Mathematical Models for Structural Reliability Analysis

Abstract

Stochastic Process Models (F. Casciati and M. Di Paola) Introduction The Orthogonal-Increment Model The Correlation-Stationary Model Time-Invariant Linear Systems Models of Common Use The Evolutionary Model Time-Invariant Linear Systems Markov Processes A Model of Common Use Ito Stochastic Differential Equation Some Examples Approximation of Mechanical Processes: Physical versus Ito Equations The Random Pulse Train Model The Delta-Correlated Model Fokker Planck and Moment Equations for Parametric Delta Correlated Input Quasi-Linear Systems Simulation of Delta Correlated Processes and Response Simulation of Normal White Noise Input and Response Orthogonal-Increment Model for Delta Correlated Processes Multidegree-of-Freedom Systems Under Parametric Delta Correlated Input Moment Equation Approach for MDOF Systems Simulation of Multivariate Delta Correlated Processes and Response Conclusions and References Appendix Characterization of Random Variables Joint Characterization of Random Variables Operation on Stochastic Processes Kronecker Algebra: Some Fundamentals Dimension Reduction and Discretization in Stochastic Problems by Regression Method (O. Ditlevsen) Introduction Linear Regression Normal Distribution Non-Gaussian Distributions and Linear Regression Marginally Transformed Gaussian Processes and Fields Discretized Fields Defined by Linear Regression on a Finite Set of Field Values Discretization Defined by Linear Regression on a Finite Set of Linear Functionals Poisson Load Field Example Stochastic Finite Element Methods and Reliability Calculations Classical versus Statistical-Stochastic Interpolation Formulated on the Basis of the Principle of Maximum Likelihood Computational Practicability of the Statistical-Stochastic Interpolation Method Field Modeling on the Basis of Measured Noisy Data Discretization Defined by Linear Regression on Derivatives at a Single Point Conditioning on Crossing Events Slepian Model Vector Processes Application of Slepian Model Processes in Stochastic Mechanics Conclusions and References Reliability of Randomly Excited Hysteretic Systems (J.B. Roberts) Introduction Models of Hysteresis Bilinear Hysteresis Curvilinear Hysteresis Backbone Models The Stochastic Averaging Method The Equation of Motion Averaging the Energy Dissipation Terms Averaging the Excitation Term The FPK Equations Stationary Solutions The Characteristic Frequency Stationary Response of the Bilinear Oscillator Response Statistics Comparison with Simulation Results Yield Statistics Stationary Response of Oscillators with Curvilinear Hysteresis Response Statistics Comparison with Experimental Results Non-Stationary Excitation and Response Numerical Solution of the FPK Equation Comparison with Simulation Results The Energy Envelope Method Calculation of the Backbone Calculation of the Area Enclosed by a Loop Calculation of T(E), C(E) and D2(E) The Loss Factor The Case b = 0 Comparison with Simulation Concluding Remarks and References Non-Parametric Estimation of Failure Probabilities (A.M. Hasofer) Introduction The Single Dimensional Case A Short Statement of Extreme Value Theory Asymptotics of the Top Order Statistics Estimation of High Quantiles for Type I A Test for Extreme Value Domain of Attraction Estimating Quantiles for Type III Estimating Quantiles for Type II The Choice of k Extension to More than One Dimension An Illustration Introducing Importance Sampling A Primer of Importance Sampling The Weissman Estimator Revisited A Modified Weissman Estimator Direct Simulation of k Upper Order Statistics Extension to the Multidimensional Case Simulation Examples The Threshold Method Serial Dependence and Seasonality Conclusions and References Response Surface Methods and Asymptotic Approximations (K. Breitung and L. Faravelli) Introduction Response Surface Methods Response Surface Model of Limit State Functions The Regression Model in a Projection Framework A Test for Lack of Fit The Calculation of Failure Probabilities The Basic Problem Analytic Approximation Methods Approximations for Non-Normal Distributions Parameter Optimization and Uncertainty Derivatives with Respect to Parameters Approximate Bayesian Analysis for Parameter Uncertainties Numerical Examples Function Approximation on Subspaces Reliability Assessment Conclusions and References Appendix A -- Analytical Details Projections and Projection Matrices Definiteness Under Constraints Quadratic Forms on Subspaces Maximum Likelihood for Non-Gaussian Distribution Improvement by Importance Sampling Methods Differential Geometry of a Surface Asymptotic Approximations Asymptotic Approximations for Multidimensional Integrals Appendix B -- Notation Stochastic Methods for Offshore Structures (R.S. Langley and S. McWilliam) Introduction Types of Offshore Structures Environmental Loading The Offshore Environment Environmental Forces Stochastic Response Analysis Overview The Environmental Model The Wave Force Model The Structural Model The Analytical Solution Technique Fixed Offshore Structures Morison-Type Wave Loading Statistics Quasi-Static Response of Linear Structures Large Floating Structures Response of Linearly Moored Structures to Non-Linear Wave Forces Response of Non-Linearly Moored Vessels to Non-Linear Wave Forces Fatigue Analysis Overview Regular Wave Analysis Narrow Band Random Analysis Wide Band Random Analysis Reliability Methods Concluding Remarks and References Index

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