## 6 Ultimate Load Carrying Capacity of strengthened T-Section Beams  
### Resistance of T-Section Beams Corresponding to Crack Initiation \(P_{cr}\)  
The bending resistance of the RC T-section beams corresponding to the tensile crack initiation of cross-section can be obtained through solving the position of the neutral axis, x, as shown in Figure 9. The assumptions made include: (1) plane sections remain plane; (2) the ultimate tensile stress of the concrete was achieved at the bottom fiber of the cross-section; (3) the strengthening materials had perfect bond with the T-section beams and no bonding failure occurred. Setting the resultant forces acting on the section equal to zero, x can be solved as follows;  
\[N_{cc}+\ N_{sc}=N_{st}+N_{ct}+N_{r} \tag{3}\]  
\[N_{cc}=\left\{\begin{array}{c}0.5E_{c}\epsilon_{c}\bigg{[}\big{(}1+\frac{x- h_{1}}{x}\big{)}B_{f}h_{t}+\frac{(x-h_{t})^{2}}{x}b\bigg{]}\\ 0.5E_{c}\epsilon_{c}B_{f}h_{t}\end{array}\right\}\qquad\left[\begin{array}{ cc}for&x>\ h_{t}\\ for&x\leq\ h_{t}\end{array}\right] \tag{4}\]  
\[N_{cs}=A_{sc}E_{s}\epsilon_{cs} \tag{5}\]  
\[N_{ct}=\left\{\begin{array}{c}0.5E_{c}\epsilon_{ct}b(h-x)\\ 0.5E_{c}\epsilon_{ct}\bigg{(}\frac{(h_{t}-x)^{2}}{h-x}B_{f}+\left[1+\frac{h_{t }-x}{h-x}\right]\bigg{)}\end{array}\right\} \tag{6}\]  
\[N_{st}=A_{st}E_{s}\epsilon_{ct} \tag{7}\]  
\[N_{R}=bt_{R}E_{R}\epsilon_{R} \tag{8}\]  
\[\epsilon_{ct}=\ \epsilon_{R}\frac{h-x}{h+0.5\ t_{R}-x}=\epsilon\frac{h-x}{h-x -a_{t}}=\epsilon_{cs}\frac{h-x}{x-a_{c}}=\epsilon_{c}\frac{h-x}{x} \tag{9}\]  
where \(\epsilon_{ct}\), \(\epsilon_{R}\), \(\epsilon_{ts}\), \(\epsilon_{cs}\), and \(\epsilon_{c}\) denote the strain at the bottom fiber of the T-section beam, strengthening plate, tensile reinforcement, compressive reinforcement, and top fiber of the T-section beam, respectively (see Figure 9); b and Bif denote the width of web and flange of the T-section beam, respectively; \(h_{t}\), \(h_{0}\), and \(h\) denote the depth of flange, effective depth, and overall depth of the cross-section, respectively.  
Figure 8: Central deflections \(u_{max}\) of tested T-section beam specimens at ultimate resistance.  
Substituting the fracture strain \(\varepsilon_{ct}=\varepsilon_{t0}\) into Equations (3)-(9), the neutral axis position, x, could be found. With x solved, the bending resistance of the cross-section could be determined through taking the moment to the neutral axis as follows;  
\[M_{cR}=\ M_{cc}+M_{sc}+M_{st}+M_{ct}+M_{R} \tag{10}\]  
\[M_{cc}=\left\{\begin{array}{cc}\frac{3\pi^{2}-3\lambda t_{1}+t_{2}^{2}}{3\pi^ {2}}E_{c}\varepsilon_{c}B_{f}h_{t}\\ \frac{2\pi^{2}}{3}E_{c}\varepsilon_{c}B_{f}\end{array}\right\}\quad\left[ \begin{array}{cc}for&x\geq h_{t}\\ \\ for&x<h_{t}\end{array}\right] \tag{11}\]  
\[M_{sc}=E_{s}\varepsilon_{cs}A_{sc}(x-a_{c}) \tag{12}\]  
\[M_{st}=E_{s}\varepsilon_{ct}A_{st}(h_{0}-x) \tag{13}\]  
\[M_{ct}=\left\{\begin{array}{cc}\frac{2}{3}(h-x)^{2}\ b\sigma_{t}\\ \frac{h+h-2x}{2(h-x)}\left[(h-x)-\frac{h+2h-3x}{3(h+h-2x)}(h-h_{1})\right] \varepsilon_{c}E_{c}B_{f}\end{array}\right\}\quad\left[\begin{array}{cc}for &x\geq h_{t}\\ \\ for&x<h_{t}\end{array}\right] \tag{14}\]  
\[M_{R}=E_{R}\varepsilon_{R}A_{R}\Big{(}h-x+\frac{t_{R}}{2}\Big{)} \tag{15}\]  
where \(\sigma_{t}\) denotes the ultimate tensile strength of the concrete \(A_{sc}\), \(A_{st}\) and \(A_{R}\) denote the area of the compressive reinforcement, tensile reinforcement, and strengthening materials attached to the T-section beam. With the developed equations, the bending resistance of the cross-section can be determined. Thus, the resistance of the T-section beam corresponding to the crack initiation can be determined as follows:  
\[P_{cr}=\frac{2M_{cr}}{L_{a}} \tag{16}\]  
where \(L_{a}\) denotes the shear span of the beam.