Hoop Stress Calculator

Determine the stresses in thin walled shells (spheres and cylinders) using the hoop stress calculator.

Hoop stress41.6667 MPapr/t, acting around the circumference — TWICE the longitudinal stress, which is why a pressurised pipe splits along its length rather than snapping across

Longitudinal stress20.8333 MPapr/2t — exactly half the hoop stress, always

Radial stress at the inner surface-1 MPacompressive, and equal to the internal pressure. Thin-wall theory neglects it, which is fine while it is small against the hoop stress — here it is 2.4% of it

Radius to thickness ratio41.6667above 10, so thin-wall theory is valid

Lamé hoop stress at the bore42.1726 MPathe thick-wall solution, which is 1.214% higher than the thin-wall figure. Stress is not uniform through a thick wall — it peaks at the bore

von Mises equivalent stress36.0844 MPacombining the two principal stresses — NOT their sum. For a cylinder it works out at 0.866 times the hoop stress, so biaxial tension is slightly less severe than the hoop figure alone suggests

Factor of safety on yield6.9282comfortable

Pressure at first yield69.282 barscaling linearly, since every stress here is proportional to pressure

Minimum wall for this pressure1.5 mmat a factor of 1.5 on yield

Why domed endsa sphere needs half the wallthe same pressure and radius give pr/2t in a sphere against pr/t in a cylinder, so a flat end would need to be far thicker than a domed one — and a flat plate carries the load in BENDING, which is worse still

The formula

σ_h = pr/t; σ_l = pr/2t; sphere σ = pr/2t

Hoop stress is twice longitudinal

Cut a cylinder lengthwise and the pressure acts on a projected area of diameter times length, resisted by two wall thicknesses. Cut it across and the pressure acts on the full circular area, resisted by the entire circumference. The geometry works out at exactly a factor of two, so the hoop stress always reaches yield first and a pressurised pipe splits along its length.

A sphere has no such asymmetry: every direction is equivalent and the membrane stress is pr/2t throughout. For the same pressure and radius a sphere therefore needs half the wall thickness, which is why pressure vessels have domed ends rather than flat ones — and why a flat end, which carries the load in bending rather than tension, is worse still.

Thin-wall theory has a limit

The pr/t formulas assume stress is uniform through the wall and neglect the radial compression entirely. That is a good approximation while the radius exceeds about ten times the thickness. Below that the stress peaks noticeably at the bore, and the Lamé thick-wall solution — computed above alongside — is the one to use.