Mohr’s circle is a graphical tool that shows how normal and shear stresses change as you rotate the plane you look at. Its centre and radius give the principal stresses and maximum shear directly.
How to use the Mohr’s Circle Calculator
- Normal stress σx
- Normal stress σy
- Shear stress τxy
- Rotation angle θ
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Enter the normal stresses σx, σy and the shear stress τxy (with signs; tension positive), and optionally a rotation angle.
Inputs explained
| Field | What to enter |
|---|---|
| Normal stress σx | Units: MPa, kPa, Pa, GPa, psi, ksi |
| Normal stress σy | Units: MPa, kPa, Pa, GPa, psi, ksi |
| Shear stress τxy | Units: MPa, kPa, Pa, GPa, psi, ksi |
| Rotation angle θ | Units: °, rad, arcmin, arcsec, mrad |
What each result means
- Centre of the circle = ([Normal stress σx] + [Normal stress σy])÷2
- Radius (maximum shear stress) = sqrt((([Normal stress σx] – [Normal stress σy])÷2)^2 + [Shear stress τxy]^2)
- Principal stress σ₁ = ([Normal stress σx] + [Normal stress σy])÷2 + sqrt((([Normal stress σx] – [Normal stress σy])÷2)^2 + [Shear stress τxy]^2)
- Principal stress σ₂ = ([Normal stress σx] + [Normal stress σy])÷2 – sqrt((([Normal stress σx] – [Normal stress σy])÷2)^2 + [Shear stress τxy]^2)
- Principal angle θp = deg(atan2(2×[Shear stress τxy], [Normal stress σx] – [Normal stress σy]))÷2
- σx′ on the rotated plane = ([Normal stress σx] + [Normal stress σy])÷2 + ([Normal stress σx] – [Normal stress σy])÷2×cos(2×rad([Rotation angle θ])) + [Shear stress τxy]×sin(2×rad([Rotation angle θ]))
- σy′ on the rotated plane = ([Normal stress σx] + [Normal stress σy])÷2 – ([Normal stress σx] – [Normal stress σy])÷2×cos(2×rad([Rotation angle θ])) – [Shear stress τxy]×sin(2×rad([Rotation angle θ]))
- τx′y′ on the rotated plane = -([Normal stress σx] – [Normal stress σy])÷2×sin(2×rad([Rotation angle θ])) + [Shear stress τxy]×cos(2×rad([Rotation angle θ]))
Example values
How the main result changes when Normal stress σx changes and everything else stays at the example values above:
| Normal stress σx (MPa) | Radius (maximum shear stress) (MPa) |
|---|---|
| 40 | 42.4264 |
| 60 | 50 |
| 80 | 58.3095 |
| 100 | 67.082 |
| 120 | 76.1577 |
| 160 | 94.8683 |
Formula
C = (σx + σy)/2, R = √(((σx − σy)/2)² + τxy²), σ₁,₂ = C ± R
σx′ = C + (σx − σy)/2·cos 2θ + τxy·sin 2θ, τx′y′ = −(σx − σy)/2·sin 2θ + τxy·cos 2θ
Worked example
Inputs — Normal stress σx: 80 MPa; Normal stress σy: -20 MPa; Shear stress τxy: 30 MPa; Rotation angle θ: 30 °.
Results — Centre of the circle = 30 MPa; Radius (maximum shear stress) = 58.30952 MPa; Principal stress σ₁ = 88.30952 MPa; Principal stress σ₂ = -28.30952 MPa; Principal angle θp = 15.4819 °; σx′ on the rotated plane = 80.98076 MPa; σy′ on the rotated plane = -20.98076 MPa; τx′y′ on the rotated plane = -28.30127 MPa.
Key terms
- Principal stresses
- Normal stresses on planes with zero shear.
Frequently asked questions
What do points on the circle represent?
The normal and shear stress on planes at different orientations.
Why is the angle doubled?
A rotation θ of the plane corresponds to 2θ on Mohr's circle.
Where is the maximum shear stress?
At the top and bottom of the circle, 45° from the principal planes.
What sign convention is used?
Tension positive; positive τxy acts in +y on the +x face.