Investigating the Uncertainty Quantification of Failure of Shallow Foundation of Cohesionless Soils Through Drucker–Prager Constitutive Model and Probabilistic FEM

Uncertainty quantification in science and engineering has become increasingly important due to advances in computational mechanics and numerical simulation techniques. In this work, the relationship between uncertainty in soil material parameters and the variability of failure loads and displacements of a shallow foundation is investigated. A Drucker–Prager constitutive law is implemented within a stochastic finite element framework. The random material variables considered are the critical state line slope c, the unload–reload path slope κ, and the hydraulic permeability k defined by Darcy’s law. The novelty of this work lies in the integrated stochastic u–p finite element framework. The framework combines Drucker–Prager plasticity with spatially varying material properties, and Latin Hypercube Sampling. This approach enables probabilistic prediction of failure loads, displacements, stresses, strains, and limit-state initiation points at reduced computational cost compared to conventional Monte Carlo simulations. Statistical post-processing of the output parameters is performed using the Kolmogorov–Smirnov test. The results indicate that, for the investigated configurations, the distributions of failure loads and displacements can be adequately approximated by Gaussian distributions, despite the presence of material nonlinearity. Furthermore, the influence of soil depth and load eccentricity on the limit-state response is quantified within the proposed probabilistic framework.