## Phase Diagram of Soil

Soil is composed of solids, liquids, and gases. Liquids and gases are mostly water and air, respectively. These two (water and air) are called voids which occupy between soil particles. The figure shown below is an idealized soil drawn into phases of solids, water, and air.

Weight-Volume Relationship from the Phase Diagram
total volume = volume of soilds + volume of voids
$V = V_s + V_v$

volume of voids = volume of water + volume of air
$V_v = V_w + V_a$

total weight = weight of solids + weight of water
$W = W_s + W_w$

## Soil Properties

Void Ratio, e
Void ratio is the ratio of volume of voids to the volume of solids.

$e = \dfrac{V_v}{V_s}$

Porosity, n
Porosity is the ratio of volume of voids to the total volume of soil.

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

Degree of Saturation, S
Degree of saturation is the ratio of volume of water to the volume of voids.

$S = \dfrac{V_w}{V_v}$

Water Content or Moisture Content, w
Moisture content, usually expressed in terms of percentage, is the ratio of the weight of water to the weight of solids.

$w = \dfrac{W_w}{W_s} \times 100\%$

Unit Weight, γ
Unit weight is the weight of soil per unit volume. Also called bulk unit weight (γ), and moist unit weight (γm).

$\gamma = \dfrac{W}{V}$

Dry Unit Weight, γd
Dry unit weight is the weight of dry soil per unit volume.

$\gamma_d = \dfrac{W_s}{V_{dry}}$

Saturated Unit Weight, γsat
Saturated unit weight is the weight of saturated soil per unit volume.

$\gamma_{sat} = \dfrac{W_{sat}}{V_{sat}}$

Effective Unit Weight, γ'
Effective unit weight is the weight of solids in a submerged soil per unit volume. Also called buoyant density or buoyant unit weight (γb).

$\gamma ' = \gamma_{sat} - \gamma_w$

Specific Gravity of Solid Particles, G
Specific gravity of solid particles of soil is the ratio of the unit weight of solids (γs) to the unit weight of water (γw).

$G = \dfrac{\gamma_{s}}{\gamma_w}$

### Formulas for Properties of Soil

Basic Formulas
Unit weight, $\gamma = s \gamma_w$

Weight, $W = \gamma V = s \gamma V$

Specific gravity, $s = \dfrac{\gamma}{\gamma_w}$

Physical Properties
Total weight, $W = W_w + W_s$

Total volume, $V = V_s + V_v$

Volume of voids, $V_v = V_w + V_a$

Void ratio, $e = \dfrac{V_v}{V_s}$,   Note: $0 \lt e \lt \infty$

Porosity, $n = \dfrac{V_v}{V}$,   Note: $0 \lt n \lt 1$

Relationship between e and n, $n = \dfrac{e}{1 + e}$   and   $e = \dfrac{n}{1 - n}$

Water content or moisture content, $w = \dfrac{W_w}{W_s} \times 100\%$,   Note: $0 \lt w \lt \infty$

Degree of saturation, $S = \dfrac{V_w}{V_v}$,   Note: $0 \le S \le 1$

Relationship between G, w, S, and e, $Gw = Se$

Moist unit weight or bulk unit weight, $\gamma_m = \dfrac{W}{V} = \dfrac{(G + Se)\gamma_w}{1 + e} = \dfrac{G( 1 + w)\gamma_w}{1 + e}$

Dry unit weight, $\gamma_d = \dfrac{W_s}{V} = \dfrac{G\gamma_w}{1 + e}$

Saturated unit weight, $\gamma_{sat} = \dfrac{(G + e)\gamma_w}{1 + e}$

Submerged or buoyant unit weight, $\gamma_b = \gamma_{sat} - \gamma_w = \dfrac{(G - 1)\gamma_w}{1 + e}$

Critical hydraulic gradient, $i_{cr} = \dfrac{\gamma_b}{\gamma_w} = \dfrac{G - 1}{1 + e}$

Relative Density, $D_r = \dfrac{e_{max} - e}{e_{max} - e_{min}} = \dfrac{\dfrac{1}{(\gamma_d)_{min}} - \dfrac{1}{\gamma_d}}{\dfrac{1}{(\gamma_d)_{min}} - \dfrac{1}{(\gamma_d)_{max}}}$

Atterberg Limits
Plasticity index, $PI = LL - PL$

Liquidity index, $LI = \dfrac{MC - PL}{PI}$

Shrinkage index, $SI = PL - SL$

Activity of clay, $A_c = \dfrac{PI}{\mu}$,   where $\mu$ = soil finer than 0.002 mm in percent

Other Formulas
Volume of voids, $V_v = \dfrac{eV}{1 + e}$

Volume of solids, $V_s = \dfrac{V}{1 + e}$

Volume of water, $V_w = \dfrac{SeV}{1 + e}$

Weight of water, $W_w = \dfrac{SeV \gamma_w}{1 + e}$

Weight of soil, $W = \dfrac{V(G + Se)\gamma_w}{1 + e}$

Dry unit weight, $\gamma_d = \dfrac{\gamma_m}{1 + w}$

### Relationship between G, w, S, and e

The relationship between specific gravity of solid particles G, water content or moisture content w, degree of saturation S, and void ratio e, is given by the following:

$Gw = Se$

The formula above can be derived as follows:
$\gamma_s = G\gamma_w$

$\dfrac{W_s}{V_s} = G\gamma_w$

$\dfrac{W_s}{V_s} \cdot \dfrac{W_w}{W_w} = G\gamma_w$

$\dfrac{W_w}{V_s} \cdot \dfrac{W_s}{W_w} = G\gamma_w$

$\dfrac{W_w}{V_s} \cdot \dfrac{1}{W_w/W_s} = G\gamma_w$

$\dfrac{\gamma_w V_w}{V_s} \cdot \dfrac{1}{w} = G\gamma_w$

$\dfrac{V_w}{V_s} \cdot \dfrac{1}{w} = G$

$\dfrac{V_w}{V_s} = Gw$

$\dfrac{V_w}{V_s} \cdot \dfrac{V_v}{V_v} = Gw$

$\dfrac{V_w}{V_v} \cdot \dfrac{V_v}{V_s} = Gw$

$Se = Gw$

Thus,   $Gw = Se$   as stated above.

### Relationship between e and n

The relationship between the the void ratio e, and the porosity n is given by:

$e = \dfrac{n}{1 - n}$   and   $n = \dfrac{e}{1 + e}$

Derivation is as follows
$e = \dfrac{V_v}{V_s}$       ← void ratio

$e = \dfrac{V_v}{V - V_v}$

$e = \dfrac{V_v}{V - V_v} \cdot \dfrac{1/V}{1/V}$

$e = \dfrac{V_v/V}{1 - V_v/V}$       → n = Vv / V

$e = \dfrac{n}{1 - n}$           (okay!)

$n = \dfrac{V_v}{V}$       ← porosity

$n = \dfrac{V_v}{V_s + V_v}$

$n = \dfrac{V_v}{V_s + V_v} \cdot \dfrac{1/V_s}{1/V_s}$

$n = \dfrac{V_v/V_s}{1 + V_v/V_s}$       → e = Vv / Vs

$n = \dfrac{e}{1 + e}$           (okay!)

## Consistency of Soil (Atterberg Limits)

Consistency is the term used to describe the ability of the soil to resist rupture and deformation. It is commonly describe as soft, stiff or firm, and hard.

Water content greatly affects the engineering behavior of fine-grained soils. In the order of increasing moisture content, a dry soil will exist into four distinct states: from solid state, to semisolid state, to plastic state, and to liquid state. The water contents at the boundary of these states are known as Atterberg limits. Between the solid and semisolid states is shrinkage limit, between semisolid and plastic states is plastic limit, and between plastic and liquid states is liquid limit.

Atterberg limits, then, are water contents at critical stages of soil behavior. They, together with natural water content, are essential descriptions of fine-grained soils.

Liquid Limit, LL
Liquid limit is the water content of soil in which soil grains are separated by water just enough for the soil mass to loss shear strength. A little higher than this water content will tend the soil to flow like viscous fluid while a little lower will cause the soil to behave as plastic.

Plastic Limit, PL
Plastic limit is the water content in which the soil will pass from plastic state to semi-solid state. Soil can no longer behave as plastic; any change in shape will cause the soil to show visible cracks.

Shrinkage Limit, SL
Shrinkage limit is the water content in which the soil no longer changes in volume regardless of further drying. It is the lowest water content possible for the soil to be completely saturated. Any lower than the shrinkage limit will cause the water to be partially saturated. This is the point in which soil will pass from semi-solid to solid state.

### Determination of Liquid, Plastic, and Shrinkage Limits

Casagrande Cup Method for Liquid Limit Test

Casagrande Cup
Courtesy of MOHAN LAL AND SONS

The semispherical brass cup is repeatedly dropped into a hard rubber base from a height of 10 mm by a cam-operated crank.

The dry powder of the soil is mixed with distilled water turning it into a paste. The soil paste is then placed into the cup to a thickness of about 12.5 mm and a groove is then cut at the center of the paste using the standard grooving tool. The crank operating the cam is turned at the rate of 2 revolutions per second lifting the cup and dropped it from a height of 10 mm. The liquid limit is the moisture content required to close a distance of 12.5 mm along the bottom of the groove after 25 blows.

The required closure in 25 blows is difficult to achieve in a single test. Four or more tests to the same soil at varying water contents are to be done for 12.5 mm closure of the groove. The results are then plotted on a semi-logarithmic graph with moisture content along the vertical axis (algebraic scale) and number of blows along the horizontal axis (logarithmic scale).

The graph is approximated by the best fit straight line, usually called the flow line and sometimes called liquid state line. The moisture content that corresponds to 25 blows is the liquid limit of the soil.

The slope of the flow line is called flow index and may be written as

Flow index,   $FI = \dfrac{w_1 - w_2}{\log (N_2/N_1)}$

where w1 and w2 are the water content corresponding to number of blows N1 and N2, respectively.

Plastic Limit Test
The plastic limit can easily be found by rolling a small soil sample into thin threads until it crumbles. The water content at which the threads break at approximately 3 mm in diameter is the plastic limit. Two or more tests are made and the average water content is taken as plastic limit. In this test, soil will break at smaller diameter when wet and breaks in larger diameter when dry.

Fall Cone Method for Liquid and Plastic Limit Tests

Cone Penetrometer
Courtesy of SAIGON ISC

Fall cone method offers more accurate result of liquid limit and plastic limit tests. In this method, a cone with a mass of 80 grams and an apex angle of 30° is suspended above so that its pointed part will just in contact with the soil sample. The cone is permitted to fall freely under its own weight for a period of 5 seconds. The water content that allows the cone to penetrate for 20 mm during this period defines the liquid limit of the soil.

Like the cup method, four or more tests are required because it is difficult to find the liquid limit in a single test. The results are then plotted into a semi-logarithmic paper with water content along the vertical axis (arithmetic scale) and penetration along the horizontal axis (logarithmic scale). The best fit straight line is then drawn and the water content that corresponds to 20 mm penetration defines the liquid limit.

The plastic limit can be found by repeating the test with a cone of similar geometry but with a mass of M2 = 240 grams. The liquid state line of this cone will be below the liquid state line of the M1 = 80 grams cone and parallel to it.

The plastic limit is given as

$PL = LL - \dfrac{2 \, \Delta w}{\log (M_2/M_1)}$

Shrinkage Limit Test
The shrinkage limit is determined as follows. A mass of wet soil, m1, is placed in a porcelain dish 44.5 mm in diameter and 12.5 mm high and then oven dried. With oven-dried soil still in the dish, the volume of shrinkage can be determined by filling the dish with mercury. The volume of mercury that fills the dish is equal to the shrinkage volume. The shrinkage limit is calculated from

$SL = \dfrac{m_1 - m_2}{m_2} - \dfrac{(V_1 - V_2)\rho_w}{m_2}$

where m1 = mass of wet soil, m2 = mass of oven-dried soil, V1 = volume of wet soil, V2 = volume of oven-dried soil, and ρw = density of water.

Other Formulas
Shrinkage ratio

$SR = \dfrac{m_2}{V_2 \rho_w}$

Specific gravity of solids

$G = \dfrac{1}{\dfrac{1}{SR} - \dfrac{SL}{100}}$

## Formulas for Unit Weights of Soil

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 of 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

## Formulas for 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:

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

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

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

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

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

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

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