The analytic and numerical form-finding of minimal surfaces and their application as shell structures

Paper in proceeding, 2021

One usually thinks of minimal surfaces as being used for fabric and cable net structures. However, they can also be used as the geometry for a shell structure, in which case the stresses under load will depart from a uniform tension, and may include both tensile and compressive stresses. Any minimal surface with principal curvature coordinates can be constructed analytically using the fact that it can be expressed by a single function of a complex variable. But the analytical approach is, in general, quite complicated. We show a quick and easy numerical approach that automatically produces a minimal surface and the principal curvature coordinates at the same time applicable to any minimal surface whose boundaries are either principal curvature or asymptotic directions, or a combination of the two. Straight lines and cable boundaries form asymptotic lines, which are oriented at 45° angle to the principal directions on a minimal surface, and a surface that is normal to a sphere has a principal curvature direction as its boundary. If the surface is materialised with members following asymptotic directions, any load acting over a small patch of the surface is transferred as a force couple acting along with the asymptotic members that bound the patch. The same happens in continuous surfaces, and if the small patch is taken to the limit, the point load is transferred as a moment along with asymptotic directions.