## 4 Simplified Design Procedures  
All the international guidelines suggest a simple approach based on the results of fracture analyses of strengthened systems and on the following assumptions:  
* The cross-section remains plane in the deformed configuration;  
Figure 4: Debonding due to vertical cracks. Figure 3: Debonding due to vertical cracks. * The elasto-plastic constitutive law is considered for concrete (in compression) and reinforcing steel;
* Any strains in the steel reinforcement and concrete are directly proportional to their distance from the neutral axis;
* There is no relative slip between the external reinforcement and the concrete core;
* There is perfect adhesion between the reinforcement and the concrete core, due to negligible shear deformation within the adhesive layer;
* The linear elastic stress-strain constitutive law applies to FRP reinforcement up to failure. More specifically, these procedures fix the design strain for debonding Modes 1 and 2 in ultimate limit states, as follows:  
\[\epsilon_{\text{fd}}=\min\biggl{\{}\eta_{\text{a}}\cdot\frac{\epsilon_{\text{ fk}}}{\gamma_{\text{f}}},\epsilon_{\text{fd,di}}\biggr{\}}, \tag{1}\]  
where \(\epsilon_{\text{fk}}\) is the FRP characteristic strain at failure, \(\eta_{\text{a}}\) is the FRP environmental reduction factor, \(\gamma_{\text{f}}\) is the FRP partial safety factor, and \(\epsilon_{\text{fd,di}}\) is the maximum strain due to debonding (for the debonding mode \(i\)-th). The maximum strain due to the two debonding modes may be given directly or indirectly by the corresponding maximum stress, \(f_{\text{fd,di}}\) (for the debonding mode \(i\)-th). In the first case, the ACI 440.2R-17 formulation [12], based on a modified form of the debonding strain equation for Mode 2, proposed by [PERSON] et al. (2003, 2004), may be considered:  
\[\epsilon_{\text{fd,d2}}=0.41\cdot\sqrt{\frac{f_{\text{fk}}}{n\cdot E_{\text{ f}}\cdot f_{\text{f}}}}, \tag{2}\]  
In the second case, the Mode 1 failure stress can be evaluated using the following relationship [15]:  
\[f_{\text{fd,d1}}=\frac{1}{\gamma_{\text{f,d}}\cdot\sqrt{\gamma_{\text{c}}}} \cdot\sqrt{\frac{2\cdot E_{\text{f}}\cdot\Gamma_{\text{Fk}}}{t_{\text{f}}}}, \tag{3}\]  
where:  
* \(E_{\text{f}}\) and \(t_{\text{f}}\) are the Young modulus of elasticity and the thickness of the FRP, respectively;
* \(\gamma_{\text{f,d}}\) is the FRP partial factor and \(\gamma_{\text{c}}\) is the partial factor for concrete;
* \(\Gamma_{\text{Fk}}\) is the specific fracture energy of the FRP-concrete interface, and may be expressed as follows:  
\[\Gamma_{\text{Fk}}=0.03\cdot k_{\text{b}}\cdot\sqrt{f_{\text{ck}}\cdot f_{ \text{ctm}}}, \tag{4}\]  
where \(f_{\text{ck}}\) and \(f_{\text{ctm}}\) are the characteristic strength and the average tensile strength of the concrete, respectively, and \(k_{\text{b}}\) is a geometric coefficient dependent on both the width of the strengthened beam and the width of the FRP system. The Mode 1 failure stress reported above is associated with a bonded length, \(l_{\text{b}}\), equal or greater than the optimal length given by [15]:  
\[l_{\text{e}}=\sqrt{\frac{E_{\text{f}}\cdot f_{\text{f}}}{2\cdot f_{\text{ctm }}}}, \tag{5}\]  
If \(l_{\text{b}}<l_{\text{e}}\), the ultimate design strength may be reduced as follows:  
\[f_{\text{fd,d1,rid}}=f_{\text{fd,d1}}\cdot\frac{l_{\text{b}}}{l_{\text{e}}} \cdot\biggl{(}2-\frac{l_{\text{b}}}{l_{\text{e}}}\biggr{)}, \tag{6}\]Mode 2 failure stress may be evaluated using the following relationship [15]:  
\[f_{\text{fd,d2}}=k_{\text{cr}}\cdot f_{\text{fdd}}=\frac{k_{\text{cr}}}{\gamma_{ \text{f,d}}\cdot\sqrt{\gamma_{\text{c}}}}\cdot\sqrt{\frac{2\cdot E_{\text{f}} \cdot\Gamma_{\text{FK}}}{t_{\text{f}}}}, \tag{7}\]  
where \(k_{\text{cr}}\) can be taken as equal to 3.0 if specific data are not available. The corresponding values of the design strain in the FRP system (for the debonding mode \(i\)-th) can be calculated according to the following expression [15]:  
\[\varepsilon_{\text{fd,di}}=\frac{f_{\text{fd,di}}}{E_{\text{f}}}. \tag{8}\]