Civil Engineering Reference
In-Depth Information
Figure 15.5
Area between two isochrones.
is no seepage flow, either because it represents the limit of consolidation for
t
n
or
because of the impermeable boundary for
t
m
, and so the isochrones must be vertical at
the base, as shown in Fig. 15.4(a). Since soil grains and water are incompressible the
velocity of the upward seepage at any level must equal the rate of settlements at that
level and
∂ρ
∂
k
γ
w
∂
u
=
(15.13)
t
∂
z
The movement of isochrones represents changes of excess pore pressure and changes
of effective stress. Figure 15.5 shows isochrones for
t
1
and
t
2
. From Eq. (15.1) the
change of thickness
δσ
. If the total stress
δ
δ
δ
=−
δ
h
of th
e
thin slice
z
is given by
h
m
v
z
δσ
=−
δ
remains constant,
u
and
δ
h
=
m
v
δ
z
δ
u
(15.14)
where
u
is the shaded area in Fig. 15.5. Summing the changes of thickness for all
thin slices in the depth
z
, the change of surface settlement between the times
t
1
and
t
2
is given by
δ
z
δ
δρ
=
m
v
×
areaOAB
(15.15)
Hence the settlement of a consolidating layer in a given time is given by
m
v
times the
area swept by the isochrone during the time interval.
15.5 Solution for one-dimensional consolidation by
parabolic isochrones
Simple and reasonably accurate solutions for the rate of settlement for one-dimensional
consolidation can be obtained by assuming that the general shapes of the isochrones in
Fig. 15.4(a) can be approximated by parabolas. It is necessary to treat the cases
t
<
t
c
and
t
t
c
separately; the ideas behind each analysis are the same but the algebra
differs slightly.
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