Library / Mechanical / MEC-49
Mechanical
Mohr’s circle centre (mean in-plane stress)
Centre of Mohr’s circle: the average of the two normal stresses on perpendicular faces.
- Ref
- MEC-49
- Rev
- v1
- Field
- Mechanical
- Result
- C [MPa]
C is the average of the normal stresses on two perpendicular faces. When the element is turned through any angle, the new pair of normal stresses moves to either side of C but always adds up to the same total, 2C, so C is the fixed centre of Mohr’s circle. The shear stress has no effect on it; the shear stress sets the radius of the circle (see the maximum in-plane shear stress entry). This is step 1 of the stress route: centre, radius, rotated stresses, principal stresses, then Tresca and von Mises.
Centre of Mohr’s circle: the average of the two normal stresses on perpendicular faces. It does not change when the element is rotated, so every transformed pair of normal stresses is spread equally either side of it.
Calculator
- Half the difference, (σx − σy)/2, the horizontal distance from C to each face stress30 MPa
- Sum σx + σy = 2C, unchanged by any rotation40 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 lecture state (σx = 50, σy = −10 MPa) has a centre of 20 MPa. Turning the element by 30° with a shear stress of 40 MPa gives 69.64 and −29.64 MPa, which still average 20 MPa.
- If σx = σy the centre equals that common stress and the circle is carried entirely by the shear stress. If they are equal and the shear is zero, the circle shrinks to a single point and every plane has the same stress.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (MPa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| σx Normal stress in x (tension positive) | −200 to 300 MPa | −105 | 145 | 250 | +10 MPa | +25 % |
| σy Normal stress in y (tension positive) | −200 to 300 MPa | −75 | 175 | 250 | +10 MPa | +25 % |
Sweep of Normal stress in x (tension positive)
| Normal stress in x (tension positive) (MPa) | Centre of Mohr’s circle (MPa) |
|---|---|
| −200 | −105 |
| −137.5 | −73.75 |
| −75 | −42.5 |
| −12.5 | −11.25 |
| 50 | 20 |
| 112.5 | 51.25 |
| 175 | 82.5 |
| 237.5 | 113.8 |
| 300 | 145 |
Sweep one variable
Variables and how each one contributes
Centre of Mohr’s circle
What the equation gives you. Everything else feeds it.
Normal stress in x (tension positive)
Normal stress on the x face, in MPa. Tension is positive and compression negative, so the slider runs from −200 to +300 MPa. Raises C by half the change, because C is an average. Raising it by 10 MPa (to 60 MPa) moves the result by +25 %, from 20 to 25 MPa.
Normal stress in y (tension positive)
Normal stress on the y face, in MPa. Tension is positive and compression negative, so the slider runs from −200 to +300 MPa. Raises C by half the change, in exactly the same way as σx, because the two stresses are simply averaged. Raising it by 10 MPa (to 0 MPa) moves the result by +25 %, from 20 to 25 MPa.
Worked example with the values above
- Add the two normal stresses: σx + σy = 50 + (−10) = 40 MPa.
- Divide by 2: 40/2.
- C = 20 MPa.
Assumptions and limits
- Both stresses act at the same point on two perpendicular faces.
- A single plane of interest, so the same centre applies to every pair of perpendicular faces in that plane.
Common mistakes
- Averaging stresses taken at two different points.
- Dropping the minus sign of a compressive stress before averaging.
Statistics and uncertainty
Because C is an average, independent errors in the two stresses partly cancel: if each carries an error of ±5 MPa, C has an error of about ±3.5 MPa (5 divided by √2). Errors with a common cause, such as a gauge calibration fault that biases both readings, do not cancel and pass straight through.
Where it is used
- Locating the circle in a hand Mohr’s circle construction.
- Checking that a rotated pair of normal stresses still adds up to the original sum.
Maths you need
- Mean (average): sum divided by the count
- Sign convention: tension positive, compression negative