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Hoop Stress in a Thin-Walled Cylinder

Also known as circumferential stress · pressure vessel stress

σh=pd2t\sigma_{h} = \frac{p d}{2 t}

Worked example: 1 m vessel, 10 mm wall, 2 MPa → 100 MPa hoop — press Try an example to run it live, then adjust anything.

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See also
Longitudinal Stress in a Thin-Walled Cylinder — the axial stress in the same shell, exactly half thisMembrane Stress in a Thin-Walled Sphere (σ = pr/2t) — the same vessel built as a sphere carries half this stress, which is why large storage vessels are sphericalLamé Hoop Stress in a Thick-Walled Cylinder — what to use once the wall is too thick to pretend the stress is uniform through itASME Required Wall Thickness (t = PR/(SE − 0.6P)) — the code version of the same check, with a joint efficiency and the 0.6P correction
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Hoop Stress in a Thin-Walled Cylinder explained

pdtσh

Slice a pressurised pipe lengthwise and the pressure acting on the projected area d × L must be held by two wall strips of area t × L: p·d·L = 2·σ·t·L, so σ = pd/2t. Edme Mariotte worked this out in the 1680s while sorting out burst pipes in the waterworks feeding the fountains at Versailles. A 1 m diameter vessel at 2 MPa with a 10 mm wall carries σ = 2 × 10⁶ × 1 ÷ (2 × 0.01) = 100 MPa. In field units, 150 psi in a 24-inch line needs t = 150 × 24 ÷ (2 × 15 000) = 0.12 inch of wall before any corrosion or mill-tolerance allowance.

Hoop stress is twice the longitudinal stress, which is why an overpressured pipe splits along its length rather than snapping in two, and why a boiled sausage always bursts lengthwise. The 1954 de Havilland Comet crashes made the point at altitude: repeated pressurisation cycles drove fatigue cracks from the corners of the fuselage cutouts, where the hoop stress concentrated far above the nominal value — the investigation rewrote how the world thinks about fatigue and stress concentration. Two limits on the formula: it is a thin-wall result, valid when d/t is greater than about 20, and it uses gauge pressure, the difference across the wall.

Hoop Stress in a Thin-Walled Cylinder formula

σh=pd2t\sigma_{h} = \frac{p d}{2 t}
Where
  • σh\sigma_{h}= Hoop stress (kPa)
  • pp= Internal gauge pressure (kPa)
  • dd= Internal diameter (mm)
  • tt= Wall thickness (mm)

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