sm7 162

________________________________________________________________________
PROBLEM (*7.162) A curved beam of channel-shaped cross section is acted upon end moments M
(Fig. P7.162). Calculate the dimension b required so that the stress at the points A and B of the beam
are equal in magnitude.
*SOLUTION
40 2(120 20) 40 4800Ab b=+ ×=+
We have 120
A
rm=m and mm280
B
r
=
. Applying Eq. (7.67):
()()()()
AB
AB AB
M
Rr MRr
Aer Aer
σσ
−− −−
=− =− =
from which
, ()(
BAA
rRr rRr−−= −
)
B280( 120) 120( 280)RR
−
−= −
or 168
R
mm=
Then A
R=
dA
r
∑∫
or 160 280
120 160
2(20)
168 bdr dr
Arr
⎡⎤
=+
⎢⎥
⎣⎦
∫∫ 81
168 ln 40ln
68
b4
⎡
⎤
=+
⎢
⎥
⎣
⎦
48.3306 3760.6181b=+
Hence
40 4800 48.3306 3760.6181bb
+
=+
or
124.8bm=m
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