## 3 Results
### Strength Prediction/Theoretical Results  
Chile has no design standards for calculating the flexural and deflection strength of beams with RA. Therefore, it was necessary to investigate whether beams with RA could be designed while applying the current design codes. For this purpose, a comparative study was conducted between the experimental results of this study and the theoretical results obtained from Eurocode 2 [44] and ACI 318-19 [45]. Tables 5 and 6 show the equations used to determine the theoretical cracking moments, and ultimate moments, capacity and deflection of the beams, respectively, while Table 7 shows the comparison with the experimental results. The end load (\(P_{u}\)) was recorded from the load-strain curve at the end failure stage for all beams (237.66 kN, 202.80 kN, and 193.95 kN for beams containing 0%, 50%, and 100% treated recycled aggregates, respectively), then the end moments were calculated using these loads. The critical load (\(P_{\sigma}\)) was determined from the experimental critical moment calculated from characteristics obtained from previous tests. Details of the calculations are provided in Appendix A. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Beam**} & \multicolumn{2}{c}{**Initial Cracking**} & \multicolumn{2}{c}{**Yielding**} & \multicolumn{2}{c}{**Ultimate Strength**} & \multicolumn{2}{c}{**Failure**} \\ \cline{2-9}  & **Load (kN)** & \begin{tabular}{c} **Deformation** \\ **(mm)** \\ \end{tabular} & **Load (kN)** & \begin{tabular}{c} **Deformation** \\ **(mm)** \\ \end{tabular} & **Load (kN)** & \
\begin{tabular}{c} **Deformation** \\ **(mm)** \\ \end{tabular} & **Load (kN)** &
\begin{tabular}{c} **Deformation** \\ **(mm)** \\ \end{tabular} \\ \hline N100 & 63.40 & 1.86 & 174.00 & 8.41 & 205.40 & 37.44 & 168.00 & 38.05 \\ CR50 & 69.41 & 2.34 & 165.30 & 8.92 & 198.61 & 38.27 & 163.36 & 38.49 \\ CR100 & 65.22 & 2.30 & 169.65 & 8.51 & 194.34 & 45.16 & 179.12 & 45.79 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Summary of results for the tested beams. Figure 5: Load–deflection curves for the tested beams. \begin{table}
\begin{tabular}{c l} \hline \hline  & **Expressions Used** \\ \hline Ultimate moment (N \(\cdot\) mm) & \(M_{u}=\alpha\cdot f_{c}^{\prime}\cdot a\cdot b\cdot(d-\frac{a}{2})+A_{s}^{\prime} \cdot f\s^{\prime}\cdot(d-d^{\prime})\) \\  & Where. \\  & \(d=\) distance from the compression end to the centroid of the tensile reinforcement (mm) \\  & \(d^{\prime}=\) distance from the compression end to the centroid of the compression reinforcement (mm) \\  & \(A_{s}^{\prime}=\) compression steel reinforcement (mm\({}^{2}\)) \\  & \(A_{s}^{\prime}=\) tensile steel reinforcement (mm\({}^{2}\)) \\  & \(f_{c}^{\prime}=\) compressive strength in a cylindrical specimen at 28 days (MPa) \\  & \(f_{c}^{\prime}=\) yield strength of steel (MPa) \\  & \(a=\) depth of the compressive block (mm) \\  & \(a=\frac{(A_{s}-d^{\prime})\cdot f_{c}}{a\cdot f_{c}^{\prime}}\) \\  & \(\alpha=0.85\) [ACI 318-19] \\  & \(\alpha=\left\{\begin{array}{ll}1\\ 1\\ 1-\frac{\ell-50}{200}\end{array}\right.\), where \(f_{c}^{\prime}\leq 50\) [Eurocode 2] \\ \hline Load capacity \((kN)\) & \(P=2\cdot(\frac{M}{2})\) \\  & Where: \\  & \(l=\) load to support distance (mm) \\  & \(M=\) critical or ultimate moment, if critical or ultimate load is desired, respectively (N \(\cdot\) mm) \\ \hline Deflection \((mm)\) & \(\delta=\frac{p_{d}}{\Delta k\cdot l_{eff}}\cdot(3L^{2}-4a^{2})=\) deflection (mm) \\  & Where: \\  & \(a=\) shear span (mm) \\  & \(l_{eff}=\) effective moment of inertia (mm\({}^{4}\)) \\  & \(L=\) effective length (mm) \\  & \(E=\) modulus of elasticity of concrete (MPa) \\ \hline \hline \end{tabular}  
\end{table} 
Table 6: List of expressions used to predict the ultimate moment, capacity, and deflection of the beams.