Comp Anal Randomness S1 V3

Proper treatment of structural behavior under severe loading - such as the performance of a high-rise building during an earthquake - relies heavily on the use of probability-based analysis and decision-making tools. Proper application of these tools is significantly enhanced by a thorough understanding of the underlying theoretical and computational concepts as provided by this book.
Detailing the computational aspects of stochastic analysis within the field of structural mechanics, this book first presents a few motivating examples that demonstrate the various random effects within the context of simple structural analysis models. It moreover briefly reviews the fundamental concepts from continuum mechanics and puts them in the perspective of modern numerical tools, such as the finite element method. More advanced topics are developed step by step while gradually increasing the complexity of the structural and probabilistic analyses.
This volume is intended for structural analysts and advanced students who wish to explore the benefits of stochastic analysis. It will provide researchers and decision makers working on structural and infrastructural systems with the necessary probabilistic information needed for strategic developments in construction, inspection and maintenance.

1 Introduction

1.1 Outline
1.2 Introductory examples
1.2.1 Outline of analysis
1.2.2 Static analysis
1.2.3 Buckling analysis
1.2.4 Dynamic analysis

2 Preliminaries in Probability Theory and Statistics

2.1 Definitions
2.2 Probabilistic models
2.2.1 Random variables
2.2.2 Some types of distributions
2.2.3 Conditional distribution
2.2.4 Functions of random variables
2.2.5 Random vectors
2.2.6 Joint probability density function models
2.2.7 Marginal and conditional distribution
2.3 Estimation
2.3.1 Basic properties
2.3.2 Confidence intervals
2.3.3 Chi-square test
2.3.4 Correlation statistics
2.3.5 Bayesian updating
2.3.6 Entropy concepts
2.4 Simulation techniques
2.4.1 General remarks
2.4.2 Crude Monte Carlo simulation
2.4.3 Latin Hypercube sampling
2.4.4 Quasirandom sequences
2.4.5 Transformation of random samples
2.4.6 Simulation of correlated variables

3 Regression and Response Surfaces

3.1 Regression
3.2 Ranking of variables
3.3 Response surface models
3.3.1 Basic formulation
3.3.2 Linear models and regression
3.3.3 First- and second-order polynomials
3.3.4 Weighted interpolation
3.3.5 Moving Least Squares Regression
3.3.6 Radial basis functions
3.4 Design of experiments
3.4.1 Transformations
3.4.2 Saturated designs
3.4.3 Redundant designs

4 Mechanical vibrations due to random excitations

4.1 Basic definitions
4.2 Markov processes
4.2.1 Upcrossing rates
4.3 Single-degree-of-freedom system response
4.3.1 Mean and variance of response
4.3.2 White noise approximation
4.4 Multi-degree-of-freedom response
4.4.1 Equations of motion
4.4.2 Covariance analysis
4.4.3 First passage probability
4.5 Monte-Carlo simulation
4.5.1 General remarks
4.5.2 Central difference method
4.5.3 Euler method
4.5.4 Newmark method
4.5.5 Digital simulation of white noise
4.6 Fokker-Planck equation
4.7 Statistical linearization
4.7.1 General concept
4.8 Dynamic stability analysis
4.8.1 Basics
4.8.2 Nonlinear stability analysis
4.8.3 Linear stability analysis

5 Response analysis of spatially random structures

5.1 Representation of Random Fields
5.1.1 Basic Definitions
5.1.2 Properties of the auto-covariance function
5.1.3 Spectral Decomposition
5.1.4 Conditional Random Fields
5.1.5 Local Averages of Random Fields
5.2 Geometrical Imperfections
5.3 Stochastic Finite Element Formulation
5.3.1 Elasticity (Plane Stress)
5.3.2 Principle of Virtual Work
5.4 Finite Element Method
5.4.1 Element Formulation
5.4.2 Structural response
5.4.3 Stochastic Stiffness Matrix
5.4.4 Integration Point Method
5.4.5 Static Response - Perturbation Method
5.4.6 Monte Carlo Simulation
5.4.7 Natural Frequencies of a Structure with Randomly Distributed Elastic Modulus

6 Computation of failure probabilities

6.1 Structural Reliability
6.1.1 Definitions
6.1.2 First Order - Second Moment Concept
6.1.3 FORM - First Order Reliability Method
6.2 Monte-Carlo-Simulation
6.2.1 Definitions and Basics
6.2.2 Importance Sampling (Weighted Simulation)
6.2.3 Directional Sampling
6.2.4 Asymptotic Sampling
6.3 Application of RSM
6.3.1 Basic concept
6.3.2 Structural examples
6.4 First Passage Failure
6.4.1 Problem formulation
6.4.2 Extension to Non-Linear Problems