In this study, a numerical model is presented for the nonlinear vibrational analysis in the symmetrical plane of the rectangular offshore floating structures moored by cables. The upper end of each mooring cable is connected to the floating structure and the other end is fixed to the sea bed. The nonlinear equations of motions of the mooring cables are derived by using nonlinear cable elements that are formulated based on the extended Hamilton principle. The floating platform is modeled as a rigid body with three degrees of freedom. The forces applied on the floating structure and cables are analyzed, formulated and expressed in details. The connection conditions between the floating structure and mooring cables are introduced to formulate the equations of motions of the system as a whole. The vibrations of the floating structure under horizontal sinusoidal excitation are analyzed numerically. The influence of different sag-to-span ratios or inclined angles of the mooring cables, and that of different current velocities on the displacements of the floating structure and maximum tensile force in the cables are studied. The displacement amplitudes of the moored floating structure and maximum cable tensile force under different current velocities are also studied for different excitation frequencies.