## 5 Analytical Prediction of Beam Capacities
Following Equations (13) and (14) gives the effective strain (\(\varepsilon_{fe}\)) and bond-reduction coefficient \(k_{v}\). Equations (15)-(17) give the factors \(k_{1}\), \(k_{2}\), and effective length \(L_{e}\) used in Equation (14). \[M_{n}=A_{s}f_{y}\Big{(}d-\frac{a}{2}\Big{)}+A_{f}f_{f}\Big{(}d_{f}-\frac{a}{2} \Big{)} \tag{2}\]4  
\[M_{n}=A_{s}f_{y}\Big{(}d-\frac{a}{2}\Big{)}+A_{f}f_{f}\Big{(}d_{f}-\frac{a}{2} \Big{)} \tag{2}\]5  
\[M_{n}=A_{s}f_{y}\Big{(}d-\frac{a}{2}\Big{)}+A_{f}f_{f}\Big{(}d_{f}-\frac{a}{2} \Big{)} \tag{2}\]6  
\[M_{n}=A_{s}f_{y}\Big{(}d-\frac{a}{2}\Big{)}+A_{f}f_{f}\Big{(}d_{f}-\frac{a}{2} \Big{)} \tag{2}\]7  
\[M_{n}=A_{s}f_{y}\Big{(}d-\frac{a}{2}\Big{)}+A_{f}f_{f}\Big{(}d_{f}-\frac{a}{2} \Big{)} \tag{2}\]8  
Table 2 gives a detailed comparison of analytical and experimental load-carrying capacities of the beam specimens. The control beam did not carry any flexural and shear CFRP reinforcement and its flexural and shear capacities were computed by ACI 318-08 [34]. The flexural and shear capacities of the CFRP strengthened beams are calculated by Equations (2) and (7), as recommended by ACI 440.2R [33]. The flexural capacity of the beam is the main function of the amount of CFRP applied at the tensile face of the beam. As beams B1, B2, B3, and B4 carry the same amount of CFRP flexural reinforcement ratio, these beams had the same analytical moment capacity of 25.35 kN-m. Similarly, beam B5 and B6 have the same analytical moment capacity of 33.41 kN-m. On the other hand, the shear capacity of the CFRP strengthened beams is proportional to the amount of CFRP shear reinforcement. Due to the different amounts of CFRP shear reinforcement, specimens B1 to B5 have different analytical shear capacities. However, beam B3 and B6 have the same CFRP shear reinforcement ratio and as a result, these have the same analytical shear capacities. From the analytical flexural and shear strengths of beams, their corresponding peak loads \(P_{m}\) for moment and \(P_{v}\) for shear are calculated and presented in \[M_{n}=A_{s}f_{y}\Big{(}d-\frac{a}{2}\Big{)}+A_{f}f_{f}\Big{(}d_{f}-\frac{a}{2} \Big{)} \tag{2}\]9 For instance, the analytical moment capacity of control beam 17.28 kN-m will result in the corresponding load capacity of (\(P_{m}=\frac{M_{h}\times 2}{L_{1}}\)) 72.8 kN. The analytical peak load of each beam is evaluated as the lesser of \(P_{m}\) and \(P_{v}\), depending upon which type of failure will govern the ultimate load carrying capacity of the beam. In order to understand the effect of CFRP flexural and shear reinforcement, the control beam was designed to have the same flexural and shear capacity. However, the \(P_{m}\) for control beam was slightly lesser than the \(P_{v}\) and it suggests that the dominant mode of failure in the beam is flexural rather than shear, or it indicates that the flexural failure will occur first,  
\[\varepsilon_{fd}=0.41\sqrt{\frac{f_{c}^{\prime}}{nE_{f}f_{f}}}\leq 0.9\varepsilon_{fu} \tag{3}\]0 Dimensional variables used in shear-strengthening calculations. followed by the shear failure of the beam specimen. Similarly, in the case of beams B1 to B4 and B6, flexural failure governed the overall design, and in the case of beam B5 the shear failure governed the peak load of the beam, as given in \[\varepsilon_{fd}=0.41\sqrt{\frac{f_{c}^{\prime}}{nE_{f}f_{f}}}\leq 0.9\varepsilon_{fu} \tag{3}\]1 A comparison of experimental and analytical peak loads has also been provided in the last column of \[\varepsilon_{fd}=0.41\sqrt{\frac{f_{c}^{\prime}}{nE_{f}f_{f}}}\leq 0.9\varepsilon_{fu} \tag{3}\]2 A very close agreement between analytical and experimental load values was achieved for all of the tested beams. In the case of beam B5, the difference is slightly more compared to the other beams. The experimental peak load of beam B5 was slightly higher compared to the analytical load.