Kinematic response of an end-bearing pile subjected to vertical P-wave considering the three-dimensional wave scattering

A rigorous three-dimensional model is developed for the investigation of the kinematic pile response under vertical P-wave excitations. The present model considers the equilibrium of soil and pile in both radial and vertical directions to account for the three-dimensional wave scattering. The closed-form series solution is obtained through an exact theoretical formulation. The effects of Poisson's ratio of pile, pile-soil modulus ratio, slenderness ratio of pile, and soil damping on the kinematic response factor and amplification factor are given. There is a slight influence of Poisson's ratio of pile on the kinematic response of the pile. The amplification effect is significant within a small range near the first resonance frequency. Long flexible piles would experience higher response amplification, and hence larger foundation input motion, compared to short pile. The equivalent dynamic kinematic Winkler modulus was also derived. It was found that the equivalent Winkler modulus is depth-dependent. The static Winkler modulus increases with depth and both the equivalent static and dynamic Winkler modulus are higher for short piles compared to the case for long pile at the same relative depth. The pile-soil modulus ratio was found to mainly affect the equivalent dynamic Winkler modulus along the upper parts of pile.