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Terramechanics and off-road mobility
Soil compaction resistance
The work per unit distance a wheel or track spends pressing soil down to depth z, found by integrating the Bekker law over sinkage.
- Ref
- TER-11
- Rev
- v1
- Field
- Terramechanics and off-road mobility
- Result
- Rc [N]
Rc is the work per unit distance done in pressing the soil down to depth z over a width bc, found by integrating the Bekker pressure over the sinkage: Rc = bc ∫ p dζ = (kc + bckφ) zn+1/(n + 1). The moduli are used in the same units as in the pressure–sinkage relation (kPa and kPa/m for n = 1), and the result is in N. It excludes bulldozing, rut side-wall friction, tyre losses and repeated-pass effects, and it applies to one wheel or track of this width.
The work per unit distance a wheel or track spends pressing soil down to depth z, found by integrating the Bekker law over sinkage. It excludes bulldozing, side-wall friction and repeated passes.
Calculator
- Pressure at the final sinkage, (kc/bc + kφ) zn50 kPa
- Resistance in kilonewtons0.5 kN
- Resistance per metre of width, Rc / bc1,250 N/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.
- Sinking 50 mm into this soil costs 500 N per wheel of width 0.4 m. Doubling the sinkage to 100 mm costs 2,000 N, four times as much, so a vehicle that sinks twice as deep spends four times as much force on compaction.
- On soft ground compaction is the main part of the motion resistance, so reducing the sinkage with a larger footprint is the best way to save thrust. For scale, 500 N is 10 % of a 5 kN wheel load.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N) | Local step | Change in result |
|---|---|---|---|---|---|---|
| z Sinkage | 0.005 to 0.3 m | 5 | 18,000 | 18,000 | +10% | +21 % |
| n Sinkage exponent | 0.4 to 1.4 | 125.7 | 4,310 | 4,184 | +10% | −29.4 % |
| k_φ Frictional sinkage modulus (kPa/m for n = 1) | 100 to 5,000 kPa/m | 300 | 2,750 | 2,450 | +10% | +5 % |
| kc Cohesive sinkage modulus (kPa for n = 1) | 0 to 1,000 kPa | 250 | 1,500 | 1,250 | +10% | +5 % |
| bc Contact width | 0.1 to 1.5 m | 312.5 | 1,188 | 875 | +10% | +5 % |
Sweep of Sinkage
| Sinkage (m) | Compaction resistance (N) |
|---|---|
| 0.005 | 5 |
| 0.04187 | 350.7 |
| 0.07875 | 1,240 |
| 0.1156 | 2,674 |
| 0.1525 | 4,651 |
| 0.1894 | 7,173 |
| 0.2263 | 10,240 |
| 0.2631 | 13,850 |
| 0.3 | 18,000 |
Sweep one variable
Variables and how each one contributes
Compaction resistance
What the equation gives you. Everything else feeds it.
Cohesive sinkage modulus (kPa for n = 1)
The cohesive modulus of sinkage, in kPa for n = 1. Raises the resistance, because a stiffer soil needs more work for the same sinkage. Raising it by 10 % (to 220) moves the result by +5 %, from 500 to 525 N.
Frictional sinkage modulus (kPa/m for n = 1)
The frictional modulus of sinkage, in kPa/m for n = 1. Raises the resistance through the term bc kφ. Raising it by 10 % (to 550) moves the result by +5 %, from 500 to 525 N.
Contact width
The width of the wheel or track in contact, in m. Raises the resistance through the term bc kφ, since a wider wheel compacts a wider strip. Raising it by 10 % (to 0.44) moves the result by +5 %, from 500 to 525 N.
Sinkage
The sinkage depth, in m. Raises the resistance as z raised to n + 1, so with n = 1 doubling the sinkage quadruples it. This is the strongest lever. Raising it by 10 % (to 0.055) moves the result by +21 %, from 500 to 605 N.
Sinkage exponent
The sinkage exponent of the soil. Changes the power of z in the resistance. For z below 1 m, a larger exponent gives a smaller resistance. Raising it by 10 % (to 1.1) moves the result by −29.4 %, from 500 to 352.9 N.
Worked example with the values above
- Soil term: kc + bc kφ = 200 + 0.4 × 500 = 400.
- Sinkage term: zn+1 / (n + 1) = 0.05^2 / 2 = 0.00125.
- Rc = 400 × 0.00125 kN = 0.5 kN = 500 N.
Assumptions and limits
- The Bekker pressure–sinkage law applies and the soil behaves as a homogeneous medium, with the full width at the final depth.
- The work of compaction is the only loss, excluding bulldozing, rut-wall friction, tyre hysteresis and repeated passes.
Common mistakes
- Using a sinkage in millimetres where the formula expects metres.
- Adding the resistance of the front and rear wheels without allowing for the rear wheels running in the front wheels’ ruts, which reduces it.
Statistics and uncertainty
The resistance is a power function of the sinkage and the soil moduli, and the sinkage is itself uncertain by 20 to 40 %, so for n = 1 a 25 % error in z gives about 56 % in Rc. The formula leaves out bulldozing and multi-pass effects, which can double the resistance in deep ruts. Use measured towing resistance for design work.
Where it is used
- Estimating the dominant motion resistance of a vehicle on soft ground.
- Comparing the benefits of wider tyres or tracks on a given soil.
Maths you need
- Integration of a power law: ∫ ζⁿ dζ = z^(n+1) / (n+1)
- Work per distance: a force