Analytical Study On The Vertical Dynamic Response Of Piles By Hankel Integral Transformations

The problem of the dynamic interactions between the soil and pile foundations plays an important role in the dynamic analysis of civil engineering,seismic engineering,transportation engineering and bridge engineering,which can provide reliable references and rapid estimation in the design of pile foundations.Many researches on the analytical solutions of dynamic response of pile foundations have been extensively developed via assuming that the pile base is resting on a rigid base and half space,and an elastic support with a Winkler model,however,such methods cannot rigorously consider the contribution of the wave effect of soil underlying the pile base to its dynamic response.This paper tries to focus on the problem of pile foundations embedded in half space and subsoil overlying a rigid base subjected to vertical harmonic loading,the governing equations of soil underlying the pile base,as well as the soil surrounding the pile shaft and pile,are established respectively,based on elastodynamic and poroelastodynamic theories.The problem is solved by Hankel integral transformations.The work done is listed as follows:(1)The pile is modeled as a one-dimensional elastic rod,while the soil is treated as a viscoelastic medium,and the soil is divided into a series of infinitesimally thin independent layer along the pile shaft and a homogeneous half space underlying the pile base.Based on the elastodynamic theory,the governing equations of soil and pile under vertical loading can be established,respectively.The Hankel integral transformations are employed,the assumptions proposed by Novak is introduced,and the mixed boundary conditions are established,thereby the pile base and shaft resistances can be obtained.Accordingly,substituting the expressions of the pile shaft and base resistance into the governing equation of the pile leads to the vertical dynamic impedance of pile top that describes the vertical dynamic response of the soil-pile system.Finally,following the verification of the derived solution against an existing solution,a parametric analysis has been carried out to investigate the influence of relevant parameters of the soil-pile system on the vertical dynamic impedance of pile top.(2)On the basis of the elastodynamic theory,the governing equations of the soil and pile are established,respectively.Considering the boundary conditions of the substratum with finite thickness,the vertical dynamic response of a pile embedded in viscoelatic soil overlying rigid bedrock is investigated analytically by the method of Hankel transformations.In order to check the validity of the proposed solution,the solution presented in this paper is compared against a published solution for a pile embedded in half space and against a rigorous solution for an end-bearing pile embedded in viscoelastic soil.Selected examples are also presented to investigate the influence of the substratum thickness on the vertical dynamic response of pile top.(3)Based on the Biot poroelastodynamic theory,we try to further delve into the vertical dynamic characteristics of piles embedded in poroelastic soil overlying a rigid base.The assumptions mentioned above are introduced into this part,and furthermore,the contact surface between the foundation base and soil is permeable.Following the methodology of validation,arithmetical examples are presented to identify the parameters that govern the vertical dynamic response of the system in poroelastic soil.(4)Considering the existence of the groundwater and the variation of its table level,an analytical solution of a large diameter pile partially-embedded in poroelastic soil overlying rigid bedrock is presented.The obtained solution is compared against an existing solution for a special case of a pile(neglecting the lateral deformation of the pile)fully-embedded in poroelastic soil,and then the influence of the embedment ratio of pile,the variation of the groundwater table level and the Poisson’s ratio of the pile material on the dynamic impedance of pile top is discussed in detail.