## 5 Design of FRP Strengthening  
Table 7 summarizes the design capacities and demand-capacity ratios for the FRP laminates and the FRP anchors used for shear strengthening of the end yielding of specimens S1A, S1B, and S5. Details on the design approach are provided in Sections 5.1 and 5.2 below. \(\mathrm{V_{e}}\) is the design shear and represents the maximum shear demand that an MRF beam would experience if it developed flexural hinges at the ends. \(\mathrm{V_{e}}\) was determined using the capacity  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Spec.**} & \multirow{2}{*}{**Load**} & \multicolumn{6}{c}{**First Yield**} & \multicolumn{6}{c}{**Effective Yield**} & \multicolumn{3}{c}{**Ultimate Strength**} \\ \cline{3-11}  & & \(\mathbf{M_{y,0}}\) & \(\mathrm{\phi_{y,0}}\) & \(\mathrm{\theta_{y,0}}\) & \(\mathrm{M_{y,eff}}\) & \(\mathrm{\phi_{y,eff}}\) & \(\mathrm{\theta_{y,eff}}\) & \(\mathrm{M_{u}}\) & \(\mathrm{\phi_{u}}\) & \(\mathrm{\phi_{u}}\) \\ \cline{3-11}  & & **[kN-m]** & **[rad/m]** & **[rad]** & **[kN-m]** & **[rad/m]** & **[rad]** & **[kN-m]** & **[rad/m]** & **[m/m]** \\ \hline \multirow{3}{*}{S1A} & down & 1914 & 0.003 & 0.006 & 2248 & 0.004 & 0.007 & 3017 & 0.15 & 0.14 \\ \cline{2-11}  & up & 1548 & 0.003 & 0.005 & 1844 & 0.004 & 0.006 & 2394 & 0.14 & 0.13 \\ \hline \multirow{3}{*}{S1B} & down & 1871 & 0.003 & 0.006 & 2251 & 0.004 & 0.007 & 3018 & 0.15 & 0.14 \\ \cline{2-11}  & up & 1512 & 0.003 & 0.005 & 1847 & 0.004 & 0.006 & 2395 & 0.14 & 0.13 \\ \hline \multirow{3}{*}{S1A} & down & 2337 & 0.004 & 0.006 & 2588 & 0.004 & 0.006 & 3419 & 0.12 & 0.11 \\ \cline{2-11}  & up & 1530 & 0.003 & 0.005 & 1989 & 0.004 & 0.006 & 2584 & 0.20 & 0.18 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Analytical flexural capacities.  
design approach in SS 18.4 of ACI 318-14; for the half-length cantilever test specimens, \(\rm V_{e}=M_{u}/L_{s}\), where \(\rm M_{u}\) is the beam's ultimate moment capacity evaluated for expected material properties (as reported in Table 5) and \(\rm L_{s}\) is the shear span. To note, due to the higher longitudinal reinforcement ratio at the top face and the presence of the slab, the beams have a higher flexural capacity in the downward-loading direction, which, for the cantilever specimens, translates into a higher shear demand when loading down. \(\rm V_{n}\) is the nominal shear capacity of the unstrengthened MRF beams at the end-yielding region for \(\rm V_{c}=0\) (Table 6). \(\rm V_{f,req}\) is the required FRP shear contribution (i.e., \(\rm V_{f,req}=(V_{e}-V_{n})/\phi_{v}/\psi_{f}\), with \(\rm\phi_{v}=1.0\)). \(\rm n_{req}\) is the required number of layers of FRP (i.e., \(\rm n_{req}=V_{f,req}/(2t_{w}t_{\rm\,}\varepsilon_{\rm\,E}t_{\rm\,E}t_{\rm \,F}/\varsigma_{\rm\,f})\), where \(\rm t_{f}\) is the nominal thickness of each layer of FRP laminate, \(\rm w_{f}\) is the width of FRP laminate, \(\rm E_{f}\) is the tensile modulus of the FRP material, \(\rm d_{fv}\) is the effective depth of the FRP shear reinforcement, and \(\rm s_{f}\) is the center-to-center spacing of the FRP strips). n is the layers provided. \(\rm F_{req}\) is the required tensile strength per anchor (i.e., \(\rm F_{req}=E_{f}\ \varepsilon_{\rm\,F}w_{f}\ n_{f}\), where \(\rm E_{f}\) is the tensile modulus of the FRP anchor and \(\rm t_{f}\) is the nominal thickness of each layer of FRP laminate). \(\rm F_{fr}\), \(\rm F_{cc}\), \(\rm F_{cb}\), and \(\rm F_{sd}\) are the anchor tensile capacities for the failure modes of anchor fracture, concrete cone, combined concrete cone and bond, and fan-to-FRP sheet debonding, respectively, per either [PERSON] (2018) or [PERSON] et al. (2019), for which the equations are provided in Appendix B. DCR is the demand capacity ratio for the anchors (i.e., \(\rm DCR=F_{req}/min\ (F_{fr},\rm F_{cc},F_{cb},\rm F_{sd})\)).