Library / Mechanical / MEC-70
Mechanical
Fully plastic limit pressure of a thick cylinder
Pressure at which yielding has spread through the whole wall in the elastic–perfectly plastic Tresca model.
- Ref
- MEC-70
- Rev
- v1
- Field
- Mechanical
- Result
- PL [MPa]
PL is the pressure at which the plastic zone has spread to the outer surface, so that no elastic outer layer is left to hold the yielded material. In the elastic–perfectly plastic Tresca model it is also the pressure at which the wall collapses. It is the upper end of the autofrettage window, and it is not an allowable operating or processing pressure.
Pressure at which yielding has spread through the whole wall in the elastic–perfectly plastic Tresca model. It is the upper bound on any autofrettage pressure and is never a safe operating or processing pressure.
Calculator
- First-yield pressure for comparison112.5 MPa
- Ratio PL/PY1.848
- Radius ratio b/a2
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.
- For b/a = 2 and σY = 300 MPa the wall collapses at 207.9 MPa, 1.85 times the first-yield pressure. An autofrettage pressure must be chosen strictly between the two.
- Tripling the outer radius raises the limit only from 207.9 to 537.5 MPa, because the pressure grows with the logarithm of the radius ratio.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (MPa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| σY Uniaxial yield strength | 100 to 1,500 MPa | 69.31 | 1,040 | 970.4 | +10% | +10 % |
| a Inner radius (bore) | 10 to 150 mm | 4.027 | 690.8 | 686.7 | +10% | −13.8 % |
| b Outer radius | 60 to 300 mm | 54.7 | 537.5 | 482.8 | +10% | +13.8 % |
Sweep of Uniaxial yield strength
| Uniaxial yield strength (MPa) | Fully plastic limit pressure (MPa) |
|---|---|
| 100 | 69.31 |
| 275 | 190.6 |
| 450 | 311.9 |
| 625 | 433.2 |
| 800 | 554.5 |
| 975 | 675.8 |
| 1,150 | 797.1 |
| 1,325 | 918.4 |
| 1,500 | 1,040 |
Sweep one variable
Variables and how each one contributes
Fully plastic limit pressure
What the equation gives you. Everything else feeds it.
Inner radius (bore)
The inner (bore) radius of the cylinder, in mm, in the same length unit as b. It must be smaller than b. Lowers PL: a larger bore reduces the radius ratio b/a and so its logarithm. Raising it by 5 mm (to 55 mm) moves the result by −13.8 %, from 207.9 to 179.4 MPa.
Outer radius
The outer radius of the cylinder, in mm, in the same length unit as a. It must be larger than a. Raises PL logarithmically: each doubling of b/a adds the same amount, σY ln 2. Raising it by 5 mm (to 105 mm) moves the result by +7.04 %, from 207.9 to 222.6 MPa.
Uniaxial yield strength
The uniaxial yield strength of the material at the autofrettage temperature, in MPa. The model treats the material as elastic and perfectly plastic and uses the Tresca criterion, so the wall yields where the hoop stress minus the radial stress reaches this value. Raises PL in direct proportion. Raising it by 25 MPa (to 325 MPa) moves the result by +8.33 %, from 207.9 to 225.3 MPa.
Worked example with the values above
- Radius ratio: b/a = 100 / 50 = 2.
- Natural logarithm: ln(2) = 0.69315.
- PL = σY × ln(b/a) = 300 × 0.69315 = 207.9 MPa.
Assumptions and limits
- A long, uniform, single-piece thick cylinder with uniform internal pressure, no external pressure, small deformation and an initially stress-free isotropic material.
- Elastic, perfectly plastic behaviour with the Tresca criterion at constant temperature, and a hoop stress that is the largest principal stress (σθ ≥ σz ≥ σr). The axial stress must be checked separately: open ends, closed ends and axial restraint give different states.
- End regions, openings and attachments are ignored. This is a teaching and preliminary-analysis model, not a qualified manufacturing specification.
Common mistakes
- Treating PL as a pressure at which the vessel can be operated or processed.
- Forgetting that real materials strain-harden, so the true collapse pressure is higher than this model gives.
Statistics and uncertainty
PL is proportional to σY, so a ±5 % scatter in yield strength gives a ±5 % scatter in the limit. Real materials harden and the Bauschinger effect changes reversal, so measured collapse pressures differ from this idealised value.
Where it is used
- Setting the upper end of an autofrettage pressure window.
- A first estimate of the collapse pressure of a thick cylinder.
Maths you need
- Thick-walled cylinder: a wall too thick for the thin-wall formula
- Tresca criterion: yield when the largest stress difference reaches σY
- Natural logarithm: the power of e that gives a number