## 3 Description of the Retrofitted Building Models and Ground Motion Datasets
### Properties of the Isolation Layer
In Equation (13), \(g\) is the acceleration due to gravity, assumed to be \(9.8\) m/s\({}^{2}\), and \({}_{s}\,\alpha_{y}\) is the yielding shear strength coefficient. In this study, the design-allowable horizontal displacement (\(\delta_{a}\)) was set to \(0.40\) m, while the value of \({}_{s}\,n\) was set to \(2\) following the design recommendation [2]. Then, the two parameters of the isolation layer, \(T_{f}\) and \({}_{s}\,\alpha_{y}\), were adjusted, such that the following condition was satisfied:  
\[\left(\frac{2\pi}{T_{f}}\right)^{2}{\delta_{a}}^{2}+16_{s}\,\alpha_{y}\delta \delta_{a}\geq{V_{l}}^{2}. \tag{14}\]  
The isolation layer below level \(0\) comprises natural rubber bearings (NRBs), elastic sliding bearings (ESBs), and steel dampers. Figure 8 shows the layout of the isolators and dampers in the isolation layer of the two retrofitted building models: Model-Tf34, shown in Figure 8a, and Model-Tf44, shown in Figure 8b. As shown in the figures, the steel dampers were placed at the perimeter frames to provide torsional resistance. The point G shown in this figure is the center of mass of the superstructure, point S\({}_{0}\) is the center of stiffness of the isolation layer calculated according to the initial stiffness of the isolators and dampers, while point S\({}_{1}\) is the center of stiffness of the isolation layer calculated according to the secant stiffness of the isolators and dampers considering their displacement (8) of \(0.40\) m.  
For Model-Tf34, \(T_{f}=3.38\) s and \({}_{s}\,\alpha_{y}=0.0448\), while for Model-Tf44, \(T_{f}=4.40\) s and \({}_{s}\,\alpha_{y}=0.0396\). Note that \(T_{f}\) was calculated according to the horizontal stiffness of the NRBs, while \({}_{s}\,\alpha_{y}\) was calculated according to the yield shear strength of the ESBs and steel dampers. As shown in Figure 8, the eccentricity at the isolation layer was non-negligible in both models. The eccentricity indices [45] of Model-Tf34, calculated according to the initial stiffnesses in the X and Y directions, were \(R_{eX}=0.285\) and \(R_{eY}=0.216\), respectively, while those calculated according to the secant stiffnesses in the X and Y directions were \(R_{eX}=0.014\) and \(R_{eY}=0.054\), respectively. Similarly, the eccentricity indices of Model-Tf44, calculated according to initial stiffnesses in the X and Y directions, were \(R_{eX}=0.155\) and \(R_{eY}=0.126\), respectively, while those calculated according to the secant stiffnesses in the X and Y directions were \(R_{eX}=0.233\) and \(R_{eY}=0.015\), respectively. In this study, the perimeter frames X\({}_{1A}\) and Y\({}_{6}\) are referred to as the \"flexible-side frames,\" while the frames X\({}_{6A}\) and Y\({}_{1}\) are referred to as the \"stiff-side frames\". Note that no optimization to  
Figure 8: Layout of the isolators and dampers in the isolation layer: (**a**) Model-Tf34 and (**b**) Model-Tf44. minimize the torsional response was made to choose the dampers in this study, because such optimization was beyond the scope of this study. Figure 9 shows envelopes of the force-deformation relationship for the isolators and dampers. The behavior of the NRBs was assumed to be linear elastic, while that of the ESBs and dampers was assumed to be bilinear. All isolators were chosen from a catalog provided by the Bridgestone Corporation [55; 56], considering that the ultimate horizontal deformation was larger than 150% of \(\delta_{x}\) (1.5 \(\times\) 0.40 m = 0.60 m). Meanwhile, the steel dampers were chosen from a catalog provided by Nippon Steel Corporation Engineering Co. Ltd. (Tokyo, Japan) [57]. Tables 2-4 list the properties of the isolators and dampers used in the two models. The yield strength of an ESB was calculated as:  
\[Q_{yD}=\mu P_{V}. \tag{15}\]  
In Equation (15), \(P_{V}\) is the vertical load of the ESB due to gravity.