Shear Slides Things Along a Face
So far the stresses pulled or pushed a face straight on, the normal stresses. But what happens when the force runs parallel to the face, along it? It does not pull; it slides. Scissors cutting paper, a bolt snapping between two plates, these are all this. The stress at work is the shear stress τ, and the skew it produces is the shear strain γ. The striking part is that the whole tension story you just saw repeats once more. Put τ where σ was, γ where ε was, and G where E was. τ = G·γ, the same shape. They are siblings, differing only in name and direction.
First, get a feel for what shear is. Picture a deck of cards or a thick book. Push the top sideways: what happens? It does not stretch; each sheet slides a little over the one below. So the once-square stack becomes a slanted parallelogram. In the widget, push the top. Each layer offsets and the whole thing tilts. This is the essence of shear: a force acting parallel to a face, sliding things along that face. If tension pulls the two ends apart, shear pushes the layers sideways and skews them.
Now turn that shear force into a stress, exactly as in A1. Take the force V sliding along the face and divide by the area A it acts over, and you get the shear stress. τ = VA. When a single pin is pulled in opposite directions top and bottom, it tries to part along the plane between them. That plane is the shear plane, and τ is what acts on it. Drag V and watch the shear plane glow red. The formula is the same as the normal stress σ = PA. The only difference is one thing: whether the force is perpendicular or parallel to the face. Perpendicular gives σ, parallel gives τ. Both are in MPa.
How does the material respond to a shear stress? It skews. The shear strain γ you glimpsed in A2 now makes its full entrance. Drag τ to shear a square element. The clean right angle collapses and the element tilts into a parallelogram. That collapsed angle is the shear strain γ. Where normal strain ε was a ratio of lengths, shear strain γ is a change in angle. It is a deformation no length ratio can measure, so it is measured by an angle. The larger τ, the larger γ. Now that we have both the stress τ and the strain γ, it is time to join them.
The Hooke's law that was σ = E·ε in tension holds the same way in shear. τ = G·γ. Drag γ. In the elastic range, τ is directly proportional to γ, and the slope of that line is the shear modulus G. G is how much the material resists sliding, its shear stiffness. Where E was the stiffness against stretching, G is the stiffness against skewing. The two are not strangers. Within one material, E, G, and the Poisson's ratio ν we meet in the next lesson are tied together in one relation. Typically G is about 40 percent of E. For steel, E is about 200 GPa and G about 80 GPa.
The place shear shows up most is bolts and pins. Join two plates with a bolt and pull from both sides, and the bolt tries to be sliced clean along the plane where the plates meet. That is shear. Compare single and double shear with the buttons. Single shear has one plane to cut, so τ = VA acts in full. Double shear wraps the middle plate from both sides, giving two planes to cut. Then the same force V is split between two planes, τ = V(2A), and the stress drops by half. That is why important connections are deliberately designed as double shear. When choosing a pin or a rivet, the key calculation is whether the shear stress stays below the material's allowable shear strength.