6.8 Thick-Walled Cylinder Subjected to Internal and External Pressures
173
Fig. 6.5 a Thick-walled cylinder, b plane stress case and c plane strain case
Here σ θ and σ r denote tangential (circumferential) and radial stresses acting
normal to the sides of the element.
Since r is the only independent variable, the above equation can be written as
d
dr
(r σ r ) − σ θ = 0.
(6.21)
From Hooke’s law,
ε r =
1
E
(σ r − νσ θ ), ε θ =
1
E
(σ θ − νσ r )
Also,
ε r =
du
dr
and ε θ =
u
r
Hence, the stresses in terms of strains are
σ r =
E
(1 − ν 2 )
(ε r + νε θ )
σ θ =
E
(1 − ν 2 )
(ε θ + νε r )
Substituting the values of ε r and ε θ in the above expressions, we get
σ r =
E
1 − ν 2
du
dr
+ ν
u
r
σ θ =
E
1 − ν 2
u
r
+ ν
du
dr