SOIL ENGINEERING IMPORTANT FORMULAS
A soil mass consists of solid particles which form a porous structure. The voids in the soil mass may be filled with air, water or partly with water and partly with water. Soil is a three phase system in general.
VOLUMETRIC RELATIONSHIPS
Void Ratio
Void ratio is the volume of voids to the volume of solids. It is denoted by ‘e’
e=Vv/Vs
It is expressed as a decimal.
Porosity
It is defined as the ratio of volume of voids to the total volume. It is denoted by ‘n’
n=Vv/V
It is generally expressed as a percentage
1/n=V/Vv= (Vv+vs)/Vv
1/n= 1+ (1/e) = (1+e)/e
n=e/ (1+e) (a)
1/e= (1/n)-1= (1-n)/n
e=n/ (1-n) (b)
In equations (a) and (b), the porosity should be expressed as a ratio and not percentage.
Degree of saturation
The degree of saturation is the ratio of the volume of water to the volume of voids. It is denoted by ‘S’.
S=Vw/Vv
The degree of saturation generally expressed as a percentage. It is equal to zero when the soil is absolutely dry and 100% when the soil is fully saturated.
Percentage air voids
It is the ratio of volume of air to the total volume.
na= Va/V
It is also expressed as a percentage.
Air content
Air content is defined as the ratio of the volume of air to the volume of voids
ac= Va/Vv
Also,
na=n ac
Water content
The water content (w) is defined as the ratio of the mass of water to the mass of soilids
w= Mw/Ms
It is also known as the moisture content (m). It is expressed as a percentage but used as a decimal in computation.
VOLUME-MASS RELATIONSHIPS
1. BULK MASS DENSITY
The bulk mass density (ρ) is defined as the total mass (M) per unit volume (V)
ρ=M/V
2. DRY MASS DENSITY
The dry mass density (ρd) is defined as the mass of solids per unit total volume
ρd= Ms/V
3. SATURATED MASS DENSITY
The saturated mass density (ρsat) is the bulk density of the soil when it is fully saturated
ρsat= Msat/V
4. SUBMERGED MASS DENSITY
When the soil exists below water, it is in a submerged condition. The submerged mass density (ρ’) of the soil is defined as the submerged mass per unit total volume.
ρ’= Msub/V
5. MASS DENSITY OF SOLIDS
The mass density of solids (ρs) is equal to the ratio of the mass of solids to the volume of solids
ρs=Ms/Vs
VOLUME-WEIGHT RELATIONSHIPS
BULK UNIT WEIGHT (γ)= W/V
DRY UNIT WEIGHT (γd) = Ws/V
SATURATED UNIT WEIGHT (γsat) = Wsat/V
SUBMERGED UNIT WEIGHT (γsub or γ’) = Wsub/V
UNIT WEIGHT OF SOIL SOLIDS (γs) = Ws/Vs
SPECIFIC GRAVITY OF SOLIDS
The specific gravity of soil particles (G) is defined as the ratio of the mass of a given volume of solids to the mass of an equal volume of water at 4° C.
G = ρs/ρw
The mass density of water ρw at 4°C is 1gm/ml, 1000 kg/m3 or 1 Mg/m3
BASIC RELATIONSHIPS
Sl No Relationship in mass density Relationship in unit weight
1 n = e/(1+e) n = e/(1+e)
2 e = n/(1-n) e = n/(1-n)
3 na = n ac na = n ac
4 ρ= (G+Se)ρw/(1+e) γ= (G+Se) γ w/(1+e)
5 ρd = Gρw/(1+e) γd = G γ w/(1+e)
6 ρsat = (G+e)ρw/(1+e) γsat = (G+e) γ w/(1+e)
7 ρ’ = (G-1)ρw/(1+e) γ’ = (G-1) γ w/(1+e)
8 e = wG/s e = wG/s
9 ρd = ρ/(1+w) γd = γ/(1+w)
10 ρd = (1-na)G ρw/(1+wG) γd = (1-na)G γ w/(1+wG)
