## 4 NSAS Effectiveness for Underground Structures  
### Measurement Definition  
Referring to Section 2.2, the NSAS isolation system is described in nondimensional form for underground structures, including damping ratio \(\xi=c_{d}/2\sqrt{m_{c}k_{iso}}\), negative-stiffness ratio \(\chi=k_{n}/k_{iso}\), and stiffness ratio \(\kappa=k_{d}/k_{iso}\); \(m_{c}\) and \(k_{iso}\) denote the mass of the central column and the stiffness of the isolator inserted between the central column and the upper slab. Dimensionless measurements are defined by comparing the seismic responses of the underground structure controlled by the NSAS isolation system or conventional isolation system (CIS, including the linear isolator in parallel with a dashpot, same as the NSAS isolation system) to those of the uncontrolled underground structure. The criteria of peak responses (represented by \(\beta\)) and root mean square (RMS, represented by \(\gamma\)) were adopted in terms of the relative displacement between the top and the bottom of the central column, and the base shear force of the column, as  
\[\left\{ \begin{array}{l}\beta_{Dis}=Dis_{peak}/Dis_{peak,0}\\ \beta_{SF}=SF_{peak}/SF_{peak,0}\\ \gamma_{Dis}=Dis_{RMS}/Dis_{RMS,0}\\ \gamma_{SF}=SF_{RMS}/SF_{RMS,0}\end{array} \right. \tag{10}\]  
where \(Dis_{peak}\) and \(Dis_{RMS}\) are the peak and RMS responses, respectively, of the relative displacement between the top and the bottom of the central column under the control of the NSAS or CIS, respectively. \(SF_{peak}\) and \(SF_{RMS}\) are the peak and RMS responses, respectively, of the shear force of the central column  
Figure 7: Models of dynamic contact behavior between soil and underground structure: (**a**) master–slave surface relationship; contact behavior in (**b**) normal and (**c**) tangential directions. under the control of the NSA or CIS. Footnote \"0\" refers to a case of the uncontrolled underground structure without any control devices. Given the established measurement system, Daikai station was adopted as a typical example [38] to demonstrate the seismic-vibration mitigation effect and the feasibility of the proposed NSAS isolation system. The obtained effectiveness and design can be used as guidelines for other typical underground structures with similar and more complex typologies. Daikai station was severely damaged during the 1995 Great Hanshin earthquake [4]. It is a typical two-span subway station with a cross-section, displayed in Figure 8. The dimensions of the central column are \(0.40\times 1.0\) m, and the space between central columns is 3.5 m. The density, Poisson's ratio and Young's modulus of the structural concrete in this model were 2500 kg/m\({}^{3}\), 0.20 and 30 GPa, respectively. The initial compression yield stress, initial tensile yield stress and limited compression yield stress of the concrete material were 18.8, 2.4 and 26.8 MPa, respectively [11]. Daikai station was built on diluvial deposits and is overlain by alluvial deposits. The soil profile can be simplified into seven horizontally layered soil layers, and the characteristics and properties for each soil layer can be found in Table 1[28]. Location between subway station and soil is shown in Figure 9. When using the Davidenkov model, outlined in Section 3.2, to simulate the dynamic behavior of soil, the fitting parameters \(A\), \(B\) and \(\gamma_{0}\) in Equations (6)-(8) for different types of soil are referred to in [39]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**No.** & **Type** & **Thickness (m)** & **Density (kg/m\({}^{3}\))** & \(V_{\mathbf{s}}\) (m/s)** & **Gmax (MPa)** & **Poisson’s Ratio** \\ \hline
1 & Filled & 1.0 & 1900 & 140 & 38 & 0.333 \\
2 & Sand & 4.1 & 1900 & 140 & 38 & 0.32 \\
3 & Sand & 3.2 & 1900 & 170 & 56 & 0.32 \\
4 & Clay & 3.1 & 1900 & 190 & 70 & 0.40 \\
5 & Clay & 5.8 & 1900 & 240 & 112 & 0.30 \\
6 & Sand & 5.0 & 2000 & 330 & 222 & 0.26 \\
7 & Sand & 11.8 & 2000 & 1000 & 2040 & 0.26 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Soil profile of Daikai station.