## 4 Analysis Results  
The equations of motion of the fixed base and isolated base buildings are established based on the input parameters presented previously. The set of ordinary differential equations is solved by the fourth-order Runge-Kutta numerical method using the ode15s function in the MATLAB software. The numerical results in terms of time-history data of the absolute acceleration and shear force for the non-isolated and isolated cases of the building subjected to three components of the excitation are shown in Figures 5-8. The peak responses of the two cases are summarized in Table 2, where the responses of the building considering only two horizontal components are also added for the comparison. Figure 4: Acceleration time history data and response spectra of the selected ground motions: (**a**) Rinaldi Receiving station and (**b**) LA-Obregon Park station. Figure 5: Time history of roof absolute acceleration of the building subjected to Record 1: (**a**) X-response and (**b**) Y-response. Figure 6: Time history of roof absolute acceleration of the building subjected to Record 2: (**a**) X-response and (**b**) Y-response. The results show the high effectiveness of the isolation system in the reduction of the seismic response of the building. For example, in most cases, a reduction of about 60% in terms of the roof acceleration is recorded when the DFP bearing system is implemented, and that figure for the 1 st-floor shear force is about 80%. The comparison of the peak responses when the isolated building is subjected to only two horizontal components (XY) and three components (XYZ) is also shown in Table 2. It can be seen that there is a considerable increase in terms of the acceleration and shear force when the vertical excitation is included in the model, resulting in the effectiveness of the bearing decreases. The increase of the seismic response compared with the bi-directional case is more significant in the case of near-source record; in detail, the difference of the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Ground** & **Response** & **Non-Isolated** & **Isolated (XY)** & \begin{tabular}{c} **Per. of** \\ **Reduc. (\%)** \\ \end{tabular} & \begin{tabular}{c} **Isolated (XYZ)** \\ \end{tabular} & \begin{tabular}{c} **Per. of** \\ **Reduc. (\%)** \\ \end{tabular} \\ \hline \multirow{4}{*}{\begin{tabular}{c} Record \(1\) \\ (near-source) \\ \end{tabular} } & Roof accel.-X (g) & 4.285 & 1.038 & 76 & 1.690 & 61 \\ \cline{2-6}  & Roof accel.-Y (g) & 1.760 & 0.608 & 65 & 1.038 & 41 \\ \cline{2-6}  & 1 st-floor & \multirow{2}{*}{\
\begin{tabular}{c} 1.331 \(\times\)\(10^{4}\) \\ shear-X (kN) \\ \end{tabular} } & \multirow{2}{*}{\(1.033\times\)\(10^{3}\)} & \multirow{2}{*}{92} & \multirow{2}{*}{\(1.409\times\)\(10^{3}\)} & \multirow{2}{*}{89} \\ \cline{2-6}  & 1 st-floor & & & & \\ \cline{2-6}  & shear-Y (kN) & & & & \\ \hline \multirow{4}{*}{
\begin{tabular}{c} Record 2 \\ (far-field) \\ \end{tabular} } & Roof accel.-X (g) & 1.741 & 0.582 & 67 & 0.617 & 65 \\ \cline{2-6}  & Roof accel.-Y (g) & 1.890 & 0.610 & 68 & 0.623 & 67 \\ \cline{2-6}  & 1 st-floor & & & & \\ \cline{1-1} \cline{2-6}  & shear-X (kN) & & & & & \\ \cline{1-1} \cline{2-6}  & 1 st-floor & & & & \\ \cline{1-1} \cline{2-6}  & shear-Y (kN) & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Peak seismic response of the building. Figure 8: Time history of the 1 st-floor shear force of the building subjected to Record 2: **(a)** X-response and **(b)** Y-response. Figure 7: Time history of the 1 st-floor shear force of the building subjected to Record 1: **(a)** X-response and **(b)** Y-response. percentage of reduction between the two cases is about 15% for the X-response of the acceleration and 24% for the Y-response. The comparative results demonstrate the necessity of considering the impact of the vertical excitation on the isolated building.