Catenary Solutions for Arches and Vaults

Abstract

An arch is a basic structural form and has been used extensively in antiquity to create openings, doorways, vaulting, buttressing, bridges, and aqueducts. Most arches are built with masonry, especially those in the past. Masonry is largely a compression material. Tensions will lead to cracking, and eventual deterioration of the arch or vault. However, catenary arches are unique structural forms, characteristically having a pure compressive shape and are, therefore, ideally suited to masonry. Thrust-line equations are derived for the catenary arch subjected to uniform loads (i.e., self-weight), which is based on the original hyperbolic function for catenary shapes. A second solution is also given—a numerical procedure composed of elements or segments. Elements are formulated to predict a pure compressive shape for any number and orientation of point loads, applied externally to a catenary arch. The thrust-line equations provide an exact solution for uniform loading, but the solution for point loads is approximate. The accuracy of the point load method is determined by the number of elements. However, the point load method is unique, in the sense that the catenary arch is grown from a single element to create a pure compressive shape. This is achieved by continuously adding elements to the model until the shape of the arch is defined. Despite being an approximate solution, the point load method was found to converge to the exact solution. The point load method may also be used to determine the thrust-line path in other arches (i.e., circular, parabolic, etc.). Example solutions are provided.