Library / Mechanical / MEC-45
Mechanical
Maximum hoop stress at the bore
Largest tensile hoop stress in a thick cylinder under internal pressure only, found at the inside surface.
- Ref
- MEC-45
- Rev
- v1
- Field
- Mechanical
- Result
- σθ,max [MPa]
Set po = 0 and pi = p in the Lamé solution. The hoop stress then falls steadily from the bore to the outside, so its maximum is at r = a. p is the internal pressure in MPa and a and b the internal and external radii in mm. The result is the largest tensile hoop stress in the wall, and the place where yielding begins.
Largest tensile hoop stress in a thick cylinder under internal pressure only, found at the inside surface. It is always greater than the pressure itself.
Calculator
- Stress ratio σθ,max / p2.6
- Thin-wall estimate pa/t, for comparison20 MPa
- Hoop stress at the outside surface16 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.
- The bore stress is 2.6 times the pressure here. It is always greater than p, however thick the wall: as b grows the factor only tends to 1. Beyond a point, thickening the wall barely helps, which is why very high pressures use shrink-fitted (compound) or autofrettaged cylinders.
- The thin-wall formula pa/t gives 20 MPa for these dimensions, 23 % below the true 26 MPa, because t/a = 0.5 is far from thin.
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 | 10.36 | 370.1 | 359.7 | +10% | +28 % |
| p Internal pressure | 1 to 100 MPa | 2.6 | 260 | 257.4 | +10% | +10 % |
| b External radius | 60 to 400 mm | 10.32 | 55.45 | 45.14 | +10% | −16.9 % |
Sweep of External radius
| External radius (mm) | Maximum hoop stress (MPa) |
|---|---|
| 60 | 55.45 |
| 102.5 | 16.25 |
| 145 | 12.7 |
| 187.5 | 11.53 |
| 230 | 10.99 |
| 272.5 | 10.7 |
| 315 | 10.52 |
| 357.5 | 10.4 |
| 400 | 10.32 |
Sweep one variable
Variables and how each one contributes
Maximum hoop stress
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 bore hoop stress by 10 %, to 28.6 MPa.
Internal radius
The internal radius (bore radius), in mm. Strong. A thinner wall carries the same pressure at higher stress: raising a by 10 % (to 55 mm) raises the bore hoop stress by about 28 %, to 33.3 MPa.
External radius
The external radius, in mm. The wall thickness is b − a. A thicker wall lowers the stress, with diminishing returns: raising b by 10 % (to 82.5 mm) lowers the bore hoop stress by about 17 %, to 21.6 MPa.
Worked example with the values above
- Squares: a² + b² = 2,500 + 5,625 = 8,125; b² − a² = 3,125 mm².
- Ratio: (a² + b²)/(b² − a²) = 2.6.
- σθ,max = 10 × 2.6 = 26 MPa.
Assumptions and limits
- Internal pressure only, with zero external pressure, on a long, homogeneous, isotropic, linearly elastic cylinder.
- A section away from the ends, nozzles and other discontinuities.
Common mistakes
- Comparing σθ,max directly with the yield stress and ignoring the compressive radial stress, which makes the equivalent stress higher.
- Using it with external pressure. This formula needs po = 0; use the general Lamé hoop stress otherwise.
Statistics and uncertainty
σθ,max is linear in p. The radii enter through a difference of squares, so at these proportions a 1 % error in a or in b gives about a 2 % error in the stress (raising a by 10 % raises it 28 %, raising b by 10 % lowers it 17 %). A yield check should use the equivalent stress at the bore, not σθ alone: with the radial stress of −p there, the Tresca value is 26 − (−10) = 36 MPa at the defaults.
Where it is used
- A quick yield check of a thick pressurised cylinder, where failure starts at the bore.
- Choosing the wall thickness for a given internal pressure and material.
Maths you need
- Lamé hoop stress: A + B/r² at r = a
- Difference of squares: b² − a²