Spectral Stochastic Finite‐Element Formulation for Reliability Analysis

Abstract

An approach for the solution of problems of structural mechanics involving material variability is proposed. The material property is modeled as a stochastic process. The Karhunen‐Loeve expansion is used to represent this process in a computationally expedient manner by means of a set of random variables. Further, the well‐established deterministic finite‐element method is used to discretize the differential equations governing the structural response. A spectral expansion of the nodal random variables is introduced involving a basis in the space of random variables. The basis consists of the polynomial chaoses that are polynomials orthogonal with respect to the Gaussian probability measure. The new formulation allows the computation of the probability distribution functions of the response variables in an expeditious manner. Two problems from structural mechanics are investigated using the proposed approach. The derived results are found in good agreement with data obtained by a Monte Carlo simulation solution of these problems.