Library / Terramechanics and off-road mobility / TER-48
Terramechanics and off-road mobility
Work input per volume swept by a penetration cone
Depth-averaged penetration resistance, from the cone tip force F measured at increasing depth z (five nodes, trapezoidal integration), divided by the projected cone base area Ac and the depth interval.
- Ref
- TER-48
- Rev
- v1
- Field
- Terramechanics and off-road mobility
- Result
- W/V [J/m³]
A penetrometer records tip force F against depth z. The work to push the cone from depth z₁ to zn is ∫F dz, and dividing by the swept volume Ac(zn − z₁) gives the depth-averaged resistance in J/m³, which equals Pa. This depth average is a single soil-strength index that is less noisy than the peak value. The page uses five nodes and the trapezoidal rule.
Depth-averaged penetration resistance, from the cone tip force F measured at increasing depth z (five nodes, trapezoidal integration), divided by the projected cone base area Ac and the depth interval. Isolate the tip force from rod friction, and use the same cone and rate when comparing soils. Unused nodes repeat the last depth.
Calculator
- Penetration work, ∫F dz37.5 J
- Depth interval, zn − z₁0.15 m
- Mean cone force over the depth interval250 N
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 soil offers 2.5 MPa on average, from 37.5 J of work over a depth of 15 cm.
- Cone indices of a few hundred kPa to a few MPa are typical of trafficable soils, so compare the number with your vehicle’s required index, using the same cone and rate.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (J/m³) | Local step | Change in result |
|---|---|---|---|---|---|---|
| z5 Depth, node 5 | 0.001 to 1 m | −2.21e8 | 3,775,000 | 2.248e8 | +10% | +5.45 % |
| F2 Cone tip force, node 2 | 0 to 5,000 N | 1,500,000 | 2.65e7 | 2.5e7 | +10% | +4 % |
| Ac Cone base area | 1e-5 to 0.001 m² | 250,000 | 2.5e7 | 2.475e7 | +10% | −9.09 % |
| F3 Cone tip force, node 3 | 0 to 5,000 N | 1,167,000 | 1.783e7 | 1.667e7 | +10% | +5.33 % |
| z2 Depth, node 2 | 0.001 to 1 m | −7,000,000 | 2,990,000 | 9,990,000 | +10% | −2 % |
| F1 Cone tip force, node 1 | 0 to 5,000 N | 2,333,000 | 1.067e7 | 8,333,000 | +10% | +0.667 % |
| z3 Depth, node 3 | 0.001 to 1 m | −3,167,000 | 3,493,000 | 6,660,000 | +10% | −4 % |
| z1 Depth, node 1 | 0 to 0.05 m | 2,500,000 | 3,000,000 | 500,000 | +10% | 0 % |
| z4 Depth, node 4 | 0.001 to 1 m | 2,500,000 | 2,500,000 | 9.313e-10 | +10% | 0 % |
| F4 Cone tip force, node 4 | 0 to 5,000 N | 2,500,000 | 2,500,000 | 0 | +10% | 0 % |
| F5 Cone tip force, node 5 | 0 to 5,000 N | 2,500,000 | 2,500,000 | 0 | +10% | 0 % |
Sweep of Cone tip force, node 3
| Cone tip force, node 3 (N) | Work per volume swept (J/m³) |
|---|---|
| 0 | 1,167,000 |
| 625 | 3,250,000 |
| 1,250 | 5,333,000 |
| 1,875 | 7,417,000 |
| 2,500 | 9,500,000 |
| 3,125 | 1.158e7 |
| 3,750 | 1.367e7 |
| 4,375 | 1.575e7 |
| 5,000 | 1.783e7 |
Sweep one variable
Variables and how each one contributes
Work per volume swept
What the equation gives you. Everything else feeds it.
Cone base area
The projected base area of the cone, in m². A larger area for the same force lowers the average pressure in inverse proportion. Raising it by 10 % (to 1.1×10−4) moves the result by −9.09 %, from 2,500,000 to 2,273,000 J/m³.
Depth, node 1
The depth of node 1, in m. Sets the start of the interval. Raising it by 10 % (to 0) moves the result by +0 %, from 2,500,000 to 2,500,000 J/m³.
Depth, node 2
The depth of node 2, in m. Changes the widths of the first two trapezoids. Raising it by 10 % (to 0.055) moves the result by −2 %, from 2,500,000 to 2,450,000 J/m³.
Depth, node 3
The depth of node 3, in m. Changes the widths of the middle trapezoids. Raising it by 10 % (to 0.165) moves the result by −4 %, from 2,500,000 to 2,400,000 J/m³.
Depth, node 4
The depth of node 4, in m. Changes the widths of the last two trapezoids. Raising it by 10 % (to 0.165) moves the result by +0 %, from 2,500,000 to 2,500,000 J/m³.
Depth, node 5
The depth of node 5, in m. Sets the end of the interval. Raising it by 10 % (to 0.165) moves the result by +5.45 %, from 2,500,000 to 2,636,000 J/m³.
Cone tip force, node 1
The cone tip force at node 1, in N. Raises the work through the first trapezoid. Raising it by 10 % (to 110) moves the result by +0.667 %, from 2,500,000 to 2,517,000 J/m³.
Cone tip force, node 2
The cone tip force at node 2, in N. Raises the work through the first two trapezoids. Raising it by 10 % (to 220) moves the result by +4 %, from 2,500,000 to 2,600,000 J/m³.
Cone tip force, node 3
The cone tip force at node 3, in N. Raises the work through the middle trapezoids. Raising it by 10 % (to 440) moves the result by +5.33 %, from 2,500,000 to 2,633,000 J/m³.
Cone tip force, node 4
The cone tip force at node 4, in N. Raises the work through the last two trapezoids. Raising it by 10 % (to 440) moves the result by +0 %, from 2,500,000 to 2,500,000 J/m³.
Cone tip force, node 5
The cone tip force at node 5, in N. Raises the work through the last trapezoid. Raising it by 10 % (to 440) moves the result by +0 %, from 2,500,000 to 2,500,000 J/m³.
Worked example with the values above
- Force–depth nodes (z m, F N): (0, 100) (0.05, 200) (0.15, 400) (0.15, 400) (0.15, 400).
- Work: ∫F dz = 37.5 J; swept volume: Ac (zn − z₁) = 1×10−4 × 0.15 = 1.5×10−5 m³.
- W/V = 37.5 / 1.5×10−5 = 2,500,000 J/m³.
Assumptions and limits
- A rigid, flat track patch or cone in homogeneous soil, with a linear pressure profile where stated.
- Constant soil strength and deformation length over the test, and the same rate and cone for comparisons.
Common mistakes
- Using the linear pressure formula beyond the one-sixth limit on the load offset.
- Mixing the left and right sign convention for yaw moment.
- Including rod friction in the cone force.
Statistics and uncertainty
Pressure cells and penetrometers carry 5 % to 10 % uncertainty, and the penetration or shear work depends on rate and soil state. Compare only tests run under the same conditions.
Where it is used
- Checking track pressure distribution, thrust sharing and steering demand.
- Comparing soil strength and shear energy between test beds.
Maths you need
- Linear pressure distribution: p = W/A (1 ± 6e/L)
- Work as an integral: W = ∫ F dz
- Moment balance: the sum of moments about the vehicle centre is zero