sm12 38 39

_______________________________________________________________________
PROBLEMS (12.38 and 12.39) A beam is supported and loaded as shown in Fig. P12.38 and Fig.
P12.39. Employ work-energy approach to determine the slope at point C due to the moment M0 .
SOLUTION (12.38)
Reactions are noted in Fig. (a).
Part AC. 0
M
x
ML =−
2
2
23 232
02
00
22
LL
AC M
M
Udxxdx
EI EIL
==
∫∫
2
0
4
81
M
L
EI
=
Part BC 0'
M
x
ML =−
2
2
33
2
02
00
''
22
LL
BC M
Mxxdx
EI EIL
==
∫∫
Ud 2
0
162
M
L
EI
=
Total strain energy: 2
18o
AC BC
M
L
UU U EI =+=
Therefore 0
1
2C
M
U θ =, 0
9
C
M
L
EI
θ =
SOLUTION (12.39)
Reactions are shown in Fig. (a).
Part AC. 0
M
M
=
2
2
20
024
L
AC
M
L
Mx
EI EI
==
∫
Ud
Part AB. We have 0
x
M
M
L
=
2
22
02
00
22
LL
AB M
M
Udxxdx
EI EIL
==
∫∫
2
0
6
M
L
EI
=
Continued on next slide
Figure (a)
x
M
M
0/L
M
0/L
C
M
0
x
A
B
x’
M
0
C
Figure (a)
M
0
x
A
B
x’
2L/3
L/
R
R
AB
3
L
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Total strain energy: 2
5
12 o
AC AB
M
L
UU U EI =+=
Hence 0
1
2C
M
U θ =, 2
0
5
6
C
M
L
EI
θ =
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or
instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other
reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States
Copyright Act without the permission of the copyright owner is unlawful.