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Civil Engineering Department

SUBJECT – GEO-TECH. ENGG

TOPIC – WEIGHT - VOLUME RELATIONSHIPS

3^rd Semester

By

ER. ANKIT JOSHI

(Asst. HOD In Civil Engineering Department)

AY:2021-2022

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WEIGHT - VOLUME RELATIONSHIPS

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• Soil deposits comprise the accumulated solid particles plus the void space between the particles

• The void spaces are partially or completely filled with water or other liquid.

• Voids space not occupied by fluid are filled with air or other gas.

• Hence soil deposits are referred to as three-phase system, i.e. Solid + Liquid (water) + Gas (air)

GENERAL

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• Properties such as strength, compressibility, permeability are directly related to the ratio and interaction of these three phases.

GENERAL (continued)

• Therefore, an understanding of the terminology and definitions relating to soil composition is fundamental to the study of soil mechanics and geotechnical engineering as a whole.

• Bulk soil as it exists in nature is a more or less random accumulation of soil particles, water, and air as shown above.

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Volumes

Weights

For purpose of study and analysis it is convenient to represent the soil mass by a PHASE DIAGRAM, with part of the diagram representing the solid particles, part representing water or liquid, and another part air or other gas.

PHASE DIAGRAM

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Phase diagram in terms of mass

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Possible Cases:

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• The total volume of a given soil sample can be expressed as:

Where

V = Total volume

V_s = Volume of soil solids

V_v = Volume of voids

V_w = Volume of water

V_a = Volume of air

• Assuming that the weight of the air is negligible, we can give the total weight of the sample as

Where W_s = weight of solids

W_w = weight of water

• In engineering practice we usually measure the total volume, V, the mass of water, M_w, and the mass of dry solid M_s.

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Volume Relationships

There are three volumetric ratios that are very useful in geotechnical engineering , and these can be determined directly from the phase diagram

1. Void Ratio

2. Porosity

3. Degree of Saturation

Porosity and degree of saturation are commonly expressed as a percentage.

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In this illustration,

e = 1

n = 50%

S = 50%

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Weight or Mass Relationships

The common term used for weight relationships are:

Moisture content (w) is also referred to as water content and is defined as the ratio of weight of water to the weight of solids in a given volume of soil:

• Moisture content

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1. Unit weight (total, wet or moist unit weight) (γ) is the weight of soil per unit volume.

Weight-Volume, Mass-Volume Relationships

2. Solid unit weight

3. Unit weight of water

I. Unit Weights (N/m³ or kN/m³)

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Weight-Volume, Mass-Volume Relationship

4. Dry unit weight

5. Saturated unit weight

6. Submerged unit weight

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II. Densities (g/cm³ or kg/m³)

• Because the Newton is a derived unit, working with mass densities ρ of soil may sometimes be convenient.

• The SI unit of mass density is kilograms per cubic meter (kg/m³). We can write the density equations by replacing weight with mass in all equations in the preceding slides.

• The density of water ρ_w varies slightly, depending on the temperature. At 4C^o when water is at its densest, exactly equal 1000 kg/m³ or 1g/cm³)

Relationship between unit weight and density

The unit weights of soil in N/m³ can be obtained from densities in kg/m³ as

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Density and Unit Weight

• Mass is a measure of a body's inertia, or its "quantity of matter". Mass does not changed at different places.
• Weight is force, the force of gravity acting on a body. The value is different at various places.
• The unit weight is more frequently used than the density is (e.g. in calculating the overburden pressure).

Note: The density/or unit weight are ratios which connects the volumetric side of the PHASE DIAGRAM with the mass/or weight side.

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Relationships Between Various Physical Properties

All the weight - volume relationships needed in soil mechanics can be derived from appropriate combinations of six fundamental definitions. They are:

1. Void ratio
2. Porosity
3. Degree of saturation
4. Water content
5. Unit weight
6. Specific gravity

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1. Relationship between void ratio and porosity

2. Relationship among Void ratio, Degree of Saturation, Water content, and Specific Gravity

Dividing the denominator and numerator of the R.H.S. by V_v yields:

This is a very useful relation for solving THREE-PHASE RELATIONSHIPS.

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Textbook derivation

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3. Relationship among Unit Weight, Void Ratio, Degree of Saturation and Specific Gravity

Notes:

• Unit weights for dry, fully saturated and submerged cases can be derived from the upper equation

• Water content can be used instead of degree of saturation.

• Submerged unit weight can be approximated as

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Various Unit Weight Relationships

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Example 1

Instead think of

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Field density testing (i.e., sand replacement method) has shown bulk density of a compacted road base to be 2.06 t/m³ with a water content of 11.6%. Specific gravity of the soil grains is 2.69. Calculate the dry density, porosity, void ratio and degree of saturation.

Example 2

The rest is vey simple

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Example 3

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Our Solution

S = 1

w =25.7

e = 0.668

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Example 4

V= 0.0282 m³

S = 56%

w = 18.5%

G_s = 2.529

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Required:

e

γ

γ_d

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Our Solution

S = 56%

w = 18.5%

G_s = 2.529

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Relative Density

• The relative density Dr, also called density index is commonly used to indicate the IN SITU denseness or looseness of granular soils.

• The relative density is the parameter that compare the volume reduction achieved from compaction to the maximum possible volume reduction.

Volume reduction from compaction of granular soil

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D_r can be expressed either in terms of void ratios or dry densities.

Why e not n?

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Remarks

• The relative density of a natural soil very strongly affects its engineering behavior.

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• The range of values of D_r may vary from a minimum of zero for very LOOSE soil to a maximum of 100% for a very DENSE soil.

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• Because of the irregular size and shape of granular particles, it is not possible to obtain a ZERO volume of voids. (Do you remember well-graded vs. poorly-graded!!)

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• ASTM test designations D-4253 and D-4254 (2007) provide procedure for determining maximum and minimum dry unit weights of granular soils.

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• Granular soils are qualitatively described according to their relative densities as shown below

• The use of relative density has been restricted to granular soils because of the difficulty of determining e_max in clayey soils. Liquidity Index in fine-grained soils is of similar use as D_r in granular soils.

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