Nonlinear Analysis for Extended Pile Shaft Under Combined Vertical and Lateral Loads

The lateral response of partially embedded single piles, referred to as extended pile shafts, subjected to combined vertical and lateral loads is investigated using a novel analytical method that explicitly accounts for the P ‐Δ effect caused by axial pile force and the layering of foundation soils. An iterative load‑transfer method is proposed to determine the axial force profile along the pile, while a modified conical strain wedge (CSW) model is developed to better characterize the nonlinear horizontal pile‑soil interaction. By coupling these two models, a nonlinear finite‑difference decomposition algorithm is established to analyze the bearing performance of the extended pile shaft subjected to combined vertical and lateral loads. The accuracy of the proposed method is validated using two published experimental and numerical cases. The validated model is utilized to perform comprehensive parametric studies to investigate the influences of soil internal friction angle, extended length ratio, and vertical load on pile responses. The results reveal that the soil internal friction angle governs the mobilization depth of soil resistance; the extended length ratio serves as the most critical geometric parameter governing lateral deformation and stability; and vertical load mainly acts as an amplification factor after lateral displacement develops. Overall, the proposed method provides a unified and rational framework for predicting the lateral response of single piles under combined loads, which can offer practical guidance for the design of offshore and geotechnical pile foundations.