sm12 24

________________________________________________________________________
PROBLEM (*12.24) A simple beam AB of length L and rectangular cross section is subjected to
a moment M0 at point A as shown in Fig. P12.24. Determine the strain energy of the beam caused by both
bending and shear.
*SOLUTION
Load, shear, and moment diagram is shown in Fig. (a).
For bending
M0
2
0'
2
L
bM
Ud
EI
=∫
B
x
a
A
22
020''
2
L
M
x
dx
EIL
=∫2
0
6
M
L
EI
=
For shear
2
02
L
sVx
AG
α
=∫
Ud
where
0
6
5
M
Abh V L α == =−
Therefore
22
00
20
33
55
L
sMM
Ud x
bhGL GbhL
==
∫
Total energy, letting 312Ibh= :
22
0
32
23
(1 )
10
sbMEh
Ebh GL
+= +
UU
0
M
L
x’
x
M0
V
Figure (a)
x
-M0/L
M
x
L
M
0
L
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