Soap Film Analogy for Anisotropically Stretched Membranes and Cable Nets

Abstract

Analogical physical models are a preferred technique to intuitively grasp complex engineering problems. It is well-known that the equilibrium minimal-surface configuration of membranes under equibiaxial tension can be visually represented by the surface of a soap film under equivalent boundary conditions, but this analogy fails when the stress state is not uniform equibiaxial. We extend to this situation the analogy with soap films. The equilibrium state of an orthotropically tensioned membrane is found by geometrically stretching the shape of a soap film, in a precise manner depending upon the applied state of stress. The procedure is easily done by elaborating digital pictures. The method is mathematically justified under the kinematic hypotheses of small strains and large rotations, and further verified in a parametric design environment. It can also provide an insight into the equilibrium configuration of cable-nets, when the stresses in the warp and weft directions are considerably different. Furthermore, this visualization favors implementing transformable shapes for membranes or orthogonal cable nets, as a consequence of a modification of the ratio of the principal stress components.