# Generalized Thrust Network Analysis of Triangular Masonry Cross Vaults Inspired by [PERSON]
## 1 Introduction  
In 1954, the Italian engineer [PERSON] (1926-1981) designed a reinforced concrete cross vault, characterized by a triangular layout and obtained by assembling three triangular webs with parabolic profiles [1]. Although the vault geometry was a unique and pioneering result of structural optimization, it remained unbuilt and its project was only mentioned in reference [2] as \"exemplary for its modular conception of the structure based on the repetition of cylindrical shells\". The smart geometry invented by [PERSON] is revisited here to explore its potential realization as a masonry structure, as first proposed in references [3, 4], and it is then taken for inspiration in considering the design of a more general class of triangular masonry cross vaults. As a reliable structural assessment methodology of masonry vaults, the static (or safe) theorem of limit analysis is adopted, following its formulation by [PERSON] under the assumptions of null tensile strength, infinite compressive strength, and no-sliding behavior [5, 6, 7]. Departing from the classical slicing technique (for a recent application, see, e.g., reference [8]), which models a typical masonry vault as a series of independent arch slices, thrust network analysis (TNA) has established itself as a powerful and automatic technique for the 3D funicular analysis of masonry vaults under gravitational loads [9, 10]. It consists of finding a discrete network of compressive forces (or thrusts) in equilibrium with the vault self-weight, as reduced to nodal forces by tributary areas, and completely contained within the vault thickness. Peculiar to the method is the introduction of a form diagram, prescribing the network in horizontal projection and entailing a specific pattern of the internal force flow. Consequently, the unknown nodal heights and force densities(i.e., the thrust-to-length ratios of the branches) are determined through a suitable nonlinear optimization problem enforcing nodal equilibrium equations and static admissibility conditions [11; 12; 13; 14; 15; 16; 17]. The TNA is mechanically interpreted by relating the network nodes to rigid masonry blocks ideally constituting the vault. Accordingly, the nodal equilibrium amounts to the equilibrium of the blocks under their self-weight and the interface forces, regarded as the thrusts in the network branches. In reference [18], [PERSON] observed that \"the equilibrium of a masonry unit, or voussoir, in the vault does not require all forces to meet at one point in 3D space\". In other words, the TNA assumption that all thrusts applied to a network node converge to that node is a sufficient, but not necessary, rotational equilibrium condition for the blocks. This observation has motivated the recent introduction of the generalized thrust network analysis (GTNA) [19]. Among its advantages, the GTNA allows an extension of the set of equilibrated and statically admissible stress states in the vault and translates the classical minimum-thrust problem for the safe assessment of the vault into a simple linear programming problem. Besides the intrinsically discrete formulations discussed above, continuous methods have been proposed for the static limit analysis of masonry vaults. The thrust surface analysis approach describes the statics of a masonry vault by searching for a compressed thrust membrane within the vault thickness in equilibrium with its self-weight [20; 21; 22; 23; 24]. A generalization of that approach is offered by shell-based static limit analysis formulations, which overcome the thrust membrane concept by assuming that general shell stress states can arise within the vault to resist external actions [25; 26; 27; 28; 29; 30; 31]. Thrust surface analysis and shell-based static limit analysis can be regarded as the counterpart continuous methods of TNA and GTNA, respectively. A general and comprehensive review of computational methods for the structural analysis of masonry structures can be found in reference [32].