## 5 Conclusions  
The study involved developing 3D finite element models to conduct a nonlinear analysis of RC beams under four-point bending. The models were validated with experimental results through various variables, including mesh sensitivity analysis, energy balance, load characteristics, and crack patterns. A parametric study was then conducted to examine the effects of diameter, spacing, and number of rebars on the beam's load-carrying capacity and crack propagation. The following conclusions are drawn from this study:  
* The suggested FE models, integrating the CDP model, consider mesh size and a crack width calculation method based on equivalent length and crack direction. This comprehensive approach enables a precise simulation of both the load-displacement behavior and the dimensions of crack patterns in reinforced concrete (RC) beams. Notably, the maximum error observed is only 10.08% at the cracking load and 18.39% at the height of crack patterns when comparing the FEA with EXP results for beam S1-4. * The 20 mm mesh size is appropriate for analyzing the RC beams's load characteristics and cracking behavior. * An investigation of total energy and the ratio of kinetic energy to internal energy shows that the ABAQUS/explicit solver without mass scaling, utilizing a time step of one second, is an effective method for simulating RC beams under quasi-static four-point bending. * The parametric study conducted in this research highlighted the importance of carefully selecting an optimal combination of rebar diameter, number, and spacing while \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Value** & **B12 (7d12)** & **REF (5d14)** & **B16 (4d16)** & **B18 (3d18)** \\ \hline Mean crack width (\(\mu\)m) & 80.85 & 86.15 & 87.96 & 90.45 \\ Minimum crack width (\(\mu\)m) & 24 & 30 & 18 & 24 \\ Maximum crack width (\(\mu\)m) & 160 & 160 & 144 & 176 \\ Mean crack height (mm) & 85.71 & 133.50 & 107.14 & 106.43 \\ Mean crack distance (mm) & 65.00 & 88.89 & 76.92 & 71.43 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Parametric beam crack behavior under varying reinforcement parameters at service load. Figure 14: Comparison of the result of parametric beams at service load: (**a**) crack patterns; (**b**) mean crack width, height, distance. keeping the cross-sectional area of the rebars constant to achieve the desired load-carrying capacity in a reinforced concrete beam. This optimization process should also consider critical factors such as durability and cost. * The parametric study further indicated that reducing the rebar diameter resulted in a corresponding decrease in spacing, leading to a smaller mean crack width observed in the RC beam. These results underscore the importance of a comprehensive approach to optimize the design of reinforced concrete beams for improved performance and longevity. This study provides deep insights into predicting the nonlinear response of RC beams concerning their load characteristics and crack development. Based on the results, effective methods can be proposed for repairing and strengthening beams, enhancing their load-carrying capacity, and limiting crack propagation. Although the CDP model performs effectively for isotropic materials, additional research is needed to account for voids and defects in concrete. This will contribute to improving the accuracy of both the simulation model and experimental outcomes. **Author Contributions:** Supervision, S.-K.K. ; methodology, X.-B.L. ; software, X.-B.L. ; validation, W.K. and X.-B.L. ; writing--review and editing, S.-K.K., W.K. and X.-B.L. ; funding acquisition, S.-K.K. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by a National Research Foundation of Korea (NRF) grant funded by the Korean government (MIST) (No. 2021R1C1C1013130). **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** The data presented in this study are available in the article.