Library / Mechanical / MEC-44
Mechanical
Axial stress in a closed-end thick cylinder
Uniform axial stress in a thick cylinder with closed ends, when the wall carries the whole pressure thrust and no other axial load.
- Ref
- MEC-44
- Rev
- v1
- Field
- Mechanical
- Result
- σz [MPa]
For closed ends the end caps pass the pressure thrust to the wall, which carries it as a uniform axial stress equal to the Lamé constant A. Open ends carry no axial load, so σz = 0 for an open-ended cylinder with no applied axial load. A cylinder held between rigid supports needs a separate compatibility calculation. a and b are the internal and external radii in mm and pi and po the pressures in MPa, entered as positive values.
Uniform axial stress in a thick cylinder with closed ends, when the wall carries the whole pressure thrust and no other axial load. It equals the Lamé constant A.
Calculator
- End thrust π(pia² − pob²)78.54 kN
- Wall cross-sectional area π(b² − a²)9,817 mm²
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 default gives 8 MPa: less than a third of the 26 MPa peak hoop stress at the bore, so the axial stress is not what sets the wall size.
- σz is the same at every radius and equals the average of the hoop and radial stresses: (26 + (−10))/2 = 8 MPa at the bore and (16 + 0)/2 = 8 MPa at the outside.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (MPa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| po External pressure | 0 to 100 MPa | −172 | 8 | 180 | +10% | 0 % |
| a Internal radius | 10 to 100 mm | 0.181 | 180 | 179.9 | +10% | +45.4 % |
| pi Internal pressure | 1 to 100 MPa | 0.8 | 80 | 79.2 | +10% | +10 % |
| b External radius | 60 to 400 mm | 0.1587 | 22.73 | 22.57 | +10% | −27.4 % |
Sweep of External radius
| External radius (mm) | Axial stress (MPa) |
|---|---|
| 60 | 22.73 |
| 102.5 | 3.123 |
| 145 | 1.35 |
| 187.5 | 0.7656 |
| 230 | 0.496 |
| 272.5 | 0.3484 |
| 315 | 0.2585 |
| 357.5 | 0.1995 |
| 400 | 0.1587 |
Sweep one variable
Variables and how each one contributes
Axial stress
What the equation gives you. Everything else feeds it.
Internal pressure
The pressure acting on the inside (bore) surface, in MPa, entered as a positive value. Directly proportional while the external pressure is zero: a 10 % rise in pi raises every stress in the wall by 10 %.
External pressure
The pressure acting on the outside surface and the outer end area, in MPa. Lowers the axial stress, because it pushes on the outer end the other way: with po = 2 MPa the axial stress falls from 8 to 4.4 MPa.
Internal radius
The internal radius (bore radius), in mm. Very strong: the end thrust grows with a² while the wall area shrinks. A 10 % rise in a (to 55 mm) raises σz by about 45 %, to 11.6 MPa.
External radius
The external radius, in mm. A larger b gives more wall area for the same thrust: a 10 % rise in b (to 82.5 mm) lowers σz by about 27 %, to 5.8 MPa.
Worked example with the values above
- End thrust: F = π(pia² − pob²) = π × (10 × 50² − 0 × 75²) = 78,540 N.
- Wall area: π(b² − a²) = π × (75² − 50²) = 9,817 mm².
- σz = 78,540 / 9,817 = 8 MPa, which is the Lamé constant A.
Assumptions and limits
- Closed ends that carry the pressure thrust, with no other axial load, at a section away from the ends where the stress is uniform.
- A homogeneous, isotropic, linearly elastic material.
Common mistakes
- Using it for open-ended pipes, where σz = 0 unless an axial load is applied.
- Taking the thrust over the outer area when the pressure acts only inside. With po = 0 the thrust is piπa², not piπb².
Statistics and uncertainty
σz is linear in the pressures. The wall area b² − a² is a difference of squares, so the radii are amplified: at these proportions a 1 % error in a gives about a 4 % error in σz, and the effect grows as the wall gets thinner. In practice the end conditions are less certain than the radii: a real vessel may not carry the whole thrust through the wall.
Where it is used
- Axial stress in a capped pressure vessel or gun chamber.
- The third principal stress when applying the Tresca or von Mises criterion to a closed-end cylinder.
Maths you need
- Force over area: stress = F/A
- Area of a ring: π(b² − a²)