6.8 Thick-Walled Cylinder Subjected to Internal and External Pressures
179
It is observed that the values of σ r and σ θ are identical to those in the plane stress
case. But in the plane stress case, σ z = 0, whereas in the plane strain case, σ z has a
constant value given by Eq. (6.34).
6.9 Rotating Discs of Uniform Thickness
The equation of equilibrium with body force component given by (Eq. 2.46)
dσ r
dr
+
σ r − σ θ
r
+ F r = 0
( a )
is used to treat the case of a rotating disc, provided that the centrifugal “inertia
force” is included as a body force. It is assumed that the stresses induced by rotation
are distributed symmetrically about the axis of rotation and also independent of
disc thickness. Thus, application of Eq. (a), with the body force per unit volume F r
equated to the centrifugal force ρw
2 r , yields
dσ r
dr
+
σ r − σ θ
r
+ ρw
2 r = 0
(6.35)
where ρ is the mass density and w the constant angular speed of the disc in rad/sec.
Equation (6.35) can be written as
d
dr
(r σ r ) − σ θ + ρw
2 r
2
= 0
(6.36)
But the strain components are given by
ε r =
du
dr
and ε θ =
u
r
(6.37)
From Hooke’s law, with σ z = 0
ε r =
1
E
(σ r − νσ θ )
(6.38)
ε θ =
1
E
(σ θ − νσ r )
(6.39)
From Eq. (6.37),
u = r ε θ
du
dr
= ε r =
d
dr
(r ε θ )