## 4 Numerical Analysis
### Numerical Results and Discussion  
The load versus mid-span displacement curves for the experimental and numerical specimens were plotted in Figure 10. The experimental result was plotted as a solid line whereas the corresponding numerical result was plotted as a dashed line. In general, the numerical models assigned with TMS plate thicknesses of 2.0 mm and 2.5 mm generated similar load-displacement curves. However, the numerical model with the 1.5 mm thick TMS plate did not. Higher initial stiffness was yielded in the 1.5 mm thick TMS plate of numerical series. This is due to the assumption that interaction between the steel bar and the surrounding concrete was perfectly bonded in the numerical models resulting in greater overall stiffness [5; 30; 31]. This phenomenon was also observed in numerical studies by [PERSON] et al. [32] and [PERSON] et al. [5]. Furthermore, assumptions made on the homogeneity of concrete material [31], the strain softening of concrete beyond the cracking strain [5; 30], and time dependent effects [33] during numerical modelling can potentially increase the difference between experimental and numerical results. The numerical models for the FP series also showed higher initial stiffness when compared with the experimental results. This indicates that no delamination of the side concrete took place at the early stage of loading. A similar phenomenon was observed in [PERSON]'s [33] work, as the numerical models were unable to simulate fine hairline crack propagation as observed in the experimental specimens. Nonetheless, the majority of developed numerical models were able to simulate load versus mid-span displacement profiles that were similar to the experimental specimens. Table 7 compares the numerical ultimate load with the experimental results for each specimen. Overall, the developed numerical models show \(\pm 10\%\) of deviation in the ultimate load as compared with the experimental specimens. This shows a reasonable level of accuracy [5] which is important for any future parametric study and further optimization work on the proposed design configuration based on the developed model. Hence, the developed model has the potential for future evaluation on the performance of the proposed beam under cyclic loadings and fixed end support conditions. In addition, it can also be used to validate the outcomes of any new design formula developed in future for proposed deep beams. \begin{table}
\begin{tabular}{c c c c c} \hline
**Steel Type** & **Density** & **Poisson’s** & **Modulus of Elasticity,** & **Yield Stress** & **Ultimate Stress** \\
**(kg/m\({}^{3}\))** & **Ratio, \(\
u\)** & **_E_ (kN/mm\({}^{2}\))** & **(N/mm\({}^{2}\))** & **(N/mm\({}^{2}\))** \\ \hline High yield steel bar - 8 mm diameter & & & 520 & 755 \\ \hline High yield steel bar - 10 mm diameter & 7850 & 0.3 & 200,000 & 622 & 776 \\ \hline High yield steel bar - 12 mm diameter & & & 636 & 808 \\ \hline Mild steel plate & & 28,556 & 284 & 350 \\ \hline \end{tabular}
\end{table} 
Table 6: Steel reinforcement properties. Figure 10: Load-displacement curve comparison. Figure 11 illustrates the comparison of the cracking developed in the experimental and numerical work for each specimen. Figure 12 shows the three general types of numerical crack pattern superimposed on the experimental results. The numerical concrete tension damage view was selected in this comparison as it represents the tension cracks formed in concrete. It is observed that all the numerical models showed a good correlation in progressive and ultimate tension crack patterns with the experimental specimens.