Abstract：Modeling soil as a 3D axisymmetric continuum and taking its 3D wave effect into account，the interaction between soil layer and pile with viscoelastic bottom boundary undergoing arbitrary vertical load is theoretically investigated. The investigation will be of significance for dynamic test of pile. The pile is assumed to be vertical，elastic and of uniform cross-section，and the soil is considered as a linear viscoelastic layer with viscous damping. With Laplace transforms，the question can be solved in Laplace domain. With the aid of two potentials，the displacement of soil is decomposed and the dynamic equilibrium equation of soil layer is uncoupled and solved first. Thus the vibration modes of the soil layer are obtained to analyze the pile response. By considering the interaction between the soil layer and the pile with boundary condition of continuity of displacement and equilibrium of force at their interface，the dynamic equilibrium equation of pile is solved and an analytical solution for the displacement function in Laplace domain is yielded，so are the corresponding analytical solutions for the mobility at the level of the pile head in frequency domain. With the convolution theorem and inverse Fourier transform，a semi-analytical solution of velocity response in time-domain subjected to a semi-sine exciting force is derived. Based on the solutions proposed herein，a parametric study of the effect of some governing dimensionless parameters on mobility curves and velocity reflection wave curves is conducted to illustrate the main features of longitudinal vibration of pile.
