Deformation control equations for engineering piles subjected to vertical and nonlinear lateral loads

The deformation of existing piles adjacent to deep excavations was governed by strongly coupled pile-soil interaction. To overcome the limitations of traditional approaches, this study developed a unified analytical model based on the total potential energy principle and the variational method. By formulating the strain energy of the pile and soil together with the external work of vertical load and excavation-induced lateral earth pressure, the coupled governing equation for pile displacement was rigorously derived and solved analytically. A laboratory-scale excavation test with four staged conditions was conducted to validate the proposed model. The predicted lateral displacement profiles exhibited strong agreement with the measured data. At key depths, the absolute prediction error generally remained within 0.02-0.03 mm, and the relative error was mostly below 10%, with local maxima up to about 20% near the pile toe. These larger deviations near the pile toe were attributed to increased boundary stiffness and soil shear effects. Overall, the results demonstrated that the proposed analytical framework accurately captured the depth-dependent deformation characteristics of piles adjacent to excavations and provided a reliable theoretical basis for deformation prediction and safety evaluation in deep excavation engineering.