Geotechnical Engineering

DHARM

232 GEOTECHNICAL ENGINEERING

Figure 7.22 does not depict how much consolidation occurs as a whole in the entire

stratum. This information is of primary concern to the geotechnical engineer and may be de-

duced from Fig. 7.22 by the following procedure:

The relation between Uz and z/H for a time factor, T = 0.848, is reproduced in Fig. 7.23.

DC

0

0.5

1.0

1.5

2.0

z/H

T=¥

T = 0.848

Consolidation

yettobe

completed

0 0.2 0.4 0.6 0.8 1.0

ABUz

Consolidationcompleted

Fig. 7.23 Average consolidation at time factor 0.848
Average degree of consolidation at this time factor is the average abscissa for the entire
depth and is therefore given by the shaded area from T = 0 to T = 0.848 divided by the total
area from T = 0 to T = ∞; this is because the abscissa from T = 0 to T = 0.848 is the consolidation
completed at T = 0.848, while that from T = 0 to T = ∞ represents complete or 100% consolida-
tion.
In this case the ratio of the shaded area to the total area is found to be 90%. Thus, the
time factor corresponding to an average degree of consolidation of 90%, denoted by T 90 , is
0.848.
If this exercise is repeated for different time factors, the relation between average de-
gree of consolidation and time factor can be established as shown by curve I in Fig. 7.24.
Alternatively, the curve could have been obtained by the direct application of Eq. 7.30.
In this figure, the relationship is also given in a few cases wherein the initial hydro-
static excess pressure is not constant with depth. Equation 7.29 must be applied and the corre-
sponding mathematical expression for ui substituted and the indicated integrations performed.
Three examples–I (b), II, and III–of variable ui are presented. It is interesting to note that the
results would be identical for all linear variations. Curve II is for a sinusoidal variation of
initial hydrostatic excess.
Case III is a particular combination of linear and sinusoidal variations. Actual field
cases may be closely approximated by such combinations. It is interesting to note, once again,
that curve III is not very different from the curve I of constant initial hydrostatic excess. This
is the reason for the generally accepted conclusion that curve I is an adequate representation
of typical cases in nature.
All these curves are applicable to the conditions of double drainage.
There are many clay strata in nature in which drainage is at the top surface only, the
bottom surface being in contact with impervious rock. All such occurrences of single drainage
may be considered to be the upper-half of a case of double drainage, the other half being a
fictitious mirror image. The drainage path in this case is the thickness of the stratum itself
and so, 2H must be substituted for H in the equations for double drainage conditions.

U at T = 0.848 equals

Shaded area

Total area ABCD

=90%

T 90 is thus T = 0.848