External Pressure Critical⚠ unverified
Mechanical / Pressure Vessels · Compute the critical external buckling pressure for a cylinder
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| Pcr | Pcr | Pa | 1.0 | Critical pressure placeholder argument, in pascals (Pa). Not used in the returned expression |
| E | E | Pa | 1.0 | Modulus of elasticity of the material |
| t | t | m | 1.0 | Wall thickness |
| L | L | m | 1.0 | Cylinder length |
| r | r | m | 1.0 | Cylinder radius |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | Pcr | Pa | Critical external pressure, in pascals (Pa). Returns 0.0 when ``r`` is not positive |
The science & history
Understanding the Parameters
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Elastic modulus $E$ — buckling is an elastic-stability phenomenon, so $E$ (not strength) sets the critical pressure. A stiffer material resists collapse; the material could be nowhere near yielding when it buckles.
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Thickness/diameter ratio $t/D$ — the dominant term, entering as $(t/D)^{2.5}$: thin shells are dramatically weaker against external pressure. Halving $t/D$ cuts $P_{cr}$ by a factor of ~5.7.
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Length/diameter ratio $L/D$ — longer unsupported cylinders buckle more easily ($P_{cr} \propto 1/L$ in the intermediate range). Stiffening rings shorten the effective $L$ and are the standard fix, which is why external-pressure vessels are ribbed.
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$(1-\nu^2)^{0.75}$ — the Poisson factor from shell theory ($\approx 0.93$ for steel). The registry's $(1-0.2)$ approximates this group.
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Long vs short shells — the formula covers the intermediate range; very long cylinders approach the length-independent ring-buckling limit $P_{cr} \approx 2E(t/D)^3/(1-\nu^2)$, and very short ones are governed by the end supports.
Derivation (Approaching a Proof)
External-pressure collapse is an elastic buckling (eigenvalue) problem: below $P_{cr}$ the round shell is stable; at $P_{cr}$ it snaps into an ovalised or multi-lobe mode. The full solution (von Mises shell-buckling theory) minimises the buckling pressure over the number of circumferential lobes $n$ and the axial half-wave, for a cylinder of length $L$, diameter $D$, thickness $t$:
$$P_{cr} = \frac{E}{(1-\nu^2)}\,f\!\left(n, \frac{L}{D}, \frac{t}{D}\right).$$
Windenburg and Trilling (1934) fitted this exact but cumbersome result with a compact closed form valid in the practical intermediate-length range:
$$P_{cr} = \frac{2.42\,E\,(t/D)^{2.5}}{(1-\nu^2)^{0.75}\left[\dfrac{L}{D} - 0.45\sqrt{t/D}\right]},$$
which underlies the ASME external-pressure charts (Section VIII, UG-28). The $(t/D)^{2.5}$ and $1/L$ dependences are the signatures of thin-shell buckling, quite different from the strength-limited internal- pressure formulas.
Dimensional check. With $t/D$, $L/D$, $\sqrt{t/D}$ all dimensionless, $P_{cr} = \dfrac{E\cdot(\text{–})} {(\text{–})} = \text{Pa}$ — a pressure, as required.
History and Development
External-pressure buckling of shells was solved by von Mises and reduced to the practical Windenburg–Trilling formula (US Experimental Model Basin, 1934) for submarine and vessel design. It is the basis of the ASME BPVC external-pressure design method (the "Factor A/B" chart procedure of UG-28) and of submarine pressure-hull design, where stiffening rings are added specifically to raise $P_{cr}$ by reducing the unsupported length $L$.
Related Concepts: Euler Buckling Load, Pressure Vessel Design, Required Thickness Thin, Thin-Wall Hoop Stress, Thick-Wall Cylinders
Notes: Buckling (stability), not strength — set by $E$, collapses below yield. $P_{cr} \propto (t/D)^{2.5}$, $\propto 1/L$ → thin/long shells weak; add stiffening rings to cut effective $L$. Registry: $P_{cr}$ input unused; $(1-0.2)$ approximates $(1-\nu^2)^{0.75}\approx0.93$. ASME UG-28 basis.