In order to accurately analyze the internal force and deformation of pile under the combined horizontal dynamic load (H(t)) and torsional vibration (T(t)) on the top of the pile, a simplified analytical model of H(t)-T(t) loaded pile in layered foundation is established based on the Pasternak model, considering the interaction of multidirectional loads. The numerical solutions of the internal forces and displacements of the pile are derived by using the finite beam element method, and the results are compared with the existing theoretical solutions, model tests and finite element simulations. Parametric analysis shows that: (1) Compared with the calculation results of Winkler model, the horizontal displacement at the top of the pile and the maximum bending moment of the pile are reduced by 11.8% and 10.5%, respectively, after considering the shear effect of the soil. (2) Increasing the dimensionless frequency of the external load reduces the horizontal displacement and bending moment of the pile, and also reduces the H(t)-T(t) coupling stiffness. (3) In the layered foundation, the surface soil has the greatest influence on the internal force and displacement of the pile, and there is a critical influence depth of the surface soil. The critical influence thickness of the surface hard soil is 3.5–6.5 times that of the surface soft soil. (4) The beam element model reduces the number of element divisions and calculation time, which can effectively improve the calculation efficiency.

There are few theoretical studies on lateral bearing capacity of flexible single pile under vertical-horizontal loading path in sand foundation and the influence of pre-applied vertical load on p-y curve is rarely considered. In view of this, the ultimate soil resistance of hyperbolic p-y curve was modified accounting for compacting effect of the sandy soil around the pile under the vertical force applied in advance. Cosine function was used to characterize the distribution of radial earth pressure on the passive side. The relationship between the maximum radial earth pressure on the passive side and the total soil resistance was proposed, and the analytical expression of the shaft resisting moment on the passive side was derived. Taking an included angle β between the direction of friction resistance and the vertical direction into account, the variation of pile axial force induced by friction resistance was corrected and its computational formula was obtained. The differential equation of pile deflection was established considering the modified p-y curve, the variation of axial force caused by pile dead weight and friction resistance, the P-Δ effect and the shaft resisting moment. The numerical solution was obtained by MATLAB program. The correctness of the proposed method was verified by comparing the calculated results with the existing simulations and measurements. On this basis, the influence of vertical force applied in advance on lateral bearing capacity of single pile was discussed. The results show that the computational formula of the variation of pile axial force deduced in this paper can more accurately describe the influence of friction resistance on pile axial force and can be used for large deformation conditions. The vertical force applied in advance can enhance the lateral bearing capacity of flexible single pile. The enhancement gradually decreases with the increase of the vertical force applied in advance.