Thin cylinders review
pr
h 
t
pr
m L 
2t
Thick cylinders
Difference between thin and thick cylinders
1 20
1 20
Thick walled cylinders: Lame’s Equations
 hoop
 long
 radial
Worked Example 13
A closed cylindrical vessel of 100 mm external diameter and 20 mm thickness
carries gas at an internal pressure of 20 MPa.
a.
b.
c.
d.
e.
Plot the variation of hoop and radial stresses across the thickness of the pipe.
Determine the longitudinal stress.
Determine the maximum shear stress and state which plane it is present in.
Determine the hoop strain at the inner radius due to the internal pressure.
Determine the longitudinal strain due to the internal pressure.
Take E = 200 GPa, and ν=0.3.
r/mm
σh /MPa
30
35
40
45
50
42.5
34.2
28.8
25.1
22.5
σr /MPa
-20
-11.7
-6.3
-2.6
0
External pressure
Worked Example 14
A hollow steel tube of external diameter 10 cm and wall thickness 1.5 cm is used
under seawater, the head of water being 3000 m. Determine the hoop stresses at
the outer and inner surfaces of the tube due to hydrostatic pressure. The pressure
on the inside of the tube is atmospheric pressure. What will be the reduction in
the external diameter due to hydrostatic pressure?
Take E = 2 (10)7 N/cm2, γ =.0102 N/cm3 , and ν = 0.3
r/cm
σh (N/cm2)
σr (N/cm2)
3.5
4
4.5
5
-12000
-10594
-9630
-8940
0
-1406
-2370
-3060
Compound Cylinders
Compound Cylinders
• In thick cylinders the hoop stress at inner radius is maximum and it governs the design.
• As a result of it, the material in most of the part is under stressed and underutilized.
• To get better stress distribution and also to make cylinders to withstand higher pressures,
compound cylinders are used.
• A compound cylinder is made by heating a cylinder with inner radius slightly less than the
outer radius of another cylinder and allowing it to cool after insertion.
• The outer cylinder then exerts external pressure to internal cylinder and in turn this is
subjected to internal pressure.

Worked Example 16
A thick cylinder has an inner radius of 200 mm and an outer radius of 250 mm.
Another tube of the same material is to be shrunk onto the first tube, the outer
radius of it being 300 mm.
If the initial pressure at the junction is 6 N/mm2, what are the final hoop stresses
after a fluid is admitted at a pressure of 80 MPa? Sketch the variation of hoop
and radial stresses.


@ r  200mm  r  0 :
@ r  250mm  r  6 :
A  16.67
B  666667
666667
r2
666667
 r  16.67 
r2
 h  16.67 
B
A
0
2
200
B
A
 6
2
250
D
@ r  250mm  r  6 :
C
0
2
250
D
@ r  300mm  r  0 : C 
 6
2
300
C  13.64
D  1227273
1227273
 h  13.64 
r2
1227273
 r  13.64 
r2