Uncertainty analysis in solid mechanics with uniform and triangular distributions using stochastic perturbation-based Finite Element Method

In this paper theoretical formulation and computational implementation of the Stochastic perturbation-based Finite Element Method (SFEM) for uncertainty analysis in solid mechanics with symmetric non-Gaussian input parameters are presented. Theoretical foundations of the method are based on the general order Taylor expansions of all uncertain input parameters and state functions including even orders only. The first four probabilistic characteristics of the structural responses have been derived for symmetrical triangular and uniform probability distributions of random input including probability distribution truncation effect. The Stochastic Finite Element Method implementation has been completed for the displacement version of the FEM using statistically optimized nodal polynomial response bases, and their coefficients are determined using the Least Squares Method using the weighted and non-weighted schemes. Structural responses of several mechanical systems are analyzed using their basic probabilistic characteristics, which have been validated using the probabilistic semi-analytical approach, and also the crude Monte-Carlo simulation. A relatively good coincidence of three probabilistic numerical techniques confirms the applicability of the Stochastic perturbation-based Finite Element Method to study boundary and initial problems in mechanics with uncertainties having uniform and/or triangular probability distributions.