Stochastic Finite Elements

Scope: The course aims at the investigation of the effect of uncertain parameters (material and geometric properties, loading) on structural response variability.

Introduction: Random variables, cumulative distribution function, probability density function, statistical moments (mean value, variance, skewness and kurtosis), covariance. Stochastic processes and fields: Definition, stationary stochastic processes, ergodicity, analysis in the frequency domain-Fourier transform: autocorrelation and spectral density functions, Gaussian stochastic processes. Representation/discretization of stochastic processes and fields using (i) Point discretization methods: midpoint, integration and nodal point methods (ii) Average discretization methods: local average and weighted integral methods (iii) Spectral representation method: simulation of stationary Gaussian stochastic processes and fields. Formulation and solution of the stochastic problem: Stochastic virtual work principle, formulation of the stochastic stiffness matrix using the local average and weighted integral methods, solution by Taylor, Neumann series expansion and by Monte Carlo simulation. Applications: Computer applications on framed structures and 2D elasticity problems: investigation of the effect of several stochastic field parameters (probability distribution, correlation length and autocorrelation function) on structural response variability.