Question

A partially - saturated soil sample has a natural moisture content of 25% and a bulk unit weight of 18.5 kN/m³. The specific gravity of soil solids is 2.65 and the unit weight of water is 9.81 kN/m³. The unit weight of the soil sample on full saturation is
1. 21.12 kN/m³
2. 19.03 kN/m3
3. 18.50 kN/m3
4. 20.12 kN/m3

Answer 1

Correct Answer - Option 2 : 19.03 kN/m3

Concept:

Bulk Unit weight (γb): 

It is defined as the ratio of the total weight of soil to the total volume of the soil mass.

\({γ _b} = \frac{W}{V} = \frac{{{W_s} + {W_w}}}{{{V_s} + {V_w} + {V_a}}}\)

Water Content (w):​

Water content or moisture content of a soil mass is defined as the ratio of the weight of water to the weight of solids (dry weight) of the soil mass.

\({\rm{w = }}\frac{{{{\rm{W}}_{\rm{w}}}}}{{{{\rm{W}}_{\rm{s}}}}};{\rm{w}} \ge 0\)

It is denoted by the w and is commonly expressed as a percentage. The minimum value for water content is 0. There is no upper limit for water content.

Dry Unit Weight (γ­d):-​

Dry unit weight is defined as the weight of soil solids per unit volume of soil. It is denoted by the letter symbol γd it has the unit of kN/m3.

\({{\rm{γ }}_{\rm{d}}}{\rm{ = }}\frac{{{{\rm{W}}_{\rm{s}}}}}{{\rm{V}}}{\rm{ = }}\frac{{{{\rm{W}}_{\rm{d}}}}}{{\rm{V}}}\)

It is used as a measure of the denseness of soil. A high value of dry unit weight indicates that more solids are packed in a unit volume of soil hence a more compact soil.

The specific gravity of solids (G):-​

The specific gravity of solids is defined as the ratio of the unit weight of solids to the unit weight of water. It is denoted by the letter G and is a unitless quantity.

\({\rm{G = }}\frac{{{{\rm{γ }}_{\rm{s}}}}}{{{{\rm{γ }}_{\rm{w}}}}}\)

Note: The relationship between the degree of saturation, water content, specific gravity, and the void ratio is:

s × e = w × G

Calculation:

Given,

w = 25%, γ = 18.5 kN/m³, G = 2.65, γ_w = 9.81 kN/m3 ​​

We know, s × e = w × G 

​\(\gamma = \frac{{\left( {G + se} \right){\gamma _w}}}{{1 + e}}\) = \( \frac{{\left( {G + wG} \right){\gamma _w}}}{{1 + e}}\) = \(\frac{{G}{\left( {1 + w} \right){\gamma _w}}}{{1 + e}}\)​

\(\gamma = \frac{{G}{\left( {1 + w} \right){\gamma _w}}}{{1 + e}}\)

18.5 = \(\frac{{2.65}\times {\left( {1 + 0.25} \right)\times {9.81}}}{{1 + e}}\)

e = 0.756

For fully saturated soil sample(s=1)

\(\gamma_s = \frac{{\left( {G + e} \right){\gamma _w}}}{{1 + e}}\)

\(\gamma_s = \frac{{\left( {2.65 + 0.756} \right)\times {9.81}}}{{1 + 0.756}}\)

\(\gamma_s = 19.03\) kN/m3

Air content:

Air content is defined as the ratio of the volume of air to the volume of voids. It is denoted by ac.

\({{\bf{a}}_{\bf{c}}} = \frac{{{{\bf{V}}_{\bf{a}}}}}{{{{\bf{V}}_{\bf{v}}}}} = \frac{{{\bf{Volume\ of\ air}}}}{{{\bf{Volume\ of\ voids}}}}\)

Porosity:

Porosity is defined as the ratio of the volume of voids to the total volume of soil. It is denoted by n. It varies between 0 and 1.

\({\bf{n}} = \frac{{{{\bf{V}}_{\bf{v}}}}}{{\bf{V}}} = \frac{{{\bf{Volume\ of\ voids}}}}{{{\bf{Total\ volume}}}}\)

Percentage air voids:

Percentage air voids are defined as the ratio of the volume of air to the total volume of soil. It is denoted by n­­a.

\({{\bf{n}}_{\bf{a}}} = \frac{{{{\bf{V}}_{\bf{a}}}}}{{\bf{V}}} \times 100 = \frac{{{\bf{Volume\ of\ air}}}}{{{\bf{Total\ volume}}}} \times 100\)

Degree of Saturation: 

The degree of Saturation of a soil mass is defined as the ratio of the volume of water in the voids to the volume of voids. It is denoted by S.

\(S = \frac{{{V_w}}}{{{V_v}}} \times 100\;\;;0 \le S \le 100\)

• For a fully saturated soil mass Vv = Vw, hence S = 100%
• For fully dry soil mass Vw = 0, hence S = 0%