Environmental Engineering Reference
In-Depth Information
models are introduced. However, the general format for
the governing partial differential equations remains in
accordance with Eqs. 13.152-13.154.
Equations 13.152-13.154 result in a smooth transition
between saturated and unsaturated conditions provided
appropriate soil properties are used. Soil behavior is dictated
by the two independent stress state tensors when soil suction
is positive (i.e., the soil is treated as an unsaturated soil).
There are distinct soil properties associated with each of the
stress state variables
D
ii
and
h
. As the soil becomes saturated
and matric suction tends towards zero,
m
1
and
m
2
become
equal to the coefficient of volume change for the soil under
saturated conditions (i.e.,
m
v
) and
h
becomes 1. Therefore,
the partial differential equations for saturated conditions
(i.e.,
h
=
atmosphere boundary conditions. The climatic driven con-
ditions may result in displacements and changes in total
stress. Pore-water and pore-air flow equations might also be
solved independent of total stresses by neglecting a num-
ber of coupling terms. The equations may also be solved in
a fully coupled manner, meaning that all possible coupling
terms are taken into consideration.
13.8 MEASUREMENT OF
STRESS-DEFORMATION PROPERTIES
FOR UNSATURATED SOILS
Volume change coefficients and indices can be experimen-
tally measured for unsaturated soils. Test procedures and
equipments are available for the measurements of the vol-
ume change coefficients and indices for unsaturated soils. In
some cases, it is necessary to modify the soil-testing equip-
ment as described in earlier chapters.
Test results on compacted silt, glacial till, and other soils
are used to illustrate the relationships between the vari-
ous volume change coefficients and indices. Most of the
results are presented in a semilogarithmic form as volume
change indices. These indices can then be converted to vol-
ume change coefficients and elastic parameters that can be
used in numerical modeling.
The test procedures and equipment used in the measure-
ment of the volume change properties are common to most
soil mechanics laboratories. Some of the equipment (e.g.,
pressure plate apparatuses) is more common to the field of
soil physics and agronomy.
The test procedures and equipment for the loading consti-
tutive surfaces are described prior to those for the unloading
constitutive surfaces. The use of oedometer test results for
assessing the in situ stress state in terms of the swelling
pressure is briefly described. Procedural corrections perti-
nent to the determination of a corrected swelling pressure
are described along with an explanation of its importance to
the prediction of soil swelling or heave.
Figure 13.67 shows three-dimensional views of the void
ratio and water content constitutive surfaces. The volume
change coefficients corresponding to the loading conditions
(i.e.,
a
t
,
a
m
,
b
t
,
b
m
) are shown. Curve
A
in Fig. 13.67 is
essentially a compression curve obtained from an oedometer
or triaxial test on a soil in a saturated condition.
The coefficient of compressibility
a
t
is equal to the
a
v
coefficient of compressibility commonly measured in satu-
rated soil mechanics. The coefficient of water content change
b
t
(Fig. 13.31b) can then be calculated once the coefficient
of compressibility
a
t
has been measured. The coefficient of
water content change
b
t
is equal to the coefficient of com-
pressibility
a
t
multiplied by the specific gravity of the soil,
G
s
.
Curve
B
in Fig. 13.67b is similar to a SWCC that can be
measured using a pressure plate apparatus. The slope of the
SWCC plotted on an arithmetic scale is equal to the coefficient
1) are a special case of the general equations (i.e.,
Eqs. 13.152-13.154), which can then be written as follows
for the
x-, y-,
and
z-
directions, respectively:
σ
x
0
+
D
11
∂u
∂z
+
du
w
∂
∂x
∂x
+
D
12
∂
v
∂y
+
D
12
∂
w
τ
xy
0
+
D
44
∂u
∂
∂y
∂
v
∂x
+
∂y
+
τ
xz
0
+
D
44
∂u
∂
∂z
∂
w
∂x
+
∂z
+
=
0
(13.155)
τ
xy
0
+
D
44
∂u
∂
∂x
∂
v
∂x
∂y
+
σ
y
0
+
D
12
∂u
∂z
+
du
w
∂
∂y
∂x
+
D
11
∂
v
∂y
+
D
12
∂
w
+
τ
yz
0
+
D
44
∂
v
∂
∂z
∂
w
∂y
+
∂z
+
+
γ
t
=
0
(13.156)
τ
xz
0
+
D
44
∂u
∂
∂x
∂
w
∂x
∂z
+
τ
yz
0
+
D
44
∂
v
∂
∂y
∂
w
∂y
+
∂z
+
σ
z
0
+
D
12
∂u
∂z
+
du
w
∂
∂z
∂x
+
D
12
∂
v
∂y
+
D
11
∂
w
+
=
0
(13.157)
Equations 13.154-13.157 show the interdependency
between changes in pore-water pressure, pore-air pressures,
displacements, and total stresses. External loads may
result in changes in pore pressures and displacements. The
fraction of load distributed to the soil structure and to the
pore-water and pore-air phases is determined by the relative
compressibility of each phase, by the speed of loading, and
by the pore-water and pore-air permeability. The solutions
of Eqs. 13.152-13.154 combined with the solution of
pore-water flow and pore-air flow equations represent an
alternative and rigorous approach to the approach involving
the calculation of pore pressure parameters described by
Fredlund and Rahardjo (1993).
An alternative condition may exist where the pore-water
and pore-air pressure changes produced are the result of
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