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

This paper presents an analytical investigation into the buckling behaviour of slender piles subjected to axial loading, incorporating the effects of nonlinear pile–soil interaction modeled through the Winkler foundation approach. The soil reaction is characterized by a depth-dependent modulus of subgrade reaction to represent the variation of soil stiffness with depth. The governing differential equation is discretized using the Finite Difference Method (FDM) and transformed into an eigenvalue problem, which is solved numerically in MATLAB to determine the critical buckling loads and associated mode shapes. A detailed parametric analysis is conducted to evaluate the influence of pile length, soil stiffness parameters, and boundary conditions on overall pile stability. The results demonstrate that the stiffness variation of the supporting soil has a dominant effect on the buckling resistance compared to the initial stiffness value. Furthermore, the study reveals that piles with higher restraint conditions exhibit greater buckling capacity, while mode shape transitions observed with increasing pile length highlight the nonlinear nature of the system. The findings emphasize the necessity of incorporating realistic soil stiffness variation and boundary conditions in the design and assessment of pile foundations to ensure structural safety and reliability.