2680
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
Soil displacements
u
r
and
u
z
in the radial and vertical
directions, generated by the applied torque
T
a
, can be assumed
to be negligible (Figure 2). The tangential displacement
u
in
soil is nonzero and is assumed to be a product of separable
variables as
p
s
p p
s
u w z
r r
z
r
(1)
where
w
p
is the displacement in the tangential direction at the
pile-soil interface (i.e., it is the tangential displacement at the
outer surface of the pile shaft),
p
is the angle of twist of the pile
cross section (
w
p
=
r
p
p
) which varies with depth
z
, and
s
is a
dimensionless function that describes how the soil displacement
varies with radial distance
r
from the center of the pile. It is
assumed that
s
= 1 at
r
r
p
, which ensures no slip between pile
and soil, and that
s
= 0 at
r =
, which ensures that the soil
displacement decreases with increasing radial distance from the
pile.
Using the above soil displacement field, the strain-
displacement and stress-strain relationships are used to obtain
the total potential energy of the pile-soil system as
2
2
2
0
0 0
2
1
1
2
2
p
p
L
p
p p
s
p
s
r
s
s
s
p p
d
d
G J
dz
G r
dz
dr
d
G r
rd drdz
dr r
p
2
2
2
0
0 0
1
2
p
p
r
p
p p
s
p
s
a p z
L
d
r
G r
G
rd drdz T
dr
r
(2)
where
J
p
(=
r
p
4
/2) is the polar moment of inertia of the pile
cross section. Minimizing the potential energy (i.e., setting
= 0 where
is the variational operator) produces the equilibrium
equations for the pile-soil system. Using calculus of variations,
the differential equations governing pile and soil displacements
under equilibrium configuration are obtained.
The differential equation governing the angle of twist of pile
cross section
p
(
z
) within any layer
i
is obtained as
2
2
1 2
0
pi
i
i pi
d
t
k
dz
(3)
where
2
2
2 2
2
1, 2, ...,
1
2
p
p
si p
s
p p r
i
sn p p
s
p p
r
G r
rdr
i
n
G J
t
G r r
rdr i n
G J
(4)
2
2 2
2
2 2
0
2
1, 2, ...,
2
ln lim ln
1
p
p
si p p
s
s
p p r
i
sn p p
s
s
p
p p
r
G r L d
rdr
i
n
G J
dr r
k
G r L
d
r
G J
dr r
i n
rdr
(5)
The boundary conditions of
p
(
z
) are given by
1
1
1 2
p
a
d
t
T
dz
(6)
at the pile head (i.e., at
z = z̃
=
0),
( 1)
pi
p i
(17a)
( 1)
1
1 2
1 2
pi
p i
i
i
d
d
t
t
dz
dz
(7)
at the interface between any two layers (i.e., at
z
=
H
i
or
z̃
= H
͂
i
),
and
1 1
1 2
2
0
pn
n
n n
d
t
k t
dz
pn
(8)
at the pile base (i.e., at
z
=
L
p
or
z̃
=
1). The dimensionless terms
in the above equations are defined as:
a
a p p
T T L G J
p
;
z̃
=
z
/
L
p
and
H
͂
i
= H
i
/
L
p
. In the above equations, the
n
th
(bottom)
layer is split into two parts, with the part below the pile denoted
by the subscript
n +
1; therefore,
H
n
=
L
p
and
H
n
+1
=
. In
equation [5],
k̃
n+
1
is not defined at
r =
0 as ln(0) is undefined;
therefore, in obtaining the expression of
k̃
n+
1
, the lower limit of
integration was changed from
r =
0 to
r =
where
is a small
positive quantity (taken equal to 0.001 m in this study).
The general solution of equation (3) is given by
( )
( )
1 1
2 2
( )
i
i
pi
z C C
(9)
where
and
are integration constants of the
i
th
layer,
and
1
and
2
are individual solutions of equation (3), given by
( )
1
i
C
( )
2
i
C
1
sinh
i
z
(10a)
2
cosh
i
z
(10b)
with
1 2
i
i
i
k
t
(11)
The constants
and
are determined for each layer
using the boundary conditions given in equations (6)-(8).
( )
1
i
C
( )
2
i
C
The governing differential equation (3) resembles that of a
column (or rod) supported by a torsional spring foundation
undergoing a twist. The parameter
i
accounts for the shear
resistance of soil in the horizontal plane and
i
represents the
shear resistance of soil in the vertical plane. The torque
T
(
z
) in
the pile at any depth is given (in dimensionless form) by
t
k
( )
1 2
p
d
T z
t
dz
(12)
where
a
a p p p
T T L G J
.
The torque
T
(
z
)
includes the shear
resistance offered by the horizontal planes of both the pile and
surrounding soil. The governing differential equation (3)
describes how the rate of change of this torque
T
with depth is
balanced by the shear resistance in the vertical planes of the
soil. The boundary conditions at the interfaces of the adjacent
layers ensure continuity of angle of twist and equilibrium of
torque across these horizontal planes. The boundary condition at
the pile head ensures that equilibrium between the torque
T
(
z =
0)
and applied torque
T
a
is satisfied. The boundary condition at
the pile base ensures equilibrium by equating the torque in the
pile and soil at a horizontal plane infinitesimally above the base
with the torque in soil at a horizontal plane infinitesimally
below the base.
The differential equation of
s
(
r
) is given by
2
2
2
2
1
1
0
s
s
s
p
d
d
r dr
r
dr
r
(13)
where
