Unit Weights and Densities of Soil

Unit Weights of Soil

Symbols and Notations
γ, γ_m = unit weight, bulk unit weight, moist unit weight
γ_d = Dry unit weight
γ_sat = Saturated unit weight
γ_b, γ' = Buoyant unit weight or effective unit weight
γ_s = Unit weight of solids
γ_w = Unit weight of water (equal to 9810 N/m³)
W = Total weight of soil
W_s = Weight of solid particles
W_w = Weight of water
V = Volume of soil
V_s = Volume of solid particles
V_v = Volume of voids
V_w = Volume of water
S = Degree of saturation
w = Water content or moisture content
G = Specific gravity of solid particles
 

Bulk Unit Weight / Moist Unit Weight
$\gamma = \dfrac{W}{V}$

$\gamma = \dfrac{W_w + W_s}{V_v + V_s}$

$\gamma = \dfrac{\gamma_w V_w + \gamma_s V_s}{V_v + V_s}$

$\gamma = \dfrac{\gamma_w V_w + G \gamma_w V_s}{V_v + V_s}$

$\gamma = \dfrac{V_w + G V_s}{V_v + V_s}\gamma_w$

$\gamma = \dfrac{S V_v + G V_s}{V_v + V_s}\gamma_w$

$\gamma = \dfrac{S (V_v/V_s) + G (V_s/V_s)}{(V_v/V_s) + (V_s/V_s)}\gamma_w$

$\gamma = \dfrac{Se + G}{e + 1}\gamma_w$

Note: Se = Gw, thus,

Moist unit weight in terms of dry density and moisture content
$\gamma = \dfrac{W}{V} = \dfrac{W_s + W_w}{V}$

$\gamma = \dfrac{W_s (1 + W_w/W_s)}{V} = \dfrac{W_s}{V}(1 + w)$

Dry Unit Weight (S = w = 0)
From $\gamma = \dfrac{(G + Se)\gamma_w}{1 + e}$ and $\gamma = \dfrac{(G + Gw)\gamma_w}{1 + e}$, S = 0 and w = 0

Saturated Unit Weight (S = 1)
From $\gamma = \dfrac{(G + Se)\gamma_w}{1 + e}$, S = 100%

Buoyant Unit Weight or Effective Unit Weight
$\gamma ' = \gamma_{sat} - \gamma_w$

$\gamma ' = \dfrac{(G + e)\gamma_w}{1 + e} - \gamma_w$

$\gamma ' = \dfrac{(G + e)\gamma_w - (1 + e)\gamma_w}{1 + e}$

$\gamma ' = \dfrac{G\gamma_w + e\gamma_w - \gamma_w - e\gamma_w}{1 + e}$

$\gamma ' = \dfrac{G\gamma_w - \gamma_w}{1 + e}$

Unit weight of water
γ = 9.81 kN/m³
γ = 9810 N/m³
γ = 62.4 lb/ft³
 

Typical Values of Unit Weight for Soils

Densities of Soil

The terms density and unit weight are used interchangeably in soil mechanics. Though not critical, it is important that we know it. To find the formula for density, divide the formula of unit weight by gravitational constant g (acceleration due to gravity). But instead of having g in the formula, use the density of water replacing the unit weight of water.
 

Basic formula for density (note: m = W/g)
$\rho = \dfrac{m}{V}$
 

The following formulas are taken from unit weights of soil:

Where
m = mass of soil
V = volume of soil
W = weight of soil
ρ = density of soil
ρ_d = dry density of soil
ρ_sat = saturated density of soil
ρ' = buoyant density of soil
ρ_w = density of water
G = specific gravity of soil solids
S = degree of saturation of the soil
e = void ratio
w = water content or moisture content
 

Density of water and gravitational constant
ρ_w = 1000 kg/m³
ρ_w = 1 g/cc
ρ_w = 62.4 lb/ft³
g = 9.81 m/s²
g = 32.2 ft/sec²
 

Relative Density

Relative density is an index that quantifies the state of compactness between the loosest and densest possible state of coarse-grained soils.
 

The relative density is written in the following formulas:

where:
D_r = relative density
e = current void ratio of the soil in-situ
e_max = void ratio of the soil at its loosest condition
e_min = void ratio of the soil at its densest condition
γ_d = current dry unit weight of soil in-situ
(γ_d)_min = dry unit weight of the soil at its loosest condition
(γ_d)_max = dry unit weight of the soil at its densest condition
 

Designation of Granular Soil Based on Relative Density