Issue 46

L.U. Argiento et alii, Frattura ed Integrità Strutturale, 46 (2018) 226-239; DOI: 10.3221/IGF-ESIS.46.21 227 material is characterized by a very high anisotropy, so that the direction of bedding surfaces and element shapes plays a central role in defining the type of failure. Such experiments, further developed by other authors as Ceradini [4], have stressed that in-plane failure in most of cases occurs by applying horizontal loads exceeding the friction coefficient, such to produce cracks along the joints without crashing of elements, while failure due to the overcoming of material strength (  ,  ) occurs for much higher applied loads. Actually, the absence of tensile strength and the presence of weak mortar joints generally cause the development of cracks which transforms a masonry block structure into a system of rigid blocks which can exhibit sliding or rocking mechanisms as well as a combination of them. The geometry of these mechanisms can be quite easily identified in post seismic scenarios, since the crack pattern has been revealed, but when the attention is focused on the prevision of the expected behaviour of the structures the most likely collapse mechanisms have to be identified. To this aim, several combinations of material and structural models are possible, e.g. FEM macro and micro-modelling [5], discrete element methods [6, 7], computational limit analysis procedure for rigid block assemblages [8, 9] or homogenization models [10]. A drawback for the use of these sophisticated models in the practical assessment of structures is the large amount of time needed for the structural model elaboration, for performing the non-linear analyses themselves and for reaching proper understanding of the results significance. Conversely, the macro-block modelling in the framework of limit analysis with kinematic approach represents a more powerful tool because it allows an easy computation of the collapse load factor and the failure mode by means of minimization routines. The pioneering works of Kooharian [11] and Heyman [12], based on infinite frictional resistances, have largely been used to apply the plasticity theory to masonry block structures. According to this standard limit analysis, the application of the static theorem provides a lower-bound or safe solution of the collapse load factor, based on equilibrium equations, while the application of the kinematic theorem leads to an upper bound multiplier. The solution that satisfies the hypotheses of both theorems, equilibrium, compatibility and material conditions is the correct solution and provides the collapse load multiplier for the specific problem. However, the uniqueness of the solution is no longer guaranteed when non associated flow rules are involved, such as in case of Coulomb’s friction at block interfaces [13, 14]. This means that a range of statically and kinematically admissible solutions can be identified and a safe solution can be represented by the minimum load multiplier computed by means of the kinematic approach [8]. Crucial aspects are also the definition of the yield domains of dry or weak mortar joints able to dissipate seismic energy [15-17] or, as an alternative, the composition of units to enforce the capability of dissipating energy during motions of walls [18]. The macro-block modelling approach has been lately developed to investigate the local in-plane and out-of-plane failure modes in masonry buildings [19-22]. According to it, each block represents a portion of masonry which remains undamaged and is separated from others by a number of localized cracking where the frictional resistances can take place. When attention is paid to structural elements having a specific role in the overall building behaviour, special macro- elements can simulate them to reduce computational and modelling efforts, e.g. architectonic elements in which the seismic behaviour is almost independent from the rest of the structure of masonry churches [23, 24] or masonry vaults typified by sets of equivalent trusses [25]. This modelling strategy could also be useful to further develop recent innovative research in the field of rocking rigid block dynamics [26, 27]. When applied to in-plane failure modes of masonry block structures with dry or weak mortar joints, generally characterized by a combined rocking-sliding mechanism, crucial to this approach is the assessment of the frictional resistances along the crack. In fact, it is not easy to identify the number of active sliding interfaces along the crack and the actual frictional resistance associated to the crack line could also be far from its maximum value. This issue has already been addressed with reference to a single-storey masonry block wall [22] and in this paper it is extended to a multi-storey wall. A proper evaluation of the in-plane frictional resistances, on the other hand, represents a relevant contribution also to the analysis of the out-of plane mechanisms; in fact these resistances can play an important role when the mechanism involves the façade wall with portions of side walls of a masonry building (complex rocking). Thus, in the following sections, considering the masonry block wall as a single leaf wall arranged in a running bond pattern, the in-plane frictional resistances are firstly evaluated, also accounting for additional loads due to horizontal structures and live loads. The originality of this part can be recognized in the criterion to evaluate the contribution of the actual frictional resistances depending on the inclination angle of the crack line and the number of storeys, in addition to other geometrical and mechanical parameters. Then, the “exact” ultimate load factor in case of rocking-sliding failure mode is computed by means of minimization routines, by taking into account variable positions of the conventional crack line. Lastly, the validity and accuracy of the novel solution procedure are investigated through a sensitivity analysis and the comparison with results and benchmark examples existing in the literature.