Civil Engineering Reference
In-Depth Information
may be considered as surrounded by concentric cylinders of soil, with shear stresses
on each cylinder. For vertical equilibrium, the magnitude of the shear stress on each
cylinder must decrease inversely with the surface area of the cylinder (Cooke, 1974;
Frank, 1974). Writing the shear stress on the pile shaft as
τ
0
, the shear stress at radius
r
(for a pile of radius,
r
0
) is given by
τ
=
τ
0
r
0
r
(4.31)
Since the mode of deformation around a pile is primarily one of shear, it is more
natural to develop the solution in terms of the shear modulus
G
and Poisson's ratio,
v
,
rather than Young's modulus,
E
(noting that the shear modulus may be related to
Young's modulus by
G
=
E
/
2(1
+
v
)). The shear strain in the soil,
γ
, will thus be given
by
G
. Since the main deformation in the soil will be vertical, the shear strain
may be written approximately as
γ
=
τ/
d
w
d
r
γ
∼
(4.32)
where
w
is the vertical deflection.
These relationships may be assembled and integrated to give
r
m
G
ln
r
m
/
r
τ
0
r
0
=
τ
0
r
0
=
w
Gr
d
r
(4.33)
r
A maximum radius,
r
m
, has been introduced at which the deflections in the soil are
assumed to become vanishingly small. Empirically, this radius has been found to be
of the order of the length of the pile (Randolph and Wroth, 1978). The deflection of
the pile shaft,
w
s
, is given by
ln
r
m
/
r
0
=
ln
2
r
m
/
d
w
s
=
ζ
τ
0
r
0
G
=
ζ
τ
0
d
ζ
=
where
(4.34)
2
G
This approximate analytical derivation of the deflection of the pile shaft in terms of the
average applied shear stress
τ
0
shows a number of important features of the response
of a pile to axial load.
1 Stress changes induced in the soil are primarily shear stresses, decreasing inversely
with distance from the pile axis; thus, only soil very close to the pile is ever highly
stressed.
2 The resulting deflections decrease with the logarithm of distance from the pile
axis; thus, significant deflections extend some distance away from the pile (up to
about one pile length).
3 The deflection of the pile shaft,
w
s
, normalized by the pile radius
r
0
,is
ζ
times the
local shear strain,
is found to vary between
3 and 5, with an average value of about 4 (Baguelin and Frank, 1979).
γ
0
=
τ
0
/
G
, in the soil: the parameter
ζ
