Lower Bound Limit Analysis of Masonry Vaults Under General Load Conditions

Here the lower-bound Theorem of Limit Analysis is applied to historic masonry vaults modeled as continuous no-tension bodies where singular stress fields can take place. To this aim, by following the line traced by Heyman [1], a unilateral membrane entirely contained into the thickness of the vault is searched. According to the approach in [2], the problem is reduced to a second order partial differential equation relating the shape function of the membrane and the Airy function associated to the equilibrium stress field. The employ of a suitable numerical procedure together with the formulation of a suitable constrained optimization problem allow us for finding solutions for vaults of general shapes and in presence of arbitrary loads. Moreover, the optimization procedure allows for exploring the entire spectrum of the load bearing capacity of the vault. The effectiveness of the proposed approach is discussed for the representative case of a barrel vault, for which the Geometric Factor of Safety (GFS) under the self-weight and the seismic capacity are studied.