The calculation of the longitudinal force (Fx) depends on the longitudinal slip condition of the tire.  Typically there are two distinct slip states:  an elastic deformation state (where the longitudinal slip is less that a critical slip value) and a pure sliding state (where the longitudinal slip exceed the critical slip value).

The critical longitudinal slip is calculated below:

In pure elastic deformation conditions:

In pure sliding conditions:

Where:

See Also:

Lateral Force

Vertical Force

Rolling Resistance Moment

Aligning Torque