## 3 Description of the Retrofitted Building Models and Ground Motion Datasets
### Properties of the Isolation Layer  
Two retrofitted building models were examined in this study to consider different seismic isolation periods (\(T_{f}\)). The properties of the isolation layer were determined as follows, based on the energy-balanced design method introduced in the design recommendation [2]. First, the mass and the moment of inertia of each floor were calculated. Table 1 shows the floor mass (\(m_{j}\)), moment of inertia (\(I_{j}\)), and the radius of gyration of the floor mass (\(r_{j}\)) for the \(j\)th floor. Here, the values of the floor mass and the moment of inertia above level 1 were taken from a previous study [44], while the weight per unit area at level 0 was assumed to be 24 \(\mathrm{kN/m^{2}}\) for the calculation of the floor mass and the moment of inertia of each floor. Therefore, the calculated total mass (\(M\)) was 5412 t.  
Figure 5: View of the former Uto City Hall after the 2016 Kumamoto earthquake [44]. Photographs were taken from (**a**) the south, (**b**) the southwest, and (**c**) the north. Figure 6: Soil profiles of the former K-NET Uto station. Figures were made from the data provided by the National Research Institute for Earth Science and Disaster Resilience (NIED). (**a**) Primary and shear wave profile (\(V_{p}\): P-wave velocity, \(V_{s}\): S-wave velocity); (**b**) density profile; (**c**) soil column. Figure 7 shows the horizontal response spectra of the recorded ground motions of the first (14 April 2016: UTO0414) and second (16 April 2016: UTO0416) earthquakes observed at the K-NET Uto station [54]. Note that the spectral acceleration (\(S_{A}\)) and velocity (\(S_{V}\)) shown in Figure 7a,b were calculated as the absolute (vector) value of the two horizontal components. The total input energy spectrum (the \(V_{I}\) spectrum) was also calculated considering the simultaneous input of the two horizontal components. The viscous damping ratio was set to 0.05 for the calculations of \(S_{A}\) and \(S_{V}\), while it was set to 0.10 for the calculation of \(V_{I}\) following the work of [PERSON] [47]. In this study, the range of the seismic isolation period (\(T_{f}\)) was considered to be from 3 to 4.5 s when determining the properties of the isolation layer. Based on Figure 7c, the equivalent velocity of the total input energy (\(V_{I}\)) was set to 1.41 m/s to determine the properties of the isolation layer. Assuming that the superstructure is a rigid body, the equation of the energy balance is expressed as:  
\[\frac{1}{2}K_{f}\delta_{\mathrm{max}}{}^{2}+4_{s}\,n_{s}Q_{y}\delta_{\mathrm{ max}}=\frac{1}{2}MV_{I}{}^{2}. \tag{11}\]  
In Equation (11), \(\delta_{\mathrm{max}}\) is the maximum horizontal displacement, \({}_{s}\)\(n\) is the equivalent number of repetitions, \(K_{f}\) is the total horizontal stiffness of the flexible element, and \({}_{s}\)\(Q_{y}\) is the total yield strength of the rigid plastic element. Equation (11) can be rewritten as:  
\[\left(\frac{2\pi}{T_{f}}\right)^{2}\delta_{\mathrm{max}}{}^{2}+8_{s}\,n_{s} \alpha_{y}g\delta_{\mathrm{max}}=V_{I}{}^{2}, \tag{12}\]  
\[\text{where }K_{f}=M\left(\frac{2\pi}{T_{f}}\right)^{2}{}_{s}\,\alpha_{y}= \frac{{}_{s}\,Q_{y}}{Mg}. \tag{13}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Floor Level** & **Floor Mass** & **Moment of Inertia** & **Radius of Gyration** \\ \(j\) & \(m_{j}\) (**t)** & \(I_{j}\) (\(\times\)10\({}^{3}\) tm\({}^{2}\))** & **of Floor Mass \(r_{j}\) (m)** \\ \hline
5 & 677.8 & 78.37 & 10.75 \\
4 & 548.5 & 62.85 & 10.70 \\
3 & 543.0 & 62.50 & 10.73 \\
2 & 581.0 & 67.12 & 10.75 \\
1 & 1208.1 & 199.0 & 12.83 \\
0 & 1853.7 & 274.9 & 12.18 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Floor mass, moment of inertia, and radius of gyration of the floor mass of each floor level. Figure 7: Horizontal response spectra of the recorded ground motions at the K-NET Uto station: (**a**) Elastic acceleration response spectrum (damping ratio: 0.05); (**b**) elastic velocity response spectrum (damping ratio: 0.05); (**c**) elastic total input energy spectrum (damping ratio: 0.10).