# Unit Weights and Densities of Soil

## Unit Weights of Soil

Symbols and Notations
γ, γm = unit weight, bulk unit weight, moist unit weight
γd = Dry unit weight
γsat = Saturated unit weight
γb, γ' = Buoyant unit weight or effective unit weight
γs = Unit weight of solids
γw = Unit weight of water (equal to 9810 N/m3)
W = Total weight of soil
Ws = Weight of solid particles
Ww = Weight of water
V = Volume of soil
Vs = Volume of solid particles
Vv = Volume of voids
Vw = 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
$\gamma = \dfrac{W}{V}$

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

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

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

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

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

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

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

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

Note: Se = Gw, thus,

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

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

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

$\gamma = \gamma_d (1 + w)$

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

$\gamma_d = \dfrac{G \gamma_w}{1 + e}$

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

$\gamma_{sat} = \dfrac{(G + e)\gamma_w}{1 + e}$

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

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

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

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

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

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

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

 Type of soil γsat (kN/m3) γd (kN/m3) Gravel 20 - 22 15 - 17 Sand 18 - 20 13 - 16 Silt 18 - 20 14 - 18 Clay 16 - 22 14 - 21

## 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:

$\rho = \dfrac{(G + Se)\rho_w}{1 + e}$

$\rho = \dfrac{(G + Gw)\rho_w}{1 + e}$

$\rho_d = \dfrac{G\rho_w}{1 + e}$

$\rho_{sat} = \dfrac{(G + e)\rho_w}{1 + e}$

$\rho ' = \dfrac{(G - 1)\rho_w}{1 + e}$

Where
m = mass of soil
V = volume of soil
W = weight of soil
ρ = density of soil
ρd = dry density of soil
ρsat = saturated density of soil
ρ' = buoyant density of soil
ρw = 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
ρw = 1000 kg/m3
ρw = 1 g/cc
ρw = 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:

$D_r = \dfrac{e_{max} - e}{e_{max} - e_{min}}$

$D_r = \dfrac{\dfrac{1}{(\gamma_d)_{min}} - \dfrac{1}{\gamma_d}}{\dfrac{1}{(\gamma_d)_{min}} - \dfrac{1}{(\gamma_d)_{max}}}$

where:
Dr = relative density
e = current void ratio of the soil in-situ
emax = void ratio of the soil at its loosest condition
emin = void ratio of the soil at its densest condition
γd = current dry unit weight of soil in-situ
(γd)min = dry unit weight of the soil at its loosest condition
(γd)max = dry unit weight of the soil at its densest condition

Designation of Granular Soil Based on Relative Density

 Dr (%) Description 0 - 20 Very loose 20 - 40 Loose 40 - 70 Medium dense 70 - 85 Dense 85 - 100 Very dense