The FASTSIM algorithm is widely used in multi-body simulation (MBS) software packages for the evaluation of the tangential wheel-rail contact forces in a steady state. As the algorithm is restricted to Hertzian contact patches, a strip-based local approach is proposed to extend FASTSIM to non-elliptical contact cases. The paper presents this tangential contact approach in detail, which was briefly introduced by Ayasse & Chollet along with the semi-Hertzian method. The contact stresses and their directions are compared with the reference results from the program CONTACT. Different settings for the traction bound are explored to determine their influence on the contact stresses, creep forces, and the limits of the saturation zone in the case of wheel-rail contact. A design of experiments is constructed for a non-Hertzian contact case, with different combinations of the longitudinal, lateral, and spin creepages. The absolute error in the normalised creep forces is used as the quantity of interest and found to be consistent with results in the literature for Hertzian contact cases using FASTSIM.