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 reexamined. Classical analytical solutions provided by J. Heyman are first rederived and explored in details. Such derivations make obviously use of equilibrium relations. These are complemented by a tangency condition of the resultant thrust force at the haunches' intrados. Later, given the same basic equilibrium conditions, the tangency condition is more correctly restated explicitly 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 nondimensional 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 systematic treatment presented here reveals the implications of an important conceptual difference, which appears to be relevant in 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.
In this paper, an analytical and numerical analysis on the collapse mode of circular masonry arches is presented. Specific reference is made to the so-called Couplet-Heyman problem of finding the minimum thickness necessary for equilibrium of a masonry arch subjected to its own weight (Heyman 1977). The note reports the results of an on-going research project at the University of Bergamo. First, analytical solutions are derived in the spirit of limit analysis, according to the classical three Heyman hypotheses and explicitly obtained in terms of the unknown angular position of the intrados hinge at the haunch, the minimum thickness to radius ratio and the non-dimensional horizontal thrust (Colasante 2007, Cocchetti et al. 2010). Results are then compared to Heyman solution. Though only the first of these three characteristics is perceptibly influenced in engineering terms, especially at increasing opening angle of the arch, the treatment settles an important conceptual difference on the use of the true line of thrust, along the line of Milankovitch work. Second, numerical simulations by the Discrete Element Method (DEM) in a Discontinuous Deformation Analysis (DDA) computational environment are provided, to further support the validity of the obtained solutions, with good overall matching of the obtained results (Rusconi 2008, Rizzi et al. 2010).
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