# Homogenization of Ancient Masonry Buildings: A Case Study
## 8 Modal Analysis
### Modal Analysis of the Global Homogenized Model  
Finally, the modal analysis of the global homogenized model is carried out. The aim is to investigate the occurrence of a global dynamic behavior rather than local mechanisms of macro-elements. It is worth noticing that such an analysis, if performed with a fine model of the whole structure, is impractical.  
First, the global LTW homogenized model is studied. It is found that the dynamic behavior is characterized mainly by out-of-plane movements of macro-elements, composed by the building arms. Frequency and modal shape of the most qualitatively significanty natural modes, namely 1,5,7, are shown in Figure 21. In particular, the natural modes 1-4 involve the long longitudinal arms with frequencies of about \(f=1.2\) Hz, the natural modes 5-6 involve the short longitudinal arms with frequencies of about \(f=1.25\) Hz and the natural modes 7-10 involve the transverse arms with frequencies of about \(f=1.3\) Hz.  
Figure 19: First natural mode of the LTW homogenized model. Natural frequency \(f=1.23\) Hz. Modal shape in the: (**a**) 3D space; (**b**) \(x-z\) plane; (**c**) \(y-z\) plane.  
Figure 20: First natural mode of the SLTW homogenized model. Natural frequency \(f=1.44\) Hz. Modal shape in the: (**a**) 3D space; (**b**) \(x-z\) plane; (**c**) \(y-z\) plane.  
Moreover, as a confirmation of the occurrence of local dynamic mechanisms, it is observed also that the first mode is about the same as that of the homogenized model representing the only longitudinal arm (see Figure 16). This specific aspect is mainly due to the geometry of the building, and allows one to directly address the study to the single arms, eventually developing finer models.  
When the localized transverse walls are added and the global SLTW homogenized model is studied, the dynamic behavior continues to be characterized mainly by out-of-plane local movements of macro-elements, consisting of the building arms. Frequency and modal shape of the natural modes 1, 5 are shown in Figure 22. The natural modes 1-4 involve the transverse arms with frequencies of about \(f=1.3\) Hz and the natural modes 5-8 involve the long longitudinal arms with frequencies of about \(f=1.33\) Hz; the other modes are the higher modes of the long longitudinal arms and, therefore, are not reported.  
Figure 21: Natural modes of the global LTW homogenized model. Modal shape in the (**a**) 3D space; (**b**) \(x-z\) plane.  
It is interesting to note that the addition of localized beams, as defined in the SLTW model, causes a reversal of the modes order. This is a consequence of the effect of the localized walls, which increase the out-of-plane modal stiffness of the arms in a different way, depending on the mode number. Indeed, compared to the frequency of the modes 1, 5, 7 of the LTW model, where localized transverse walls are absent, for the SLTW, it is increased of 10.8%, 39% and 0%, respectively. As a further observation, the natural frequencies of the global SLTW homogenized model are very close to each other, so as to result in being very sensitive to the definition of the localized elements. This aspect could induce, for some configurations, the occurrence of simultaneous modes, which might be worthy of investigation.