Soil-pile interaction in the pile vertical vibration considering true three-dimensional wave effect of soil

The dynamic response of an end bearing pile embedded in a linear visco-elastic soil layer with hysteretic type damping is theoretically investigated when the pile is subjected to a time-harmonic vertical loading at the pile top. The soil is modeled as a three-dimensional axisymmetric continuum in which both its radial and vertical displacements are taken into account. The pile is assumed to be vertical, elastic and of uniform circular cross section. By using two potential functions to decompose the displacements of the soil layer and utilizing the separation of variables technique, the dynamic equilibrium equation is uncoupled and solved. At the interface of soil-pile system, the boundary conditions of displacement continuity and force equilibrium are invoked to derive a closed-form solution of the vertical dynamic response of the pile in frequency domain. The corresponding inverted solutions in time domain for the velocity response of a pile subjected to a semi-sine excitation force applied at the pile top are obtained by means of inverse Fourier transform and the convolution theorem. A comparison with two other simplified solutions has been performed to verify the more rigorous solutions presented in this paper. Using the developed solutions, a parametric study has also been conducted to investigate the influence of the major parameters of the soil-pile system on the vertical vibration characteristics of the pile.