## 4 Finite Element Analyses
### Numerical Results  
In Figure 6, analytical-numerical comparisons on the trends of the \(\psi\)_function_ with the shear strain are shown for the different values of the primary and secondary shape factors (Figure 6a,b) and shear modulus of the rubber (Figure 6c,d). In each graph, the numerical results are compared with the analytical ones obtained considering the different approaches for the effective compressive modulus (Figure 6a,c) and for the buckling load (Figure 6b,d). In Figure 7, a focus on the trends of the \(\psi\) function with the shear strain and some values of the primary (Figure 7c,d), secondary shape factors (Figure 7a,b) and the shear modulus of the rubber (Figure 7e,f), are shown. These figures highlight the good agreement between numerical and analytical results, with the proposed analytical formulation able to predict the reduction of the vertical stiffness with increasing horizontal deformations. Figure 6: Analytical numerical comparison on the trends of the \(\psi\) function with the shear strain and: (**a,b**) primary and secondary shape factors; (**c,d**) shear modulus of the rubber. The numerical results confirm the trends of the _\(\psi\) function_ with the main geometric and mechanical parameters of the U-FREIs as:  
* Great reductions of \(\psi(\delta_{H})\) with the horizontal deformation are related to smaller values of the primary and secondary shape factor (Figure 6a,b). When \(S_{2}=1.0\) (\(S_{1}=5.5\)), the \(\psi\) function drops in a reduced range of shear strain (i.e., \(\gamma_{H}\leq 50\%\)), while for \(S_{2}=4.0\) (\(S_{1}=22\)), the minimum value of \(\psi(\delta_{H})\) is around 0.5 at \(\gamma_{H}=200\%\). Figure 7: The 2D analytical vs. numerical trends of the \(\psi\) function for different values of the primary (**c,d**) and secondary (**a,b**) shape factors and of the shear modulus of the rubber (**e,f**). * The shear modulus of the rubber plays a minor role compared to the geometric parameters of the U-FREI (Figure 6c,d). A good agreement is found between numerical and analytical results, but for larger values of \(G_{e}\), the numerical trends return increasing values of \(\psi(\delta_{H})\). This is the effect of the full-rollover condition that is prominent for hard rather than soft compounds. Equation (15) does not take into account this phenomenon. From Figure 6a,d, it can be seen how the pressure approach fits the numerical results with good agreement at smaller levels of \(S_{1}-S_{2}\) and \(G_{e}\), while increasing these factors, the best fitting is obtained with the pressure solution. As for the buckling load, the best agreements were found using Equation (8), while Equations (6) and (7) appear to underestimate and overestimate, respectively, the \(\psi(\delta_{H})\) trends. The influence of the different analytical models for the effective compressive modulus and for the buckling load was studied in terms of percentage difference between numerical models and analytical predictions, defined as:  
\[E_{B_{p}}=\frac{\text{abs}\big{[}\,\psi(\overline{\tau}_{H})\big{|}_{num}- \psi\big{(}\overline{\tau}_{H},\overline{E}_{c},\overline{F}_{v,cr}\big{)} \big{|}_{an}\big{]}}{\psi\big{(}\overline{\tau}_{H},\overline{E}_{c},\overline {F}_{v,cr}\big{)}\big{|}_{an}}\times 100 \tag{21}\]  
where \(E_{B_{p}}\) is the difference referred to a generic bearing's parameter (e.g., shear modulus, primary or secondary shape factor), \(\psi(\overline{\tau}_{H})\big{|}_{num}\) is the \(\psi\)_function_ at \(\overline{\tau}_{H}\) obtained from the numerical analyses, and \(\psi(\overline{\tau}_{H})\big{|}_{an}\) is the \(\psi\)_function_ calculated with Equation (15) at \(\overline{\tau}_{H}\), with either one of the effective compressive moduli (\(\overline{E}_{c}\)) and buckling loads (\(\overline{F}_{v,cr}\)) equations proposed in Section 2. Similar to Section 3.3, four shear strain thresholds are considered, namely \(\gamma_{H}=50\%\), \(\gamma_{H}=100\%\), \(\gamma_{H}=150\%\) and \(\gamma_{H}=200\%\).