Structural Reliability
Approaches from Perspectives of Statistical Moments
1. Auflage April 2021
656 Seiten, Hardcover
Wiley & Sons Ltd
ISBN:
978-1-119-62081-5
John Wiley & Sons
Weitere Versionen
STRUCTURAL RELIABILITY
Discover a new and innovative approach to structural reliability from two authoritative and accomplished authors
The subject of structural reliability, which deals with the problems of evaluating the safety and risk posed by a wide variety of structures, has grown rapidly over the last four decades. And while the First-Order Reliability Method is principally used by most textbooks on this subject, other approaches have identified some of the limitations of that method.
In Structural Reliability: Approaches from Perspectives of Statistical Moments, accomplished engineers and authors Yan-Gang Zhao and Dr. Zhao-Hui Lu, deliver a concise and insightful exploration of an alternative and innovative approach to structural reliability. Called the Methods of Moment, the authors' approach is based on the information of statistical moments of basic random variables and the performance function. The Methods of Moment approach facilitates structural reliability analysis and reliability-based design and can be extended to other engineering disciplines, yielding further insights into challenging problems involving randomness.
Readers will also benefit from the inclusion of:
* A thorough introduction to the measures of structural safety, including uncertainties in structural design, deterministic measures of safety, and probabilistic measures of safety
* An exploration of the fundamentals of structural reliability theory, including the performance function and failure probability
* A practical discussion of moment evaluation for performance functions, including moment computation for both explicit and implicit performance functions
* A concise treatment of direct methods of moment, including the third- and fourth-moment reliability methods
Perfect for professors, researchers, and graduate students in civil engineering, Structural Reliability: Approaches from Perspectives of Statistical Moments will also earn a place in the libraries of professionals and students working or studying in mechanical engineering, aerospace and aeronautics engineering, marine and offshore engineering, ship engineering, and applied mechanics.
Preface
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 Uncertainty 3
1.2.3 Treatment of Uncertainties 4
1.2.4 Design and Decision Making Under Uncertainties 7
1.3 Deterministic Measures of Safety 8
1.4 Probabilistic Measure of Safety 8
1.5 Summary 9
2 Fundamentals of Structural Reliability Theory 10
2.1 The Fundamental Case 10
2.2 Performance Function and Failure Probability 14
2.2.1 Performance Function 14
2.2.2 Probability of Failure 14
2.2.3 Reliability Index 16
2.3 Monte Carlo Simulation 20
2.3.1 Introduction 20
2.3.2 Generation of Random Numbers 21
2.3.3 Direct Sampling 24
2.4 A Brief Review on Structural Reliability Theory 28
2.5 Summary 30
3 Moment Evaluation for Performance Functions 31
3.1 Introduction 31
3.2 Moment Computation for Some Simple Functions 33
3.2.1 Moment Computation for Linear Sum of Random Variables 33
3.2.2 Moment Computation for Multiply of Random Variables 35
3.2.3 Moment Computation for Power of a Lognormally Distributed Random Variable 37
3.2.4 Moment Computation for Power of an Arbitrarily Distributed Random Variable 41
3.2.5 Moment Computation for Reciprocal of an Arbitrary Distributed Random Variable 43
3.3 Point Estimate for a Function of One Random Variable 44
3.3.1 Rosenblueth's Two-Point Estimate 44
3.3.2 Gorman's Three-Point Estimate 45
3.4 Point Estimates in Standardized Normal Space 50
3.4.1 Basic ideas 50
3.4.2 Two- and Three-point Estimates in the Standard Normal Space 52
3.4.3 Five-Point Estimate in Standard Normal Space 53
3.4.4 Seven-Point Estimate in Standard Normal Space 54
3.4.5 General Expression of Estimating Points and Their Corresponding Weights 57
3.4.6 Accuracy of the Point Estimate 59
3.5 Point Estimates for a Function of Multiple Variables 61
3.5.1 General Expression of Point Estimate for a Function of n Variables 61
3.5.2 Approximate Point Estimate for a Function of n Variables 63
3.5.3 Dimension Reduction Integration 67
3.6 Point Estimates for a Function of Correlative Random Variables 71
3.7 Hybrid Dimension-Reduction Based Point Estimate Method 75
3.8 Summary 78
4 Direct Methods of Moment 79
4.1 Basic Concept of Methods of Moment 79
4.1.1 Introduction 79
4.1.2 The Second-Moment Method 79
4.1.3 General Expressions for Methods of Moment 82
4.2 Third-Moment Reliability Method 83
4.2.1 Introduction 83
4.2.2 Third-Moment Reliability Indices 85
4.2.3 Empirical Applicable Range of Third-Moment Reliability Method 88
4.2.4 Simplification of Third-Moment Reliability Method 91
4.2.5 Applicable Range of the Second-Moment method 94
4.3 Fourth-Moment Reliability Method 99
4.3.1 Introduction 99
4.3.2 Fourth-Moment Reliability Index on the Basis of the Pearson System 101
4.3.3 Fourth-Moment Reliability Index Based on Third-Order Polynomial Transformation 104
4.3.4 Applicable range of Fourth-Moment method 106
4.3.5 Simplification of Fourth-Moment reliability index 110
4.4 Summary 112
5 Methods of Moment Based on First/Second Order Transformation 113
5.1 Introduction 113
5.2 First-Order Reliability Method 113
5.2.1 The Hasofer-Lind reliability index 113
5.2.2 First Order Reliability Method 115
5.2.3 Numerical Solution for FORM 119
5.2.4 The Weakness of FORM 124
5.3 Second Order Reliability Method 127
5.3.1 Introduction 127
5.3.2 Second Order Approximation of the Performance Function 127
5.3.3 Failure probability for Second Order Performance Function 138
5.3.4 Methods of Moment for Second Order Approximation 143
5.3.5 Applicable Range of FORM 154
5.4 Summary 156
6 Structural Reliability Assessment based on the Information of Moments of Random Variables 158
6.1 Introduction 158
6.2 Direct Methods of Moment Without Using Probability Distribution 159
6.2.1 Second-Moment Formulation 159
6.4.2 Third-Moment Formulation 160
6.4.3 Fourth-Moment Formulation 161
6.3 First-Order Second-Moment Method 161
6.4 First-Order Third-Moment Method 166
6.4.1 First-Order Third-Moment Method in Reduced Space 166
6.4.2 First-Order Third-Moment Method in Pseudo Standard Normal Space 167
6.5 First-Order Fourth-Moment Method 180
6.5.1 First-Order Fourth-Moment Method in Reduced Space 180
6.5.2 First-Order Fourth-Moment Method in Pseudo Standard Normal Space 180
6.6 Monte Carlo Simulation Using Moment of Random Variables 191
6.7 Subset Simulation Using Statistical Moments of Random Variables 200
6.8 Summary 205
7 Transformation of Non-Normal Variables to Independent Normal Variables 206
7.1 Introduction 206
7.2 The Normal Transformation for a Single Random Variable 206
7.3 The Normal Transformation for Correlated Random Variable 207
7.3.1 Rosenblatt Transformation 207
7.3.2 Nataf Transformation 208
7.4 Pseudo Normal Transformations for a Single Random Variable 215
7.4.1 Concept of Pseudo Normal Transformation 215
7.4.2 Third Moment Pseudo Normal Transformation 217
7.4.3 Fourth Moment Pseudo Normal Transformation 223
7.5 Pseudo Normal Transformations of Correlated Random Variables 238
7.5.1 Introduction 238
7.5.2 Third Moment Pseudo Normal Transformation for Correlated Random Variables 240
7.5.3 Fourth Moment Pseudo Normal Transformation for Correlated Random Variables 243
7.6 Summary 249
8 System Reliability Assessment by the Method of Moments 251
8.1 Introduction 251
8.2 Basic Concepts of System Reliability 251
8.2.1 Multiple Failure Modes 251
8.2.2 Series and Parallel Systems 252
8.3 System Reliability Bounds 260
8.3.1 Uni-Modal Bounds 260
8.3.2 Bi-Modal Bounds 262
8.3.3 Correlation Between a Pair of Failure Modes 264
8.3.4 Bound Estimation of the Joint Failure Probability of a Pair of Failure Modes 265
8.3.5 Point Estimation of the Joint Failure Probability of a Pair of Failure Modes 268
8.4 Moment Approach for System Reliability 277
8.4.1 Performance Function for a system 277
8.4.2 Method of Moments for System Reliability 280
8.5 Methods of Moment for System Reliability Assessment of Ductile Frame Structure 289
8.5.1 Introduction 289
8.5.2 Performance Function Independent of Failure Modes 290
8.5.3 Limit Analysis 292
8.5.4 Methods of Moment for System Reliability of Ductile Frames 293
8.6 Summary 299
9 Determination of Load and Resistance Factors by Methods of Moment 300
9.1 Introduction 300
9.2 Basic Concept of Load and Resistance Factors 301
9.2.1 Basic Concept 301
9.2.2 Determination of LRFs by Second-Moment Method 301
9.2.3 Determination of LRFs under Lognormal Assumption 303
9.2.4 Determination of LRFs by FORM 304
9.2.5 Practical Method for the Determination of LRFs 311
9.3 Load and Resistance Factors by Third-Moment Method 312
9.3.1 Determination of LRFs using Third-Moment Method 312
9.3.2 Estimation of the Mean Value of Resistance 314
9.4 General Expressions of Load and Resistance Factors using Method of Moments 319
9.5 Determination of Load and Resistance Factors Using Fourth-Moment Method 320
9.5.1 Basic Formulas 320
9.5.2 Determination of the Mean Value of Resistance 321
9.6 Summary 325
10 Methods of Moment for Time-Variant Reliability 326
10.1 Introduction 326
10.2 Simulating Stationary Non-Gaussian Process using The Fourth-Moment Transformation 326
10.2.1 Introduction 326
10.2.2 Transformation for Marginal Probability Distributions 327
10.2.3 Transformation for Correlation Functions 328
10.2.4 Methods to Deal with the Incompatibility 331
10.2.5 Scheme of Simulating Stationary Non-Gaussian Random Processes 332
10.3 First Passage Probability Assessment of Stationary Non-Gaussian Processes Using Fourth-Moment Transformation 340
10.3.1 Introduction 340
10.3.2 Formulation of the First Passage Probability of Stationary Non-Gaussian Structural Responses 341
10.3.4 Computational Procedure for the First Passage Probability of Stationary Non-Gaussian Structural Responses 343
10.4 Fast Integration Algorithms for Time-Dependent Structural Reliability Analysis Considering Correlated Random Variables 344
10.4.1 Introduction 344
10.4.2 Formulation of Time-Dependent Failure Probability 345
10.4.3 Fast Integration Algorithms for the Time-Dependent Failure Probability 347
10.5 Summary 357
11 Methods of Moment for Structural Reliability with Hierarchical Modeling of Uncertainty 358
11.1 Introduction 358
11.2 Formulation Description of the Structural Reliability with Hierarchical Modeling of Uncertainty 359
11.3 Overall Probability of Failure Due to Hierarchical Modeling of Uncertainty 360
11.3.1 Evaluating Overall Probability of Failure Based on FORM 360
11.3.2 Evaluating Overall Probability of Failure Based on Methods of Moment 363
11.3.3 Evaluating Overall Probability of Failure Based on Direct Point Estimate Method 364
11.4 The Quantile of the Conditional Failure Probability 368
11.5 Application to Structural Dynamic Reliability Considering Parameters Uncertainties 375
11.6 Summary 381
12 Structural Reliability Analysis Based on the First Few L-Moments 382
12.1 Introduction 382
12.2 Definition of L-moments 382
12.3 Structural Reliability Analysis Based on the First Three L-Moments 384
12.3.1 Transformation for Independent Random Variables 384
12.3.2 Transformation for Correlated Random Variables 385
12.3.3 Reliability Analysis Using the First Three L-moments and Correlation Matrix 388
12.4 Structural Reliability Analysis Based on the First Four L-Moments 395
12.4.1 Transformation for Independent Random Variables 395
12.4.2 Transformation for Correlated Random Variables 401
12.4.3 Reliability Analysis using the First Four L-Moments and Correlation Matrix 404
12.5 Summary 406
13 Methods of Moment with Box-Cox Transformation 407
13.1 Introduction 407
13.2 Methods of Moment with Box-Cox Transformation 407
13.2.1 Criterion for Determining the Box-Cox Transformation Parameter 407
13.2.2 Procedure of the Methods of Moment with Box-Cox Transformation for Structural Reliability 408
13.3 Summary 419
Appendix A Basic probability theory 420
A.1 Events and Probability 420
A.1.1 Introduction 420
A.1.2 Events and Their Combinations 420
A.1.3 Mathematical Operations of Sets 421
A.1.4 Mathematics of Probability 422
A.2 Random Variables and Their Distributions 423
A.3 Main Descriptors of a Random Variable 425
A.3.1 Measures of Location 426
A.3.2 Measures of Dispersion 427
A.3.3 Measures of Asymmetry 428
A.3.4 Measures of Sharpness 429
A.4 Moments and Cumulants 431
A.4.1 Moments 431
A.4.2 Moment and Cumulant Generating Functions 432
A.5 Normal and Lognormal Distributions 434
A.5.1 The Normal Distribution 434
A.5.2 The Logarithmic Normal Distribution 437
A.6 Commonly Used Distributions 439
A.6.1 Introduction 439
A.6.2 Rectangular Distribution 440
A.6.3 Bernoulli Sequences and the Binomial Distribution 440
A.6.4 The Geometric Distribution 441
A.6.5 The Poisson Process and Poisson Distribution 441
A.6.6 The Exponential Distribution 442
A.6.7 The Gamma Distribution 442
A.7 Extreme Value Distributions 443
A.7.1 Introduction 443
A.7.2 The Asymptotic Distributions 445
A.7.3 The Gumbel Distribution 446
A.7.4 The Frechet Distribution 447
A.7.5 The Weibull Distribution 448
A.8 Multiple Random Variables 450
A.8.1 Joint and Conditional Probability Distribution 450
A.8.2 Covariance and Correlation 452
A.9 Functions of Random Variables 453
A.9.1 Function of a Single Random Variable 453
A.9.2 Function of Multiple Random Variables 456
A.10 Summary 458
Appendix B Three-Parameter Distributions 459
B.1 Introduction 459
B.2 The 3P Lognormal Distribution 460
B.2.1 Definition of the Distribution 460
B.2.2 Simplification of the Distribution 463
B.3 Square Normal Distribution 464
B.3.1 Definition of the Distribution 464
B.3.2 Simplification of the Distribution 466
B.4 Comparison of the 3P Distributions 468
B.5 Applications of the 3P Distributions 469
B.5.1 Statistical Data Analysis 469
B.5.2 Representations of One and Two-Parameter Distributions 471
B.5.3 Distributions of Some Random Variables used in Structural Reliability 472
B.6 Summary 473
Appendix C Four-Parameter Distributions 474
C.1 Introduction 474
C.2 The Pearson System 475
C.2.1 Definition of the system 475
C.2.2 Various types of the PDF in Pearson system 476
C.3 Cubic Normal Distribution 481
C.3.1 Definition of the distribution 481
C.3.2 Representative PDFs of the distribution 487
C.3.3 Application in data analysis 487
C.3.5 Simplification of the distribution 490
C.4 Summary 491
Appendix D Basic Theory of Stochastic Process 492
D.1 General Concept of Stochastic Process 492
D.2 Time Domain Description of Stochastic Processes 492
D.2.1 Probability Distributions of Stochastic Processes 492
D.2.2 Moment Functions of Stochastic Processes 494
D.2.3 Stationary and Nonstationary Process 495
D.2.4 Ergodicity of a Stochastic Process 496
D.3 Frequency Domain Description of Stochastic Processes 497
D.3.1 Power Spectral Density Function 497
D.3.2 Wide-and Narrow-Band Processes 497
D.4 Special Processes 498
D.4.1 White Noise Process 498
D.4.2 Markov Process 498
D.4.3 Poisson Process 499
D.4.4 Gaussian Process 499
D.5 Spectral Representation Method 499
D.6 Summary 500
References 5
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 Uncertainty 3
1.2.3 Treatment of Uncertainties 4
1.2.4 Design and Decision Making Under Uncertainties 7
1.3 Deterministic Measures of Safety 8
1.4 Probabilistic Measure of Safety 8
1.5 Summary 9
2 Fundamentals of Structural Reliability Theory 10
2.1 The Fundamental Case 10
2.2 Performance Function and Failure Probability 14
2.2.1 Performance Function 14
2.2.2 Probability of Failure 14
2.2.3 Reliability Index 16
2.3 Monte Carlo Simulation 20
2.3.1 Introduction 20
2.3.2 Generation of Random Numbers 21
2.3.3 Direct Sampling 24
2.4 A Brief Review on Structural Reliability Theory 28
2.5 Summary 30
3 Moment Evaluation for Performance Functions 31
3.1 Introduction 31
3.2 Moment Computation for Some Simple Functions 33
3.2.1 Moment Computation for Linear Sum of Random Variables 33
3.2.2 Moment Computation for Multiply of Random Variables 35
3.2.3 Moment Computation for Power of a Lognormally Distributed Random Variable 37
3.2.4 Moment Computation for Power of an Arbitrarily Distributed Random Variable 41
3.2.5 Moment Computation for Reciprocal of an Arbitrary Distributed Random Variable 43
3.3 Point Estimate for a Function of One Random Variable 44
3.3.1 Rosenblueth's Two-Point Estimate 44
3.3.2 Gorman's Three-Point Estimate 45
3.4 Point Estimates in Standardized Normal Space 50
3.4.1 Basic ideas 50
3.4.2 Two- and Three-point Estimates in the Standard Normal Space 52
3.4.3 Five-Point Estimate in Standard Normal Space 53
3.4.4 Seven-Point Estimate in Standard Normal Space 54
3.4.5 General Expression of Estimating Points and Their Corresponding Weights 57
3.4.6 Accuracy of the Point Estimate 59
3.5 Point Estimates for a Function of Multiple Variables 61
3.5.1 General Expression of Point Estimate for a Function of n Variables 61
3.5.2 Approximate Point Estimate for a Function of n Variables 63
3.5.3 Dimension Reduction Integration 67
3.6 Point Estimates for a Function of Correlative Random Variables 71
3.7 Hybrid Dimension-Reduction Based Point Estimate Method 75
3.8 Summary 78
4 Direct Methods of Moment 79
4.1 Basic Concept of Methods of Moment 79
4.1.1 Introduction 79
4.1.2 The Second-Moment Method 79
4.1.3 General Expressions for Methods of Moment 82
4.2 Third-Moment Reliability Method 83
4.2.1 Introduction 83
4.2.2 Third-Moment Reliability Indices 85
4.2.3 Empirical Applicable Range of Third-Moment Reliability Method 88
4.2.4 Simplification of Third-Moment Reliability Method 91
4.2.5 Applicable Range of the Second-Moment method 94
4.3 Fourth-Moment Reliability Method 99
4.3.1 Introduction 99
4.3.2 Fourth-Moment Reliability Index on the Basis of the Pearson System 101
4.3.3 Fourth-Moment Reliability Index Based on Third-Order Polynomial Transformation 104
4.3.4 Applicable range of Fourth-Moment method 106
4.3.5 Simplification of Fourth-Moment reliability index 110
4.4 Summary 112
5 Methods of Moment Based on First/Second Order Transformation 113
5.1 Introduction 113
5.2 First-Order Reliability Method 113
5.2.1 The Hasofer-Lind reliability index 113
5.2.2 First Order Reliability Method 115
5.2.3 Numerical Solution for FORM 119
5.2.4 The Weakness of FORM 124
5.3 Second Order Reliability Method 127
5.3.1 Introduction 127
5.3.2 Second Order Approximation of the Performance Function 127
5.3.3 Failure probability for Second Order Performance Function 138
5.3.4 Methods of Moment for Second Order Approximation 143
5.3.5 Applicable Range of FORM 154
5.4 Summary 156
6 Structural Reliability Assessment based on the Information of Moments of Random Variables 158
6.1 Introduction 158
6.2 Direct Methods of Moment Without Using Probability Distribution 159
6.2.1 Second-Moment Formulation 159
6.4.2 Third-Moment Formulation 160
6.4.3 Fourth-Moment Formulation 161
6.3 First-Order Second-Moment Method 161
6.4 First-Order Third-Moment Method 166
6.4.1 First-Order Third-Moment Method in Reduced Space 166
6.4.2 First-Order Third-Moment Method in Pseudo Standard Normal Space 167
6.5 First-Order Fourth-Moment Method 180
6.5.1 First-Order Fourth-Moment Method in Reduced Space 180
6.5.2 First-Order Fourth-Moment Method in Pseudo Standard Normal Space 180
6.6 Monte Carlo Simulation Using Moment of Random Variables 191
6.7 Subset Simulation Using Statistical Moments of Random Variables 200
6.8 Summary 205
7 Transformation of Non-Normal Variables to Independent Normal Variables 206
7.1 Introduction 206
7.2 The Normal Transformation for a Single Random Variable 206
7.3 The Normal Transformation for Correlated Random Variable 207
7.3.1 Rosenblatt Transformation 207
7.3.2 Nataf Transformation 208
7.4 Pseudo Normal Transformations for a Single Random Variable 215
7.4.1 Concept of Pseudo Normal Transformation 215
7.4.2 Third Moment Pseudo Normal Transformation 217
7.4.3 Fourth Moment Pseudo Normal Transformation 223
7.5 Pseudo Normal Transformations of Correlated Random Variables 238
7.5.1 Introduction 238
7.5.2 Third Moment Pseudo Normal Transformation for Correlated Random Variables 240
7.5.3 Fourth Moment Pseudo Normal Transformation for Correlated Random Variables 243
7.6 Summary 249
8 System Reliability Assessment by the Method of Moments 251
8.1 Introduction 251
8.2 Basic Concepts of System Reliability 251
8.2.1 Multiple Failure Modes 251
8.2.2 Series and Parallel Systems 252
8.3 System Reliability Bounds 260
8.3.1 Uni-Modal Bounds 260
8.3.2 Bi-Modal Bounds 262
8.3.3 Correlation Between a Pair of Failure Modes 264
8.3.4 Bound Estimation of the Joint Failure Probability of a Pair of Failure Modes 265
8.3.5 Point Estimation of the Joint Failure Probability of a Pair of Failure Modes 268
8.4 Moment Approach for System Reliability 277
8.4.1 Performance Function for a system 277
8.4.2 Method of Moments for System Reliability 280
8.5 Methods of Moment for System Reliability Assessment of Ductile Frame Structure 289
8.5.1 Introduction 289
8.5.2 Performance Function Independent of Failure Modes 290
8.5.3 Limit Analysis 292
8.5.4 Methods of Moment for System Reliability of Ductile Frames 293
8.6 Summary 299
9 Determination of Load and Resistance Factors by Methods of Moment 300
9.1 Introduction 300
9.2 Basic Concept of Load and Resistance Factors 301
9.2.1 Basic Concept 301
9.2.2 Determination of LRFs by Second-Moment Method 301
9.2.3 Determination of LRFs under Lognormal Assumption 303
9.2.4 Determination of LRFs by FORM 304
9.2.5 Practical Method for the Determination of LRFs 311
9.3 Load and Resistance Factors by Third-Moment Method 312
9.3.1 Determination of LRFs using Third-Moment Method 312
9.3.2 Estimation of the Mean Value of Resistance 314
9.4 General Expressions of Load and Resistance Factors using Method of Moments 319
9.5 Determination of Load and Resistance Factors Using Fourth-Moment Method 320
9.5.1 Basic Formulas 320
9.5.2 Determination of the Mean Value of Resistance 321
9.6 Summary 325
10 Methods of Moment for Time-Variant Reliability 326
10.1 Introduction 326
10.2 Simulating Stationary Non-Gaussian Process using The Fourth-Moment Transformation 326
10.2.1 Introduction 326
10.2.2 Transformation for Marginal Probability Distributions 327
10.2.3 Transformation for Correlation Functions 328
10.2.4 Methods to Deal with the Incompatibility 331
10.2.5 Scheme of Simulating Stationary Non-Gaussian Random Processes 332
10.3 First Passage Probability Assessment of Stationary Non-Gaussian Processes Using Fourth-Moment Transformation 340
10.3.1 Introduction 340
10.3.2 Formulation of the First Passage Probability of Stationary Non-Gaussian Structural Responses 341
10.3.4 Computational Procedure for the First Passage Probability of Stationary Non-Gaussian Structural Responses 343
10.4 Fast Integration Algorithms for Time-Dependent Structural Reliability Analysis Considering Correlated Random Variables 344
10.4.1 Introduction 344
10.4.2 Formulation of Time-Dependent Failure Probability 345
10.4.3 Fast Integration Algorithms for the Time-Dependent Failure Probability 347
10.5 Summary 357
11 Methods of Moment for Structural Reliability with Hierarchical Modeling of Uncertainty 358
11.1 Introduction 358
11.2 Formulation Description of the Structural Reliability with Hierarchical Modeling of Uncertainty 359
11.3 Overall Probability of Failure Due to Hierarchical Modeling of Uncertainty 360
11.3.1 Evaluating Overall Probability of Failure Based on FORM 360
11.3.2 Evaluating Overall Probability of Failure Based on Methods of Moment 363
11.3.3 Evaluating Overall Probability of Failure Based on Direct Point Estimate Method 364
11.4 The Quantile of the Conditional Failure Probability 368
11.5 Application to Structural Dynamic Reliability Considering Parameters Uncertainties 375
11.6 Summary 381
12 Structural Reliability Analysis Based on the First Few L-Moments 382
12.1 Introduction 382
12.2 Definition of L-moments 382
12.3 Structural Reliability Analysis Based on the First Three L-Moments 384
12.3.1 Transformation for Independent Random Variables 384
12.3.2 Transformation for Correlated Random Variables 385
12.3.3 Reliability Analysis Using the First Three L-moments and Correlation Matrix 388
12.4 Structural Reliability Analysis Based on the First Four L-Moments 395
12.4.1 Transformation for Independent Random Variables 395
12.4.2 Transformation for Correlated Random Variables 401
12.4.3 Reliability Analysis using the First Four L-Moments and Correlation Matrix 404
12.5 Summary 406
13 Methods of Moment with Box-Cox Transformation 407
13.1 Introduction 407
13.2 Methods of Moment with Box-Cox Transformation 407
13.2.1 Criterion for Determining the Box-Cox Transformation Parameter 407
13.2.2 Procedure of the Methods of Moment with Box-Cox Transformation for Structural Reliability 408
13.3 Summary 419
Appendix A Basic probability theory 420
A.1 Events and Probability 420
A.1.1 Introduction 420
A.1.2 Events and Their Combinations 420
A.1.3 Mathematical Operations of Sets 421
A.1.4 Mathematics of Probability 422
A.2 Random Variables and Their Distributions 423
A.3 Main Descriptors of a Random Variable 425
A.3.1 Measures of Location 426
A.3.2 Measures of Dispersion 427
A.3.3 Measures of Asymmetry 428
A.3.4 Measures of Sharpness 429
A.4 Moments and Cumulants 431
A.4.1 Moments 431
A.4.2 Moment and Cumulant Generating Functions 432
A.5 Normal and Lognormal Distributions 434
A.5.1 The Normal Distribution 434
A.5.2 The Logarithmic Normal Distribution 437
A.6 Commonly Used Distributions 439
A.6.1 Introduction 439
A.6.2 Rectangular Distribution 440
A.6.3 Bernoulli Sequences and the Binomial Distribution 440
A.6.4 The Geometric Distribution 441
A.6.5 The Poisson Process and Poisson Distribution 441
A.6.6 The Exponential Distribution 442
A.6.7 The Gamma Distribution 442
A.7 Extreme Value Distributions 443
A.7.1 Introduction 443
A.7.2 The Asymptotic Distributions 445
A.7.3 The Gumbel Distribution 446
A.7.4 The Frechet Distribution 447
A.7.5 The Weibull Distribution 448
A.8 Multiple Random Variables 450
A.8.1 Joint and Conditional Probability Distribution 450
A.8.2 Covariance and Correlation 452
A.9 Functions of Random Variables 453
A.9.1 Function of a Single Random Variable 453
A.9.2 Function of Multiple Random Variables 456
A.10 Summary 458
Appendix B Three-Parameter Distributions 459
B.1 Introduction 459
B.2 The 3P Lognormal Distribution 460
B.2.1 Definition of the Distribution 460
B.2.2 Simplification of the Distribution 463
B.3 Square Normal Distribution 464
B.3.1 Definition of the Distribution 464
B.3.2 Simplification of the Distribution 466
B.4 Comparison of the 3P Distributions 468
B.5 Applications of the 3P Distributions 469
B.5.1 Statistical Data Analysis 469
B.5.2 Representations of One and Two-Parameter Distributions 471
B.5.3 Distributions of Some Random Variables used in Structural Reliability 472
B.6 Summary 473
Appendix C Four-Parameter Distributions 474
C.1 Introduction 474
C.2 The Pearson System 475
C.2.1 Definition of the system 475
C.2.2 Various types of the PDF in Pearson system 476
C.3 Cubic Normal Distribution 481
C.3.1 Definition of the distribution 481
C.3.2 Representative PDFs of the distribution 487
C.3.3 Application in data analysis 487
C.3.5 Simplification of the distribution 490
C.4 Summary 491
Appendix D Basic Theory of Stochastic Process 492
D.1 General Concept of Stochastic Process 492
D.2 Time Domain Description of Stochastic Processes 492
D.2.1 Probability Distributions of Stochastic Processes 492
D.2.2 Moment Functions of Stochastic Processes 494
D.2.3 Stationary and Nonstationary Process 495
D.2.4 Ergodicity of a Stochastic Process 496
D.3 Frequency Domain Description of Stochastic Processes 497
D.3.1 Power Spectral Density Function 497
D.3.2 Wide-and Narrow-Band Processes 497
D.4 Special Processes 498
D.4.1 White Noise Process 498
D.4.2 Markov Process 498
D.4.3 Poisson Process 499
D.4.4 Gaussian Process 499
D.5 Spectral Representation Method 499
D.6 Summary 500
References 5
Yan-Gang Zhao, Dr. Eng., is Professor at Kanagawa University, Japan and a Foreign Associate of the Engineering Academy of Japan.
Zhao-Hui Lu, Dr. Eng., is Professor at Beijing University of Technology and former Professor at Central South University, China.
Zhao-Hui Lu, Dr. Eng., is Professor at Beijing University of Technology and former Professor at Central South University, China.
