Hi Andrés, thank you!

I have found with experience that a formula to calculate the binding correction for a saddle stitch binding, not more than 4 pages in a signature (only one fold in the middle) usually is 1/2 the circumference of a circle with radius r, being r the thickness of the paper times the amount of physical signatures. Something like: \frac{π \times t \times s}{2}. I have not tried a formula for more folds in a signature. 

The thing is, you have to recalculate for each signature fold. The innermost will have displacement zero, the second one a little bit more, the third a little bit more, and on and on until we reach the outermost signature. 

However, it is necessary to say that unless you are using a particularly thick paper, or you are using too many pages for a saddle stitch bind (my personal and professional opinion would be no more than 80, but better 60), the displacement is usually negligible. 

For example, let's say the paper thickness is 0.1 mm (which is 0.0001 meters) and there are 40 pages in 10 signatures. The radius would be 0.0001 meters per signature * 10 signatures = 0.001 meters. Then, half the circumference would be 1/2 * 3.14159 * 0.001 meters ≈  0.0016 meters or 1.6mm at the last signature. Unless there was a displacement of more than, say, 4 mm, I wouldn't worry too much about it. 

E.g. our Bonn architectural guides (https://www.dreiviertelhaus.de/reihen/afb/) are saddle-stitched booklets, usually have 60 pages and are printed on 115g coated paper. With coated paper you can’t really derivate the thickness from grammage. The horizontal page shift is visible if you look carefully. But since the paper is not very transparent, you only see the problem if you look for it. I don’t arrange the pages myself, don’t know the signature size, and I can’t request specials, since we must print cheaply to keep the low price (had to raise it from 5 to 8 € anyway, half of it goes to the wholesaler).

Hraban