J. Cent. South Univ. (2016) 23: 891-897
DOI: 10.1007/s11771-016-3136-7
Vertical vibration of a large diameter pipe pile considering transverse inertia effect of pile
ZHENG Chang-jie(郑长杰), LIU Han-long(刘汉龙), DING Xuan-ming(丁选明), ZHOU Hang(周航)
Key Laboratory of New Technology for Construction of Cities in Mountain Area
(Chongqing University), Chongqing 400045, China
Central South University Press and Springer-Verlag Berlin Heidelberg 2016
Abstract: Considering the transverse inertia effect of pile, the vertical dynamic response of a large diameter pipe pile in viscoelastic soil layer is studied. The wave propagations in the outer and inner soil are simulated by three-dimensional elastodynamic theory and those in the pile are simulated by Rayleigh-Love rod theory. The vertical and radial displacements of the outer and inner soil are obtained by utilizing Laplace transform technique and differentiation on the governing equations of soils. Then, based on the continuous conditions between the pile and soils, the displacements of the pile are derived. The frequency domain velocity admittance and time domain velocity response of the pile top are also presented. The solution is compared to a classical rod model solution to verify the validity. The influences of the radii and Poisson ratio of pile on the transverse inertia effect of pile are analyzed. The parametric study shows that Poisson ratio and outer radius of pile have significant influence on the transverse inertia effect of large diameter pipe piles, while the inner radius has little effect.
Key words: large diameter pipe pile; vertical vibration; Rayleigh-Love rod; transverse inertia effect
1 Introduction
Studies on the pile–soil interaction have important practical applications in the engineering design for pile foundations subjected to dynamic loads. The Winkler model is convenient to practical application and has been used extensively to investigate the vibrations of piles [1–6]. However, it neglects the real coupled vibration between the pile and soil. NOVAK [7] developed a plane strain model to simulate the vertical vibration of a pile in viscoelastic soil medium in which the strain of soil in the vertical direction was neglected. Afterwards, NOGAMI and NOVAK [8] studied the vertical soil–pile interaction considering the strain of soil in the vertical direction. However, the radial displacement of the soil was neglected, and therefore the effects of the longitudinal waves were neglected. WU et al [9] developed a more rigorous solution for the vertical response of an end bearing pile in viscoelastic soil layer by considering both vertical and radial displacements of the soil. The above mentioned studies all consider the pile governed by 1D classical rod theory, and thus, the radial displacement of the pile was neglected.
Large diameter piles have been widely used with the increasing need of bearing capacity of pile foundations. The three-dimensional effect of large diameter piles has significant influence on the wave propagation in pile [10–11]. LI and GONG [12] discussed the three- dimensional effect of large diameters and pointed out that if the slenderness ratio of a pile is not larger than 10, the 1D classical rod theory is no more applicable and three-dimensional effect should be considered. Furthermore, they suggested that the three-dimensional wave equation was too complicated, and the Rayleigh- Love rod theory considering the transverse inertia effect of pile can be used to simulate the large diameter piles. LI et al [13] applied the Rayleigh-Love rod model to investigate the vertical vibration of an end-bearing large diameter pile in a saturated soil layer. YANG and TANG [14] studied the vertical vibration of pile with transverse inertia effect in viscoelastic saturated soil based on the Novak layer method. L
et al [15] also used this model to investigate the longitudinal vibration of a large diameter pile in layered soil. They found that the reflected wave signal results calculated by using the Rayleigh-Love rod model are closer to the actually measured results than the results based on the classical 1D rod model.
A new type of pile called cast-in-situ concrete large diameter pipe pile (referred to as PCC pile) has been developed and extensively applied in China [16–18]. Besides PCC pile, many other large-diameter pipe piles are also widely used in practical engineering [19]. DING et al [20] studied the vertical dynamic response of large diameter pipe piles in layered soil. ZHENG et al [21] investigated the vertical vibration of large diameter pipe piles in viscoelastic soil considering the real three- dimensional wave effect of soil. However, they used the classical 1D rod theory to simulate the large diameter pipe pile, and thus, the radial displacement of pile was not considered. In this work, the Rayleigh-Love rod theory is adopted to study the vertical vibration of a large diameter pipe pile in viscoelastic soil. A parametric study is further conducted to discuss the transverse inertia effect of large diameter pipe piles.
2 Equations and boundary conditions
The computational model is shown in Fig. 1. A vertical 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. f1 and f2 are the vertical resistances of the outer and inner soil, respectively. It is assumed that the outer soil and inner soil are homogeneous and viscoelastic with frequency- independent material damping of the hysteretic type. The soil layers overlie a rigid base and the normal stresses at the surfaces of the soil layers are zero. The pile is modeled as a Rayleigh- Love rod and the supporting medium is rigid. The pile has perfect contacts with the outer and inner soil. The initial displacements and stresses of the pile–soil system are considered as zero.
Fig. 1 Computational model
2.1 Governing equations of soil
Based on the elastodynamic theory, the governing equations of the soil in axisymmetric cylindrical coordinate system can be expressed as
(1)
(2)
where 
urj(z, r, t) and uzj(z, r, t) are the radial and vertical displacements of the soil, respectively;
the dilatation of the soil; λj and μj are the complex Lame’s constants of the soil, and 
where ξj is the hysteretic damping and 
ρj is the density of the soil; j=1, 2; when j=1, the above parameters and equations correspond to the outer soil and when j=2 they correspond to the inner soil.
2.2 Governing equation of large diameter pipe pile
Considering the transverse inertia effect of the large diameter pipe pile, the vertical displacement of the pile is governed by the Rayleigh-Love rod theory [22]:
(3)
where up(z,t) is the vertical displacement of the pile; 
Ep, A, vp, ρp and Ip are the elastic modulus, mass density, section area, Poisson ratio and polar moment of inertia of the pile, respectively.
2.3 Boundary conditions
The normal stresses of the outer and inner soil at the top surfaces are zero:
(4)
(5)
where σz1 and σz2 are the normal stresses of the outer and inner soil, respectively.
The vertical displacements of the outer and inner soil at the bottoms are zero:
(6)
(7)
The displacements and stresses of the outer soil at the infinity in the horizontal direction are zero:
(8)
The displacements and stresses of the inner soil at r=0 are limited values:
(9)
The boundary condition on the top of the pile can be expressed as
(10)
The bottom of the pile is fixed, where the boundary condition can be expressed as
(11)
The displacement and stress continuity conditions r=r1 and r=r2 are as follows:
(12)
(13)
(14)
where τzr1 and τzr2 are the shear stresses of the outer and inner soil, respectively.
3 Solutions for equations
3.1 Solutions for governing equations of outer soil
Performing Laplace transform on Eqs. (1) and (2) yields
(15)
(16)
where E1, Ur1 and Uz1 are the Laplace transforms of e1, ur1 and uz1; s denotes a complex variable in Laplace domain.
The differentiation of Eqs. (15) and (16) yields
(17)
The general solution for Eq. (17) can be easily obtained as
(18)
where 
I0(·) and K0(·) are the first and second kinds of modified Bessel functions of order zero, respectively; A1, B1, C1 and D1 are coefficients to be determined from the boundary conditions.
Substituting Eq. (18) into Eqs. (4), (6) and (8) yields
(19)
(20)
Hence, E1 can be written as
(21)
By substituting Eq. (21) into Eqs. (15) and (16), the solutions for Ur1 and Uz1 can be obtained as
(22)
(23)
where
A2n and A3n are undetermined coefficients.
In the view that 
it can be obtained that
(24)
The radial and vertical displacements of the outer soil at the interface between the outer soil and pile can be expressed as
(25)
(26)
where 
The vertical resistance of the outer soil, as defined in Eq. (14), can be obtained as
(27)
where
3.2 Solutions for governing equations of inner soil
A similar solving procedure of the outer soil can be followed for the inner soil. In the view of the boundary conditions of the inner soil, the solutions can be obtained as
(28)
(29)
(30)
where 
The radial and vertical displacements of the inner soil at interface between the inner soil and pile can be expressed as
(31)
(32)
where 
The vertical resistance of the inner soil can be obtained as
(33)
where
.
3.3 Solution for governing equation of large diameter pipe pile
It is noted that 
Given 
and substituting Eqs. (27) and (33) into Eq. (3), one obtains
(34)
where Up is the Laplace transform of up.
The solution for Eq. (34) can be easily obtained as
(35)
where
Substituting Eq. (35) into Eqs. (10) and (11) yields
(36)
(37)
where P(s) is the Laplace transform of p(t).
Substituting Eq. (35) into Eq. (12) yields
(38)
(39)
The strain of the pile can be expressed as
(40)
It is assumed that the radial displacement of the pile at the center of the pile wall is zero. Then, the radial displacements of pile at the interfaces between the pile and soils can be expressed as
(41)
(42)
where 
In the view of Eq. (13), it is obtained that
(43)
(44)
The coefficients A1n, A2n, B1n and B2n can be obtained from Eqs. (38), (39), (43) and (44). Due to the complexity, the expressions of them are not given here. Substituting the determined coefficients into Eq. (35), the vertical displacement of the pile can be obtained.
By setting s=iω, the frequency domain impedance function of the pile at the pile top is defined as
(45)
Furthermore, the frequency domain velocity admittance at the pile top is defined by
(46)
When a semisinusoidal impulse force p(t)= 
is applied on the pile top, the time domain velocity response of the pile top v(t) can be obtained by the inverse Fourier transform method:
(47)
4 Comparison and parametric studies
In this section, numerical results are presented to verify the validity of the solution and analyze the transverse inertia effect of the large diameter pipe pile. Unless otherwise specified, the following parameter values are used: H=5 m; r1=0.5 m; r2=0.25 m; Ep=25 GPa; ρp=2.5 g/cm3; vp=0.2; ρ1=ρ2=2 g/cm3; l1=l2= m1=m2=5 MPa; x1=x1=0.1.
By setting vp=0, the present solution can be reduced to the classical 1D rod model solution. The reduced solution is compared with the solution of ZHENG et al [21], as shown in Fig. 2. It can be noted that great agreement is obtained between the two solutions. This demonstrates the validity of the present solution.
Figure 3 shows the influence of the Poisson ratio of pile on the frequency domain velocity admittance of thelarge diameter pipe pile at the pile top, where θ is the dimensionless frequency and θ=ωH/Cp. It can be noted that the frequency differences between two oscillation amplitudes in classical 1D rod (vp=0) are constant. However, the frequency differences show a decreasing trend with increasing frequency when the transverse inertia effect is considered. This decreasing trend becomes more pronounced as the Poisson ratio increases. Figure 4 shows the influence of the Poisson ratio of pile on the time domain velocity response of the pile top, where τ is the dimensionless time and τ=tCp/H. The impulse width T=0.001 s. It is found that the peak time of the reflected waves is slightly delayed with the increase of the Poisson ratio of pile. It is indicated that the wave velocity of the large diameter pipe pile is smaller than that of the classical 1D rod. Moreover, the oscillations after the reflected signals become more obvious with the increase of the Poisson ratio, because of the wave dispersion in the large diameter pipe pile. These differences will affect the results of pile integrity testing such as the determination of the pile length.
Fig. 2 Comparison with solution of ZHENG et al [21]
Fig. 3 Influence of Poisson ratio of pile on velocity admittance at pile top
Figure 5 shows the influence of the outer radius of pile on the velocity admittance of the pile top. With the increase of the outer radius, the oscillation amplitudes of the velocity admittance increase. Meanwhile, the transverse inertia effect becomes more pronounced. The decrease of the resonant frequencies of the velocity admittance becomes larger. This is because the slenderness ratio of pile decreases with the increase of the outer radius, and therefore, the wave dispersion becomes more obvious. Figure 6 shows the influence of the outer radius of pile on the velocity response of the pile top. With the increase of the outer radius of pile, the peak time of the reflected waves decreases slightly, and the oscillations after the reflected signals become more obvious.
Fig. 4 Influence of Poisson ratio of pile on reflected wave signals at pile top
Fig. 5 Influence of outer radius of pile on velocity admittance at pile top
Figures 7 and 8 show the influences of the inner radius of pile on the frequency domain velocity admittance and time domain velocity response of the pile top, respectively. It can be seen from Fig. 7 that the oscillation amplitudes decrease as the inner radius increases. But unlike the outer radius, the decrease of the resonant frequencies shows little change with the change of the inner radius. This means that the inner radius has negligible influence on the transverse inertia effect of large diameter pipe piles. The curves in Fig. 8 further validate the above phenomena. The peak time differences of the reflected waves between the curves vp=0.2 and vp=0 remain constant with the change of the inner radius.
Fig. 6 Influence of outer radius of pile on reflected wave signals at pile top
Fig. 7 Influence of inner radius of pile on velocity admittance at pile top
Fig. 8 Influence of inner radius of pile on reflected wave signals at pile top
5 Conclusions
Considering the transverse inertia effect of pile, the vertical vibration of a large diameter pipe pile in viscoelastic soil is theoretically investigated. The frequency domain solutions are derived by utilizing Laplace transform technique, and the time domain results are obtained by the inverse Fourier transform method. A parametric study is conducted to analyze the influences of the Poisson ratio and radii of pile on the transverse inertia effect.
When the transverse inertia effect of pile is considered, the frequency differences between two oscillation amplitudes of the velocity admittance decrease with increasing frequency. The peak time of the reflected waves increases and the oscillations after the reflected signals become more obvious. The wave velocity of the Rayleigh-Love rod is smaller than that of the classical 1D rod because of the wave dispersion. The transverse inertia effect should be considered in pile integrity testing of large diameter pipe piles. The parametric study shows that the Poisson ratio and outer radius of pile have significant influence on the transverse inertia effect of large diameter pipe piles, while the inner radius has little effect.
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(Edited by YANG Bing)
Foundation item: Project(U1134207) jointly supported by the National Natural Science Foundation and High Speed Railway Key Program of China; Project(NCET-12-0843) supported by the Program for New Century Excellent Talents in University of China; Projects(51378177, 51420105013) supported by the National Natural Science Foundation of China; Projects(2015B05014, 2014B02814) supported by the Fundamental Research Funds for the Central Universities, China
Received date: 2015-01-12; Accepted date: 2015-04-10
Corresponding author: DING Xuan-ming, Professor, PhD; Tel: +86-23-65126279; E-mail: dxmhhu@163.com
