Library / Mechanical / MEC-76
Mechanical
Residual hoop stress at the bore after autofrettage
Hoop stress at the inside surface once the autofrettage pressure has been removed and the wall has unloaded elastically.
- Ref
- MEC-76
- Rev
- v1
- Field
- Mechanical
- Result
- σθ,res(a) [MPa]
With the bore at r = a the unloading result simplifies: the peak hoop stress there is σY − PA and the Lamé response to PA is PA(b² + a²)/(b² − a²), leaving σθ,res(a) = σY − 2PAb²/(b² − a²) = σY(1 − PA/PY). It is negative (compression) whenever PA exceeds the first-yield pressure PY. It is valid only if the unloading stays elastic, which at the bore requires PA ≤ 2PY.
Hoop stress at the inside surface once the autofrettage pressure has been removed and the wall has unloaded elastically. It is negative (compressive) whenever the autofrettage pressure exceeds the first-yield pressure.
Calculator
- First-yield pressure PY112.5 MPa
- Ratio PA/PY1.577
- Compression at the bore as a share of the yield strength57.73 %
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.
- With PA = 177.4 MPa against a first-yield pressure of 112.5 MPa, the bore is left at −173.2 MPa, a compression of 57.7 % of the yield strength.
- The bore residual stress passes through zero when PA equals PY and becomes more compressive as the pressure rises. Beyond 2PY (225 MPa here) the formula cannot be trusted, because reverse yielding sets in.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (MPa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| a Inner radius (bore) | 10 to 150 mm | −13,100 | −58.47 | 13,040 | +10% | +20.6 % |
| PA Peak autofrettage pressure | 50 to 600 MPa | −1,300 | 166.7 | 1,467 | +10% | +27.3 % |
| σY Uniaxial yield strength | 100 to 1,500 MPa | −373.2 | 1,027 | 1,400 | +10% | −17.3 % |
| b Outer radius | 60 to 300 mm | −861.4 | −65.02 | 796.4 | +10% | −14.9 % |
Sweep of Peak autofrettage pressure
| Peak autofrettage pressure (MPa) | Residual hoop stress at the bore (MPa) |
|---|---|
| 50 | 166.7 |
| 118.8 | −16.67 |
| 187.5 | −200 |
| 256.3 | −383.3 |
| 325 | −566.7 |
| 393.8 | −750 |
| 462.5 | −933.3 |
| 531.3 | −1,117 |
| 600 | −1,300 |
Sweep one variable
Variables and how each one contributes
Residual hoop stress at the bore
What the equation gives you. Everything else feeds it.
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 the residual stress one for one at a fixed pressure. In practice a stronger material is treated at a higher pressure, so check PA/PY as well. Raising it by 25 MPa (to 325 MPa) moves the result by −14.4 %, from −173.2 to −148.2 MPa.
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. Lowers the residual stress, making the bore more compressive: each extra MPa of pressure gives exactly 2b²/(b² − a²) MPa more compression. Raising it by 5 MPa (to 182.4 MPa) moves the result by +7.7 %, from −173.2 to −186.5 MPa.
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 the residual stress (more compression) at a fixed pressure, because the factor 2b²/(b² − a²) grows as the bore widens relative to the wall. Raising it by 5 mm (to 55 mm) moves the result by +20.6 %, from −173.2 to −208.8 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 the residual stress (less compression), because the factor 2b²/(b² − a²) falls towards 2 as the wall gets thicker. Raising it by 5 mm (to 105 mm) moves the result by −8.21 %, from −173.2 to −159 MPa.
Worked example with the values above
- Elastic term: 2PAb²/(b² − a²) = 2 × 177.4 × 10,000 / 7,500 = 473.2 MPa.
- Subtract from the yield strength: 300 − 473.2.
- σθ,res(a) = −173.2 MPa. The same value follows from σY(1 − PA/PY) = −173.2 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 at the bore, PA ≤ 2PY.
Common mistakes
- Applying it when PA is more than 2PY, where reverse yielding invalidates the formula.
- Reading the compression as fatigue life: a favourable stress is not a life prediction.
Statistics and uncertainty
The bore residual stress depends on the yield strength and on the pressure that was actually reached, and the two partly offset each other in the ratio PA/PY. Pressure measurement accuracy therefore matters as much as the material data.
Where it is used
- The headline figure of an autofrettage treatment: the compression carried by the bore.
- Comparing treatment pressures against the compression they give.
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