Thin-Wall Longitudinal Stress⚠ unverified
Mechanical / Pressure Vessels · Longitudinal 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 | σl | Pa | Longitudinal stress |
The science & history
Understanding the Parameters
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Internal pressure $P$ — acts on the closed ends; the resulting axial force is carried by the cylindrical wall in tension. Linear in $P$.
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Radius $r$ — the end-cap force scales with the enclosed area ($\pi r^2$) while the resisting wall area scales with circumference ($2\pi r$), so the stress scales with $r$, as for hoop stress.
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Wall thickness $t$ — inversely proportional to stress; but because $\sigma_l$ is only half of $\sigma_\theta$, the hoop stress governs the cylindrical wall thickness, not the longitudinal.
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Where it governs — longitudinal stress matters for circumferential (girth) welds and for combined loading (pressure plus axial force or bending from supports/wind), where it adds to other axial stresses.
Derivation (Approaching a Proof)
Cut the cylinder with a transverse plane perpendicular to its axis, isolating one end. The internal pressure acts on the circular end area $\pi r^2$, giving an axial force
$$F_P = P\,(\pi r^2).$$
This is resisted by the longitudinal stress acting on the ring-shaped wall cross-section, area $\approx 2\pi r\,t$ (circumference times thickness):
$$F_\sigma = \sigma_l\,(2\pi r t).$$
Balancing $F_P = F_\sigma$: $P\pi r^2 = \sigma_l\,2\pi r t$, and cancelling $\pi r$:
$$\sigma_l = \frac{P r}{2 t} = \frac{\sigma_\theta}{2}.$$
The factor of 2 versus hoop stress comes from the geometry: the end cap "sees" the area $\pi r^2$ resisted by a perimeter $2\pi r$, whereas the longitudinal split sees area $2rL$ resisted by $2tL$.
Dimensional check. $\sigma_l = \dfrac{P r}{2 t} = \dfrac{\text{Pa}\cdot\text{m}}{\text{m}} = \text{Pa}$ — a stress, as required.
History and Development
The 2:1 hoop-to-longitudinal ratio is a classical membrane-theory result, central to pressure-vessel and piping codes since the 19th century. It dictates practical detail: spiral-welded pipe orients the seam to balance the two stresses, and code joint-efficiency factors distinguish longitudinal seams (which see the higher hoop stress) from circumferential seams (which see the lower longitudinal stress). Combined with the longitudinal stress from wind, weight, and support reactions, it forms the axial side of a full vessel stress analysis.
Related Concepts: Thin-Wall Hoop Stress, Spherical Vessel Stress, Principal Stresses, Von Mises Stress, Pressure Vessel Design, Required Thickness Thin
Notes: $\sigma_l = \tfrac12\sigma_\theta$ — hoop governs the cylindrical wall. Longitudinal stress governs girth welds and combines with axial/bending loads. Biaxial state ($\sigma_\theta$, $\sigma_l$, $\sigma_r \approx 0$) → check with Von Mises Stress.