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6 Answers to Supplementary Problems
Fig. 6.11 Convergence rate of the finite difference approximation for a simply supported Timoshenko beam with distributed load
u Z (0.5L) = −
5408E I Y + 563k s AG L
2
L
2 q 0
4k s AG
k s AG L 2 + 10816E I Y
(53 nodes) ,
(6.182)
u Z (0.5L) = −
31104E I Y + 3239k s AG L
2
L
2 q 0
12k s AG
k s AG L 2 + 20736E I Y
(73 nodes) ,
(6.183)
u Z (0.5L) = −
124848E I Y + 13003k s AG L
2
L
2 q 0
24k s AG
k s AG L 2 + 41616E I Y
(103 nodes) .
(6.184)
The convergence rate is shown in Fig. 6.11.
4.4 Convergence of finite difference approximation of a cantilevered Timoshenko beam
(a) Solutions for dim. = 2 × (domain nodes − 2). The following general solutions
consider that the load is opposed to the positive Z -direction.
u Z (L) = −
4096E
2 I
2
Y + 1248E I Y k s AG L
2
+ 25k
2
s A
2 G
2 L
4
L
2 q 0
4k s AG
32E I Y + k s AG L 2
k s AG L 2 + 64E I Y
(5 nodes) ,
(6.185)
u Z (L) = −
32E I Y + 11k s AG L
2
L F 0
k s AG
32E I Y + k s AG L 2
(5 nodes) .
(6.186)
The convergence rate is shown in Fig. 6.12.