Stability of Slopes

This presentation is all about various approaches used to find stability of slope

1. STABILITY OF
SLOPES
Prepared By:
Arbaz Kazi
Asst. Professor
VCET, Vasai (W)

2. CONTENTS
•Slope –Definition
•Types of Slope
•Causes of failure of slopes
•Patterns of Slope failure
•Aims of Slope stability
•Types of factor of safety
•Types of slope failure
•Stability Analysis of Slope
❑Infinite Slope
❑Finite Slope
•Taylor’s Stability Number
•Method’s of Preventing Slope failure

3. ➢Asurfaceofwhichoneendorsideisatahigher
levelthananother;arisingorfallingsurface.
➢Aslopecanbedefinedasaninclinedbodybetween
groundandair.
➢Anearthslopeisinclinedorunsupportedpartofa
soilmass.
Slope -Definition

4. SLOPES OF EARTH ARE OF TWO TYPES
1. Natural slopes
-slopes exist in hilly areas

5. 2. Man made slopes
❖The sides of cuttings
❖The slopes of embankments constructed for roads railway lines, canals etc.
❖The slopes of earth dams constructed for storing water.

6. THESLOPESWHETHERNATURALORARTIFICIALMAYBE
1.Infiniteslopes
▪Theterminfiniteslopeisusedtodesignateaconstant
slopeofinfiniteextent.
▪Thelongslopeofthefaceofamountain
2.Finiteslopes
▪Finiteslopesarelimitedinextent.
▪Theslopesofembankmentsandearthdamsare
examplesoffiniteslopes.

7. CAUSES OF FAILURE OF SLOPES
✓Gravitationalforce
✓Forceduetoseepagewater
✓Erosionofthesurfaceofslopesduetoflowingwater
✓Thesuddenloweringofwateradjacenttoaslope
✓Forcesduetoearthquakes

8. PATTERNS OF SLOPE FAILURE

9. Aims of Slope Stability Analysis
❑Toassessthestabilityofslopesundershort-term(often
duringconstruction)andlong-termconditions.
❑Toassessthepossibilityoflandslidesinvolvingnaturalor
existingengineeredslopes.
❑Toanalyzelandslidesandtounderstandfailuremechanisms
andtheinfluenceofenvironmentalfactors.
❑Toenabletheredesignoffailedslopesandtheplanningand
designofpreventiveandremedialmeasures,where
necessary.
❑Tostudytheeffectofseismicloadingsonslopesand
embankments.

10. WHAT CAUSES SLOPE TO FAIL?
•Thefailureofasoilmassoccursalongaplaneoracurvedsurface
whenalargemassofsoilslideswithrespecttoremainingmass.
•Inotherwordswecansaythereisdownwardandoutward
movementofsoilmassincaseofslopefailures.
•Aslopefailureoccurswhentheforcescausingfailurearegreater
thantheshearingresistance(shearstrength)developedalong
criticalsurfaceoffailure.
•Thefactorsleadingtoslopefailureisdividedintwotypes:
➢Thefactorswhichcausesincreaseinshearstresses.
➢Thefactorswhichcausesdecreaseinshearstrength.

11. •InslopestabilityanalysiswedeterminetheFactorof
Safetyasaratioofresistingforcestodrivingforces
Fs = Resisting / Driving
•Theoretically,anyslopewithaFactorofSafetyless
thanonewillfailandanyslopewithafactorofsafety
greaterthanonewillnot.
•Designfocusesonthesoilparametersandgeometry
thatwillprovidethemaximumfactorofsafety.
SLOPE STABILITY

12. Sometimes,theanalysisofanexistingslopewillbewhat
iscalledaparametricstudy–thatisestablishingafactor
ofsafetyandperformingananalysisthatbackcalculates
thestrengthparameters.
•Theengineerwillthendeterminehis/herconfidence
levelastowhetherornotthesoilhasthatstrength
throughexperience,lab,and/orfielddata.
SLOPE STABILITY

13. TYPES OF FACTOR OF SAFETY
❑Factorofsafetywithrespecttoshearingstrength(Fs):
Itisdefinedastheratioofavailableshearstrengthto
mobilisedshearstrength.
??????
�=??????/??????
�
❑Factorofsafetywithrespecttocohesion(Fc):Itis
definedastheratioofavailablecohesiontomobilised
shearcohesion.
??????
�=??????/??????
�
❑Factorofsafetywithrespecttofriction(FØ):Itis
definedastheratioofavailablefrictiontomobilised
friction.
??????
∅=∅/∅
�

14. Types of Slope Failures
Slope Failure Toe Failure
Base Failure

15. ASSUMPTIONS MADE IN ANALYSIS
•Stresssystemisassumedtobe2-dimensional
•Stressesin3
rd
directionaretakenaszero
•Coulomb’sequationisusedfordeterminingshear
strength.i.e.(s=c+??????∗tan∅)
•Itisassumedthatseepageconditionsandwaterlevel
areknownandporewaterpressurecanbeestimated
•Theshearingstrengthatallpointsareenoughto
mobilizeallavailableshearstrength.

16. Stability Analysis of Infinite Slope
Cohesionless Soil
1. Dry State
Thefactorofsafetyofinfiniteslopesfordrystatemaybe
writtenas
??????
�= Τtan∅tan??????
2. Submerged State
Thefactorofsafetyofinfiniteslopesforsubmergedstate
maybewrittenas
??????
�= Τtan∅tan??????
3. Steady Seepage along slope
Thefactorofsafetyofinfiniteslopesforsteadyseepage
alongslopemaybewrittenas
??????
�=??????
′
∗ Τtan∅??????
�??????�∗tan??????

17. Stability Analysis of Infinite Slope
Purely Cohesive Soil
1. Dry State
Thefactorofsafetyofinfiniteslopesfordrystatemaybe
writtenas
??????
�=Τ????????????∗cos??????∗sin??????∗??????
2. Submerged State
Thefactorofsafetyofinfiniteslopesforsubmergedstate
maybewrittenas
??????
�=Τ????????????
′
∗cos??????∗sin??????∗??????
3. Steady Seepage along slope
Thefactorofsafetyofinfiniteslopesforsteadyseepage
alongslopemaybewrittenas
??????
�=Τ????????????
�??????�∗cos??????∗sin??????∗??????

18. Stability Analysis of Infinite Slope
Cohesive Friction Soil
1. Dry State
Thefactorofsafetyofinfiniteslopesfordrystatemaybe
writtenas
??????
�= Τ??????+??????∗??????∗??????????????????
2
??????∗tan∅??????∗cos??????∗sin??????∗??????
2. Submerged State
Thefactorofsafetyofinfiniteslopesforsubmergedstate
maybewrittenas
??????
�= Τ??????+??????
′
∗??????∗??????????????????
2
??????∗tan∅??????
′
∗cos??????∗sin??????∗??????
3. Steady Seepage along slope
Thefactorofsafetyofinfiniteslopesforsteadyseepage
alongslopemaybewrittenas
??????
�= Τ??????+??????
′
∗??????∗??????????????????
2
??????∗tan∅??????
�??????�∗cos??????∗sin??????∗??????

19. Stability Analysis of Finite Slope
Followingarethemethodswhichareusedfor
stabilityanalysisoffiniteslope:
1.Wedgefailuremethod
2.Frictioncirclemethod
3.Swedishcirclemethod(methodofslices)
•Analysisofpurelycohesivesoil(Ø=0)
•Analysisofc-¢soil

20. Wedge Failure Method
Awedgefailureoccurswhenasoildeposithasaspecificplaneof
weakness.Thestratifieddepositsgenerallyfailalongtheinterface.Fig
showsasoilmassrestingonaninclinedlayerofimpermeablesoil.There
isatendencyoftheuppermasstoslidedownwardsalongtheplaneof
contactAB

21. •Theforcestryingtocauseslidingisthetangentialcomponentsof
theweight(W)alongtheplaneofcontact
�=??????∗sin??????
Whereα=anglewhichplaneAbmakeswithhorizontal
•Theforcetendingtoresisttheslidingdependsuponthecohesion
‘c’andthefrictionalforceandisgivenby
�=??????∗�+??????∗cos??????∗tan∅
WhereL=lengthoffailuresurface.
•FOSisgivenby:
??????=
??????∗�+??????∗cos??????∗tan∅
??????∗sin??????

22. Thefrictioncirclemethodisbasedonthefactthattheresultantreaction
betweenthetwoportionsofthesoilmassintowhichthetrialslipcircle
dividestheslopewillbetangentialtoaconcentricsmallercircleof
radiusrsinØ,sincetheobliquityoftheresultantatfailureistheangleof
internalfriction(Ø)(This,ofcourse,impliestheassumptionthatfriction
ismobilisedinfull).
Thissmallercircleiscalledthe‘frictioncircle’or‘Ø-circle’.
Theforcesactingontheslidingwedgeare:
i.weightWofthewedgeofsoil
ii.reactionRduetofrictionalresistance,and
iii.cohesiveforce??????
�mobilisedalongtheslipsurface
Friction Circle Method

23. Friction Circle Method

24. Friction Circle Method
FollowingprocedureisadoptedforfindingFOSwithfrictioncircle
method:
1.Consider slope as shown in fig.
A
B
Toe of a Slope
Crest of a Slope
2.Draw trial slip surface (Arc AC) passing through toe with “O” as
the centre and “R” as the radius.

25. 3.Find the centroid of the sliding soil mass ABCA and calculate it’s
weight “W”

26. 4.Lettheslipcirclebeconsideredtobemadeupofanumberof
elementaryarcseachoflength“ΔL”

27. 5.Letthecohesiveforceacting
alongthiselementopposing
theslidingofsoilbe??????
??????*ΔL
6.TotalCohesiveforcealongarc
ACformsaforcePolygon,
alsotheclosingsideACof
polygonrepresentsmagnitude
anddirectionofresultant
cohesiveforce

28. 7.LetthelengthofchordACbe�
�andlengthofArcACbe‘L’.
ThemagnitudeofResultantforceC=??????
??????*�
�.Thepositionof
resultantcanbeobtainedusingVarigno’stheorem
C*a=∑??????
??????*ΔL*R
??????
??????*??????
??????*a=??????
??????*L*R
a=
??????
????????????
*R
L>??????
??????
Hencea>R

29. 8.Onmobilizationof
frictionalresistance.
LetPbethesoil
reactionopposingthe
slidingofthesoilmass
asshown.Reactiondue
toFrictionalresistance
‘P’isinclinedatan
angle‘Ø’tothenormal
atthepointofactionas
shown

30. 9.Byknowingthemagnitudeanddirectionof‘W’anddirectionand
lineofactionofotherforces,theforcetrianglecanbecompleted.
Measuringthemagnitudeof‘C’theFOScanbecomputedas
shown.
Themobilisedcohesion??????
�=Τ??????�
�
FOSw.r.tcohesionisgivenby
10.TheminimumFOScanbeobtainedbylocatingcriticalslip
surface.
??????
�=??????/??????
�

31. •ThismethodwasdevelopedbySwedish
engineersassumingfailuresurfacetobeanarcof
acircle
•Twocasesareconsideredinthismethod
1.Analysisofpurelycohesivesoil(Ø=0)
2.Analysisofc-Øsoil
Swedish Circle Method

32. ANALYSIS OF PURELY COHESIVE
SOIL

33. •Inthismethod,no.oftrialslipcirclesareconsideredandfactorofsafetyforeachis
workedout.
•Thecirclecorrespondingtominimumfactorofsafetyiscalledascriticalslipcircle
•LetADbetrialslipcircle,withRasradiusandOascentreofrotation.LetWbethe
weightofwedgeABDAactingthroughitscentroid
•DrivingMoment=Wxe
•LetCubetheunitcohesionandL’bethelengthofsliparcAD=
2??????�??????
360
,hencethe
shearresistancedevelopedalongslipsurfacewillbeequaltoCuxL’whichactsat
radialdistanceRfromO.
•ResistingMoment=CuxL’xR
•F=Mr/Md=CuxL’xR/Wxe
•LetCm=mobilisedshearresistancenecessarytokeepequilibrium
•Wxd=CmxL’xR,thereforeCm=Wxe/L’xR
•F=Cu/Cm=CuxL’xR/Wxe

34. ANALYSIS OF COHESIVE
FRICTION SOIL
❑Inordertotestthestabilityoftheslopeofacohesive
frictionalsoil(c-Øsoil),trialslipcircleisdrawn.
❑Thematerialabovetheassumedslipcircleisdivided
intoaconvenientnumberofverticalstripsorslicesas
showninFig.
❑Theforcesbetweentheslicesareneglectedandeach
sliceisassumedtoactindependentlyasacolumnof
soilofunitthicknessandofwidthb.

36. ❑TheweightWofeachsliceisassumedtoactatits
centre.
❑Iftheweightofeachsliceisresolvedintonormal
(N)andtangential(T)components,thenormal
componentwillpassthroughthecentreofrotation
O,andhencedohotcauseanydrivingmomenton
theslice.
❑However,thetangentialcomponentTcausesa
drivingmoment�
�=TxR

37. ✓WhereRistheradiusoftheslipcircle.Thetangential
componentsofafewslicesthebasemaycauseresisting
moment,inthatcaseTisconsiderednegative.
✓cisunitcohesionandΔListhecurvedlengthofeachslice
thentheresistingforcefromCoulomb'sequationisequalto
(c*ΔL+N*tan∅).
✓Anumberoftrialslipcirclesarechosenandfactorofsafety
ofeachiscomputed.
✓Theslipcirclewhichgivestheminimumfactorofsafetyis
thecriticalslipcircle.

39. Methods of calculating ∑N and ∑T:
Method-1:
ThevalueofW,∑Nand∑Tmaybefoundbytabulating
thevaluesforallslicesasindicatedbelow:

40. Method-2 :
➢∑Nand∑Tmayalsobeobtainedgraphically.verticalline
drawnthroughthecentreofgravityofthesliceand
intersectingthetopandbottomsurfacesoftheslicemaybe
assumedtorepresenttheweightoftheslice.
➢Thismayberesolvedgraphicallyintonormalandtangential
components.Thesecomponentsforalltheslicesareplotted
separatelyasordinatesontwohorizontalbaselines.
➢Theplottedpointsarejoinedbysmoothcurvesasshownin
Fig.TheyarecalledN-curveandT-curverespectively.

41. Theareasunderthesecurvesrepresent∑Nand∑T,The
areasunderthesecurvesaremeasuredbyplanimeterand
multipliedbytheunitweightofthesoiltoobtain∑Nand
∑T.

42. Method-3 :
❖Asimplifiedrectangularplotmethodhasbeensuggested
bySingh(1962)todetermine∑Nand∑T.
❖Inthismethodtheendordinateofeachsliceisassumedto
representAsweightofthesliceanditisresolvedinto
normalandtangentialcomponents

43. Taylors Stability Number
✓Iftheslopeanglei,heightofembankmentH,theeffectiveunit
weightofmaterial,angleofinternalfriction',andunit
cohesionc'areknown,thefactorofsafetymaybedetermined.
✓Taylor(1937)conceivedtheideaofanalyzingthestabilityofa
largenumberofslopesthroughawiderangeofslopeanglesand
anglesofinternalfriction,andthenrepresentingtheresultsbyan
abstractnumberwhichhecalledthe"stabilitynumber".This
numberisdesignatedasSn.
�
�=
??????
??????
�∗??????∗??????

46. Methods of Preventing Slope failure
VIBROFLOTATION PROVISION OF BERM AT TOE
VEGETATION COVER PROVISION OF SHEET PILE