Split Ring Stress Analysis: Mechanics of Materials Problem

________________________________________________________________________
PROBLEM (*7.165) A split ring having an inner radius ri , outer radius ro , and a trapezoidal cross
section is loaded as shown in Fig. P7.165. Determine:
(a) The stress at point A.
(b) The stress at point B.
Given: ri = 4 in., ro = 10 in., b1 = 3 in., b2 = 2 in., P = 10 kips
*SOLUTION
Locate centroid :
11 22
12
Ar Ar
rAA
+
=+
9(6) (6)(8) 6.8 .
96 in
+
==
+
C
2 in.
O
A
1
212
12 12
1()
2
()ln(
o
oii
hb b
Rr
br br hb b
r
+
=
−−−
)
2
(0.5)(6) (3 2) 6.3567 .
10
[(3)(10) (2)(4)]ln (6)(3 2)
4
in
+
==
−−−
0.4433 .erR in=−=
(a) Use Eq.(7.71a),
() ()
[1 ]
AA
AAA
M
Rr rRr
PP
AAer A er
σ
−−
=− − =− +
10 (6.8)(6.3567 4)
16
.
15 (0.4433)(4) ksi
⎡⎤
−
=− + =−
⎢⎥
⎣⎦
69
(b) From Eq.(7.71b):
()
[1 ]
B
BB
rR r
P
Aer
σ
−
=− +
10 (6.8)(6.3567 10)
13
15 0.4433(10) ksi
⎡⎤
−
=− + =−
⎢⎥
⎣⎦
.06
A2
r
6 in.
3 in.
4 in.
P
A
O
P
M
=
P
r
B
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