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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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