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Terramechanics and off-road mobility
Mohr–Coulomb parameters from two shear tests
Cohesion c, and (in the derived rows) friction angle φ in degrees, from two shear-strength tests at different effective normal stresses σ₁ and σ₂, using the same stress and drainage basis for both.
- Ref
- TER-39
- Rev
- v1
- Field
- Terramechanics and off-road mobility
- Result
- c [Pa]
Two shear-strength tests at different normal stresses fix a straight Mohr–Coulomb envelope τ = c + σ tan φ. The slope between them gives tan φ = (τ₂ − τ₁)/(σ₂ − σ₁), and the intercept is c = τ₁ − σ₁ tan φ. A negative c means the envelope extrapolates to a negative strength at zero stress, which usually shows scatter or a curved envelope rather than a real property.
Cohesion c, and (in the derived rows) friction angle φ in degrees, from two shear-strength tests at different effective normal stresses σ₁ and σ₂, using the same stress and drainage basis for both. Two points cannot show the fit uncertainty.
Calculator
- Friction angle, arctan((τ₂ − τ₁)/(σ₂ − σ₁))26.57 °
- Slope of the envelope, tan φ0.5
- Strength at 50 kPa normal stress, c + 50 000 tan φ28,000 Pa
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 envelope has a friction angle of 26.57° and a cohesion of 3,000 Pa.
- At a 50 kPa normal stress it predicts a strength of 28,000 Pa, but use it only inside the stress range tested, between 10,000 and 30,000 Pa.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Pa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| τ1 Shear strength, test 1 | 100 to 100,000 Pa | −8,850 | 141,000 | 149,900 | +10% | +40 % |
| τ2 Shear strength, test 2 | 100 to 100,000 Pa | −38,000 | 11,950 | 49,950 | +10% | −30 % |
| σ1 Normal stress, test 1 | 1,000 to 20,000 Pa | −12,000 | 7,655 | 19,660 | +10% | −26.3 % |
| σ2 Normal stress, test 2 | 30,000 to 200,000 Pa | 3,000 | 7,474 | 4,474 | +10% | +21.7 % |
Sweep of Shear strength, test 2
| Shear strength, test 2 (Pa) | Cohesion (Pa) |
|---|---|
| 100 | 11,950 |
| 237.1 | 11,880 |
| 562.3 | 11,720 |
| 1,334 | 11,330 |
| 3,162 | 10,420 |
| 7,499 | 8,251 |
| 17,780 | 3,109 |
| 42,170 | −9,085 |
| 100,000 | −38,000 |
Sweep one variable
Variables and how each one contributes
Cohesion
What the equation gives you. Everything else feeds it.
Normal stress, test 1
The normal stress of test 1, in Pa. Raising it moves the intercept point and lowers c for a given slope. Raising it by 10 % (to 11,000) moves the result by −26.3 %, from 3,000 to 2,211 Pa.
Normal stress, test 2
The normal stress of test 2, in Pa. A wider stress separation makes the slope better conditioned. Raising it by 10 % (to 33,000) moves the result by +21.7 %, from 3,000 to 3,652 Pa.
Shear strength, test 1
The shear strength of test 1, in Pa. Raises the intercept c directly and lowers the slope. Raising it by 10 % (to 8,800) moves the result by +40 %, from 3,000 to 4,200 Pa.
Shear strength, test 2
The shear strength of test 2, in Pa. Raises the slope and so lowers the intercept. Raising it by 10 % (to 19,800) moves the result by −30 %, from 3,000 to 2,100 Pa.
Worked example with the values above
- Slope: tan φ = (18,000 − 8,000) / (30,000 − 10,000) = 0.5, so φ = 26.565°.
- Slope term at test 1: σ₁ tan φ = 10,000 × 0.5 = 5,000 Pa.
- c = τ₁ − σ₁ tan φ = 8,000 − 5,000 = 3,000 Pa.
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
- Mixing plate widths, or using width where the radius is needed.
- Using test points from different soil states or different loading branches.
- Using a shear point at or beyond the plateau, where the deformation length cannot be found.
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