sm12 53 55

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PROBLEMS (12.53 through 12.55) For the beam and loading shown in Figs. P12.53 through
P12.55, employ Castigliano's theorem to determine:
(a) The deflection at point C.
(b) The slope at point C.
SOLUTION (12.53)
x
x’
A
Pa
L
+
a C C
B
L
C
L
P
Segment AB
M1 = − P
a
C
x− x
L
L
Segment BC
M 2 = − Px '− C
(a) For this case C=0:
vC =
1
EI
∫
L
0
(−
Pax
ax
1
)(− )dx +
L
L
EI
L
∫
a
0
− Px '(− x ')dx '
a
Pa 2 x 3
P x '3
Pa 2
=
+
=
( L + a) ↓
EIL2 3 0 EI 3 0 3EI
(b)
∂M 1
x
=− ,
L
∂C
∂M 2
= −1.
∂C
For C=0, we have :
1 L Pax
x
1
θC =
(−
)(− )dx +
∫
EI 0
L
L
EI
L
Pa x 3
P x '2
=
+
EIL2 3 0 EI 2
∫
a
0
− Px '(− x ')dx '
a
0
Pa
=
(2 L + 3a)
6 EI
Continued on next slide
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instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other
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SOLUTION (12.54)
x
Q
M0
A
M0
3a
− Q3
2a
C
x’
a
B
M0
3a
+ 23Q
Segment AC
M1 = −
M0
Q
x+ x
3a
3
Segment CB
M2 = −
M 0 x ' 2Q
+
x'
3a
3
(a) We have
∂M 1 x
∂M 2 2 x '
Q=0
=
=
3
∂Q 3
∂Q
Thus,
1 2a M 0 x x
1 a M 0 x ' 2x '
vC = −
dx +
dx '
∫
EI 0 3a 3
EI ∫0 3a 3
2 M 0a2
=
↑
9 EI
(b) In this case Q=0:
∂M 1
∂M 2 x '
x
=− ,
=
3a
∂M 0
∂M 0 3a
So,
1 2a M 0 x x
1
θC =
dx +
∫
EI 0 3a 3a
EI
M a
= 0
3EI
∫
a
0
M0x ' x '
dx '
3a 3a
Continued on next slide
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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.
SOLUTION (12.55)
x
x’
A
M0
L
+
Qa
L
L
Q
M0
a C
B
M0
L
− Q ( LL+ a )
Segment AB
M1 = −
M0
Qa
x−
x
L
L
Segment BC
M 2 = M 0 − Qx '
(a) We have
∂M 1
∂M 2
ax
=−
= − x ' , and Q = 0 .
∂Q
L
∂Q
Hence,
1 L M 0 x ax
1 a
(− )dx +
M 0 (− x ')dx '
vC =
∫
EI 0 L
L
EI ∫0
M a2
= 0 (2 L + 3a) ↑
6 EI
(b) Now Q=0:
∂M 1 x
∂M 2
= ,
= 1.
∂M 0 L
∂M 0
Thus,
a
1 L M0x x
dx + ∫ M 0 (1)dx '
θC =
∫
0
0
EI
L L
M
= 0 ( L + 3a)
3EI
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
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