Computers and Geotechnics 36 (2009) 427–434
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Computers and Geotechnics
journal homepage: www.elsevier.com/locate/compgeo
Boundary element analysis of axially loaded piles embedded in a multi-layered soil
Z.Y. Ai a,*, J. Han b
a
Department of Geotechnical Engineering, Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University,
1239 Siping Road, Shanghai 200092, China
Civil, Environmental, and Architectural Engineering (CEAE) Department, The University of Kansas, Lawrence, KS 66045, USA
b
a r t i c l e
i n f o
Article history:
Received 5 January 2008
Received in revised form 1 June 2008
Accepted 2 June 2008
Available online 21 July 2008
Keywords:
Analytical solutions
Axially loaded piles
Boundary element method
Integral transforms
Multi-layered soil
a b s t r a c t
In a field, piles are likely installed in a multi-layered soil. Analysis of axially loaded piles in a multi-layered soil is complicated and deserves more attention. A boundary element method is used in this study to
analyze an axially loaded single pile in a multi-layered soil using the solution for vertical and horizontal
axisymmetric ring loads in a multi-layered elastic medium. Good and reasonable agreement is obtained
between the proposed and published solutions for a single pile in a homogenous soil, a finite soil, and a
Gibson soil. The proposed solution is also used to evaluate an axially loaded single pile in a multi-layered
(8 layers) soil.
Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction
Piles have been commonly used in practice to support superstructures. A number of methods have been developed in the past
to analyze axially loaded piles, including simplified analytical
methods [1,2], load-transfer methods [3–5], boundary element
methods [6,7], finite element methods [8–13], infinite layer methods [14,15], finite layer methods [16–18], and ‘‘hybrid” type methods [19–21]. Among them, the boundary element method is one of
the commonly used methods to analyze a single pile and pile
groups.
Poulos and Davis [6] and Butterfield and Banerjee [7] employed
the Mindlin solutions [22] and boundary integral equations to analyze axially loaded piles embedded in a linearly elastic homogeneous half space. Their results have been widely used for
practical applications. Combined with the Mindlin solutions, Poulos and Davis [6] proposed an approximate method for piles
embedded in a finite depth soil by utilizing the Steinbrenner [23]
approximation, while Poulos and Mattes [24] employed a mirrorimage technique for piles seated on a firm base.
In reality, natural ground is often non-homogeneous and multi-layered so that the Mindlin solutions based on a linearly elastic
homogeneous half space are not suitable for the analysis of piles
in non-homogeneous and multi-layered soils. To account for
non-homogeneity of soils, Poulos [25] proposed an empirical
* Corresponding author. Tel.: +86 21 65982201; fax: +86 21 65985210.
E-mail address: zhiyongai@mail.tongji.edu.cn (Z.Y. Ai).
0266-352X/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved.
doi:10.1016/j.compgeo.2008.06.001
method, in which a modified soil modulus is employed in the
Mindlin solutions. Chan et al. [26] and Davies and Banerjee [27]
proposed analytical solutions for an interior point load, which is
applied at the interior and interface of a two-layered soil, respectively. Utilizing the solution of Chan et al. [26], Lee et al. [28] and
Chin et al. [29] studied the behavior of axially loaded piles and
pile groups in layered soils with a simplified elastic continuum
boundary element method, respectively. Based on the solutions
of Davies and Banerjee [27], Banerjee and Davies [30,31] analyzed
the behavior of pile groups embedded in a Gibson soil and an axially and laterally loaded single pile embedded in non-homogeneous soils. However, these solutions are only appropriate for
axially loaded piles in a two-layered soil and they can not be used
for a more general situation, such as piles in a multi-layered soil.
Therefore, an analytical solution for axially loaded piles in a multilayered soil is needed.
The first author with others [32,33] obtained a more general solution for a multi-layered elastic medium subjected to both axisymmetric and asymmetric loads applied either on an external surface
or in the interior of the medium. This solution is an extension to
the Sneddon and Muki solutions [34–36]. The extended Sneddon
and Muki solution [33] is more appropriate for a multi-layered or
non-homogeneous soil than the approximation method based on
the Mindlin solutions and the solutions [26,27] for a two-layered
soil. The objective of this paper is to apply the extended Sneddon
and Muki solution with a boundary integral equation method to
study the behavior of axially loaded piles embedded in a multi-layered soil.
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Z.Y. Ai, J. Han / Computers and Geotechnics 36 (2009) 427–434
2. Solution for axisymmetric ring loads in the multi-layered soil
Prior to the development of a solution for piles embedded in a
multi-layered soil, the solution for axisymmetric ring loads in the
multi-layered soil should be obtained.
Take an n-layered elastic medium with a vertical unit ring load
or a radial unit ring load applied in the interior of the m-th layer
as shown in Fig. 1. On the z plane, which is paralleled to the ground
surface, the unknown displacements and stresses are u(r, z), w(r, z),
srz(r, z), and rz(r, z).
z ðn; zÞ be the Hankel transform
ðn; zÞ, wðn;
rz ðn; zÞ, and r
Let u
zÞ, s
of u(r, z), w(r, z), srz(r, z), and rz(r, z), the Hankel transform and its
inversion have the following relations [37]:
Z
ðn; zÞ ¼
u
1
uðr; zÞrJ 1 ðnrÞdr;
Z0
zÞ ¼
wðn;
r z ðn; zÞ ¼
wðr; zÞrJ0 ðnrÞdr;
Z0
srz ðn; zÞ ¼
ð1aÞ
1
ð1bÞ
1
Z 01
0
srz ðr; zÞrJ1 ðnrÞdr;
ð1cÞ
rz ðr; zÞrJ0 ðnrÞdr
ð1dÞ
If a point at a given depth z in the i-th layer is above the loading
surface at which the vertical or radial ring load is applied, the stresses and displacements in the transform domain can be derived
from the equations [32,33]
Gðn; zÞ ¼ ½aij 44 Gðn; 0Þ;
ð5Þ
where
ðn; zÞ; wðn;
zÞ; s
zr ðn; zÞ; r
z ðn; zÞT ;
Gðn; zÞ ¼ ½u
ð6aÞ
T
zr ðn; 0Þ; r
z ðn; 0Þ ;
ðn; 0Þ; wðn;
0Þ; s
Gðn; 0Þ ¼ ½u
ð6bÞ
½aij 44 ¼ Uðn; z Hi1 ÞUðn; DHi1 Þ . . . Uðn; DH1 Þ;
ð6cÞ
in which U(n, z) is the transfer matrix and its elements can be found
in the published paper by Ai et al. [33], DHi = Hi Hi1 (i = 1, 2, . . . n),
and Hi is the distance from the bottom of the i-th layer to the surface of the first layer.
If a point at a given depth z in the j th layer is below the loading
surface, we have
Gðn; zÞ ¼ ½bij 44 Gðn; Hn Þ;
ð7Þ
where
zr ðn; Hn Þ; r
z ðn; Hn ÞT ;
ðn; Hn Þ; wðn;
Hn Þ; s
Gðn; Hn Þ ¼ ½u
ð8aÞ
and
uðr; zÞ ¼
Z
wðr; zÞ ¼
ðn; zÞnJ 1 ðnrÞdn;
u
Z0
srz ðr; zÞ ¼
rz ðr; zÞ ¼
1
ð2aÞ
1
Gðn; 0Þ and Gðn; Hn Þ can be determined from Eqs. (3) and (4)
[32,33] and the following equation:
srz ðn; zÞnJ1 ðnrÞdn;
ð2cÞ
g41 ;
Gðn; Hn Þ ¼ ½fij 44 Gðn; 0Þ ½sij 44 fp
r z ðn; zÞnJ0 ðnrÞdn;
ð2dÞ
1
Z 01
0
ð8bÞ
ð2bÞ
zÞnJ0 ðnrÞdn;
wðn;
Z0
½bij 44 ¼ Uðn; z Hj ÞUðn; DHjþ1 Þ . . . Uðn; DHn1 ÞUðn; DHn Þ:
in which J0(n, r) and J1(n, r) are zeroth and first order Bessel functions, respectively.
On the surface of the elastic medium (i.e., z = 0), the boundary
conditions are
rz ðr; 0Þ ¼ srz ðr; 0Þ ¼ 0 or r z ðn; 0Þ ¼ srz ðn; 0Þ ¼ 0:
ð3Þ
The bottom of the multi-layered elastic medium (i.e., z = Hn)
usually can be treated as a fixed boundary (if the n-th layer represents an elastic half-space, Hn can be assigned to a large value, for
example 10,000 m). The fixed boundary has
ðn; Hn Þ ¼ wðn;
Hn Þ ¼ 0:
uðr; Hn Þ ¼ wðr; Hn Þ ¼ 0 or u
ð4Þ
a
E1 , v1
E2 , v2
Em , vm
½fij 44 ¼ Uðn; DHn ÞUðn; DHn1 Þ . . . Uðn; DH1 Þ;
ð10aÞ
½sij 44 ¼ Uðn; DHn ÞUðn; DHn1 Þ . . . Uðn; DHm2 Þ;
ð10bÞ
in which DHm2 = Hm Hm1 and Hm1 is the distance from the plane at
which the vertical or radial load is applied, to the surface of the first
layer.
The vertical unit ring load can be expressed as
fpg41 ¼ ½ 0; 0; 0; dðr r0 Þ=2pr T :
T
g41 ¼ ½ 0; 0; 0; J 0 ðnr0 Þ=2p :
fp
ð11bÞ
r
ΔH 1
ΔH 1
Δ H2
E 2 , v2
ΔH 2
Δ Hm2
E m , vm
Δ Hm
Δ H m1
ro
ΔH m 2
E m+ 1 , vm+ 1
Δ H m +1
E m +1 , v m +1
Δ H m +1
E n −1 , v n −1
Δ Hn − 1
E n −1 , v n −1
Δ H n −1
E n , vn
ΔHn
E n , vn
ΔH n
z
ð11aÞ
Taking a zeroth order Hankel transform of Eq. (11a) yields
E1 , v1
Δ H m1
ro
where
b
r
ð9Þ
z
Fig. 1. Ring unit loads in a multi-layered soil: (a) vertical ring unit load and (b) radial ring unit load.
ΔH m
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Z.Y. Ai, J. Han / Computers and Geotechnics 36 (2009) 427–434
The radial unit ring load is
3. Solution for axially loaded piles in the multi-layered soil
T
fpg41 ¼ ½ 0; 0; dðr r 0 Þ=2pr; 0 :
ð12aÞ
Taking the first order Hankel transform of Eq. (12a) yields:
g41 ¼ ½ 0; 0; J 1 ðnr 0 Þ=2p; 0 T :
fp
ð12bÞ
The unknown variables of Gðn; 0Þ and Gðn; Hn Þ can be solved as follows: under the vertical unit ring load,
1 ðn; 0Þ ¼
u
1 ðn; 0Þ ¼
w
1
f22 s14 f12 s24
J ðnr0 Þ
;
2p 0
f12 f21 þ f11 f22
ð13aÞ
1
f21 s14 f11 s24
J ðnr 0 Þ
;
2p 0
f12 f21 f11 f22
ð13bÞ
Fig. 2 presents an axially loaded pile embedded in the multilayered elastic soil. Using the analytical solutions obtained in the
previous section, the boundary integral equation for the pile-soil
interface can be written as follows:
wðr; zÞ ¼
Z
L
ws w1 ðr a ; h; r; zÞdh þ
0
þ
Z
L
wr w2 ðr a ; h; r; zÞdh
0
Z
rb
wb w1 ðr1 ; L; r; zÞdr1 ;
0
ð17aÞ
srz1 ðn; Hn Þ ¼
1
f22 f31 s14 f21 f32 s14 f12 f31 s24 þ f11 f32 s24 þ f12 f21 s34 f11 f22 s34
J ðnr0 Þ
;
2p 0
f12 f21 þ f11 f22
ð13cÞ
r z1 ðn; Hn Þ ¼
1
f22 f41 s14 f21 f42 s14 f12 f41 s24 þ f11 f42 s24 þ f12 f21 s44 f11 f22 s44
J ðnr0 Þ
;
2p 0
f12 f21 þ f11 f22
ð13dÞ
under the radial unit ring load,
2 ðn; 0Þ ¼
u
1
f22 s13 f12 s23
J ðnr0 Þ
;
2p 1
f12 f21 þ f11 f22
1
f21 s13 f11 s23
2 ðn; 0Þ ¼
J ðnr 0 Þ
;
w
2p 1
f12 f21 f11 f22
ð14aÞ
uðr; zÞ ¼
Z
0
ð14bÞ
þ
L
ws u1 ðra ; h; r; zÞdh þ
Z
Z
L
wr u2 ðr a ; h; r; zÞdh
0
rb
wb u1 ðr 1 ; L; r; zÞdr 1 ;
ð17bÞ
0
srz2 ðn; Hn Þ ¼
1
f22 f31 s13 f21 f32 s13 f12 f31 s23 þ f11 f32 s23 þ f12 f21 s33 f11 f22 s33
J ðnr0 Þ
;
2p 1
f12 f21 þ f11 f22
ð14cÞ
r z2 ðn; Hn Þ ¼
1
f22 f41 s13 f21 f42 s13 f12 f41 s23 þ f11 f42 s23 þ f12 f21 s43 f11 f22 s43
J ðnr0 Þ
;
2p 1
f12 f21 þ f11 f22
ð14dÞ
Taking an inverse Hankel transform of Gðn; zÞ in Eqs. (5) and (7),
the solution for displacements in the multi-layered elastic medium
subjected to a vertical or radial ring load can be obtained as follows: under the vertical unit ring load,
w1 ðr0 ; Hm ; r; zÞ
(R1
1 ðn; 0Þ þ a22 w
1 ðn; 0ÞÞJ0 ðnrÞdn
nða21 u
¼ R01
z1 ðn; 0ÞÞJ 0 ðnrÞdn
nðb23 srz1 ðn; 0Þ þ b24 r
0
for z < Hm ;
for z > Hm ;
ð15aÞ
u1 ðr 0 ; Hm ; r; zÞ
(R1
1 ðn; 0Þ þ a12 w
1 ðn; 0ÞÞJ1 ðnrÞdn
nða11 u
¼ R01
z1 ðn; 0ÞÞJ 1 ðnrÞdn
nðb
s
ðn;
0Þ
þ
b
13 rz1
14 r
0
for z < Hm ;
for z > Hm ;
ð15bÞ
E1 , v1
r
ra
Δ H1
h
dh
Δ H2
L
for z < Hm ;
for z > Hm ;
ð16aÞ
u2 ðr 0 ; Hm ; r; zÞ
(R1
2 ðn; 0Þ þ a12 w
2 ðn; 0ÞÞJ1 ðnrÞdn
nða11 u
0
¼ R1
z2 ðn; 0ÞÞJ1 ðnrÞdn
nðb13 srz2 ðn; 0Þ þ b14 r
0
P
E 2 , v2
under the radial unit ring load,
w2 ðr 0 ; Hm ; r; zÞ
(R1
2 ðn; 0Þ þ a22 w
2 ðn; 0ÞÞJ0 ðnrÞdn
nða21 u
0
¼ R1
z2 ðn; 0ÞÞJ0 ðnrÞdn
nðb23 srz2 ðn; 0Þ þ b24 r
0
in which ws is the vertical fictitious stress acting along the pile
shaft; wr is the radial fictitious stress acting along the pile shaft;
wb is the vertical fictitious stress acting on the pile base. L is the pile
length, ra is the pile radius, and rb is the pile base radius.
Eqs. (17a) and (17b) are a rigorous mathematical treatment of
axially loaded piles embedded in a multi-layered soil. Because of
the complexity of the analysis associated with the integral equations, analytical solutions for the above equations are not available.
For practical applications with enough accuracy, the numerical
E i , vi
rb
Δ Hi
for z < Hm ;
for z > Hm ;
ð16bÞ
The numerical computational methods for the solution of the
multi-layered elastic medium are presented in the published paper
by Ai et al. [33].
E n −1 , v n −1
Δ H n −1
E n , vn
ΔH n
z
Fig. 2. A pile embedded in the multi-layered soil.
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Z.Y. Ai, J. Han / Computers and Geotechnics 36 (2009) 427–434
solutions can be obtained by dividing the pile shaft into n equal
segments and the base into m rings, on which a uniform load is
distributed. Therefore, the discrete form of the vertical and radial
displacements of any element i on the shaft can be derived as
follows:
ðws Þi ¼
n
X
n
m
X
X
ðws Þj ðK WW Þij þ
ðwr Þj ðK UW Þij þ
ðwb Þj ðK BW Þij ;
j¼1
j¼1
j¼1
Pi ¼
ð18aÞ
j¼1
n
n
m
X
X
X
ðws Þj ðK WU Þij þ
ðwr Þj ðK UU Þij þ
ðwb Þj ðK BU Þij ;
ðus Þi ¼
j¼1
ð18bÞ
j¼1
ð18cÞ
j¼1
in which i = 1, 2, 3, . . . m. The elements of the K matrices can be
found in the appendix.
Combining Eqs. (18a), (18b), and (18c) yields the following
equation:
½W ¼ ½K½W;
ð19Þ
T
½W ¼ ½K1 ½W:
ð20Þ
To solve the above equation, the following differential equations
should be considered:
ow
Pz
;
¼
oz
Ap Ep
ow
u ¼ lp
ra ;
oz
ð21Þ
ð22Þ
in which w and u are the vertical and radial displacements of the
pile, respectively, Ap ¼ pr 2a , Ep is Young’s modulus of the pile material, lp is Poisson’s ratio of the pile material, and Pz is the axial force
along the piles at the depth z and can be obtained as follows:
Z
L
z
ws dh þ
Z
rb
0
j¼1
wb dr:
ð23Þ
If a unit displacement at the head of the pile is given, Eq. (21) can be
re-written in a finite difference form as follows:
3 2 w1 3
0
1
0
0 0
0
0
0
6 w2 7
6 1 0
1
0 0
0
0
0 7
7
76
6
7
76
6
w3 7
1 0
0
0
0 76
6 0 1 0
6
76 . 7
6
6 76 . 7
76 . 7
6
7
7
6
7
0
0
0 1 0
1
0 76
6 0
wn2 7
76
6
7
6
5
4 0
0
0
0 0 1 0
1 4 wn1 5
0
0
0
0 0
0 2 2
wn
3 2 3
2
P1
1
6 P2 7 6 0 7
7 6 7
6
7 6 7
6
6 P3 7 6 0 7
7 6 7
2L 6
6 .. 7 6 .. 7
¼
6 . 7 þ 6 . 7;
7 6 7
nEp Ap 6
7 6 7
6
6 Pn2 7 6 0 7
7 6 7
6
4 Pn1 5 4 0 5
0
ð25Þ
j¼1
wi1 þ wi
i ¼ 1; 2; 3 . . . n;
2
wi wi1
lp ra i ¼ 1; 2; 3 . . . n:
ðus Þi ¼ L=n
ð26aÞ
ð26bÞ
The displacement at the base of the pile can be expressed as
ðwb Þi ¼ wn
i ¼ 1; 2; 3 . . . m:
ð26cÞ
Using an iterative scheme proposed by Butterfield and Banerjee
[7], the solution for Eq. (20) can be obtained. Once the fictitious
stresses are obtained, the actual stresses and displacements
(including those on the real pile boundaries) can be obtained using
the approach described in Butterfield and Banerjee [7].
4. Results of analysis
4.1. A pile in a homogeneous elastic half-space
To verify the proposed solution, a simple problem with an axially loaded pile in a homogeneous elastic half-space is first selected. This homogeneous elastic half-space is considered
comprising of several elastic layers with identical elastic parameters (i.e., one homogeneous layer) with a large depth (for example
10,000 m). Fig. 3 presents a comparison of the normalized pile
head stiffness (P/GWD) by Butterfield and Banerjee [7] and the proposed solution, in which W is the settlement at the pile head, L is
the length of the pile, D is the diameter of the pile, k = Ep/G, Ep is
the elastic modulus of the pile, G is the shear modulus of the soil,
and P is the applied load on the pile head. Fig. 4 presents a comparison between the proposed method and Butterfield and Banerjee
[7] on the distribution of shear stress s along the shaft with depth
z. In this comparison, the pile is in the homogeneous elastic halfspace and has a slenderness ratio of L/D = 80 and different pile-soil
modulus ratios. Both Figs. 3 and 4 show that the results from the
proposed method are in good agreement with those of Butterfield
and Banerjee [7], which verifies that the proposed solution for a
multi-layered soil can be used for a homogeneous soil.
120
2
Pn
ni
X
ðws Þðnþ1jÞ L=n:
T
where
½W ¼ ½½ws ½us ½wb ,
½W ¼ ½½ws ½wr ½wb ,
½K ¼
2
3
½K WW ½K UW ½K BW 4 ½K WU ½K UU ½K BU 5. The unknown fictitious stresses can be ex½K WB ½K UB ½K BB pressed as follows:
Pz ¼
ðwb Þj r b =m þ
ðws Þi ¼
in which i = 1, 2, 3, . . . n.
The vertical displacements (wb)i at the pile base can also be
written as
j¼1
m
X
If wi is obtained using Eq. (24), the displacement of the pile segment i can then be written as follows (w0 = 1):
j¼1
n
n
m
X
X
X
ðwb Þi ¼
ðws Þj ðK WB Þij þ
ðwr Þj ðK UB Þij þ
ðwb Þj ðK BB Þij ;
in which wi1 and wi are the displacements at top and bottom of the
pile segment i, respectively. Pi1 and Pi are the axial forces acting on
top and bottom of the pile segment i, respectively, and they can be
obtained by Eq. (23) and expressed as follows:
Butterfield and Banerjee [7] λ=60000
The authors λ=60000
Butterfield and Banerjee [7] λ=6000
The authors λ=6000
P/GDW
80
ð24Þ
40
0
0
20
40
L/D
60
80
Fig. 3. Normalized pile head stiffness in an elastic half-space.
100
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Z.Y. Ai, J. Han / Computers and Geotechnics 36 (2009) 427–434
1.6
0.2
Chin et al [29]h/L=1.2
Poulos [38] h/L=1.2
The authors h/L=1.2
Chin et al [29] h/L=2
Poulos [38] h/L=2
The authors h/L=2
0.16
WEsD/P
τ/(P/πDL)
1.2
0.8
0.12
0.08
The authors λ= 60000
The authors λ= 6000
0.4
0.04
Butterfield and Banerjee [7] λ=60000
Butterfield and Banerjee [7] λ=6000
0.0
0.0
0.2
0.4
z/L
0
0.6
0.8
0
1.0
5
10
15
20
25
30
L/D
Fig. 4. Distribution of shear stress along the pile in an elastic half-space.
Fig. 6. Settlement influence factor of a single pile in a finite soil layer.
4.2. A floating pile in a soil with a finite depth
32
Lee and Small [16] K=1000
Lee and Small [16] K=10000
24
The authors K=1000
P/WEsD
The authors K=10000
16
8
0
0
10
20
30
40
50
60
L/D
Fig. 7. Normalized pile head stiffness of a single pile in a finite soil layer.
1
The authors K=10000
Chow [12] K=10000
0.8
0.6
Pz/P
The second problem is selected in this study containing a floating pile embedded in a homogeneous or Gibson soil with a finite
depth as shown in Fig. 5. This problem was studied by Poulos
[6,38] based on the Mindlin solutions and the Steinbrenner
approximation technique, and by Chin et al. [29] based on the Chan
et al. [26] solutions for a two-layered elastic medium. Therefore,
both methods are used here for comparisons. In the Chin et al.
[29] method, the rigid base was obtained by specifying an arbitrary
large modulus ratio E2/E1 (say 109), where E2 and E1 are Young’s
moduli of the second layer (i.e., the rigid base) and the first layer
(i.e., the finite soil layer). This problem is also analyzed using the
proposed method in this study.
Fig. 6 presents the comparisons of the settlement influence factors I0 (i.e. WEsD/P) for a single pile in a homogenous soil with a finite depth obtained by Poulos [38], Chin et al. [29], and the
authors. In this problem, the pile-soil modulus ratio is K = Ep/
Es = 1000 and Poisson’s ratio of the soil was 0.499. The comparisons
show a good agreement among all the results, especially for h/L = 2
(h is the finite depth of the soil layer and L is the pile length). However, when h/L = 1.2, Poulos [38] calculated lower factors and Chin
et al. [29] calculated higher factors than those of the proposed
method.
In addition, Lee and Small [16] presented a finite layer approach
for the analysis of axially loaded piles. Chow [12] adopted an axisymmetric finite element method to analyze the piles in a crossanisotropic elastic soil. Figs. 7 and 8 present the normalized pile
head stiffness (P/WEsD) and the axial load ratios for a single pile
in a homogenous soil with a finite depth obtained by Lee and Small
[16], Chow [12], and the present method. In these analyses, the ra-
0.4
0.2
a
b
P
P
0
0
L
L
h
h
0.2
0.4
0.6
0.8
1
z/L
D
Es (L)
D
Fig. 5. A floating pile in a soil with a finite depth: (a) the homogeneous soil and (b)
the Gibson soil.
Fig. 8. Axial load ratio of a single pile in a finite soil layer.
tio of the depth h to the pile length L was 2, the slender ratio L/D
was 20, and Poisson’s ratio of the soil was 0.499. Pz in Fig. 8 is
the load in the pile at depth z. Figs. 7 and 8 both show that the finite layer approach, the finite element method, and the proposed
method produce almost identical results.
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Z.Y. Ai, J. Han / Computers and Geotechnics 36 (2009) 427–434
When a single pile is embedded in a Gibson soil (i.e., the modulus increasing linearly with depth), the Gibson soil can be considered having several elastic layers (for example, 20 layers used in
this study) with increasing elastic moduli. Fig. 9 presents the comparisons of the normalized pile head stiffness (P/WEs (L)D) for a
single pile in a Gibson soil with a finite depth obtained by Lee
and Small [16], Chin et al. [29], Banerjee and Davies [30], and the
proposed method. In the analyses, the ratio of the depth h to the
pile length L was 2, the pile-soil modulus ratio K = Ep/
Es(L) = 10,000, and Poisson’s ratio of the soil was 0.499. The comparisons show that the results of the proposed method agree well
with those from the finite layer method obtained by Lee and Small
[16]. However, Chin et al. [29] and Banerjee and Davies [30] underestimate the results as compared with those obtained by the proposed method.
Fig. 10 shows the axial load ratios for a single pile in a Gibson
soil with a finite depth obtained by Lee and Small [16] and the proposed method. In these analyses, the slender ratio L/D was 25,
h/L = 2, K = Ep/Es(L) = 10,000, and Poisson’s ratio of the soil was
0.499. This comparison shows that the results of the proposed
method agree well with those of the finite layer method obtained
by Lee and Small [16].
The above comparisons clearly show that the proposed method
for an axially loaded pile in a multi-layered soil can be used to evaluate a pile in a homogenous or Gibson soil with a finite depth.
4.3. A pile in a multi-layered soil
As mentioned earlier, in a field, piles are installed in a soil,
which is likely multi-layered. For a demonstration purpose, a pile
driven into an eight-layered soil is selected and shown in Fig. 11.
P
Es1, vs1
h1
Es2, vs2
h2
Es3, vs3
h3
Es4, vs4
h4
D
Es5, vs5
h5
Es6, vs6
h6
Es7, vs7
h7
Es8, vs8
h8
Fig. 11. A pile in a multi-layered soil.
20
Lee and Small [16] K=10000
a
The authors K=10000
16
35
homogeneous soil K=1650
30
Chin et al [29] K=10000
Gibson soil K=1650
12
layered soil K=1650
25
P/WEsavgD
P/WEs(L)D
Banerjee and Davies [30] K=10000
8
4
20
15
10
0
5
15
25
35
45
5
L/D
0
Fig. 9. Normalized pile head stiffness of a single pile in Gibson soil with a finite
depth.
5
15
20
25
30
35
40
45
25
30
35
40
45
L/D
b
1
10
35
homogeneous soil K=16500
30
Gibson soil K=16500
0.8
layered soil K=16500
P/WEsavgD
25
P/Pz
0.6
20
15
0.4
10
The authors K=10000
0.2
5
Lee and Small [16] K=10000
0
0
0
0.2
0.6
0.4
0.8
1
z/L
Fig. 10. Axial load ratio of a floating pile in the Gibson soil for L/D = 25 and vs =
0.499.
5
10
15
20
L/D
Fig. 12. Normalized pile head stiffness at different slenderness and pile-soil
modulus ratios in the eight-layered soil.
Z.Y. Ai, J. Han / Computers and Geotechnics 36 (2009) 427–434
a
method is used in this study to develop a solution for an axially
loaded single pile in a multi-layered soil. The proposed method
by the authors is used to analyze a single pile in homogenous
and finite soils and obtain reasonably close results to the existing
solutions by others.
The parametric study for a single pile in a multi-layered soil
(i.e., an eight-layered soil) shows that the normalized pile head
stiffness increases with the slenderness ratio at a larger pile-soil
modulus ratio but has little change at a smaller ratio. The results
also show that the pile with a larger pile-soil modulus ratio has
higher normalized pile head stiffness, lower shear stresses in its
upper portion, and higher shear stresses in its lower portion. The
distribution of shear stresses depends on the pile-soil modulus ratio and the modulus ratio of the individual soil layers.
3
homogenous soil K=1650
Gibson soil K=1650
layered soil K=1650
τ/(P/π DL)
2
1
0
0
0.2
433
0.4
0.6
0.8
1
Acknowledgement
z/L
b
The work reported here is supported by the National Natural
Science Foundation of China (Grant No. 50578121). The authors
wish to express their gratitude for this financial support.
3
homogenous soil K=16500
Gibson soil K=16500
layered soil K=16500
Appendix
2
τ/(P/π DL)
The elements of the K matrices in Eq. (18) are expressed as
follow:
Z
ðK WW Þij ¼
1
ðK UW Þij ¼
0
0
0.2
0.4
0.6
0.8
1
ðK WU Þij ¼
ðK UU Þij ¼
This pile has a shaft diameter D and is loaded by an axial force P at
its top. In this analysis, each soil layer have identical Poisson’s ratio
of 0.499 and thickness of 0.25L (L is the length of the pile and
equals to h1 + h2 + h3 + h4). Young’s modulus ratio from layers 1
to 8 is Es1:Es2:Es3:Es4:Es5:Es6:Es7:Es8 = 1:4:2:3:6:4:3:10.
Fig. 12 shows the normalized pile head stiffness P/WEsavgD, for
piles at different slenderness ratios and different pile-soil modulus
P
ratios K = Ep/Esavg in the eight-layered soil, here Esavg ¼ 4i¼1 Ei DHi =L.
It is shown that the pile with a larger pile-soil modulus ratio has
higher normalized pile head stiffness, which increases with the
slenderness ratio. However, the normalized pile head stiffness of
the pile with a smaller pile-soil modulus ratio changes insignificantly with the slenderness ratio. The curves in Fig. 12 indicate that
the pile has the largest normalized pile head stiffness in a Gibson
soil but the smallest in a homogenous soil.
Fig. 13 shows the distribution of shear stress s along the shaft
with a slenderness ratio of L/D = 30 at different pile-soil modulus
ratios of K = Ep/Esavg. It is shown that the pile with a larger pile-soil
modulus ratio has lower shear stresses in its upper portion but
higher shear stresses in its lower portion. Fig. 13 also shows that
the distribution of shear stresses along the pile in the layered soil
is very different from that in the homogeneous and Gibson soil.
5. Conclusions
Based on the solution for vertical and horizontal axisymmetric
ring loads in a multi-layered elastic medium, a boundary element
ðK BU Þij ¼
ða:1Þ
Z
w2 ðra ; h; r a ; zÞdh;
ða:2Þ
w1 ðr 1 ; L; ra ; zÞdr 1 ;
ða:3Þ
u1 ðr a ; h; ra ; zÞdh;
ða:4Þ
u2 ðr a ; h; ra ; zÞdh;
ða:5Þ
u1 ðr 1 ; L; r a ; zÞdr1 ;
ða:6Þ
jd
ðj1Þd
jD
ðj1ÞD
z/L
Fig. 13. Distribution of shear stress along the shaft at the slenderness ratio L/D = 30
and different pile-soil modulus ratios in the eight-layered soil.
w1 ðr a ; h; r a ; zÞdh;
ðj1Þd
Z
ðK BW Þij ¼
jd
Z
jd
ðj1Þd
Z
jd
ðj1Þd
Z
jD
ðj1ÞD
where d ¼ nL, D ¼ rmb , z = (i 0.5)d. Also
ðK WB Þij ¼
ðK UB Þij ¼
ðK BB Þij ¼
Z
Z
Z
jd
w1 ðra ; h; r; LÞdh;
ða:7Þ
w2 ðr a ; h; r; LÞdh;
ða:8Þ
w1 ðr1 ; L; r; LÞdr 1 ;
ða:9Þ
ðj1Þd
jd
ðj1Þd
jD
ðj1ÞD
where r = (i 0.5)D.
Some Explanations are needed regarding the kernel functions in
the above-mentioned K matrices in the computation. When the
load point is not coincident with the calculation point, the integral
kernels are limitary and the general Guass–Legendre integral can
be used to obtain the reliable numerical integration results. When
the load point and the calculation point coincide, the kernel functions are of singularity. In order to obtain reliable values, the following techniques can be used:
(1) dividing the kernel functions into two segments by taking
the singular point as the boundary;
(2) taking the general Gauss–Legendre integral to calculate the
two segments numerically;
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Z.Y. Ai, J. Han / Computers and Geotechnics 36 (2009) 427–434
(3) subdividing the two segments into four segments and then
take the general Gauss–Legendre integral to integrate them
once more; and
(4) comparing the integral results in step (2) with those in step
(3).
If the difference between the results in step (2) and (3) is smaller than a given value (for example 106), it is reasonable to believe
that the reliable results have been obtained. If not, the segments
are subdivided and integrated over again until the difference between two seriate results is smaller than the given value.
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