Issue 51

C. Anselmi et alii, Frattura ed Integrità Strutturale, 51 (2020) 486-503; DOI: 10.3221/IGF-ESIS.51.37 501 that cuts the oculus in half. Mod2 involves a slight increase in the volume of the oculus with a consequent slight decrease in the weight of the drum blocks, and this leads to the negligible growth of the α multiplier value. Finally, with regard to point (3), for a slice of dome equal to 1/8 of the entire dome , corresponding to a single sail of the dome subjected to its own weight and to that of the lantern on top, in [19] is shown the construction of the pressure curve that, starting from the upper end of the section of the voussoir at the crown, is tangent to the intrados of the dome near the haunches, and intercepts the center of the section at the drum base (Fig.20a). Instead, still using the program implemented in Excel, which analyzes a segment equal to 1/16 of the dome , two similar pressure curves were obtained using two different objectives in the optimization problem: for the first, it was imposed that the multiplier has value zero, in compliance with the condition that the curve intercepts the center of the section at the base of the drum (Fig.20b); for the second, it was imposed that the horizontal thrust transferred from the dome to the drum assumes the value of 2000 kN, in accordance with that of 400 t estimated by Como [19] (Fig.20c). The results of the analyses referred to in points (1) and (2) are summarized in Tab. 5 (in it the ratios Wc1/Wa1 and Wc2/Wa2 provide indicative coefficients on collapse safety), while the most significant collapse mechanisms are shown in Figs. 18 and 19. Instead, for the comparison referred to in point (3), the pressure curve in [19] and those obtained in this paper pursuing two different objectives are shown in Fig. 20. dome slice analysis Mod1  multiplier Actual weight Wa1(kN) Weight at collapse Wc1 (kN) Ratio Wc1/Wa1 Mod2  multiplier Actual weight Wa2(kN) Weight at collapse Wc2 (kN) Ratio Wc2/Wa2 without and with cracks 30.096 23,900 122,621 5.131 30.226 27,272 126,419 4.635 Table 5 : Comparison between the values obtained with Mod1 and Mod2 Figure 18 : Collapse mechanism of S. Maria del Fiore dome (Mod1) Figure 19 : Collapse mechanism of S. Maria del Fiore dome (Mod2)