Paper published in the "International Journal of Space Structures": "Weingarten surfaces, the Codazzi equations and the membrane theory for the formfinding of tension structures, shells and vaults"

Our paper, “Weingarten surfaces, the Codazzi equations and the membrane theory for the formfinding of tension structures, shells and vaults”, written together with Chris Williams, has just been published in the International Journal of Space Structures in December 2025.

It is freely available here: https://journals.sagepub.com/doi/10.1177/09560599251388389?utm_source=researchgate.net&utm_medium=article

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 we show they are identical with the in plane components of the membrane equilibrium equations for the case when there is no tangential load.

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 also present shell forms in which forces are distributed within the surface itself, avoiding the need for substantial edge beams, arches, or cables and aligning more closely with structural behaviour observed in nature. It is for instance important for masonry bridges with weak edges.

Many thanks to the Lars Erik Lundberg Scholarship Foundation and Åke och Greta Lissheds Stiftelse for their support of this work.