Computational Analysis of Randomness in Structural Mechanics

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