Engineering
Thin-Walled Pressure Vessel Stress formulas: Hoop (Pr/t) vs Axial (Pr/2t)
Calculate pressure-only membrane stresses in an ideal thin-walled closed cylinder and distinguish a yield-to-hoop ratio from a design safety assessment.
State the closed-cylinder membrane assumptions
For a uniform thin-walled closed cylinder under internal-minus-external pressure P, away from ends and openings, hoop membrane stress is approximately P*r/t and axial membrane stress P*r/(2*t). The 2:1 relation applies to these pressure-only components, not arbitrary combined loading or open-ended pipes.
P and yield strength use MPa; radius and thickness both use meters here. The input gate r/t>=10 is a rough thin-wall screening convention, not proof of adequate accuracy or design compliance. External-pressure buckling, thick walls, joints, fatigue, corrosion and local stresses are outside this example.
The ratio yield strength/hoop stress is only a comparison with one stress component. It does not perform a combined-stress yield criterion or determine a safe vessel rating. Zero pressure gives zero membrane stress and an undefined finite yield-to-hoop ratio. Zero thickness is invalid, never zero stress.
| Vessel Design Parameter | Cell / Formula | Cylindrical Tank Input | Stress / Safety Output |
|---|---|---|---|
| Internal Pressure P & Radius r | B2 = 2.4 MPa, B3 = 0.50 m | Wall thickness B4 t = 0.012 m | r / t = 41.67 (passes screening threshold) |
| Circumferential Hoop Stress | (B2 * B3) / B4 | (2.4 * 0.50) / 0.012 | 100.00 MPa (hoop component) |
| Longitudinal Axial Stress | (B2 * B3) / (2 * B4) | (2.4 * 0.50) / 0.024 | 50.00 MPa (Half of hoop) |
| Yield Strength S_y & FoS | B5 = 250 MPa -> B5 / sigma_h | 250 / 100 (or 1.25 at 4.8 MPa) | Yield/hoop ratio = 2.50 |
Pressure-only hoop membrane stress in MPa
When P and S_y are entered in MPa and r and t share the same length unit (meters or mm), stress outputs directly in MPa.
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(OR(B2<0,B3<=0,B4<=0,B5<=0),"Invalid inputs",IF(B3/B4<10,"Outside thin-wall model",ROUND((B2*B3)/B4,2))))Yield-strength-to-hoop-stress ratio
The ratio uses the unrounded hoop stress and is not a complete factor-of-safety assessment.
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(OR(B2<0,B3<=0,B4<=0,B5<=0),"Invalid inputs",IF(B3/B4<10,"Outside thin-wall model",IF(B2=0,"Undefined (zero pressure)",ROUND(B5 / ((B2 * B3) / B4), 2)))))Axial membrane stress for closed ends
This pressure-only axial component follows from the end-cap force; additional axial loading is not included.
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(OR(B2<0,B3<=0,B4<=0,B5<=0),"Invalid inputs",IF(B3/B4<10,"Outside thin-wall model",ROUND((B2 * B3) / (2 * B4), 2))))Try the example
Use the original values or change them to test your understanding.