# Density Index and relative compaction

Learn Density Index, relative compaction, density index and void ratio relationship,charectristics of granular soils in dense and loose states,Relative density

## Density Index

The term density index ID or relative density or degree of density is used to express the relative compactness (or degree of compaction) of a natural cohesionless soil deposit.

The density index is defined as the ratio of the difference between the voids ratio of the soil in its loosest state emax and its natural voids ratio e to the difference between the voids ratios in the loosest and densest states:

\[I_{D} = \frac{e_{max}-e}{e_{max}-e_{min}}. (1)\]

where

e_{max} = voids ratio in the loosest state

e_{min} = voids ratio in the densest state

e = natural voids ratio of the deposit.

This term is used for cohesionless soil only. This term is not applicable to cohesive soil because of uncertaintities in the laboratory determination of the voids ratio in the loosest state of the soil (e_{max}). When the natural state of the cohesionless soil is in its loosest form,

e = e_{max} and hence ID = 0.

When the natural deposit is in its densest state, e = e_{min} and hence ID = 1.

For any intermediate state, the density index varies between zero and one.

Equation 1, defining density index, can be easily derived by noting the fact that the density index is a function of voids ratio expressed by:

ID = f (e) …(i)

### Relationship between density index and void ratio relationship

This relationship between ID and e may be represented graphically, as shown in fig.

The slope of the straight line AB, representing the relationship between ID and e is given by

\[tanθ = \frac{1}{e_{max}-e_{min}}\]

\[cotθ = (e_{max }– e_{min}) …(ii)\]

Now, for an intermediate value e we have,

(e_{max} – e) = ID cotθ or

\[ID = \frac{e_{max}-e}{cotθ} …(iii)\]

Substituting the value of cotθ from equation ii, we get

\[I_{D} = \frac{e_{max}-e}{e_{max}-e_{min}}\]

From fig., we observe that when e = e_{max}, ID = 0 and when e = e_{min}, ID = 1. Now from equation, we have

\[e=\frac{G\gamma_{w}}{\gamma_{d}}-1\]

\[e_{max}=\frac{G\gamma_{w}}{\gamma_{d,min}}-1\]

\[e_{min}=\frac{G\gamma_{w}}{\gamma_{d,max}}-1\]

\[[I_{D} = \frac{\gamma_{d}-\gamma_{d,min}}{\gamma_{d,max}-\gamma_{d,min}}][\frac{\gamma_{dmax}}{\gamma_{d}}]\]

The above equation gives density index in terms of densities.

Density index is also expressed in terms of porosity as follows.

\[I_{d} = \frac{(n_{max}-n)(1-n_{min})}{(n_{max}-n_{min})(1-n)}\]

where,

ϒd = in-situ dry density;

η = in-situ porosity.

ϒd, max = maximum dry density or dry density corresponding to most compact state.

ϒd, min = minimum dry dnsity or dry density corresponding to most loosest state.

ηmax = maximum porosity at loosest state;

ηmin = minimum porosity at densest state.

### Table 1 Charactristics of granular soils in dense and loose states

gives the maximum and minimum voids ratio, porosity and dry unit weight of some typical granular soils.

[table id = 10\]

### Table 2.Relative density

gives the characteristics of density of granular soils on the basis of relative density.

[table id=11 /]

## Relative Compaction (RC)

Degree of compaction is also sometimes expressed in terms of an index called relative compaction (RC) defined as follows:

\[R_{C} = \frac{\gamma_{d}}{\gamma_{damas}}\]

where,

ϒd,max is the maximum dry density from compaction test.

In recent years, the use of the above index has become a generally accepted practice for judging the measure of compaction of both coarse-grained as well as cohesive soils. Since ϒd = ϒs (1 + e),in general, we have

\[R_{C} = \frac{1+e_{min}}{1+e}\]

Relative compaction (RC) can also be expressed in terms of relative density (ID) as follows

\[R_{C} = \frac{R_{0}}{1-I_{D}(1-R_{0}}\]

where,

\[R_{0} = \frac{ϒ_{d,min}}{ϒ_{d,max}}\]

and RC and ID are in fraction form.

Lee and Singh (1971) give the following approximate relation between RC and ID

RC = 80 + 0.2 ID

(where both RC and ID are in percent form) When the soil is in looset form, ID = 0, which gives minimum value of RC as 80% from equation . When the soil is in densest form, ID = 100% corresponding to which RC = 100% from equation Thus, relative compaction varies from 80% to 100% according to equation.