# Seismic Response Effect on Base-Isolated Rigid Structures by Mass Eccentricity in Nuclear Plants
## 3 Sensitivity Analysis of Base Isolation Response Due to Mass Eccentricity
### Seismic Response Sensitivity of Base-Isolated Structures by Mass Eccentricity  
In this paper, the response due to structural stiffness eccentricity is not considered. Therefore, a small base-isolation bearing designed for equipment, developed in previous studies, was applied, assuming a constant support stiffness. Consequently, a base-isolated cubic design with lumped mass was selected as the analysis model for the upper structure, which facilitates setting the mass eccentricity sensitivity of the seismic response.  
To reduce the significant computational time required for seismic time history analysis when modeling small base isolators at full scale and performing nonlinear analysis, this study simplified the small base isolators by representing them as spring elements for sensitivity analysis.  
For reference, Figure 2a depicts the configuration where the four 1-ton base isolators developed earlier are positioned at the corners of the 4-ton cubic mass model's underside for support. Figure 2b shows the simplification for analysis efficiency, where each base isolator is modeled with one vertical (Z) and two horizontal (X, Y) spring elements, totaling 12 spring elements. The stiffnesses of the springs in the horizontal and vertical directions are calculated as follows.  
\[\mathrm{K_{h}}=\left(G\times A\right)\,/\,T_{R} \tag{2}\]  
\[\mathrm{K_{v}}=\left(E_{C}\times A\right)\,/\,T_{R} \tag{3}\]  
(\(\mathrm{K_{h}}\): Horizontal stiffness, \(\mathrm{K_{v}}\): Vertical stiffness. \(T_{R}\): The total thickness of the rubber layer in the base isolator, \(G\): The Shear modulus of the rubber, \(E_{C}\): The elastic modulus adjusted based on compression characteristics, \(A\): Effective design area).  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Properties** & **Values** \\ \hline Outer Diameter (mm) & 100 mm \\ Design Horizontal Freq. & 2.3 Hz \\ Shape Factors, S1/S2 & 9.9/5.0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Design specifications for 1-ton LRB for NPP equipment.  
Figure 1: LRB cross-sectional shape.  
To investigate the changes in horizontal dynamic characteristics and response sensitivity due to the one-axis mass eccentricity of the structure, seismic analysis was conducted using code-based methods [14]. In the sensitivity analysis, as shown in Figure 3, the mass eccentricity was defined as 0% and 100% when the mass center of the upper structure (Dummy Mass) was positioned at the center and at the edge, respectively. In this context, the analysis was theoretically extended up to 100% eccentricity for the purpose of the study. However, in actual structural design, eccentricities exceeding 50% are very unusual.  
It should be noted that the points from A1 to A4 in Figure 3 represent the corner positions of the upper structure, which are the actual locations of the acceleration sensors used in subsequent vibration testing, as described later. By varying the upper structure's mass eccentricity from 0% to 100% in 1% intervals, modal analysis and response spectrum analysis were conducted to observe the impact on displacement and acceleration seismic responses.