Bend allowance
A Socratic walk-through of bend allowance — reasoned out one step at a time, not lectured.
The question we started with
THE QUESTION #Why is the flat blank cut shorter than the sum of the sides it will be folded into?
You want a right-angled bracket with two 50 mm legs, in 2 mm sheet. Obvious: cut a strip 100 mm long, fold it in the middle. The shop cuts it at about 96.5 mm.
Three and a half millimetres have gone missing, and nothing was removed — no swarf, no trimming, one operation from strip to finished part. Metal is not compressible by any such amount. So either the finished legs are not 50 mm, or we have been measuring something that is not there. Which do you suspect?
Reasoning it through
REASONING #Ask where the "50 mm" is measured to. On a drawing it runs to the corner — but look at the actual part: there is no corner. A press brake cannot produce a mathematical vertex; it produces an arc of some inside radius, because sheet folded sharply enough simply cracks. The outside corner on the drawing is an apex the metal never reaches.
How much phantom length does that apex hold? For a 90° bend with inside radius R on sheet of thickness T, each flat face stops where the curve begins, and the apex sits R + T beyond that tangent point. With R = 2 mm and T = 2 mm that is 4 mm a side, so "50 + 50" quietly claims 8 mm of material at a corner occupied instead by an arc.
So the sum is really: flat + flat + arc. Each flat is 50 − 4 = 46 mm. The only remaining question is how long the arc is — and here the problem stops being drawing geometry and becomes a question about the metal.
Consider what the bend does through the thickness. The outer surface has further to travel round the curve than the inner, so it is stretched and the inner compressed. Somewhere between lies a surface that is neither — whose length in the finished part equals its length in the flat strip. That is the neutral axis, and it is the whole of the accounting: blank length is conserved along that one fibre and nowhere else, which is why measuring the outside of the part against the outside of the blank never reconciles.
Where is the neutral surface? The tempting answer is mid-thickness. It is not: bending also thins the sheet, compression on the inside not simply mirroring tension on the outside, so the surface sits closer to the inside face. Fabricators express its position as a fraction k of the thickness measured from that face, and the values in common use sit below half. Take k = 0.44 as an illustration (a recalled typical value, not a constant of nature — it varies with material, with the ratio of radius to thickness, and with whether the bend is air-bent or bottomed).
Then the arc's length, on that fibre, for a quarter turn of radius R + kT:
(π/2) × (2 + 0.44 × 2) = 1.5708 × 2.88 = 4.52 mm.
And the blank is 46 + 4.52 + 46 = 96.52 mm. The missing 3.48 mm is simply 8 mm of apex we counted twice against 4.52 mm of arc we forgot to count once.
Is the shop's table of deductions — one line per material, thickness and tool — craft superstition? No: k cannot be predicted reliably enough to trust, depending as it does on the alloy's behaviour past yield and on the tooling. So the shop bends a test piece, measures it, and back-calculates. That is empirical calibration of a genuine constant of their setup, which is why a fabricator will not work from someone else's table.
The analogy
THE ANALOGY #Think of measuring a walking route on a map where two streets meet at a right angle. The map says 50 metres up to the corner and 50 metres back down — but the walker rounds the corner on a curve and covers less ground, because the corner point is on the map and not under anyone's feet.
the walker chooses their line freely and nothing about them is stretched, whereas in the metal every fibre is forced to a length set by its distance from the bend centre — so the shortfall is not a shortcut taken but a bookkeeping fact about which fibre we agreed to count.
Clarifying the model
THE MODEL #The point most often mangled is that bend allowance and bend deduction are not two names for one number. The allowance is the arc's real length along the neutral axis — what you add when building the blank up from tangent point to tangent point. The deduction is what you subtract from the sum of the outside dimensions: the doubled setback minus the allowance, 2 × 4 − 4.52 = 3.48 mm here. They differ because they start from different measurements of the same part.
One consequence worth holding: because the setback grows with R + T while the arc grows only as (π/2)(R + kT), the deduction gets larger as the radius or the sheet gets thicker — so a job re-run in heavier gauge on the same tooling needs a new blank, and a change of nothing but the die opening changes the correct blank length while every drawing dimension stays identical.
Here is the test. If the mechanism really is neutral-axis arc geometry, a single fixed deduction cannot stay correct across radii, thicknesses and materials; it must scale with them in the way above. Bend the same part in two thicknesses and measure: if the deduction does not grow, the account is wrong. The thinning story has its own falsifier — if careful measurement placed the neutral surface at or beyond mid-thickness (k ≥ 0.5) for ordinary bends, the explanation for why it sits inboard would have to be abandoned.
A picture of it
THE PICTURE #How to readThis is a packet diagram repurposed as a length map — read each numbered unit as one millimetre of strip, not one bit. Walk left to right along the blank as the guillotine would cut it: two flat regions of 46 mm, each 50 mm less the 4 mm of setback belonging to the corner, with the bend arc of about 4.5 mm between them (rounded to whole millimetres; the arithmetic gives 96.52 mm). The missing 3.48 mm is then visible as a subtraction — the drawing charges 8 mm for two outside corners and the metal supplies 4.5 mm of arc in their place.
What became clearer
WHAT CLEARED #Nothing goes missing. A drawing dimensions a part to corners the metal replaces with an arc, and a folded part conserves its length along exactly one surface through the thickness — the neutral axis, nearer the inside face because bending thins the sheet. Cut the blank to the length of that fibre and the legs come out right. The shop's deduction table is that conservation law, calibrated on their own tooling because its one term that matters cannot be looked up.
Where to go next
ONWARD #- Why springback means the tool must over-bend, and why the amount is again a property of the shop's setup rather than the drawing.
- How the minimum bend radius before cracking depends on grain direction in rolled sheet, and why layouts specify which way the part lies on the strip.
Key terms
TERMS #| Term | What it means |
|---|---|
| Neutral axis | the surface through the thickness whose length is unchanged by bending; the fibre along which blank length is conserved. |
| k-factor | the position of the neutral axis expressed as a fraction of sheet thickness measured from the inside face. |
| Bend allowance | the true arc length of the bend along the neutral axis, added between the two tangent points. |
| Bend deduction | the amount subtracted from the sum of the outside dimensions to give blank length. |
Every term the collection defines is gathered in the glossary.