plane stress transformation
Earlier, stress was split into the part perpendicular to a face and the part along the face. Which way you set that face is actually a choice. At the same point, turn the face at a slant and the normal stress and the shear stress on it change completely. It sounds odd, and it is natural. Stress is defined only once there is a face, and the direction of the face was changed. So the stress state at a point is not one number. It is a bundle that changes with angle. Handling that is stress transformation. Turn the angle and the normal and shear stresses sway like a sine, and among them a special angle appears where the shear is 0. There the normal stress is largest, and that direction is where the material actually splits. This lesson shows that sway directly. The direction of the split is that special angle.
The standard way to draw the stress state at a point is the stress element. Picture a very small square around the point and mark the stresses on the four faces with arrows. The left and right faces carry a horizontal normal stress, the top and bottom faces carry a vertical normal stress, and a shear stress slides along the faces. Move the shear and the shear arrows appear on all four faces at once. Shear never acts on one face alone. Equilibrium would break. It always comes as a pair, on all four faces together, complementary shear. This one small element holds the whole stress state of the point. Three numbers decide a plane stress completely. Three numbers fix one point, and the arrows show them.
Now the heart. Look at the same point and only turn the element at a slant. Move the angle and the square rotates, and the normal stress σ′ and the shear stress τ′ on the new faces change together. The point is the same and only the direction of the face changed, yet the stress changes. It is confusing at first, and on reflection it is natural. The same bundle of force, cut by a face of a different direction, splits into normal and parallel parts in a different ratio. The more slanted the face, the more the force skims past it, and the shear grows. So σ′ and τ′ are functions of the angle. A point does not have one stress value. It has a different set for every angle you look from. Then the natural question is which angle is the most dangerous.
Write that sway as a formula and it is the stress transformation. σ′ = (σx+σy)2 + (σx−σy)2·cos2θ + τxy·sin2θ, and τ′ = −(σx−σy)2·sin2θ + τxy·cos2θ. It looks long and the heart is simple. Both are sines and cosines of 2θ, smooth waves. Move the angle and the two points on the σ′ curve and the τ′ curve move together. Two things to notice. First, it is 2θ and not the angle itself, so the period is 180 degrees. Turn the element 180 degrees and it is home again, which is natural. Second, the average of σ′, (σx+σy)2, does not change no matter the angle. This sum is a quantity preserved under rotation. With only these two curves in hand, finding the dangerous angle is reading the peaks and the zeros of the curves.
Turn the element and an angle appears where the shear stress τ′ is exactly 0. That face is a principal plane, and the normal stress on it is a principal stress. Move the angle until τ′ sits on 0, and at that moment the shear arrows vanish and what remains is a pure push and pull. At this very angle the normal stress is largest. The largest principal stress σ1 and the smallest σ2 come out on these faces. It matters because a material usually splits perpendicular to the direction of this largest principal stress. Twist a stick of chalk until it breaks and the section splits in a slanted helix, which is the break following a principal plane. To predict failure, finding σ1 and its direction, the principal stress and the principal angle, is the first job. The direction of the split is that angle, and finding it is the first job.
Now, the other way, find the angle where the shear stress is largest. Move the angle and feel for where τ′ is maximum, and it is exactly 45 degrees turned from a principal plane. It is always so. The plane of maximum shear differs from a principal plane by 45 degrees. The size of that maximum shear is half the difference of the two principal stresses, τmax = (σ1−σ2)2. It matters because a ductile material such as steel yields to shear before it yields to normal stress. So pull a tension specimen and it does not slide on a clean cross section. It slides and lengthens along a face tilted 45 degrees. Those slanted streaks are the trace of the plane of maximum shear. On a principal plane the shear is 0 and the normal stress is extreme, and 45 degrees from there the shear is extreme. These two special angles are the conclusion of stress transformation.