Geoscience Reference
In-Depth Information
vanished, the settlement is denoted by
S
&
. In practice, one adopts the so-called
degree of consolidation
U
, which is defined in terms of settlement, according to
U =
(
S - S
i
)/(
S
&
- S
i
)
(6.3)
The beginning of consolidation is then expressed by
U
= 0% and the end by
U
=
100%.
The elaboration of the uniaxial consolidation process (after Terzaghi) is as
follows. During
dt
the decrease of pore pressure -
du
in segment
dz
results in an
increase of effective stress
d
'
, and the corresponding volume compaction of the
segment
dz
is therefore
29
-
t
)
dzdt
, which must be equal to the
volume of expelled water
dq
during
dt
of segment
dz
, which is (
dzdu
= -
(
u
/
q
/
z
)
dzdt
. This
leads to (
q
/
z
) +
(
u
/
t
) = 0, at every
dt
and
dz
. Inserting Darcy's law
q =
-
(
k
/
w
)(
u
/
z
-
w
) provides the consolidation equation in terms of pore pressure
u
,
according to
2
u/
z
2
=
)
[m
2
/s]
u/
t
with
c
v
= k
/(
(6.4)
c
v
w
Here,
c
v
is the consolidation coefficient, assuming uniform and constant
permeability and compressibility (homogenous soil layer). In reality, during the
consolidation process, when with compaction the pore volume decreases, the
permeability and the compressibility will decrease. It is sometimes stated that the
consolidation coefficient, being proportional to
k
and reciprocal to
, will be yet
more or less a constant. Applying the relation between the pore pressure
u
and the
settlement
S
according to (4.1), (5.4) and (5.17), and solving (6.4) for the
oedometer test, assuming that
c
v
is a constant, the following expression for the
degree of consolidation
U
is obtained
2
)
j
{ exp[-(2
j-
1)
2
T
v
2
/4] / (2
j
-1)
2
} with
T
v
= c
v
t/h
2
U =
1 - (8
/
%
(6.5)
Here,
h
is the sample height. For large times,
T
v
> 0.2, the higher order terms
(for
j
> 1) vanish and (6.5) reduces to
2
) exp(
-T
v
2
/4)
U =
1 - (8
/
(6.6)
For small time,
T
v
< 0.2, (6.5) is parabolic with time and can be approximated by
)
0.5
U =
(4
T
v
/
(6.7)
It appears that for
T
v
= 0.2 the degree of consolidation
U
becomes 50.5%, and
for
T
v
= 2 it becomes 99.4%. That implies that consolidation is practically
completed for
T
v
= T
hd
=
2
h
2
/c
v
(6.8)
29
=
'+ u;
0
= d
= d
'+ du; du =
d
' =
d
/
; dV = dzd
=
dzd
'=
dudz
Search WWH ::
Custom Search