Civil Engineering Reference
In-Depth Information
Equation (9.2) may also be solved numerically by using the characteristic solutions,
which are of the form
w
=
g
(
z
−
ct
)
+
h
(
x
+
ct
)
(9.3)
where
g
and
h
are unspecified functions which represent downward (increasing
z
) and
upward travelling waves respectively. Taking downward displacement and compres-
sive strain and stress as positive, the force,
F
, and particle velocity,
v
, in the pile are
given by
(
AE
)
p
g
+
h
(
AE
)
p
∂
w
∂
F
=−
=−
(9.4)
z
c
g
−
h
=
∂
w
∂
v
=−
(9.5)
t
where the prime denotes the derivative of the function with respect to its argument.
The velocity and force can each be considered as made up of two components, one
due to the downward travelling wave (represented by the function
g
) and one due to
the upward travelling wave (represented by the function
h
)
.
Using subscripts
d
and
u
for these two components, the velocity is
cg
+
ch
v
=
v
d
+
v
u
=−
(9.6)
The force
F
is similarly expressed as:
(
AE
)
p
g
−
(
AE
)
p
h
=
F
=
F
d
+
F
a
=−
Z
(
v
d
−
v
u
)
(9.7)
where
Z
c
and is referred to as the pile impedance. [Note, some authors have
referred to the pile impedance as
Z
=
(
AE
)
p
/
c
, relating axial stress and velocity rather than
force and velocity. The more common definition of pile impedance as
Z
=
E
/
=
(
AE
)
p
/
c
will be adopted here.]
The relationships given above may be used to model the passage of waves down
and up piles of varying cross-section, allowing for interaction with the surrounding
soil. It is helpful to consider the pile as being made up of a number of elements, each
of length
z
, with the soil resistance acting at nodes at the mid-point of each element
(see Figure 9.28). Numerical implementation of the characteristic solutions involves
tracing the passage of the downward and upward travelling waves from one element
interface to the next. The time increment,
t
, is chosen such that each wave travels
across one element in the time increment (giving
t
=
z
/
c
).
1, the soil resistance may be taken as
T
i
, the value of which
will depend on the local soil displacement and velocity (see later). Taking
T
i
as positive
when acting upwards on the pile (that is, with the soil resisting downward motion of
the pile), the soil resistance will lead to upward and downward waves of magnitude
Between nodes
i
and
i
+
F
u
=−
F
d
=
T
i
/
2
(9.8)
These waves will lead to modification of the waves propagating up and down the pile.
