Environmental Engineering Reference
In-Depth Information
equations governing unsaturated soil behavior. Equations
13.143-13.148 can be used for the derivation of governing
partial differential equations. Equations 13.143-13.148
form relatively simple and straightforward relationships to
the compressibility of soils.
where:
σ
i
0
,τ
ij
0
=
initial stresses,
D
11
=
E(
1
−
μ)/
[
(
1
+
μ)(
1
−
2
μ)
],
D
12
=
Eμ/
[
(
1
+
μ)(
1
−
2
μ)
],
D
44
=
E/
[2
(
1
+
μ)
],
13.7.7 Partial Differential Equations
for Stress-Deformation Analysis
Partial differential equations for stress-deformation analysis
are required for determining stress and strain distributions
within an unsaturated soil. The stress distribution can be
used for certain types of analyses (e.g., slope stability anal-
ysis) and the strain distributions can be used in settlement,
heave, or other ground movement analyses. The prediction
of total stresses in saturated-unsaturated soil masses are con-
siderably easier to predict than are strains and displacements.
The partial differential equations for a stress-deformation
analysis can be written by combining the differential
equations governing force equilibrium (i.e., Eqs. 13.132,
13.133, and 13.134), the stress-strain constitutive rela-
tionships, and the relationship between strains and small
displacements (i.e., Eqs. 13.135 to 13.140). The stress-strain
relationships adopted largely determine the complexity of
the partial differential equations. The simplest equations
that can be derived are based on a generalization of
Hooke's law written in terms of independent stress state
variables (e.g., Eqs. 13.143-13.148). The partial differential
equation for the
x-
direction can be written as
h
=
E/
[
H(
1
−
2
μ)
],
γ
t
=
body force acting downward [
γ
t
=
γ
s
(
1
−
n)
+
γ
w
nS
,kN/m
3
], and
specific weight of soil particles, kN/m
3
.
γ
s
=
The unit weight of the soil,
γ
t
, shown in Eq. 13.155
corresponds to the vertical body force (i.e., unit weight).
The initial stresses and pore pressures presented in Eqs.
13.152-13.154 can be removed in the case where an incre-
mental transient approach is considered (i.e., taking deriva-
tives with respect to time of
u
,
v
,
w
,
u
w
, and
u
a
). The
unsaturated soil properties shown in Eqs. 13.152-13.154
vary with soil suction and net stresses. In this sense, the
solution of the partial differential equations is nonlinear.
Equations 13.152-13.154 show that the equilibrium
of forces within a representative elemental volume can
be expressed as a function of five primary variables: the
displacements in the
x
-,
y
-, and
z
-direction (i.e.,
u
,
v
,
and
w
), the pore-water pressure
u
w
, and the pore-air pressure
u
a
. Therefore, two additional equations are required in order
to render the system of equations solvable.
The choice of primary variables (i.e.,
u
,
v
,
w
,
u
w
, and
u
a
) is not arbitrary. These five primary variables result in
meaningful
boundary conditions
for unsaturated soil prob-
lems. Integration by parts of the second-order derivatives
of Eqs. 13.152-13.154 can be used to obtain a solution.
Integration results in a surface integral corresponding to a
Neumann-type boundary condition. The Neumann boundary
condition associated with Eqs. 13.152-13.154 corresponds
to external loads (i.e., forces) in the
x
-,
y
-, and
z
-directions,
respectively. Another type of boundary condition that can be
applied to Eqs. 13.152-13.154 is a predetermined displace-
ment value (i.e., Dirichlet-type boundary conditions with
u
,
v
, and
w
as primary variables).
The three-dimensional equilibrium of forces results in
three differential equations. Each equation is formed by three
main partial derivatives of the stress tensor components with
respect to the
x
-,
y
-, and
z
-directions. The partial derivatives
are a result of the assumption concerning the forces and
stresses acting on the REV shown in Fig. 13.65. The stresses
are continuously distributed in space. The spatial distribution
of the variables can be described using the partial derivative
of each force or stress component for a given direction.
Various terms within each of the main partial derivatives
are a function of the constitutive law adopted to describe
the stress-strain behavior of the saturated-unsaturated soil.
The partial differential equations governing the equilibrium
of forces acting on an unsaturated soil element become
increasingly complex as more
∂
x
σ
x
0
+
D
11
∂u
∂z
−
hd
u
a
−
u
w
∂
∂x
+
D
12
∂
v
∂y
+
D
12
∂
w
τ
xy
0
+
D
44
∂u
+
du
a
+
∂
∂y
∂
v
∂x
∂y
+
(13.152)
τ
xz
0
+
D
44
∂u
∂
∂z
∂
w
∂x
+
∂z
+
=
0
the partial differential equation for the
y-
direction as
τ
xy
0
+
D
44
∂u
σ
yo
+
D
12
∂u
∂x
∂
∂x
∂
v
∂x
∂
∂y
∂y
+
+
∂z
−
hd
u
a
−
u
w
+
du
a
+
D
11
∂
v
∂y
+
D
12
∂
w
(13.153)
τ
yz
0
+
D
44
∂
v
∂
∂z
∂
w
∂y
+
∂z
+
+
γ
t
=
0
and the partial differential equation for the
z-
direction as
τ
xz
0
+
D
44
∂u
∂y
τ
yz
0
∂
∂x
∂
w
∂x
∂
∂z
+
+
+
D
44
∂
v
σ
z
0
+
D
12
∂u
∂x
∂
w
∂y
∂
∂z
∂z
+
+
(13.154)
∂z
−
hd(u
a
−
u
w
)
+
du
a
+
D
12
∂
v
∂y
+
D
11
∂
w
=
0
elaborate
stress-strain
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