Case III.

Case III:  and

Similar to Case I and II, slop ratios due to the critical values of the inclination angle and the slip ratio are obtained as:

                                           (1)

                (2)

The nondimensional slip ratio, is expressed as:

                                           (3)

where,

For the longitudinal force, the lateral force, and the self-aligning torque, two sub-cases should also be considered similar to Cases I and II. A slip ratio due to the critical value of the slip angle is not needed here since the required condition for Case III, , replaces the critical condition of the slip angle.

1.  Elastic Deformation State:  and

In the elastic deformation state,  and  may be written:

                               (4)

        (5)

                                                        (6)

         (7)

                                                  (8)

2.  Complete Sliding State:  or

                                                                   (9)

                                                                   (10)

                                                               (11)

                                                             (12)

                                                                 (13)

Respectively, the Longitudinal force , the lateral force , and three components of the self-aligning torques , , and  always have positive values, but they can be transformed to have positive or negative values depending on the slip ratio , the slip angle , and the inclination angle  in the Contact coordinate system.