Funicular analysis of masonry vaults under general loading conditions through a constrained force density method

Funicular analysis is extensively adopted to assess masonry vaults and domes, especially those of historical interest. Surface structures (e.g., shells) are modelled as statically indeterminate networks experiencing only compressive stresses. The no-tension networks are supported along the boundary and are in equilibrium with vertical and horizontal loads applied at the nodes. Following the lower bound theorem of limit analysis, the structure is safe if a thrust network lying between the extrados and the intrados of the vault can be found. In this paper, a minimization problem is formulated for networks of general shape with fixed plan geometry, subjected to arbitrary loads. The Maxwell number (i.e., the sum of the force-times-length products for all the edges in the network) is adopted as objective function, whereas both the so-called force densities and the coordinates of the constrained nodes are minimization unknowns. Suitable constraints enforce bounds on the vertical coordinates of the network nodes, along with the non-positivity of the axial forces in the network members. The ensuing problem is solved through mathematical programming techniques. Applications addressing the equilibrium of curvilinear structures are shown, namely a cross vaults and a circular dome. The optimal thrust networks are compared with those found when adopting a suitable norm of the horizontal thrusts as objective function. Eventually, curvilinear structures subjected to both vertical and horizontal actions occurring in earthquake-prone areas are also dealt with.