# Seismic Analysis of Slender Monumental Structures: Current Strategies and Challenges
## 5 Modelling Strategies  
One final aspect deserving attention is the numerical modelling of slender masonry structures, which introduces different challenges due to the inherent difficulties associated with the description of the complex behavior of the masonry material. According to the level of accuracy desired, three main modelling strategies can be adopted, see Figure 10: (1) detailed micro-modelling, in which units and mortar are represented by continuum elements whereas unit-mortar interfaces are represented by discontinuous elements accounting for potential crack and slip planes; (2) simplified micro-modelling, in which expanded units are represented by continuum elements while the behavior of mortar joints and unit-mortar interfaces is lumped into \"average\" discontinuous elements; (3) macro-modelling, where no distinction among units, mortar and unit-mortar interfaces exists but masonry is treated as a homogeneous anisotropic continuum [69]. Several efforts have also been made in the last decade to improve the so-called homogenization techniques, trying to link micro- and macro-modelling approaches by deriving the constitutive relations of masonry from the individual characteristics of the components of a selected elementary cell [22, 28, 70, 71, 72, 73, 74, 75]. Regardless of the modelling strategy, a thorough experimental description of the material is always required. As for micro-modelling, this must be obtained from laboratory tests in the masonry constituents and small masonry samples, whereas, for macro-modelling, tests must be performed in masonry specimens of sufficient size under homogeneous states of stress and strain [27]. In either case, the task is not trivial considering that the mechanical behavior of masonry is influenced by a large number of factors (see also Section 2), such as the material properties of units and mortar, the shape and dimensions of the units, the bond arrangement, the inner core, the joint thickness, the quality of workmanship, the degree of curing, environment and age. Figure 9: Qutb Minar tower: (**a**) exterior view and (**b**) detail of the fluted shaft. In addition to the different modelling approaches in terms of material, three main methods can be adopted for the numerical simulation of heterogeneous masonry constructions, including slender monumental structures: (1) the macro element approach; (2) the discrete element method; and (3) the finite element method. The difference among the various strategies is mainly related to the assumptions about the material and structural behavior, input parameters, modelling effort and computational time required. The macro element approach relies on the assumption of a no-tension masonry material, usually with infinite compressive strength and no sliding between units. It is a very simple tool that can be applied to evaluate the failure mechanisms of historical masonry structures (kinematic limit analysis), allowing a good estimation of the collapse load factor and displacement capacity of monolithic macro-blocks in which the structure can be ideally subdivided according to the most probable collapse mechanisms. The analysis is not computationally demanding, but it requires an a priori knowledge of the possible disconnections between the rigid blocks composing the kinematic chain, which is not straightforward when a large variety of mechanisms are possible in the structure. Usually, these are selected on the basis of post-earthquake surveyed crack patterns or based on the analyst experience [4, 76, 77]; yet, an incorrect evaluation of the collapse mechanisms can lead to neglect the mechanism with the lowest load factor and, consequently, to overestimate the real maximum capacity of the structure [70]. Conversely, an incorrect evaluation of the collapse mechanisms may also lead to incorrect overly conservative interventions, if the mechanism with the lowest safety factor assumed by the analyst cannot occur. Moreover, loading history and crack evolution can be hardly included.