As the literal meaning of this word appear, relative density is the measure of looseness in relation with something. So you might be thinking why this relative word means. This relative word is used because we are measuring with respect to loosest state of the soil. It is denoted by \(D_r(I_D)\). Hence, this is very important engineering parameter to work on coarse grained soil. To determine this in the laboratory IS code IS-2720-Part-15 is used.

## Definition

So, if you want to define the relative density for cohesion less soils is the ratio of the difference between the void ratio of a cohesion- less soil in the loosest state and in-situ void ratio to the difference between its void ratios in the loosest and the densest states which is written as:

$$D_r or I_D = \frac {e_{max}-e_{min}}{e_{max}-e_{min}}\times100\%$$

where :

\(e_{max}\) = void ratio in the loosest state

\(e_{min}\) = void ratio in the densest state

\(e_{nat}\) = void ratio obtained in the field in the natural state

## Significance of relative density

It is very good indicator of the denseness of coarse grained soil deposit, in comparison to void ratio hence used frequently in the filed. So, if we have two sand deposits which are having same grain shape and size characteristics can show quite different engineering behavior if they have different \(D_r\).

If the \(D_r\) is high the granular soil will be dense and it will have high shear strength and low compressibility. Small \(D_r\) granular soil is unstable.

Also read: void ratio, permeability of soil, scope of geotechnical engineering, water content of soil, specific gravity,

## Limit for granular soil

You must be thinking the use of \(D_r\) in the field, so we generally categorize the coarse grain soil on the basis of \(D_r\) in the following categories:

Relative density % | Classification |

<15 | Very loose |

15-35 | Loose |

35-65 | Medium |

65-85 | Dense |

>85 | Very dense |

## Relative density in terms of density

To determine \(D_r\) in terms of density we must recall some basic relations,

$$\mathrm{e}=\mathrm{G} \gamma_{\mathrm{w}} / \gamma_{\mathrm{d}}-1$$

Similarly, we can find the void ratio also for loosest and densest state –

$$ \mathrm{e}_{max}=\mathrm{G} \gamma_{\mathrm{w}} / \gamma_{\mathrm{d}min}-1\\ \mathrm{e}_{min}=\mathrm{G} \gamma_{\mathrm{w}} / \gamma_{\mathrm{d}max}-1 $$

Hence, substituting these values in above equation, we obtain the expression for relative density as

$$

D_{\mathrm{r}}=\frac{\left[\left(G \gamma_{\mathrm{w}} / \gamma_{\mathrm{d} \min }\right)-\left(G \gamma_{\mathrm{w}} / \gamma_{\mathrm{d}}\right)\right]}{\left[\left(G \gamma_{\mathrm{w}} / \gamma_{\mathrm{d} \min }\right)-\left(G \gamma_{\mathrm{w}} / \gamma_{\mathrm{d} \max }\right)\right]} \\\Rightarrow D_{\mathrm{r}} =\frac{\left(1 / \gamma_{\mathrm{d} \min }\right)-\left(1 / \gamma_{\mathrm{d}}\right)}{\left[\left(1 / \gamma_{\mathrm{d} \min }\right)-\left(1 / \gamma_{\mathrm{d} \max }\right)\right]} \\\Rightarrow D_{\mathrm{r}} =\frac{\left(\gamma_{\mathrm{d}}-\gamma_{\mathrm{d} \min } / \gamma_{\mathrm{d} \min } \cdot \gamma_{\mathrm{d}}\right)}{\left(\gamma_{\mathrm{d}_{m}}-\gamma_{\mathrm{d} \min } / \gamma_{\mathrm{d} \min } \cdot \gamma_{\mathrm{d} \max }\right)} \\\Rightarrow D_{\mathrm{r}} =\frac{\gamma_{\mathrm{d} \max }}{\gamma_{\mathrm{d}}}\left[\frac{\gamma_{\mathrm{d}}-\gamma_{\mathrm{dmin}}}{\gamma_{\mathrm{dmax}}-\gamma_{\mathrm{dmin}}}\right]

$$

Finally,we can use this relation to determine the relative density in the laboratory.

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