# The Performance Study of a Novel Vertical Variable Stiffness Disc Spring Seismic Isolation Bearing
## 4 Test Results and Discussion
### Results and Analysis of the Cyclic Load Test
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Load and Stiffness** & \(\mathbf{F_{1}}\) (\(\mathbf{\kappa}\)) & \(\mathbf{F_{3}}\) (\(\mathbf{\kappa}\)) & \(\mathbf{K_{1}}\) (\(\mathbf{\kappa}\)/mm) & \(\mathbf{K_{3}}\) (\(\mathbf{\kappa}\)/mm) \\ \hline Theoretical results & 346.03 & 712.36 & 11.09 & 33.92 \\ Test results & 363.92 & 764.15 & 11.66 & 37.06 \\ Relative errors (\%) & 4.92 & 6.78 & 4.89 & 8.47 \\ \hline \hline \end{tabular} Note: \(K_{1}=F_{1}/x_{1}\) is the loading stiffness of the isolation stage of the bearing, and \(K_{3}=(F_{3}-F_{1})/(x_{3}-x_{1})\) is the loading stiffness of the protection stage of the bearing, where \(F_{1}\) and \(F_{3}\) are the peak loads of the isolation and protection stages, and \(x_{1}\) and \(x_{3}\) are the displacements corresponding to \(F_{1}\) and \(F_{3}\).  
\end{table} 
Table 3: Peak loads and loading stiffnesses of the VVSDSI bearing during the isolation and protection stages. stiffress \(K_{e}\) is the slope of the bearing's hysteresis curve, which can be determined using the following equation:  
\[K_{e}=\frac{F_{\text{max}}-F_{\text{min}}}{D_{\text{max}}-D_{\text{min}}} \tag{11}\]  
where \(D_{\text{max}}\) and \(D_{\text{min}}\) are the maximum displacement values in the loading and unloading directions of the hysteresis curve, respectively, and \(F_{\text{max}}\) and \(F_{\text{min}}\) are the loads corresponding to \(D_{\text{max}}\) and \(D_{\text{min}}\), respectively. Figure 10: Energy dissipation of the VVSDSI bearing under different preloading displacements and loading amplitudes. (**a**) Different preloading displacements: (**b**) different loading amplitudes. Figure 9: Load-displacement curves for VVSDSI bearing. (**a**) Preloading displacement 16 mm; (**b**) preloading displacement 23 mm; (**c**) preloading displacement 30 mm. According to the theoretical relationship between damping and energy loss, the formula for computing the equivalent damping ratio is as follows:  
\[\zeta=\frac{W}{4\pi\omega_{S}}=\frac{2W}{\pi K_{e}D^{2}} \tag{12}\]  
where \(W\) is the area of the hysteresis curve envelope, indicating the energy loss per cycle of the curve; \(\omega_{S}\) is the elastic strain energy: and \(D\) is the relative displacement of the hysteresis curve (\(D=D_{\text{max}}-D_{\text{min}}\)). Table 4 presents the experimental and theoretical analysis results of hysteresis curves under different preloading displacements and loading amplitudes. The data in Table 4 show that the maximum errors of \(F_{\text{max}}\) and \(F_{\text{min}}\) are 7.13% and 3.45%, respectively, with average errors of 2.20% and 1.23%. The average error for equivalent stiffness values reaches 6.53%. It is worth noting that when the preloading displacement is 30 mm and the loading amplitudes are 1 mm and 2 mm, the equivalent stiffness error is relatively large. This phenomenon occurs because, at this stage, the upper disc spring group is approaching its limit state, leading to an increase in the nonlinearity of the disc springs, which in turn results in greater discrepancies. In summary, the experimental results align well with the theoretical solutions, indicating that the proposed variable-stiffness seismic isolation bearing (VVSDSI) hysteresis model can effectively predict the hysteretic behavior of the bearing. Figure 11 and Table 5 compare the equivalent stiffnesses of the bearings under different preloading displacements and loading amplitudes. The results indicate that the vertical equivalent stiffness of the bearings ranges from 12.63 kN/mm to 52.42 kN/mm, with the minimum stiffness value of 12.63 kN/mm occurring at a preloading displacement of 16 mm and a loading amplitude of 6 mm. As shown in Figure 11a, when the preloading displacement is 16 mm, the equivalent stiffness decreases as the loading amplitude increases, with values of 24.26 kN/mm, 18.16 kN/mm, 14.02 kN/mm, and 12.63 kN/mm, respectively. This demonstrates that, for the same preloading displacement, the equivalent stiffness of the bearing decreases with an increasing loading amplitude.