200
6 Two-Dimensional Problems in Elasticity …
σ r = −
2
r
P
π
sin θ
σ θ = 0
τ r θ = 0
Example 6.2 A thick cylinder of inner radius 10 cm and outer radius 15 cm is
subjected to an internal pressure of 12 MPa. Determine the radial and hoop stresses
in the cylinder at the inner and outer surfaces.
Solution: The radial stress in the cylinder is given by
σ r =
p i a
2
− p o b
2
b 2 − a 2
−
p i − p o
b 2 − a 2
a
2 b
2
r 2
The hoop stress in the cylinder is given by
σ θ =
p i a
2
− p o b
2
b 2 − a 2
+
p i − p o
b 2 − a 2
a
2 b
2
r 2
As the cylinder is subjected to internal pressure only, the above expressions reduce
to
σ r =
p i a
2
b 2 − a 2
−
p i
b 2 − a 2
a
2 b
2
r 2
and
σ θ =
p i a
2
b 2 − a 2
+
p i
b 2 − a 2
a
2 b
2
r 2
Stresses at inner face of the cylinder (i.e. at r = 10 cm):
Radial stress = σ r =
12 × (0.1)
2
(0.15) 2 − (0.1) 2
−
(0.15)
2
(0.1)
2
(0.1) 2
12
(0.15) 2 − (0.1) 2
= 9.6 − 21.6
or σ r = −12 MPa.
Hoop stress = σ θ =
12 × (0.1)
2
(0.15) 2 − (0.1) 2
+
12
(0.15) 2 − (0.1) 2
(0.15)
2
(0.1)
2
(0.1) 2
= 9.6 + 21.6
or σ θ = 31.2 MPa.
Stresses at outer face of the cylinder (i.e. at r = 15 cm):