This paper presents an analytical approach for a three dimensional tire model used for the steady-state simulations of vehicles. The functional relationship of slip ratios of the SAE definition and new definition suitable for vehicle dynamics analysis is first studied. New formula for friction is used and a contact pressure change is considered during driving/braking. The compliances of carcass, braker, and tread are employed for longitudinal and lateral elastic deformation. Rolling resistance force, plysteer, and conicity are also included in the explicit formulations of longitudinal force, lateral force, and aligning torque due to comprehensive slips.