Solids-Water-A ir Relationships and Index Properties of Soils 11

ne diagrammatic representation of thedifferent phases in a soil mass is called the phase diagram. Figure
a) shows a three-phase system which is applicable for a partially saturated soil. When all the voids are
Thed with water, thesample becomes saturated and thus the gaseous phase is absent whereas in a dry soil, the
Ihquid phase is missing [Fig. 2. 1(b)). Thus, both the saturated and dry states of soil have
two phases only.
In Fig. 2.1, the volumes and weights of various constituents of the soil mass are designated
by and W.
respectively. The subscripts a, w, s and vstand for air, water, solids and voids
respectively.
he weight of air, for all praçtical purposes, may be assumed to be zero. The
equal to the weight of water WThe weight of solids isrepresented by W, which is weight of the yoids is thus
dry soil. The total weight of a moist Sample is thusW= W.+ W The total evidently the weight of a
volume
namly. ()volume of solids, V, (i) volume of water, V,, and (iii) volume of air, V AsVthe consists of three parts,
by water or air or both, the volume of voids V, = V, + V,. Thus V= V,+V,= voids may be filled
note that the term 'voids is used to include both the filled and the V,+V,+Vg It is important to
by the soil grains. unfilled portions of the pore spaces enclosed

22 SIMPLEDEFINITIONS
As mentioned earlier, the relative proportions of solids, water and air
present in a mass influence itsphysical
properties. Certain terms related to volumetric and gravimetric compositions soil of soils are frequently used in
soilengineering. These terms are described in the following
sections.
Water Content

Water content, w, also called the moisture content, is defined as the ratio of
the weight of water to the weight
of solids in a given mass of soil. It is usually expressed in
percentage.
W
W= X 100
W, (2.1)
In nature, fine-grained soils have higher values of natural moisture content
compared to the coarse-grained
soils. In general, it can be written that w >0, since there can be no upper limit to water
content.
Void Ratio -rato
Void ratio, e of a soil sample is defined as the ratio of the volume of voids to the volume of solids.
Thus,
symbolically.
e=
Vy
V (2.2)
In nature, even though the individual void sizes are larger in coarse-grained soils, the void ratios of fine
grained soils are generally. higher than those of coarse-grained soils, In general, it can be written that e> 0,
since a soil has to contain some voids but there cannot be an upper limit to the void volume.

Porosity %
Porosity,n of a soil sample is defined as the ratio of the volume of voids in the sample to the total volume of
the soil, expressed as a percentage. Thus, symbolically,

(2.3)
X Contents

and
Applied Soil Mechanics
Basic

<n<
100.
that 0
written with respect to the
itcan be
former

engineering, void
in soil|
Thus, voids-the

12 cent. of
100per of
volume However,
mass is a direct
cannot
exceed p r o p o r t i o n

of soil. volume of
asoil
Porosity of
asoil express
the volume
in the remain the
sarme. Hence
thetotal
and porosity
change

Both void ratio with

respectto
factthat
any
volume of
solids
whenthe
volume of asoil
the latter the the
solids, and due to while changes
volume of This is voids (V)
favoured for use. volume of numerator
change.
ratio is more change in
the onlythe (V)
of a similar in which
denominator

consequence
voidratio (V)and
the
convenient to use numerator
it is more boththe usually
while in porosity, voids. It is
changes volume of
watertothe
Degree ofSaturation volume of
ratioofthe
defined as the
saturation, Sor
S., is (2.4)
Degree of Thus,
percentage.
expressedas a Vex 100 water, for a
S=
is filled with
voids which thus
volume of
perfectly dry
soil, V =0;
portion of
indicates the for a cent and 100
of saturation
100 per cent. Similarly, between O per
hence S =| or
degree varies
Since the saturation
saturatedsoil, V,
= V, and saturated soil. the degree of
fully partially
S = 0 o r 0per
cent. For a
0 <Ss 100.
i cent.That is,
Air Content volume of voids.
air vojds to the
ratio of the volume of (2.5)
Air content,a, is the
a,-=1-S

Percentage Air Voids percentage of the total volume

expressed as a
volume of air voids
Percentage air voids, n,
is the ratio of the
Symbolically, (2.6)
of thesoil mass.
n,=x100
(2.7)
It canbe seen that n, =n a

Unit Weight
regard to
is its weight per unit of volume. The unit weight must be expressed with due
Unit weight of asoil unit weight are important in relation to soils:
the state of thesoil. The following definitions of as
Wper unit of total volume, V. It is also known
Bulk unit weight, y,is the total weight of a soil mass,
the total unit weight. Symbolically,
W W, + W (2.8)
Y=yV, +Vut Va
units, it is convenient to express it in kN/m'. Incgs/MKS units, it is expressed in gmf/cc, kgt/m
or
In SI
tUm'. In this text, the suffix f in the units gmf and kgf is dropped and the units g and kg are used to express
weights in cgs and MKS system.
 Solids-Water-A ir Relationships and Index Properties of 13
Soils
Dry unit wweight, Ya is the weight of solids, W, per unit of total volume. Symbolically,
W
Y= (2.9)
The dry unit weight is used as
sa measure ofthe denseness of asoil., Ahigh value of dry unit weight indicates
that more solids are packed in a unit volumeof the soil and hence a more compact soil.
Saturated unit weight, Ywn is the ratio of the total weight of a fully saturated soilsample, Wa to
V. In other words, it is the bulk unit weight of a its total volume
soil when it is completely saturated.
Sai
Yvar (2.10)
Submerged unit weight or Buoyant unit weight y'or y, is the submerged
When asoil mass is submerged below theground water table a
weightof soil solids per unit volume.
equal in magnitude to the weight of water displaced by the solids.buoyant force acts on the soil solids which is
The net weight of the
reduced weight is known as the submerged or solids is reduced;the
the buoyant weight.
Submerged unit weight is equal in magnitude tothe saturated unit
of water, Yw weight, y minus the unit weight

Y'= W(uh) (2.11)

The unit weight of water, Yw depends on its temperature.
weight of water is taken to be constant at 1.0 glcc or 9.8 However, for all practical purposes, the unit
kN/m.
The submerged unitweight is roughly one half of the saturated
unit weight.
Unit weight of solids, y, is the ratio of the weight of solids, W, to the
volume of solids, V,
W,
Y =
V. (2.12)
Specific Gravity
Specific gravity ofsolids, G, (sometimes written as G) is defined as the ratio of the
of solids to the weight of an equivalent volume of water at 4°C. weight of a given volume

W,
(2.13)
Specific gravity of solids can also be defined as the ratio of the unit weight of solids to that of water.
Thus,
G, = (2.14)

At 4°C,y=1g/cc or 9.8 kN/m'

The value of specific gravity of soil grains is required in the determination of unit weight of solids
(Y, =G, Y), unit weight of soil, void ratio,degree of saturation, water content by the
in several soil tests.
pycnometer method and
 and Applied Soil
14 Basic
Mechanics
The value of G, for a majority of soils lies between 2.65 and 2.80. Lower values are for coarSe-grained
soils. The presence of organic matter leads to very low values. Soils high in iron or mica exhibit high values,

Table 2.l gives typical values of G, for different soils.

Table 2.1 Typical Values of G,
Specific gravity
Soil type
2.65-2.68
Clean sands and gravel
2.66 - 2.70
Silt andsilty sands
2.70- 2.80
Inorganic clays
2.75-2.85
Soils high in mica, iron
Ouite variable: may
Organic soils fall below 2.0

used in soil engineering. Mass specific gravity is the

Pparent or Mass specificgravity, G is also sometimes ratio of the total weight of a given mass of soilto the
Speciti gravity ofthe the soil mass and is defined
as the
bulk unit weight of the soj to theunit weight
Weight of anequivalent volume of water. It is also the ratio of the
of water

Gm =
W_Y (2.15)
V
of unit weight' is more logical in relation
The authors. however,do not favour the use ofthis term. The use
to a soil mass.
2.3. SOMEIMPORTANT RELATIONSHIPS
defined in Section 2.2 can now be
Afew importantand often used relationships between various quantities involved. Alternatively,
established. This can doneby starting with the basic definitions of one of the parameters
as unity (1)and the volumes and
a phase diagram can be used in which one of the volumes or weights is taken using one or the other
weights of all other quantities are then worked out in terms of this assumed value by
this method. Both methods
ratio defined carlier. Most commonly, the volume of solids is assumed as unity in
are demonstrated here.

Relation between W,,Wand w:

W= W,+W, = W,(1+ W/W)
W (2.16)
W, 1+w
Relation between e
and n

(2.17)
 Solids-Water-Air Relationships and Index Properties of Soils 15

Or (2.18)
FIg. 2.2 (a) shows the three-phase diagranm in which the volume of solids is shown as unity.

(A) (A)
(W)
+ (S)
V=1 (W)

(S)

(a) (b)

Fig. 2.2 Phase diagram.

V, = 1; hence V,=e and V=V,+ V,=l+e

n=
V, e

V 1+e
Fig. 2.2 (b) shows the phase diagram in which the total volume is
taken as unity.
V=1; hence V, =n and V,=V-V,=|-n
e =
V,n
V, 1-n
Itcan be readily seen from Fig. 2.2 (a) that in a total volume of soil equal to (l +e), the
is Iand the volume of voids is e. Thus, if the total volume of a soil is V,
volume of solids
e
V, = 1+e and =
V, V
Relation between e, w,G, and S

Vy Vw
e= V. V,V,

V W/w
W,/Y
V, Wy G, Yw
V W,
-wG,
WG,
or e = (2.19)
X Contents Soil Mechanics
Applied

Basic and

16

(A)
fo
(W)
se , orwo,

(S)

diagram.
Phase
Fig. 2.3

Fig. 2.3,
phase diagram of G, Y
Alternatively, from the
V =Se and or W =WX W,=w
V,= 1; , =e; W, = V, xY, =Se
V,xY=lxG, %.;
W, = Se
Ww
W =
W, G, Y
(2.20)
wGs
or e S
e=w G, saturated soil.
saturated soil, S= 1; completely
a
For a completely ratio of
calculate the void
frequently used to
bq. 2.20 is
and Y
terms of G, e, w
Bulk Unit Weight, Y, in
W W,+ W, W, (1 + W/W,)
Y=y V,+ V, V, (1+V/V)

Since
Www V-e
W, V, and V,
(2.21)
G,1, (1+w)
1+e
Se
Since wG
G,+Se) (2.22)
=1+e
Alternatively, from the phase diagram
of Fig. 2.3,
w W,+ W G,Y t Se Yor G, Y1+e
t wG, Y
=y V, +V, 1+e

G,Y (1+ w)
1+e
Solids-Water-A ir Relationships and Index Properties 17
of Soils
Saturated Unit Weight, Yet in terms
of G, e and Yw
Fig. 2.4 shows the phase
diagram for a saturated soil in which only solids and water are present.
W

(W) eYw

(S)

Fig. 2.4 Phase diagram (saturated soil)

W W,+ Ww G, Yu teYw
V V,+ V, 1+e

Ysar = (2.23)
The same equation would result by substituting
S= 1 in Eq. 2.22.
Dry Unit Weight, Ya in terms of G,, e and Yw

Fig. 2.5 shows the phase diagram fora dry soil in which only
solids and air are present.

(A)

(S) Gs Yw

Fig. 2.5 Phase diagram (dry soil)

W, W
VV V,+ V,
G, Y
1+e
(2.24)

The same equation would result by substituting S = 0 in Eq. 2.22.

X Contes

Applied Soil Mechanics

Basic and

18

and Y
Ya in terms of G, W, S
WG,
From Bq. 2.19, e =
Substituting for e in Eq. 2.24. (2.25)
G S
dry unit
weight for such a
content, the
given water
saturated (S = 1) at a
fully
When the soil becomes
equation (2.26)
condition is given by the
G, Yw
compaction
behaviour of
found useful in
the study of
by Ea. 2.26 is
unitweight given
Cero air voids'
soils.
and Y
e in terms of Y, G,
From Eq. 2.24,
(1+e) Yu = GYw (2.27)
or e=
GYw1
Yu

determine void ratio of

an insitu soildeposit.
Eq. 2.27 is often used to

Unit Weight, y' in terms

of Ge and y
Submerged
Y= Yst - Yw

or y=
(G,- 1). (2.28)

Relation between Y, Y and w

W W, + W W, (1+ W/W)-=Ya (l+w)

(2.29)
or atw
Solids- Wate r-Air Relationships and Index 19
Properties of Soils
Relation between Ya,G, Wand n, Qc

V= V,+ V +V
V
V V

Vw
+ na

V, V,
V

W/G, Y W/Y
V

wW/Yw
G, Yw
WYa
G, Yw
w+

(1-n) G, Yw
1+ wG (2.30)

Eq. 2.30 expresses the relationship between dry unit weight and the percentage air voids. This will be
useful in the study of compaction behaviour in soils.
Ya when n, =0, that is, when the soil becomes fully saturated at a given water content, is given by the
equation

Ya = 1+ wG

which is the same as Eq. 2.26. In other words, when n, =0 andYa when S= l represent the same condition.
Some of the useful relationships which were derived above are listed below for ready reference.
W
i) W, = 1+w

(ii) V,= I+e

(iii) n= 1
+e

(iv) e=
WG,
S
 Basic and Applied
Soil Mechanics
20

G, +Se) G, (1 + w) Y
(v) 1+e

G, +e
(vi) Ysar

G,
(vi) 1+e

G
(vii)

(ix) e
(G. Y -1
Ya

Y
() Ya =
1+w

G (1-n) G, Ye
(xi) Ya =
1+ wG/s 1+wG,
RMINATION