## 4 Non-Linear Numerical Modelling and Structural Analysis
### Definition of the Sesmic Inputs and Related Demand  
The seismic demand has been determined by adopting as an intensity measure (IM) the value of the pseudo-acceleration \(\mathrm{S_{a}}\) in the elastic regime. The site-related response spectrum associated with 50 years as a reference period for limit state of life safety has been adopted in line with [3]. The values of the damping coefficient \(\xi\) have been distinguished \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline
**Type** & _FPS1_ & _FPS2_ & _FPS3_ & _FPS4_ & _FPS5_ & _FPS6_ & _FPS7_ & _FPS8_ & _FPS9_ \\ \hline \(R\) [m] & & & & 1.5 & & & & \\ \(K_{2}\) [\(\mathrm{kN/m}\)] & 355 & 442 & 471 & 293 & 413 & 400 & 431 & 335 & 382 \\ \(K_{1}\) [\(\mathrm{kN/m}\)] & 18,080 & 22,517 & 23,996 & 14,957 & 21,038 & 20,381 & 21,958 & 17,093 & 19,460 \\ \(\mu_{Slow}\) [\%] & & & & 1 & & & & \\ \(\mu_{Fast}\) [\%] & & & & 3 & & & & \\ \(\alpha\) [\%] & & & & & 30 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Properties of FPS devices in the function of the axial load. Figure 9: In-plane location of the single-concave FPS devices with different modeling characteristics. Reprinted/adapted with permission from [26]. between fixed-base and base-isolated structures, adopting 5% for the former and 2% for the latter one [12]. Figure 11 shows the adopted elastic response spectra. The set of ground motion inputs to realize the NL dynamic simulations is composed of 21 natural records composed of 3 acceleration components along the in-plane and vertical directions, as used in [12] and further in [26]. In [12], the selection of the 21 inputs has been performed from the ESM (European-Strong-Motion database) [27]. The structural response of both fixed-base and base-isolated buildings has been investigated by means of modal analysis. Figure 12 reports the summary of the first three eigenmodes (and related periods of vibration) associated with the different structural configurations, including the influence of the infills. It can be highlighted that no significant variations in the modal shapes occur due to the presence of infills for both fixed-base and base-isolated structures. In the case of a fixed-base structure, as expected, the presence of the infills significantly reduces the periods of vibration due to their stiffening effects. On the contrary, the periods of vibration of the base-isolated structures are not strongly affected by the infills. Figure 11: Adopted response spectra in elastic regime according to limit state of life safety [3]. Reprinted/adapted with permission from [26]. Figure 10: NL numerical idealization for fixed-base (**a**) and base-isolated (**b**) structures without infills and fixed-base (**c**) and base-isolated (**d**) structures with infills using SAP 2000. With the aim of ensuring spectrum compatibility with the spectra presented in Figure 11, each record has been properly scaled for what concerns the X-direction compo  
Figure 12: Representation of the first three eigenmodes for the fixed-base structure with (**a**) and without infills (**c**) and for the base-isolated structure with (**b**) and without infills (**d**). nent. In particular, the IM evaluated in concomitance with the fundamental period T\({}_{1}\) of the structure related to the elastic spectrum of the specific record has been scaled to the IM of the design spectra of Figure 11. This operation has been performed for both fixed-base and base-isolated structures, including and not including the effect of infills made of masonry. The response spectra related to the selected natural records and associated with the different structural configurations are shown in Figures 13 and 14. Note that for the base-isolated infilled frame, the scaled records are almost equal to the ones corresponding to the base-isolated bare frame (i.e., without accounting for the infills). This demonstrates the minor influence of the presence of infills on the modification of the natural frequency of the base-isolated building.