# Seismic Analysis of Slender Monumental Structures: Current Strategies and Challenges
## 5 Modelling Strategies
Remarkable advances have been made towards the automatic assessment of partial failure mechanisms in masonry structures [78, 79, 80], but further efforts are needed to fully convert kinematic limit analysis into an automatized process free of arbitrary decisions, which replicates the real conditions of the building. The distinct element method (DEM) idealizes masonry as a discontinuum material consisting of an assemblage of distinct rigid or deformable bodies whose joints are modelled as contact surfaces with suitable interface laws. Originated from the study of rock mechanics in the 1980s [81], this approach enables to simulate various sources of nonlinearities in masonry structures, including large displacements caused by sliding at the joints with consequent contact update, or the complete separation between blocks. The natural field  
Figure 10: Modelling strategies for masonry structures (adapted from [27]): (a) detailed micro-modelling, (b) simplified micro-modelling, (c) macro-modelling. of application of DEM is composed of structures formed by regularly shaped masonry or stone blocks, such as stone bridges, columns and pillars, arches, and vaults. DEM is also appropriate for modelling the out-of-plane behavior of multi-leaf masonry walls taking into account the real units' arrangement. Some applications to masonry structures can be found in [82; 83; 84; 85; 86; 87]. The application to complex structures of relevant size remains controversial, mainly because of the high number of block elements needed to obtain a realistic discretization of the construction and of the large number of contact points required for the accurate representation of interface stresses, which make the analysis computationally non-viable, especially for 3D problems [23; 28]. The finite element method (FEM) remains the most used tool for the advanced analysis of the seismic behavior of masonry constructions, allowing to use different element types and material modelling strategies, including discrete micro-models or continuous macro-models. As mentioned previously, micro-modelling considers the behavior of units, mortar and unit-mortar interfaces separately, thus leading to more accurate results, but mesh preparation can be unwieldy. Owing to the high level of refinement required, its application is mostly limited to the analysis of small structural portions, the computational effort being extremely demanding at large scale [74; 88; 89; 90]. By contrast, macro-modelling considers masonry as a homogeneous continuum and requires a reduced computational effort, thereby being preferred when dealing with large structures and a compromise between accuracy and efficiency is sought [91]. However, due to the impossibility to execute ad-hoc experimental tests for the characterization of the various parameters necessary to describe the anisotropic behavior of the material in the inelastic range, isotropic macro-models can be adopted for the non-linear analysis of masonry structures, which is an acceptable simplification at the structural level [92; 93; 94; 95; 96]. Some recent applications of FEM-based modelling approaches to slender masonry structures can be found in [33; 38]. Figures 11 and 12 report the results of the blind predictions provided by several experts on the out-of-plane behavior of two U-shaped masonry structures tested on the shaking table of the LNEC (National Laboratory of Civil Engineering), Lisbon, Portugal, the first specimen being built with granite stones and the second specimen with fired clay bricks, together with lime-based mortar [97]. Several types of analyses were adopted to evaluate the seismic capacity of the structures in terms of failure mechanisms and PGA at collapse, using the three afore-mentioned modelling approaches (Figure 11). The same input parameters, i.e., geometry, material properties, normalized acceleration response spectrum and accelerogram envelope, were provided to the experts for both specimens, see [98].