## 4 NSAS Effectiveness for Underground Structures
### Displacement Control  
Dealing with the improvement of the NSAS isolation system compared with the CIS, the CIS stiffness and damping parameters were designed by referring to [10]. CIS stiffness \(k_{iso}\) was 18,000 kN/m, and the damping coefficient \(c_{d}\) was 40 kNs/m. The maximal shear force under CIS control was 308 kN,  
Figure 10: Time series of input acceleration for Daikai station (horizontal direction). (**a**) Port Island and (**b**) HIK station records.  
Figure 9: Location between subway station and soil.  
potentially presenting about a 60% decrease of shear force response. The maximal displacement of the top relative to the bottom of the central column was 54 mm.  
Given the same isolator and dashpot designed for CIS, the NSAS isolation system was further designed with variable negative-stiffness ratio \(\chi\) and positive-stiffness ratio \(\kappa\). Analysis results of \(\beta_{Dis,NSAS}/\beta_{Dis,CIS}\) and \(\gamma_{Dis,NSAS}/\gamma_{Dis,CIS}\) are plotted in Figure 11, for \(\kappa\) = 0.30, 0.35, 0.40 and 0.50, respectively, and \(\chi\) = \(-\)0.10, \(-\)0.15, \(-\)0.20 and \(-\)0.25, respectively. Furthermore, the red, green, blue and light blue bars are used to represent different parameter sets of \(\chi=-\)0.10, \(-\)0.15, \(-\)0.20 and \(-\)0.25. These values are given in Tables 2 and 3, respectively. For the majority of the parameter sets of \(\chi\) and \(\kappa\), \(\beta_{Dis,NSAS}/\beta_{Dis,CIS}\) and \(\gamma_{Dis,NSAS}/\gamma_{Dis,CIS}\) were lower than 1.0. In the sense of peak- and RMS response-based evaluation, the NSAS isolation system exhibited improved capacity for seismic-vibration control of the relative displacement of the central column in comparison with that of the CIS. Specifically, when \(\kappa\ =\ 0.35\) and \(\chi=-\)0.20, the values of \(\beta_{Dis,NSAS}/\beta_{Dis,CIS}\) and \(\gamma^{Dis,NSAS}/\gamma^{Dis,CIS}\) were 0.85 and 0.87, respectively, which indicated about a 15% decrease in the relative displacement of the central column. On the basis of the enhanced energy-dissipation effect introduced in Section 2.3, the damping coefficient of the NSAS isolation system, the same as that of the CIS, provided a more effective mitigation of the relative displacement of the central column of the underground structure. As an intuitive illustration, the relative displacement responses of the central column were plotted for the uncontrolled underground structure, and the structures controlled with the NSAS isolation system (\(\kappa\) = 0.35, \(\chi=-\)0.20) and CIS, as shown in Figure 12. With the installation of the NSAS isolation system and CIS, the maximal displacement of the central column was increased from 38 to 46 and to 54 mm, respectively. Compared with the case of the original uncontrolled central column (38 mm), the maximal relative displacement of the central column was decreased from 16 (CIS) to 8 mm (NSAS isolation system), resulting in about a 50% reduction using the NSAS isolation system. At the end of the earthquake, the column exhibited permanent displacement in the uncontrolled underground structures, and in underground structures with CIS and NSAS isolation systems. This phenomenon was caused by the damage of the underground structure and the plastic soil deformations that had accumulated during the earthquake loading cycles.  
Figure 11: Ratios of the relative displacements of the central column of the underground structures controlled with a negative-stiffness amplification-system (NSAS)-based isolation system, and a conventional isolation system (CIS): (**a**) \(\beta_{Dis,NSAS}/\beta_{Dis,CIS}\):(**b**) \(\gamma_{Dis,NSAS}/\gamma_{Dis,CIS}\).