## 4 Vertical Spring Design for Vertical Seismic Isolation Device
### Disc Spring Design  
The disc spring used in this study is a cone-shaped, thin steel structure with outer diameter (\(D\)), inner diameter (\(d\)), thickness (\(t\)), and height (\(H\)) as shown in Figure 7. When a vertical force is applied, the disc spring stably undergoes compression deformation and has a constant stiffness value within the operating displacement. Once the design stiffness value is determined, it can be designed to have the appropriate load capacity and displacement range by stacking the required number of disc springs in series or in parallel. The relationship between the applied load (\(F\)) and the corresponding displacement (\(y\)) in the disc spring in Figure 7 can be described as the following equation [24]:  
\[F=\frac{2 Et}{(1-\
u^{2})ZD^{2}}\left[y^{3}-3 hy^{2}+2y\left(h^{2}-t^{2}\right)\right] \tag{1}\]  
where  
\[Z=\frac{1}{\pi}\left(\frac{C-1}{C}\right)^{2}\left[\left(\frac{C+1}{C-1} \right)-\ln\frac{2}{C}\right]^{-1} \tag{2}\]  
\[C=\frac{D}{d} \tag{3}\]  
In the above equation, \(E\) and \(\
u\) represent the elastic modulus and Poisson's ratio of the material, respectively. When the disc springs are stacked in series, the total displacement and equivalent stiffness are proportional to \(1/n\) (\(n\) is number of disc springs). Therefore, when the design VIF is determined, it can be implemented by connecting an appropriate number of disc springs in series. Table 5 presents the design parameters of the disc spring designed in this study. The plate spring material used in the design is JIS SUP10 spring steel. In Table 5 above, the shut displacement (\(h\)) refers to the maximum displacement that the disc spring can accommodate. The disc spring is mounted on the upper flange of LRB, which functions as a horizontal seismic isolator. If the outer diameter of the disc spring is larger than the outer diameter of the LRB, bending deformation may occur in the upper flange of LRB due to the load reaction force applied to the disc spring by the seismic response of the superstructure. In this study, considering the outer diameter of 100 mm of the LRB, the outer diameter of the disc spring is determined to be 80 mm. In general, the design-allowable displacement of the disc spring assumes 75% of the shut displacement. In this case, the design-allowable displacement for a single disc spring of Table 5 is 1.65 mm, considering the shut displacement (h = 2.2 mm). Therefore,  
Figure 7: Schematics of the disc spring configurations and dimensional design parameters. assuming that the relationship between displacement and reaction force is linear within the design-allowable displacement of the single disc spring, the load capacity of the single disc spring becomes 22.9 kN from Equation (1) above, and, accordingly, the spring stiffness is determined to be 13,879 kN/m. Figure 8 compares the stiffness test results for six single disc springs manufactured according to the design parameters in Table 5 and the stiffness values calculated from Equation (1). As shown in the figure, it was confirmed that the stiffness test results and calculation results were well matched up to the design-allowable displacement of 1.65 mm, and linearity was guaranteed. Table 6 presents the results of design summary for the set of disc spring that satisfies the design VIF. As shown in the table, when the design VIF is lower, the required number of disc springs greatly increases. This results in a significant increase in the total height of the spring and acts as a burden in the housing design. When the design VIF is set to 2.0 Hz or less, the required housing height of the vertical seismic isolation device is more than 500 mm. Therefore, when using a disc spring, it is desirable to determine the design VIF at least larger than 2.5 Hz with consideration of an appropriate height. Figure 8: Comparison of the linearity characteristics between tests and calculation values by formula for a single disc spring.