Civil Engineering Reference
In-Depth Information
12.11 Consolidation by drainage in two and three dimensions
The majority of settlement analyses are based on the frequently incorrect assumption that the flow of
water in the soil is one dimensional, partly for ease of calculation and partly because in most cases knowl-
edge of soil compression values in three dimensions is limited. There are occasions when this assumption
can lead to significant errors (as in the case of an anisotropic soil with a horizontal permeability so much
greater than its vertical value that the time-settlement relationship is considerably altered) and when
dealing with a foundation which is relatively small compared with the thickness of the consolidating layer
some form of analysis allowing for lateral drainage becomes necessary. For an isotropic, homogeneous
soil the differential equation for three-dimensional consolidation is:
u
x
2
2
u
2
u
u
t
∂
∂
+
∂
∂
+
∂
∂
=
∂
c
v
2
y
2
z
2
∂
For two dimensions one of the terms in the bracket is dropped.
12.12 Numerical determination of consolidation rates
When a consolidating layer of clay is subjected to an irregular distribution of initial excess pore water
pressure, the theoretical solutions are not usually applicable unless the distribution can be approximated
to one of the cases considered. In such circumstances the use of a numerical method is fairly common. A
spreadsheet can be used for such a purpose and Example
12.5
illustrates the use of a spreadsheet to find
the solution.
A brief revision of the relevant mathematics is set out below.
Maclaurin's series
Assuming that f(x) can be expanded as a power series:
y
=
f x
(
)
= +
a
a x
+
a x
2
+
a x
3
+
a x
n
n
0
1
2
3
dy
dx
= ′ = +
f x
(
)
a
2
a x
+
3
a x
2
+
4
a x
3
+
na x
n
n
−
1
1
2
3
4
2
d y
dx
=
′′
=
f x
(
)
2
a
2 3
.
a x
3 4
.
a x
2
n n
(
1
)
a x
n
n
−
2
+
+
+
−
2
3
4
2
d y
dx
3
=
′′′
=
n
−
3
f
(
x
)
2 3
.
a
+
2 3 4
.
.
a x
+
n n
(
−
1
)(
n
−
2
)
a x
n
3
4
3
If we put x
=
0 in each of the above:
=
′′
=
′′′
f
(
0
)
f
(
0
)
a
f
(
0
);
a
=
′
f
(
0
);
a
;
a
;
etc
.
=
0
1
2
3
2
!
3
!
Generally
f
n
(
0
)
a
=
n
n
!
