sm7 99

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PROBLEM (*7.99) A round copper tube of outside diameter d and an aluminum core of
diameter d/3 are bonded together to form a composite beam of the cross section shown in Fig.
P7.99. The allowable stress in the bronze is b
σ
. Determine the maximum bending moment M
that can be supported by the beam, in terms of Ea , Eb, b
σ
, and d, as needed. Calculate the value
of M for: Ea = 70 GPa, Eb = 110 GPa, b
σ
= 520 MPa, d = 75 mm.
*SOLUTION
1.571
ba
nEE==
Transform to bronze:
4
44
5
[()]
64 3 324
bdd
Id
π
π
=−=
Aluminum Core: Each element of area has width reduced in ratio n.
4
4
1()
64 3 5,184
add
Inn
ππ
==
and
44
11
(5 ) (80 )
324 16 5184
tba dd
III nn
ππ
=+= + = +
We have
31 , (80 )
2 2592
bt
bb
t
I
Mc d
M
I
dn
σπ
σσ ===+
Introducing the given data:
36
(0.075) 1
(520 10 )(80 )
2592 1.571
M
π
=×+
21.44 kN m=⋅
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