## 5 Design Equations for the Shear Strength of RC Beams  
### Strength of RC Beams before and after Retrofitting  
The shear strength of beam with shear steel reinforcement only (V\({}_{\text{Rd}}\)) can be evaluated by the two methods proposed in Eurocode 2 [70]:  
1. The Standard Method (Equation (9)) that considers the steel (V\({}_{\text{Rd},s}\)) and the concrete contribution (V\({}_{\text{Rd},c}\)); The truss model has the compressive strut inclination angle (\(\theta\)) equal to 45\({}^{\circ}\). 2. The Variable Inclination Method (Equation (10)) not containing the concrete contribution; The truss model has a variable compressive strut inclination angle (\(\theta\)) lower than 45\({}^{\circ}\) \[\text{V}_{\text{Rd}}=\text{min}\{\text{V}_{\text{Rd},s}+\text{V}_{\text{Rd },c};\text{ V}_{\text{Rd},\text{max}}\}\] (9) \begin{table}
\begin{tabular}{c c c c} \hline  & **V\({}_{\text{Rd},s\text{ exp}}\)** & **V\({}_{\text{Rd},f\text{ exp}}\)** & **V\({}_{\text{Rd},e\text{ exp}}\)** \\ \hline DB1-R & 81.55 & 72.14 & 60.06 \\ DB2-R & 73.96 & 60.12 & 78.67 \\ \hline \end{tabular}
\end{table} 
Table 4: Experimental contribution to the shear resistance of the transverse steel reinforcement (V\({}_{\text{Rd},s\text{ exp}}\) ), the CFRP reinforcement (V\({}_{\text{Rd},f\text{ exp}}\) ) and the concrete (V\({}_{\text{Rd},c\text{ exp}}\) ) [kN]. \[\mathrm{V_{Rd}}=\min\{\mathrm{V_{Rd,s}};\;\mathrm{V_{Rd,max}}\} \tag{10}\]  
The design value of the shear force, which can be sustained by the yielding shear reinforcement (\(\mathrm{V_{Rd,s}}\)), can be calculated by Equation (11). \[\mathrm{V_{Rd,s}}=\frac{0.9\cdot\mathrm{d}\cdot\mathrm{f_{ywd}}\cdot\mathrm{A_{ sw}}}{\mathrm{s}}\cdot\cot\Theta \tag{11}\]  
where \(\mathrm{f_{ywd}}\), \(\mathrm{A_{sw}}\) and \(\mathrm{s}\) are the design yield stress, the cross-sectional area and the spacing of the shear steel reinforcement. The design shear resistance of the member without shear steel reinforcement (\(\mathrm{V_{Rd,c}}\)) can be calculated by Equation (12) assuming member axial load equal to zero. \[\mathrm{V_{Rd,c}}=0.18/\gamma_{\mathrm{c}}\cdot\left(1+\sqrt{\frac{200}{ \mathrm{d}}}\right)\cdot\left(100\cdot\mathrm{\rho_{l}}\cdot\mathrm{f_{ck}} \right)^{\frac{1}{3}}\cdot\mathrm{d}\cdot\mathrm{b_{w}}>\mathrm{v_{min}}\cdot \mathrm{d}\cdot\mathrm{b} \tag{12}\]  
in which \(\gamma_{\mathrm{c}}\) is the partial factory, \(\mathrm{\rho_{l}}\) is the reinforcement ratio for the longitudinal reinforcement, \(\mathrm{b_{w}}\) is the smallest width of the cross-section in the tensile area, \(\mathrm{f_{ck}}\) is the characteristic compressive cylinder strength of concrete at 28 days and the parameter \(\mathrm{v_{min}}\) may be found in Country National Annex. The term \(\mathrm{V_{Rd,max}}\) is the maximum shear capacity of the section calculated by Equation (13) that depends on the compressive strength of the concrete strut ([PERSON]'s truss analogy model). \[\mathrm{V_{Rd,max}}=0.9\cdot\mathrm{d}\cdot\mathrm{b_{w}}\cdot\alpha_{\mathrm{ cw}}\cdot\mathrm{v_{1}}\cdot\mathrm{f_{cd}}\cdot\left(\frac{\cot\Theta+\cot \alpha}{1+\cot\Theta^{2}}\right) \tag{13}\]  
in which the coefficients \(\alpha_{\mathrm{cw}}\) and \(\mathrm{v_{1}}\) for use in a Country may be found in its National Annex, \(\mathrm{f_{cd}}\) is the design value of concrete compressive strength and \(\alpha\) is the angle between shear reinforcement and the beam axis perpendicular to the shear force. From a physical point of view, the Standard Method is unsatisfactory because the concrete contribution is empirical and includes different resisting mechanisms (compressed concrete zone contribution, aggregate interlock, and dowel action of the longitudinal rebars crossing the shear cracks) difficult to estimate. Furthermore, the experimental evidence shows a redistribution of the forces in the beam web, which results in compressive concrete strut inclination angles smaller than 45\({}^{\circ}\). The number of stirrups crossed by the shear cracks can be greater and more stirrups can contribute to the beam shear strength.