Library / Mechanical / MEC-47
Mechanical
Thin-walled cylinder hoop stress
Hoop stress in a cylinder whose wall is thin compared with its radius, under internal gauge pressure.
- Ref
- MEC-47
- Rev
- v1
- Field
- Mechanical
- Result
- σθ [MPa]
p is the internal gauge pressure with negligible external pressure, a the internal radius (so di = 2a) and t the wall thickness, all in the units shown. The formula follows from the Lamé result by setting b = a + t and neglecting higher powers of t/a. It is also a force balance on half the cylinder: the pressure over a width 2a per unit length against two walls, each carrying σt. The radial stress varies from −p to zero and is small beside the hoop stress when t/a is small.
Hoop stress in a cylinder whose wall is thin compared with its radius, under internal gauge pressure. It is twice the axial stress of a closed-end cylinder.
Calculator
- Thickness-to-radius ratio t/a0.02
- Exact Lamé bore hoop stress (b = a + t)101 MPa
- Axial stress with closed ends, pa/(2t)50 MPa
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The hoop stress is 100 MPa, twice the 50 MPa axial stress, so a closed pipe under pressure splits along its length rather than round its circumference.
- Check validity with t/a. At the defaults the exact bore stress is 101.0 MPa, 1 % above this value. At t/a = 0.1 the formula is about 5 % low, and at t/a = 0.5 (the thick-wall example) it is 23 % too low.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (MPa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| p Internal gauge pressure | 0.1 to 20 MPa | 5 | 1,000 | 995 | +10% | +10 % |
| t Wall thickness | 1 to 50 mm | 20 | 1,000 | 980 | +10% | −9.09 % |
| a Internal radius | 100 to 2,000 mm | 20 | 400 | 380 | +10% | +10 % |
Sweep of Wall thickness
| Wall thickness (mm) | Hoop stress (MPa) |
|---|---|
| 1 | 1,000 |
| 7.125 | 140.4 |
| 13.25 | 75.47 |
| 19.38 | 51.61 |
| 25.5 | 39.22 |
| 31.63 | 31.62 |
| 37.75 | 26.49 |
| 43.88 | 22.79 |
| 50 | 20 |
Sweep one variable
Variables and how each one contributes
Hoop stress
What the equation gives you. Everything else feeds it.
Internal gauge pressure
The internal gauge pressure, in MPa. Directly proportional: a 10 % rise in p raises the hoop stress by 10 %.
Internal radius
The internal radius, in mm. For the diameter formula use di = 2a. Directly proportional: a 10 % rise in a (to 550 mm) raises the hoop stress by 10 %, to 110 MPa. Bigger vessels need thicker walls for the same pressure.
Wall thickness
The wall thickness, in mm. Inversely proportional: a 10 % rise in t (to 11 mm) lowers the hoop stress by 9 %, to 90.9 MPa. Thickness is the main design lever.
Worked example with the values above
- Check the wall is thin: t/a = 10 / 500 = 0.02, below about 0.1, so the thin-wall formula applies.
- Force per unit length: p a = 2 MPa × 500 mm = 1,000 N/mm.
- σθ = 1,000 / 10 = 100 MPa.
Assumptions and limits
- A thin wall, with t/a well under 0.1, so that the hoop stress is uniform through the thickness.
- Internal gauge pressure with negligible external pressure, in a long cylinder away from the ends, supports and nozzles, where local stresses are higher.
Common mistakes
- Mixing up the diameter and radius forms: pa/t uses the radius, while pdi/(2t) uses the diameter.
- Using the formula for a thick wall, where it underestimates the bore stress.
Statistics and uncertainty
σθ = pa/t, so errors in p and a pass through proportionally and thickness is inverse. Thickness is usually the least certain input: a manufacturing tolerance of ±10 % on t gives about ±10 % on the stress, and corrosion removes metal during service. Codes therefore add a corrosion allowance and a factor of safety on top of this nominal stress.
Where it is used
- Wall thickness of boilers, gas bottles, pipes and tanks.
- Estimating the pressure at which a thin vessel will yield, given the yield stress of the material.
Maths you need
- Force balance: pressure force on half the cylinder equals the wall force
- Pressure: force per unit area