Explanation to Compound Cylindrical Shells

According to Lame’s equation, the thickness of a cylindrical shell is given by:

From this equation, we see that if the internal pressure (p) acting on the shell is equal to or greater than the allowable working stress (σt) for the shell material, no thickness of the shell will prevent failure. Thus it is impossible to design a cylinder to withstand an internal pressure equal to or greater than the allowable working stress.

This problem is overcome by inducing an initial compressive stress on the wall of cylindrical shell. This can be done by two ways:

1. By using compound cylindrical shells, and 
2. By using the theory of plasticity.

In the compound cylindrical shell as shown in the figure, the outer cylinder (smaller inside diameter than the outer diameter of the inner cylinder) is fitted over the inner cylinder by heating and cooling. Upon cooling, contact pressure is developed at the junction of the two cylinders, which causes a compressive tangential stress to the material of the inner cylinder and a compressive tangential stress to the material of the outer cylinder. When the cylinder is loaded, the compressive stress is first relieved and then the tensile stress is induced. Thus, a compound cylinder is effective for resisting higher internal pressure than a single cylinder with the same overall dimensions. The principle of compound cylinder is used in the design of gun tubes.

According to the theory of plasticity, a temporary high internal pressure is applied to the inside of the cylinder wall till the plastic stage is reached. This results in residual compressive stress upon the removal of the internal pressure, making the cylinder more effective to withstand high internal pressure.

Reference:
Textbook of machine design by khurmi, gupta