# Seismic Response and Recentering Behavior of Reinforced Concrete Frames: A Parametric Study
## 5 Nonlinear Time History Analysis on a Multi-Story RC Building
Considering the actual values of \(ALR\), \(p_{l,i}\), and \(p_{s,i}\) at each storey level \(i\), we can compare the numerical results from the NTHAs with the predictions from the parametric study for the consistent combination of the parameters. To check the reliability and generality of the contour plots constructed in the previous section, the characteristic behavioral indexes \(F_{0}/F_{y}\), \(F_{max}/F_{y}\), and \(\mu\) are compared with those identified from the inter-story drift versus the normalized story shear force graphs at each story level. All four of the representative story levels considered have an identical volumetric shear reinforcement percentage \(p_{s}\) (roughly equal to 0.3%); therefore, a reference has been made to the contours pertinent to \(p_{s}=0.40\%\). Different \(ALR\) and \(p_{l}\) values are associated with each story level; in particular, the longitudinal reinforcement percentage \(p_{l}\) for story level 2 is equal to 0.68%, while for story levels 3, 7, and 8, \(p_{l}\) is equal to 0.50%. Consequently, a reference has been made to the minimum value considered in the parametric study, that is, \(p_{l}=1\%\). Regarding \(ALR\), it should be noted that the four columns of each story level do not have the same axial load intensity. In particular, for the gravity loads, the frame is symmetrical and uniformly loaded, which makes it reasonable to assume that the internal columns have twice the axial load as compared with the external columns. However, for practical purposes, a single average value of \(ALR\) per story level has been defined to facilitate the use of the contour and surface plots of the parametric study. Such an average \(ALR\) value is equal to 0.40 for the story level 2, 0.34 for the story level 3, 0.10 for the story level 7, and 0.05 for the story level 8. Once the values of \(ALR\), \(p_{l}\), and \(p_{s}\) had been determined, these were used to estimate the three behavioral indexes from the contours (shown in Figure 22), and then they were compared with the actual behavioral indexes evaluated from the hysteretic loops listed in Figure 23.  
Figure 22: Behavioral indexes identified from the contour plots relative to \(p_{s}=0.40\%\). Based on the comparison reported in Figure 23, we note that there is an excellent agreement between the estimates of the behavioral indexes from the contour plots of the parametric study (carried out on a single-story prototype RC frame) and those from the NTHAs on a multi-story building with different geometric and mechanical characteristics. This comparison emphasizes the usefulness of the contour plots/surface plots drawn in Section 4 in anticipating the overall cyclic behavior of a generic RC building, depending on the actual mechanical parameters of the RC sections at each story level. The authors believe that these behavioral indexes are important parameters that can be used in practical design contexts. As an example, the recentering index is strictly related to the ability of each story level to return to its original position at the end of the earthquake shaking. To prove this, the NTHAs have been repeated by adding at the end of the 10 cycles a zero-acceleration segment of duration 20 s, thus making it possible to investigate the free vibration response of the structure. This augmented time history is purposely considered to check whether the RC frame is able to return to its original configuration or has accumulated a permanent residual displacement at the end of the seismic event due to its poor recentering attitude. The inter-story drift time histories obtained for story level 3 and 8 are shown below. As can be seen from Figure 24, in floor 3, a permanent residual drift equal to 0.12% has been accumulated, whereas negligible residual displacement was noted in floor 8.