Formulas - Phase Relationship PDF

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Phase Relationship formulas and Soil Compaction...

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Phase Relationship and Soil Compaction (By Behzad Fatahi) Part A: Phase Relationship Basic soil relationship: Vt = Va + Vw + Vs = Vv + Vs

(cm3 or m3)

Mt = Ms + Mw

(g or t)

Wt = Ws + Ww

(N or kN)

Ma = Wa = 0

(insignificant)

Some simple expansions:

Simple soil properties: V

e=

e = Vv

Void ratio:

s

e

n = 1+e

Vv Vt

Porosity:

n=

Degree of saturation:

Sr = Vw (usually %)

Air content:

A = V a (%)

Density:

ρ=

M (g/cm3 V

Unit weight:

γ=

W = V

Specific gravity:

γ ρ Gs = ρ s = γs w w

Moisture content:

w=

Va = Vv (1 − Sr ) = eVs (1 − Sr ) (cm3 or m3)

V

Vw = eVs Sr (cm3 or m3)

v

V

Ms = Vs Gs ρw (g or t)

t

Mw Ms

Ws = Vs Gs γw (N or kN)

or t/m3)

Mw = eSr Vs ρw = Vw ρw (g or t)

ρg (kN/m3)

=

=

Ww Ws

Ww = eSr Vs γw = Vw γw (N or kN)

Ws

γsub = γsat − γw (kN/m3)

Vs γw

γdry =

(%)

Relationship of soil properties:

Gs w = S r e

Bulk unit weight (kN/m3):

γbulk =

Dry unit weight (kN/m3):

γdry =

γbulk 1+w

Saturated unit weight (kN/m3):

γsat =

(Gs +e)γw 1+e

(1+w)Gs γw 1+e

=

Gs γw 1+e

γdry −γdry(min)

γdry(max)

dry(max) −γdry(min)

γdry

Dr(%) = e

=

(Gs +Sr e)γw

=

1+e eSr γw (1+e)w

Ws Vt

(kN/m3); ρdry =

Ms Vt

(t/m3)

= [Gs (1 − n) + nSr ]γw = Gs γw (1 − n)(1 + w)

= Gs γw (1 − n)

= [Gs (1 − n) + n]γw = γdry + nγw = γdry (1 + w)

Relative density (Dr) based on dry unit weight: Dr(%) = γ

n 1−n

× 100%; or

emax −e max −emin

Relative Density (%) 0 - 15 15 - 35 35 - 65 65 - 85 85 - 100

Consistency Term Very loose Loose Medium dense Dense Very dense

Some constant values:

ρw = 1 t/m3 = 1 g/cm3 ; g = 9.81 N/kg; γw = 9.81 kN/m3 ; and Gs = 2.65 (if not given)

Unit conversions:

1m3 = 103 L = 106 cm3 ; For water only: 1L = 1kg = 9.81N; From density to unit weight, 1t/m3 = 1g/cm3 = 9.81kN/m3 1

Prepared and Copyright by Behzad Fatahi Shrinkage limit (wSL) (Tutorial 1B):

Fig. 1

Definition (refer to Fig. 1): wSL =

MSL Ms

=

Fig. 2

WSL Ws

Volumes characterised by shrinkage limit: Vw = Vv thus, Va = 0 and Sr = 1 Water content at shrinkage limit (refer to Fig. 1): wSL =

W1 −Ws −(V1 −VSL )γw Ws

(Note: ‘W’ is measured in N; and ‘V’ is measured in L)

Part B: Soil Compaction Definition: Mechanical densification of soil that involves expulsion of air from void spaces within a soil (no.of blows/layer)×(no.of layers)×(weight of hammer in N)×(height of drops in m) Volume of mould in m3

Compaction energy:

E=

Relative compaction (RC):

RC = (γ

(γdry )

field

dry )max. lab

× 100%

Other relationships: ρw G s 1+e

=

Dry density of the soil:

ρdry =

Zero air voids dry density:

ρw G s ρzav = 1+wG s

ρwet 1+w

2...