Principal Stress & Mohr's Circle Calculator

Transform a plane (2D) stress state into its principal stresses. Enter the normal stresses σ_x, σ_y and the shear stress τ_xyto get the principal stresses, maximum in-plane shear, principal angle, and a scale-correct Mohr's circle you can rotate.

What Are Principal Stresses?

At any point in a loaded body, the stress depends on the orientation of the plane you examine. Principal stresses are the maximum and minimum normal stresses found by rotating that plane, and on the planes where they act the shear stress is zero. The principal directions are therefore the natural axes of the stress state, and the principal stresses are what most failure criteria are built on.

For a plane stress state defined by σ_x, σ_y, and τ_xy, there are two in-plane principal stresses, σ₁ (major) and σ₂ (minor). The third principal stress is zero out of plane.

Principal Stress Formulas

The average normal stress σ_avgis the centre of Mohr's circle and the radius R is the maximum in-plane shear stress. The principal stresses lie at the ends of the horizontal diameter, and the maximum shear acts on planes rotated 45° from the principal planes.

Stress Transformation

To find the stresses on any plane rotated by an angle θ from the x-axis, use the transformation equations. Drag the rotation slider in the calculator to see these update live and watch the corresponding point sweep around Mohr's circle:

A physical rotation of θ corresponds to a rotation of 2θ around Mohr's circle, which is why the principal planes (where the shear is zero) are reached when 2θ brings the point onto the horizontal axis.

Reading Mohr's Circle

Mohr's circle is a graphical map of every plane orientation. Each point on the circle gives the normal stress (horizontal axis) and shear stress (vertical axis) on one plane. The circle is centred at σ_avgwith radius R. The amber points mark the stresses on the x-face and y-face you entered; they sit at opposite ends of a diameter. As you rotate the element, the emerald diameter shows the new x′ and y′ faces, and the green arc marks the 2θ sweep.

Sign Conventions

Tensile normal stress is positive and compression is negative. A positive τ_xy acts in the conventional sense on the positive x-face. The principal angle θ_p is reported as the rotation from the x-axis to the major principal direction, positive counter-clockwise. Because the maximum-shear planes are always 45° from the principal planes, θ_s = θ_p− 45°.

From Principal Stresses to Yield

Once you have the principal stresses, the von Mises equivalent stress for plane stress is σ_v = √(σ₁² − σ₁σ₂ + σ₂²), reported above. Compare it with the material yield strength to check for yielding, or use the dedicated von Mises calculator for full 3D states.

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