Mohr's circle explained

First, move the stress state onto a point. On a plane whose horizontal axis is normal stress and whose vertical axis is shear stress, one pair of stresses on a face becomes one point. The x face of the element is (σx, τxy), and the y face turned 90 degrees from it is (σy, −τxy). Move the shear and the two points travel up and down, and they always sit opposite each other. The x face and the y face differ by 90 degrees on the element, yet on the plane they are plotted 180 degrees apart. A factor of 2 has already appeared. The segment joining the two points is the diameter of the circle. Seeing a stress state as a place on a plane, rather than as a bundle of numbers, is Mohr's first leap. The two points fix the diameter.

Now turn the element a little at a time and keep plotting (σ′, τ′) for each angle. Move the angle and the point does not scatter. It travels along exactly one circle. Whatever angle you cut, that stress lies somewhere on this circle, and every stress state possible at the point is held in this one circle. The center of the circle sits on the horizontal axis, at the average σ = (σx+σy)2. The average that stayed fixed under rotation in D1 is exactly that center. The radius is R = √(((σx−σy)/2)² + τxy²), the same R from D2. The transformation that used to sway compresses into one circle of radius R about the average. The circle stands in for the formula. The average holds the center.

Mohr's circle carries one agreement. Turn the real element through an angle, and the point on the circle turns through twice that angle, 2θ. Turn the element on the left and the point on the circle on the right moves through 2θ. The reason for the doubling is that the transformation was entirely a function of 2θ. That 2θ becomes a real angle of rotation once it is on the circle. Keep this one agreement and Mohr's circle is a protractor. To know the stress on a face, see how far that face has turned from the x face, and read the point turned through twice that angle on the circle. There is no sign to memorize and no sine to memorize. Rotation is visible as rotation. Twice the element's angle is the angle on the circle.

Now read the answers off the circle, starting with the principal stresses. The circle meets the horizontal axis, the line τ=0, in two places. They are the right-hand end and the left-hand end of the circle, and the shear there is 0, so those faces are the principal planes. The normal stress at the right-hand end is the largest principal stress σ1, and the left-hand end is the smallest, σ2. The center is the average and the radius is R, so σ1 = average + R and σ2 = average − R. What D2 asked you to memorize as a formula is, in the picture, simply the two ends of the circle. Bring the point to an end and, the moment it touches the horizontal axis, the shear vanishes. Finding the principal stresses is marking the two points that reach farthest left and right on the circle. The two ends are the two principal stresses.

Last is the maximum shear. The place where the shear is largest on the circle is the top, and the height up to it is exactly the radius R. So the maximum shear stress is τ_max = R. D2 had τ_max = (σ1−σ2)2, and since σ1−σ2 is the diameter, half of it is the radius, which says the same thing. In the picture it is enough to measure the radius of the circle. The top stands 90 degrees along the circle from the principal stresses at the two ends. 90 degrees on the circle is half of that on the real element, 45 degrees, and that 45 degrees from D2 comes out here on its own. Set the point on the top and the relation shows itself. A stress state at a point is one circle, the principal stresses are the left and right ends, the maximum shear is the top and the bottom, and turning the element is a rotation of 2 times on the circle. Mohr's circle turns the whole stress transformation into one picture. One picture stands in for the formula.