Civil Engineering Reference
In-Depth Information
depths greater than
n
. Using Eq. (15.20) and the geometry of a parabola it is possible
to calculate the excess pore pressure at any depth and at any time
t
t
c
.
In practice, the most important thing to calculate is the surface settlement
<
ρ
t
after
a time
t
<
t
c
; and this is found by substituting for
n
into Eq. (15.16), giving
12
c
v
t
1
3
m
v
ρ
=
σ
(15.21)
t
The final surface settlement
ρ
∞
will occur after a long time when all excess pore
pressures have dissipated and
σ
=
σ
. Hence, from Eq. (15.1),
ρ
∞
=
m
v
H
σ
(15.22)
Combining Eqs. (15.21) and (15.22),
c
v
t
H
2
ρ
2
√
3
t
ρ
∞
=
(15.23)
Equation (15.23) may be written in terms of a dimensionless degree of consolidation
U
t
and a dimensionless time factor
T
v
given by
=
ρ
t
ρ
∞
U
t
(15.24)
c
v
t
H
2
T
v
=
(15.25)
and the general solution becomes
√
3
T
v
2
U
t
=
(15.26)
This solution is va
lid un
til the point N in Fig. 15.6(a) reached D when
t
=
t
c
; at this
=
√
12
c
v
t
so
T
v
1
instant
n
=
H
=
12
and
U
t
=
0.33. For
t
>
t
c
the isochrone no longer
touches ED and a new analysis is required.
(b) t
=
t
m
>t
c
Figure
1
5.6(b) shows an isochrone for
t
m
; it intersects the base orthogonally at M
where
u
=
m
σ
. Making use of the geometry of a parabola and proceeding as before,
H
1
3
m
2
ρ
=
m
v
σ
−
(15.27)
t
d
ρ
2
3
m
v
σ
H
d
m
k
γ
2
m
σ
H
t
d
t
=−
d
t
=
(15.28)
w
m
d
m
3
c
v
H
2
1
t
3
T
v
d
t
=−
=−
(15.29)
