Thin-Wall Pressure Vessel Stress Calculator

Understanding the thin-wall pressure vessel stress calculator — concept and principle

Pipes, tanks, boilers, gas cylinders and even soda cans hold pressure by stretching their walls. This calculator gives the two membrane stresses in a thin-walled cylinder, the hoop (circumferential) stress and the longitudinal (axial) stress, plus the stress in a thin sphere, and checks whether the thin-wall assumption is reasonable for your dimensions.

Hoop stress 62.5 MPa is twice the 31.25 MPa longitudinal stress.

How it is calculated

Hoop stress: σ_h = P × D ÷ (2t).
Longitudinal stress: σ_L = P × D ÷ (4t).
Sphere: σ = P × D ÷ (4t), the same in every direction.
Validity: thin-wall theory is accepted when D/t > 20 (equivalently r/t > 10); otherwise use Lamé's thick-wall equations.

P is in MPa and D, t in mm, so stresses come out in MPa.

The principle

Cut a cylinder of length L in half along its axis. Pressure pushes the two halves apart with force P × D × L, and two wall strips of area t × L hold them together, so 2σ_h tL = PDL, giving PD/(2t). Now cut it across: pressure acts on the end area πD²/4, resisted by the wall ring of area about πDt, giving PD/(4t). The hoop stress is therefore twice the longitudinal stress, which is why overpressured pipes and sausages split lengthwise. A sphere has the hoop situation in every direction but only the end-cap geometry, so its stress equals the longitudinal value; for the same material and pressure it needs half the wall thickness of a cylinder.

Worked example

Inputs: P = 2 MPa (20 bar), D = 500 mm, t = 8 mm.
Hoop: 2 × 500 ÷ (2 × 8) = 1,000 ÷ 16 = 62.5 MPa.
Longitudinal: 1,000 ÷ 32 = 31.25 MPa.
Sphere: also 31.25 MPa.
D/t = 500 ÷ 8 = 62.5 > 20, so thin-wall theory is valid: yes.

Each point is the calculator re-run with only “Wall thickness (mm)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Wall thickness (mm)	Hoop stress (MPa)
2	250
4	125
6	83.33333333
8	62.5
10	50
12	41.66666667
16	31.25

How to read the result

Compare the hoop stress, the larger one, with the allowable stress of the material and weld joint. Using the mean diameter (D + t = 508 mm) gives a slightly higher hoop stress, 63.5 MPa, which is why design codes add correction terms to the inner-radius formula.

Common mistakes

Using bar or psi without converting (1 MPa = 10 bar ≈ 145 psi).
Using radius in place of diameter, which halves the stress.
Ignoring corrosion allowance, weld efficiency and the extra stresses at heads, nozzles and supports.

Keep in mind: These are membrane stresses for an ideal thin shell under internal pressure only. Pressure vessels are safety-critical and regulated; they must be designed to the applicable code (for example ASME BPVC Section VIII or EN 13445) and checked by a qualified engineer.

Frequently asked questions

Why do pipes burst along their length?

The hoop stress, which pulls the wall apart around the circumference, is twice the longitudinal stress, so cracks open along the axis.

How do I find the required wall thickness?

Rearranging the hoop formula gives t = PD/(2σ_allow), before code corrections, corrosion allowance and weld efficiency.

What changes for external pressure?

The wall is in compression and can buckle, so external-pressure design uses buckling rules, not these formulas alone.

Sources: ASME Boiler and Pressure Vessel Code, Section VIII, Division 1 · R. C. Hibbeler, Mechanics of Materials, Pearson

Checked 8 Nov 2026. Standard textbook formulas in SI units, computed in your browser. For learning and estimates — real designs must follow your local codes and be checked by a qualified engineer.

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