Theory of pile vibration considering true three-dimensional wave effect of soil and its check on the approximate theories

By modeling soil as a three-dimensional axisymmetric continuum and taking its radial and vertical displacement into account, the interaction between a soil layer and an integral end bearing pile undergoing vertical harmonic load is theoretically investigated. The pile is assumed to be vertical, elastic and of uniform cross-section, and the soil is considered as a linear visco-elastic layer with hysteretic type damping. With the aid of two potentials, the displacement of soil layer is decomposed and then its dynamic equilibrium equation is uncoupled and solved first. Thus the resistance factor and vibration modes of the soil layer are obtained and used to analyze the pile response. By considering the interaction between the soil layer and the pile with boundary condition of displacement continuity and force equilibrium at the interface of them, the dynamic equilibrium equation of pile is solved and an analytical solution for the pile response in frequency domain is yielded, which is used to define complex stiffness of the pile head and the soil local complex stiffness. When governing parameters are varying, a comparison is made with simplified solutions obtained at the condition of plane strain hypothesis or neglecting radial displacement. The comparison involves the soil resistance factor, the soil local complex stiffness and the complex stiffness of the pile head. It proves that in the case of low frequency, stiff soil and slender pile, there can be significant discrepancies between them, but with frequency increasing, the simplified solutions converge to the same.