Structural Reliability: Approaches from Perspectives of Statistical Moments

ISBN-13: 9781119620815 / Angielski / Twarda / 2021 / 656 str.

Yan-Gang Zhao; Zhao-Hui Lu

Spis treści

Preface1 Measures of Structural Safety 11.1 Introduction 11.2 Uncertainties in Structural Design 11.2.1 Uncertainties in the Properties of Structures and Their Environment 11.2.2 Sources and Types of Uncertainty 31.2.3 Treatment of Uncertainties 41.2.4 Design and Decision Making Under Uncertainties 71.3 Deterministic Measures of Safety 81.4 Probabilistic Measure of Safety 81.5 Summary 92 Fundamentals of Structural Reliability Theory 102.1 The Fundamental Case 102.2 Performance Function and Failure Probability 142.2.1 Performance Function 142.2.2 Probability of Failure 142.2.3 Reliability Index 162.3 Monte Carlo Simulation 202.3.1 Introduction 202.3.2 Generation of Random Numbers 212.3.3 Direct Sampling 242.4 A Brief Review on Structural Reliability Theory 282.5 Summary 303 Moment Evaluation for Performance Functions 313.1 Introduction 313.2 Moment Computation for Some Simple Functions 333.2.1 Moment Computation for Linear Sum of Random Variables 333.2.2 Moment Computation for Multiply of Random Variables 353.2.3 Moment Computation for Power of a Lognormally Distributed Random Variable 373.2.4 Moment Computation for Power of an Arbitrarily Distributed Random Variable 413.2.5 Moment Computation for Reciprocal of an Arbitrary Distributed Random Variable 433.3 Point Estimate for a Function of One Random Variable 443.3.1 Rosenblueth's Two-Point Estimate 443.3.2 Gorman's Three-Point Estimate 453.4 Point Estimates in Standardized Normal Space 503.4.1 Basic ideas 503.4.2 Two- and Three-point Estimates in the Standard Normal Space 523.4.3 Five-Point Estimate in Standard Normal Space 533.4.4 Seven-Point Estimate in Standard Normal Space 543.4.5 General Expression of Estimating Points and Their Corresponding Weights 573.4.6 Accuracy of the Point Estimate 593.5 Point Estimates for a Function of Multiple Variables 613.5.1 General Expression of Point Estimate for a Function of n Variables 613.5.2 Approximate Point Estimate for a Function of n Variables 633.5.3 Dimension Reduction Integration 673.6 Point Estimates for a Function of Correlative Random Variables 713.7 Hybrid Dimension-Reduction Based Point Estimate Method 753.8 Summary 784 Direct Methods of Moment 794.1 Basic Concept of Methods of Moment 794.1.1 Introduction 794.1.2 The Second-Moment Method 794.1.3 General Expressions for Methods of Moment 824.2 Third-Moment Reliability Method 834.2.1 Introduction 834.2.2 Third-Moment Reliability Indices 854.2.3 Empirical Applicable Range of Third-Moment Reliability Method 884.2.4 Simplification of Third-Moment Reliability Method 914.2.5 Applicable Range of the Second-Moment method 944.3 Fourth-Moment Reliability Method 994.3.1 Introduction 994.3.2 Fourth-Moment Reliability Index on the Basis of the Pearson System 1014.3.3 Fourth-Moment Reliability Index Based on Third-Order Polynomial Transformation 1044.3.4 Applicable range of Fourth-Moment method 1064.3.5 Simplification of Fourth-Moment reliability index 1104.4 Summary 1125 Methods of Moment Based on First/Second Order Transformation 1135.1 Introduction 1135.2 First-Order Reliability Method 1135.2.1 The Hasofer-Lind reliability index 1135.2.2 First Order Reliability Method 1155.2.3 Numerical Solution for FORM 1195.2.4 The Weakness of FORM 1245.3 Second Order Reliability Method 1275.3.1 Introduction 1275.3.2 Second Order Approximation of the Performance Function 1275.3.3 Failure probability for Second Order Performance Function 1385.3.4 Methods of Moment for Second Order Approximation 1435.3.5 Applicable Range of FORM 1545.4 Summary 1566 Structural Reliability Assessment based on the Information of Moments of Random Variables 1586.1 Introduction 1586.2 Direct Methods of Moment Without Using Probability Distribution 1596.2.1 Second-Moment Formulation 1596.4.2 Third-Moment Formulation 1606.4.3 Fourth-Moment Formulation 1616.3 First-Order Second-Moment Method 1616.4 First-Order Third-Moment Method 1666.4.1 First-Order Third-Moment Method in Reduced Space 1666.4.2 First-Order Third-Moment Method in Pseudo Standard Normal Space 1676.5 First-Order Fourth-Moment Method 1806.5.1 First-Order Fourth-Moment Method in Reduced Space 1806.5.2 First-Order Fourth-Moment Method in Pseudo Standard Normal Space 1806.6 Monte Carlo Simulation Using Moment of Random Variables 1916.7 Subset Simulation Using Statistical Moments of Random Variables 2006.8 Summary 2057 Transformation of Non-Normal Variables to Independent Normal Variables 2067.1 Introduction 2067.2 The Normal Transformation for a Single Random Variable 2067.3 The Normal Transformation for Correlated Random Variable 2077.3.1 Rosenblatt Transformation 2077.3.2 Nataf Transformation 2087.4 Pseudo Normal Transformations for a Single Random Variable 2157.4.1 Concept of Pseudo Normal Transformation 2157.4.2 Third Moment Pseudo Normal Transformation 2177.4.3 Fourth Moment Pseudo Normal Transformation 2237.5 Pseudo Normal Transformations of Correlated Random Variables 2387.5.1 Introduction 2387.5.2 Third Moment Pseudo Normal Transformation for Correlated Random Variables 2407.5.3 Fourth Moment Pseudo Normal Transformation for Correlated Random Variables 2437.6 Summary 2498 System Reliability Assessment by the Method of Moments 2518.1 Introduction 2518.2 Basic Concepts of System Reliability 2518.2.1 Multiple Failure Modes 2518.2.2 Series and Parallel Systems 2528.3 System Reliability Bounds 2608.3.1 Uni-Modal Bounds 2608.3.2 Bi-Modal Bounds 2628.3.3 Correlation Between a Pair of Failure Modes 2648.3.4 Bound Estimation of the Joint Failure Probability of a Pair of Failure Modes 2658.3.5 Point Estimation of the Joint Failure Probability of a Pair of Failure Modes 2688.4 Moment Approach for System Reliability 2778.4.1 Performance Function for a system 2778.4.2 Method of Moments for System Reliability 2808.5 Methods of Moment for System Reliability Assessment of Ductile Frame Structure 2898.5.1 Introduction 2898.5.2 Performance Function Independent of Failure Modes 2908.5.3 Limit Analysis 2928.5.4 Methods of Moment for System Reliability of Ductile Frames 2938.6 Summary 2999 Determination of Load and Resistance Factors by Methods of Moment 3009.1 Introduction 3009.2 Basic Concept of Load and Resistance Factors 3019.2.1 Basic Concept 3019.2.2 Determination of LRFs by Second-Moment Method 3019.2.3 Determination of LRFs under Lognormal Assumption 3039.2.4 Determination of LRFs by FORM 3049.2.5 Practical Method for the Determination of LRFs 3119.3 Load and Resistance Factors by Third-Moment Method 3129.3.1 Determination of LRFs using Third-Moment Method 3129.3.2 Estimation of the Mean Value of Resistance 3149.4 General Expressions of Load and Resistance Factors using Method of Moments 3199.5 Determination of Load and Resistance Factors Using Fourth-Moment Method 3209.5.1 Basic Formulas 3209.5.2 Determination of the Mean Value of Resistance 3219.6 Summary 32510 Methods of Moment for Time-Variant Reliability 32610.1 Introduction 32610.2 Simulating Stationary Non-Gaussian Process using The Fourth-Moment Transformation 32610.2.1 Introduction 32610.2.2 Transformation for Marginal Probability Distributions 32710.2.3 Transformation for Correlation Functions 32810.2.4 Methods to Deal with the Incompatibility 33110.2.5 Scheme of Simulating Stationary Non-Gaussian Random Processes 33210.3 First Passage Probability Assessment of Stationary Non-Gaussian Processes Using Fourth-Moment Transformation 34010.3.1 Introduction 34010.3.2 Formulation of the First Passage Probability of Stationary Non-Gaussian Structural Responses 34110.3.4 Computational Procedure for the First Passage Probability of Stationary Non-Gaussian Structural Responses 34310.4 Fast Integration Algorithms for Time-Dependent Structural Reliability Analysis Considering Correlated Random Variables 34410.4.1 Introduction 34410.4.2 Formulation of Time-Dependent Failure Probability 34510.4.3 Fast Integration Algorithms for the Time-Dependent Failure Probability 34710.5 Summary 35711 Methods of Moment for Structural Reliability with Hierarchical Modeling of Uncertainty 35811.1 Introduction 35811.2 Formulation Description of the Structural Reliability with Hierarchical Modeling of Uncertainty 35911.3 Overall Probability of Failure Due to Hierarchical Modeling of Uncertainty 36011.3.1 Evaluating Overall Probability of Failure Based on FORM 36011.3.2 Evaluating Overall Probability of Failure Based on Methods of Moment 36311.3.3 Evaluating Overall Probability of Failure Based on Direct Point Estimate Method 36411.4 The Quantile of the Conditional Failure Probability 36811.5 Application to Structural Dynamic Reliability Considering Parameters Uncertainties 37511.6 Summary 38112 Structural Reliability Analysis Based on the First Few L-Moments 38212.1 Introduction 38212.2 Definition of L-moments 38212.3 Structural Reliability Analysis Based on the First Three L-Moments 38412.3.1 Transformation for Independent Random Variables 38412.3.2 Transformation for Correlated Random Variables 38512.3.3 Reliability Analysis Using the First Three L-moments and Correlation Matrix 38812.4 Structural Reliability Analysis Based on the First Four L-Moments 39512.4.1 Transformation for Independent Random Variables 39512.4.2 Transformation for Correlated Random Variables 40112.4.3 Reliability Analysis using the First Four L-Moments and Correlation Matrix 40412.5 Summary 40613 Methods of Moment with Box-Cox Transformation 40713.1 Introduction 40713.2 Methods of Moment with Box-Cox Transformation 40713.2.1 Criterion for Determining the Box-Cox Transformation Parameter 40713.2.2 Procedure of the Methods of Moment with Box-Cox Transformation for Structural Reliability 40813.3 Summary 419Appendix A Basic probability theory 420A.1 Events and Probability 420A.1.1 Introduction 420A.1.2 Events and Their Combinations 420A.1.3 Mathematical Operations of Sets 421A.1.4 Mathematics of Probability 422A.2 Random Variables and Their Distributions 423A.3 Main Descriptors of a Random Variable 425A.3.1 Measures of Location 426A.3.2 Measures of Dispersion 427A.3.3 Measures of Asymmetry 428A.3.4 Measures of Sharpness 429A.4 Moments and Cumulants 431A.4.1 Moments 431A.4.2 Moment and Cumulant Generating Functions 432A.5 Normal and Lognormal Distributions 434A.5.1 The Normal Distribution 434A.5.2 The Logarithmic Normal Distribution 437A.6 Commonly Used Distributions 439A.6.1 Introduction 439A.6.2 Rectangular Distribution 440A.6.3 Bernoulli Sequences and the Binomial Distribution 440A.6.4 The Geometric Distribution 441A.6.5 The Poisson Process and Poisson Distribution 441A.6.6 The Exponential Distribution 442A.6.7 The Gamma Distribution 442A.7 Extreme Value Distributions 443A.7.1 Introduction 443A.7.2 The Asymptotic Distributions 445A.7.3 The Gumbel Distribution 446A.7.4 The Frechet Distribution 447A.7.5 The Weibull Distribution 448A.8 Multiple Random Variables 450A.8.1 Joint and Conditional Probability Distribution 450A.8.2 Covariance and Correlation 452A.9 Functions of Random Variables 453A.9.1 Function of a Single Random Variable 453A.9.2 Function of Multiple Random Variables 456A.10 Summary 458Appendix B Three-Parameter Distributions 459B.1 Introduction 459B.2 The 3P Lognormal Distribution 460B.2.1 Definition of the Distribution 460B.2.2 Simplification of the Distribution 463B.3 Square Normal Distribution 464B.3.1 Definition of the Distribution 464B.3.2 Simplification of the Distribution 466B.4 Comparison of the 3P Distributions 468B.5 Applications of the 3P Distributions 469B.5.1 Statistical Data Analysis 469B.5.2 Representations of One and Two-Parameter Distributions 471B.5.3 Distributions of Some Random Variables used in Structural Reliability 472B.6 Summary 473Appendix C Four-Parameter Distributions 474C.1 Introduction 474C.2 The Pearson System 475C.2.1 Definition of the system 475C.2.2 Various types of the PDF in Pearson system 476C.3 Cubic Normal Distribution 481C.3.1 Definition of the distribution 481C.3.2 Representative PDFs of the distribution 487C.3.3 Application in data analysis 487C.3.5 Simplification of the distribution 490C.4 Summary 491Appendix D Basic Theory of Stochastic Process 492D.1 General Concept of Stochastic Process 492D.2 Time Domain Description of Stochastic Processes 492D.2.1 Probability Distributions of Stochastic Processes 492D.2.2 Moment Functions of Stochastic Processes 494D.2.3 Stationary and Nonstationary Process 495D.2.4 Ergodicity of a Stochastic Process 496D.3 Frequency Domain Description of Stochastic Processes 497D.3.1 Power Spectral Density Function 497D.3.2 Wide-and Narrow-Band Processes 497D.4 Special Processes 498D.4.1 White Noise Process 498D.4.2 Markov Process 498D.4.3 Poisson Process 499D.4.4 Gaussian Process 499D.5 Spectral Representation Method 499D.6 Summary 500References 5

O autorze

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.