Head Thickness Ellipsoidal⚠ unverified
Mechanical / Pressure Vessels · Compute the required thickness of a 2:1 ellipsoidal head
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| P | P | Pa | 1.0 | Internal pressure |
| D | D | m | 1.0 | Head diameter |
| S | S | Pa | 1.0 | Allowable stress of the material |
| E | E | — | 1.0 | Joint efficiency factor, dimensionless. Default is 1.0 |
| SF | SF | — | 1.0 | Safety factor applied to the allowable stress, dimensionless. Default is 1.0 |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | t | m | Required head thickness, in metres (m). Returns 0.0 when ``S * E / SF`` does not exceed ``0.1 * P`` |
The science & history
Understanding the Parameters
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Diameter $D$ — the head spans the full vessel diameter; note this formula uses $D$ (like a cylinder's hoop formula with the factor of 2), whereas the hemispherical-head formula uses the radius.
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The factor $2SE$ — a 2:1 ellipsoidal head with the standard geometry factor $K = 1$ carries stress comparable to a cylinder's longitudinal direction, so its allowable term is $2SE$ (twice the cylindrical shell's), making the head thinner than the shell it caps.
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Joint efficiency $E$ — the weld derating (dimensionless), not modulus; and $SF$ an extra divisor.
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The $0.2P$ term — the thick-wall/mean-surface correction for the ellipsoidal geometry (smaller than the cylinder's $0.6P$ because the head is less highly stressed).
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Knuckle stresses — the simple formula covers the membrane region; the knuckle (the tight-radius transition to the cylinder) sees local bending and compression not captured here, which is why very thin heads can buckle at the knuckle and codes add checks for it.
Derivation (Approaching a Proof)
An ellipsoidal head's membrane stresses vary from crown to knuckle; ASME condenses the analysis into an equivalent-cylinder form using a shape factor $K$. For the standard 2:1 head, $K = 1$, and the maximum membrane stress is comparable to the longitudinal stress of a cylinder of the same diameter. Requiring that stress not exceed the effective allowable $SE/SF$ and inverting for thickness, with the mean-surface correction giving the $0.2P$ denominator term:
$$t = \frac{P D\,K}{\dfrac{2 S E}{SF} - 0.2\,P} \xrightarrow{K=1} \frac{P D}{\dfrac{2 S E}{SF} - 0.2\,P}.$$
The $2SE$ in the denominator (versus $SE$ for the cylindrical shell) is why an ellipsoidal head is thinner than the shell it closes.
Dimensional check. $t = \dfrac{P D}{2SE/SF - 0.2P} = \dfrac{\text{Pa}\cdot\text{m}}{\text{Pa}} = \text{m}$ — a length, as required.
History and Development
The 2:1 ellipsoidal head is the industry-standard closure, standardised in ASME BPVC Section VIII (UG-32) because it approximates the stress efficiency of a hemisphere while being far shallower and cheaper to form. Its formula sits between the flat-head (bending-dominated, thick) and hemispherical-head (membrane-dominated, thin) cases. Knuckle-region stress concentration and buckling under internal or external pressure are handled by supplementary code rules beyond this membrane sizing.
Related Concepts: Head Thickness Hemispherical, Required Thickness Thin, Spherical Vessel Stress, Pressure Vessel Design, Thin-Wall Hoop Stress
Notes: ASME UG-32, 2:1 head ($K=1$). Uses diameter $D$ (hemispherical uses radius). $E$ = joint efficiency, not modulus. $2SE$ term makes the head thinner than the shell. Membrane only — knuckle bending/buckling need separate checks.