§1Thin cylinders
Cut the shell in imagination and balance what the pressure pushes against what the wall pulls. Two cuts, two stresses — and the hoop one is double.
Hoop governs — which is why pressurised tubes split along their length, never around it, and why a longitudinal weld carries twice the duty of a circumferential one (weld efficiency enters as a divisor on the allowable stress).
Ø600 mm receiver at 1.0 MPa, allowable 80 MPa: t = pr/σ = 1.0 × 300/80 = 3.75 mm; adding a 1 mm corrosion allowance and rounding, t = 5 mm, running at a comfortable 60 MPa hoop. Pressure equipment is code territory — the mechanics above is the skeleton the codes clothe with rules, allowances and inspection.
§2Thin spheres
A sphere carries the same pressure at half the cylinder’s wall (the receiver above would need only 1.9 mm) — the shape’s whole surface works in the efficient longitudinal mode. That efficiency is why gas storage trends spherical and why cylinder ends are dished rather than flat: a hemispherical head is a sphere doing a lid’s job.
Contents§3Thin or thick?
The thin formulas assume the stress is uniform through the wall. That holds while the wall is a small fraction of the diameter.
Working rule: treat the shell as thin while t < d/20. Beyond that the inner fibres carry visibly more than the outer — the thin formula under-reports the bore stress and Lamé takes over. Hydraulic cylinders, gun-drill bushes and high-pressure fittings live on the thick side almost by definition.
Contents§4Thick cylinders — Lamé
In a thick wall the hoop stress varies through the thickness, peaking at the bore. For internal pressure p on radii rᵢ (bore) and r₀ (outside):
Bore Ø80 mm at 35 MPa, allowable 110 MPa: r₀ = 40√(145/75) = 55.6 mm → wall 15.6 → use 16 mm (r₀ = 56: bore stress 107.9 MPa ✓). Note the formula’s warning: as p approaches σ the required r₀ runs to infinity — past that point no amount of wall helps, and the answers are stronger material, autofrettage or compound (shrink-fitted) construction.
§5Flat circular plates
A flat cover doesn’t stretch like a shell — it bends like a plate, and its stress grows with the square of the radius-to-thickness ratio.
Ø300 mm opening at 0.5 MPa, edges effectively fixed by the bolted flange, allowable 90 MPa: (r/t)² = 90/(0.75 × 0.5) = 240, so r/t = 15.5 and t = 150/15.5 = 9.7 → use 10 mm. A flat lid this size needs 10 mm where a dished end would need 2 — flatness is bought with thickness, which is the whole argument for dished ends.
§6Quick reference
The working core of the page on one card rack.
Thin cylinder
σ_h = pd/2t · σ_l = pd/4t
hoop governs, ×2
Thin sphere
σ = pd/4t
half the cylinder’s wall
Regime
thin while t < d/20
Thick (Lamé)
σ_bore = p(r₀²+rᵢ²)/(r₀²−rᵢ²)
r₀ = rᵢ√((σ+p)/(σ−p))
Flat plate
fixed: 0.75 p(r/t)²
supported: ≈1.24 p(r/t)²