Prediction of structural dynamics of the Electric Arc Furnace (EAF) is rather difficult, because of a number of phenomena occurring during the scrap melting process. Three large electrodes, corresponding to each phase of a AC circuit, are lowered by the main mast towards the scrap to activate the melting process, induced by the electric arc. Electric current fed to each electrode produces a strong magnetic field and applies an electromechanical force on the other electrodes. Arc voltage looks irregular upon time, even because of the scrap motion within the vessel and temperature growth. The vertical position of the mast is controlled by an hydraulic actuator. Nevertheless, a heavy vibration of the structures affects the regularity of the melting process. A fully coupled model of the whole system is herein proposed, through a multi-physics approach. A first analytical approach, describing the electric circuit of the whole system, is implemented into a Multi Body Dynamics (MBD) model of the EAF, while a reduced Finite Element Method (FEM) model of the flexible structures is used for a preliminary optimization of the design parameters. Electromechanical forces due to the mutual induction among the electrodes are computed and the dynamic response of the system is investigated. Proposed model allows a first refinement of the EAF design, although a complete experimental validation on the real machine has to be performed, in spite of problems due the extremely difficult accessibility of structures during the melting process.