## 3 Numerical Simulation and Results
### Simulation Results and Discussion
#### 3.3.2 Shear Modulus of Elastomeric Bearing  
The effect of the three M-R models on the compression and shear of the elastomer bearing was evaluated through the finite element analysis. In order to analyze the effect of three M-R models on the compression of the bearing, the compression forces according to the vertical displacements of the bearing for 2P, 5P, and 9P were found and provided in Table 7. The values of Table 7 are plotted in Figure 11 as the force-displacement graph. Similarly, in order to analyze the effect of three M-R models on the shear of the bearing, the shear force according to the horizontal displacement of the bearing under compression of 392 ton\({}_{\text{f}}\) (corresponding to a pressure of 6 MPa) was found, and the values are provided in Table 8. The values of Table 8 are plotted in Figure 12 as force-displacement graphs. Figure 11 shows the compressive load-displacement graph obtained with the three M-R models. The load-displacement graphs obtained with the 2P and 5P models are almost linear, and the average percentage error between the results obtained with 2P and 5P models is around 2.23% (more details of the error are provided in Table 7), which is negligible. In comparison, we observe a nonlinear trend for the compressive load-displacement graph for the 9P model. The deviation between 2P and 9P model results is around 11.83% (from Table 7), which means that the 9P M-R model is not suitable for simulating compression behavior.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Mooney-Rivlin Models**} & \multicolumn{2}{c}{**Steel Plate**} & \multicolumn{2}{c}{**Elastomer**} \\ \cline{2-5}  & **Stress (MPa)** & **Difference Rate** & **Stress (MPa)** & **Safety Factor** \\ \hline
2-Parameter & 76.4 & 0.0\% & 3.36 & 0.0\% \\
5-Parameter & 76.0 & \(-\)0.5\% & 3.94 & +17.3\% \\
9-Parameter & 86.1 & +12.7\% & 9.16 & +172.6\% \\ \hline \hline \end{tabular}
\end{table}
Table 6: Maximum equivalent stresses and its difference rate of the elastomeric bearing with respect to the Mooney–Rivlin model under compression and shear.  
Figure 10: Stress distribution of elastomeric bearing with 9P M-R model as shear strain is 0.5 under the vertical pressure of 6 MPa: (**a**) for all domain; (**b**) for elastomer only.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Dispacement** & \multicolumn{3}{c}{**Compression Forces (Ton\({}_{\text{f}}\))**} \\ \hline
**(mm)** & **2-Parameter** & **5-Parameter** & **9-Parameter** \\ \hline
0 & 0.0 & 0.0 & 0.0 \\
1 & 122.7 & 111.6 & 96.4 \\
2 & 231.0 & 229.8 & 207.2 \\
3 & 355.2 & 355.1 & 343.4 \\
4 & 485.3 & 487.8 & 515.3 \\
5 & 620.6 & 627.2 & 731.6 \\ \hline
**Difference * (\%)** & N/A & 2.23\% & 11.83\% \\ \hline \hline \end{tabular}  
* Difference (\%) is the difference rate between the 5P or 9P value and the 2P value.  
\end{table}
Table 7: Compression force and displacement with respect to the three M-R models through the numerical simulation.  
Figure 11: Compression force-displacement graph with respect to the three M-R models through the numerical simulation.  
Figure 12 shows the shear force-displacement graph obtained with the three M-R models. All three models almost predict the linear behavior in the shear load, although a slight nonlinear trend is found for the 9P model when the shear strain rate is below 0.5. From Table 8, the deviation between the results of shear force and displacement from 2P and 5P models is almost zero (around 0.38%). On the other hand, the percentage error between 2P and 9P results is nearly about 4.82%. Overall, the third-order 9P M-R model showed a nonlinear curve for both compression and shear load graphs. However, the second-order 5P and the first-order 2P M-R model showed a linear trend for force-displacement graphs with negligible difference between the two results. Therefore, we can say that the 2P M-R model can be successfully applied for simulating the nonlinear behavior of elastomeric bearing under compression and shear loading.