# Strengthening Design of RC Columns with Direct Fastening Steel Jackets
## 3 Design Procedure
### Estimating Lateral Load Capacity of Strengthened RC Column  
According to the assumption that the plane section remains plane after deformation, the strain profile and stress profile of the longitudinal rebar and steel plates of the strengthened RC column are defined, respectively (see Figure 7). It should be noted that the side plate is divided into \(n\) equal small parts to accurately define the stress in the side plate. Moreover, the buckling of the steel plate under a compressive state is embedded.  
\[\left\{\begin{array}{l}\sigma_{pt}=E_{p}(d_{c}+\frac{t_{p}}{2}-x_{c})\frac{ \epsilon_{cut}}{x_{c}}\leq f_{py}\\ \sigma_{pc}=E_{p}(\frac{t_{p}}{2}+x_{c})\frac{\epsilon_{cut}}{x_{c}}\leq\sigma _{p,critical}\\ -\sigma_{p,critical}\leq\sigma_{pside\_i}=E_{p}(d_{c}-\frac{d_{c}-d_{p}}{2}- \frac{\Delta_{i}}{2}-(i-1)\Delta_{i}-x_{c})\frac{\epsilon_{cut}}{x_{c}}\leq f _{py}\\ -f_{yl}\leq\sigma_{\uparrow 1}=E_{s}(d_{c}-c-\phi_{st}-\frac{\phi_{t}}{2}-x_{c})\frac{ \epsilon_{cut}}{x_{c}}\leq f_{yl}\\ -f_{yl}\leq\sigma_{lj}=E_{s}(d_{c}-c-\phi_{st}-\frac{\phi_{t}}{2}-(j-1)s_{l}-x_ {c})\frac{\epsilon_{cut}}{x_{c}}\leq f_{yl}\\ -f_{yl}\leq\sigma_{lm}=E_{s}(d_{c}-c-\phi_{st}-\frac{\phi_{t}}{2}-(m-1)s_{l}-x_ {c})\frac{\epsilon_{cut}}{x_{c}}\leq f_{yl}\end{array}\right. \tag{26}\]  
where \(i\) represents the \(i\)th partitioned part of the steel plate parallel to the lateral load counting from the tensile side; \(\sigma_{pside\_i}\) is the stress of the \(i\)th partitioned part of the steel plate parallel to the lateral load; \(\Delta_{i}\) is the length of the equal small part and is equal to \(d_{p}\)/\(n_{i}\) and \(\sigma_{p,critical}\) is the ultimate compressive stress of the steel plate, which can be defined by Equation (8).  
Figure 7: Vertical strain and stress profiles of strengthened RC column: (**a**) strain profile; (**b**) stress profile.  
The compressive depth of the strengthened RC column is then determined by the axial equilibrium.  
\[0=N_{0}+\frac{\sum\limits_{j=1}^{m}k_{j}A_{l}\sigma_{lj}}{\gamma_{s}}+\frac{\eta _{i}\sigma_{pt}A_{p}-A_{p}\sigma_{pc}+2\sum\limits_{i=1}^{m}t_{p}\Delta_{i} \sigma_{pside\_i}}{\gamma_{s}}-\frac{\alpha f_{c}^{\prime}d_{c}\beta x_{c}}{ \gamma_{c}} \tag{27}\]  
By taking moment to the extreme compressive fiber of the strengthened RC column, the moment capacity is given as:  
\[\begin{array}{ll}M_{cap,stren}=&N_{0}\frac{d_{c}}{2}-\frac{af_{c}^{\prime}d_ {c}\beta x_{c}}{\gamma_{c}}+\frac{\sum\limits_{j=1}^{m}k_{j}A_{l}\sigma_{lj}(d_ {c}-c-\phi_{pt}-\frac{\phi_{p}}{2}-(j-1)s_{l})}{\gamma_{s}}\\ &&+\frac{\eta_{i}\sigma_{pt}A_{p}(d_{c}+\frac{lp}{2})+A_{p}\sigma_{pc}+2\sum \limits_{i=1}^{m}t_{p}\Delta_{i}\sigma_{pside\_i}(d_{c}-\frac{d_{c}-d_{p}}{2}- \frac{\Delta_{i}}{2}-(i-1)\Delta_{i})}{\gamma_{s}}\end{array} \tag{28}\]  
Under the assumption the stiffness of the upper and lower joints is comparable, the lateral load resistance is then given by:  
\[V_{cap,stren}=\frac{M_{cap,stren}}{0.5L} \tag{29}\]  
Then, the relationship between the ultimate lateral load and the lateral load resistance of the strengthened RC column can be checked.