# Strengthening Design of RC Columns with Direct Fastening Steel Jackets
## 3 Design Procedure
### Estimating Lateral Load Capacity of Damaged RC Columns  
The deficiency of the RC column may occur for a variety of reasons (e.g., fire). It is necessary to estimate the lateral load capacity of the damaged RC column and compare this with the load demand. According to the assumption that the plane section remains plane after deformation, the strain profile and the stress of the longitudinal rebar can be defined (Figure 6).  
\[\left\{\begin{array}{c}-f_{yl}\leq\sigma_{l1}=E_{s}(d_{c}-c-\phi_{st}-\frac {\phi_{l}}{2}-x_{c})\frac{\varepsilon_{lm}}{x_{c}}\leq f_{yl}\\ -f_{yl}\leq\sigma_{lj}=E_{s}(d_{c}-c-\phi_{st}-\frac{\phi_{l}}{2}-(j-1)s_{l}-x_ {c})\frac{\varepsilon_{lm}}{x_{c}}\leq f_{yl}\\ -f_{yl}\leq\sigma_{lm}=E_{s}(d_{c}-c-\phi_{st}-\frac{\phi_{l}}{2}-(m-1)s_{l}-x _{c})\frac{\varepsilon_{lm}}{x_{c}}\leq f_{yl}\end{array}\right. \tag{20}\]  
where \(s_{l}\) is the longitudinal rebar spacing; subscript \(lj\) represents the \(j\)th row longitudinal rebar counting from the tensile sides (\(1\leq j\leq m\)); \(m\) is the total rows of longitudinal rebars which is equal to the stirrup legs; \(\phi_{l}\) is the diameter of the longitudinal rebar; \(\phi_{st}\) is the diameter of the stirrup; \(c\) is the cover thickness of the column; \(x_{c}\) is the compressive depth of the column; \(\varepsilon_{cu}\) is the ultimate strain of concrete and is taken as 0.003 in accordance with ACI 318 [31],\(f_{yl}\) is the yield strength of the longitudinal reinforcement, and \(\sigma_{lj}\) is the stress of the \(j\)th row longitudinal rebar.  
The stress profile of the concrete is quantified using the equivalent rectangular stress block. The compressive depth is then determined by the axial equilibrium:  
\[N_{0}+\frac{\sum\limits_{j=1}^{m}k_{j}A_{l}\sigma_{lj}}{\gamma_{s}}-\frac{\alpha f _{c}^{\prime}d_{c}\beta x_{c}}{\gamma_{c}}=0 \tag{21}\]  
where \(k_{j}\) is the number of the \(j\)th row longitudinal rebar; \(\gamma_{s}\) is the partial safety factor of rebar; \(\gamma_{c}\) is the partial safety factor of concrete; \(N_{0}\) is the axial load; \(A_{l}\) is the cross-sectional area of longitudinal rebar; \(f_{c}^{\prime}\) is the concrete strength; and \(\alpha\) and \(\beta\) are the two factors used to determine the equivalent rectangular stress block.  
Figure 6: Vertical strain and stress profiles of RC column: (**a**) strain profile; (**b**) stress profile.  
By taking moment to the extreme compressive fiber of the RC column, the moment capacity is given as:  
\[M_{cap}=N_{0}\frac{d_{c}}{2}+\frac{\sum\limits_{i=1}^{\frac{n}{\sum\limits_{i=1}^{ k}{f_{i}A_{i}\sigma_{ij}(d_{c}-c-\phi_{st}-\frac{\phi_{t}}{2}-(j-1)s_{i})}}}}{\gamma_{s}}- \frac{\alpha_{rf}d_{c}\beta_{X_{c}}\frac{\beta_{X_{c}}}{\gamma_{c}}}{\gamma_{c}} \tag{22}\]  
Under the assumption that the stiffness of the upper and lower joints is comparable, the lateral load resistance is then given by:  
\[V_{cap}=\frac{M_{cap}}{0.5L} \tag{23}\]  
where \(L\) is the length of the column.  
If lateral load capacity cannot satisfy lateral load demand, the RC column can be strengthened using the direct fastening steel jackets.