Library / Terramechanics and off-road mobility / TER-05
Terramechanics and off-road mobility
Approach or departure angle
The steepest ramp a vehicle can drive onto (approach) or off (departure) before the front or rear bumper or body touches.
- Ref
- TER-05
- Rev
- v1
- Field
- Terramechanics and off-road mobility
- Result
- α [°]
h is the height above the ground of the limiting point at the front (for the approach angle) or the rear (for the departure angle), and ℓ is its horizontal distance from the tyre contact point. The angle is the steepest ramp the vehicle can climb or leave before that point touches. It is a rigid geometric estimate: it uses the clearance with the vehicle at its static ride height, and ignores suspension compression, tyre deflection and the movement of the vehicle on soft ground.
The steepest ramp a vehicle can drive onto (approach) or off (departure) before the front or rear bumper or body touches. It comes from the height of the limiting point and its distance from the tyre contact.
Calculator
- Equivalent gradient, 100 h / ℓ55.56 %
- Angle in radians0.5071 rad
- Height needed to clear a 30° ramp at this overhang, ℓ tan 30°0.5196 m
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 bumper 0.5 m high and 0.9 m ahead of the tyre contact gives an approach angle of 29.05°, a 56 % ramp. Cutting the overhang to 0.7 m raises it to 35.54°.
- Typical road cars have approach angles of 12° to 20°, and off-road vehicles 30° to 45° or more. A short, high front end matters more than a high ground clearance under the body.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (°) | Local step | Change in result |
|---|---|---|---|---|---|---|
| h Height of the limiting point above the ground | 0.2 to 1.5 m | 12.53 | 59.04 | 46.51 | +10% | +2.37 ° |
| ℓ Horizontal distance from the tyre contact to that point | 0.4 to 3 m | 9.462 | 51.34 | 41.88 | +10% | −2.26 ° |
Sweep of Horizontal distance from the tyre contact to that point
| Horizontal distance from the tyre contact to that point (m) | Approach or departure angle (°) |
|---|---|
| 0.4 | 51.34 |
| 0.725 | 34.59 |
| 1.05 | 25.46 |
| 1.375 | 19.98 |
| 1.7 | 16.39 |
| 2.025 | 13.87 |
| 2.35 | 12.01 |
| 2.675 | 10.59 |
| 3 | 9.462 |
Sweep one variable
Variables and how each one contributes
Approach or departure angle
What the equation gives you. Everything else feeds it.
Height of the limiting point above the ground
The height of the limiting point (bumper, tow hitch or body edge) above the ground, in m. Raises the angle: a higher bumper clears a steeper ramp, with a slightly smaller effect than proportional at larger angles. Raising it by 10 % (to 0.55) moves the result by +8.17 %, from 29.05 to 31.43 °.
Horizontal distance from the tyre contact to that point
The horizontal distance from the tyre contact point to the limiting point, in m: the overhang. Lowers the angle: a long overhang strikes a ramp sooner. Raising it by 10 % (to 0.99) moves the result by −7.77 %, from 29.05 to 26.8 °.
Worked example with the values above
- Ratio: h / ℓ = 0.5 / 0.9 = 0.5556.
- Angle: α = tan⁻¹(0.5556) in radians = 0.5071 rad.
- α = 29.05°, equivalent to a gradient of 55.56 %.
Assumptions and limits
- A rigid vehicle at its static ride height, with the limiting point as a single sharp feature.
- A flat ground contact and a straight, firm ramp. Deformable ground and tyre deflection are ignored.
Common mistakes
- Measuring the distance ℓ from the axle centre instead of from the tyre contact point.
- Using the height of the body underside when a lower bumper, tow hitch or skid plate is the limiting point.
Statistics and uncertainty
Both lengths can be measured to a centimetre or two, so the angle is accurate to about 1°. The larger uncertainty is physical: the measured angle applies at static ride height with the nominal load, and a heavy load, a loaded suspension or soft ground reduces the real clearance by 5 to 15 cm, which can lower the angle by several degrees.
Where it is used
- Comparing off-road vehicles and judging whether a vehicle can climb a bank or ramp.
- Specifying vehicle geometry for loading ramps and ship or landing-craft use.
Maths you need
- Trigonometry: tan⁻¹ of a ratio of lengths
- Gradient: angle and percentage slope