Pressure Vessel Stress Calculator

Pressure vessels (tanks, pipes, boilers, and cylinders) develop stress in their walls when pressurized. Two principal stresses arise from the geometry of a cylinder.

Thin-wall assumption. When the diameter-to-thickness ratio D/t exceeds 20, the thin-wall (membrane) equations are sufficiently accurate:

Hoop stress (circumferential): sigma_h = P x r / t = P x D / (2t)

Longitudinal stress (axial, for closed ends): sigma_l = P x r / (2t) = P x D / (4t)

where P is internal gauge pressure, r is the inner radius, and t is the wall thickness.

The hoop stress is always twice the longitudinal stress. This is why cylindrical pressure vessels typically fail by splitting along the longitudinal seam rather than at the end caps.

Von Mises equivalent stress combines the two principal stresses into a single scalar that can be compared to material yield strength:

sigma_vm = sqrt(sigma_h^2 - sigma_h x sigma_l + sigma_l^2)

Safety factor. This calculator reports yield strength divided by Von Mises stress, which is the ordinary mechanical-design check. Be careful not to read it as an ASME number. ASME Section VIII Division 1 sets its allowable stress as the lesser of UTS/3.5 and (2/3) x Sy, so its headline factor is against ultimate tensile strength, not yield. The two tests are not interchangeable, and for a ductile steel the ASME rule is usually the stricter of the two. Use the figure here for sizing and sanity-checking, and use the code itself for anything that has to be stamped.

Worked example. A 500 mm bore air receiver at 2 MPa (about 290 psi) with a 10 mm wall:

• Hoop stress = 2 x 250 / 10 = 50 MPa
• Longitudinal stress = 2 x 250 / 20 = 25 MPa
• Von Mises = sqrt(2500 - 1250 + 625) = 43.3 MPa
• Against mild steel at Sy = 250 MPa, the safety factor is 5.8

Note that Von Mises comes out below the hoop stress, not above it. Biaxial tension in the same sign is less damaging than the same stress applied alone, which surprises people the first time they see it. Sizing on hoop stress alone is therefore conservative, and that is exactly why the codes do it.

When thin-wall fails. If D/t < 20, use thick-wall (Lame) equations. The thin-wall formula underestimates the peak stress at the bore, and the error grows fast. Measured against the true Lame value, the membrane figure reads about 5% low at D/t = 20, 10% low at D/t = 10, and 19% low at D/t = 5. The calculator works out the Lame stress for you at every ratio and shows the gap.

There is a second thing the membrane formula hides at low D/t: the stress stops being uniform through the wall. It peaks at the bore and falls off toward the outside surface, so a thick vessel is only fully stressed on its inner skin. That is the reason gun barrels and high-pressure tubing are autofrettaged, which means over-pressurising them once during manufacture to yield the bore and leave it in residual compression when the pressure comes off.

For spherical pressure vessels: sigma = P x r / (2t) in all directions. That is equal biaxial tension, which is why spheres are the strongest shape per unit material.