sm10 88

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PROBLEM (10.88) Before the loads are applied, a small gap δB, exists between beam AC and
support B, as seen in Fig. P10.88. Subsequent to loading, the gap closes and reactions develop at each
support. Find these reactions.
SOLUTION
Statics:
0: 2
yABC
FRRRwa=++=
∑ (1)
5
0: 3 2 2
CAB
M
RR wa=+=
∑ (2)
2
'' ( ) 2
2
AB w
EIv R x R wa x a x a=+−<−>−<−>
223
1
11
'() 2
22 6
AB w
EIv R x R wa x a x a C=+−<−>−<−>+
334
12
11
() 2
66 24
AB w
EIv R x R wa x a x a C x C=+−<−>−<−>++
Boundary conditions:
2
(0) 0: 0vC==
223
13433
(3 ) 0: 0
2972
AB
va C Ra Ra wa==−+− =
333 4
13433
() : 62972
BA A B B
va Ra Ra Ra wa EI δδ =− − − + =− (3)
From Eq.(2):
53
42
BA
R
wa R=− (4)
Substitute this into Eq.(3) to obtain:
3
37
248
B
AEI
R
wa
a
δ
=−
Equation (4) gives
3
47 9
32 4 B
BEI
Rwa
a
δ
=− −
Equation (1) yields
3
65 3
96 4 B
CEI
Rwa
a
δ
=− +
wa
a
R
A
A
C
B
y
RB
w
x
aa
D
RC