Minimum thickness of elliptical masonry arches

In this paper, we compute the minimum thickness and the location of the imminent intrados hinge of symmetric elliptical masonry arches when subjected to their weight. While this problem (Couplet's problem) was solved rigorously for semicircular arches more than a century ago, no results have been available for elliptical arches. Motivated from the recent growing interest in identifying the limit equilibrium states of historic structures, this paper first computes two neighboring physically admissible thrust-lines which can just be located in elliptical arches by adopting either a polar or a cartesian coordinate system. These two distinguishable, physically admissible thrust-lines are neighboring thrust-lines to Hooke's catenary which is not a physically admissible thrust-line as has been shown recently. Accordingly, the paper shows that the answer for the minimum thickness of symmetric elliptical masonry arches is not unique and that it depends on the coordinate system adopted and the associated stereotomy exercised. This result is confirmed by developing a variational formulation after selecting the appropriate directions of the rupture that initiates at the intrados hinge. The paper concludes that Hooke's limiting catenary, although not a physically admissible thrust-line, offers a conservative value for the minimum thickness in most practical configurations.