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

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 into Nth 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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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 Figure 1, 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 h_j 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 k_stj and k_sbj, which represent the contact stress (τ_θz) 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 omitted; refer to PDF]

The dynamic equilibrium equation of the jth (1 ≤ j ≤ N) saturated soil subjected to torsional load can be expressed in terms of a cylindrical coordinate system as [26]$$\begin{array}{}\text{(1)}& G_{sj}{\nabla }^2{u}_{\theta j}r,z,t-\frac{u_{\theta j}r,z,t}{r^2}{G}_{sj}={\rho }_j\frac{{\partial }^2{u}_{\theta j}r,z,t}{\partial t^2}+{\rho }_{fj}\frac{{\partial }^2{w}_{\theta j}r,z,t}{\partial t^2},\end{array}$$where u_θj(r, z, t) and $w$_θj(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; ρ_j = (1 − n_j)ρ_sj + n_jρ_fj is the bulk density of the jth soil layer; for the jth soil layer, n_j is the porosity of the soil; ρ_fj is the density of the pore fluid; G_sj and ρ_sj are the shear modulus and density of the soil skeleton, respectively; and ${\nabla }^2={\partial }^2/\partial r^2+1/r\partial /\partial r+{\partial }^2/\partial z^2$.

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$$\begin{array}{}\text{(2)}& \frac{{\rho }_{fj}g}{k_{dj}}\frac{\partial w_{\theta j}r,z,t}{\partial t}+{\rho }_{fj}\frac{{\partial }^2{u}_{\theta j}r,z,t}{\partial t^2}+\frac{{\rho }_{fj}}{n_j}\frac{{\partial }^2{w}_{\theta j}r,z,t}{\partial t^2}=0,\end{array}$$where k_dj 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, u_θj(r, z, t) = u_θj(r, z)e^iωt and $w$_θj(r, z, t) = $w$_θj(r, z)e^iωt, ω is the circular frequency of excitation, and t is the time variable. Equations (1) and (2) can, therefore, be expressed as$$\begin{array}{}\text{(3)}& G_{sj}{\nabla }^2{u}_{\theta j}r,z-\frac{u_{\theta j}r,z}{r^2}{G}_{sj}=-{\rho }_j{\omega }^2{u}_{\theta j}r,z-{\rho }_{fj}{\omega }^2{w}_{\theta j}r,z,\text{(4)}& \frac{i{\rho }_{fj}g}{k_{dj}}{w}_{\theta j}r,z-{\rho }_{fj}\omega u_{\theta j}r,z-\frac{{\rho }_{fj}\omega w_{\theta j}r,z}{n_j}=0.\end{array}$$

Substituting equation (4) into equation (3), the governing equation of the jth saturated soil layer can be written as$$\begin{array}{}\text{(5)}& \frac{{\partial }^2{u}_{\theta j}r,z}{\partial r^2}+\frac{1}{r}\frac{\partial u_{\theta j}r,z}{\partial r}+\frac{{\partial }^2{u}_{\theta j}r,z}{\partial z^2}-\frac{u_{\theta j}r,z}{r^2}=-\frac{{\omega }^2}{G_{sj}}{\rho }_j+\frac{n_j\omega {\rho }_{fj}}{i{b}_j/{\rho }_{fj}-\omega }{u}_{\theta j}r,z.\end{array}$$

The one-dimensional equation of motion of the jth pile segment subjected to a time-harmonic torsional load can be expressed as$$\begin{array}{}\text{(6)}& G_{pj}\frac{{\partial }^2{\varphi }_{j}z{e}^{i\omega t}}{\partial z^2}+4\frac{f_{j}z{e}^{i\omega t}}{r_j^2}={\rho }_{pj}\frac{{\partial }^2{\varphi }_{j}z{e}^{i\omega t}}{\partial t^2},\end{array}$$where G_pj, ρ_pj, r_j, and ϕ_j(z) are the shear modulus, density, radius, and the twist angle amplitude of the jth pile segment, respectively; f_j(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 the jth soil layer can be written in terms of the local coordinate system as$$\begin{array}{}\text{(7)}& u_{\theta j}r⟶\infty ,z=0,\hspace{1em}1\le j\le N,\text{(8)}& {\frac{\partial u_{\theta j}r,z}{\partial z}-\frac{k_{stj}}{G_{sj}}{u}_{\theta j}r,z|}_{z=0}=0,\hspace{1em}1\le j\le N,\text{(9)}& {u_{\theta j}r,z|}_{z=h_j}=0,\hspace{1em}j=1,\text{(10)}& {\frac{\partial u_{\theta j}r,z}{\partial z}+\frac{k_{sbj}}{G_{sj}}{u}_{\theta j}r,z|}_{z=h_j}=0,\hspace{1em}2\le j\le N,\end{array}$$where k_stj and k_sbj are the distributed reaction coefficients at the top and bottom of the jth soil layer, respectively. It is noted that k_stN is equal to zero due to the free surface of the soil.

The continuity condition of the interface between the jth and (j + 1)th (1 ≤ j ≤ N − 1) soil layers can be written as$$\begin{array}{}\text{(11)}& k_{stj}=k_{sbj+1}.\end{array}$$

The boundary conditions at the base and top of the jth pile segment can be expressed in terms of the local coordinate system as$$\begin{array}{}\text{(12)}& {\frac{\text{d}{\varphi }_{j}z}{\text{d}z}|}_{z=0}=-\frac{T_j}{G_{pj}{I}_{pj}},\hspace{1em}1\le j\le N,\text{(13)}& {{\varphi }_{j}z|}_{z=h_j}=0,\hspace{1em}j=1,\text{(14)}& {\frac{\text{d}{\varphi }_{j}z}{\text{d}z}+\frac{{\varphi }_{j}z{k}_{pbj}}{G_{pj}{I}_{pj}}|}_{z=h_j}=0,\hspace{1em}2\le j\le N,\end{array}$$where T_j 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, T_N denotes the torque amplitude applied at the pile top; I_pj is the polar moment of inertia of the jth pile segment; and k_pbj is the reaction coefficient at the bottom of the jth pile segment.

At the interface between the jth and (j + 1)th pile segments (z = H_j), the circumferential displacement and torque of the pile satisfy the following continuity conditions (see Figure 3):$$\begin{array}{}\text{(15)}& \begin{array}{c}{\varphi }_{j}z={\varphi }_{j+1}z,\hspace{1em}j=\mathrm{1,2},\dots N,\\ G_{pj}{I}_{pj}\frac{d{\varphi }_{j}z}{dz}=G_{pj+1}{I}_{pj+1}\frac{d{\varphi }_{j+1}z}{dz}.\end{array}\end{array}$$

[figures omitted; refer to PDF]

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 the jth 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$$\begin{array}{}\text{(16)}& {k_{Tj}=\frac{T_{j}z}{{\varphi }_{j}z}|}_{z=H_j}=k_{pbj+1}.\end{array}$$

The continuity conditions of the displacement and stress of the pile-soil interface can expressed as$$\begin{array}{}\text{(17)}& u_{\theta j}{r}_j,z={\varphi }_{j}z{r}_j,\text{(18)}& {f_{j}z={\tau }_{r\theta j}r,z|}_{r=r_j}=G_{sj}{\frac{\partial u_{\theta j}r,z}{\partial r}-\frac{u_{\theta j}r,z}{r}|}_{r=r_j}.\end{array}$$

3. Solution of the Pile-Soil System

3.1. Solution of the Soil Layer

Substituting a single-variable function u_θj(r, z) = R_j(r)Z_j(z) into equation (5), we have$$\begin{array}{c}\text{(19)}& \frac{{\text{d}}^2{Z}_{j}z}{\text{d}{z}^2}+J_j^2{Z}_{j}z=0,r^2\frac{{\text{d}}^2{R}_{j}r}{\text{d}{r}^2}+r\frac{\text{d}{R}_{j}r}{\text{d}r}-1+q_j^2{r}^2{R}_{j}r=0,\end{array}$$where$$\begin{array}{}\text{(20)}& q_j^2=J_j^2-\frac{{\omega }^2}{G_{sj}}{\rho }_j+\frac{n_j{\rho }_{fj}\omega }{i{b}_j/{\rho }_{fj}-\omega }.\end{array}$$

The general solutions of equation (20) can be expressed as$$\begin{array}{c}\text{(21)}& Z_{j}z=C_j\sin \begin{array}{c}{J}_{j}z\end{array}+D_j\cos J_{j}z,R_{j}r=A_j{K}_1{q}_{j}r+B_j{I}_1{q}_{j}r,\end{array}$$where I₁(q_jr) and K₁(q_jr) are the modified Bessel functions of the first and second kind of first order, respectively; A_j, B_j, C_j, and D_j are the constants to be determined by the boundary conditions.

To satisfy the boundary condition given in equation (7), the constant B_j = 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$$\begin{array}{}\text{(22)}& \tan h_j{J}_j=\frac{-h_j{J}_j}{{\overline{k}}_{stj}},\hspace{1em}j=1,\end{array}$$where ${\overline{k}}_{stj}=k_{stj}{h}_j/G_{sj}$ denotes the dimensionless reaction coefficient at the top of the jth soil layer.

For the jth soil layer (2 ≤ j ≤ N), substituting the boundary conditions given in equations (8) and (10) into equation (21) yields$$\begin{array}{}\text{(23)}& \tan h_j{J}_j=\frac{{\overline{k}}_{sbj}+{\overline{k}}_{stj}{h}_j{J}_j}{{h_j{J}_j}^2-{\overline{k}}_{sbj}{\overline{k}}_{stj}},\hspace{1em}2\le j\le N,\end{array}$$where ${\overline{k}}_{sbj}=k_{sbj}{h}_j/G_{sj}$ denotes the dimensionless reaction coefficients at the base of the jth soil layer. It is noted that J_j 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$$\begin{array}{}\text{(24)}& u_{\theta j}r,z={\displaystyle \sum _{m=1}^{\infty }{A}_{mj}{K}_1{q}_{mj}r\sin J_{mj}z+{\phi }_{mj}},\end{array}$$where$$\begin{array}{}\text{(25)}& {\phi }_{mj}=\arctan \frac{J_{mj}{h}_j}{{\overline{k}}_{stj}};\hspace{1em}{q}_{mj}^2=J_{mj}^2-\frac{{\rho }_{sj}{\omega }^2}{G_{sj}}.\end{array}$$

The shear stress amplitude τ_rθj corresponding to equation (24) can be expressed as$$\begin{array}{}\text{(26)}& {\tau }_{r\theta j}=G_{sj}\frac{\partial u_{\theta j}r,z}{\partial r}-\frac{u_{\theta j}r,z}{r}=-G_{sj}{\displaystyle \sum _{m=1}^{\infty }{D}_{mj}{q}_{mj}{K}_2{q}_{mj}r\sin J_{mj}z+{\phi }_{mj}},\end{array}$$where K₂(q_mjr) denotes the modified Bessel functions of the second kind of the second order.

3.2. Solution of the Pile Segment

For the jth (1 ≤ j ≤ N) pile segment, by utilizing the stress continuity condition that is given in equation (18), substituting equation (26) into equation (6) yields$$\begin{array}{}\text{(27)}& {\varphi }_{j}z={\alpha }_{1j}\cos {\lambda }_{j}z+{\alpha }_{2j}\sin {\lambda }_{j}z+{\displaystyle \sum _{m=1}^{\infty }{\psi }_{mj}\sin J_{mj}z+{\phi }_{mj}},\end{array}$$in which$$\begin{array}{}\text{(28)}& {\psi }_{mj}=-\frac{4G_{sj}{q}_{mj}{K}_2{q}_{mj}{r}_j{A}_{mj}}{G_{pj}{r}_j^2{J}_{mj}^2-{\lambda }_j^2};\hspace{1em}{\lambda }_j=\sqrt{\frac{{\rho }_{pj}}{G_{pj}}}\omega ,\end{array}$$where α_1j and α_2j are the undetermined coefficients.

By using the displacement continuity condition given in equation (17), substituting equations (24) and (27) into equation (17) results in$$\begin{array}{}\text{(29)}& r_j{\alpha }_{1j}\cos {\lambda }_{j}z+{\alpha }_{2j}\sin {\lambda }_{j}z+{\displaystyle \sum _{m=1}^{\infty }{\psi }_{mj}\sin J_{mj}z+{\phi }_{mj}}={\displaystyle \sum _{m=1}^{\infty }{A}_{mj}{K}_1{q}_{mj}r\sin J_{mj}z+{\phi }_{mj}}.\end{array}$$

By invoking the orthogonality of eigenfunctions sin(J_mjz + φ_mj) (m = 1, 2, 3, …), multiplying sin(J_mjz + φ_mj) on both sides of equation (29), and then, integrating over the interval z = [0, h_j], the undetermined coefficient A_mj is found to be$$\begin{array}{}\text{(30)}& A_{mj}=\frac{1}{L_{mj}{E}_{mj}}{\displaystyle {\int }_0^{h_j}{P}_j\sin J_{mj}z+{\phi }_{mj}\text{d}z},\end{array}$$where$$\begin{array}{c}\text{(31)}& L_{mj}={\displaystyle {\int }_0^{h_j}{\sin }^2{J}_{mj}z+{\phi }_{mj}\text{d}z};\hspace{1em}{P}_j={\alpha }_{1j}\cos {\lambda }_{j}z+{\alpha }_{2j}\sin {\lambda }_{j}z;E_{mj}=\frac{1}{r_j}{K}_1{q}_{mj}{r}_j+\frac{4G_{sj}{q}_{mj}{K}_2{q}_{mj}{r}_j}{G_{pj}{r}_j{J}_{mj}^2-{\lambda }_j^2}.\end{array}$$

The amplitude of the twist angle of the jth pile segment is then given by$$\begin{array}{}\text{(32)}& {\varphi }_{j}z={\alpha }_{1j}\cos {\lambda }_{j}z+{\displaystyle \sum _{m=1}^{\infty }{\xi }_{mj}\sin J_{mj}z+{\phi }_{mj}}+{\alpha }_{2j}\sin {\lambda }_{j}z+{\displaystyle \sum _{m=1}^{\infty }{\zeta }_{mj}\sin J_{mj}z+{\phi }_{mj}},\end{array}$$where$$\begin{array}{c}\text{(33)}& {\overline{\lambda }}_j={\lambda }_j{h}_j;{\overline{J}}_{mj}=h_j{J}_{mj};\\ {\overline{q}}_{mj}=h_j{q}_{mj};\\ {\overline{r}}_j=\frac{r_j}{h_j};\\ {\overline{\mu }}_j=\frac{G_{pj}}{G_{sj}};\\ {\overline{E}}_{mj}=r_j{E}_{mj};\\ {\overline{L}}_{mj}=\frac{L_{mj}}{h_j};\\ {\nu }_{mj}=\frac{-2{\overline{q}}_{mj}{K}_2{\overline{q}}_{mj}{\overline{r}}_j}{{\overline{r}}_j{\overline{\mu }}_j{\overline{J}}_{mj}^2-{\overline{\lambda }}_j^2{\overline{E}}_{mj}{\overline{L}}_{mj}};\\ {\xi }_{mj}={\nu }_{mj}\frac{\cos {\phi }_{mj}-\cos {\overline{J}}_{mj}+{\overline{\lambda }}_j+{\phi }_{mj}}{{\overline{J}}_{mj}+{\overline{\lambda }}_j}+\frac{\cos {\phi }_{mj}-\cos {\overline{J}}_{mj}-{\overline{\lambda }}_j+{\phi }_{mj}}{{\overline{J}}_{mj}-{\overline{\lambda }}_j};\\ {\zeta }_{mj}={\nu }_{mj}\frac{\sin {\phi }_{mj}-\sin {\overline{J}}_{mj}+{\overline{\lambda }}_j+{\phi }_{mj}}{{\overline{J}}_{mj}+{\overline{\lambda }}_j}+\frac{\sin {\phi }_{mj}-\sin {\overline{J}}_{mj}-{\overline{\lambda }}_j+{\phi }_{mj}}{{\overline{\lambda }}_j-{\overline{J}}_{mj}}.\end{array}$$

Based on the boundary conditions of the jth (1 ≤ j ≤ N) pile segment, substituting equations (12)–(14) into equation (32), the variables α_1j and α_2j 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$$\begin{array}{}\text{(34)}& k_{Tj}=\frac{T_j}{{\varphi }_{j}z=0}=\frac{G_{pj}{I}_{pj}}{r_j}-{\overline{r}}_j\frac{{\alpha }_{1j}/{\alpha }_{2j}{\displaystyle {\sum }_{m=1}^{\infty }{\xi }_{mj}{\overline{J}}_{mj}\cos {\phi }_{mj}}+{\overline{\lambda }}_j+{\displaystyle {\sum }_{m=1}^{\infty }{\zeta }_{mj}{\overline{J}}_{mj}\cos {\phi }_{mj}}}{{\alpha }_{1j}/{\alpha }_{2j}1+{\displaystyle {\sum }_{m=1}^{\infty }{\xi }_{mj}\sin {\phi }_{mj}}+{\displaystyle {\sum }_{m=1}^{\infty }{\zeta }_{mj}\sin {\phi }_{mj}}},\end{array}$$where$$\begin{array}{c}\text{(35)}& \frac{{\alpha }_{1j}}{{\alpha }_{2j}}=-\frac{\sin {\overline{\lambda }}_1+{\displaystyle {\sum }_{m=1}^{\infty }{\zeta }_{m1}\sin {\overline{J}}_{m1}+{\phi }_{m1}}}{\cos {\overline{\lambda }}_1+{\displaystyle {\sum }_{m=1}^{\infty }{\xi }_{m1}\sin {\overline{J}}_{m1}+{\phi }_{m1}}},\hspace{1em}j=1,\frac{{\alpha }_{1j}}{{\alpha }_{2j}}=-\frac{{\overline{\lambda }}_j\cos {\overline{\lambda }}_j+{\displaystyle {\sum }_{m=1}^{\infty }{\zeta }_{mj}{\overline{J}}_{mj}\cos {\overline{J}}_{mj}+{\phi }_{mj}+h_j{k}_{Tj-1}/G_{pj}{I}_{pj}\sin {\overline{\lambda }}_j+{\displaystyle {\sum }_{m=1}^{\infty }{\zeta }_{mj}\sin {\overline{J}}_{mj}+{\phi }_{mj}}}}{-{\overline{\lambda }}_j\sin {\overline{\lambda }}_j+{\displaystyle {\sum }_{m=1}^{\infty }{\xi }_{mj}{\overline{J}}_{mj}\cos {\overline{J}}_{mj}+{\phi }_{mj}+h_j{k}_{Tj-1}/G_{pj}{I}_{pj}\cos {\overline{\lambda }}_j+{\displaystyle {\sum }_{m=1}^{\infty }{\xi }_{mj}\sin {\overline{J}}_{mj}+{\phi }_{mj}}}},\hspace{1em}2\le j\le N.\end{array}$$

Through further recursion from the 1st pile segment to the Nth pile segment, the torsional impedance function at the pile head can be expressed as$$\begin{array}{}\text{(36)}& k_T=k_{TN}=\frac{T_N}{{\varphi }_{N}z=0}=\frac{G_{pN}{I}_{pN}}{r_N}-{\overline{r}}_N\frac{{\alpha }_{1N}/{\alpha }_{2N}{\displaystyle {\sum }_{m=1}^{\infty }{\xi }_{mN}{\overline{J}}_{mN}\cos {\phi }_{mN}}+{\overline{\lambda }}_N+{\displaystyle {\sum }_{m=1}^{\infty }{\zeta }_{mN}{\overline{J}}_{mN}\cos {\phi }_{mN}}}{{\alpha }_{1N}/{\alpha }_{2N}1+{\displaystyle {\sum }_{m=1}^{\infty }{\xi }_{mN}\sin {\phi }_{mN}}+{\displaystyle {\sum }_{m=1}^{\infty }{\zeta }_{mN}\sin {\phi }_{mN}}}.\end{array}$$

Based on the definition in [26], the dimensionless torsional impedance at the pile top can be expressed as$$\begin{array}{}\text{(37)}& k_T^{\prime }=\frac{3T_N}{16G_{sN}{r}_N^3\varphi z=0}=-\frac{3\pi {\overline{\mu }}_N{\overline{r}}_{{}_N}R}{32},\end{array}$$where$$\begin{array}{}\text{(38)}& R=\frac{{\alpha }_{1N}/{\alpha }_{2N}{\displaystyle {\sum }_{m=1}^{\infty }{\xi }_{mN}{\overline{J}}_{mN}\cos {\phi }_{mN}}+{\overline{\lambda }}_N+{\displaystyle {\sum }_{m=1}^{\infty }{\zeta }_{mN}{\overline{J}}_{mN}\cos {\phi }_{mN}}}{{\alpha }_{1N}/{\alpha }_{2N}1+{\displaystyle {\sum }_{m=1}^{\infty }{\xi }_{mN}\sin {\phi }_{mN}}+{\displaystyle {\sum }_{m=1}^{\infty }{\zeta }_{mN}\sin {\phi }_{mN}}}.\end{array}$$

The frequency response function of the twist angle of the pile top can be written as$$\begin{array}{}\text{(39)}& H_{\theta }\omega =\frac{1}{k_T}=\frac{-r_N}{G_{pN}{I}_{pN}}\frac{1}{{\overline{r}}_{{}_N}}\frac{{\alpha }_{1N}/{\alpha }_{2N}1+{\displaystyle {\sum }_{m=1}^{\infty }{\xi }_{mN}\sin {\phi }_{mN}}+{\displaystyle {\sum }_{m=1}^{\infty }{\zeta }_{mN}\sin {\phi }_{mN}}}{{\alpha }_{1N}/{\alpha }_{2N}{\displaystyle {\sum }_{m=1}^{\infty }{\xi }_{mN}{\overline{J}}_{mN}\cos {\phi }_{mN}}+{\overline{\lambda }}_N+{\displaystyle {\sum }_{m=1}^{\infty }{\zeta }_{mN}{\overline{J}}_{mN}\cos {\phi }_{mN}}}.\end{array}$$

Then, the admittance function of angular velocity of the arbitrary point at the pile top can be further expressed as$$\begin{array}{}\text{(40)}& H_v\omega =i\omega r_s{H}_{\theta }\omega =\frac{r_s}{v_{psN}{I}_{pN}{\rho }_{pN}}{H}_v^{\prime }\omega ,\end{array}$$where r_s 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$$\begin{array}{}\text{(41)}& H_v^{\prime }\omega =-i{\overline{\lambda }}_N\frac{{\alpha }_{1N}/{\alpha }_{2N}1+{\displaystyle {\sum }_{m=1}^{\infty }{\xi }_{mN}\sin {\phi }_{mN}}+{\displaystyle {\sum }_{m=1}^{\infty }{\zeta }_{mN}\sin {\phi }_{mN}}}{{\alpha }_{1N}/{\alpha }_{2N}{\displaystyle {\sum }_{m=1}^{\infty }{\xi }_{mN}{\overline{J}}_{mN}\cos {\phi }_{mN}}+{\overline{\lambda }}_N+{\displaystyle {\sum }_{m=1}^{\infty }{\zeta }_{mN}{\overline{J}}_{mN}\cos {\phi }_{mN}}}.\end{array}$$

By applying the inverse Fourier transform into equation (40), the response function of unit pulse torque in the time domain can be written as$$\begin{array}{}\text{(42)}& ht=IFT{H}_v\omega =\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{+\infty }\frac{r_s}{v_{psN}{I}_{pN}{\rho }_{pN}}}{H}_v^{\prime }\omega e^{i\tilde{\omega }\overline{t}}\text{d}\tilde{\omega },\end{array}$$where $\tilde{\omega }=\omega T_c$ and $T_c={\displaystyle {\sum }_{j=1}^N{h}_j/v_{psj}}$ denote the nondimensional frequency and propagation time of elastic shear wave propagating from the pile top to pile tip, respectively; $\overline{t}=t/T_c$ is the nondimensional time variable.

If the Fourier transform of the arbitrary exciting torque T(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)$H_v$(ω)]. In particular, when the exciting torque is a half-sine pulse, we have$$\begin{array}{}\text{(43)}& Tt=\begin{array}{cc}{T}_{\max }\sin \frac{\pi t}{t_0},& t<t_0,\\ 0,& t\ge t_0,\end{array}\end{array}$$where t₀ and T_max 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$$\begin{array}{}\text{(44)}& Vt=T_{\max }IFT\frac{r_s}{v_{psN}{I}_{pN}{\rho }_{pN}}{H}_v^{\prime }\omega \frac{\pi t_0}{{\pi }^2-t_0^2{\omega }^2}1+e^{-i\omega t_0}.\end{array}$$

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$$\begin{array}{}\text{(45)}& V^{\prime }t=\frac{Vt}{\max Vt},\end{array}$$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 stress τ_θz in the depth direction [1, 31], thus resulting in inevitable error in the dynamic response. However, when considering the effect of the shear stress τ_θz, 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 ρ_pj = 2300 kg/m³, $v_{psj}$ = 2450 m/s, G_pj = 1.38 × 10¹⁰ Pa, E_pj = 3.31 × 10¹⁰ Pa, r_j = r₀ = 0.4 m (j = 1, 2, …, N), and the Poisson’s ratio is 0.2; the soil properties are ρ_sj = 2650 kg/m³, ρ_fj = 1000 kg/m³, n_j = 0.4, and G_sj = 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 ${\overline{k}}_{stj}$ (the distributed reaction coefficient at the top of the jth soil layer) is within the range from 0.001 to 0.1. When ${\overline{k}}_{\text{st}j}=10$, 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 ${\overline{k}}_{stj}$ has a negligible influence on the torsional impedance in the higher frequency range.

[figures omitted; refer to PDF]

Figure 5 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 ${\overline{k}}_{stj}$ 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 ${\overline{k}}_{stj}$ has a negligible influence on the dynamic response in the high frequency range. Consequently, the influence of ${\overline{k}}_{stj}$ on the velocity response is negligible, and ${\overline{k}}_{stj}$ is fixed at 0.01 in the following analysis.

[figure omitted; refer to PDF]

Figure 7 shows influence of the neck width (W_RN) 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 = t_RS1 = 0.00284s (t_RS1 is the time of arrival of the first reflected signal) and, thus, the calculated L_E = $v_{ps}$ × t_RS1/2 = 3.48 m with error = 0.57% ($v_{ps}$ 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 t_RS1, 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 = t_RSE = 0.00899s and, thus, H = $v_{ps}$ × t_RSE/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 W_RN. This result indicates that the amplitude of the reflected signal increases with increasing degree of defect.

[figure omitted; refer to PDF]

Figure 8 depicts the influence of the bulb width (W_RB) 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 = t_RS1 = 0.00284s and, thus, L_E = $v_{ps}$ × t_RS1/2 = 3.48 m with error = 0.57%. When the time is approximately equal to two times t_RS1, 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 = t_RSE = 0.00899s and, thus, H = $v_{ps}$ × t_RSE/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 W_RB, which indicates that higher amplitude of the reflected signal is associated with larger value of the bulb width.

[figure omitted; refer to PDF]

Figure 11 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 L_L = 2.0 m, another obvious reflected signal (ARS1) arrives when t = t_ARS1 = 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 omitted; refer to PDF]

Figure 12 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 L_L = 2.0 m. Moreover, the time of arrival of RSE shows a marked increase with the increase of L_L.

[figure omitted; refer to PDF]

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 with G_s2 = 5.52 × 10⁷ Pa (i.e., G_s1/G_s2 = 0.25); Case 2: the soil is homogeneous with G_s2 = 1.38 × 10⁷ Pa (i.e., G_s1/G_s2 = 1.0); and Case 3: the soil has a soft interlayer with G_s2 = 3.45 × 10⁶ Pa (i.e., G_s1/G_s2 = 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 omitted; refer to PDF]

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.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant nos. 51878619 and 52078465).

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