I read a lot of threads here one BPL before and during the design of the tarp. Hoverer I don't know how many of the contributors actually have experience in real material engineering - on the other hand, they might have practical experience. And to be fair, I think this is mostly theory, as I have yet to hear of very many actual tarp failures.
One thread I found while researching is this one
Skip down to Lance Marshall's post Re: "Still, the math is interesting" on 10/18/2009 21:55:47 MDT
On the materials. Cuben fiber is close to a stable fabric, naturally it too has some elongation or creep if put under constant pressure, but I think it will hold up well in a shelter situation, in comparison to the first intended use, yacht sails. Silnylon on the other hand have more stretch so one needs to do separate calculations for either fabric.
I talked about the subject with my math doctor last year while still in school. We planned on visiting the architect teacher who had the appropriate software. Sadly we ended up not doing it as I don't have the material specs nor an exact model of the tarp. With that in mind, we concluded that the most important part for a "catenary cut tarp" is the smooth curve of the ridgeline and edges. The calculations would be done under controlled circumstances while the tarp would be set up on uneven ground, attached to trees and subjected to uniform winds, also providing I could produce an exact replica of the model tarp. Therefore I think trial and error, i.e. manufacturing experience, is a more suitable approach. Non the less, I am sad I didn't put the hours into it, as I've now finished the polytechnic university and would have loved the whole experiment - after all, I am an engineer and thus gear and experiment freak..
That kind of took up the subject of patch design. But if one would make a force model of a tarp in action I'd estimate that for a well built caternary tarp, the forces spread along the edges. For a flat tarp it would depend more on the pitch, the reason for the usually sloppier look. Basically it comes down to Newton I, where the force vector goes along the taut tieout line to the ground and secondly into the tarp. There's just too many factors in a real world situation. Hence, uniform 1/4, half circles.