# Geotechnical Engineering

November 24, 2017 | Author: Fred Anthony A. Uera | Category: Density, Porosity, Soil, Natural Materials, Materials Science

formulas...

#### Description

Physical Properties 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 of Soil total volume = volume of soilds + volume of voids

volume of voids = volume of water + volume of air

total weight = weight of solids + weight of water

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

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

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

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.

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

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

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

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).

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).

Formulas for Properties of Soil Symbols and Notations = void ratio = porosity = moisture content, water content = specific gravity of any substance = specific gravity of solids = degree of saturation = volume of soil mass = volume of air = volume of water = volume of solids = volume of voids = total weight of soil = weight of water = weight of solids = relative density = unit weight of soil mass, moist unit weight, bulk unit weight = unit weight of soil solids = unit weight of water = = buoyant unit weight, submerged unit weight = = dry unit weight = saturated unit weight = liquid limit = plastic limit = liquidity index = plasticity index

= group index

Basic Formulas Unit weight, Weight, Specific gravity,

Physical Properties of Soil Total weight, Total volume, Volume of voids,

Void ratio,

Porosity,

, Note:

, Note:

Relationship between e and n,

and

Water content or moisture content,

Degree of saturation,

, Note:

, Note:

Relationship between G, w, S, and e, Moist unit weight or bulk unit weight, , also

Dry unit weight,

Saturated unit weight,

or

and

Submerged or buoyant unit weight,

Relative Density,

or

or

or

Atterberg Limits Plasticity index,

Liquidity index, Shrinkage index,

Activity of clay,

Other Formulas Volume of voids,

Volume of solids,

Volume of water,

Weight of water,

Weight of soil, Dry unit weight,

, where = soil finer than 0.002 mm in percent

Relationship between specific gravity of solids, moisture content, degree of saturation, and void ratio The relationship between

,

,

, and

G = specific gravity of solid particles w = moisture content or water content S = degree of saturation e = void ratio

The formula above can be derived as follows:

is given by the following

Thus,

as stated above.

Relationship between void ratio and porosity The relationship between

and

is given by and

Derivation is as follows → void ratio

→ n = Vv / V (okay!)

→ porosity

→ e = Vv / Vs (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 (see Figure 2 below), 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,

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

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

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

Specific gravity of solids

Unit Weights and Densities of Soil Unit Weights of Soil Symbols and Notations γ = Unit weight, bulk unit weight γm = 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

Note: Se = Gw, thus,

Moist unit weight in terms of dry density and moisture content

Dry Unit Weight (S = w = 0) From

and

, S = 0 and w = 0

Saturated Unit Weight (S = 1) From

, S = 100%

Buoyant Unit Weight or Effective Unit Weight

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 γsat (kN/m3)

γd (kN/m3)

Gravel

20 - 22

15 - 17

Sand

18 - 20

13 - 16

Type of soil

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)

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 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:

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