Why bending is mostly stretching and squeezing
What happens inside a beam, and where the neutral axis sits.
Bending is mostly stretching and squeezing because the two faces of anything being bent are forced apart into a longer curve on the outside of the bend and pushed together into a shorter curve on the inside, and it is that stretching on one face and squeezing on the other that actually does the work of resisting the bend.
A plastic ruler pushed from both ends stays straight and then suddenly does not, always bowing the same way, and the first time I watched that happen slowly enough to see the outer face pale and thin while the inner face bunched, bending stopped looking like a single simple thing to me.
A longer path outside, a shorter one inside
Picture a beam bent into a gentle curve, and picture the two faces of that beam, the one on the outside of the curve and the one on the inside. The outside face is now following a longer path than it was when the beam was straight, because it has been carried further from the centre of the curve, and the only way a fixed length of material can cover a longer path is to stretch. The inside face has the opposite problem, following a shorter path than before, and the only way it can do that without buckling out of the way entirely is to squeeze together, compressing along its own length. Somewhere between those two faces, at the beam's own centre, sits a line that is neither stretched nor squeezed at all, following exactly the same length of path it always did, called the neutral axis, and every layer of material between that line and the outer surfaces is doing some blend of stretching or squeezing depending on how far it sits from the neutral line. For a simple, symmetrical cross-section the neutral axis sits exactly halfway between the two outer faces. For a lopsided shape, with more material above the centre than below it, the neutral axis shifts toward wherever the extra material sits, since it always settles at the point where the stretching effect and the squeezing effect balance each other out exactly.
Bending a rubber eraser
Bending a thick rubber eraser between two fingers makes this visible without any instruments at all. The outer face of the bend visibly stretches and pales as the bend deepens, the rubber thinning slightly as it is pulled along its own length to cover the longer path the bend has put it on. The inner face does the opposite at the same moment, visibly bunching and wrinkling as it is squeezed together to fit the shorter path on that side of the bend. Nothing about the eraser as a whole has changed length, since one face stretching and the other squeezing by roughly matching amounts is exactly what lets the eraser bend at all without any part of it tearing or being crushed, and letting go allows both faces to spring straight back to their original, unstretched and unsqueezed state. Running a fingernail lightly along the eraser's flat side while it is held bent, from one end to the other, reveals the transition directly, moving from a stretched, thinned region near one edge, through an unchanged strip at the centre, into a bunched, thickened region near the other edge.
Why the stretched face gets the most care
Seeing bending as stretching on one face and squeezing on the other explains why a beam's cross-sectional shape matters so much to how well it resists bending, since material sitting near the neutral axis is doing very little stretching or squeezing and therefore very little work, while material sitting at the outer faces, furthest from the neutral axis, is doing the most stretching or squeezing and carrying the most load. It also explains why a crack or a flaw on the outer, stretched face of a bent part is far more dangerous than the same flaw sitting near the neutral axis, since a flaw sitting exactly where the material is already being pulled apart the hardest has the least margin left before it grows. This is the reasoning behind polishing or shot-peening the tension face of a component that will be bent repeatedly in service, deliberately treating the one surface where a small flaw does the most damage.
Permanent bends and materials that crack in tension
This picture holds well for a gentle, elastic bend, the kind whose stiffness is described by a material's Young's modulus, that the material can spring back from once the load is removed, but it stops describing the whole story once a bend is pushed far enough that the outer face's stretching exceeds what the material can recover from, at which point that face starts to yield permanently rather than springing back, the beginning of a bend that stays bent. It also assumes the material behaves the same way in stretching as it does in squeezing, which holds for many common materials. Some materials tolerate being squeezed far better than being stretched, and in those the stretched face of a bend is the one at risk of failing first. Cast iron is a well-known example, comfortably tolerating squeezing but cracking at a much lower stretching load, which is why cast iron beams were once cast with a much larger flange on the stretched side than on the squeezed side, putting extra material where the weaker behaviour needed it.