# Seismic Response and Recentering Behavior of Reinforced Concrete Frames: A Parametric Study
## 5 Nonlinear Time History Analysis on a Multi-Story RC Building  
This section is aimed at demonstrating the usefulness of the results illustrated in the parametric study for describing the cyclic behavior of the RC frames. To this aim, an eight-story RC frame with non-seismic details, reported in [PERSON] and Vona [22] and designed in accordance with the 1972 Italian Building Code [23], was analyzed. This represents an archetypal building, representative of the substandard RC frames built in the 1960s-1970s in Italy. The plan view and front view, along with the section details and reinforcement arrangement, are depicted in Figure 17. The material parameters considered for calibrating the nonlinear material behavior via fiber-hinge elements in the RC frame, directly taken from the reference paper [22], are listed in Table 3. Considering the regularity in the plan and in the elevation, nonlinear time history analyses (NTHAs) were carried out on a simplified planar model of the internal frame (highlighted with a dashed red rectangle in Figure 17), as per EC8 \(\lx@sectionsign\)4.2.3 [24]. From the analysis for gravitational loads, the values of the axial load acting on the columns (prior to the application of the seismic loads) were obtained based on which values of \(ALR\) were calculated for each column. The longitudinal reinforcement percentage \(p_{l}\) and volumetric shear reinforcement percentage \(p_{s}\) were evaluated from the reinforcement arrangement details reported above. Once the \(ALR\), \(p_{l}\), and \(p_{s}\) values were determined, the pivot hysteresis model was calibrated for each structural member according to the equations of [PERSON] et al. [8]. As dynamic input, a sinusoidal acceleration of period \(T=0.64\) s, amplitude \(a_{g}=0.25\) g (selected to reach a sufficiently high story drift in all story levels, so as to describe the post-elastic cyclic behavior), and the number of cycles \(n=10\)  
Figure 16: Three-dimensional surface and contour plots of the ductility index \(\mu\) for \(p_{s}=1.20\%\) and \(1.60\%\). (to investigate the recentering behavior after the repetition of multiple cycles) was applied at the base of the frame. It is worth noting that in the case of a multi-story building, the seismic response and recentering behavior of each story level \(j\) could be described in terms of the inter-story drift \(\delta_{j}\) and story shear force \(V_{j}\); hence, the characteristic points \(P_{0}\), \(P_{y}\), and \(P_{max}\) were identified in the \(V/V_{max}-\delta\) plane (the shear force was normalized with respect to its maximum value for comparison purposes among the story levels). Consequently, the previously defined indexes for the \(j^{th}\) story level were calculated: recentering index \(V_{0,j}/V_{y,j}\); hardening index \(V_{max,j}/V_{y,j}\); and ductility index \(\mu=\delta_{max,j}/d_{y,j}\). Based on these considerations, the inter-story drift versus the normalized story shear force diagrams and the related bilinear idealized curve for story levels 2, 3, 7, and 8, along with related characteristic points and behavioral indexes, are shown in Figures 18-21. It is worth noting that, for the sake of brevity, only four of the eight story levels are considered here to illustrate some representative results associated with the different \(ALR\) values. Figure 19: Inter-story drift versus normalized story shear force (**a**) and bilinear idealization (**b**) for story level 3. Figure 20: Inter-story drift versus normalized story shear force (**a**) and bilinear idealization (**b**) for story level 7. Figure 18: Inter-story drift versus normalized story shear force (**a**) and bilinear idealization (**b**) for story level 2. It is clearly seen that the shape of the hysteretic loops is different (in terms of pinching, ductility, and hardening factors) depending on the story level considered. This is ascribed to the different mechanical parameters \(ALR\), \(p_{l}\), and \(p_{s}\) characterizing each story level.