## 4 Optimal Layout for Shear Connectors  
### Uniformly Distributed Shear Connectors  
In most of practical composite beams, the shear connectors are uniformly distributed along the beam, since this type of distribution reduces the interface slip to some extent, and it is also convenient to construct. To reveal the disadvantages of this approach, a composite beam with uniformly distributed shear connectors is investigated with regard to the distribution of the interface slip along the longitudinal direction of beam. Four different types of load (as shown in Figure 6) are applied to a simply supported composite beam with partial interaction, i.e., stiffness of shear connector \(k\) = 3.46 \(\times\) 10\({}^{4}\) N/mm and connector distance \(a\) = 200 mm. The shear force diagrams of the four load cases are shown in Figure 6, and the cross-section size and material parameters of the composite beam are the same as those given in Figure 5. Because it is symmetrical, only one-half of the composite beam is studied, and the results for the interface slip of the half-span beam under four different load cases are depicted in Figure 7.  
Figure 6: Shear force and moment diagram of four load cases.  
As can be observed in Figure 6, the distribution of the cross-sectional shear force along the beam becomes increasingly uneven from the first type of load to the fourth type of load. It is known from [PERSON] [25] that the distribution of shear flow \(v\) at the interface is the same as the distribution of cross-sectional shear force \(V\) along the beam in terms of \(v=VS_{z}/I_{z}\), as shown in Figure 8, in which \(\alpha=S_{z}/I_{z}\) is a constant value for a given cross-section. Meanwhile, according to Assumption 5 in Section 2.1, the distribution of interface slip is similar to the distribution of shear flow at the interface. Thus, the distribution of interface slip is similar to the distribution of cross-sectional shear force. For this reason, with the increasingly uneven distribution of cross-sectional shear force along the beam from the first type of load to the fourth type of load, a more uneven distribution of interface slip occurs, as can be seen in Figure 7. For example, the fourth type of load (in Figure 6) is an extreme case, in which the maximum cross-sectional shear force occurs at the two ends of the beam and the cross-sectional shear force is at its minimum at mid-span. Correspondingly, the distribution of interface slip is the most uneven, as shown in Figure 7 and Table 3. Essentially, the role of the shear connectors is to resist interface slip. Therefore, for cases of uneven distribution of shear force, the number of shear connectors in the mid-span region is too great to be utilized fully, while the number of shear connectors in the region of the supports is not sufficient to resist the interface slip. Nevertheless, the shear force varies along the beam for most load cases. Therefore, uniform distribution may not be the best way to distribute shear connectors.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Load Types** & **I** & **II** & **III** & **IV** \\ \hline Maximum slip \(s_{\text{max}}\) (mm) & 1.31 & 1.62 & 1.90 & 2.11 \\ \(s_{\text{max}}/s_{\text{max}}\) & 1.0 & 1.24 & 1.45 & 1.61 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of maximum slip at the end of the beam under four load cases.  
Figure 7: Distribution of interface slip along the half-span composite beam under four different types of load.