## 3 Results and Discussion
### Evaluation of the Seismic Response of a Base-Isolated Structure
According to Equations (3) and (4), these factors modify the properties of the isolators by more than 40%, which, in the most critical case, leads to higher drifts, shear forces, and demand-to-capacity ratio (DCR). DCR is the ratio of demand to capacity of a structural element and is less than 1. The values found in this research provide a more realistic approach to the factors that can be applied in the design of isolated structures based on laboratory tests performed on UN-FREI and UR-FREI materials. Figure 18 shows the inter-story drifts for UN-FREI and UR-FREI, which represent the relative lateral displacement between two consecutive floors. This parameter is a relevant factor in evaluating the seismic response of buildings, as excessive drifts can lead to structural and non-structural damage. According to FEMA 450 [54], the allowable limit for isolated buildings analyzed using spectral methods is 1.5%. The drift for the isolated building without modification factors was close to 1.0% because, as the column dimensions continued to be reduced, the DCR was exceeded. In the case of UN-FREI, \(\lambda_{\text{(ae,min)}}\) for Kh, \(\beta\), and Kv were close to 1, which explains why the drifts with DTM and DTM min are very similar. However, when applying the factor \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{3}{c}{**UN-FREI**} & \multicolumn{5}{c}{**UR-FREI**} \\ \hline  & \(\lambda_{\text{(ae,max)}}\) & & \(\lambda_{\text{(ae,min)}}\) & \(\lambda_{\text{(ae,max)}}\) & & \(\lambda_{\text{(ae,min)}}\) & \\ \hline Kh & 1.07 & Kh & 1.00 & Kh & 1.00 & Kh & 0.94 \\ \(\beta\) & 1.03 & \(\beta\) & 0.98 & \(\beta\) & 1.12 & \(\beta\) & 0.94 \\ Kv & 1.00 & Kv & 0.99 & Kv & 1.11 & Kv & 1.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Factors modifying environmental conditions. Figure 18: Comparison of drifts between D\({}_{\text{TM}}\), D\({}_{\text{TM}}\) max, and D\({}_{\text{TM}}\) min: (**a**) UN-FREI, and (**b**) UR-FREI. \(\lambda_{\text{(ae,max)}}\), the drifts increased. For UR-FREI, it is observed that \(\lambda_{\text{(ae,max)}}\) increases the drifts, while \(\lambda_{\text{(ae,min)}}\) decreases them. The fundamental period of a structure with seismic isolation at the base is largely determined by the horizontal stiffness of the isolated system and the total mass of the structure (Equation (5)). As Kh increases, the fundamental period of the isolated system decreases, leading to an increase in the stiffness of the structure. A shorter period in the response spectrum results in higher accelerations, which in turn increases the forces transmitted to the building and consequently the inter-story drifts. \[T_{M}=2\pi\sqrt{\frac{M}{K_{h}}} \tag{5}\]  
Table 8 presents the periods in the first mode of vibration obtained for DTM, DTM max, and DTM min in both materials. The shear forces per story are shown in Figure 19. DTM and DTM min exhibit nearly identical shear forces per story because the modification factor is close to 1. When applying the factor \(\lambda_{\text{(ae,max)}}\), the shear forces per story increased by 545 kN for UN-FREI and 373 kN for UR-FREI. As Kh increases, the base shear also increases because the seismic forces transmitted to the base become greater. Figure 20 shows the DCRs of two columns in the first story (Figure 7a). The forces should not exceed the structural element's capacity, as indicated by the DCRs remaining below 1.0. Figure 19: Comparison of shear per story between DTM, DTM max, and DTM min: (**a**) UN-FREI, and (**b**) UR-FREI. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Period** & **UN-FREI** & **UR-FREI** \\ \hline T DTM (s) & 2.42 & 2.42 \\ T DTM min (s) & 2.43 & 2.50 \\ T DTM max (s) & 2.34 & 2.39 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Periods according to modification factors. When \(\lambda_{\text{(ae,max)}}\) is applied, the DCRs increase in the columns of the building with UN-FREL, meaning the element is working closer to its maximum capacity, which increases the risk of building failure.