIA and PMA Solution Schemes 357
11.3.4 Solution of Nested Optimization Problems 358
11.3.5 Example of RBDO Using RIA and PMA 358
11.3.6 Decoupling Techniques for Solving RBDO Problems 361
11.3.6.1 Decoupling: Serial Single Loop Methods 361
11.3.6.2 Decoupling: Uni-levelMethods 361
11.3.6.3 Sequential Approximate Programming (SAP) 361
11.4 RBDO with System Reliability Constraints 362
11.4.1 Formulation of System RBDO 362
11.4.2 Structural Systems RBDO with Component Reliability Constraints 363
11.4.3 Structural System RBDO--solution Schemes 363
11.5 Simulation-based Design Optimization 363
11.5.1 Introduction 363
11.5.2 Problem Formulation 364
11.5.3 Remarks About Solutions 365
11.6 Life-cycle Cost and Risk Optimization 367
11.6.1 Introduction 367
11.6.2 Optimal Structural Design Under Stochastic Loads 367
11.6.3 Optimal Structural Design Considering Inspections and Maintenance 368
11.7 Discussion and Conclusion 368
A Summary of Probability Theory 371
A.1 Probability 371
A.2 Mathematics of Probability 371
A.2.1 Axioms 371
A.2.2 Derived Results 372
A.2.2.1 Multiplication Rule 372
A.2.2.2 Complementary Probability 372
A.2.2.3 Conditional Probability 372
A.2.2.4 Total ProbabilityTheorem 372
A.2.2.5 Bayes'Theoremx 372
A.3 Description of Random Variables 373
A.4 Moments of Random Variables 373
A.4.1 Mean or Expected Value (First Moment) 373
A.4.2 Variance and Standard Deviation (Second Moment) 374
A.4.3 Bounds on the Deviations from the Mean 374
A.4.4 Skewness 1 (Third Moment) 374
A.4.5 Coefficient 2 of Kurtosis (Fourth Moment) 375
A.4.6 Higher Moments 375
A.5 Common Univariate Probability Distributions 375
A.5.1 Binomial B(n, p) 375
A.5.2 Geometric G(p) 376
A.5.3 Negative Binomial NB(k, p) 376
A.5.4 Poisson PN( t) 377
A.5.5 Exponential EX( ) 377
A.5.6 Gamma GM(k, ) [and Chi-squared 2(n)] 378
A.5.7 Normal (Gaussian) N( , ) 379
A.5.8 Central LimitTheorem 381
A.5.9 Lognormal LN( , ) 381
A.5.10 Beta BT(a, b, q, r) 383
A.5.11 Extreme Value Distribution Type I EV - I( , ) [Gumbel distribution] 385
A.5.12 Extreme Value Distribution Type II EV - II(u, k) [Frechet Distribution] 386
A.5.13 Extreme Value Distribution Type III EV - III( , u, k) [Weibull] 388
A.5.14 Generalized Extreme Value distribution GEV 390
A.6 Jointly Distributed Random Variables 390
A.6.1 Joint Probability Distribution 390
A.6.2 Conditional Probability Distributions 391
A.6.3 Marginal Probability Distributions 391
A.7 Moments of Jointly Distributed Random Variables 392
A.7.1 Mean 392
A.7.2 Variance 393
A.7.3 Covariance and Correlation 393
A.8 Bivariate Normal Distribution 393
A.9 Transformation of Random Variables 397
A.9.1 Transformation of a Single Random Variable 397
A.9.2 Transformation of Two or More Random Variables 397
A.9.3 Linear and Orthogonal Transformations 398
A.10 Functions of Random Variables 398
A.10.1 Function of a Single Random Variable 398
A.10.2 Function of Two or More Random Variables 398
A.10.3 Some Special Results 399
A.10.3.1 Y = X1 + X2 399
A.10.3.2 Y = X1 X2 399
A.11 Moments of Functions of Random Variables 400
A.11.1 Linear Functions 400
A.11.2 Product of Variates 400
A.11.3 Division of Variates 401
A.11.4 Moments of a Square Root [Haugen, 1968] 401
A.11.5 Moments of a Quadratic Form [Haugen, 1968] 402
A.12 Approximate Moments for General Functions 402
B Rosenblatt and Other Transformations 403
B.1 Rosenblatt Transformation 403
B.2 Nataf Transformation 405
B.3 Orthogonal Transformation of Normal Random Variables 407
B.4 Generation of Dependent Random Vectors 410
C Bivariate and Multivariate Normal Integrals 415
C.1 Bivariate Normal Integral 415
C.1.1 Format 415
C.1.2 Reductions of Form 417
C.1.3 Bounds 417
C.2 Multivariate Normal Integral 419
C.2.1 Format 419
C.2.2 Numerical Integration of Multi-Normal Integrals 419
C.2.3 Reduction to a Single Integral 420
C.2.4 Bounds on the Multivariate Normal Integral 420
C.2.5 First-OrderMulti-Normal (FOMN) Approach 421
C.2.5.1 Basic Method: B-FOMN 421
C.2.5.2 Improved Method: I-FOMN 424
C.2.5.3 Generalized Method: G-FOMN 425
C.2.6 Product of Conditional Marginals (PCM) Approach 426
D Complementary Standard Normal Table 429
D.1 Standard Normal Probability Density Function (x) 432
E Random Numbers 433
F Selected Problems 435
References 457
Index 497
Preface to the Second Edition xvii
Preface to the First Edition xviii
Acknowledgements xx
1 Measures of Structural Reliability 1
1.1 Introduction 1
1.2 DeterministicMeasures of Limit State Violation 2
1.2.1 Factor of Safety 2
1.2.2 Load Factor 3
1.2.3 Partial Factor ('Limit State Design') 4
1.2.4 A Deficiency in Some SafetyMeasures: Lack of Invariance 5
1.2.5 Invariant SafetyMeasures 8
1.3 A Partial Probabilistic SafetyMeasure of Limit State Violation--The Return
Period 8
1.4 Probabilistic Measure of Limit State Violation 12
1.4.1 Introduction 12
1.4.2 The Basic Reliability Problem 14
1.4.3 Special Case: Normal Random Variables 17
1.4.4 Safety Factors and Characteristic Values 19
1.4.5 Numerical Integration of the Convolution Integral 23
1.5 Generalized Reliability Problem 24
1.5.1 Basic Variables 24
1.5.2 Generalized Limit State Equations 25
1.5.3 Generalized Reliability Problem Formulation 26
1.5.4 Conditional Reliability Problems* 27
1.6 Conclusion 29
2 Structural Reliability Assessment 31
2.1 Introduction 31
2.2 Uncertainties in Reliability Assessment 33
2.2.1 Identification of Uncertainties 33
2.2.2 Phenomenological Uncertainty 34
2.2.3 Decision Uncertainty 34
2.2.4 Modelling Uncertainty 34
2.2.5 Prediction Uncertainty 35
2.2.6 Physical Uncertainty 36
2.2.7 Statistical Uncertainty 36
2.2.8 Uncertainties Due to Human Factors 37
2.2.8.1 Human Error 37
2.2.8.2 Human Intervention 40
2.2.8.3 Modelling of Human Error and Intervention 43
2.2.8.4 Quality Assurance 44
2.2.8.5 Hazard Management 45
2.3 Integrated Risk Assessment 45
2.3.1 Calculation of the Probability of Failure 45
2.3.2 Analysis and Prediction 47
2.3.3 Comparison to Failure Data 48
2.3.4 Validation--a Philosophical Issue 50
2.3.5 The Tail Sensitivity 'Problem' 50
2.4 Criteria for Risk Acceptability 51
2.4.1 Acceptable Risk Criterion 51
2.4.1.1 Risks in Society 51
2.4.1.2 Acceptable or Tolerable Risk Levels 53
2.4.2 Socio-economic Criterion 54
2.5 Nominal Probability of Failure 56
2.5.1 General 56
2.5.2 Axiomatic Definition 56
2.5.3 Influence of Gross and Other Errors 57
2.5.4 Practical Implications 58
2.5.5 Target Values for Nominal Failure Probability 59
2.6 Hierarchy of Structural ReliabilityMeasures 60
2.7 Conclusion 61
3 Integration and Simulation Methods 63
3.1 Introduction 63
3.2 Direct and Numerical Integration 63
3.3 Monte Carlo Simulation 65
3.3.1 Introduction 65
3.3.2 Generation of Uniformly Distributed Random Numbers 65
3.3.3 Generation of Random Variates 66
3.3.4 Direct Sampling ('Crude' Monte Carlo) 68
3.3.5 Number of Samples Required 69
3.3.6 Variance Reduction 72
3.3.7 Stratified and Latin Hypercube Sampling 73
3.4 Importance Sampling 73
3.4.1 Theory of Importance Sampling 73
3.4.2 Importance Sampling Functions 75
3.4.3 Observations About Importance Sampling Functions 76
3.4.4 Improved Sampling Functions 79
3.4.5 Search or Adaptive Techniques 80
3.4.6 Sensitivity 81
3.5 Directional Simulation* 82
3.5.1 Basic Notions 82
3.5.2 Directional Simulation with Importance Sampling 84
3.5.3 Generalized Directional Simulation 85
3.5.4 Directional Simulation in the Load Space 87
3.5.4.1 Basic Concept 87
3.5.4.2 Variation of Strength with Radial Direction 89
3.5.4.3 Line Sampling 90
3.6 Practical Aspects of Monte Carlo Simulation 90
3.6.1 Conditional Expectation 90
3.6.2 Generalized Limit State Function - Response Surfaces 91
3.6.3 Systematic Selection of Random Variables 92
3.6.4 Applications 92
3.7 Conclusion 93
4 Second-Moment and Transformation Methods 95
4.1 Introduction 95
4.2 Second-Moment Concepts 95
4.3 First-Order Second-Moment (FOSM) Theory 97
4.3.1 The Hasofer-Lind Transformation 97
4.3.2 Linear Limit State Function 98
4.3.3 Sensitivity Factors and Gradient Projection 101
4.3.4 Non-Linear Limit State Function--General Case 102
4.3.5 Non-Linear Limit State Function--Numerical Solution 106
4.3.6 Non-Linear Limit State Function--HLRF Algorithm 106
4.3.7 Geometric Interpretation of Iterative Solution Scheme 109
4.3.8 Interpretation of First-Order Second-Moment (FOSM) Theory 110
4.3.9 General Limit State Functions--Probability Bounds 112
4.4 The First-Order Reliability (FOR) Method 112
4.4.1 Simple Transformations 112
4.4.2 The Normal Tail Transformation 114
4.4.3 Transformations to Independent Normal Basic Variables 116
4.4.3.1 Rosenblatt Transformation 117
4.4.3.2 Nataf Transformation 118
4.4.4 Algorithm for First-Order Reliability (FOR) Method 121
4.4.5 Observations 124
4.4.6 Asymptotic Formulation 125
4.5 Second-Order Reliability (SOR) Methods 126
4.5.1 Basic Concept 126
4.5.2 EvaluationThrough Sampling 126
4.5.3 EvaluationThrough Asymptotic Approximation 127
4.6 Application of FOSM/FOR/SOR Methods 128
4.7 Mean Value Methods 129
4.8 Conclusion 130
5 Reliability of Structural Systems 131
5.1 Introduction 131
5.2 Systems Reliability Fundamentals 132
5.2.1 Structural System Modelling 132
5.2.1.1 Load Modelling 132
5.2.1.2 MaterialModelling 133
5.2.1.3 System Modelling 135
5.2.2 Solution Approaches 136
5.2.2.1 Failure Mode Approach 136
5.2.2.2 Survival Mode Approach 137
5.2.2.3 Upper and Lower Bounds--Plastic Theory 138
5.2.3 Idealizations of Structural Systems 139
5.2.3.1 Series Systems 139
5.2.3.2 Parallel Systems--General 141
5.2.3.3 Parallel Systems--Ideal Plastic 143
5.2.3.4 Combined and Conditional Systems 146
5.3 Monte Carlo Techniques for Systems 147
5.3.1 General Remarks 147
5.3.2 Importance Sampling 147
5.3.2.1 Series Systems 147
5.3.2.2 Parallel Systems 149
5.3.2.3 Search-Type Approaches in Importance Sampling 150
5.3.2.4 Failure Modes Identification in Importance Sampling 151
5.3.3 Directional Simulation 151
5.3.4 Directional Simulation in the Load Space 151
5.4 System Reliability Bounds 153
5.4.1 First-Order Series Bounds 153
5.4.2 Second-Order Series Bounds 154
5.4.3 Second-Order Series Bounds by Loading Sequences 157
5.4.4 Series Bounds by Modes and Loading Sequences 158
5.4.5 Improved Series Bounds and Parallel System Bounds 158
5.4.6 First-Order Second-Moment Method in Systems Reliability 159
5.4.7 Correlation Effects 164
5.4.8 Bounds by Matrix Operations and Linear Programming* 164
5.5 Implicit Limit States 168
5.5.1 Introduction 168
5.5.2 Response Surfaces 169
5.5.2.1 Basics of Response Surfaces 169
5.5.2.2 Fitting the Response Surface 170
5.5.3 Applications of Response Surfaces 172
5.5.4 Other Techniques for Obtaining Surrogate Limit States 173
5.6 Functionally Dependent Limit States 173
5.6.1 Effect of Order of Loading 173
5.6.2 Failure Mode Enumeration and Reduction 174
5.6.3 Reduction of Number of Limit States--Truncation 175
5.6.4 Applications 176
5.7 Conclusion 177
6 Time-Dependent Reliability 179
6.1 Introduction 179
6.2 Time-Integrated Approach 182
6.2.1 Basic Notions 182
6.2.2 Conversion to a Time-Independent Format* 184
6.3 Discretized Approach 185
6.3.1 Known Number of Discrete Events 185
6.3.2 Random Number of Discrete Events 187
6.3.3 Return Period 188
6.3.4 Hazard Function 189
6.4 Stochastic Process Theory 191
6.4.1 Stochastic Process 191
6.4.2 Stationary Processes 192
6.4.3 Derivative Process 193
6.4.4 Ergodic Processes 194
6.4.5 First-Passage Probability 194
6.4.6 Distribution of Local Maxima 196
6.5 Stochastic Processes and Outcrossings 196
6.5.1 Discrete Processes 196
6.5.1.1 Borges Processes 196
6.5.1.2 Poisson Counting Process 197
6.5.1.3 Filtered Poisson process 198
6.5.1.4 Poisson Spike Process 199
6.5.1.5 Poisson SquareWave Process 200
6.5.1.6 Renewal Processes 201
6.5.2 Continuous Processes 202
6.5.3 Barrier (or Level) Upcrossing Rate 202
6.5.4 Outcrossing Rate 205
6.5.4.1 Generalization from Barrier Crossing Rate 205
6.5.4.2 Outcrossings for Discrete Processes 207
6.5.4.3 Outcrossings for Continuous Gaussian Processes 209
6.5.4.4 General Regions and Processes 213
6.5.5 Numerical Evaluation of Outcrossing Rates 214
6.6 Time-Dependent Reliability 215
6.6.1 Introduction 215
6.6.2 SamplingMethods for Unconditional Failure Probability 216
6.6.2.1 Importance and Conditional Sampling 216
6.6.2.2 Directional Simulation in the Load Process Space 217
6.6.3 FOSM/FOR Methods for Unconditional Failure Probability 218
6.6.4 Summary for Time-Dependent Reliability Estimation 225
6.7 Load Combinations 226
6.7.1 Introduction 226
6.7.2 General Formulation 226
6.7.3 Discrete Processes 228
6.7.4 Simplifications 230
6.7.4.1 Load Coincidence Method 230
6.7.4.2 Borges Processes 231
6.7.4.3 Deterministic Load Combination--Turkstra's Rule 233
6.8 Ensemble Crossing Rate and Barrier Failure Dominance 234
6.8.1 Introduction 234
6.8.2 Ensemble Crossing Rate Approximation 234
6.8.3 Application to Turkstra's Rule and the Point Crossing Formula 235
6.8.4 Barrier Failure Dominance 236
6.8.5 Validity 237
6.9 Dynamic Analysis of Structures 237
6.9.1 Introduction 237
6.9.2 Frequency Domain Analysis 238
6.9.3 Reliability Analysis 240
6.10 Fatigue Analysis 241
6.10.1 General Formulation 241
6.10.2 The S-N Model 242
6.10.3 Fracture Mechanics Models 243
6.11 Conclusion 244
7 Load and Load EffectModelling 247
7.1 Introduction 247
7.2 Wind Loading 248
7.3 Wave Loading 252
7.4 Floor Loading 255
7.4.1 General 255
7.4.2 Sustained Load Representation 256
7.4.3 Equivalent Uniformly Distributed Load 260
7.4.4 Distribution of Equivalent Uniformly Distributed Load 263
7.4.5 Maximum (Lifetime) Sustained Load 265
7.4.6 Extraordinary Live Loads 267
7.4.7 Total Live Load 268
7.4.8 Permanent and Construction Loads 269
7.5 Conclusion 271
8 ResistanceModelling 273
8.1 Introduction 273
8.2 Basic Properties of Hot-Rolled Steel Members 273
8.2.1 Steel Material Properties 273
8.2.2 Yield Strength 274
8.2.3 Moduli of Elasticity 277
8.2.4 Strain-Hardening Properties 278
8.2.5 Size Variation 278
8.2.6 Properties for Reliability Assessment 279
8.3 Properties of Steel Reinforcing Bars 280
8.4 Concrete Statistical Properties 281
8.5 Statistical Properties of Structural Members 284
8.5.1 Introduction 284
8.5.2 Methods of Analysis 284
8.5.3 Second-moment Analysis 284
8.5.4 Simulation 287
8.6 Connections 290
8.7 Incorporation of Member Strength in Design 290
8.8 Conclusion 292
9 Codes and Structural Reliability 293
9.1 Introduction 293
9.2 Structural Design Codes 294
9.3 Safety-Checking Formats 296
9.3.1 Probability-Based Code Rules 296
9.3.2 Partial Factors Code Format 297
9.3.3 Simplified Partial Factors Code Format 299
9.3.4 Load and Resistance Factor Code Format 300
9.3.5 Some Observations 300
9.4 Relationship Between Level 1 and Level 2 SafetyMeasures 301
9.4.1 Derivation from FOSM / FORTheory 302
9.4.2 Special Case: Linear Limit State Function 303
9.5 Selection of Code Safety Levels 304
9.6 Code Calibration Procedure 305
9.7 Example of Code Calibration 310
9.8 Observations 315
9.8.1 Applications 315
9.8.2 SomeTheoretical Issues 316
9.9 Performance-Based Design 317
9.10 Conclusion 319
10 Probabilistic Evaluation of Existing Structures 321
10.1 Introduction 321
10.2 Assessment Procedures 323
10.2.1 Overall Procedure 323
10.2.2 Service-Proven Structures 325
10.2.3 Proof Loading 326
10.3 Updating Probabilistic Information 327
10.3.1 Bayes Theorem 327
10.3.2 Updating Failure Probabilities for Proof Loads 328
10.3.3 Updating Probability Density Functions 328
10.3.4 Pre-Posterior Analysis 332
10.4 Analytical Assessment 333
10.4.1 General 333
10.4.2 Models for Deterioration 334
10.5 Acceptance Criteria for Existing Structures 338
10.5.1 Nominal Probabilities 338
10.5.2 Semi-Probabilistic Safety Checking Formats 339
10.5.3 Probabilistic Criteria 340
10.5.4 Decision-Theory-Based Criteria 340
10.5.5 Life-Cycle Decision Approach 342
10.6 Conclusion 343
11 Structural Optimization and Reliability 345
11.1 Introduction 345
11.2 Types of Reliability-based Optimization Problems 346
11.2.1 Introduction 346
11.2.2 Deterministic Design Optimization (DDO) 347
11.2.2.1 Formulation 347
11.2.2.2 Example of DDO Using FOSM 348
11.2.3 Reliability-Based Design Optimization (RBDO) 349
11.2.3.1 Formulation 349
11.2.3.2 Example of RBDO using FOSM 350
11.2.4 Life-Cycle Cost and Risk Optimization (LCRO) 351
11.2.4.1 Formulation 351
11.2.4.2 Example of LCRO using FOSM 352
11.2.5 Comparison, Summary and Outlook 353
11.3 Reliability Based Design Optimization (RBDO) Using First Order Reliability (FOR) 354
11.3.1 Introduction 354
11.3.2 Alternative Robust Solutions Schemes 354
11.3.3 Comparison Between RIA and PMA Solution Schemes 357
11.3.4 Solution of Nested Optimization Problems 358
11.3.5 Example of RBDO Using RIA and PMA 358
11.3.6 Decoupling Techniques for Solving RBDO Problems 361
11.3.6.1 Decoupling: Serial Single Loop Methods 361
11.3.6.2 Decoupling: Uni-levelMethods 361
11.3.6.3 Sequential Approximate Programming (SAP) 361
11.4 RBDO with System Reliability Constraints 362
11.4.1 Formulation of System RBDO 362
11.4.2 Structural Systems RBDO with Component Reliability Constraints 363
11.4.3 Structural System RBDO--solution Schemes 363
11.5 Simulation-based Design Optimization 363
11.5.1 Introduction 363
11.5.2 Problem Formulation 364
11.5.3 Remarks About Solutions 365
11.6 Life-cycle Cost and Risk Optimization 367
11.6.1 Introduction 367
11.6.2 Optimal Structural Design Under Stochastic Loads 367
11.6.3 Optimal Structural Design Considering Inspections and Maintenance 368
11.7 Discussion and Conclusion 368
A Summary of Probability Theory 371
A.1 Probability 371
A.2 Mathematics of Probability 371
A.2.1 Axioms 371
A.2.2 Derived Results 372
A.2.2.1 Multiplication Rule 372
A.2.2.2 Complementary Probability 372
A.2.2.3 Conditional Probability 372
A.2.2.4 Total ProbabilityTheorem 372
A.2.2.5 Bayes'Theoremx 372
A.3 Description of Random Variables 373
A.4 Moments of Random Variables 373
A.4.1 Mean or Expected Value (First Moment) 373
A.4.2 Variance and Standard Deviation (Second Moment) 374
A.4.3 Bounds on the Deviations from the Mean 374
A.4.4 Skewness 1 (Third Moment) 374
A.4.5 Coefficient 2 of Kurtosis (Fourth Moment) 375
A.4.6 Higher Moments 375
A.5 Common Univariate Probability Distributions 375
A.5.1 Binomial B(n, p) 375
A.5.2 Geometric G(p) 376
A.5.3 Negative Binomial NB(k, p) 376
A.5.4 Poisson PN( t) 377
A.5.5 Exponential EX( ) 377
A.5.6 Gamma GM(k, ) [and Chi-squared 2(n)] 378
A.5.7 Normal (Gaussian) N( , ) 379
A.5.8 Central LimitTheorem 381
A.5.9 Lognormal LN( , ) 381
A.5.10 Beta BT(a, b, q, r) 383
A.5.11 Extreme Value Distribution Type I EV - I( , ) [Gumbel distribution] 385
A.5.12 Extreme Value Distribution Type II EV - II(u, k) [Frechet Distribution] 386
A.5.13 Extreme Value Distribution Type III EV - III( , u, k) [Weibull] 388
A.5.14 Generalized Extreme Value distribution GEV 390
A.6 Jointly Distributed Random Variables 390
A.6.1 Joint Probability Distribution 390
A.6.2 Conditional Probability Distributions 391
A.6.3 Marginal Probability Distributions 391
A.7 Moments of Jointly Distributed Random Variables 392
A.7.1 Mean 392
A.7.2 Variance 393
A.7.3 Covariance and Correlation 393
A.8 Bivariate Normal Distribution 393
A.9 Transformation of Random Variables 397
A.9.1 Transformation of a Single Random Variable 397
A.9.2 Transformation of Two or More Random Variables 397
A.9.3 Linear and Orthogonal Transformations 398
A.10 Functions of Random Variables 398
A.10.1 Function of a Single Random Variable 398
A.10.2 Function of Two or More Random Variables 398
A.10.3 Some Special Results 399
A.10.3.1 Y = X1 + X2 399
A.10.3.2 Y = X1 X2 399
A.11 Moments of Functions of Random Variables 400
A.11.1 Linear Functions 400
A.11.2 Product of Variates 400
A.11.3 Division of Variates 401
A.11.4 Moments of a Square Root [Haugen, 1968] 401
A.11.5 Moments of a Quadratic Form [Haugen, 1968] 402
A.12 Approximate Moments for General Functions 402
B Rosenblatt and Other Transformations 403
B.1 Rosenblatt Transformation 403
B.2 Nataf Transformation 405
B.3 Orthogonal Transformation of Normal Random Variables 407
B.4 Generation of Dependent Random Vectors 410
C Bivariate and Multivariate Normal Integrals 415
C.1 Bivariate Normal Integral 415
C.1.1 Format 415
C.1.2 Reductions of Form 417
C.1.3 Bounds 417
C.2 Multivariate Normal Integral 419
C.2.1 Format 419
C.2.2 Numerical Integration of Multi-Normal Integrals 419
C.2.3 Reduction to a Single Integral 420
C.2.4 Bounds on the Multivariate Normal Integral 420
C.2.5 First-OrderMulti-Normal (FOMN) Approach 421
C.2.5.1 Basic Method: B-FOMN 421
C.2.5.2 Improved Method: I-FOMN 424
C.2.5.3 Generalized Method: G-FOMN 425
C.2.6 Product of Conditional Marginals (PCM) Approach 426
D Complementary Standard Normal Table 429
D.1 Standard Normal Probability Density Function (x) 432
E Random Numbers 433
F Selected Problems 435
References 457
Index 497
ROBERT E. MELCHERS, PhD, is a Professor in the Department of Civil Engineering at The University of Newcastle, Australia. His main areas of research expertise are in structural engineering risk and reliability analyses, probabilistic modelling of engineering systems, corrosion and deterioration modeling, and investigation of structural failures.
ANDRÉ T. BECK, PhD, is an Associate Professor in the Department of Structural Engineering at the University of São Paulo, Brazil. His research interests include structural mechanics and structural safety.
ANDRÉ T. BECK, PhD, is an Associate Professor in the Department of Structural Engineering at the University of São Paulo, Brazil. His research interests include structural mechanics and structural safety.
