Our flat sheet is what it is - it is NOT two different lengths (to achieve desired OD and ID).
It is one absolute length.
We can visually see an measure and calculate that the OD and ID are not equal.
We can measure (or calculate) the OD and ID and see that there is a difference of 6.28mm.
These numbers are all verifiable and logical.
So how did the flat sheet go from being one length to being two lengths (with a difference in length of 6.28mm) when bent?
A. The material compresses on the inside of the bend and stretches on the outside of the bend.
A1. The neutral plane (neither stretching or compressing) is generally not midplane in the thickness, but usually closer the inside of the bend than to the outside of the bend.
A2. This means that the material generally stretches more than it compresses.
From the Machinery's Handbook - we see that this stretching/compression is dependent on
Thickness
Material Properties (for example Aluminum is different than Steel. One Steel is different than another Steel...)
Bend Radius (by convention - the inside radius)
Bend Angle (by convention - the "outside" angle, the total angle of bending)
(Also Grain, but we will assume the material is isotropic).
So, we have to calculate the k-factor or the Bend Allowance based on all of these variables.
In addition to the variables listed above - the machine tooling has an effect.
We use k-factor when the tolerances are not particularly close - run a sample on the machine and set the appropriate k-factor.
When the tolerances are tight - then we run multiple tests and record a Bend Allowance Table that might need to be updated month-to-month as the machine tooling wears.
The starting material is shorter and stretched and compressed during bending to get the desired size.
As you say, this question has come up here many times - I have tried to lay out logically so that others searching in the future can work their way though the logic.

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