Library / Terramechanics and off-road mobility / TER-04
Terramechanics and off-road mobility
Traction limit on a slope
Largest thrust the ground can support on a slope for an all-wheel-drive vehicle with uniform grip: the friction coefficient times the normal load, which falls as the slope steepens.
- Ref
- TER-04
- Rev
- v1
- Field
- Terramechanics and off-road mobility
- Result
- Hmax [kN]
μ is the friction coefficient between the running gear and the ground, m the vehicle mass and θ the slope angle. For an all-wheel-drive vehicle with uniform adhesion and no other normal loads, the driven normal load is m g cos θ, so the largest thrust the ground can support is μ m g cos θ. Compare it with the thrust needed to climb, m g (sin θ + Crr cos θ). Soft soil needs a shear model and a motion-resistance calculation, because the thrust is then limited by the soil strength and the slip.
Largest thrust the ground can support on a slope for an all-wheel-drive vehicle with uniform grip: the friction coefficient times the normal load, which falls as the slope steepens.
Calculator
- Weight component along the slope, m g sin θ17.03 kN
- Thrust left for climbing and acceleration, Hmax − m g sin θ40.93 kN
- Steepest slope the friction can hold, tan⁻¹ μ30.96 °
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- On a 10° slope the ground can supply 57.97 kN to this 10 t vehicle, of which 17.03 kN is needed just to hold it against gravity, leaving 40.93 kN for climbing and acceleration.
- The slope where the friction can no longer hold the vehicle is tan⁻¹ μ = 30.96° for μ = 0.6. On wet clay (μ = 0.3) it is only 16.7°, and at that angle there is no thrust left to climb.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (kN) | Local step | Change in result |
|---|---|---|---|---|---|---|
| m Vehicle mass | 2,000 to 60,000 kg | 11.59 | 347.8 | 336.2 | +10% | +10 % |
| μ Friction coefficient between tyre or track and ground | 0.1 to 1.2 | 9.661 | 115.9 | 106.3 | +10% | +10 % |
| θ Slope angle | 0 to 40 ° | 45.09 | 58.86 | 13.77 | +10% | −0.323 % |
Sweep of Slope angle
| Slope angle (°) | Maximum thrust (kN) |
|---|---|
| 0 | 58.86 |
| 5 | 58.64 |
| 10 | 57.97 |
| 15 | 56.85 |
| 20 | 55.31 |
| 25 | 53.35 |
| 30 | 50.97 |
| 35 | 48.22 |
| 40 | 45.09 |
Sweep one variable
Variables and how each one contributes
Maximum thrust
What the equation gives you. Everything else feeds it.
Friction coefficient between tyre or track and ground
The friction coefficient between the tyre or track and the ground: about 0.6 to 0.8 on dry gravel, 0.3 on wet grass, 0.2 on snow and 0.1 on ice. Directly proportional: the thrust the ground can supply scales with the friction available. Raising it by 10 % (to 0.66) moves the result by +10 %, from 57.97 to 63.76 kN.
Vehicle mass
The vehicle mass, in kg. All of it is assumed to rest on the driven wheels or tracks. Directly proportional, since a heavier vehicle presses harder on the ground. Raising it by 10 % (to 11,000) moves the result by +10 %, from 57.97 to 63.76 kN.
Slope angle
The slope angle, in degrees. Lowers the normal load as cos θ, so the effect is small at moderate slopes but grows as the slope steepens. Raising it by 10 % (to 11) moves the result by −0.323 %, from 57.97 to 57.78 kN.
Worked example with the values above
- Normal load on the slope: m g cos θ = 10,000 × 9.81 × 0.9848 = 96,610 N.
- Multiply by the friction coefficient: μ × 96,610 = 0.6 × 96,610.
- Hmax = 57.97 kN.
Assumptions and limits
- All-wheel drive with a uniform friction coefficient, and the whole weight on driven wheels or tracks.
- No other normal loads (no downforce, towing or winch loads), and no soil sinkage or deformation.
Common mistakes
- Using the weight m g in place of the normal load m g cos θ, which overstates the limit on steep slopes.
- Using the same friction coefficient on soft ground as on a hard surface. Soft soil has its own thrust limit.
Statistics and uncertainty
The friction coefficient is the weak input: it varies by ±20 to 30 % even over one field, with moisture, vegetation and loose material, and the result is directly proportional to it. A bound on the limit of climbing is therefore only as good as the friction estimate, so for safety cases use the low end of the likely range.
Where it is used
- Deciding whether slope climbs are limited by grip rather than by engine power.
- Choosing tyres, tracks or chains for slippery slopes.
Maths you need
- Friction: F ≤ μ N
- Resolving forces on a slope: normal load m g cos θ