## 3 Collapse Performance of Seismically Isolated Buildings
### Analysis Models of Isolated Buildings
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c c c} \hline \hline \multicolumn{11}{c}{\multirow{3}{*}{-}} & \multicolumn{11}{c}{**(b) Building at Low-to-Moderate Seismicity Region**} \\ \cline{2-11}  & \multicolumn{4}{c}{**Under Strong Windy Load**} & \multicolumn{4}{c}{**Under Weak Windy Load**} \\ \cline{2-11}  & \multicolumn{4}{c}{**3-Story**} & \multicolumn{4}{c}{**5-Story**} & \multicolumn{4}{c}{**10-Story**} & \multicolumn{4}{c}{**3-Story**} & \multicolumn{4}{c}{**5-Story**} & \multicolumn{4}{c}{**10-Story**} \\ \cline{2-11}  & \multicolumn{4}{c}{**3L\({}_{\text{E}}\)40C**} & \multicolumn{4}{c}{**5L\({}_{\text{E}}\)40C**} & \multicolumn{4}{c}{**10L\({}_{\text{E}}\)40C**} & \multicolumn{4}{c}{**3L\({}_{\text{E}}\)30B**} & \multicolumn{4}{c}{**5L\({}_{\text{E}}\)30B**} & \multicolumn{4}{c}{**10L\({}_{\text{E}}\)30B**} \\ \cline{2-11}  & **LB** & **M** & **UB** & **LB** & **M** & **UB** & **LB** & **M** & **UB** & **LB** & **M** & **UB** & **LB** & **M** & **UB** & **LB** & **M** & **UB** \\ \hline \(T_{1}\) & 0.70 & & 1.00 & & 1.49 & & & & 0.70 & & & 1.00 & & & 1.49 & \\ \(T_{M}\) & 2.29 & 1.78 & 1.46 & 2.15 & 1.86 & 1.46 & 2.22 & 1.91 & 1.51 & 3.14 & 2.76 & 2.27 & 3.12 & 2.73 & 2.27 & 3.02 & 2.74 & 2.20 \\ \(T_{1}/T_{M}\) & 0.31 & 0.39 & 0.48 & 0.47 & 0.54 & 0.68 & 0.67 & 0.78 & 0.99 & 0.22 & 0.25 & 0.31 & 0.32 & 0.37 & 0.44 & 0.49 & 0.54 & 0.68 \\ \(\beta_{\text{eq}M}\) & 2.29 & 2.42 & 2.65 & 2.77 & 2.81 & 2.86 & 25.9 & 25.7 & 23.8 & 1.81 & 1.84 & 2.25 & 1.38 & 14.7 & 16.6 & 1.69 & 14.3 & 17.7 \\ \(B_{M}\) & 1.56 & 1.58 & 1.63 & 1.65 & 1.66 & 1.67 & 1.62 & 1.61 & 1.58 & 1.44 & 1.45 & 1.55 & 1.32 & 1.34 & 1.45 & 1.41 & 1.33 & 1.75 \\ \hline \multicolumn{11}{c}{Note: LB is the lower bound property, M is the mean property, UB is the upper bound property, \(T_{1}\) is the 1 st modal} \\ \multicolumn{11}{c}{period of superstructures (s), \(T_{M}\) is the effective period at the \(D_{M}\) (s), \(\beta_{\text{eq}M}\) is the effective damping ratio under MCE,} \\ \multicolumn{11}{c}{\(B_{M}\) is the damping coefficient at the \(D_{M}\), \(D_{M}\) is the maximum displacement.} \\ \end{tabular}
\end{table} 
Table 3: _Cont._prototype isolated buildings with symmetric planes have no eccentricity. No overturning is also assumed to occur in the superstructure during seismic loading. As shown in Figure 5, the SLMM consists of the multi-spring assemblage located on the top of a shear spring representing the hysteretic behavior for an isolation system. The spring assemblage representing the entire superstructure consists of in-series connected transverse springs with the stiffness, \(K_{s,i}\), and the yield strength, \(F_{s,i}\). The structural properties of the yield strength, initial stiffness, and post-yield stiffness of the transverse spring representing the story in the superstructure are determined by the procedure as follows:  
1. With a lateral load pattern corresponding to the yield displacement of the superstructure, \(\delta_{y}\) is then obtained. Then, the yield displacement of the \(i\)th story, \(d_{yi}\), is calculated by multiplying the relative eigenvector \(\{f_{i+1}-f_{i}\}\) and the value of \(\delta_{y}\). 2. The stiffness of spring elements, \(K_{s,i}\), is determined by introducing the coefficient for modifying the stiffness, \(\rho_{i}\) to the seismic weight over relative eigenvector, \(W_{i}/[f_{i+1}-f_{i}]\). The value of \(\rho_{i}\) can be obtained throughout iterations by changing it until the difference between \(T_{1}\) and \(\{f_{i}\}\) of the SLMM and those of the detailed model is sufficiently small. 3. The yield strength of spring elements, \(Q_{s,i}\) is determined by multiplying \(K_{s,i}\) and \(d_{yi}\). Until the difference between the strength obtained from the SLMM and detailed model is sufficiently small, the \(Q_{s,i}\) value is modified by introducing the coefficient for modifying the strength, \(\gamma_{i}\). Figure 6a shows the comparison of the first mode periods and mode shape vectors obtained from the detailed model and the SLMN of the superstructure.