Pressure vessels · Structural
Hoop Stress Calculator
Size circumferential stress in thin-walled cylinders — enter pressure and geometry, pick a material, and watch the vessel diagram respond as safety margins appear.
Material properties
Result
Hoop stress
σₕ = P·Dm / (2t)
Enter pressure, outside diameter, and wall thickness — the vessel diagram tracks your inputs.
Thin-walled cylinder theory — formulas, assumptions, and design context.
Hoop stress (circumferential stress) acts around the circumference of a cylindrical pressure vessel. Internal pressure tries to expand the diameter, putting the wall in tension. In thin-walled cylinders, hoop stress is about twice the longitudinal stress — often the critical design parameter.
This calculator uses mean diameter Dm = outside diameter − wall thickness, consistent with common thin-wall pressure vessel practice.
- Hoop stress (σₕ)σₕ = (P × Dm) / (2 × t)Circumferential — typically the governing stress
- Longitudinal stress (σₗ)σₗ = (P × Dm) / (4 × t)Half of hoop stress in closed cylindrical shells
- Radial stress (inner) (σᵣ)σᵣ ≈ −P (at inner wall)Compressive at the bore; often small in thin walls
- Von Mises (σᵥₘ)σᵥₘ = √(σₕ² + σₗ² + σᵣ² − σₕσₗ − σₕσᵣ − σₗσᵣ)Combined stress for yield comparison
Valid when wall thickness is small compared to diameter:
- t/r ≤ 0.1 (radius basis)
- t/D ≤ 0.05 (diameter basis)
Beyond these limits, stress varies through the wall thickness — use Lamé thick-wall theory or FEA instead.
Pressure pipelines
Oil, gas, and water transmission
Storage tanks
Industrial fluid containment
Boilers
High-pressure steam generation
Gas cylinders
Compressed gas storage
Hydraulic cylinders
Power-system accumulators
Process vessels
Chemical reactors and columns
Compare von Mises stress to yield strength. Typical interpretation:
| Factor | Interpretation |
|---|---|
| < 1.0 | Unsafe |
| 1.0 – 1.5 | Marginal |
| 1.5 – 2.0 | Acceptable minimum |
| 2.0 – 4.0 | Typical design |
| > 4.0 | Conservative |
- Uniform internal pressure only
- Linear elastic material behaviour
- No stress concentrations (nozzles, welds, threads)
- No external loads or thermal stress
- Cylindrical geometry with closed ends (longitudinal stress included)
Worked example: Hoop stress in a pipe: P = 2 MPa, D = 1000 mm, t = 10 mm
- Mean diameter Dm = D − t = 1000 − 10 = 990 mm
- σₕ = P·Dm / (2t) = 2 × 990 / (2 × 10)
- Longitudinal σₗ = P·Dm / (4t) = half of the hoop stress
σₕ ≈ 99 MPa (σₗ ≈ 49.5 MPa)
What is hoop stress?
Hoop (circumferential) stress is the tensile stress that acts around the circumference of a cylindrical pressure vessel as internal pressure tries to expand its diameter. In a thin-walled cylinder it is about twice the longitudinal stress, so it usually governs the design.
Which diameter does the calculator use?
You enter the outside diameter, but the thin-wall formulas use the mean diameter Dm = D − t for accuracy. For genuinely thin walls the difference between outside, mean and inside diameters is small.
When is the thin-walled assumption valid?
Thin-wall theory applies when the wall thickness is no more than about 1/10 of the radius (t/r ≤ 0.1, i.e. t/D ≤ 0.05). Above that, stress varies significantly through the wall and you should use thick-wall (Lamé) theory instead.
How is the safety factor calculated?
When you pick a material, the safety factor is the material's yield strength divided by the von Mises stress. As a guide, 1.5–2 is acceptable for controlled applications and 2–4 is typical for general engineering.
Related calculator
For axial, bending, and combined stress in general components, try the Stress Calculator.