Stochastic Finite Element Modeling of Laminated Fiber-Reinforced Composite Beams Under Transverse Loading
Authors: Chen Boyang , Jones Simon , Riley Matt 
Published: January 2022

Abstract
It is common in analytic fiber-reinforced composite theory to assume uniformly distributed material properties across the fiber direction to minimize computational expense. However, manufacturing processes introduce imperfections during the construction of composite materials, such as localized delamination, non-uniform distribution in matrix and fibers, pre-existing stress, and tolerance issues [1]. These imperfections make it more difficult to predict the behavior of composite materials under loading. As a result, manufacturers and designers must use conservative estimates of material strength.
This study aims to quantify the uncertainty in laminated fiber-reinforced composite beams subjected to cantilever loads on a macroscopic scale and to provide an all-inclusive introduction to stochastic composite modeling using the finite element method. This introduction is intended for upper undergraduates or new graduate students how are already familiar with structural mechanics and the finite element method. The goal of the paper is to introduce the key topics related to stochastic composite modeling and have validation material with which they can develop and verify custom finite element code.
The system investigated herein is a composite cantilever beam subjected to a transverse tip displacement. Classical Lamination Theory (CLT) is first employed to predict the transverse tip displacement of a beam composed of four lamina at adjustable fiber orientations. A finite element model is then created using a CLT approach to simulate the composite beam’s deformation under tip loading. The Euler-Bernoulli beam elements contain two nodes with two degrees of freedom each: transverse deflection and rotation. These elements are relatively simplistic relative to other composite finite elements, but are sufficient to demonstrate the effect of stochastic material property variation on the overall response of the beam without obfuscating the approach. The finite element results are validated against the analytic predictions for multiple fiber direction layups to ensure the numerical predictions are accurate.
The stochastic approach for varying material properties is then added to the validated finite element code. A Karhunen–Loève expansion of a modified exponential kernel is used to produce spatially-varying elastic modulus profiles for each lamina in the composite beam. The predicted tip displacement for the beam with varying properties is computed, and then CLT is used to determine the effective uniform elastic modulus that is required to produce the same tip displacement. This comparison allows the reader to quantify the impact of the spatially varying properties to a single design property: the effective flexural modulus. A Monte Carlo simulation of 1000 composite beams is then used to determine the statistical distribution of the effective flexural modula. Results suggest that the “averaging effect” of bonding multiple laminas with varying material properties together into composite beams produces effective flexural modula for the beams that do not vary as significantly as the laminas’ elastic modula. Standard deviations of the effective flexural modula are found to be an order of magnitude smaller than that of the variation imposed on the laminas’ elastic modulus.