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Optimized quadrature-based finite element for efficient probabilistic analysis of laterally loaded piles in spatially variable soils
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DOI：10.1016/j.compgeo.2026.108496.png)
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Abstract
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Soil properties often exhibit strong spatial variability, which introduces substantial uncertainties in pile performance. Probabilistic analysis is essential to quantify these uncertainties. Traditional discrete spring element (DSE) methods discretize soil resistance as springs at element nodes, neglecting its variations within elements. Thus, a large number of elements with small sizes are demanded to model the pile-soil interaction in spatially variable soils, especially with short scale of fluctuations (SOF). When combined with the large number of evaluations required for probabilistic analysis, the computational cost is massive. This study proposes an optimized quadrature-based element that employs Gauss-Legendre quadrature to capture spatially variable soil resistance within the element. The element soil stiffness matrix and soil resistance vector are formulated with an arbitrary number of Gaussian points. A formula is derived to establishes the optimal number of Gaussian points based on the normalized SOF of the soil. This novel approach enables the element to capture the spatial variability of soil resistance across different SOF values by adjusting the number of Gauss points, which has minimal impact on computation time. Using the proposed element, the total number of elements required can be significantly reduced, as only 1 to 4 elements are sufficient to model piles ranging from rigid to long-flexible types. Four case studies demonstrate that the proposed method achieves accuracy comparable to DSE models, while the computation time for 10,000 Monte Carlo simulation (MCS) realizations is reduced from hours to minutes. Finally, a systematic parametric study is conducted to evaluate the effect of soil spatial variability and develop a rapid preliminary assessment approach.
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