Civil Engineering Reference
In-Depth Information
Errors associated with the explicit equation
Errors fall into two main groups: truncation errors (due to ignoring the higher derivatives) and rounding-off
errors (due to working to only a certain number of decimal places). The size of the space increment,
Δ
z,
affects both these errors but in different ways: the smaller
Δ
z is, the less the truncation error that arises
but the greater the round-off error tends to become.
The value of r is also important. For stability r must not be greater than 0.5 and, for minimum truncation
errors, should be 1/6; the usual practice is to take r as near as possible to 0.5. This restriction means that
the time interval must be short and a considerable number of iterations become necessary to obtain the
solution for a large time interval. With present software this is not a problem, but if necessary use can be
made of either the implicit finite difference equation or the relaxation method.
Example 12.5:
Degree of consolidation by finite difference
method
A layer of clay 4 m thick is drained on its top surface and has a uniform initial excess
pore pressure distribution. The consolidation coefficient of the clay is 0.1 m
2
/month.
Using a numerical method, determine the degree of consolidation that the layer will
have undergone 24 months after the commencement of consolidation. Check your
answer by the theoretical curves of Fig.
12.4.
Solution:
In a numerical solution the grid must first be established: for this example the layer has
been split into four layers each of
Δ
z
=
1.0 m (it is important to remember that since
Simpson's rule is being applied to determine the degree of consolidation, the layer
should be divided into an even number of strips). The initial excess pore pressure values
have been taken everywhere throughout the layer as equal to 100 units.
In 24 months:
c t
z
0 1 24
1 0
.
×
v
r
=
=
=
2 4
.
∆
2
.
For the finite difference equation r must not be greater than 0.5, so use five time incre-
ments, i.e.
Δ
t
=
4.8 months and
0 1 4 8
1 0
.
×
.
r
=
=
0 48
.
.
Owing to the instantaneous dissipation at the drained surface the excess pore pressure
during the iteration process are also given). The finite difference formula is applied to
each point of the grid, except at the drained surface:
u
+
=
r u
(
+
u
−
2
u
)
+
u
0
,
k
1
2
,
k
4
,
k
0
,
k
0
,
k
For example, with the first time increment the point next to the drained surface has u
=
0.48(0
+
100
−
2
×
100)
+
100
=
52.0. Note that at the undrained surface the finite
difference equation alters.
