Library / Mechanical / MEC-46
Mechanical
Hoop stress at the outside surface
Hoop stress at the outer surface of a thick cylinder under internal pressure only.
- Ref
- MEC-46
- Rev
- v1
- Field
- Mechanical
- Result
- σθ(b) [MPa]
The same Lamé result evaluated at r = b with po = 0 and pi = p. The radial stress at the outside surface is zero, which makes this the one place a strain gauge can read the hoop stress directly. The bore value then follows by multiplying by (a² + b²)/(2a²). p is the internal pressure in MPa and a and b the radii in mm.
Hoop stress at the outer surface of a thick cylinder under internal pressure only. It is the lowest hoop stress in the wall.
Calculator
- Ratio of bore to outside hoop stress (a² + b²)/(2a²)1.625
- Maximum hoop stress at the bore26 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 the outside the hoop stress is 16 MPa, about 38 % below the 26 MPa at the bore (a ratio of 1.6).
- A strain gauge on the outside reads the hoop stress directly, since the radial stress there is zero. The bore value follows by multiplying by (a² + b²)/(2a²) = 1.63, so a proof test can infer the stress at the bore, where it cannot be measured.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (MPa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| a Internal radius | 10 to 100 mm | 0.362 | 360.1 | 359.7 | +10% | +45.4 % |
| p Internal pressure | 1 to 100 MPa | 1.6 | 160 | 158.4 | +10% | +10 % |
| b External radius | 60 to 400 mm | 0.3175 | 45.45 | 45.14 | +10% | −27.4 % |
Sweep of External radius
| External radius (mm) | Hoop stress at the outside (MPa) |
|---|---|
| 60 | 45.45 |
| 102.5 | 6.245 |
| 145 | 2.699 |
| 187.5 | 1.531 |
| 230 | 0.9921 |
| 272.5 | 0.6968 |
| 315 | 0.5169 |
| 357.5 | 0.399 |
| 400 | 0.3175 |
Sweep one variable
Variables and how each one contributes
Hoop stress at the outside
What the equation gives you. Everything else feeds it.
Internal pressure
The internal pressure, in MPa. Directly proportional: a 10 % rise in p raises the outside hoop stress by 10 %, to 17.6 MPa.
Internal radius
The internal radius (bore radius), in mm. Very strong: a 10 % rise in a (to 55 mm) raises the outside hoop stress by about 45 %, to 23.3 MPa, because the wall carries the load with less material.
External radius
The external radius, in mm. A thicker wall lowers the outside stress quickly: a 10 % rise in b (to 82.5 mm) lowers it by about 27 %, to 11.6 MPa.
Worked example with the values above
- Wall term: b² − a² = 75² − 50² = 3,125 mm².
- Numerator: 2pa² = 2 × 10 × 50² = 50,000.
- σθ(b) = 50,000 / 3,125 = 16 MPa.
Assumptions and limits
- Internal pressure only, on a long, homogeneous, isotropic, linearly elastic cylinder.
- A section away from the ends and any discontinuities, with a uniform temperature.
Common mistakes
- Taking this value as the peak stress. It is the lowest hoop stress in the wall; the maximum is at the bore.
- Using it when an external pressure acts. Use the general Lamé hoop stress for that case.
Statistics and uncertainty
A gauge on the outside measures strain, and converting it to stress needs E and Poisson’s ratio: a 3 % uncertainty in E gives about 3 % in the stress. The radii amplify errors through b² − a²: at these proportions a 1 % error in a gives about a 4 % error here. A strain gauge reading is therefore usually the more accurate route to the bore stress than calculating from measured radii.
Where it is used
- Strain-gauge checks on the outside of a pressure vessel in a proof test.
- Estimating how much the bore stress exceeds the stress at the outside surface.
Maths you need
- Lamé hoop stress: A + B/r² at r = b
- Ratios: bore value over outside value