## 4 Results and Discussion  
### Mesh Sensitivity  
A comparative analysis of mesh sizes was conducted to assess their ability to predict an RC beam's ultimate load and displacement accurately. Mesh sizes of 50 mm, 30 mm, 20 mm, and 10 mm were employed to examine mesh convergence at the load service, defined at approximately 0.6 times the ultimate load. The corresponding results are presented in Table 3. The results show that a mesh size of 20 mm had the highest level of accuracy compared to the experimental results, with errors of only 1.04% and 5.39% for ultimate load and corresponding displacement, respectively. Specifically, in the experiment, the ultimate load and displacement were recorded as 226.32 KN and 20.333 mm, whereas in the FEA results, the corresponding results for ultimate load and displacement were 256.293 KN and 19.238 mm, respectively.  
Figure 6: Equivalent length (\(h_{\mathrm{eq}}\)) based on element dimensions and crack direction.  
The crack patterns obtained using the 20 mm and 10 mm mesh sizes closely resembled the experimental results, as demonstrated in Figure 7. In addition, Figure 8 illustrates that the ultimate load of the RC beam began to converge from the mesh of 20 mm. Consequently, a mesh size of 20 mm was chosen to analyze the RC beam's nonlinear working and cracking behavior.  
In theory, employing a finer mesh size theoretically enhances accuracy. However, for beam S1-2, a mesh size of 20 mm demonstrates superior accuracy in line with experimental outcomes compared to a 10 mm mesh size. Several factors account for this phenomenon. Initially, the presence of initial defects or voids [29] in the concrete beam induces alterations in the load-displacement curve of the RC beam when utilizing a solid concrete beam. Secondly, deviations in the experimental process contribute to variations in the experimental results compared to the idealized experiment.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Method** & **Mesh Size** &
\begin{tabular}{c} **Total** \\ **Element** \\ \end{tabular} & **Time** & **Pu** & **Pu Error** & **u** & **u Error** \\ \hline \hline \multirow{4}{*}{FEA} & **(mm)** & & **(hours)** & **(KN)** & **(\%)** & **(mm)** & **(\%)** \\ \cline{2-8}  & 10 & 294,224 & 16.313 & 215.078 & 4.97\% & 18.774 & 7.67\% \\ \cline{2-8}  & 20 & 53,144 & 1.805 & 228.673 & 1.04\% & 19.238 & 5.39\% \\ \cline{2-8}  & 30 & 28,604 & 0.899 & 256.293 & 13.24\% & 19.383 & 4.67\% \\ \cline{2-8}  & 50 & 21,080 & 0.919167 & 263.046 & 16.23\% & 17.368 & 14.58\% \\ \hline EXP [28] & & & & 226.32 & & 20.333 & \\ \hline \hline \end{tabular} Note: Pu and u are the ultimate load and corresponding displacement at the midspan of the RC beam, respectively.  
\end{table}
Table 3: Comparing simulation and experiment results for ultimate load and displacement at different mesh sizes.  
Figure 7: Comparing crack patterns under service load conditions of (**a**) experimental result [16], (**b**) 50 \(\times\) 50 mm mesh size, (**c**) 30 \(\times\) 30 mm mesh size, (**d**) 20 \(\times\) 20 mm mesh size, and (**e**) 10 \(\times\) 10 mm mesh size.  
When performing the identical analysis, confirm that the 20 mm mesh size remains suitable for assessing mesh sensitivity in the case of beam S1-4.