A constrained force density method for the funicular analysis and design of arches, domes and vaults

A computational approach is proposed to cope with the assessment/design of arcuated structures through funicular analysis. As investigated in the literature, the equilibrium of funicular networks can be conveniently handled through the force density method, using independent sets of branches in the case of networks with fixed plan geometry. In this contribution, the minimization/maximization of the horizontal thrusts of funicular networks is stated both in terms of any set of independent force densities and of the height of the restrained nodes. The dependent force densities are written as a function of the independent ones by handling the horizontal equilibrium through the reduced row echelon form of its augmented matrix. A suitable set of local constraints is formulated to prescribe lower and upper bounds for the vertical coordinates of both restrained and unrestrained vertices of the network, and to enforce no-tension members. Due to its peculiar form, this problem can be efficiently solved through techniques of sequential convex programming that were originally conceived to handle multi–constrained formulations of size optimization for elastic structures. Both vertical and horizontal loads are considered to assess the predictions of the proposed algorithm that can handle networks with general topology and boundary conditions.