Issue 51

C. Anselmi et alii, Frattura ed Integrità Strutturale, 51 (2020) 486-503; DOI: 10.3221/IGF-ESIS.51.37 487 Maria del Fiore dome) the dome shows a much greater load-bearing capacity compared to the case of interlocking between the bricks completely missing (Santa Maria dell’Umiltà dome in Pistoia). In several previous works [1, 2, 3, 4] the authors had already dealt with rotation domes subject to vertical and horizontal loads; by using the static theorem of the limit analysis applied to the dome discretized in rigid macro-blocks of variable shape aligned along the parallels and the meridians, several optimization models have been implemented in order to search for the load collapse multiplier, thus to evaluate the degree of structural safety. It should be noted that many different modelling approaches in the literature were used to investigate the behaviour of masonry constructions, and particularly of domes. Among those, it is worth mentioning the works by O’Dwyers on the funicular analysis of masonry vaults [5], and those using analytical and finite element models [6, 7, 8, 10, 11, 12]. Rigid block models based on limit analysis and contact dynamics were also adopted [9, 13]. In the previously papers, as in this, the yield domain conditions for the quadrilateral interfaces between the macro-blocks are expressed in terms of stress resultants, in observance of the following mechanical assumptions: inability to carry tension, unlimited or limited compressive strength (in the second case the domain is suitably linearized), frictional sliding with dilatancy. This last hypothesis, even if not realistic, has been assumed because in some cases the influence of failure for sliding is negligible in the kinematic mechanism of the domes; that allows us to base the formulation on simple optimization problems rather than more involved nonlinear analyses, with considerable reduction of computational costs. The yield domain conditions for the quadrilateral interfaces between the macro-blocks are expressed in terms of stress resultants, in observance of the following mechanical assumptions: inability to carry tension, unlimited or limited compressive strength (in the second case the domain is suitably linearized), frictional sliding with dilatancy. This last hypothesis, even if not realistic, has been assumed because in some cases the influence of sliding failure is negligible in the kinematic mechanism of the domes; that allows us to base the formulation on simple optimization problems rather than more involved nonlinear analyses, with considerable reduction of computational costs. It should be pointed out that the sliding failure was taken into account assuming a friction coefficient fc = 0.75. This value is perhaps excessive when compared with those taken by other authors in the most recent literature [14, 15, 16, 17] where, for different materials, coefficients varying from 0.63 to 0.69 are used. However, even when a value included into that range is taken into account, the results - static and kinematic - are almost identical to those which are here presented (at least for the tested cases 7 and 11 of Tab. 1). Furthermore, analyzing the response even in the case of a further reduction in the friction coefficient, it was found that the result remains unchanged up to the value fc = 0.2 - 0.22 for which the value of the collapse multiplier decreases and the kinematic mechanism shows an evident failure for sliding. Finally, for fc = 0.18 - 0.20 there is a further reduction of the multiplier, but the collapse mechanism is qualitatively equivalent. Compared to previous tackled problems, in this work there are two new aspects: a different type of dome, because it is set on octagonal and not circular drum, and the role played by the interlocking between the bricks in correspondence of the ribs on the bearing capacity of the dome. As we consider vertical loads, we studied a dome segment - corresponding to one sixteenth - between two contiguous meridian symmetry planes (Fig. 1). In this context, we have developed a program through Excel and its Solver to solve the static optimization problem that provides the collapse multiplier. Once the collapse multiplier has been defined, again using Excel, the corresponding kinematic problem is implemented to represent the failure mechanism at the instant in which the collapse is reached. In order to simulate the lack of interlocking between the bricks at the ribs or its presence, an advantageous aspect in the present modeling is that the yield conditions on the related interfaces of the macro-blocks can be either introduced (the failure is possible) or not (the failure is not possible). Although for the considered application cases has always been used a discretization in only eight macro-elements (six for the dome and two for the drum), the program implemented in Excel appears sufficiently versatile and, in addition to the mechanical characteristics, allows to define the intrados profile, the thickness variability, as well as to insert any window opening in the drum, the lantern at the top and the hoops at each level. Some results of application cases are shown and, finally, a first approach was made to the analysis of the Santa Maria del Fiore dome in Florence, built by Brunelleschi. Of course, models with reduced block dimensions could be implemented to take into account failure mechanisms which were not considered in the present work. However, it should be noted that the configuration of the macro-blocks taken into consideration aimed at reproducing the main features of the crack patterns which are frequently observed on masonry domes, such as those investigated in this manuscript. As the authors pointed out in the manuscript, such crack patterns usually involve the ribs and the webs along radial and meridian interfaces which correspond to the ones considered in the model that was presented. However, it is clear that further analysis could be carried out to investigate the effect of interlocking and block size on the obtained failure mechanism.