# Seismic Response and Recentering Behavior of Reinforced Concrete Frames: A Parametric Study
## 3 Model Validation
The comparison between the model load-displacement curves obtained with \(\eta=0\) and \(\eta=1\) is shown in Figure 4.  
As can be seen from Figure 4, the two model curves are identical; therefore, it seems that the \(\eta\) parameter did not produce a significant effect on the load-displacement curve. This could be due to the number of cycles considered; the effects of the \(\eta\) parameter on the load-displacement curve with a greater number of cycles have not been investigated. The present paper is focused on the determination of the seismic behavior of the RC frames through synthetic and practical indicators that can be used for design purposes. To this aim and to simplify the obtained cyclic behavior of the frame, first, the backbone branch enveloping the peaks for each cycle is obtained, then a bilinear curve with hardening \begin{table}
\begin{tabular}{c c c c} \hline  & **Experiment** & **Numerical Model** & **Relative Error (\%)** \\ \hline \(F_{max}\) (+) (kN) & 300.6 & 299.3 & 0.4 \\ \(F_{max}\) (\(-\)) (kN) & 294.0 & 295.9 & 0.6 \\ \(F_{0}\) (+) (kN) & 26.7 & 26.4 & 1.1 \\ \(F_{0}\) (\(-\)) (kN) & 30.4 & 30.1 & 1.0 \\ \(E_{diss}\) (kJ) & 27.2 & 26.6 & 2.2 \\ \hline \end{tabular}
\end{table} 
Table 1: Maximum strength \(F_{max}\) (positive and negative), maximum strength at zero displacement \(F_{0}\) (positive and negative), and dissipated energy \(E_{diss}\) values related to experimental and numerical model curves. Figure 4: Comparison between model load–displacement curves with \(\eta=0\) and \(\eta=1\). Figure 3: Comparison of load–displacement curves between experimental and pivot models. idealization has been associated to this backbone load-displacement curve. The elastic branch of the bilinear curve passes through the origin and the point corresponding to a force value equal to 60% of the maximum force \(F_{max}\), while the hardening branch has a slope such that it passes through the point corresponding to the maximum force (\(d_{max},F_{max}\)) and satisfies the energy equivalence condition (equal to the enclosed area) between the bilinear and backbone curves in the range \(0-d_{max}\). The observation of the idealized bilinear curve has made it possible to identify three main characteristic points that are sufficient to describe not only the loading branch of the seismic response (backbone) but also the unloading branch and the associated recentering behavior. In particular, based on the backbone and the associated bilinear curves, it is possible to obtain different parameters, such as the yielding point \(P_{y}=\left(d_{y},F_{y}\right)\), maximum point \(P_{max}=\left(d_{max},F_{max}\right)\), and unloading residual force point \(P_{0}=\left(0,F_{0}\right)\). From those parameters, it is possible to determine three synthetic parameters describing the cyclic response and recentering behavior of the RC frame, namely a \"_recentering index_\" \(F_{0}/F_{y}\), a \"_hardening index_\" \(F_{max}/F_{y}\), and a \"_ductility index_\" \(\mu=d_{max}/d_{y}\). The bilinear curve with identification of the above-introduced characteristic points is illustrated in Figure 5, while the corresponding parameters are listed in Table 2.