Mathematical Models for Structural Reliability Analysis

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
Itô Stochastic Differential Equation
Some Examples
Approximation of Mechanical Processes: Physical versus Itô 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

Biography

Fabio Casciati, Brian Roberts