12:30 Jan 9 

I think the "cintres de très grands rayons" refers to the curvature of the pipe cross-section, as it immediately follows a table showing pipe diameters against surface area. There is no reference to bending, as in a beam. On the other hand, it seems odd to use this method for something as easy to calculate as the surface area of a cylinder, but as you rightly point out, the painter may not be much of a mathematician.

12:04 Jan 9 

One also has to take into account that the painting contractor estimating the job is unlikely to be either a mathematician or an engineer.

09:35 Jan 9 

In the context of painting, it kind of makes sense, since you need to know what area will be covered. Mathematically, you might describe the fibre as the "path", and the length of the path as the line integral along the path. The area is of course the surface integral. As for translating "fibre", "effective length" might work.

09:22 Jan 9 

We do understand "fibre" the same way, though I had been thinking simply of the fibre staying on the surface of the pipe when it went round a bend particularly because of the bit about "surface des éléments rectilignes". However, if "cintres de très grands rayons" means the radius of the pipe, rather than, as I understand it, the arc described by the length of pipe, then you are right.

22:40 Jan 8 

The connection with beams is, as indicated, "a convenient descriptive shorthand"; the idea is that the imaginary fibres are contained within the beam/structure even when bent or twisted, so the length is measured, as it were, without leaving the surface (or interior) of the object, rather than being a straight line in Cartesian space. For a straight cylinder, this would be either a spiral or a circle. If my understanding is correct, this would make "length" or "dimension" a bit too general, though by no means incorrect.

17:39 Jan 8 

So "fibre" (-er) then.

14:51 Jan 8 

In the theory of beams, such structures are imagined to be composed of infinitesimal fibres. These have no physical reality (except in composite materials), but "fibre" (or US fiber) is used as a convenient descriptive shorthand. See eg http://en.wikipedia.org/wiki/Bending

14:35 Jan 8 

something to do with the longest GRAIN