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Principles of Geotechnical Enginee...

9th Edition
Braja M. Das + 1 other
ISBN: 9781305970939

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BuyFindarrow_forward

Principles of Geotechnical Enginee...

9th Edition
Braja M. Das + 1 other
ISBN: 9781305970939
Textbook Problem

For a given soil, show that,

a. γ sat = γ d + ( e 1 + e ) γ w

b. γ d = e S γ w ( 1 + e ) w

c. e = γ sat γ d γ d γ sat + γ w

d. w sat = n γ w γ sat n γ w

(a)

To determine

Show that γsat=γd+(e1+e)γw.

Explanation

Express the formula for saturated unit weight (γsat) as follows.

γsat=(Gs+e)γw1+e (1)

Here, Gs is the specific gravity of the soil, e is the void ratio, and γw is the unit weight of the water.

Rearrange equation (1) as shown below.

γsat=(Gsγw+eγw)1+e=Gsγw1+e+eγw1+e

Here, (Gsγw1+e) is the equation of dry unit weight (γd) of the soil.

Substitute γd for Gsγw1+e.

γsat=γd+(e1+e)γw

Hence, the saturated unit weight of the soil γsat=γd+(e1+e)γw_ is proved.

(b)

To determine

Show that γd=eSγw(1+e)w.

Explanation

Express the formula for dry unit weight of the soil (γd) as follows.

γd=Gsγw1+e (2)

Here, Gs is the specific gravity of soil solids.

Express the relation between specific gravity of soil solids (Gs), degree of saturation (S), void ratio (e) and moisture content (w) as shown below.

Gs=Sew

Substitute Sew for Gs in Equation (2).

γd=(Sew)γw1+e=eSγw(1+e)w

Hence, the dry unit weight of the soil γd=eSγw(1+e)w_ is proved.

(c)

To determine

Show that e=γsatγdγdγsatγw.

Explanation

Refer to part (a).

The saturated unit weight of the soil γsat=γd+(e1+e)γw.

Rearrange Equation as shown below.

γsat(1+e)=γd(1+e)+eγwγsat+γsate=γd+γde+eγwγsatγd=γde+eγwγsateγsatγd=e(γdγsat+γw)

e=γsatγdγdγsat+γw

Hence, the void ratio e=γsatγdγdγsat+γw_ is proved.

(d)

To determine

Show that wsat=nγwγsatnγw.

Explanation

Express the formula for saturated unit weight (γsat) as follows.

γsat=(1+wsat)Gsγw1+e (3)

Express the formula for specific gravity of soil solids (Gs) as shown below.

Gs=Sew

The degree of saturation S=1 for saturated sample.

Substitute 1 for S.

Gs=ewsat

Substitute ew for Gs in Equation (3).

γsat=(1+wsat)(ewsat)γw1+e=(1+wsat)wsate(1+e)γw

Here, e(1+e) is the equation of porosity (n) of the soil.

Substitute n for e(1+e).

γsat=(1+wsat)wsatnγwγsatwsat=nγw+nγwwsat(γsatnγw)wsat=nγwwsat=nγwγsatnγw

Hence, the moisture content wsat=nγwγsatnγw_ is proved.

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