Application of Spectral Stochastic Finite Element via Galerkin Method in Beam Bending Theories

The chapter delves into the application of Spectral Stochastic Finite Element via Galerkin Method in beam bending theories, addressing the critical issue of portraying irregular disparities in mechanical properties. It explores the propagation of uncertainty through experimental measurements and the use of epistemic uncertainty assumptions. The text introduces several numerical methods, including the Perturbation method, Karhunen-Loeve decomposition, and the Galerkin method, to solve stochastic differential equations. It also compares the performance of Monte Carlo Simulation (MCS), Neumann-based Monte Carlo (NMC), and Spectral Stochastic Finite Element Method (SSFEM) in quantifying uncertainty. The chapter presents a variational problem for a stationary beam based on Euler-Bernoulli, Timoshenko, and Levinson-Bickford theories, and discusses the numerical simulations and results for the statistical moments of the displacement field. The unique aspects of the trigonometric series model and the advantages of the Levinson-Bickford theory are highlighted. The chapter concludes with a comparison of processing times among the different uncertainty quantification methodologies.