## 2 Review of the Analytical Solutions for the Vertical Response of FREIs
### Axial Deformation under Vertical and Lateral Loads  
A closed-form analytical solution was proposed for the overall vertical deformation of strip-shaped elastomeric bearings under axial and shear loads [40,42]. The total vertical displacement \(\delta_{v}^{t}\) is seen as the sum of two quantities: the first one due to pure compression \(\delta_{v}^{c}\), while the second one is the effect of the shear deformation \(\delta_{v}^{s}\) (Figure 1b):  
\[\delta_{v}^{t}=\delta_{v}^{c}+\delta_{v}^{s}=\frac{F_{v}}{K_{v}^{\delta_{H}=0} }+\frac{\pi G_{e}A_{c}}{4F_{v,cr}}\left[\frac{\pi f_{v}-\sin(\pi f_{v})}{1- \cos(\pi f_{v})}\right]\frac{\delta_{H}^{2}}{H} \tag{5}\]  
where \(F_{v,cr}\) is the buckling load of the bearing, and \(f_{v}=F_{v}/F_{v,cr}\) is the dimensionless ratio of the vertical applied load to the buckling load. Formula (5) was initially derived for bonded elastomeric bearings with rigid reinforcement, but recent studies demonstrated how it can be used to predict the total vertical deformation of unbonded strip-shaped bearings with flexible reinforcement with good accuracy [43].  
Different analytical formulations were proposed for the buckling load \(F_{v,cr}\). In Equation (5), the following equation for bearings with rigid reinforcements was first used [44]:  
\[F_{v,cr,1}^{rig}=\frac{4\pi G_{e}a^{3}}{\sqrt{15}t_{f}t_{e}} \tag{6}\]  
A solution for the buckling load of bearings with flexible reinforcements was also proposed. \(F_{v,cr}\) can be obtained solving the following non-linear equation [45]:  
\[C_{1}\left(\frac{F_{v,cr}^{flex}}{G_{e}A_{c}}\right)^{3}+C_{2}\left(\frac{F_{ v,cr}^{flex}}{G_{e}A_{c}}\right)^{2}+C_{2}\left(\frac{F_{v,cr}^{flex}}{G_{e}A_{c}} \right)-C_{3}=0 \tag{7}\]  
where \(C_{1}-C_{4}\) are constants depending on the geometric and mechanical properties of the bearing as follows:  
* \(C_{1}=\frac{(EI)_{eff}}{(EI)_{eff}}\left[\frac{f_{c}}{A_{c}}-\left(\frac{f_{B }}{A_{c}}\right)^{2}\right]\)
* \(C_{2}=\frac{(EI)_{eff}}{(EI)_{eff}}\left[\frac{C}{A_{c}}-\frac{f_{c}}{A_{c}}- \frac{2f_{B}B}{A_{c}}\right]+\pi^{2}\rho\left[1+\frac{(EI)_{eff}}{(EI)_{eff}} \left(\frac{f_{B}}{A_{c}}\right)^{2}\right]\)
* \(C_{3}=\frac{(EI)_{eff}}{(EI)_{eff}}\left[\frac{C}{A_{c}}-\left(\frac{B}{A_{c}} \right)^{2}\right]+\pi^{2}\rho\left[1+\frac{(EI)_{eff}}{(EI)_{eff}}\left(2\frac {f_{B}B}{A_{c}^{2}}-\frac{f_{c}}{A_{c}}\right)^{2}\right]\)
* \(C_{4}=\pi^{2}\rho\frac{(EI)_{eff}}{(EI)_{eff}}\left[\frac{C}{A_{c}}-\left( \frac{B}{A_{c}}\right)^{2}\right]+\pi^{4}\rho^{2}\)\[\bullet\quad(EI)_{eff}\approx a^{3}G_{e}S_{1}^{2}\frac{8}{15}\left[1- \frac{2}{21}(\alpha a)^{2}\right]\] \[\bullet\quad(EI)_{eff}\approx aG_{e}S_{1}^{2}\bigg{\{}\frac{32}{3675} \left[1-\frac{2}{77}(\alpha a)^{2}+\frac{1}{2\lambda S_{1}^{3}\left(a/t_{f} \right)^{3}}\right]\bigg{\}}\] \[\bullet\quad B\approx\frac{8}{7}\left[1-\frac{1}{210}(\alpha a)^{2}\right]\] \[\bullet\quad C\approx\frac{1}{a}\left\{\frac{552}{245}\left[1- \frac{1}{210}(\alpha a)^{2}\right]+\frac{2S_{1}}{\lambda\left(a/t_{f}\right)^ {3}}\right\}\] \[\bullet\quad f_{B}\approx\frac{12}{35}\left[1+\frac{16}{315}( \alpha a)^{2}\right]\] \[\bullet\quad f_{\mathrm{C}}\approx\frac{216}{245a}\left[1+\frac{2 6}{945}(\alpha a)^{2}\right]\] \[\bullet\quad\rho\approx\frac{4}{5}51^{2}\left(\frac{a}{H}\right)^ {2}\left[1-\frac{2}{21}(\alpha a)^{2}\right]\]  
When the reinforcement tends to be rigid, the solution of Equation (7) tends to [28]:  
\[F_{v,\sigma,2}^{rig.}=G_{e}A_{c}\left(\frac{-1+\sqrt{1+4\pi^{2}\rho}}{2}\right) \tag{8}\]  
Vertical Stiffness and Effective Compressive Modulus under Axial and Shear Loads: Analytical Solution