## 3 Numerical Study
### Results and Discussion of Finite Element Modelling
#### 3.4.1 Verification of the Numerical Model by Using Experimental Results
Extensive cracking was observed for the compression-shear failure and flexural failure of the BCSP-6 and BCSP-10 beams. The yielding region expanded to the beam when the loading increased. The stress concentration increased at the two loading points during later loads, after which plastic strains developed in the entire section of the beam. The von Mises stresses, which represent the inelastic behavior of the beam sections, are shown in Figure 13. It was observed that the maximum stress was concentrated at the bolts. For all the BCSP beams, the maximum von Mises stress was 756.6 MPa, located at the bolts under the two points of loading. This maximum principal stress of the BCSP beams could have led to the ductile failure of the BCSP beams via the spalling of concrete under the loading points. The minimum principal stress contour plots of the concrete under the finite element model's stress states during peak load. A comparison was made between the FEA predicted and experimentally measured applied load-strain curves. Figure 6 includes comparisons of the experimentally measured applied load, including the average concrete strain, the longitudinal strain at tension, and the strain at the top of the steel plate. As shown, the analytical prediction compares very well with the experimental results. These comparisons are excellent at the global level (Figure 13), where the yielding of the bolts at the two load points, the steel plate, and the steel bars occur, respectively. The energy absorption (EA) of the BCSP-6, BCSP-10, and BCSP-15 beams increased by about 300%, 327%, and 462% as compared to the CB, respectively. This result indicates that the BCSP beams developed increased stiffness and energy absorption with the increased thicknesses of their steel plates. The values for the test results and FE modelling of the EA for all the models are shown in Table 12 and Figure 14. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Specimens** & **P\({}_{\text{peak,Exp}}\)** & **P\({}_{\text{peak,FE}}\)** & **P\({}_{\text{peak, Exp}}\)/P\({}_{\text{peak, FE}}\)** & **\(\Delta_{\text{Exp}}\) (mm)** & **\(\Delta_{\text{Exp}}\)/\(\Delta_{\text{FE}}\)** \\
**Designation** & **(kN)** & **(kN)** & **(\%)** & **(\%)** & **(mm)** & **(mm)** \\ \hline
**CB** & 375.5 & 420 & 0.89 & 10.98 & 11.5 & 1.09 \\ \hline
**BCSP-6** & 652.3 & 759.6 & 0.86 & 11.7 & 17.34 & 0.79 \\ \hline
**BCSP-10** & 837.6 & 831.2 & 1.01 & 12.5 & 18.2 & 0.69 \\ \hline
**BCSP-15** & 935.3 & 1000.4 & 0.93 & 15.5 & 13.07 & 1.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Comparison between the results derived from the experiment and FEM. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Specimens** & **EA\({}_{\text{EXP}}\)** & **EA\({}_{\text{FE}}\)** & **EA\({}_{\text{Exp}}\)/EA\({}_{\text{FE}}\)** & **EA\({}_{\text{EXP}}\) (\%)** & **EA\({}_{\text{FE}}\) (\%)** & **EA\({}_{\text{Exp}}\)/EA\({}_{\text{FE}}\)** \\
**Designation** & **(kN-m)** & **(kN-m)** & **(\%)** & **Increase** & **Increase** & **Increase** \\ \hline
**CB** & 4.03 & 4.3 & 0.94 & - & - & - \\ \hline
**BCSP-6** & 13.9 & 17.2 & 0.8 & 247.5 & 300 & 0.83 \\ \hline
**BCSP-10** & 16.7 & 18.4 & 0.91 & 317.5 & 327 & 0.97 \\ \hline
**BCSP-15** & 22.5 & 24.2 & 0.93 & 458.3 & 462 & 0.99 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Energy absorption: The comparison between the experiment and FEM for all of the over-reinforced concrete specimens. EA: Energy absorption. Figure 12: Comparison of the load-displacement curves derived from the experimental and numerical studies. Figure 13: The von Mises stresses for all the beam specimens.