Hoop Stress Calculator
Compute the circumferential (hoop) stress in a thin-walled pressure vessel — or solve backwards for pressure, radius, or wall thickness. Includes longitudinal stress, maximum shear, von Mises, thin-wall validity check (r/t ≥ 10), and a safety-factor check against typical yield strengths.
Calculator
All calculations run in your browser. Math is done in SI units (Pa, m); unit labels are never mixed.
Formulas
σ_hoop = P · r / t · σ_long = P · r / (2t) · τ_max = σ_hoop / 2 · σ_vM = σ_hoop · √3/2Hoop stress is the tensile stress pulling the wall apart around the circumference. The inverse forms solve for any variable: P = σ·t/r, r = σ·t/P, t = P·r/σ. The longitudinal stress assumes closed ends (end caps or flanges); the von Mises value follows from the biaxial state (σ_hoop, σ_hoop/2).
Worked example (verified)
P = 1 MPa, inner radius r = 100 mm, wall t = 5 mm (r/t = 20, thin-wall OK):
| Quantity | Value |
|---|---|
| Hoop stress σ = P·r/t | 20 MPa |
| Longitudinal stress | 10 MPa |
| Maximum shear τ_max | 10 MPa |
| von Mises | 17.3205 MPa |
| Safety factor vs A36 steel (250 MPa), Tresca | 12.5 |
Imperial check: 100 psi × 6 in / 0.25 in = 2,400 psi = 16.5474 MPa — identical physics, converted with 1 psi = 6,894.757293168 Pa exactly.
Thin-wall limit vs Lamé (thick wall)
The membrane formulas assume stress is uniform through the wall (r/t ≥ 10). Below that, the exact Lamé solution for internal pressure gives the peak hoop stress at the inner surface: σ = P·(ro² + ri²)/(ro² − ri²).
| Case | Thin-wall P·r/t | Lamé (exact) | Thin-wall error |
|---|---|---|---|
| P=10 MPa, ri=50 mm, wall=5 mm (r/t=10) | 100 MPa | 105.24 MPa | ≈5% low |
| P=10 MPa, ri=50 mm, wall=10 mm (r/t=5) | 50 MPa | 55.45 MPa | ≈10% low |
The thin-wall formula always reads low versus Lamé — unconservative for thick walls. This calculator flags any r/t below 10.
Frequently asked questions
What is hoop stress?
Hoop stress (circumferential stress) is the tensile stress in the wall of a pressurized cylinder acting around its circumference. For a thin-walled vessel it is σ = P·r/t, where P is internal pressure, r is the inner radius, and t is the wall thickness. Example: P = 1 MPa, r = 100 mm, t = 5 mm gives σ = 20 MPa.
Why is hoop stress twice the longitudinal stress?
Cut the cylinder lengthwise: pressure acting on the projected area (2·r·L) must be balanced by the stress in the two wall cross-sections (2·t·L), so σ_hoop = P·r/t. Cut it across: pressure on the end cap (π·r²) is balanced by the annular wall area (≈2·π·r·t), giving σ_long = P·r/(2t) — exactly half. That is why the hoop direction always governs in a thin-walled vessel.
When does the thin-wall formula apply?
The thin-wall formulas assume the stress is nearly uniform through the wall, which holds when r/t ≥ 10 (inner radius over thickness). At exactly r/t = 10 the thin-wall result reads about 5% low versus the exact Lamé solution (100 MPa vs 105.24 MPa for a 10 MPa, 50 mm bore, 5 mm wall case). This calculator warns you whenever r/t drops below 10.
What if the wall is thick (r/t below 10)?
Use the Lamé thick-wall equations. For internal pressure only, the peak hoop stress (at the inner surface) is σ = P·(ro² + ri²)/(ro² − ri²). Example: P = 10 MPa, ri = 50 mm, ro = 60 mm gives 55.45 MPa, while the thin-wall formula gives only 50 MPa — about 10% low and unconservative. For certified designs, thick-wall analysis follows ASME BPVC Section VIII.
What is the maximum shear stress in the wall?
On the outer surface of the vessel the radial stress is zero, so the two in-plane principal stresses are σ_hoop and σ_long = σ_hoop/2. The maximum shear is half their difference from zero: τ_max = σ_hoop/2. Example: with σ_hoop = 20 MPa, τ_max = 10 MPa. Under the Tresca criterion, yielding begins when τ_max reaches half the tensile yield strength.
How do I avoid mixing up MPa, Pa, and psi?
Convert everything to consistent units before multiplying: 1 MPa = 1,000 kPa = 1,000,000 Pa, 1 bar = 100,000 Pa, and 1 psi = 6,894.757293168 Pa exactly. This calculator does all math in SI (Pa, m) and only relabels the result, so MPa and Pa can never be mixed up. For lengths: 1 in = 25.4 mm exactly.
How is the safety factor calculated?
The safety factor compares the governing stress against the material's yield strength. This calculator shows both the Tresca factor (yield / σ_hoop) and the von Mises factor (yield / 0.866·σ_hoop). Example: 20 MPa hoop stress in A36 steel (250 MPa yield) gives a Tresca factor of 12.5. The built-in yield values (250, 205, 276, 70 MPa) are typical textbook figures for comparison only — never use them as design allowables.
Should I use the inner or the outer radius?
The thin-wall formula uses the inner radius, where the pressure acts. Using the outer radius instead overestimates the stress slightly, which is conservative but inconsistent with the published formula. At r/t ≥ 10 the difference between inner, mean, and outer radius is under 10% anyway. Enter the inner (bore) radius for the textbook result.
Can I use this to design a real pressure vessel?
No. This is an educational calculator for the thin-wall membrane formulas only. Real pressure vessels are designed to codes such as ASME BPVC Section VIII (or EN 13445), which add weld joint efficiencies, corrosion allowances, external loads, buckling checks, and inspection regimes — none of which are modeled here. Any vessel that can injure people or damage property must be designed and stamped by a qualified engineer.
What about open-ended pipes with no end caps?
The longitudinal stress σ_long = P·r/(2t) assumes closed ends (caps, flanges, or a boiler). In an open-ended pipe the end caps are missing, so there is no axial pressure load and σ_long ≈ 0 — but the hoop stress σ = P·r/t is unchanged, because it comes from the side wall equilibrium alone. If your pipe has open ends, read the hoop and shear results and ignore the longitudinal line.
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