Bend Allowance

Bend a strip of metal and the outside of the bend is stretched, the inside is squashed, and somewhere between them is a surface that is neither — the neutral axis. That surface keeps its original length, so the length of material a bend consumes is simply the arc the neutral axis travels through. That is the whole of the bend allowance: $BA = \theta(R + Kt)$, an arc length, with $\theta$ in radians, $R$ the inside radius, and $Kt$ the distance from the inside surface out to the neutral axis.

Everything except $K$ is circle geometry that nobody can argue with. $K$ is where the argument lives.

The K-factor is not a constant, and this is the single most useful thing on this page. If the material simply bent about its own middle, $K$ would be 0.5 and there would be nothing to discuss. It does not. Bending pushes the neutral axis towards the inside of the bend, because the compressed inner material resists being shortened more than the outer material resists being stretched, and the metal thins slightly through the bend. So $K$ sits below 0.5 — roughly 0.3 to 0.5 across ordinary work, and it moves with three things. It moves with the material: a soft aluminium and a high-strength steel put the neutral axis in different places. It moves with the ratio of bend radius to thickness, climbing towards 0.5 as the radius gets large relative to the thickness, because a gentle bend barely disturbs the metal, and falling towards 0.3 at tight radii where the inside surface is heavily upset. And it moves with the forming method: air bending, bottoming and coining are three different processes and give three different K-factors in the same sheet.

Which is why a shop that bends accurately does not take $K$ from a table. It measures its own. The procedure takes twenty minutes and pays for itself the first day. Bend a coupon of the real material to a known angle over the real punch in the real die. Measure the two flanges to the outside and add them; measure the flat blank you started with. The difference between the blank and the summed outside flanges is the bend deduction, and from the deduction and the outside setback the allowance follows, and from the allowance this equation run backwards gives $K$. Write it on the tooling cabinet against the material, the thickness, the punch radius, the die opening and the method. Change any one of those and you have a different number.

Now the trap that catches everybody once. $\theta$ here is the angle of bend — how far the material is rotated — not the included angle between the finished flanges. They are supplementary: included $=180° - \theta$. For a square corner both are 90°, which is precisely why the mistake survives so long. Every right-angle bend in the shop checks out, and then the first 45° flange comes off the brake wrong by a visible amount. If a drawing calls out the angle between the flanges, subtract it from 180 before it goes in here.

The second trap is the radius. $R$ is the radius on the inside of the bend. The outside radius is larger by exactly one material thickness, and a print that dimensions the outside will quietly give you a bend allowance one thickness too long at every bend. Worth knowing too: in air bending you do not really choose the inside radius. The punch nose sets a lower bound, but the radius the material actually takes is governed by the die opening — a wider die makes a larger radius with the same punch in the same sheet. That is why changing the die to reduce tonnage silently changes the radius, the K-factor and therefore every flat pattern developed for the old setup.

And springback is not in this equation. Nothing on this page is. The allowance describes the geometry while the punch is down and the material is loaded. When the ram lifts, the elastic portion of the strain recovers: the angle opens, the radius grows, and the part you measure is not the part the tool made. It is a bigger effect in high-strength steels and in aluminium than most people expect, and it is dealt with by overbending, by bottoming or coining, or by measuring the recovery on your own material and compensating — never by arithmetic on this page.