Effect of radial homogeneity on low-strain integrity detection of a pipe pile in a viscoelastic soil layer

Abstract

This article proposes a new analytical model for the low-strain integrity detection of a pipe pile embedded in a viscoelastic soil layer with radial inhomogeneity by extending Novak’s plane-strain model and transfer method of complex stiffness to consider viscous-type damping. The analytical solutions for the complex impedance, the velocity admittance and the reflected wave signal of velocity at the pile head are also derived. Extensive parametric analyses are further conducted to investigate the effects of the disturbance degree and the disturbance range of surrounding soil due to construction operation on the velocity admittance and the reflected wave signal of velocity at the pile head. It is demonstrated that the proposed model and the obtained solutions can provide extensive scope of application, compared with the relevant existing solutions.

Keywords

Integrity detection pipe pile construction disturbance pile–soil interaction vertical vibration

Introduction

The dynamic response of pile–soil interaction is an important topic for geotechnical engineering and soil dynamics. It is well known that the construction operations may disturb the soil in the immediate vicinity around the pile, that is, the pile shaft is surrounded by an annular disturbed zone. In the early works of the problem of pile vibration, the surrounding soil is generally assumed to be radially homogeneous. However, this assumption may not be realistic when the construction disturbance is significant. To consider the radial inhomogeneity of the surrounding soil, Novak and colleagues^1,2 proposed an analytical model to investigate the effect of reduced resistance of the soil around the pile by defining a massless annular boundary zone in their pioneering studies. Subsequently, Veletsos and Dotson^3,4 extended Novak’s plane-strain model to further consider the inertia effect of the annular boundary zone. To describe the inhomogeneity of the boundary zone, Dotson and Veletsos⁵ and Han and Sabin^6,7 developed analytical models assuming a linear and a quadratic variation of shear modulus within the disturbed boundary zone, respectively. Furthermore, Nogami and Konagai^8,9 and El Naggar and Novak¹⁰ considered the radial inhomogeneity of the surrounding soil by obtaining the coefficients of the complex stiffness at the interface between the disturbed boundary zone and the ultimate undisturbed zone based on Novak’s model.

Furthermore, El Naggar¹¹ proposed an analytical model by dividing the boundary zone into N annular sub-zones to account for the radial inhomogeneity within the disturbed zone. However, Wang et al.¹² found that the effect of surrounding soil cannot be considered in El Naggar’s model. Subsequently, Wang et al.¹³ and Yang et al.¹⁴ proposed a new plane-strain model of solid pile embedded in viscoelastic soil with radial inhomogeneity by combining the complex stiffness transfer method and hysteretic-type damping model.

The aforementioned studies are mainly devoted to the solid pile. In addition, some substantial developments of the dynamic model for pipe pile have also been made by some investigators. For example, Ding and colleagues^15–17 obtained an analytical solution for the vibration characteristics of large-diameter pipe piles in a viscoelastic soil under vertical excitation. Zheng et al.¹⁸ proposed an analytical model for the vertical dynamic response of pipe piles in homogeneous viscoelastic soil accounting for the three-dimensional wave effect. Moreover, Li et al.¹⁹ proposed an analytical solution for the dynamic impedance at the head of large-diameter pipe piles in radially inhomogeneous soil using a hysteretic-type damping model.

It is noted that hysteretic-type damping is independent of frequency to represent the material damping, and it has limitations in describing the dynamic response of related problems subjected to non-harmonic loads in the time domain. In contrast, the viscous-type damping is suitable for describing the dynamic response of pile vibrations subjected to generalized modes of dynamic load in the time domain.^20,21 However, based on an extensive review of the literature, for the moment, no study has been conducted to the dynamic behaviour of a pipe pile in a viscoelastic soil layer with radial inhomogeneity using the viscous-type damping model. Consequently, the main purpose of this article is to propose a new analytical model for the low-strain integrity detection of a pipe pile embedded in a viscoelastic soil with radial inhomogeneity using the viscous-type damping model. In addition, the corresponding analytical solutions for the velocity admittance and the reflected signal of the wave velocity at the pile head are determined. Extensive parametric analyses have also been performed to discuss the effects of the disturbance degree and the disturbance range of surrounding soil due to construction operation on the velocity admittance and the reflected signal of the wave velocity at the pile head.

Formulation of the governing equations

The analytical model of the pile–soil interaction system is shown in Figure 1(a) and (b). The thickness of the soil layer is H. The inner and outer radius of the pile 0 are $r_{0}$ and $r_{1}$ , respectively. The radius and radial thickness of the disturbed zone within the soil layer are $r_{n + 1}$ and b, respectively. The disturbed zone is subdivided into n concentric sub-zones numbered 1, 2, …, i, …, n along the radial direction. The radius of the interface between the $(i - 1) th$ and ith sub-zones is represented by $r_{i}$ .

Figure 1.

Analytical model: (a) section view and (b) vertical view.

The undisturbed zone of the surrounding soil is homogeneous, isotropic and viscoelastic with viscous-type damping. Within the disturbed zone, $G_{i} (r)$ and $η_{i} (r)$ are given by the expressions (1) and (2)

G_{i} (r) = {\begin{matrix} G_{1} & r = r_{1} \\ G_{n + 1} \times f (r) & r_{1} < r < r_{n + 1} \\ G_{n + 1} & r > r_{n + 1} \end{matrix}

(1)

η_{i} (r) = {\begin{matrix} η_{1} & r = r_{1} \\ η_{n + 1} \times f (r) & r_{1} < r < r_{n + 1} \\ η_{n + 1} & r > r_{n + 1} \end{matrix}

(2)

where $G_{i} (r)$ and $η_{i} (r)$ are the shear modulus and viscous damping coefficient of the ith annular sub-zone, respectively. $f (r)$ denotes the parabolic variation of the material properties within the disturbed zone of the ith sub-zone.^6,11,12 The shear modulus and viscous damping coefficient of each sub-zone are considered to be homogeneous and can be determined by equations (1) and (2). The mechanical coefficients of the Kelvin–Voigt model are denoted by δ_p and k_p. The uniform distributed excitation at the pile head is represented by $p (t)$ .

In this article, the following assumptions are used:

The pile is linear elastic with a circular cross section.

The outer surrounding soil of the interaction system consists of two annular zones, an inner disturbed zone and an outer undisturbed zone with semi-infinity.

The inner annular zone of disturbed soil is divided into a series of sub-zones. The soil properties within different annular sub-zones are radially inhomogeneous.

The frequency-dependent viscous-type damping is used to describe the inside and outside soil of pipe pile.

The deformations of the pile and surrounding soil are small. The equilibrium of shear stress and the continuity of displacement are both satisfied at the interfaces between pipe pile, adjacent annular zones and sub-zones of the surrounding soil.

Based on Novak’s plane-strain model, the governing equation for the ith inner annular sub-zone of disturbed zone is given by equation (3)

\begin{matrix} G_{i} \frac{\partial^{2} u_{i} (r, t)}{\partial r^{2}} + η_{i} \frac{\partial^{3} u_{i} (r, t)}{\partial t \partial r^{2}} + \frac{G_{i}}{r} \frac{\partial u_{i} (r, t)}{\partial r} \\ + \frac{η_{i}}{r} \frac{\partial^{2} u_{i} (r, t)}{\partial t \partial r} = ρ_{i} \frac{\partial^{2} u_{i} (r, t)}{\partial t^{2}} \end{matrix}

(3)

where $u_{i} (r, t)$ represents the vertical displacement of the ith annular sub-zone of the disturbed soil. $G_{i}$ , $η_{i}$ and $ρ_{i}$ are the shear modulus, the viscous damping coefficient and the mass density of the ith annular sub-zone of the disturbed zone, respectively.

Similarly, the governing equation for the ith layer of inside soil can be written as follows

\begin{matrix} G_{0} \frac{\partial^{2} u_{0} (r, t)}{\partial r^{2}} + η_{0} \frac{\partial^{3} u_{0} (r, t)}{\partial t \partial r^{2}} + \frac{G_{0}}{r} \frac{\partial u_{0} (r, t)}{\partial r} \\ + \frac{η_{0}}{r} \frac{\partial^{2} u_{0} (r, t)}{\partial t \partial r} = ρ_{0} \frac{\partial^{2} u_{0} (r, t)}{\partial t^{2}} \end{matrix}

(4)

where $u_{0} (r, t)$ denotes the vertical displacement of inner soil; $ρ_{0}$ , $G_{0}$ and $η_{0}$ are the mass density, the shear modulus and the viscous damping coefficient of the inner soil, respectively.

The governing equation for the ith pile segment is expressed as follows

\frac{\partial^{2} u_{p} (z, t)}{\partial z^{2}} - \frac{2 π r_{0} f_{0}}{E_{p} A_{p}} - \frac{2 π r_{1} f_{1}}{E_{p} A_{p}} = \frac{ρ_{p}}{E_{p}} \frac{\partial^{2} u_{p} (z, t)}{\partial t^{2}}

(5)

where $u_{p} (z, t)$ is the vertical displacement of the ith pile segment. $ρ_{p}$ , $E_{p}$ and $A_{p}$ are the mass density, Young’s modulus and the cross-sectional area, respectively. $f_{0}$ and $f_{1}$ are the shear stresses exerted by the inner soil and surrounding soil on the pile shaft, respectively.

Solutions of the governing equations

Applying the Laplace transform to equation (3) yields

\begin{matrix} G_{i} \frac{\partial^{2} U_{i} (r, s)}{\partial r^{2}} + η_{i} s \frac{\partial^{2} U_{i} (r, s)}{\partial r^{2}} + \frac{G_{i}}{r} \frac{\partial U_{i} (r, s)}{\partial r} \\ + \frac{η_{i} s}{r} \frac{\partial U_{i} (r, s)}{\partial r} = ρ_{i} s^{2} U_{i} (r, s) \end{matrix}

(6)

where $U_{i} (r, s)$ is the Laplace transform of $u_{i} (r, t)$ .

After rearranging the terms in equation (6), equation (7) is obtained as follows

\frac{\partial^{2} U_{i} (r, s)}{\partial r^{2}} + \frac{1}{r} \frac{\partial U_{i} (r, s)}{\partial r} = q_{i}^{2} U_{i} (r, s)

(7)

where $q_{i}^{2} = ρ_{i} s^{2} / G_{i} + η_{i} s$ .

The general solution of equation (7) can be expressed as follows

U_{i} (r, s) = A_{i} K_{0} (q_{i} r) + B_{i} I_{0} (q_{i} r)

(8)

where $I_{0} (q_{i} r)$ and $K_{0} (q_{i} r)$ are modified Bessel functions of the first and second kind of order zero, respectively. A_i and B_i are undetermined coefficients.

The displacement at the outermost zone of the outer soil diminishes at infinity and can be written as follows

lim_{r \to \infty} u_{n + 1} (r, t) = 0

(9)

Then, it yields

B_{n + 1} = 0

(10a)

U_{n + 1} (r, s) = A_{n + 1} K_{0} (q_{n + 1} r)

(10b)

The vertical shear stress of the undisturbed zone can be expressed as follows

\begin{matrix} τ_{n + 1} = (G_{n + 1} + η_{n + 1} s) \frac{\partial U_{n + 1} (r, s)}{\partial r} \\ = - (G_{n + 1} + η_{n + 1} s) q_{n + 1} A_{n + 1} K_{1} (q_{n + 1} r) \end{matrix}

(11)

Then, the vertical complex stiffness at the interface between the disturbed zone and the undisturbed zone can be conveniently expressed as follows

\begin{matrix} K K_{n + 1} = \frac{2 π r_{n + 1} τ_{n + 1} (r_{n + 1})}{U_{n + 1} (r, s)} \\ = 2 π r_{n + 1} (G_{n + 1} + η_{n + 1} s) q_{n + 1} \frac{K_{1} (q_{n + 1} r_{n + 1})}{K_{0} (q_{n + 1} r_{n + 1})} \end{matrix}

(12)

The shear stress of the ith annular sub-zone can be given by

\begin{matrix} τ_{i} = (G_{i} + η_{i} s) \frac{\partial U_{i} (r, s)}{\partial r} \\ = - (G_{i} + η_{i} s) q_{i} [A_{i} K_{1} (q_{i} r) - B_{i} I_{1} (q_{i} r)] \end{matrix}

(13)

Therefore, the vertical complex stiffness at the outer boundary (r=r_i₊₁) of the ith sub-zone is expressed as follows

\begin{matrix} K K_{i + 1} = \frac{2 π r_{i + 1} τ_{i} (r_{i + 1})}{U_{i} (r_{i + 1}, s)} \\ = 2 π r_{i + 1} q_{i} (G_{i} + η_{i} s) \frac{A_{i} K_{1} (q_{i} r_{i + 1}) - B_{i} I_{1} (q_{i} r_{i + 1})}{A_{i} K_{0} (q_{i} r_{i + 1}) + B_{i} I_{0} (q_{i} r_{i + 1})} \end{matrix}

(14)

Similarly, the vertical complex stiffness at the inner boundary (r = r_i)

\begin{matrix} K K_{i} = \frac{2 π r_{i} τ_{i} (r_{i})}{U_{i} (r_{i}, s)} \\ = 2 π r_{i} q_{i} (G_{i} + η_{i} s) \frac{A_{i} K_{1} (q_{i} r_{i}) - B_{i} I_{1} (q_{i} r_{i})}{A_{i} K_{0} (q_{i} r_{i}) + B_{i} I_{0} (q_{i} r_{i})} \end{matrix}

(15)

Thus, the recursion formula of the vertical stiffness can be easily given by

K K_{i} = 2 π r_{i} q_{i} (G_{i} + η_{i} s) \frac{C_{i} + E_{i} K K_{i + 1}}{D_{i} + F_{i} K K_{i + 1}}

(16)

where

\begin{matrix} C_{i} = 2 π r_{i + 1} q_{i} (G_{i} + η_{i} s) \\ [I_{1} (q_{i} r_{i + 1}) K_{1} (q_{i} r_{i}) - K_{1} (q_{i} r_{i + 1}) I_{1} (q_{i} r_{i})] \\ D_{i} = 2 π r_{i + 1} q_{i} (G_{i} + η_{i} s) \\ [I_{1} (q_{i} r_{i + 1}) K_{0} (q_{i} r_{i}) + K_{1} (q_{i} r_{i + 1}) I_{0} (q_{i} r_{i})] \\ E_{i} = I_{0} (q_{i} r_{i + 1}) K_{1} (q_{i} r_{i}) + K_{0} (q_{i} r_{i + 1}) I_{1} (q_{i} r_{i}) \\ F_{i} = I_{0} (q_{i} r_{i + 1}) K_{0} (q_{i} r_{i}) - K_{0} (q_{i} r_{i + 1}) I_{0} (q_{i} r_{i}) \end{matrix}

Applying the Laplace transform in equation (4), it yields

\begin{matrix} G_{0} \frac{\partial^{2} U_{0} (r, s)}{\partial r^{2}} + η_{0} s \frac{\partial^{2} U_{0} (r, s)}{\partial r^{2}} + \frac{G_{0}}{r} \frac{\partial U_{0} (r, s)}{\partial r} \\ + \frac{η_{0} s}{r} \frac{\partial U_{0} (r, s)}{\partial r} = ρ_{0} s^{2} U_{0} (r, s) \end{matrix}

(17)

where $U_{0} (r, s)$ is the Laplace transform of $u_{0} (r, t)$ .

After rearranging the terms in equation (17), it yields

\frac{\partial^{2} U_{0} (r, s)}{\partial r^{2}} + \frac{1}{r} \frac{\partial U_{0} (r, s)}{\partial r} = q_{0}^{2} U_{0} (r, s)

(18)

where $q_{0}^{2} = ρ_{0} s^{2} / G_{0} + η_{0} s$ .

The general solution of equation (18) can be given by

U_{0} (r, s) = A_{0} K_{0} (q_{0} r) + B_{0} I_{0} (q_{0} r)

(19)

The displacement of inner soil is a limited value if r = 0, namely, ${u_{0} (r, t) |}_{r \to 0} < \infty$ . Hence, A₀ = 0.

Then, equation (19) can be reduced to

U_{0} (r, s) = B_{0} I_{0} (q_{0} r)

(20)

Combining the displacement continuity at the interface between the pile shaft and the inner soil $u_{0} (r_{0}, t) = u_{p} (r_{0}, t)$ , the vertical complex stiffness at the interface between the inner soil and the pile shaft can be obtained as follows

K K_{0} = - \frac{2 π r_{0} τ_{0} (r_{0})}{U_{p}} = - 2 π r_{0} (G_{0} + η_{0} s) \frac{I_{1} (q_{0} r_{0})}{I_{0} (q_{0} r_{0})}

(21)

Performing the Laplace transform in equation (5) and combining with equations (14) and (21), it gives

\frac{\partial^{2} U_{p} (z, s)}{\partial z^{2}} - α^{2} U_{p} (z, s)

(22)

where $α^{2} = (K K_{1} / E_{p} A_{p}) - (K K_{0} / E_{p} A_{p}) + (ρ_{p} / E_{p}) s^{2}$ , $U_{p} (z, s)$ is the Laplace transform of $u_{p} (z, t)$ .

The general solution of equation (22) can be expressed as follows

U_{p} (z, s) = C e^{α z} + D e^{- α z}

(23)

The boundary conditions at the pile toe and the pile head can be given by equations (24a) and (24b), respectively

E_{p} A_{p} {\frac{\partial u_{p} (z, t)}{\partial z} |}_{z = 0} = p (t)

(24a)

E_{p} {\frac{\partial u_{p} (z, t)}{\partial z} |}_{z = H} = - (k_{p} u_{p} (z, t) + δ_{p} \frac{\partial u_{p} (z, t)}{\partial t})

(24b)

Applying the Laplace transform in equations (24a) and (24b) and combining with equation (23), it yields

C = \frac{ξ P (s)}{α (ξ - 1) E_{p} A_{p}}

(25a)

D = \frac{P (s)}{α (ξ - 1) E_{p} A_{p}}

(25b)

where $ξ = α E_{p} - (k_{p} + δ_{p} s) / (α E_{p} + (k_{p} + δ_{p} s)) e^{- 2 α H}$ .

Substituting equations (25a) and (25b) into equation (23), it can be given by

U_{p} (z, s) = \frac{(ξ e^{α z} + e^{- α z})}{α (ξ - 1) E_{p} A_{p}} P (s)

(26)

Given s = iω, the complex impedance of the vertical displacement at the pile head can be expressed as follows

K_{d} = {\frac{P (i ω)}{U_{p} (z, i ω)} |}_{z = 0} = \frac{α (ξ - 1) E_{p} A_{p}}{(ξ + 1)} = \frac{E_{p} A_{p}}{H} K'_{d}

(27a)

K'_{d} = \frac{\bar{α} (ξ - 1)}{(ξ + 1)}

(27b)

where $\bar{α} = α H$ , $P (i ω)$ is the Fourier transform of p(t); $K'_{d}$ is the dimensionless complex stiffness at the pile head.

Furthermore, $K'_{d}$ can be further rewritten as follows

K'_{d} = K_{r} + i K_{i}

(28)

where $K_{r}$ and $K_{i}$ denote the true stiffness and the equivalent damping, respectively.

The frequency response function of the vertical displacement at the pile head can be given by

H_{v} = \frac{i ω U_{p} (z, i ω)}{P (i ω)} = \frac{i ω (ξ + 1)}{α (ξ - 1) E_{p} A_{p}} = \frac{1}{ρ_{p} A_{p} V_{p}} H'_{v}

(29)

H'_{v} = \frac{i θ (ξ + 1)}{\bar{α} (ξ - 1)}

(30)

where $H'_{v}$ is a dimensionless frequency response function of the vertical velocity at the pile cap, $V_{p}^{2} = E_{p} / ρ_{p}$ , $θ = ω T_{c}$ , $T_{c} = H / V_{p}$ .

Taking a transient semi-sine wave $(0 ⩽ t ⩽ T)$ as the transient excitation applied on the pile head, the time-domain function of the velocity response at the pile head can be derived by inverse Fourier transform

\begin{matrix} g (t) = Q_{max} IFT [- \frac{1}{ρ_{p} A_{p} V_{p}} {H'}_{v} \frac{π T}{π^{2} - T^{2} ω^{2}} (1 + e^{- i ω T})] \\ = - \frac{Q_{max}}{ρ_{p} A_{p} V_{p}} {V'}_{v} \end{matrix}

(31)

V'_{v} = \frac{1}{2 π} \int_{- \infty}^{\infty} [{H'}_{v} \frac{T'}{π^{2} - {T'}^{2} θ^{2}} (1 + e^{- i θ T'})] e^{i θ t'} d θ

(32)

where Q_max and T are the excitation amplitude and impulse width, respectively; $T' = T / T_{c}$ is the dimensionless impulse width.

Results and discussion

In this section, numerical results are presented to investigate the vertical dynamic response for integrity testing of a pipe pile embedded in a viscoelastic soil layer with radial inhomogeneity.

Unless otherwise specified, the following parameter values are used: H = 6 m, r₀ = 0.38 m, r₁=b=0.5 m, $ρ^{p} = 2500 kg / m^{3}$ , $ρ_{i} = ρ_{0} = 2000 kg / m^{3}$ , $E^{p} = 25 GPa$ ; k_p=1000 kN/m³, δ_p = 100 kN s/m², $V_{n + 1} = 50 m / s$ , $η_{n + 1} = 5000 Ns / m^{3}$ , the coefficient of disturbance degree $β = V_{1} / V_{n + 1} = 2.0$ . The shear modulus $G_{0}$ and the viscous damping coefficient $η_{0}$ of the inner soil are identical with the corresponding parameters of the first annular sub-zone.

Figure 2 shows the effect of soil weakening $(β < 1)$ due to construction disturbance on the dynamic response at the pile head. It can be observed that the degree of weakening of the surrounding soil due to construction disturbance has an obvious effect on the dynamic response at the pile head. Specifically, with the increase in degree of soil weakening in the disturbed zone, the oscillation amplitudes and resonance frequencies of the velocity admittance and the amplitudes of the reflected signal increase.

Figure 2.

Effect of soil weakening due to construction disturbance on dynamic response at the pile head: (a) velocity admittance and (b) reflected signal.

Figure 3 displays the effect of the weakening range on the velocity admittance and the reflected wave signal of velocity at the pile head when β = 0.4. It can be seen that the oscillation amplitudes of the velocity admittance and the reflected signal increase with an increase in the weakening zone of the surrounding soil. In contrast, the change in the resonance frequencies of the velocity admittance can be practically ignored when there is an increase in the weakening range b. Furthermore, the more weakening range b of surrounding soil, the corresponding effect on the dynamic response is less at the pile head.

Figure 3.

Effect of weakening range due to construction disturbance on dynamic response at the pile head (β = 0.4): (a) velocity admittance and (b) reflected signal.

Figure 4 shows the effect of soil strengthening $(β < 1)$ due to construction disturbance on the velocity admittance and the reflected wave signal of velocity at the pile head. It is clear that the dynamic response at the pile head depends significantly on the degree of soil strengthening due to construction disturbance. In contrast, with the increase in the degree of soil strengthening of the disturbed zone, the oscillation amplitudes and the resonance frequencies of the velocity admittance and the amplitudes of the reflected signal decrease.

Figure 4.

Effect of soil strengthening due to construction disturbance on dynamic response at the pile head: (a) velocity admittance and (b) reflected signal.

Figure 5 shows the effect of the strengthening range of the surrounding soil on the velocity admittance and the reflected wave signal of velocity at the pile head when β = 2.0. It is indicated that the oscillation amplitudes of the velocity admittance and the reflected wave signal increase with a decrease in the strengthening range. In contrast, the effect of the strengthening range on the resonance frequencies of the velocity admittance can also be practically ignored. In addition, the larger the strengthening range of the surrounding soil, the corresponding effect on the dynamic response is less at the pile head.

Figure 5.

Effect of strengthening range due to construction disturbance on dynamic response at the pile head (β = 2.0): (a) velocity admittance and (b) reflected signal.

Conclusion

Based on Novak’s plane-strain model, a new analytical model for the low-strain integrity detection of a pipe pile embedded in a viscoelastic soil layer with radial inhomogeneity is proposed by extending transfer method of complex stiffness to consider viscous-type damping. The analytical solutions for the complex impedance, the velocity admittance and the reflected wave signal of velocity at the pile head are also derived.

Extensive parametric analyses are further performed to investigate the effects of the coefficient of disturbance degree, the weakening or strengthening range of surrounding soil on the velocity admittance and the reflected wave signal of velocity at the pile head. The results show that (1) the oscillation amplitudes and the resonance frequencies of velocity admittance and the amplitudes of the reflected wave signal decrease with an increase in the coefficient of disturbance degree and (2) the effect of the disturbance range on the resonance frequencies of the velocity admittance can be practically ignored.

The proposed model and obtained solutions can be reduced to analyse the integrity detection of an end-bearing pile in a viscoelastic soil described in related previous studies. In addition, the obtained solutions can also be conveniently reduced to investigate the vertical vibration problem of solid pile or pipe pile embedded in radially homogeneous medium.

Footnotes

Acknowledgements

The first author (C.C.) would like to acknowledge the support from Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering, Hohai University.

Handling Editor: Zhiwei Gao

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 no. 3132018110).

References

Novak

Sheta

. Approximate approach to contact problems of piles. In: ASCE conference on dynamic response of structures: experimentation, observation, prediction and control, Florida, 15–16 January 1980, pp.53–79. New York: ASCE.

Novak

Han

YC.

Impedances of soil layer with boundary zone. J Geotech Eng1990; 116(6): 1008–1014.

Veletsos

Dotson

KW.

Impedances of soil layer with disturbed boundary zone. J Geotech Eng1986; 112(3): 363–368.

Veletsos

Dotson

KW.

Vertical and torsional vibration of foundations in inhomogeneous media. J Geotech Eng1988; 114(9): 1002–1021.

Dotson

Veletsos

AS.

Vertical and torsional impedances for radially inhomogeneous viscoelastic soil layers. Soil Dyn Earthq Eng1990; 9(3): 110–119.

Han

Sabin

GCW

. Impedances for radially inhomogeneous viscoelastic soil media. J Eng Mech1995; 121(9): 939–947.

Han

YC.

Dynamic vertical response of piles in nonlinear soil. J Geotech Geoenviron1997; 123(8): 710–716.

Nogami

Konagai

Dynamic response of vertically loaded nonlinear pile foundations. J Geotech Eng1987; 113(2): 147–160.

Nogami

Konagai

Time domain flexural response of dynamically loaded single piles. J Eng Mech1988; 114(9): 1512–1525.

10.

El Naggar

Novak

. Nonlinear axial interaction in pile dynamics. J Geotech Eng1994; 120(4): 678–696.

11.

El Naggar

. Vertical and torsional soil reactions for radially inhomogeneous soil layer. Struct Eng Mech2000; 10(4): 299–312.

12.

Wang

Yang

Zhang

ZQ.

Study on dynamic response of pile based on complex stiffness transfer model of radial multizone plane strain. Chin J Rock Mech Eng2008; 27(4): 825–831.

13.

Wang

Yang

Zhang

et al . A new approach for vertical impedance in radially inhomogeneous soil layer. Int J Numer Anal Met2012; 36(6): 697–707.

14.

Yang

Wang

Zhang

et al . Vertical dynamic response of pile in a radially heterogeneous soil layer. Int J Numer Anal Met2009; 33(8): 1039–1054.

15.

Ding

Liu

HL.

Time-domain analytical solution of the vibration response of a large-diameter pipe pile subjected to transient concentrated load. Chin J Geotech Eng2013; 35(6): 1010–1017.

16.

Ding

Liu

Kong

et al . Time-domain analysis of velocity waves in a pipe pile due to a transient point load. Comput Geotech2014; 58(5): 101–116.

17.

Ding

Luan

Zheng

et al . Influence of the second-order effect of axial load on lateral dynamic response of a pipe pile in saturated soil layer. Soil Dyn Earthq Eng2017; 103: 86–94.

18.

Zheng

Ding

Liu

et al . Analytical solution to vertical vibration of cast-in-place concrete large-diameter pipe piles by considering 3D wave effect of soils. Chin J Geotech Eng2013; 35(12): 2247–2254.

19.

Wang

et al . Vertical vibration of a large-diameter pipe pile considering the radial inhomogeneity of soil caused by the construction disturbance effect. Comput Geotech2017; 85: 90–102.

20.

Nogami

Novak

Soil-pile interaction in vertical vibration. Earthq Eng Struct D1976; 4(3): 277–293.

21.

Militano

Rajapakse

RKND

. Dynamic response of a pile in a multi-layered soil transient torsional and axial loading. Géotechnique1999; 49(1): 91–109.