What is weight?

Weight is defined as the quantity of matter contained in a body. Weight can also be defined as the force acting on a body due to gravity. Mathematically, weight can be defined as-

W = m.g

where

W is the weight of the body.

m is the mass of the body.

g is the acceleration due to gravity.

The unit of weight in the SI unit system is Newton (N).

Volume

Volume is defined as the amount of space an object occupies.

The unit of volume in the SI unit system is cubic meters.

Weight-Volume Relationships

Weight-Volume relationships have significant importance in the field of soil mechanics and foundation engineering. The weight-volume relationship is a basic concept upon which the other concepts of geotechnical engineering are dependent. The basic weight-volume relationships are derived from the three-phase system diagram.

Three-phase system

A three-phase system is the most basic concept of Geotechnical Engineering, used for the analysis of soil. A three-phase system is a diagram that consists of solid particles, known as soil grains, air, and water. The void space between the solid particles is filled with water and air. But, if dry soil mass is considered, then the voids are filled only with air. In reality, all the three components, solid particles, air, and water of a soil mass do not occupy separate spaces, but they are mixed to form a single soil mass. But, for calculations and observations, all these components are assumed to occupy separate spaces.

The weight and volume in a three-phase system diagram are distributed as the weight of air (W_a), the weight of water (W_w), and the weight of soil solids (W_s) and the volume of air (V_a), the volume of water (V_w), and the volume of soil solids (V_s) respectively. W_a is considered as zero. The summation of V_a and V_w is known as V_v. Where, V_v is the volume of voids present in the soil. One important parameter of the three-phase system is moisture content or water content. Moisture content is an important factor, mostly constituted with particles of clayey soils.

Moisture content or water content (w) is defined as the ratio of the weight of water to the weight of soil solids. Hence, water content,  $w=\frac{W_w}{W_s}$. Since, $W_{w }=W_{p }- W_s$, the formula for water content can also be stated as,

$W_w=\frac{W_{p }- W_s}{W_s}$

where W_p is the total weight of soil.

Another important parameter of the three-phase system is the void ratio. Void ratio (e) is the ratio of the volume of voids and volume of soil solids. Hence, $e=\frac{V_v}{V_s}$.

The moisture content and void ratios are important for the analysis of soil and deriving many general relationships like permeability, plasticity, compaction, consolidation, and so on.

Given below are the weight-volume relationships derived from the three-phase system of a soil mass.

Unit weight of soil

The unit weight of soil is defined as the weight per unit volume of the soil mass. The unit weight can be calculated using the formula $\frac{W_p}{V}$, where, W_p is the total weight of soil and V is the volume of soil.

Dry unit weight of soil

The dry unit weight of soil is defined as the weight of soil solids per unit of its total volume. The dry unit weight can be calculated using the formula $\frac{W_d}{V}$, where, W_d is the dry weight of soil and V is the volume of soil.

Unit weight of solids

The unit weight of soil solids is defined as the weight of soil solids per unit volume of soil solids. The unit weight of solids can be calculated using the formula $\frac{W_d}{V_s}$, where, V_s is the volume of soil solids.

Saturated unit weight of soil

A soil is said to be saturated when all the voids of soil get filled with water and saturation occurs. So, the saturated unit weight of soil is defined as the ratio of the weight of the saturated soil sample to the total volume of the soil sample. The saturated unit weight can be calculated using the formula, $\frac{W_{sat}}{V}$, where, W_sat is the saturated weight of soil.

Density

Density is defined as mass per unit volume of a substance. Density can be calculated using the formula M/U, where, M is the mass of the body and U is the volume of the body.

As weight is mass x acceleration due to gravity, in order to convert density into unit weight, a multiplication factor of 9.81 should be used.

Hence, $unit weight=density \times 9.81$

Specific Gravity

Specific gravity is defined as the ratio of the unit weight of soil solids to the unit weight of water. Specific gravity is calculated as per the recommendations of ASTM. As per ASTM, the specific gravity should be calculated at the temperature of 27 degrees Celcius. Hence, specific gravity can be calculated using the following equation-

$Specific Gravity =\frac{Unit weight of soil solids}{Unit weight of water}$

Porosity

Porosity is defined as the ratio of the volume of voids to the volume of soil. Porosity can be calculated as-

$n=\frac{V_v}{V}$

where, n is the porosity

Degree of saturation

The degree of saturation is defined as the ratio of the volume of water to the volume of voids. The degree of saturation can be calculated as-

$S_r=\frac{V_w}{V_v}$

where S_r is the degree of saturation

Relative density and Relative compaction

The term relative density is mostly associated with particles of cohesionless granular soils. Relative density is used to denote the degree of compaction of granular soils. The relative density is calculated using the following equation-

$I_d=\frac{e_{max}-e}{e_{max}-e_{min}}$

where I_d is the relative density, e_max is the maximum void ratio, and e_min is the minimum void ratio

The degree of compaction for granular soils can sometimes be denoted using relative compaction. It can be calculated using the following equation-

$Relative Compaction=\frac{Dry weight of soil}{Maximum dry weight of soil from compaction test}$

The relative compaction varies from 80% to 100%. When the particles of soil are packed loosely, the relative compaction is less and when the particles are packed densely, the relative compaction is more.

General Relationships

Between Unit weight, G_s, void ratio, degree of saturation, and unit weight of water

The general relationships between unit weight, G_s, void ratio, degree of saturation, and unit weight of water are important for the analysis and measurement of various important parameters in soil mechanics. The general relationship is given below-

$Unit weight of soil = \frac{(G_s+e.S_r).(Unit weight of water)}{(1+e)}$

For unit weight of saturated soil, S_r = 1 and for dry unit weight of soil, S_r = 0.

Between G_s, void ratio, degree of saturation, and moisture content

The general relationships between G_s, void ratio, degree of saturation, and water content of water are important for the analysis and measurement of various important parameters in soil mechanics. The general relationship is given below-

S_r.e = w.G_s

Context and Applications

• Bachelors in Technology (Civil Engineering)
• Masters in Technology (Geotechnical Engineering)

Practice Problems

Q1. As per the American Society for Testing and Materials (ASTM), what is the standard temperature for calculating specific gravity?

a) $27°C$

b) $17°C$

c) $40°C$

d) $5°C$

Answer– a

Explanation– As per ASTM, the specific gravity is calculated at the temperature of $27°C$.

Q2. What is the relationship between density and unit weight?

a) Unit weight = 9.81$\times$Density

b) Unit weight = 8.91$\times$Density

c) Unit weight = 7.81$\times$Density

d) Unit weight = 8.81$\times$Density

Answer– a

Explanation– The relationship between the density and unit weight is, Unit weight = 9.81$\times$Density

Q3. What is defined as the ratio of the unit weight of soil solids to the unit weight of water?

a) Porosity

b) Void ratio

c) Specific Gravity

d) Relative density

Answer– c

Explanation- Specific gravity is defined as the ratio of the unit weight of soil solids to the unit weight of water.

Q4. What is the formula for the degree of saturation?

a) $\frac{V_v}{V_s}$

b) $\frac{V_w}{V_v}$

c) $\frac{V_w}{V_s}$

d) $\frac{V_w}{V_v}$

Answer– d

Explanation– The formula for the degree of saturation is $\frac{V_w}{V_v}$.

Q5. What is the formula for moisture content?

a) $\frac{W_w}{W_s}$

b) $\frac{W_a}{W_s}$

c) $\frac{V_w}{V_s}$

d) $\frac{W_w}{W_p}$

Answer– a

Explanation– The formula for moisture content is $\frac{W_w}{W_s}$.