Mohr's Circle Stress Visualizer

Plane-stress transformation

Enter the plane-stress state σx, σy, and τxy, and the tool draws the corresponding Mohr's circle alongside the physical stress element. Rotating the plane by θ moves the plotted point 2θ around the circle, and the sweep control animates that relationship so it is visible rather than memorised.

• Principal stresses σ₁ and σ₂, and the principal angle θp.
• Maximum in-plane shear stress τmax and the plane it acts on.
• The transformed stresses σx′, σy′, and τx′y′ at any rotation angle.
• Sign convention follows the standard Hibbeler / Beer textbook convention.

Equations used

The circle is centred at the average normal stress and has a radius equal to the maximum in-plane shear:

• σavg = (σx + σy) / 2
• R = √(((σx − σy)/2)² + τxy²)
• σ₁, σ₂ = σavg ± R
• τmax = R
• tan(2θp) = 2·τxy / (σx − σy)
• σx′ = σavg + ((σx − σy)/2)·cos 2θ + τxy·sin 2θ

Frequently asked questions

What are principal stresses?

Principal stresses are the normal stresses on the one orientation of the element where the shear stress is exactly zero. They are the largest and smallest normal stresses at that point, σ₁ = σavg + R and σ₂ = σavg − R, and they sit at the two ends of the horizontal diameter of Mohr's circle.

Why does the element rotate by θ but the circle by 2θ?

The stress-transformation equations depend on cos 2θ and sin 2θ rather than cos θ and sin θ, because rotating a plane by 180° returns the same physical plane. That factor of two is exactly why Mohr's circle works: a physical rotation of θ corresponds to an angular movement of 2θ around the circle.

Where does maximum shear stress occur?

The maximum in-plane shear stress equals the radius R of Mohr's circle, and it acts on planes rotated 45° from the principal planes — that is, at θp + 45°. The normal stress on those planes is not zero: it equals the average stress σavg, which is the horizontal position of the circle's centre.

What sign convention does Mohr's circle use for shear?

This tool uses the common textbook convention in which normal stress is positive in tension and the τ axis is drawn positive downward, with the x-face point plotted at (σx, +τxy). With that orientation a counter-clockwise physical rotation maps to a counter-clockwise sweep on the circle, which keeps the picture and the algebra consistent.