sm12 64

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PROBLEM (12.64) A uniform bar of flexural rigidity EI is fixed at one end and loaded at the other
end as shown in Fig. P12.64. Use Castigliano's theorem to obtain:
(a) The vertical deflection at point D.
(b) The slope at point D.
SOLUTION
C
A
P
x
P
2a
B
D
x
P
a-
C
C
a
x
a
(a) C=0.
Segment CD
1
1M
Px x
P
∂
=− =−
∂
M
Segment BC
2
2M
Pa a
P
∂
==
∂
M
Segment AB
2
3M
M
Px Pa x a
P
∂
=− =−
∂
Thus
i
Di
M
EI M dx
P
δ
∂
=∂
∫
2
22 2 2
000
(2 )
aaa
P x dx Pa dx P x ax a dx=++−+
∫∫∫
2
3
32
0
1
(1)
33
a
x
Pa P ax a x
2
+−+=+
33
48
(42
33
Pa Pa )−+=+
or 3
2
D
δ
Pa
EI
=↓
Continued on next slide
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(b) 11
1MCPx MC=− − ∂ ∂ =−
22
1MPaC MC=+ ∂ ∂=
33
1MPxPaC MC=−− ∂ ∂=−
Let C=0:
2
000
()
aaa
i
Di
M
EI M dx P xdx Pa dx P x a dx
C
θ
∂
==+−−
∂
∫∫∫∫ 2
3
2Pa=
or 2
3
2
DPa
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
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