Library / Mechanical / MEC-48
Mechanical
Thin-walled cylinder axial stress
Axial stress in a thin-walled cylinder with closed ends carrying the pressure thrust.
- Ref
- MEC-48
- Rev
- v1
- Field
- Mechanical
- Result
- σz [MPa]
With closed ends the pressure acts on the end area πa² and the thin wall, of area about 2πat, carries the thrust: σz = pπa²/(2πat) = pa/(2t). p is the internal gauge pressure, a the internal radius (di = 2a) and t the wall thickness. For open ends with no axial load, σz = 0. It is half the thin-wall hoop stress, which is why the hoop stress sets the wall thickness.
Axial stress in a thin-walled cylinder with closed ends carrying the pressure thrust. It is half the hoop stress. For open ends with no axial load it is zero.
Calculator
- Pressure thrust on the end, pπa²1,571 kN
- Hoop stress pa/t, twice the axial stress100 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 axial stress of 50 MPa is exactly half the 100 MPa hoop stress, so the hoop stress sets the design thickness and the axial stress is a secondary check.
- The end thrust is large: about 1571 kN at the defaults, carried by a wall area of only about 31 400 mm². The end caps, flanges and bolts must carry the same force.
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 | 2.5 | 500 | 497.5 | +10% | +10 % |
| t Wall thickness | 1 to 50 mm | 10 | 500 | 490 | +10% | −9.09 % |
| a Internal radius | 100 to 2,000 mm | 10 | 200 | 190 | +10% | +10 % |
Sweep of Wall thickness
| Wall thickness (mm) | Axial stress (MPa) |
|---|---|
| 1 | 500 |
| 7.125 | 70.18 |
| 13.25 | 37.74 |
| 19.38 | 25.81 |
| 25.5 | 19.61 |
| 31.63 | 15.81 |
| 37.75 | 13.25 |
| 43.88 | 11.4 |
| 50 | 10 |
Sweep one variable
Variables and how each one contributes
Axial 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 axial stress by 10 %.
Internal radius
The internal radius, in mm. Directly proportional: a 10 % rise in a (to 550 mm) raises the axial stress by 10 %, to 55 MPa, since the end area grows as a² while the wall grows as a.
Wall thickness
The wall thickness, in mm. Inversely proportional: a 10 % rise in t (to 11 mm) lowers the axial stress by 9 %, to 45.5 MPa.
Worked example with the values above
- End thrust: F = pπa² = 2 × π × 500² = 1,571,000 N.
- Wall area: 2πat = 2π × 500 × 10 = 31,420 mm².
- σz = 1,571,000 / 31,420 = 50 MPa, which equals pa/(2t).
Assumptions and limits
- A thin wall (t/a well under 0.1) with closed ends that carry the pressure thrust and no other axial load.
- A section away from the ends and nozzles, with uniform stress through the wall thickness.
Common mistakes
- Applying it to an open-ended pipe, which has no axial stress unless an axial load acts on it.
- Forgetting the factor 2 and quoting the hoop stress as the axial stress.
Statistics and uncertainty
σz = pa/(2t) has the same sensitivity as the hoop stress: errors in p and a pass through proportionally and t enters inversely, so ±10 % wall thickness gives about ±10 % in σz. In practice the pressure often has peaks above the nominal value, which matters more than the small errors in the dimensions.
Where it is used
- Sizing end caps, flanges and bolts for the end thrust of a pressure vessel.
- Axial stress in tanks and gas bottles, as an input to a Tresca or von Mises check.
Maths you need
- Force balance: end thrust equals wall force
- Area of a thin ring: 2πat