# Relative Compaction | Properties of Soil | Soil Mechanics

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## Relative Compaction (Rc)

It is used to indicate the degree of denseness/compactness in absolute terms as density index (ID), but it can be applied for both cohesionless & cohesive soil.

$$\mathrm{Rc}=\frac{\gamma_{d}}{\gamma_{\text {dmax }}} \times 100$$

Relative Compaction is unit less.

It is defined as the ratio of the dry unit weight of soil in the existing/natural state to the dry unit weight of soil in the densest state.

$$\gamma_{d}$$= Dry unit weight of soil in existing / natural state.

$$\gamma_{dmax}$$= Dry unit weight of soil in densest state.

$$\begin{array}{l} \mathrm{Rc}=\frac{G \gamma_{w}\left(1+e_{\min }\right)}{(1+e) G_{\psi}} \times 100 \\ \mathrm{Rc}=\frac{\left(1+e_{\min }\right)}{(1+e)} \times 100 \end{array}$$

Rc= 80 + 0.2ID

In loosest state Id=0% ⇒ Rc⇒ 80%.

In densest state Id=100% ⇒ Rc⇒ 100%.

## Packing of Uniform Sphere

• Packing of the uniform sphere is the only system, void ratio of porosity of which can be found mathematically.
• Spheres of uniform size attain stable loosest packing in a “cubical array“.
• Spheres of uniform size attain stable densest packing in “Rhombohedral array“.

### 1. Loosest State

Here each particle is in contact with 6 more similar particles.

Volume of soil V=1*1*1  = 1m³.

Volume of solids in soil = (1/d)*(1/d)*(1/d) = 1/d³

The volume of one solid = πd³/6.

Total volume of solids Vs = $$\frac{\pi d^{3}}{6} \times \frac{1}{d^{3}}=\frac{\pi}{6}$$

Volume of voids Vv= V-Vs=1-(π/6)=0.476m³.

$$e_{\max }$$= Vv/Vs. = .476/(π/6) =.91

$$η_{\max }$$= $$e_{\max }$$/(1+$$e_{\max }$$) = .91/(1+.91) =.47

### 2. Densest  State

Here each particle is in contact with 12 more similar particles.

Volume of unit rhombohedral

V=$$\sqrt{1+2 \cos \alpha} \times(1-\cos \alpha)$$

Volume of soil α=60º

V=$$\sqrt{1+2 \cos60º } \times(1-\cos60º)$$=0.707m³

Volume of solids Vs = π/6.

Volume of voids Vv= V-Vs= 0.707-(π/6)=0.183m³.

$$e_{\min }$$= Vv/Vs. = .183/(π/6) =.35

$$η_{\min }$$= $$e_{\min }$$/(1+$$e_{\min }$$) = .35/(1+.35) =.26

If $$e_{\min }$$,$$e_{\max }$$ is not given in Question, then apply

$$e_{\min }$$=0.35,

$$e_{\min }$$=0.91.

 Subject Soil Mechanics Unit Soil Formation & Properties of Soil Topic Relative Compaction Next Topic Interrelationship Between the Properties of Soil Previous Topic Density Index

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