Free · Physics models
Plane-Stress Mohr-Circle Parameters
Compute the center, radius, principal values, plane-stress von Mises value and principal direction from Cartesian stresses.
- Formula & worked example
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Calculator inputs
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How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- Use the worked example to check the method. Reset restores the starting fields.
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Formula and method
For 100 and 40 MPa normal stress with 30 MPa shear, the Mohr circle has center 70 MPa and radius about 42.426 MPa.
Center = (σx+σy)/2; radius = √[((σx−σy)/2)²+τ²]; σ1,2 = center±radius; σv = √(σx²−σxσy+σy²+3τ²).
Worked example
Enter these known values and leave the other values blank.
- Normal stress in the X direction (σ==xx==)
- 100 MPa
- Normal stress in the Y direction (σ==yy==)
- 40 MPa
- Shear stress (τ==xy==)
- 30 MPa
- Shear stress (τ==yx==)
- 30 MPa
- Principal stress (σ==1==)
- 112.43 MPa
- Principal stress (σ==2==)
- 27.574 MPa
- Maximum shear stress (τ==max==)
- 42.43 MPa
- von Mises stress (σ==Mises==)
- 101.49 MPa
- Angle of orientation (θ)
- 0.3927 rad
- Mean stress (σ==m==)
- 70 MPa
Assumptions and limitations
- This is a plane-stress calculation with out-of-plane normal and shear stresses set to zero for the von Mises result.
- The radius is maximum in-plane shear. The absolute three-dimensional maximum shear can differ when the zero out-of-plane principal stress is included.
- Cartesian tensor shear components are symmetric, τyx = τxy. Plotting opposite-face traction signs on a Mohr circle is a separate graphical convention.
- The maximum-principal direction uses atan2 and is defined modulo 180 degrees. Equal normal stresses and zero shear leave that direction undetermined.
- No yield strength or failure criterion threshold is supplied. The von Mises scalar alone does not certify safety.
- Enter known quantities and leave the unknown blank. Editable quantities support inverse solutions and locks; inconsistent locked quantities are reported. Unit selectors convert the quantity independently of its display.
- Reviewed hash-pinned model correction: This is a plane-stress calculation with out-of-plane normal and shear stresses set to zero for the von Mises result. The radius is maximum in-plane shear. The absolute three-dimensional maximum shear can differ when the zero out-of-plane principal stress is included. Cartesian tensor shear components are symmetric, τyx = τxy. Plotting opposite-face traction signs on a Mohr circle is a separate graphical convention. The maximum-principal direction uses atan2 and is defined modulo 180 degrees. Equal normal stresses and zero shear leave that direction undetermined. No yield strength or failure criterion threshold is supplied. The von Mises scalar alone does not certify safety. Enter known quantities and leave the unknown blank. Editable quantities support inverse solutions and locks; inconsistent locked quantities are reported. Unit selectors convert the quantity independently of its display.
Common questions
Is the circle radius always the absolute maximum shear?
It is maximum in-plane shear. In plane stress, the zero out-of-plane principal stress may give a larger three-dimensional maximum shear.
Does von Mises stress certify a safe design?
No. A suitable material strength, loading context and applicable design criterion are also required.
How do I use the inverse calculation?
Clear the quantity you want to find and enter the remaining known values in their selected units. The model can only solve branches with enough independent information.
References
Use a symmetric Cartesian stress tensor, and atan2 for the maximum principal direction including equal-normal and reversed-normal cases. Hydrostatic states have no unique principal direction.