1.3 Soil Phase Relationships.pdf

Soil mechanics

1. 1.3 Soil Phase Relationships
Mr. Shade S. Muluti
[email protected]
School of Engineering & The Built Environment
Department of Civil and Mining Engineering
Soil Mechanics
TCVG3682/2021

2. 2
LearningOutcomes
TCVG3682/2021 –Mr. S.S. Muluti
✓Determinetheproportionsofthemainconstituentsina
soil.
✓Understandanddifferentiatebetweenthetwo-phase
systemsofsoils.
✓Determineandunderstandtherelationshipsbetween
differentvariablesofsoils.
▪Uponcompletionofthischapter,studentsshouldbeableto
dothefollowing:

3. 3
PhaseSystemofSoils
TCVG3682/2021 –Mr. S.S. Muluti
▪Soilisnotacoherentsolidmateriallikesteelandconcrete
butisaparticulatematerial.
▪Soils,astheyexistinnature,consistofsolidparticles
(mineralgrains,rockfragments)withwaterandairinthe
voidsbetweentheparticles.
▪Thewaterandaircontentsarereadilychangedbychanges
inambientconditionsandlocation.
▪Astherelativeproportionsofthethreephasesvaryinany
soildeposit,itisusefultoconsiderasoilmodelwhichwill
representthesephasesdistinctlyandproperlyquantifythe
amountofeachphase.

4. 4
PhaseSystemofSoils
TCVG3682/2021 –Mr. S.S. Muluti
Soil Model
Source: Adopted from Chappidi(2017)

5. 5
PhaseSystemofSoils
TCVG3682/2021 –Mr. S.S. Muluti
▪Thecompositionsofnaturalsoilsmayincludediverse
componentswhichmaybeclassifiedintothreelarge
groups:
❑SolidPhase(minerals,cementation&organicmaterial)
❑LiquidPhase(waterwithdissolvedsalts)
❑GaseousPhase(airorothersomegas)
▪Thespacesbetweenthesolids(solidparticles)arecalled
voids.
▪Waterisoftenthepredominantliquidandairisthe
predominantgas.
▪Wewillusethetermswaterandairinsteadofliquidand
gases.

6. 6
PhaseSystemofSoils
TCVG3682/2021 –Mr. S.S. Muluti
▪Soilsaremadeupofeithertwo-phaseorthree-phase
systems:
Two Phase System
1)FullySaturatedSoil-madeupoftwophasessolid
particlesandporewater(noaircontent).
2)CompletelyDrySoil-madeupoftwophasessolid
particlesandporeair(nowatercontent).
ThreePhaseSystem
3)PartiallySaturatedSoil-madeupofthreephases-
solidparticles,porewaterandporeair.
PartiallySaturatedsoilisoftenattributedafourthphase

7. 7
TwoPhaseSystem
TCVG3682/2021 –Mr. S.S. Muluti
1. Fully Saturated Soil
Mineral Skeleton Fully Saturated
Source: Adopted from Chappidi(2017)

8. 8
TwoPhaseSystem
TCVG3682/2021 –Mr. S.S. Muluti
2. Completely Dry Soil
Mineral Skeleton Dry Soil
Source: Adopted from Chappidi(2017)

9. 9
ThreePhaseSystem
TCVG3682/2021 –Mr. S.S. Muluti
3. Partially Saturated Soil
Mineral Skeleton Partially Saturated
Source: Adopted from Chappidi(2017)

10. 10
PhaseRelationsofSoils
TCVG3682/2021 –Mr. S.S. Muluti
▪Thesoilmodelisgivendimensionalvaluesforthesolid,
waterandaircomponents:
Source: Adopted from Chappidi(2017)

11. 11
PhaseRelationsofSoils
TCVG3682/2021 –Mr. S.S. Muluti
▪Thevolumeofthevariousconstituentsofasoilis
quantifiedbyfollowingquantities:
oV
T=TotalVolumeofasoil.
oV
v=Volumeofthevoids(pores).
oV
s=Volumeofthesoilsolids.
oV
a=Volumeoftheairinthevoids.
oV
w=Volumeofthewater.
▪Thus,forallsoils:
V
T= V
v+ V
s= (V
a+ V
w) + V
s

12. 12
PhaseRelationsofSoils
TCVG3682/2021 –Mr. S.S. Muluti
▪Theweightofthevariousconstituentsofasoilisquantified
byfollowingquantities:
oW
T=TotalWeightofasoil.
oW
a=Weightoftheairinthevoids(pores)≈0.
oW
w=Weightofthewater.
oW
s=Weightofthesoilsolids.
▪Thus,forallsoils:
W
T= W
w+ W
s
❖IfV
v=V
w(→V
a=0)andW
w≠0,thesoilissaidtobe
saturated;otherwise,itisunsaturated.
NOTE:

13. 13
PhaseRelationsofSoils
TCVG3682/2021 –Mr. S.S. Muluti
▪Forthepurposeofengineeringanalysisanddesign,itis
necessarytoexpressrelationsbetweentheweightsand
thevolumesofthethreephases.
▪Thevariousrelationscanbegroupedinto:
1)Weightrelations
2)Volumerelations
3)Weight-Volumerelations
Source: Adopted from Chappidi(2017)

14. 14
1.WeightRelationships
TCVG3682/2021 –Mr. S.S. Muluti
▪Thefollowingarethebasicweightrelations:
a)WatercontentorMoisturecontent(w)
b)Specificgravity(G
s)
W
T= W
w+ W
s
Where:
oW
T=TotalWeightofasoil.
oW
a=Weightoftheair≈0.
oW
w=Weightofthewater.
oW
s=Weightofthesoilsolids.
Source: Adopted from Chappidi(2017)

15. 15
1.WeightRelationships
TCVG3682/2021 –Mr. S.S. Muluti
a)WatercontentorMoisturecontent(w)
▪Theratioofthemassofwaterpresenttothemassof
solidparticlesiscalledthewatercontent(w),or
sometimesthemoisturecontent.
▪Thewatercontentofasoilisfoundbyweighinga
sampleofthesoilandthenplacingitinanovenat
110±5°Cuntiltheweightofthesampleremains
constant,thatis,alltheabsorbedwaterisdrivenout.
??????=
�
�
�
�
×100%

16. 16
1.WeightRelationships
TCVG3682/2021 –Mr. S.S. Muluti
b)Specificgravity(G
s)
▪Thespecificgravityofasolidsubstanceistheratioof
theweightofagivenvolumeofmaterialtotheweight
ofanequalvolumeofwater(at20°C).
▪G
sisusefulbecauseitenablesthevolumeofsolid
particlestobecalculatedfrommassorweight.
▪Thespecificgravityofsoilsolidsisoftenneededfor
variouscalculationsinsoilmechanics.
▪Formostinorganicsoils,thevalueofG
sliesbetween
2.60and2.80.
??????
�=
�
�
�
�
=
γ
�∙�
�
γ
�∙�
�
=
γ
�
γ
�
∴??????
�=
γ
�
γ
�

17. 17
2.VolumeRelationships
TCVG3682/2021 –Mr. S.S. Muluti
▪Thefollowingarethebasicvolumerelations:
a)Voidratio(e)
b)Porosity(n)
c)Degreeofsaturation(S)
d)Aircontent(a)
Where:
oV
T=TotalVolumeofasoil.
oV
v=Volumeofthevoids(pores).
oV
s=Volumeofthesoilsolids.
oV
a=Volumeoftheairinthevoids.
oV
w=Volumeofthewater.
V
T= V
s+V
v+ V
s
V
v = V
a+ V
w
Source: Adopted from Chappidi(2017)

18. 18
2.VolumeRelationships
TCVG3682/2021 –Mr. S.S. Muluti
a)Voidratio(e)
▪Isdefinedastheratioofthevolumeofvoids(V
v)tothe
volumeofsoilsolids(V
s)andisexpressedasa
decimal.
▪Thevoidratioofrealcoarse-grainedsoilsvary
between0.3and1.0.
▪Claysoilscanhavevoidratiogreaterthanone.
??????=
�
�
�
�
??????=
�
�
�
�
=
�
�
�
??????−�
�
=
�
�
�
??????(1−
�
�
�
??????
)
=
�
1−�
∴??????=
�
1−�

19. 19
2.VolumeRelationships
TCVG3682/2021 –Mr. S.S. Muluti
b)Porosity(n)
▪Isdefinedastheratioofthevolumeofvoids(V
v)tothe
totalvolumeofsoil(V
T)andisexpressedasa
percentage.
▪Voidratioandporosityareinter-relatedtoeachother
asfollows:
�=
�
�
�
�
×100%
�=
�
�
�
�
=
�
�
�
�+�
�
=
�
�
�
�(1+
�
�
�
�
)
=
??????
1+??????
∴�=
??????
1+??????

20. 20
2.VolumeRelationships
TCVG3682/2021 –Mr. S.S. Muluti
c)DegreeofSaturation(S)
▪Isdefinedastheratioofthevolumeofwater(V
w)to
thevolumeofvoids(V
v)andisexpressedasa
percentage.
▪Thedegreeofsaturationtelluswhatpercentageof
thevolumeofvoidscontainswater.
oForfullysaturatedsoil:S=1or100%(V
v=V
w).
oFordrysoil: S=0.
oForpartiallysaturatedsoil:1<S<0
??????=
&#3627408457;
&#3627408484;
&#3627408457;
&#3627408483;
×100%

21. 21
2.VolumeRelationships
TCVG3682/2021 –Mr. S.S. Muluti
d)AirContent(a)
▪Isdefinedastheratioofairvolume(V
a)tototalvolume
(V
T)andisexpressedasapercentage.
▪Theairvoids(V
a),isthatpartofthevoidsspacenot
occupiedbywater.
oForacompletelydrysoil:a=n
oForasaturatedsoil: a=0
??????=
&#3627408457;
&#3627408462;
&#3627408457;
??????
×100%
??????=&#3627408475;(1−??????)

22. 22
3.Weight-VolumeRelationships
TCVG3682/2021 –Mr. S.S. Muluti
▪Thefollowingarethebasicweight-volumerelations:
a)UnitWeight(γ)
b)DryUnitWeight(γ
d)
c)Saturatedunitweight(γ
sat)
d)EffectiveunitWeight(γ')
Source: Adopted from Chappidi(2017)

23. 23
3.Weight-VolumeRelationships
TCVG3682/2021 –Mr. S.S. Muluti
a)UnitWeight(??????)
▪Theunitweightofasoilistheratiooftheweightofsoil
tothetotalvolume.
γ=
&#3627408458;
??????
&#3627408457;
??????
▪Densityisameasureofthequantityofmassinaunit
volumeofmaterial.
▪Unitweightisameasureoftheweightofaunitvolume
ofmaterial.
▪Bothcanbeusedinterchangeably.Theunitsofdensity
areton/m³,kg/m³org/cm³.
▪TheunitofunitweightiskN/m³.

24. 24
3.Weight-VolumeRelationships
TCVG3682/2021 –Mr. S.S. Muluti
b)DryUnitWeight(??????
d)
▪Isdefinedastheratiooftheweightofsolidstothetotal
volume.
γ
&#3627408465;=
&#3627408458;
&#3627408480;
&#3627408457;
??????
▪Thedryunitweightcanalsobedeterminedas:
γ=
&#3627408458;
??????
&#3627408457;
??????
=
&#3627408458;
&#3627408480;1+
&#3627408458;
&#3627408484;
&#3627408458;
&#3627408480;
&#3627408457;
??????
=γ
&#3627408465;(1+??????) ∴γ
&#3627408465;=
γ
1+??????
γ
&#3627408465;=
&#3627408458;
&#3627408480;
&#3627408457;
&#3627408480;(1+??????)
=
γ
&#3627408480;
(1+??????)
=
γ
&#3627408484;∙??????
&#3627408480;
1+??????
∴γ
&#3627408465;=
γ
&#3627408484;∙??????
&#3627408480;
1+??????

25. 25
3.Weight-VolumeRelationships
TCVG3682/2021 –Mr. S.S. Muluti
c)SaturatedUnitweight(??????
sat)
▪Forasaturatedsoil,theunitweightbecomes:
γ
&#3627408480;&#3627408462;&#3627408481;=
&#3627408458;
??????
&#3627408457;
??????
γ
&#3627408480;&#3627408462;&#3627408481;=
&#3627408458;
&#3627408480;1+
&#3627408458;
&#3627408484;
&#3627408458;
&#3627408480;
&#3627408457;
&#3627408480;(1+??????)
=
γ
&#3627408480;1+
??????
??????
&#3627408480;
1+??????
=
γ
&#3627408484;??????
&#3627408480;1+
??????
??????
&#3627408480;
1+??????
=
γ
&#3627408484;??????
&#3627408480;+??????
1+??????
▪Thesaturatedunitweightcanalsobedeterminedas:
∴γ
&#3627408480;&#3627408462;&#3627408481;=
γ
&#3627408484;??????
&#3627408480;+??????
1+??????

26. 26
3.Weight-VolumeRelationships
TCVG3682/2021 –Mr. S.S. Muluti
d)EffectiveUnitWeight(??????')
▪Theeffectiveorsubmergedunitweightofthesoilis
givenas:
γ
&#3627408480;&#3627408482;&#3627408463;=γ′=γ
&#3627408480;&#3627408462;&#3627408481;−γ
&#3627408484;
Ground Surface
G.W.T
γ
&#3627408480;&#3627408462;&#3627408481;
γ
&#3627408465;&#3627408479;??????
γ
γ
&#3627408480;&#3627408482;&#3627408463;
??????=&#3627409358;
??????=&#3627409358;&#3627408533;??????&#3627409359;
??????=&#3627409359;
??????
??????=&#3627409367;.&#3627409366;&#3627409359;????????????/??????
&#3627409361;
Source: Adopted from Chappidi(2017)

27. 27
Density-VolumeRelationships
TCVG3682/2021 –Mr. S.S. Muluti
▪IntheSIsystem,theunitusedforunitweightiskilo
Newtonspercubicmeter(kN/m
3
).
▪BecausetheNewtonisaderivedunit,workingwithmass
densities(ρ)ofsoilmaysometimesbeconvenient.
▪TheSIunitofmassdensityiskg/m
3
.
▪Wecanwritethedensityequationsasfollows:
??????=
??????
??????
&#3627408457;
??????
??????
&#3627408465;=
??????
&#3627408480;
&#3627408457;
??????
Dry Density:
Buoyant Density:
Density:
??????′=??????−??????
&#3627408484; ??????
??????=&#3627409359;&#3627409358;&#3627409358;&#3627409358;????????????/??????
&#3627409361;

28. 28
Density-VolumeRelationships
TCVG3682/2021 –Mr. S.S. Muluti
▪Thedensity-volumerelationsmaybederivedbyreferringto
thesoilelementinwhichthevolumeofsoilsolidsisequal
to1andthevolumeofvoidsisequaltoe:
??????
&#3627408484;=????????????
&#3627408480;??????
&#3627408484;
??????
&#3627408480;=??????
&#3627408480;??????
&#3627408484; &#3627408457;
&#3627408480;=1
&#3627408457;
&#3627408483;=??????
Mass Volume
Source: Adopted from Chappidi(2017)

29. 29
Density-VolumeRelationships
TCVG3682/2021 –Mr. S.S. Muluti
▪Asmentionedbefore,duetotheconvenienceofworking
withdensitiesintheSIsystem,thefollowingequations,
similartounit–weightrelationshipswillbeuseful:
??????=
1+????????????
&#3627408480;??????
&#3627408484;
1+??????
??????
&#3627408465;=
??????
&#3627408480;??????
&#3627408484;
1+??????
Dry Density:
Saturated Density:
Density:
??????
&#3627408480;&#3627408462;&#3627408481;=
(??????
&#3627408480;+??????)??????
&#3627408484;
1+??????
Moisture Content:??????=
????????????
??????
&#3627408480;

30. 30
Relationshipsamongγ,e,w,andG
s
TCVG3682/2021 –Mr. S.S. Muluti
▪ConsiderthefollowingthreephasediagramwhereV
s=1:
&#3627408458;
&#3627408484;=????????????
&#3627408480;γ
&#3627408484;
&#3627408458;
??????
&#3627408458;
&#3627408480;=??????
&#3627408480;γ
&#3627408484;
&#3627408457;
??????=1+??????
&#3627408457;
&#3627408484;=w??????
&#3627408480;
&#3627408457;
&#3627408480;=1
&#3627408457;
&#3627408483;=??????
Source: Adopted from Chappidi(2017)

31. 31
Relationshipsamongγ,e,w,andG
s
TCVG3682/2021 –Mr. S.S. Muluti
▪Theweightsofsoilsolidsandwatercanbegivenas:
&#3627408458;
&#3627408480;=??????
&#3627408480;γ
&#3627408484;
&#3627408458;
&#3627408484;=??????&#3627408458;
&#3627408480;=????????????
&#3627408480;γ
&#3627408484;
▪IfG
sisexpressesas:
??????
&#3627408480;=
&#3627408458;
&#3627408480;
&#3627408457;
&#3627408480;γ
&#3627408484;
▪Hence:
γ
&#3627408465;=
&#3627408458;
&#3627408480;
&#3627408457;
??????
=
??????
&#3627408480;γ
&#3627408484;
1+??????
e=
??????
&#3627408480;γ
&#3627408484;
γ
&#3627408465;
−1OR
γ=
&#3627408458;
??????
&#3627408457;
??????
=
&#3627408458;
&#3627408480;+&#3627408458;
&#3627408484;
&#3627408457;
??????
=
??????
&#3627408480;γ
&#3627408484;+????????????
&#3627408480;γ
&#3627408484;
1+??????
=
1+????????????
&#3627408480;γ
&#3627408484;
1+??????
Unit Weight:
Dry Unit Weight:

32. 32
Relationshipsamongγ,e,w,andG
s
TCVG3682/2021 –Mr. S.S. Muluti
▪Iftheweightandvolumeofwaterofthesoilcanbegiven
as:
&#3627408458;
&#3627408484;=??????&#3627408458;
&#3627408480;=????????????
&#3627408480;γ
&#3627408484;
▪Hence:
e=????????????
&#3627408480;
γ
&#3627408480;&#3627408462;&#3627408481;=
&#3627408458;
??????
&#3627408457;
??????
=
&#3627408458;
&#3627408480;+&#3627408458;
&#3627408484;
&#3627408457;
??????
=
??????
&#3627408480;γ
&#3627408484;+??????γ
&#3627408484;
1+??????
=
??????
&#3627408480;+??????γ
&#3627408484;
1+??????
Saturated Unit Weight:
Void ratio:
&#3627408457;
&#3627408484;=
&#3627408458;
&#3627408484;
γ
&#3627408484;
=
????????????
&#3627408480;γ
&#3627408484;
γ
&#3627408484;
=????????????
&#3627408480;
??????=
&#3627408457;
&#3627408484;
&#3627408457;
&#3627408483;
=
????????????
&#3627408480;
??????
????????????=????????????
&#3627408480;ORDegree of Saturation:

33. 33
Relationshipsamongγ,e,w,andG
s
TCVG3682/2021 –Mr. S.S. Muluti
▪Ifthesoilsampleissaturated—thatis,thevoidspacesarecompletely
filledwithwater,thesoilelementbecomesatwo-phasediagramwhere
V
s=1:
&#3627408457;
&#3627408483;=&#3627408457;
&#3627408484;=??????&#3627408458;
&#3627408484;=??????γ
&#3627408484;
&#3627408458;
??????
&#3627408458;
&#3627408480;=??????
&#3627408480;γ
&#3627408484;
&#3627408457;
&#3627408480;=1
&#3627408457;
??????=1+??????
Source: Adopted from Chappidi(2017)

34. 34
Relationshipsamongγ,nandw
TCVG3682/2021 –Mr. S.S. Muluti
▪ConsiderthefollowingthreephasediagramwhereV
T=1:
&#3627408458;
&#3627408484;=????????????
&#3627408480;γ
&#3627408484;(1−&#3627408475;)
&#3627408458;
&#3627408480;=??????
&#3627408480;γ
&#3627408484;(1−&#3627408475;)
&#3627408457;
??????=1
&#3627408457;
&#3627408480;=1−&#3627408475;
&#3627408457;
&#3627408483;=&#3627408475;
Source: Adopted from Chappidi(2017)

35. 35
Relationshipsamongγ,nandw
TCVG3682/2021 –Mr. S.S. Muluti
▪IfV
Tisequalto1,thenV
visequalton,soV
s=1−n.
▪Therefore,theweightofsoilsolids(W
s)andtheweightof
water(W
w)canthenbeexpressedasfollows:
&#3627408458;
&#3627408480;=??????
&#3627408480;γ
&#3627408484;(1−&#3627408475;)
&#3627408458;
&#3627408484;=??????&#3627408458;
&#3627408480;=????????????
&#3627408480;γ
&#3627408484;(1−&#3627408475;)
&#3627408475;=
&#3627408457;
&#3627408480;
&#3627408457;
??????
▪Recall:

36. 36
Relationshipsamongγ,nandw.
TCVG3682/2021 –Mr. S.S. Muluti
▪Hence:
??????
&#3627408480;&#3627408462;&#3627408481;=
&#3627408458;
&#3627408484;
&#3627408458;
&#3627408480;
=
&#3627408475;γ
&#3627408484;
(1−&#3627408475;)γ
&#3627408484;??????
&#3627408480;
=
&#3627408475;
(1−&#3627408475;)??????
&#3627408480;
γ
&#3627408480;&#3627408462;&#3627408481;=
&#3627408458;
??????
&#3627408457;
??????
=
&#3627408458;
&#3627408480;+&#3627408458;
&#3627408484;
&#3627408457;
??????
=
(1−&#3627408475;)??????
&#3627408480;γ
&#3627408484;+??????γ
&#3627408484;
1+??????
=[(1−&#3627408475;)??????
&#3627408480;+&#3627408475;]γ
&#3627408484;
Saturated Unit Weight:
Moisture Content:
γ=
&#3627408458;
??????
&#3627408457;
??????
=
&#3627408458;
&#3627408480;+&#3627408458;
&#3627408484;
&#3627408457;
??????
=??????
&#3627408480;γ
&#3627408484;(1−&#3627408475;)(1+??????)Unit Weight:
γ
&#3627408465;=
&#3627408458;
&#3627408480;
&#3627408457;
??????
=
??????
&#3627408480;γ
&#3627408484;(1−&#3627408475;)
1
=??????
&#3627408480;γ
&#3627408484;(1−&#3627408475;)Dry Unit Weight:

37. 37
Relationshipsamongγ,nandw.
TCVG3682/2021 –Mr. S.S. Muluti
▪Ifthesoilsampleissaturated—thatis,thevoidspacesarecompletely
filledwithwater,thesoilelementbecomesatwo-phasediagramwhere
V
T=1:
&#3627408457;
&#3627408483;=&#3627408457;
&#3627408484;=&#3627408475;
&#3627408457;
&#3627408480;=1−&#3627408475;
&#3627408457;
??????=1
&#3627408458;
&#3627408484;=&#3627408475;γ
&#3627408484;
&#3627408458;
&#3627408480;=??????
&#3627408480;γ
&#3627408484;(1−&#3627408475;)
Source: Adopted from Chappidi(2017)

38. 38
VariousUnitWeightRelationships
TCVG3682/2021 –Mr. S.S. Muluti
Note:The3
rd
columnisaspecialcaseofthe1
st
columnwhenS=1.

39. 39
Typicalvaluesofe,w,andγ
TCVG3682/2021 –Mr. S.S. Muluti
▪Sometypicalvaluesofvoidratio,moisturecontentina
saturatedcondition,anddryunitweightforsoilsinanatural
statearegivenintheTablebelow.

40. 40
Summary-PhaseRelations
TCVG3682/2021 –Mr. S.S. Muluti
▪Insummary,fortheeasysolutionofphaserelation
problems,youdon’thavetomemorizelotsofcomplicated
formulas.
▪Mostofthemcaneasilybederivedfromthephase
diagram.
▪Justrememberthefollowingsimplerules:
(a)Rememberthebasicdefinitionsofproperties.
(b)Drawaphasediagram.
(c)AssumeeitherV
S=1orV
T=1.
(d)SolvetheProblem.

41. 41
Example1-PhaseRelations
TCVG3682/2021 –Mr. S.S. Muluti
▪Forasoilcoresample,youaregiventhefollowingdata:
▪Weightofsoilsample,W
T=1013g
▪Vol.ofsoilsample,V
s=585.0cm
3
▪SpecificGravity,G
s=2.65
▪Dryweightofsoil,W
s=904.0g
i.MoistureContent
ii.Voidratio
iii.Porosity
iv.DegreeofSaturation
v.DryunitWeight
▪Determine:

42. 42
Example1-PhaseRelations
TCVG3682/2021 –Mr. S.S. Muluti
&#3627408457;
&#3627408480;=
&#3627408458;
&#3627408480;
??????
&#3627408480;??????
&#3627408484;
=
(904)
(2.65)(1.0)
=341.1??????&#3627408474;
3
&#3627408458;
??????=1013g &#3627408458;
&#3627408480;=904g &#3627408457;
??????=585g▪Given:
&#3627408457;
&#3627408483;=&#3627408457;
??????−&#3627408457;
&#3627408480;=585−341.1=243.9??????&#3627408474;
3
&#3627408457;
&#3627408484;=
&#3627408458;
&#3627408484;
??????
&#3627408484;
=
(109)
(1.0)
=109??????&#3627408474;
3
&#3627408457;
&#3627408462;=&#3627408457;
&#3627408483;−&#3627408457;
&#3627408484;=243.9−109=134.9??????&#3627408474;
3
&#3627408458;
&#3627408484;=&#3627408458;
??????−&#3627408458;
&#3627408480;=1013−904=109g

43. 43
Example1-PhaseRelations
TCVG3682/2021 –Mr. S.S. Muluti
▪Fromthethree-phasediagram,wecandetermine:
i.MoistureContent(w):
ii.Voidratio(e):
iii.Porosity(n):
iv.DegreeofSaturation(S):
v.DryunitWeight(γ
d):
??????=
&#3627408458;
&#3627408484;
&#3627408458;
&#3627408480;
=
(109)
(904)
×100=&#3627409359;&#3627409360;.&#3627409359;%
??????=
&#3627408457;
&#3627408483;
&#3627408457;
&#3627408480;
=
(243.9)
(341.1)
=&#3627409358;.&#3627409365;&#3627409359;&#3627409363;
&#3627408475;=
&#3627408457;
&#3627408483;
&#3627408457;
??????
=
(243.9)
(585)
×100=&#3627409362;&#3627409359;.&#3627409365;%
??????=
&#3627408457;
&#3627408484;
&#3627408457;
&#3627408483;
=
(109)
(243.9)
×100=&#3627409362;&#3627409362;.&#3627409365;%
??????
&#3627408465;=
&#3627408458;
&#3627408480;
&#3627408457;
??????
=
(904)
(585)
=&#3627409359;.&#3627409363;&#3627409363;??????/????????????
&#3627409361;

44. 44
Example2-PhaseRelations
TCVG3682/2021 –Mr. S.S. Muluti
▪Forasaturatedsoil,showthat:
??????
&#3627408480;&#3627408462;&#3627408481;=
1+??????
&#3627408480;&#3627408462;&#3627408481;
1+??????
&#3627408480;&#3627408462;&#3627408481;??????
&#3627408480;
??????
&#3627408480;??????
&#3627408484;
Solution:
??????
&#3627408480;&#3627408462;&#3627408481;=
&#3627408458;
??????
&#3627408457;
??????
=
&#3627408458;
&#3627408484;+&#3627408458;
&#3627408480;
&#3627408457;
??????
=
&#3627408458;
&#3627408480;&#3627408462;&#3627408481;&#3627408458;
&#3627408484;+&#3627408458;
&#3627408480;
&#3627408457;
??????
=(1+&#3627408458;
&#3627408484;)
&#3627408458;
&#3627408480;
&#3627408457;
??????
&#3627408457;
&#3627408483;=&#3627408457;
&#3627408484;=??????&#3627408458;
&#3627408484;=??????γ
&#3627408484;
&#3627408458;
??????
&#3627408458;
&#3627408480;=??????
&#3627408480;γ
&#3627408484;
&#3627408457;
&#3627408480;=1
&#3627408457;
??????=1+??????
Recall:

45. 45
Example2-PhaseRelations
TCVG3682/2021 –Mr. S.S. Muluti
▪Recall:
&#3627408458;
&#3627408480;=??????
&#3627408480;&#3627408457;
&#3627408480;??????
&#3627408484;
??????+1=
&#3627408457;
&#3627408483;+&#3627408457;
&#3627408480;
&#3627408457;
&#3627408480;
=
&#3627408457;
??????
&#3627408457;
&#3627408480;
▪Substitutingweget:
??????
&#3627408480;&#3627408462;&#3627408481;=(1+&#3627408458;
&#3627408480;&#3627408462;&#3627408481;)
??????
&#3627408480;??????
&#3627408484;
1+??????
▪Recallagain:
??????=&#3627408458;
&#3627408480;&#3627408462;&#3627408481;??????
&#3627408480;
▪Finally,substitutingweget:
∴??????
&#3627408532;??????&#3627408533;=
&#3627409359;+??????
&#3627408532;??????&#3627408533;
&#3627409359;+??????
&#3627408532;??????&#3627408533;??????
&#3627408532;
??????
&#3627408532;??????
??????

46. 46
Homework-PhaseRelations
TCVG3682/2021 –Mr. S.S. Muluti
▪Provethefollowingrelationships::
a)
b)
c)
d)
e)
f)
??????
&#3627408465;=(1−n)??????
&#3627408484;??????
&#3627408480;
??????
&#3627408480;&#3627408462;&#3627408481;=[(??????
&#3627408480;−n??????
&#3627408480;−1]??????
&#3627408484;
&#3627408458;
(&#3627408480;&#3627408462;&#3627408481;)=
&#3627408475;??????
&#3627408484;
??????
&#3627408480;&#3627408462;&#3627408481;−&#3627408475;??????
&#3627408484;
??????
&#3627408480;=
??????
&#3627408480;&#3627408462;&#3627408481;
??????
&#3627408484;−&#3627408458;
&#3627408464;(??????
&#3627408480;&#3627408462;&#3627408481;−??????
&#3627408484;)
??????=
??????
&#3627408480;&#3627408462;&#3627408481;−??????
&#3627408465;
??????
&#3627408465;−??????
&#3627408480;&#3627408462;&#3627408481;+??????
&#3627408484;)
??????
&#3627408480;&#3627408462;&#3627408481;=&#3627408475;
1+??????
&#3627408480;&#3627408462;&#3627408481;
??????
&#3627408480;&#3627408462;&#3627408481;
??????
&#3627408480;&#3627408462;&#3627408481;

47. 47TCVG3682/2021 –Mr. S.S. Muluti
▪Das,B.,M.(2014),“PrinciplesofGeotechnical
Engineering”,EighthEdition,CENGAGELearning,ISBN-
13:978-1-133-10867-2
▪Budhu,M.(2015),“SoilMechanicsFundamentals”,First
Edition,JohnWiley&Sons,ISBN:978-1-119-01965-7.
▪Chebet,F.(2020),GeotechnicalEngineering1-Lecture
Notes,UniversityofCapeTown(UCT).
▪Orabi,A.(2016),303322SoilMechanics–Lecture1:
Introduction&PropertiesofSoil,InternationalUniversityfor
Science&Technology(IUST).
▪Ahmed,S.S.(n.d),CE210SoilMechanicsandFoundation
EngineeringI-LectureNotes:Chapter2-SoilDefinition,
ClassificationandProperties,CollegeofEngineering-MU.
References