Library / Terramechanics and off-road mobility / TER-02
Terramechanics and off-road mobility
Thrust to climb a grade
Tractive thrust needed to climb a constant slope at steady speed without towing: the weight component along the slope plus rolling resistance.
- Ref
- TER-02
- Rev
- v1
- Field
- Terramechanics and off-road mobility
- Result
- H [kN]
The grade G is entered as a percentage: 100 × tan θ, so a 10 % grade is a slope of 5.7° and a 100 % grade is 45°. At steady speed with no towing, the thrust must balance the weight component along the slope, m g sin θ, and the rolling resistance, Crr m g cos θ, which acts on the normal load and so falls slightly on a slope. The power in the derived rows is the thrust times the climbing speed, at the ground, before drivetrain losses.
Tractive thrust needed to climb a constant slope at steady speed without towing: the weight component along the slope plus rolling resistance. The grade G is entered as a percentage (100 × tan θ).
Calculator
- Slope angle, θ5.711 °
- Weight component along the slope, m g sin θ9.761 kN
- Rolling resistance, Crr m g cos θ3.905 kN
- Power at the ground at the climbing speed, H U68.33 kW
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.
- A 10 t vehicle on a 10 % grade needs 13.67 kN of thrust, of which 9.761 kN is gravity and 3.905 kN rolling resistance. At 5 m/s this is 68.33 kW at the ground, before drivetrain losses.
- The thrust more than doubles on a 30 % grade, reaching 31.95 kN, and on soft sand (Crr = 0.15) the same 10 % grade needs 24.4 kN, because the surface adds more resistance than the slope does.
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 | 2.733 | 82 | 79.26 | +10% | +10 % |
| G Grade | 0 to 60 % | 3.924 | 53.84 | 49.91 | +10% | +7.03 % |
| Crr Rolling resistance coefficient | 0.01 to 0.3 | 10.74 | 39.05 | 28.31 | +10% | +2.86 % |
| U Climbing speed | 0.5 to 20 m/s | 13.67 | 13.67 | 0 | +10% | 0 % |
Sweep of Grade
| Grade (%) | Thrust required (kN) |
|---|---|
| 0 | 3.924 |
| 7.5 | 11.25 |
| 15 | 18.43 |
| 22.5 | 25.36 |
| 30 | 31.95 |
| 37.5 | 38.12 |
| 45 | 43.84 |
| 52.5 | 49.07 |
| 60 | 53.84 |
Sweep one variable
Variables and how each one contributes
Thrust required
What the equation gives you. Everything else feeds it.
Vehicle mass
The vehicle mass, in kg. Directly proportional: both the gravity and the rolling terms scale with the mass. Raising it by 10 % (to 11,000) moves the result by +10 %, from 13.67 to 15.03 kN.
Grade
The slope as a grade in per cent (100 × tan of the slope angle): the rise per 100 m travelled horizontally. The gravity term grows almost in direct proportion to the grade, so it quickly dominates the rolling term. Raising it by 10 % (to 11) moves the result by +7.03 %, from 13.67 to 14.63 kN.
Rolling resistance coefficient
The rolling resistance coefficient of the surface: 0.01 to 0.02 on tarmac, 0.04 on firm gravel, 0.1 or more on soft sand or snow. Raises only the rolling part. On this 10 % grade it is a smaller part than gravity, so the effect is modest. Raising it by 10 % (to 0.044) moves the result by +2.86 %, from 13.67 to 14.06 kN.
Climbing speed
The steady climbing speed, in m/s. It does not change the thrust here, only the power in the derived rows. Does not change the thrust, because this force balance has no speed-dependent term (drag is left out). It sets the power needed: doubling the speed doubles the power for the same thrust.
Worked example with the values above
- Slope angle: θ = tan⁻¹(10 / 100) = 5.711°, so sin θ = 0.0995 and cos θ = 0.995.
- Gravity term: m g sin θ = 10,000 × 9.81 × 0.0995 = 9,761 N. Rolling term: Crr m g cos θ = 0.04 × 98,100 × 0.995 = 3,905 N.
- H = 9,761 + 3,905 = 13,670 N = 13.67 kN. At 5 m/s the power is 68.33 kW.
Assumptions and limits
- Constant speed, no towed load, and no aerodynamic drag (which is small at climbing speed).
- A constant rolling resistance coefficient and a firm, uniform slope. On soft soil use a sinkage-based resistance instead.
Common mistakes
- Entering the slope angle in degrees where the grade in per cent is expected. A 10 % grade is 5.7°, not 10°.
- Treating the thrust as the engine force. Drivetrain losses and the grip limit still apply.
Statistics and uncertainty
The gravity term is exact for a given mass and slope; the weak term is the rolling resistance coefficient, which is uncertain by 20 to 30 % on natural ground. On a 10 % grade at the defaults, rolling is about 29 % of the thrust, so a 30 % error in Crr gives roughly 9 % error in the thrust. Grade measurement errors matter too: 1 percentage point of grade is about 10 % of the gravity term at a 10 % grade.
Where it is used
- Checking that a vehicle can climb a given slope, and the power that takes.
- Setting gradeability specifications for military and off-road vehicles.
Maths you need
- Slope and grade: grade % = 100 tan θ
- Resolving forces on a slope: m g sin θ and m g cos θ