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: