Environmental Engineering Reference
In-Depth Information
The total stresses within the soil mass can be determined
from a finite element stress analysis using any particular
soil behavioral model. It can be assumed that stresses
are constant within a small grid element if the density of
the search grid is sufficiently fine. The constant stresses
are signified by stresses at the center points of the grid
element. The resisting and actuating forces acting on the
i
th segment of a slip surface can be calculated as follows
(Fig. 12.81):
(Bellman, 1957), the optimal function
H
i
+
1
(k)
obtained at
state point
{
k
}
located in stage [
i
+
1] is defined as
H
i
+
1
(k)
=
H
i
(j)
+
G
i
(j,k)
(12.101)
where:
G
i
(j,k)
=
return function calculated from state point
{
j
}
of stage [
i
] to state point
{
k
}
of stage [
i
+
1].
ne
ne
At the initial stage, the value of the optimal function
H
1
(j)
is equal to zero:
R
i
=
R
ij
=
τ
f
ij
l
ij
ij
=
1
ij
=
1
ne
H
1
(j)
=
0
j
=
1
,...,
NP
1
(12.102)
[
c
ij
+
(σ
i
n
u
a
)
tan
φ
ij
+
u
w
)
tan
φ
ij
]
l
ij
=
−
(u
a
−
ij
=
1
where:
(12.99)
ne
ne
NP
1
=
number of state points in the initial stage.
S
i
=
S
ij
=
τ
ij
l
ij
(12.100)
ij
=
1
ij
=
1
1), the optimal function
H
n
+
1
(k)
must be equal to the minimum value of the return
function
G
m
, that is;
At the final stage (i.e.,
i
=
n
+
where:
=
a grid element traveled by the
i
th segment,
ij
τ
f
ij
,τ
ij
=
shear strength and shear stress acting at the
center point of
ij
, respectively,
n
H
n
+
1
(j)
=
G
m
=
min
(R
i
−
F
s
S
i
)j
=
1
,...,
NP
n
+
1
c
ij
,φ
ij
,φ
ij
=
strength parameters of
the
saturated-
i
=
1
(12.103)
unsaturated soil within
ij
,
where:
ne
=
number of
ij
, and
l
ij
=
length of the
i
th segment limited by the
boundary of
ij
.
NP
n
+
1
=
number of state points located in the final stage.
}
located in stage [
i
] is introduced. The optimal function
H
i
(j)
is defined as the minimum of the return function
G
calcu-
lated from a state point for the initial stage to state point
An
optimal function
H
i
(j)
obtained at state point
{
j
The
optimal point
in the final stage is defined as the state
point at which the calculated optimal function is a minimum.
From the optimal state point
{
k
}
found in the final stage, the
}
located in stage [
i
]. According to
the principle of optimality
{
j
optimal state point
located in the previous stage is also
determined. The
optimal path
defined by connecting optimal
{
j
}
stage i
stage i + 1
stage i
stage i + 1
stage i
stage i + 1
j
j
j
τ
f
ij
τ
ij
Sij
(ij)
σ
ij
Rij
τ
f
ij
τ
ij
Sij
(ij)
σ
ij
Rij
S
i
τ
f
ij
R
i
τ
ij
Sij
Rij
(ij)
σ
ij
τ
ij
S
ij
τ
f
ij
R
ij
σ
ij
(ij)
θ
θ
θ
k
k
k
stage i
stage i + 1
stage i
stage i + 1
stage i
stage i + 1
Figure 12.81
Actuating and resisting forces acting on
i
th segment between segments.
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