Weingarten surfaces, the Codazzi equations and the membrane theory for the formfinding of tension structures, shells and vaults

Journal article, 2025

The formfinding of shell, masonry vault and fabric or cable net structures involves obtaining a geometry and a stress state in equilibrium under a dominant load case. In so doing we can give the formfinding model very different properties to those of the finished structure, for example modelling a fabric structure by a soap film. In order to obtain a formfound geometry we need to make a number of decisions about the geometrical and structural properties we aim to achieve. These decisions can be couched in a number of ways, and in this paper we concentrate on statements about the principal curvatures of the surface and the principal membrane stresses. A Weingarten surface has a functional relationship between the mean of the two principal curvatures and their product – the Gaussian curvature. Examples include minimal surfaces, surfaces of constant mean curvature and surfaces of constant Gaussian curvature. It is well known that the Codazzi equations enable us to obtain a spacing of the principal curvature lines on a Weingarten surface that is not arbitrary, but determined by the principal curvatures. The Codazzi equations are purely geometric but they are identical with the in plane components of the membrane equilibrium equations for the case when there is no tangential load. Zero-length or close-coiled springs are used in the Anglepoise lamp and have a length that is proportional to the tension, once the tension is sufficient for the coils to separate. We demonstrate that surfaces constructed from a fine grid of zero-length springs have a membrane stress such that the product of the principal stress is constant. If we add an isotropic stress we arrive at a condition similar to that for the curvature of a linear Weingarten surface. We provide a number of examples of the application of these ideas.