Designing funicular grids with planar quads using isotropic Linear-Weingarten surfaces

Abstract

The design of curved structural building envelopes such as gridshells or cable-net roofs is challenging as it requires to account for a wide variety of constraints. In particular, the chosen shape must be mechanically efficient, fabricable, and fit the site geometry. This article shows how a family of surfaces, called isotropic Linear Weingarten (iLW) surfaces, may fulfil all these constraints together, and be used as an intuitive design tool. We start by showing that these shapes are funicular for a uniform vertical load, and that principal projected stress lines form a conjugate net. This allows in particular for the design of gridshells with planar faces and low bending moments or for the design of self-stressed cable-nets cladded by planar glass panels. We then propose a discrete model of iLW surfaces based on recent advances in discrete differential geometry. We use this model to propose an optimization-based generation method, for which the inputs are the boundary curves and two control parameters. We demonstrate the shape potential on several examples.