NSSFEM: Nonlinear Spectral Stochastic Finite Element Method for Analysis of Structures with Elastoplastic Material

Abstract

The stochastic finite element method is one of the most effective tools for analyzing systems with uncertainty in computational stochastic mechanics. In this research, a novel approach called the nonlinear spectral stochastic finite element method (NSSFEM) was developed to analyze structures with nonlinear materials. The proposed NSSFEM incorporates uncertainty in both loads and elastic modulus. In the first step, the input random variables are modeled using the operators of NSSFEM, and an appropriate number of terms from the Karhunen-Loève expansion is selected. Next, the stiffness matrix is formed, assuming linear material behavior as the problem-solving begins. Subsequently, the responses are modeled as random processes and expanded using polynomial chaos. During each increment of the solution process, the stress state at the Gaussian points is checked before completing a sub-increment. If the material yielding criteria are activated, the stresses are modified according to plasticity conditions, thereby correcting the solutions. This iterative process continues until the problem is fully resolved and the desired solution is achieved. The displacements obtained through the proposed NSSFEM demonstrate an impressive accuracy of 97% when compared with results from the Monte Carlo method. The source code of the proposed NSSFEM is available at https://github.com/seyedsajadmousavi/NSSFEM