# 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/m^{3})

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/m^{3}

γ = 9810 N/m^{3}

γ = 62.4 lb/ft^{3}

**Typical Values of Unit Weight for Soils**

Type of soil | γ_{sat} (kN/m^{3}) |
γ_{d} (kN/m^{3}) |

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:

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

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

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

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

Where

m = mass of soil

V = volume of soil

W = weight of soil

ρ = density of soil

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/m^{3}

ρ_{w} = 1 g/cc

ρ_{w} = 62.4 lb/ft^{3}

g = 9.81 m/s^{2}

g = 32.2 ft/sec^{2}

## 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:

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

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**

D_{r} (%) |
Description |

0 - 20 | Very loose |

20 - 40 | Loose |

40 - 70 | Medium dense |

70 - 85 | Dense |

85 - 100 | Very dense |

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