Singular stress fields for masonry-like vaults

In the present paper, we apply the theorems of limit analysis to vaults modeled as masonry-like materials, that is, unilateral continuous bodies. On allowing for singular stresses, we consider statically admissible stress field concentrated on surfaces lying inside the masonry. Such structures are unilateral membranes, whose geometry is described a la Monge, and the equilibrium of them, under vertical loads, is formulated in the Pucher form. The problem is reduced to a singular partial differential equation of the second order where the shape f and the stress function F appear symmetrically. The unilateral restrictions require that the membrane surface lies in between the extrados and intrados surfaces of the vault and that the stress function be concave. Such a constraint is, in general, not satisfied on a given shape for given loads: in such a case, the shape has to be modified to fit the constraint. In a sense, the unilateral assumption renders the membrane an underdetermined structure that must adapt its shape in order to satisfy the unilateral restrictions. A number of simple examples are presented to illustrate how the method works.