This research paper presents a finite element approach for the analysis of composite (two-layer) beams, with a steel bottom layer and a concrete top layer. These layers function as a planar composite beam interconnected by connectors. Issues arise at the interfaces of different materials (steel and concrete), leading to slip and uplift. The connector element consists of two-directional coupled springs: a horizontal spring parallel to the contact surface captures slip, while a vertical spring orthogonal to the contact surface captures uplift. The vertical spring activates only if there is separation between the layers; otherwise, contact forces exist. The analysis employs a geometrically nonlinear finite element formulation for planar composite structures, considering non-penetration conditions. The co-rotational method decomposes the element's motion into rigid body motion and small deformations. When material behavior becomes nonlinear,  the finite element method based on the displacement-based formulation is commonly used. To prevent penetration between the layers, contact resolution methods such as augmented Lagrangian methods with Uzawa updating schemes are employed. Finite element models are used to conduct a parametric study to investigate the performance of the displacement-based formulation and the influence of using the interaction effect of shear and tensile behavior of the connector (coupled connector model) with the non-penetration condition. The performance of the proposed formulation is assessed through numerous numerical applications