# Strengthening Design of RC Columns with Direct Fastening Steel Jackets
## 2 Review of Previous Work
### Axial Strengthening of RC Columns by Direct Fastening Steel Jackets  
From the experimental study on the axial strengthening of RC columns by the direct fastening of steel jackets [23], the reliability and effectiveness of the proposed method were observed. Critical parameters (i.e., vertical spacing between adjacent connections, thickness of the steel plate and number of fasteners in each connection) affecting load bearing performance and deformation behavior were identified. In the proposed strengthening method, steel plates can directly sustain axial load. The axial load contribution of the steel plates is determined by buckling strength as buckling occurs prior to the yield strength of steel plates. The buckling strength is presented in the following section. Direct fastening connections behave in the manner of transverse reinforcement (i.e., stirrups). Due to the transverse dilation of the column, passive confinement stress is mobilized in stirrups and direct fastening connections, which enhances the strength of the column. To determine confined concrete strength, the equivalent passive confinement stress shown in Equation (3) should be determined. \[f_{est}=\frac{ma_{st}f_{yst}A_{st}}{s_{st}l_{st}} \tag{4}\]  
\[f_{ed}=\frac{2\alpha_{d}n_{f}F_{b}}{(s_{d}+d_{d})d_{c}} \tag{5}\]  
where \(m\) is the stirrup legs; \(l_{st}\) is the center-to-center distance of the peripheral stirrup; \(s_{st}\) is the stirrup spacing; \(f_{yst}\) is the yield strength of the stirrup; \(A_{st}\) is the cross-sectional area of the stirrup; \(s_{d}\) is the clear vertical spacing of the adjacent connections; \(d_{d}\) is the length of the right steel angle bracket; \(d_{c}\) is the column width; \(a_{st}\) is the stress ratio of the stirrup; and \(\alpha_{d}\) is the shear force ratio of the direct fastening connection. A theoretical study was carried out to examine the stress ratio of the stirrup and the shear force ratio of the direct fastening connection [24]. Based on an extensive parameter study, a lower bound value of 0.34 is used to represent the stress ratio of the stirrups with a yield strength of 400 MPa. To consider the effect of the yield strength, a yield strength factor of \(\gamma_{fyst}\) was proposed. The yield strength factors for yield strengths of 500 MPaand 600 MPa are 0.8 and 0.66, respectively. The shear force ratio of the connections is obtained using:  
\[\alpha_{d}=\gamma_{nf}(0.82-0.64\lambda_{d}) \tag{6}\]  
where \(\lambda_{d}\) is the normalized connection spacing (\(\lambda_{d}=s_{d}/d_{c}\)); and \(\gamma_{nf}\) represents a factor of the number of fasteners on the shear force ratio of the connections. When the number of fasteners is 2 and 4, the factor is equal to 0.72 and 0.47, respectively. The failure criterion proposed in [29] is used to derive the confined concrete strength of \(f_{cc^{\prime}}\). The failure criterion is given by:  
\[\tau_{oct}^{*}=6.9638\left(\frac{0.09-\sigma_{oct}^{*}}{c-\sigma_{oct}^{*}} \right)^{0.9297} \tag{7}\]  
\[c=12.2445(\cos 1.5\theta)^{15}+7.3319(\sin 1.5\theta)^{2} \tag{8}\]  
\[\cos\theta=\frac{\sqrt{3}}{2}\frac{s_{1}}{\sqrt{l_{2}}} \tag{9}\]  
\[\tau_{oct}^{*}=\frac{\tau_{oct}}{f_{c}^{\prime}} \tag{10}\]  
\[\sigma_{oct}^{*}=\frac{\sigma_{oct}}{f_{c}^{\prime}} \tag{11}\]  
where \(\tau_{oct}\) and \(\sigma_{oct}\) are the octahedral shear and normal stress, respectively; \(\theta\) is the direction of the deviatoric stress on the deviatoric plane; \(s_{1}\) is the first deviatoric stress; \(l_{2}\) represents the second invariant of the deviatoric stress tensor; and \(f_{c}^{\prime}\) is the compressive strength of the unconfined concrete.