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Terramechanics and off-road mobility
Track thrust with finite slip
Thrust of one flat track on soil with uniform pressure, when the soil is sheared gradually along the contact.
- Ref
- TER-18
- Rev
- v1
- Field
- Terramechanics and off-road mobility
- Result
- H [kN]
For one flat track with uniform normal pressure and soil parameters, driving straight, the shear displacement along the contact is j = s x, where x is the distance from the front of the contact. Integrating the Janosi–Hanamoto law τ = τmax(1 − e−j/Ks) over the length gives H = (c bc ℓc + Wc tan φs)[1 − (Ks/(s ℓc))(1 − e−sℓc/Ks)]. As s → 0 the thrust tends to zero, and the slip must be greater than zero for the formula to be used. Ks is entered in mm. Cohesion in kPa, widths and lengths in m and the load in kN give the thrust in kN.
Thrust of one flat track on soil with uniform pressure, when the soil is sheared gradually along the contact. It rises towards the maximum soil thrust as slip grows and is smaller at low slip.
Calculator
- Maximum (fully mobilised) thrust, c bc ℓc + Wc tan φs117.2 kN
- Fraction of the maximum developed93.75 %
- Shear displacement at the rear of the contact, s ℓc400 mm
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 10 % slip the track develops 93.8 % of its 117.2 kN maximum, 109.8 kN. At 2 % slip it develops only 70 % (82.04 kN) and at 40 % slip 115.3 kN.
- The thrust reaches its plateau when s ℓc is a few times Ks. A long track reaches the plateau at a lower slip: halving the length to 2 m lowers the thrust at 10 % slip to 100.8 kN, because the soil at the front of the contact has sheared less.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (kN) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Wc Load carried by this track | 30 to 600 kN | 19.99 | 328.5 | 308.5 | +10% | +9.66 % |
| φs Soil friction angle | 5 to 45 ° | 19.83 | 187.5 | 167.7 | +10% | +12.1 % |
| s Track slip | 0.01 to 0.6 | 58.72 | 115.9 | 57.22 | +10% | +0.606 % |
| c Soil cohesion | 0 to 20 kPa | 106.1 | 143.6 | 37.5 | +10% | +0.341 % |
| Ks Shear deformation parameter | 5 to 100 mm | 88.41 | 115.7 | 27.29 | +10% | −0.667 % |
| ℓc Track contact length | 2 to 7 m | 100.8 | 115.9 | 15.1 | +10% | +0.95 % |
| bc Track width | 0.25 to 1.2 m | 108 | 115.1 | 7.125 | +10% | +0.341 % |
Sweep of Track slip
| Track slip | Track thrust (kN) |
|---|---|
| 0.01 | 58.72 |
| 0.08375 | 108.4 |
| 0.1575 | 112.5 |
| 0.2313 | 114 |
| 0.305 | 114.8 |
| 0.3787 | 115.2 |
| 0.4525 | 115.5 |
| 0.5262 | 115.8 |
| 0.6 | 115.9 |
Sweep one variable
Variables and how each one contributes
Track thrust
What the equation gives you. Everything else feeds it.
Soil cohesion
The soil cohesion, in kPa. Raises the maximum thrust through the area term bc ℓc c, a small part of the total at this load. Raising it by 10 % (to 2.2) moves the result by +0.341 %, from 109.8 to 110.2 kN.
Track width
The width of the track, in m. Raises only the cohesive part (c bc ℓc), so it has little effect on frictional soil. Raising it by 10 % (to 0.55) moves the result by +0.341 %, from 109.8 to 110.2 kN.
Track contact length
The length of the track in contact with the ground, in m. Raises the cohesive part and also the shear displacement s ℓc at the rear of the contact, so the soil is better mobilised. Raising it by 10 % (to 4.4) moves the result by +0.95 %, from 109.8 to 110.9 kN.
Load carried by this track
The load carried by this track, in kN. Raises the frictional part in direct proportion. It is the biggest term for frictional soil. Raising it by 10 % (to 215.6) moves the result by +9.66 %, from 109.8 to 120.4 kN.
Soil friction angle
The soil friction angle, in degrees. Raises the frictional part through its tangent. Raising it by 10 % (to 33) moves the result by +12.1 %, from 109.8 to 123.1 kN.
Shear deformation parameter
The shear deformation parameter of the soil, in mm. Lowers the thrust at a given slip, because the soil needs more shear displacement to develop its strength. Raising it by 10 % (to 27.5) moves the result by −0.667 %, from 109.8 to 109.1 kN.
Track slip
The track slip, from 0 to 1. Raises the thrust, quickly at first, then gradually as it approaches the maximum. Here the default slip is already near the plateau. Raising it by 10 % (to 0.11) moves the result by +0.606 %, from 109.8 to 110.5 kN.
Worked example with the values above
- Maximum thrust: c bc ℓc + Wc tan φs = 4 + 113.2 = 117.2 kN.
- Displacement ratio: s ℓc / Ks = 0.1 × 4 / 0.025 = 16. Bracket: 1 − (1 − e−16) / 16 = 0.9375.
- H = 117.2 × 0.9375 = 109.8 kN.
Assumptions and limits
- One flat track with uniform normal pressure, uniform soil parameters, straight driving and j = s x along the contact.
- A monotonic Janosi–Hanamoto soil law with no softening, and s greater than zero.
Common mistakes
- Using zero slip. The formula has no value at s = 0, where the thrust is zero.
- Using Ks in metres instead of millimetres.
Statistics and uncertainty
The thrust inherits the uncertainty of the maximum soil strength (±15 to 30 %) and of Ks (±30 to 50 %). At the plateau only the first one matters, but at low slip the term in Ks dominates and the thrust varies nearly in direct proportion to s ℓc / Ks. The assumption of uniform pressure is also optimistic, since real tracks have pressure peaks under the road wheels.
Where it is used
- Estimating track thrust at a realistic slip rather than assuming the full soil strength.
- Choosing the track length and the slip at which a tracked vehicle can deliver a given drawbar pull.
Maths you need
- Integration of an exponential: ∫ (1 − e^(−x/K)) dx
- Mohr–Coulomb strength: maximum shear stress