Pipes, boilers, scuba tanks and balloons hold pressure by tension in their walls. In a cylinder the hoop stress is twice the longitudinal stress, which is why overpressurised pipes split lengthwise.
How to use the Hoop Stress Calculator
- Vessel shape
- Inner radius r
- Wall thickness t
- Internal pressure P
- Joint efficiency η
- Young’s modulus E
- Poisson’s ratio ν
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Choose the vessel shape, enter the inner radius, wall thickness, internal pressure and joint efficiency (1 for seamless), and the material’s E and ν for deformations. Valid for r/t > 10.
Inputs explained
| Field | What to enter |
|---|---|
| Vessel shape | Choose one: Cylinder, Sphere |
| Inner radius r | Units: mm, cm, m, in, µm |
| Wall thickness t | Units: mm, cm, m, in, µm |
| Internal pressure P | Units: MPa, kPa, Pa, GPa, psi, ksi |
| Joint efficiency η | A number |
| Young’s modulus E | Units: GPa, MPa, Pa, psi, Mpsi |
| Poisson’s ratio ν | A number |
What each result means
- Longitudinal stress σ_l = [Internal pressure P]×[Inner radius r]÷(2×[Wall thickness t]×[Joint efficiency η])
- Radial stress at the inner wall = -[Internal pressure P]
- r/t ratio (thin wall if > 10) = [Inner radius r]÷[Wall thickness t]
Example values
How the main result changes when Inner radius r changes and everything else stays at the example values above:
| Inner radius r (mm) | Hoop stress σ_h (MPa) |
|---|---|
| 250 | 50 |
| 375 | 75 |
| 500 | 100 |
| 625 | 125 |
| 750 | 150 |
| 1,000 | 200 |
Formula
Cylinder: σ_h = P·r/(t·η), σ_l = P·r/(2t·η) · Sphere: σ = P·r/(2t·η)
Δd/d = (σ_h − ν·σ_l)/E, ΔV/V = Pr(5 − 4ν)/(2tE) (cylinder) or 3Pr(1 − ν)/(2tE) (sphere)
Worked example
Inputs — Vessel shape: Cylinder; Inner radius r: 500 mm; Wall thickness t: 10 mm; Internal pressure P: 2 MPa; Joint efficiency η: 1; Young's modulus E: 200 GPa; Poisson's ratio ν: 0.3.
Results — Hoop stress σ_h = 100 MPa; Longitudinal stress σ_l = 50 MPa; Radial stress at the inner wall = -2 MPa; Change in diameter = 0.425 mm; Relative change in volume = 0.095 %; r/t ratio (thin wall if > 10) = 50.
Key terms
- Hoop stress
- Circumferential tension in a pressurised wall.
Frequently asked questions
Why do pipes burst lengthwise?
The hoop stress is twice the longitudinal stress.
When is the thin-wall formula valid?
When the radius is more than about 10 times the wall thickness.
What is joint efficiency?
The relative strength of a welded or riveted seam (1 for seamless).
Why are pressure vessels spherical?
A sphere has half the wall stress of a cylinder of the same radius.