A Mathematical Solution to Predict the Shape of a Catenary Arch or Shell Subjected to Non-uniform Loads Mitchell Gohnert School of Civil and Environmental Engineering, University of the Witwatersrand, P.O. Wits, 2050, South Africa
[email protected] Keywords: Shells, arches, catenary, finite element, form finding, low-cost housing Abstract. Thin shell vaults have been used extensively in antiquity. These structures were constructed primarily of masonry and are therefore only capable of resisting a compressive force. In recent times, these types of structures are being applied to arches and shells, especially in the design of low-cost housing using earth bricks. Similarly, these types of structures are only capable of resisting compressive forces. However, the problem is defining the shape. The mathematical equation for a catenary curve is well known, but the equation is only valid for a uniformly distributed load (i.e., self-weight). The design of catenary shell structures are therefore severely limited, since reality requires shells and arches to withstand point and varying uniform loads— maintenance, snow and wind loads are examples of non-uniform distributed loads that must be accounted for in the design. The proposed method requires the catenary shell or arch to be divided into elements, similar to finite elements. However, the first element at a support is solved using equilibrium equations, which determines the magnitude and direction of the compressive thrust- lines in successive elements. This approach allows the curve to naturally grow and form into a pure compression structure, without prior knowledge of the required shape. This method differs radically from conventional finite element methods, which requires the shape to be known and defined at the start of the analysis.