Analysis of pile groups in a poroelastic medium subjected to horizontal vibration

Computers and Geotechnics 36 (2009) 406–418

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Analysis of pile groups in a poroelastic medium subjected to horizontal vibration Xiang-Lian Zhou *, Jian-Hua Wang Department of Civil Engineering, Shanghai Jiao Tong University, Shanghai 200030, China

a r t i c l e

i n f o

Article history: Received 1 February 2008 Received in revised form 20 August 2008 Accepted 21 August 2008 Available online 1 October 2008 Keywords: Poroelastic medium Pile groups Fredholm integral equation Biot’s theory

a b s t r a c t This work investigates the dynamic response of pile groups embedded in a poroelastic medium subjected to horizontal loading. The dynamic response is analyzed using the Muki and Sternberg Method. The load transfer problem is formulated in terms of a second-kind Fredholm integral. The dynamic impedance of the pile groups is computed using the pile–soil–pile dynamic interaction factors. The shear force, bending moment and pore pressure is obtained using the superposition method. Numerical results indicate that the pile flexibility ratio and the pile distance have considerable influence on the dynamic response of the piles and the poroelastic medium. Crown Copyright Ó 2008 Published by Elsevier Ltd. All rights reserved.

1. Introduction Piles in foundations are often submitted to strong horizontal forces, as in the case of piles in the foundations of bridges, high buildings, offshore structures and supporting walls. To ensure that the force or settlement experienced by the pile foundation satisfies practical requirements, various approaches have been developed involving a pile or a group of piles in a single-phase elastic media. The earliest formulation was proposed by Poulos and Davies [1] and Poulos [2], for the analysis of a single pile under vertical loading. To date, several methodologies have been proposed to calculate the dynamic response of pile foundations subjected to vertical and horizontal loading [3,4]. These methods are essentially numerical in nature and involve discretization of the domain (FEM) or its boundary (BEM). For example, Sen et al. [5] used the boundary element formulation for the dynamic analysis of the pile groups. Dobry and Gazetas [6] proposed a simple model for calculating the dynamic interaction factor between the piles in a homogeneous elastic half-space. Mammon et al. [7] used the hybrid boundary element formulation and the rigorous boundary element formulation to analyze the dynamic response of the pile groups. Kaynia and Kausel [8] provided the elastic solutions for the pile groups in a layered half-space based on the boundary integral techniques. Gazetas et al. [9] used the beam on the Dynamic Winkler Foundation simplified model and Green’s function based on the rigorous method to determine the dynamic response of the pile groups. Wu and Finn [10] used the finite element method to analyze the dynamic response of the pile foundations. Cairo et al. [11] carried out the direct small strain analysis of the pile groups * Corresponding author. Tel./fax: +86 21 62932915. E-mail addresses: [email protected], [email protected] (X.-L. Zhou).

under vertical harmonic vibration. The method made use of the closed form stiffness matrices derived by Kausel and Roëset [12] for the study of wave propagation problems in layered media. However, existing models for the dynamic analysis of the pile foundations subjected to vertical and horizontal loading have been limited to the single-phase case, i.e., the surrounding media of the pile foundations are treated as a single-phase media. In fact, some pile foundations are embedded in saturated porous media. Obviously, the single-phase model is not appropriate for the design of pile foundations embedded in poroelastic media. The main drawback of the single-phase model lies in the fact that it cannot predict the influence of pore pressure on the dynamic response of the pile foundations. However, pore pressure near the pile foundations due to seismic loading is crucial for understanding and exploring mechanisms of the liquefaction of the surrounding soil. Biot [13,14] pioneered the development of a three-dimensional elastodynamic theory for saturated soil. Zeng and Rajapakse [15] used the classical Muki and Sternberg formulation [16] to analyze the steady state dynamic response of an axially loaded elastic bar partially embedded in a poroelastic medium. Lu [17] used the Muki method to study the horizontal dynamic response of a single pile in the saturated soil. Maeso et al. [18] presented a three-dimensional time-harmonic boundary element model for the dynamic stiffness coefficients of piles and pile groups embedded in a saturated soil. The objective of this paper is to study the effect of the pile groups under horizontal loading in the poroelastic medium. The pile is represented by the compressible beam column element and the poroelastic medium uses Biot’s three-dimensional elastodynamic theory. The fundamental solutions for the poroelastic medium subjected to horizontal load can be obtained by using the Hankel integral transform. Following the Muki and Sternberg

0266-352X/$ - see front matter Crown Copyright Ó 2008 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2008.08.013

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X.-L. Zhou, J.-H. Wang / Computers and Geotechnics 36 (2009) 406–418

method, the pile–soil–pile interaction problem is divided into two systems: namely, the extended half-space and the fictitious piles with the Young’s modulus equal to the difference between the Young’s moduli of the real piles and the poroelastic medium. The governing equation is found to be a Fredholm integral of the second-kind, and solved by the appropriate numerical method. The pile–soil–pile interaction effects are approximated by introducing the dynamic interaction factor and using the superposition method. Selected numerical solutions are given in this paper.

According the Muki [19] method, the expressions of the fundamental solutions can be written in the following forms

ur ðr; h; z; tÞ ¼

1 X

urm ðr; z; tÞ cosðmhÞ

ð5aÞ

uhm ðr; z; tÞ sinðmhÞ

ð5bÞ

uzm ðr; z; tÞ cosðmhÞ

ð5cÞ

pfm ðr; z; tÞ cosðmhÞ

ð5dÞ

m¼0

uh ðr; h; z; tÞ ¼

1 X m¼0

uz ðr; h; z; tÞ ¼

1 X m¼0

2. Governing equations and the general solutions According to the consolidation theory of Biot for a poroelastic medium, the governing equations can be expressed in terms of the displacement and the pore pressure as follows [14]

opf oe G ouh €r Gr2 ur þ ðk þ GÞ  2 ð2 þ ur Þ  ¼ qu or r oh or 1 oe G our 1 opf €h  ðuh  2 Þ ¼ qu Gr uh þ ðk þ GÞ r oh r 2 r oh oh 2

Gr2 uz þ ðk þ GÞ

oe opf €z  ¼ qu oz oz

opf 1 _ r þ qf u €r  ¼ 0 w or kd

0 kd

2

¼ kd =ðqf gÞ

ð2aÞ ð2bÞ

1 X

rzzm ðr; z; tÞ cosðmhÞ

ð5fÞ

1 X

rzrm ðr; z; tÞ cosðmhÞ

ð5gÞ

rzh ðr; h; z; tÞ ¼

1 X

rzhm ðr; z; tÞ sinðmhÞ

ð5hÞ

m¼0

qz ðr; h; z; tÞ ¼

1 X

qzm ðr; z; tÞ cosðmhÞ

where subscript m denotes the order of the term in the series. The motion is assumed to be time harmonic with a factor of ei-t, pffiffiffiffiffiffiffi where - is the frequency of the motion and i ¼ 1. For brevity, the term ei-t is suppressed from all following expressions. DifferÞ þ ð1dÞ þ 1r oð1eÞ þ oð1f and with the entiation of Eqs. (1d)–(1f), oð1dÞ or r oh oz aid of Eq. (4)

r2 pf ¼

1 _ € 0 e  qf e kd

ð6Þ

Differentiation of Eqs. (1a)–(1c),

oð1aÞ or

þ ð1aÞ þ 1r r

oð1bÞ oh

þ oð1cÞ yield oz

r2 pf ¼ Mr2 e  q€e

ð7Þ

where M = k + 2G Substitution of Eq. (6) into Eq. (7) result in

r2 e ¼

q  qf 1 €e e_ þ 0 M kd M

ð8Þ

Substitution of Eqs. (5a)–(5e) into Eqs. (1a) and (1b) can obtain

ð2cÞ surface of half-space

ouz  pf oz   ou ou rzr ¼ G r þ z oz or   ouh ouz rzh ¼ G þ oz roh

ð3bÞ

_z qz ¼ w

ð3dÞ

x

θ r

ð3aÞ

h D1 plane

N0e

iωt

y

ð3cÞ D2 plane

where qz is the fluid discharge. Assuming a solid skeleton and an incompressible fluid gives the continuity equation as

_ r 1 ow _r w _ h ow _z ou_ r u_ r 1 ou_ h ou_ z ow þ þ þ þ þ þ þ ¼0 r oh r oh or r oz or r oz

ð5iÞ

m¼0

The constitutive relations of a homogenous poroelastic medium can be expressed as

rzz ¼ ke þ 2G

ð5eÞ

m¼0

ð1cÞ

where ur, uh and uz are the displacements of the solid matrix in the r, h and z directions, respectively; wr, wh and wz are the fluid displacements relative to the solid matrix in the r, h and z directions, respectively; pf is the pore pressure; q, qs and qf are the mass densities of the porous medium, the solid skeleton and the pore fluid, respectively, and have the following relation, q = (1  u)qs + uqf; u is the porosity of the porous medium; k and G are Lame constants; 0 kd is the dynamic permeability; 52 is the Laplace operator; e is the dilatation of the solid skeleton; overdots denote the derivatives of field variables with respect to time t and

our ur 1 ouh ouz þ þ þ r oh or r oz

em ðr; z; tÞ cosðmhÞ

m¼0

rzr ðr; h; z; tÞ ¼

ð1fÞ

e¼

1 X

ð1bÞ

opf 1 _ z þ qf u €z  ¼ 0 w oz kd

o 1 o 1 o o þ þ þ or2 r or r 2 oh2 oz2

eðr; h; z; tÞ ¼

rzz ðr; h; z; tÞ ¼

ð1eÞ

r2 ¼

m¼0

ð1aÞ

ð1dÞ

2

1 X

m¼0

1 opf 1 _ h þ qf u €h  ¼ 0 w r oh kd

2

pf ðr; h; z; tÞ ¼

saturated half-space z

ð4Þ

Fig. 1. Model of a poroelastic half-space subjected to a horizontal load.

408

o2 ðurm þ uhmÞ 1 oðurm þ uhm Þ ðm þ 1Þ2 þ ðurm þ uhm Þ  r or or2 r2 !     opfm m o2 ðurm þ uhm Þ oem m  em   pfm þ ðk þ GÞ þ 2 oz r r or or €rm þ u €hm Þ ¼ qðu

G

M02eiωt

o2 ðurm  uhm Þ 1 oðurm  uhm Þ ðm  1Þ2 þ ðurm  uhm Þ  or 2 r or r2 !     opfm m o2 ðurm  uhm Þ oem m þ em  þ pfm þ ðk þ GÞ þ 2 oz r r or or € rm  u € hm Þ ¼ qðu

ð9aÞ

M01eiωt

ð9bÞ

M0neiωt

N02eiωt

N0neiωt

N 01eiωt

r s

θ r

L

s

s

d

Fig. 2. Model of a group of piles in the poroelastic medium.

[M 0i-M *i (0)]eiωt

M *i (0)e iωt

M 0ie iωt

iωt

[N 0i-N*i (0)]eiωt

N*i (0)e

N0ie iωt

x D0 qi (z)eiωt

y

z Li

Li

qi (z)eiωt

d

Dz elastic pile

Dξ DL

porous half-space

N*i (L i )e iωt

M *i (L i )eiωt

(a) The fictitious pile

iωt

N*i (L i )e

M *i (L i )eiωt

(b) The extended porous half-space

Fig. 3. Model of the ith pile in the poroelastic medium.

Present result M ylonakis [23]

2.5

Present result Mylonakis [23]

2.5

2.0

s/d=5

2.0 s/d=5

1.5

Im(Kg)

Re(Kg)

G

X.-L. Zhou, J.-H. Wang / Computers and Geotechnics 36 (2009) 406–418

1.0

0.5

1.5

s/d=2 1.0

0.5 s/d=2

0.0 0.0

0.0 0.2

0.4

0.6

a0

0.8

1.0

0.0

0.2

0.4

0.6

a0

Fig. 4. The impedance for the 3  3 pile group (L/d = 20, Ep/Es = 1000, m = 0.4, qp/qs = 1.5, bs = 0.05).

0.8

1.0

409

X.-L. Zhou, J.-H. Wang / Computers and Geotechnics 36 (2009) 406–418

Application of the mth order Hankel integral transforms in Eqs. (8), (6), and (1c) result in 2 d ~em m dz2

 a2 ~em m ¼ 0

ð10aÞ

2 m ~fm d p

~m ~m  n2 p fm ¼ bem dz2 2 m ~m ~ zm d u ðk þ GÞ d~em 1 dp fm m ~m  d2 u þ zm ¼  2 G G dz dz dz

a

d

ðqqf Þ M

2

ð10cÞ

mþ1 ~ rm ~mþ1 d ðu þu ðk þ GÞ m 1 m hm Þ ~mþ1 ~ mþ1 ~  d2 ðu n~em  np rm þ uhm Þ ¼ 2 dz G G fm

ð11aÞ

0.22 s/d=5

0.20

s/d=10

0.18

0.16

0.16

0.14

0.14

0.12

0.12

Im(Kg)

0.18

0.10 0.08

s/d

poroelastic medium one-phase s/d=5

0.10 0.08

s/d=2

0.06

0.06

0.04

0.04 poroelastic medium one-phase

0.02

s/d

0.02

0.00

0.00

-0.02

-0.02

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

s/d=10

s/d=2

0.0

0.1

0.2

0.3

0.4

a0

b

-2 ; b = Mg + q-2; d2 ¼ n2  qG -2 .

In all manipulations, a tilde () represents the Hankel transform; the superscript m denotes the mth order Hankel transformation; n is the Hankel transform parameter. Application of the (m + 1)th and (m  1)th order Hankel transforms in Eqs. (9a) and (9b) yields

0.22 0.20

Re(Kg)

ð10bÞ

where a2 = n2 + g; g ¼ k0-M 

0.60

0.6

0.7

0.8

0.9

1.0

a0 0.60

poroelastic medium one-phase

0.55

0.5

0.55

s/d=5

0.50

s/d

poroelastic medium one-phase

0.50 0.45

0.40

0.40

0.35

0.35

Im(Kg)

Re(Kg)

s/d=10

0.45

0.30 0.25

0.30 0.25

0.20

0.20

0.15

0.15

0.10

0.10

0.05

s/d=5

0.05

s/d

s/d=2

0.00

0.00

-0.05

s/d=2

s/d=10

-0.05 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

a0

c

poroelastic medium one-phase

1.4 1.2

s/d=10 s/d=5

0.8

Im(Kg)

Re(Kg) Kg

1.0

0.6 0.4 0.2 0.0 s/d=2

s/d

-0.2 -0.4 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

a0

1.8 1.6

0.5

0.6

0.7

0.8

0.9

1.0

1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3

poroelastic medium s/d

one-phase

s/d=5

s/d=10

s/d=2

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

a0 Fig. 5. For Ep/Es = 100, the impedances of pile groups in the poroelastic medium: (a) case for n  n = 1  2, (b) case for n  n = 2  2 and (c) case for n  n = 3  3.

410

X.-L. Zhou, J.-H. Wang / Computers and Geotechnics 36 (2009) 406–418

2

~ m1 ~ m1 d ðu ðk þ GÞ m 1 m m1 rm  uhm Þ ~ rm ~ m1 ~  d2 ðu u ð11bÞ n~em þ np hm Þ ¼  2 dz G G fm

Application of the (m + 1)th and (m  1)th Hankel transforms in Eqs. (3b) and (3c) yield

Application of the mth Hankel transforms in Eqs. (3a) and (3d) yields

~ mþ1 r~ mþ1 zrm þ rzhm ¼ G

ð13aÞ

ð12aÞ

~ m1 r~ m1 zrm  rzhm

ð13bÞ

ð12bÞ

Eqs. (10a)–(10c) and Eqs. (11a) and (11b) can be solved and the solutions expressed as

r~ mzzm ¼ 2G 0 ~m q zm ¼ kd

a

~m du zm ~m þ k~em m  pfm dz ! ~m op qf -2 u~ mzm  fm oz

  mþ1 ~ rm ~mþ1 dðu þu hm Þ ~m  nu zm dz   m1 ~ ~m1 dðurm  u hm Þ ~m ¼G þ nu zm dz

0.8 0.7

0.6

0.6 0.5

s/d=10

0.5

s/d=5 0.4

0.3

Im(Kg)

0.4

Re(Kg)

s/d

poroelastic medium one-phase

s/d=5

0.2

0.2

s/d=2

0.1 0.0

0.3

0.1

poroelastic medium

s/d=10

s/d

one-phase

-0.1

s/d=2

0.0

-0.2 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

a0

b

poroelastic medium one-phase

1.2

s/d=10

1.0

Im(Kg)

Re(Kg)

0.8 0.6 s/d=5

0.4 0.2 s/d=2

0.0 s/d

-0.2 -0.4 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1

1.0

0.8

0.9

1.0

s/d

one-phase

s/d=5

s/d=10

s/d=2

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

a0 5.5

5.0 4.5

5.0

s/d

poroelastic medium one-phase

4.0

poroelastic medium one-phase

4.5

s/d

4.0

s/d=5

3.5

3.5

3.0

s/d=5

s/d=10

3.0

Im(Kg)

2.5

Re(Kg)

0.7

poroelastic medium

a0

c

0.6

a0

1.6 1.4

0.5

2.0 1.5 1.0

2.5

s/d=10

2.0 1.5 1.0

0.5

0.5

0.0

0.0 s/d=2

-0.5

s/d=2

-0.5

-1.0

-1.0 0.0

0.1

0.2

0.3

0.4

0.5

a0

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

a0

Fig. 6. For Ep/Es = 10,000, the impedances of pile groups in the poroelastic medium: (a) case for n  n = 1  2, (b) case for n  n = 2  2 and (c) case for n  n = 3  3.

411

X.-L. Zhou, J.-H. Wang / Computers and Geotechnics 36 (2009) 406–418

~em ¼ A1 eaz þ B1 eaz   b ~m ðA1 eaz þ B1 eaz Þ þ A2 enz þ B2 enz p f ¼

ð14aÞ ð14bÞ

g

r~ mzz ¼ b1 ðA1 eaz þ B1 eaz Þ þ b2 ðA2 enz þ B2 enz Þ þ 2GdðA3 edz  B3 edz Þ

  a ðA1 eaz  B1 eaz Þ þ a1 ðA2 enz  B2 enz Þ þ A3 edz þ B3 edz

~m u z ¼

ð17aÞ

g

ð14cÞ ~ mþ1 ~ hmþ1 u þu r

  n ðA1 eaz þ B1 eaz Þ  a1 ðA2 enz þ B2 enz Þ ¼

~ m1 ~ hm1 ¼ u u r

az az ~m Þ þ d2 ðA2 enz  B2 enz Þ  d3 ðA3 edz þ B3 edz Þ q z ¼ d1 ðA1 e  B1 e

ð17bÞ ~ mþ1 r~ mþ1 þr ¼ c1 ðA1 eaz  B1 eaz Þ  ðb2 þ 1ÞðA2 enz  B2 enz Þ zr zh

g

þ A4 edz þ B4 edz

 GnðA3 edz þ B3 edz Þ þ GdðA4 edz  B4 edz Þ

ð14dÞ

  n ðA1 eaz þ B1 eaz Þ þ a1 ðA2 enz þ B2 enz Þ þ A5 e þ B5 e

ð17cÞ

~ m1 r~ m1 r ¼ c1 ðA1 eaz  B1 eaz Þ þ ðb2 þ 1ÞðA2 enz  B2 enz Þ zr zh þ c2 ðA3 edz þ B3 edz Þ þ GdðA4 edz  B4 edz Þ

g

dz

dz

ð14eÞ

~m du n mþ1 z ~ ~ mþ1 ~ rm1  u ~ m1 þ ½u þu  ðu Þ h h 2 r dz

where 2Gd2 f

Substitution of Eqs. (10c), (11a), and (11b) into Eq. (2b) yields

~em ¼

Substitution of Eqs. (14b)–(14e) and (16a-16b) into Eqs. 12a, 12b, 13a, and 13b yields

ð15Þ

2Ga2

n

b

a1 ¼ q-2 ; b1 ¼ g þ k  g ; b2 ¼ 2a1 nG  1; c1 ¼ aqf -2 nqf ab 2 g þ g ; d2 ¼  q þ n; d3 ¼ qf -

ð17dÞ 2Gaf

g

; c2 ¼

þ fG; d1 ¼ 

3. Boundary conditions and continuity conditions in saturated half-space

Substitution of Eqs. (14a), (14c), (14d), and (14e) into Eq. (15) gives

2d A5 ¼ A 3 þ A4 n

a

2d B5 ¼  B3 þ B4 n

ð16a-16bÞ

1.1

Consider the poroelastic half-space with a cylindrical polar coordinate system (r, h, z) defined as shown in Fig. 1. The D1 plane is bounded by 0 6 z 6 h and the D2 plane is bounded by

b

1.0 poroelastic medium 0.9

1.0 poroelastic medium

0.9

one-phase

0.8

0.8

0.7

0.7

0.6

one-phase

0.6 s/d=5

0.5

1

Re(pile1)

0.4

Re(pile1) s/d=5

0.5

N(z)

N(z)

1.1

0.3 0.2

1

0.4 0.3 0.2

0.1

0.1

Im(pile1)

0.0

0.0

-0.1

-0.1

-0.2

-0.2

Im(pile1)

-0.3

-0.3 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.0

1.0

0.1

0.2

0.3

0.4

z/L

c

0.5

0.6

0.7

0.8

0.9

1.0

z/L

d

1.2

poroelastic medium

1.4 poroelastic medium 1.2

one-phase

one-phase

1.0

Re(pile1) 1.0 s/d=5

0.8

s/d=5

5 1

0.6

0.6

Re(pile1)

N(z)

N(z)

0.8

0.4 Re(pile5) 0.2

Im(pile1)

1

Re(pile6)

6

0.4 Im(pile6)

0.2

Im(pile1) 0.0

0.0

-0.2

-0.2

Im(pile5) -0.4

-0.4 0.0

0.1

0.2

0.3

0.4

0.5

z/L

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

z/L

Fig. 7. The shear force of pile groups in the poroelastic medium when N0 = 1.0  105 N, M0 = 0.0: (a) case for n  n = 1  2, (b) case for n  n = 2  2, (c) case for n  n = 3  3 and (d) case for n  n = 4  4.

412

X.-L. Zhou, J.-H. Wang / Computers and Geotechnics 36 (2009) 406–418

h < z < 1. The boundary conditions for the free and permeable surface at z = 0 and 0 6 r < 1 are

pf 1 ðr; h; 0Þ ¼ 0

ð18aÞ

rzz1 ðr; h; 0Þ ¼ 0 rzr1 ðr; h; 0Þ ¼ 0 rzh1 ðr; h; 0Þ ¼ 0

ð18bÞ ð18cÞ ð18dÞ

4. Dynamic analysis of pile groups in a poroelastic half-space

The continuity conditions at z = h and 0 6 r < 1 are

uz1 ðr; h; hÞ  uz2 ðr; h; hÞ ¼ 0

ð19aÞ

ur1 ðr; h; hÞ  ur2 ðr; h; hÞ ¼ 0

ð19bÞ

uh1 ðr; h; hÞ  uh2 ðr; h; hÞ ¼ 0

ð19cÞ

rzz1 ðr; h; hÞ  rzz2 ðr; h; hÞ ¼ 0 rzr1 ðr; h; hÞ  rzr2 ðr; h; hÞ ¼ r1 rzh1 ðr; h; hÞ  rzh2 ðr; h; hÞ ¼ r2

ð19dÞ ð19eÞ ð19fÞ

pf 1 ðr; h; hÞ  pf 2 ðr; h; hÞ ¼ 0

ð19gÞ

qz1 ðr; h; hÞ  qz2 ðr; h; hÞ ¼ 0

ð19hÞ

In all manipulations, a subscript i (i = 1, 2) is used to denote the domain number of the D1 and D2 planes. Arbitrary functions in the D1 plane are expressed as A11 to A15 and B11 to B15 . Arbitrary functions in D2 plane are expressed as A21 to A25 and B21 to B25 . Note that for the D2 plane, arbitrary functions A21 to A25  0 ensure the regularity of the solutions at infinity. The arbitrary functions can be determined

a

from the above boundary and continuity conditions. For a homogeh neous circular load, r1 ¼ N0 cos and r 2 ¼  N0 Asin h. A The final step to obtain the solutions is to invert the integral transform. The inverse integral of the Lipschitz–Hankel type involving the product of the Bessel function can be determined by various formulas developed by Eason et al. [20].

0.350

4.1. Shear force, pore pressure and bending moment of pile groups We now consider the elastic pile groups of radius a and length L (a/L << 1) embedded in a poroelastic half-space as shown in Fig. 2. The pile and the surrounding porous medium are assumed to be fully bounded. Following the Muki and Sternberg method, the system is described as an extended poroelastic half-space and the fictitious piles as shown in Fig. 3. In our solution, we treat the extended embedded medium as a three-dimensional poroelastic continuum. In contrast, the ith fictitious pile is regarded as a one-dimensional elastic continuum as far as its constitutive laws and equilibrium conditions are concerned. Halpern and Christiano [21] found that the difference between the vertical compliances and the load transfer mechanism of impermeable and fully permeable rigid plates on a poroelastic half-space in the low-frequency range is negligible. This allows the assumption to be made that

b

0.325 0.300

0.35

poroelastic medium

poroelastic medium

0.30

0.275 0.250

0.25

0.225

s/d=5

0.20

s/d=5

Re(pile1)

0.150

1

Pf(z)

0.175

Pf(z)

1

Re(pile1)

0.200

0.125 0.100

0.15 0.10

0.075 0.050

Im(pile1)

0.05

Im(pile1)

0.025 0.000

0.00

-0.025 -0.050

-0.05 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.0

1.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

z/L

z/L

c

d

poroelastic medium

0.4

0.3

s/d=5

5

0.4

poroelastic medium

s/d=5

0.3

1 Re(pile1)

Pf(z)

Re(pile5)

0.1

1

0.2

Pf(z)

0.2

6

Re(pile1) 0.1 Re(pile6)

Im(pile1)

Im(pile1) 0.0

0.0

Im(pile6)

Im(pile5) -0.1

-0.1 0.0

0.1

0.2

0.3

0.4

0.5

z/L

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

z/L

Fig. 8. The pore pressure of pile groups in the poroelastic medium when N0 = 1.0  105 N, M0 = 0.0: (a) case for n  n = 1  2, (b) case for n  n = 2  2, (c) case for n  n = 3  3 and (d) case for n  n = 4  4.

413

X.-L. Zhou, J.-H. Wang / Computers and Geotechnics 36 (2009) 406–418

ð20aÞ

the angle of the top of the ith pile; upi(0) is the horizontal displacement of the top of the ith pile. If the additional assumption of a small cross sectional rotation of the pile is made, the analysis can be simplified further since the direct moment transfer effects at the ends become negligible. In such circumstances, it is reasonable to assume that

ð20bÞ

Mi ð0Þ ¼ M 0i

ð22aÞ

Mi ðLi Þ ¼ 0

ð22bÞ

the response is not significantly influenced by the exact hydraulic boundary condition at the contact pile–soil surface [22]. The hydraulic boundary condition on the pile–soil interface is therefore neglected in this study.

qi ðzÞ ¼ 

dNi ðzÞ þ qpi Ai -2 upi ðzÞ ði ¼ 1; 2; . . . nÞ dz

dMi ðzÞ ¼ Ni ðzÞ dz * p i,

where q qpi and qsi are the densities of the fictitious pile, the real pile and the poroelastic medium, respectively, and have the following relation qp*i = qpi  (1  u)qsi; upi(z) is the horizontal displacement of the ith fictitious pile; qi(z) is the contact force per unit length on the shaft of the ith fictitious pile; N*i(z) and M*i(z) are the shear force and the bending moment of the ith fictitious pile, respectively; Ai is the area of the ith pile; - is frequency. Based on the response of the reinforcement, the response of the embedded ith pile can be obtained

M i ðzÞ ¼ M i ð0Þ þ

Z

hsi ðzÞ ¼ hpi ðzÞ 0 < z < Li

Ni ðnÞdn ði ¼ 1; 2; . . . ; nÞ Z z M i ð0Þz 1 hpi ðzÞ ¼ þ ðz  nÞNi ðnÞdn þ h0i Epi Ii Epi Ii 0 Z z Mi ð0Þz2 1 upi ðzÞ ¼ þ ðz  nÞ2 Ni ðnÞdn þ h0i z þ upi ð0Þ 2Epi Ii 0 2Epi Ii

ð21aÞ

hsi ðzÞ ¼

j¼1

þ

Lj

^ ji ðn; zÞdn qj ðnÞ/

1.6 poroelastic medium one-phase

1.4

one-phase

1.2

1.4 Re(pile1)

1.2

1.0

Re(pile1)

s/d=5

1.0

s/d=5

1

1

0.8

M(z)

M(z)

ð24Þ

^ ji ðn; zÞ is the axial angle influence function; / ^ ji ðn; zÞ is found where / to possess merely finite jump discontinuities at n = z in the interval [0, L].

b

poroelastic medium

Z 0

ð21cÞ

2.0

1.6

n X ^ ji ð0; zÞ þ N j ðLj Þ/ ^ ji ðLj ; zÞ ½ðN 0j ð0Þ  Nj ð0ÞÞ/

ð21bÞ

where Ep*i = Epi  Esi, Ep*i, Epi and Esi are the Young’s modulus of the fictitious pile, the real pile and the solid skeleton, respectively; h0i is

1.8

ð23Þ

Foregoing the fundamental solutions and the superposition method, the angle hsi(z) in the extended half-space can be written as

z 0

a

With the aid of the Muki and Sternberg method, the condition that the axial angle of the fictitious pile be equal to the axial angle of the cross section of the extended half-space is necessary; i.e.,

0.8 0.6

0.6 0.4

0.4

Im(pile1)

0.2

0.2

Im(pile1)

0.0

0.0

-0.2

-0.2

-0.4 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

z/L

c

1.8

poroelastic medium

1.6

one-phase

d

0.7

0.8

0.9

1.0

1.8

poroelastic medium one-phase

Re(pile6)

1.2

Re(pile5) 1.2

5 1.0

1.0

s/d=5

0.8 0.6

1 Re(pile1)

Im(pile6)

0.4

M(z)

M(z)

0.6

2.0 1.6 1.4

1.4

0.8

0.5

z/L

0.6 0.4 0.2

0.2 0.0 -0.2 -0.4

0.0

-0.6

-0.2

-0.8

-0.4 Im(pile5)

0.0

0.1

0.2

s/d=5

-1.0 -1.2

Im(pile1)

-0.6

Im(pile1)

Re(pile1)

1

-1.4

0.3

0.4

0.5

z/L

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

6 0.9

1.0

z/L

Fig. 9. The bending moment of the pile groups in the poroelastic medium when N0 = 1.0  105 N, M0 = 0.0: (a) case for n  n = 1  2, (b) case for n  n = 2  2, (c) case for n  n = 3  3 and (d) case for n  n = 4  4.

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X.-L. Zhou, J.-H. Wang / Computers and Geotechnics 36 (2009) 406–418

^ ii ðz ; zÞ ¼ 1=GAi ^ ii ðzþ ; zÞ  / /

ð25Þ

With the aid of the superposition method and the application of Eq. (21b), the axial angle of the ith pile can be written as

 Z z n  X M j ð0Þz 1 hpi ðzÞ ¼ þ ðz  nÞN j ðnÞdn þ h0j Epj Ij Epj Ij 0 j¼1

ð26Þ

Adopting the superposition method, the conditions that the axial angle of the fictitious pile be equal to the average of the extensional axial angle of the cross section of the extended half-space are used to solve the pile–soil–pile interaction problem. Manipulation of Eqs. (21), (23), (24) and (26) together with the axial angle compatibility condition, the Fredholm’s integral equation of the second-kind can be expressed as

^ ii ðnþ ; zÞ  / ^ ii ðn ; zÞ þ Ni ðzÞ½/

Lj

Nj ðnÞ

0

j¼1

^ ji ðn; zÞ o/ dn on

f3j ðzÞ ¼ qpj Aj -2

Nj ðgÞðn  gÞ2 dg

Z

z

ð28aÞ

^ ji ðn; zÞdn  1 n/

ð28bÞ

^ ji ðn; zÞdn /

ð28cÞ

0

Z

z

0

^ ji ðn; zÞ ¼ Pn where / j¼1

h

ourj oz

cos h2ji þ

ouhj oz

i sin h2ji ; urj and uhj are the dis-

placements of the jth pile in the r and h direction, respectively; hji is the angle between the jth pile and the ith pile. Utilizing the displacement influence function, the horizontal displacement in the extended half-space can be written as n  X

^ ji ð0; 0Þ þ N0j u

Z

Lj

Nj ðnÞ

ð29Þ

^ ji ðn; zÞ is the horizontal displacement influence function. where u Substitution of Eq. (21c) into Eq. (29), and taking usi(0) = upi(0), the horizontal displacement of the top of the ith pile can be expressed as

b

0.050 0.025

0.000

Im(pile1)

0.000 poroelastic medium -0.025

one-phase

-0.050 -0.075 -0.100

1

-0.125

-0.050

poroelastic medium

-0.075

one-phase

-0.100

s/d=5

N(z)

N(z)

0

0

0.025

Im(pile1)

n

f2j ðzÞ ¼ qpj Aj -2

z

-0.025

Z

^ ji ðn; zÞ ou dn on 0 j¼1  Z Lj ^ ji ðn; zÞdn upj ðnÞu þqpj Aj -2

ð27Þ

a

1 2Epj Ij

f1j ðnÞ ¼

usi ðzÞ ¼

Z Lj ^ ji ðn; zÞdn ðz  nÞNj ðnÞdn þ qpj Aj -2 f1j ðnÞ/ Epj Ij 0 0  þh0j f2j ðzÞ þ upj ð0Þf3j ðzÞ " # n X qpj Aj -2 Mj ð0Þ Z Lj 2 ^ Mj ð0Þz ^  N0j /ji ð0; zÞ  n /ji ðn; zÞdn ¼ 2Epj Ij Ep Ij 0 j¼1 1



Z

"Z n X

where

Re(pile1)

-0.125 -0.150

-0.150

s/d=5

Re(pile1)

1

-0.175 -0.175 -0.200 -0.200

-0.225

-0.225

-0.250 -0.275

-0.250

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.0

1.0

0.1

0.2

0.3

0.4

c

0.5

0.6

0.7

0.8

0.9

1.0

z/L

z/L 0.10

d

Im(pile1) 0.05

0.15 Im(pile1)

0.10 Im(pile5) 0.05

0.00

Im(pile6)

0.00 -0.05

-0.05 -0.10

5

-0.15

s/d=5

1 -0.20

s/d=5

-0.15

N(z)

N(z)

-0.10

Re(pile1)

-0.20 Re(pile1)

-0.25

-0.25

1

6

-0.30 -0.35

-0.30

Re(pile5)

poroelastic medium

-0.35

one-phase

-0.40

Re(pile6)

-0.40

poroelastic medium

-0.45

one-phase

-0.50

0.0

0.1

0.2

0.3

0.4

0.5

z/L

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

z/L

Fig. 10. The shear force of the pile groups in the poroelastic medium when N0 = 0, M0 = 1.0  105 Nm: (a) case for n  n = 1  2, (b) case for n  n = 2  2, (c) case for n  n = 3  3 and (d) case for n  n = 4  4.

415

X.-L. Zhou, J.-H. Wang / Computers and Geotechnics 36 (2009) 406–418

"

n X q A-2 Mj ð0Þ Z Lj 2 1 ^ji ð0; 0Þ þ pj ^ ji ðn; 0Þdn N0j u n u 2Epj Ij c1j 0 j¼1 Z Lj Z Lj ^ ji ðn; 0Þ ou ^ ji ðn; 0Þdn dn þ qpj Aj -2 þ Nj ðnÞ f1 ðnÞu on 0 0 # Z

The total bending moment is

upi ð0Þ ¼

M0 ¼

^ ji ðn; 0Þdn nu

ð30Þ

0

RL ^ ji ðn; 0Þdn. where c1j ¼ 1  qpj Aj -2 0 j u To generate group stiffness for the case of a rigid cap, the displacements of all heads must be equal to unity

uG ð0Þ ¼ upi ð0Þ ¼

n X

uij ¼

j¼1

1 X aij N0j ks

n X

ð31Þ

Ni ðzÞ ¼

n X

"

N0i

2EP I

where N0 is the total load of the top of the group piles and N0i is the load of the top of the ith pile.

^ji ðn; zÞdn n2 p

0

Z

Lj

Nj ðnÞ

0

0

ð34aÞ Mi ð0Þ ¼

n  Z X 

poroelastic medium

Lj

 Nj ðnÞdn

ð34bÞ

0

j¼1

b

0.09

Lj

Z Lj ^ji ðn; zÞ op ^ji ðn; zÞdn fj1 ðnÞp dn þ qp A-2 on 0 0  Z Lj Z Lj ^ji ðn; zÞdn þ upj ð0Þqp A-2 ^ji ðn; zÞdn np p þh0j qp A-2

þ

ð32Þ

0.08

ð33Þ

qp A-2 Mj ð0Þ Z

^ji ð0; zÞ þ N0j p

j¼1

i¼1

a

ði ¼ 1; 2; . . . ; nÞ

Solving Eqs. (20b), (27), (32) and (33), yield the shear force N*(z), the axial angle of pile top h0, the horizontal displacement up(0) and the bending moment M0. Once the shear force of fictitious pile N*(z) is known, the shear force, bending moment and pore pressure of the ith pile can be determined by

where uG(0) is the total displacement of the top of the group piles; ks is the normalized horizontal dynamic impedance of the single pile; N0j is the load acting on the top of the jth pile; aij is the dynamic interaction factor between the jth and ith pile. For a rigid cap and the loads not equal to each other, the total load of the pile groups is

N0 ¼

Mi

i¼1

Lj

þh0;j qpj Aj -2

n X

0.10

poroelastic medium

0.07

0.08 0.06 s/d=5

0.05

s/d=5

Re(pile1)

1

Re(pile1)

Pf(z)

0.04

Pf(z)

0.06

1

0.03 0.02

0.04 0.02

0.01

Im(pile1)

Im(pile1)

0.00

0.00 -0.01

-0.02

-0.02 -0.03

-0.04

-0.04

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

c

0.5

0.6

0.7

0.8

0.9

1.0

z/L

z/L 0.12

d

poroelastic medium

0.15

poroelastic medium

0.10

s/d=5

0.10 0.08

s/d=5

5 0.06

1

Pf(z)

Pf(z)

1

0.05

0.04

Im(pile1)

0.02

Im(pile5)

0.00

6

Im(pile1) 0.00

Im(pile6)

-0.02

-0.05

Re(pile1)

-0.04

Re(pile1) -0.06

Re(pile5)

Re(pile6)

-0.10

-0.08

0.0

0.1

0.2

0.3

0.4

0.5

z/L

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

z/L

Fig. 11. The pore pressure of the pile groups in the poroelastic medium when N0 = 0, M0 = 1.0  105 Nm: (a) case for n  n = 1  2, (b) case for n  n = 2  2, (c) case for n  n = 3  3 and (d) case for n  n = 4  4.

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X.-L. Zhou, J.-H. Wang / Computers and Geotechnics 36 (2009) 406–418

pfi ðzÞ ¼

n X

" ^fji ð0; zÞ þ N0j p

qp A-2 Mj ð0Þ Z 2EP I

j¼1

Lj

^fji ðn; zÞdn n p

0

Z lj ^fji ðn; zÞ op ^fji ðn; zÞdn dn þ qp A-2 f1j ðnÞp on 0 0  Z Lj Z Lj ^fji ðn; zÞdn þ upj ð0Þqp A-2 ^fji ðn; zÞdn np þh0j qp A-2 p

þ

Z

Lj

N j ðnÞ

0

 p 2 sin h 2  1=2     2s s 1 Vs aij ðs; 0Þ  exp ðbs þ iÞ  a0 d d 2 V La       1=2  p 2s s 1 aij s;  exp ðbs þ iÞ  a0 d d 2 2

aij ðs; hÞ  aij ðs; 0Þ cos2 h þ aij s;

2

0

ð35aÞ ð35bÞ ð35cÞ

ð34cÞ ^ji ðn; zÞ is the shear force influence function and p ^fji ðn; zÞ is the where p pore pressure influence function. 4.2. Dynamic impedance of pile groups The pile–soil–pile system shown in Fig. 2 is assumed to be subject to horizontal loading and a bending moment applied at the pile head. The superposition method is used to calculate the dynamic impedance of n identical piles connected through a rigid cap by superimposing the interaction factors between individual pile pairs. Complex interaction factors can be used to obtain the approximate dynamic group effects. In terms of the Mylonakis and Gazetas [23] method for horizontal dynamic analysis in the single-phase elastic medium, the horizontal dynamic interaction factors aij are written as

where aij(s, h) is the dynamic interaction factors between the jth  and ith pile, aii = 1; aij(s, 0), aij s; p2 are the dynamic interaction factors corresponding to waves traveling along and perpendicular to the direction of loading, respectively; s is the distance between the jth and ith pile; VLa is the so-called ‘‘Lysmer’s analogue” wave 3:4V s velocity, V La ¼ ð1vÞ p; Vs is the shear wave velocity; bs is the hysteretic soil damping; m is poisson’s ratio; a0 ¼ -V sd. In order to obtain the dynamic impedance of the pile group in the poroelastic medium, we used the Mylonakis formulation for simplicity. Eqs. (31) and (32) were solved to obtain the displacement of the pile group uG(0) and the shear load N0i of the single pile. The dynamic impedance of the pile group is

kG ¼

1.1

N0 uG

ð36Þ

1.1

1.0

poroelastic medium

1.0

0.9

one-phase

0.9

0.8

0.8

0.7

0.7

poroelastic medium one-phase

Re(pile1) s/d=5

0.6

s/d=5

Re(pile1)

0.6

1

0.5

M(z)

M(z)

0.5

1

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

Im(pile1)

0.0

0.0

-0.1

-0.1

-0.2

Im(pile1)

-0.2 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.0

1.0

0.1

0.2

0.3

0.4

z/L 1.6 poroelastic medium

1.4

one-phase Re(pile5)

1.2 1.0

s/d = 5

1

Im(pile1)

M(z)

M(z)

5 0.8 0.6 0.4 Im(pile5) 0.2 0.0 -0.2

Im(pile1)

0.0

0.1

0.2

0.3

0.4

0.5

z/L

0.5

0.6

0.7

0.8

0.9

1.0

z/L

0.6

0.7

0.8

0.9

1.0

1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3

poroelastic medium one-phase

Re(pile6)

s/d=5

1

6

Re(pile1)

Im(pile6)

Im(pile1) 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

z/L

Fig. 12. The bending moment of the pile groups in the poroelastic medium when N0 = 0, M0 = 1.0  105 Nm: (a) case for n  n = 1  2, (b) case for n  n = 2  2, (c) case for n  n = 3  3 and (d) case for n  n = 4  4.

X.-L. Zhou, J.-H. Wang / Computers and Geotechnics 36 (2009) 406–418

417

5. Numerical results and discussions

6. Conclusion

If the porosity and the density of the pore fluid are assumed to be zero, then the porous medium is reduced to a conventional single-phase elastic medium. To verify the proposed method for calculation of pile groups’ responses to horizontal loading, our results are compared with those of Mylonakis and Gazetas [23], as illustrated in Fig. 4. The impedance kg (real and imaginary parts) has been normalized with respect to the respective single pile static stiffness multiplied by the number of piles in the pile groups. The accuracy of the present formulation and its numerical results are confirmed by the above comparisons. In what follows, the influences of the pile distance s/d and the pile flexibility ratio Ep/Es on the dynamic response of the pile groups in the poroelastic medium are investigated. Figs. 5 and 6 show the variation in the impedance of 1  2, 2  2 and 3  3 pile groups embedded in the poroelastic medium with respect to the normalized frequency parameter a0 for the two cases: Ep/Es = 100 and Ep/Es = 10,000. The properties are m = 0.25; k ¼ G ¼ G0 ð1 þ 2bs iÞ; G0 = 1.0  106; bs = 0.05; L/d = 20; u = 0.3; qp/ qs = 1.5; qf/qs = 0.5; s/d = 2, 5, 10. Fig. 5 shows the horizontal impedance of 1  2, 2  2 and 3  3 pile groups with a pile flexibility ratio Ep/Es = 100. It is observed that for the pile group, the dynamic response is a complex relation between the pile distances and the medium properties. This causes the impedance to increase or decrease as the pile distances change. The results for Ep/Es = 10,000 are presented in Fig. 6. Over the entire frequency range, impedance increase as Ep/Es increases. The impedance also increases with the number of piles. The impedance in the poroelastic medium is greater than in the single-phase elastic medium. The influence of the number of piles on the shear force, the bending moment and the pore pressure of group piles, along pile length in the poroelastic medium and the single-phase elastic medium are shown in Figs. 7–9. The normalized properties are: m = 0.25; k ¼ G ¼ G0 ð1 þ 2bs iÞ; G0 ¼ 1:0  106 ; bs = 0.05; L/d = 20; Ep/Es = 1000; u = 0.3; qp/qs = 1.5; qf/qs = 0.5; s/d = 5; a0 = 0.8. Fig. 7 shows the profiles of the normalized shear force, the pore pressure and the bending moment along the pile length with N0 = 1.0  105 N and M0 = 0.0. The normalized shear force, bending  ; moment and pore pressure are expressed as NðzÞ ¼ nNðzÞ N0 nApf   ; p ðzÞ ¼ . MðzÞ ¼ 2nMðzÞ f N0 d N0 In these graphs, the shear force, pore pressure and bending moment are plotted at four different numbers of piles, n  n = 1  2, n  n = 2  2, n  n = 3  3 and n  n = 4  4. It can be seen that the corner piles carry greater shear force and pore pressure than the center piles when N0 = 1.0  105 N and M0 = 0.0. In contrast, the center piles have a greater bending moment than the corner piles, which is a result of the pile–soil–pile interaction. The shear forces and pore pressure along the pile length rapidly decrease. The shear force and pore pressure are mostly confined to the upper part of the piles. The shear force and pore pressure seem to be less sensitive to the medium properties, while the bending moment in the poroelastic medium is smaller than in the single-phase medium. Figs. 10–12 show the profiles of the normalized shear force, pore pressure and bending moment along the pile length with N0 = 0.0 and M0 = 1.0  105 Nm. The normalized shear force, pore  ; pressure and bending moment are expressed as NðzÞ ¼ ndNðzÞ Mð0Þ 3 nAd p nMðzÞ  f ðzÞ ¼ Mð0Þ f . MðzÞ ¼ Mð0Þ ; p In these graphs, the head-load transmitted onto the pile through the cap would produce a response atop the piles that would be different from the response of the piles when N0 = 1.0  105 N and M0 = 0. The center piles carry greater shear force, bending moment and pore pressure than the corner piles.

A simple and computationally efficient solution was developed for pile groups embedded in a poroelastic medium that were subjected to a horizontal harmonic load. The response of the piles embedded in the porous half-space were analyzed using the Muki and Sternberg formulation. This problem can be divided into the Fredholm integral equation of the second-kind solved by the Hankel integral transform, and the numerical inverse transform. The dynamic interaction factors and the superposition method used with the proposed method are in convincing agreement with more rigorous solutions. Numerical results demonstrate a considerable difference between pile responses of single-phase and poroelastic mediums. Numerical solutions for the impedance values indicate a significant dependence on the pile distances s/d and the pile flexibility ratio Ep/Es. The dynamic impedance response was more complex. The shear force, pore pressure and bending moment have a strong dependence on the number of piles and the loading path. The solution presented may be extended to analyze the dynamic response of the pile groups in a layered porous half-space. Acknowledgements The research was financially supported by Shanghai Leading Academic Discipline Project No. B208. The project was also supported by National Natural Science Foundation of China with Grant No.50679041.

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