sm12 58 61

________________________________________________________________________
PROBLEMS (12.58 through 12.61) A beam is loaded and supported as shown in Figs. P12.58
through P12.61. Apply Castigliano's theorem to determine the deflection at point C.
SOLUTION (12.58)
A
C
P
2a
x
B
y
a
03
2
M
Pa=
Segment CB
10
M
M=
Segment AC
20
()
M
MPxa=− −
We have: 12 0()
M
PMPx∂∂= ∂∂=−−a
Hence,
33
22
0() (2 )
aa
Caa
EIv M x a dx P x ax a dx=− − + − +
∫∫
33
22
22
023
aa
aa
xx
M
ax P ax a x=− + + −+
33
31 1
[(91)2] [(271)82
]
22 3
Pa Pa−++ −−+
=− 3
3
Pa
=−
or 3
3
CPa
v
EI
=↑
SOLUTION (12.59)
a
B
x
”
A
D
R
A=Q/2
x
RB=2wa+Q/2
a a
C
w
x’
Q
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 AC
1
11
22
M
x
MQ xQ
∂
==
∂
Let Q=0:
1
1
0
10
aM
Mdx
EI Q
∂
=
∂
∫
Segment CB
2
211
(')' '
22
M
Qa x Qx wx=+−−
2
1
(')
22
Qax wx
'−=−
21('
2
Max
Q
∂=−
∂)
Let Q=0:
2
2
20
11'1
'()(')
'
22
a
Mwx
M
dx a x dx
EI Q EI
∂=− −
∂
∫∫ 4
48
wa
EI
=−
Segment BD
23
31'' 0
2M
xQ
∂
Mw
=− =
∂
Therefore
4
48
i
Ci
Mwa
vM dx
QEI
∂
==↑
∂
∫
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 (12.60)
Add force Q at point C as shown
in Fig. (a).
2
02
LM
Ud
EI
=∫x
0
1L
CUM
M
dx
QEI Q
δ
∂
∂
==
∂∂
∫
Part AC. ,0
M
MPx Q
∂
=− =
∂
Part CB. (), (
22
LM
MPxQx x
Q)
L
∂
=− − − =− −
∂
Let Q=0:
2
02
11
()(0) ()[( )]
2
LL
CL
L
Px dx Px x dx
EI EI
δ =− +−−−
∫∫
2332
2
11 1 1
()[()()()(
2 3 32 22 222
L
L
PLP LLL
xxdx L L
EI EI 2
)]
L
=−−+
∫
=−
3
5
48 PL EI
=
Q
SOLUTION (12.61)
Segment AC
1
1()
82 2
M
wL Q x
Q
Mx
∂
=+ =
∂
Segment CB
22
23''
()'
82 2 2
M
wL Q wx x
Q
∂
=+− =
∂
Mx
Let Q=0:
2
22
00
3' ''
()
82 8 2 2
LL
CwLx x wLx wx x
EIv dx dx=+−
∫∫ '
223
'dx
22
00
3' '
()
16 16 4
LL
wLx wLx wx
wdxw −
∫∫
=+ 4
96
wL
=
Continued on next slide
P
C
x
B
Figure (a)
L/
2
L/
2
L/
2
B
A
x
L/
2
C
w
x’
Q
82
Q
wL +382
Q
wL
+
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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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or
4
96
CwL
vEI =↓
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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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