## 1 Introduction  
The most common composite constructions in buildings, bridges and other large span structures are steel-concrete composite beams due to their virtues, such as high strength, high stiffness and good ductility. A steel beam is connected to a concrete slab using shear connectors so that they act compositely. In consideration of composite action between the steel beam and the concrete slab, shear connectors are arranged at the interface between the steel beam and the concrete slab of composite beams, and these transfer the transverse shear stress from one component to the other, thus leading to composite action. For a composite beam with full interaction between the concrete slab and the steel beam, there is no relative slip at the interface. Full interaction is achieved when there is a sufficient quantity of shear connectors in the composite beam for additional shear connectors to have no impact on calculating the beam strength. Full interaction in composite beams is desirable, because the deflection and slip are lower in these cases than in cases with partial interaction under normal service load, even under extreme conditions such as fire [1]. However, it may not always be possible or essential to have full interaction. For instance, the number of shear connectors required to achieve full interaction may be so great that there are difficulties in arranging them in the beam (e.g., cost, labor), or the applied load on the beam could be safely supported with fewer shear connectors than would be required by full interaction. Furthermore, full interaction cannot be obtained in practice because of the deformability of shear connectors. As a result, partial interaction is observed in most composite beams either due to the limitation of the number of shear connectors that the top flange of steel beam can accommodate, or for an optimal design. Compared to a composite beam with full interaction, a composite beam with partial interaction has a greater deflection, since interface slip will reduce the composite action and stiffness of composite beams. It is therefore unsafe to overlook the impact of interface slip on the deflection of composite beams with partial interaction [2; 3; 4]; even for a full composite beam, the impact of interface slip may lead to a stiffness decrease of up to 17% for short span composite beams [5]. The decrease in stiffness could be owing to the fact that the shear connectors are deformable, allowing some interface slip or a reduction of the interaction between the concrete slab and the steel beam, even though their strength is adequate for full interaction [6]. Thus, it is increasingly important to decrease interface slip in order to improve the composite effects and reduce the deflection of composite beams. In view of the concept of full interaction, the most direct way to reduce the interface slip of composite beams with partial interaction is to increase the number of shear connectors at the interface between the two components of the composite beam. In this case, although the interface slip is decreased, the number of shear connectors may be so great that there are difficulties in arranging them in the composite beam. Because of the limitation of the amount of shear connectors that the top flange of steel beam can accommodate, to reduce the interface slip, some scholars have proposed optimal methods of shear connector distribution in recent years, as opposed to the uniform distribution of shear connectors, as is adopted in most of practical composite beams, since this type of distribution can reduce interface slip to some extent. [PERSON] [7] proposed a precise solution for the deflection of composite beams with a triangular distribution of connectors along the span. The results indicate that the number of shear connectors may be too great to be accommodated at the region of the supports, although the stiffness of the composite beam was increased.