# Horizontal Vibration of a Large-Diameter Pipe Pile in Viscoelastic Soil

Authors: Changjie Zheng; Hanlong Liu; Xuanming Ding; Qiang Fu
Journal: Mathematical Problems in Engineering
 (2013)

---

## Abstract

An analytical solution is developed in this paper to investigate the horizontal dynamic response of a large-diameter pipe pile in viscoelastic soil layer. Potential functions are applied to decouple the governing equations of the outer and inner soil. The analytical solutions of the outer and inner soil are obtained by the method of separation of variables. The horizontal dynamic response and complex dynamic stiffnesses of the pipe pile are then obtained based on the continuity conditions between the pile and the outer and inner soil. To verify the validity of the solution, the derived solution in this study is compared with an existing solution for a solid pile. Numerical examples are presented to analyze the vibration characteristics of the pile and illustrate the effects of major parameters on the stiffness and damping properties.


---

## Body

## 1.Introduction

Many studies have been devoted to the horizontal dynamic response of pile foundation in the recent years. The Winkler model is a popular method for the analysis of horizontal response of piles. Novak [1] studied the dynamic response of pile subjected to horizontal, vertical, and rocking loads  for the frequency domain. Based on Novak’s frequency domain solution, Nogami and Konagai [2] developed a time domain Winkler model consisting of series of springs, dashpots, and masses units with frequency independent parameters. Yao and Nogami [3] proposed an analytical solution for the low-frequency cyclic response of a pile in a linear viscoelastic Winkler subgrade. A nonlinear Winkler model was developed by Nogami and Chen [4] in the frequency domain and by Nogami et al. [5] in the time domain for pile lateral response. The Winkler model theory is simple and practical, but it neglects the coupled vibration between the pile and soil, and the impedance parameters of the model are hard to determine. Assuming the soil as an elastic half space and the pile as a virtual rod, Rajapakse and Shah [6, 7] proposed the integral solution of the soil and pile considering the real pile-soil reaction. However, this method is only suitable for a floating pile, and the corresponding numerical calculation requires considerable computational effort. Nogami and Novak [8, 9] developed a method in which the soil around the pile was considered as a homogeneous and isotropic layer. A closed-form solution of an end-bearing pile was proposed based on the continuity assumption of the displacement and stress between the pile and soil layer.A new type of pile called cast-in-situ concrete large-diameter pipe pile (referred to as PCC pile) has been developed and widely applied in China [10–13]. Besides PCC piles, many other large-diameter pipe piles are also widely used in practical engineering, such as prestressed concrete pipe piles and large-diameter steel pipe piles [14–16]. For large-diameter pipe piles, apart from the soil around the pile, soil exists inside the pile as well. The way in which the inner soil interacts with the pipe pile is worth studying. Only the interaction between the outer soil and the pile was considered in previous studies, and therefore their solutions cannot be used to analyze the dynamic response of large-diameter pipe piles. In this paper, an analytical solution for the horizontal dynamic response of a large-diameter pipe pile in viscoelastic soil layer is developed with the interaction of the pipe pile between both the outer and inner soil being taken into account. Numerical results are presented to analyze the vibration characteristics of the pile-soil system.


## 2.Basic Assumptions and Computational Model

The main assumptions adopted in this paper are as follow.(1) The outer and inner soil layers are viscoelastic, homogeneous, and isotropic, and the material damping is of the frequency independent hysteresis type; (2) the surfaces of the outer and inner soil are free, and the bottoms of the outer and inner soil are fixed; (3) the pile is elastic, and the pile tip is clamped; (4) the deformation of the pile-soil system is small; (5) the pile has perfect contacts with the outer and inner soil; (6) the vertical displacements of the pile and soil are zero.The computational model is shown in Figure1. The pile is subjected to a time-harmonic horizontal force or rocking moment at the pile head. H is the pile length. r1 and r2 are the outer and inner radii of the pile section, respectively. f1(z)eiωt and f2(z)eiωt are the soil resistances of the outer and inner soil, respectively.Figure 1
Computational model.


## 3. Governing Equations and Boundary Conditions

### 3.1. Dynamic Equilibrium Equations of the Outer Soil

The dynamic equilibrium equations of the outer soil in polar coordinate system can be expressed as(1)[(λ1+2G1)+i(λ1′+2G1′)]∂∂r(e1eiωt)-1r(G1+iG1′)∂∂θ(w1eiωt)+(G1+iG1′)∂2∂z2(u1eiωt)=ρ1∂2∂t2(u1eiωt),(2)[(λ1+2G1)+i(λ1′+2G1′)]∂r∂θ(e1eiωt)+(G1+iG1′)∂∂r(w1eiωt)+(G1+iG1′)∂2∂z2(v1eiωt)=ρ1∂2∂t2(v1eiωt),
where i=-1, e1=∂u1/∂r+u1/r+(1/r)(∂v1/∂θ), and w1=∂v1/∂r+v1/r-(1/r)(∂u1/∂θ). In addition, r, θ, and z are the radial, circumferential, and vertical directions of the column coordinate, respectively; u1 and v1 are the amplitudes of the radial and circumferential displacements of the outer soil, respectively; λ1, G1 and λ1′, G1′ are the real and imaginary parts of the complex Lame's constants of the outer soil, respectively; ρ1 is the mass density of the outer soil; ω is the excitation frequency.


### 3.2. Dynamic Equilibrium Equations of the Inner Soil

The dynamic equilibrium equations of the inner soil in polar coordinate system can be expressed as(3)[(λ2+2G2)+i(λ2′+2G2′)]∂∂r(e1eiωt)-1r(G2+iG2′)∂∂θ(w2eiωt)+(G2+iG2′)∂2∂z2(u2eiωt)=ρ2∂2∂t2(u2eiωt),(4)[(λ2+2G2)+i(λ2′+2G2′)]∂r∂θ(e2eiωt)+(G2+iG2′)∂∂r(w2eiωt)+(G2+iG2′)∂2∂z2(v2eiωt)=ρ2∂2∂t2(v2eiωt),
where e2=∂u2/∂r+u2/r+(1/r)(∂v2/∂θ) and w2=∂v2/∂r+v2/r-(1/r)(∂u2/∂θ). In addition, u2 and v2 are the amplitudes of the radial and circumferential displacements of the inner soil, respectively; λ2, G2 and λ2′, G2′ are the real and imaginary parts of the complex Lame’s constants of the inner soil, respectively; ρ2 is the mass density of the inner soil.


### 3.3. Dynamic Equilibrium Equation of the Pile

The horizontal displacement of the pileup(z)eiωt is governed by the following equation:
(5)EpIp∂4∂z4(upeiωt)+m∂2∂t2(upeiωt)+f1(z)eiωt+f2(z)eiωt=0,
where Ep is the Young’s modulus of the pile, Ip is the second moment of area of the pile section, and m is the mass of the pile per unit length.


### 3.4. Boundary Conditions and Initial Conditions

The boundary conditions at the tops of the outer and inner soil are:(6)∂u1∂z|z=0=0,(7)∂u2∂z|z=0=0.The boundary conditions at the bottoms of the outer and inner soil are(8)u1|z=H=v1|z=H=0,(9)u2|z=H=v2|z=H=0.The boundary conditions at the bottom of the pile are(10)up|z=H=0,∂up∂z|z=H=0.The continuity conditions of displacements on the outer interface are(11)u1(r1,θ,z,t)=up(z,t)cosθ,(12)v1(r1,θ,z,t)=-up(z,t)sinθ.The continuity conditions of displacements on the inner interface are(13)u2(r2,θ,z,t)=up(z,t)cosθ,(14)v2(r2,θ,z,t)=-up(z,t)sinθ.


## 3.1. Dynamic Equilibrium Equations of the Outer Soil

The dynamic equilibrium equations of the outer soil in polar coordinate system can be expressed as(1)[(λ1+2G1)+i(λ1′+2G1′)]∂∂r(e1eiωt)-1r(G1+iG1′)∂∂θ(w1eiωt)+(G1+iG1′)∂2∂z2(u1eiωt)=ρ1∂2∂t2(u1eiωt),(2)[(λ1+2G1)+i(λ1′+2G1′)]∂r∂θ(e1eiωt)+(G1+iG1′)∂∂r(w1eiωt)+(G1+iG1′)∂2∂z2(v1eiωt)=ρ1∂2∂t2(v1eiωt),
where i=-1, e1=∂u1/∂r+u1/r+(1/r)(∂v1/∂θ), and w1=∂v1/∂r+v1/r-(1/r)(∂u1/∂θ). In addition, r, θ, and z are the radial, circumferential, and vertical directions of the column coordinate, respectively; u1 and v1 are the amplitudes of the radial and circumferential displacements of the outer soil, respectively; λ1, G1 and λ1′, G1′ are the real and imaginary parts of the complex Lame's constants of the outer soil, respectively; ρ1 is the mass density of the outer soil; ω is the excitation frequency.


## 3.2. Dynamic Equilibrium Equations of the Inner Soil

The dynamic equilibrium equations of the inner soil in polar coordinate system can be expressed as(3)[(λ2+2G2)+i(λ2′+2G2′)]∂∂r(e1eiωt)-1r(G2+iG2′)∂∂θ(w2eiωt)+(G2+iG2′)∂2∂z2(u2eiωt)=ρ2∂2∂t2(u2eiωt),(4)[(λ2+2G2)+i(λ2′+2G2′)]∂r∂θ(e2eiωt)+(G2+iG2′)∂∂r(w2eiωt)+(G2+iG2′)∂2∂z2(v2eiωt)=ρ2∂2∂t2(v2eiωt),
where e2=∂u2/∂r+u2/r+(1/r)(∂v2/∂θ) and w2=∂v2/∂r+v2/r-(1/r)(∂u2/∂θ). In addition, u2 and v2 are the amplitudes of the radial and circumferential displacements of the inner soil, respectively; λ2, G2 and λ2′, G2′ are the real and imaginary parts of the complex Lame’s constants of the inner soil, respectively; ρ2 is the mass density of the inner soil.


## 3.3. Dynamic Equilibrium Equation of the Pile

The horizontal displacement of the pileup(z)eiωt is governed by the following equation:
(5)EpIp∂4∂z4(upeiωt)+m∂2∂t2(upeiωt)+f1(z)eiωt+f2(z)eiωt=0,
where Ep is the Young’s modulus of the pile, Ip is the second moment of area of the pile section, and m is the mass of the pile per unit length.


## 3.4. Boundary Conditions and Initial Conditions

The boundary conditions at the tops of the outer and inner soil are:(6)∂u1∂z|z=0=0,(7)∂u2∂z|z=0=0.The boundary conditions at the bottoms of the outer and inner soil are(8)u1|z=H=v1|z=H=0,(9)u2|z=H=v2|z=H=0.The boundary conditions at the bottom of the pile are(10)up|z=H=0,∂up∂z|z=H=0.The continuity conditions of displacements on the outer interface are(11)u1(r1,θ,z,t)=up(z,t)cosθ,(12)v1(r1,θ,z,t)=-up(z,t)sinθ.The continuity conditions of displacements on the inner interface are(13)u2(r2,θ,z,t)=up(z,t)cosθ,(14)v2(r2,θ,z,t)=-up(z,t)sinθ.


## 4.Solutions for the Governing Equations

### 4.1. Solutions for the Dynamic Equilibrium Equations of the Outer Soil

For the amplitudesu1 and v1, (3) can be expressed as
(15)[(λ1+2G1)+i(λ1′+2G1′)]∂e1∂r-1r(G1+iG1′)∂w1∂θ+(G1+iG1′)∂2u1∂z2=-ρ1ω2u1,[(λ1+2G1)+i(λ1′+2G1′)]∂e1r∂θ+(G1+iG1′)∂w1∂r+(G1+iG1′)∂2v1∂z2=-ρ1ω2v1.
The potential functions ϕ1(r,θ,z,t) and ψ1(r,θ,z,t) are introduced as
(16)u1=∂ϕ1∂r+1r∂ψ1∂θv1=1r∂ϕ1∂θ-∂ψ1∂r.
It is easily obtained that
(17)e1=∇2ϕ1,w1=-∇2ψ1.
Substituting (16) and (17) into (15), one obtains
(18)[(λ1+2G1)+i(λ1′+2G1′)]∇2ϕ1+[ρ1ω2+(G1+iG1′)∂2∂z2]ϕ1=0,(G1+iG1′)∇2ψ1+[ρ1ω2+(G1+iG1′)∂2∂z2]ψ1=0,
where ∇2=∂2/∂r2+(1/r)(∂/∂r)+(1/r2)(∂2/∂θ2).Equations (18) can be written as
(19){η12+i[(η12-2)Dv1+2Ds1]}∇2ϕ1+[(ωvs1)2+(1+iDs1)∂2∂z2]ϕ1=0,(20)(1+iDs1)∇2ψ1+[(ωvs1)2+(1+iDs1)∂2∂z2]ψ1=0,
where η=vl1/vs1=[(λ1+2G1)/G1], Dv1=λ1′/λ1, and Ds1=G1′/G1. In addition, vl1 and vs1 are the longitudinal and shear wave velocities of the outer soil, respectively; Dv1 and Ds1 are the hysteretic damping ratios of the outer soil.Using the method of separation of variables, givenϕ1=R11(r)Θ11(θ)Z11(z), (19) can be split into the following three equations:
(21)1R11d2R11dr2+1r1R11dR11dr-1r2m12=q112,1Z11d2Z11dz2=-g12,1Θ11d2Θ11dθ2=-m12,
where q112=((1+iDs1)g12-(ω/vs1)2)/(η12+i[(η12-2)Dv1+2Ds1]).The solutions for (21) can be easily obtained as
(22)R11=A11Km1(q11r)+B11Im1(q11r)Z11=C11sin(g1z)+D11cos(g1z)Θ11=E11sin(m1θ)+F11cos(m1θ),
where Km1(q11r) and Im1(q11r) are modified Bessel functions of the first and second kind of order m1, respectively, and A11, B11, C11, D11, E11, and F11 are undetermined coefficients.The potential functionϕ1 is expressed as
(23)ϕ1=[A11Km1(q11r)+B11Im1(q11r)]×[C11sin(g1z)+D11cos(g1z)]×[E11sin(m1θ)+F11cos(m1θ)].
Similarly, the potential function ψ1 can be obtained as
(24)ψ1=[A12Km1(q12r)+B12Im1(q12r)]×[C12sin(g1z)+D12cos(g1z)]×[E12sin(m1θ)+F12cos(m1θ)],
where q122=((1+iDs1)g12-(ω/vs1)2)/(1+iDs1) and A12, B12, C12, D12, E12, and F12 are undetermined coefficients.The displacement and stress of the outer soil vanish to zero whenr→∞. Hence, B11=B12=0.It is found from (11) and (12) that u1 is an even function of θ, and v1 is an odd function of θ. Thus, E11=F12=0 and m1=1.Substituting (23) and (24) into (6) and (8), one obtains
(25)C11=C12=0,g1n=(2n-1)π2H,n=1,2,3,….
Then, the potential functions ϕ1 and ψ1 are written as
(26)ϕ1=∑n=1∞A1nK1(q11nr)cos(g1nz)cosθ,ψ1=∑n=1∞A2nK1(q12nr)cos(g1nz)sinθ.
Thereafter, the displacements of the outer soil can be expressed as follows:
(27)u1=∑n=1∞{-A1n[1rK1(q11nr)+q11nK0(q11nr)]ssssssssss+A2n1rK1(q12nr)}cos(g1nz)cosθ(28)v1∑n=1∞{-A1n1rK1(q11nr)+A2nxxxxx×[1rK1(q12nr)+q12nK0(q12nr)]}cos(g1nz)sinθ.
Substituting (27) and (28) into (11) and (12), respectively, yields
(29)∑n=1∞{-A1n[1r1K1(q11nr1)+q11nK0(q11nr1)]ssss+A2n1r1K1(q12nr1)}cos(g1nz)=up,(30)∑n=1∞{-A1n1r1K1(q11nr1)+A2nxxx×[1r1K1(q12nr1)+q12nK0(q12nr1)]}cos(g1nz)=-up.
It can be obtained from (29) and (30) that
(31)A2n=γ1nA1n,
where γ1n=((2/r1)K1(q11nr1)+q11nK0(q11nr1))/((2/r1)K1(q12nr1)+q12nK0(q12nr1)).The radial displacement of the outer soil on the outer interface can be expressed as(32)u1|r=r1=∑n=1∞η1nA1ncos(g1nz)cosθ,
where η1n=-[(1/r1)K1(q11nr1)+q11nK0(q11nr1)]+γ1n(1/r1)K1(q12nr1).The horizontal resistance of the outer soil can be obtained as(33)f1=-∫02π(σr1cosθ-τrθ1sinθ)|r=r1r1dθf1=πr1G1∑n=1∞ξ1nA1ncos(g1nz),
where ξ1n=-{η12+i[Dv1(η12-2)+2Ds1]}q11n2K1(q11nr1)-(1+iDs1)γ1nq12n2K1(q12nr1).


### 4.2. Solutions for the Dynamic Equilibrium Equations of the Inner Soil

For the amplitudesu2 and v2, (4) can be expressed as
(34)[(λ2+2G2)+i(λ2′+2G2′)]∂e2∂r-1r(G2+iG2′)∂w2∂θ+(G2+iG2′)∂2u2∂z2=-ρ2ω2u2,[(λ2+2G2)+i(λ2′+2G2′)]∂e2r∂θ+(G2+iG2′)∂w2∂r+(G2+iG2′)∂2v2∂z2=-ρ2ω2v2.
The potential functions ϕ2(r,θ,z,t) and ψ2(r,θ,z,t) are introduced as
(35)u2=∂ϕ2∂r+1r∂ψ2∂θ,v2=1r∂ϕ2∂θ-∂ψ2∂r.
Thus,
(36)e2=∇2ϕ2,w2=-∇2ψ2.
Substituting (35) and (36) into (34), (34) can be expressed as:
(37){η22+i[(η22-2)Dv2+2Ds2]}∇2ϕ2+[(ωvs2)2+(1+iDs2)∂2∂z2]ϕ2=0,(1+iDs2)∇2ψ2+[(ωvs2)2+(1+iDs2)∂2∂z2]ψ2=0,
where η=vl2/vs2=[(λ2+2G2)/G2], Dv2=λ2′/λ2, and Ds2=G2′/G2. In addition, vl2 and vs2 are the longitudinal and shear wave velocities of the inner soil, respectively; Dv2 and Ds2 are the hysteretic damping ratios of the inner soil.Using the method of separation of variables, the potential functionsϕ2 and ψ2 are obtained as
(38)ϕ2=[A21Km2(q21r)+B21Im2(q21r)]×[C21sin(g2z)+D21cos(g2z)]×[E21sin(m2θ)+F21cos(m2θ)],ψ2=[A22Km2(q22r)+B22Im2(q22r)]×[C22sin(g2z)+D22cos(g2z)]×[E22sin(m2θ)+F22cos(m2θ)],
where q212=((1+iDs2)g22-(ω/vs2)2)/(η22+i[(η22-2)Dv2+2Ds2]) and q222=((1+iDs2)g22-(ω/vs2)2)/(1+iDs2).The displacement and stress of the inner soil are limited values whenr=0. Hence, A21=A22=0.u
2 is an even function of θ, and v2 is an odd function of θ. Thus, E21=F22=0 and m2=1.Substituting (38) into (7) and (9), one obtains
(39)C21=C22=0,g2n=(2n-1)π2H.
Then, the potential functions ϕ2 and ψ2 are written as
(40)ϕ2=∑n=1∞B1nI1(q21nr)cos(g2nz)cosθ,ψ2=∑n=1∞B2nI1(q22nr)cos(g2nz)sinθ.
Thereafter, the displacements of the inner soil can be expressed as follows:
(41)u2=∑n=1∞{B1n[q21nI0(q21nr)-1rI1(q21nr)]∑n=1∞u2=x+B2n1rI1(q22nr)}cos(g2nz)cosθ,v2=∑n=1∞{-B1n1rI1(q21nr)+B2nxxxxxxx×[1rI1(q22nr)-q22nI0(q22nr)]}cos(g2nz)sinθ.Substituting (41) into (13) and (14) yields
(42)∑n=1∞{B1n[q21nI0(q21nr2)-1r2I1(q21nr2)]xxxx+B2n1r2I1(q22nr2)}cos(g2nz)=up,∑n=1∞{-B1n1r2I1(q21nr2)xxxx+B2n[1r2I1(q22nr2)-q22nI0(q22nr2)]}xxxx×cos(g2nz)=-up.
It can be obtained from (42) that
(43)B2n=γ2nB1n,
where γ2n=(q21nI0(q21nr2)-(2/r2)I1(q21nr2))/(q22nI0(q22nr2)-(2/r2)I1(q22nr2)).The radial displacement of the inner soil on the inner interface can be expressed as(44)u2|r=r2=∑n=1∞η2nB1ncos(g2nz)cosθ,
where η2n=q21nI0(q21nr2)-(1/r2)I1(q21nr2)+γ2n(1/r2)I1(q22nr2).The horizontal resistance of the inner soil can be obtained as(45)f2=∫02π(σr2cosθ-τrθ2sinθ)|r=r2r2dθ=πr2G2∑n=1∞ξ2nB1ncos(g2nz),
where ξ2n=-{η22+i[Dv2(η22-2)+2Ds2]}q21n2I1(q21nr2)-(1+iDs2)γ2nq22n2I1(q22nr2).


### 4.3. Solution for the Dynamic Equilibrium Equation of the Pile

The amplitudeup can be expressed as
(46)EpIpd4updz4-mω2up=-f1-f2.
Substituting (33) and (45) into (46), one obtains
(47)d4updz4-β4up=-πEpIp[r1G1∑n=1∞ξ1nA1ncos(g1nz)d4updz4-β4up=-πEpIpx+r2G2∑n=1∞ξ2nB1ncos(g2nz)],
where β4=mω2/EpIp.The solution for (47) can be obtained as
(48)up=N1sin(βz)+N2cos(βz)+N3sinh(βz)+N4cosh(βz)-∑n=1∞ζ1nA1ncos(g1nz)-∑n=1∞ζ2nB1ncos(g2nz),
where
(49)ζ1n=πr1G1ξ1nEpIp(g1n4-β4),ζ2n=πr2G2ξ2nEpIp(g2n4-β4),
where N1, N2, N3, and N4 are undetermined coefficients.It can be obtained from (11), (13), (32), and (44) that
(50)N1sin(βz)+N2cos(βz)+N3sinh(βz)+N4cosh(βz)-∑n=1∞ζ1nA1ncos(g1nz)-∑n=1∞ζ2nB1ncos(g2nz)=∑n=1∞η1nA1ncos(g1nz),(51)N1sin(βz)+N2cos(βz)+N3sinh(βz)+N4cosh(βz)-∑n=1∞ζ1nA1ncos(g1nz)-∑n=1∞ζ2nB1ncos(g2nz)=∑n=1∞η2nB1ncos(g2nz).
Equations (50) and (51) can also be expressed as
(52)∑n=1∞η1nA1ncos(g1nz)=∑n=1∞η2nB1ncos(g2nz).It is found thatg1n=g2n. Given gn=g1n=g2n, it is obtained from (52) that
(53)B1n=η1nA1nη2n.Multiplyingcos(gnz) on both sides of (50) and then integrating on the interval [0,H], one obtains
(54)η1nA1nH2=∫0H[N1sin(βz)+N2cos(βz)+N3sinh(βz)xxxxxxxxx+N4cosh(βz)]cos(gnz)dzxxxxxxxxx-(ζ1n+ζ2nη1nη2n)A1nH2.
Thus,
(55)A1n=η2n(δ1nN1+δ2nN2+δ3nN3+δ4nN4),
where
(56)δ1n=2∫0Hsin(βz)cos(gnz)dz(η1nη2n+ζ1nη2n+ζ2nη1n)H,δ2n=2∫0Hcos(βz)cos(gnz)dz(η1nη2n+ζ1nη2n+ζ2nη1n)H,δ3n=2∫0Hsinh(βz)cos(gnz)dz(η1nη2n+ζ1nη2n+ζ2nη1n)H,δ4n=2∫0Hcosh(βz)cos(gnz)dz(η1nη2n+ζ1nη2n+ζ2nη1n)H.
Substituting (53) into (55) yields
(57)B1n=η1n(δ1nN1+δ2nN2+δ3nN3+δ4nN4).
Equation (48) can be expressed as
(58)up=N1[sin(βz)-∑n=1∞κ1ncos(gnz)]+N2[cos(βz)-∑n=1∞κ2ncos(gnz)]+N3[sinh(βz)-∑n=1∞κ3ncos(gnz)]+N4[cosh(βz)-∑n=1∞κ4ncos(gnz)],
where κ1n=(η2nζ1n+η1nζ2n)δ1n, κ2n=(η2nζ1n+η1nζ2n)δ2n, and κ3n=(η2nζ1n+η1nζ2n)δ3n, κ4n=(η2nζ1n+η1nζ2n)δ4n.With the displacement of the pile described by (58), the angle of rotation φ, the bending moment M, and the shear force Q are obtained as follows:(59)[upφMEpIpQEpIp]=[sin(βz)-∑n=1∞κ1ncos(gnz)cos(βz)-∑n=1∞κ2ncos(gnz)sinh(βz)-∑n=1∞κ3ncos(gnz)cosh(βz)-∑n=1∞κ4ncos(gnz)βcos(βz)+∑n=1∞κ1ngnsin(gnz)-βsin(βz)+∑n=1∞κ2ngnsin(gnz)βcosh(βz)+∑n=1∞κ3ngnsin(gnz)βsinh(βz)+∑n=1∞κ4ngnsin(gnz)-β2sin(βz)+∑n=1∞κ1ngn2cos(gnz)-β2cos(βz)+∑n=1∞κ2ngn2cos(gnz)β2sinh(βz)+∑n=1∞κ3ngn2cos(gnz)β2cosh(βz)+∑n=1∞κ4ngn2cos(gnz)-β3cos(βz)-∑n=1∞κ1ngn3sin(gnz)β3sin(βz)-∑n=1∞κ2ngn3sin(gnz)β3cosh(βz)-∑n=1∞κ3ngn3sin(gnz)β3sinh(βz)-∑n=1∞κ4ngn3sin(gnz)]×{N1N2N3N4}.Assuming the displacement, the angle of rotation, the bending moment, and the shear force at the pile head as U0, φ0, M0, and Q0, respectively, one obtains
(60)-N1∑n=1∞κ1n+N2(1-∑n=1∞κ2n)-N3∑n=1∞κ3n+N4(1-∑n=1∞κ4n)=U0N1+N3=φ0βN1∑n=1∞κ1ngn2-N2(β2-∑n=1∞κ2ngn2)+N3∑n=1∞κ3ngn2+N4(β2+∑n=1∞κ4ngn2)=M0EpIp-N1+N3=Q0EpIpβ3.
It can be obtained from (60) that
(61)N1=a1φ0-a2Q0,N2=a3U0+a4φ0+a5M0+a6Q0,N3=a7φ0+a8Q0,N4=a7U0+a8φ0+a9M0+a10Q0,
where
(62)a1=12β;a2=12EpIpβ3a3=γ(β2+∑n=1∞κ4ngn2);a4=[(β2+∑n=1∞κ4ngn2)(∑n=1∞κ1n+∑n=1∞κ3n)a4=x+(1-∑n=1∞κ4n)(∑n=1∞κ1ngn2+∑n=1∞κ3ngn2)]γa1;a5=(∑n=1∞κ4n-1)γEpIp;a6=[(β2+∑n=1∞κ4ngn2)(∑n=1∞κ3n-∑n=1∞κ1n)+(1-∑n=1∞κ4n)(∑n=1∞κ3ngn2-∑n=1∞κ1ngn2)]γa2a7=1+(∑n=1∞κ2n-2)a31-∑n=1∞κ4n;a8=(∑n=1∞κ1n+∑n=1∞κ3n)a1+(∑n=1∞κ2n-1)a41-∑n=1∞κ4n;a9=(∑n=1∞κ2n-1)a51-∑n=1∞κ4n;a10=(∑n=1∞κ2n-1)a6+(∑n=1∞κ3n-∑n=1∞κ1n)a21-∑n=1∞κ4n;γ=1((1-∑n=1∞κ2n)(β2+∑n=1∞κ4ngn2)+(β2-∑n=1∞κ2ngn2)(1-∑n=1∞κ4n))-1.Substituting (61) into (58), one obtains
(63)up=U0b1(z)+φ0b2(z)+M0b3(z)+Q0b4(z),
where
(64)b1(z)=[cos(βz)-∑n=1∞κ2ncos(gnz)]a3+[cosh(βz)-∑n=1∞κ4ncos(gnz)]a7,b2(z)=[sin(βz)-∑n=1∞κ1ncos(gnz)+sinh(βz)x-∑n=1∞κ3ncos(gnz)]a1+[cos(βz)-∑n=1∞κ2ncos(gnz)]a4+[cosh(βz)-∑n=1∞κ4ncos(gnz)]a8,b3(z)=[cos(βz)-∑n=1∞κ2ncos(gnz)]a5+[cosh(βz)-∑n=1∞κ4ncos(gnz)]a9,b4(z)=[sinh(βz)-∑n=1∞κ3ncos(gnz)-sin(βz)+∑n=1∞κ1ncos(gnz)]a2+[cos(βz)-∑n=1∞κ2ncos(gnz)]a6+[cosh(βz)-∑n=1∞κ4ncos(gnz)]a10.
Substituting (63) into (10), one obtains
(65)U0b1(H)+φ0b2(H)+M0b3(H)+Q0b4(H)=0,U0b1′(H)+φ0b2′(H)+M0b3′(H)+Q0b4′(H)=0.
It can be obtained from (65) that
(66)M0=b4(H)b2′(H)-b2(H)b4′(H)b3(H)b4′(H)-b4(H)b3′(H)φ0+b4(H)b1′(H)-b1(H)b4′(H)b3(H)b4′(H)-b4(H)b3′(H)U0,Q0=b3(H)b1′(H)-b1(H)b3′(H)b4(H)b3′(H)-b3(H)b4′(H)U0+b3(H)b2′(H)-b2(H)b3′(H)b4(H)b3′(H)-b3(H)b4′(H)φ0.
The horizontal complex dynamic stiffness Kh, rocking complex dynamic stiffness Kr, and horizontal-rocking complex dynamic stiffness Khr are expressed as
(67)Kh(iω)=b3(H)b1′(H)-b1(H)b3′(H)b4(H)b3′(H)-b3(H)b4′(H),Kr(iω)=b4(H)b2′(H)-b2(H)b4′(H)b3(H)b4′(H)-b4(H)b3′(H),Khr(iω)=b3(H)b2′(H)-b2(H)b3′(H)b4(H)b3′(H)-b3(H)b4′(H).


## 4.1. Solutions for the Dynamic Equilibrium Equations of the Outer Soil

For the amplitudesu1 and v1, (3) can be expressed as
(15)[(λ1+2G1)+i(λ1′+2G1′)]∂e1∂r-1r(G1+iG1′)∂w1∂θ+(G1+iG1′)∂2u1∂z2=-ρ1ω2u1,[(λ1+2G1)+i(λ1′+2G1′)]∂e1r∂θ+(G1+iG1′)∂w1∂r+(G1+iG1′)∂2v1∂z2=-ρ1ω2v1.
The potential functions ϕ1(r,θ,z,t) and ψ1(r,θ,z,t) are introduced as
(16)u1=∂ϕ1∂r+1r∂ψ1∂θv1=1r∂ϕ1∂θ-∂ψ1∂r.
It is easily obtained that
(17)e1=∇2ϕ1,w1=-∇2ψ1.
Substituting (16) and (17) into (15), one obtains
(18)[(λ1+2G1)+i(λ1′+2G1′)]∇2ϕ1+[ρ1ω2+(G1+iG1′)∂2∂z2]ϕ1=0,(G1+iG1′)∇2ψ1+[ρ1ω2+(G1+iG1′)∂2∂z2]ψ1=0,
where ∇2=∂2/∂r2+(1/r)(∂/∂r)+(1/r2)(∂2/∂θ2).Equations (18) can be written as
(19){η12+i[(η12-2)Dv1+2Ds1]}∇2ϕ1+[(ωvs1)2+(1+iDs1)∂2∂z2]ϕ1=0,(20)(1+iDs1)∇2ψ1+[(ωvs1)2+(1+iDs1)∂2∂z2]ψ1=0,
where η=vl1/vs1=[(λ1+2G1)/G1], Dv1=λ1′/λ1, and Ds1=G1′/G1. In addition, vl1 and vs1 are the longitudinal and shear wave velocities of the outer soil, respectively; Dv1 and Ds1 are the hysteretic damping ratios of the outer soil.Using the method of separation of variables, givenϕ1=R11(r)Θ11(θ)Z11(z), (19) can be split into the following three equations:
(21)1R11d2R11dr2+1r1R11dR11dr-1r2m12=q112,1Z11d2Z11dz2=-g12,1Θ11d2Θ11dθ2=-m12,
where q112=((1+iDs1)g12-(ω/vs1)2)/(η12+i[(η12-2)Dv1+2Ds1]).The solutions for (21) can be easily obtained as
(22)R11=A11Km1(q11r)+B11Im1(q11r)Z11=C11sin(g1z)+D11cos(g1z)Θ11=E11sin(m1θ)+F11cos(m1θ),
where Km1(q11r) and Im1(q11r) are modified Bessel functions of the first and second kind of order m1, respectively, and A11, B11, C11, D11, E11, and F11 are undetermined coefficients.The potential functionϕ1 is expressed as
(23)ϕ1=[A11Km1(q11r)+B11Im1(q11r)]×[C11sin(g1z)+D11cos(g1z)]×[E11sin(m1θ)+F11cos(m1θ)].
Similarly, the potential function ψ1 can be obtained as
(24)ψ1=[A12Km1(q12r)+B12Im1(q12r)]×[C12sin(g1z)+D12cos(g1z)]×[E12sin(m1θ)+F12cos(m1θ)],
where q122=((1+iDs1)g12-(ω/vs1)2)/(1+iDs1) and A12, B12, C12, D12, E12, and F12 are undetermined coefficients.The displacement and stress of the outer soil vanish to zero whenr→∞. Hence, B11=B12=0.It is found from (11) and (12) that u1 is an even function of θ, and v1 is an odd function of θ. Thus, E11=F12=0 and m1=1.Substituting (23) and (24) into (6) and (8), one obtains
(25)C11=C12=0,g1n=(2n-1)π2H,n=1,2,3,….
Then, the potential functions ϕ1 and ψ1 are written as
(26)ϕ1=∑n=1∞A1nK1(q11nr)cos(g1nz)cosθ,ψ1=∑n=1∞A2nK1(q12nr)cos(g1nz)sinθ.
Thereafter, the displacements of the outer soil can be expressed as follows:
(27)u1=∑n=1∞{-A1n[1rK1(q11nr)+q11nK0(q11nr)]ssssssssss+A2n1rK1(q12nr)}cos(g1nz)cosθ(28)v1∑n=1∞{-A1n1rK1(q11nr)+A2nxxxxx×[1rK1(q12nr)+q12nK0(q12nr)]}cos(g1nz)sinθ.
Substituting (27) and (28) into (11) and (12), respectively, yields
(29)∑n=1∞{-A1n[1r1K1(q11nr1)+q11nK0(q11nr1)]ssss+A2n1r1K1(q12nr1)}cos(g1nz)=up,(30)∑n=1∞{-A1n1r1K1(q11nr1)+A2nxxx×[1r1K1(q12nr1)+q12nK0(q12nr1)]}cos(g1nz)=-up.
It can be obtained from (29) and (30) that
(31)A2n=γ1nA1n,
where γ1n=((2/r1)K1(q11nr1)+q11nK0(q11nr1))/((2/r1)K1(q12nr1)+q12nK0(q12nr1)).The radial displacement of the outer soil on the outer interface can be expressed as(32)u1|r=r1=∑n=1∞η1nA1ncos(g1nz)cosθ,
where η1n=-[(1/r1)K1(q11nr1)+q11nK0(q11nr1)]+γ1n(1/r1)K1(q12nr1).The horizontal resistance of the outer soil can be obtained as(33)f1=-∫02π(σr1cosθ-τrθ1sinθ)|r=r1r1dθf1=πr1G1∑n=1∞ξ1nA1ncos(g1nz),
where ξ1n=-{η12+i[Dv1(η12-2)+2Ds1]}q11n2K1(q11nr1)-(1+iDs1)γ1nq12n2K1(q12nr1).


## 4.2. Solutions for the Dynamic Equilibrium Equations of the Inner Soil

For the amplitudesu2 and v2, (4) can be expressed as
(34)[(λ2+2G2)+i(λ2′+2G2′)]∂e2∂r-1r(G2+iG2′)∂w2∂θ+(G2+iG2′)∂2u2∂z2=-ρ2ω2u2,[(λ2+2G2)+i(λ2′+2G2′)]∂e2r∂θ+(G2+iG2′)∂w2∂r+(G2+iG2′)∂2v2∂z2=-ρ2ω2v2.
The potential functions ϕ2(r,θ,z,t) and ψ2(r,θ,z,t) are introduced as
(35)u2=∂ϕ2∂r+1r∂ψ2∂θ,v2=1r∂ϕ2∂θ-∂ψ2∂r.
Thus,
(36)e2=∇2ϕ2,w2=-∇2ψ2.
Substituting (35) and (36) into (34), (34) can be expressed as:
(37){η22+i[(η22-2)Dv2+2Ds2]}∇2ϕ2+[(ωvs2)2+(1+iDs2)∂2∂z2]ϕ2=0,(1+iDs2)∇2ψ2+[(ωvs2)2+(1+iDs2)∂2∂z2]ψ2=0,
where η=vl2/vs2=[(λ2+2G2)/G2], Dv2=λ2′/λ2, and Ds2=G2′/G2. In addition, vl2 and vs2 are the longitudinal and shear wave velocities of the inner soil, respectively; Dv2 and Ds2 are the hysteretic damping ratios of the inner soil.Using the method of separation of variables, the potential functionsϕ2 and ψ2 are obtained as
(38)ϕ2=[A21Km2(q21r)+B21Im2(q21r)]×[C21sin(g2z)+D21cos(g2z)]×[E21sin(m2θ)+F21cos(m2θ)],ψ2=[A22Km2(q22r)+B22Im2(q22r)]×[C22sin(g2z)+D22cos(g2z)]×[E22sin(m2θ)+F22cos(m2θ)],
where q212=((1+iDs2)g22-(ω/vs2)2)/(η22+i[(η22-2)Dv2+2Ds2]) and q222=((1+iDs2)g22-(ω/vs2)2)/(1+iDs2).The displacement and stress of the inner soil are limited values whenr=0. Hence, A21=A22=0.u
2 is an even function of θ, and v2 is an odd function of θ. Thus, E21=F22=0 and m2=1.Substituting (38) into (7) and (9), one obtains
(39)C21=C22=0,g2n=(2n-1)π2H.
Then, the potential functions ϕ2 and ψ2 are written as
(40)ϕ2=∑n=1∞B1nI1(q21nr)cos(g2nz)cosθ,ψ2=∑n=1∞B2nI1(q22nr)cos(g2nz)sinθ.
Thereafter, the displacements of the inner soil can be expressed as follows:
(41)u2=∑n=1∞{B1n[q21nI0(q21nr)-1rI1(q21nr)]∑n=1∞u2=x+B2n1rI1(q22nr)}cos(g2nz)cosθ,v2=∑n=1∞{-B1n1rI1(q21nr)+B2nxxxxxxx×[1rI1(q22nr)-q22nI0(q22nr)]}cos(g2nz)sinθ.Substituting (41) into (13) and (14) yields
(42)∑n=1∞{B1n[q21nI0(q21nr2)-1r2I1(q21nr2)]xxxx+B2n1r2I1(q22nr2)}cos(g2nz)=up,∑n=1∞{-B1n1r2I1(q21nr2)xxxx+B2n[1r2I1(q22nr2)-q22nI0(q22nr2)]}xxxx×cos(g2nz)=-up.
It can be obtained from (42) that
(43)B2n=γ2nB1n,
where γ2n=(q21nI0(q21nr2)-(2/r2)I1(q21nr2))/(q22nI0(q22nr2)-(2/r2)I1(q22nr2)).The radial displacement of the inner soil on the inner interface can be expressed as(44)u2|r=r2=∑n=1∞η2nB1ncos(g2nz)cosθ,
where η2n=q21nI0(q21nr2)-(1/r2)I1(q21nr2)+γ2n(1/r2)I1(q22nr2).The horizontal resistance of the inner soil can be obtained as(45)f2=∫02π(σr2cosθ-τrθ2sinθ)|r=r2r2dθ=πr2G2∑n=1∞ξ2nB1ncos(g2nz),
where ξ2n=-{η22+i[Dv2(η22-2)+2Ds2]}q21n2I1(q21nr2)-(1+iDs2)γ2nq22n2I1(q22nr2).


## 4.3. Solution for the Dynamic Equilibrium Equation of the Pile

The amplitudeup can be expressed as
(46)EpIpd4updz4-mω2up=-f1-f2.
Substituting (33) and (45) into (46), one obtains
(47)d4updz4-β4up=-πEpIp[r1G1∑n=1∞ξ1nA1ncos(g1nz)d4updz4-β4up=-πEpIpx+r2G2∑n=1∞ξ2nB1ncos(g2nz)],
where β4=mω2/EpIp.The solution for (47) can be obtained as
(48)up=N1sin(βz)+N2cos(βz)+N3sinh(βz)+N4cosh(βz)-∑n=1∞ζ1nA1ncos(g1nz)-∑n=1∞ζ2nB1ncos(g2nz),
where
(49)ζ1n=πr1G1ξ1nEpIp(g1n4-β4),ζ2n=πr2G2ξ2nEpIp(g2n4-β4),
where N1, N2, N3, and N4 are undetermined coefficients.It can be obtained from (11), (13), (32), and (44) that
(50)N1sin(βz)+N2cos(βz)+N3sinh(βz)+N4cosh(βz)-∑n=1∞ζ1nA1ncos(g1nz)-∑n=1∞ζ2nB1ncos(g2nz)=∑n=1∞η1nA1ncos(g1nz),(51)N1sin(βz)+N2cos(βz)+N3sinh(βz)+N4cosh(βz)-∑n=1∞ζ1nA1ncos(g1nz)-∑n=1∞ζ2nB1ncos(g2nz)=∑n=1∞η2nB1ncos(g2nz).
Equations (50) and (51) can also be expressed as
(52)∑n=1∞η1nA1ncos(g1nz)=∑n=1∞η2nB1ncos(g2nz).It is found thatg1n=g2n. Given gn=g1n=g2n, it is obtained from (52) that
(53)B1n=η1nA1nη2n.Multiplyingcos(gnz) on both sides of (50) and then integrating on the interval [0,H], one obtains
(54)η1nA1nH2=∫0H[N1sin(βz)+N2cos(βz)+N3sinh(βz)xxxxxxxxx+N4cosh(βz)]cos(gnz)dzxxxxxxxxx-(ζ1n+ζ2nη1nη2n)A1nH2.
Thus,
(55)A1n=η2n(δ1nN1+δ2nN2+δ3nN3+δ4nN4),
where
(56)δ1n=2∫0Hsin(βz)cos(gnz)dz(η1nη2n+ζ1nη2n+ζ2nη1n)H,δ2n=2∫0Hcos(βz)cos(gnz)dz(η1nη2n+ζ1nη2n+ζ2nη1n)H,δ3n=2∫0Hsinh(βz)cos(gnz)dz(η1nη2n+ζ1nη2n+ζ2nη1n)H,δ4n=2∫0Hcosh(βz)cos(gnz)dz(η1nη2n+ζ1nη2n+ζ2nη1n)H.
Substituting (53) into (55) yields
(57)B1n=η1n(δ1nN1+δ2nN2+δ3nN3+δ4nN4).
Equation (48) can be expressed as
(58)up=N1[sin(βz)-∑n=1∞κ1ncos(gnz)]+N2[cos(βz)-∑n=1∞κ2ncos(gnz)]+N3[sinh(βz)-∑n=1∞κ3ncos(gnz)]+N4[cosh(βz)-∑n=1∞κ4ncos(gnz)],
where κ1n=(η2nζ1n+η1nζ2n)δ1n, κ2n=(η2nζ1n+η1nζ2n)δ2n, and κ3n=(η2nζ1n+η1nζ2n)δ3n, κ4n=(η2nζ1n+η1nζ2n)δ4n.With the displacement of the pile described by (58), the angle of rotation φ, the bending moment M, and the shear force Q are obtained as follows:(59)[upφMEpIpQEpIp]=[sin(βz)-∑n=1∞κ1ncos(gnz)cos(βz)-∑n=1∞κ2ncos(gnz)sinh(βz)-∑n=1∞κ3ncos(gnz)cosh(βz)-∑n=1∞κ4ncos(gnz)βcos(βz)+∑n=1∞κ1ngnsin(gnz)-βsin(βz)+∑n=1∞κ2ngnsin(gnz)βcosh(βz)+∑n=1∞κ3ngnsin(gnz)βsinh(βz)+∑n=1∞κ4ngnsin(gnz)-β2sin(βz)+∑n=1∞κ1ngn2cos(gnz)-β2cos(βz)+∑n=1∞κ2ngn2cos(gnz)β2sinh(βz)+∑n=1∞κ3ngn2cos(gnz)β2cosh(βz)+∑n=1∞κ4ngn2cos(gnz)-β3cos(βz)-∑n=1∞κ1ngn3sin(gnz)β3sin(βz)-∑n=1∞κ2ngn3sin(gnz)β3cosh(βz)-∑n=1∞κ3ngn3sin(gnz)β3sinh(βz)-∑n=1∞κ4ngn3sin(gnz)]×{N1N2N3N4}.Assuming the displacement, the angle of rotation, the bending moment, and the shear force at the pile head as U0, φ0, M0, and Q0, respectively, one obtains
(60)-N1∑n=1∞κ1n+N2(1-∑n=1∞κ2n)-N3∑n=1∞κ3n+N4(1-∑n=1∞κ4n)=U0N1+N3=φ0βN1∑n=1∞κ1ngn2-N2(β2-∑n=1∞κ2ngn2)+N3∑n=1∞κ3ngn2+N4(β2+∑n=1∞κ4ngn2)=M0EpIp-N1+N3=Q0EpIpβ3.
It can be obtained from (60) that
(61)N1=a1φ0-a2Q0,N2=a3U0+a4φ0+a5M0+a6Q0,N3=a7φ0+a8Q0,N4=a7U0+a8φ0+a9M0+a10Q0,
where
(62)a1=12β;a2=12EpIpβ3a3=γ(β2+∑n=1∞κ4ngn2);a4=[(β2+∑n=1∞κ4ngn2)(∑n=1∞κ1n+∑n=1∞κ3n)a4=x+(1-∑n=1∞κ4n)(∑n=1∞κ1ngn2+∑n=1∞κ3ngn2)]γa1;a5=(∑n=1∞κ4n-1)γEpIp;a6=[(β2+∑n=1∞κ4ngn2)(∑n=1∞κ3n-∑n=1∞κ1n)+(1-∑n=1∞κ4n)(∑n=1∞κ3ngn2-∑n=1∞κ1ngn2)]γa2a7=1+(∑n=1∞κ2n-2)a31-∑n=1∞κ4n;a8=(∑n=1∞κ1n+∑n=1∞κ3n)a1+(∑n=1∞κ2n-1)a41-∑n=1∞κ4n;a9=(∑n=1∞κ2n-1)a51-∑n=1∞κ4n;a10=(∑n=1∞κ2n-1)a6+(∑n=1∞κ3n-∑n=1∞κ1n)a21-∑n=1∞κ4n;γ=1((1-∑n=1∞κ2n)(β2+∑n=1∞κ4ngn2)+(β2-∑n=1∞κ2ngn2)(1-∑n=1∞κ4n))-1.Substituting (61) into (58), one obtains
(63)up=U0b1(z)+φ0b2(z)+M0b3(z)+Q0b4(z),
where
(64)b1(z)=[cos(βz)-∑n=1∞κ2ncos(gnz)]a3+[cosh(βz)-∑n=1∞κ4ncos(gnz)]a7,b2(z)=[sin(βz)-∑n=1∞κ1ncos(gnz)+sinh(βz)x-∑n=1∞κ3ncos(gnz)]a1+[cos(βz)-∑n=1∞κ2ncos(gnz)]a4+[cosh(βz)-∑n=1∞κ4ncos(gnz)]a8,b3(z)=[cos(βz)-∑n=1∞κ2ncos(gnz)]a5+[cosh(βz)-∑n=1∞κ4ncos(gnz)]a9,b4(z)=[sinh(βz)-∑n=1∞κ3ncos(gnz)-sin(βz)+∑n=1∞κ1ncos(gnz)]a2+[cos(βz)-∑n=1∞κ2ncos(gnz)]a6+[cosh(βz)-∑n=1∞κ4ncos(gnz)]a10.
Substituting (63) into (10), one obtains
(65)U0b1(H)+φ0b2(H)+M0b3(H)+Q0b4(H)=0,U0b1′(H)+φ0b2′(H)+M0b3′(H)+Q0b4′(H)=0.
It can be obtained from (65) that
(66)M0=b4(H)b2′(H)-b2(H)b4′(H)b3(H)b4′(H)-b4(H)b3′(H)φ0+b4(H)b1′(H)-b1(H)b4′(H)b3(H)b4′(H)-b4(H)b3′(H)U0,Q0=b3(H)b1′(H)-b1(H)b3′(H)b4(H)b3′(H)-b3(H)b4′(H)U0+b3(H)b2′(H)-b2(H)b3′(H)b4(H)b3′(H)-b3(H)b4′(H)φ0.
The horizontal complex dynamic stiffness Kh, rocking complex dynamic stiffness Kr, and horizontal-rocking complex dynamic stiffness Khr are expressed as
(67)Kh(iω)=b3(H)b1′(H)-b1(H)b3′(H)b4(H)b3′(H)-b3(H)b4′(H),Kr(iω)=b4(H)b2′(H)-b2(H)b4′(H)b3(H)b4′(H)-b4(H)b3′(H),Khr(iω)=b3(H)b2′(H)-b2(H)b3′(H)b4(H)b3′(H)-b3(H)b4′(H).


## 5.Numerical Results and Analysis

In this section, numerical results are presented to verify the validity of the solution and analyze the horizontal vibration characteristics of the pile-soil system. In the numerical procedure, the summation ofn is 20. Unless otherwise specified, the following parameter values are used: H=10 m, r1=0.5m, r2=0.38 m, Ep=25 GPa, ρp=2.5 g/cm3, ρ1=ρ2=1.8 g/cm3, G1=G2=10 MPa, Ds1=Dv1=Ds2=Dv2=0.02, and v1=v2=0.3.


### 5.1. Verification

This solution is verified by being compared with Nogami's solution (1977) for a solid pile. Givenr2→0, the solution for the pipe pile of this study is simplified to that of a solid pile. Figure 2 indicates that the simplified solution of this study agrees well with the solution proposed by Nogami (1977) for the horizontal response of a solid pile. The stiffness oscillates in the low-frequency range and then diminishes to zero and attains negative values in the high-frequency range. The damping approaches to zero at first and increases almost linearly with the frequency in the high-frequency range.Comparison of the horizontal complex stiffness of this solution with Nogami's solution (1977).
(a)
Real part
(b)
Imaginary part


### 5.2. Analysis of the Vibration Characteristics of the Pile-Soil System

The complex dynamic stiffness on the pile head is often used to analyze the vibration characteristics of the pipe pile. Three types of complex stiffnesses are given in (67). The real parts of the complex stiffnesses represent the real stiffness, while the imaginary parts reflect the damping of the pile-soil system. The complex stiffnesses are influenced by many parameters such as the pile length, radii of pile section, and shear modulus of soil.Figures3, 4, and 5 show the influence of the pile length on the complex stiffnesses on the pile top. The stiffnesses decrease in low-frequency range (about 0~150 Hz) but increase in high-frequency range (150~300 Hz) as the pile length increases. In the low-frequency range, the dynamic stiffness of pile mainly depends on the static stiffness of pile. The static stiffness of pile decreases with the increase of the pile length since the pile tip is clamped. With increasing frequency, the pile-soil coupled vibration provides more soil resistances to the pile, so the stiffnesses become higher as the pile length increases in high-frequency range. The dampings increase with the increase of the pile length in the whole frequency range. However, when the pile length reaches a critical value, the stiffnesses and dampings show little change.Variation of horizontal complex stiffness of the pipe pile with the pile length.
(a)
Real part
(b)
Imaginary partVariation of rocking complex stiffness of the pipe pile with the pile length.
(a)
Real part
(b)
Imaginary partVariation of horizontal-rocking complex stiffness of the pipe pile with the pile length.
(a)
Real part
(b)
Imaginary partFigures6, 7, and 8 show the influence of the pile radii on the complex stiffnesses on the pile top. With the increase of r1 or decrease of r2, the stiffnesses and dampings increase. When r2=0, the stiffness and damping are close to those of the pile with r2=0.2 m. It shows that, when r2<0.2 m, the effect of the inner radius is negligible. Furthermore, it is seen that the stiffnesses and dampings of the pile with r1=0.6 m and r2=0.38 m are larger than those of the pile with r1=0.5 m and r2=0.2 m, while the section areas of the two piles are approximately equal. It shows that the dynamic stiffness of a pipe pile increases with the increase of the average radius of the pile section.Variation of horizontal complex stiffness of the pipe pile with the pile radii.
(a)
Real part
(b)
Imaginary partVariation of rocking complex stiffness of the pipe pile with the pile radii.
(a)
Real part
(b)
Imaginary partVariation of horizontal-rocking complex stiffness of the pipe pile with the pile radii.
(a)
Real part
(b)
Imaginary partFigures9, 10 and 11 show the influence of the shear modulus of soil on the complex stiffnesses on the pile top. The stiffnesses increase steeply with the increase of G1. The horizontal and horizontal-rocking damping increase, but the rocking damping decreases with the increase of G1. In the low-frequency range, G2 has negligible influence on the stiffnesses and dampings. In the high-frequency range (about 200~300 Hz), the stiffnesses increase, but the dampings decrease slightly with the increase of G2.Variation of horizontal complex stiffness of the pipe pile with the shear modulus of soil.
(a)
Real part
(b)
Imaginary partVariation of rocking complex stiffness of the pipe pile with the shear modulus of soil.
(a)
Real part
(b)
Imaginary partVariation of horizontal-rocking complex stiffness of the pipe pile with the shear modulus of soil.
(a)
Real part
(b)
Imaginary part


## 5.1. Verification

This solution is verified by being compared with Nogami's solution (1977) for a solid pile. Givenr2→0, the solution for the pipe pile of this study is simplified to that of a solid pile. Figure 2 indicates that the simplified solution of this study agrees well with the solution proposed by Nogami (1977) for the horizontal response of a solid pile. The stiffness oscillates in the low-frequency range and then diminishes to zero and attains negative values in the high-frequency range. The damping approaches to zero at first and increases almost linearly with the frequency in the high-frequency range.Comparison of the horizontal complex stiffness of this solution with Nogami's solution (1977).
(a)
Real part
(b)
Imaginary part


## 5.2. Analysis of the Vibration Characteristics of the Pile-Soil System

The complex dynamic stiffness on the pile head is often used to analyze the vibration characteristics of the pipe pile. Three types of complex stiffnesses are given in (67). The real parts of the complex stiffnesses represent the real stiffness, while the imaginary parts reflect the damping of the pile-soil system. The complex stiffnesses are influenced by many parameters such as the pile length, radii of pile section, and shear modulus of soil.Figures3, 4, and 5 show the influence of the pile length on the complex stiffnesses on the pile top. The stiffnesses decrease in low-frequency range (about 0~150 Hz) but increase in high-frequency range (150~300 Hz) as the pile length increases. In the low-frequency range, the dynamic stiffness of pile mainly depends on the static stiffness of pile. The static stiffness of pile decreases with the increase of the pile length since the pile tip is clamped. With increasing frequency, the pile-soil coupled vibration provides more soil resistances to the pile, so the stiffnesses become higher as the pile length increases in high-frequency range. The dampings increase with the increase of the pile length in the whole frequency range. However, when the pile length reaches a critical value, the stiffnesses and dampings show little change.Variation of horizontal complex stiffness of the pipe pile with the pile length.
(a)
Real part
(b)
Imaginary partVariation of rocking complex stiffness of the pipe pile with the pile length.
(a)
Real part
(b)
Imaginary partVariation of horizontal-rocking complex stiffness of the pipe pile with the pile length.
(a)
Real part
(b)
Imaginary partFigures6, 7, and 8 show the influence of the pile radii on the complex stiffnesses on the pile top. With the increase of r1 or decrease of r2, the stiffnesses and dampings increase. When r2=0, the stiffness and damping are close to those of the pile with r2=0.2 m. It shows that, when r2<0.2 m, the effect of the inner radius is negligible. Furthermore, it is seen that the stiffnesses and dampings of the pile with r1=0.6 m and r2=0.38 m are larger than those of the pile with r1=0.5 m and r2=0.2 m, while the section areas of the two piles are approximately equal. It shows that the dynamic stiffness of a pipe pile increases with the increase of the average radius of the pile section.Variation of horizontal complex stiffness of the pipe pile with the pile radii.
(a)
Real part
(b)
Imaginary partVariation of rocking complex stiffness of the pipe pile with the pile radii.
(a)
Real part
(b)
Imaginary partVariation of horizontal-rocking complex stiffness of the pipe pile with the pile radii.
(a)
Real part
(b)
Imaginary partFigures9, 10 and 11 show the influence of the shear modulus of soil on the complex stiffnesses on the pile top. The stiffnesses increase steeply with the increase of G1. The horizontal and horizontal-rocking damping increase, but the rocking damping decreases with the increase of G1. In the low-frequency range, G2 has negligible influence on the stiffnesses and dampings. In the high-frequency range (about 200~300 Hz), the stiffnesses increase, but the dampings decrease slightly with the increase of G2.Variation of horizontal complex stiffness of the pipe pile with the shear modulus of soil.
(a)
Real part
(b)
Imaginary partVariation of rocking complex stiffness of the pipe pile with the shear modulus of soil.
(a)
Real part
(b)
Imaginary partVariation of horizontal-rocking complex stiffness of the pipe pile with the shear modulus of soil.
(a)
Real part
(b)
Imaginary part


## 6.Conclusions

By considering the coupled vibration between the pile and both the outer and inner soil, the analytical solution of the horizontal response of a large-diameter pipe pile in viscoelastic soil layer has been derived in this paper. The validity of the solution proposed in this study is verified by being compared with Nogami's solution for a solid pile. A parametric study has been conducted to investigate the vibration characteristics and the effects of major parameters. The calculated results reveal that(1) the stiffnesses decrease in low-frequency range but increase in high-frequency range with the increase of the pile length. The dampings increase with the pile length in the whole frequency range. However, when the pile length reaches a critical value, the stiffnesses and dampings with different pile lengths have a little difference. (2) With the increase of the outer radius or decrease of the inner radius, the stiffnesses and dampings all increase. However, when the inner radius is smaller than 0.2 m, the effect of the inner radius is negligible. Moreover, the dynamic stiffness of a pipe pile increases with the increase of the average radius of the pile section. (3) The stiffnesses and the horizontal and horizontal-rocking damping increase, but the rocking damping decreases with the increase of the shear modulus of the outer soil. In the low-frequency range, the shear modulus of the inner soil has negligible influence on the stiffnesses and dampings. In the high-frequency range, the stiffnesses increase but the dampings decrease slightly with the increase of the shear modulus of the inner soil.In this paper, the pile tip is clamped. However, there may be some other conditions at the pile tip, such as pinned, free, and elastic supporting. The present solution can be easily extended to other boundary conditions. Further study is needed to investigate the horizontal response of pipe pile in these boundary conditions.



---

*Source: 269493-2013-11-13.xml*