# 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  
The results on IIS behaviour discussed above have been derived from linear response spectrum analyses. In the next step, some simplified assumptions and limitations are removed in order to assess the effectiveness of IIS through more accurate models and methods. Hence, the three-dimensional (3D) FE models of the selected solutions (Figure 5b) are built in the SAP2000 computer code [43] and analysed through linear and nonlinear time history analyses (THA). The cases of standalone lower structure and of conventional vertical extension, consisting of US attached to the LS without any isolation system (designated as A in Figure 3b), have also been considered. In the nonlinear THA, the 3D FE model of the LS already utilized for the push-over analyses (see Section 3.2) is retained. However, in the 3D models of the IIS design solutions, the values of the ultimate bending moment for the flexural plastic hinges at the ends of masonry piers are updated to consider the compression due to the gravity load of the vertical extension. This additional compression has a general stabilizing effect for the LS, which is reflected in a reduction in the tensile stress arising in the walls for lateral load. The Pivot-type hysteretic model is adopted for describing the cyclic response of both hinges in bending and shear (Figure 6a), with the values of the parameters \(\alpha_{1}\), \(\alpha_{2}\), \(\beta_{1}\), \(\beta_{2}\), and \(\eta_{Pivot}\) defined to control the shape of the hysteresis loops and the stiffness and strength degradation. In particular, \(\alpha\)1 defines the position of the pivot points P1 and P2, whereas \(\alpha\)2 defines the position of the pivot points P3 and P4 for unloading from positive and negative force, respectively. \(\beta_{1}\) and \(\beta_{2}\) define the position of the pinching pivot points PP1 and PP2 for reverse loading from zero toward positive and negative force, respectively. The parameter \(\eta_{Pivot}\) determines the amount of degradation of the elastic slopes after plastic deformation. According to [60], the model parameters \(\alpha_{1}\), \(\alpha_{2}\), \(\beta_{1}\), and \(\beta_{2}\), are here set equal to 0.45, while the parameter \(\eta_{Pivot}\) is set equal to zero. More details on the modelling approach adopted here for the hysteretic behaviour of masonry are provided in [31; 60]. Concerning the isolation system (Figure 5c), the hysteretic behaviour of HDRBs can be reasonably described by the bilinear constitutive model calibrated as follows [61] (Figure 6b). If the values are known of global secant stiffness \(k_{ISO}\), design displacement \(S_{De}\) (from the displacement design spectrum at \(T_{ISO}\)), and equivalent viscous damping ratio \(\zeta_{ISO}\), number of isolators \(n\), then, the values of secant stiffness \(k_{eff}\), initial stiffness \(k_{1}\), post-yield stiffness  
Figure 6: (**a**) Pivot-type model for plastic hinges in masonry piers of LS, (**b**) bilinear model of isolator. \(k_{2}\), intersection between the prolongation of post-yield branch and the \(y\)-axis \(Q\), and yield displacement \(D_{y}\), can be derived according the following formulas:  
\[k_{eff}=\frac{k_{ISO}}{n} \tag{3}\]  
\[\xi_{ISO}=2\cdot Q\cdot\frac{\left(S_{De}-D_{y}\right)}{\pi\,k_{eff}\cdot S_{De} ^{2}} \tag{4}\]  
\[Q^{(1)}=\frac{\pi}{2}\cdot k_{eff}\cdot S_{De}\cdot\xi_{ISO},\quad k_{2}{}^{(1 )}=k_{eff}-\frac{Q^{(1)}}{S_{De}},\quad k_{1}{}^{(1)}=10\cdot k_{2}{}^{(1)} \tag{5}\]  
\[D_{y}=\frac{Q^{(1)}}{k_{1}{}^{(1)}-\,k_{2}{}^{(1)}} \tag{6}\]  
\[Q=\frac{\pi\cdot k_{eff}\,S_{De}^{2}\cdot\xi_{ISO}}{2\cdot\left(S_{De}-D_{y} \right)},\quad k_{2}=k_{eff}-\frac{Q}{S_{De}},\quad k_{1}=10\cdot k_{2} \tag{7}\]  
The secant stiffness \(k_{eff}\) (Equation (3)) is obtained by dividing the global secant stiffness \(k_{ISO}\) for the number of isolators (\(n=10\)).