Issue 51

C. Anselmi et alii, Frattura ed Integrità Strutturale, 51 (2020) 486-503; DOI: 10.3221/IGF-ESIS.51.37 499 Figure 17 : Collapse mechanism for case 11 (failure at ribs / Lf=2 / Nh=0 / sT=2.5 / multiplier α=13.046) The case of S. Maria del Fiore To validate the optimization program developed in Excel, also on a actual dome, a first approach was made to the analysis of the most famous dome on octagonal drum: the one built by Brunelleschi for the Cathedral of Florence between 1420 and 1436, of which numerous researchers have dealt [19-22]. Obviously - as has already been done by other authors who have carried out various types of analysis - it is assumed that the drum is a regular octagon, and with it also the dome; this allows us to study only one sixteenth of the dome, as for the theoretical one examined above. Actually, compared to the theoretical model assumed, the octagonal dome shows some irregularities concerning geometry and static behavior: - the sides have different lengths each-other, - the drum rests alternately on four solid walls (inclined at 45° with respect to the axis of the nave) and on four ogival arches (two parallel to the axis of the nave and two to that of the transept), with a consequent different crack pattern, also determined from the different structural elements that surround it. Moreover, the architectural reliefs carried out over the centuries present notable differences between them. To overcome the uncertainties related to the dimensions of some elements that characterize the structure, we have taken the dimensions assumed by M. Como [19] using some rounding done by G. Conti [20]. In the Excel program, which for the dome (excluding the drum) provided a discretization in six elements, an further block was introduced on top, at the base of the lantern, in analogy to what was done by Como [19], while the other 30 voussoirs of its discretization have been incorporated by us, in groups of 5, into our 6 blocks. However, despite using the average value of the unit weight of the brick masonry assumed by him, we have taken into account the different unit weight of the portions of masonry made of sandstone present both in the part at the base of the dome and in the ring on top, named serraglio . Anyway, the weight of the dome (including the lantern) estimated by us is 29,272 t, quite close to the value reported by other scholars. The hypothesis that, despite the octagonal form, Brunelleschi have built a rotational dome (and therefore without the use of a supporting framework), remained secret for 500 years, but in recent times shared by all the most recent scholars of masonry structures, starting by Di Pasquale [21], supports the conviction that in correspondence of the ribs there is a continuity of masonry texture, with good interlocking between the bricks. This has allowed us to introduce the assumption that in the (j+1) faces (see Fig.2) of the blocks of the analyzed segment belonging to the meridian plane passing through the rib there is no failure. Furthermore, having the purpose of making a comparison with the pressure curve shown by Como [19], that starting from the top of the dome involves the whole upper part of the drum, 13 meters high, we had to make a second modeling. In fact, the limitation imposed by the program implemented in Excel - just two blocks to discretize the drum - led us to