In an internally pressurized thick cylinder

Concept:

In case of thin cylinders, the hoop stress is determined by assuming it to be uniform across the thickness of the cylinder but in thick cylinders, the hoop stress is not uniform across the thickness, it varies from a maximum value at the inner circumference to a minimum value at the outer circumference.

For a thick cylinder hoop stress (tangential stress) is given by,

\({σ _h} = \frac{{P\;{{\left( {{r_i}} \right)}^2}}}{{{{\left( {{r_0}} \right)}^2} - {{\left( {{r_i}} \right)}^2}}}\left[ {1 + \frac{{{{\left( {{r_0}} \right)}^2}}}{{{x^2}}}} \right]\)

For a thick cylinder Radial stress is given by,

\({σ _r} = \frac{{P\;{{\left( {{r_i}} \right)}^2}}}{{{{\left( {{r_0}} \right)}^2} - {{\left( {{r_i}} \right)}^2}}}\left[ {1 - \frac{{{{\left( {{r_0}} \right)}^2}}}{{{x^2}}}} \right]\)

ri ≤ x ≤ ro

Where, ri and ro are inner and outer radius, respectively

The Variation of the hoop stress and the radial stress from the figure can be concluded as 

• The hoop stress varies parabolically from the maximum value at the inner radius to the minimum value at the outer radius 
• The radial stress also varies parabolically but the maximum value is at the inner radius and at the outer radius the value is zero.