Mohr's Circle Calculator

What the Mohr's circle calculator does

At a single point in a loaded material, the stress depends on which direction you look. Mohr's circle is the classic way to handle this. From the normal and shear stresses on one set of faces, this calculator finds the largest and smallest normal stresses, the maximum shear stress, the von Mises stress, and the stresses on a plane rotated to any angle.

Below is why stress changes with direction, what the circle represents, the equations behind it, and a worked example.

How to use it

1. Enter the normal stresses on the x and y faces, and the shear stress between them.
2. Enter a rotation angle measured from the principal axes, to see the stresses on a plane turned to that angle.
3. Press Calculate for the principal stresses, maximum shear, von Mises stress, and the rotated stresses, or Reset to clear it.

Why stress depends on direction

Stress at a point is richer than a single number, because the same loading presses differently on differently angled planes cut through that point. Imagine a small block deep inside a stressed part. Its vertical faces might feel one combination of pushing and sliding, while faces cut at an angle feel another. Turn the block, and the split between normal stress, the pushing, and shear stress, the sliding, shifts.

This matters because materials fail in particular ways: some by being pulled apart, where the largest normal stress is what counts, others by sliding, where the largest shear stress does. To know whether a part is safe, you cannot just look at the stresses on one convenient set of faces; you have to find the worst stresses over every possible direction. Mohr's circle is the tool that does exactly that.

The circle and what it shows

The insight behind Mohr's circle is that, as you rotate the imaginary block, the normal and shear stresses on its faces trace out a circle when plotted against each other. Every orientation is a point on that circle, so the whole circle captures the complete stress state at the point, every direction at once.

Once you have the circle, the key answers are just features of it. The centre sits at the average of the two normal stresses, and its radius is set by how the stresses are split. The leftmost and rightmost points of the circle, where the shear vanishes, are the largest and smallest normal stresses, called the principal stresses. The top and bottom, the highest and lowest points, give the maximum shear stress. Reading the design off the circle replaces a great deal of trigonometry.

The equations it uses

With σ_x and σ_y the normal stresses and τ_xy the shear stress, the centre of the circle is their average and the radius combines the difference and the shear:

centre = (σ_x + σ_y) ÷ 2 and R = √( ((σ_x − σ_y)/2)² + τ_xy² )

The principal stresses are the centre plus and minus the radius, and the maximum shear stress is the radius itself:

σ₁, σ₂ = centre ± R and τ_max = R

The stresses on a plane rotated by an angle θ from the principal direction follow the circle around by twice that angle, as centre + R·cos(2θ) for the normal stress and R·sin(2θ) for the shear.

Principal stress, max shear, and von Mises

The three results matter for different failure modes. The principal stresses are the most extreme pulling or pushing the point feels, and the larger one warns of failure in brittle materials, which tend to break apart under tension. The maximum shear stress warns of sliding failure, the way ductile metals tend to yield.

The von Mises stress, which the calculator also gives, rolls the stress state into a single number designed to predict yielding in ductile materials. It combines the principal stresses so that the result can be compared directly against the material's yield strength: if the von Mises stress reaches the yield strength, the material is expected to start yielding. It is the figure most widely used in modern design to judge whether a part under combined stresses is safe.

Units and precision

The calculator works in megapascals, the usual unit for engineering stress, for both the inputs and the results. The rotation angle is in degrees, measured from the principal axes. Tension is taken as positive and compression as negative, so signs carry meaning and should be entered with care. Results are shown to two decimal places, which suits the precision of real stress analysis.

A worked example

Take a point with a normal stress of 80 MPa on the x faces, 20 MPa on the y faces, and a shear stress of 30 MPa.

The centre is (80 + 20) ÷ 2 = 50 MPa, and the radius is √((30)² + (30)²) = √1,800 ≈ 42.4 MPa. So the principal stresses are 50 ± 42.4, giving 92.4 MPa and 7.6 MPa, and the maximum shear stress is 42.4 MPa. The von Mises stress works out to about 88.9 MPa, the figure you would compare against the material's yield strength to check for safety.

Questions people ask

What is Mohr's circle used for?

It finds how stress varies with direction at a point, giving the largest and smallest normal stresses, the maximum shear stress, and the stresses on any rotated plane, all from the stresses on one set of faces.

What are principal stresses?

They are the largest and smallest normal stresses at a point, found on the planes where the shear stress is zero. They mark the most extreme pulling and pushing the material feels there.

What is the maximum shear stress?

It is the greatest sliding stress over all orientations, equal to the radius of Mohr's circle. It governs yielding in ductile materials and occurs at 45 degrees to the principal directions.

What is von Mises stress?

It is a single combined stress that predicts yielding in ductile materials. When it reaches the material's yield strength, the material is expected to begin yielding, which makes it a standard design check.

References

A quick note on where this comes from. Mohr's circle, the principal stresses and maximum shear from stress transformation, and the von Mises yield criterion are standard mechanics of materials, described in the Wikipedia article on Mohr's circle and in textbooks such as Beer and Johnston's Mechanics of Materials. The pascal and the other SI units follow the US National Institute of Standards and Technology.

1. Wikipedia, Mohr's circle. https://en.wikipedia.org/wiki/Mohr%27s_circle
2. Beer, F. P., and Johnston, E. R., Mechanics of Materials (stress transformation and Mohr's circle).
3. National Institute of Standards and Technology (NIST), Special Publication 811, Guide for the Use of the International System of Units (SI). https://www.nist.gov/pml/special-publication-811