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Abstract

A new mathematical model of a floating pile embedded in a viscoelastic saturated soil for low-strain integrity detection is proposed using the theory of porous media. The saturated surrounding soil is divided into Novak’s thin layers along the pile length and a homogeneous half-space underlying the pile base to consider the effect of excitation frequency on the dynamic stiffness of the soil beneath the pile base. And the corresponding analytical solutions for the dynamic impedance and the reflected wave signal of vertical velocity at the pile head are derived and verified by comparing its reduced solution with the existing solutions for the end-bearing pile. In addition, an extensive parametric analysis is further conducted to investigate the effects of the slenderness ratio of pile, the modulus ratio of pile to the surrounding soil, and the permeability coefficient of saturated soil on the vertical vibration characteristics of the pile–soil interaction system.

Keywords

Integrity detection reflected wave signal dynamic impedance pile–soil interaction vertical vibration porous medium

Introduction

Problems related to the vertical steady-state vibration of floating pile in the soil layer have attracted increasing attention due to their significance in the field of civil engineering and solid mechanics.^1–4 For the vibration problem of the pile–soil system, it is generally important to determine the dynamic impedance and the velocity response subjected to harmonic load at the pile head which can provide theoretical reference to the integrity detection of pile in practice.

In the dynamic system of pile–soil interaction, various mathematical models have been developed by investigators.⁵ The Winkler model is extensively employed due to its simplicity in which soil layers are represented by a series of springs and dashpots.^6–9 However, the Winker model is limited in describing the mechanism of wave propagation within the pile–soil system.^10,11 Novak et al.^12–14 presented the plane-strain model for the pile–soil interaction system to simplify the surrounding soil as a linear viscoelastic layer with the hysteretic-type damping. Manna and Baidya¹⁵ investigated the possible reasons for the unsatisfactory performance of the Novak model. Furthermore, various axisymmetric continuum models were also proposed by researchers, in which the three-dimensional (3D) wave effect of energy propagation can be considered.^16–18

The surrounding soil of pile in the aforementioned studies of the pile–soil system is assumed as a single-phase medium. However, soil is generally a multiphase medium that can be modeled as a liquid-filled porous medium. Biot^19,20 developed a theory to describe the deformation and wave propagation in a fluid-saturated porous elastic medium. Based on Biot’s theory for porous elastic solid, Rajapakse and Senjuntichai²¹ explicitly derived rigorous analytical solutions for the vertical vibration of pile which describes the relationship between the generalized displacement and the force of multilayered porous media in the Fourier frequency space. Zhou et al.²² investigated the dynamic response of a pile embedded in a saturated half-space subjected to transient vertical loading by adopting Biot’s porous elastodynamic equations. Li et al.²³ used Biot’s elastodynamic theory as the basis to present an analytical solution for the vertical vibration of an end-bearing pile in a saturated soil layer. Subsequently, Zheng et al.²⁴ presented an analytical method to study the vertical vibration of a floating pile embedded in porous elastic soil using Biot’s theory, in which the saturated elastic soil is divided into two independent parts: a half-space underlying the pile base and a series of Novak’s thin layers along the pile shaft. However, some soil parameters defined in Biot’s theory^18,19 are not clear enough and difficult to determine, which leads to inconvenience for its application in engineering, to some extent.

Moreover, Bowen²⁵ proposed the theory of porous media (TPM) by integrating the continuum theory of mixtures with the concept of volume fractions. In contrast to Biot’s theory, the TPM is also proven to provide a comprehensive and extensive modeling framework.^26,27 Substantial developments with respect to the TPM were extended to geomechanical problems and were contributed by the pioneering studies of de Boer et al.^28–30 In the subject of pile–soil dynamic interaction, Liu and Yang³¹ extended the governing equations of the TPM to obtain analytical solutions that describes the vertical vibration of an end-bearing pile in saturated poroelastic soil. In addition, Yang and Pan³² and Zhang et al.³³ deduced the frequency solutions for the vertical vibration of an end-bearing pile in the saturated porous viscoelastic soil using the variable separation method.

Based on an extensive review of the literature, it can be seen that little work has been conducted on the dynamic response of a floating pile in a viscoelastic saturated soil using the TPM. The main purpose of this article is to propose a new mathematical model for both velocity response and dynamic impedance of a floating pile embedded in a viscoelastic saturated soil which can provide theoretical reference to the low-strain integrity detection of pile in engineering practice. The saturated surrounding soil is divided into Novak’s thin layers along the pile length and a homogeneous half-space underlying the pile base to consider the effect of excitation frequency on the dynamic stiffness of the soil beneath the pile base. And the corresponding analytical solutions for the dynamic impedance and the velocity response at the pile head are derived and verified by comparing its reduced solution with the existing solutions for the end-bearing pile. In addition, an extensive parametric analysis further needs to be conducted to investigate the effects of the slenderness ratio of pile, the modulus ratio of the pile to the surrounding soil, and the permeability coefficient of saturated soil on the vertical vibration characteristics of the pile–soil interaction system.

Statement of problem and basic assumption

The problem under consideration is shown in Figure 1. An elastic solid pile is embedded in a liquid-saturated porous viscoelastic half-space subjected to a vertical time-harmonic load $p (t) = P e^{i ω t}$ at the pile head (where $i = \sqrt{- 1}$ and $ω$ is the angular frequency). $f_{0}$ and R denote the soil reactions against the pile shaft and the pile base, respectively. The mechanical parameters of the pile–soil system are listed in Table 1.

Figure 1.

Conceptual model of a floating pile embedded in the saturated porous viscoelastic half-space.

Table 1.

Variables and symbols of the material parameters.

G	Shear modulus of soil skeleton	$n^{S} = 1 - n^{L}$	Volume fraction of soil skeleton
$υ$	Possion’s ratio of soil skeleton	$k^{L}$	Darcy’s permeability coefficient of saturated soil
$ξ$	Viscous damping ratio of soil	$γ^{LR}$	Specific weight of liquid
$μ^{S} = G (1 + i ξ)$	Lame constant of soil skeleton	$E_{p}$	Elastic modulus of pile
$λ^{S} = 2 υ μ^{S} / (1 - 2 υ)$	Lame constant of soil skeleton	$ρ_{p}$	Density of pile
$ρ^{S}$	Soil density	$A_{p}$	Cross-sectional area of pile
$ρ^{L}$	Liquid density	L	Length of pile
$n^{L}$	Volume fraction of pore liquid	$r_{0}$	Radius of pile

It is assumed that the equilibrium of shear stress and the continuity of displacement are both satisfied at the interfaces; the deformation of the soil–pile system is infinitesimal; the soil is divided into two parts along the pile length: the upper part consists of a series of independent Novak’s thin layers, while the lower part is simplified as a porous viscoelastic half-space.

Formulation of the governing equations

On the basis of the TPM proposed by Bowen,²⁵ the 3D dynamic governing equations for the saturated porous viscoelastic soil layer in axisymmetric cylindrical coordinates can be formulated as

$(λ^{S} + μ^{S}) graddiv U_{S} + μ^{S} divgrad U_{S} - n^{S} grad p - ρ^{S} {\overset{..}{U}}_{S} + S_{v} ({\dot{U}}_{L} - {\dot{U}}_{S}) = 0$ (1a)

$- n^{L} grad p - ρ^{L} {\overset{\cdot\cdot}{U}}_{L} - S_{v} ({\overset{\cdot}{U}}_{L} - {\overset{\cdot}{U}}_{S}) = 0$ (1b)

$div (n^{S} {\overset{\cdot}{U}}_{S} + n^{L} {\overset{\cdot}{U}}_{L}) = 0$ (1c)

where $U_{S}$ are $U_{L}$ are the displacement vectors for the soil skeleton and pore water, respectively; p is the pore pressure of the incompressible pore fluid; $S_{v} = (n^{L})^{2} γ^{LR} / k^{L}$ , in equations (1a) and (1b), denotes the coupled interaction between the soil skeleton and the pore liquid; and the symbols “grad” and “div” represent the gradient and divergence operator, respectively.

As shown in Figure 1, the interaction system of a floating pile embedded in the saturated porous viscoelastic half-space is subjected to an axial time-harmonic load $P e^{i ω t}$ . Thus, all field variables are time harmonic with the term $e^{i ω t}$ , that is

$\begin{matrix} u_{S} = U_{S} e^{i ω t}, u_{L} = U_{L} e^{i ω t}, w_{S} = W_{S} e^{i ω t} \\ w_{L} = W_{L} e^{i ω t}, w_{P} = W_{P} e^{i ω t} \end{matrix}$ (2)

where $u_{S}$ and $u_{L}$ represent the radial displacements of the soil skeleton and pore liquid, respectively; $w_{S}$ and $w_{L}$ represent the vertical displacements of the soil skeleton and pore liquid, respectively; and $w_{P}$ represents the vertical displacement of the pile.

Analytical solution for the dynamic impedance of a floating pile

Combining the cylindrical symmetry conditions of the interaction system and Novak’s plain-strain model, equations (1a)–(1c) can be expressed in the frequency domain by equations (3a) and (3b)

$μ^{S} (\frac{\partial^{2} W_{S}}{\partial r^{2}} + \frac{1}{r} \frac{\partial W_{S}}{\partial r}) + ω^{2} ρ^{s} W_{S} + i ω S_{v} (W_{L} - W_{S}) = 0$ (3a)

$(i ω S_{v} - ρ^{L} ω^{2}) W_{L} = i ω S_{v} W_{S}$ (3b)

Substituting equation (3a) into equation (3b), equation (4) can be obtained

$\frac{\partial^{2} W_{S}}{\partial r^{2}} + \frac{1}{r} \frac{\partial W_{S}}{\partial r} - q^{2} W_{S} = 0$ (4)

where $q^{2} = (i ω S_{v} + (ω S_{v}^{2} / (i S_{v} - ρ^{L} ω)) - ω^{2} ρ^{S}) / μ^{S}$ .

The general solution of equation (4) is given by

$W_{S} = A I_{0} (qr) + B K_{0} (qr)$ (5)

where $I_{0} (qr)$ and $K_{0} (qr)$ are the modified zero-order Bessel functions of the first and second kinds, respectively, and A and B are undetermined coefficients.

Taking into account the boundary conditions expressed in equations (6a) and (6b)

${W_{S} |}_{r \to \infty} = 0$ (6a)

$w_{S} (r_{0}, t) = W_{S}^{0} e^{i ω t} (W_{S}^{0} = {W_{S} |}_{r \to r_{0}})$ (6b)

it is obtained that A = 0 and $B = W_{S}^{0} / K_{0} (q r_{0})$ .

The solution of equation (4) can be further expressed as

$W_{S} = \frac{W_{S}^{0}}{K_{0} (q r_{0})} K_{0} (qr)$ (7a)

$w_{S} = \frac{W_{S}^{0}}{K_{0} (q r_{0})} K_{0} (qr) e^{i ω t}$ (7b)

Therefore, the shear stresses at the pile–soil interface in the saturated soil layer can be written as

$τ_{rz} |_{r = r_{0}} = μ^{S} \frac{\partial w_{S}}{\partial r} |_{r = r_{0}} = - \frac{W_{S}^{0} μ^{S} q}{K_{0} (q r_{0})} K_{1} (qr) e^{i ω t}$ (8)

Integrating ${\bar{τ}}_{rz} |_{\bar{r} = 1}$ along the cylindrical interface between the floating pile and the saturated soil, the corresponding solution of soil reaction against the pile shaft can be given by

$f_{0} = 2 π r_{0} \frac{W_{S}^{0} μ^{S} q}{K_{0} (q r_{0})} K_{1} (qr) e^{i ω t} = K_{s} W_{S}^{0} e^{i ω t}$ (9)

where $K_{s} = (2 π r_{0} μ^{S} q / K_{0} (q r_{0})) K_{1} (qr)$ is the complex stiffness of the surrounding soil.

Furthermore, the dynamic equilibrium equation of the floating pile can be expressed as

$\bar{m} \frac{\partial^{2} w_{p} (z, t)}{\partial t^{2}} - E_{p} A_{p} \frac{\partial^{2} w_{p} (z, t)}{\partial z^{2}} + K_{s} w_{p} (z, t) = 0$ (10)

where $\bar{m} = ρ_{p} A_{p}$ .

Inserting $w_{P} (z, t) = W_{P} e^{i ω t}$ into equation (10) gives

$\frac{\partial^{2} W_{p}}{\partial z^{2}} + b W_{p} = 0$ (11)

where $b = (\bar{m} ω^{2} - K_{s}) / E_{p} A_{p}$ .

Thus, the solution of equation (11) can be given by

$W_{p} = A_{1} \cos (Λ z) + B_{1} \sin (Λ z)$ (12)

where $Λ = \sqrt{(\bar{m} ω^{2} - K_{s}) / E_{p} A_{p}}$ and $A_{1}$ and $B_{1}$ are undetermined coefficients.

The boundary conditions at the head and base of the pile can be expressed using equations (13) and (14), respectively

$\frac{\partial W_{p}}{\partial z} |_{z = - L} = \frac{P}{E_{p} A_{p}}$ (13)

$\frac{\partial W_{p}}{\partial z} |_{z = 0} = \frac{R}{E_{p} A_{p}}$ (14)

where R is the soil reaction at the pile base.

The soil reaction at the pile base R can be expressed using equations (15) and (16) based on the procedure proposed by Cai and Hu³⁴

$R = - G r_{0} K_{b} {W_{p} |}_{z = 0}$ (15)

$K_{b} = \frac{4}{1 - υ} \int_{0}^{1} Φ (x) d x$ (16)

where $K_{b}$ is the complex stiffness at the pile base and $Φ (x)$ is an undetermined function.

The undetermined function $Φ (x)$ can be obtained by solving the following equivalent integration equation of the second Fredholm type

$Φ (x) + \frac{1}{π} \int_{0}^{1} F (x, y) Φ (y) d y = 1$ (17)

The kernel function $F (x, y)$ can be defined as

$F (x, y) = 2 \int_{0}^{\infty} H (ξ) \cos (ξ x) \cos (ξ y) d ξ$ (18)

where

$\begin{matrix} f (ξ) = \frac{c_{41} - c_{21} c_{32} - \frac{ξ}{s} D_{4}}{λ^{*} ξ c_{31} - (λ^{*} + 2) q c_{41} + 2 ξ c_{21} c_{32} + 2 ξ D_{4}} \\ H (ξ) = \frac{ξ f (ξ)}{υ - 1} - 1, D_{4} = s \frac{2 q c_{31} - 2 ξ c_{21} c_{32}}{ξ^{2} + s^{2}}, c_{21} = \frac{D_{2}}{D_{3}} \\ c_{31} = ξ \frac{λ^{*} + 1 - c_{21} (1 + D_{1})}{a_{0}^{2} + ρ^{*} a_{0}^{2} D_{1} + D_{3}}, c_{32} = \frac{- ξ (1 + D_{1})}{a_{0}^{2} (1 + ρ^{*} D_{1})} \\ c_{41} = \frac{ξ c_{31} - 1}{q}, q = \sqrt{ξ^{2} + D_{3}}, s = \sqrt{ξ^{2} - a_{0}^{2} (1 + ρ^{*} D_{1})} \\ D_{2} = \frac{1 + D_{1}}{D_{1}} ρ^{*} a_{0}^{2}, D_{3} = \frac{ρ^{*} / D_{1} + 2 ρ^{*} - 1}{λ^{*} + 2} a_{0}^{2}, \\ D_{1} = \frac{n^{L} ρ^{*} a_{0}}{i b^{*} n^{L} - ρ^{*} a_{0}} \\ a_{0} = \sqrt{\frac{ρ}{G}} r_{0} ω, ρ^{*} = \frac{ρ^{L}}{ρ}, b^{*} = \frac{b}{\sqrt{ρ G}} r_{0}, λ^{*} = \frac{λ^{S}}{G}, \\ b = \frac{n^{L} ρ^{L} g}{k^{L}}, ρ = ρ^{S} + ρ^{L} \end{matrix}$ (18)

Substituting equation (12) into equations (13) and (14), equations (19) and (20) can be obtained in the following forms, respectively

$A_{1} = \frac{P [\cos (Λ L) - a \sin (Λ L)]}{Λ E_{P} A_{P} [\sin (Λ L) + a \cos (Λ L)]}$ (19)

$B_{1} = \frac{P}{Λ E_{P} A_{P}}$ (20)

where $a = G r_{0} K_{b} / Λ E_{p} A_{p}$ .

Therefore, the vertical dynamic impedance at the pile head can be expressed as

$K_{v} = {\frac{P}{w_{p}} |}_{z = - L} = \frac{Λ E_{p} A_{p} [\sin (Λ L) + a \cos (Λ L)]}{\cos (Λ L) - a \sin (Λ L)}$ (21)

The dimensionless vertical dynamic impedance at the pile head can be further written as

$K_{v}^{*} = \frac{K_{v}}{E_{p} A_{p}} = Re (K_{v}^{*}) + i Im (K_{v}^{*})$ (22)

where the real part $Re (K_{v}^{*})$ and the imaginary part $Im (K_{v}^{*})$ represent the real dynamic stiffness at the pile head and the equivalent damping of the soil–pile system, respectively.

Furthermore, the frequency response function of velocity at the pile head can be obtained as

$H (i ω) = \frac{i ω}{K_{z} (i ω)} = \frac{H' (i ω)}{V_{p} E_{p} A_{p}}$ (23)

where $H' (i ω)$ is the dimensionless frequency response function of velocity at the pile head and $V_{p} = \sqrt{E_{p} / ρ_{p}}$ is the longitudinal wave speed of the pile.

Using the convolution theorem and inverse Fourier transformation, the velocity response in the time domain at the pile head can be written as

$V (t) = q (t) * IFT [H (i ω)] = IFT [Q (i ω) \cdot H (i ω)]$ (24)

where $q (t)$ is the vertical load applied on the pile head and $Q (ω)$ is the Fourier transformation of $q (t)$ .

For the integrity detection of pile foundation, the hammer excitation $q (t)$ to the pile head is generally simplified as a half-sine pulse load, which is given by

$q (t) = Q_{\max} \sin \frac{π t}{T}, t \in (0, T)$ (25)

where $Q_{\max}$ denotes the peak value of the pulse and T is the pulse width, as shown in Figure 2.

Figure 2.

A half-sine pulse excitation.

Combining equations (24) and (25), the dimensionless frequency response function of velocity at the pile head $V'$ can be expressed as

$V^{'} = \frac{V (t) ρ_{p} V_{p} A_{p}}{Q_{\max}} = \frac{1}{2} \int_{- \infty}^{\infty} H^{'} (\bar{ω}) \frac{\bar{T}}{π^{2} - {\bar{T}}^{2} {\bar{ω}}^{2}} (1 + e^{- i \bar{T} \bar{ω}}) e^{i \bar{t} \bar{ω}} d \bar{ω}$ (26)

where $\bar{ω} = T_{c} ω$ is the dimensionless excitation frequency, $T_{c} = L / V_{p}$ , $\bar{T} = T / T_{c}$ , and $\bar{t}$ is the dimensionless time variable that satisfies $\bar{t} = t / T_{c}$ .

Results and discussion

This section presents the numerical results to demonstrate the validity of the obtained analytical solutions and to investigate the vertical dynamic response of the floating pile in the saturated porous viscoelastic soil. Unless otherwise specified, the parameter values in Table 2 are used. For the convenience of comparison, the dynamic impedance in the following figures is normalized as $K_{v}^{*} / K_{0}^{*}$ , where $K_{0}^{*} = lim_{a_{0} \to 0} K_{v}^{*}$ denotes the dimensionless static stiffness at the pile head.

Table 2.

Parameter values of the surrounding soil and the solid pile.

G	60 MPa	$ρ^{L}$	1000 kg/m³
$υ$	0.25	$k^{L}$	1 × 10⁻⁴ m/s
$ξ$	0.1	$E_{p}$	20 GPa
$n^{L}$	0.4	$ρ_{p}$	2500 kg/m³
$ρ^{S}$	1800 kg/m³	$r_{0}$	0.25 m

The impedance solution expressed in equation (22) for a floating pile can be reduced to describe the vertical vibration of an end-bearing pile in the saturated viscoelastic soil by setting the complex impedance at the pile base $K_{b} \to \infty$ . Therefore, based on the same parameters, the solution of $K_{v}^{*}$ is verified by comparing with the existing solution for an end-bearing pile embedded in the saturated viscoelastic soil presented by Yu et al.,³⁵ as shown in Figure 3. It can be seen that the present solution of vertical dynamic impedance is in very good agreement with that proposed by Yu et al.³⁵ Thus, it is indicated that the accuracy of the present solution is validated with the independent comparison.

Figure 3.

Comparison of the reduced vertical impedance with the solution of Yu et al. for an end-bearing pile.

Dynamic impedance at the pile head

The dynamic impedance at the pile head to time-harmonic loads is generally used to investigate the vibration characteristics of the pile–soil system. Figures 4 –6 show the effects of the slenderness ratio of pile, the modulus ratio of pile to soil, and the permeability coefficient of soil on the dynamic impedance at the pile head, respectively.

Figure 4.

Effect of the pile slenderness ratio on the vertical impedance.

Figure 5.

Effect of the modulus ratio of pile to soil on the vertical impedance $(l = L / r_{0} = 20)$ .

Figure 6.

Effect of the permeability coefficient on the vertical impedance $(l = L / r_{0} = 20)$ .

Figure 4 shows the dimensionless vertical dynamic impedance $K_{v}^{*}$ at the pile head versus the dimensionless frequency $a_{0}$ with different values of the pile slenderness ratio l $(l = L / r_{0})$ . It is evident that the vertical dynamic impedance of the floating pile depends significantly on the frequency $a_{0}$ and shows similar patterns for different values of l. With the increase of the pile slenderness ratio, the resonance frequencies and oscillation amplitudes of the dynamic impedance at resonance frequencies decrease. In addition, the effect of the pile slenderness ratio on the real part of the vertical impedance is greater than that on the imaginary part. Furthermore, the larger the slenderness ratio of the pile is, the less the fluctuation of the dynamic impedance is.

Figure 5 shows the dimensionless vertical dynamic impedance $K_{v}^{*}$ at the pile head versus the dimensionless frequency $a_{0}$ with different values of the modulus ratio of pile to soil $(E_{p} / G)$ . It is observed that with the increase of the modulus ratio of pile to soil the resonant frequency of the soil–pile system and the amplitudes of the dynamic impedance at the resonant frequencies increase. Moreover, the more the modulus ratio of pile to soil is, the greater the fluctuation of the dynamic impedance is.

Figure 6 shows the dimensionless vertical dynamic impedance $K_{v}^{*}$ at the pile head versus the dimensionless frequency $a_{0}$ with different values of the permeability coefficient. It can be seen that the effect of the permeability coefficient on the real part of the vertical impedance is larger than that on the imaginary part. Furthermore, the resonance frequencies are almost unchanged and the oscillation amplitudes of the dynamic impedance at resonance frequencies slightly decrease with the increase of the permeability coefficient within the high-frequency range. However, the effect of the permeability coefficient on the real part of the vertical impedance is negligible within the low-frequency range. The effect of the permeability coefficient on the imaginary part of the vertical impedance can be ignored in the entire frequency range. In addition, the dynamic impedance for the pile in saturated soil is distinct from that in single-phase soil.

Velocity response at the pile head

Due to the impact at the pile head, it generates the downward stress wave along the pile shaft. When the downward stress wave encounters a change in cross section or in concrete quality, the reflected wave of velocity response can be observed at the pile head. The analytical solution for the velocity response at the pile head can depict the typical patterns of the reflected signal corresponding to different pile defects, which is able to provide theoretical reference to the low-strain integrity detection of pile in engineering practice.

Figure 7 shows the dimensionless reflected wave signal of vertical velocity in the time domain at the pile head $V'$ versus the dimensionless frequency $a_{0}$ with different values of the pile slenderness ratio l. It is clear that the pile slenderness ratio has considerable effect on the dynamic response of the floating pile. The oscillation amplitudes of vertical velocity at the pile head increase with the increasing pile slenderness ratio l.

Figure 7.

Effect of the pile slenderness ratio on the reflected signal.

Figure 8 shows the dimensionless reflected wave signal of vertical velocity in the time domain at the pile head $V'$ versus the dimensionless frequency $a_{0}$ with different values of the modulus ratio of pile to soil. It can be seen that the amplitude of the reflected signal increases with the increasing relative modulus of pile to soil. Figure 9 shows the dimensionless reflected wave signal of vertical velocity in the time domain at the pile head $V'$ versus the dimensionless frequency $a_{0}$ with different values of the permeability coefficient. It can be observed that the effect of the permeability coefficient on the reflected wave signal of vertical velocity at the pile head can practically be ignored, since it has little influence on the propagation speed of signals. In addition, the oscillation amplitude of the reflective velocity increases with the increasing permeability coefficient of the surrounding saturated soil. It is noted that the propagation speed of the reflected signal in saturated soil is less than that in single-phase soil.

Figure 8.

Effect of the modulus ratio of pile to soil on the reflected signal $(l = L / r_{0} = 20)$ .

Figure 9.

Effect of the permeability coefficient on the reflected signal $(l = L / r_{0} = 20)$ .

Conclusion

Based on the TPM, a new mathematical model of a floating pile embedded in a viscoelastic saturated soil under vertical time-harmonic load is proposed for the evaluation of low-strain integrity detection. The corresponding analytical solutions for the dynamic impedance and velocity response at the pile head are derived and then verified. The proposed analytical solution for velocity response at the pile head can also be reduced to investigate the vertical vibration problem of end-bearing piles in saturated soil and solid piles embedded in single-phase half-space. Moreover, it should be noted for the limitations of the obtained solution that the transverse isotropy of the surrounding soil, the vertical inhomogeneity of soil layers around the pile, and the discontinuous effect of the pile–soil interface are not considered in the proposed model.

An extensive parametric analysis is further conducted to investigate the effects of the pile slenderness ratio, the relative modulus of pile to soil, and permeability of the saturated soil on the dynamic impedance and the reflected wave signal of vertical velocity. The parametric analysis shows that (1) with the increase of the pile slenderness ratio and the decrease of the modulus of pile to soil the resonance frequencies decrease and the oscillation amplitudes of the dynamic impedance at resonance frequencies decrease; (2) the effect of the permeability coefficient on the velocity response at the pile head can be practically negligible; (3) only within the high-frequency range the resonance frequencies increase and the oscillation amplitudes of the dynamic impedance at resonance frequencies slightly decrease with the increase of the permeability coefficient.

Footnotes

Handling Editor: Lalit Borana

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research,authorship,and/or publication of this article: This work was financially supported by the National Natural Science Foundation of China (Grant Nos 51578100,51722801,and 51878109) and the Fundamental Research Funds for the Central Universities (Grant Nos 3132014326 and 3132018110). The first author would like to acknowledge the support from the Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering,Hohai University.

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