DHARM

230 GEOTECHNICAL ENGINEERING

It may be shown that if there exists a single drainage face only for the layer, one of the

boundary conditions gets modified as

∂

∂

u

z = 0 at the impermeable boundary. Noting the fact

that the maximum drainage path in this case is the total thickness of the layer itself and

designating the latter as H, instead of 2H, we shall arrive at the same solution as indicated by

Eq. 7.27. That is to say, the effect of double drainage or single drainage may be easily ac-

counted for by substituting for H in the solution the length of the maximum drainage path.

The average degree of consolidation over the depth of the stratum at any time during

the consolidation process may be determined as follows:

The average initial hydrostatic excess pressure may be written as:

1

(^20)
2
H
udzi
H
z.
Similarly, the average hydrostatic excess pressure at any time t during consolidation is
1
(^20)
2
H
udz
H
z.
The average consolidation ratio U is the average value of Uz(= 1 – u/ui) over the depth of
the stratum. It may be written as
U =^1
0
2
0
− 2
z
z
udz
udz
H
i
H
.
...(Eq. 7.28)
Substituting for u from Eq. 7.26, we have
U = 1
2
0
2
0
2
0
2
−
z
z
∑
−

∞ u
Mz
H
dz
Mudz
e
i
H
i
H
MT
m
.sin.
.

. ...(Eq. 7.29)

In the special case of constant initial hydrostatic excess pressure, this reduces to

*U =^1

2

2

0

2

− −

=

∞

∑ M e

MT

m

. ...(Eq. 7.30)

A numerically equivalent procedure may also be employed for arriving at the average
degree of consolidation from the graphical modes of presentation of results as indicated in the
following section.

*The following approximate expressions have been found to yield values for T with good degree

of precision:

When U < 60%, T = (π/4)U^2

When U > 60%, T = – 0.9332 log 10 (1 – U) – 0.0851.