210
6 Two-Dimensional Problems in Elasticity …
.
A
B O
1 5 0
W
160
.
80
100
160
Y B
Y A
20
20
20
R
Fig. 6.19 Loaded circular ring with unsymmetrical I-section
Now, A = 20 × 100 + 120 × 20 + 80 × 20 = 6000 mm
2
y B =
(100 × 20 × 10) + (120 × 20 × 80) + (80 × 20 × 150)
6000
= 75.33 mm.
∴ y A = (160 − 75.33) = 84.67 mm.
Also, R = (150 + 75.33) = 225.33 mm.
∴ m = −1 +
225.33
6000
80 ln(225.33 + 84.67) + (20 − 80) ln(225.33 + 64.67)+
(100 − 20) ln(225.33 − 55.33) − 100 ln(225.33 − 75.33)
∴ m = 0.072.
Moment = M = P R = 80 × 1000 × 225.33 = 1.803 × 10
7 N − mm.
Now, Stress at point B = σ B =
P
A
+
M
AR
1 +
y B
m(R+y B )
∴ σ B = −
80,000
6000
+
1.803 × 10
7
6000 × 225.33
1 +
(−75.33)
0.072(225.33 − 75.33)
∴ σ B = −93.02 N/mm
2
( Compression )
Stress at point
A = σ A =
P
A
+
M
AR
1 +
y A
m(R + y A )
= −
80,000
6000
+
1.803 × 10
7
6000 × 225.33
1 +
84.67
0.072(225.33 + 84.67)
∴ σ A = 50.6 N/mm
2
(Tension)