sm7 166

________________________________________________________________________
PROBLEM (7.166) A machine component with an elliptical cross section is fixed at one end and
supports a concentrated load P at the free end ( Fig. P7.166). Calculate the stress at points A and B.
Given: ri = 4 in., ro = 7 in. , a = 3 in., b = 1.5 in., P = 10 kips
SOLUTION
4. 7
AB
rinri==
.n
11
( ) (7 4) 5.5 .
22
oi
rrr i=+=+=n
From Table 7.1
2
(3)(1.5) 4.5 .Aab in ππ π == =
P
P
M
=-Pr B
A
22
2[] dA b rra
ra
π
=−−
∫
22
2(1.5)
[5.5 5.5 3 ] 2.7967 .
3in
π
=−−=
Hence
4.5 5.0549 .
2.7967
A
A
R
in
dA
r
π
== =
∫
0.4451 .erR in=−=
Equation (7.70) with
M
PF=− :
3
()
10(10 ) 5.5(5.0549 4)
[1 ] 1
4.5 (0.4451)(4)
A
AA
rR r
P
Aer
σπ
⎡
⎤
−−
=+ = +
⎢
⎥
⎣
⎦
3.01 ksi
=
3
()
10(10 ) 5.5(5.0549 7)
[1 ] 1
4.5 (0.4451)(7)
B
BB
rR r
P
Aer
σπ
⎡
⎤
−−
=+ = +
⎢
⎥
⎣
⎦
1.721 ksi
=−
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