On the analysis of minimum thickness in circular masonry arches

In this paper, the so-called Couplet-Heyman problem of finding the minimum thickness necessary for equilibrium of a circular masonry arch, with general opening angle, subjected only to its own weight is re-examined. Classical analytical solutions provided by J. Heyman are first re-derived and explored in details. Such derivations make obviously use of equilibrium relations. These are complemented by a tangency condition of the resultant thrust at the haunches’ intrados. Later, given the same basic equilibrium conditions, the tangency condition is more correctly re-stated in terms of the true line of thrust, i.e. the locus of the centers of pressure of the resultant internal forces at each theoretical joint of the arch. Explicit solutions are obtained for the unknown position of the intrados hinge at the haunches, the minimum thickness to radius ratio and the non-dimensional horizontal thrust. As expected from quoted Coulomb’s observations, only the first of these three characteristics is perceptibly influenced, in engineering terms, by the analysis. This occurs more evidently at increasing opening angle of the arch, especially for over-complete arches. On the other hand, the treatment presented here sets an important conceptual difference, which appears to be relevant, in the analysis of the statics of masonry arches. Finally, similar trends are confirmed as well for a Milankovitch-type solution that accounts for the true self-weight distribution along the arch.