This document discusses soil phase systems and relationships between various soil properties. It describes soil as having either a 3-phase or 2-phase system, depending on whether it is partially or fully saturated/dry. The 3-phase system includes volumes and weights of solids, water, and air. Key relationships defined include water content, void ratio, porosity, degree of saturation, dry density, bulk density, and specific gravity. Density index and relative compaction are also explained. Functional relationships are presented between various properties like void ratio, degree of saturation, dry density, specific gravity, and unit weights.

1. CE6405 SOIL MECHANICS
V.Nageshwaran, M.E.,
Assistant Professor,
Department of Civil Engineering,
UCET

2. PHASE SYSTEM

3. SOIL PHASE SYSTEM
 Soil – 3 Phase System – Partial
 Soil – 2 Phase System – Dry or Wet System

4. 3 PHASE SYSTEM – PARTIALLY SATURATED
 Vs – Volume of Solids
 Vw – Volume of Water
 Va – Volume of Air
 Vv – Volume of Voids
 V – Total Volume of Soil
 Ws – Weight of Solids
 Ww – Weight of Water
 Wa – Weight of Air = 0
 Wv – Weight of Voids
 W – Total Weight of Soil
Volume Relationships
 V = Vs+Vw+Va
 Vv = Vw+Va
Weight Relationships
 W = Ws+Ww+Wa
 Wv = Ww+Wa
 Wv = Ww [Since, Wa = 0]
Note: Wd – Weight of Solids

5. 2 PHASE SYSTEM – DRY (COMPLETELY) SYSTEM
 Vs – Volume of Solids
 Vw – Volume of Water = 0
 Va – Volume of Air
 Vv – Volume of Voids
 V – Total Volume of Soil
 Ws – Weight of Solids
 Ww – Weight of Water = 0
 Wa – Weight of Air = 0
 Wv – Weight of Voids
 W – Total Weight of Soil
Volume Relationships
 V = Vs+Vw+Va
 Vv = Vw+Va [Since, Vw = 0]
Weight Relationships
 W = Ws+Ww+Wa
 Wv = Ww+Wa
 Wv = 0 [Since, Ww & Wa =0]
Note: Wd – Weight of Solids

6. 2 PHASE SYSTEM – WET (COMPLETELY) SYSTEM
 Vs – Volume of Solids
 Vw – Volume of Water
 Va – Volume of Air = 0
 Vv – Volume of Voids
 V – Total Volume of Soil
 Ws – Weight of Solids
 Ww – Weight of Water
 Wa – Weight of Air = 0
 Wv – Weight of Voids
 W – Total Weight of Soil
Volume Relationships
 V = Vs+Vw+Va
 Vv = Va+Vw [Since, Va = 0]
Weight Relationships
 W = Ws+Ww+Wa
 Wv = Ww+Wa
 Wv = Ww [Since, Wa =0]
Note: Wd – Weight of Solids

7. VOLUME-MASS/WEIGHT RELATIONSHIPS

8. VOLUME-MASS/WEIGHT RELATIONSHIPS
 Water Content, w – Ratio of Weight of Water, Ww to the
Weight of Solids, Ws or Wd – Expressed as a %.
or100
W
W
w
s
w
 100
W
W
w
d
w

1001
W
W
100
W
WW
w
ss
s









or100
M
M
w
s
w
 100
M
M
w
d
w

1001
M
M
100
M
MM
w
ss
s









Where,

9. CONT…
Density of Soil – Ratio of Mass of Soil, M to the Unit Volume
of Soil, V.
 Bulk Density, ρ – M/V [Total Vol.]
 Dry Density, ρd – Md/V [Vol. prior to drying]
 Density of Solids, ρs – Md/Vs [Vol. of Solids]
 Saturated Density, ρsat – Msat/V
 Submerged Density, ρsub – (Md)sat/V
ρ sub = ρsat – ρw

10. CONT…
Unit Weight of Soil – Ratio of Weight of Soil, W to the Unit
Volume of Soil, V.
 Bulk Unit Weight , γ – W/V [Total Vol.]
 Dry Unit Weight , γd – Wd/V [Vol. prior to drying]
 Unit Weight of Solids, γs – Wd/Vs [Vol. of Solids]
 Saturated Unit Weight , γsat – Wsat/V
 Submerged Unit Weight , γsub – (Wd)sat/V
γsub = γsat – γw
 Conversion between Density & Unit Weight

11. CONT…
 Specific Gravity, Gs – Ratio of Weight of a given Volume of
Soil Solids at a given temperature to the Weight of an equal
volume of distilled water at that temperature, both Weights
being taken in air.
 Indian Standard specifies 27ºC as the Standard Temperature
for reporting the Specific Gravity.
 Apparent or Mass or Bulk Specific Gravity, Gm – Specific
Gravity of Soil Mass.
w
s
s
γ
γ
G 
w
m
γ
γ
G 

12. CONT…

13. SUMMARY OF VOLUME-MASS/WEIGHT RELATIONSHIPS
w
s
s
γ
γ
G 
w
m
γ
γ
G 
Unit WeightsDensities
V
W
γ 
V
W
γ d
d 
s
d
s
V
W
γ 
V
W
γ sat
sat 
 
V
W
γ satd
sub 
wsatsub γγγ 
V
M
ρ 
V
M
ρ d
d 
s
d
s
V
M
ρ 
V
M
ρ sat
sat 
 
V
M
ρ satd
sub 
wsatsub ρρρ 

14. TERMS AND TERMINOLOGIES

15. VOID RATIO, POROSITY & DEGREE OF SATURATION
 Void Ratio, e – Ratio of Volume of Voids to the Volume of
Soil Solids in the given Soil mass.
 Porosity, n – Ratio of Volume of Voids to the Total Volume of
the given Soil mass.
 Degree of Saturation, S – Ratio of Volume of Water present
in a given Soil mass to the Total Volume of Voids in it.
s
v
V
V
e 
V
V
n v

v
w
V
V
S 

16. DEGREE OF SATURATION
 Fully Saturated Soil – Vw = Vv S = 1
 Perfectly Dry Soil – Vw = 0 S = 0

17. % AIR VOIDS & AIR CONTENT
 % Air Voids, na – Ratio of Volume of Air Voids to the Total
Volume of Soil mass.
 Air Content, ac – Ratio of Volume of Air Voids to the Volume
of Voids.
 Degree of Saturation, S – Ratio of Volume of Water present
in a given Soil mass to the Total Volume of Voids in it.
V
V
n a
a
wva
v
a
c VVV;
V
V
a 
v
w
V
V
S 
S1
V
V
1a
v
w
c 

18. CONT…
 ID = 0; When e = emax; i.e. Soil is in its Loosest State.
 ID = 1; When e = emin; i.e. Soil is in its Densest State.
 ID = 0 to 1; When e ≠ emax or emin; i.e. Soil is not in its Loosest
or Densest State.
minmax
max
D
ee
ee
I



d
d.max
d.mind.max
d.mind
D
γ
γ
.
γγ
γγ
I



 
  n1nn
)n(1nn
I
minmax
minmax
D



StateLoosestMostatDensityDryγd.min 
StateCompactMostatDensityDryγd.max 

19. CONT…

20. DENSITY INDEX
 Density Index/Relative Density/Degree of Density, ID –
Ratio of the Difference between the Voids ratio of the Soil in
its Loosest State (emax) and its Natural Voids ratio (e) to the
Difference between the Voids ratio in the Loosest and
Densest State (emin).
 Expresses the Relative Compactness or Degree of
Compaction of a Natural Cohesionless Soil Deposit.
 Not applicable to Cohesive Soil – Uncertainties –
Determination of Void ratio in Loosest State.
minmax
max
D
ee
ee
I




21. RELATIVE COMPACTION
 Relative Compaction/Degree of Compaction, RC – Ratio
of Dry Density of Soil at Natural State to the Dry Density of
Soil at its Compact State.
e1
e1
R min
C



d.max
d
C
γ
γ
R 
  DO
O
C
I1R1
R
R


d.max
d.min
O
γ
γ
R 
 Lee & Singh (1971) approximate eqn. DC 0.2I80R 

22. SUMMARY OF TERMS AND TERMINOLOGIES
e1
e1
R min
C



d.max
d
C
γ
γ
R 
minmax
max
D
ee
ee
I



d
d.max
d.mind.max
d.mind
D
γ
γ
.
γγ
γγ
I



 
  n1nn
)n(1nn
I
minmax
minmax
D



V
V
n a
a
v
w
c
V
V
1a 
s
v
V
V
e 
V
V
n v

v
w
V
V
S 
v
a
c
V
V
a 
S1ac  DC 0.2I80R 

23. FUNCTIONAL RELATIONSHIPS
(i) Relation between e, Gs, w & S
e.Sew 
e
e
e
e
V
V
S w
v
w
v
w

.1γ
γe
W
W
w
s
ww
d
w

wss
w
s
s .γGor γ
γ
γ
G 
RatioVoidWaterew 
s
w
ws
ww
G
e
.γG
.γe
w 
sw w.Ge 
S
w.G
e s
 ssat.Gwe 

24. CONT…
(ii) Relation between e, S & na
wwva eeVVV 
V
V
n a
a 
e.SeBut;
e1
ee
n w
w
a 



 
e1
S1e
na



e1VVV vs 

25. CONT…
(iii) Relation between na, ac & n
v
a
c
V
V
a 
V
V
n v

c
a
a n.a
V
V
n 

26. CONT…
(iv) Relation between d, Gs & e (or n)
V
W
γ d
d 
 e1V&1Vs 
e1
.1γ
d
s
γ 
γ
V
.Vγ
γ ss
d 
e1
.γG
d
ws
γ 
wss .γGγBut, 
1
)1(.γG
d
ws
γ n

1e d
ws
γ
.γG

1V&n1Vs 

27. CONT…
(v) Relation between , Gs & e (or n)
V
.Vγ.Vγ
V
WW
V
W
γ wwsswdsat
sat




 e1V&,eV1,V wws 
e1
.eγ.γG
e1
.eγ.1γ
sat
wwswws
γ 




satγ
 
e1
.γeG
sat
ws
γ 


.nγn)(1.γGγ wwssat 
1V&n,Vn,1V ws 
    .nγ1.γ
1
.nγ1.γ
γ ws
ws
sat 

 n
n

28. CONT…
(vi) Relation between , Gs, e, & S
V
.Vγ.Vγ
V
WW
V
W
γ wwsswd 



 e1V&,eV1,V wws 
 
e1
.γe.SG
e1
.e.Sγ.γG
e1
.eγ.1γ wswwswws
γ 






γ
e1
.γG
d
ws
γ 
 
e1
.γe.SG ws
γ 


 
e1
.γeG
sat
ws
γ 



29. CONT…
'
γ
 
e1
.γ1G
γ ws'



(vii) Relation between , Gs & e
wsat
'
γγγ 
 
w
ws'
γ
e1
.γeG
γ 




30. CONT…
(viii) Relation between , , & w
d
W
w
W
w 
w1
W
Wd


dγ
 .Vw1
W
V
d
W
d
γ


w1
γ
d
γ


γ
d
W
W
d
W
d
W
w
W
w1 



31. CONT…
(ix) Relation between , , & n
  wwswssubd γ.γG1.γ1.γw 
'
γ
  wd
'
γn1γγ 
d
γ
e1
γG ws


dγ
    wssubd .γ1Gw  e1Vand
   
e1
.γ1G
V
w
γ wssubd'


 or
e1
γ
e1
.γG
γ wws'




n

1
e1
1and

32. CONT…
(x) Relation between , , & Sγ dγ
 
e1
.γ.eG
γ ws



S
satγ
e1
γ.e
e1
γ.G
γ wws



 Sor
 











e1
.γG
e1
.γeG
Sγγ wsws
d
 dsatd γγSγγ 

33. CONT…
(xi) Relation between , Gs, w & Sdγ
e1
γ.G
γ ws
d

 But
S
sw.G
e 
S
w.G
1
γ.G
γ
s
ws
d


ssat
ws
d
.Gw1
.γG
γ

;1S 

34. CONT…
(xii) Relation between , Gs, w & nadγ
or
 
s
wa
d
G
1
w
.γn1
γ


 or
swa VVVV 
s
s
w
w
a
γ
W
γ
W
VV or
s
d
w
da
s
d
w
da
γ
γ
γ
w.γ
V
V
V.γ
W
.Vγ
w.W
V
V
1 
ws
d
w
da
.γG
γ
γ
w.γ
V
V
1    






sw
d
G
1
γ
γ
n1 wa
 
s
wsa
d
w.G1
.γ.Gn1
γ




35. SUMMARY OF FUNCTIONAL RELATIONSHIPS
e
e
e
e
V
V
S w
v
w
v
w

wss
w
s
s .γGor γ
γ
γ
G 
S
w.G
e s

v
a
c
V
V
a 
V
V
n v

c
a
a n.a
V
V
n 
e1
ee
n w
a



 
e1
S1e
na



 dsatd γγSγγ 
e1
w
.γe.S
s
G
γ











S
s
w.G
1
w
γ.
s
G
γd


e1
w
.γ
s
G
γd


w1
γ
γd


n)(1.γGγ wsd 

36. CONT…
 
G
1
w
.γn1
γ
s
wa
d



  wd
'
.γn1γγ 
 
e1
γ.1G
γ ws'



wsat
'
γγγ 
e1
w
.γe
s
G
sat
γ











 
s
wsa
d
w.G1
.γ.Gn1
γ


