Library / Terramechanics and off-road mobility / TER-03
Terramechanics and off-road mobility
Maximum climbable grade
The steepest slope a vehicle can climb at steady speed on firm ground with a given available thrust, for small angles.
- Ref
- TER-03
- Rev
- v1
- Field
- Terramechanics and off-road mobility
- Result
- Gmax [%]
H is the thrust available at the ground, m the vehicle mass and Crr the rolling resistance coefficient. For steady climbing without towing, H = m g (sin θ + Crr cos θ). For small slopes sin θ ≈ tan θ and cos θ ≈ 1, giving tan θ ≈ H / (m g) − Crr, and the result is the grade 100 tan θ in per cent. At steep slopes the approximation slightly underestimates the grade. The thrust must be less than the grip limit, and a negative result means the thrust does not even overcome rolling resistance on level ground.
The steepest slope a vehicle can climb at steady speed on firm ground with a given available thrust, for small angles. It is the thrust-to-weight ratio less the rolling resistance coefficient.
Calculator
- Thrust-to-weight ratio, H / m g25.48 %
- Slope angle for this grade, tan⁻¹(G / 100)12.13 °
- Grade from the exact balance sin θ + Crr cos θ = H / m g22.1 %
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.
- With 25 kN of thrust on a 10 t vehicle (a thrust-to-weight ratio of 25.5 %) and Crr = 0.04, the steepest grade is 21.48 %. The exact balance gives 22.1 %, so the small-angle formula is slightly conservative.
- On soft ground the loss is large: with Crr = 0.15 the limit falls to 10.48 %, and traction, not thrust, usually decides first (see the slope traction limit).
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (%) | Local step | Change in result |
|---|---|---|---|---|---|---|
| m Vehicle mass | 2,000 to 60,000 kg | 0.2474 | 123.4 | 123.2 | +10% | −10.8 % |
| H Available tractive thrust | 5 to 100 kN | 1.097 | 97.94 | 96.84 | +10% | +11.9 % |
| Crr Rolling resistance coefficient | 0.01 to 0.3 | −4.516 | 24.48 | 29 | +10% | −1.86 % |
Sweep of Available tractive thrust
| Available tractive thrust (kN) | Maximum grade (%) |
|---|---|
| 5 | 1.097 |
| 16.88 | 13.2 |
| 28.75 | 25.31 |
| 40.63 | 37.41 |
| 52.5 | 49.52 |
| 64.38 | 61.62 |
| 76.25 | 73.73 |
| 88.13 | 85.83 |
| 100 | 97.94 |
Sweep one variable
Variables and how each one contributes
Maximum grade
What the equation gives you. Everything else feeds it.
Available tractive thrust
The tractive thrust available at the ground, in kN: the smaller of the engine force and the traction limit. Raises the grade in direct proportion to H / (m g), so the result responds more than in proportion, since the rolling term is subtracted. Raising it by 10 % (to 27.5) moves the result by +11.9 %, from 21.48 to 24.03 %.
Vehicle mass
The vehicle mass, in kg. Lowers the thrust-to-weight ratio and so the grade the vehicle can climb. Raising it by 10 % (to 11,000) moves the result by −10.8 %, from 21.48 to 19.17 %.
Rolling resistance coefficient
The rolling resistance coefficient of the surface. Subtracts directly from the thrust-to-weight ratio. A soft surface costs a lot of climbing ability. Raising it by 10 % (to 0.044) moves the result by −1.86 %, from 21.48 to 21.08 %.
Worked example with the values above
- Thrust-to-weight ratio: H / (m g) = 25,000 / (10,000 × 9.81) = 0.2548.
- Subtract the rolling coefficient: tan θ ≈ 0.2548 − 0.04 = 0.2148.
- Grade = 100 × 0.2148 = 21.48 %.
Assumptions and limits
- Steady speed, no towing, no aerodynamic drag, and the thrust fully available (not grip limited).
- A small-angle approximation, valid below about 25 to 30 % grade, and a constant rolling coefficient.
Common mistakes
- Using the engine force at the tyre without checking that the tyres can transmit it.
- Reading the result as a slope angle in degrees. It is a grade in per cent.
Statistics and uncertainty
The grade is a difference of the thrust-to-weight ratio (known to ±5 to 10 %) and the rolling coefficient (±30 %), so for a modest thrust margin the error is large: at the defaults a 10 % error in H shifts the grade by about 2.5 percentage points. The small-angle formula adds a systematic error of about 3 % of the grade at 20 %. Test climbs on the actual surface are the only reliable rating.
Where it is used
- Gradeability ratings for military, agricultural and off-road vehicles.
- Checking if a vehicle can meet a required slope, or how much a load reduces it.
Maths you need
- Thrust-to-weight ratio: force divided by weight
- Grade and slope angle: grade % = 100 tan θ