Computational Analysis of Randomness in Structural Mechanics Structures and Infrastructures Book Series, Vol. 3

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

Biography

Christian Bucher