# Strengthening Design of RC Columns with Direct Fastening Steel Jackets
## 5 Worked Example
### Evaluating Lateral Load Capacity of Strengthened RC Column  
Since the required enhancement ratio is around 1.4 and the longitudinal rebar ratio is 0.015, 4 mm steel plate is utilized to strengthen the fire-damaged RC column based on the findings in Figure 8. Additionally, four fasteners on the connection are used.  
(a) Determining connection spacing  
Flexural failure is the desirable failure mode. To realize this, Equation (24) should be satisfied.  
\[300\ \text{kN}\ =\ V\leq V_{cap,stren}\leq 0.6V_{stren}\]  
According to Equation (25), the shear capacity of the strengthened fire-damaged RC column is given as:  
\[\begin{array}{ll}V_{stren}&=V_{d}+V_{c,pf}+V_{s,pf}\\ &=0.5\frac{d_{c}}{s_{d}+d_{d}}\frac{2n_{f}F_{d}}{7z}+\frac{0.17(1+\frac{N_{0}} {14K})\sqrt{f_{c,pf}^{\prime}d_{c}\omega}}{7z_{c}}+\frac{mA_{ff_{st,pf}}d_{c}}{ 7z_{s}s_{d}}\\ &=0.5\times\frac{500}{g_{d}}\times 2\times\frac{4\times 1.6\times 1.35\times 1.17\times 4\times 4\times 400}{12}+\frac{0.17(1+\frac{440000}{1.45000})}{1.5}\\ &\times\sqrt{15}\times 500\times(500-c-\phi_{id}-\frac{\phi_{l}}{2})+\frac{4\times 3 14K^{2}\times 376}{1500}\times(500-c-\phi_{id}-\frac{\phi_{l}}{2})\\ \end{array}\]Solving the above two equations, the connection spacing is given as:  
\[s_{d}\leq 160\ \text{mm}\]  
Here, the connection spacing is taken as 100 mm. The corresponding shear capacity of the strengthened column is 623 kN.  
Determining ultimate compressive stress in compressive steel plate  
The slenderness ratio of the steel plate is given as:  
\[\lambda_{sr}=\frac{s_{d}}{t_{p}}=\frac{100}{4}=25\]  
The initial imperfection factor is given by (19).  
\[\alpha_{i}=1.046-0.0073\lambda_{sr}=1.046-0.0073\times 25=0.86\]  
The ultimate compressive stress of the steel plate is determined by (18).  
\[\sigma_{p,critical} =\frac{4\pi^{2}Dd_{p}}{s_{d}^{2}A_{p}}(1-\alpha_{i})=\frac{4\pi^{ 2}E_{p}J_{p}^{2}}{12s_{d}^{2}(1-\mu_{j}^{2})}(1-\alpha_{i})\] \[=\frac{4\times 3.144^{2}\times 200\times 10^{3}\times 4^{2}}{12 \times 100^{3}\times(1-0.3^{2})}\times(1-0.86)\] \[=162\ \text{MPa}\ <\ f_{yp}=300\ \text{MPa}\]  
Determining lateral load capacity of strengthened RC column  
The stresses in steel plate and longitudinal rebar are given by Equation (26). The compressive depth of the strengthened fire-damaged RC column is determined by Equation (27).  
\[0= N_{0}+\frac{4A_{d}\sigma_{1}+2A_{d}\sigma_{2}+2A_{d}\sigma_{1}+4A_{d} \sigma_{1}}{12\cdot 2}\] \[+\frac{0.6\sigma_{p1}A_{p}-A_{p}\sigma_{p}\sigma_{1}+2\sum\limits _{i=1}^{L}t_{p}\lambda_{i}\sigma_{pside,i}}{1.2}-\frac{a_{f,ep}^{f}d_{i}\beta \kappa_{c}}{15}\]  
The compressive height of the plane section of the strengthened fire-damaged RC column is solved as:  
\[x_{c}=302\ \text{mm}\]  
The moment capacity of the strengthened fire-damaged RC column is derived by Equation (28).  
\[\begin{array}{ll}M_{cap,stren}&=N_{0}\frac{d_{c}}{2}-\frac{\alpha f^{\prime }c_{c,p}d_{c}g\kappa_{c}}{1.5}\\ &+\frac{4A_{d}\sigma_{1}(d_{c}-\phi_{d,x}-\frac{\phi_{d}}{2})+2A_{d}\sigma_{p2} (d_{c}-c-\phi_{d,x}-\frac{\phi_{d}}{2}-s_{l})}{+2A_{d}\sigma_{1}(d_{c}-c-\phi _{d,x}-\frac{\phi_{d}}{2}-s_{l})}\\ &+\frac{2A_{d}\sigma_{3}(d_{c}-c-\phi_{d,x}-\frac{\phi_{d}}{2}-2s_{l})+4A_{d} \sigma_{1}(d_{c}-c-\phi_{d,x}-\frac{\phi_{d}}{2}-3s_{l})}{12}\\ &+\frac{0.6\sigma_{p1}A_{p}(d_{c}+\frac{\eta}{2})+A_{p}\sigma_{p1}\frac{\eta} {2}+\frac{\eta}{2}+\frac{\eta}{2}\cdot t_{p}\lambda_{i}\sigma_{pside,i}(d_{c}- \frac{d_{c}-d_{p}}{2}-\frac{\Delta_{i}}{2}-(i-1)\Delta_{i})}{1.2}\\ &=530\ \text{kN}\cdot\text{m}\end{array}\]  
The lateral load resistance of the strengthened fire-damaged RC column is given by Equation (29).  
\[V_{cap,stren}=\frac{M_{cap,stren}}{0.5L}\ =\ \frac{530\times 10^{3}}{0.5\times 3 000}\ =\ 353\ \text{kN}\ >\ V=300\ \text{kN}\]  
The condition in (24) is rechecked herein.  
\[V_{cap,stren}=353\ \text{kN}\ <\ 0.6V_{stren}=374\ \text{kN}\]