sm12 51 52

________________________________________________________________________
PROBLEMS (12.51 and 12.52) For the beam and loading shown in Fig. P12.51 and P12.52, use
Castigliano's theorem to determine the slope
θB
at point B.
SOLUTION (12.51)
x
w
C
A
B
L
1
M = −C − wx 2
∂M ∂C = −1
2
M ∂M
1 L
wx 2
dx =
C
+
θB = ∫
(
)dx
2
EI ∂C
EI ∫0
Letting C=0:
θB =
1
EI
∫
L
0
wx 2
wL3
dx =
2
6 EI
SOLUTION (12.52)
y
A
P
P
a D
2
P+C/4a
Segment AD
M1 = ( P +
C
)x
4a
E a
C
B
P-C/4a
∂M 1 ∂M 2 ∂M 3
x
=
=
=
4a
∂C
∂C
∂C
Segment DE
M 2 = (P +
C
) x − P( x − a)
4a
Segment EB
M 3 = (P +
C
) x − P( x − a) − P( x − 3a)
4a
Let C = 0 :
∂M i
1
θB =
Mi
dx
∫
EI
∂C
2
3a
4a
4 a Px
1 a
x
x
x
=
dx]
[ ∫ Px( )dx + ∫ Pa( )dx + ∫ 4 Pa( )dx − ∫
3a
3 a 4a
a
4a
4a
4a
EI 0
Continued on next slide
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.
Integrating
Pa 3 a 3 9a 3 a 3 64a 3 36a 3 64a 3 27 a 3
θB =
− +
−
+
−
[ +
]
4aEI 3
2
2
2
2
3
3
3 Pa 2
=
2 EI
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.