## 5 Results and Discussion
### Displacement at the Top of the Structures  
Figure 12 illustrates the horizontal displacement at the top of each building in the longitudinal and transversal direction for the seven accelerograms. The percentage of reduction in the responses for the low-rise building (Figure 12a,b) equipped with friction dampers, in comparison with the structure without dampers, generally exceeds 70.29% in both directions, and reaches 86.10% with ChiChi earthquake excitation in the transversal direction. Although the reduction in the longitudinal direction with viscous dampers is bigger than those obtained with friction dampers, it is limited in the transversal direction, reaching a maximum of 39.49%. As for the damped case with TMD, the percentage of reduction does not exceed 12.94% for all earthquake records except Samos. By evaluating the mean value of percentage reduction in both directions, which is equal to 6.72% and 9.75% with a tuned mass damper, 91.11% and 30.82% with viscous dampers, and 71.44% and 76.87 % with friction dampers, it can be seen that friction dampers perform better than the two other types in the response reduction in the low-rise building. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Building Type**} & \multirow{2}{*}{**Direction**} & \multicolumn{2}{c}{**Fundamental Period**} & \multicolumn{2}{c}{**Top Roof Displacement**} & \multicolumn{2}{c}{**Base Shear**} \\  & & **[s]** & & **[cm]** & **[km]** & **[km]** \\  & & **Altern. 1** & **Altern. 2** & **Altern. 1** & **Altern. 2** & **Altern. 1** & **Altern. 2** \\ \hline \multirow{2}{*}{Low-rise} & Longitudinal & 0.158 & 0.158 & 0.228 & 0.290 & 6971.2 & 7296.0 \\  & Transversal & 0.218 & 0.218 & 0.126 & 0.057 & 1525.1 & 2000.0 \\ \multirow{2}{*}{Mid-rise} & Longitudinal & 0.707 & 0.708 & 1.682 & 1.775 & 14,030.2 & 14,639.4 \\  & Transversal & 0.998 & 0.999 & 1.124 & 0.298 & 3100.0 & 3500.0 \\ \multirow{2}{*}{High-rise} & Longitudinal & 2.016 & 2.016 & 6.915 & 7.787 & 12,373.0 & 13,950.1 \\  & Transversal & 2.239 & 2.239 & 4.349 & 2.672 & 3494.58 & 4000.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Responses of the three investigated buildings for the two alternatives of FDs placement. Figure 11: Modeling configuration of FDs. The maximum roof displacement value of the mid-rise building for each history record is plotted in Figure 12 with and without dampers. According to the results, it can be seen that utilizing viscous dampers reduces the displacement the most in both horizontal directions, which goes beyond 91.46%. For comparison purposes, the maximum displacement values with friction dampers in an 87.26% and 79.25% reduction and with tuned mass damper in a 31.71% and 19.24% reduction, respectively, in longitudinal and transversal direction. Since the viscous and friction dampers have the greatest impact on the displacement by evaluating the mean value of percentage reduction in both directions, which is equal to 6.58% and 6.30% with a tuned mass damper, 93.70% and 98.96% with viscous dampers, and 81.14% and 63.29% with friction dampers, both systems seem to perform well under all earthquake records for the mid-rise building. From the results of the high-rise building, all three types of dampers contribute to significant reduction in terms of displacement. One could observe that the percentage of reduction for the high-rise building equipped with friction dampers reaches a maximum of 90.36% in the longitudinal direction, and a maximum of 69.50% in the transversal direction, which is considered high. Viscous dampers also provide high values of reduction, reaching a maximum of 91.89% and 76.96% in both horizontal directions, respectively. Moreover, the pendulum configuration of PTMD offers a great reduction of 52.10% and 65.75% for the  
Figure 12: Horizontal displacement at the top of (**a**,**b**) the low-rise building, (**c**,**d**) the mid-rise building, and (**e**,**f**) the high-rise building. longitudinal and transversal direction, respectively.