How accurate are the reliability assessments conducted by the finite element method?

Abstract

The present paper investigates the accuracy of the finite element method (FEM) in stochastic setting. The performance of the FEM for solving the transversal vibration eigenvalue problem of a uniform, homogeneous beam in presence of uncertainties is considered aiming to establish how accurate the method is in predicting the beam’s reliability as well as its probability of failure. An explicit solution is first provided for the approximate fundamental frequency of the beam as a function of the number of elements, for different boundary conditions when the mesh is uniform along the length of the beam allowing an analytical evaluation of the structural reliability and the probability of failure when, e.g., the random uncertainty in the Young modulus of the beam is considered. The exact solution of the vibration problem derived within Bernoulli-Euler beam theory is exploited to evaluate the actual reliability as well as the actual probability of failure which, being compared with required reliability or allowed probability of failure thresholds, permits to verify the accuracy of the FEM in the probabilistic context and to warn about “unreliability of reliability conclusions”.