sm10 72

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PROBLEM (*10.72) Figure P10.72 shows a nonprismatic propped cantilever beam AB with
flexural rigidity EI from A to C and 2EI from C to B. The beam supports a uniform load of intensity w.
Using a direct-integration approach, find :
(a) The support reactions (RA, MA, and RB).
(b) The mid span deflection if w = 60 lb/ft, L = 16 ft, and EI = 200 x 106 lb ⋅ in.2
*SOLUTION
w
(a)
MA
A
L/2
C
L/2
RA
B
RB
w
MA
M
A
C
L/2
RA
x
V
1
M 1 = − M A + RA x − wx 2
2
L
L
M 2 = − M A + RA [ + ( x − )]
2
2
wL L
L
1
L
−
[ + ( x − )] − w( x − ) 2
2 4
2
2
2
Segment AC
1
EIv1 '' = − M A + RA x − wx 2
2
1
1
EIv1 ' = − M A x + RA x 2 − wx 3 + C1
2
6
1
1
1
EIv1 = − M A x 2 + RA x3 − wx 4 + C1 x + C2
2
6
24
(1)
Segment CB
M
M
R L wL2 1
wL
L w
L
)( x − ) − ( x − ) 2
EIv2 '' = 2 = − A + A −
+ ( RA −
2
2
4
16 2
2
2
4
2
2
M x R Lx wL x 1
wL
L
w
L
)( x − ) 2 − ( x − )3 + C3
EIv2 ' = − A + A −
+ ( RA −
2
4
16
4
2
2
12
2
2
2
2 2
M x
R Lx wL x
wL
L
w
L
1
EIv2 = − A + A
−
+ ( RA −
)( x − )3 − ( x − ) 4 + C3 x + C4
4
8
32
12
2
2
48
2
Continued on next slide
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Boundary & continuity conditions:
v1 '(0) = 0 : C1 = 0
v1 (0) = 0 :
C2 = 0
M L R L2 wL3
M L R L2 wL3
L
L
v1 '( ) = v2 '( ) : − A + A −
=− A + A −
+ C3
2
2
2
8
48
4
8
32
M L wL3
C3 = − A +
4
96
2
3
M A L RA L wL4
M A L2 RA L3 wL4
L
L
v1 ( ) = v2 ( ) : −
+
−
=−
+
−
2
2
8
48 384
16
32 128
M L2 wL4
− A +
+ C4
8
192
M L2 R L3
C4 = − A − A
16
96
7M A 7
7
1
7
v2 ( L) = 0 : − M A L2 + RA L3 −
wL4 = 0 , RA =
+ wL
16
8
256
2 L 32
Statics:
∑M
B
= 0: MA =
9
wL2
80
and
RA =
49
wL ↑
80
∑F
= 0 : RB =
y
31
wL ↑
80
(b) Equation (1) becomes
w0
v1 =
(−27 L2 x 2 + 49 x3 − 20 x 4 )
480 EI
Letting x=L/2:
wL4
vC =
↓
256 EI
Substitute the given data:
60(16) 4
vC =
= 0.077 in.
256(200 ×103 )
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
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.