This paper proposes a composite frame element for the simulation of the inelastic response of structural members made up of two or more components with different materials, such as reinforced concrete (RC), steel-concrete composite members, prestressed members, and members with fiber-reinforced polymer (FRP) reinforcement. The element accounts for the relative slip at the interface between the components. Nonlinear geometry effects are accounted for through the corotational formulation, which permits the response simulation of composite frame elements under large displacements. The element formulation enhances the standard Hu-Washizu variational principle with fields describing the bond-slip behavior between components. Three alternatives for the mixed formulation of the element are derived in this paper, which focuses on theory and implementation: mixed-displacement, mixed-force, and mixed-mixed. The paper presents the benefits and shortcomings of the formulation alternatives for modeling the pullout failure of reinforcing bars and the fixed-end rotation of RC members and discusses the numerical ramifications of each alternative. A companion paper discusses the convergence performance of the different mixed formulations and validates the proposed element through correlation studies with available experimental results.