Environmental Engineering Reference
In-Depth Information
The PDEs for a stress-deformation analysis can be writ-
ten as
the use of numerical modeling. Problems can involve the
solution of either a single PDE or more than one PDE in
an uncoupled or coupled manner. The examples are typ-
ical of problems encountered in geotechnical engineering.
Automatic, optimized mesh generation and grid refinement
techniques are used in solving each of the unsaturated soil
mechanics problems.
The saturated-unsaturated software programs from Soil-
Vision Systems were used for the solution of the example
problems in this topic (M.D. Fredlund, 2005). The two-
and three-dimensional seepage problems were solved using
the software called SVFlux. The stress analysis and slope
stability example problem was solved using SVSolid and
SVSlope. The volume change problem involving expansive
soils was solved using SVSolid and SVFlux. The unsat-
urated soil properties were estimated using the SoilVision
database (M.D. Fredlund, 2003).
D
11
D
44
∂u
∂
∂x
∂u
∂x
+
∂ν
∂y
∂
∂y
∂ν
∂x
D
12
+
∂y
+
=
0
(1.5)
D
44
∂u
D
12
∂
∂x
∂ν
∂x
∂
∂y
∂u
∂x
+
∂ν
∂y
∂y
+
+
D
11
+
γ
=
0
(1.6)
where:
u
=
displacement in
x-
direction,
=
v
displacement in
y-
direction,
D
11
=
−
μ
)/[(1
+
μ
)(1
−
2
μ
)],
E
(1
D
12
=
E
μ
/[(1
+
μ
)(1
−
2
μ
)],
D
44
=
E
/[2(1
+
μ
)],
E
=
Young modulus, kPa,
μ
=
Poisson ratio, and
1.6.6 Example of Two-Dimensional Seepage Analysis
Figure 1.20a shows the final optimized mesh for the compu-
tation of steady-state saturated-unsaturated seepage through
a two-dimensional earthfill dam. Concentrations of finite
elements occur at locations where refinement of the mesh
is required to obtain an accurate solution (e.g., locations
of increased gradient). Figure 1.20b shows the computed
hydraulic heads for steady-state seepage through the earth-
fill dam. A parametric study showed that variations in the
permeability function for the shell of the dam had little effect
on the phreatic line and the equipotential lines. Solving
the entire saturated-unsaturated region removes the need to
make any assumption regarding the location of the phreatic
surface.
γ
=
total unit weight or body force
acting in
y
-direction (i.e., vertical), kN/m
3
.
If the computed stress states are optimized with respect
to failure conditions in the soil, the shape and location of
the critical slip can be determined. This is, in essence, a
slope stability analysis where the shape of the potential slip
surface is not prescribed. A similar type of analysis is also
possible for the calculation of bearing capacity and earth
pressures.
Stress-versus-deformation analyses become more chal-
lenging when the objective of the analysis is to compute
deformations. Predictions of deformation require accuracy
for the nonlinear constitutive soil properties. The stress-
deformation analysis needs to start with the calculation of
the initial stress states and pore-water pressures.
It is possible to simultaneously solve more than one par-
tial differential equation in a coupled or uncoupled manner.
Some studies have shown that uncoupled solutions yield sat-
isfactory results for most geotechnical engineering purposes
(Vu 2003). It is also possible to use the AGR approach
when solving two or more PDEs. In other words, there can
be independent optimized grid (or mesh) being used for the
solution of the seepage equation while a different mesh is
simultaneously being used for the stress analysis. The two
solvers (i.e., two meshes) are able to communicate with one
another as the overall solution moves toward convergence.
The above equations are simply meant to illustrate how the
classic areas of saturated-unsaturated soil mechanics can be
solved in a similar manner through the solution of a limited
number of PDEs.
1.6.7 Finite Element Mesh for Three-Dimensional
Tailings Pond
The generation of an appropriate finite element mesh for
a three-dimensional problem is an extremely difficult task.
Even when a three-dimensional mesh can be generated, there
is no assurance that it will meet all the requirements for a
correct solution to the nonlinear partial differential seep-
age equation. Consequently, it is extremely important that
mathematically satisfied criteria be used in the automatically
generated and refined meshes that are part of the three-
dimensional solution.
Figure 1.21 illustrates the automatically generated mesh
for steady-state seepage through the waste pond. A total of
nine soil units were identified and permeability functions
were input for all soils near the ground surface. The solu-
tion to this problem also involved the use of time-dependent
flux boundary conditions that varied from one location to
another. Further details related to data input and the solution
of this problem can be found in Rykaart et al. (2001a). The
primary intent of this example is to illustrate the optimized
finite element mesh.
1.6.5 Numerical Modeling of Saturated-Unsaturated
Soils
A series of example problems involving unsaturated soils are
presented to illustrate solutions that can be obtained through
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