# Generalized Thrust Network Analysis of Triangular Masonry Cross Vaults Inspired by [PERSON]
## 4 Structural Analysis of the Musmeci Vault Reinvented as a Masonry Structure  
In this section, a masonry instance of the Musmeci vault is considered and its structural safety is investigated using GTNA. For complying with the original geometry, the vault thickness is assumed to be the only design parameter. A uniform normalized thickness \(h/2a=0.02\) is initially considered, corresponding to a thickness \(h=25\,\text{cm}\). Outward horizontal settlements are considered in the rib planes at the corner supports for a minimum-thrust investigation. To that end, the beam network topologies depicted in plan view in Figure 6 are adopted for the discretization of the vault mid-surface. The network refinement is controlled by a refinement parameter \(N\), coinciding with the number of beams along the half-web (the case \(N=4\) is illustrated in the figure). Panels (a) and (c), respectively, refer to orthogonal and fan-like networks, obtained by adapting the form diagrams proposed for a classical TNA of cross vaults with a square layout in reference [17]. Distinct patterns of internal force flows underlie the two network topologies. In particular, the orthogonal topology presupposes that the self-weight of the webs is first conveyed to the ribs, which are then in charge of its transmission to the corner supports. Conversely, a direct flow of the self-weight of the webs toward the corner supports is envisaged within the fan-like topology. Motivated by the potential blending of the two internal force flow patterns, the orthogonal and fan-like topologies are conveniently interpreted as the two limit cases, respectively, corresponding to \(\lambda=0\) and \(\lambda=1\), of a continuous sequence of network topologies indexed by a shape parameter \(\lambda\in[0,1]\). By way of example, panel (b) of Figure 6 shows the beam network for the choice \(\lambda=0.5\). Thanks to the problem symmetry, in numerical simulations, one-sixth of the vault is considered and suitable symmetry conditions are accounted for on the symmetry planes. At the corner support, additional static admissibility conditions are prescribed, requiring the line of action of the resultant constraint reaction to be contained within the springing section. The left and mid columns of Figure 7 show the generalized thrust networks of minimum thrust for the Musmeci vault obtained assuming, from top to bottom, the values \(\lambda=\{0,\,0.5,\,1\}\) for the network shape parameter. A refinement parameter \(N=16\) is considered. For the typical network beam \(b\), a pipe with axis on the straight line \(\tau^{b}\) and diameter proportional to \(\tilde{N}^{b}\) (Section 3) is plotted. Green [resp., blue] dots are used to identify the beam end-sections where the generalized thrust network is tangent to the vault extrados [resp., intrados]. The orthogonal network prompts the formation of a series of web arches supported by the ribs, which thus play a fundamental static role (Figure 7a,b). Conversely, the fan-like network is characterized by the formation of fan arches, that interact on the vault ridge line, and directly rest on the corner supports (Figure 7g,h). An intermediate static regime is obtained for an intermediate value of the shape parameter, with the ribs that are only loaded in the regions close to the corner supports (Figure 7d,e). The right column of Figure 7 shows the settlement mechanisms of minimum thrust, which are kinematically dual to the obtained generalized thrust networks. A progressive transformation is recognized, from one driven by the settlement of the rib arches and triggering a settlement of the web arches for the orthogonal network, to one due to the settlement of the fan arches for the fan-like network. The obtained values of the normalized thrust are \(H/V=\{1.039,\,1.054,\,1.079\}\) for \(\lambda=\{0,\,0.5,\,1\}\), respectively. Accordingly, the orthogonal topology emerges as the most efficient pattern for the internal force flow. The minimum thickness of the Musmeci vault reinvented as a masonry structure is then investigated.