Library / Mechanical / MEC-77
Mechanical
Reverse-yielding screening ratio at the bore
Ratio of the autofrettage pressure to twice the first-yield pressure.
- Ref
- MEC-77
- Rev
- v1
- Field
- Mechanical
- Result
- UR
During unloading the bore stress state moves from the peak (hoop σY − PA, radial −PA) back towards zero. Reverse yielding at the bore occurs if the radial–hoop Tresca difference exceeds σY on the way down, which in the ideal model happens when PA > 2PY. UR = PA/(2PY) is therefore a screen: 1 or below passes this single check. It is a screening condition at one point and radial–hoop pair, not proof that unloading is acceptable everywhere. The full Tresca measure, including the axial stress, must also be checked. Real Bauschinger behaviour can cause reverse yielding earlier.
Ratio of the autofrettage pressure to twice the first-yield pressure. Unloading is purely elastic at the bore only while the radial–hoop Tresca measure stays within yield, which requires this ratio to be 1 or less.
Calculator
- First-yield pressure PY112.5 MPa
- Largest pressure that passes, 2PY225 MPa
- Pressure margin remaining, 2PY − PA47.56 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.
- A 177.4 MPa treatment on a cylinder with PY = 112.5 MPa has UR = 0.7886, below 1, so the bore-surface screen passes. This does not replace the full unloading check.
- The limit pressure is 2PY = 225 MPa, so there is 47.56 MPa of margin. For comparison the fully plastic limit is 207.9 MPa.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| a Inner radius (bore) | 10 to 150 mm | 0.5974 | 22.33 | 21.73 | +10% | +7.53 % |
| PA Peak autofrettage pressure | 50 to 600 MPa | 0.2222 | 2.667 | 2.444 | +10% | +10 % |
| σY Uniaxial yield strength | 100 to 1,500 MPa | 0.1577 | 2.366 | 2.208 | +10% | −9.09 % |
| b Outer radius | 60 to 300 mm | 0.6084 | 1.936 | 1.327 | +10% | −5.47 % |
Sweep of Peak autofrettage pressure
| Peak autofrettage pressure (MPa) | Reverse-yielding screening ratio |
|---|---|
| 50 | 0.2222 |
| 118.8 | 0.5278 |
| 187.5 | 0.8333 |
| 256.3 | 1.139 |
| 325 | 1.444 |
| 393.8 | 1.75 |
| 462.5 | 2.056 |
| 531.3 | 2.361 |
| 600 | 2.667 |
Sweep one variable
Variables and how each one contributes
Reverse-yielding screening ratio
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. Raises UR: a larger bore lowers the first-yield pressure that UR is measured against. Raising it by 5 mm (to 55 mm) moves the result by +7.53 %, from 0.7886 to 0.848.
Outer radius
The outer radius of the cylinder, in mm, in the same length unit as a. It must be larger than a. Lowers UR: a thicker wall raises the first-yield pressure. Raising it by 5 mm (to 105 mm) moves the result by −3.01 %, from 0.7886 to 0.7649.
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. Lowers UR in inverse proportion at a fixed pressure: a stronger material has a larger first-yield pressure. Raising it by 25 MPa (to 325 MPa) moves the result by −7.69 %, from 0.7886 to 0.728.
Peak autofrettage pressure
The peak internal pressure applied during autofrettage, in MPa. For a partial treatment it must lie between the first-yield pressure and the fully plastic limit. Raises UR in direct proportion. Raising it by 5 MPa (to 182.4 MPa) moves the result by +2.82 %, from 0.7886 to 0.8109.
Worked example with the values above
- First-yield pressure: PY = (300/2)(1 − 2,500/10,000) = 112.5 MPa, so 2PY = 225 MPa.
- Divide the autofrettage pressure by it: 177.4 / 225.
- UR = 0.7886, so this check passes.
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 a value at or below 1 as proof of acceptable unloading everywhere in the wall.
- Ignoring the axial stress in the full Tresca measure.
- 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
Because the screen is a simple ratio, it inherits the scatter of the yield strength (which cancels in part, since PA and PY both depend on it) and of the pressure actually applied. A value close to 1 should be treated as marginal, not as a pass.
Where it is used
- A quick check before relying on elastic residual-stress formulas.
- Deciding whether an elastoplastic unloading analysis is needed.
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
- Bauschinger effect: earlier yielding on load reversal