Dynamic responses of a pile embedded in a layered poroelastic half-space to harmonic lateral loads

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech. 2010; 34:493–515Published online 8 June 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nag.814

Dynamic responses of a pile embedded in a layered poroelastichalf-space to harmonic lateral loads

Bin Xu1, Jian-Fei Lu2,∗,† and Jian-Hua Wang1

1Department of Civil Engineering, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China2Department of Civil Engineering, Jiangsu University, Zhenjiang, Jiangsu 212013, People’s Republic of China

SUMMARY

A single pile embedded in a layered poroelastic half-space subjected to a harmonic lateral load isinvestigated in this study. Based on Biot’s theory, the frequency domain fundamental solution for ahorizontal circular patch load applied in the layered poroelastic half-space is derived via the transmissionand reflection matrices method. Utilizing Muki and Sternberg’s method, the second kind of Fredholmintegral equation describing the dynamic interaction between the layered half-space and the pile subjectedto a top harmonic lateral load is constructed. The proposed methodology is validated by comparing resultsof this paper with some existing results. Numerical results show that for a two-layered half-space, thethickness of the upper softer layer has pronounced influences on the dynamic response of the pile and thehalf-space. For a three-layered half-space, the presence of a softer middle layer in the layered half-spacewill enhance the compliance for the pile significantly, while a stiffer middle layer will diminish thedynamic compliance of the pile considerably. Copyright q 2009 John Wiley & Sons, Ltd.

Received 20 June 2008; Revised 31 March 2009; Accepted 9 April 2009

KEY WORDS: layered poroelastic half-space; pile; harmonic lateral load; transmission and reflectionmatrices (TRM) method; Fredholm integral equation

1. INTRODUCTION

Dynamic responses of a pile embedded in a uniform or a layered half-space and subjected totime-harmonic loads have received considerable attention during the past few decades. Very fewexact treatment of the level of Pak and Ji [1] and Abedzadeh and Pak [2] has been achieved for thepile problem. Most solutions in the literature are approximate or numerical in nature. For instance,Kuhlemeyr [3] investigated the static and dynamic laterally loaded floating piles using a finite

∗Correspondence to: Jian-Fei Lu, Department of Civil Engineering, Jiangsu University, Zhenjiang, Jiangsu 212013,People’s Republic of China.

†E-mail: [email protected]

Contract/grant sponsor: National Natural Science Foundation of China; contract/grant number: 50578071Contract/grant sponsor: Chinese Education Ministry

Copyright q 2009 John Wiley & Sons, Ltd.

494 B. XU, J.-F. LU AND J.-H. WANG

element method with a viscous energy-absorbing boundary layer. Within the realm of Muki andSternberg’s approximate approach [4, 5] for the spatial load-transfer process, Apirathvorakij andKarasudhi [6] proposed an integral equation method with a discontinuous integrand for the caseof an elastic bar embedded in a saturated elastic half-space under lateral loads. In an improvedmathematical formulation, Pak and Jennings [7] were able to show that the corresponding laterallyloaded pile problem in elastodynamics can be elegantly cast as a standard Fredholm integral equa-tion of the second kind with a continuous kernel. Rajapakse and Shah examined the applicabilityof Muki and Sternberg’s approach [4, 5] for the dynamic case by using a non-uniform body forcemodel [8]. They concluded that the fictitious pile model with both fictitious modulus and densitycan produce reasonable results at the low-frequency range. Sen et al. [9] proposed a boundaryelement method (BEM) wherein the soil and the pile are treated as separate substructures withtheir coupling being enforced at discrete points. Moreover, a comprehensive review on the subjectwas presented by Novak [10]. The dynamic analysis of piles embedded in a poroelastic half-spacewas first addressed by Zeng and Rajapakse [11] for the case of an axially loaded pile via Mukiand Sternberg’s approach. Jin et al. [12] applied the similar approach as those of Apirathvorakijand Karasudhi [6] and Pak and Jennings [7] to investigate the lateral dynamic compliance of apile embedded in a homogeneous poroelastic half-space.

As soil inhomogeneity has a significant influence on the response of embedded structures, thestudy concerning the dynamic response of the layered half-space to external loads has also beenconducted for a very long time. Early in 1950s, the propagator matrix method was developed byHarkrider [13] and Haskell [14]. Based on the finite element method, Valliappan et al. [15] presenteda numerical solution to the two-dimensional wave propagation in a fluid-saturated half-spacesubjected to a surface strip harmonical load. Besides, the exact stiffness matrix method, which isbased on integral transformmethod, was put forward by Senjuntichai and Rajapakse [16]. Moreover,the transmission and reflection matrix (TRM) method established by Luco and Apsel [17, 18] is avery important method for solving dynamic problems for a layered half-space. The advantage ofthe method is that the mismatched exponential terms are eliminated in all the terms of the TRM. Asa result, the TRM method can be valid for the high frequency and the small layer thickness cases,which are difficult to be addressed by the conventional propagator matrix method. Consequently,the TRM method has been used widely in solving layered structure problems [19–21]. Recently,Lu and Hanyga [22] derived the fundamental solution for a layered porous half-space subjectedto a vertical point force or a point fluid source by the TRM method. More recently, the dynamicresponses of a layered poroelastic half-space to moving loads were tackled via the TRM byXu et al. [23].

To analyze the behavior of piles embedded in a layered soil, Lee and Small [24] investigatedthe statically loaded pile in a layered soil on the basis of rigorous elastic load-transfer theory.Southcott and Small [25] used the finite layer technique to analyze vertically loaded piles and pilegroup embedded in a layered soil. Moreover, the dynamic response of a pile embedded in theGabson soil was considered by Militano using the exact stiffness matrix method [26].

As the TRM method is usually known in the seismology community rather than geotechnicalcommunity, thus, the TRM method has not been applied to the problem of piles embedded in alayered poroelastic half-space. As a result, the main contribution of this study is to apply the TRMmethod to handle the dynamic response of a pile embedded in a layered poroelastic half-space to aharmonical lateral top load (Figure 1). The pile in this paper is idealized as a one-dimensional beamand described by the Bernoulli–Euler beam theory, while the layered half-space is described byBiot’s theory. Based on Biot’s theory and the TRMmethod, the fundamental solution for a horizontal

Copyright q 2009 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2010; 34:493–515DOI: 10.1002/nag

DYNAMIC RESPONSES OF A PILE 495

Figure 1. A cylindrical elastic pile embedded in a layered poroelastic half-space and subjectedto a time-harmonic lateral load at the top.

circular patch load applied in the layered poroelastic half-space is derived first. Following Mukiand Sternberg’s method [4, 5], the frequency domain Fredholm integral equation accounting forthe dynamic interaction between the pile and the layered half-space is then constructed. Numericalsolution of the integral equation yields the dynamic response of the pile-half-space system to theharmonical top load. By the proposed model, the influences of some parameters on the dynamicresponse of the pile-half-space system are analyzed.

2. FUNDAMENTAL SOLUTION FOR A LAYERED POROELASTIC HALF-SPACE

2.1. Biot’s theory

According to Biot’s theory, equations of motion for the bulk material and the pore fluid are expressedin terms of the solid displacement (ui ) and the infiltration displacement (wi ) as follows [27–29]:

�ui, j j +(�+�2M+�)u j, j i +�Mw j, j i = �ui +�fwi (1)

�Mu j, j i +Mw j, j i = �fui +mwi +bpwi (2)

where � and � are Lame constants of the solid skeleton; � is the bulk density of the porous medium,which is equal to �=(1−�)�s+��f (where �s is the density of the solid skeleton and �f is thedensity of the pore fluid); � is the porosity of the poroelastic medium; m=a∞�f/� and a∞ isthe tortuosity of the porous medium; bp=�/k,� and k represent the viscosity of the pore fluidand the permeability of the porous medium, respectively; the superimposed dot above a variabledenotes the time derivative.



According to Biot’s theory, the constitutive relation for a homogeneous porous medium has theform [27–29]

�i j = 2�i j +�i j e−�i j pf (3)

pf = −�Me+Mϑ (4)

where �i j is the stress of the bulk material; i j denotes the strain tensor of the solid skeleton;pf is the excess pore fluid pressure and i j is the Kronecker delta. In (3)–(4), the dilatation of thesolid skeleton e and fluid volume increment ϑ are defined as

e=ui,i , ϑ=−wi,i (5)

To derive the general solutions for Biot’s equations, the Fourier transform with respect to timeand frequency is involved, which has the following definition [30]:

f (�) =∫ +∞

−∞f (t)e−i�t dt (6)

f (t) = 1

2�

∫ +∞

−∞f (�)ei�t d� (7)

where f (t) represents a function in the time domain, f (�) is the Fourier transform of f (t); t and� denote time and frequency, respectively; and a bar over a variable denote the Fourier transform.

Since the half-space is axisymmetric with respect to the z axis, it is more convenient to treat theproblem in the cylindrical coordinate system (r, , z). To derive the general solution for the porousmedium, the Hankel integral transform is required, which has the following definition [30]:

f [m](�) =∫ +∞

0r f (r)Jm(�r)dr (8)

f (r) =∫ +∞

0� f [m](�)Jm(�r)d� (9)

where Jm(∗) denotes the mth order first kind of Bessel function and � denotes the horizontalwavenumber.

Based on the Helmholtz decomposition method, the general solutions of the displacements, thestresses and the pore pressure in the frequency wavenumber domain can be obtained in termsof the potential theory and the Hankel transform with respect to the radial coordinate r . Explicitsolutions for the displacements, the stresses and the pore pressure in the frequency wavenumberdomain are available in [31].

2.2. The fundamental solution for a layered poroelastic half-space subjected to a circularpatch load

In the ensuing sections dealing with the dynamic response of the pile-half-space system, thefundamental solution corresponding to a uniform horizontal patch load applied over a circular areawith a radius R in the layered poroelastic half-space is required. As mentioned above, this solutionis derived by the TRM method in this study.



The model for the layered poroelastic half-space with N horizontal poroelastic layers overlyinga poroelastic half-space is illustrated in Figure 1. The j th porous layer is denoted by the symbolL( j) and the bottom layer is denoted by the symbol L(N+1). The thickness of the j th layer ish( j) = z j −z j−1 and z j−1, z j denote the depth of the upper and lower boundary of the j th layer.

To develop the fundamental solution, the frequency–wavenumber domain expressions ofdisplacements, stresses and the pore pressure for the j th porous layer are organized in thefollowing matrix form:

W( j)(�, z,�)8×1 =⎡⎣D( j)

d (�,�) D( j)u (�,�)

S( j)d (�,�) S( j)

u (�,�)

⎤⎦ [W( j)

d (�, z,�)T W( j)u (�, z,�)T] (10)

W( j)(�, z,�)8×1 = [ ˜w( j)[1]z

˜u( j)[1]z ( ˜u( j)[2]

r + ˜u( j)[2] )( ˜u( j)[0]

r − ˜u( j)[0] )

× ˜p( j)[1]f ( ˜�( j)[0]

zr − ˜�( j)[0]z )( ˜�( j)[2]

zr + ˜�( j)[2]z ) ˜�( j)[1]

zz ]T (11)

W( j)d (�, z,�) = [b( j)e− j

f (z−z j−1) d( j)e− js (z−z j−1) f ( j)e− j

t (z−z j−1) h( j)e− jt (z−z j−1)]T (12)

W( j)u (�, z,�) = [a( j)e− j

f (z j−z) c( j)e− js (z j−z) e( j)e− j

t (z j−z) g( j)e− jt (z j−z)]T (13)

where the superscript j denotes the j th porous layer; a( j)(�,�)∼h( j)(�,�) are the arbitraryconstants; j

f , js and j

t are the complex related to the wavenumbers for the P1, P2 and S wave of

the j th porous layer and are given in [31]; D( j)d (�,�), D( j)

u (�,�), S( j)d (�,�) and S( j)

u (�,�) are the

4×4 matrix, which can be derived by the method in [22]. The vectors W( j)d (�, z,�), W( j)

u (�, z,�)

are termed as the down-going and the up-going wave vector.It follows from Equations (12) and (13) that for each porous layer there are eight arbitrary

constants to be determined. As a result, there are totally 8×(N+1) unknowns to be determinedfor the layered half-space. Using the continuity condition at the layer interfaces, the free surfaceboundary condition as well as the radiation condition of the bottom half-space, these unknown canbe determined.

The up-going and the down-going wave vectors at the j th porous layer above the source layerL(l) have the following expressions [22]:

W( j)u (�, z j ,�) = g

Tue

( j)(�,�)

gTue

( j+1)(�,�) . . .

gTue

(l−2)(�,�)

gTu

(l−1)(�,�)W(l)

u (�, zl−1,�) (14)

W( j)d (�, z j−1,�) = g

Rue

( j−1)(�,�)W( j)

u (�, z j ,�), j =1,2, . . . , l−1 (15)

wheregTue

( j),

gRue

( j)are the generalized transmission and reflection matrices for the up-going waves

incident on the j th interface;W(l)u is the up-going wave vector of the source layer and the expression

forgTu

( j)is given in [22].



The up-going and the down-going wave vectors at the j th porous layer below the sourcelayer (L(l)) have the following expressions [22]:

W( j)d (�, z j−1,�) = g

Tde

( j−1)(�,�)

gTde

( j−2)(�,�) . . .

gTde

(l+1)(�,�)

gTd

(l)(�,�)W(l)

d (�, zl ,�) (16)

W( j)u (�, z j ,�) = g

Rd

( j)(�,�)W( j)

d (�, z j ,�), j = l, l+1, . . . ,N (17)

wheregTde

( j),

gRde

( j)are the generalized transmission and reflection matrices for the down-going

incident on the j th interface; W(l)d is the down-going wave vector of the source layer and the

expression forgRd

( j)is available in [22].

For the layered poroelastic half-space subjected to a horizontal circular patch load in the source

layer L(l) with the depth s, the variables ˜w(l)[1]z , ˜u(l)[1]

z , ( ˜u(l)[2]r + ˜u(l)[2]

), ( ˜u(l)[0]r − ˜u(l)[0]

), ˜p(l)[1]f ,

˜�(l)[1]zz , (�(l)[2]

zr + ˜�(l)[2]z ) should be continuous when across the load plane. However, the stress

combination ( ˜�(l)[0]zr − ˜�(l)[0]

z ) should satisfy the following jump condition at the load plane locatedat z=s:

( ˜�(l)[0]zr − ˜�(l)[0]

z )|z=s+ −( ˜�(l)[0]zr − ˜�(l)[0]

z )|z=s− = 2RJ1(R�)

A�(18)

The wave vectors in the source layer involved in (14) and (16) can be determined using thecontinuity condition and the transmission and reflection matrices for the layered poroelastic half-space [22], while the wave vectors in an arbitrary layer can be evaluated using (14)–(17). Oncethe wave vectors for an arbitrary layer are determined by (14)–(17), displacements, stresses andthe pore pressure in the frequency wavenumber domain can be evaluated via (10). The frequencydomain fundamental solution can be obtained by inversion of the Hankel transform numerically.

3. THE FREDHOLM INTEGRAL EQUATION FOR A PILE EMBEDDEDIN A LAYERED POROELASTIC HALF-SPACE

As shown in Figure 1, a cylindrical elastic pile with diameter d(d=2R) and length L(d/L�1)embedded in a layered poroelastic half-space is considered in this study. The pile top is subjectedto a horizontal time-harmonical load Q0ei�t or a bending moment M0ei�t . Following Muki andSternberg [4, 5] and Pak and Jennings [7], the present problem is decomposed into two sub-problems: an extended layered poroelastic half-space and a fictitious pile (Figure 2). The responseof the extended layered poroelastic half-space is characterized by Biot’s theory, while the fictitiouspile is described by the 1-D Bernoulli–Euler beam theory.

3.1. The Fredholm integral equation for the pile

It has been reported by Halpern and Christiano [32] that the difference of the compliances betweenan impermeable and fully permeable rigid plates on a poroelastic half-space in the low-frequencyrange is small. Also, using BEM, Maeso et al. analyzed the dynamical response of a single pile



x

y

z

*( ) /xM L A

o

z

z

0

( ) /xq z A

the extended layered

poroelastic half-space

the fictitious pile

h

L

0 *[ (0)] /xQ Q A *(0)xQ

L

*( )xQ L

0 *[ (0)] /xM M A

*( ) /xQ L A

( )xq z

*( )xM L

*(0)xM

L

Figure 2. Decomposition of a pile embedded in a layered half-space into an extended layered poroelastichalf-space and a fictitious pile.

and pile group embedded in a poroelastic medium [33]. Similarly, they also found that the exacthydraulic boundary condition on the pile–soil interface has a very small influence on the responseof the piles. Thus, it is reasonable to assume that the influence of the exact hydraulic boundarycondition at the pile–soil contact surface is not significant for the response of the pile. Consequently,the exact hydraulic boundary condition at the pile–soil interface is ignored in this study.

Moreover, for simplicity, the Fredholm integral equation of the pile is only constructed for atwo-layered poroelastic half-space. However, it is straightforward to extend the Fredholm integralequation for the current two-layered case to the half-space with an arbitrary number of layers.

It is supposed that Lame’s constants and the density of the upper layer of the half-space are�(1), �(1), �(1), respectively, and its thickness is h; also, Lame’s constants and the density for thebottom half-space are �(2), �(2), �(2), respectively. Accordingly, the fictitious pile is divided intothe two parts: the upper part and the bottom part. The Young’s modulus and the density for theupper part and the bottom part are given by [4, 5]

E (1)p∗ = Ep−E (1)

s (19)

�(1)p∗ = �p−�(1) (20)

E (2)p∗ = Ep−E (2)

s (21)

�(2)p∗ = �p−�(2) (22)



where Ep, E(i)s (i=1,2) are the Young’s modulus for the pile and the two porous layers with

E (i)s =�(i)(3�(i)+2�(i))/(�(i)+�(i))(i=1,2); �p, �(i)(i=1,2) are densities for the pile and the

two porous layers.As shown in Figure 2, the shear force and the bending moment for the fictitious pile are denoted

by Qx∗(z) and Mx∗(z), respectively. The fictitious pile is subjected to the horizontal distributedload qx (z) along the pile shaft. The top and the bottom of the fictitious pile are subjected toQx∗(0), Mx∗(0), Qx∗(L), Mx∗(L) (Figure 2), respectively. The extended layered poroelastic half-space are subjected to the following loads (Figure 2(a)): qx (z), which is uniformly distributedover the region occupied by the pile; Q0− Qx∗(0),M0− Mx∗(0) and Qx∗(L), Mx∗(L), which areapplied over the circular domains �0 and �L , respectively. As suggested by Pak and Jennings [7],for piles with a larger length–radius ratio, it is reasonable to assume that moments applied at twoends of the fictitious pile fulfill the following relations:

M0− Mx∗(0) = 0 (23)

Mx∗(L) = 0 (24)

For the fictitious pile, the displacement u(p)x∗ (z), the distributed horizontal force qx (z), the shear

force Qx∗(z) and the bending moment Mx∗(z) satisfy the following relations:

qx (z) = −dQx∗(z)

dz+�p∗ Au

(p)x∗ (z)�2 (25)

dMx∗(z)

dz= Qx∗(z) (26)

where u(p)x∗ (z) is the horizontal displacement of the fictitious pile; �p∗ =�(1)

p∗ for 0�z<h,�p∗ =�(2)p∗

for h<z�L; A is the cross section area of the pile. Note that as the Bernoulli–Euler beam theoryis used in this study, the rotary inertia of the pile is ignored in (26).

Using the Bernoulli–Euler beam theory, the following relations for the bending moment Mx∗(z),

the rotary angle (p)x∗ (z) and the horizontal displacement u(p)

x∗ (z) for the fictitious pile are obtained:

Mx∗(z) =∫ z

0Qx∗(�)d�+ Mx∗(0) (27)

(p)x∗ (z) = 1

Ep∗ Ix

∫ z

0(z−�)Qx∗(�)d�+ Mx∗(0)z

Ep∗ Ix+

(p)x∗ (0) (28)

u(p)x∗ (z) = 1

2Ep∗ Ix

∫ z

0(z−�)2 Qx∗(�)d�+ Mx∗(0)z2

2Ep∗ Ix+

(p)x∗ (0)z+ u(p)

x∗ (0) (29)

where Ix is the second moment of the pile cross section; Ep∗ =E (1)p∗ for 0�z<h, while Ep∗ =E (2)

p∗for h<z�L . In terms of (24) and (27), the following relation for the shear force of the fictitiouspile and the moment at the top of the fictitious pile is obtained:∫ L

0Qx∗(�)d�=−Mx∗(0) (30)



Many compatibility conditions can account for the compatibility between the fictitious pile andthe extended half-space [11]. However, for the pile with a large length–radius ratio in this study,we follow the condition proposed by Pak and Jennings [7]. By this condition, the compatibilitybetween the pile and the layered poroelastic half-space is accomplished by requiring the rotary

angle of the fictitious pile ( (p)x∗ (z)) and that of the extended poroelastic half-space along the z-axis

( (s)x (z)) to be equal , i.e.

(p)x∗ (z)=

(s)x (z), 0�z<h, h<z�L (31)

In view of the discontinuity at the interface between the two layers, the rotary angle of theextended layered poroelastic half-space along the z-axis reads as follows:

(s)x (z) = Q0�

(G)

x (0, z)+ Qx∗(z)[�(G)

x (z+, z)−�(G)

x (z−, z)]

+[∫ h−

0Qx∗(�)

��(G)

x (�, z)

��d�+

∫ L

h+Qx∗(�)

��(G)

x (�, z)

��d�

]

+�p∗ A�2∫ L

0u px∗(�)�

(G)

x (�, z)d�, 0�z<h, h<z�L (32)

where is an infinitesimal, �(G)

x (�, z) represents the rotary angle at the center of �z due to a unit

patch load applied on �� (Figure 2); �(G)

x (z−, z), �(G)

x (z+, z) denote the rotary angle at the centerof �z when the patch load �� is located at z−, z+, respectively.

Using (28), (29), (31) and (32), the Fredholm integral equation describing the horizontal inter-action between the pile and the two-layered half-space has the form

Qx∗(z)[�(G)

x (z+, z)−�(G)

x (z−, z)]

+[∫ h−

0Qx∗(�)

��(G)

x (�, z)

��d�+

∫ L

h+Qx∗(z)

��(G)

x (�, z)

��d�

]

− 1

Ep∗ Ix

∫ z

0(z−�)Qx∗(�)d�+�p∗ A�2

∫ L

0�1(�)�

(G)

x (�, z)d�+ (p)x∗ (0)�2(z)+ u(p)

x∗ (0)�3(z)

= Mx∗(0)z

Ep∗ Ix−Q0�

(G)

x (0, z)− �p∗ A�2Mx∗(0)

2Ep∗ Ix

∫ L

0�2�

(G)

x (�, z)d�, 0�z<h, h<z�L (33)

where

�1(�) = 1

2Ep∗ Ix

∫ �

0Qx∗(�)(�−�)2 d� (34)

�2(z) = �p∗ A�2∫ �

0��

(G)

x (�, z)d�−1 (35)

�3(z) = �p∗ A�2∫ �

0�

(G)

x (�, z)d� (36)



The displacement of the extended layered poroelastic half-space at the center of �z along thex direction has the following expression:

u(s)x (z) = Q0U

(G)x (0, z)+

[∫ h−

0Qx∗(�)

�U (G)x (�, z)

��d�+

∫ L

h+Qx∗(z)

�U (G)x (�, z)

��d�

]

+�p∗ A�2∫ L

0u(p)x∗ (�)U (G)

x (�, z)d� (37)

where U (G)x (�, z) denotes the horizontal displacement at the center of �z due to a unit uniform

load at the ��.

Assuming u(s)x (0)= u(p)

x∗ (0) and using (29) and (37), u(p)x∗ (0) in (33) has the following expression:

u(p)x∗ (0) = 1

�

{Q0U

(G)x (0,0)+

[∫ h−

0Qx∗(�)

�U (G)x (�,0)

��d�+

∫ L

h+Qx∗(z)

�U (G)x (�,0)

��d�

]

+�p∗ A�2Mx∗(0)

2Ep∗ Ix

∫ L

0�2�

(G)

x (�,0)d�+�p∗ A�2∫ L

0�1(�)U

(G)x (�,0)d�

+ (p)x∗ (0)�p∗ A�2

∫ L

0�U (G)

x (�,0)d�

}(38)

where �=1−�p∗ A�2∫ L0 U (G)

x (�,0)d�.

3.2. The shear force of the pile and the pore pressure along the pile side

Once the shear force, the top rotary angle and the top horizontal displacement of the fictitious pileare obtained, the shear force of the real pile can be evaluated by the superposition of the shearforce of the fictitious pile and the shear force acting over the circular domain �z , which is

Qx (z) = Q0 f(G)(0, z)

+�p∗ A�2[∫ h−

0�1(�) f

(G)(�, z)d�+∫ L

h+�1(�) f

(G)(�, z)d�

]

+�p∗ A�2Mx∗(0)

2Ep∗ Ix

∫ L

0�2 f (G)(�, z)d�

+[∫ h−

0Qx∗(�)

� f (G)(�, z)

��d�+

∫ L

h+Qx∗(z)

� f (G)(�, z)

��d�

]

+ (p)x∗ (0)�p∗ A�2

∫ L

0� f (G)(�, z)d�

+u(p)x∗ (0)�p∗ A�2

∫ L

0f (G)(�, z)d�, 0�z<h, h<z�L (39)



0.0 0.1 0.2 0.3 0.4 0.5 0.60.0

0.2

0.4

0.6

0.8

1.0

*

the real part of present the imaginary part of presentthe real part of [7] the imaginary part of [7]

0.0 0.1 0.2 0.3 0.4 0.5 0.60.0

0.2

0.4

0.6

0.8

1.0

*

the real part of present the imaginary part of presentthe real part of [7] the imaginary part of [7]

(c)

(b)

(a)

(d)

0.0 0.2 0.4 0.6 0.8 1.0-1.5

0.0

1.5

3.0

4.5

6.0 the real part of this paper

the imaginary part of this paper

the real part of [7]

the imaginary part of [7]

z/L

0.0 0.2 0.4 0.6 0.8 1.0-0.2

0.0

0.2

0.4

0.6

0.8

1.0

z/L

the real part of this paper the imaginary part of this paperthe real part of [7] the imaginary part of [7]

Figure 3. Comparison of the present result for the dynamic response of a pile with those of [7]: (a) Chh/Chh0versus normalized frequency �∗ =0.0∼0.6; (b) Cmm/Cmm0 versus normalized frequency �∗ =0.0∼0.6;(c) dimensionless bending moment M∗

x (z)= Mx (z)/[4��(1)R3] along the axis of the pile for �∗ =0.2when the pile top subjected to a time-harmonic horizontal load Q0ei�t ; and (d) dimensionless bendingmoment M∗

x (z)= Mx (z)/[4��(1)R3] along the axis of the pile for �∗ =0.2 when the pile top subjectedto a time-harmonic bending moment M0ei�t .

where f (G)(�, z) is equal to �(G)zr (�, z)A, �(G)

zr (�, z) represents the shear stress at the center of�z due to a uniform patch load at ��. Note that when deriving the above equation, the relationf (G)(z+, z)− f (G)(z−, z)=−1.0 is used.Besides, the pore pressure along the right side of the pile has the following expression:

pf(z) = Q0 p(G)f (0, z)

+�p∗ A�2[∫ h−

0�1(�) p

(G)f (�, z)d�+

∫ L

h+�1(�) p

(G)f (�, z)d�

]

+�p∗ A�2Mx∗(0)

2Ep∗ Ix

∫ L

0�2 p(G)

f (�, z)d�

+[∫ h−

0Qx∗(�)

� p(G)f (�, z)

��d�+

∫ L

h+Qx∗(z)

� p(G)f (�, z)

��d�

]



0.0 0.2 0.4 0.6 0.8 1.0

-0.3

0.0

0.3

0.6

0.9

1.2

1.5

*

*hh

C

imaginary part h=0.0 mh=5.0 mh=10.0 m

h=0.0 mh=5.0 mh=10.0 m

real part

0.0 0.2 0.4 0.6 0.8 1.0-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

*mhC

h=0.0 mh=5.0 mh=10.0 m

*

imaginary part

h=0.0 mh=5.0 mh=10.0 m

real part

0.0 0.2 0.4 0.6 0.8 1.0-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

*hm

C

h=0.0 mh=5.0 mh=10.0 m

*

imaginary part

h=0.0 mh=5.0 mh=10.0 m

real part

0.0 0.2 0.4 0.6 0.8 1.0

-0.08

-0.06

-0.04

-0.02

0.00

*mm

C

*

imaginary part

h=0.0 mh=5.0 mh=10.0 m

h=0.0 mh=5.0 m

h=10.0 m

real part

(a)

(b)

(c)

(d)

Figure 4. Dimensionless compliance for the pile embedded in the two-layered poroelastic half-space versusnormalized frequency �∗ =0.0∼1.0 with h=0.0, 5.0 and 10.0m, respectively: (a) the dimensionless C∗

hh ;(b) the dimensionless C∗

mh ; (c) the dimensionless C∗hm ; and (d) the dimensionless C∗

mm .

+ (p)x∗ (0)�p∗ A�2

∫ L

0� p(G)

f (�, z)d�

+u(p)x∗ (0)�p∗ A�2

∫ L

0p(G)f (�, z)d�, 0�z<h, h<z�L (40)

where p(G)f (�, z) represents the pore pressure at the right side of the �z due to a unit horizontal

patch load at ��.

4. NUMERICAL METHOD AND RESULTS

Integral Equation (33) can be solved numerically. The methodology for solving integralequation (33) is detailed in [6]. Numerical solution of the integral Equations (33) and (38)yields solutions for the shear force, the top rotary angle and the horizontal displacementat the fictitious pile top. Using the shear force, the top rotary angle and the top horizontaldisplacement of the fictitious pile, the response of the pile and the half-space can be eval-uated by the corresponding equations. To validate our model, in Section 4.1, a special case



.00.0 0.2 0.4 0.6 0.8 1-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

i0

tQ e

z /L

imaginary part

h=0.0 mh=5.0 mh=10.0 m

h=0.0 mh=5.0 m

h=10.0 m

real part

.00.0 0.2 0.4 0.6 0.8 1-0.20

-0.15

-0.10

-0.05

0.00

0.05

*(

)x

Qz

0i tM e

z/L

imaginary part

h=0.0 mh=5.0 mh=10.0 m

h=0.0 mh=5.0 m

h=10.0 m

real part

*(

)x

Qz

(a)

(b)

Figure 5. The dimensionless shear force Q∗x (z) along the axis of the pile for the normalized frequency

�∗ =0.4 when h=0.0, 5.0 and 10.0m, respectively: (a) a time-harmonic horizontal load Q0ei�t appliedat the top of the pile and (b) a time-harmonic bending moment M0ei�t applied at the top of the pile.

of our model is compared with existing results. Also, to show the capacity of the proposedmodel, in Sections 4.2 and 4.3, some numerical examples and corresponding analysis arepresented.

4.1. Comparison of our results with existing results

As the first example, a special case of our model is compared with known results. In this example,the pile embedded in a poroelastic half-space with two poroelastic layers overlying a bottomporoelastic half-space and subjected to a time-harmonic horizontal top load Q0ei�t and a topmoment M0ei�t is considered (see Figure 1). To compare our results with those of [7], theparameters M , a∞, �, bp, �, �f for each poroelastic layer are assumed to tend to zero as in [12],and all the other parameters for different layers assume the same values. In this way, the layeredporoelastic half-space is reduced to a homogeneous quasi-elastic half-space. It should be notedthat for an elastic medium there are singularities in the path of the integration when performingthe inverse Hankel transform to calculate the fundamental solution. However, some researcherstend to circumvent this difficulty using a special viscoelastic model. In the viscoelastic model, the



0.0 0.2 0.4 0.6 0.8 1.0-0.5

0.0

0.5

1.0

1.5

i0

tQ e

*(

)x

Mz

z/L


h=0.0 mh=5.0 mh=10.0 m

real part

0.0 0.2 0.4 0.6 0.8 1.0-0.5

0.0

0.5

1.0

1.5

2.0

*(

)x

Mz

0i tM e

z/L

imaginary part

h=0.0 mh=5.0 mh=10.0 m

h=0.0 mh=5.0 mh=10.0 m

real part

(a)

(b)

Figure 6. Dimensionless bending moment M∗x (z) along the axis of the pile for the normalized

frequency �∗ =0.4 when h=0.0, 5.0 and 10.0m, respectively: (a) a time-harmonic horizontalload Q0ei�t applied at the top of the pile and (b) a time-harmonic bending moment M0ei�t

applied at the top of the pile.

material damping is taken into account by using complex Lame’s constants i.e. �=�0(1+ i�s) and�=�0(1+ i�s), where �s denotes the damping ratio. In this paper we use a damping ratio of �s=0.05

for the soil and �( j)0 =2.0×107N/m2, �( j)

0 =2.0×107N/m2, �( j)s =2.0×103 kg/m3 ( j=1,2,3).

The thicknesses of the two layers are h(1) =h(2) =3.0m, respectively. The parameters for the piletake the following values: R=0.5m, L=25.0m, Ep=1.0×1012N/m2, �p=3.4×103 kg/m3. The

frequency � is normalized as �∗ =√

�(1)s /�(1)

0 �R.In Reference [7], the compliance matrix is defined as follow:[

�

]=

[Chh Chm

Cmh Cmm

][Q0

M0

](41)

where �= ux (0), =�ux (0)/�z.Figure 3(a), (b) shows the comparison of results of the current study for compliances Chh/Chh0,

Cmm/Cmm0 with those of [7], respectively, where Chh0,Cmm0 are compliances for the static



0.0 0.2 0.4 0.6 0.8 1-0.08

-0.04

0.00

0.04

0.08

0.12

0.16

.0

i0

tQ e

z /L

*(

)f

pz


h=0.0 mh=5.0 m

h=10.0 mreal part

0.0 0.2 0.4 0.6 0.8 1.0

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0i tM e

z/L

imaginary part

h=0.0 mh=5.0 mh=10.0 m

h=0.0 mh=5.0 mh=10.0 m

real part

*(

)f

pz

(a)

(b)

Figure 7. Dimensionless pore pressure p∗f (z) along the side of the pile for the normalized

frequency �∗ =0.4 when h=0.0, 5.0 and 10.0m, respectively: (a) a time-harmonic horizontalload Q0ei�t applied at the top of the pile and (b) a time-harmonic bending moment M0ei�t

applied at the top of the pile.

case. Figures 3(c), (d) show the comparison of the dimensionless bending moment M∗x (z)=

Mx (z)/[4��(1)0 R3] along the pile axis for the dimensionless frequency �∗ =0.2 when the pile

is subjected to a unit horizontal load and a unit bending moment separately. It follows fromFigure 3(a)–(d) that our solutions are in a good agreement with those of [7].

4.2. Pile embedded in a two-layered poroelastic half-space

In this section, a cylindrical pile embedded in a softer upper layer overlying a stiffer poroelastichalf-space and subjected a time-harmonic horizontal load Q0ei�t and a bending moment M0ei�t

is considered. The material parameters for the layered poroelastic half-space assume the followingvalues: �(1) =4.0×106N/m2, �(2) =10×�(1), �(1) =4.0×106N/m2, �(2) =10×�(1), �( j) =0.4,�( j)s =2.0×103 kg/m3, a( j)∞ =2.0, �( j)

f =1.0×103 kg/m3, �( j) =0.97, b( j)p =1.94×108 kg/m3 s,

M ( j) =2.44×108N/m2 ( j=1,2). The parameters for the pile are: R=0.5m, L=20.0m,

�p=3.0×103 kg/m3, Ep=1.0×1010N/m2.In calculation, the time-harmonic horizontal load Q0ei�t and the time-harmonic bending

moment M0ei�t are applied at the top of the pile separately. When the time-harmonic horizontal



0.0 0.2 0.4 0.6 0.8 1.0-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

imaginary part

real part

*

*hh

C

0.0 0.2 0.4 0.6 0.8 1.0-0.3

-0.2

-0.1

0.0

0.1

*mhC

imaginary part

real part

*

0.0 0.2 0.4 0.6 0.8 1.0-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

*mm

C

imaginary part

real part

*

(a) (b)

(c)

Figure 8. Dimensionless compliance for the pile embedded in a two-layeredporoelastic half-space versus the normalized frequency �∗ =0.0∼1.0 withb( j)p =bp=1.0×10−4 kg/m3 s,1.94×106 kg/m3 s and 1.94×108 kg/m3 s ( j =1,2),respectively: (a) dimensionless C∗

hh ; (b) dimensionless C∗mh ; and (c) dimensionless C∗

mm .

load Q0ei�t is acted on the pile top, the shear force Qx (z), the bending moment Mx (z) andthe pore pressure pf(z) are normalized as Q∗

x (z)= Qx (z)/Q0, M∗x (z)= Mx (z)/(Q0R) and

p∗f (z)=�R2 pf(z)/Q0. Also, the compliances Chh , Cmh in Equation (41) are normalized as

C∗hh =�(2)Rux (0)/Q0 and C∗

mh =�(2)R2 x (0)/Q0, respectively. When the pile top is subjectedto the time-harmonic bending moment M0ei�t , the shear force Qx (z), the bending momentMx (z) and the pore pressure pf(z) are normalized as Q∗

x (z)= Qx (z)R/M0, M∗x (z)= Mx (z)/M0

and p∗f (z)= R3 pf(z)/M0. The compliances Chm Cmm in Equation (1) is normalized as C∗

hm =�(2)R2ux (0)/M0,C∗

mm =�(2)R3 x (0)/M0. The frequency of the loads is normalized as �∗ =√�(2)/�(2)�R.

4.2.1. Influences of the thickness of the upper softer layer. In this example, three different valuesof h are considered: h=0.0, 5.0 and 10.0m. The parameters for the poroelastic layers as well asthose for the pile and the loads take the same values as above.

Figure 4 shows the variation of the C∗hh , C

∗mh , C

∗hm and C∗

mm versus the normalized frequency�∗ =0.0∼1.0 for three values of h=0.0, 5.0 and 10.0m, respectively. It follows from Figure 4(a)



0.0 0.2 0.4 0.6 0.8 1.0-0.04

0.00

0.04

0.08

0.12

i0

tQ e

imaginary part

real part

z/L

*(

)f

pz

0.0 0.2 0.4 0.6 0.8 1.0

-0.02

-0.01

0.00

0.01

0.02

0.030

i tM e

imaginary part

real part

z/L

*(

)f

pz

(a)

(b)

Figure 9. Dimensionless pore pressure p∗f (z) along the side of the pile for the normalized frequency �∗ =0.4

when b( j)p =bp=1.0×10−4 kg/m3 s,1.94×106 kg/m3 s and 1.94×108 kg/m3 s ( j =1,2), respectively: (a)

a time-harmonic horizontal load Q0ei�t applied at the top of the pile and (b) a time-harmonic bendingmoment M0ei�t applied at the top of the pile.

that the magnitude of the C∗hh increases with increasing thickness of the upper softer layer.

Figure 4(b), (c) and (d) show that when the frequency �∗>0.4, the variation of C∗mh , C

∗hm and

C∗mm for three values of h is similar to that of C∗

hh . One can also see from Figure 4(b), (c) thatthe values of C∗

hm is equal to C∗mh , which is consistent with the reciprocal theorem. As a result,

the result for C∗hm will not be presented in what follows.

Figures 5–7 show the profiles for the dimensionless shear force Q∗x (z), the bending moment

M∗x (z) along the axis of the pile and the pore pressure p∗

f (z) along the side of the pile with�∗ =0.4 for three different thicknesses of the upper layer. One can see from Figure 5 that the shearforce Q∗

x (z) has a peak at the interface between the upper layer and the bottom layer. Also themagnitude of the peak around the interface increases with increasing thickness of the upper layer.

Figure 6(a) shows that the maximum bending moment M∗x (z) increases considerably with

increasing thickness of the upper soft layer. Also, Figure 6(a) and (b) show that the bendingmoment M∗

x (z) at the lower part of the pile increases with increasing thickness of the upper softlayer. Figure 7 shows that for the two-layered poroelastic half-space, the pore pressure p∗

f (z) alsohas a peak near the interface between the upper layer and the bottom layer. Also, the real part



0.0 0.2 0.4 0.6 0.8 1.0-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

imaginary part

real part *hh

C

*.00.0 0.2 0.4 0.6 0.8 1

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

*mhC imaginary part

real part

*

0.0 0.2 0.4 0.6 0.8 1.0-0.04

-0.03

-0.02

-0.01

0.00

0.01

*mm

C

imaginary part

real part

*

(a) (b)

(c)

Figure 10. Dimensionless compliance for the pile embedded in the three-layered poroelastic half-spaceversus normalized frequency �∗=0.0∼1.0 for the Case A, the Case B and the Case C: (a) the dimensionless

C∗hh ; (b) the dimensionless C∗

mh ; and (c) the dimensionless C∗mm .

of the pore pressure p∗f (z) around the interface increases with increasing thickness of the upper

soft layer.

4.2.2. Influences of the permeability of the poroelastic layers. To examine the influence of thepermeability of the poroelastic layers on the dynamic responses of the pile, three values of bp are

used for the two-layered poroelastic half-space: b( j)p =bp=1.0×10−4 kg/m3 s, 1.94×106 kg/m3 s

and 1.94×108 kg/m3 s ( j =1,2), respectively. The other parameters for the poroelastic layers aswell as for the pile and the loads assume the same values as above, while the thickness of theupper layer is h=5.0m.

Figure 8 shows the variation of C∗hh , C

∗mh and C∗

mm versus the normalized frequency �∗ =0.0∼1.0 for three values of b( j)

p , respectively. It is observed that with increasing bp, the magnitudesof the real and the imaginary part of C∗

hh , C∗mh and C∗

mm decrease. Also, with increasing bp, C∗hh

has a more obvious variation than C∗mh and C∗

mm . One can also see that the permeability of theporoelastic layers almost has no influence on C∗

hh , C∗hm and C∗

mm when the frequency of the loadstends to zero.

Figure 9 shows the profiles of the dimensionless pore pressure p∗f (z) along the side of the pile

for the normalized frequency �∗ =0.4 under the three different values of bp. One can see that thepore pressure for larger bp are larger than that for smaller bp.



0.0 0.2 0.4 0.6 0.8 1.0

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

i0

tQ e

*(

)x

Qz

*(

)x

Qz

z/L

imaginary part

real part

0.0 0.2 0.4 0.6 0.8 1.0-0.20

-0.15

-0.10

-0.05

0.00

0.05

z/L

0i tM e

(a)

(b)

Figure 11. Dimensionless shear force Q∗x (z) along the axis of the pile with the normalized frequency

�∗ =0.5 for the pile embedded in the three layered half-space for the Case A, the Case B and the CaseC: (a) a time-harmonic horizontal load Q0ei�t applied at the top of the pile and (b) a time-harmonic

bending moment M0ei�t applied at the top of the pile.

4.3. Pile embedded in a three-layered poroelastic half-space

To investigate the influence of the soil inhomogeneity on the dynamic responses of the pile, thecylindrical pile embedded in a three-layered poroelastic half-space and subjected to a horizontaltime-harmonic load Q0ei�t and a bending moment M0ei�t at the pile top separately is consideredin this section. In this example, the layered poroelastic half-space consists of two overlying porouslayers and a bottom poroelastic half-space. The ratio of the shear modulus between the threelayers are: (A) �(1) :�(2) :�(3) =1 :1 :1; (B) �(1) :�(2) :�(3) =1 :0.1 :1; (C) �(1) :�(2) :�(3)=1 :10 :1,where �(1) =4.0×107N/m2. Also, for the three cases, Lame’s constant � of each layer is �( j) =�( j), j =1,2,3. The thicknesses for the two upper layers are h(1) =4.5m and h(2) =7.5m. Theremaining parameters for the layered poroelastic half-space take the same values as those inSection 4.2. The parameters for the pile are: R=0.5m, L=30.0m, Ep=1.0×1011N/m2. Theshear force Qx (z), the bending moment Mx (z) and the pore pressure pf(z) are normalized inthe same way as in Section 4.2. The frequency of the loads is normalized as �∗ =√

�(1)/�(1)�Rand Chh , Cmh , Cmm in (41) are normalized as C∗

hh =�(1)Rux (0)/Q0,C∗mh =�(1)R2 x (0)/Q0 and

C∗hm =�(1)R2ux (0)/M0,C∗

mm =�(1)R3 x (0)/M0, respectively.



0.0 0.2 0.4 0.6 0.8 1.0

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

i0

tQ e

*(

)x

Mz

z/L

imaginary part

real part

0.0 0.2 0.4 0.6 0.8 1.0-0.5

0.0

0.5

1.0

1.5

2.0

z/L

imaginary part

real part

0i tM e

*(

)x

Mz

(a)

(b)

Figure 12. Dimensionless bending moment M∗x (z) along the axis of the pile with the normalized frequency

�∗ =0.5 for the pile embedded in the three-layered half-space for the Case A, the Case B and the Case C:(a) a time-harmonic horizontal load Q0ei�t applied at the top of the pile and (b) a time-harmonic bending

moment M0ei�t applied at the top of the pile.

Figure 10 shows the variation of the C∗hh , C

∗mh and C∗

mm versus the normalized frequency�∗ =0.0∼1.0 for the three cases (case A, case B and case C), respectively. One can see fromFigure 10 (a) that the presence of a softer middle layer in the layered half-space (case B) enhancesthe real part of C∗

hh slightly. On the contrary, the stiffer middle layer of the layered half-space(case C) diminishes the real part of C∗

hh . Figure 10(b) and (c) show that the variation of C∗mh and

C∗mm is similar to that of C∗

hh in Figure 10(a).Figures 11–13 show the dimensionless shear force Q∗

x (z), the bending moment M∗x (z), the pore

pressure p∗f (z) for the case A, the case B and the case C when the normalized frequency �∗ =0.5.

From Figure 11, one can see that for the shear force Q∗x (z), a peak occurs at the interface of

different layers. At the interface between the upper layer and the middle layer, the dimensionlessshear force Q∗

x (z) for the case C is larger than those of the other cases, while at the interfacebetween the middle layer and the bottom layer, the dimensionless shear force Q∗

x (z) for the caseB is larger than those for the other cases.

Figure 12(a), (b) show that for the lower part of the pile, the dimensionless bending momentM∗

x (z) for the stiffer middle layer case (Case C) is lager than those for the other cases. It follows



0.0 0.2 0.4 0.6 0.8 1.0-0.08

-0.04

0.00

0.04

0.08

0.12

i0

tQ e

*(

)f

pz

z /L

imaginary part

real part

.00.0 0.2 0.4 0.6 0.8 1

-0.02

-0.01

0.00

0.01

0.02

*(

)f

pz

z /L

imaginary part

real part

0i tM e

(b)

(a)

Figure 13. Dimensionless pore pressure p∗f (z) along the side of the pile with the normalized frequency

�∗ =0.5 for the pile embedded in the three-layered half-space for the Case A, the Case B and the Case C:(a) a time-harmonic horizontal load Q0ei�t applied at the top of the pile and (b) a time-harmonic bending

moment M0ei�t applied at the top of the pile.

from Figure 13(a), (b) that the pore pressure has a peak at the interface between different layers.Also, for the Case B and the Case C, the real part of the pore pressure has a sign change at theinterface between the upper layer and the middle layer.

5. CONCLUSIONS

The dynamic response of a pile embedded in a layered poroelastic half-space to a harmonic lateralload is addressed in this study. The fundamental solution for a horizontal circular patch loadapplied in the layered poroelastic half-space is established via the TRM method. By using Mukiand Sternberg’s method, the second kind of Fredholm integral equation describing the dynamicinteraction between the pile and the layered poroelastic half-space is obtained. Numerical solutionof the integral equation yields the shear force, the bending moment, the displacement of the pileas well as the response of the layered poroelastic half-space. Results of this paper are comparedwith existing results, which shows that our solution is in a good agreement with existing results.

The numerical results of this study demonstrate that the soil inhomogeneity has a significantinfluence on the response of the pile: for the two-layered half-space, a larger thickness of the upper



softer layer will lead to a larger compliance. At the interface between the upper layer and thebottom layer, a shear force peak occurs and the magnitude of the peak increases with increasingthickness of the upper soft layer. The bending moment at the lower part of the pile also increaseswith increasing thickness of the upper soft layer. For the three-layered poroelastic half-space, thepresence of a softer middle layer enhances the compliance of the pile significantly, while a stiffermiddle layer will diminish the compliance of the pile.

ACKNOWLEDGEMENTS

The project is supported by the National Natural Science Foundation of China with grant number 50578071.Also, the research is supported by the returned oversea scholar funding from Chinese Education Ministry.Besides, the constructive comments from the referees are highly appreciated.

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Dynamic responses of a pile embedded in a layered poroelastic half-space to harmonic lateral loads