Limit equilibrium analysis and the minimum thickness of circular masonry arches to withstand lateral inertial loading

In this paper, we compute the location of the imminent hinges and the minimum thickness, t, of a circular masonry arch with mid-thickness radius, R, and embracing angle, beta, which can just sustain its own weight together with a given level of a horizontal ground acceleration, epsilon g. Motivated from the recent growing interest in identifying the limit equilibrium states of historic structures in earthquake prone areas, this paper shows that the value of the minimum horizontal acceleration that is needed to convert an arch with slenderness (t/R, beta) into a four-hinge mechanism depends on the direction of the rupture at the imminent hinge locations. This result is obtained with a variational formulation and the application of the principle of stationary potential energy, and it is shown that a circular arch becomes a mechanism with vertical ruptures when subjected to a horizontal ground acceleration that is slightly lower than the horizontal acceleration needed to create a mechanism with radial ruptures. The paper explains that the multiplicity on the solution for the minimum uplift acceleration is a direct consequence of the multiple possible ways that a masonry arch with finite thickness may rupture at a given location. The paper further confirms that the results obtained with commercially available distinct element software are in very good agreement with the rigorous solution.