Analysis of Beams with a Three-dimensional Random Field of the Modulus of Elasticity Using the Stochastic Finite Element Method

This study proposes a stochastic finite element method (SFEM) for analyzing the static response of beams with material properties modeled as three-dimensional spatial random fields. The method employs weighted integration to discretize spatial variations in Young’s modulus and utilizes a perturbation approach for efficient statistical response computation. Validation is performed using Monte Carlo simulations (MCs) with the spectral representation method to establish a benchmark dataset, showing strong agreement between the two methods, particularly for large correlation distances. The results demonstrate that spatial variability in Young’s modulus significantly affects beam displacement. Shorter correlation lengths reduce displacement variability, while longer correlation lengths lead to greater deflection dispersion. Additionally, an enhancement in the standard deviation of Young's elastic modulus correlates with a higher coefficient of variation (COV) of displacement, confirming structural sensitivity to material randomness. The COV of displacement shows a nearly proportional relationship to the COV of Young’s modulus, which provides key insights into the predictability of stochastic structural behavior. While SFEM is computationally more efficient than MCs, its first-order perturbation formulation limits accuracy in highly nonlinear cases. Future work should explore higher-order stochastic approximations, non-Gaussian random fields, and nonlinear extensions. These findings contribute to advancing stochastic structural analysis by extending SFEM to 3D random fields, providing a foundation for uncertainty quantification in engineering design and highlighting the importance of spatially varying material properties.