Library / Terramechanics and off-road mobility / TER-09
Terramechanics and off-road mobility
Bekker pressure–sinkage relation
The standard empirical law for how deep a loaded plate sinks into soil: contact pressure equals a width-dependent stiffness times the sinkage raised to the soil exponent n.
- Ref
- TER-09
- Rev
- v1
- Field
- Terramechanics and off-road mobility
- Result
- p [kPa]
p is the contact pressure under a plate or footprint of width bc that has sunk to depth z. kc and kφ are the empirical cohesive and frictional pressure–sinkage moduli of the soil and n is the sinkage exponent, all found from plate tests. Their units depend on n: [kc] = Pa m1−n and [kφ] = Pa m−n. For the default n = 1 they are kPa and kPa/m, the units used here (and with z in m and bc in m). kc is not the shear cohesion c. For a rectangular plate use the smaller dimension as bc.
The standard empirical law for how deep a loaded plate sinks into soil: contact pressure equals a width-dependent stiffness times the sinkage raised to the soil exponent n.
Calculator
- Aggregate stiffness, kc/bc + kφ1,000 kPa/mⁿ
- Sinkage in millimetres50 mm
- Pressure at twice the sinkage100 kPa
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.
- With k = kc/bc + kφ = 1,000 kPa/m, a 0.4 m plate sinking 50 mm needs 50 kPa. For n = 1 the pressure is simply proportional to the sinkage, so doubling the depth doubles it (100 kPa at 100 mm).
- A wider plate sinks further at the same pressure: widening the plate to 0.8 m cuts the stiffness to 750 kPa/m, so the pressure needed for 50 mm of sinkage falls to 37.5 kPa.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (kPa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| z Sinkage | 0.005 to 0.3 m | 5 | 300 | 295 | +10% | +10 % |
| n Sinkage exponent | 0.4 to 1.4 | 15.09 | 301.7 | 286.6 | +10% | −25.9 % |
| k_φ Frictional sinkage modulus (kPa/m for n = 1) | 100 to 5,000 kPa/m | 30 | 275 | 245 | +10% | +5 % |
| kc Cohesive sinkage modulus (kPa for n = 1) | 0 to 1,000 kPa | 25 | 150 | 125 | +10% | +5 % |
| bc Contact (plate) width | 0.1 to 1 m | 35 | 125 | 90 | +10% | −4.55 % |
Sweep of Sinkage
| Sinkage (m) | Contact pressure (kPa) |
|---|---|
| 0.005 | 5 |
| 0.04187 | 41.87 |
| 0.07875 | 78.75 |
| 0.1156 | 115.6 |
| 0.1525 | 152.5 |
| 0.1894 | 189.4 |
| 0.2263 | 226.3 |
| 0.2631 | 263.1 |
| 0.3 | 300 |
Sweep one variable
Variables and how each one contributes
Contact pressure
What the equation gives you. Everything else feeds it.
Cohesive sinkage modulus (kPa for n = 1)
The cohesive modulus of sinkage: the part of the soil’s stiffness that depends on the width of the plate. In kPa for n = 1. Raises the pressure for a given sinkage, with less effect on a wide plate, since it is divided by the width. Raising it by 10 % (to 220) moves the result by +5 %, from 50 to 52.5 kPa.
Frictional sinkage modulus (kPa/m for n = 1)
The frictional modulus of sinkage: the part of the soil’s stiffness independent of the plate width. In kPa/m for n = 1. Raises the pressure for a given sinkage. It is typically the larger term on sandy soils. Raising it by 10 % (to 550) moves the result by +5 %, from 50 to 52.5 kPa.
Contact (plate) width
The width of the plate or footprint, in m (the smaller dimension for a rectangle). Lowers the pressure needed for a given sinkage, through kc/bc, so a wide footprint sinks more at the same pressure. Raising it by 10 % (to 0.44) moves the result by −4.55 %, from 50 to 47.73 kPa.
Sinkage
The depth of sinkage, in m. Raises the pressure as z raised to the power n, so with n = 1 the relation is a straight line. Raising it by 10 % (to 0.055) moves the result by +10 %, from 50 to 55 kPa.
Sinkage exponent
The sinkage exponent of the soil: about 0.5 to 1 for dry sand and sandy loam, and 1 or higher for clay and snow. Changes the curve’s shape. For sinkage below 1 m, a larger exponent gives a lower pressure at the same sinkage. Raising it by 10 % (to 1.1) moves the result by −25.9 %, from 50 to 37.06 kPa.
Worked example with the values above
- Aggregate stiffness: kc/bc + kφ = 200 / 0.4 + 500 = 1,000.
- Sinkage term: zn = 0.05^1 = 0.05.
- p = 1,000 × 0.05 = 50 kPa.
Assumptions and limits
- A homogeneous soil with a power-law pressure–sinkage curve, a rigid plate and a uniform pressure.
- Parameters measured with a plate of similar size, at a similar soil density and moisture.
Common mistakes
- Mixing the units of kc and kφ. Their units depend on n, and a pair from a table may be in a different length system.
- Confusing kc with the shear cohesion c.
Statistics and uncertainty
The moduli come from plate tests, and are typically scattered by ±20 to 40 % between tests on the same site, since soil density and moisture vary. The exponent n is fitted over a limited range of sinkage and has an uncertainty of ±0.1, which matters when extrapolating to deeper sinkage. Use the parameters only for the soil state, plate size and sinkage range of the tests.
Where it is used
- Characterising soil from plate tests and predicting the sinkage of wheels, tracks and footings.
- Lunar and planetary rover design, where the soil parameters are fitted from simulants.
Maths you need
- Power laws: z^n
- Pressure: force per unit area