Equation of three phase system:

Let,

V = Total volume of soil

V_s = Volume of solid particle

V_a = Volume of air

V_w = Volume of water

V_v = Volume of void

So, from the definition of

Total volume of soil is equal to the sum of volume of air, water and solid particle respectively.

i.e. V = V_w + V_a + V_s

or, V = V_s + V_v [∴V_w + V_a ]

Similarly,

For weight

Total weight (W) = Weight of solid (W_s) + Weight of water (W_w) + Weight of air (W_a)

i.e W = W_s + W_a + W_w

Also,

Weight of void (W_v) = W_a + W_w

or, W_v = W_w [∴ Weight of air is negligible, so W_a =0 ]

∴ W_v = W_w

Total weight of soil mass,

W = W_s + W_w

Similarly, in terms of mass

M = M_s + M_w

Some important terms:

1. Void ratio (e) = Volume of void (V_v) / Volume of solid (V_s)

2. Porosity (n) = Volume of void (V_v) / Total volume (V)

3. Degree of saturation (S or S_r)

or, S = Volume of water (V_w) / Volume of void (V_v)

4. Air content (a_c) = Volume of air (V_a) / Volume of void (V_v)

5. Percentage of air void (n_a) = Volume of air (V_a) / Total volume (V)

6. Water content or Moisture content (w) = Weight of water (W_w) / Weight of solid (W_s)

7. Unit weight (γ) = Weight (W) / Volume (V)

a. Bulk unit weight (γ) = W / V

or, γ = (W_s + W_v) / ( V_s + V_v)

b. Dry unit weight (γ_d) = W_s / V

c. Saturated unit weight (γ_sat) = W_sat / V

d. Submerged unit weight (γ_sub) = W_sub / V

or, γ_sub = γ_sat – γ_w

or, γ^I = γ_sat – γ_w

e. Unit weight of solid (γ_s) = W_s / V_s

8. Specific gravity (G) = (M_s / M_w)

or, G = (M_s / M_w) = (ρ_s / ρ_w) = (γ_s / γ_w)

At 4^o C

ρ_w = 1000 kg/m³ = 1g/ml

γ_w = 9810 N/m³ = 9.81 KN/m³

Some important relation:

a. Relationship between e,w,S_r and G

We know,

    e = V_v / V_s

or, e = (V_v / V_w) * (V_w / V_s)

or, e = (V_v / V_w) * (W_w / γ_w) * (γ_s / W_s)

or, e = (V_v / V_w) * (W_w / W_s) * (γ_s / γ_w)

or, e = (V_v / V_w) * w * G [∴ w = (W_w / W_s), G = (γ_s / γ_w) ]

or, e = (1/V_w / V_v ) * G * w

or, e = (1 / S_r) * w * G

or, S_r * e = w * G

∴ S_r * e = w * G

b. Relationship between e and n

   n = V_v / V = V_v / (V_v + V_s)

or, 1/n = (V_v + V_s) / V_v

or, 1/n = 1 + (V_s / V_v)

or, 1/n = 1 + 1/V_v /V_s  [∴ e = V_v / V_s]

or, 1/n = 1 + 1/e

or, 1/n = (e+1)/e

or, n = e / (e+1)

or, n(e+1) = e

or, e = ne + n

or, e = n / (1-n)

∴ e = n / (1-n)

c. Relationship between unit weight (γ) in terms of terms of water content (w), void ratio (e), specific gravity (G) and γ_w.

We know,

   γ = W / V = ( W_s + W_v) / ( V_a + V_s + V_w)

or, γ = ( W_s + W_w) / ( V_v + V_s) [∴ W_w = W_v]

or, γ = (W_s / V_s) * {(1 + W_w / W_s) / (1 + V_v / V_s)}

or, γ = γ_s * (1 + w) / (1+ e) ————–(1)

[∴w = (W_w / W_s), e = V_v / V_s ]

Also,

   G = γ_s / γ_w

or, γ_s  = G γ_w

Now, From equation (1) become,

∴γ= G γ_w (1 + w) / (1+ e)

d. Relationship between γ, G, S_r, e and γ_w.

We know,

     γ = W / V

or, γ = (W_s + W_v) / (V_s + V_v)

or, γ = (W_s / V_s) * { (1 + W_v / W_s) / ( 1 + V_v / V_s) }

or, γ = γ_s * (1 + w) / (1 + e)

or, γ = G γ_w * (1 + w) / (1 + e)

or, γ = γ_w * { (G + Gw) / (1+e) } ————–(1)

We know,

S_re = wG ————-(2)

From equation (1) and (2), we get

∴ γ = γ_w * { (G + S_re) / (1+e) }

Condition:

• If soil is fully saturated S_r = 1 (100%)

γ as γ_sat

∴ γ_sat = γ_w * { (G + e) / (1+e) }

• If soil is dry then S_r = 0 (0%)

γ as γ_d

∴ γ_d = (γ_w * G)/ (1+e)

e. Relation between dry unit weight (γ_d), Bulk unit weight (γ) and water content (w).

We have,

w = W_w / W_s

Adding one on both side

   w + 1 = (W_w + W_s) / W_s

or, w + 1 = (W_w + W_s) / W_s

or, w + 1 = W / W_s

or, W_s = W / (w + 1) ———–(1)

Also,

γ_d = W_s / V ————(2)

From equation (1) and (2)

  γ_d = W / (w + 1) * 1 / V

or, γ_d = W / V (1 + w)

or, γ_d = W / V * 1 / (1+w)

or, γ_d = γ* 1 / ( 1 + w) [∴γ = W / V]

∴ γ_d = γ/ (1 + w)

f. Relationship between submerged unit weight (γ^I), specific gravity of solid (G) and void ratio (e).

We have,

γ_sub or γ^I = γ_sat – γ_w

But,

γ_sat = {(G + e) γ_w} / (1+e)

Now,

   γ^I = {(G + e) γ_w / (1+e)} – γ_w

or, γ^I = {(G+e) γ_w – γ_w(1+e)} / (1+e)

or, γ^I =  γ_w(G-1) / (1+e)

∴ γ^I =  γ_w(G-1) / (1+e)

g. Relationship between dry unit weight (γ_d), specific gravity of solid (G), water content (w) and percentage of air void (n_a).

We have,

   V = V_s + V_a + V_w

or,V = V_a + (W_w / γ_w) + (W_s / γ_s)

[∴γ = W_w / V_w, Simillarly γ_s = W_s / V_s]

Dividing both side by V, we get

1 = (V_a / V) + (W_w / Vγ_w) + (W_s / Vγ_s)

But,

    w = W_w / W_s

or,W_w = w * W_s

Now,

1 = n_a + {(w * W_s) / ( V * γ_w) } + W_s / V * γ_s

[∴ n_a = V_a / V]

or, 1 – n_a  = (W_s / V) * {(w / γ_w) + (1 / γ_s)}

or, 1 – n_a  = (W_s / γ_w * V) * (w + γ_w / γ_s)

or, 1 – n_a  = (W_s / γ_w * V) * (w + 1 / G)

[∴ G = γ_w / γ_s]

or, 1 – n_a  = (γ_d / γ_s) *(w + 1 / G)

[∴γ_d = W_s / V]

or, γ_d = {(1-n_a) G γ_w} / (1 + wG)

When n_a = 0 (Fully saturated)

∴ γ_d = G γ_w / (1 + wG)

1.2 Index properties and their determination for coarse and fine grained soil

The properties of soil which helps to know engineering behaviors of soil and also helps to determine the classification of soil accurately is called index properties of soil.

The list of index properties:

1. Water content
2. Specific gravity of soil
3. Field ( In – situ) density
4. Particle size distribution
5. Consistency limit and indices
6. Density index

1.2.1 Determination of Various Index Properties:

1. Determination of water content

a. Oven drying method:

Process:

• Take a clean container and weight the container.
• Take about 100 gm wet sample and weight it again.
• Keep the soil with container in oven for overnight by maintain 104^o C temperature.
• After drying find the weight of dry soil and container.

Calculation:

Weight of container = W₁

Weight of container + Wet soil = W₂

Weight of container + Dry soil = W₃

Water content (w) = (W_w / W_s) * 100%

or, w = {(W₂ – W₁) / (W₃ – W₁)} * 100%

2. Determination of specific gravity

a. Pycnometer method:

Process:

• Take empty pycnometer.
• Take about 200 gm of dry sample in pycanometer and weight it.
• Add water in pycnometer and weight it.
• Take empty pycnometer and fill with water and weight it.

Calculation:

M₁ = Mass of empty pycnometer

M₂ = Mass of dry sample fill in pycnometer

M₃ = Mass of wet sample ( saturated sample)

M₄ = Mass of water in pycnometer

M_s= Mass of dry soil

Now,

    M₄ = M₃ – M_s + M_w

or, M₄ = M₃ – M_s + V_s *ρ_w

or, M₄ = M₃ – M_s + M_s * (1 / G) ————(1)

[∴ G = ρ_s / ρ_w]

Also,

M_s = M₂ – M₁

Putting the value in equation (1).

   M₄ = M₃ – M₂+ M₁ + (M₂ – M₁) / G

or, M₄ – M₃ + M₂ – M₁ = (M₂ – M₁) / G

or, G = {( M₂ – M₁) / (M₄ – M₃ + M₂ – M₁)}

or, G =  ( M₂ – M₁) / (M₄ + M₂) – (M₁ + M₃)

       ∴ G =  ( M₂ – M₁) / (M₄ + M₂) – (M₁ + M₃)

3. Determination of field density

a. Core cutter method

Process:

• Take a clean core cutter.
• Clean and level the ground where density is to be measured.
• Press cylinder cutter into the ground to its full depth with the help of rammer.
• Remove the core cutter from the soil.
• Take the mass of cutter and soil.

Calculation:

M₁ = Mass of clean core cutter

M₂ = Mass of cutter with soil

H = Height of cutter

d = Diameter

A = Area of cutter

Now,

Mass of soil (M) = M₂ – M₁

Volume of soil = A * H = (πd²/4)* H

Where,

H = 130 mm

d = 100 mm

Then,

Density (ρ ) = M/V

b. Sand replacement method:

Process:

• The site is cleaned and a square tray with a central hole is place on the surface.
• The hole of diameter is equal to the diameter of hole in tray.
• The excavated soil is collected in tray and weighted and water content is determined.
• The pouring cylinder is fitted with sand to about ¾ capacity and weight is taken and also its density is found out.
• The cylinder is then place over the hole and tap is opened and sand is allowed to run to fill the excavated whole and conical end.
• When no further flow of sand takes place, the tap is closed and bottle with remaining sand is weighted. The weight of sand filling the cone of bottle is taken separately.  

Calculation:

Weight of soil in the hole = W₁

Weight of pouring cylinder + sand before pouring = W₂

Weight of cylinder + sand after pouring = W₃

Weight of sand filling conical funnel = W₄

Weight of sand filling hole = W₂ – W₃ – W₄

Unit weight of sand = γ

Now,

Volume of sand = (W₂ – W₃ – W₄) / γ

                          = Volume of hole = V

Bulk density (γ) = W₁ / (1 + w)

Dry density (γ_d) = γ / (1+w)

4. Particle Size Distribution:

a. Sieve analysis:

• Sieve is utensil made up of spun brass having size of square opening in mm or microns (80mm – 75µ) and diameter of sieve is 15-20 cm and used for separating coarse and fine grained soil.
• Sieve analysis is done for coarse grained soil.
  • Gravel: Size greater than 4.75mm.
  • Sand: Size in between 75 µ to 4.75mm.
•  The sieves are stacked one over the other with decreasing size from top to bottom. The sieve of largest opening is kept at the top to bottom. The sieve of largest opening is kept at the top. A lid or cover is place at the top of largest sieve. A receiver known as pan which has no opening is placed at the bottom of smallest sieve.
• There are two type of sieve analysis:
  • Dry sieve analysis
  • Wet sieve analysis

b. Sedimentation analysis:

Sedimentation analysis or wet mechanical analysis is conducted on soul fraction finer than 75micron which is kept in suspension in a liquid medium usually water.

Sedimentation analysis is based in Stoke’s Law according to which gives the terminal velocity of small sphere setting in a fluid of infinite extent.

Now,

Terminal velocity (v) = {gD² * (G – 1) ρ_w} / 18η ———(1)

If a particle fall through height H_e in “t” minutes,

v = H_e / 60t ————-(2)

Combining equation (1) and (2)

D = M (H_e / t)^1/2

Where,

M = {30η / g(G – 1) ρ_w }

Uses of Particle size distribution curve:

• Classification of coarse grained soils.
• Determination of coefficient of permeability.
• To know susceptibility of soil to frost action.
• Design of drainage filter.
• Determination of shear strength of soil.
• Compressibility of soil.
• Used for soil stabilization.
• Design of pavement.
• Indicate mode of deposition of soil.
• Age of soil deposit.

5. Consistency of soil:

• The physical state of fine grained soil at particular water content is known as consistency.
• Mostly used for fine grained.
• The limiting water content at which a soil passes from one state of consistency to another state is called consistency limit.

Determination of consistency limit:

a. Liquid limit (LL):

At liquid limit, the soil possesses a small value of shear strength. The liquid limit is the minimum water content at which the soil is still in liquid state but has a small shearing strengh agaist flowing.

b. Plastic Limit (PL):

In plastic limit soil can be moulded into any shape.

w_p = W_w / W_s = (W₁ – W₂) / W₂

where,

W₁ = Weight of soil before drying

W₂ = Weight of soil after drying

c. Shrinkage limit:

In shrinkage limit water content is lowest.

w_s = {(M₁ – M_d) – (V₁ – γ_d) γ_w} / M_d

Where,

M₁ = Mass of wet soil pat

V₁ = Volume of wet soil pat

M_d = Mass of fry soil pat

V_d = Volume of dry soil pat

Or,

w_s = {(V_d γ_w) / M_d} – 1/G