** Interrelationship Between the Properties of Soil **

- s·e=w·G
- \(γ=\frac{(G+e·s)γ_w}{1+e}\)
- \(γ_{dry}=\frac{G·γ_w}{1+e}\)
- \(γ_{sat}=\frac{(G+e)·γ_w}{1+e}\)
- \(γ′=\frac{(G-1)·γ_w}{1+e}\)
- \(γ′=γ_{dry}\)+(η-1)\(γ_w\)
- \(γ_{dry}\)=\(\frac{γ}{1+w}\)
- \(γ_d=\frac{G(1-η_a)·γ_w}{1+w·G}\)
- \(η=\frac{1}{1+e}\)
- \(Ws=\frac{Wt}{1+w}\)
- \(Vs=\frac{Vt}{1+e}\)
- (Vs)mix=Vs1+Vs2,(Ws)mix=Ws1+Ws2 In the problem of mixing, excavation, transportation of soil Weight of solid, volume of solid not change.

**1. Interrelationship Between Voids ratio (e), Water Content (w), Specific Gravity (G), Degree of Saturation(s)**

We know that:

\(\mathrm{e}=\frac{V v}{Vs}=\left(\frac{V v}{V s}\right)\left(\frac{V w}{V s}\right)\) Multiply by Vw in Numerator Denominator.

\(e=\frac{1·Vw}{s·Vs}\) {\(Vw=\frac{Ww}{γ_w}\)}

\(e=\frac{Ww}{s·Vs·γ_w}\){\(G=\frac{γ_s}{γ_w}=\frac{Ws}{Vsγ_w}\)}

\(e=\frac{Ww·G·γ_w}{s·γ_w·Ws}\) \(e=\frac{wG}{s}\)**s·e=w·G**

**2. Interrelationship Between Voids ratio (e), Unit Weight (γ), Specific Gravity (G), Degree of Saturation(s), Unit Weight of Water(\(γ_w\))**

We know that

\(γ=\frac{W}{V}\) \(γ=\frac{Ws+Ww}{Vv+Vs}\)(G=ϒs/ϒw)& (ϒs=Ws/Vs)

\(γ=\frac{VsGγ_w+Vwγ_w}{Vs(1+\frac{Vv}{Vs})}\) \(γ=\frac{Vsγ_w(G+Vw/Vs)}{Vs(e+1)}\) \(\frac{Vw}{Vs}=\frac{Vw·Vv}{Vs·Vv}=e·s\)**\(γ=\frac{(G+e·s)γ_w}{1+e}\)**

**3. Interrelationship Between Voids ratio (e),Dry ****Unit Weight (\(γ_{dry}\))****, Specific Gravity (G), Unit Weight of Water(\(γ_w\))**

We know that:

For dry soil, s=0,γ=**\(γ_{dry}\)**

**\(γ_{dry}=\frac{G·γ_w}{1+e}\)**

**4. Interrelationship Between Voids ratio (e), Saturated ****Unit Weight (\(γ_{sat}\))****, Specific Gravity (G),Unit Weight of Water(\(γ_w\))**

We know that:

For saturated soil, s=1,γ=**\(γ_{sat}\)**

**\(γ_{sat}=\frac{(G+e)·γ_w}{1+e}\)**

**5. Interrelationship Between Voids ratio (e), Submerged Unit Weight γ’, Specific Gravity (G),Unit Weight of Water (\(γ_w\))**

We know that:

γ’=**\(γ_{sat}\)–****(\(γ_w\))**

γ’=**\(\frac{(G+e)·γ_w}{1+e}\)–****(\(γ_w\))**

γ’=**\(\frac{(G+e)·γ_w-γ_w(1+e)}{1+e}\)**

\(γ′=\frac{(G-1)·γ_w}{1+e}\)

**6. Interrelationship Between Submerged ****Unit Weight γ’****, Porosity(η), ****Dry ****Unit Weight (\(γ_{dry}\)), ****Unit Weight of Water(\(γ_w\)) **

We know that:

γ’=\(\frac{(G-1)·γ_w}{1+e}\)

γ’=\(\frac{G·γ_w}{1+e}\frac{γ_w}{1+e}\)

e=\(\frac{η}{1-η}\)

1+e=\(\frac{η}{1-η}\)+1

1+e=\(\frac{1}{1-η}\)

\(\frac{1}{1+e}\)=1-η

γ’=\(γ_{dry}\)-(1-η)\(γ_w\)

\(γ′=γ_{dry}\)+(η-1)\(γ_w\)

**7. Interrelationship Between Dry ****Unit Weight (\(γ_{dry}\))****, Water Content (w), Unit Weight(γ) **

**\(γ_{dry}\)=**\(\frac{Wd}{V}=\frac{Ws}{V}=\frac{W}{V(1+w)}\)

**\(γ_{dry}\)=**\(\frac{γ}{1+w}\)

**8. Interrelationship Between Dry ****Unit Weight (\(γ_{dry}\))****, Specific Gravity (G), Unit Weight of Water(\(γ_w\)), Water Content(w), ****Percentage Air Voids (ηa) **

**V=Vv+Vs= Va+Vw+Vs**

Divided by V both side.

**\(\frac{V}{V}=\frac{Va}{V}+\frac{Vw}{V}+\frac{Vs}{V}\)**

**\(1=η_a+\frac{Ww}{V·γ_w}+\frac{Ws}{G·γ_w·V}\)**

**\(1-η_a=\frac{w·Ws}{V·γ_w}+\frac{Ws}{G·γ_w·V}\)**

**\(1-η_a=\frac{w·Wd}{V·γ_w}+\frac{Wd}{G·γ_w·V}\)**

**\(1-η_a=\frac{w·γ_d}{γ_w}+\frac{γ_d}{G·γ_w·}\)**

**\(1-η_a=\frac{γ_d}{γ_w}(w+\frac{1}{G}\)**

**\(γ_d=\frac{G(1-η_a)·γ_w}{1+w·G}\) **

“This relationship is also significant to understand the compactness of soil”

**\(γ_d=\frac{G(1-η_a)·γ_w}{1+w·G}\)**

Subject | Soil Mechanics |

Unit | Soil Formation & Properties of Soil |

Topic | Interrelationship Between the Properties of Soil |

Next Topic | Test of Specific Gravity |

Previous Topic | Relative Compaction |