Civil Engineering Reference
In-Depth Information
By eliminating
β
in formula (
14
) with formula (
15
), we can get a new Eq. (
16
):
δ
s
e
2
δ
c
I
2
a
+
r
I
a
e
r
=
(16)
ark
Then the optimization problem P2 can be changed into optimization problem P3:
+
δ
s
e
2
2
−
δ
c
I
2
kI
a
e
r
min
−
(17)
2
δ
s
e
2
δ
c
I
2
a
+
r
I
a
e
r
st
.
=
(18)
ark
1
r
−
1
2
)
δ
s
e
2
+(
≥
F
U
(19)
1
a
−
1
2
)
δ
c
I
2
(
−
F
≥
0
(20)
To solve the problem, we can put formula (
18
) aside for the time being and follow
Kuhn-Tucker (K-T) method to get the solution. The Kuhn-Tucker conditions of
optimization problem
P
3 are the following formulas from Eqs. (
21
)to(
25
) with
the precondition:
r
1
,
r
2
≥
0:
1
a
−
1
2
)
δ
c
I
akI
a
−
1
e
r
−
+
δ
c
I
−
2
r
2
(
=
0
(21)
1
r
−
1
2
)
δ
s
e
rkI
a
e
r
−
1
−
+
δ
s
e
−
2
r
1
(
=
0
(22)
−
=
r
2
r
1
0
(23)
1
r
−
1
2
)
δ
s
e
2
[
+(
−
]=
r
1
F
U
0
(24)
1
a
−
1
2
)
δ
c
I
2
r
2
[(
−
F
]=
0
(25)
We can get the following Eq. (
26
) through solving the K - T problem:
4
1
r
−
1
2
)
a
2
s
(
δ
I
e
=
(26)
1
a
−
1
2
)
(
r
2
δ
c
Then we turn back to take formula (
18
) into consideration, by plugging equa-
tion (
26
) into formula (
18
), we can get the final solution:
1
a
−
2
a
+
r
−
2
ln
(
2
+
1
)
δ
c
r
−
a
−
2
)
r
2
c
ln
(
δ
a
−
2
+
ak
4
(
a
+
r
−
2
)
r
−
2
)
s
a
2
(
δ
e
∗
=
ex p
(27)
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