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Mechanical
Autofrettage pressure for a target bore compression
Pressure needed to leave a chosen compressive hoop stress C at the bore, found by inverting the elastic-unloading result.
- Ref
- MEC-78
- Rev
- v1
- Field
- Mechanical
- Result
- PA [MPa]
The elastic-unloading result σθ,res(a) = σY(1 − PA/PY) can be inverted. If the target is a compressive hoop stress of magnitude C at the bore, then −C = σY(1 − PA/PY), giving PA = PY(1 + C/σY). The plastic radius for that pressure can then be found with the bisection entry and checked to lie between a and b. The inversion is invalid if unloading yields, which needs C ≤ σY, or if the material law or axial state does not apply. It is not a fatigue-based optimisation rule.
Pressure needed to leave a chosen compressive hoop stress C at the bore, found by inverting the elastic-unloading result. It is a way to turn a stress target into a treatment pressure, not a fatigue optimum.
Calculator
- First-yield pressure PY112.5 MPa
- Fully plastic limit PL207.9 MPa
- Plastic zone radius that this pressure produces66.31 mm
- Plastic fraction of the wall, (c − a)/(b − a)0.3263
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.
- To leave 150 MPa of compression at the bore of a 50 mm bore, 100 mm outer radius cylinder in 300 MPa steel, the pressure is 168.8 MPa, 58.9 % of the way from first yield to the fully plastic limit.
- That pressure yields the wall out to 66.31 mm, a plastic fraction of 0.3263. The pressure is only valid if this lies inside the wall and unloading stays elastic.
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 | 93.75 | 618.8 | 525 | +10% | +6.67 % |
| a Inner radius (bore) | 10 to 150 mm | 5.96 | 222.8 | 216.8 | +10% | −7 % |
| b Outer radius | 60 to 300 mm | 68.75 | 218.8 | 150 | +10% | +5.79 % |
| C Target compressive hoop stress at the bore (magnitude) | 0 to 400 MPa | 112.5 | 262.5 | 150 | +10% | +3.33 % |
Sweep of Target compressive hoop stress at the bore (magnitude)
| Target compressive hoop stress at the bore (magnitude) (MPa) | Autofrettage pressure (MPa) |
|---|---|
| 0 | 112.5 |
| 50 | 131.3 |
| 100 | 150 |
| 150 | 168.8 |
| 200 | 187.5 |
| 250 | 206.3 |
| 300 | 225 |
| 350 | 243.8 |
| 400 | 262.5 |
Sweep one variable
Variables and how each one contributes
Autofrettage 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 PA through PY: a larger bore leaves a thinner wall that yields at lower pressure. Raising it by 5 mm (to 55 mm) moves the result by −7 %, from 168.8 to 156.9 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 PA through PY: a thicker wall yields at higher pressure. Raising it by 5 mm (to 105 mm) moves the result by +3.1 %, from 168.8 to 174 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 PA, but less than in proportion: PA = (σY + C)(1 − a²/b²)/2, so only the σY part of the sum grows. Raising it by 25 MPa (to 325 MPa) moves the result by +5.56 %, from 168.8 to 178.1 MPa.
Target compressive hoop stress at the bore (magnitude)
The compressive hoop stress wanted at the bore after unloading, entered as a positive magnitude in MPa. The slider runs from 0 to 400 MPa. Raises PA in direct proportion to C, by PY/σY for each MPa. Raising it by 25 MPa (to 175 MPa) moves the result by +5.56 %, from 168.8 to 178.1 MPa.
Worked example with the values above
- First-yield pressure: PY = 112.5 MPa.
- Bore factor: 1 + C/σY = 1 + 150/300 = 1.5.
- PA = 112.5 × 1.5 = 168.8 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.
- Elastic unloading, so C must not exceed σY.
Common mistakes
- Entering the target compression as a negative number: C is a magnitude.
- Treating the result as a fatigue-optimal pressure: it only meets the stated target.
- Using these cylinder formulas for a sphere, a non-uniform wall, nozzles, heads, welded or layered construction, significant axial restraint or large deformation. Those need an elastoplastic finite-element model of the real geometry and load path.
Statistics and uncertainty
The result is proportional to the first-yield pressure, so it inherits the scatter in the yield strength. A target compression close to σY leaves little room for error, because the margin to reverse yielding is then small.
Where it is used
- Choosing a treatment pressure to meet a stated bore compression.
- Comparing the pressure needed for different compression targets before checking the plastic zone.
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