Stochastic Finite Element Analysis for Static Bending Beams with a Two-Dimensional Random Field of Material Properties

This study presents the development and application of the stochastic finite element method (SFEM) to analyze the static response of beams with a two-dimensional (2D) spatially varying elastic modulus. A 2D stationary stochastic field is employed to model the elastic modulus, capturing the material heterogeneity along both the longitudinal and vertical directions of the beam. The weighted integral method is applied to represent the random field as random variables and to compute the element stiffness matrices, while a first-order perturbation technique is utilized to estimate the statistical moments of the nodal displacement vector, including the mean and covariance matrix. This method enhances both computational efficiency and accuracy in capturing material heterogeneity compared to traditional approaches. The precision and effectiveness of the developed SFEM are evaluated through comparisons with Monte Carlo simulations (MCs), demonstrating strong agreement in the analysis of the coefficient of variation (COV) of displacement. A sensitivity analysis is conducted to examine the influence of the correlation length and dispersion of the stochastic field on the COV. The results indicate that the COV generally increases as these parameters grow, with the most significant variations occurring at small correlation lengths. As the correlation length becomes very large, the COV of displacement converges toward the standard deviation of the input stochastic field. Furthermore, the study reveals that the correlation length along the beam’s longitudinal axis has a more pronounced effect on the COV of displacement compared to the vertical correlation length.