## 4 Observation of LRB Failure Mode through Dynamic Test  
### Simulation of Dynamic Test through FE Analysis  
Response spectrum analysis was performed for the stress evaluation of small-sized LRBs. The modeling was performed as shown in Figure 7 using a commercial program, ANSYS 2021 R2 [14]. A finite element (FE) model was prepared and analyzed to simulate the seismic test using a shaking table. As shown in Figure 7a, the 4-ton dummy mass model is supported by the four LRBs shown in Figure 7b at each square corner. It is a laminated structure composed of thin steel discs on every other rubber disc in the middle, similar to the model in Figure 7b, with thick steel end plates and rectangular flanges at the top and bottom, and a lead core inserted in the vertical center hole. All of these were modeled with a 3D solid element.  
As boundary conditions for analysis, all the degrees of freedom of the bottom surfaces of the LRB model in the four corners are fixed. In addition, it is assumed that the lead core, steel plates, and rubber plates are bonded together.  
The results of the modal analysis show that the dynamic characteristics of the LRB for small facilities, as shown in Table 3, are in good agreement with the design frequencies of 1 st horizontal and 4 th vertical modes.  
Figure 5: Hysteresis curve for OPT2.  
Figure 6: Section view of LRB after ultimate static test: (**a**) vertical section; (**b**) horizontal section.  
Figure 8 shows a representative result of the response spectrum analysis for a BDBE of 0.6 g. Regarding the lead core of the LRB, it is concluded that large plastic deformation occurred in excess of the yield strength in all areas. Shear deformation contributes to the seismic energy dissipation of the LRB system. In the case of the steel layer, it was calculated that plastic deformation occurred at the edges facing around the lead core and was similar to the trend of the actual buckled LRB. In the case of the rubber layer, the yield strength of rubber does not appear clearly because of the characteristics of non-metallic materials. Here, with reference to the literature, the shear strength of natural and vulcanized rubber has a wide application range of approximately 1 to 6 MPa [4].  
Table 4 describes the maximum stresses of the rubber and steel layers under three different seismic BDBE inputs of 0.4, 0.5, and 0.6 g. It can be roughly estimated that rubber and steel experienced yielding near the edge of the lead core at 0.6 g of DBE, as shown in Figure 8. The maximum shear strength at rubber failure can be estimated using [PERSON]'s law of shear modulus in Table 2 times the maximum shear strain in Figure 5, to be approximately 1.8 MPa. Considering the results described in Table 4 with the reference value and estimation of shear strength, it is concluded analytically that local buckling or damage may have already started from the case of a BDBE of 0.4 g.  
Figure 8: Analysis results of LRB simulating shaking table tests (DBE: 0.6 g): (a) steel layer; (b) rubber layer.  
Figure 7: FE analysis models for simulation of dynamic test: (a) isolated dummy mass model; (b) middle portion of LRB model.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Mode No.** & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline Modal analysis results [Hz] & 2.3 & 2.3 & 3.4 & 37.4 & 51.3 & 54.0 \\ Design natural frequencies [Hz] & 2.3 & - & - & 36.9 & - & - \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of designed LRB natural modes with analysis results.