Effect of stereotomy on the lower bound value of minimum thickness of semi-circular masonry arches

Constructing structural systems with the minimal required cross section of its members was a strong motivation for progress whole along the history of building. This article investigates the effect of stereotomy on the minimum thickness value of a semi-circular arch made of masonry: a material with negligible tensile strength. The arch is modeled with its center line to which the loading is assigned according to arch length. Here, instead of some scalar parameters such as the rupture angle and the location of the middle hinge on the intrados, the geometry of the stereotomy is associated with a continuous function. Stereotomy-related constraints are introduced to keep the results feasible from an engineering point of view, and it is also demonstrated that they are essential for a well-posed constrained optimization problem. The necessary condition for a non-vanishing lower bound of minimum thickness values is derived analytically considering the assumptions of limit state analysis, infinite friction, and a no-tension material model. The lower bound thickness to radius ratio (t/R) is found to be t/R = 0.0819. A numerical method is introduced to demonstrate the existence of a valid stereotomy at the lower bound. Multiple admissible stereotomy patterns are presented at various minimum thickness values (higher than the lower bound) to demonstrate that suitable stereotomy for a fixed (t/R) ratio is far not unique—in general even the location of the middle hinge and the rupture angle might vary.