# Seismic Response and Recentering Behavior of Reinforced Concrete Frames: A Parametric Study
## 4 Parametric Study  
### Selection of Parameter Ranges  
To generalize the results for a range of practical reinforcement configurations and to provide some guidelines that can be useful to describe the seismic response and recentering behavior of the RC frames, a parametric study has been performed considering different values of \(ALR\), \(p_{l}\), and \(p_{s}\). Based on the model defined in Section 3, different configurations of axial load and reinforcement details were analyzed within the following ranges:  
* \(ALR=\left(0.05;0.20;0.40;0.60;0.80\right)\);
* \(p_{l}=\left(1\%;2\%;3\%;4\%\right)\);
* \(p_{s}=\left(0.40\%;0.80\%;1.20\%;1.60\%\right)\).  
The values selected for \(ALR\) cover almost the entire spectrum of possible values for RC frames, taking into account both weakly and heavily loaded configurations in relation to the compressive strength capacity of the frame columns. Similarly, the range selected for \(p_{l}\) covers almost every possible configuration for the RC columns; for example, the Italian Building Code [17; 18] sets the lower and upper bounds of longitudinal reinforcement  
\begin{table}
\begin{tabular}{c c c|c c c} \hline  & **Prototype RC Frame** & & **Point** & \(d\) **(mm)** & \(F\) **(kN)** \\ \hline \(F_{0}/F_{y}\) & \(F_{max}/F_{y}\) & \(\mu\) & \(P_{0}\) & 0 & 26.4 \\ \((-)\) & \((-)\) & \((-)\) & \(P_{y}\) & 30 & 283.3 \\
0.09 & 1.06 & 2.17 & \(P_{max}\) & 65 & 299.3 \\ \hline \end{tabular}
\end{table}
Table 2: Characteristic parameters obtained for the prototype RC frame.  
Figure 5: Bilinear curve obtained for the prototype RC frame.  
ratios for the RC columns' design equal to 1% and 4%, respectively. Finally, the range of \(p_{s}\) depends on stirrup spacing and cross ties. Having in mind the RC columns with non-seismic construction details, such as stirrups with no additional cross ties and wide spacing, values in the order of 0.40% (lower bound) might be reasonable, whereas values in the order of 1.60% (upper bound) reflect the amounts of transverse reinforcement typically observed in the critical regions of the well-designed RC frames in the seismic area. Cyclic pushover analyses have been performed, as described in Section 3, applying a loading protocol with an increasing drift reaching up to \(F_{max}\) or up to failure if it occurred at a lower drift. In the analyses, the appropriate \(ALR\) value has been obtained by applying forces to the columns at the beam-column joints. Considering all the combinations of the \(ALR\), \(p_{l}\), and \(p_{s}\) values in the above intervals, 80 scenarios have been obtained. The results are shown in terms of both the force-drift curves (where the forces are displayed in a normalized form as \(F/F_{max}\)) and 3D surface and contour plots of the three indexes \(F_{0}/F_{y}\), \(F_{max}/F_{y}\), and \(\mu\). It has been found that the behavior of the frame in terms of the cyclic response and recentering properties is very different depending on the combination of the \(ALR\), \(p_{l}\), and \(p_{s}\) values considered. Some representative examples are given in the following subsection and critically commented.