The mathematical description of tires does not usually consider frictional interaction between the tire and the surface over which it moves. An approach for doing so is presented in the present paper for a homogeneous tire without grooves and which is loaded axially. The tire footprint area is divided into smaller areas that either stick or slide at the interface between the tire and its supporting structure when frictional forces are applied. Discretization of the contact problem into finite elements leads to a nonlinear system of equations for the nodal displacements. The algorithm applied to compute the normal and tangential forces in the contact zone is described. The Newton‐Raphson method and the modern quasi‐Newton methods BFGS (Broyden, Fletcher, Goldfarb, Shanno) and DFP (Davidon, Fletcher, Powell) proved to be the most effective. Normal and tangential forces in the footprint area were computed by the Lagrange multiplier method or directly by calculation of the stress tensor in the contact zone.