Can't find a forum that would specifically address this, so...
I'm envisioning a little earthbag roundhouse cabin and would like to make a domed, monolithic roof. I don't like the conventional earthbag "conehead" domes, I'd like something with a shallow spherical curve that I can gutter to harvest rainwater or maybe do a living roof on. I know geodesic domes have awesome free-supporting strength, but I don't want an observatory or grain silo... maybe a small arc-section of a much larger dome would give the desired shallow curve... like a wok, not a hemisphere.
But I'm no physicist (I don't even play one on TV), and I fear a shallow arc from a geodesic wouldn't have the sort of stability and strength a full hemisphere dome would have. Maybe it would need trusses like a regular roof. I don't know. I just thought the geodesic would be a good strong framework to skin with ferrocrete for a solid, monolithic roof that had a gradual enough slope to do rain collection or a green roof on. I know it would be easier to just get a metal silo roof but I'd rather go monolithic than metal; less noise and more tornado-resistant.
Anybody know much about the physics of geodesic domes? Would this "lens" design have enough strength to be self-supporting?
For supporting the weight of anything above it, including a living roof, a parabolic or catenary curve are ideal for the strongest shape. I believe a catenary is best for self-supporting structures, while a parabola is best if there is a vertical load atop it. A "pointy" ogive arch (or dome, if done in 3 dimensions) is close to both of those shapes.
The next best "ideal" shape is a semi-circle or hemi-sphere-- like a Roman arch-- but that is still a bit weaker for bearing a load.
A flatter dome provides a little strength with the shape, but due to the curves, I would think that one might just as well over-engineer it and span the distance with straight beams, or go for a triangular/conical roof using straight-beam trusses. The complexities of curved surfaces might not be worth the small gains you would make.