Enhanced Random Finite Element Limit Analysis framework for assessing probabilistic failure: Application to eccentrically loaded shallow foundations

This paper presents an efficient probabilistic framework for evaluating the failure probability of eccentrically loaded shallow foundations on undrained soils with spatially variable shear strength. The approach combines Random Finite Element Limit Analysis (RFELA) with a reduced-dimension Monte Carlo formulation in which uncertainty in load magnitude and effective eccentricity is formulated analytically, with the resulting integrals evaluated using Gauss–Hermite quadrature, while soil variability is sampled numerically. This reduces the computational burden associated with conventional Monte Carlo simulation, which requires simultaneous sampling of both loads and soil properties. The method explicitly accounts for the influence of load eccentricity induced by random load combinations and incorporates a discretized representation of the ultimate bearing capacity derived from precomputed failure mechanisms. The proposed framework is validated against brute-force Monte Carlo simulations using up to 1 0 9 realizations, showing excellent agreement across a wide range of failure probabilities. Results show that the proposed approach achieves accurate estimates of probabilities as low as 1 0 − 9 while reducing the computational cost by several orders of magnitude. The framework is therefore particularly suitable for large-scale reliability assessments of footings under spatial variability and variable load eccentricity.