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Terramechanics and off-road mobility

Rigid wheel resistance on clay

Motion resistance of a smooth rigid wheel rolling on clay, from the load on the wheel, its size and the clay strength.

Ref: TER-21
Rev: v1
Field: Terramechanics and off-road mobility
Result: R_t [kN]

R_{t} = \frac{W^{2}}{5.7 d b c_{u}}

For the lecture’s smooth rigid wheel on clay, the motion resistance is R_t = W²/(5.7 d b c_u). W is the load on that wheel, d its diameter and b its width; in consistent kN, m and kPa the result is in kN. The load enters squared, so resistance grows faster than load. The relation represents the lecture’s approximation and is not a general tyre model. The constant 5.7 is the strip bearing factor.

Motion resistance of a smooth rigid wheel rolling on clay, from the load on the wheel, its size and the clay strength. The load enters squared, so resistance grows faster than load.

Calculator

Solve for

R_tWheel resistanceComputed

WLoad on the wheel

2100

dWheel diameter

0.31.5

bWheel width

0.10.8

c_uUndrained shear strength

kPa

5200

Resistance as a fraction of the wheel load, R_t/W0.3655
Wheel load that would give a resistance of 10 % of the load5.472 kN

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.

Wheel resistance (kN) current value 7.31 kN

WLoad on the wheelKey driver

0.0731183

c_uUndrained shear strength

1.4658.5

bWheel width

2.7421.9

dWheel diameter

3.919.5

A 20 kN load on a 0.8 m by 0.3 m wheel in 40 kPa clay meets a resistance of 7.31 kN, which is 36.5 % of the load. Halving the load to 10 kN cuts the resistance to 1.827 kN, a quarter, not a half.
Doubling the wheel width and diameter cuts the resistance by a factor of four: 1.827 kN. This is why large, wide, low-pressure tyres help in soft ground.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (kN)	Local step	Change in result
W Load on the wheel	2 to 100 kN	0.0731	182.7	182.7	+10%	+21 %
c_u Undrained shear strength	5 to 200 kPa	1.462	58.48	57.02	+10%	−9.09 %
b Wheel width	0.1 to 0.8 m	2.741	21.93	19.19	+10%	−9.09 %
d Wheel diameter	0.3 to 1.5 m	3.899	19.49	15.59	+10%	−9.09 %

Sweep of Load on the wheel

Load on the wheel (kN)	Wheel resistance (kN)
2	0.0731
14.25	3.711
26.5	12.83
38.75	27.44
51	47.53
63.25	73.11
75.5	104.2
87.75	140.7
100	182.7

Sweep one variable

Vary

Wheel resistance (kN)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

R_t

Wheel resistance

Result · kN

What the equation gives you. Everything else feeds it.

Load on the wheel

kN · typical 2 to 100

The load on the wheel, in kN. Raises the resistance with the square of the load: a 10 % rise gives about a 21 % rise. Raising it by 10 % (to 22) moves the result by +21 %, from 7.31 to 8.845 kN.

Wheel diameter

m · typical 0.3 to 1.5

The wheel diameter, in m. Lowers the resistance in inverse proportion: a larger wheel sinks less and rolls more easily. Raising it by 10 % (to 0.88) moves the result by −9.09 %, from 7.31 to 6.645 kN.

Wheel width

m · typical 0.1 to 0.8

The wheel width, in m. Lowers the resistance in inverse proportion: a wider wheel spreads the load. Raising it by 10 % (to 0.33) moves the result by −9.09 %, from 7.31 to 6.645 kN.

c_u

Undrained shear strength

kPa · typical 5 to 200

The undrained shear strength of the clay, in kPa. Lowers the resistance in inverse proportion: firmer clay offers less resistance. Raising it by 10 % (to 44) moves the result by −9.09 %, from 7.31 to 6.645 kN.

Worked example with the values above

Numerator: W² = 20² = 400 kN².
Denominator: 5.7 d b c_u = 5.7 × 0.8 × 0.3 × 40 = 54.72 kN.
R_t = 400 / 54.72 = 7.31 kN.

Assumptions and limits

A smooth rigid wheel on saturated clay, in the lecture’s approximation.
Consistent units: kN, m and kPa give a resistance in kN.

Common mistakes

Using the total vehicle weight instead of the load on one wheel.
Applying the relation to a pneumatic tyre or to firm ground.

Statistics and uncertainty

The relation is quadratic in load and inverse in strength, so a 30 % uncertainty in c_u alone gives a 30 % uncertainty in resistance, and a 10 % load error gives 21 %. Treat it as an estimate, not a prediction.

Where it is used

Estimating how much force a wheel needs to roll through soft clay.
Comparing wheel sizes for soft-ground vehicles.

Maths you need

Undrained strength: strength of clay under quick loading
Squares: the load enters squared

Related formulas

Practise this formula