## 2 Review of the Analytical Solutions for the Vertical Response of FREIs
### Variability of the \(\psi\) Function  
This section shows the trends of the \(\psi\)_function_ with the main geometric and mechanical parameters of the bearing. Equation (15) is applied to strip-shaped FREIs using the following approach. For the effective compressive modulus \(E_{c}^{\delta_{H}}\), either Equation (3) or (4) are used, while Equations (6)-(8) are used to determine the buckling load \(F_{v,cr}\). The solutions for \(E_{c}\) allow us to estimate the differences between the assumption of compressible and incompressible elastomeric layers, while using the different equations for \(F_{v,cr}\) return results on the influence of rigid or flexible reinforcements. Figure 2 shows the trends of the \(\psi\)_function_ with the shear strain (i.e., \(\gamma_{H}=\delta_{H}/t_{r}\)) and with the:  
* Primary shape factor (Figure 2a,b),
* Secondary shape factor (Figure 2c,d),
* Shear modulus of the rubber (Figure 2e,f),  
considering the two values of effective compressive modulus (Figure 2a,c,e) and the three values of the buckling load (Figure 2b,d,f) given in Sections 2.1 and 2.2. Tables 1-3 report the percentage reductions of the \(\psi\)_function_ at four significant levels of shear strains, i.e., \(\gamma_{H}=50\%\), \(\gamma_{H}=100\%\), \(\gamma_{H}=150\%\) and \(\gamma_{H}=200\%\), with the secondary shape factor, primary shape factor and shear modulus of the rubber, respectively. In each plot, a maximum level of shear strain equal to \(200\%\) was used, as this level of horizontal deformation approximately corresponds to the full-rollover threshold of the U-FREIs [53; 54; 55]. In Figure 3, the trends of the \(\psi\) function are shown for the limit values of the primary (Figure 3c,d) and secondary shape factor (Figure 3a,b), as well as the shear modulus of the rubber (Figure 3e,f), accounting for the different formulations for the effective compressive modulus (Figure 3a,c,e) and for the buckling load (Figure 3c,d,f). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{S_{2}}\) & \(\mathbf{1-\psi(\gamma_{H}=50\%)}\) & \(\mathbf{1-\psi(\gamma_{H}=100\%)}\) & \(\mathbf{1-\psi(\gamma_{H}=150\%)}\) & \(\mathbf{1-\psi(\gamma_{H}=200\%)}\) \\ \hline
1 & 74.6\% & 91.8\% & 96.4\% & 97.9\% \\
2 & 25.6\% & 56.6\% & 75.6\% & 84.3\% \\
3 & 10.3\% & 30.4\% & 50.9\% & 64.3\% \\
4 & 4.80\% & 16.1\% & 31.2\% & 44.0\% \\
5 & 2.53\% & 8.95\% & 18.9\% & 28.8\% \\
6 & 1.48\% & 5.39\% & 11.9\% & 19.0\% \\
7 & 0.955\% & 3.53\% & 7.99\% & 13.1\% \\
8 & 0.673\% & 2.50\% & 5.74\% & 9.54\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Percentage reduction of the \(\psi\)_function_ with the shear strain for different values of the secondary shape factor. For low values of \(S_{2}\), \(\psi(\delta_{H})\) greatly reduces, starting from relatively small shear strains, while for larger values of the secondary shape factors little reductions of the \(\psi\)_function_ are obtained at large lateral deformations (Figure 2a,b). This function almost drops to zero for a combination of lower \(S_{2}\) and larger \(\gamma_{H}\). Table 1 shows how the transition value of \(S_{2}\) appears to be 2.5. For \(S_{2}\leq 2.5\), a reduction of the order of 60.5% (on average on \(\gamma_{H}\)) is obtained, while for \(S_{2}>2.5\), the same reduction is of the order of 39%. This transition threshold matches that previously found for the stable/unstable response of U-FREIs under axial and shear load [15; 17; 18] and is also highlighted in Figure 4, where the percentage reductions of the \(\psi\) function with the secondary shape factor is shown at different levels of shear strain. This figure shows how, for relatively low values of the horizontal deformation, the vertical bearing capacity of the FREI is greatly reduced (\(1-\psi(0)/\psi(\gamma_{H})>80\%\)) when \(S_{2}\leq 2.5\), while greater values of the secondary shape factors (i.e., \(S_{2}>2.5-3\)) ensure slight reductions of the vertical stiffness of the bearing (\(10\%<1-\psi(0)/\psi(\gamma_{H})<60\%\)).