Civil Engineering (CE) Exam > Civil Engineering (CE) Notes > Soil Mechanics > Three-Phase System & Phase Relationships in Soil Mechanics
Three-Phase System & Phase Relationships in Soil Mechanics
Three-Phase System & Phase Relationships in Soil Mechanics
1. Introduction
Soil is a three-phase material consisting of:
- Solid - mineral soil grains (S)
- Liquid - pore water (W)
- Gas - air or water vapour (A)
The total volume: V = Vs + Vv = Vs + Vw + Va
In a fully saturated soil → Va = 0 (voids filled with water only) In a completely dry soil → Vw = 0
2. Volume Relationships
Void Ratio (e)
Ratio of volume of voids to volume of solids.
e = Vv / Vs
- Dense sand: e ≈ 0.3 - 0.6
- Loose sand: e ≈ 0.6 - 0.9
- Soft clay: e can exceed 1.0
Porosity (n)
Ratio of volume of voids to total volume.
n = Vv / V = e / (1 + e)
e = n / (1 - n)
Range: 0 < n < 1 (expressed as % or fraction)
Degree of Saturation (S)
Ratio of volume of water to volume of voids.
S = Vw / Vv × 100 (%)
- S = 0% → Completely dry
- S = 100% → Fully saturated
- 0 < S < 100% → Partially saturated
Air Content (ac) and Air Void Ratio (av)
ac = Va / V = n(1 - S)
av = Va / Vs = e(1 - S)
3. Weight / Mass Relationships
Total weight: W = Ws + Ww (weight of air is negligible)
Water Content (w)
Ratio of weight of water to weight of solids.
w = Ww / Ws × 100 (%)
- Expressed as a percentage
- Can exceed 100% for soft marine clays
4. The Most Important Formula (GATE Star Formula)
S × e = w × Gs
Where S = degree of saturation (decimal), e = void ratio, w = water content (decimal), Gs = specific gravity of solids.
This single formula connects all four key parameters. Memorise it without fail.
5. Unit Weight Relationships
γw = unit weight of water = 9.81 kN/m³ ≈ 10 kN/m³ (for GATE approximations)
Bulk Unit Weight (γt)
γt = W / V = Gs·γw·(1 + w) / (1 + e)
Dry Unit Weight (γd)
γd = Ws / V = Gs·γw / (1 + e) = γt / (1 + w)
Saturated Unit Weight (γsat)
γsat = (Gs + e)·γw / (1 + e)
Submerged / Buoyant Unit Weight (γ')
γ' = γsat - γw = (Gs - 1)·γw / (1 + e)
Typical Ranges
| Type | Formula | Range (kN/m³) |
|---|---|---|
| γdry | Gs·γw / (1+e) | 14 - 18 |
| γbulk | Gs·γw·(1+w) / (1+e) | 16 - 22 |
| γsat | (Gs+e)·γw / (1+e) | 18 - 23 |
| γ' | γsat - γw | 8 - 13 |
6. Specific Gravity (Gs)
Gs = γs / γw = Ws / (Vs × γw)
| Soil Type | Typical Gs |
|---|---|
| Sand | 2.63 - 2.67 |
| Gravel | 2.65 - 2.68 |
| Clay | 2.68 - 2.80 |
| Organic soil / Peat | 1.30 - 1.90 |
| Default assumed value | 2.65 or 2.67 |
7. Complete Formula Summary
| Parameter | Primary Formula | Secondary Formula |
|---|---|---|
| Void Ratio (e) | e = Vv/Vs | e = n/(1-n) |
| Porosity (n) | n = Vv/V | n = e/(1+e) |
| Saturation (S) | S = Vw/Vv | Se = wGs |
| Water Content (w) | w = Ww/Ws | w = Se/Gs |
| Air Content (ac) | ac = Va/V | ac = n(1-S) |
| Dry Unit Weight (γd) | γd = Ws/V | γd = γt/(1+w) |
| Bulk Unit Weight (γt) | γt = W/V | γt = Gs·γw·(1+w)/(1+e) |
| Saturated (γsat) | (Gs+e)·γw/(1+e) | γd + n·γw |
| Buoyant (γ') | γsat - γw | (Gs-1)·γw/(1+e) |
8. Relative Density (Dr)
Used only for granular soils (sands and gravels). Also called Density Index (ID).
Dr = (emax - e) / (emax - emin) × 100 (%)
Also expressed in terms of dry unit weight:
Dr = [(γd - γd,min) / (γd,max - γd,min)] × (γd,max / γd) × 100 (%)
| Dr (%) | State |
|---|---|
| 0 - 15 | Very Loose |
| 15 - 35 | Loose |
| 35 - 65 | Medium Dense |
| 65 - 85 | Dense |
| 85 - 100 | Very Dense |
Note: emax → γd,min (loosest state); emin → γd,max (densest state)
The document Three-Phase System & Phase Relationships in Soil Mechanics is a part of the Civil Engineering (CE) Course Soil Mechanics.
All you need of Civil Engineering (CE) at this link: Civil Engineering (CE)
FAQs on Three-Phase System & Phase Relationships in Soil Mechanics
| 1. What are volume relationships in soil mechanics? |  |
Ans. Volume relationships in soil mechanics refer to the relationships between the volumes of solids, water, and air present in a soil sample. These relationships help in understanding the behaviour of soil under various conditions, including the concepts of total volume, void ratio, and degree of saturation.
| 2. How do weight and mass relationships affect soil mechanics? | |
Ans. Weight and mass relationships are crucial in soil mechanics as they relate to the gravitational forces acting on soil particles. The weight of soil influences its stability and load-bearing capacity, while mass affects the soil's density and compaction characteristics. Understanding these relationships aids in the analysis of soil behaviour under different loading conditions.
| 3. What is the significance of the GATE Star Formula in soil mechanics? | |
Ans. The GATE Star Formula is significant in soil mechanics as it provides a comprehensive method for calculating important parameters such as unit weight, moisture content, and specific gravity. It serves as a vital tool for students and professionals in assessing soil properties and predicting performance in engineering applications.
| 4. How is specific gravity (Gs) defined in the context of soil? | |
Ans. Specific gravity (Gs) in the context of soil is defined as the ratio of the density of a soil solid to the density of water at a specified temperature. It is an important property as it helps in determining the weight of soil particles, which is essential for various calculations in soil mechanics and geotechnical engineering.
| 5. What is relative density (Dr) and why is it important in soil mechanics? | |
Ans. Relative density (Dr) is a measure of the compactness of a soil mass compared to its loosest and densest states. It is important in soil mechanics as it indicates the stability and shear strength of granular soils. Higher relative density usually correlates with greater strength and reduced settlement potential in engineering applications.
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