## 2 Numerical Study
### Two Three-Story Models of Buildings: Non-Isolated and Isolated with Rubber Bumpers  
In this analysis, an isolated building was equipped with four bumpers that were square in shape. Rubber bumpers were modeled as elements with a square cross-section with dimensions of 40 \(\times\) 40 cm, and their arrangement is presented in Figure 6. The stiffness of the bumpers was assumed to be 0.12 kN/mm, while the post yield impact stiffness was equal to 500 kN/mm. Figure 5: Lateral displacement time histories of non-isolated and isolated building with a 19 cm gap. Figure 6: Schematic diagram of two three-story isolated models and the locations of rubber bumpers. The dynamic response of the three-story non-isolated and the isolated buildings was determined, and the effectiveness of the application of rubber bumpers in reducing the negative effects of earthquake-induced bounding between base-isolated buildings was verified. As can be seen from Figure 7, some collisions between the two models during seismic excitation were observed. Moreover, the rubber bumpers were activated after impact. Figure 7 indicates that the first story exhibited an overlap of 1.62 cm, the second story an overlap of 4.78 cm, and the third story one of 6.52 cm. The maximum impact forces in each story were calculated to be 2.38 \(\times\) 10\({}^{5}\) kN, 6.9 \(\times\) 10\({}^{5}\) kN and 10.05 \(\times\) 10\({}^{5}\) kN for the first, second and the third story, respectively (see Figure 8). Moreover, time histories for the impact forces are presented in Figure 9.  
Figure 7: Lateral displacement time histories of non-isolated and the isolated buildings with an 8 cm size gap: (**a**) base level; (**b**) first story; (**c**) second story; (**d**) third story. Another case was analyzed in which the gap size between the two adjacent buildings was reduced to 5 cm. Two Three-Story Models of Buildings: Non-Isolated and Isolated (Gap Size = 5 cm) Exposed to Different Earthquakes  
To investigate the effectiveness of rubber bumpers during seismic excitation more intensively, four different earthquake records were considered: Kobe (1995), Parkfield (1966), San Fernando (1971) and Tabas (1978). Their peak ground accelerations were 817 cm/s\({}^{2}\), 462 cm/s\({}^{2}\), 1202 cm/s\({}^{2}\) and 344 cm/s\({}^{2}\), respectively. A model of three-story buildings with a seismic gap of 5 cm was used in the analysis. The lateral displacement time histories for each story of the buildings during seismic excitation are presented in Figure 10. As can be seen from Figure 10, the peak values of the lateral displacements of buildings were equal to 20.99 cm for the Parkfield earthquake, 10.19 cm for the Kobe earthquake, 20.00 cm for the San Fernando earthquake and 37.00 cm for the Tabas earthquake. Moreover, the peak drifts of all models were calculated to be 0.606 cm, 0.257 cm, 0.577 cm and 0.345 cm for the Parkfield, Kobe, San Fernando and Tabas earthquakes, respectively. Figure 8: Hysteresis loops of rubber bumpers during impact: (**a**) base level; (**b**) first story; (**c**) second story; (**d**) third story. The hysteresis loops of the rubber bumpers during impact, when the base level exceeds a relative displacement of 5 cm, are shown in Figure 11. The results indicate that the energy was absorbed by bumpers in order to control the lateral displacement of all stories. The peak values of the forces induced in bumpers were equal to: 15.55 \(\times\) 10\({}^{3}\) kN, 8.15 \(\times\) 10\({}^{3}\) kN, 4.85 \(\times\) 10\({}^{3}\) kN and 10.40 \(\times\) 10\({}^{3}\) kN for the Parkfield, Kobe, San Fernando and Tabas earthquakes, respectively. As can be clearly seen, impact occurred after the critical distance (5 cm) and energy had dissipated, as depicted by the hysteresis loops in Figure 11.  
Figure 9: Impact force time histories obtained for each story: (**a**) base level; (**b**) first story; (**c**) second story; (**d**) third story.