sm12 78 80

________________________________________________________________________
PROBLEMS (12.78 through *12.80) A beam is supported and loaded as shown in Figs. P12.78
through P12.80. Use Castigliano's theorem to determine the reactions.
SOLUTION (12.78)
Consider
A
M
as redundant.
Due to symmetry: 1
2
AB
R
Rw== L
2
1
2
AA
M
Rx M wx=−−
1
A
M
M
∂
=
−
∂
2
0
1
0( )(1
2
L
AAA
wx )
R
xM dx
EI
θ == − − −
∫
or
233
0
3
2646
L
AA A
0
x
wx wL wL
RMx M=− + + =− + +
Solving
2
12
AwL
M =
Statics: 2
12
BwL
M=
SOLUTION (12.79)
Consider
A
R
and A
M
as redundants.
Segment AC: 1AA
M
Rx M=−
Segment CB: 20AA
M
Rx M M=− − −
Continued on next slide
A
B
L
x
w
M
R
A
RB
A
MBB
A
B
L/2
x
R
A
RB
M
A
MB
L/2
y
M0
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Thus,
0
i
Ai
A
M
EI M M
θ
∂
==
∂
∫
or
2
0
02
0( )(1) ( )(1)
LL
AA AA
L
R
x M dx R x M M dx=−−+−−−
∫∫
2
22
0
02
22
LL
AA
AA
L
Rx Rx
M
xMxMx+ +− + +
0
=−
After simplifying
(1)
0
2
AA
MMRL+− =
Similarly,
0
i
Ai
A
M
EIv M R
∂
=
=
∂
∫
or
2
0
02
0( ) ( )
LL
AA AA
L
R
xMxdx RxM Mxdx=−+−−
∫∫
22
32 32
0
02
32 322
L
L
AA AA
L
Mx
Rx Mx Rx Mx
=+ ++ −
Simplifying,
0
390
28
AA
RL M M−−= (2)
Solving Eqs. (1) and (2):
0
03
1
42
AA
M
MM RL
=
=↑
Statics:
AB AB
R
RMM=− =
SOLUTION (*12.80)
Consider
A
R
as redundant.
R
A
aa
2E
I
P
P
EI
a
B
R
B =2RA-P
D
A
E
R
C =P-RA
C
a
x’
x
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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Segment AD:
11AA
M
Rx M R x=− ∂ ∂ =−
Segment BD:
22
()
AA
M
Rx Px a M R x=− + − ∂ ∂ =−
Segment BE:
33
()'(') ', '
M
AA A
PRxPxa RxPa M R x=− − −=− + ∂ ∂=−
Segment EC:
44
()', '
M
AA
PRx M R x=− ∂ ∂=−
Therefore
3
2
1
00
1
22
aa
AA
A6
M
RR
dx x dx
EI R EI EI
∂==
∂
∫∫
a
22
2
2
11
[()](
22
aa
A
aa
A
M)
M
dx R x P x a x dx
EI R EI
∂=−+−−
∂
∫∫
33
75
612
A
Ra Pa
EI EI
=−
22
2
3
3
11
(' ')
'
aa
A
aa
A
M
M
dx R x Pax dx
EI R EI
∂=−
∂
∫∫
33
73
32
A
Ra Pa
EI EI
=−
22
4
4
00
11
(' ')
aa
A
A
M'
M
dx Pax R x dx
EI R EI
∂=−+
∂
∫∫
33
33
A
Ra Pa
EI EI
=−
Since
0
i
Ai
A
M
vM
R
∂
==
∂
∑∫
We have
1771 5 31
()(
6633 1223
A
RP+++ − ++ =
)0
Solving
9
16
A
R
P=↓
Statics:
17
81
BC
6
R
PRP=↑ = ↑
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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
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