# Vertical Dynamic Response of Pile Embedded in Layered Transversely Isotropic Soil

Authors: Wenbing Wu; Guosheng Jiang; Shenggen Huang; Chin Jian Leo
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

The dynamic response of pile embedded in layered transversely isotropic soil and subjected to arbitrary vertical harmonic force is investigated. Based on the viscoelastic constitutive relations for a transversely isotropic medium, the dynamic governing equation of the transversely isotropic soil is obtained in cylindrical coordinates. By introducing the fictitious soil pile model and the distributed Voigt model, the governing equations of soil-pile system are also derived. Firstly, the vertical response of the soil layer is solved by using the Laplace transform technique and the separation of variables technique. Secondly, the analytical solution of velocity response in the frequency domain and its corresponding semianalytical solution of velocity response in the time domain are derived by means of inverse Fourier transform and convolution theorem. Finally, based on the obtained solutions, a parametric study has been conducted to investigate the influence of the soil anisotropy on the vertical dynamic response of pile. It can be seen that the influence of the shear modulus of soil in the vertical plane on the dynamic response of pile is more notable than the influence of the shear modulus of soil in the horizontal plane on the dynamic response of pile.


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## Body

## 1. Introduction

The dynamic interaction of soil-pile system is a complicated contact problem owing to the complexity of soil layer. Therefore, many investigators have paid their attention to develop theoretical models to analyze the soil layer for the dynamic interaction of soil-pile system. Many of these various theoretical models fall into four main categories. (1) The dynamic Winkler model [1–7], which represents the soil as a series of distributed springs attached to the pile, is the first category. The stiffness of the soil “springs” is calculated by assuming that wave energy can only propagate outwards under plane-strain conditions [8]. (2) The plane-strain model [9–16], which assumes that the soil is made up of an infinite number of infinitesimally thin, independent, horizontal, elastic layers that extend to infinity, is the second category. It can be noted that the plane-strain model neglects the strain in the vertical direction and the waves propagate only in the horizontal direction. (3) Various three-dimensional axisymmetric continuum models [17–22], in which the soil layer is three-dimensional and the wave effect of soil is considered, is the third category. Due to the complexity of the three-dimensional axisymmetric continuum models, it is very difficult to promote the use of these models in layered soil. (4) The radially inhomogeneous model [23–26], in which the soil region is assumed as a linear viscoelastic medium composed of two concentric regions, an inner annular region of disturbed medium, and an outer semi-infinite undisturbed region, is the fourth category. It can be seen that the model does not simulate the interaction of soil zones rigorously and it is only an approximate simplified model that may lead to the argument whether the results adequately reflect the real interaction of the soil zones [27].Most of the previous investigations on the dynamic interaction of soil-pile system had treated soil as an isotropic medium. However, in practice soil deposits possess a certain degree of anisotropy owing to their deposition history resulting in properties that are different in the horizontal and vertical direction. The vertical and horizontal differences can be simulated by virtue of a transversely isotropic soil model [28] which a simple isotropic model cannot allow for. Tsai [29] studied the torsional dynamic response of a circular disk on a transversely isotropic half-space by utilizing integral transform technique. It is worth noting that the anisotropic material constants have obvious influence on the resonant amplitude and frequency of vibration. Liu and Novak [30] investigated the dynamic response of single pile embedded in transversely isotropic layered media by means of FEM combining with dynamic stiffness matrices of the soil derived from Green’s functions for ring loads. Chen et al. [31] proposed a model of transversely isotropic layered elastic media to study the dynamic horizontal impedances of double piles embedded in the ground and subjected to harmonic sway-rocking loadings at the head of the two piles. Chen et al. [32] studied the transient torsional dynamic response of a pile embedded in transversely isotropic saturated soil by utilizing the Laplace transform. Wang et al. [33] investigated the dynamic response of an end bearing pile embedded in transversely isotropic saturated soil when there was a time-harmonic torsional loading acting on the pile head. Notwithstanding the work above, it can be seen that no investigation corresponding to the vertical dynamic response of a pile embedded in layered transversely isotropic elastic soil has been reported in existing literatures until now.Based on the above literature review, the main objective of this paper is to develop an extended soil-pile interaction model to simulate the vertical dynamic response of pile embedded in layered transversely isotropic soil. Utilizing the fictitious soil pile model to simulate the dynamic interaction between the pile and pile end soil [34], the governing equations of soil-pile system are established when the vertical wave effect of surrounding soil is taken into account. Then, the analytical solution of velocity response in the frequency domain and its corresponding semianalytical solution of velocity response in the time domain are derived by means of integral transform technique and separation of variables technique. Based on these solutions, a parametric study has been conducted to assess the influence of the soil anisotropy on the vibration behavior of the pile.


## 2. Governing Equations

### 2.1. Computational Model and Assumptions

The problem studied in this paper is the vertical vibration of viscoelastic pile embedded in layered transversely isotropic soil and the geometric model is shown in Figure1. Based on the fictitious soil pile model, the soil-pile system is discretized into a total of m segments along the vertical direction, which are numbered by 1,2,…,j,…,m. The thickness of the jth (1≤j≤m) soil-pile segment is denoted by lj and the depth of the jth (1≤j≤m) soil layer top is denoted by hj. If the numbers of the soil-pile segments are enough, the properties of soil and pile within each segment can be assumed to be homogeneous, respectively. The pile length is denoted by Hp and the thickness of pile end soil is denoted by Hs. q(t) is an arbitrary vertical harmonic force acting on the pile head.Figure 1
Schematic of pile-soil interaction model.The following assumptions are adopted during the analysis.(1)
The surrounding soil of pile is layered, transversely isotropic and viscoelastic. The damping force of soil is proportional to the strain rate, and the proportional coefficient of thejth soil layer is denoted by ηjs.(2)
The top surface of soil layer is assumed to be free boundary without normal and shear stresses, and the bottom surface of pile end soil is assumed to be rigid boundary without displacements.(3)
The dynamic interaction of the adjacent soil layers is simulated by using the distributed Voigt model. The spring constant and damping coefficient of the distributed Voigt model between thejth soil layer and its upper adjacent soil layer are denoted by kjs and δjs, and the corresponding values between the jth soil layer and its lower adjacent soil layer are denoted by kj-1s and δj-1s, respectively.(4)
During vibration, the vertical wave effect of surrounding soil is taken into account, but the radial wave effect of surrounding soil is ignored.(5)
Both the pile and the fictitious soil pile are vertical, viscoelastic, and circular in cross-section and have a perfect contact with the surrounding soil during vibration.(6)
During vibration, the soil-pile system is subjected to small deformations and strains, and the conditions of displacement continuity and force equilibrium are satisfied at the interface of the adjacent pile (including fictitious soil pile) segments.


### 2.2. Governing Equations of Soil-Pile System

#### 2.2.1. Dynamic Equation of Soil

Combining with the viscoelastic constitutive relations for a transversely isotropic medium proposed by Ding [35], the dynamic equilibrium equation of the transversely isotropic soil in cylindrical coordinates can be derived as follows
(1)[C11(∂2∂r2+1r∂∂r-1r2)+C661r2∂2∂θ2+C44∂2∂z2]ur+[C11-C66r∂2∂r∂θ-C11+C66r2∂∂θ]uθ+(C13+C44)∂2w∂r∂z=ρs∂2ur∂t2,[(C11-C66)1r∂2∂r∂θ+(C11+C66)1r2∂∂θ]ur+(C13+C44)1r∂2w∂θ∂z+[C66(∂2∂r2+1r∂∂r-1r2)+C111r2∂2∂θ2+C44∂2∂z2]uθ=ρs∂2uθ∂t2,(C13+C44)(∂2∂r∂z+1r∂∂z)ur+(C13+C44)1r∂2uθ∂θ∂z+[C44(∂2∂r2+1r∂∂r+1r2∂2∂θ2)+C33∂2∂z2]w=ρs∂2w∂t2,
where ur, uθ, and w denote the radial displacement, circumferential displacement, and vertical displacement, respectively. The coefficients of the above equations should satisfy the following equations
(2)llllllllC11=C22=Ehs(1-μhvsμvhs)(1+μhhs)(1-μhhs-2μhvsμvhs),C12=Ehs(μhhs+μhvsμvhs)(1+μhhs)(1-μhhs-2μhvsμvhs),C13=C23=Ehsμhhs1-μhhs-2μhvsμvhs,llllllC33=Ev(1-μhhs)1-μhhs-2μhvsμvhs,lllllllllllllllC44=C55=Gvs,lllllllllllllllllllllllllllllC66=(C11-C22)2,lllllllllllllllllllllllll
where Ehs and Evs denote the horizontal and vertical elastic modulus, respectively. Gvs denotes the shear modulus in the vertical plane. μhvs is the Poisson’s ration in the vertical direction caused by the horizontal stress and μvhs is the Poisson’s ratio in the horizontal direction caused by the vertical stress and μhvs and μvhs should satisfy μvhs/Evs=μhvs/Ehs. μhhs is the Poisson’s ratio in the orthogonal direction of the horizontal strain caused by the horizontal stress.Owing to the assumption that only the vertical wave effect of surrounding soil is taken into consideration, the equilibrium equation of the transversely isotropic soil for the axisymmetric problem can be further rewritten as follows(3)C33∂2w∂z2+C44[∂2w∂r2+1r∂w∂r]=ρs∂2w∂t2.Based on (3) and taking into account the viscosity of soil, the governing equation of the jth transversely isotropic soil layer for the axisymmetric problem can be established as
(4)δjs∂2wj∂z2+ηjsGvjs∂∂t(∂2wj∂z2)+(1r∂wj∂r+∂2wj∂r2)+ηjsGvjs∂∂t(1r∂wj∂r+∂2wj∂r2)=ρjsGvjs∂2wj∂t2,
where wj=wj(r,z,t) is the vertical displacement of the jth soil layer. ηjs, ρjs, and Gvjs denote the viscous damping coefficient, density, and shear modulus in the vertical plane of the jth soil layer. δjs=2(1-μhhjs)(1+μvhjs)/(1-μhhjs-2(μvhjs)2κjs). κjs=(1+μhhjs)Ghjs/(1+μvhjs)Gvjs denotes the ratio of the elastic modulus of the jth soil layer in the horizontal direction to that in the vertical direction; Ghjs, μvhjs, and μhhjs denote the shear modulus in the horizontal plane, the Poisson’s ratio in the horizontal direction caused by the vertical stress, and the Poisson’s ratio in the orthogonal direction of the horizontal strain caused by the horizontal stress of the jth soil layer, respectively.


#### 2.2.2. Dynamic Equation of Pile

Denotinguj=uj(z,t) to be the vertical displacement of the jth pile (including fictitious soil pile) segment and according to the Euler-Bernoulli rod theory, the dynamic equilibrium equation of pile can be established as
(5)EjpAjp∂2uj∂z2+Ajpηjp∂3uj∂t∂z2-mjp∂2uj∂t2-2πrjpτrzjs(rjp,z,t)=0,
where Ejp=ρjp(Vjp)2, Ajp=π(rjp)2, rjp, mjp, ρjp, Vjp, and ηjp denote the elastic modulus, cross-section area, radius, mass per unit length of pile (including fictitious soil pile), density, elastic longitudinal wave velocity, and viscous damping coefficient of the jth pile (including fictitious soil pile) segment, respectively. τrzjs(rjp,z,t) is the frictional force of the jth soil layer acting on the surface of the jth pile shaft and can be expressed as
(6)τrzjs(rjp,z,t)=Gvjs∂wj(rjp,z,t)∂r+ηjs∂2wj(rjp,z,t)∂t∂r.Combining with the assumptions, the boundary and initial conditions of soil-pile system can be established as follows.(
1) Boundary Conditions of Soil. At the top surface of the jth soil layer,(7a)Evjs∂wj∂z|z=hj=(kjswj+δjs∂wj∂t)|z=hj.At the bottom surface of thejth soil layer,
(7b)Evjs∂wj∂z|z=hj+lj=-(kj-1swj+δj-1s∂wj∂t)|z=hj+lj.At an infinite radial distance of thejth soil layer,
(7c)σj(∞,z)=0,wj(∞,z)=0.(2) Boundary Conditions of Pile. At the top surface of the jth pile (including fictitious soil pile) segment,(8a)∂uj∂z|z=hj=-Zj(s)ujEjpAjp|z=hj.At the bottom surface of thejth pile (including fictitious soil pile) segment,
(8b)∂uj∂z|z=hj+lj=-Zj-1(s)ujEjpAjp|z=hj+lj,whereZj(s) and Zj-1(s) denote the displacement impedance function at the top and bottom surfaces of the jth pile (including fictitious soil pile) segment. s is the Laplace transform parameter.(3) Boundary Condition at the Interface of Soil-Pile System
(9)
w
(
r
j
p
,
z
,
t
)
=
u
j
(
z
,
t
)
.(
4) Initial Conditions of Soil-Pile System. Initial conditions of the jth soil layer are as follows:(10a)wj|t=0=0,∂wj∂t|t=0=0.Initial conditions of thejth pile (including fictitious soil pile) segment are as follows:
(10b)uj|t=0=0,∂uj∂t|t=0=0.


## 2.1. Computational Model and Assumptions

The problem studied in this paper is the vertical vibration of viscoelastic pile embedded in layered transversely isotropic soil and the geometric model is shown in Figure1. Based on the fictitious soil pile model, the soil-pile system is discretized into a total of m segments along the vertical direction, which are numbered by 1,2,…,j,…,m. The thickness of the jth (1≤j≤m) soil-pile segment is denoted by lj and the depth of the jth (1≤j≤m) soil layer top is denoted by hj. If the numbers of the soil-pile segments are enough, the properties of soil and pile within each segment can be assumed to be homogeneous, respectively. The pile length is denoted by Hp and the thickness of pile end soil is denoted by Hs. q(t) is an arbitrary vertical harmonic force acting on the pile head.Figure 1
Schematic of pile-soil interaction model.The following assumptions are adopted during the analysis.(1)
The surrounding soil of pile is layered, transversely isotropic and viscoelastic. The damping force of soil is proportional to the strain rate, and the proportional coefficient of thejth soil layer is denoted by ηjs.(2)
The top surface of soil layer is assumed to be free boundary without normal and shear stresses, and the bottom surface of pile end soil is assumed to be rigid boundary without displacements.(3)
The dynamic interaction of the adjacent soil layers is simulated by using the distributed Voigt model. The spring constant and damping coefficient of the distributed Voigt model between thejth soil layer and its upper adjacent soil layer are denoted by kjs and δjs, and the corresponding values between the jth soil layer and its lower adjacent soil layer are denoted by kj-1s and δj-1s, respectively.(4)
During vibration, the vertical wave effect of surrounding soil is taken into account, but the radial wave effect of surrounding soil is ignored.(5)
Both the pile and the fictitious soil pile are vertical, viscoelastic, and circular in cross-section and have a perfect contact with the surrounding soil during vibration.(6)
During vibration, the soil-pile system is subjected to small deformations and strains, and the conditions of displacement continuity and force equilibrium are satisfied at the interface of the adjacent pile (including fictitious soil pile) segments.


## 2.2. Governing Equations of Soil-Pile System

### 2.2.1. Dynamic Equation of Soil

Combining with the viscoelastic constitutive relations for a transversely isotropic medium proposed by Ding [35], the dynamic equilibrium equation of the transversely isotropic soil in cylindrical coordinates can be derived as follows
(1)[C11(∂2∂r2+1r∂∂r-1r2)+C661r2∂2∂θ2+C44∂2∂z2]ur+[C11-C66r∂2∂r∂θ-C11+C66r2∂∂θ]uθ+(C13+C44)∂2w∂r∂z=ρs∂2ur∂t2,[(C11-C66)1r∂2∂r∂θ+(C11+C66)1r2∂∂θ]ur+(C13+C44)1r∂2w∂θ∂z+[C66(∂2∂r2+1r∂∂r-1r2)+C111r2∂2∂θ2+C44∂2∂z2]uθ=ρs∂2uθ∂t2,(C13+C44)(∂2∂r∂z+1r∂∂z)ur+(C13+C44)1r∂2uθ∂θ∂z+[C44(∂2∂r2+1r∂∂r+1r2∂2∂θ2)+C33∂2∂z2]w=ρs∂2w∂t2,
where ur, uθ, and w denote the radial displacement, circumferential displacement, and vertical displacement, respectively. The coefficients of the above equations should satisfy the following equations
(2)llllllllC11=C22=Ehs(1-μhvsμvhs)(1+μhhs)(1-μhhs-2μhvsμvhs),C12=Ehs(μhhs+μhvsμvhs)(1+μhhs)(1-μhhs-2μhvsμvhs),C13=C23=Ehsμhhs1-μhhs-2μhvsμvhs,llllllC33=Ev(1-μhhs)1-μhhs-2μhvsμvhs,lllllllllllllllC44=C55=Gvs,lllllllllllllllllllllllllllllC66=(C11-C22)2,lllllllllllllllllllllllll
where Ehs and Evs denote the horizontal and vertical elastic modulus, respectively. Gvs denotes the shear modulus in the vertical plane. μhvs is the Poisson’s ration in the vertical direction caused by the horizontal stress and μvhs is the Poisson’s ratio in the horizontal direction caused by the vertical stress and μhvs and μvhs should satisfy μvhs/Evs=μhvs/Ehs. μhhs is the Poisson’s ratio in the orthogonal direction of the horizontal strain caused by the horizontal stress.Owing to the assumption that only the vertical wave effect of surrounding soil is taken into consideration, the equilibrium equation of the transversely isotropic soil for the axisymmetric problem can be further rewritten as follows(3)C33∂2w∂z2+C44[∂2w∂r2+1r∂w∂r]=ρs∂2w∂t2.Based on (3) and taking into account the viscosity of soil, the governing equation of the jth transversely isotropic soil layer for the axisymmetric problem can be established as
(4)δjs∂2wj∂z2+ηjsGvjs∂∂t(∂2wj∂z2)+(1r∂wj∂r+∂2wj∂r2)+ηjsGvjs∂∂t(1r∂wj∂r+∂2wj∂r2)=ρjsGvjs∂2wj∂t2,
where wj=wj(r,z,t) is the vertical displacement of the jth soil layer. ηjs, ρjs, and Gvjs denote the viscous damping coefficient, density, and shear modulus in the vertical plane of the jth soil layer. δjs=2(1-μhhjs)(1+μvhjs)/(1-μhhjs-2(μvhjs)2κjs). κjs=(1+μhhjs)Ghjs/(1+μvhjs)Gvjs denotes the ratio of the elastic modulus of the jth soil layer in the horizontal direction to that in the vertical direction; Ghjs, μvhjs, and μhhjs denote the shear modulus in the horizontal plane, the Poisson’s ratio in the horizontal direction caused by the vertical stress, and the Poisson’s ratio in the orthogonal direction of the horizontal strain caused by the horizontal stress of the jth soil layer, respectively.


### 2.2.2. Dynamic Equation of Pile

Denotinguj=uj(z,t) to be the vertical displacement of the jth pile (including fictitious soil pile) segment and according to the Euler-Bernoulli rod theory, the dynamic equilibrium equation of pile can be established as
(5)EjpAjp∂2uj∂z2+Ajpηjp∂3uj∂t∂z2-mjp∂2uj∂t2-2πrjpτrzjs(rjp,z,t)=0,
where Ejp=ρjp(Vjp)2, Ajp=π(rjp)2, rjp, mjp, ρjp, Vjp, and ηjp denote the elastic modulus, cross-section area, radius, mass per unit length of pile (including fictitious soil pile), density, elastic longitudinal wave velocity, and viscous damping coefficient of the jth pile (including fictitious soil pile) segment, respectively. τrzjs(rjp,z,t) is the frictional force of the jth soil layer acting on the surface of the jth pile shaft and can be expressed as
(6)τrzjs(rjp,z,t)=Gvjs∂wj(rjp,z,t)∂r+ηjs∂2wj(rjp,z,t)∂t∂r.Combining with the assumptions, the boundary and initial conditions of soil-pile system can be established as follows.(
1) Boundary Conditions of Soil. At the top surface of the jth soil layer,(7a)Evjs∂wj∂z|z=hj=(kjswj+δjs∂wj∂t)|z=hj.At the bottom surface of thejth soil layer,
(7b)Evjs∂wj∂z|z=hj+lj=-(kj-1swj+δj-1s∂wj∂t)|z=hj+lj.At an infinite radial distance of thejth soil layer,
(7c)σj(∞,z)=0,wj(∞,z)=0.(2) Boundary Conditions of Pile. At the top surface of the jth pile (including fictitious soil pile) segment,(8a)∂uj∂z|z=hj=-Zj(s)ujEjpAjp|z=hj.At the bottom surface of thejth pile (including fictitious soil pile) segment,
(8b)∂uj∂z|z=hj+lj=-Zj-1(s)ujEjpAjp|z=hj+lj,whereZj(s) and Zj-1(s) denote the displacement impedance function at the top and bottom surfaces of the jth pile (including fictitious soil pile) segment. s is the Laplace transform parameter.(3) Boundary Condition at the Interface of Soil-Pile System
(9)
w
(
r
j
p
,
z
,
t
)
=
u
j
(
z
,
t
)
.(
4) Initial Conditions of Soil-Pile System. Initial conditions of the jth soil layer are as follows:(10a)wj|t=0=0,∂wj∂t|t=0=0.Initial conditions of thejth pile (including fictitious soil pile) segment are as follows:
(10b)uj|t=0=0,∂uj∂t|t=0=0.


## 2.2.1. Dynamic Equation of Soil

Combining with the viscoelastic constitutive relations for a transversely isotropic medium proposed by Ding [35], the dynamic equilibrium equation of the transversely isotropic soil in cylindrical coordinates can be derived as follows
(1)[C11(∂2∂r2+1r∂∂r-1r2)+C661r2∂2∂θ2+C44∂2∂z2]ur+[C11-C66r∂2∂r∂θ-C11+C66r2∂∂θ]uθ+(C13+C44)∂2w∂r∂z=ρs∂2ur∂t2,[(C11-C66)1r∂2∂r∂θ+(C11+C66)1r2∂∂θ]ur+(C13+C44)1r∂2w∂θ∂z+[C66(∂2∂r2+1r∂∂r-1r2)+C111r2∂2∂θ2+C44∂2∂z2]uθ=ρs∂2uθ∂t2,(C13+C44)(∂2∂r∂z+1r∂∂z)ur+(C13+C44)1r∂2uθ∂θ∂z+[C44(∂2∂r2+1r∂∂r+1r2∂2∂θ2)+C33∂2∂z2]w=ρs∂2w∂t2,
where ur, uθ, and w denote the radial displacement, circumferential displacement, and vertical displacement, respectively. The coefficients of the above equations should satisfy the following equations
(2)llllllllC11=C22=Ehs(1-μhvsμvhs)(1+μhhs)(1-μhhs-2μhvsμvhs),C12=Ehs(μhhs+μhvsμvhs)(1+μhhs)(1-μhhs-2μhvsμvhs),C13=C23=Ehsμhhs1-μhhs-2μhvsμvhs,llllllC33=Ev(1-μhhs)1-μhhs-2μhvsμvhs,lllllllllllllllC44=C55=Gvs,lllllllllllllllllllllllllllllC66=(C11-C22)2,lllllllllllllllllllllllll
where Ehs and Evs denote the horizontal and vertical elastic modulus, respectively. Gvs denotes the shear modulus in the vertical plane. μhvs is the Poisson’s ration in the vertical direction caused by the horizontal stress and μvhs is the Poisson’s ratio in the horizontal direction caused by the vertical stress and μhvs and μvhs should satisfy μvhs/Evs=μhvs/Ehs. μhhs is the Poisson’s ratio in the orthogonal direction of the horizontal strain caused by the horizontal stress.Owing to the assumption that only the vertical wave effect of surrounding soil is taken into consideration, the equilibrium equation of the transversely isotropic soil for the axisymmetric problem can be further rewritten as follows(3)C33∂2w∂z2+C44[∂2w∂r2+1r∂w∂r]=ρs∂2w∂t2.Based on (3) and taking into account the viscosity of soil, the governing equation of the jth transversely isotropic soil layer for the axisymmetric problem can be established as
(4)δjs∂2wj∂z2+ηjsGvjs∂∂t(∂2wj∂z2)+(1r∂wj∂r+∂2wj∂r2)+ηjsGvjs∂∂t(1r∂wj∂r+∂2wj∂r2)=ρjsGvjs∂2wj∂t2,
where wj=wj(r,z,t) is the vertical displacement of the jth soil layer. ηjs, ρjs, and Gvjs denote the viscous damping coefficient, density, and shear modulus in the vertical plane of the jth soil layer. δjs=2(1-μhhjs)(1+μvhjs)/(1-μhhjs-2(μvhjs)2κjs). κjs=(1+μhhjs)Ghjs/(1+μvhjs)Gvjs denotes the ratio of the elastic modulus of the jth soil layer in the horizontal direction to that in the vertical direction; Ghjs, μvhjs, and μhhjs denote the shear modulus in the horizontal plane, the Poisson’s ratio in the horizontal direction caused by the vertical stress, and the Poisson’s ratio in the orthogonal direction of the horizontal strain caused by the horizontal stress of the jth soil layer, respectively.


## 2.2.2. Dynamic Equation of Pile

Denotinguj=uj(z,t) to be the vertical displacement of the jth pile (including fictitious soil pile) segment and according to the Euler-Bernoulli rod theory, the dynamic equilibrium equation of pile can be established as
(5)EjpAjp∂2uj∂z2+Ajpηjp∂3uj∂t∂z2-mjp∂2uj∂t2-2πrjpτrzjs(rjp,z,t)=0,
where Ejp=ρjp(Vjp)2, Ajp=π(rjp)2, rjp, mjp, ρjp, Vjp, and ηjp denote the elastic modulus, cross-section area, radius, mass per unit length of pile (including fictitious soil pile), density, elastic longitudinal wave velocity, and viscous damping coefficient of the jth pile (including fictitious soil pile) segment, respectively. τrzjs(rjp,z,t) is the frictional force of the jth soil layer acting on the surface of the jth pile shaft and can be expressed as
(6)τrzjs(rjp,z,t)=Gvjs∂wj(rjp,z,t)∂r+ηjs∂2wj(rjp,z,t)∂t∂r.Combining with the assumptions, the boundary and initial conditions of soil-pile system can be established as follows.(
1) Boundary Conditions of Soil. At the top surface of the jth soil layer,(7a)Evjs∂wj∂z|z=hj=(kjswj+δjs∂wj∂t)|z=hj.At the bottom surface of thejth soil layer,
(7b)Evjs∂wj∂z|z=hj+lj=-(kj-1swj+δj-1s∂wj∂t)|z=hj+lj.At an infinite radial distance of thejth soil layer,
(7c)σj(∞,z)=0,wj(∞,z)=0.(2) Boundary Conditions of Pile. At the top surface of the jth pile (including fictitious soil pile) segment,(8a)∂uj∂z|z=hj=-Zj(s)ujEjpAjp|z=hj.At the bottom surface of thejth pile (including fictitious soil pile) segment,
(8b)∂uj∂z|z=hj+lj=-Zj-1(s)ujEjpAjp|z=hj+lj,whereZj(s) and Zj-1(s) denote the displacement impedance function at the top and bottom surfaces of the jth pile (including fictitious soil pile) segment. s is the Laplace transform parameter.(3) Boundary Condition at the Interface of Soil-Pile System
(9)
w
(
r
j
p
,
z
,
t
)
=
u
j
(
z
,
t
)
.(
4) Initial Conditions of Soil-Pile System. Initial conditions of the jth soil layer are as follows:(10a)wj|t=0=0,∂wj∂t|t=0=0.Initial conditions of thejth pile (including fictitious soil pile) segment are as follows:
(10b)uj|t=0=0,∂uj∂t|t=0=0.


## 3. Solutions of the Governing Equations

### 3.1. Vibrations of the Soil Layer

DenotingWj(r,z,s)=∫0+∞wj(r,z,t)e-stdt to be the Laplace transform with respect to time of wj(r,z,t) and associating with the initial condition (10a), (4) is transformed by using the Laplace transform technique as follows
(11)(δjs+ηjs·sGvjs)∂2Wj∂z2+(1+ηjs·sGvjs)(1r∂Wj∂r+∂2Wj∂r2)=(sVvjs)2Wj,
where Vvjs=Gvjs/ρjs is the shear wave velocity of the jth soil layer in the vertical direction.By virtue of the separation of variables technique and denotingWj(r,z,s)=Rj(r,s)Zj(z,s), (11) can be decoupled as follows
(12)d2Rj(r,s)dr2+1rdRj(r,s)dr-ξj2Rj(r,s)=0,(13)d2Zj(z,s)dz2+βj2Zj(z,s)=0,
where constants ξj and βj must satisfy the following equation
(14)ξj2=(δjs+ηjs·s/Gvjs)βj2+(s/Vvjs)2(1+ηjs·s/Gvjs).It can be seen that (12) is Bessel equation and (13) is ordinary differential equation of second order whose general solutions can be easily obtained. Associating with these general solutions, the vertical displacement of the jth soil layer, Wj(r,z,s) can be derived as
(15)Wj(r,z,s)=[AjK0(ξjr)+BjI0(ξjr)]×[Cjsin(βjz)+Djcos⁡(βjz)],
where I0(·) and K0(·) denote the modified Bessel functions of order zero of the first and second kind, respectively. Aj, Bj, Cj, and Dj are constants determined by the boundary conditions.Convertingz=hj and z=hj+lj in the global coordinates into z′=0 and z′=lj in the local coordinates, (7a), (7b), and (7c) are transformed by using the Laplace transform technique and can be rewritten as follows(16a)[(kjs+δjs·s)EvjsWj(r,z′,s)-∂Wj(r,z′,s)∂z′]|z′=0=0,(16b)[(kj-1s+δj-1s·s)EvjsWj(r,z′,s)+∂Wj(r,z′,s)∂z′]|z′=lj=0,(16c)σ-j(∞,z′)=0,Wj(∞,z′)=0.According to the properties of the modified Bessel functions, that is, ifr→∞, then In(·)→∞ and Kn(·)→0, it can be obtained that Bj=0 from boundary conditions (16c). By means of boundary conditions (16a) and (16b), the following equation can be obtained
(17)tan⁡(βjlj)=(kjs+δjs·sEvjslj+kj-1s+δj-1s·sEvjslj)βjlj×((βjlj)2-(kjs+δjs·sEvjslj)(kj-1s+δj-1s·sEvjslj))-1=(K-j+K-j′)βjlj(βjlj)2-K-jK-j′,
where K-j=((kjs+δjs·s)/Evjs)lj and K-′j=((kj-1s+δj-1s·s)/Evjs)lj denote the dimensionless complex stiffness of the upper surface and lower surface of the jth soil layer. Substituting s=iω into (17) and solving it by using bisection method in the frequency domain, a series of eigenvalues, βjn, can be derived. ξjn can also be derived by substituting βjn into (14).Then, the vertical displacement of thejth soil layer can be rewritten as
(18)Wj(r,z′,s)=∑n=1∞AjnK0(ξjnr)sin(βjnz′+ϕjn),
where ϕjn=arctan(βjnlj/K-j) and Ajn are a series of constants determined by the boundary conditions which reflect the dynamic interaction of soil and pile.


### 3.2. Vibrations of the Pile

DenotingUj(z,s) to be the Laplace transform with respect to time of uj(z,t), (5) is transformed by using the Laplace transform technique and can be rewritten combining with (6) and (18)
(19)(Vjp)2(1+ηjpEjp·s)∂2Uj∂z′2-s2Uj-2πrjpρjpAjp(Gvjs+ηvjs·s)∑n=1∞AjnξjnK1(ξjnrjp)llllllllllllllllllllllllllllllllllllllllllllll×sin(βjnz′+ϕjn)=0.It is not difficult to obtain that the general solution of (19) can be expressed as
(20)Uj=Mj[cos⁡(λ-jz′lj)+∑n=1∞χjn′sin(βjnz′+ϕjn)]+Nj[sin⁡(λ-jz′lj)+∑n=1∞χjn′′sin⁡(βjnz′+ϕjn)],
where
(21)χjn′=χjn[cos⁡⁡(β-jn+λ-j+ϕjn)-cos⁡ϕjnβ-jn+λ-jllllllllllllllllllllllllllllllllllllll+cos⁡⁡(β-jn-λ-j+ϕjn)-cos⁡ϕjnβ-jn-λ-j],χjn′′=χjn[sin⁡(β-jn+λ-j+ϕjn)-sinϕjnβ-jn+λ-jlllllllllllllllllllllllllllllllllll-sin⁡(β-jn-λ-j+ϕjn)-sinϕjnβ-jn-λ-j],χjn=(Gvjs+ηjs·s)ξ-jnK1(ξ-jnr-jp)tj2ρjpljr-jp[β-jn2(1+(ηjp/Ejp)·s)+s2tj2]φjnLjn,llllllllllφjn=K0(ξ-jnr-jp)+2(Gvjs+ηjs·s)ξ-jnK1(ξ-jnr-jp)tj2ρjplj2r-jp[β-jn2(1+(ηjp/Ejp)·s)+s2tj2],Ljn=∫0ljsin2(βjnz′+ϕjn)dz′,llllllllllllllllllllllllllllllllllll
where λ¯j=-s2tj2/(1+(ηjp/Ejp)·s), β-jn=βjnlj, ξ-jn=ξjnlj, and r-jp=rjp/lj are all dimensionless parameters. tj=lj/Vjp denotes the propagation time of elastic longitudinal wave in the jth pile segment. K1(·) is the modified Bessel functions of order one of the second kind.Combining with the boundary conditions (8a) and (8b), the displacement impedance function at the head of the jth pile segment can be derived in the local coordinates as follows
(22)Zj(s)=-EjpAjp(∂Uj/∂z′)|z′=0Uj|z′=0=-EjpAjplj×(MjNj∑n=1∞χjn′β-jncos⁡ϕjnlllllllll+λ-j+∑n=1∞χjn′′β-jncos⁡ϕjn)×(MjNj(1+∑n=1∞χjn′sinϕjn)llllllllll+∑n=1∞χjn′′sinϕjn)-1,
where
(23)MjNj=(∑n=1∞χjn′′β-jncos⁡(β-jn+ϕjn)+λ-jcos⁡λ-jlllwlllll+Zj-1(s)ljEjpAjp[sinλ-j+∑n=1∞χjn′′sin(β-jn+ϕjn)])w×(∑n=1∞χjn′β-jncos⁡⁡(β-jn+ϕjn)-λ-jsinλ-jllllllwlllll+Zj-1(s)ljEjpAjp[cos⁡λ-j+∑n=1∞χjn′sin(β-jn+ϕjn)])-1,
where Zj-1(s) denotes the displacement impedance function at the head of the (j-1)th pile segment which can be obtained by using boundary conditions. Then, following the method of recursion typically used in the transfer function technique, the displacement impedance function at the head of pile can be derived as
(24)Zm(s)=-EmpAmp(∂Um/∂z′)|z′=0Um|z′=0=-EmpAmplm×(MmNm∑n=1∞χmn′β-mncos⁡ϕmnllllllll+λ-m+∑n=1∞χmn′′β-mncos⁡ϕmn)×(MmNm(1+∑n=1∞χmn′sinϕmn)llllllll+∑n=1∞χmn′′sinϕmn)-1,
where
(25)MmNm=-(∑n=1∞χmn′′β-mncos⁡(β-mn+ϕmn)+λ-mcos⁡λ-mllllllll+Zm-1(s)lmEmpAmp[sinλ-m+∑n=1∞χmn′′sin⁡(β-mn+ϕmn)])×(∑n=1∞χmn′β-mncos⁡⁡(β-mn+ϕmn)-λ-msinλ-mlllllllll+Zm-1(s)lmEmpAmp[cos⁡λ-m+∑n=1∞χmn′sin(β-mn+ϕmn)])-1,χmn′=χmn[cos⁡(β-mn+λ-m+ϕmn)-cos⁡ϕmnβ-mn+λ-mllllllllllllll+cos⁡⁡(β-mn-λ-m+ϕmn)-cos⁡ϕmnβ-mn-λ-m],χmn′′=χmn[sin(β-mn+λ-m+ϕmn)-sinϕmnβ-mn+λ-mlllllllllllll-sin⁡(β-mn-λ-m+ϕmn)-sinϕmnβ-mn-λ-m],χmn=(Gvms+ηms·s)ξ-mnK1(ξ-mnr-mp)tm2ρmplmr-mp[β-mn2(1+(ηmp/Emp)·s)+s2tm2]φmnLmn,φmn=K0(ξ-mnr-mp)lllllllllllllllllllllllllllllllllllllllllllllllllll+2(Gvms+ηms·s)ξ-mnK1(ξ-mnr-mp)tm2ρmplm2r-mp[β-mn2(1+(ηmp/Emp)·s)+s2tm2],Lmn=∫0lmsin2(βmnz′+ϕmn)dz′,lllllllllllllllllllllllllll
where λ¯m=-s2tm2/(1+(ηmp/Emp)·s), β-mn=βmnlm, ξ-mn=ξmnlm, and r-mp=rmp/lm are all dimensionless parameters. tm=lm/Vmp denotes the propagation time of elastic longitudinal wave in the mth pile segment. ϕmn and βmn can be obtained from the following equations
(26)ϕmn=arctan(βmnlmK-m),tan(βmlm)=(K-m+K-m′)βmlm(βmlm)2-K-mK-m′,
where K-m=((kms+δms·s)/Evms)lm and K-m′=((km-1s+δm-1s·s)/Evms)lm denote the dimensionless complex stiffness of the upper surface and lower surface of the mth soil layer.Then, the velocity transfer function at the head of pile can be obtained as(27)Gv(s)=sZm(s)=-lm·sEmpAmp(MmNm(1+∑n=1∞χmn′sinϕmn)lllllllllllllllll+∑n=1∞χmn′′sinϕmn)×(MmNm∑n=1∞χmn′β-mncos⁡ϕmnlllllllll+λ-m+∑n=1∞χmn′′β-mncos⁡ϕmn)-1.Lettings=iω and substituting it into (27), the velocity response in the frequency domain at the head of pile can be obtained as
(28)Hv(iω)=iωZ2(iω)=-1ρmpAmpVmpHv′,
where Hv′ is the dimensionless velocity admittance at the pile head which can be expressed as
(29)Hv′=iωtm(MmNm(1+∑n=1∞χmn′sinϕmn)lllllllllllllll+∑n=1∞χmn′′sinϕmn)×(MmNm∑n=1∞χmn′β-mncos⁡ϕmnlllllllllll+λ-m+∑n=1∞χmn′′β-mncos⁡ϕmn)-1.By means of the inverse Fourier transform and convolution theorem, the velocity response in the time domain at the head of pile can be expressed asV(t)=IFT[Q(ω)Hv(iω)], where Q(ω) denotes the Fourier transform of q(t), which is the vertical excitation acting on the pile head.In particular, the excitation can be regarded as a half-sine pulse in the nondestructive detection of pile foundation as follows(30)q(t)={Qmax⁡sin(πTt),t∈(0,T),0,t≥T,
where T and Qmax⁡ denote the duration of the impulse or impulse width and the maximum amplitude of the vertical excitation, respectively. Then, the velocity response in the time domain at the head of pile can be expressed as
(31)V(t)=q(t)*IFT[Hv(iω)]=IFT[Q(iω)·Hv(iω)]=-Qmax⁡ρmpAmpVmpVv′,
where Vv′ is the dimensionless velocity response which can be expressed as
(32)Vv′=12∫-∞∞iω-t-mllllllllll×(MmNm(1+∑n=1∞χmn′sinϕmn)+∑n=1∞χmn′′sinϕmn)llllllllll×(MmNm∑n=1∞χmn′β-mncos⁡ϕmn+λ-mlllllllllllllllllll+∑n=1∞χmn′′β-mncos⁡ϕmn)-1llllllllll×T-π2-T-2ω-2·(1+e-iω-T-)eiω-t-dω-,
where ω-=Tcω denotes the dimensionless frequency. T- denotes the dimensionless pulse width which should satisfy T-=T/Tc. t- denotes the dimensional time variable which should satisfy t-=t/Tc.


## 3.1. Vibrations of the Soil Layer

DenotingWj(r,z,s)=∫0+∞wj(r,z,t)e-stdt to be the Laplace transform with respect to time of wj(r,z,t) and associating with the initial condition (10a), (4) is transformed by using the Laplace transform technique as follows
(11)(δjs+ηjs·sGvjs)∂2Wj∂z2+(1+ηjs·sGvjs)(1r∂Wj∂r+∂2Wj∂r2)=(sVvjs)2Wj,
where Vvjs=Gvjs/ρjs is the shear wave velocity of the jth soil layer in the vertical direction.By virtue of the separation of variables technique and denotingWj(r,z,s)=Rj(r,s)Zj(z,s), (11) can be decoupled as follows
(12)d2Rj(r,s)dr2+1rdRj(r,s)dr-ξj2Rj(r,s)=0,(13)d2Zj(z,s)dz2+βj2Zj(z,s)=0,
where constants ξj and βj must satisfy the following equation
(14)ξj2=(δjs+ηjs·s/Gvjs)βj2+(s/Vvjs)2(1+ηjs·s/Gvjs).It can be seen that (12) is Bessel equation and (13) is ordinary differential equation of second order whose general solutions can be easily obtained. Associating with these general solutions, the vertical displacement of the jth soil layer, Wj(r,z,s) can be derived as
(15)Wj(r,z,s)=[AjK0(ξjr)+BjI0(ξjr)]×[Cjsin(βjz)+Djcos⁡(βjz)],
where I0(·) and K0(·) denote the modified Bessel functions of order zero of the first and second kind, respectively. Aj, Bj, Cj, and Dj are constants determined by the boundary conditions.Convertingz=hj and z=hj+lj in the global coordinates into z′=0 and z′=lj in the local coordinates, (7a), (7b), and (7c) are transformed by using the Laplace transform technique and can be rewritten as follows(16a)[(kjs+δjs·s)EvjsWj(r,z′,s)-∂Wj(r,z′,s)∂z′]|z′=0=0,(16b)[(kj-1s+δj-1s·s)EvjsWj(r,z′,s)+∂Wj(r,z′,s)∂z′]|z′=lj=0,(16c)σ-j(∞,z′)=0,Wj(∞,z′)=0.According to the properties of the modified Bessel functions, that is, ifr→∞, then In(·)→∞ and Kn(·)→0, it can be obtained that Bj=0 from boundary conditions (16c). By means of boundary conditions (16a) and (16b), the following equation can be obtained
(17)tan⁡(βjlj)=(kjs+δjs·sEvjslj+kj-1s+δj-1s·sEvjslj)βjlj×((βjlj)2-(kjs+δjs·sEvjslj)(kj-1s+δj-1s·sEvjslj))-1=(K-j+K-j′)βjlj(βjlj)2-K-jK-j′,
where K-j=((kjs+δjs·s)/Evjs)lj and K-′j=((kj-1s+δj-1s·s)/Evjs)lj denote the dimensionless complex stiffness of the upper surface and lower surface of the jth soil layer. Substituting s=iω into (17) and solving it by using bisection method in the frequency domain, a series of eigenvalues, βjn, can be derived. ξjn can also be derived by substituting βjn into (14).Then, the vertical displacement of thejth soil layer can be rewritten as
(18)Wj(r,z′,s)=∑n=1∞AjnK0(ξjnr)sin(βjnz′+ϕjn),
where ϕjn=arctan(βjnlj/K-j) and Ajn are a series of constants determined by the boundary conditions which reflect the dynamic interaction of soil and pile.


## 3.2. Vibrations of the Pile

DenotingUj(z,s) to be the Laplace transform with respect to time of uj(z,t), (5) is transformed by using the Laplace transform technique and can be rewritten combining with (6) and (18)
(19)(Vjp)2(1+ηjpEjp·s)∂2Uj∂z′2-s2Uj-2πrjpρjpAjp(Gvjs+ηvjs·s)∑n=1∞AjnξjnK1(ξjnrjp)llllllllllllllllllllllllllllllllllllllllllllll×sin(βjnz′+ϕjn)=0.It is not difficult to obtain that the general solution of (19) can be expressed as
(20)Uj=Mj[cos⁡(λ-jz′lj)+∑n=1∞χjn′sin(βjnz′+ϕjn)]+Nj[sin⁡(λ-jz′lj)+∑n=1∞χjn′′sin⁡(βjnz′+ϕjn)],
where
(21)χjn′=χjn[cos⁡⁡(β-jn+λ-j+ϕjn)-cos⁡ϕjnβ-jn+λ-jllllllllllllllllllllllllllllllllllllll+cos⁡⁡(β-jn-λ-j+ϕjn)-cos⁡ϕjnβ-jn-λ-j],χjn′′=χjn[sin⁡(β-jn+λ-j+ϕjn)-sinϕjnβ-jn+λ-jlllllllllllllllllllllllllllllllllll-sin⁡(β-jn-λ-j+ϕjn)-sinϕjnβ-jn-λ-j],χjn=(Gvjs+ηjs·s)ξ-jnK1(ξ-jnr-jp)tj2ρjpljr-jp[β-jn2(1+(ηjp/Ejp)·s)+s2tj2]φjnLjn,llllllllllφjn=K0(ξ-jnr-jp)+2(Gvjs+ηjs·s)ξ-jnK1(ξ-jnr-jp)tj2ρjplj2r-jp[β-jn2(1+(ηjp/Ejp)·s)+s2tj2],Ljn=∫0ljsin2(βjnz′+ϕjn)dz′,llllllllllllllllllllllllllllllllllll
where λ¯j=-s2tj2/(1+(ηjp/Ejp)·s), β-jn=βjnlj, ξ-jn=ξjnlj, and r-jp=rjp/lj are all dimensionless parameters. tj=lj/Vjp denotes the propagation time of elastic longitudinal wave in the jth pile segment. K1(·) is the modified Bessel functions of order one of the second kind.Combining with the boundary conditions (8a) and (8b), the displacement impedance function at the head of the jth pile segment can be derived in the local coordinates as follows
(22)Zj(s)=-EjpAjp(∂Uj/∂z′)|z′=0Uj|z′=0=-EjpAjplj×(MjNj∑n=1∞χjn′β-jncos⁡ϕjnlllllllll+λ-j+∑n=1∞χjn′′β-jncos⁡ϕjn)×(MjNj(1+∑n=1∞χjn′sinϕjn)llllllllll+∑n=1∞χjn′′sinϕjn)-1,
where
(23)MjNj=(∑n=1∞χjn′′β-jncos⁡(β-jn+ϕjn)+λ-jcos⁡λ-jlllwlllll+Zj-1(s)ljEjpAjp[sinλ-j+∑n=1∞χjn′′sin(β-jn+ϕjn)])w×(∑n=1∞χjn′β-jncos⁡⁡(β-jn+ϕjn)-λ-jsinλ-jllllllwlllll+Zj-1(s)ljEjpAjp[cos⁡λ-j+∑n=1∞χjn′sin(β-jn+ϕjn)])-1,
where Zj-1(s) denotes the displacement impedance function at the head of the (j-1)th pile segment which can be obtained by using boundary conditions. Then, following the method of recursion typically used in the transfer function technique, the displacement impedance function at the head of pile can be derived as
(24)Zm(s)=-EmpAmp(∂Um/∂z′)|z′=0Um|z′=0=-EmpAmplm×(MmNm∑n=1∞χmn′β-mncos⁡ϕmnllllllll+λ-m+∑n=1∞χmn′′β-mncos⁡ϕmn)×(MmNm(1+∑n=1∞χmn′sinϕmn)llllllll+∑n=1∞χmn′′sinϕmn)-1,
where
(25)MmNm=-(∑n=1∞χmn′′β-mncos⁡(β-mn+ϕmn)+λ-mcos⁡λ-mllllllll+Zm-1(s)lmEmpAmp[sinλ-m+∑n=1∞χmn′′sin⁡(β-mn+ϕmn)])×(∑n=1∞χmn′β-mncos⁡⁡(β-mn+ϕmn)-λ-msinλ-mlllllllll+Zm-1(s)lmEmpAmp[cos⁡λ-m+∑n=1∞χmn′sin(β-mn+ϕmn)])-1,χmn′=χmn[cos⁡(β-mn+λ-m+ϕmn)-cos⁡ϕmnβ-mn+λ-mllllllllllllll+cos⁡⁡(β-mn-λ-m+ϕmn)-cos⁡ϕmnβ-mn-λ-m],χmn′′=χmn[sin(β-mn+λ-m+ϕmn)-sinϕmnβ-mn+λ-mlllllllllllll-sin⁡(β-mn-λ-m+ϕmn)-sinϕmnβ-mn-λ-m],χmn=(Gvms+ηms·s)ξ-mnK1(ξ-mnr-mp)tm2ρmplmr-mp[β-mn2(1+(ηmp/Emp)·s)+s2tm2]φmnLmn,φmn=K0(ξ-mnr-mp)lllllllllllllllllllllllllllllllllllllllllllllllllll+2(Gvms+ηms·s)ξ-mnK1(ξ-mnr-mp)tm2ρmplm2r-mp[β-mn2(1+(ηmp/Emp)·s)+s2tm2],Lmn=∫0lmsin2(βmnz′+ϕmn)dz′,lllllllllllllllllllllllllll
where λ¯m=-s2tm2/(1+(ηmp/Emp)·s), β-mn=βmnlm, ξ-mn=ξmnlm, and r-mp=rmp/lm are all dimensionless parameters. tm=lm/Vmp denotes the propagation time of elastic longitudinal wave in the mth pile segment. ϕmn and βmn can be obtained from the following equations
(26)ϕmn=arctan(βmnlmK-m),tan(βmlm)=(K-m+K-m′)βmlm(βmlm)2-K-mK-m′,
where K-m=((kms+δms·s)/Evms)lm and K-m′=((km-1s+δm-1s·s)/Evms)lm denote the dimensionless complex stiffness of the upper surface and lower surface of the mth soil layer.Then, the velocity transfer function at the head of pile can be obtained as(27)Gv(s)=sZm(s)=-lm·sEmpAmp(MmNm(1+∑n=1∞χmn′sinϕmn)lllllllllllllllll+∑n=1∞χmn′′sinϕmn)×(MmNm∑n=1∞χmn′β-mncos⁡ϕmnlllllllll+λ-m+∑n=1∞χmn′′β-mncos⁡ϕmn)-1.Lettings=iω and substituting it into (27), the velocity response in the frequency domain at the head of pile can be obtained as
(28)Hv(iω)=iωZ2(iω)=-1ρmpAmpVmpHv′,
where Hv′ is the dimensionless velocity admittance at the pile head which can be expressed as
(29)Hv′=iωtm(MmNm(1+∑n=1∞χmn′sinϕmn)lllllllllllllll+∑n=1∞χmn′′sinϕmn)×(MmNm∑n=1∞χmn′β-mncos⁡ϕmnlllllllllll+λ-m+∑n=1∞χmn′′β-mncos⁡ϕmn)-1.By means of the inverse Fourier transform and convolution theorem, the velocity response in the time domain at the head of pile can be expressed asV(t)=IFT[Q(ω)Hv(iω)], where Q(ω) denotes the Fourier transform of q(t), which is the vertical excitation acting on the pile head.In particular, the excitation can be regarded as a half-sine pulse in the nondestructive detection of pile foundation as follows(30)q(t)={Qmax⁡sin(πTt),t∈(0,T),0,t≥T,
where T and Qmax⁡ denote the duration of the impulse or impulse width and the maximum amplitude of the vertical excitation, respectively. Then, the velocity response in the time domain at the head of pile can be expressed as
(31)V(t)=q(t)*IFT[Hv(iω)]=IFT[Q(iω)·Hv(iω)]=-Qmax⁡ρmpAmpVmpVv′,
where Vv′ is the dimensionless velocity response which can be expressed as
(32)Vv′=12∫-∞∞iω-t-mllllllllll×(MmNm(1+∑n=1∞χmn′sinϕmn)+∑n=1∞χmn′′sinϕmn)llllllllll×(MmNm∑n=1∞χmn′β-mncos⁡ϕmn+λ-mlllllllllllllllllll+∑n=1∞χmn′′β-mncos⁡ϕmn)-1llllllllll×T-π2-T-2ω-2·(1+e-iω-T-)eiω-t-dω-,
where ω-=Tcω denotes the dimensionless frequency. T- denotes the dimensionless pulse width which should satisfy T-=T/Tc. t- denotes the dimensional time variable which should satisfy t-=t/Tc.


## 4. Analysis of Vibration Characteristics

According to the derivation process shown in the previous section, it can be seen that the difference between the shear modulus in the vertical plane and the shear modulus in the horizontal plane reflects the soil anisotropy. Therefore, based on the solutions, the influence of these two kinds of shear modulus of pile surrounding soil and pile end soil on the dynamic response of pile is studied in detail. Unless otherwise specified, the length, radius, density, and longitudinal wave velocity of pile are15m, 0.5m, 2500kg/m3, and 3800m/s, respectively. The spring constant of the distributed Voigt model is equal to the elastic modulus of the lower soil layer, and the damping coefficient of the distributed Voigt model is 10000N·m-3·s.


### 4.1. Influence of the Anisotropy of Pile Surrounding Soil on the Dynamic Response of Pile

Firstly, the influence of the shear modulus of pile surrounding soil in the vertical plane on the dynamic response of pile is investigated. Parameters of pile end soil are as follows: the thickness is three times that of pile diameter; the soil density is2000kg/m3; both the shear modulus in the vertical plane and the shear modulus in the horizontal plane are 120MPa; both the Poisson’s ratio in the horizontal direction caused by the vertical stress and the Poisson’s ratio in the orthogonal direction of the horizontal strain caused by the horizontal stress are 0.35; the damping coefficient is 1000N·m-3·s. Parameters of pile surrounding soil are as follows: the soil density is 1800kg/m3; both the Poisson’s ratio in the horizontal direction caused by the vertical stress and the Poisson’s ratio in the orthogonal direction of the horizontal strain caused by the horizontal stress are 0.4; the damping coefficient is 1000N·m-3·s; the shear modulus in the horizontal plane is 60MPa; the shear modulus in the vertical plane is Gv2s=20MPa,40MPa,60MPa,80MPa,100MPa, respectively.Figure2 shows the influence of the shear modulus of pile surrounding soil in the vertical plane on the dynamic response of pile. As shown in Figure 2(a), it can be noted that the velocity admittance curves oscillate about a mean amplitude as the frequency increases. As the shear modulus of pile surrounding soil in the vertical plane increases, the amplitude of resonance peaks gradually decreases, but the resonance frequency of velocity admittance curves almost remains unchanged. As shown in Figure 2(b), it is observed that the amplitude of the incident pulses and reflective wave signals decreases with the increase of the shear modulus of pile surrounding soil in the vertical plane. As the shear modulus of pile surrounding soil in the vertical plane increases, the raising phenomenon between the incident pulses and the primary reflective wave signals will be gradually aggravated, and the declining phenomenon between the primary reflective wave signals and the secondary reflective wave signals will also be gradually intensified.Influence of the shear modulus of pile surrounding soil in the vertical plane on the dynamic response of pile.
(a)
Velocity admittance curves
(b)
Reflected wave signal curveAfter that, the influence of the shear modulus of pile surrounding soil in the horizontal plane on the dynamic response of pile is studied. Parameters of pile surrounding soil are as follows: the shear modulus in the vertical plane is60MPa, and the shear modulus in the horizontal plane is Gh2s=20MPa,40MPa,60MPa,80MPa,100MPa, respectively. The other parameters of soil-pile system are the same as those shown in the previous case.Figure3 shows the influence of the shear modulus of pile surrounding soil in the horizontal plane on the dynamic response of pile. As shown in Figure 3(a), it can be seen that the amplitude of resonance peaks gradually increases with the increase of the shear modulus of pile surrounding soil in the horizontal plane, but the change of resonance frequency can be ignored. As shown in Figure 3(b), it can be seen that the amplitude of the reflective wave signals decreases with the increase of the shear modulus of pile surrounding soil in the horizontal plane, but the decreasing ratio is small. The raising phenomenon between the primary reflective wave signals and the secondary reflective wave signals will be gradually intensified with the increase of the shear modulus of pile surrounding soil in the horizontal plane.Influence of the shear modulus of pile surrounding soil in the horizontal plane on the dynamic response of pile.
(a)
Velocity admittance curves
(b)
Reflected wave signal curves


### 4.2. Influence of the Anisotropy of Pile End Soil on the Dynamic Response of Pile

In this section, the influence of the shear modulus of pile end soil in the vertical plane on the dynamic response of pile is firstly investigated. Parameters of pile surrounding soil are as follows: the soil density is1800kg/m3; both the shear modulus in the horizontal plane and the shear modulus in the vertical plane are 60MPa; both the Poisson’s ratio in the horizontal direction caused by the vertical stress and the Poisson’s ratio in the orthogonal direction of the horizontal strain caused by the horizontal stress are 0.4; the damping coefficient is 1000N·m-3·s. Parameters of pile end soil are as follows: the thickness is three times that of pile diameter; the soil density is 2000kg/m3; both the Poisson’s ratio in the horizontal direction caused by the vertical stress and the Poisson’s ratio in the orthogonal direction of the horizontal strain caused by the horizontal stress are 0.35; the damping coefficient is 1000N·m-3·s; the shear modulus in the horizontal plane is 120MPa; the shear modulus in the vertical plane is Gv1s=80MPa, 100MPa, 120MPa, 140MPa, 160MPa, respectively.Figure4 shows the influence of the shear modulus of pile end soil in the vertical plane on the dynamic response of pile. As shown in Figure 4(a), as the shear modulus of pile end soil in the vertical plane increases, the amplitude of resonance peaks gradually decreases with the same resonance frequency, but the decreasing ratio is small. As shown in Figure 4(b), it can be seen that the amplitude of the reflective wave signals decreases with the increase of the shear modulus of pile surrounding soil in the vertical plane.Influence of the shear modulus of pile end soil in the vertical plane on the dynamic response of pile.
(a)
Velocity admittance curves
(b)
Reflected wave signal curvesThen, the influence of the shear modulus of pile end soil in the horizontal plane on the dynamic response of pile is studied. Parameters of pile end soil are as follows: the shear modulus in the vertical plane is120MPa, and the shear modulus in the horizontal plane is Gh1s=80MPa, 100MPa, 120MPa, 140MPa, 160MPa, respectively. The other parameters of soil-pile system are the same as those shown in the previous case.Figure5 shows the influence of the shear modulus of pile end soil in the horizontal plane on the dynamic response of pile. It can be seen that the influence of the shear modulus of pile end soil in the horizontal plane on the dynamic response of pile can be ignored.Influence of the shear modulus of pile end soil in the horizontal plane on the dynamic response of pile.
(a)
Velocity admittance curves
(b)
Reflected wave signal curves


## 4.1. Influence of the Anisotropy of Pile Surrounding Soil on the Dynamic Response of Pile

Firstly, the influence of the shear modulus of pile surrounding soil in the vertical plane on the dynamic response of pile is investigated. Parameters of pile end soil are as follows: the thickness is three times that of pile diameter; the soil density is2000kg/m3; both the shear modulus in the vertical plane and the shear modulus in the horizontal plane are 120MPa; both the Poisson’s ratio in the horizontal direction caused by the vertical stress and the Poisson’s ratio in the orthogonal direction of the horizontal strain caused by the horizontal stress are 0.35; the damping coefficient is 1000N·m-3·s. Parameters of pile surrounding soil are as follows: the soil density is 1800kg/m3; both the Poisson’s ratio in the horizontal direction caused by the vertical stress and the Poisson’s ratio in the orthogonal direction of the horizontal strain caused by the horizontal stress are 0.4; the damping coefficient is 1000N·m-3·s; the shear modulus in the horizontal plane is 60MPa; the shear modulus in the vertical plane is Gv2s=20MPa,40MPa,60MPa,80MPa,100MPa, respectively.Figure2 shows the influence of the shear modulus of pile surrounding soil in the vertical plane on the dynamic response of pile. As shown in Figure 2(a), it can be noted that the velocity admittance curves oscillate about a mean amplitude as the frequency increases. As the shear modulus of pile surrounding soil in the vertical plane increases, the amplitude of resonance peaks gradually decreases, but the resonance frequency of velocity admittance curves almost remains unchanged. As shown in Figure 2(b), it is observed that the amplitude of the incident pulses and reflective wave signals decreases with the increase of the shear modulus of pile surrounding soil in the vertical plane. As the shear modulus of pile surrounding soil in the vertical plane increases, the raising phenomenon between the incident pulses and the primary reflective wave signals will be gradually aggravated, and the declining phenomenon between the primary reflective wave signals and the secondary reflective wave signals will also be gradually intensified.Influence of the shear modulus of pile surrounding soil in the vertical plane on the dynamic response of pile.
(a)
Velocity admittance curves
(b)
Reflected wave signal curveAfter that, the influence of the shear modulus of pile surrounding soil in the horizontal plane on the dynamic response of pile is studied. Parameters of pile surrounding soil are as follows: the shear modulus in the vertical plane is60MPa, and the shear modulus in the horizontal plane is Gh2s=20MPa,40MPa,60MPa,80MPa,100MPa, respectively. The other parameters of soil-pile system are the same as those shown in the previous case.Figure3 shows the influence of the shear modulus of pile surrounding soil in the horizontal plane on the dynamic response of pile. As shown in Figure 3(a), it can be seen that the amplitude of resonance peaks gradually increases with the increase of the shear modulus of pile surrounding soil in the horizontal plane, but the change of resonance frequency can be ignored. As shown in Figure 3(b), it can be seen that the amplitude of the reflective wave signals decreases with the increase of the shear modulus of pile surrounding soil in the horizontal plane, but the decreasing ratio is small. The raising phenomenon between the primary reflective wave signals and the secondary reflective wave signals will be gradually intensified with the increase of the shear modulus of pile surrounding soil in the horizontal plane.Influence of the shear modulus of pile surrounding soil in the horizontal plane on the dynamic response of pile.
(a)
Velocity admittance curves
(b)
Reflected wave signal curves


## 4.2. Influence of the Anisotropy of Pile End Soil on the Dynamic Response of Pile

In this section, the influence of the shear modulus of pile end soil in the vertical plane on the dynamic response of pile is firstly investigated. Parameters of pile surrounding soil are as follows: the soil density is1800kg/m3; both the shear modulus in the horizontal plane and the shear modulus in the vertical plane are 60MPa; both the Poisson’s ratio in the horizontal direction caused by the vertical stress and the Poisson’s ratio in the orthogonal direction of the horizontal strain caused by the horizontal stress are 0.4; the damping coefficient is 1000N·m-3·s. Parameters of pile end soil are as follows: the thickness is three times that of pile diameter; the soil density is 2000kg/m3; both the Poisson’s ratio in the horizontal direction caused by the vertical stress and the Poisson’s ratio in the orthogonal direction of the horizontal strain caused by the horizontal stress are 0.35; the damping coefficient is 1000N·m-3·s; the shear modulus in the horizontal plane is 120MPa; the shear modulus in the vertical plane is Gv1s=80MPa, 100MPa, 120MPa, 140MPa, 160MPa, respectively.Figure4 shows the influence of the shear modulus of pile end soil in the vertical plane on the dynamic response of pile. As shown in Figure 4(a), as the shear modulus of pile end soil in the vertical plane increases, the amplitude of resonance peaks gradually decreases with the same resonance frequency, but the decreasing ratio is small. As shown in Figure 4(b), it can be seen that the amplitude of the reflective wave signals decreases with the increase of the shear modulus of pile surrounding soil in the vertical plane.Influence of the shear modulus of pile end soil in the vertical plane on the dynamic response of pile.
(a)
Velocity admittance curves
(b)
Reflected wave signal curvesThen, the influence of the shear modulus of pile end soil in the horizontal plane on the dynamic response of pile is studied. Parameters of pile end soil are as follows: the shear modulus in the vertical plane is120MPa, and the shear modulus in the horizontal plane is Gh1s=80MPa, 100MPa, 120MPa, 140MPa, 160MPa, respectively. The other parameters of soil-pile system are the same as those shown in the previous case.Figure5 shows the influence of the shear modulus of pile end soil in the horizontal plane on the dynamic response of pile. It can be seen that the influence of the shear modulus of pile end soil in the horizontal plane on the dynamic response of pile can be ignored.Influence of the shear modulus of pile end soil in the horizontal plane on the dynamic response of pile.
(a)
Velocity admittance curves
(b)
Reflected wave signal curves


## 5. Conclusions

By considering a pile embedded in layered transversely isotropic soil as a dynamic soil-pile interaction problem, the governing equations of soil-pile system are established when there is arbitrary vertical harmonic force acting on the pile head. Then, an analytical solution for the velocity response in the frequency domain and its corresponding semianalytical solution for the velocity response in the time domain have been derived by virtue of the transform technique and the separation of variables technique. An extensive parameter study has been undertaken to investigate the influence of the soil anisotropy on the vertical dynamic response of pile and the following conclusions have been obtained.(1)
Whether for the pile surrounding soil or for the pile end soil, it can be seen that the influence of the shear modulus in the vertical plane on the dynamic response of pile is more notable than the influence of the shear modulus in the horizontal plane on the dynamic response of pile. Therefore, the shear modulus of soil in the vertical plane plays a leading role in the dynamic response of pile when only the vertical wave effect of soil is taken into account.(2)
As the shear modulus of pile surrounding soil in the vertical plane increases, both the amplitude of the resonance peaks of velocity admittance curves and the reflective wave signals of reflected wave signal curves gradually decrease. As the shear modulus of pile surrounding soil in the horizontal plane increases, the amplitude of the resonance peaks of velocity admittance curves gradually increases, but the reflective wave signals of reflected wave signal curves gradually decrease.(3)
As the shear modulus of pile end soil in the vertical plane increases, both the amplitude of the resonance peaks of velocity admittance curves and the reflective wave signals of reflected wave signal curves gradually decrease. The influence of the shear modulus of pile end soil in the horizontal plane on the dynamic response of pile can be ignored.



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*Source: 126916-2014-07-17.xml*