
The equilibrium of stresses in cylindrical polar coordinates is given as:
, the body force, points radially outwards. It generally equals zero.
The Lamé Equations
These describe the variation in radial and hoop stresses throughout the thickness of the wall:
and
are constants that can be found from the boundary consitions:

The ratio between the inner and outer radii is . The larger the value, the more inaccurate the thin-walled assumption is.
Internal Pressure Only
When there is only an internal pressure, , the coefficients of the Lamé equations are:
Therefore, the radial and hoop stresses are:
External Pressure Only
When there is only an external pressure, , (like a submarine) the coefficients of the Lamé equations are:
Therefore, the radial and hoop stresses are:
Internal & External Pressure
These problems can be solved using superposition.
- The Lamé equations describe the variation in radial and hoop stresses through the cylinder wall:
- For internal pressure only:
- For external pressure only:
- For internal and external pressure: