# 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
Expressed in polar coordinates, the local hoop strain \(\Delta\epsilon_{j,ch}^{crit}\) is defined as the tangential strain of a thick cylinder of radius \(\mathrm{R_{ch}}=\mathrm{r}+0.5\mathrm{D_{b}}\) where \(\mathrm{r}\) is the radius of chamfered specimen corners [12]:  
\[\Delta\epsilon_{j,ch}^{crit}=w/(r+0.5D_{b}) \tag{7}\]  
At corners, the total strain of the jacket is the sum of Equations (6) and (7) with upper limit being the strain capacity of the jacket \(\epsilon_{ju}=0.025\) as  
\[\epsilon_{ju} =\epsilon_{j}^{dil,crit}+\Delta\epsilon_{j,ch}^{crit}=\
u \epsilon_{s,crit}+\Delta\epsilon_{j,ch}^{crit} \tag{8}\]  
Assuming symmetric outwards buckling for the four corner bars, then the mean lateral strain of the jacket calculated over the straight part of the specimen side \(\mathrm{b_{s}}\) (here \(\mathrm{b_{s}}=\mathrm{b}\)\(-2\mathrm{r}=200\)\(-2\times 25=150\) mm) is \(\Delta\epsilon_{j}^{crit}=2\mathrm{w/b_{s}}\), which along with Equations (7) and (8) results in  
\[\Delta\epsilon_{j}^{crit}=2\Delta\epsilon_{j,ch}^{buckl}(r+0.5D_{b})/b_{s}=2 \big{(}\epsilon_{ju}-\
u\epsilon_{s,crit}\big{)}(r+0.5D_{b})/b_{s} \tag{9}\]  
The total mean lateral strain of the jacket upon bar instability by also considering concrete core dilation is  
\[\epsilon_{j}^{buckl,crit}=\epsilon_{j}^{dil,crit}+\Delta\epsilon_{j}^{crit}= \
u\epsilon_{s,crit}+2\big{(}\epsilon_{ju}-\
u\epsilon_{s,crit}\big{)}(r+0.5D _{b})/b_{s} \tag{10}\]  
Equation (9) defines the strain reserve of the jacket upon buckling which corresponds to axial strength reserve of the encased concrete, \(\Delta f_{cc}\), from Equations (1) and (2) as follows [12]:  
\[\Delta f_{cc}=3\Delta\sigma_{lat}=3\alpha^{SRG}\rho_{v}^{SRG}E_{f}\big{(} \epsilon_{ju}-\
u\epsilon_{s,crit}\big{)}(r+0.5D_{b})/b_{s} \tag{11}\]  
As the bars become unstable, they transfer to the encased concrete core overload \(\Delta f_{ax}\) equal to  
\[\Delta f_{ax}=(f_{s,crit}-f_{s,res})\frac{As}{\big{(}A_{gross}-A_{s}\big{)}}, \ f_{s,res}=6f_{y}/\bigg{(}\frac{s}{D_{b}}\bigg{)} \tag{12}\]  
where \(\mathrm{f_{s,res}}\) is the bar residual capacity at an axial strain \(\varepsilon\geq\epsilon_{s,crit}\). For the conservatism of the estimations from Equations (9)-(11), parameter \(\
u\) is taken equal to unit. Based on [3], an arrangement of \(\mathrm{s/D_{b}}<5\) enables a compression bar to develop a stress-strain response identical to the tensile one. This implies that bars after critical conditions continue to develop stress due to compression and do not release load to the concrete core, and thus, \(\mathrm{f_{s,res}}=0\); this is valid for the group C of the present study. If the overload on the bar, represented by \(\Delta f_{ax}\), exceeds the strength reserve of the core, \(\Delta f_{cc}\), failure caused by buckling is expected to occur immediately. However, if the overload does not surpass the strength reserve, the risk of buckling-induced failure is mitigated, allowing the core to utilize its complete axial strain capacity, \(\epsilon_{ccu}^{buckl}\), at higher strain levels. In the latter case, the mean lateral strain of the jacket \(\epsilon_{j}^{buckl}\) is described by Equation (10) where \(\epsilon_{j}^{dil,crit}\) is substituted by \(\epsilon_{j,eff}\) calculated by Equation (1) (see Section 4.1). The relationship between the total mean jacket lateral strain, \(\epsilon_{j}^{buckl}\), and the associated axial strain capacity, \(\epsilon_{ccu}^{buckl}\), is defined by adopting the Poisson ratio \(\
u_{u}\) from \(\epsilon_{j}^{buckl}=\
u_{u}\epsilon_{ccu}^{buckl}\). When the reserve is close to the overload (\(\Delta f_{cc}-\Delta f_{ax}<5\)\(MPa\)), the \(\
u_{u}\) can be considered equal to 0.5, else 1.