Environmental Engineering Reference
In-Depth Information
Lines through a point on the soil structure constitutive sur-
face have slopes of
m
1
and
m
2
with respect to the net normal
stress and matric suction, respectively:
It is suggested that simpler approximations can be applied
for the estimation of the shape of the constitutive surfaces.
For example, let us assume that the constitutive surfaces
are approximately planar at a particular void ratio or water
content, as shown in Figs. 13.16 and 13.17. Therefore, every
point on the surface corresponding to an equal void ratio or
water content plane has the same
a
t
and
a
m
coefficients or
b
t
and
b
m
coefficients, respectively. As a result,
a
t
and
b
t
coefficients obtained from the saturation plane (i.e.,
u
a
−
u
w
equal to zero plane) can be used for other points on
the surface as long as the void ratio (or water content) is
constant. Similarly,
a
m
and
b
m
coefficients obtained from
the zero net normal stress plane (i.e.,
σ
−
u
a
equal to zero
plane) can also be used for other points on the surface along
a constant void ratio or water content plane. In this way, the
values for
a
t
,
a
m
,
b
t
, and
b
m
obtained on the saturation and
zero net normal stress planes are assumed to apply for the
entire constitutive surface.
The suggested approximation method constitutes a crude
estimation procedure for determining the volumetric coef-
ficients over the entire constitutive surface. The estimation
procedure is inferior to directly measuring the volumetric
coefficients at individual state points. The required level of
accuracy for the problem at hand must be borne in mind
when solving practical engineering problems.
The following discussion pertains to the laboratory tests
which can be used to obtain
a
t
,
a
m
,
b
t
, and
b
m
coeffi-
cients. The
a
t
coefficients can be obtained from curve
A
in Fig. 13.16, which shows the results of a consolidation
test on a soil which has been saturated. Figure 13.18 shows
a typical compression curve for a compacted soil which has
been saturated while maintaining a constant volume prior to
its decrease in volume due to loading. The
b
t
∂
V
v
/V
0
∂
σ
−
u
a
m
1
=
(13.66)
∂
V
v
/V
0
∂
u
a
−
u
w
m
2
=
(13.67)
Lines through a point on the water phase constitutive sur-
face have slope of
m
1
and
m
2
with respect to the net normal
stress and matric suction, respectively:
∂
V
w
/V
0
∂
σ
−
u
a
m
1
=
(13.68)
∂
V
w
/V
0
∂
u
a
−
u
w
m
2
=
(13.69)
The
m
1
,
m
2
,
m
1
, and
m
2
coefficients are related to the
a
t
,
a
m
,
b
t
, and
b
m
coefficients as follows:
a
t
m
1
=
(13.70)
1
+
e
0
a
m
m
2
=
(13.71)
1
+
e
0
b
t
G
s
1
m
1
=
(13.72)
+
e
0
b
m
G
s
1
m
2
=
(13.73)
+
e
0
coefficient in
where:
Fig. 13.17 can be computed as the
a
t
coefficient divided by
the specific gravity of the solids.
Curve
B
in Fig. 13.17 is a SWCC that can be obtained
using a pressure plate type of test. Figure 13.19 illustrates
the
b
m
coefficient on both the drying and wetting portions
of a typical SWCC.
The
a
m
coefficient can be related to the
b
m
coefficient
through use of a shrinkage test. A shrinkage test relates the
void ratio of a soil to the water content at various matric
suctions while drying the soil under zero external load.
Figure 13.20 shows a shrinkage curve associated with a
shrinkage-type test. The slope of the shrinkage curve (i.e.,
de/
dw
or [
∂e/∂(u
a
−
u
w
)
]
/
[
∂
w
/∂(u
a
−
u
w
)
]) defines the
ratio between
a
m
and
b
m
coefficients at all soil suction values.
e
0
=
initial void ratio prior to deformation.
The
m
1
is essentially equal to
m
1
on the saturation plane
(i.e.,
S
=
100%) since
a
t
=
b
t
G
s
.The
m
1
coefficient for
the saturated condition is commonly called the coefficient
of volume change,
m
v
.
13.5.3 Laboratory Tests Used to Obtain Volumetric
Deformation Coefficients
The
a
t
,
a
m
,
b
t
, and
b
m
coefficients are used to discuss the
relationships that exist between the volumetric deformation
coefficients. These coefficients vary from one state point
to another along a nonlinear constitutive surface. A direct
method to determine these coefficients at a specific state
point is to measure their magnitude at the stress point under
consideration. The experimental measurements required are
similar to those conducted for the verification of the consti-
tutive surfaces. Numerous specimens and a long period of
testing are generally required to experimentally define the
entire constitutive surface.
13.5.4 Relationships among Volumetric Deformation
Coefficients for Unloading Surfaces
Hysteresis causes the constitutive surfaces obtained when
loading (and drying) a soil to be different from the surfaces
obtained when unloading (and wetting). Consequently,
the volumetric deformation coefficients associated with
each constitutive surface will be different. For example,
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