The invention relates to the technical field of structural engineering, and discloses a prestressed support friction damping assembled steel structure system and a design method. The system is formedby stacking layered frame structures up and down, each layered frame structure includes steel columns, steel beams and floor plates, the steel columns between the upper and lower adjacent layered frame structures are disconnected at the connection positions of the floor plates on the lower layer, connection through self friction is achieved, and meanwhile, high-strength cables are connected between the adjacent layered frame structures; and the top ends of the steel columns are provided with upper flange plates so as to be fixedly connected to the steel beams, the bottom ends of the steel columns are provided with lower flange plates, the lower flange plates are in contact with the upper surfaces of the floor plates of the adjacent layered frame structure on the lower layer, the upper surfaces of the floor plates are provided with friction coating layers and limiting devices, and the limiting devices are steel plate strips arranged along the edges of the floor plates. By means of the system, the layers are connected through the high-strength cables and the mutual friction between the columns, construction is convenient and efficient to achieve, and the stress performance is good.