Mohr Circle Center⚠ unverified
Mechanical / Stress Analysis · Compute the centre (average normal stress) of Mohr's circle
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| sigma_x | σx | Pa | 1.0 | Normal stress along the x-axis |
| sigma_y | σy | Pa | 1.0 | Normal stress along the y-axis |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | σavg | Pa | Average normal stress at the centre of Mohr's circle, in pascals (Pa) |
The science & history
Understanding the Parameters
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Normal stresses $\sigma_x$, $\sigma_y$ — the direct stresses on two perpendicular faces of the stress element. Their average is the circle centre; their difference drives the radius.
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Why the average is invariant — $\sigma_x + \sigma_y$ is the trace of the 2D stress tensor, a rotation invariant: rotate the element to any angle and the two face-normal stresses still sum to the same value. So the centre never moves as you transform axes — only the point on the circle does.
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Relation to principals — the principal stresses are $\sigma_{1,2} = \sigma_{avg} \pm R$. The centre is their midpoint; a purely hydrostatic (equal biaxial) state collapses the circle to this point ($R = 0$).
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Shear absent from the centre — the centre lies on the normal-stress axis, so it carries no shear; the shear information is entirely in the radius.
Derivation (Approaching a Proof)
Under a rotation of the axes by angle $\theta$, the transformed normal stress on the new x-face is
$$\sigma_{x'} = \frac{\sigma_x + \sigma_y}{2} + \frac{\sigma_x - \sigma_y}{2}\cos 2\theta + \tau_{xy}\sin 2\theta.$$
The first term is constant; the last two are a sinusoid in $2\theta$ with amplitude $R = \sqrt{[(\sigma_x-\sigma_y)/2]^2 + \tau_{xy}^2}$. Writing $\sigma_{x'}$ and the transformed shear $\tau_{x'y'}$ parametrically in $2\theta$ traces a circle in the $(\sigma, \tau)$ plane centred at
$$\sigma_{avg} = \frac{\sigma_x + \sigma_y}{2}$$
with radius $R$. The constant term is the centre precisely because it is the $2\theta$-independent part of the transformation — the average of the two face stresses.
Dimensional check. $\sigma_{avg} = \dfrac{\sigma_x + \sigma_y}{2} = \dfrac{\text{Pa} + \text{Pa}}{2} = \text{Pa}$ — a stress, as required.
History and Development
Otto Mohr introduced his stress circle in 1882, turning the algebra of stress transformation into a compact geometric construction that dominated strength-of-materials teaching for a century (and remains the clearest way to see principal stresses and maximum shear). The centre-and-radius description — $\sigma_{avg}$ here, $R$ in Mohr's Circle Radius — is the modern analytic distillation of that circle.
Related Concepts: Mohr's Circle Radius, Principal Stresses, Max Principal Stress, Min Principal Stress, Von Mises Stress, Tresca Stress
Notes: Centre at $(\sigma_{avg}, 0)$; principal stresses $= \sigma_{avg} \pm R$. $\sigma_x+\sigma_y$ is a rotation invariant (tensor trace) — the centre is fixed under axis rotation. 2D (plane stress) construction.