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A Second Iteration
2 ,
To observe the effect of further iterations, return to (4) and determine
w
x
x
0 (
C
60
2
4 x 6
15 Lx 5
15 L 2 x 4
2 L 3 x 3
6 L 4 x 2
w
=
+
+
)
dx dx
0
2 L 3 L
x
0 (
x 3
4 x 6
15 Lx 5
15 L 2 x 4
2 L 3 x 3
6 L 4 x 2
+
+
+
)
dx dx
0
2 L 2 L
0
x
0 (
dx dx
3 x 2
4 x 6
15 Lx 5
15 L 2 x 4
2 L 3 x 3
6 L 4 x 2
+
+
)
C
8400 (
10 x 8
50 Lx 7
70 L 2 x 6
14 L 3 x 5
70 L 4 x 4
13 L 5 x 3
39 L 6 x 2
=
+
+
+
)
(16)
From Eq. (11.74),
4 x 4
15 Lx 3
15 L 2 x 2
2 L 3 x
6 L 4
140
(
+
+
)
EI
L 2
P cr =−
(17)
(
10 x 6
50 Lx 5
+
70 L 2 x 4
+
14 L 3 x 3
70 L 4 x 2
13 L 5 x
+
39 L 6
)
This can be evaluated, giving
x
0
L
/
4
L
/
2
3 L
/
4
L
(18)
P cr L 2
EI
21.54
21.03
20.40
20.08
20.00
% Error
6.67
4.17
1.06
0
.
55
0
.
94
The mean value would be
P cr L 2
EI
1
5 (
=
21
.
54
+
21
.
03
+
20
.
40
+
20
.
08
+
20
.
00
) =
20
.
61
(
2
.
09% Error
)
(19)
From (11)
38 EI
L 2
P cr =
20
.
(
0
.
92% Error
)
(20)
and from the Rayleigh quotient of (14),
196 EI
L 2
P cr =
20
.
(
0
.
007% Error
)
(21)
Finite Differences
As is to be expected, the finite difference expressions of Chapter 8 can be employed to
compute the critical loading. Simply use the finite difference relations to discretize the gov-
erning equations [Eq. (11.31)]. Apply the boundary conditions and then find the critical load
from a characteristic equation formed of the determinant of the finite difference equations.
EXAMPLE 11.16 A Fixed-Hinged Column
Use finite difference equations to compute the critical load for the fixed-hinged column of
Fig. 11.28.
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