Library / Terramechanics and off-road mobility / TER-10
Terramechanics and off-road mobility
Sinkage under uniform pressure
Sinkage of a rectangular footprint carrying load Wc with uniform pressure, found by inverting the Bekker law.
- Ref
- TER-10
- Rev
- v1
- Field
- Terramechanics and off-road mobility
- Result
- z [mm]
Wc is the load on the footprint, bc and ℓc its width and length. The mean pressure Wc/(bcℓc) is set equal to the Bekker pressure (kc/bc + kφ) zn, and solving for z gives the sinkage in mm. It assumes uniform pressure and a prescribed rectangular footprint. A wheel’s contact length changes as it sinks, so for a wheel the load equilibrium must be solved with a compatible geometry. The soil parameters are used in the same units as in the pressure–sinkage relation.
Sinkage of a rectangular footprint carrying load Wc with uniform pressure, found by inverting the Bekker law. It assumes the footprint is prescribed and does not change as the vehicle sinks.
Calculator
- Mean contact pressure, Wc / (bc ℓc)50 kPa
- Aggregate stiffness, kc/bc + kφ1,000 kPa/mⁿ
- Sinkage in metres0.05 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 20 kN load on a 0.4 m by 1 m footprint presses with 50 kPa and sinks 50 mm in this soil. Doubling the load to 40 kN doubles the sinkage to 100 mm for n = 1.
- With a lower exponent (n = 0.7) the same load gives 13.85 mm, and in a stiffer soil (kφ = 1000) it gives 33.33 mm. The sinkage is very sensitive to both exponent and stiffness.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (mm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Wc Load carried by the footprint | 2 to 200 kN | 5 | 500 | 495 | +10% | +10 % |
| ℓc Footprint length | 0.3 to 5 m | 10 | 166.7 | 156.7 | +10% | −9.09 % |
| n Sinkage exponent | 0.4 to 1.4 | 0.559 | 117.7 | 117.1 | +10% | +31.3 % |
| kc Cohesive sinkage modulus (kPa for n = 1) | 0 to 1,000 kPa | 16.67 | 100 | 83.33 | +10% | −4.76 % |
| k_φ Frictional sinkage modulus (kPa/m for n = 1) | 100 to 5,000 kPa/m | 9.091 | 83.33 | 74.24 | +10% | −4.76 % |
| bc Footprint width | 0.1 to 1.5 m | 21.05 | 80 | 58.95 | +10% | −4.76 % |
Sweep of Load carried by the footprint
| Load carried by the footprint (kN) | Sinkage (mm) |
|---|---|
| 2 | 5 |
| 26.75 | 66.88 |
| 51.5 | 128.8 |
| 76.25 | 190.6 |
| 101 | 252.5 |
| 125.8 | 314.4 |
| 150.5 | 376.3 |
| 175.3 | 438.1 |
| 200 | 500 |
Sweep one variable
Variables and how each one contributes
Sinkage
What the equation gives you. Everything else feeds it.
Load carried by the footprint
The load carried by the footprint, in kN. Raises the sinkage. With n = 1 it is directly proportional, and with n below 1 the sinkage rises faster than the load. Raising it by 10 % (to 22) moves the result by +10 %, from 50 to 55 mm.
Footprint width
The width of the footprint, in m. A wider footprint spreads the load, but it also reduces the soil stiffness through kc/bc, so the net effect is small. Raising it by 10 % (to 0.44) moves the result by −4.76 %, from 50 to 47.62 mm.
Footprint length
The length of the footprint, in m. Lowers the sinkage in inverse proportion for n = 1: a longer footprint carries the same load at a lower pressure. Raising it by 10 % (to 1.1) moves the result by −9.09 %, from 50 to 45.45 mm.
Cohesive sinkage modulus (kPa for n = 1)
The cohesive modulus of sinkage, in kPa for n = 1. A stiffer soil sinks less. Raising it by 10 % (to 220) moves the result by −4.76 %, from 50 to 47.62 mm.
Frictional sinkage modulus (kPa/m for n = 1)
The frictional modulus of sinkage, in kPa/m for n = 1. A stiffer soil sinks less. Raising it by 10 % (to 550) moves the result by −4.76 %, from 50 to 47.62 mm.
Sinkage exponent
The sinkage exponent of the soil. For a pressure ratio below 1, the usual case, a larger exponent gives a larger sinkage, because the ratio is raised to the power 1/n. Raising it by 10 % (to 1.1) moves the result by +31.3 %, from 50 to 65.65 mm.
Worked example with the values above
- Mean pressure: Wc / (bc ℓc) = 20 / (0.4 × 1) = 50 kPa. Aggregate stiffness: kc/bc + kφ = 1,000.
- Ratio: 50 / 1,000 = 0.05, raised to the power 1/n = 1 gives 0.05 m.
- z = 50 mm.
Assumptions and limits
- Uniform pressure under a prescribed rectangular footprint, and a homogeneous soil following the Bekker law.
- A static load, with no dynamic, slip or repeated-pass effects.
Common mistakes
- Using a wheel’s footprint length that does not match its sinkage. The two must be solved together for a wheel.
- Mixing the pressure units (kPa) with the stiffness units from a table in other units.
Statistics and uncertainty
Sinkage depends on the load, the footprint and two to three fitted soil parameters, which together have an uncertainty of 20 to 40 %. For n below 1 the error is amplified, because the result varies with the parameters raised to the power 1/n: with n = 0.7 a stiffness 20 % too low gives a sinkage 37 % too large. Treat computed ruts as a guide that needs checking against field data.
Where it is used
- Estimating the rut depth a vehicle will make on a given soil.
- Checking whether a footprint is large enough to keep the sinkage within the vehicle’s clearance.
Maths you need
- Pressure-sinkage relation: p = (k_c/b + k_φ) z^n
- Powers and roots: solving z^n = x