Library / Terramechanics and off-road mobility / TER-41
Terramechanics and off-road mobility
Accumulated shear displacement around a driven wheel
Circumferential soil shear displacement at contact angle θ beneath a rigid wheel, from the entry angle θf, radius r and driving slip i (0 ≤ i ≤ 1).
- Ref
- TER-41
- Rev
- v1
- Field
- Terramechanics and off-road mobility
- Result
- j [m]
For a wheel with driving slip i the wheel’s rim moves backwards relative to the soil. The shear displacement at contact angle θ is j(θ) = r[(θf − θ) − (1 − i)(sin θf − sin θ)]: the first term is the rim arc travelled since entry, and the second is the ground distance covered by the wheel centre over the same time. At i = 0 the displacement is the difference of arc and chord, and at full slip it equals the whole arc.
Circumferential soil shear displacement at contact angle θ beneath a rigid wheel, from the entry angle θf, radius r and driving slip i (0 ≤ i ≤ 1). Simple rigid-wheel kinematics with constant slip and zero displacement at entry. Angles are in radians and increase towards the entry point.
Calculator
- Shear displacement at the exit, θ = 00.05296 m
- Arc length of contact, r θf0.18 m
- Displacement at the exit as a share of the arc29.42 %
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.
- At θ = 0.2 rad the soil has been sheared 37.66 mm, and by the exit it reaches 52.96 mm.
- The displacement grows towards the exit, so shear stress, which follows the Janosi curve, is mobilised mostly in the rear part of the contact.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| θf Entry angle | 0.2 to 1.5 rad | 0 | 0.2103 | 0.2103 | +10% | +18.8 % |
| r Wheel radius | 0.05 to 1.5 m | 0.006276 | 0.1883 | 0.182 | +10% | +10 % |
| i Driving slip | 0 to 1 | 0.01021 | 0.12 | 0.1098 | +10% | +7.29 % |
| θ Contact angle of interest | 0 to 0.2 rad | 0.03766 | 0.05296 | 0.0153 | +10% | −4.25 % |
Sweep of Driving slip
| Driving slip | Shear displacement (m) |
|---|---|
| 0 | 0.01021 |
| 0.125 | 0.02393 |
| 0.25 | 0.03766 |
| 0.375 | 0.05138 |
| 0.5 | 0.0651 |
| 0.625 | 0.07883 |
| 0.75 | 0.09255 |
| 0.875 | 0.1063 |
| 1 | 0.12 |
Sweep one variable
Variables and how each one contributes
Shear displacement
What the equation gives you. Everything else feeds it.
Wheel radius
The wheel radius, in m. Raises the shear displacement in direct proportion. Raising it by 10 % (to 0.33) moves the result by +10 %, from 0.03766 to 0.04142 m.
Entry angle
The entry angle, in rad. A larger entry angle lengthens the arc and raises the displacement. Raising it by 10 % (to 0.66) moves the result by +18.8 %, from 0.03766 to 0.04475 m.
Contact angle of interest
The contact angle of interest, in rad. Moving the point closer to the exit (θ → 0) raises the displacement. Raising it by 10 % (to 0.22) moves the result by −4.25 %, from 0.03766 to 0.03606 m.
Driving slip
The driving slip, from 0 to 1. Higher slip raises the displacement towards the full arc length. Raising it by 10 % (to 0.275) moves the result by +7.29 %, from 0.03766 to 0.0404 m.
Worked example with the values above
- Arc term: r(θf − θ) = 0.3 × 0.4 = 0.12 m.
- Ground term: r(1 − i)(sin θf − sin θ) = 0.3 × 0.75 × 0.36597 = 0.082344 m.
- j = 0.12 − 0.082344 = 0.037656 m.
Assumptions and limits
- A rigid wheel of constant radius and width, with stresses uniform across the width and constant driving slip.
- Contact angle measured from the downward vertical, with angles increasing from exit to entry, and an arc without soil bulging.
Common mistakes
- Entering angles in degrees instead of radians.
- Ordering the nodes from entry to exit, which reverses the sign of the integral.
- Mixing the normal-stress and shear-stress sign conventions.
Statistics and uncertainty
Stress sensors on a wheel rim read to perhaps 5 % to 10 %, and the trapezoidal sum over five nodes is coarse, so a smooth stress peak between nodes can be missed. Add nodes where stress changes quickly.
Where it is used
- Checking measured wheel loads and thrusts against force-cell readings.
- Building wheel–soil models from measured stress data.
Maths you need
- Contact angle: angle at the wheel centre from the vertical to a contact point
- Trapezoidal rule: area under sampled data
- Janosi–Hanamoto shear: τ = τmax(1 − e^(−j/K))