## 3 Materials and Methods
### Description of the Problem and Retrofit Alternatives
\[V_{n}=V_{c}+V_{s}+\psi_{f}V_{f} \tag{53}\]The contribution of the shear stress of the FRP is given by Equation (54):  
\[V_{n}=V_{c}+V_{s} \tag{45}\]0  
where for rectangular sections, \(A_{fo}\) is determined by Equation (55):  
\[V_{n}=V_{c}+V_{s} \tag{45}\]1  
where \(n\) is the number of FRP reinforcement layers, \(t_{f}\) is the nominal thickness of a reinforcing sheet, \(w_{f}\) is the nominal width of a reinforcing sheet, \(f_{fe}\) is the effective stress in the FRP reinforcement, \(\alpha\) is the angle formed by the extreme fiber in traction with the FRP strip, \(d_{fo}\) the depth of the FRP shear reinforcement, and sf the distance between the centers of the FRP strips. Specific preventive maintenance measures based on material properties were incorporated to ensure that the retrofit alternatives achieve the expected service life of 50 years from installation or until structural damage due to an earthquake occurs. Table 1 summarizes the general characteristics of the beam analyzed and the shear strength values of the five retrofit alternatives considered. The beam requiring retrofit is assumed to be part of a reinforced concrete building in northern Quito, classified as a high seismic hazard zone under Ecuador's current seismic regulations. The following transport distances were considered: 8.0 km from the concrete factory, 42.3 km for steel, 40.2 km for structural adhesives and materials used in preventive maintenance, and 13.2 km to the landfill for waste disposal. For CFRP and epoxy resin, transport distances include 3800 km by air and 37.3 km by land, as these materials are imported into Ecuador. Figure 2 shows the schematics of the retrofitting alternatives. A detailed description of the alternatives to retrofit is provided. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{5}{c}{Original beam} \\ \hline A (m) & B (m) & E (m) & DL (mm) & Number \\
0.4 & 0.6 & 0.2 & 16 & 8 \\ DT (mm) & S a (m) & f\({}^{\prime}\)c (Mpa) & fy (Mpa) & V (kN) \\
10 & 0.1 & 24 & 421 & 558.79 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: General information about the original beam and the retrofitting alternatives. CJ Alternative: A 75 mm thick layer of concrete is applied to the bottom and side faces of the beam. The concrete, with a compressive strength of 24 MPa and a soft consistency, is pre-mixed at the plant and pumped on-site. Before application, a continuous layer of two-component, thixotropic, epoxy resin-based structural adhesive is applied to the hardened concrete surface. The reinforcement consists of four longitudinal bars 12 mm in diameter and stirrups 10 mm in diameter, spaced at 150 mm and anchored to the slab using chemicals. Preventive maintenance includes the application of an anti-carbonation paint. SCJ Alternative: A 75 mm thick shotcrete jacket is applied to the bottom and side faces of the beam. The shotcrete has a compressive strength of 24 MPa and fluid consistency. It is applied using a wet spraying technique. The reinforcement is identical to that used in the CJ alternative. Preventive maintenance consists of the application of anti-carbonation paint. STE Alternative: 4 mm thick A572 Gr50 hot-rolled steel plates are applied to the lateral faces of the beam and secured using a two-component thixotropic epoxy resin adhesive. Preventive maintenance includes the application of a two-component epoxy resin resin-based anticorrosive paint and passive fire protection achieved by spraying cement-based fireproof mortar. STA Alternative: 4 mm thick A572 Gr50 hot-rolled steel plates are secured to the beam's lateral faces using mechanical anchorage with bolts. Preventive maintenance includes the application of two-component epoxy resin-based anticorrosive paint and passive fire protection provided by the projection of cement-based fireproof mortar. CFRP Alternative: The CFRP is applied using a U-wrap method with two layers of unidirectional carbon fabric, each 0.165 mm thick.