# Generalized Thrust Network Analysis of Triangular Masonry Cross Vaults Inspired by [PERSON]
## 2 Musmeci Triangular Parabolic Cross Vault  
The structure referred to here as the Musmeci vault consists of a reinforced concrete parabolic cross vault with an equilateral triangular bay. It was conceived by [PERSON], in collaboration with the architect [PERSON], in 1954, as a module forming, through four-fold repetition, the cover of a rural market in southern Italy [1; 2]. The cover, which accommodated the rhomboidal layout of the market space, was the result of [PERSON]'s early investigations on structural design optimization. An original sketch of the planar view and a reconstructed perspective view of the whole structure are depicted in Figure 1. In detail, each vault was formed by three triangular webs with parabolic profile, characterized by external arches with a rise \(f=3.65\,\mathrm{m}\) and span \(2a=12.50\,\mathrm{m}\). The intersection of the three webs generated the ribs of the vault as semi-parabolic arches with a rise \(f\) and span \(b=a/\cos\left(30^{\circ}\right)=2a/\sqrt{3}=7.22\,\)m. In the following, the equilibrium design approach developed within the original Musmeci structural model [1] is briefly described, as also reviewed in references [3; 4]. For the typical triangular vault to be considered, the Cartesian reference frame shown in Figure 2 is introduced. A slicing technique is adopted to find an equilibrated membrane stress state in the parabolic webs. By using a symmetry argument, only the web whose ridge line is parallel to the \(y\)-axis is considered. It is discretized into a family of independent parabolic web arches of infinitesimal width \(\mathrm{d}y\), with a span linearly increasing from \(0\) to \(2a\), and with mid-curve parameterized as follows:  
\[z(x)=\frac{f}{a^{2}}x^{2}. \tag{1}\]  
Figure 1: Musmeci triangular parabolic cross vault: (**a**) original sketch of the planar view, reproduced from reference [1], and (**b**) reconstructed three-dimensional view. Figure 2: Musmeci triangular parabolic cross vault: structural model [4]. Motivated by the web shallownness, the web self-weight is assumed to be approximately uniform in horizontal projection and with a magnitude \(q\) per unit surface. It follows that the vertical and horizontal actions (per unit \(\mathrm{d}y\) in horizontal projection) transmitted by the web arch of span \(2x\) to the rib arch are, respectively, given by:  
\[V_{\mathrm{web}}(x)=qx,\quad H_{\mathrm{web}}=\frac{qa^{2}}{2f}. \tag{2}\]  
Consequently, the rib arch, which is parameterized in terms of the abscissa \(\xi=x/\cos(30^{\circ})=2x/\sqrt{3}\) as  
\[z(\xi)=\frac{f}{b^{2}}\xi^{2}, \tag{3}\]  
is subjected to the vertical and horizontal loads (per unit \(\mathrm{d}\xi\) in horizontal projection), respectively, given by:  
\[p(\xi)=2V_{\mathrm{web}}(x(\xi))\frac{\mathrm{d}y}{\mathrm{d}\xi}=\frac{\sqrt {3}}{2}q\xi,\quad s(\xi)=2H_{\mathrm{web}}\cos{(30^{\circ})}\frac{\mathrm{d}y} {\mathrm{d}\xi}=\frac{3\sqrt{3}}{16}\frac{qb^{2}}{f}. \tag{4}\]  
Since the rib is subjected to a vertical load \(p\) linearly varying from the crown to the abutment, [PERSON] inferred that the thrust line differs from the parabolic rib mid-curve. However, he realized that the constant horizontal load \(s\) was capable of bringing the former very close to the latter, the maximum eccentricity resulting in a few centimeters [1]. A structural analysis of the ribs was carried out by observing that, \"Since the flexural stiffness of rib is practically negligible at the abutment, where the section is theoretically reduced to a point, and at the crown, where the angle between webs becomes zero, the behavior of the rib can be considered similar to that of a three-hinged semi-arch\".