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From Figure 2, a trial slip surface consists of
n
+1 vertices, each of which is identi-
fied by the x- and y- coordinates of
x
i
,y
i
, where
i
ranges from 1 to
n
+1. The exit point
and entry points (
V
1
and
V
n+1
) are controlled by using the upper and lower limits
given by
x
l
,x
u
and
x
L
,x
U
. The x-coordinates of vertices from 2 to
n
are calculated by
using the following Eq. 3.
+
−
=+
x
x
xx
n
1
1
×−+
(
i
1
ζ
),
i
=
2, ...,
n
(3)
i
1
i
n
where ζ
i
is a variable used to determine the x-coordinates of
V
i
and is subjected to -
0.5 <
i
< 0.5. Given the value of
x
2
, we can calculate
y
2max
and
y
2min
using Eq. 2. A
variable χ
2
within the range of 0 to 1 is introduced to represent the y-coordinates of
V
2
.
y
2
can be obtained using Eq. 4.
ζ
(
)
(4)
yy
=
+
y
−
y
×
χ
2
2 min
2 max
2 min
2
Two lines are obtained by connecting
V
1
and
V
n+1
,
V
1
and
V
2
, respectively. The
angles of the lines from the downward direction
V
1
are denoted as β
2
,…,
β
n+1
respec-
tively.
n
-2 values are randomly generated within the range of β
2
to β
n+1
using
n
-2 val-
ues of χ
3
,…,
χ
n
in the range of 0 to 1.0 as given by Eq. 5 and they are sorted by
ascending order and are related to β
3
to β
n
, respectively. The y-coordinates of
V
3
to
V
n
are determined by Eq. 6.
(
)
βχ β β
+
=×
−
(5)
i
i
n
1
2
(
)
π
(
)
yy
=+
tan
β
−
× −
xx
(6)
2
i
1
i
i
1
where
i
= 3,…,
n
.
Each trial slip surface obtained by this procedure can be represented mathemati-
cally by the vector
(
)
V
, in which 2 variables are used to
determine the slip surface. The lower bound and upper bound to each element in
V
are
denoted as
l
and
u
, respectively, where
i
ranges from 1 to 2 . The proposed pro-
cedure cannot guarantee that all the generated trial slip surfaces are admissible, so the
treatment as shown in Figure 3 will be necessary.
For the generated failure surface as shown in Figure 3, CD can be considered as
unacceptable as this will be a restraint to the slope failure. CD can be kept for analy-
sis if necessary but it will not be the critical solution in most cases so that a more ef-
ficient search is to correct CD to CD' so that the failure mechanism is acceptable. D'
is determined by extrapolating BC to D', where D' and D have the same x-ordinate.
Vertices D should be moved to D', and the same check should be performed for
every adjacent three vertices until the whole failure surface is acceptable. If D is
higher than D', CD will be acceptable and no modification is required. In view of the
=
x
,
ζχ
,
;...,
ζχ
,
;
x
1
2
2
n
n
n
+
1
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