Thick Wall Pressurized Sphere

Hoop and radial stress through the wall of a pressurised sphere

σθ(a) = (pᵢ(2a³ + b³) − 3pₒb³) / (2(b³ − a³))

Section through a thick wall sphere, showing inner radius a, outer radius b, internal and external pressure, and the hoop and radial stresses.

Enter units, internal pressure, inner diameter, wall thickness and external pressure to work out the hoop and radial stress through the wall of a pressurised sphere.

About Thick Wall Sphere

Pressure inside a sphere pulls the wall apart the same amount in every direction, so there is one hoop stress and it makes no difference where you cut. It peaks at the internal surface and eases towards the outside. Radial stress does the reverse: minus the pressure at the inside face, zero at the outer one.

For the same diameter and wall, a sphere sees roughly half the hoop stress a cylinder does. Hence the domed ends on pressure vessels, and hence spherical storage tanks.

Assumes an even wall of isotropic material, elastic throughout, with stress still in step with strain. Openings, nozzles and welds all lift the local stress above these numbers. It is not a code check: where a vessel has to be certified, the code sets the allowable stress and the factors.