Library / Terramechanics and off-road mobility / TER-34
Terramechanics and off-road mobility
Relative density of a granular test bed
Where the void ratio e of a cohesionless soil lies between its loosest state emax and densest state emin, as a fraction.
- Ref
- TER-34
- Rev
- v1
- Field
- Terramechanics and off-road mobility
- Result
- Dr
Relative density describes where a granular soil sits between its loosest state (emax) and its densest state (emin): Dr = (emax − e)/(emax − emin). A value of 0 is the loosest state and 1 the densest; below 0.35 is usually called loose, above 0.65 dense. The reference values are found by standard tests, so Dr depends on the test method used.
Where the void ratio e of a cohesionless soil lies between its loosest state emax and densest state emin, as a fraction. It uses reference values determined for that material and test method; report values outside 0 to 1 for review.
Calculator
- Relative density in percent62.5 %
- Void ratio range, emax − emin0.4
- Void ratio at a relative density of 50 %0.7
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 bed is at a relative density of 62.5 %, which is medium dense.
- The void ratio would have to fall to 0.7 to reach 50 %, so the bed is denser than that midpoint.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| e Measured void ratio | 0.5 to 1.5 | −1.5 | 1 | 2.5 | +10% | −26 % |
| emin Minimum void ratio (densest state) | 0.2 to 0.5 | 0.3571 | 0.625 | 0.2679 | +10% | +14.3 % |
| emax Maximum void ratio (loosest state) | 0.9 to 1.5 | 0.625 | 0.85 | 0.225 | +10% | +11 % |
Sweep of Measured void ratio
| Measured void ratio | Relative density |
|---|---|
| 0.5 | 1 |
| 0.625 | 0.6875 |
| 0.75 | 0.375 |
| 0.875 | 0.0625 |
| 1 | −0.25 |
| 1.125 | −0.5625 |
| 1.25 | −0.875 |
| 1.375 | −1.187 |
| 1.5 | −1.5 |
Sweep one variable
Variables and how each one contributes
Relative density
What the equation gives you. Everything else feeds it.
Maximum void ratio (loosest state)
The maximum void ratio, the loosest state. A larger emax gives a larger relative density for the same e. Raising it by 10 % (to 0.99) moves the result by +11 %, from 0.625 to 0.6939.
Measured void ratio
The measured void ratio of the bed. A looser bed (larger e) lowers the relative density in direct proportion. Raising it by 10 % (to 0.715) moves the result by −26 %, from 0.625 to 0.4625.
Minimum void ratio (densest state)
The minimum void ratio, the densest state. A smaller emin widens the range and lowers the relative density for the same e. Raising it by 10 % (to 0.55) moves the result by +14.3 %, from 0.625 to 0.7143.
Worked example with the values above
- Numerator: emax − e = 0.9 − 0.65 = 0.25.
- Denominator: emax − emin = 0.9 − 0.5 = 0.4.
- Dr = 0.25 / 0.4 = 0.625.
Assumptions and limits
- Representative, intact specimens of known mass or volume, with the same units for all masses and densities.
- Solid particles of a single particle density, and fully drained, non-swelling soil.
Common mistakes
- Including the container mass in the masses.
- Entering water content in percent where a fraction is needed.
- Using a particle density that does not represent the soil, which shifts the void ratio and the saturation.
Statistics and uncertainty
Soil index quantities depend on sampling and handling: water content is typically repeatable to 1 % to 2 % of the value, and void ratio to about 0.02, but sampling disturbance or a non-representative specimen can dominate the error.
Where it is used
- Recording the state of a test bed before a plate, shear or wheel test.
- Checking soil data for consistency before fitting strength models.
Maths you need
- Phase diagram: solids, water and air volumes and masses
- Water content: mass of water over mass of dry solids
- Particle density: mass of solids per unit volume of solids