Calculator · Structural Mechanics
Mohr's Circle Calculator
Mohr's circle is the transformation law drawn as geometry: every plane through the point plots as one position on a circle, so the principal stresses and the maximum shear can be read off directly.
Stress state
σ is tensile positive. τxy acts on the +x face in the +y direction. θ is a counter-clockwise rotation of the axes in physical space.
Mohr's circle
Circle geometry
| Centre σavg | 30 MPa |
|---|---|
| Radius R | 58.31 MPa |
| σ₁ = centre + R | 88.31 MPa |
| σ₂ = centre − R | -28.31 MPa |
| Maximum in-plane shear τmax = R | 58.31 MPa |
| x-face plotted at | (80, 30) MPa |
| y-face plotted at | (-20, -30) MPa |
| x′-face at θ = 30° | (80.981, -28.301) MPa |
Open in the Lab
Sweep θ in the Lab and watch the plotted face travel around this circle.
Technical reference
centre C = (σx + σy)/2
radius R = √( ((σx−σy)/2)² + τxy² )
(σ − C)² + τ² = R²
- C
- centre of the circle on the σ axis, the average normal stress[Pa]
- R
- radius, equal to the maximum in-plane shear stress[Pa]
τ is plotted positive UPWARD on this chart and the x-face plots at (σx, +τxy). Both axes use one shared scale, so the circle is a true circle and distances can be compared directly by eye.
A counter-clockwise physical rotation of the element by θ moves the plotted point CLOCKWISE around the circle by 2θ. Angles on the circle are always double the physical angles.
Diametrically opposite points are perpendicular physical faces; that is why σx + σy is constant and equals twice the centre.
Assumptions
- · Plane stress: the out-of-plane face is free, σz = 0, so this is the in-plane circle only.
- · The full three-dimensional picture has three circles; when σ₁ and σ₂ share a sign, the largest of them is not the one drawn here.
- · The circle is a graphical statement of equilibrium and holds regardless of material.