## 1 Introduction  
In the last decade, impressive advancements have been made with respect to the preservation of the structural integrity of historic masonry structures [1]. In the literature, there are several advanced methods of analysis that can be implemented in both numerical methods or analytical tools [2]. The majority of such studies address the structural analysis and safety problem of masonry structures by using numerical models based either on the Finite Element Method (FEM) [3, 4, 5] or Discrete Element Method (DEM) [6]. In the case of FE models, the masonry is typically modelled following either a continuum approach (designated as macro-modelling) [7] or discretizing explicitly both units and joints (designated as micro-modelling). In the case of DE models, the masonry is typically modelled as the assemblage of rigid blocks inter-connected through interfaces of a given stiffness, which makes it possible to capture their large displacement response and the opening and closing of joints [6]. It is worth noting that both of these methodsrequire a large set of input parameters [8] and can be time-consuming and computationally expensive in the case of modelling failure, which becomes even more relevant in the case of dynamic analysis. Hence, powerful analytical tools based on the theorems of limit analysis can be an appropriate alternative. The main advantage of limit analysis approaches is that they require less knowledge of the mechanical behavior of materials and are, therefore, more practical. Specifically, the Upper Bound theorem of limit analysis is a useful tool for the rapid assessment of historic masonry structures [9]. However, the computation of the load multiplier depends on the macroblocks' geometry that strongly influences the assessment of their structural integrity. Therefore, multiple (theoretically infinite) failure mechanisms need to be considered to evaluate the minimum of the kinematically compatible load multipliers [10]. Some researchers proposed using optimization routines that solve the minimization problem that is constrained under specific hypotheses [11]. [PERSON] et al. [12; 13] proposed a macro-block model coupled with a simplified procedure for predicting the collapse load and the failure mechanism of in-plane loaded masonry walls with non-associative frictional contact interfaces. The same authors [14] have recently revisited the previous macro-block approach implementing the frictional resistance computation. [PERSON] et al. [14] implemented a novel two-step analysis method in a visual programming environment, which can manage the data arising from both non-linear static or dynamic analysis to detect the most likely collapse mechanism through the Control Surface Method (CSM). In this framework, the macro-block geometry's parameterization allows for the exploration of the domain of possible solutions using the upper bound method of limit analysis coupled with an optimization tool based on Genetic Algorithms. Due to their relatively low computational cost, upper bound limit-analysis procedures have been implemented in building codes [15; 16] in order to provide rapid verifications of the stability of masonry structures subjected to earthquakes. Such approaches include both strength and displacement based procedures, which, while rapid, tend to incorporate limited dynamic effects and thus yield predictions that are generally on the more conservative side [17] and that can consequently result in expensive, and occasionally unnecessary, retrofitting solutions. As an alternative, non-linear dynamic analysis, whereby the equations of motion of the macro-blocks are directly integrated, could be used to more accurately model the dynamic behaviour of these masonry structural assemblies. Specifically, by modelling the macro-block as a SDOF system undergoing rocking motion, better agreement has been found with experimental results, with substantially less scatter presented as compared to the corresponding code-based predictions [17].