Application of Galerkin Method to Kirchhoff Plates Stochastic Bending Problem

Abstract

In this paper, the Galerkin method is used to obtain approximate solutions for Kirchhoff plates stochastic bending problem with uncertainty over plates flexural rigidity coefficient. The uncertainty in the rigidity coefficient is represented by means of parameterized stochastic processes. A theorem of Lax-Milgram type, about existence and uniqueness of the theoretical solutions, is presented and used in selection of the approximate solution space. The Wiener-Askey scheme of generalized polynomials chaos (gPC) is used to model the stochastic behavior of the displacement solutions. The performance of the approximate Galerkin solution scheme developed herein is evaluated by comparing first and second order moments of the approximate solution with the same moments evaluated from Monte Carlo simulation. Rapid convergence of approximate Galerkin's solution to the first and second order moments is observed, for the problems studied herein. Results also show that using the developed Galerkin's scheme one gets adequate estimates for accrued probability function to a random variable generated by the stochastic process of displacement.