Library / Terramechanics and off-road mobility / TER-07
Terramechanics and off-road mobility
Force to push a rigid wheel over a step
The horizontal force at the axle needed to roll an unpowered rigid wheel over a sharp step, from the wheel load, wheel radius and step height.
- Ref
- TER-07
- Rev
- v1
- Field
- Terramechanics and off-road mobility
- Result
- Faxle [kN]
Ww is the load on the wheel, Rw its radius and ho the height of a sharp step. At the moment the wheel is about to lift off the ground, the contact moves to the step corner. Balancing moments about that corner for an unpowered rigid wheel pushed horizontally at its axle gives F = Ww √(2Rwho − ho²) / (Rw − ho). Driven torque, tyre deformation and soil failure change this. It applies for 0 < ho < Rw; at ho = Rw the force becomes infinite for a pushed wheel.
The horizontal force at the axle needed to roll an unpowered rigid wheel over a sharp step, from the wheel load, wheel radius and step height. It rises steeply as the step approaches the wheel radius.
Calculator
- Horizontal force per unit wheel load, F / Ww1.02
- Step height as a fraction of the wheel radius, ho / Rw0.3
- Angle of the line from the step corner to the axle, from the vertical45.57 °
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 0.5 m wheel carrying 5 kN needs 5.101 kN of horizontal push to mount a 0.15 m step, about 1.02 times its load. The push equals the load when the step height is about 29 % of the wheel radius.
- For a step half the wheel radius (0.25 m) the push is 8.66 kN, and at 0.4 m (80 % of the radius) it is 24.49 kN. A wheel with a radius of at least three times the step height climbs it with a push of about 1.1 times its load.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (kN) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ho Step height | 0.02 to 0.4 m | 1.458 | 24.49 | 23.04 | +10% | +8.61 % |
| Ww Wheel load | 1 to 20 kN | 1.02 | 20.4 | 19.38 | +10% | +10 % |
| Rw Wheel radius | 0.25 to 1 m | 3.099 | 11.46 | 8.358 | +10% | −7.5 % |
Sweep of Step height
| Step height (m) | Horizontal force at the axle (kN) |
|---|---|
| 0.02 | 1.458 |
| 0.0675 | 2.9 |
| 0.115 | 4.143 |
| 0.1625 | 5.465 |
| 0.21 | 7.023 |
| 0.2575 | 9.016 |
| 0.305 | 11.81 |
| 0.3525 | 16.19 |
| 0.4 | 24.49 |
Sweep one variable
Variables and how each one contributes
Horizontal force at the axle
What the equation gives you. Everything else feeds it.
Wheel load
The load carried by the wheel, in kN. Directly proportional: a heavier load needs a proportionally bigger push. Raising it by 10 % (to 5.5) moves the result by +10 %, from 5.101 to 5.611 kN.
Wheel radius
The wheel radius, in m. Lowers the force: a large wheel meets the step at a shallower angle, which is why big wheels roll over obstacles more easily. Raising it by 10 % (to 0.55) moves the result by −7.5 %, from 5.101 to 4.719 kN.
Step height
The height of the step, in m. It must be less than the wheel radius. Raises the force very quickly, since the geometry gets steeper as the step approaches the wheel radius. Raising it by 10 % (to 0.165) moves the result by +8.61 %, from 5.101 to 5.54 kN.
Worked example with the values above
- Geometry: 2Rwho − ho² = 2 × 0.5 × 0.15 − 0.15² = 0.1275 m², whose square root is 0.3571 m.
- Denominator: Rw − ho = 0.5 − 0.15 = 0.35 m.
- Faxle = 5 × 0.3571 / 0.35 = 5.101 kN.
Assumptions and limits
- A rigid, unpowered wheel pushed horizontally at the axle over a sharp step, quasi-statically.
- 0 < ho < Rw, with no slip at the step corner, and no soil failure or tyre deformation.
Common mistakes
- Using a step height equal to or larger than the wheel radius, where the formula has no finite answer.
- Applying a rigid-wheel result to a soft tyre. A tyre wraps the step and needs less force.
Statistics and uncertainty
The geometry is exact for a sharp step and a rigid wheel, so the result is as accurate as the step height and radius, but a small error in ho near the wheel radius makes a huge difference: at ho/Rw = 0.8 a 5 % error in the height gives roughly 26 % in the force. Real tyres, which deform over the edge, need much less force than this rigid estimate.
Where it is used
- Selecting wheel size for carts, trolleys and robotic vehicles that must cross obstacles.
- Estimating the push or tow force needed to move a wheeled load over a kerb or rock.
Maths you need
- Moments about a point: moment balance about the step corner
- Pythagoras: chord geometry of a circle