Environmental Engineering Reference
In-Depth Information
m
2
pressure dissipation in the unsaturated soil was found to be
a time-dependent process which could be simulated using
the water flow differential equation.
=
coefficient of water volume change with
respect to a change in matric suction
d(u
a
−
u
w
)
during
K
0
loading.
16.2.2 Physical Relations Required for Formulation
The one-dimensional consolidation equation for saturated
soils is derived by equating the time derivative of the vol-
ume change constitutive equation to the divergence of the
water flow rate. The volume change constitutive equation
for a saturated soil relates the change in void ratio
de
to the
change in effective stress
d(σ
The air phase constitutive relation is equal to the differ-
ence between the soil structure and the water phase constitu-
tive equations. The air phase constitutive equation can also
be expressed in a general form as a function of the stress
state variables:
dV
a
V
0
=
m
1
k
d(σ
y
−
m
2
d(u
a
−
u
a
)
+
u
w
)
(16.4)
u
w
)
. The rate of water flow
through a soil mass is described by Darcy's law.
The one-dimensional consolidation equation for an
unsat-
urated
soil can similarly be formulated by satisfying the
continuity requirement for a multiphase material. In this
case, the change in the total volume of the soil element
must be equal to the sum of the changes in the volume of
water and the volume of air in the element. The soil solids
are assumed to be incompressible. One-dimensional consol-
idation under
K
0
loading has the following soil structure
constitutive equation:
−
where:
dV
a
/V
0
=
change in the volume of air in the soil ele-
ment with respect to the initial volume of the
element,
V
a
=
volume of air in the element,
m
1
k
=
coefficient of air volume change with respect
to a change in net normal stress
d(σ
y
−
u
a
)
,
and
m
2
=
coefficient of air volume change with respect
to a change in matric suction
d(u
a
−
dV
v
V
0
=
u
w
)
.
m
1
k
d(σ
y
−
m
2
d(u
a
−
u
a
)
+
u
w
)
(16.2)
Figure 16.1 shows the relationship among the three con-
stitutive surfaces described by the above equations. The
coefficients of volume change must satisfy the continuity
requirement at any stress point on the constitutive surfaces:
where:
dV
v
/V
0
=
volume change of the soil element with
respect to the initial volume of the element
(also referred to as the change in volumetric
strain
dε
v
),
m
1
k
=
m
1
k
−
m
1
k
(16.5)
m
2
=
m
2
−
m
2
(16.6)
V
v
=
volume of soil voids in the element,
V
0
=
initial overall volume of the element,
When the soil is saturated (i.e., degree of saturation
S
at
100%), the four coefficients of volume change,
m
1
k
,m
2
,m
1
k
,
and
m
2
, are equal to the volume change modulus
m
v
for a sat-
urated soil. The volume change modulus
m
v
can be obtained
from a one-dimensional consolidation test on a saturated soil
specimen. The
m
1
k
and
m
2
modulus values are equal to zero
under saturated conditions (Eqs. 16.5 and 16.6). As the soil
becomes unsaturated (i.e.,
S<
100%), the absolute values of
the
m
1
k
,m
2
,m
1
k
, and
m
2
coefficients take on independent
values (Fig. 16.1). The volume change coefficients have a
negative sign, which indicates that an increase in the stress
state variable causes a decrease in volume. Under unsaturated
soil conditions, the absolute magnitude of the
m
2
water phase
coefficient will be greater than the absolute magnitude of the
m
2
soil structure coefficient under similar stress state con-
ditions. In other words, an increase in matric suction causes
a larger change in the volume of water removed from the
soil than the overall volume change of the soil element. As
a result, an increase in matric suction results in a decrease
in the degree of saturation. The magnitude of the
m
1
k
soil
structure coefficient, on the other hand, is greater than the
magnitude of the
m
1
k
water phase coefficient under similar
m
1
k
=
coefficient of volume change with respect to
a change in net normal stress
d(σ
y
−
u
a
)
for
K
0
loading, and
m
2
=
coefficient of volume change with respect to
a change in matric suction
d(u
a
−
u
w
)
during
K
0
loading.
The water phase constitutive relation for
K
0
loading can
be written as
dV
w
V
0
=
m
1
k
d(σ
y
−
m
2
d(u
a
−
u
a
)
+
u
w
)
(16.3)
where:
dV
w
/V
0
=
change in the volume of water in the soil ele-
ment with respect to the initial volume of the
element,
V
w
=
volume of water in the element,
m
1
k
=
coefficient of water volume change with
respect to a change in the net normal stress
d(σ
y
−
u
a
)
for
K
0
loading, and
Search WWH ::
Custom Search