202
6 Two-Dimensional Problems in Elasticity …
σ r =
p i a
2
− 0
b 2 − a 2
−
p i − 0
b 2 − a 2
a
2 b
2
a 2
=
p i a
2
− p i b
2
b 2 − a 2
=
− p i (b
2
− a
2
)
(b 2 − a 2 )
Therefore, σ r = −p i .
Similarly,
Hoop stress = σ θ =
p i a
2
− p 0 b
2
b 2 − a 2
+
p i − p 0
b 2 − a 2
a
2 b
2
r 2
At r = a and p 0 = 0,
σ θ =
p i a
2
− 0
b 2 − a 2
+
p i − 0
b 2 − a 2
a
2 b
2
a 2
σ θ =
p i (a
2
+ b
2
)
(b 2 − a 2 )
Here the maximum and minimum stresses are.
σ 3 = −p i and σ 1 = σ θ .
But the maximum shear stress = τ max =
1
2 (σ 1 − σ 3 ).
i.e.
τ max =
1
2
p i
a
2
+ b
2
b 2 − a 2
+ p i
=
1
2
p i a
2
+ p i b
2
+ p i b
2
− p i a
2
b 2 − a 2
35 =
p i b
2
(b 2 − a 2 )
i.e.
35 =
8 × b
2
(b 2 − a 2 )
35b
2
− 35a
2
= 8b
2
35b
2
− 8b
2
= 35a
2
35b
2
− 8b
2
= 35(0.5)
2