Issue 51

G. Cocchetti et alii, Frattura ed Integrità Strutturale, 51 (2020) 356-375; DOI: 10.3221/IGF-ESIS.51.26 373 Further remarks on the present study, its implications, practical value and perspectives may be outlined as follows: • Issue of dilatancy at the blocks’ interfaces. Matter of dilatancy may be pertinent, like as due to interlocking arising among masonry joints, for the mutual surfaces of the chunks being not perfectly planar, with a relative roughness that shall locally be present there. Then, a little relative normal displacement among the joints may accompany a tangential relative displacement, when sliding is activated, the resulting friction angle being the arctangent of the relevant normal to sliding displacement ratio. This shall indeed introduce a further characteristic parameter of the joint, within the analysis, beyond the already considered main one, namely the friction coefficient. This further effect may likely be secondary in the analysis, explicitly focusing on the role of friction, in the considered context of masonry arches; a main novelty, in analytical terms. Possibly, the dilatancy effect may induce some variations in the threshold friction coefficient values that are derived within the present paper, for the potential sliding onset and the transitions among the various possible collapse modes, and for the recorded collapse characteristics. Moreover, it looks hard to state, a priori, if such a dilatancy effect may lay on the safe or the unsafe approximation side, for the critical arch behaviour. As a conjecture, the dilatancy process may lead to further resources of the arch to withstand under self-weight, if confinement within the arch shall hold true (like with firm supports). In that sense, the present analysis neglecting dilatancy may turn out to lay on the conservative side, meaning that the predicted least-thickness condition may anyway be safe (even a lower arch thickness may suffice for arch equilibrium, due to dilatancy). Certainly, dilatancy shall anyhow lead to another, different source of non-normality, and possible related effects. Thus, the perspective handling of the issue of dilatancy may certainly be appropriate, for a complete understanding of arch collapse but shall require a dedicated and comprehensive treatment, and may then constitute the subject of future research work. • Issue of real friction values, in practical contexts, as pertinent to a reducing friction, apt to prevent/induce sliding. The present investigation is mainly and firstly endowed of a theoretical value. The results look useful in terms of the overall behaviour of the considered mechanical system; a full recognition of all its possible features. However, there may be reasons that may induce or be associated to a reducing friction effect among the blocks, as possibly connected to external conditions, like loosening of the joints, fading or spreading supports, percolation of rain, humidity, rubbish and mud within the joints, loosening of the mortar, if present, in cemented joints, and possible damage within that, existing interfaces between different composing materials (concrete, soil, earth, stone, masonry, etc.) among the arch’s blocks, bearing interface on underlying pier, wall or ground with underfilled loosened material, like earth, natural sediments, etc. These effects may lead the friction coefficient values to reduce and possibly approach the critical ranges where sliding may be triggered, so that a verification about that shall anyway be attempted. Moreover, arches of different opening angles or of varied morphological shapes, symmetric or unsymmetric, do display a variation of the critical friction coefficient value possibly leading to the onset of sliding. Indeed, the discussion in Section 2.3, still concerning symmetric arches but with variable opening angles, already shows that the transition value of friction apt to prevent any sliding within the arch raises up to a value of friction coefficient around 1.4 (friction angle about 55°), for the limit case where the range of mixed mode vanishes (limit horseshoe arch). For this case, and other overcomplete arch cases, the critical value of friction leading to possible sliding onset may be nearing the range for practical applications, where, with the further possible intervention of some of the above effects, may really come to induce arch collapse with sliding. Thus, the issue of quantifying the amount of friction that shall be necessary to avoid sliding seems anyway much important, also in practical terms, for investigating the whole equilibrium ranges of the arch. • Arch discretisation. The analysis actually refers to that of a continuous arch, i.e. in strict Heyman sense, where rupture joint may appear at critical locations, precisely where they shall be. This leads to the definition of the true least-thickness critical condition that nature by the gravitational field will find to let the arch to first collapse once thickness is gradually reduced. If fracture joints shall instead occur only at pre-defined locations, due to specific block arrangements, this should lead to lower critical thicknesses, for the arch to withstand. This aspect was widely discussed in previous works [6,8], in the context of infinite friction, and shall apply as well in the present, finite friction one. Thus, specific block patterns, within the arch, may lead to some variations, of the general characteristics of a continuous arch, though the latter truly mark the real underlying least-thickness condition, at variable friction. A CKNOWLEDGEMENTS his work has been carried-out at the University of Bergamo, School of Engineering (Dalmine). The financial support by ministerial (MIUR) funding “Fondi di Ricerca d’Ateneo ex 60%” at the University of Bergamo is gratefully acknowledged. T