For thick cylinders such as guns, pipes to hydraulic presses, high pressure hydraulic pipes the wall thickness is relatively large and the stress variation across the thickness is also significant.
The problem may be solved by considering an axisymmetry about z-axis and solving the differential equations of stress equilibrium in polar co-ordinates. In general the stress equations of equilibrium without body forces can be given as
For axisymmetry about z-axis and this gives
In a plane stress situation if the cylinder ends are free to expand σz = 0 and due to uniform radial deformation and symmetry
The equation of equilibrium reduces to
If we consider a general case with body forces such as centrifugal forces in the case of a rotating cylinder or disc then the equations reduce to
It is convenient to solve the general equation so that a variety of problems may be solved.