# Delaying the Occurrence of Bar Buckling in RC Columns Confined with SRG Jacketing
## 4 Analysis of the Axial and Lateral Response Due to SRG Confinement
### Criteria for the Effectiveness of SRG Jacketing in Delaying Bar Buckling  
In the present experimental investigation, the application of SRG jackets to RC columns with sparsely spaced stirrups managed to delay but not to preclude buckling of compression reinforcement. According to the mechanics of longitudinal bars embedded in concrete prismatic members, when the bar reaches critical conditions (i.e., instability), it bends laterally to maintain compatibility with the increasing axial strain of the supporting concrete core [4]. Therefore, the concrete core becomes overstressed, and crushing occurs. The critical axial concrete strain of the SRG confined columns is in direct relationship with the mobilized lateral confining pressure. At the same time, the longitudinal bar buckles symmetrically at a stress f\({}_{\rm s,crit}\) which is related to the available s/D\({}_{\rm b}\) ratio [8,12,23,29,32,33]:  
\[\frac{s}{D_{b}}=\Psi\sqrt{\frac{E_{hi}}{f_{s,crit}}} \tag{5}\]  
where \(\psi\) accounts for the buckling length; \(\psi=\pi/4\) when assuming a bar of pinned ends as in the case of yielded ties of poor anchorage ends and \(\psi=\pi/2\) of fixed ends as in the case of stiff ties of good anchorage conditions. In the present study, since stirrups are of small diameter (6 mm) and yielded at corners, it is considered \(\psi=\pi/4\). The term E\({}_{\rm hi}\) is the tangent modulus of steel at point (\(\varepsilon_{\rm s,crit}\), f\({}_{\rm s,crit}\)) into the hardening branch of the stress-strain relationship (see Figure 8a), defined as per Equation (A2) in Appendix A[2,4]. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{} & \multicolumn{4}{c}{**Experimental Data**} & \multicolumn{10}{c}{**Analytical Values**} \\ \hline \multirow{3}{*}{**Name**} & \multirow{3}{*}{**f\({}_{\rm cc}\) MPa**} & \multirow{3}{*}{\(\varepsilon_{\rm cc}\)} & \multirow{3}{*}{\(\varepsilon_{\rm ccu}\)} & \multirow{3}{*}{\(\sigma_{\rm lat}\)/f\({}_{\rm o}\)} & \multirow{3}{*}{\(\varepsilon_{\rm cc}\)P/\(\varepsilon_{\rm cc}\)} & \multirow{3}{*}{\(\varepsilon_{\rm ccu}\)} & \multirow{3}{*}{\(\varepsilon_{\rm ccu}\)/f\({}_{\rm cccu}\)} & \multirow{3}{*}{\(\Delta\varepsilon_{\rm cc}\)} & \multirow{3}{*}{\(\varepsilon_{\rm jeff}\)} & \multirow{3}{*}{\(\varepsilon_{\rm scu}\)} & \multirow{3}{*}{\(\varepsilon_{\rm jeff}\)} & \multirow{3}{*}{\(\varepsilon_{\rm scu}\)} & \multirow{3}{*}{\(\Delta\varepsilon_{\rm cc}\)} & \multirow{3}{*}{\(\varepsilon_{\rm j}^{\rm bancyl}\)} & \multirow{3}{*}{\(\varepsilon_{  
Equation (5) is also used when steel is into its plateau of zero stiffness, as \(\mathrm{E_{hi}}\) in Equation (5) is taken equal to the secant modulus \(\mathrm{E_{sec}=f_{y}/\varepsilon_{s,crit}}\), with \(\varepsilon_{s,crit}\) ranges between \(\varepsilon_{y}\) and \(\varepsilon_{h}\). The strain values at yielding, at the end of the yielding plateau and at ultimate (\(\varepsilon_{y}\), \(\varepsilon_{h}\) and \(\varepsilon_{u}\), respectively), along with the stress values at yielding and ultimate (\(\mathrm{f_{y}}\) and \(\mathrm{f_{u}}\), respectively), for the longitudinal reinforcement used in this study appear in Figure 8a. Note that the term \(\varepsilon_{s,crit}\) corresponds to the axial strain at which the bar becomes unstable. For the post-elastic stress-strain law of the longitudinal bars used in the tests (Figure 8a), the bar compressive strain ductility, \(\mu_{es}=\varepsilon_{s,crit}/\varepsilon_{y}\), is plotted against the ratio \(s/\mathrm{D_{b}}\) in Figure 8b; in this buckling curve, the horizontal jump occurs at strain \(\varepsilon_{s,crit}\) equal to \(\varepsilon_{h}\) as a result of the difference in \(\mathrm{E_{ho}}\) and \(\mathrm{E_{sec}}\) at that strain. Additionally, the experimental values (the average of identical specimens was considered) of the columns' compressive strain ductility \(\mu=\varepsilon_{\mathrm{ccu}}/\varepsilon_{y}\) are plotted (depicted in circular dots in Figure 8b).