Mathematical model of a moment-less arch

This paper presents a mathematical model for predicting the geometrical shapes of rigid, two-pin, moment-less arches of constant cross section. The advancement of this work lies in the inclusion of arch self-weight and the ability to produce moment-less arch forms for any span/rise ratio, and any ratio of uniformly distributed load per unit span, w, to uniformly distributed arch weight per unit arch length, q. The model is used to derive the shapes of two classical ‘moment-less’ arch forms: parabolic and catenary, prior to demonstrating a general case, not restricted by the unrealistic load assumptions (absence of q, in the case of a parabolic form, or no w, in the case of a catenary arch). Using the same value of span/rise ratio, and w/q>1, the behaviour of the moment-less and parabolic arches under permanent loading, (w+q), is analysed. Results show the former to be developing much lower stresses than its parabolic rival, even when there are relatively small differences in the two geometries; for a medium span/rise ratio of 4 and w/q=2, differences in the parabolic and moment-less arch geometries would, in practical terms, be viewed as insignificant, but the stresses in them are different.