sm10 161

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PROBLEM (*10.161) Before the load is applied to the beam shown in Fig. P10.161, a small gap δC
exists between beam AB and the support at C. Following the application of load, the gab closes and
reactions develop at each support. Determine the reactions for the beam.
*SOLUTION
Consider
C
R
as redundant.
2
11(4 )4 8
2BB
ARaaRa==
2
219
(3 )3
22
APaaPa=− =−
2
31(2 )2 2
2CC
ARaaRa==
33 3
/1 2 3
423294
() () ()
33323
AB B C
aa
EIt A A a A R a Pa R a=++= −+
/11
[(2 )2][( )]
223
CB B a
EIt R a a Pa a=− 33
41
36
B
R
aPa=−
We have 2( ) :
CB C AB
tt
δ
+=
3
2
425
836 C
BC EI
RR P
a
δ
=+ − = (1)
0:
A
M=
∑ 13
24
BC
R
RP+= (2)
From Eqs.(1) and (2): 33
33 9 3
84 168
CC
CB
EI EI
RP R P
aa
δ
δ
=− = +
Then, 0:
y
F=
∑
3
13
16 8 C
AEI
RP
a
δ
=+
x
M
tan
g
ent at B
B
M
A
2
A
P
a
2a 2a
x
-
2RCa
P
A
a
4RBa
2a/3 C
A
3
R
A RB
RC
y
δ
c
tC/B
A
1
4a/3
B
A
2a
2RBa
δ
c
tA/B=2( tC/B+ δ c)
x