Required Thickness Thin⚠ unverified
Mechanical / Pressure Vessels · Compute the minimum wall thickness for a thin-walled cylinder
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| P | P | Pa | 1.0 | Internal pressure |
| r | r | m | 1.0 | Cylinder 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 | Minimum required wall thickness, in metres (m). Returns 0.0 when ``S * E / SF`` does not exceed ``P`` |
The science & history
Understanding the Parameters
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Allowable stress $S$ — the code maximum allowable stress, already derived from the material's yield and ultimate with the code's built-in safety margins. $S/SF$ applies any additional factor.
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Joint efficiency $E$ — a derating (0.7–1.0, dimensionless — this is not Young's modulus) for welded seams: fully radiographed butt welds get $E = 1.0$, spot-examined less, so a lower-quality weld demands a thicker wall. It multiplies the allowable stress.
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Safety factor $SF$ — an extra divisor on $SE$ beyond the code's own margins; at $SF = 1$ the formula is the standard ASME expression.
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The $0.6P$ term — a correction that shifts the reference from inner to roughly mean radius, extending the thin-wall formula's accuracy to moderate thickness. It makes the required thickness slightly smaller than the naïve $Pr/(SE)$ and matches the exact thick-wall solution better.
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Blow-up condition — if $SE/SF \le 0.6P$ the denominator is non-positive: the pressure is too high for a thin-wall design and a thick-wall vessel is required.
Derivation (Approaching a Proof)
Start from the thin-wall hoop stress $\sigma_\theta = Pr/t$ (Thin-Wall Hoop Stress) and require it not to exceed the effective allowable $SE/SF$:
$$\frac{P r}{t} \le \frac{S E}{SF} \;\Longrightarrow\; t \ge \frac{P r}{SE/SF}.$$
The ASME code refines this by referencing the stress to the mean radius rather than the inner radius. For a thin wall the mean radius is $r + t/2$; carrying the algebra through, the $t/2$ correction converts to the empirical $-0.6P$ in the denominator (ASME uses $0.6$ for the circumferential-stress formula):
$$t = \frac{P r}{\dfrac{SE}{SF} - 0.6\,P}.$$
At low pressure ($0.6P \ll SE$) this reduces to the simple $Pr/(SE/SF)$; the correction only matters as $P$ approaches the allowable.
Dimensional check. $t = \dfrac{P r}{SE/SF - 0.6P} = \dfrac{\text{Pa}\cdot\text{m}}{\text{Pa}} = \text{m}$ — a length ($E$, $SF$, $0.6$ are dimensionless), as required.
History and Development
The $t = PR/(SE - 0.6P)$ formula is the workhorse of ASME BPVC Section VIII Division 1 (UG-27), descended from the boiler codes that followed 19th-century boiler explosions. The joint efficiency concept codifies decades of weld-inspection practice, and the $0.6P$ correction (with $0.4P$ for the longitudinal-stress check) bridges thin- and thick-wall theory. Real designs add a corrosion allowance to this computed $t$.
Related Concepts: Thin-Wall Hoop Stress, Head Thickness Ellipsoidal, Head Thickness Hemispherical, Pressure Vessel Design, Burst Pressure, Thick-Wall Cylinders
Notes: ASME UG-27 (hoop governing). $E$ = joint efficiency (0.7–1.0), not Young's modulus. $0.6P$ extends thin-wall accuracy (mean-radius correction). Add corrosion allowance to $t$. Denominator $\le 0$ → pressure too high for thin-wall design.