## 4 Optimal Layout for Shear Connectors
### Stepwise Uniform Distribution for Shear Connectors  
As can be seen from the above section, the uniform distribution of shear connectors has its own disadvantages. The stiffness of the shear connectors in the mid-span region is not being fully utilized to reduce interface slip. However, interface slip has a significant influence on deflection, as described in Section 3, so it is increasingly important to reduce interface slip in order to improve the composite action and reduce the deflection of composite beams. It was made known by [PERSON] and [PERSON] [8] that the distribution of interface slip is directly related to the distribution of shear connector density. Although the beam can effectively reduce the interface slip under a triangular distribution of shear connector density [8], such a distribution of shear connectors is inconvenient for engineering applications, since each connector has a different stiffness along the half-span. As can be observed in Table 1, the interface slip along the beam is equal to zero when the two components of the composite beam achieve full interaction. Therefore, the most direct and effective way to reduce the interface slip of composite beams with partial interaction is to increase the stiffness of shear connectors in the regions with the largest interface slip. For the purpose of achieving this goal, there are three different ways of increasing the stiffness: (1) an increase in the number of shear connectors; (2) an increase in the stiffness value of every shear connector; and (3) adopting a stepwise uniform distribution for shear connectors without increasing the total number of shear connectors. The first two methods have their own deficiencies, not only with regard to the stiffness of shear connectors being incompletely utilized at the mid-span, but also with respect to the increase of cost. Furthermore, it is increasingly difficult to accommodate the shear connectors in the beam when the total number of shear connectors increases. Therefore, this paper will adopt the third method for the developed composite beam element, which is to rearrange the shear connectors to achieve an optimal layout without increasing the total number of shear connectors. The optimal layout method adopts a stepwise uniform distribution of shear connectors--namely, 2-stepped rectangles, 3-stepped rectangles and 5-stepped rectangles (as shown in Figure 9)--to approximate the triangular distribution of shear connector density along the beam [6]. For the purpose of effectively utilizing the shear connectors and reducing the interface slip, a greater number of shear connectors are distributed in the region of the beam ends, and a lower number of shear connectors are distributed in the mid-span region. Meanwhile, the total number of shear connectors is assumed to remain constant before and after redistribution. The efficiency of the proposed connector layout method is evaluated by comparing the interface slip before and after the redistribution. The comparison is performed on a simply supported composite beam with partial interaction, i.e., with the stiffness value of a shear connector \(k\) = 3.46 \(\times\) 10\({}^{4}\) N/mm. The loads applied on the composite beam are shown in Figure 6, and the cross-sectional size and corresponding elastic material parameters are the same as those shown in Figure 5. Figure 9 shows the redistribution of the shear connectors along the composite beam, which is divided into 20 elements based on a distribution using different stepped rectangles. The slip distribution curves obtained by using different types of distribution for the shear connectors are compared in Figure 10a--d. The corresponding slips of the composite beam end are shown in Table 4.  
Figure 8: Distribution of the cross-sectional shear force and shear flow at the interface along the beam. It can be observed from Figure 9(a)-d and Table 4 that, with an increase in the number of stepped rectangles, the interface slip of the beam end decreases.