Mathematical model of a moment-less arch

Abstract

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.