Geotechnical Engineering

DHARM

COMPOSITION OF SOIL TERMINOLOGY AND DEFINITIONS 19

∴ wsat = e/G or e = wsat.G ...(Eq. 2.22)

na =

V

V

VV

VV

v

v

v

v

v

v

e

v

v

e

avw

sv

v

s

w

s

v

s

w

= − s

+

=

−

+

=

−

1 1 +

But S.e = Vw/Vs

∴ na =

eSe

e

eS

e

−

+

= −

+

.()

1

1

1

...(Eq. 2.23)

Also na = ()ee^1 + (1 – S) = n(1 – S) ...(Eq. 2.24)

ac = Va/Vv

S = Vw/Vv

ac + S =

()VV

V

aw

v

+

= Vv/Vv = 1

∴ ac = (1 – S) ...(Eq. 2.25)

In view of Eq. 2.25, Eq. 2.24 becomes na = n.ac, which is Eq. 2.20.

2.3.2 Relationships Involving Unit Weights, Grain Specific Gravity, Void

Ratio, and Degree of Saturation

γ = W/V =

WW

VV

WWW

VVV

sw

sv

sws

svs

+

+

= +

+

(/)

(/)

1

1

But Ww/Ws = w, as a fraction ;

V

V

v

s

= e ; and

W

V

s

s

= γs = G.γw

∴ γ = G w

w e

γ ()

()

1

1

+

+

(w as a fraction) ...(Eq. 2.26)

Further, γ =

()

()

GwG

e w

+

1 +

γ

But w.G = S.e

∴γ =

(.)

()

.

GSe

e w

+

1 +

γ (S as a fraction) ...(Eq. 2.27)

This is a general equation from which the unit weights corresponding to the saturated

and dry states of soil may be got by substituting S = 1 and S = 0 respectively (as a fraction).

∴γsat =

Ge

e w

+

+

F

HG

I

1 KJ

.γ ...(Eq. 2.28)

and γd =

G

e

. w
()

γ

1 +

...(Eq. 2.29)

Note. γsat and γd may be derived from first principles also in just the same way as γ.

The submerged unit weight γ′ may be written as :

γ′ = γsat – γw ...(Eq. 2.12)