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From (7), we find
x
0
L
/
4
L
/
2
3
L
/
4
L
P
cr
L
2
EI
(9)
30
.
00
23
.
53
20
.
87
20
.
00
20
.
00
% Error
48.58
16.53
3.18
−
0
.
94
−
0
.
94
The mean value would be
1
5
(
EI
L
2
=
88
EI
L
2
P
cr
=
30
+
23
.
53
+
20
.
87
+
20
.
00
+
20
.
00
)
22
.
(10)
which differs from the correct value by 13.32%.
One method of avoiding the problem of Eq. (11.74) leading to a buckling load that depends
on
x
, is to take the integral of
n
−
1
n
. Then
w
and
w
EI
0
w
n
−
1
dx
P
cr
=
(11)
L
0
w
n
dx
In the case of this example,
C
1
L
5
L
0
w
C
L
5
4
+
2
5
3
20
CL
5
0
3
L
2
x
2
5
Lx
3
2
x
4
=
0
(
−
+
)
=
−
=
dx
dx
L
0
w
L
0
(
−
=
−
C
60
1
6
x
2
L
4
2
x
3
L
3
15
x
4
L
2
15
x
5
L
4
x
6
+
+
−
+
)
dx
dx
(12)
L
7
=
−
C
60
1
2
+
5
2
+
4
7
C
140
L
7
−
2
+
3
−
=
so that from (11),
EI
L
w
0
dx
3
·
140
20
EI
L
2
=
00
EI
L
2
0
P
cr
=
dx
=
21
.
(13)
0
w
1
which is 4.01% in error.
Another possibility is provided by the
Rayleigh quotient
L
0
w
2
dx
EI
P
cr
=
(14)
L
0
w
2
dx
1
which is derived for eigenvalue problems in elementary vibration textbooks. Insert
w
of
(6) in (14) to obtain
L
0
12
EI
L
0
w
2
L
4
L
3
x
15
L
2
x
2
25
Lx
3
10
x
4
2
EI
dx
(
−
+
+
−
+
)
dx
=
w
2
dx
=
P
cr
L
0
L
0
(
−
12
L
4
x
+
6
L
3
x
2
+
60
L
2
x
3
−
75
Lx
4
+
24
x
5
)
2
dx
243
EI
L
2
=
20
.
(15)
The error associated with this approximation is 0.11%.
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