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<p>This study investigated the buckling behavior of slender piles fully embedded in soil under axial compression, considering depth-dependent soil stiffness within a Winkler-type foundation model. The governing differential equation was discretized using the finite difference method and formulated as a generalized eigenvalue problem to determine the critical buckling loads and corresponding mode shapes. A parametric analysis was conducted to examine the influence of pile length, initial subgrade stiffness, stiffness gradient, and boundary conditions. The results showed that soil–structure interaction significantly modifies the classical buckling response, leading to nonlinear variation of the critical load and convergence behavior for increasing pile length. The relative influence of soil stiffness parameters was found to depend on the boundary conditions, with the stiffness gradient dominating in highly restrained configurations and the initial subgrade stiffness becoming more influential in free-head cases. The study provides a consistent framework for interpreting the combined effects of soil stiffness variation and end restraints on pile stability.</p>

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A Finite Difference Approach to the Buckling Analysis of Fully Embedded Piles under Various End Restraints

  • 期刊论文

  • 发表日期：

    2026年5月22日

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  • Vlora Shatri,
  • Burbuqe Shatri,
  • Bajram Shefkiu

摘要

This study investigated the buckling behavior of slender piles fully embedded in soil under axial compression, considering depth-dependent soil stiffness within a Winkler-type foundation model. The governing differential equation was discretized using the finite difference method and formulated as a generalized eigenvalue problem to determine the critical buckling loads and corresponding mode shapes. A parametric analysis was conducted to examine the influence of pile length, initial subgrade stiffness, stiffness gradient, and boundary conditions. The results showed that soil–structure interaction significantly modifies the classical buckling response, leading to nonlinear variation of the critical load and convergence behavior for increasing pile length. The relative influence of soil stiffness parameters was found to depend on the boundary conditions, with the stiffness gradient dominating in highly restrained configurations and the initial subgrade stiffness becoming more influential in free-head cases. The study provides a consistent framework for interpreting the combined effects of soil stiffness variation and end restraints on pile stability.