Mohr's circle is the classic graphical tool for stress transformation: instead of re-deriving the stress-transformation equations every time you need the normal and shear stress on a rotated plane, you plot one circle and read the answer off its geometry. This calculator does the same math numerically, returning the principal stresses, maximum shear stress, and principal angle directly from your σx, σy, and τxy inputs.
How the circle encodes every possible orientation
Any 2D stress state can be described by three numbers: the normal stresses on two perpendicular faces (σx, σy) and the shear stress between them (τxy). As you imagine rotating the stress element, the normal and shear stress on its faces trace out a circle in σ-τ space — Mohr's circle. The circle is centered at (σx + σy)/2 on the σ-axis, with a radius of √(((σx − σy)/2)² + τxy²). Every point on the circle is a valid (σ, τ) pair for some rotation angle, which is why the circle contains all the information needed for stress transformation without repeating the trigonometry by hand.
Why the principal stresses matter
The two points where the circle crosses the σ-axis — σ1 (maximum) and σ2 (minimum) — are the principal stresses: the normal stresses on the orientation where shear stress drops to zero. These are usually the numbers compared against a material's yield or ultimate strength in a failure check, since a material typically fails under the largest normal stress it experiences, regardless of which direction that stress happens to point in the original x-y coordinate system. See the companion Von Mises Stress Calculator for combining principal stresses into a single equivalent-stress failure criterion.
Maximum shear stress and the principal angle
The circle's radius is also the maximum in-plane shear stress, τmax — it occurs 45° away from the principal planes, at the top and bottom of the circle. The principal angle, θp, is the physical rotation (measured from the x-axis) that aligns the stress element with its principal planes. Many failure modes — welds, adhesive joints, ductile-material yielding — are governed by shear rather than normal stress, so τmax can be the more important number depending on the material and the failure mode you're checking. Compare against the standalone Shear Stress Calculator for the simpler single-force case.
Limits of this 2D model
This calculator handles plane stress (2D) — the common case for thin plates, shells, and surface-stress analysis where the out-of-plane stress is zero or negligible. A fully general 3D stress state has three principal stresses and needs three Mohr's circles (or a full eigenvalue analysis) to characterize completely. For a purely uniaxial or biaxial check without shear, see the Stress Calculator; for the full 3D failure criterion, use the Von Mises Stress Calculator's component-stress mode.