Weight to Volume relationships in soils

This calculator computes the volume and weight relationships in soils, which are useful relationships used in soil mechanics.
CalculationAssumptions
Unit weight of water, ﻿γw​=9.81
 kN/m3
Weight of air is zero, therefore ﻿Wt​=Ww​+Ws​
﻿
The calculator assumes the known parameters are the soil sample mass ﻿Mt​
, the over-dry soil sample mass ﻿Ms​
, the volume of the soil sample ﻿Vt​
 (based on the test tube dimensions) and soil specific gravity ﻿G
 (determined from lab tests). This is a typical situation.
InputSoil sample values:
﻿
2446​
5777​
3523​
2446​
1077​
OutputWeight-volume relationships:
﻿
0.61​
0.38​
69.4%​
15.6%​
1.95​
19.1​
1.68​
16.52​
2.71​
26.59​
2.06​
20.23​
1.06​
10.42​
ExplanationWeight-Volume RelationshipsA soil mass with total volume ﻿V
 and total weight ﻿W
 is shown below with its three phases of soil mass (soil solids, air and water).
﻿
Soil mass and it's three phases [Adapted from Source]
In most cases you will know the soil sample mass ﻿Mt​
, the over-dry soil sample mass ﻿Ms​
, the volume of the soil sample ﻿Vt​
 (based on the test tube dimensions) and soil specific gravity ﻿G
 (determined from lab tests). 
👉Note, you can either work in mass (with units ﻿g
) or weight (with units ﻿N
). If you work in mass, your densities will be in ﻿g/cm3
 while if you work in weight, your densities will be in ﻿kN/m3
. Remember also if you use mass, take ﻿γw​=1 g/cm3
 and if you use weight, take ﻿γw​=9.81 kN/m3
.
We can then determine the remaining mass and volume quantities:
Volume of soil solids, ﻿Vs​=Ms​/Gγw​
 which is derived using the definition of solid unit weight ﻿γs​=Ms​/Vs​
 and the definition of soil specific gravity, ﻿G=γs​/γw​
. Therefore, ﻿γs​=Ms​/Vs​→Gγw​=Ms​/Vs​→Vs​=Ms​/Gγw​
﻿
👉Note, ﻿G
 is typically within the range 2.65 (average from granular soils) and 2.80 (average for cohesive soils).
Volume of voids, ﻿Vv​=Vt​−Vs​
﻿
Volume of water, ﻿Vw​=Ms​γw​
﻿
Volume of air, ﻿Va​=Vv​−Vw​=Vt​−Vw​−Vs​
﻿
We now know all our mass and volume quantities (﻿Ms​,Mw​,Vs​,Va​,Vw​
). We can then define the following weight-volume relationships commonly used in soil mechanics:
Void ratio, ﻿e=Vv​/Vs​
﻿
Porosity, ﻿n=Vv​/Vt​
 
Degree of saturation, ﻿s=Vw​/Vv​
 which can range between zero for a completely dry soil, to 1 for a fully saturated soil
Moisture content, ﻿w=Mw​/Ms​
﻿
Unit weight, ﻿γt​=Mt​/Vt​
, also known as moist unit weight or total unit weight since it includes weight of water and soil solids
Dry unit weight, ﻿γd​=Ms​/Vt​
﻿
Solid unit weight, ﻿γs​=Ms​/Vs​
﻿
Saturated unit weight, ﻿γsat​=γt​
 for ﻿s=100%
 (i.e. ﻿Va​=0
) is when the soil is completely saturated
Submerged (buoyant) unit weight, ﻿γb​=γsat​−γw​
 for ﻿s=100%
 is when the soil is below the ground water table
Weight-Volume Relationships for ﻿Vs​=1
﻿Useful weight-volume relationships can be developed by considering a soil sample with ﻿Vs​=1
 as shown below.
Therefore, the volume of voids ﻿Vv​=e/Vs​=e
.
﻿
Weight-volume relationships for ﻿Vs​=1
 [Adapted from Source]
Weight-volume relationships can then be derived in terms of ﻿e
. Here are some common forms: 
Moist unit weight, ﻿γt​=γw​×G(1+w)/(1+e)
﻿
Dry unit weight, ﻿γd​=γw​(G)/(1+e)
﻿
Degree of saturation, ﻿S=wG/e
﻿
Saturated unit weight, ﻿γsat​=γw​(G+e)/(1+e)
﻿
Submerged unit weight, ﻿γb​=γw​(G−1)/1+e)
﻿
The derivation of the above equations are found in Advanced Soil Mechanics by Braja M. Das.
Weight-Volume Relationships for ﻿Vt​=1
﻿Useful weight-volume relationships can also be developed by considering a soil sample with ﻿Vt​=1
 as shown below.
Therefore, the volume of voids ﻿Vv​=nVt​=n
.
﻿
Weight-volume relationships for ﻿V=1
 [Adapted from Source]
Weight-volume relationships can then be derived in terms of ﻿n
. Here are some common forms: 
Moist unit weight, ﻿γt​=γw​×G(1−n)(1+w)
﻿
Dry unit weight, ﻿γd​=γw​ G(1−n)
﻿
Saturated unit weight, ﻿γsat​=γw​[G−n(G−1)]
﻿
The derivation of the above equations are found in Advanced Soil Mechanics by Braja M. Das.
ReferencesAdvanced Soil Mechanics by Braja M. Das
Soil Phase Diagram and Relationship Formulas
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Weight to Volume relationships in soils

Calculation

Assumptions

Input

Output

Explanation

Weight-Volume Relationships

Weight-Volume Relationships for

$V_{s} = 1$

Weight-Volume Relationships for

$V_{t} = 1$

References

Related Resources

		0.61
		0.38
		69.4%
		15.6%
		1.95
		19.1
		1.68
		16.52
		2.71
		26.59
		2.06
		20.23
		1.06
		10.42




		2446

		5777
		3523
		2446
		1077

Weight to Volume relationships in soils

Calculation

Assumptions

Input

Output

Explanation

Weight-Volume Relationships

Weight-Volume Relationships for ﻿Vs​=1﻿

Weight-Volume Relationships for ﻿Vt​=1﻿

References

Related Resources

Weight-Volume Relationships for

$V_{s} = 1$

Weight-Volume Relationships for

$V_{t} = 1$