## 5 Analytical Prediction of Beam Capacities  
Several researchers compared the experimental and analytical flexural and shear capacities of different CFRP strengthened RC beams [21; 22; 23; 24; 25; 26; 27; 28]. However, most of them [22; 26; 27; 28; 37] found close analytical values when ACI 440.2R-08 [33] was used. In the current study, ACI 440.2R-08 [33] was adopted to analytically obtain the flexural and shear capacities of the beams. The flexural strength of CFRP strengthened beams depends upon the governing mode of failure. Reinforced concrete beams strengthened with CFRP can experience three major failure modes: (i) tension failure of CFRP strip; (ii) debonding of CFRP from the concrete surface; and (iii) the crushing of concrete on the compression sides. ACI 440.2R-08 [33] considers all three failure modes to satisfy strain compatibility and force equilibrium conditions. Figure 7 illustrates the internal strain and stress distribution of a CFRP strengthened rectangular RC beam under flexure. The ultimate nominal strength of beam (\(M_{n}\)) is obtained by satisfying strain compatibility and force equilibrium conditions and also considers the governing mode of failure. Equations (2)-(6) give the flexural or ultimate moment capacity of CFRP strengthened beams. In order  
Figure 6: Moment-curvature relation for control specimen C1, CFRP-B3, and CFRP-B5. to check the debonding failure of CFRP strengthened beams, the effective strain in the CFRP strips (\(\varepsilon_{f}\)) should be limited to debonding strain level (\(\varepsilon_{fd}\)), as given by Equation (4). \[M_{n}=A_{s}f_{y}\Big{(}d-\frac{a}{2}\Big{)}+A_{f}f_{f}\Big{(}d_{f}-\frac{a}{2} \Big{)} \tag{2}\]  
\[\varepsilon_{fd}=0.41\sqrt{\frac{f_{c}^{\prime}}{nE_{f}f_{f}}}\leq 0.9\varepsilon_{fu} \tag{3}\]  
\[\varepsilon_{f}=\varepsilon_{cu}\Big{(}\frac{d_{f}-c}{c}\Big{)}\leq\varepsilon _{fd} \tag{4}\]  
\[f_{f}=\varepsilon_{f}E_{f} \tag{5}\]  
\[a=\beta_{1}c \tag{6}\]  
According to ACI 440.2R-08, the shear strength of CFRP strengthened beams can be computed by Equation (7), as follows. \[V_{n}=V_{c}+V_{s}+\Psi_{f}V_{f} \tag{7}\]  
where \(V_{n}\) is the nominal strength of a CFRP strengthened beam, \(V_{s}\) is shear strength provided by the steel and calculated by ACI 318-08 [34], as given by Equation (8), and \(V_{c}\) is the shear strength provided by the concrete, which can be computed by ACI 318-08 [34], as given by Equation (9). \[V_{s}=\frac{A_{v}f_{y}d}{s} \tag{8}\]  
\[V_{c}=0.17\sqrt{f_{c}^{\prime}}b_{w}d \tag{9}\]  
where \(V_{f}\) is the shear strength provided by the CFRP shear strips and \(\Psi_{f}\) = 0.85 is a reduction factor for CFRP shear strips applied on the sides of beams. ACI 440.2R-08 [33] recommends Equation (10) to calculate the shear strength provided by the FRP strips. The effective stress in FRP and area of CFRP in shear can be calculated by using Equations (11) and (12), respectively. The dimensional variables used in Equations (10) and (12) can be seen in Figure 8.  
\[M_{n}=A_{s}f_{y}\Big{(}d-\frac{a}{2}\Big{)}+A_{f}f_{f}\Big{(}d_{f}-\frac{a}{2} \Big{)} \tag{2}\]0  
\[M_{n}=A_{s}f_{y}\Big{(}d-\frac{a}{2}\Big{)}+A_{f}f_{f}\Big{(}d_{f}-\frac{a}{2} \Big{)} \tag{2}\]1  
\[M_{n}=A_{s}f_{y}\Big{(}d-\frac{a}{2}\Big{)}+A_{f}f_{f}\Big{(}d_{f}-\frac{a}{2} \Big{)} \tag{2}\]2  
\[M_{n}=A_{s}f_{y}\Big{(}d-\frac{a}{2}\Big{)}+A_{f}f_{f}\Big{(}d_{f}-\frac{a}{2} \Big{)} \tag{2}\]3 Stain, stress, and force distribution diagram of CFRP strengthened RC beam. For bonded face plies or CFRP strips, which is a strengthening system that does not enclose the entire section of the beam, failure of CFRP shear strips is mainly attributed to the delamination of CFRP from the concrete surface. Due to this reason, the bond stress was analyzed by [PERSON] [39] to determine the usefulness of these systems and proposed the effective strain level that can be achieved. The effective strain (\(\varepsilon_{fe}\)) is calculated using a bond-reduction coefficient \(k_{v}\).