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.

在本文中，提出了基于随机微扰的有限元方法 (SFEM) 的理论公式和计算实现，用于具有对称非高斯输入参数的固体力学中的不确定性分析。该方法的理论基础是基于所有不确定输入参数和仅包括偶数阶的状态函数的一般阶泰勒展开式。对于随机输入的对称三角形和均匀概率分布，包括概率分布截断效应，已经导出了结构响应的前四个概率特征。使用统计优化的节点多项式响应基为 FEM 的位移版本完成了随机有限元方法的实现，并且它们的系数是使用加权和非加权方案使用最小二乘法确定的。几个机械系统的结构响应使用它们的基本概率特性进行分析，这些特性已经使用概率半分析方法和粗蒙特卡罗模拟进行了验证。三种概率数值技术的相对良好的重合证实了基于随机微扰的有限元方法的适用性，用于研究具有均匀和/或三角形概率分布的不确定性的力学中的边界和初始问题。已使用概率半分析方法以及粗略的 Monte-Carlo 模拟进行了验证。三种概率数值技术的相对良好的重合证实了基于随机微扰的有限元方法的适用性，用于研究具有均匀和/或三角形概率分布的不确定性的力学中的边界和初始问题。已使用概率半分析方法以及粗略的 Monte-Carlo 模拟进行了验证。三种概率数值技术的相对良好的重合证实了基于随机微扰的有限元方法的适用性，用于研究具有均匀和/或三角形概率分布的不确定性的力学中的边界和初始问题。