# Dynamic Response of an Inhomogeneous Elastic Pile in a Multilayered Saturated Soil to Transient Torsional Load

Authors: Kaifu Liu; Zhiqing Zhang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, we solve the dynamic response of an inhomogeneous elastic pile embedded in a multilayered saturated soil and subjected to a transient torsional load via a semianalytical method. To portray the inhomogeneity of the pile and the stratification of surrounding soil, the pile-soil system is subdivided intoNth layers along the depth direction in view of the variation of shear modulus or cross-sectional dimension of the pile or differences in soil properties. Then, the vibration displacement solution with undermined constants for any saturated soil layer subjected to the time-harmonic torsional load is obtained by virtue of the separation of variables scheme. To establish the connection of adjacent longitudinal soil layers, the circumferential contact traction at the interface of the adjacent layers is treated as the distributed Winkler subgrade model independent of the radial distance. Then, by utilizing the continuity conditions of the pile-soil system and the method of recursion typically used in the transfer function technique, the torsional impedance of the pile top can be derived in the frequency domain. By virtue of inverse Fourier transform and convolution theorem, the velocity response of an inhomogeneous pile subjected to a transient half-sine exciting torque and embedded in a layered saturated soil is gained in the time domain. Finally, selected numerical results are gained to investigate the influence of typical defects in pile and soil layering on the velocity response of the pile top in the time domain.


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## Body

## 1. Introduction

Pile foundations of machinery, bridges, offshore structures, and towers are commonly subjected to the dynamic torsional loads induced by the eccentricity in applied lateral loads, such as running machinery, moving traffic, wind, earthquake, and impact. Therefore, the analysis of the dynamic response of piles under torsional loads is essential in understanding the distribution of the displacements and forces in piles and can also provide valuable guidelines for dynamic foundation design and nondestructive detection of piles. In the past, various analytical, semianalytical, and numerical methods were proposed to solve the pile-soil dynamic interaction problem.In the pile-soil dynamic analysis, the key challenge is the theoretical model of the surrounding soil. Over the past several decades, the simplified continuum model was widely utilized to model the surrounding soil, and the corresponding dynamic response of the pile-soil system can be solved by an analytical or semianalytical method. For instance, Militano and Rajapakse [1] investigated the time history of twist angle and vertical displacement at the top end of a pile embedded in a multilayered soil and subjected to transient torsional and axial loads via a semianalytical method. Zhao and Xiang [2] investigated the torsional vibration of pipe pile in unsaturated soil based on the mixture theory. Wu et al. [3] proposed an analytical solution to calculate the apparent phase velocity of the pipe pile segment with soil plug filling inside based on the additional mass model. Luan et al. [4] presented a new analytical model for calculating the horizontal dynamic impedance of pile groups with arbitrary members connected with a rigid pile cap. Guan et al. [5] investigated the torsional vibration characteristics of tapered pile considering both the compaction effect of the pile surrounding soil and the stress diffusion effect of pile end soil. Moreover, based on this simplified continuum model, the analytical solution for the dynamic torsional and vertical response of a radially inhomogeneous soil layer can also be derived (the work of Veletsos and Dotson [6], Dotson and Veletsos [7], Han and Sabin [8], and Zhang et al. [9]). Compared to the simplified continuum model, the continuum model is rigorous in theory and can simulate correctly the mechanism of the soil-structure dynamic interaction. Hence, this rigorous model has been comprehensively applied into the study of the soil-rigid circular disc/footing interaction [10, 11] and pile-soil interaction [12–15]. Furthermore, various numerical methods are also employed to study the pile-soil interaction problem. Liu and Novak [16] investigated the dynamic response of a single pile embedded in transversely isotropic layered media using the finite element method (FEM). Tham et al. [17] studied the torsional vibration of an elastic pile embedded in a layered half-space via a coupled FEM-BEM method. Chen et al. [18] proposed a three-dimensional wave-pile-soil coupling FEM to investigate the deformation mechanism of monopile under current and fifth-order Stokes wave. Xu et al. [19] developed a fully coupled dynamic effective-stress FEM to study the effects of the frequency content of input motion and the amplitude of both horizontal and vertical components of input motion on the settlement of the pile group in saturated sand deposits. By virtue of the discrete element method (DEM), Li et al. [20] examined the sand plug behavior inside an open-ended pile considering the pile driving process. Since Biot [21, 22] established the general theory of elastic wave propagation in a fluid saturated poroelastic medium, the dynamic response of the buried loads [23, 24] and vibration of pile and rigid circular disc in the poroelastic medium were investigated in detail [25–29].Most of the existing studies are mainly concerned on the vibration characteristics of an intact pile, which is commonly used in the dynamic foundation design. However, in engineering practice, no matter what construction method is employed in pile formation, various defects in pile, such as bulging, necking, and weak concrete, are inevitable, and this kind of defective pile is generally called defective or inhomogeneous pile. It is noted that the defects in a pile could greatly affect the static and dynamic bearing capacities of the pile. Hence, in order to evaluate the potential defects in a pile, the related works were carried out by researchers. For instance, Wu et al. [30] studied the vertical vibration characteristics of a variable cross-section pile and variable modulus pile and gave an important insight into the evaluation of the construction quality of the pile. Liu et al. [31] developed a solution to the torsional response of a multidefective pile in layered soil based on the simplified continuum model and found that shear wave has a special advantage in detecting the shallow defect in a pile.However, the two abovementioned studies are limited to the purely elastic soil. Therefore, the objective of the present paper is to develop a semianalytical solution to investigate the dynamic response of an inhomogeneous pile embedded in a layered saturated soil. At first, the time-harmonic torsional vibration solution of the pile-soil system is obtained in the frequency domain using the separation of variables scheme and transfer function technique. Then, the velocity response at the pile top is derived by virtue of inverse Fourier transform and convolution theorem. Using the solution developed, the selected numerical results are obtained to analyze the influence of the typical defects in a pile and soil layering on the velocity response of the pile top in the time domain.


## 2. Statement of the Boundary-Value Problem

### 2.1. Basic Assumptions and Governing Equations

As shown in Figure1, the problem considered in this paper is the dynamic torsional response of an inhomogeneous elastic pile embedded in a layered saturated soil. Allowing for the variation of shear modulus or cross-sectional dimension of the pile or differences in soil properties of a layered saturated soil, the pile-soil system is subdivided into a total of N segments (layers) numbered by 1, 2, …, j, …, N from pile tip to pile top. The thickness of the jth (1 ≤ j ≤ N) soil layer is equal to the length of the jth (1 ≤ j ≤ N) pile segment and is denoted by hj such that the jth pile segment is completely embedded in the jth soil layer. In view of the symmetry of the torsional vibration problem, the cylindrical coordinate system (r, θ, z) is adopted in the present study. In order to derive the solution for the present problem, the following assumptions are made during the analysis: (1) The soil is a linear elastic isotropic saturated layer, and the pile is vertical, elastic, end bearing, and circular in cross section. The pile and soil layer properties are assumed to be homogeneous within each segment or layer, respectively, but may change from segment to segment or layer to layer; (2) the pile-soil system is subjected to small deformations and strains during the vibration. The pile has a perfect contact with the surrounding soil during the vibration; (3) the free surface of the soil has no normal and shear stresses, and the soil is infinite in the radial direction; (4) there is no displacement occurring at the bottom of the layer due to the fixed bottom boundary; (5) to establish the connection of adjacent longitudinal soil layers, the circumferential contact traction at the interface of adjacent soil layers is treated as the distributed Winkler subgrade model independent of the radial distance. Moreover, the circumferential contact tractions acting at the jth soil layer due to (j − 1)th and (j + 1)th soil layers are, respectively, denoted by the distributed reaction coefficients as kstj and ksbj, which represent the contact stress (τθz) per unit of the circumferential displacement along the adjacent soil interface. The reliability of this assumption will be proved in the following section, and the key derivation procedure is illustrated in Figure 2.Figure 1
Interaction between an inhomogeneous elastic pile and the layered saturated soil.Figure 2
Flow chart of the proposed semianalytical method.The dynamic equilibrium equation of thejth (1 ≤ j ≤ N) saturated soil subjected to torsional load can be expressed in terms of a cylindrical coordinate system as [26](1)Gsj∇2uθjr,z,t−uθjr,z,tr2Gsj=ρj∂2uθjr,z,t∂t2+ρfj∂2wθjr,z,t∂t2,where uθj(r, z, t) and wθj(r, z, t) are, respectively, the circumferential displacements of the soil skeleton in the jth saturated soil layer and that of the pore fluid relative to the soil skeleton; ρj = (1 − nj)ρsj + njρfj is the bulk density of the jth soil layer; for the jth soil layer, nj is the porosity of the soil; ρfj is the density of the pore fluid; Gsj and ρsj are the shear modulus and density of the soil skeleton, respectively; and ∇2=∂2/∂r2+1/r∂/∂r+∂2/∂z2.According to the study by Zienkiewicz et al. [32], the equation of motion of the pore fluid in a saturated soil can be written as(2)ρfjgkdj∂wθjr,z,t∂t+ρfj∂2uθjr,z,t∂t2+ρfjnj∂2wθjr,z,t∂t2=0,where kdj denotes the dynamic permeability coefficient containing the viscosity of the liquid; g is the gravitational acceleration.To gain the solution of the transient response of the pile-soil system, we solve the time-harmonic response first. For the time-harmonic motion,uθj(r, z, t) = uθj(r, z)eiωt and wθj(r, z, t) = wθj(r, z)eiωt, ω is the circular frequency of excitation, and t is the time variable. Equations (1) and (2) can, therefore, be expressed as(3)Gsj∇2uθjr,z−uθjr,zr2Gsj=−ρjω2uθjr,z−ρfjω2wθjr,z,(4)iρfjgkdjwθjr,z−ρfjωuθjr,z−ρfjωwθjr,znj=0.Substituting equation (4) into equation (3), the governing equation of the jth saturated soil layer can be written as(5)∂2uθjr,z∂r2+1r∂uθjr,z∂r+∂2uθjr,z∂z2−uθjr,zr2=−ω2Gsjρj+njωρfjibj/ρfj−ωuθjr,z.The one-dimensional equation of motion of thejth pile segment subjected to a time-harmonic torsional load can be expressed as(6)Gpj∂2ϕjzeiωt∂z2+4fjzeiωtrj2=ρpj∂2ϕjzeiωt∂t2,where Gpj, ρpj, rj, and ϕj(z) are the shear modulus, density, radius, and the twist angle amplitude of the jth pile segment, respectively; fj(z) denotes the amplitude of the contact traction along the jth pile-soil interface.


### 2.2. The Boundary and Continuity Conditions of the Pile-Soil System

The boundary conditions of thejth soil layer can be written in terms of the local coordinate system as(7)uθjr⟶∞,z=0,1≤j≤N,(8)∂uθjr,z∂z−kstjGsjuθjr,z|z=0=0,1≤j≤N,(9)uθjr,z|z=hj=0,j=1,(10)∂uθjr,z∂z+ksbjGsjuθjr,z|z=hj=0,2≤j≤N,where kstj and ksbj are the distributed reaction coefficients at the top and bottom of the jth soil layer, respectively. It is noted that kstN is equal to zero due to the free surface of the soil.The continuity condition of the interface between thejth and (j + 1)th (1 ≤ j ≤ N − 1) soil layers can be written as(11)kstj=ksbj+1.The boundary conditions at the base and top of thejth pile segment can be expressed in terms of the local coordinate system as(12)dϕjzdz|z=0=−TjGpjIpj,1≤j≤N,(13)ϕjz|z=hj=0,j=1,(14)dϕjzdz+ϕjzkpbjGpjIpj|z=hj=0,2≤j≤N,where Tj denotes the torque amplitude of the (j + 1)th pile segment acting on the top end of the jth pile segment when 1 ≤ j ≤ N ‒ 1; for the Nth pile segment, TN denotes the torque amplitude applied at the pile top; Ipj is the polar moment of inertia of the jth pile segment; and kpbj is the reaction coefficient at the bottom of the jth pile segment.At the interface between thejth and (j + 1)th pile segments (z = Hj), the circumferential displacement and torque of the pile satisfy the following continuity conditions (see Figure 3):(15)ϕjz=ϕj+1z,j=1,2,…N,GpjIpjdϕjzdz=Gpj+1Ipj+1dϕj+1zdz.Figure 3
Pile segment used in derivation of impedance (in the local coordinate system): (a) (j + 1)th pile segment and (b) jth pile segment.
(a)(b)Due to the continuity conditions of the twist angle and torque at the interface of the adjacent pile segment, the torsional impedance is continuous at the corresponding interface. Accordingly, the torsional impedance of thejth pile segment is equal to the reaction coefficient at the base of the (j + 1)th pile segment, and the continuity conditions of the jth pile segment can then be written as(16)kTj=Tjzϕjz|z=Hj=kpbj+1.The continuity conditions of the displacement and stress of the pile-soil interface can expressed as(17)uθjrj,z=ϕjzrj,(18)fjz=τrθjr,z|r=rj=Gsj∂uθjr,z∂r−uθjr,zr|r=rj.


## 2.1. Basic Assumptions and Governing Equations

As shown in Figure1, the problem considered in this paper is the dynamic torsional response of an inhomogeneous elastic pile embedded in a layered saturated soil. Allowing for the variation of shear modulus or cross-sectional dimension of the pile or differences in soil properties of a layered saturated soil, the pile-soil system is subdivided into a total of N segments (layers) numbered by 1, 2, …, j, …, N from pile tip to pile top. The thickness of the jth (1 ≤ j ≤ N) soil layer is equal to the length of the jth (1 ≤ j ≤ N) pile segment and is denoted by hj such that the jth pile segment is completely embedded in the jth soil layer. In view of the symmetry of the torsional vibration problem, the cylindrical coordinate system (r, θ, z) is adopted in the present study. In order to derive the solution for the present problem, the following assumptions are made during the analysis: (1) The soil is a linear elastic isotropic saturated layer, and the pile is vertical, elastic, end bearing, and circular in cross section. The pile and soil layer properties are assumed to be homogeneous within each segment or layer, respectively, but may change from segment to segment or layer to layer; (2) the pile-soil system is subjected to small deformations and strains during the vibration. The pile has a perfect contact with the surrounding soil during the vibration; (3) the free surface of the soil has no normal and shear stresses, and the soil is infinite in the radial direction; (4) there is no displacement occurring at the bottom of the layer due to the fixed bottom boundary; (5) to establish the connection of adjacent longitudinal soil layers, the circumferential contact traction at the interface of adjacent soil layers is treated as the distributed Winkler subgrade model independent of the radial distance. Moreover, the circumferential contact tractions acting at the jth soil layer due to (j − 1)th and (j + 1)th soil layers are, respectively, denoted by the distributed reaction coefficients as kstj and ksbj, which represent the contact stress (τθz) per unit of the circumferential displacement along the adjacent soil interface. The reliability of this assumption will be proved in the following section, and the key derivation procedure is illustrated in Figure 2.Figure 1
Interaction between an inhomogeneous elastic pile and the layered saturated soil.Figure 2
Flow chart of the proposed semianalytical method.The dynamic equilibrium equation of thejth (1 ≤ j ≤ N) saturated soil subjected to torsional load can be expressed in terms of a cylindrical coordinate system as [26](1)Gsj∇2uθjr,z,t−uθjr,z,tr2Gsj=ρj∂2uθjr,z,t∂t2+ρfj∂2wθjr,z,t∂t2,where uθj(r, z, t) and wθj(r, z, t) are, respectively, the circumferential displacements of the soil skeleton in the jth saturated soil layer and that of the pore fluid relative to the soil skeleton; ρj = (1 − nj)ρsj + njρfj is the bulk density of the jth soil layer; for the jth soil layer, nj is the porosity of the soil; ρfj is the density of the pore fluid; Gsj and ρsj are the shear modulus and density of the soil skeleton, respectively; and ∇2=∂2/∂r2+1/r∂/∂r+∂2/∂z2.According to the study by Zienkiewicz et al. [32], the equation of motion of the pore fluid in a saturated soil can be written as(2)ρfjgkdj∂wθjr,z,t∂t+ρfj∂2uθjr,z,t∂t2+ρfjnj∂2wθjr,z,t∂t2=0,where kdj denotes the dynamic permeability coefficient containing the viscosity of the liquid; g is the gravitational acceleration.To gain the solution of the transient response of the pile-soil system, we solve the time-harmonic response first. For the time-harmonic motion,uθj(r, z, t) = uθj(r, z)eiωt and wθj(r, z, t) = wθj(r, z)eiωt, ω is the circular frequency of excitation, and t is the time variable. Equations (1) and (2) can, therefore, be expressed as(3)Gsj∇2uθjr,z−uθjr,zr2Gsj=−ρjω2uθjr,z−ρfjω2wθjr,z,(4)iρfjgkdjwθjr,z−ρfjωuθjr,z−ρfjωwθjr,znj=0.Substituting equation (4) into equation (3), the governing equation of the jth saturated soil layer can be written as(5)∂2uθjr,z∂r2+1r∂uθjr,z∂r+∂2uθjr,z∂z2−uθjr,zr2=−ω2Gsjρj+njωρfjibj/ρfj−ωuθjr,z.The one-dimensional equation of motion of thejth pile segment subjected to a time-harmonic torsional load can be expressed as(6)Gpj∂2ϕjzeiωt∂z2+4fjzeiωtrj2=ρpj∂2ϕjzeiωt∂t2,where Gpj, ρpj, rj, and ϕj(z) are the shear modulus, density, radius, and the twist angle amplitude of the jth pile segment, respectively; fj(z) denotes the amplitude of the contact traction along the jth pile-soil interface.


## 2.2. The Boundary and Continuity Conditions of the Pile-Soil System

The boundary conditions of thejth soil layer can be written in terms of the local coordinate system as(7)uθjr⟶∞,z=0,1≤j≤N,(8)∂uθjr,z∂z−kstjGsjuθjr,z|z=0=0,1≤j≤N,(9)uθjr,z|z=hj=0,j=1,(10)∂uθjr,z∂z+ksbjGsjuθjr,z|z=hj=0,2≤j≤N,where kstj and ksbj are the distributed reaction coefficients at the top and bottom of the jth soil layer, respectively. It is noted that kstN is equal to zero due to the free surface of the soil.The continuity condition of the interface between thejth and (j + 1)th (1 ≤ j ≤ N − 1) soil layers can be written as(11)kstj=ksbj+1.The boundary conditions at the base and top of thejth pile segment can be expressed in terms of the local coordinate system as(12)dϕjzdz|z=0=−TjGpjIpj,1≤j≤N,(13)ϕjz|z=hj=0,j=1,(14)dϕjzdz+ϕjzkpbjGpjIpj|z=hj=0,2≤j≤N,where Tj denotes the torque amplitude of the (j + 1)th pile segment acting on the top end of the jth pile segment when 1 ≤ j ≤ N ‒ 1; for the Nth pile segment, TN denotes the torque amplitude applied at the pile top; Ipj is the polar moment of inertia of the jth pile segment; and kpbj is the reaction coefficient at the bottom of the jth pile segment.At the interface between thejth and (j + 1)th pile segments (z = Hj), the circumferential displacement and torque of the pile satisfy the following continuity conditions (see Figure 3):(15)ϕjz=ϕj+1z,j=1,2,…N,GpjIpjdϕjzdz=Gpj+1Ipj+1dϕj+1zdz.Figure 3
Pile segment used in derivation of impedance (in the local coordinate system): (a) (j + 1)th pile segment and (b) jth pile segment.
(a)(b)Due to the continuity conditions of the twist angle and torque at the interface of the adjacent pile segment, the torsional impedance is continuous at the corresponding interface. Accordingly, the torsional impedance of thejth pile segment is equal to the reaction coefficient at the base of the (j + 1)th pile segment, and the continuity conditions of the jth pile segment can then be written as(16)kTj=Tjzϕjz|z=Hj=kpbj+1.The continuity conditions of the displacement and stress of the pile-soil interface can expressed as(17)uθjrj,z=ϕjzrj,(18)fjz=τrθjr,z|r=rj=Gsj∂uθjr,z∂r−uθjr,zr|r=rj.


## 3. Solution of the Pile-Soil System

### 3.1. Solution of the Soil Layer

Substituting a single-variable functionuθj(r, z) = Rj(r)Zj(z) into equation (5), we have(19)d2Zjzdz2+Jj2Zjz=0,r2d2Rjrdr2+rdRjrdr−1+qj2r2Rjr=0,where(20)qj2=Jj2−ω2Gsjρj+njρfjωibj/ρfj−ω.The general solutions of equation (20) can be expressed as(21)Zjz=CjsinJjz+DjcosJjz,Rjr=AjK1qjr+BjI1qjr,where I1(qjr) and K1(qjr) are the modified Bessel functions of the first and second kind of first order, respectively; Aj, Bj, Cj, and Dj are the constants to be determined by the boundary conditions.To satisfy the boundary condition given in equation (7), the constant Bj = 0. For the 1st soil layer (i.e., j = 1), substituting the boundary conditions given in equations (8) and (9) into equation (21) results in(22)tanhjJj=−hjJjk¯stj,j=1,where k¯stj=kstjhj/Gsj denotes the dimensionless reaction coefficient at the top of the jth soil layer.For thejth soil layer (2 ≤ j ≤ N), substituting the boundary conditions given in equations (8) and (10) into equation (21) yields(23)tanhjJj=k¯sbj+k¯stjhjJjhjJj2−k¯sbjk¯stj,2≤j≤N,where k¯sbj=ksbjhj/Gsj denotes the dimensionless reaction coefficients at the base of the jth soil layer. It is noted that Jj in equations (22) and (23) can be solved by the numerical method (e.g., bisection method).Then, the solution of equation (5) can be written as(24)uθjr,z=∑m=1∞AmjK1qmjrsinJmjz+φmj,where(25)φmj=arctanJmjhjk¯stj;qmj2=Jmj2−ρsjω2Gsj.The shear stress amplitudeτrθj corresponding to equation (24) can be expressed as(26)τrθj=Gsj∂uθjr,z∂r−uθjr,zr=−Gsj∑m=1∞DmjqmjK2qmjrsinJmjz+φmj,where K2(qmjr) denotes the modified Bessel functions of the second kind of the second order.


### 3.2. Solution of the Pile Segment

For thejth (1 ≤ j ≤ N) pile segment, by utilizing the stress continuity condition that is given in equation (18), substituting equation (26) into equation (6) yields(27)ϕjz=α1jcosλjz+α2jsinλjz+∑m=1∞ψmjsinJmjz+φmj,in which(28)ψmj=−4GsjqmjK2qmjrjAmjGpjrj2Jmj2−λj2;λj=ρpjGpjω,where α1j and α2j are the undetermined coefficients.By using the displacement continuity condition given in equation (17), substituting equations (24) and (27) into equation (17) results in(29)rjα1jcosλjz+α2jsinλjz+∑m=1∞ψmjsinJmjz+φmj=∑m=1∞AmjK1qmjrsinJmjz+φmj.By invoking the orthogonality of eigenfunctions sin(Jmjz + φmj) (m = 1, 2, 3, …), multiplying sin(Jmjz + φmj) on both sides of equation (29), and then, integrating over the interval z = [0, hj], the undetermined coefficient Amj is found to be(30)Amj=1LmjEmj∫0hjPjsinJmjz+φmjdz,where(31)Lmj=∫0hjsin2Jmjz+φmjdz;Pj=α1jcosλjz+α2jsinλjz;Emj=1rjK1qmjrj+4GsjqmjK2qmjrjGpjrjJmj2−λj2.The amplitude of the twist angle of thejth pile segment is then given by(32)ϕjz=α1jcosλjz+∑m=1∞ξmjsinJmjz+φmj+α2jsinλjz+∑m=1∞ζmjsinJmjz+φmj,where(33)λ¯j=λjhj;J¯mj=hjJmj;q¯mj=hjqmj;r¯j=rjhj;μ¯j=GpjGsj;E¯mj=rjEmj;L¯mj=Lmjhj;νmj=−2q¯mjK2q¯mjr¯jr¯jμ¯jJ¯mj2−λ¯j2E¯mjL¯mj;ξmj=νmjcosφmj−cosJ¯mj+λ¯j+φmjJ¯mj+λ¯j+cosφmj−cosJ¯mj−λ¯j+φmjJ¯mj−λ¯j;ζmj=νmjsinφmj−sinJ¯mj+λ¯j+φmjJ¯mj+λ¯j+sinφmj−sinJ¯mj−λ¯j+φmjλ¯j−J¯mj.Based on the boundary conditions of thejth (1 ≤ j ≤ N) pile segment, substituting equations (12)–(14) into equation (32), the variables α1j and α2j are obtained, and the torsional impedance function at the top end of the jth pile segment can be written in terms of the local coordinate system as(34)kTj=Tjϕjz=0=GpjIpjrj−r¯jα1j/α2j∑m=1∞ξmjJ¯mjcosφmj+λ¯j+∑m=1∞ζmjJ¯mjcosφmjα1j/α2j1+∑m=1∞ξmjsinφmj+∑m=1∞ζmjsinφmj,where(35)α1jα2j=−sinλ¯1+∑m=1∞ζm1sinJ¯m1+φm1cosλ¯1+∑m=1∞ξm1sinJ¯m1+φm1,j=1,α1jα2j=−λ¯jcosλ¯j+∑m=1∞ζmjJ¯mjcosJ¯mj+φmj+hjkTj−1/GpjIpjsinλ¯j+∑m=1∞ζmjsinJ¯mj+φmj−λ¯jsinλ¯j+∑m=1∞ξmjJ¯mjcosJ¯mj+φmj+hjkTj−1/GpjIpjcosλ¯j+∑m=1∞ξmjsinJ¯mj+φmj,2≤j≤N.Through further recursion from the 1st pile segment to theNth pile segment, the torsional impedance function at the pile head can be expressed as(36)kT=kTN=TNϕNz=0=GpNIpNrN−r¯Nα1N/α2N∑m=1∞ξmNJ¯mNcosφmN+λ¯N+∑m=1∞ζmNJ¯mNcosφmNα1N/α2N1+∑m=1∞ξmNsinφmN+∑m=1∞ζmNsinφmN.Based on the definition in [26], the dimensionless torsional impedance at the pile top can be expressed as(37)kT′=3TN16GsNrN3ϕz=0=−3πμ¯Nr¯NR32,where(38)R=α1N/α2N∑m=1∞ξmNJ¯mNcosφmN+λ¯N+∑m=1∞ζmNJ¯mNcosφmNα1N/α2N1+∑m=1∞ξmNsinφmN+∑m=1∞ζmNsinφmN.The frequency response function of the twist angle of the pile top can be written as(39)Hθω=1kT=−rNGpNIpN1r¯Nα1N/α2N1+∑m=1∞ξmNsinφmN+∑m=1∞ζmNsinφmNα1N/α2N∑m=1∞ξmNJ¯mNcosφmN+λ¯N+∑m=1∞ζmNJ¯mNcosφmN.Then, the admittance function of angular velocity of the arbitrary point at the pile top can be further expressed as(40)Hvω=iωrsHθω=rsvpsNIpNρpNHv′ω,where rs denotes the radial distance to the arbitrary point, and it is commonly selected as r0 (i.e., the radius of the homogeneous pile); the dimensionless admittance function of angular velocity of the pile top is(41)Hv′ω=−iλ¯Nα1N/α2N1+∑m=1∞ξmNsinφmN+∑m=1∞ζmNsinφmNα1N/α2N∑m=1∞ξmNJ¯mNcosφmN+λ¯N+∑m=1∞ζmNJ¯mNcosφmN.By applying the inverse Fourier transform into equation (40), the response function of unit pulse torque in the time domain can be written as(42)ht=IFTHvω=12π∫−∞+∞rsvpsNIpNρpNHv′ωeiω˜t¯dω˜,where ω˜=ωTc and Tc=∑j=1Nhj/vpsj denote the nondimensional frequency and propagation time of elastic shear wave propagating from the pile top to pile tip, respectively; t¯=t/Tc is the nondimensional time variable.If the Fourier transform of the arbitrary exciting torqueT(t) acting at the pile top is denoted by T(ω), then the velocity response of the pile top in the time domain is given via the inverse Fourier transform and convolution theorem as V(t) = T(t)∗h(t) = IFT[T(ω)Hv(ω)]. In particular, when the exciting torque is a half-sine pulse, we have(43)Tt=Tmaxsinπtt0,t<t0,0,t≥t0,where t0 and Tmax denote the duration of the impulse and the maximum amplitude of the exciting torque, respectively.Then, the velocity response of the pile top in the time domain can be written as(44)Vt=TmaxIFTrsvpsNIpNρpNHv′ωπt0π2−t02ω21+e−iωt0.It is noted that the infinite integral involved in equation (44) can be numerically determined with finite terms (e.g., the lower and upper limits of the integral are from ‒4000 to 4000).To facilitate analysis, it is useful to introduce the normalized velocity response(45)V′t=VtmaxVt,where max[V(t)] denotes the maximum value of V(t).


## 3.1. Solution of the Soil Layer

Substituting a single-variable functionuθj(r, z) = Rj(r)Zj(z) into equation (5), we have(19)d2Zjzdz2+Jj2Zjz=0,r2d2Rjrdr2+rdRjrdr−1+qj2r2Rjr=0,where(20)qj2=Jj2−ω2Gsjρj+njρfjωibj/ρfj−ω.The general solutions of equation (20) can be expressed as(21)Zjz=CjsinJjz+DjcosJjz,Rjr=AjK1qjr+BjI1qjr,where I1(qjr) and K1(qjr) are the modified Bessel functions of the first and second kind of first order, respectively; Aj, Bj, Cj, and Dj are the constants to be determined by the boundary conditions.To satisfy the boundary condition given in equation (7), the constant Bj = 0. For the 1st soil layer (i.e., j = 1), substituting the boundary conditions given in equations (8) and (9) into equation (21) results in(22)tanhjJj=−hjJjk¯stj,j=1,where k¯stj=kstjhj/Gsj denotes the dimensionless reaction coefficient at the top of the jth soil layer.For thejth soil layer (2 ≤ j ≤ N), substituting the boundary conditions given in equations (8) and (10) into equation (21) yields(23)tanhjJj=k¯sbj+k¯stjhjJjhjJj2−k¯sbjk¯stj,2≤j≤N,where k¯sbj=ksbjhj/Gsj denotes the dimensionless reaction coefficients at the base of the jth soil layer. It is noted that Jj in equations (22) and (23) can be solved by the numerical method (e.g., bisection method).Then, the solution of equation (5) can be written as(24)uθjr,z=∑m=1∞AmjK1qmjrsinJmjz+φmj,where(25)φmj=arctanJmjhjk¯stj;qmj2=Jmj2−ρsjω2Gsj.The shear stress amplitudeτrθj corresponding to equation (24) can be expressed as(26)τrθj=Gsj∂uθjr,z∂r−uθjr,zr=−Gsj∑m=1∞DmjqmjK2qmjrsinJmjz+φmj,where K2(qmjr) denotes the modified Bessel functions of the second kind of the second order.


## 3.2. Solution of the Pile Segment

For thejth (1 ≤ j ≤ N) pile segment, by utilizing the stress continuity condition that is given in equation (18), substituting equation (26) into equation (6) yields(27)ϕjz=α1jcosλjz+α2jsinλjz+∑m=1∞ψmjsinJmjz+φmj,in which(28)ψmj=−4GsjqmjK2qmjrjAmjGpjrj2Jmj2−λj2;λj=ρpjGpjω,where α1j and α2j are the undetermined coefficients.By using the displacement continuity condition given in equation (17), substituting equations (24) and (27) into equation (17) results in(29)rjα1jcosλjz+α2jsinλjz+∑m=1∞ψmjsinJmjz+φmj=∑m=1∞AmjK1qmjrsinJmjz+φmj.By invoking the orthogonality of eigenfunctions sin(Jmjz + φmj) (m = 1, 2, 3, …), multiplying sin(Jmjz + φmj) on both sides of equation (29), and then, integrating over the interval z = [0, hj], the undetermined coefficient Amj is found to be(30)Amj=1LmjEmj∫0hjPjsinJmjz+φmjdz,where(31)Lmj=∫0hjsin2Jmjz+φmjdz;Pj=α1jcosλjz+α2jsinλjz;Emj=1rjK1qmjrj+4GsjqmjK2qmjrjGpjrjJmj2−λj2.The amplitude of the twist angle of thejth pile segment is then given by(32)ϕjz=α1jcosλjz+∑m=1∞ξmjsinJmjz+φmj+α2jsinλjz+∑m=1∞ζmjsinJmjz+φmj,where(33)λ¯j=λjhj;J¯mj=hjJmj;q¯mj=hjqmj;r¯j=rjhj;μ¯j=GpjGsj;E¯mj=rjEmj;L¯mj=Lmjhj;νmj=−2q¯mjK2q¯mjr¯jr¯jμ¯jJ¯mj2−λ¯j2E¯mjL¯mj;ξmj=νmjcosφmj−cosJ¯mj+λ¯j+φmjJ¯mj+λ¯j+cosφmj−cosJ¯mj−λ¯j+φmjJ¯mj−λ¯j;ζmj=νmjsinφmj−sinJ¯mj+λ¯j+φmjJ¯mj+λ¯j+sinφmj−sinJ¯mj−λ¯j+φmjλ¯j−J¯mj.Based on the boundary conditions of thejth (1 ≤ j ≤ N) pile segment, substituting equations (12)–(14) into equation (32), the variables α1j and α2j are obtained, and the torsional impedance function at the top end of the jth pile segment can be written in terms of the local coordinate system as(34)kTj=Tjϕjz=0=GpjIpjrj−r¯jα1j/α2j∑m=1∞ξmjJ¯mjcosφmj+λ¯j+∑m=1∞ζmjJ¯mjcosφmjα1j/α2j1+∑m=1∞ξmjsinφmj+∑m=1∞ζmjsinφmj,where(35)α1jα2j=−sinλ¯1+∑m=1∞ζm1sinJ¯m1+φm1cosλ¯1+∑m=1∞ξm1sinJ¯m1+φm1,j=1,α1jα2j=−λ¯jcosλ¯j+∑m=1∞ζmjJ¯mjcosJ¯mj+φmj+hjkTj−1/GpjIpjsinλ¯j+∑m=1∞ζmjsinJ¯mj+φmj−λ¯jsinλ¯j+∑m=1∞ξmjJ¯mjcosJ¯mj+φmj+hjkTj−1/GpjIpjcosλ¯j+∑m=1∞ξmjsinJ¯mj+φmj,2≤j≤N.Through further recursion from the 1st pile segment to theNth pile segment, the torsional impedance function at the pile head can be expressed as(36)kT=kTN=TNϕNz=0=GpNIpNrN−r¯Nα1N/α2N∑m=1∞ξmNJ¯mNcosφmN+λ¯N+∑m=1∞ζmNJ¯mNcosφmNα1N/α2N1+∑m=1∞ξmNsinφmN+∑m=1∞ζmNsinφmN.Based on the definition in [26], the dimensionless torsional impedance at the pile top can be expressed as(37)kT′=3TN16GsNrN3ϕz=0=−3πμ¯Nr¯NR32,where(38)R=α1N/α2N∑m=1∞ξmNJ¯mNcosφmN+λ¯N+∑m=1∞ζmNJ¯mNcosφmNα1N/α2N1+∑m=1∞ξmNsinφmN+∑m=1∞ζmNsinφmN.The frequency response function of the twist angle of the pile top can be written as(39)Hθω=1kT=−rNGpNIpN1r¯Nα1N/α2N1+∑m=1∞ξmNsinφmN+∑m=1∞ζmNsinφmNα1N/α2N∑m=1∞ξmNJ¯mNcosφmN+λ¯N+∑m=1∞ζmNJ¯mNcosφmN.Then, the admittance function of angular velocity of the arbitrary point at the pile top can be further expressed as(40)Hvω=iωrsHθω=rsvpsNIpNρpNHv′ω,where rs denotes the radial distance to the arbitrary point, and it is commonly selected as r0 (i.e., the radius of the homogeneous pile); the dimensionless admittance function of angular velocity of the pile top is(41)Hv′ω=−iλ¯Nα1N/α2N1+∑m=1∞ξmNsinφmN+∑m=1∞ζmNsinφmNα1N/α2N∑m=1∞ξmNJ¯mNcosφmN+λ¯N+∑m=1∞ζmNJ¯mNcosφmN.By applying the inverse Fourier transform into equation (40), the response function of unit pulse torque in the time domain can be written as(42)ht=IFTHvω=12π∫−∞+∞rsvpsNIpNρpNHv′ωeiω˜t¯dω˜,where ω˜=ωTc and Tc=∑j=1Nhj/vpsj denote the nondimensional frequency and propagation time of elastic shear wave propagating from the pile top to pile tip, respectively; t¯=t/Tc is the nondimensional time variable.If the Fourier transform of the arbitrary exciting torqueT(t) acting at the pile top is denoted by T(ω), then the velocity response of the pile top in the time domain is given via the inverse Fourier transform and convolution theorem as V(t) = T(t)∗h(t) = IFT[T(ω)Hv(ω)]. In particular, when the exciting torque is a half-sine pulse, we have(43)Tt=Tmaxsinπtt0,t<t0,0,t≥t0,where t0 and Tmax denote the duration of the impulse and the maximum amplitude of the exciting torque, respectively.Then, the velocity response of the pile top in the time domain can be written as(44)Vt=TmaxIFTrsvpsNIpNρpNHv′ωπt0π2−t02ω21+e−iωt0.It is noted that the infinite integral involved in equation (44) can be numerically determined with finite terms (e.g., the lower and upper limits of the integral are from ‒4000 to 4000).To facilitate analysis, it is useful to introduce the normalized velocity response(45)V′t=VtmaxVt,where max[V(t)] denotes the maximum value of V(t).


## 4. Numerical Results and Discussion

### 4.1. Comparison with Other Solutions

It is noted that the existed solutions for an elastic pile subjected to a harmonic torsional loading and embedded in layered soil neglect the gradient of the shear stressτθz in the depth direction [1, 31], thus resulting in inevitable error in the dynamic response. However, when considering the effect of the shear stress τθz, it is difficult to propose a rigorous solution due to the unknown continuity conditions of the adjacent soil layers. Accordingly, the reaction coefficient of adjacent soil layers is proposed in this paper to establish the connection of the adjacent soil layers and to derive the corresponding solution. Apparently, the value of the reaction coefficient has an important influence on the reliability and applicability of the developed solution. In the following section, unless otherwise specified, the pile properties are ρpj = 2300 kg/m3, vpsj = 2450 m/s, Gpj = 1.38 × 1010 Pa, Epj = 3.31 × 1010 Pa, rj = r0 = 0.4 m (j = 1, 2, …, N), and the Poisson’s ratio is 0.2; the soil properties are ρsj = 2650 kg/m3, ρfj = 1000 kg/m3, nj = 0.4, and Gsj = 1.38 × 107 Pa (j = 1, 2, …, N).To verify the solution developed in the present paper, the pile-soil layer is divided into 3 layers (N = 3), and the pile and soil are assumed to be homogeneous. Figure 4 shows the comparison of the torsional impedance of the present study with that of Wang et al. [26]. In Figure 4, Re( ) and Im( ) denote the real and imaginary parts of the physical quantity, respectively. It can be seen from Figure 4 that the real and imaginary parts of the torsional impedance of the present solution are close to those of Wang et al. [26] when k¯stj (the distributed reaction coefficient at the top of the jth soil layer) is within the range from 0.001 to 0.1. When k¯stj=10, the real part of the torsional impedance of the present solution has an obvious error in the relatively low frequency range when compared to the rigorous solution. However, the error of the torsional impedance of the present solution is slight, as the excitation frequency is in the higher frequency range. This result indicates that k¯stj has a negligible influence on the torsional impedance in the higher frequency range.Figure 4
Comparison of the torsional impedance of a homogeneous pile embedded in a homogeneous saturated soil (N = 3, h1 = 7 m, h2 = 0.5 m, and h3 = 3.5 m).
(a)(b)Figure5 shows the comparison of the normalized velocity response of a homogeneous pile obtained from the present study with that of Wang et al. [26]. It can be observed from Figure 5 that the velocity response curve of the present solution agrees well with that of Wang et al. [26] when k¯stj changes from 0.001 to 10. This is due to the face that the majority of data used in the calculation of the velocity response belongs to high frequency data, and k¯stj has a negligible influence on the dynamic response in the high frequency range. Consequently, the influence of k¯stj on the velocity response is negligible, and k¯stj is fixed at 0.01 in the following analysis.Figure 5
Comparison of the normalized velocity response of for a homogeneous pile embedded in a homogeneous saturated soil (N = 3, h1 = 7 m, h2 = 0.5 m, and h3 = 3.5 m).


### 4.2. Influence of the Neck Width and Blub Width

In practical engineering, cast-in-place piles are widely used, and the corresponding defect can generally be modelled as pile with a neck, pile with a bulb, and pile with a weak concrete segment (see Figure6). As shown in Figure 6, it is the typically inhomogeneous pile with one defect, and the pile can be divided into 3 segments. WRN (neck width) and WRB (bulb width) denote the radii of the neck and bulb, respectively. LE and LL denote, respectively, the embedment depth and length of the defect. In addition, r0 is the radius of a homogeneous pile.Figure 6
Geometry of a typically inhomogeneous pile with one defect: (a) pile with a neck; (b) pile with a bulb; and (c) pile with a weak concrete segment.Figure7 shows influence of the neck width (WRN) on the normalized velocity response of the pile top. It can be seen from Figure 7 that compared to the homogeneous pile, the first reflected signal (i.e., RS1) of the neck arrives at the pile top when t = tRS1 = 0.00284s (tRS1 is the time of arrival of the first reflected signal) and, thus, the calculated LE = vps × tRS1/2 = 3.48 m with error = 0.57% (vps is the one-dimensional elastic shear wave velocity in a homogeneous pile and taken as 2450 m/s). The main reason for the error is the fact that the boundary of reflected signal is difficult to accurately determine in theory. When the time is approximately equal to two times tRS1, the second reflected signal (i.e., RS2) of the neck arrives at the pile top with the corresponding amplitude of the reflected signal decreasing markedly due to the energy dissipation in the propagating process. The reflected signal from the pile end (i.e., RSE) arrives at the pile top when t = tRSE = 0.00899s and, thus, H = vps × tRSE/2 = 11.01 m with error = 0.09%. It is also observed from Figure 7 that the phase of RS1 is the same as that of the incident pulse first and then opposite to that of the incident pulse. The phase of the RSE is opposite to that of the incident pulse for the fixed bottom boundary. Moreover, the amplitude of the RS1 increases markedly with the decrease of WRN. This result indicates that the amplitude of the reflected signal increases with increasing degree of defect.Figure 7
Influence of the neck width WRN on the normalized velocity response of the pile top (N = 3, h1 = 7 = m, LL = h2 = 0.5 m, and LE = h3 = 3.5 m).Figure8 depicts the influence of the bulb width (WRB) on the normalized velocity response of the pile top. It can be seen from Figure 8 that compared to the homogeneous pile, RS1 arrives at the pile top when t = tRS1 = 0.00284s and, thus, LE = vps × tRS1/2 = 3.48 m with error = 0.57%. When the time is approximately equal to two times tRS1, RS2 arrives at the pile top and the amplitude of RS2 decreases markedly because of the energy dissipation. RSE arrives at the pile top when t = tRSE = 0.00899s and, thus, H = vps × tRSE/2 = 11.01 m with error = 0.09%. It is also observed from Figure 8 that the phase of RS1 is the opposite to that of the incident pulse first and then the same with that of the incident pulse. However, the phase of RS2 is the same as that of the incident pulse. Moreover, the amplitude of RS1 increases markedly with the increase of WRB, which indicates that higher amplitude of the reflected signal is associated with larger value of the bulb width.Figure 8
Influence of the bulb widthWRB on the normalized velocity response of the pile top (N = 3, h1 = 7 m, LL = h2 = 0.5 m, and LE = h3 = 3.5 m).


### 4.3. Effect of Weak Concrete

Generally, if the pile has a weak concrete segment, the shear modulus of the pile will decrease greatly. Hence, we use the shear wave velocity to simulate the weak concrete in theory. The influence of the shear wave velocity in the weak concrete segment (vps2) on the normalized velocity response of the pile top is shown in Figure 9. The changing trend of the velocity response at the top end of a pile with a weak concrete segment is similar to that of the pile with a neck. When t = tRS1 = 0.00284s, RS1 arrives at the pile top and LE = vps × tRS1/2 = 3.48 m with error = 0.57%. When t is approximately equal to two times tRS1, RS2 arrives at the pile top. However, compared to the defect of neck, the amplitude of RS2 is small. The phase of RS1 is opposite to that of the incident pulse first and then the same as that of incident pulse. The amplitude of RS1 increases markedly with the decrease of vps2. Furthermore, the time of arrival of RSE increases obviously with the decrease of vps2. This result indicates that RSE will need much more time to arrive at the pile top as vps2 decreases. As a result, the estimated pile length will be longer than the actual length. For instance, when t = tRSE = 0.00918s, RSE of the pile with vps2 = 1600 m/s arrives at the pile top and the pile length can be estimated as H = vps × tRSE/2 = 11.25 m with error = 2.27%.Figure 9
Influence of shear wave velocity in weak concrete on the normalized velocity response of the pile top (N = 3, h1 = 7 m, LL = h2 = 0.5 m, and LE = h3 = 3.5 m).


### 4.4. Effect of Length of the Defect

Figure10 portrays the influence of the neck length on the normalized velocity response of the pile top. It can be seen from Figure 10 that the neck length has marked influence on the velocity response of the pile top. As the neck length increases from 0.5 m to 2.0 m, the width of RS1 and RS2 shows an obvious increase, and another reflected signal (ARS1) arrives at the pile top when t = 0.00448s. It should be mentioned again that there exist two abrupt changes in the cross-sectional dimension for a pile with a neck, one is at the top of the neck, and the other is at the bottom of the neck. When the neck length is long enough and the shear wave velocity is relatively slow, the reflected signal of these two abrupt changes does not overlap in the time domain. Furthermore, the embedded depth corresponding to the bottom of the neck is 5.5 m, and the corresponding calculated depth is 0.00448 × 2450/2 = 5.49 m with error = 0.18%.Figure 10
Influence of neck length on the normalized velocity response of the pile top (N = 3, WRN = 0.3 m, and LE = h3 = 3.5 m).Figure11 shows the influence of the blub length on the normalized velocity response of the pile top. It can be seen from Figure 11 that the blub length has marked influence on the velocity response of the pile top. The width of RS1 and RS2 also increases with the increase of blub length. For the pile with LL = 2.0 m, another obvious reflected signal (ARS1) arrives when t = tARS1 = 0.00448s, and the phase of ARS1 is the same as that of the incident impulse. It is worth noting that this signal is easy to be identified as a defect by mistake due to the similarity with the reflected signal of the defect.Figure 11
Influence of bulb length on the normalized velocity response of the pile top (N = 3, WRB = 0.45 m, and LE = h3 = 3.5 m).Figure12 shows the influence of the weak concrete length on the normalized velocity response of the pile top. It can be seen from Figure 12 that the influence of the weak concrete length on the velocity response is similar to that of the neck length. The width of RS1 and RS2 also increases with the increase of the weak concrete length, and there exists an individual ARS1 when LL = 2.0 m. Moreover, the time of arrival of RSE shows a marked increase with the increase of LL.Figure 12
Influence of weak concrete length on the normalized velocity response of the pile top (N = 3, vps2 = 1600 m/s, and LE = h3 = 3.5 m).


### 4.5. Influence of Soil Layering

In some cases, the soil properties may change greatly in certain embedment depth. In order to explicitly identify the influence of the variation of the surrounding soil properties, the pile is assumed to be homogeneous, and three cases of soil properties are investigated. Case 1: the soil has a stiff interlayer withGs2 = 5.52 × 107 Pa (i.e., Gs1/Gs2 = 0.25); Case 2: the soil is homogeneous with Gs2 = 1.38 × 107 Pa (i.e., Gs1/Gs2 = 1.0); and Case 3: the soil has a soft interlayer with Gs2 = 3.45 × 106 Pa (i.e., Gs1/Gs2 = 4). It can be seen from Figure 13 that the phase of the reflected signal of the interface of adjacent soil layers is, respectively, opposite to and same as that of the incident pulse for Case 1 and Case 3. However, when the soil is homogeneous, the velocity curve is smooth and the reflected signal of the soil interface does not emerge. It is also observed from Figure 13 that the shape of the reflected signal induced by the soil with a soft or stiff interlayer is in a half-sine form, which is obviously different from that of the defect. Moreover, the amplitude of the reflected signal induced by the soil interface is much lower than that of the defect.Figure 13
Influence of soil layering on the normalized velocity response of the pile (N = 3, h1 = 7 m, h2 = 0.5 m, and h3 = 3.5 m).


## 4.1. Comparison with Other Solutions

It is noted that the existed solutions for an elastic pile subjected to a harmonic torsional loading and embedded in layered soil neglect the gradient of the shear stressτθz in the depth direction [1, 31], thus resulting in inevitable error in the dynamic response. However, when considering the effect of the shear stress τθz, it is difficult to propose a rigorous solution due to the unknown continuity conditions of the adjacent soil layers. Accordingly, the reaction coefficient of adjacent soil layers is proposed in this paper to establish the connection of the adjacent soil layers and to derive the corresponding solution. Apparently, the value of the reaction coefficient has an important influence on the reliability and applicability of the developed solution. In the following section, unless otherwise specified, the pile properties are ρpj = 2300 kg/m3, vpsj = 2450 m/s, Gpj = 1.38 × 1010 Pa, Epj = 3.31 × 1010 Pa, rj = r0 = 0.4 m (j = 1, 2, …, N), and the Poisson’s ratio is 0.2; the soil properties are ρsj = 2650 kg/m3, ρfj = 1000 kg/m3, nj = 0.4, and Gsj = 1.38 × 107 Pa (j = 1, 2, …, N).To verify the solution developed in the present paper, the pile-soil layer is divided into 3 layers (N = 3), and the pile and soil are assumed to be homogeneous. Figure 4 shows the comparison of the torsional impedance of the present study with that of Wang et al. [26]. In Figure 4, Re( ) and Im( ) denote the real and imaginary parts of the physical quantity, respectively. It can be seen from Figure 4 that the real and imaginary parts of the torsional impedance of the present solution are close to those of Wang et al. [26] when k¯stj (the distributed reaction coefficient at the top of the jth soil layer) is within the range from 0.001 to 0.1. When k¯stj=10, the real part of the torsional impedance of the present solution has an obvious error in the relatively low frequency range when compared to the rigorous solution. However, the error of the torsional impedance of the present solution is slight, as the excitation frequency is in the higher frequency range. This result indicates that k¯stj has a negligible influence on the torsional impedance in the higher frequency range.Figure 4
Comparison of the torsional impedance of a homogeneous pile embedded in a homogeneous saturated soil (N = 3, h1 = 7 m, h2 = 0.5 m, and h3 = 3.5 m).
(a)(b)Figure5 shows the comparison of the normalized velocity response of a homogeneous pile obtained from the present study with that of Wang et al. [26]. It can be observed from Figure 5 that the velocity response curve of the present solution agrees well with that of Wang et al. [26] when k¯stj changes from 0.001 to 10. This is due to the face that the majority of data used in the calculation of the velocity response belongs to high frequency data, and k¯stj has a negligible influence on the dynamic response in the high frequency range. Consequently, the influence of k¯stj on the velocity response is negligible, and k¯stj is fixed at 0.01 in the following analysis.Figure 5
Comparison of the normalized velocity response of for a homogeneous pile embedded in a homogeneous saturated soil (N = 3, h1 = 7 m, h2 = 0.5 m, and h3 = 3.5 m).


## 4.2. Influence of the Neck Width and Blub Width

In practical engineering, cast-in-place piles are widely used, and the corresponding defect can generally be modelled as pile with a neck, pile with a bulb, and pile with a weak concrete segment (see Figure6). As shown in Figure 6, it is the typically inhomogeneous pile with one defect, and the pile can be divided into 3 segments. WRN (neck width) and WRB (bulb width) denote the radii of the neck and bulb, respectively. LE and LL denote, respectively, the embedment depth and length of the defect. In addition, r0 is the radius of a homogeneous pile.Figure 6
Geometry of a typically inhomogeneous pile with one defect: (a) pile with a neck; (b) pile with a bulb; and (c) pile with a weak concrete segment.Figure7 shows influence of the neck width (WRN) on the normalized velocity response of the pile top. It can be seen from Figure 7 that compared to the homogeneous pile, the first reflected signal (i.e., RS1) of the neck arrives at the pile top when t = tRS1 = 0.00284s (tRS1 is the time of arrival of the first reflected signal) and, thus, the calculated LE = vps × tRS1/2 = 3.48 m with error = 0.57% (vps is the one-dimensional elastic shear wave velocity in a homogeneous pile and taken as 2450 m/s). The main reason for the error is the fact that the boundary of reflected signal is difficult to accurately determine in theory. When the time is approximately equal to two times tRS1, the second reflected signal (i.e., RS2) of the neck arrives at the pile top with the corresponding amplitude of the reflected signal decreasing markedly due to the energy dissipation in the propagating process. The reflected signal from the pile end (i.e., RSE) arrives at the pile top when t = tRSE = 0.00899s and, thus, H = vps × tRSE/2 = 11.01 m with error = 0.09%. It is also observed from Figure 7 that the phase of RS1 is the same as that of the incident pulse first and then opposite to that of the incident pulse. The phase of the RSE is opposite to that of the incident pulse for the fixed bottom boundary. Moreover, the amplitude of the RS1 increases markedly with the decrease of WRN. This result indicates that the amplitude of the reflected signal increases with increasing degree of defect.Figure 7
Influence of the neck width WRN on the normalized velocity response of the pile top (N = 3, h1 = 7 = m, LL = h2 = 0.5 m, and LE = h3 = 3.5 m).Figure8 depicts the influence of the bulb width (WRB) on the normalized velocity response of the pile top. It can be seen from Figure 8 that compared to the homogeneous pile, RS1 arrives at the pile top when t = tRS1 = 0.00284s and, thus, LE = vps × tRS1/2 = 3.48 m with error = 0.57%. When the time is approximately equal to two times tRS1, RS2 arrives at the pile top and the amplitude of RS2 decreases markedly because of the energy dissipation. RSE arrives at the pile top when t = tRSE = 0.00899s and, thus, H = vps × tRSE/2 = 11.01 m with error = 0.09%. It is also observed from Figure 8 that the phase of RS1 is the opposite to that of the incident pulse first and then the same with that of the incident pulse. However, the phase of RS2 is the same as that of the incident pulse. Moreover, the amplitude of RS1 increases markedly with the increase of WRB, which indicates that higher amplitude of the reflected signal is associated with larger value of the bulb width.Figure 8
Influence of the bulb widthWRB on the normalized velocity response of the pile top (N = 3, h1 = 7 m, LL = h2 = 0.5 m, and LE = h3 = 3.5 m).


## 4.3. Effect of Weak Concrete

Generally, if the pile has a weak concrete segment, the shear modulus of the pile will decrease greatly. Hence, we use the shear wave velocity to simulate the weak concrete in theory. The influence of the shear wave velocity in the weak concrete segment (vps2) on the normalized velocity response of the pile top is shown in Figure 9. The changing trend of the velocity response at the top end of a pile with a weak concrete segment is similar to that of the pile with a neck. When t = tRS1 = 0.00284s, RS1 arrives at the pile top and LE = vps × tRS1/2 = 3.48 m with error = 0.57%. When t is approximately equal to two times tRS1, RS2 arrives at the pile top. However, compared to the defect of neck, the amplitude of RS2 is small. The phase of RS1 is opposite to that of the incident pulse first and then the same as that of incident pulse. The amplitude of RS1 increases markedly with the decrease of vps2. Furthermore, the time of arrival of RSE increases obviously with the decrease of vps2. This result indicates that RSE will need much more time to arrive at the pile top as vps2 decreases. As a result, the estimated pile length will be longer than the actual length. For instance, when t = tRSE = 0.00918s, RSE of the pile with vps2 = 1600 m/s arrives at the pile top and the pile length can be estimated as H = vps × tRSE/2 = 11.25 m with error = 2.27%.Figure 9
Influence of shear wave velocity in weak concrete on the normalized velocity response of the pile top (N = 3, h1 = 7 m, LL = h2 = 0.5 m, and LE = h3 = 3.5 m).


## 4.4. Effect of Length of the Defect

Figure10 portrays the influence of the neck length on the normalized velocity response of the pile top. It can be seen from Figure 10 that the neck length has marked influence on the velocity response of the pile top. As the neck length increases from 0.5 m to 2.0 m, the width of RS1 and RS2 shows an obvious increase, and another reflected signal (ARS1) arrives at the pile top when t = 0.00448s. It should be mentioned again that there exist two abrupt changes in the cross-sectional dimension for a pile with a neck, one is at the top of the neck, and the other is at the bottom of the neck. When the neck length is long enough and the shear wave velocity is relatively slow, the reflected signal of these two abrupt changes does not overlap in the time domain. Furthermore, the embedded depth corresponding to the bottom of the neck is 5.5 m, and the corresponding calculated depth is 0.00448 × 2450/2 = 5.49 m with error = 0.18%.Figure 10
Influence of neck length on the normalized velocity response of the pile top (N = 3, WRN = 0.3 m, and LE = h3 = 3.5 m).Figure11 shows the influence of the blub length on the normalized velocity response of the pile top. It can be seen from Figure 11 that the blub length has marked influence on the velocity response of the pile top. The width of RS1 and RS2 also increases with the increase of blub length. For the pile with LL = 2.0 m, another obvious reflected signal (ARS1) arrives when t = tARS1 = 0.00448s, and the phase of ARS1 is the same as that of the incident impulse. It is worth noting that this signal is easy to be identified as a defect by mistake due to the similarity with the reflected signal of the defect.Figure 11
Influence of bulb length on the normalized velocity response of the pile top (N = 3, WRB = 0.45 m, and LE = h3 = 3.5 m).Figure12 shows the influence of the weak concrete length on the normalized velocity response of the pile top. It can be seen from Figure 12 that the influence of the weak concrete length on the velocity response is similar to that of the neck length. The width of RS1 and RS2 also increases with the increase of the weak concrete length, and there exists an individual ARS1 when LL = 2.0 m. Moreover, the time of arrival of RSE shows a marked increase with the increase of LL.Figure 12
Influence of weak concrete length on the normalized velocity response of the pile top (N = 3, vps2 = 1600 m/s, and LE = h3 = 3.5 m).


## 4.5. Influence of Soil Layering

In some cases, the soil properties may change greatly in certain embedment depth. In order to explicitly identify the influence of the variation of the surrounding soil properties, the pile is assumed to be homogeneous, and three cases of soil properties are investigated. Case 1: the soil has a stiff interlayer withGs2 = 5.52 × 107 Pa (i.e., Gs1/Gs2 = 0.25); Case 2: the soil is homogeneous with Gs2 = 1.38 × 107 Pa (i.e., Gs1/Gs2 = 1.0); and Case 3: the soil has a soft interlayer with Gs2 = 3.45 × 106 Pa (i.e., Gs1/Gs2 = 4). It can be seen from Figure 13 that the phase of the reflected signal of the interface of adjacent soil layers is, respectively, opposite to and same as that of the incident pulse for Case 1 and Case 3. However, when the soil is homogeneous, the velocity curve is smooth and the reflected signal of the soil interface does not emerge. It is also observed from Figure 13 that the shape of the reflected signal induced by the soil with a soft or stiff interlayer is in a half-sine form, which is obviously different from that of the defect. Moreover, the amplitude of the reflected signal induced by the soil interface is much lower than that of the defect.Figure 13
Influence of soil layering on the normalized velocity response of the pile (N = 3, h1 = 7 m, h2 = 0.5 m, and h3 = 3.5 m).


## 5. Conclusions

Based on Biot’s poroelastodynamic theory, we derive a semianalytical solution for the dynamic response of an inhomogeneous elastic pile embedded in a multilayered saturated soil and subjected to a transient torsional load. The time-harmonic torsional vibration solution of the pile-soil system is derived first in the frequency domain. Then, the time domain solution corresponding to the pile under transient torsional load is gained by virtue of inverse Fourier transform and convolution theorem. After validating the accuracy and reliability of the proposed reaction coefficient, selected numerical results are gained to investigate the influence of the typical defects in pile and soil layering on the velocity response in the time domain. The main conclusions are as follows:(1)
For the end-bearing pile, no matter it is homogeneous or inhomogeneous, the phase of the reflected signal from the pile end is opposite to that of the incident pulse.(2)
For a pile with a neck, the phase of the first and second reflected signals of the defect is the same as that of the incident pulse. The amplitude of reflected signal increases greatly with the decrease of the neck width. The width of reflected signal increases with the increase of the neck length.(3)
For a pile with a bulb, the phase of the first reflected signal is opposite to that of the incident pulse. However, the phase of the second reflected signal is the same as that of the incident pulse. The amplitude of the reflected signal increases greatly with the increase of bulb width.(4)
The characteristic of the reflected signal of the pile with a weak concrete segment is similar to that of the pile with a neck. Furthermore, as the length of the defect (neck, bulb, or weak concrete segment) is long enough, the reflected signal of the upper and lower interfaces of the defect will not overlap in the time domain.(5)
The abrupt change in adjacent soil layer properties has certain influence on the velocity response of the pile top. The shape of the reflected signal of the soil with a soft or stiff interlayer is in a half-sine form, and the amplitude of the reflected signal of soil interface is much lower than that of the defect.



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*Source: 5528237-2021-04-12.xml*