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Terramechanics and off-road mobility
Mohr–Coulomb soil shear strength
The maximum shear stress soil can carry before it fails: a cohesion term plus the effective normal stress times the tangent of the soil friction angle.
- Ref
- TER-12
- Rev
- v1
- Field
- Terramechanics and off-road mobility
- Result
- τmax [kPa]
c is the soil cohesion, σ′ the effective normal stress on the shear surface (total stress minus pore-water pressure for saturated soil) and φs the soil friction angle. The shear strength is the cohesion plus the friction that the normal stress mobilises. Use total-stress or effective-stress parameters to match the test conditions. In terramechanics this sets the maximum shear stress a soil can give under a wheel or track, so it limits the thrust. Dry sand has c ≈ 0, while saturated clay has a small φs and a large c.
The maximum shear stress soil can carry before it fails: a cohesion term plus the effective normal stress times the tangent of the soil friction angle. It sets the limit of soil thrust under a wheel or track.
Calculator
- Frictional strength, σ′ tan φs28.87 kPa
- Cohesion as a share of the strength6.479 %
- Friction coefficient of the soil, tan φs0.5774
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 c = 2 kPa, σ′ = 50 kPa and φs = 30° the soil can carry 30.87 kPa of shear, of which 6.48 % is cohesion. The soil is mostly frictional, so its strength rises with the load on it.
- A cohesive soil behaves differently: with c = 25 kPa and φs = 0 the strength is 25 kPa whatever the load, so adding weight gives no extra thrust on saturated clay, and the vehicle can only increase thrust by increasing the contact area.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (kPa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| σ′ Effective normal stress | 5 to 300 kPa | 4.887 | 175.2 | 170.3 | +10% | +9.35 % |
| c Soil cohesion | 0 to 50 kPa | 28.87 | 78.87 | 50 | +10% | +0.648 % |
| φs Soil friction angle | 0 to 45 ° | 2 | 52 | 50 | +10% | +11.7 % |
Sweep of Effective normal stress
| Effective normal stress (kPa) | Maximum shear strength (kPa) |
|---|---|
| 5 | 4.887 |
| 41.88 | 26.18 |
| 78.75 | 47.47 |
| 115.6 | 68.76 |
| 152.5 | 90.05 |
| 189.4 | 111.3 |
| 226.3 | 132.6 |
| 263.1 | 153.9 |
| 300 | 175.2 |
Sweep one variable
Variables and how each one contributes
Maximum shear strength
What the equation gives you. Everything else feeds it.
Soil cohesion
The soil cohesion, in kPa: near 0 for dry sand, 1 to 5 kPa for sandy loam and 10 to 100 kPa or more for clay. Adds directly. It matters most in clay and in light-load conditions where the normal stress is small. Raising it by 10 % (to 2.2) moves the result by +0.648 %, from 30.87 to 31.07 kPa.
Effective normal stress
The effective normal stress on the shear surface, in kPa: the contact pressure less any pore-water pressure. Raises the friction part in direct proportion, so a heavier vehicle on the same area can get more thrust from frictional soils. Raising it by 10 % (to 55) moves the result by +9.35 %, from 30.87 to 33.75 kPa.
Soil friction angle
The soil friction angle, in degrees: 25° to 35° for sands and 15° to 25° for clays. Raises the friction part through its tangent, which is the strongest lever for frictional soils. Raising it by 10 % (to 33) moves the result by +11.7 %, from 30.87 to 34.47 kPa.
Worked example with the values above
- Friction coefficient: tan φs = tan 30° = 0.5774.
- Frictional term: σ′ tan φs = 50 × 0.5774 = 28.87 kPa.
- τmax = c + 28.87 = 2 + 28.87 = 30.87 kPa.
Assumptions and limits
- A homogeneous soil with linear (Mohr–Coulomb) strength and parameters that match the drainage conditions of the loading.
- The normal stress σ′ is the effective stress on the shear surface, and strength does not soften after the peak.
Common mistakes
- Using total stress for saturated, drained soil. Subtract the pore pressure first.
- Confusing the shear cohesion c with the Bekker sinkage modulus kc.
Statistics and uncertainty
Soil parameters are rarely known precisely. The friction angle varies by ±3° to ±5° between tests on the same soil, which is a ±10 to 15 % change in tan φs, and the cohesion varies even more with moisture, by a factor of two or more. The strength therefore carries an uncertainty of about 15 to 30 %, which carries straight through to the thrust a vehicle can develop on that soil.
Where it is used
- Predicting the thrust available on soil, and slope and foundation stability.
- Choosing tyre or track design: more contact area for clay, more load for sand.
Maths you need
- Friction: τ = c + σ tan φ
- Effective stress: σ′ = σ − u