Maxwell–Rankine stress functions of membrane shells and their relation to that of planar funicular gridshells

Abstract

This paper investigates the correlations among 3D graphic statics, Maxwell–Rankine stress function, and Pucher's equation (the governing equation of membrane shells). When there is only vertical load, a solution of Pucher's equation can be converted into a Maxwell–Rankine stress function. The resulting stress function can be further discretized into Rankine reciprocal diagrams. The analogous membrane shell will simultaneously be discretized into a gridshell, which has planar polygonal faces and prismatoidal Rankine diagrams. A family of analytical free-edge membrane shells is also presented. Once these membrane shells are discretized, they can have co-apex pyramidal Rankine diagrams. The resulting discretized Rankine diagrams provide an intuitive way for structure designers to perceive the equilibrium of shell structures.