Question

If Y_d = dry density, Y = bulk density and w = water content, then the correct relationship of these will be
1. \(Y = \dfrac{Y_d}{1-w}\)
2. \(Y_d = \dfrac{Y}{1+w}\)
3. \(Y_d = \dfrac{Y}{1-w}\)
4. \(Y = \dfrac{Yd}{1 + w}\)

Answer 1

Correct Answer - Option 2 : \(Y_d = \dfrac{Y}{1+w}\)

Concept: 

Water Content ( W ):

It is defined as the ratio of the weight of water to the weight of solid in a given soil mass. It is usually expressed as a percentage.

 \(w = \;\frac{{{W_W}}}{{{W_S}}} \times 100 = \;\frac{{W - {W_S}}}{{{W_S}}} \times 100\)

Bulk density:

It is defined as the ratio of the total mass ( M ) of the soil to the total volume ( V ) of the soil. It is expressed in Kg / m³

\(\rho = \;\frac{M}{V}\)

Dry density:

It is defined as the ratio of the mass of Solids to the total Volume of the soil ( in moist condition ).

\({\rho _d} = \;\frac{{{M_S}}}{V}\)

Relation between dry unit weight, bulk unit weight, and water content ( w ).

We know that,

\(w = \frac{{{W_W}}}{{{W_S}}}\)

\(w + 1 = \;\frac{{{W_W}}}{{{W_S} + 1}}\) =\(\frac{{{W_{W + {W_S}}}}}{W}\)

\(w + 1 = \;\frac{W}{{{W_S}}}\)

\({W_S} = \;\frac{W}{{1 + w}}\)

Dividing both sides by V, we get

\(\frac{W_s}{V} = \frac{W}{V(1+w)}\)

\({\gamma _{d\;}} = \;\frac{\gamma }{{1 + w}}\;\)