Tractive resistance explained

Tractive resistance is the total resistance that opposes the motion of a vehicle on the road. The tractive effort (the force at the driving wheels) must be at least equal to it for the vehicle to move or maintain its speed.

Components

1. Air resistance: $R_a = \tfrac12\rho C_d A v^2$, rises with the square of speed; dominant at high speed.
2. Rolling resistance: $R_r = f_r W$ due to tyre deformation, road surface, and bearing friction; $f_r$ depends on the surface (0.01-0.02 on good paved road, 0.04-0.1 on gravel/earth) and the tyre pressure.
3. Gradient resistance: $R_g = W\sin\theta \approx WG$; positive on upgrades and negative (helps) on downgrades.
4. Curve resistance: the extra resistance on a horizontal curve due to lateral slip and off-tracking; large on sharp curves; compensated for in gradient by grade compensation.
5. Inertia (acceleration) resistance: $R_i = \gamma\dfrac{W}{g}a$, which is the force required to accelerate the vehicle and its rotating parts.
6. Transmission losses in the engine, gear box and axle (about 10-20% of the engine power).

$$T_{required} = R_a + R_r + R_g + R_c + R_i$$

Limit by adhesion

The tractive effort delivered can not be more than the friction between the driving tyres and the road: $T \le \mu\,W_d$ ($W_d$ = weight on driving wheels). Otherwise the wheels spin.

Uses in highway design

Tractive resistance fixes the engine power needed, the maximum gradient (ruling gradient), the grade compensation on curves, the length of the climbing lane and the speed of heavy vehicles on grades.