# Seismic Response and Recentering Behavior of Reinforced Concrete Frames: A Parametric Study
## 4 Parametric Study
### Three-Dimensional Surface and Contour Plots of Synthetic Behavioral Indexes  
Based on the entire set of results for the 80 cases of the parametric study, 3D surfaces describing the variation of \(F_{0}/F_{y}\), \(F_{max}/F_{y}\), and \(\mu\) depending on \(ALR\), \(p_{l}\), and \(p_{s}\) have been obtained by cubic interpolation of the discrete values obtained. These 3D surfaces are useful for synthetically describing the trends of the seismic response and recentering capacity of the analyzed set of the RC frames based on the loading and reinforcement configuration. From the 3D surfaces derived above, we also construct contours for the evaluation of \(F_{0}/F_{y}\), \(F_{max}/F_{y}\), and \(\mu\) as a function of \(ALR\) and \(p_{l}\) for a given value of \(p_{s}\), which can be used for practical design purposes, as illustrated in the sequel to this paper. It is worth noting that the \(F_{0}\) values for the cases with \(ALR\geq 0.60\), affected by a significant variation between the cycles and asymmetric cyclic behavior, as observed in the previous examples, is assumed as the average value of \(F_{0}^{+}\) and \(F_{0}^{-}\). For the cases with \(ALR<0.60\), this averaging process is not necessary since the trend of the cycles is regular and symmetrical; consequently, the variations mentioned above are negligible. Therefore, in these cases, only the positive value \(F_{0}^{+}=F_{0}\) has been reported. The 3D surfaces and related contour plots mentioned above are shown and commented on below. By observing the trends depicted in Figures 11-16, the following general considerations can be drawn:  
1. The recentering index is much more sensitive to the \(ALR\) than to the longitudinal and transverse reinforcement. In particular, for given \(p_{l}\) and \(p_{s}\) values, variations up to 100% of the \(F_{0}/F_{y}\) parameter are observed by spanning the range of \(ALR\) investigated. Such variations are more marked for low amounts of steel reinforcement (see, e.g., the variability of \(F_{0}/F_{y}\) corresponding to \(p_{s}=0.40\%\) and \(p_{l}\cong 1-1.25\%\) shown in the left part of Figure 11) and tend to diminish for higher amounts of steel reinforcement. The peak values of \(F_{0}/F_{y}\) (poorest recentering behavior) are generally observed in the range of \(ALR=0.35-0.40\), which characterizes most of the RC column configurations encountered in the existing buildings. The sensitivity of \(F_{0}/F_{y}\) with respect to \(p_{l}\) is relatively modest, while the presence of higher amounts of transverse reinforcement  
Figure 10: Force–drift curve (**a**) and bilinear idealization (**b**) for \(ALR=0.80-p_{l}=2\%-p_{s}=1.20\%\). slightly increases the recentering index. It is worth noting that these trends consistently reflect, at a structural (macroscopic) scale, the empirical relationships expressed at the material (microscopic) level by the calibration parameters of the pivot model derived by [PERSON] et al. (2012) and reported in Equations (3)-(6). 2. The hardening index \(F_{max}/F_{y}\) is instead more correlated (than \(F_{0}/F_{y}\)) with the longitudinal reinforcement since, as reasonably expected, higher values of \(p_{l}\) lead to an increase in the overall flexural capacity of the RC frame. The peak values of the \(F_{max}/F_{y}\) parameter are identified in a range of the \(ALR\) value that somehow depends on the transverse reinforcement, i.e., it is close to \(ALR=0.3-0.4\) for lower amounts of transverse reinforcement (\(p_{s}=0.40\%\)) and close to \(ALR=0.5-0.6\) for higher amounts (\(p_{s}=1.60\%\)). This indicates that there is a reciprocal influence between the hardening behavior and the loading condition in terms of axial load, as reasonably expected. 3. The deformation capacity of the RC frame is expressed in this work by the ductility index \(\mu\).