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Terramechanics and off-road mobility

Limiting cone index, wheeled vehicle

Cone index the soil must have for a wheeled vehicle to keep moving, from its weight and the tyre dimensions and deflection.

Ref: TER-27
Rev: v1
Field: Terramechanics and off-road mobility
Result: CI_L,wheel [kPa]

C I_{L,wheel} = \frac{1.85 W}{2 n b^{0.8} d^{0.8} δ^{0.4}}

The lecture’s limiting cone index for a wheeled vehicle is CI_L = 1.85 W/(2n b^0.8 d^0.8 δ^0.4), with W the total weight in kN, n the number of axles per side, b the tyre width, d the unloaded diameter and δ the tyre deflection, all in m. The lecture’s dimensional convention produces kPa. It is an empirical correlation, not a guaranteed pass or fail prediction. A bigger tyre, a wider one, or more deflection (lower pressure) all lower the strength the soil needs.

Cone index the soil must have for a wheeled vehicle to keep moving, from its weight and the tyre dimensions and deflection. Like the tracked version it is an empirical correlation, not a guarantee.

Calculator

Solve for

CI_L,wheelLimiting cone indexComputed

kPa

WVehicle weight

20600

nAxles per side

bTyre width

0.150.7

dUnloaded tyre diameter

0.51.6

δTyre deflection

0.020.3

Limiting cone index in psi52.61 psi
Cone index required with tyre deflection doubled274.9 kPa

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.

Limiting cone index (kPa) current value 362.7 kPa

WVehicle weightKey driver

48.41,450

nAxles per side

121725

bTyre width

232795

δTyre deflection

234690

dUnloaded tyre diameter

249632

A 150 kN four-wheeler on 0.4 m wide, 1 m diameter tyres with 0.1 m deflection needs soil of 362.7 kPa. Doubling the deflection by lowering the tyre pressure reduces that to 274.9 kPa, a fall of 24.2 %.
A wider tyre helps almost as much: increasing the width from 0.4 to 0.6 m gives 262.2 kPa.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (kPa)	Local step	Change in result
W Vehicle weight	20 to 600 kN	48.36	1,451	1,402	+10%	+10 %
n Axles per side	1 to 6	120.9	725.4	604.5	+1	−33.3 %
b Tyre width	0.15 to 0.7 m	231.8	794.9	563.1	+10%	−7.34 %
δ Tyre deflection	0.02 to 0.3 m	233.7	690.5	456.7	+10%	−3.74 %
d Unloaded tyre diameter	0.5 to 1.6 m	249	631.5	382.5	+10%	−7.34 %

Sweep of Tyre deflection

Tyre deflection (m)	Limiting cone index (kPa)
0.02	690.5
0.055	460.7
0.09	378.3
0.125	331.7
0.16	300.5
0.195	277.7
0.23	259.9
0.265	245.6
0.3	233.7

Sweep one variable

Vary

Limiting cone index (kPa)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

CI_L,wheel

Limiting cone index

Result · kPa

What the equation gives you. Everything else feeds it.

Vehicle weight

kN · typical 20 to 600

The total weight of the vehicle, in kN. Raises the limiting cone index in proportion. Raising it by 10 % (to 165) moves the result by +10 %, from 362.7 to 399 kPa.

Axles per side

· typical 1 to 6

The number of axles on each side, a whole number. Lowers the limiting cone index in inverse proportion. Raising it by 1 (to 3) moves the result by −33.3 %, from 362.7 to 241.8 kPa.

Tyre width

m · typical 0.15 to 0.7

The tyre width, in m. Lowers the limiting cone index, with the power 0.8 of the width. Raising it by 10 % (to 0.44) moves the result by −7.34 %, from 362.7 to 336.1 kPa.

Unloaded tyre diameter

m · typical 0.5 to 1.6

The unloaded tyre diameter, in m. Lowers the limiting cone index, with the power 0.8 of the diameter. Raising it by 10 % (to 1.1) moves the result by −7.34 %, from 362.7 to 336.1 kPa.

Tyre deflection

m · typical 0.02 to 0.3

The tyre deflection, in m: how much the tyre flattens under load. More deflection (lower pressure) spreads the load. Lowers the limiting cone index, with the power 0.4 of the deflection. Raising it by 10 % (to 0.11) moves the result by −3.74 %, from 362.7 to 349.1 kPa.

Worked example with the values above

Numerator: 1.85 W = 1.85 × 150 = 277.5 kN.
Denominator: 2n b^0.8 d^0.8 δ^0.4 = 4 × 0.4804 × 1 × 0.3981 = 0.7651.
CI_L = 362.7 kPa.

Assumptions and limits

A wheeled vehicle on fine-grained soil, with the lecture’s correlation constants.
All dimensions in m and the weight in kN, giving kPa.

Common mistakes

Using the loaded diameter instead of the unloaded diameter.
Using the wheel count in place of the axle count per side.

Statistics and uncertainty

The empirical exponents (0.8, 0.8, 0.4) fit data from a limited range of tyres and soils. Treat a difference of less than 10 % between two designs as no real difference, and extrapolate with care.

Where it is used

Choosing tyres or tyre pressures for soft ground.
Comparing wheeled vehicles with tracked ones on the same soil.

Maths you need

Cone index: penetrometer resistance of soil
Powers: fractional exponents on tyre dimensions

Related formulas

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