Arch and Cable Suspended Bridges

An arch bridge is a mainly horizontal deck with an arch as its load-carrying structure. The arch works by transferring the weight and other external vertical loads into a horizontal thrust on the abutments placed at both sides. When the main load on the arch is the homogeneously distributed deck weight, and the arch is properly shaped, the arch rib is subjected to pure compression along its whole extension. Both tensile forces along the arch rib and shear forces transversal to it are absent. Bending moments vanish as well. This has made possible the use of masonry for arch bridges, which uses materials such as stone, brick, and concrete that are strong in compression but very weak to tension and shearing. In the following sections, we determine mathematically this optimal arch shape, which turns out to be a parabola. The novelty is that these well-known results are obtained using just basic algebra and trigonometry, without any mention of infinitesimal calculus, including in the demonstration of the absence of shearing forces. Subsequently, the method is extended to cable suspended bridges.