Find hoop and axial stress in a pressurised cylinder or pipe.
A stress check is not a pressure vessel design. Vessels and piping follow a code such as ASME VIII or B31.3, which sets the allowable stress, joint efficiency and corrosion allowance.
Example Result
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Pressure inside a cylinder tries to split it along its length, and the wall resists with hoop stress. For a thin wall, σ_H = P × D ÷ (2 × t). Enter the pressure, the diameter and the wall thickness, and choose whether your diameter is inside, outside or mean. When the wall is too thick for that formula, the calculator uses the Lamé equation instead.
Enter the internal pressure.
Enter the diameter, and say whether it is inside, outside or mean.
Enter the wall thickness.
Add the yield strength for a simple margin.
Press Calculate. A thick wall switches to the Lamé equation by itself.
By O.E. — Partner, chairman of the board and CEO of a metal products manufacturer for 21 years; the company exports to 45 countries.
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Take a cylinder with a mean diameter of 300 mm, a 10 mm wall and 5 MPa inside.
Hoop stress is twice the axial stress. That is why a pipe that fails from pressure usually splits along its length, and why a long seam matters more than a round seam. In US units the formula is the same: 500 psi in a 12 in cylinder with a ⅜ in wall gives 8,000 psi.
The formula is exact only for a thin ring. At the usual limit of t = D ÷ 20, here is how each diameter compares with the exact Lamé peak, which is 10.03 × P:
| Diameter in the formula | Result | Error |
|---|---|---|
| Inside | 9.5 × P | −5.2% |
| Mean | 10.0 × P | −0.25% |
| Outside | 10.5 × P | +4.7% |
So the mean diameter is the accurate choice. Using the outside diameter errs on the safe side, which is why Barlow's formula for pipe uses it.
Stress is not even through a thick wall. It peaks at the inside surface:
σ_max = P × (r_o² + r_i²) ÷ (r_o² − r_i²)
| Cylinder | Pressure | Thin-wall result | Lamé peak | Thin-wall error |
|---|---|---|---|---|
| 300 ID, 320 OD | 5 MPa | 77.5 MPa | 77.6 MPa | −0.1% |
| 100 ID, 150 OD | 20 MPa | 50 MPa | 52 MPa | −3.8% |
| 60 ID, 140 OD | 10 MPa | 12.5 MPa | 14.5 MPa | −14% |
The thin-wall answer is low, so it is on the unsafe side. A very thick wall also gives little extra strength for the added metal, since the outer layers carry far less stress than the bore.
Yield strength divided by hoop stress gives a simple margin. Design codes are stricter and also cap the stress by tensile strength. ASME Section VIII Division 1 takes the lower of tensile strength ÷ 3.5 and two-thirds of yield strength, using values from Section II Part D.
For a steel with 70,000 psi tensile and 34,200 psi yield at temperature, that is the lower of 20,000 psi and 22,800 psi, so 20,000 psi.
A stress check is not a pressure vessel design. Vessels and piping follow a code such as ASME VIII or B31.3, which sets the allowable stress, joint efficiency and corrosion allowance. To size a wall to one of those formulas, use the pressure vessel wall thickness calculator. A pressure test or a failure releases stored energy, so have pressurised equipment designed and inspected by qualified people.
Pipe work often rearranges the same idea around the outside diameter: P = 2 × S × t ÷ D. A 4 in Schedule 40 pipe has an outside diameter of 4.5 in and a wall of 0.237 in. At S = 20,000 psi that gives 2 × 20,000 × 0.237 ÷ 4.5 = 2,107 psi. Check the wall for your pipe in the pipe schedule chart, and remember that mill tolerance and threads reduce it. Using the tensile strength in place of S gives a rough burst pressure, not a working pressure.
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