## 4 Numerical Results  
Although traditional linear models of HDRBs (e.g., VEL model) are usually considered sufficient to predict the seismic response of base-isolated buildings, more complex models can become necessary in particular circumstances. Indeed, various developments can crop up, such as coupling of horizontal and vertical motions, change of critical buckling load due to large horizontal displacement, tension buckling, cavitation, and post-cavitation phenomena affecting the rubber layers in tension and strength degradation in cyclic tensile loading. Advanced nonlinear models of shear and axial laws of HDRBs, like the CM1, CM1C and CM2 CR models mentioned in Section 2, take into account these phenomena, providing more accurate expressions of their horizontal and vertical stiffnesses. Herein par \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \(\mathbf{\kappa}_{K0}\) & \(\mathbf{T}_{1H}\) & \(\mathbf{T}_{1V}\) & \(\mathbf{K}_{H0}\) & \(\mathbf{K}_{V0}\) & \(\mathbf{D}\) & \(\mathbf{t_{e}}\) & \(\mathbf{t_{i}}\) & \(\mathbf{S}_{1}\) & \(\mathbf{S}_{2}\) & \(\mathbf{\gamma}_{tot\_max}\) & (\(\mathbf{P}_{max}\)/\(\mathbf{P}_{cr}\)) & \(\mathbf{\sigma_{s}}\) \\ \hline
400 & 2.749 & 0.125 & 7.69 & 3074 & 77.12 & 21.27 & 2.248 & 8.57 & 3.63 & 5 & 0.48 & 20.70 \\
1200 & 2.730 & 0.072 & 7.69 & 9224 & 65.31 & 15.25 & 0.980 & 15.26 & 4.28 & 5 & 0.32 & 15.26 \\
2400 & 2.726 & 0.051 & 7.69 & 18448 & 61.10 & 13.35 & 0.507 & 30.15 & 4.58 & 4.2 & 0.20 & 9.84 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Properties of HDRBs for the B7 test structure (dimensions in kN, cm and s). \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \(\mathbf{\kappa}_{K0}\) & \(\mathbf{T}_{1H}\) & \(\mathbf{T}_{1V}\) & \(\mathbf{K}_{H0}\) & \(\mathbf{K}_{V0}\) & \(\mathbf{D}\) & \(\mathbf{t_{e}}\) & \(\mathbf{t_{i}}\) & \(\mathbf{S}_{1}\) & \(\mathbf{S}_{2}\) & \(\mathbf{\gamma}_{tot\_max}\) & (ticular attention is paid to the study of the CM2 CR model used to include the consequences of tension buckling and cavitation and post-cavitation phenomena. To assess the reliability of this model when the seismic response of base-isolated buildings subjected to strong EQs is simulated, the incremental dynamic analysis (IDA) of the test structures described in Section 3 is carried out with reference to different values of the nominal stiffness ratio, comparing results provided by linear (i.e., VEL) and nonlinear (i.e., CM1, CM1C, CM2 CR) models. For this purpose, a homemade computer code based on a lumped plasticity approach [34; 35] has been developed, using a bending moment-axial load interaction domain for the columns and adopting a bilinear moment-curvature law with a hardening ratio of 5% for the sections of all frame members where inelastic deformations occur. Moreover, each beam is discretized in four elements of equal length and each column in a single element, since the inelastic deformations can appear at the end, quarter-span and mid-span sections, or at the end sections, respectively. An equivalent viscous damping equal to 2% is used for the superstructure following the Rayleigh approach, with reference to the fundamental vibration modes in the horizontal (Y) and vertical (Z) directions. As for the seismic input considered in the IDAs, three near-fault EQs are selected from the PEER database [36; 37] ensuring that each one has long-duration horizontal velocity pulses and agrees with main design hypotheses (i.e., high-risk seismic region and medium subsoil class). Table 8 lists names and dates of the EQs, the recording station, magnitude (\(M_{w}\)), epicentral distance (\(\Delta\)), horizontal (\(PGA_{H}\)), and vertical (\(PGA_{V}\)) peak ground accelerations of the selected near-fault ground motions whose horizontal and vertical response spectra are shown in Figure 3a,b, respectively. Figure 3: Horizontal (**a**) and vertical (**b**) acceleration spectra.