## 2 High-Performance Fiber Reinforced Concrete to Repair RC Components
### Code Equation to Determine the Tensile Strength  
It's well known that fiber reinforced concrete (FRC) shows a good tensile strength. In the retrofitting technology proposed in this research, FRC is used to increase the shear resistance of the pier and so to improve its seismic capacity. In the literature, many models have been proposed to predict the shear resistance. The one presented in CNR-DT 204/2006 guideline [55] and described in Section 3 can be a valid tool to calculate the shear strength contribution given by the fibers. This model is calibrated on the base of the ultimate residual tensile strength \(\mathrm{f_{Flu}}\)--using the equations presented in this sub-section. This guideline calculates the tensile residual strength using a linear elastic or a rigid-plastic model for the softening material behavior. The linear elastic model identifies two values, \(\mathrm{f_{FPs}}\) and \(\mathrm{f_{Flu}}\), based on serviceability (SLE) and ultimate state (SLU) limit, respectively. They can be defined by the equivalent values of flexural resistances (\(\mathrm{f_{eq1}}\), \(\mathrm{f_{eq2}}\)) through Equations (2) and (3)  
\[\mathrm{f_{Fts}}=0.45\cdot\mathrm{f_{eq1}} \tag{2}\]  
\[\mathrm{f_{Ftu}}=\mathrm{k}\cdot\left[\mathrm{f_{Fts}}-\frac{\mathrm{w_{u}}}{ \mathrm{w_{i2}}}.\left(\mathrm{f_{Fts}}-0.5\cdot\mathrm{f_{eq2}}+0.2\cdot \mathrm{f_{eq1}}\right)\right]\geq 0 \tag{3}\]  
where \(\mathrm{f_{eq1}}\) and \(\mathrm{f_{eq2}}\) are the significant post-cracking equivalent resistances for the SLE and for the SLU, respectively, \(\mathrm{k}\) is a coefficient equal to 0.7 if the section is entirely subjected to traction, otherwise it is equal to 1, \(\mathrm{w_{i2}}\) is the mean value of the crack opening corresponding to \(\mathrm{f_{eq2}}\), and \(\mathrm{w_{u}}\) is the ultimate crack opening value. For the four-points bending test on notched samples, according to UNI 11039 [52], the characteristic values of equivalent strengths, \(\mathrm{f_{eq1}}\) and \(\mathrm{f_{eq2}}\), are evaluated at crack opening intervals \(0\leq\mathrm{w}\leq 0.6\) mm and \(0.6\leq\mathrm{w}\leq 3.0\) mm. The equivalent strengths can be obtained by Equations (4) and (5) in UNI 11039 [52]. \[\mathrm{f_{eq1}}=\frac{\mathrm{L}}{\mathrm{b(h-a)}^{2}}.\frac{\mathrm{U_{1}}} {0.6} \tag{4}\]  
\[\mathrm{f_{eq2}}=\frac{\mathrm{L}}{\mathrm{b(h-a)}^{2}}.\frac{\mathrm{U_{2}}} {2.4} \tag{5}\]  
where \(\mathrm{b}\) is the section beam width; \(\mathrm{L}\), \(\mathrm{a}\), and \(\mathrm{h}\) are the beam geometries in Figure 2b; and \(\mathrm{U_{1}}\)[\(10-3\) J] and \(\mathrm{U_{2}}\)[\(10-3\) J] are the areas underlined beneath the load-CTOD curve in the CTOD range of 0.025-0.625 mm and 0.625-3.025 mm, respectively, as shown in Figure 6a,b. Such equivalent strengths are \(\mathrm{f_{q11}}\) and \(\mathrm{f_{eq2}}\), which correspond, respectively, to crack openings. \(\mathrm{w_{i1}}\) is equal to 0.3 mm (used for the SLE, according to Eurocode 2 [57]) and \(\mathrm{w_{i2}}\) is equal to 1.8 mm, corresponding to the average values in the selected ranges (employed for the SLU condition). This study aims to define a simple tool useful to designers to predict the shear strength improvement given by the new concrete jacket adopted in the retrofitting technology. For that reason, the rigid-plastic model introduced in CNR DT 204/2006 [55] is selected as it allows defining the \(\mathrm{f_{Flu}}\) on the base of the \(\mathrm{f_{eq2}}\) using Equation (6). The predicted \(\mathrm{f_{Flu}}\) value resulted smaller than the one predicted by Equation (3), and therefore, the predicted shear strength of the concrete jacket with fiber is safer. \[\mathrm{f_{Flu}}=\frac{\mathrm{f_{eq2}}}{3} \tag{6}\]  
The characteristic tensile strength \(\mathrm{f_{ctk}}\) is given by Equation (7), which is computed starting from the average tensile strength \(\mathrm{f_{ctm}}\) obtained by Equation (8) for concrete with \(\mathrm{f_{ck}}>50\) MPa, as indicated in Italian construction code NTC 2018 [58]. \[\mathrm{f_{ctk}}=0.7\cdot\mathrm{f_{ctm}} \tag{7}\]