## 3 Materials and Methods
### Description of the Problem and Retrofit Alternatives  
This study aims to determine the optimum seismic retrofitting solution for a reinforced concrete beam with shear deficiencies. The analyzed beam has a span of 6.0 m. It presents material properties and longitudinal and transverse reinforcement, typical of beams built according to obsolete standards or non-designed practices. In order to comply with current seismic regulations, the retrofit of the beam includes a 22% increase in shear strength within the intended plastic hinge zone. The analysis assumes a service life of 50 years for the retrofit beam, which ensures compliance with modern seismic requirements and improves the durability of the structural element. The seismic strengthening alternatives are evaluated using a \"cradle-to-grave\" life cycle analysis, which incorporates economic, environmental, and social dimensions while assessing functionality. Optimal seismic retrofitting is selected from five alternatives: concrete jacketing (CJ), shotcrete jacketing (SCJ), steel jacketing with structural epoxy adhesive (STE), steel jacketing with anchor bolts (STA), and use of carbon fiber reinforced polymer (CFRP). The methods for the design of retrofitting alternatives are considered to be an additive approach. Figure 1: Methodology of the study. The equations in the Ecuadorian standard NEC-15 [39], which are based on ACI-318, were used for the concrete jacketing alternatives. The nominal shear strength considers the contribution of the concrete (\(V_{c}\)) and the shear reinforcement (\(V_{s}\)), according to Equation (45). \[V_{n}=V_{c}+V_{s} \tag{45}\]  
The contribution of concrete to shear was determined using Equation (46). \[V_{c}=0.17\sqrt{f^{{}^{\prime}}_{c}}b_{w}d \tag{46}\]  
where \(f^{{}^{\prime}}_{c}\) is the compressive strength of the concrete cylinder in MPa, \(b_{w}\) is the width of the beam, and \(d\) is the effective depth of the beam. For the contribution of the vertical stirrup, ACI-318 considers the truss model represented by Equation (47). \[V_{s}=\frac{A_{v}f_{y}d}{s} \tag{47}\]  
where \(A_{v}\) is the effective area of the stirrup legs, \(f_{y}\) is the steel yield strength, and \(s\) is the spacing between the stirrups. For the alternatives incorporating steel plates, equation 48 followed the reference guidelines [40]. This expression considers the shear strength of reinforced concrete beams with continuous horizontal steel plates attached to the web, the addition of the contributions of the concrete, the internal shear reinforcement, and the external steel plates. Equation (49) determines the contribution of shear stress (\(Vp\)), considering the sum of the shear stresses along its depth and thickness. \[V_{n}=V_{c}+V_{s}+V_{p} \tag{48}\]  
\[V_{p}=\tau_{avg}h_{p}t_{p} \tag{49}\]  
where \(\tau_{avg}\) is the average shear stress in the steel plates, \(h_{p}\) is the depth of the plates, and \(t_{p}\) is the thickness of the plates on either side of the beam's web. For the quadratic distribution of shear stress over the depth of the plate, the average stress is calculated using Equation (50):  
\[\tau_{avg}=\frac{2}{3}\tau_{max} \tag{50}\]  
\(\tau_{max}\) is the maximum shear stress in a steel plate, which can be determined from [PERSON]'s maximum shear stress theory. This theory considers the yield strength of the material \(f_{yp}\), as indicated in Equation (51):  
\[\tau_{max}=\frac{1}{2}f_{yp} \tag{51}\]  
Thus, when considering Equations (49)-(51), the contribution of the plates becomes Equation (52):  
\[V_{p}=\frac{1}{3}f_{yp}h_{p}t_{p} \tag{52}\]  
For the design of the CFRP, the Guide for the Design and Construction of Externally Bonded FRP Systems for Reinforcing Concrete Structures (ACI 440) was used, where the contribution of the FRP shear stress (\(Vf\)) in Equation (53) is affected by the coefficient \(\psi_{f}\) with a value of 0.85 for U-shaped wraps.