## 6 Ultimate Load Carrying Capacity of strengthened T-Section Beams
### Yielding Resistance and Ultimate Resistance of Strengthened T-Section Beams  
The assumptions made to predict the yielding and ultimate resistance of the T-section beams strengthened by FRP include: (1) plane sections remain plane; (2) plastic stress was fully developed in the top fiber of the cross-section; (3) tensile strength of the concrete was ignored in the cross-section; and (4) the strengthening CFRP materials were assumed to be fully attached to the T-section beam. Figure 10 shows the strain and stress distributions in the cross-section corresponding to the yielding and ultimate resistance of the T-section beams. The position of the neutral axis in the cross-section could be found via equating the resultant internal forces acting on the cross-section to zero, i.e.,  
\[N_{cc}+N_{sc}=N_{st}+N_{R} \tag{17}\]  
Figure 9: Strain and stress distribution on cross-section of T-section beam before concrete cracking. \[N_{cc}=\left\{ \begin{array}{c}nf_{c}B_{h}h_{t}+nf_{c}b(\lambda x-h_{t})\\ nf_{c}B_{f}\lambda x\end{array} \right\}\quad\left[ \begin{array}{cc}for&x>h_{t} \\ for&x\leq h_{t}\end{array} \right] \tag{18}\]  
\[N_{cs}=min\left(f_{yc},E_{s},\epsilon_{cs}\right)A_{sc} \tag{19}\]  
\[N_{st}=min\left(f_{yt},E_{s},\epsilon_{ct}\right)A_{st} \tag{20}\]  
\[N_{R}=bt_{R}\;min\;(f_{R},E_{R}\epsilon_{R}) \tag{21}\]  
\[\epsilon_{cu}=\;\epsilon_{ts}\frac{h-x}{h-x-\;a_{t}}=\epsilon_{cs}\frac{h-x}{x -\;a_{c}}=\epsilon_{c}\frac{h-x}{x} \tag{22}\]  
where \(N_{cc}\), \(N_{sc}\), \(N_{st}\), \(N_{R}\), denote the internal resultant forces acting on the concrete under compression, compressive reinforcement, tensile reinforcement, and strengthening materials, respectively (see Figure 7); \(\lambda=0.8\) for fck \(\leq 50\) MPa, \(\lambda=0.8-(\text{fck}-50)/400\) for \(50<\text{fck}\leq 90\) MPa; \(\eta=1.0\) for fck \(\leq 50\) MPa; \(\eta=1.0-(\text{fck}-50)/200\) for \(50<\text{fck}\leq 90\) MPa [16]; \(B_{f}\), \(b\), \(h_{t}\) denote the width of the flange, width of the web, and height of the flange of the T-section beam, respectively;\(f_{yc}\),\(f_{yt}\) denote the yield strength of the compressive and tensile steel reinforcement, respectively; \(A_{sc}\), \(A_{st}\) and \(A_{R}\) denote the area of compressive reinforcement, tensile reinforcement, and strengthening CFRP attached to the T-section beam; \(h_{t}\), \(h_{0}\) denote the depth of flange, effective depth, and overall depth of the cross-section, respectively; \(\epsilon_{cu}\) denotes the ultimate strain of the concrete under compression. Corresponding values for different strength of concrete could be determined as per the Eurocode 2 [16]. Thus, with the solved x, the plastic bending moment of the cross-section corresponding to yielding of the tensile reinforcement could be determined as follows;  
\[M_{py}=M_{pc}+M_{psc}+M_{pts}+M_{pR} \tag{23}\]  
\[M_{pc}=\left\{ \begin{array}{c}nf_{c}B_{f}h_{t}(x-0.5h_{t})+nf_{c}(\lambda x -h_{t})b[x-h_{t}-0.5(\lambda x\geq h_{t})]\\ \lambda xnf_{c}B_{f}(x-0.5\lambda x)\end{array} \right\}\quad\left[ \begin{array} []{cc}for&x\geq h_{t}\\ for&x<h_{t}\end{array} \right] \tag{24}\]  
\[M_{psc}=min\left(f_{yc},E_{s}\epsilon_{cs}\right)A_{sc}(x-a_{c}) \tag{25}\]  
\[M_{pst}=min\left(f_{yt},E_{s}\epsilon_{ct}\right)A_{st}(h_{0}-x) \tag{26}\]  
\[M_{pR}=bt_{R}min\;(f_{R}E_{R}\epsilon_{R})\left(h-x+\frac{t_{r}}{2}\right) \tag{27}\]  
where \(f_{c}\) denotes the compressive strength of the concrete. Regarding the bending resistance of the cross-section corresponding to the ultimate tensile strength of the flexural reinforcement \(M_{pu}\) the ultimate strength of the tensile steel reinforcement \(f_{u}\) can be used  
Figure 10: Strain and stress distribution on cross-section of T-section beams corresponding to yielding and ultimate resistance. for the calculation.