sm12 76

________________________________________________________________________
PROBLEM (12.76) Using the unit-load method, find the support reactions RA and MA for the
beam of Fig. P12.76.
SOLUTION
Let
A
R
and A
M
be redundant.
b
a
1( ) ' 1maxm
=
−+ =−
C
B
P
M
Px=−
1 lb
⋅
in.
C
B
P
δ
B
1 lb
L
P
x
R
A
RB
C
M
A
MB
x
'
AA
Mm Mm
vdx
EI EI
θ ==
∫∫
dx
2
0
1()[( )] (
23
b
AP Pb a b)xaxdx
EI EI
=−−+= +
∫
vP
2
0
1()(1) 2
b
AP Pb
Px dx
EI EI
θ =−−=
∫
Let now A
R
and A
M
are applied.
A
R
A
1 lb
⋅
in.
B
B
1 lb
x
x
M
A
MR '
AA 1xM mx m
−==
=
32
0
()
1()
32
LAA A A
AR
R
xMx RL ML
vdx
EI EI
−
==
∫−
2
0
()1
1()
2
LAA A
AR A
Rx M RL
dx M L
EI EI
θ
−
==
∫−
Continued on next slide
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Since
AP AR AP AR
vv
θ
θ ==
or
3221
()(
23 3 2
AA
)
R
LML
Pb a b
EI EI
+= −
22 1()
23
AA
RL
Pb
M
L
EI EI
=−
Solving,
22
32
(3 )
AA
Pb Pab
RabM
LL
=+↑=
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.