Civil Engineering Reference
In-Depth Information
(a) t
=
t
n
<t
c
Figure 15.6(a) shows an isochrone for time
t
n
; the slope is vertical at
N
and no consol-
idation has occurred below a depth
n
. From Eq. (15.15) (noting that the area below a
parabola is
3
×
base
×
height), the surface settlement is given by
1
3
m
v
n
ρ
=
×
=
σ
m
v
area AEN
(15.16)
t
Differentiating Eq. (15.16) and noting that
m
v
and
σ
are assumed to be constants
during consolidation, the rate of settlement is given by
ρ
d
1
3
m
v
d
n
d
t
t
=
σ
(15.17)
d
t
The rate of surface settlement is also related to the gradient of the isochrone at A.
From Eq. (15.13) and noting that from the geometry of a parabola the gradient at A
is 2
σ
/
n
, we have
d
ρ
k
γ
2
σ
n
t
d
t
=
(15.18)
w
Hence, equating the rates of surface settlement from Eqs. (15.17) and (15.18),
n
d
n
d
t
k
m
v
=
6
w
=
6
c
v
(15.19)
γ
and, integrating with the boundary condition
n
=
0at
t
=
0,
12
c
v
t
n
=
(15.20)
Equation (15.20) gives the rate at which the effects of consolidation progress into the
soil from the drainage boundary; no dissipation of excess pore pressure will occur at
Figure 15.6
Geometry of parabolic isochrones.
