Published: 30 September 2015

Coupled vibration of a concrete pipe pile with saturated soil due to longitudinal loading

Changjie Zheng1
Xuanming Ding2
Shuhong An3
Yuming Fan4
1, 3, 4College of Civil and Transportation Engineering, Hohai University, Nanjing, 210098, China
2Key Laboratory of New Technology for Construction of Cities in Mountain Area, Chongqing University, Chongqing, 400045, China
3Water Conservancy and Civil Engineering College, Shandong Agricultural University, Taian, 271018, China
Corresponding Author:
Xuanming Ding
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Abstract

This paper considers the longitudinal coupled vibration of an elastic bearing concrete pipe pile with the saturated soil. The outer and inner saturated soil are governed by the dynamic consolidation theory originally presented by Biot. The governing equations of soil are transferred to ordinary differential equations by Laplace transform. The volumetric strain and pore pressure of soil are obtained by directly solving the coupling equations of soil without introducing potential functions. The analytical expressions of the displacements and shear stresses of the soil are then obtained. The pile response is derived on the basis of 1D elastic theory and the perfect contacts between the pile and soils. The displacement and velocity of the pile in time domain are obtained by using numerical inverse transformation. Selected numerical results are presented to portray the influence of the existences of soils, pile geometry and dynamic permeability coefficients of soils on the vibration characteristics of the pipe pile. At last, the displacement response between a pipe pile and solid pile are compared.

1. Introduction

Many theories for pile dynamics have been developed to provide useful guidelines for engineering design of pile foundations and techniques for dynamic testing of piles. Nogami and Konagai [1, 2] used an approximate time domain method to compute the response of single piles subjected to harmonic excitation, and then extended the method to the response of pile groups. Gazetas et al. [3] developed a simple method for computing the dynamic steady-state axial response of floating pile groups embedded in homogeneous and non-homogeneous soil. Wang et al. [4] proposed an analytical solution for the vertical dynamic response of an inhomogeneous viscoelastic pile. The interaction between the pile and soil were simulated by Voigt model, therefore, the real coupled vibration between the pile and surrounding soil were not considered. Novak et al. [5] developed a plane strain model assuming an elastic layer consisting of many independent infinitesimally thin horizontal layers. This model reflected the fundamental characteristics of the interaction between the pile and the surrounding soil but neglected the strain of the soil in the vertical direction. Nogami and Novak [6] developed a more rigorous model of vertically loaded end-bearing pile by assuming that the surrounding soil has no radial displacement and considering the pile-soil interaction in the vertical direction. Militano and Rajapakse [7] used Laplace transform method to deduce the frequency domain solutions for an elastic pile subjected to transient torsional and axial loadings in multi-layered elastic soil, and then obtained the time domain solutions by using a numerical Laplace inversion procedure. Hung et al. [8] studied the Seismic behavior of pile in liquefiable soil by centrifuge shaking table tests.

The above studies are based on the assumption that the soil around pile is a single phase continuum. However, the soil is generally a two-phase material consisting of a solid skeleton and pores, which are saturated with fluid. The first theory of wave propagation in saturated medium was established by Biot [9, 10]. Based on this theory, Zhou et al. [11]studied the vertical transient dynamic response of a pile embedded in a poroelastic half-space. Using the differential operator splitting and variable separation method, Yu et al. [12] investigated the time-harmonic response of an end-bearing pile in elastic saturated soil layer. Wang et al. [13] studied the longitudinal vibration of an elastic bearing pile in saturated soil layer by using potential function method.

Most of the previous studies focus on the solid pile. The concrete pipe piles are also widely used in practical engineering, such as prestressed concrete pipe piles and large-diameter cast-in-situ concrete pipe piles [14-16]. The dynamic response of a pipe pile is different from that of a solid pile because of the interaction between the pile and the inner soil. In this paper, an analytical solution for the longitudinal response of an elastic bearing concrete pipe pile in saturated soil is proposed. The frequency domain solution is obtained by directly solving the governing equations of soils without introducing potential functions. The time domain displacement and velocity responses of the pipe pile are obtained using numerical inverse transformation method. Numerical results are presented to analyze the longitudinal vibration characteristics of the concrete pipe pile.

2. Governing equations

The computational model is shown in Fig. 1. A dynamic longitudinal uniform pressure p(t) is applied on the pile head. H is the pile length. r1 and r2 are the outer and inner radii of the pile section, respectively. λ1, λ2, μ1 and μ2 are the complex Lame’s constants of the outer and inner soil, respectively. ρ1, ρ2, n1, n2, kd1 and kd2 are the densities, porosities and permeability coefficients of the outer and inner soil, respectively. f1 and f2 are the outer and inner soil resistances, respectively. It is assumed that the outer and inner soil are homogeneous and viscoelastic with frequency-independent material damping of the hysteretic type, and the soil medium is two-phase material consisting of incompressible soil grains and fluid. The pile is modeled by 1D elastic theory and has perfect contact with the outer and inner soil. The soil resistances under the soils and pile are modeled by the Winkler model, where the elastic coefficients under the outer soil, inner soil and pile are k1, k2 and kp, respectively. The initial displacements and stresses of the pile-soil system are considered as zero.

Fig. 1Computational model

Computational model

2.1. Governing equations of the outer soil

Zienkiewicz et al. [17, 18] pointed out that the convective acceleration terms of the fluid can be neglected for computation convenience. The governing equations of the outer soil undergoing axisymmetric motion, in absence of compression of the solid matrix and pore fluid, can be written as [10]:

1
μ12ur1+λ1+μ1e1r-μ1ur1r2-p1r=ρ1u¨r1,
2
μ12uz1+λ1+μ1e1z-p1z=ρ1u¨z1,
3
-p1r=ρfu¨r1+w˙r1k'd1,
4
-p1z=ρfu¨z1+w˙z1k'd1,
5
u˙r1r+u˙r1r+u˙z1z+w˙r1r+w˙r1r+w˙z1z=0,

where:

2=2r2+1rr+2z2, e1=ur1r+ur1r+uz1z,

ur1 and uz1 are the radial and vertical displacements of the solid phase of the outer soil, respectively; wr1 and wz1 are the radial and vertical relative displacements between the fluid and solid phase of the outer soil, respectively; p1 is the pore pressure of the outer soil; kd1' is the dynamic permeability of the outer soil; ρf is the density of the pore fluid.

2.2. Governing equations of the inner soil

Similarly, the governing equations of the inner soil can be expressed as:

6
μ22ur2+λ2+μ2e2r-μ2ur2r2-p2r=ρ2u¨r2,
7
μ22uz2+λ2+μ2e2z-p2z=ρ2u¨z2,
8
-p2r=ρfu¨r2+w˙r2k'd2,
9
-p2r=ρfu¨r2+w˙r2k'd2,
10
u˙r2r+u˙r2r+u˙z2z+w˙r2r+w˙r2r+w˙z2z=0,

where:

e2=ur2r+ur2r+uz2z,

ur2 and uz2 are the radial and vertical displacements of the solid phase of the inner soil, respectively; wr2 and wz2 are the radial and vertical relative displacements between the pore fluid and solid phase of the inner soil, respectively; p2 is the pore pressure of the inner soil; kd2' is the dynamic permeability of the inner soil.

2.3. Wave equation of the concrete pipe pile

The motion of the concrete pipe pile is governed by the 1D elastic theory [19]:

11
EpA2upz2-2πr1f1-2πr2f2=ρpA2upt2,

where up(z,t) is the longitudinal displacement of the pile; Ep, ρp and A are the Young’s modulus, mass density and section area of the pile, respectively.

3. Solutions for the equations

3.1. Solutions for the governing equations of the outer soil

The differentiations (1)r+(1)r+(2)z and (3)r+(3)r+(4)z yield:

12
2p1=λ1+2μ12e1-ρ1e¨1,
13
2p1=e˙1k'd1-ρfe¨1.

Performing Laplace transform on Eqs. (12) and (13) and combining them, one obtains:

14
2E1-β1E1=0,

where β1=kd1'(ρ1-ρf)s2+s/kd1'(λ1+2μ1); E1 is the Laplace transform of e1.

The solution for Eq. (14) can be easily obtained as:

15
E1=A11K0(q11r)+B11I0(q11r)C11cos(g11z)+D11sin(g11z),

where q112=β1+g112; I0 and K0 are the first and second kind of modified Bessel functions of order zero, respectively.

The boundary conditions for the outer soil can be expressed as follows:

16
σz1 z=0=p1 z=0=0,
17
λ1+2μ1uz1z+k1uz1z=H=0,
18
ur1 r=uz1 r=σz1 r=0,
19
ur1 r=r1=wr1 r=r1=0,

where σz1 is the normal stress of the outer soil.

It can be obtained from Eqs. (16)-(18) that:

20
B11=C11=0,

and g11=g1n satisfy the following transcendental equation:

21
tang1nH=k1g1nλ1+μ1.

Thus E1 can be expressed as:

22
E1=n=1A11nK0(q11nr)sin(g1nz).

P1, the Laplace transform of p1 can be obtained from Eq. (12):

23
P1=n=1χ1A11nK0(q11nr)+A12nK0(q12nr)sin(g1nz),

where χ1=(λ1+2μ1)β1-ρ1s2; q12n=g1n.

Substituting Eqs. (22) and (23) into Eqs. (1)-(5), the solutions for the displacements of the outer soil can be obtained as:

24
Ur1=n=1ζ11nA11nK1(q11nr)+ζ12nA12nK1(q12nr)+A13nK1(q13nr)sin(g1nz),
25
Uz1=n=1ζ13nA11nK0(q11nr)+ζ14nA12nK0(q12nr)-q13ng1nA13nK0(q13nr)cos(g1nz),
26
Wr1=n=1δ11nA11nK1q11nr+δ12nA12nK1q12nr
-kd1'ρfs2A13nK1q13nrsing1nz,
27
Wz1=n=1[δ13nA11nK0(q11nr)+δ14nA12nK0q12nr
+k'd1ρfs2q13ng1nA13nK0q13nrcosg1nz,

where:

ζ11n=q11n(λ1+μ1-χ1)μ1β1-ρ1s2, ζ12n=q12nρ1s2, ζ13n=g1n(χ1-λ1-μ1)μ1β1-ρ1s2, ζ14n=-g1nρ1s2,
δ11n=kd1'χ1q11n-ζ11nρfs2, δ12n=kd1'q12n-ζ12nρfs2,
δ13n=-kd1'χ1g1n+ζ13nρfs2, δ14n=-kd1'g1n+ζ14nρfs2, q13n2=ρ1s2μ1+g1n2,

Ur1, Uz1, Wr1 and Wz1 are the Laplace transforms of ur1, uz1, wr1 and wz1, respectively.

Substituting Eqs. (24) and (26) into Eq. (19) yields:

28
A11n=ϑ11nA13n,
29
A12n=ϑ12nA13n,

where:

ϑ11n=(δ12n+ζ12nkd1'ρfs2)K1(q13nr1)(δ11nζ12n-ζ11nδ12n)K1(q11nr1), ϑ12n=(δ11n+ζ11nkd1'ρfs2)K1(q13nr1)(ζ11nδ12n-δ11nζ12n)K1(q12nr1).

The vertical displacement of the outer soil at the interface between the outer soil and the pile can be expressed as:

30
Uz1 r=r1=n=1ξ1nA13ncos(g1nz),

where ξ1n=-(q13n/g1n)K0(q13nr1)+ζ13nϑ11nK0(q11nr1)+ζ14nϑ12nK0(q12nr1).

The resistance of the outer soil, equal to the shear stress of the outer soil at the interface between the outer soil and the pile, can be obtained as:

31
F1=μ1n=1A13nκ1nK1(q13nr1)cos(g1nz),

where:

κ1n=-q13n2g1n+ζ13nq11nδ12n+ζ12nkd1'ρfs2δ11nζ12n-ζ11nδ12n+ζ14nq12nδ11n+ζ11nkd1'ρfs2ζ11nδ12n-δ11nζ12n,

F1(z,s) is the Laplace transform of f1(z,t).

3.2. Solutions for the governing equations of the inner soil

A similar solving procedure can be followed for the inner soil. The boundary conditions of the inner soil are as follows:

32
σz2 z=0=p2 z=0=0,
33
λ2+2μ2uz2z+k2uz2z=H=0,
34
ur2 r=r2=wr2 r=r2=0,
35
ur2 r=0<, uz2 r=0<, σz2 r=0<,

where σz2 is the normal stress of the inner soil.

In view of the above boundary conditions of the inner soil, the solutions for the inner soil can be derived as follows:

36
E2=n=1A23nϑ21nI0(q21nr)+ϑ22nI0(q22nr)sin(g2nz),
37
P2=n=1A23nϑ21nχ2I0(q21nr)+ϑ22nχ22nI0(q22nr)sin(g2nz),
38
Ur2=n=1A23nϑ21nζ21nI1(q21nr)+ϑ22nζ22nI1(q22nr)+I1(q23nr)sin(g2nz),
39
Uz2=n=1A23nϑ21nζ23nI0(q21nr)+ϑ22nζ24nI0(q22nr)+q23ng2nI0(q23nr)cos(g2nz),
40
Wr2=n=1A23nϑ21nδ21nI1(q21nr)+ϑ22nδ22nI1(q22nr)-kd2'ρfs2I1(q23nr)sin(g2nz),
41
Wz2=n=1ϑ21nδ23nI0(q21nr)+ϑ22nδ24nI0(q22nr)
-kd2'ρfs2q23ng2nI0q23nrcosg2nz,

where:

q21n2=β2+g2n2, q22n=g2n, q23n2=ρ2s2μ2+g2n2,
ϑ21n=(δ22n+ζ22nkd2'ρfs2)I1(q23nr2)(δ21nζ22n-ζ21nδ22n)I1(q21nr2), ϑ22n=(δ21n+ζ21nkd2'ρfs2)I1(q23nr2)(ζ21nδ22n-δ21nζ22n)I1(q22nr2),
β2=kd2'(ρ2-ρf)s2+skd2'(λ2+2μ2), tang2nH=k2g2nλ2+2μ2, ζ21n=q21n(χ2-λ2-μ2)μ2β2-ρ2s2,
ζ22n=-q22nρ2s2, ζ23n=g2nλ2+μ2-χ2μ2β2-ρ2s2, ζ24n=g2nρ2s2, χ2=λ2+2μ2β2-ρ2s2,
δ21n=-kd2'χ2q21n+ζ21nρfs2, δ22n=-kd2'q22n+ζ22nρfs2,
δ23n=kd2'χ2g2n-ρfs2ζ23n, δ24n=kd2'g2n-ρfs2ζ24n,

E2, P2, Ur2, Uz2, Wr2 and Wz2 are the Laplace transforms of e2, p2, ur2, uz2, wr2 and wz2, respectively.

The vertical displacement of the inner soil at the interface between the inner soil and the pile is expressed as:

42
Uz2 r=r2=n=1ξ2nA23ncos(g2nz),

where ξ2n=(q23n/g2n)I0(q23nr2)+ϑ21nζ23nI0(q21nr2)+ϑ22nζ24nI0(q12nr2).

The resistance of the inner soil can be expressed as:

43
F2=μ2n=1A23nκ2nI1(q23nr2)cos(g2nz),

where:

κ2n=q23n2g2n+ζ23nq21nδ22n+ζ22nkd2'ρfs2δ21nζ22n-ζ21nδ22n+ζ24nq22nδ21n+ζ21nkd2'ρfs2ζ21nδ22n-δ21nζ22n,

F2(z,s) is the Laplace transform of f2(z,t).

3.3. Solution for the wave equation of the pile

Performing Laplace transform on Eq. (11) and substituting Eqs. (30) and (43) into Eq. (11), one obtains:

44
2Up(z,s)z2-s2Cp2Up(z,s)=2πr1EpAμ1n=1A13nκ1nK1(q13nr1)cos(g1nz)
+2πr2EpAμ2n=1A23nκ2nI1(q23nr2)cos(g2nz),

where Cp=Ep/ρp; Up(z,s) is the Laplace transform of up(z,t).

The solution for Eq. (44) can be easily derived as:

45
Up=M1esCpz+M2e-sCpz+n=1θ1nA13ncosg1nz+n=1θ2nA23ncosg2nz,

where:

θ1n=2πr1μ1κ1nK1(q13nr1)ρpA(Cp2g1n2-s2), θ2n=2πr2μ2κ2nI1(q23nr2)ρpA(Cp2g2n2-s2).

The displacements of the pile and the soils at the pile-soil interfaces are continuous:

46
Uz1 r=r1=Up,
47
Uz2 r=r2=Up.

Using (46) and (47), it can be obtained that:

48
n=1ξ1nA13ncos(g1nz)=n=1ξ2nA23ncos(g2nz).

Assuming λ1=λ2, μ1=μ2 and k1=k2, then g1n=g2n. Given gn=g1n=g2n, one can obtain:

49
A23n=ξ1nA13nξ2n.

By invoking the orthogonality of eigenfunctions cosh(gnz), A13n and A23n can be obtained:

50
A13n=ξ2nγ1nM1+γ2nM2,
51
A23n=ξ1nγ1nM1+γ2nM2,

where:

γ1n=0HesCpzcos(gnz)dz(ξ1nξ2n-θ1nξ2n-θ2nξ1n)Ln, γ2n=0He-sCpzcos(gnz)dz(ξ1nξ2n-θ1nξ2n-θ2nξ1n)Ln,
Ln=0Hcos(gnz)cos(gnz)dz.

The boundary conditions at the top and bottom of the pile are as follows:

52
Upzz=0=-PsEp,
53
z=H0,

where P(s) is the Laplace transform of p(t).

Substituting Eq. (45) into Eqs. (52) and (53), M1 and M2 can be obtained:

54
M1=CpP(s)(γ-1)sEp,
55
M2=γCpP(s)(γ-1)sEp,

where:

γ=-kp+EpAsCpesCpH+n=1θ1nξ2n+θ2nξ1nγ1nkpcosgnH-EpAgnsingnHkp-EpAsCpe-sCpH+n=1θ1nξ2n+θ2nξ1nγ2nkpcosgnH-EpAgnsingnH.

All of the coefficients have been determined. Substituting the above coefficients into Eq. (45), the vertical displacement response of the pile can be expressed as:

56
Upz,s=Cppsγ-1sEpesCpz+γe-sCpz+n=1θ1nξ2n+θ2nξ1nγ1n+γ2nγcosgnz.

The velocity response of the pile in Laplace domain Vp can be written as:

57
Vpz,s=sUp
=Cppsγ-1EpesCpz+γe-sCpz+n=1θ1nξ2n+θ2nξ1nγ1n+γ2nγcosgnz.

Given s=iω, the Laplace transform can be considered as one-side Fourier transform. The frequency domain velocity admittance of the pipe pile at the pile head is defined as [4, 20]:

58
Hv=iωUp(0,iω)P(iω)=Cpγ+1+n=1(θ1nξ2n+θ2nξ1n)(γ1n+γ2nγ)Ep(γ-1).

The time-domain displacement and velocity of the pile can be obtained by numerical inverse Fourier transform:

59
upz,t=12π-+Upz,iωeiωtdω,
60
vpz,t=12π-+Vpz,iωeiωtdω.

4. Numerical results and discussions

Numerical results are presented to verify the correctness of this solution and analyze the vibration characteristics of the pile-soil system. In the following analysis, the velocity admittance in frequency domain as well as the displacement and velocity responses in time domain of the pile are expressed as dimensionless values.

4.1. Velocity admittance in frequency domain

To verify the accuracy of the solution derived in this paper, the present solution is compared with that for an end bearing solid pile. By setting r2=0 and kp=k1=k2=, the elastic bearing pipe pile can be reduced to an end bearing solid pile. Fig. 2 shows the comparison of the velocity admittance between the reduced solution in this paper with the solution proposed by Yu et al. [13], where the frequency is transformed to dimensionless values by order of θ=ωH/Cp. It can be noted that great agreement is obtained between the two solutions.

The difference between the pipe pile and solid pile is the existence of the inner soil. The outer soil and inner soil both provide resistance to the pile under dynamic loading. Fig. 3 shows the influence of the existences of the outer and inner soil on the velocity admittance of pile. It is noted that the oscillation amplitudes when only the outer soil exists are larger than those when both of the outer and inner soil exist. It shows that the existence of the inner soil has valuable influence, and the interaction between the pile and the inner soil cannot be neglected. Furthermore, the oscillation amplitudes of the velocity admittance of pile when only the inner soil exists are larger than those when only the outer soil exists, especially at the initial frequencies. It indicates that the outer soil provides more resistance than the inner soil, and the influence of the outer soil is more obvious than the inner soil.

Fig. 2Comparison of the velocity admittance between the present solution and Yu et al. (2008)

Comparison of the velocity admittance between the present solution and Yu et al. (2008)

Fig. 3The influence of the existences of the outer and inner soil on the velocity admittance

The influence of the existences of the outer and inner soil on the velocity admittance

The influence of the geometry of the pipe pile on the velocity admittance is shown in Fig. 4. It is noted that with the decrease of H/r1, the oscillation amplitudes increase. That means the smaller slenderness ratio, the larger oscillation amplitudes of velocity admittance. It is because the impedance of pile and radiation damping increase as the slenderness ratio increases. It is also found that the oscillation amplitudes decrease with the decrease of r1/r2 . That means the smaller thickness of the pile wall, the smaller oscillation amplitudes of velocity admittance. The influence of the dynamic permeability coefficients of the soils on the velocity admittance is shown in Fig. 5. It is noted that the oscillation amplitudes of the velocity admittance increase slightly with the increase of the dynamic permeability coefficients of the soils.

Fig. 4The influence of the pile geometry on the velocity admittance

The influence of the pile geometry on the velocity admittance

Fig. 5The influence of the dynamic permeability coefficients of the soils on the velocity admittance

The influence of the dynamic permeability coefficients of the soils on the velocity admittance

4.2. Velocity response in time domain

The semisinusoidal impulse load is usually used to simulate the loading in dynamic pile testing [4, 20]. The time history of the semisinusoidal impulse is shown in Fig. 6(a), where P0 is the amplitude of the impulse load; T0 is the pulse width; τ is the dimensionless time, and τ=tCp/H. Firstly the results of the proposed solution are compared with 3D dynamic finite element analysis performed with the software ABAQUS. Eight-node, isoparametric finite elements are used to model the pipe pile and soil (Fig. 7). Infinite elements were placed on the surrounding boundary to reduce the total number of elements and provide “quiet” boundaries to prevent stress wave reflected at the far boundaries coming back to the domain of interest. The comparison of the velocity response at the pile head subjected to a semisinusoidal impulse is shown in Fig. 8. It can be seen that the proposed analytical solution are in close agreement with the FEM result.

The influences of the soil existences, pile geometry and dynamic permeability coefficients of soils on the time domain velocity response at the pile head subjected to a semisinusoidal impulse are presented in Figs. 9-11. It can be found from Fig. 9 that the reflected wave signals of the pile top in the velocity curves when only the outer soil exists are larger than those when both the outer and inner soil exist, while those when only the inner soil exists are much larger than the two others. Fig. 10 indicates that the reflected wave signals increase with the decrease of H/r1 and decrease with the decrease of r1/r2. Fig. 11 shows that the reflected wave signals increase slightly with the increase of the dynamic permeability coefficients of the soils.

Fig. 6Transient loading considered in the analysis

Transient loading considered in the analysis

a) Semisinusoidal impulse

Transient loading considered in the analysis

b) Triangular impulse

Fig. 7The 3D FEM mesh used for the numerical simulation of the problem

The 3D FEM mesh used for the numerical simulation of the problem

Fig. 8Comparison between the analytical solution and FEM result

Comparison between the analytical solution and FEM result

Fig. 9The influence of the existences of the outer and inner soil on the reflected wave signals

The influence of the existences of the outer and inner soil on the reflected wave signals

4.3. Displacement response in time domain

The triangular pulse is one of the commonly used types of transient loading in vibroengineering, which is shown in Fig. 6(b). Fig. 12 shows the time histories of non-dimensional displacement of the pile head of a pipe pile, a solid pile with the same area and a solid pile with the same perimeter. It is found that the displacement of a pipe pile is much smaller than that of a solid pile. The peak displacement of the pipe pile is about 23.5 % smaller than that of the solid pile with the same area, and almost half of that of the solid pile with the same perimeter. It can be concluded that use of pipe piles can efficiently decrease the vibration range of pile foundations.

Fig. 10The influence of the pile geometry on the reflected wave signals

The influence of the pile geometry on the reflected wave signals

Fig. 11The influence of the dynamic permeability coefficients of the soils on the reflected wave signals

The influence of the dynamic permeability coefficients of the soils on the reflected wave signals

Fig. 12Time histories of non-dimensional displacement of the pile head between pipe piles and solid piles

Time histories of non-dimensional displacement of  the pile head between pipe piles and solid piles

5. Conclusions

The longitudinal coupled vibration of a concrete pipe pile with saturated soil is studied in this paper. The frequency domain solution is obtained by directly solving the governing equations of soils without introducing potential functions. The solving process and the solution are more concise, and the physical conceptions are more precise. The time domain results are obtained by using numerical inverse transformation. Some selected numerical results are given to analyze the effects of pile geometry and soil existences on the coupled vibration characteristics of the concrete pipe pile and saturated soil. The following conclusions can be obtained:

1) For a pipe pile, the inner soil also provides resistance during the vibration. The dynamic interaction between the pile and the inner soil must be considered. Nevertheless, the influence of the outer soil is much larger than that of the inner soil.

2) The oscillation amplitudes of velocity admittance and reflected wave signals increase with the decrease of slenderness ratio, while they decrease with the decrease of the outer and inner radius ratio. Furthermore, they increase with the increase of the dynamic permeability coefficients of the soils.

3) The peak displacement of the pipe pile is about 23.5 % smaller than that of the solid pile with the same area, and almost half of that of the solid pile with the same perimeter. Use of pipe piles can efficiently decrease the vibration range of pile foundations.

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About this article

Received
30 December 2014
Accepted
11 April 2015
Published
30 September 2015
SUBJECTS
Mechanical vibrations and applications
Keywords
concrete pipe pile
coupled vibration
saturated soil
velocity response
Acknowledgements

This work was supported by the National Natural Science Joint High Speed Railway Key Program Foundation of China (Grant No. U1134207), the Program for New Century Excellent Talents in University (Grant No. NCET-12-0843) and the National Natural Science Foundation of China (Grant No. 51378177).