sm7 11

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PROBLEM (7.11) An aluminum beam with an hollow circular cross section is acted upon by
a bending moment M about the z axis (Fig.P7.11). Calculate:
(a) The normal stress at point A.
(b) The normal stress at point B.
(c) The radius of curvature 1
ρ
of the beam of a transverse cross section.
Given: M = 400 N⋅m, D = 50 mm, d = 30 mm, E = 70 GPa,
ν
= 0.29
SOLUTION
44 4 4
1()(503
0)
64 64
IDd
π
=−= − 34 267.035(10 ) mm=
(a) 25 0.025 .
A
y
mm m==
9
(400)(0.025) 37.45
267.035(10 )
A
AMy
M
Pa
I
σ
−
== =
(b) 15 0.015 .
B
y
mm m==
9
(400)(0.015) 22.47
267.035(10 )
B
BMy
M
Pa
I
σ
−
== =
(c) 1
M
EI ρ
=
99
400
70(10 )(267.035 10 )
−
=×
y
M
z
B
A
C
d
D
0.0214=
46.73 m
ρ
=
Hence,
146.73 161.14
0.29 m
ρ
ρν =− =− =−
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