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Terramechanics and off-road mobility
Maximum soil thrust under a flat contact
Upper limit of the thrust a flat contact patch can develop on soil when the soil is fully sheared: cohesion times contact area plus the load times the tangent of the friction angle.
- Ref
- TER-15
- Rev
- v1
- Field
- Terramechanics and off-road mobility
- Result
- Hmax [kN]
c is the soil cohesion, Ac the contact area, Wc the load carried by the contact and φs the soil friction angle. The thrust is the shear strength integrated over the area: cohesion over the whole area plus friction from the normal load. It assumes fully mobilised, uniform soil parameters over a flat contact, with the normal stress equal to Wc/Ac. Real thrust is lower because the shear displacement is finite (see the Janosi–Hanamoto and finite-slip equations).
Upper limit of the thrust a flat contact patch can develop on soil when the soil is fully sheared: cohesion times contact area plus the load times the tangent of the friction angle.
Calculator
- Cohesive part, c Ac0.8 kN
- Frictional part, Wc tan φs11.55 kN
- Maximum thrust-to-load ratio, Hmax / Wc0.6174
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 20 kN load on 0.4 m² of soil with c = 2 kPa and φs = 30° can develop at most 12.35 kN of thrust, 0.617 of the load. Friction supplies 93.5 % of it.
- Sand (c ≈ 0) gives 11.55 kN: the frictional term only. Wet clay with c = 25 kPa and φs = 0 gives 10 kN from the cohesive term, which depends on the area and not on the weight.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (kN) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Wc Load carried by the contact | 2 to 200 kN | 1.955 | 116.3 | 114.3 | +10% | +9.35 % |
| c Soil cohesion | 0 to 50 kPa | 11.55 | 31.55 | 20 | +10% | +0.648 % |
| φs Soil friction angle | 0 to 45 ° | 0.8 | 20.8 | 20 | +10% | +11.7 % |
| Ac Contact area | 0.1 to 10 m² | 11.75 | 31.55 | 19.8 | +10% | +0.648 % |
Sweep of Load carried by the contact
| Load carried by the contact (kN) | Maximum soil thrust (kN) |
|---|---|
| 2 | 1.955 |
| 26.75 | 16.24 |
| 51.5 | 30.53 |
| 76.25 | 44.82 |
| 101 | 59.11 |
| 125.8 | 73.4 |
| 150.5 | 87.69 |
| 175.3 | 102 |
| 200 | 116.3 |
Sweep one variable
Variables and how each one contributes
Maximum soil thrust
What the equation gives you. Everything else feeds it.
Soil cohesion
The soil cohesion, in kPa. Adds a term proportional to the contact area. It matters more on cohesive soils and with large contact areas. Raising it by 10 % (to 2.2) moves the result by +0.648 %, from 12.35 to 12.43 kN.
Contact area
The area of the contact, in m². Raises only the cohesive term, so a larger footprint helps clay but does little on sand. Raising it by 10 % (to 0.44) moves the result by +0.648 %, from 12.35 to 12.43 kN.
Load carried by the contact
The load carried by the contact, in kN. Raises the frictional term in direct proportion, which is the main source of thrust in frictional soil. Raising it by 10 % (to 22) moves the result by +9.35 %, from 12.35 to 13.5 kN.
Soil friction angle
The soil friction angle, in degrees. Raises the frictional term through its tangent. Raising it by 10 % (to 33) moves the result by +11.7 %, from 12.35 to 13.79 kN.
Worked example with the values above
- Cohesive part: c Ac = 2 × 0.4 = 0.8 kN.
- Frictional part: Wc tan φs = 20 × 0.5774 = 11.55 kN.
- Hmax = 0.8 + 11.55 = 12.35 kN.
Assumptions and limits
- Fully mobilised shear strength over the whole contact, with uniform soil parameters and uniform normal stress.
- A flat contact, with the normal load Wc evenly spread over the area Ac.
Common mistakes
- Treating the maximum as the thrust at a given slip. At low slip the real thrust is much lower.
- Using the footprint area of a wheel as flat when its real contact is curved and its normal stress varies.
Statistics and uncertainty
The two soil parameters have uncertainties of ±15 to 30 % (see the Mohr–Coulomb equation), and the thrust inherits the one that dominates, in this case the friction angle: a 3° error in φs at 30° is about ±12 % in tan φs and in the thrust. Because it is a maximum, the true thrust at a normal slip is 60 to 95 % of this value, so treat this as an upper bound.
Where it is used
- A quick upper bound on wheel or track traction on soil, before applying a shear mobilisation law.
- Comparing the traction of vehicles with different weights and footprints.
Maths you need
- Mohr–Coulomb strength: τ = c + σ tan φ
- Pressure and area: force = stress × area