## 3 Numerical Simulation and Results  
### Analysis Model  
In the numerical simulation of elastomeric bearings, it is very important to consider a proper M-R model for finding the nonlinear behavior of materials accurately. In the case of hyperelastic materials such as rubber, the Mooney-Rivlin (M-R) model is generally used, and, depending on the number of terms in the series, there are mainly three kinds of M-R models, such as 2P (first-order), 5P (second-order), and 9P (third-order). In order to find the  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\epsilon_{qx}\) & \(D_{x}\) (mm) & \(F_{x}\) (Ton\({}_{\mu}\)) & \(F_{x}\) (kN) & \(\tau_{s}\) (MPa) \\ \hline
0.27 & 35.1 & 12.75 (= 25.5/2) & 125.0 & 0.195 \\
0.58 & 75.4 & 26.95 (= 53.9/2) & 264.1 & 0.413 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Shear modulus test conditions.  
Figure 4: Mean shear force-displacement graph of elastomeric bearing specimen measured through the experiment. The thickness of elastomer in this specimen (T\({}_{\rm q}\)) is 130 mm.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Hardness (Shore A)** & 50 + 5 & 60 + 5 & 70 + 5 \\ \hline
**Shear modulus (MPa)** & 0.7 & 0.9 & 1.15 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Shear modulus (G\({}_{\rm g}\)) and hardness (Shore A): EN 1337-3 Annex D: Shear Modulus Comments.  
most suitable model for numerical simulation of elastomeric bearings, the finite element analysis (FEA) for the compression and shear tests were performed with the above three types of M-R models and compared with experimental results. Numerical simulations were performed on the same domain size (800 \(\times\) 800 \(\times\) 200 mm) as the experimental specimen.  
The finite element model for structural analysis of the elastomeric bearing was created so that the length of one side did not exceed 25 mm using an 8-node brick element with 3 degrees of freedom. The number of elements along the thickness direction of the elastomer layer between the reinforcing plates should be set to three or more because elastomer is a nonlinear material. Since the thickness of the elastomer layer is 32 mm, a finite element model was created by dividing the elastomer layer between the reinforcing plates into three, so that the aspect ratio of the brick element is 0.4 or more. Only half of the domain was modeled as the geometric shape and boundary conditions are symmetrical. The finite element models for numerical simulation (FEA) are shown in Figure 5.  
The elastomeric bearing is a laminated structure of neoprene synthetic rubber and steel plate. In the experimental results for shear modulus in Section 2.3, it is confirmed that this value corresponds to an elastomer with a Shore hardness of 50A. Accordingly, the neoprene synthetic rubber with the shore hardness of 50A was used for the elastomeric bearings in this analysis, and S355 structural steel was used for the steel plate [12; 20]. The mechanical properties for S355 structural steel are summarized in Table 4. The Mooney-Rivlin constants of the 2P, SP, and 9P M-R models for the neoprene rubber are provided in Table 5, and the material behavior curve for the Mooney-Rivlin hyperelastic model is shown in Figure 6.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Items** & **Sign** & **Units** & **Values** \\ \hline Young’s modulus & E & GPa & 200 \\ Poisson’s ratio & \(\
u\) & & 0.29 \\ Yield strength & \(\sigma_{Y}\) & MPa & 355 \\ Ultimate strength & \(\sigma_{U}\) & MPa & 510–680 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Mechanical properties for S355 structural steel (EN 10025-2).  
Figure 5: Half finite element model of elastomeric bearing (800 \(\times\) 800 \(\times\) 200 mm).