Head Thickness Hemispherical⚠ unverified
Mechanical / Pressure Vessels · Compute the required thickness of a hemispherical head
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| P | P | Pa | 1.0 | Internal pressure |
| r | r | m | 1.0 | Head radius |
| 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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Radius $r$ — the crown (spherical) radius; the formula uses radius, giving a thickness that is about half the ellipsoidal head's (which uses diameter with the same $2SE$ denominator).
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The $2SE$ denominator — reflects the sphere's efficient biaxial stress: for the same $P$, $r$, and $S$, a hemisphere is the thinnest head. It has no knuckle, so it avoids the local bending that penalises ellipsoidal and torispherical heads.
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Joint efficiency $E$ and safety factor $SF$ — as elsewhere, the weld derating (dimensionless, not modulus) and an extra allowable-stress divisor.
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Thickness discontinuity — because the hemisphere is so much thinner than the cylinder, joining them creates a thickness mismatch and a local bending discontinuity at the junction; codes require a tapered transition to manage it.
Derivation (Approaching a Proof)
The membrane stress in a thin hemisphere is $\sigma = Pr/2t$ (Spherical Vessel Stress), uniform in all directions. Requiring it not to exceed the effective allowable $SE/SF$:
$$\frac{P r}{2 t} \le \frac{S E}{SF} \;\Longrightarrow\; t \ge \frac{P r}{2\,SE/SF} = \frac{P r}{2SE/SF}.$$
Applying the code's mean-surface correction adds the $-0.2P$ term to the denominator:
$$t = \frac{P r}{\dfrac{2 S E}{SF} - 0.2\,P}.$$
The single factor of 2 in the denominator — absent from the cylindrical shell formula — is the direct consequence of the sphere carrying half the cylinder's governing stress.
Dimensional check. $t = \dfrac{P r}{2SE/SF - 0.2P} = \dfrac{\text{Pa}\cdot\text{m}}{\text{Pa}} = \text{m}$ — a length, as required.
History and Development
Hemispherical heads give the lowest possible wall stress and are used where material savings justify the higher forming cost — cryogenic tanks, high-pressure reactors, and gas spheres. ASME BPVC Section VIII (UG-32) codifies the thickness, and the well-known shell-to-head thickness discontinuity at the junction (the hemisphere being ~half the shell thickness) is a classic discontinuity-stress problem requiring a tapered weld transition.
Related Concepts: Spherical Vessel Stress, Head Thickness Ellipsoidal, Required Thickness Thin, Pressure Vessel Design, Thin-Wall Hoop Stress
Notes: ASME UG-32. Uses radius $r$ (ellipsoidal uses diameter) → ~half the ellipsoidal thickness, thinnest head. $E$ = joint efficiency, not modulus. No knuckle, but a shell-junction thickness discontinuity needs a tapered transition.