Laguerre geometry for optimization of gridshell with specified force distribution

TL;DR

This paper introduces a novel method using Laguerre geometry and L-isothermic surfaces to optimize gridshells for specified force distributions, ensuring equilibrium with minimal bending deformation.

Contribution

It develops a geometric approach leveraging Laguerre transformations on L-isothermic surfaces to optimize gridshells for desired force distributions without re-optimizing after transformation.

Findings

Force distributions can be adjusted by parameter tuning on the reference surface.

Laguerre transformation preserves key geometric and static properties.

The method effectively minimizes bending deformation in gridshell design.

Abstract

For the design of gridshells consisting of continuous beams in two directions and quadrilateral faces to cover a large space of architecture, it is important to arrange each row and column of beams along a planar curve and ensure planar faces for constructability and cross-sectional compatibility at joints. It is also important that the gridshell is in equilibrium mainly with axial forces against the design loads; i.e., bending deformation should be avoided. In this study, we first find a continuous shell surface where the principal curvature lines coincide with the principal directions of membrane forces. For this purpose, we utilize the L-isothermic surface, which is a kind of membrane O-surface. Specifically, the generalized Dupin cyclide is used as the reference surface, which has an explicit form of membrane forces with a single arbitrary parameter against normal loads. Various force distributions are obtained as the parameter is adjusted without changing the surface shape. Next, the shell is discretized into a gridshell, and target axial forces are obtained from the section length of the covering region of each node. The axial forces are adjusted by optimizing the cross-sectional radii of the beams with pipe sections to realize the specified target force distribution. Since the Laguerre transformation preserves geometric and static properties of the L-isothermic surface, the force can be adjusted by a simple process without re-optimization to obtain an approximate optimal solution of the gridshell after transformation. The ratio of out-of-plane shear force to the normal load at the node is also evaluated to investigate the effect of deformation on the force distribution.