Civil Engineering Reference
In-Depth Information
∂
∂
2
u
z
=
∂
∂
u
t
⇒
c
v
2
where c
v
=
the coefficient of consolidation and equals
k
k
m
(
1
+ =
e
)
γ
a
γ
w
w
v
In the foregoing theory, z is measured from the top of the clay and complete drainage is assumed at
both the upper and lower surfaces, the thickness of the layer being taken as 2H. The initial excess pore
pressure, u
i
,
=
−
dp.
The boundary conditions can be expressed mathematically:
when z
=
0, u
=
0
when z
=
2H, u
=
0
when t
=
0, u
=
u
i
A solution for
∂
∂
2
u
z
=
∂
∂
u
t
c
v
2
that satisfies these conditions can be obtained and gives the value of the excess pore pressure at depth
z at time t, u
z
:
m
=∞
∑
2
u
M
Mz
H
2
i
u
=
sin
e
−
M T
z
m
=
0
where
u
i
=
the initial excess pore pressure, uniform over the whole depth
M
1
2
=
π
(
2
m
+
1
)
where m is a positive integer varying from 0 to
∞
T
=
c t
H
v
2
, known as the time factor.
Owing to the drainage at the top and bottom of the layer the value of ui
i
will immediately fall to zero
at these points. With the mathematical solution it is possible to determine, u at time t for any point within
the layer. If these values of pore pressures are plotted, a curve (known as an isochrone) can be drawn
through the points (Fig.
12.3b
). The maximum excess pore pressure is seen to be at the centre of the
layer and, for any point, the applied pressure increment,
∆
′
σ
= +
∆
σ
. After a considerable time u will
u
1
1
become equal to zero and
Δ
σ
1
will equal
∆
′
σ
1
.
Fig. 12.3
Variation of excess pore pressure with depth and time.
