The data in Figs 2–8.

In practical engineering, pile foundation will be subject to the combined action of vertical and Rayleigh wave, and the existence of vertical dynamic load will cause the second-order effect that will lead to the increase of horizontal displacement. This paper presents a computational model for the effect of vertical load on the lateral response of monopole under Rayleigh wave. The pile cap is equivalent to a rigid block, and the constraint of pile head is regarded as a flexible constraint. By means of operator decomposition theory and variable separation method, the horizontal dynamic response of single-phase soil in uniform free field under Rayleigh wave propagation is obtained. The closed form solution of soil resistance under the combined action of Rayleigh wave and vertical load is obtained by means of operator decomposition theory and variable separation method. Based on Timoshenko beam theory, the dynamic differential equation of pile considering vertical load is established. A comparison with an existing solution is performed to verify the proposed solution. Through numerical examples, the effects of the vertical load, flexible constraints, dimensionless frequency and Poisson’s ratio on the lateral response of monopile are assessed.在实际工程场景中，桩基会同时承受竖向荷载与瑞利波（Rayleigh wave）的联合作用，而竖向动荷载的存在会引发二阶效应，进而导致水平位移增大。本文针对瑞利波作用下单桩的横向响应受竖向荷载影响的效应，建立了相应的计算模型。将桩帽等效为刚性块，将桩头约束视作柔性约束。借助算子分解理论与分离变量法，推导得到均匀自由场中瑞利波传播时单相土体的水平动力响应；通过算子分解理论与分离变量法，得到瑞利波与竖向荷载联合作用下的土抗力闭合解。基于铁木辛柯梁（Timoshenko beam theory）理论，建立考虑竖向荷载的桩体动力微分方程。将本文所提解法与已有解析解进行对比验证，确保其正确性。通过数值算例，分析了竖向荷载、柔性约束、无量纲频率以及泊松比（Poisson’s ratio）对单桩横向响应的影响规律。