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Terramechanics and off-road mobility
Sinkage exponent from two plate observations
Exponent of the power-law pressure–sinkage relation p = Bzⁿ from two pressure and sinkage pairs measured with the same plate width and terrain preparation, on the same loading branch.
- Ref
- TER-35
- Rev
- v1
- Field
- Terramechanics and off-road mobility
- Result
- n
The Bekker pressure–sinkage law p = B·zⁿ plots as a straight line on log–log axes, so two points (z₁, p₁) and (z₂, p₂) fix the exponent n = ln(p₂/p₁)/ln(z₂/z₁). Values of n near 1 mean a stiff, elastic-like soil; larger values mean a soil that stiffens with depth, and values below 1 mean one that yields. It needs both points from the same plate and loading branch.
Exponent of the power-law pressure–sinkage relation p = Bzⁿ from two pressure and sinkage pairs measured with the same plate width and terrain preparation, on the same loading branch. A multi-point log fit is better for noisy data.
Calculator
- Pressure ratio, p₂/p₁4
- Sinkage ratio, z₂/z₁2
- Coefficient B from the first point, p₁/z₁ⁿ5×107 Pa/mⁿ
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.
- The exponent is 2: the pressure rises faster than the sinkage, so the soil stiffens with depth.
- Doubling the sinkage multiplies the pressure by 2^2 = 4, which is what the plate test shows on a log–log plot as a straight line.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| p1 Pressure at the first sinkage | 1,000 to 1,000,000 Pa | −3.644 | 6.322 | 9.966 | +10% | −6.88 % |
| p2 Pressure at the second sinkage | 1,000 to 1,000,000 Pa | −4.322 | 5.644 | 9.966 | +10% | +6.88 % |
| z2 Second sinkage | 0.04 to 0.3 m | 0.5119 | 2 | 1.488 | +10% | −12.1 % |
| z1 First sinkage | 0.005 to 0.02 m | 0.6667 | 2 | 1.333 | +10% | +15.9 % |
Sweep of Pressure at the second sinkage
| Pressure at the second sinkage (Pa) | Sinkage exponent |
|---|---|
| 1,000 | −4.322 |
| 2,371 | −3.076 |
| 5,623 | −1.83 |
| 13,340 | −0.5848 |
| 31,620 | 0.661 |
| 74,990 | 1.907 |
| 177,800 | 3.152 |
| 421,700 | 4.398 |
| 1,000,000 | 5.644 |
Sweep one variable
Variables and how each one contributes
Sinkage exponent
What the equation gives you. Everything else feeds it.
Pressure at the first sinkage
The pressure at the first sinkage, in Pa. A higher first pressure at the same second pressure lowers the exponent. Raising it by 10 % (to 22,000) moves the result by −6.88 %, from 2 to 1.862.
Pressure at the second sinkage
The pressure at the second sinkage, in Pa. A higher second pressure raises the exponent. Raising it by 10 % (to 88,000) moves the result by +6.88 %, from 2 to 2.138.
First sinkage
The first sinkage, in m. A larger first sinkage brings the points closer together in sinkage and raises the exponent for the same pressures. Raising it by 10 % (to 0.022) moves the result by +15.9 %, from 2 to 2.319.
Second sinkage
The second sinkage, in m. A larger second sinkage lowers the exponent for the same pressure ratio. Raising it by 10 % (to 0.044) moves the result by −12.1 %, from 2 to 1.758.
Worked example with the values above
- Pressure ratio: ln(p₂/p₁) = ln(4) = 1.3863.
- Sinkage ratio: ln(z₂/z₁) = ln(2) = 0.69315.
- n = 1.3863 / 0.69315 = 2.
Assumptions and limits
- The same soil state, plate or ring geometry and loading rate for each point.
- A power-law pressure–sinkage law, a monotonic exponential shear curve and a linear Mohr–Coulomb envelope in the tested range.
Common mistakes
- Using points from the unloading branch together with one from the loading branch.
- Taking the two points too close in sinkage, which amplifies the noise in the logarithms.
- Using sinkages measured from different datum surfaces.
Statistics and uncertainty
Soil strength and stiffness parameters from two-point methods carry the full measurement scatter of two points, which is typically 10 % to 30 %, and give no estimate of the fit uncertainty. Use a least-squares fit when more tests are available.
Where it is used
- Fitting Bekker and Janosi shear parameters from bevameter and ring-shear tests.
- Getting first-estimate strength parameters from a pair of simple field tests.
Maths you need
- Bekker relation: p = (kc/b + kφ) zⁿ
- Janosi–Hanamoto curve: τ = τmax(1 − e^(−j/K))
- Mohr–Coulomb strength: τ = c + σ tan φ