Four Catenary Architectures to Triangularly Stretch a Circular Membrane Aperture

Abstract

Theoretically discussed and analytically compared are four structural paradigms for the isotropic stretching of circular membrane apertures with three-point tensioning of regular (120 deg) symmetry. These are 1) a film triangle with parabolic catenary systems on the sides, 2) a circular arc-triangular sheet with edge cables, 3) the latter with circular corner cutouts, and 4) directly tensioning the film disk around the circumference with radial catenaries, as here named. The last two are newly proposed to increase structural efficiency: to reduce suspension forces for a given support system footprint (equivalently, to reduce the support footprint for given suspension loads). Mechanics and geometries are rigorously formulated with key design relations symbolically derived. Of the four options, the radial catenary is found with a wide margin to be the most efficient, followed by the arc-circular perimeter with corner cutouts. The parabolic catenary is next, with simple arc-circular perimeter cables being the least efficient. Finally, derivative configuration options and some hardware engineering considerations are briefly touched upon. The latter include quantitative impact on support structure mass, physical feasibility, and system design concerns.