Proof That The Submerged Weight
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teshProof That The Submerged Weight
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tesh1.
Proof that the submerged weight, Ws of a unit volume of soil is given by Ws = γw(1-n) (G-1),
in which is γw the unit weight of water, n is porosity of the soil material, and G is specific
gravity of soil particles.
Answer:
From the relationship of soil property parameters like: unit weight ( γ ¿, specific gravity (G),
void ratio(e), porosity(n), degree of saturation(S), moisture content(w), volume(V), and
weight (W) of the soil we can express
Ws = γw(1-n) (G-1)
( )
Ww
1+
W Ws+Ww = Ws Ws Ww Vv Vv Vw n
γ= = ; but =w ; =e , n= , =S∧e=
V Vs+Vv Vs Vv Ws Vs V Vv 1−n
1+
Vs
Ws
=γs=G γw
Vs
1+w
γ=G γw( )…………………. *1
1+e
Ww Vw S∧Vv
From =w , = =e
Ws Vv Vs
Vw
Se= ∧¿
Vs
Se=wG … … … … … … … … … … … … …∗2
Substitute equation *2 into equation *1
G+ Se ………………………………………………...*3
γ=γw ( )
1+ e
This equation (*3) is the general equation in which for the saturation case S =1 and for dry case
S = 0 then the equation becomes
γsat=γw ( G+e
1+e ) ; for S=1and γdry=γw (
¿ ; for S=0
1+e )
In our case, it is submerged unit weight ( γ ' )in a unit volume of soil.
γ ' =γsat−γw
= γw ( G+e
1+e )
−γw
= γw[ ( )
G+e
1+e
−1]
= γw ( G−1
1+ e )
n
From the porosity void ratio relationship e=
1−n
( )
G−1
'
γw
γ = n
1+
1−n
γ ' =γw(G−1)(1−n)