Stochastic Finite Element Formulation of a Three-Node Quadratic Bar Element with Non-Uniform Cross-Section Based on the Perturbation Method for Simultaneously Non-Deterministic Elastic Modulus and Applied Load 2024-26-0470

The finite element method is one of the most robust tools in structural analysis. Typically, the input parameters in a finite element model are assumed to be deterministic. However, in practice, almost all material and geometrical properties, including the load, possess randomness. The consideration of the probabilistic nature of these quantities is essential to effectively designing a system that is robust against the uncertainties arising due to the variation in the input parameters, the significance of which has been documented by NASA in “Probabilistic Risk Assessment Procedures Guide for NASA Managers and Practitioners”, 2011. Among the various techniques applicable for stochastic analysis, the perturbation method, which is based on a sound mathematical foundation derived from Taylor’s series expansion, is widely acknowledged for its much higher efficiency compared to the well-known Monte-Carlo method. With this motivation, this work presents a stochastic finite element formulation of a bar element, commonly used in aerospace structural simulations. To the best of the authors’ knowledge, this is the first time a formulation will be put forth for the case of a three-node quadratic bar element with a non-uniform cross-section based on the perturbation method. Interestingly, the method will be formulated for a simultaneously non-deterministic elastic modulus and distributed axial load. Subsequently, the finite element implementation of the formulated method shall be elucidated with an investigation of the behaviour of the computed structural response in a sample problem for varying randomness of the inputs, along with a results-based discussion on the performance of the method compared to the Monte Carlo method, endorsing its high efficiency.