Issue 51

C. Anselmi et alii, Frattura ed Integrità Strutturale, 51 (2020) 486-503; DOI: 10.3221/IGF-ESIS.51.37 500 adopt a different modeling of the oculus (circular hole). However, as for the first modeling, the same horizontal maximum width of 5.8 meters assumed in [19] as average diameter was utilized: modeling 1 (later called Mod1). The oculus has been modeled with a hexagonal shape, but the drum is not considered in all its height, because the basic section of our calculation model was placed in the horizontal plane that cuts in half the aforementioned oculus. To this plan belong the faces (i+1) (Fig.2), i.e. those that have the smallest area and are therefore more prone to crushing failures. modeling 2 (later called Mod2). To be able to consider the whole upper part of the drum, the oculus has been modeled with a lozenge shape having vertical axis equal to the height of the drum (13 m), and the horizontal one equal to 5.8 m. The mechanical and geometrical characteristics assumed for the dome are listed below in Tabs. 3 and 4. Mechanical data Values and units of measure friction coefficient f c 0.75 masonry density γ m 18.5 kN/m 3 sandstone density γ s 25 kN/m 3 density of covering material (as overload) γ c 10 kN/m 3 limit compressive strength σ 0 -4000 kN/m 2 lantern weight on a segment of dome PL 469 kN Table 3 : Mechanical data of S. Maria del Fiore dome. Geometric data In the plane passing from ribs dimensions (m) In the plane that cuts the sails dimensions (m) Heights and other dimensions (m) outer diameter 54 49.88 internal diameter 45 41.57 radius of arch pointed-fifth 36 dome thickness at the base 4.5 4.15 outside diameter of the lantern base 7 6.47 drum thickness 5 4.6 dome height 35.75 drum height for Mod1 7.5 height of the drum upper block for Mod1 4.6 height of the drum lower block for Mod1 2.9 drum height for Mod2 13 height of the drum upper block for Mod2 7.5 height of the drum lower block for Mod2 5.5 average diameter of the oculus 5.8 Table 4 : Geometric data of S. Maria del Fiore dome. With the program implemented in Excel, for Mod1 and Mod2, three different analyses were performed: (1) the classical one already applied to the theoretical dome (see previous sub-paragraph 7.1); (2) a similar analysis but applied to the dome in the current state, taking into account the cracks already present; (3) a comparison between the pressure curve obtained from M.Como and shown in [19] and those obtained with Mod2. More precisely, as regards points (1) and (2), the value of the collapse load multiplier obtained is the same, so this value can provide an indication, even if approximate, about the safety of the actual dome subject to vertical loads. On the other hand, the almost coincident values obtained with Mod2 are justified by the fact that, as for the Mod1, the failure occurs by crushing at the lower area section, corresponding to interface between the two blocks of the drum, belonging to the plane