Environmental Engineering Reference
In-Depth Information
Solving Eq. 14.49 for mean net normal stress and substi-
tuting into Eq. 14.52 gives
where:
σ
ij
=
components of the net total stress tensor and
dV
w
V
0
=
b
i
=
components of the body force vector.
β
w
1
dε
ν
+
β
w
2
d(u
a
−
u
w
)
(14.53)
Substituting the strain-displacement Eq. 14.45 and the
stress-strain Eq. 14.48 into the equilibrium equation,
14.56 gives the following governing equations for general
three-dimensional problems (i.e., equations for
x
-,
y
- and
z
-directions):
where:
m
1
m
1
m
1
E
3
(
1
β
w
1
=
,
or
β
w
1
−
2
u)
m
1
m
2
m
1
m
1
E
m
2
−
β
w
2
m
2
−
β
w
2
=
,
or
(
1
−
2
μ)H
G
∂ε
ν
∂x
i
−
β
∂(u
a
−
u
w
)
∂u
a
∂x
i
+
2
u
i
+
G
∇
+
b
i
=
0
The unloading constitutive relationship for the water phase
is presented graphically in a form of a constitutive surface
in Fig. 14.53b. The coefficients of water volume change,
m
1
and
m
2
, indicate the amount of water taken on or released
by the soil as a result of a change in the net normal stress
and matric suction. Therefore, the slopes on the water phase
constitutive surface can be obtained by differentiating the
surface with respect to net normal stress and matric suction,
respectively:
1
−
2
μ
∂x
i
(14.57)
where:
m
2
E/H
1
β
=
m
1
=
−
2
μ
∂
2
∂x
2
+
∂
2
∂y
2
+
∂
2
∂z
2
2
∇
=
(the Laplace operator)
∂u
i
∂u
∂x
+
∂ν
∂y
+
∂
w
∂z
ε
ν
=
∂x
i
=
∂θ
∂(σ
mean
−
m
1
=
(14.54)
E
u
a
)
G
=
2
(
+
μ)
∂θ
∂(u
a
−
m
2
=
(14.55)
Equation 14.57 forms the stress analysis portion in three
dimensions for an unsaturated, swelling soil with a contin-
uous air phase. The equation has essentially the same form
as that presented by Biot (1941) for a soil with occluded air
bubbles.
u
w
)
where:
θ
=
V
w
/V
0
, volumetric water content.
14.6.6 Governing Partial Differential for Soil
Structure Equilibrium for Two-Dimensional Analysis
The stress analysis portion in two dimensions for an unsat-
urated, swelling soil can be written as follows:
14.6.4 Quantification of Boundary Conditions
for Water Phase
Moisture flux and head boundary conditions are most com-
monly placed on the boundaries of the problem. It is nec-
essary to perform an independent (coupled or uncoupled)
seepage analysis if a moisture flux boundary condition is
considered. The results of the seepage analysis would then
need to be combined with a stress-deformation analysis.
Moisture flux boundary conditions would need to be deter-
mined through an independent transient moisture flow mod-
eling (see Chapter 16).
Hydraulic head (or pore-water pressure) boundary con-
ditions are considered in this section when using a stress-
deformation analysis to predict total heave in an expansive
soil. In other words, the intent is to illustrate the independent
stress-deformation process as it relates to an expansive soil.
c
11
∂u
∂y
+
∂
∂x
∂u
∂x
+
∂ν
∂y
∂
∂y
∂ν
∂x
c
12
+
c
33
∂(u
a
−
u
w
)
−
d
s
+
b
x
=
0
(14.58)
∂x
∂u
∂y
+
c
12
∂
∂x
∂ν
∂x
∂
∂y
∂u
∂x
+
∂ν
∂y
c
33
+
c
22
∂(u
a
−
u
w
)
−
d
s
+
b
y
=
0
(14.59)
∂y
where
(
1
−
μ)E
c
11
=
c
22
=
(
1
+
μ)(
1
−
2
μ)
14.6.5 Governing Partial Differential for Soil
Structure Equilibrium for Three-Dimensional Analysis
The equations of overall static equilibrium for an unsaturated
soil can be written as follows:
∂σ
ij
∂x
j
+
μE
c
12
=
(
1
+
μ)(
1
−
2
μ)
E
c
33
=
2
(
1
+
μ)
E
b
i
=
0
(14.56)
d
s
=
(
1
−
2
μ)H
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