Thin-Wall Hoop Stress⚠ unverified
Mechanical / Pressure Vessels · Hoop (circumferential) stress in a thin-walled cylinder
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| P | P | Pa | 2000000.0 | Internal pressure |
| r | r | m | 0.5 | Radius |
| t | t | m | 0.01 | Wall thickness |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| sigma | σ_θ | Pa | Hoop stress |
The science & history
Understanding the Parameters
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Internal pressure $P$ — the gauge pressure (above ambient); it pushes outward on the wall. Hoop stress is linear in $P$.
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Radius $r$ — larger vessels are more highly stressed for the same wall and pressure ($\sigma_\theta \propto r$): a big tank needs a proportionally thicker wall. Use the inner or mean radius; for thin walls the distinction is negligible.
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Wall thickness $t$ — the load-bearing thickness. Thickness is inversely proportional to stress, so it is the design variable: invert to $t = Pr/\sigma_{allow}$ for sizing (Required Thickness Thin).
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Thin-wall validity — the membrane assumption (uniform stress through the wall) holds for $t/r \lesssim 0.1$; thicker walls have a stress gradient and need Lamé thick-cylinder theory (Thick-Wall Cylinders).
Derivation (Approaching a Proof)
Cut the cylinder with a longitudinal plane through its axis, isolating a half-shell of length $L$. The pressure acts on the projected diametral area $2rL$, giving an outward force
$$F_P = P\,(2 r L).$$
This is resisted by hoop stress acting on the two wall edges exposed by the cut, each of area $tL$:
$$F_\sigma = \sigma_\theta\,(2\,t L).$$
Force balance $F_P = F_\sigma$ gives $P(2rL) = \sigma_\theta(2tL)$, and the lengths and factors of 2 cancel:
$$\sigma_\theta = \frac{P r}{t}.$$
The projected-area trick (pressure on a curved surface equals pressure times the flat projected area) is what makes this a one-line result.
Dimensional check. $\sigma_\theta = \dfrac{P r}{t} = \dfrac{\text{Pa}\cdot\text{m}}{\text{m}} = \text{Pa}$ — a stress, as required.
History and Development
Hoop stress analysis dates to the early study of boilers and cannon; the factor-of-two relationship between hoop and longitudinal stress, and the failure of cylinders by longitudinal splitting, were understood by the 19th century after catastrophic boiler explosions drove the creation of the ASME Boiler & Pressure Vessel Code (1914). The thin-wall membrane result remains the starting point of every vessel and pipe design, refined by joint efficiency, corrosion allowance, and the code's $0.6P$ thick-wall correction.
Related Concepts: Thin-Wall Longitudinal Stress, Spherical Vessel Stress, Required Thickness Thin, Burst Pressure, Pressure Vessel Design, Thick-Wall Cylinders
Notes: $\sigma_\theta = 2\sigma_l$ — cylinders split lengthwise. Uses gauge pressure; $\sigma_\theta \propto r$. Thin-wall valid for $t/r \lesssim 0.1$ (else Thick-Wall Cylinders). Sphere stress is half (Spherical Vessel Stress).