sm12 58 61

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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)
y
P
x
2a
A
C
a B
M0 =
3
Pa
2
Segment CB
M1 = M 0
Segment AC
M 2 = M 0 − P( x − a)
We have:
Hence,
∂M 1 ∂P = 0
∂M 2 ∂P = −( x − a)
3a
3a
EIvC = − ∫ M 0 ( x − a)dx + P ∫ ( x 2 − 2ax + a 2 )dx
a
a
3a
2
3a
2
x
x
= M 0 − + ax + P
− ax 2 + a 2 x
2
3
a
a
3Pa 3 1
1
Pa 3
[− (9 − 1) + 2] + Pa 3 [ (27 − 1) − 8 + 2] = −
2
2
3
3
3
Pa
vC =
↑
3EI
=
or
SOLUTION (12.59)
Q
A
RA=Q/2
a
x
x”
w
C a
x’
B
a D
RB=2wa+Q/2
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
∂M 1 x
1
M 1 = Qx
=
2
∂Q 2
Let Q=0:
∂M 1
1 a
M1
dx = 0
∫
0
EI
∂Q
Segment CB
1
1
M 2 = Q(a + x ') − Qx '− wx '2
2
2
Q
1
= (a − x ') − wx '2
2
2
∂M 2 1
= (a − x ')
∂Q 2
Let Q=0:
∂M 2
1
1 a wx '2 1
wa 4
=
−
−
'
(
)
(
')
M
dx
a
x
dx
=
−
'
2
∂Q
EI ∫
EI ∫0
48EI
2 2
Segment BD
∂M 3
1
M 3 = − wx ''2
=0
2
∂Q
Therefore
∂M i
wa 4
↑
vC = ∫ M i
dx =
∂Q
48EI
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
L M
U =∫
dx
0 2 EI
∂U
1 L ∂M
=
δC =
M
dx
∂Q EI ∫0
∂Q
Q
P
x
C
B
L/2
L/2
Figure (a)
∂M
=0
∂Q
L
M = − Px − Q( x − ),
2
M = − Px,
Part AC.
Part CB.
∂M
L
= −( x − )
∂Q
2
Let Q=0:
1 L2
1 L
L
(− Px)(0)dx +
(− Px)[−( x − )]dx
∫
∫
0
L
2
EI
EI
2
P L 2 L
P 1 3 1 L 3 L 1 2 L 1 L 2
( x − x)dx =
[ L − ( ) −( ) L +( ) ( ) ]
=
EI ∫L 2
2
EI 3
3 2
2 2
2 2 2
3
5 PL
=
48 EI
δC =
SOLUTION (12.61)
Q
L/2
A
wL
8
C
+ Q2
wL Q
+ )x
8
2
L/2
x’
x
Segment AC
M1 = (
w
B
3 wL
8
+ Q2
∂M 1 x
=
∂Q 2
Segment CB
M2 = (
wx '2
3wL Q
+ ) x '−
8
2
2
∂M 2 x '
=
∂Q
2
Let Q=0:
L 2 3wLx '
wLx x
wx '2 x '
dx + ∫ (
−
) dx '
0
0
8 2
8
2 2
2
2
L 2 wLx
L 2 3wLx '
wx '3
wL4
dx + w∫ (
−
)dx ' =
= w∫
0
0
16
16
4
96
Continued on next slide
EIvC = ∫
L2
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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
wL4
vC =
↓
96 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
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