Environmental Engineering Reference
In-Depth Information
The transient water flow equation 16.81 along with the
equation for the SWCC (Eq. 16.82) and the permeability func-
tion (Eq. 16.83) can be used to compute pore-water pressure
profiles (i.e., matric suctions) at various times during an infil-
tration process. The matric suction profiles are then used in
an uncoupled manner with the stress-deformation equation
to calculate overall volume changes. Deformations due to
changes in matric suction at any elapsed time are calculated
based on the initial suction profile and the matric suctions at
subsequent times.
16.85 can be written as follows for two-dimensional plane
strain problems:
c
11
∂u
∂y
+
∂
∂x
∂u
∂x
+
∂
v
∂y
∂
∂y
∂
v
∂x
c
12
+
c
33
∂
u
a
−
u
w
−
d
s
+
b
x
=
0
(16.86)
∂x
∂u
∂y
+
c
12
∂
∂x
∂
v
∂x
∂
∂y
∂u
∂x
+
∂
v
∂y
c
33
+
c
12
∂
u
a
−
u
w
−
d
s
+
b
y
=
0
(16.87)
16.8.2 Partial Differential Equations for
Two-Dimensional Expansive Soil Example
as Coupled Analysis
The transient saturated-unsaturated seepage equation can also
be fully coupled with a stress-deformation analysis. In this
case, the equations of overall static equilibrium for the soil
must be solved. Relevant stress-deformation equations for the
coupled formulation are presented in vector notation:
∂y
where
(1
−
μ
)
E
c
11
=
c
22
=
(1
+
μ
)(1
−
2
μ
)
μE
c
22
=
(1
+
μ
)(1
−
2
μ
)
E
c
33
=
2(1
+
μ
)
∂σ
ij
∂x
j
+
b
i
=
0
(16.84)
E
d
s
=
(1
−
2
μ
)
H
where:
Equations 16.86 and 16.87 can be used to compute dis-
placements in the horizontal and vertical directions under an
applied load and/or due to changes in matric suction. The
solution of the seepage equation 16.80 and the soil struc-
ture equilibrium equations 16.86 and 16.87 can be obtained
using either an uncoupled or coupled procedure.
The solutions for uncoupled and coupled volume change
problems in expansive soils have been studied for various
example problems byVu (2003) andVu and Fredlund (2003a).
Both uncoupled and coupled solutions were presented for the
case where the elasticity parameters
E
,
H
, the coefficients of
water volume change
m
1
and
m
2
, and the coefficient of per-
meability
k
w
were functions of both stress state variables. The
coupled and uncoupled solutions appeared to yield essentially
the same answers for the amount of heave in a swelling soil.
An uncoupled solution is used when solving the following
example problem involving heave below a floor slab built on
grade. The initial and subsequent boundary conditions must
be designated when solving the stress-deformation equations
(i.e., Eqs. 16.86 and 16.87). The boundary conditions include
the initial matric suctions, initial total stress conditions, and
elasticity parameter functions associated with the volume
change of the soil. The results from a seepage analysis (i.e.,
for changes in soil suction) can be used as input to the stress-
deformation solution.
σ
ij
=
components of the net total stress tensor and
b
i
=
components of the body force vector.
Substituting the strain-displacement relationship and the
stress-strain relationship into the equilibrium equation 16.84
gives the following governing equations for general three-
dimensional problems (i.e., equations for
x
-,
y
-, and
z
-
directions):
β
∂
u
a
−
u
w
G
∂ε
v
∂x
i
−
∂u
a
∂x
i
+
2
u
i
+
G
∇
+
b
i
=
0
1
−
2
μ
∂x
i
(16.85)
where
m
2
E/H
1
β
=
m
1
=
−
2
μ
∂
2
∂x
2
+
∂
2
∂y
2
+
∂
2
∂z
2
2
∇
=
(Laplace operator)
∂u
i
∂x
i
=
∂u
∂x
+
∂
v
∂y
+
∂w
∂z
ε
v
=
E
G
=
2
(
1
+
μ)
16.8.3 Case History of Slab-on-Grade with Shallow
Perimeter Footings on Regina Clay
The methodology for the prediction of heave in an expansive
soil was applied to data collected by the Prairie Regional Sta-
tion of the Division of Building Research (DBR), National
Equations 16.79 and 16.85 form a system of coupled
equations for the theory of swelling in three dimensions for
an unsaturated, swelling soil with a continuous air phase.
The equations have a similar form to those presented by
Biot (1941) for a soil with occluded air bubbles. Equation
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