## 3 Numerical Study
### Results and Discussion of Finite Element Modelling  
#### 3.4.1 Verification of the Numerical Model by Using Experimental Results  
The experimental and numerical load-deflection curves for all the beam specimens are shown in Figure 12. As a result of other imperfections, such as gaps between the components of experimental specimens that cannot be fully simulated by the numerical models, an ascending segment of the load-displacement curves was observed in the numerical results. The comparison results show that the FE load-deflection curves are generally expected to be stiffer than in the experimental plots. The stiffness of the FE models was even higher than that of the experimental beams after first cracking. Several factors may have contributed to this. The micro-cracks that exist in the concrete are the most important factor in this experimental design. These micro-cracks could be produced by drying shrinkage in the concrete and/or as a result of the handling of the beams, since the FE models do not include micro-cracks, which could reduce the stiffness of all experimental beams. Moreover, the FE load-displacement curves almost follow the same design trend as the experimental results. The load-deflection relationship of the FE models is in good agreement with the experimental results, prior to the failure load for all specimens. The mean load ratios of the experimental to FE peak loads were 0.87%, 0.86%, 1.01% and 0.98%. Also, the average deflection ratios of the experimental to FE deflection at peak load were 1.09%, 0.79%, 0.69% and 1.2% for the CB, BCSP-6, BCSP-10, and BCSP-15 beam specimens, respectively, as shown in Table 10.  
The design recommendations of the ACI-code and CSA [28; 36], which depend on stress-strain diagrams, were used to calculate the nominal load capacities (P\({}_{\text{n}}\)). The ratios of the normal load (P\({}_{\text{n}}\)) to the spalling load of the experimental (P\({}_{\text{Spall,EXP}}\)) beam specimens were 0.99%, 0.98%, and 1%. Those equal to the ratio of the normal load to the spalling load of the FE (P\({}_{\text{n}}\)/P\({}_{\text{Spall,FE}}\)) were 0.89%, 1.03%, and 1.06%. Those equal to the ratio of the spalling load of the experimental to the spalling load  
Figure 11: Comparison between the mesh sizes for all FE models. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mesh Size (mm)** & **P\({}_{\text{peak}}\) (kN)** & **Max Stress (S), Mises (MPa)** & **FE Model (Compressive Damage)** \\ \hline
10 & 614 & 741.4 & \\ \hline
25 & 813.2 & 748.6 & \\ \hline
50 & 985.01 & 747.0 & \\ \hline
75 & 1079.9 & 742.5 & \\ \hline
100 & 1082 & 714.6 & \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Comparative (compressive damage of the BCSP-10 beam) mesh size convergence study. of the FE (P\({}_{\text{spall,EXP}}\)/P\({}_{\text{spall,FE}}\)) were 1.11%, 0.95%, and 0.94% for the CB, BCSP-6, and BCSP-10 beam specimens, respectively, while the BCSP-15 beam specimen experienced flexural failure, as shown in Table 11. The finite element analysis gave accurate values of P\({}_{\text{spall}}\) for all of the beams. Both the experimental and predicted results were in very good agreement. Also, the load-deflection curve from the experiment was in agreement with the FE analysis. The results indicate that the constitutive models of this study can be used to accurately predict the behavior of over-reinforced concrete with steel plates without buckling failure. Increasing the area of the steel plate induced a sharp increase in the ductility of the BCSP beams. Small displacements caused the failure of the beams when compared with the CB. The stress response of the over-RC beams for four different analyses is demonstrated in Figure 13. From this figure, it can be concluded that, with reference to the stress response, the difference of the stress for the four models can be presented. The steel plate of the BCSP beam yielded first. In the finite element analysis, yielding of the tie and steel bars in tension was observed without failure.