# Mathematical Models for Structural Reliability Analysis

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## Book Description

Mathematical Models for Structural Reliability Analysis offers mathematical models for describing load and material properties in solving structural engineering problems. Examples are provided, demonstrating how the models are implemented, and the limitations of the models are clearly stated. Analytical solutions are also discussed, and methods are clearly distinguished from models. The authors explain both theoretical models and practical applications in a clear, concise, and readable fashion.

## Table of Contents

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