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Mechanical
Residual radial stress at radius r after autofrettage
Radial stress left in the wall after unloading.
- Ref
- MEC-75
- Rev
- v1
- Field
- Mechanical
- Result
- σr,res [MPa]
The residual radial stress is the peak radial stress minus the elastic Lamé radial response to the autofrettage pressure, σr,res = σr,A − [PAa²/(b² − a²)](1 − b²/r²). In the yielded zone σr,A = −PA + σY ln(r/a); in the ring it is (σYc²/2b²)(1 − b²/r²). With no pressure left it must be zero at the bore and at the outside, so the result is a useful check on the whole calculation. It is small compared with the residual hoop stress.
Radial stress left in the wall after unloading. It is zero at both surfaces, as it must be with no pressure applied, and small compared with the residual hoop stress.
Calculator
- Peak radial stress at this radius, σr,A−76.5 MPa
- Elastic unloading term, PAa²/(b² − a²) × (1 − b²/r²)−61.56 MPa
- Residual radial stress at the bore (must be zero)0 MPa
- Residual radial stress at the outside (must be zero)0 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.
- At 70 mm the residual radial stress is −14.94 MPa. It is zero at both surfaces, as it must be with no pressure applied.
- That is small beside the residual hoop stress of −173.2 MPa at the bore: the hoop stress dominates the benefit and the risk.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (MPa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| b Outer radius | 60 to 300 mm | −121.8 | −1.664 | 120.1 | +10% | +49.7 % |
| σY Uniaxial yield strength | 100 to 1,500 MPa | −74.69 | −4.98 | 69.71 | +10% | +10 % |
| f Target plastic fraction of the wall | 0.05 to 0.95 | −34.74 | −0.256 | 34.48 | +10% | +18.3 % |
| a Inner radius (bore) | 10 to 150 mm | −26.98 | −4.215 | 22.76 | +10% | −15.1 % |
| r Radius where the stress is wanted | 10 to 300 mm | −16.52 | −0.9015 | 15.62 | +10% | −34 % |
Sweep of Radius where the stress is wanted
| Radius where the stress is wanted (mm) | Residual radial stress at radius r (MPa) |
|---|---|
| 10 | — |
| 46.25 | — |
| 82.5 | −6.735 |
| 118.8 | — |
| 155 | — |
| 191.3 | — |
| 227.5 | — |
| 263.8 | — |
| 300 | — |
Sweep one variable
Variables and how each one contributes
Residual radial stress at radius r
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. Changes the result through the pressure and the Lamé response; the sign of the effect depends on r. Raising it by 5 mm (to 55 mm) moves the result by −15.1 %, from −14.94 to −12.69 MPa.
Outer radius
The outer radius of the cylinder, in mm, in the same length unit as a. It must be larger than a. Changes the result through the pressure and the Lamé response; the sign of the effect depends on r. Raising it by 5 mm (to 105 mm) moves the result by +25.1 %, from −14.94 to −18.69 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. Scales every term, so the residual stress changes in direct proportion at a fixed fraction f. Raising it by 25 MPa (to 325 MPa) moves the result by +8.33 %, from −14.94 to −16.18 MPa.
Target plastic fraction of the wall
The fraction of the wall thickness (b − a) that is yielded at peak pressure, a plain number between 0 and 1. A value of 0 means yielding only just begins and 1 means the whole wall has yielded. The slider runs from 0.05 to 0.95. Changes the stress through the pressure and the position of the front. Raising it by 0.05 (to 0.45) moves the result by +22.7 %, from −14.94 to −18.32 MPa.
Radius where the stress is wanted
The radius at which the stress is wanted, in mm, anywhere in the wall from the bore (a) to the outside (b). Moves the point through the wall; the stress is zero at the two surfaces and largest in between. Raising it by 5 mm (to 75 mm) moves the result by −25.3 %, from −14.94 to −11.16 MPa.
Worked example with the values above
- Peak radial stress at r = 70 mm (yielded zone, interface at 70 mm): σr,A = −76.5 MPa.
- Elastic unloading term: PAa²/(b² − a²) = 59.15 MPa, multiplied by 1 − b²/r² = −1.041, gives −61.56 MPa.
- σr,res = −76.5 − (−61.56) = −14.94 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.
- Unloading is purely elastic.
Common mistakes
- Expecting a non-zero radial stress at the surfaces: it must be zero with no pressure.
- Confusing the radial result with the hoop result: the two differ greatly in size.
Statistics and uncertainty
Like the hoop result, the radial residual stress is a difference of two large terms and is sensitive to the pressure. Its value at the two surfaces, which must be zero, is a built-in check: if a calculation does not return zero there, an error has crept in.
Where it is used
- Verifying a residual-stress calculation: the radial stress must vanish at the bore and the outside surface.
- Completing the stress state when the full Tresca measure is checked.
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