Relation Between Dry Unit Weight, Specific Gravity, Percentage of Air Voids and Water Content

19th April 2020 0 By Malay Sautya

Relation Between Dry Unit Weight, Specific Gravity, Percentage of Air Voids and Water Content

In this article, we shall make the formula or relation between the dry unit weight( $\gamma _{d}$ ), specific gravity(G), percentage of air voids and water content(w).

[Note: Dry unit weight means dry density]

Soil three-phase diagram is shown in the above picture. From this diagram, we can write,

Total Volume = Volume of solids ( $V_{s}$ )+ Volume of water ( $V_{w}$ ) + Volume of air ( $V_{a}$ )

$V = V_{s} + V_{w} + V_{a}$
[Both sides divided by V]
Or, $\frac{V}{V}= \frac{V_{s}}{V} + \frac{V_{w}}{V} + \frac{V_{a}}{V}$

[We know, $\frac{V_{a}}{V} = n_{a}$ ]

Or, $1 = \frac{V_{s}}{V} + \frac{V_{w}}{V} + n_{a}$
Or, $1 - n_{a} = \frac{V_{s}}{V} + \frac{V_{w}}{V}$

[ We know, $\gamma_{s} = \frac{W_{s}}{V_{s}}$ , and Specific Gravity(G) $= \frac{\gamma_{s}}{\gamma_{w} } = \frac{W_{s}}{V_{s}\gamma_{w} }$ ]

[From this equation, $G = \frac{W_{s}}{V_{s}\gamma_{w} }$ , we can write $V_{s} = \frac{W_{s}}{G\gamma _{w}}$ . Now, place the value of $V_{s}$ in the above equation.]

Or, $1 - n_{a} = \frac{W_{s}/G\gamma_{w}}{V} + \frac{W_{w}/\gamma _{w}}{V}$

[We know, Density of solids $(\gamma_{d}) = \frac{W_{s}}{V}$ ]

Or, $1 - n_{a} = \frac{\gamma_{d}}{G\gamma_{w} } + \frac{wW_{s}/\gamma _{w}}{V}$
Or, $1 - n_{a} = \frac{\gamma_{d}}{G\gamma_{w} } + \frac{w\gamma_{d} }{\gamma_{w} }$
Or, $1 - n_{a} = \frac{\gamma_{d}}{\gamma_{w} }\left [ w + \frac{1}{G} \right ]$
$\gamma_{d} = \frac{(1 - n_{a})G \gamma_{w}}{1 + wG}$