# Intermediate Isolation System with Nonlinear Lower Structure and Isolation System
## 6 Time History Analyses on 3D Models and Comparison to IIS Design Spectrum
### Modelling Approaches and Analysis Types
By specifying the value of the equivalent viscous damping ratio (Equation (4), \(\xi_{ISO}=0.15\)) and neglecting \(D_{y}\), we can calculate a first estimate for \(Q^{(1)}\), \(k_{2}{}^{(1)}\) and \(k_{1}{}^{(1)}\) from Equation (5). From these values, we can derive the value of \(D_{y}\) through Equation (6) and recalculate \(Q\), \(k_{2}\), and \(k_{1}\) (Equation (7)). Data on the bilinear constitutive model of the isolators utilized in the design solutions B, C, D, E are reported in Table 2.  
In this procedure, the definition of the design displacements \(S_{De}\)(T) is an issue to be further investigated. The idea is to derive floor spectra in terms of displacement for the design of the isolation system. The seismic isolators can be preliminarily sized from data sheets of producers, e.g., [62]. As an example for the solution B (\(T_{ISO}=0.5\) s), set the values of design displacement and secant stiffness close to \(S_{de}\) and \(k_{eff}\), respectively, and by choosing hard rubber (G = 1.4 MPa), an isolator diameter of 900 mm and total rubber thickness of 60 mm are derived. For the configuration E (\(T_{ISO}=5\) s), a solution with quite small isolators (200 mm diameter, 75 mm total rubber thickness) is derived even when considering a soft rubber (G = 0.4 MPa). Therefore, a mixed solution (isolators plus sliders) could be a better choice. Because the design solutions are non-classically damped systems (see Section 4), the fully populated damping matrix, which leads to coupled equation of motions, is accounted for through direct integration THA [43]. Direct integration THA is also developed for the 3D FE model of the as-is configuration. The method of superposition of modal damping matrices [43; 63] has been used for constructing the viscous damping matrix, as recommended by [64]. In both model types, representing the as-is configuration and the design configurations (A-E), energy dissipation sources other than that associated with the yielding of structural elements have been accounted for through viscous damping. For the masonry LS in the as-is configuration, a single value of the modal damping ratio is specified, equal to 5%. For the IIS configurations, different values of modal damping ratios have been attributed to the modes based on the structural portion (LS, ISO, US) that contributes \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \(T_{ISO}\) & \(k_{eff}\) & \(k_{1}\) [N/mm] & \(k_{2}\) [N/mm] & \(Q\) & \([\textbf{N}]\) & \(S_{de}\) [mm] \\  & [**s**] & [**N/mm**] & & & & \\ \hline
**B** & 0.5 & 14,844 & 112,223 & 11,222 & 128,343 & 36 \\
**C** & 1 & 3711 & 28,056 & 2806 & 64,506 & 71 \\
**D** & 1.5 & 1649 & 12,469 & 1247 & 43,004 & 107 \\
**E** & 5 & 148 & 1122 & 112 & 5800 & 160 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Data of the bilinear constitutive model of the isolators utilized in the IIS design solutions. more significantly to structural response. Therefore, a modal damping ratio equal to 0 is attributed to the first three modes, which manly involves the isolated upper structure, since the energy dissipating mechanisms usually accounted through viscous damping do not arise either at the isolated level or in the US (which follows the isolation system almost rigidly). For the modes from the fourth to the twelfth, mainly involving the LS, the values of the modal damping ratios are 5%. Higher modes involving the US are not present, due to its high stiffness. Seven earthquake records are selected by means of the software REXEL v 3.5 [65] from the SIMBAD database [66] (for PGA = [0.1, 0.3] g and T = [0.15, 2] s); the record set is selected to derive an average spectrum close to the response spectrum of the site (Figure 3a), with upper and lower tolerance of 30% and 10% (Figure 7). The major data of the ground motion records are provided in Table 3.  
The horizontal component of the ground motion has been applied only in the weakest direction (i.e., Y-direction) and only the nonlinear behaviour of in-plane masonry piers in the Y-Direction has been considered.