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: 