Form-finding structures

Abstract

Inherently characterized by the interaction of geometry and forces, the unique nature of long span dome, shell, and membrane structures readily allows collaboration between architects and engineers in the examination of their optimal form. Through the elimination of bending and shear forces in the structure, less material and reinforcement is needed. By minimizing the use of materials, a form that is economical, sustainable and aesthetically attractive emerges. However, this optimization must be done through formfinding methods, whereby the structure itself defines its own shape based on its figure of equilibrium under applied loads. Unlike free forms which are defined mathematically, form-finding shapes rely on the structure and loads themselves for definition. Before the use of computers, these equilibrium shapes could only be found through cumbersome physical models. As technology has advanced, numerical methods have evolved to solve for the optimal shape. This paper presents a brief history of physical methods formerly used, as well as common applications for these structures. Two numerical methods, the Pucher's equation method and the force-density method (FDM), are then presented. Pucher's equation relies on a prescribed stress resultant throughout the structure, while the forcedensity method relies on prescribed force-to-length ratios in each bar or cable, leading to a single system of linear equations. Advantages and disadvantages of both methods are discussed, as well as examples illustrating the types of structures that can be formed. These methods are shown to be powerful tools that can be generalized to a number of situations with minimal input required by the designer. The structures are able to define themselves, leading to extremely rational and beautiful forms.