Weight Relations & Inter Relations - Soil Mechanics - Civil Engineering (CE)

Weight Relations

Density is the mass per unit volume of a material. Unit weight (also called unit mass when gravity is explicitly included) is the weight per unit volume of a material. In practice for soil mechanics the terms density and unit weight are used according to context; unit weight is commonly used in geotechnical work. Common units are kg/m³, g/cm³ for density and kN/m³ or N/m³ for unit weight. The following are the basic weight relations used in soil engineering.

1. Water content (w) - the ratio of the mass of water present to the mass of solid particles.

Water content is usually expressed as a decimal or as a percentage. Its value is 0% for completely dry soil and may exceed 100% for very organic or highly porous soils where water mass is large compared with the mass of solids.

2. The specific gravity of soil solids (G_s) (sometimes referred to as particle unit weight when multiplied by unit weight of water) describes the density of the solid grains relative to water.

Here the symbol shown in the figure represents the unit weight of water (commonly written as γ_w). For most inorganic mineral soils G_s lies between 2.60 and 2.80. The presence of organic matter reduces the value of G_s.

3. Dry unit weight (γ_d) - the weight of solids per unit total volume when all water is removed from the soil.

The dry unit weight is a measure of the amount of solid particles per unit volume and is fundamental for mass-volume relations and strength calculations.

4. Bulk (total) unit weight (γ) - the weight of solids plus water (and air) per unit total volume.

Bulk unit weight can be measured in the field or laboratory and varies with water content and degree of compaction.

5. Saturated unit weight (γ_sat) - the bulk unit weight when all voids are filled with water (degree of saturation S_r = 100%).

6. Buoyant (submerged) unit weight (γ') - the effective weight per unit volume of soil when the soil is submerged (i.e., submerged under standing water or below the groundwater table).

Buoyant unit weight is the difference between the saturated unit weight and the unit weight of water; it is used when calculating effective stresses in submerged soils.

Inter-Relations

When soil specimens arrive in the laboratory it is important to record their physical state (water content, bulk unit weight, etc.) because these properties can change during transport and storage. Several physical state properties are not measured directly but are calculated from other measured quantities. The following inter-relations are standard and form the basis of many laboratory and field calculations.

The relations below use the usual symbols:

• M_s = mass of solids
• M_w = mass of water
• V = total volume
• V_s = volume of solids
• V_v = volume of voids
• e = void ratio = V_v/V_s
• n = porosity = V_v/V = e/(1+e)
• S_r = degree of saturation = V_w/V_v
• w = water content = M_w/M_s
• G_s = specific gravity of solids = ρ_s/ρ_w
• γ = bulk unit weight = total weight / V
• γ_d = dry unit weight = weight of solids / V
• γ_w = unit weight of water

Key algebraic inter-relations (derived from mass-volume definitions) are:

Relation between bulk unit weight, G_s, e and w

Bulk unit weight γ can be written as a function of specific gravity, void ratio and water content:

γ = G_s·γ_w·(1 + w) / (1 + e)

Derivation (one logical line per relation):

Mass of solids = M_s.

Volume of solids V_s = M_s / (G_s·ρ_w).

Total volume V = V_s·(1 + e).

Bulk density ρ = total mass / V = M_s(1 + w) / V.

Replace V and rearrange to obtain ρ = G_s·ρ_w·(1 + w)/(1 + e).

Multiply by g or express in unit-weight notation to obtain γ = G_s·γ_w·(1 + w)/(1 + e).

Dry unit weight

Dry unit weight γ_d is obtained by setting w = 0 in the previous relation, or directly from bulk unit weight:

γ_d = γ / (1 + w)

and

γ_d = G_s·γ_w / (1 + e)

Relation for void ratio from γ_d

Rearrange γ_d = G_s·γ_w / (1 + e) to obtain

1 + e = G_s·γ_w / γ_d

and therefore

e = (G_s·γ_w / γ_d) - 1

Saturated unit weight

When S_r = 1 (fully saturated), the water content at saturation is w_sat = e / G_s.

Substitute into the bulk relation to get:

γ_sat = γ_w·(G_s + e) / (1 + e)

Buoyant (submerged) unit weight

Submerged unit weight γ' (also written as γ_sub) is the effective weight per unit volume when soil is underwater:

γ' = γ_sat - γ_w

Using γ_sat expression,

γ' = γ_w·(G_s - 1) / (1 + e)

Relation between degree of saturation, water content, void ratio and G_s

From definitions, S_r = V_w/V_v and V_w = M_w/ρ_w, so

S_r = (w·G_s) / e

or equivalently

w = e·S_r / G_s

These relations allow conversion between measured quantities (mass-based water content, bulk unit weight) and volume-based state parameters (void ratio, degree of saturation).

Practical notes and uses:

• Measure water content and bulk unit weight on receipt of samples to record the state of the specimen before any drying or disturbance.
• Use γ_d = γ/(1+w) to convert measured bulk unit weight to dry unit weight for compaction or density specifications.
• Use γ_sat and γ' when calculating subsurface stresses, stability under submerged conditions and buoyancy effects on light structures.
• Use e = (G_s·γ_w/γ_d) - 1 to compute void ratio from dry unit weight when G_s is known.
• Use S_r = w·G_s/e to check saturation state; S_r ≤ 1 must hold for physically realistic soils.

These weight relations and inter-relations form the basis for laboratory testing (e.g., specific gravity tests, water content determination, unit weight determinations, and compaction tests) and for field calculations of stresses, settlement and stability.