Hoop Stress Calculator
Find the hoop and longitudinal stress in a thin-walled pressure vessel from pressure, radius, and wall thickness.
Thin-wall assumption holds: r/t = 50 (≳ 10). Longitudinal stress is half the hoop stress.
Hoop (circumferential) stress in a thin-walled pressure vessel is σ_hoop = p·r ÷ t, and longitudinal stress is σ_long = p·r ÷ 2t. For p = 2 MPa, r = 500 mm, t = 10 mm, σ_hoop = 2×500 ÷ 10 = 100 MPa and σ_long = 50 MPa. Hoop stress is always twice the longitudinal stress.
Why hoop stress governs pressure-vessel design
Internal pressure pushes outward on every part of a cylinder’s wall. That outward push creates two distinct membrane stresses. The hoop stress (σ_hoop), also called circumferential stress, acts around the circumference and tries to burst the cylinder open along its length. The longitudinal stress (σ_long) acts along the axis and tries to pull the end caps off. For a cylinder the hoop stress is exactly twice the longitudinal stress, so it is the hoop direction that sets the limit on how much pressure the vessel can hold.
This is why a sausage splits along its length before its ends blow off, and why a burst water pipe tears open in a long axial seam rather than snapping in two. The wall is working twice as hard in the hoop direction, so failure follows the line of highest stress — a crack running along the pipe, perpendicular to the hoop stress that drives it.
p = internal gauge pressure, r = inside radius (d = 2r), t = wall thickness. With p in MPa and r, t in mm, both stresses come out in MPa. Valid for thin walls where r/t ≳ 10.
Worked example
A cylindrical steel vessel holds gas at 2 MPa gauge pressure. Its inside radius is 500 mm and the wall is 10 mm thick. Find the hoop and longitudinal stress, and confirm the thin-wall assumption is valid.
- 1 Gather p, r, and t in consistent units. Use p = 2 MPa, r = 500 mm, t = 10 mm. Keeping pressure in MPa and lengths in mm makes the mm units cancel, so both stresses come out directly in MPa.
- 2 If you have the diameter, halve it. The radius is what enters the formula: r = d ÷ 2. A 1000 mm inside diameter gives r = 500 mm.
- 3 Compute the hoop stress. σ_hoop = p·r ÷ t = 2 × 500 ÷ 10 = 100 MPa. This is the circumferential stress that tries to split the cylinder along its length.
- 4 Compute the longitudinal stress. σ_long = p·r ÷ 2t = 2 × 500 ÷ (2 × 10) = 50 MPa — exactly half the hoop stress.
- 5 Check the thin-wall ratio. r ÷ t = 500 ÷ 10 = 50, which is comfortably above 10, so the thin-wall (membrane) formulas apply and the stress is essentially uniform through the wall.
Hoop vs longitudinal stress
The two membrane stresses in a thin-walled cylinder, plus the ratio that tells you when these formulas are valid.
| Quantity | Formula | Notes |
|---|---|---|
| Hoop (circumferential) stress | σ_hoop = p·r ÷ t = p·d ÷ 2t | Largest stress; acts around the circumference and drives axial (lengthwise) cracks |
| Longitudinal (axial) stress | σ_long = p·r ÷ 2t | Exactly half the hoop stress; acts along the axis |
| Stress ratio | σ_hoop ÷ σ_long = 2 | Hoop stress is always twice the longitudinal stress in a cylinder |
| Thin-wall validity | r ÷ t ≳ 10 | Membrane formulas assume a thin wall; below ~10 use thick-wall (Lamé) equations |
| Spherical vessel | σ = p·r ÷ 2t | A sphere carries equal stress in all directions — half a cylinder’s hoop stress |
When the thin-wall assumption breaks down
These formulas treat the wall as a thin membrane in which the stress is uniform from the inside surface to the outside surface. That approximation is accurate while the radius-to-thickness ratio r/t is roughly 10 or more. Below that the wall is “thick”: the stress varies significantly across it, peaking on the inside surface, and the simple membrane values understate the true maximum. For thick-walled cylinders you need the Lamé equations, which give the radial and hoop stress as functions of position through the wall.
A few other assumptions are worth remembering. The pressure must be gauge pressure — the internal pressure above the surrounding atmosphere, since that difference is what the wall resists. The formulas describe the membrane stress well away from discontinuities; near end caps, nozzles, welds, and supports, local stress concentrations can push the real stress well above these values, which is exactly where real vessels tend to crack. Always compare the hoop stress against the material strength with an appropriate factor of safety before trusting a design.