Library / Terramechanics and off-road mobility / TER-37
Terramechanics and off-road mobility
Mean shear stress inferred from annular shear torque
Mean shear stress on the face of an annular shear ring of outer radius ro and inner radius ri, from the torque T after instrument drag is removed.
- Ref
- TER-37
- Rev
- v1
- Field
- Terramechanics and off-road mobility
- Result
- τ [Pa]
A ring of outer radius ro and inner radius ri, pressed on the soil and turned, shears an annulus. For a uniform shear stress τ on the face the torque is T = ∫τ·r·2πr dr = 2πτ(ro³ − ri³)/3, so τ = 3T/(2π(ro³ − ri³)). The torque must be corrected for instrument friction, and the stress is only meaningful if the shear is uniform, which is a reasonable approximation at the peak.
Mean shear stress on the face of an annular shear ring of outer radius ro and inner radius ri, from the torque T after instrument drag is removed. It assumes uniform mobilised shear on the horizontal face and excludes sidewall and grouser forces.
Calculator
- Face area of the ring, π(ro² − ri²)0.02356 m²
- Equivalent shear force on the face, τ × area128.6 N
- Effective radius of the shear, 2(ro³ − ri³)/(3(ro² − ri²))0.07778 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 torque of 10 N m on the ring means a mean shear stress of 5,457 Pa, or 5.457 kPa.
- The equivalent shear acts at an effective radius of 0.07778 m, which is why the stress varies with the cube of the radius.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Pa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| T Net torque | 0.1 to 500 N m | 54.57 | 272,800 | 272,800 | +10% | +10 % |
| ro Outer radius | 0.06 to 0.5 m | 38.24 | 52,470 | 52,430 | +10% | −27.4 % |
| ri Inner radius | 0 to 0.059 m | 4,775 | 6,009 | 1,234 | +10% | +4.96 % |
Sweep of Net torque
| Net torque (N m) | Mean shear stress (Pa) |
|---|---|
| 0.1 | 54.57 |
| 0.29 | 158.2 |
| 0.8409 | 458.9 |
| 2.438 | 1,331 |
| 7.071 | 3,858 |
| 20.5 | 11,190 |
| 59.46 | 32,450 |
| 172.4 | 94,090 |
| 500 | 272,800 |
Sweep one variable
Variables and how each one contributes
Mean shear stress
What the equation gives you. Everything else feeds it.
Net torque
The net torque on the ring, in N m. Raises the stress in direct proportion. Raising it by 10 % (to 11) moves the result by +10 %, from 5,457 to 6,002 Pa.
Outer radius
The outer radius of the ring, in m. A larger outer radius spreads the torque over a larger area and lowers the stress with the cube. Raising it by 10 % (to 0.11) moves the result by −27.4 %, from 5,457 to 3,959 Pa.
Inner radius
The inner radius of the ring, in m. A larger inner radius removes the inner area and raises the stress slightly. Raising it by 10 % (to 0.055) moves the result by +4.96 %, from 5,457 to 5,728 Pa.
Worked example with the values above
- Cubes: ro³ − ri³ = 0.001 − 1.25×10−4 = 8.75×10−4 m³.
- Denominator: 2π × 8.75×10−4 = 0.0054978.
- τ = 3T / 0.0054978 = 3 × 10 / 0.0054978 = 5,456.7 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