Hoop stress (thin wall)
-P rm / tHoop stress at bore (Lamé)
-Von Mises stress at bore
-Safety factor
-Sy / von MisesMinimum wall thickness
-Yield pressure
-Bore reaches yieldAll results
| Inside radius ri / outside radius ro | - |
| Thickness to inside radius t / ri | - |
| Hoop stress, thin wall, mean radius | - |
| Longitudinal stress, thin wall | - |
| Hoop stress at outer surface (Lamé) | - |
| Radial stress at bore | - |
| Longitudinal stress (Lamé) | - |
| Thin-wall error against Lamé bore stress | - |
| Yield strength used | - |
| Thin-wall minimum thickness for the safety factor | - |
Method and formulas
Let ri and ro be the inside and outside radii, P the internal gauge pressure, t = ro − ri and rm = ri + t/2. These are classical elasticity results (Lamé 1852) found in any strength-of-materials textbook, so you can check them by hand.
| Quantity | Formula |
|---|---|
| Thin-wall cylinder, hoop | σh = P rm / t |
| Thin-wall cylinder, longitudinal (closed ends) | σl = P rm / (2 t) |
| Thin-wall sphere | σ = P rm / (2 t) in every direction |
| Thick cylinder, hoop at radius r | σh(r) = P ri² (1 + ro²/r²) / (ro² − ri²) |
| Thick cylinder, radial at radius r | σr(r) = P ri² (1 − ro²/r²) / (ro² − ri²), equal to −P at the bore |
| Thick cylinder, longitudinal | σl = P ri² / (ro² − ri²) for closed ends, 0 for open ends |
| Thick sphere, tangential at radius r | σt(r) = P ri³ (1 + ro³/(2r³)) / (ro³ − ri³) |
| Thick sphere, radial | σr(r) = P ri³ (1 − ro³/r³) / (ro³ − ri³) |
| Von Mises | σ' = √(½[(σh − σr)² + (σr − σl)² + (σl − σh)²]) |
| Safety factor | n = Sy / σ' at the bore, where the stress is highest |
The minimum wall thickness is found by increasing ro until the von Mises safety factor at the bore equals your target. Thick-wall vessels have a limit: however thick the wall, the von Mises stress at the bore of a cylinder never falls below about 1.73 P (1.5 P for a sphere), so a safety factor cannot be reached if Sy/n is below that. The calculator tells you when this happens. The yield pressure is the pressure at which the von Mises stress at the bore equals Sy. Yield strengths are typical handbook values; use the certified value for your material.
Worked example: 100 mm bore cylinder, 5 mm wall, 10 MPa
ri = 50 mm, ro = 55 mm, so ro² − ri² = 525 mm². Thin wall with the mean radius 52.5 mm: σh = 10 × 52.5 / 5 = 105.0 MPa. Lamé at the bore: σh = 10 × (3025 + 2500) / 525 = 105.2 MPa, radial stress −10 MPa, longitudinal 10 × 2500 / 525 = 47.6 MPa. Von Mises = √(½[115.2² + 57.6² + 57.6²]) = 99.8 MPa. A steel with Sy = 250 MPa gives a safety factor of about 2.5.
Questions
How do you calculate the wall thickness of a pressure vessel?
For a thin-walled cylinder the hoop stress is σ = P·r/t and the longitudinal stress (closed ends) is P·r/(2t), so the wall needed for an allowable stress σa is t = P·r/σa. For thicker walls the Lamé equations apply: the hoop stress at the bore is σ = P(ro² + ri²)/(ro² − ri²), which gives ro = ri·√((σa + P)/(σa − P)). Example: a 100 mm bore cylinder with a 5 mm wall at 10 MPa has a Lamé bore hoop stress of 105.2 MPa and a von Mises stress of 99.8 MPa, so a steel with 250 MPa yield has a safety factor of about 2.5.
When is a vessel thin-walled?
The thin-wall formulas assume the stress is uniform through the thickness. They are accurate to a few percent when the wall is less than about one tenth of the bore radius (t/ri < 0.1). Above that the Lamé thick-wall solution should be used, and this calculator always shows both so you can see the difference.
What is the difference between hoop stress and longitudinal stress?
Hoop (circumferential) stress acts around the circumference and tries to split the vessel along its length. Longitudinal (axial) stress acts along the axis and tries to pull the end caps off. In a closed thin cylinder the hoop stress is twice the longitudinal stress, which is why pressure pipes split along a seam rather than around it. A sphere carries P·r/(2t) in every direction, so for the same pressure and radius it needs half the wall of a cylinder.
Is this calculation the same as ASME Section VIII or EN 13445?
No. Pressure vessel codes add allowable stress rules, joint efficiency, corrosion allowance, minimum thickness, nozzle reinforcement, head shape factors and testing requirements, and require a qualified design. This calculator gives the underlying mechanics of a plain pressure-containing cylinder or sphere, useful for sizing tubes, hydraulic cylinders, accumulators and housings. It is not a code calculation and must not be used to certify a pressure vessel.
What is the yield pressure of a cylinder?
It is the internal pressure at which the von Mises stress at the bore reaches the yield strength, so yielding begins there. The calculator solves this exactly from the Lamé stresses. It is not the burst pressure: a ductile cylinder can carry more pressure after first yield, but design practice keeps working pressure well below it.
Does it include external pressure, fatigue or temperature?
No. Only static internal pressure is considered. External pressure can cause buckling at a much lower load, and pulsating pressure (hydraulic cylinders, pumps) makes fatigue the governing limit. Temperature changes the material strength. Check these separately.
Engineering aid for guidance. It is not a pressure vessel code calculation (ASME, EN 13445, PD 5500) and does not cover joint efficiency, corrosion allowance, heads and nozzles, external pressure, fatigue, temperature or fracture. Pressure equipment is regulated: a qualified person must design and certify it. Verify critical designs; you remain responsible for the design. Nothing you enter leaves your browser except through a share link you copy yourself.