A Stress State Is a Circle, Rotating the Element Is Rotating on It

First, move the stress state onto a point. Draw a plane whose horizontal axis is the normal stress σ and vertical axis is the shear stress τ. Then the stress (σ, τ) on one face becomes a single point in this plane. The x-face of the stress element is (σx, τxy), and the y-face, 90 degrees around, is (σy, −τxy). Drag τxy. The two points move up and down, and if you look closely they are always on exactly opposite sides. The x-face and y-face are 90 degrees apart, yet on the plane they plot 180 degrees apart. The doubling is already here. The segment joining these two points will become the diameter of the circle. Seeing the stress state as a position on a plane rather than a bundle of numbers, this is Mohr's first leap.

Now the magic unfolds. Rotate the element a little at a time and keep plotting the (σ′, τ′) at each step. Drag θ. Where does the point go? It does not scatter; it travels along exactly one circle. Cut at any angle and that stress is somewhere on this circle. Every possible stress state at the point is held in this one circle. The circle's center is on the horizontal axis, at σ = (σx+σy)2, the average. That average, which stayed fixed under rotation in D1, is exactly the center. The circle's radius is R = √(((σx−σy)/2)² + τxy²), the very R from D2. The complicated swinging transformation equations are compressed into one circle, centered on the average with radius R.

Here is one promise of Mohr's circle: rotate the actual element by θ, and the point on the circle goes around by twice that, 2θ. Rotate the left element by θ. The point on the right circle moves by 2θ. Why double? Because the transformation equations were all functions of 2θ. On the circle that 2θ becomes a real rotation angle. Keep this one promise and Mohr's circle is just a protractor. To find the stress on any face, see how many degrees that face is turned from the x-face, and read off the point rotated twice that much on the circle. No signs, no sine and cosine to memorize. Rotation becomes rotation, right before your eyes.

Now read the answers off the circle. First, the principal stresses. There are two places where the circle meets the horizontal axis, the τ=0 line: the far right end and the far left end of the circle. There the shear is zero, so those are the principal planes. The σ value at the right end is the maximum principal stress σ1, the left end is the minimum σ2. And since the center is the average and the radius is R, naturally σ1 = average + R and σ2 = average − R. What you memorized as a formula in D2 is, in the picture, simply the two ends of the circle. Drag θ to bring the point to an end. The instant the point touches the horizontal axis, the shear vanishes. Finding the principal stresses is nothing more than marking the two points that reach farthest left and right on the circle.

Last is the maximum shear. Where on the circle is τ largest? At the top, of course. The height up to there is exactly the radius R. So the maximum shear stress τ_max = R. In D2 we wrote τ_max = (σ1−σ2)2, and since σ1−σ2 is the diameter, half of it is the radius, the same statement. In the picture you just measure the circle's radius. And the top sits 90 degrees from the ends (the principal points) along the circle. Ninety degrees on the circle is half that, 45 degrees, on the real element. The 45 degrees from D2 comes out automatically here. Drag θ to lift the point to the top. To sum up, the stress state at a point is one circle, the principal stresses are its left and right ends, the maximum shear is its top and bottom, and rotating the element is a doubled rotation on the circle. Mohr's circle is the tool that turns the whole of stress transformation into a single picture.