The walls of pressure vessels generally undergo triaxial loading. For cylindrical pressure vessels, the normal loads on a wall element are longitudinal stress, circumferential (hoop) stress and radial stress.

The radial stress for a thick-walled cylinder is equal and opposite to the gauge pressure on the inside surface, and zero on the outside surface. The circumferential stress and longitudinal stresses are usually much larger for pressure vessels, and so for thin-walled instances, radial stress is usually neglected.

The radial stress for a thick walled pipe at a point ${\displaystyle r}$  from the central axis is given by

${\displaystyle \sigma _{r}(r)={\frac {p_{i}r_{i}^{2}-p_{o}r_{o}^{2}}{r_{o}^{2}-r_{i}^{2}}}+{\frac {r_{i}^{2}r_{o}^{2}(p_{o}-p_{i})}{r^{2}(r_{o}^{2}-r_{i}^{2})}}\ }$ 

where ${\displaystyle r_{i}}$  is the inner radius, ${\displaystyle r_{o}}$  is the outer radius, ${\displaystyle p_{i}}$  is the inner absolute pressure and ${\displaystyle p_{o}}$  is the outer absolute pressure.^[1] Maximum radial stress occurs when ${\displaystyle r=r_{i}}$  (at the inside surface) and is equal to gauge pressure, or ${\displaystyle p_{i}-p_{o}}$ .^[2]