## 3 Finite-Element Modeling (FE) and Validating
Table 1 presents the variations in shank diameters. In the validated model, the shank diameter was set to 22 mm. Figures 11 and 12 illustrate the effect of changes in stud diameter on the load-displacement response and ultimate load of the steel-concrete composite connection. Alterations in stud diameter have a substantial impact on the overall behavior of the connection. These findings are consistent with prior research and align with the guidelines specified in the standard, which includes calculations for stud failure. Table 2 presents the variations in the height of the stud shank, with the validated model utilizing a stud height of 115 mm. Figure 13 depicts the results obtained from the numerical models incorporating these variations. While significant differences were anticipated due to the \(\alpha\) correction factor specified in the standard, the results do not reveal substantial deviations when the stud height is altered. In the validated model, a C40/50 concrete strength was employed, in accordance with the experimental test, which was designed using the same materials and geometry as those in the bridge, and constructed by the bridge contractor in a manner identical to that of the bridge itself. \begin{table}
\begin{tabular}{c c} \hline
**Variations** & **Diameter [mm]** \\ \hline D1 & 16 \\ D2 & 18 \\ D3 & 20 \\ D4 & 25 \\ \hline \end{tabular}
\end{table} 
Table 1: Shank diameter variations. Figure 11: The effect of stud diameter on the load–displacement behavior of steel–concrete connections. Figure 12: The effect of the shank diameter of the stud on the ultimate loads. Previous studies [3,4] have demonstrated that, with increasing concrete strength, the influence of stud length diminishes, particularly when using UHPC. In such cases, stud failure becomes the dominant mode of failure. Conversely, for concrete grades typically used in bridge construction, where the concrete zone experiences considerably higher levels of damage (as compared to UHPC), the selection of stud length exerts a markedly reduced effect. In the research, several variations were considered, including the geometry of the stud head. In the validated model, the head diameter was set to 31 mm, with the variations presented in Table 3. Figure 14 illustrates the results, where a negligible impact on the behavior of the steel-concrete connection was observed. \begin{table}
\begin{tabular}{c c c} \hline
**Variations** & \multicolumn{2}{c}{**Height [mm]**} \\ \hline H1 & 3d & 66 \\ H2 & 4d & 88 \\ H3 & 6d & 132 \\ H4 & 7d & 154 \\ H5 & 8d & 176 \\ \hline \end{tabular}
\end{table} 
Table 2: Shank height variations. Figure 13: The effect of stud height on the load–displacement behavior of steel–concrete connections. Several variations were examined in the investigation, as the height of the head of the stud. Table 4 summarizes these variations, with the stud height set to 31 mm in the validated model. The results, shown in Figure 15, indicate that the effects on the behavior of the steel-concrete composite structure were minimal. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variations** & \multicolumn{2}{c}{**Head Height Variations [mm]**} \\ \hline HH1 & 0.5d & 11 \\ HH2 & 0.6d & 13.2 \\ HH3 & 0.7d & 15.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Head height variations. Figure 14: The effect of the head diameter on the load–displacement behavior of steel–concrete connection. Figure 15: The effect of the head height on the load–displacement behavior of steel–concrete connection. In this research, longitudinal spacing was examined. Several studies, as introduced earlier, have addressed this topic, with the following being particularly noteworthy. [PERSON] et al. [29] conducted an analysis of the shear strength of grouped-stud connectors, highlighting the effect of longitudinal spacing. Their analysis, which considered varying spacings and concrete strengths, showed that the maximum load occurred when stud spacing was 13d. Similarly, [PERSON] et al.