Stability Assessment of a Critical Rock Slope

8/3/2019 Stability Assessment of a Critical Rock Slope

1/25

Stability Assessment of aCritical Rock Slope

Final Project Report

GroupParticipants: StudentNumber:

RebecaBarja 71116107

PriyadarshiHem 71071104

PabloUrrutia 36638088

ShujingZhang 71180103

Course: MINE590W

Instructor: DirkVanZyl

Date: Dec.14,2010


2/25

2/21

TableofContents

1 Summary....................................................................................................................................3

2 Objective.....................................................................................................................................3

3 Introduction.................................................................................................................................3

4 Statistics.....................................................................................................................................4

4.1 NormalDistribution..............................................................................................................4

4.2 BetaDistribution..................................................................................................................5

4.3 ExponentialDistribution.......................................................................................................5

5 GoldSimModel...........................................................................................................................6

5.1 BaseCase...........................................................................................................................8

5.2 RemedialMeasures...........................................................................................................13

6 Costs........................................................................................................................................14

7 DiscussionofResultsandConslusions.....................................................................................15

8 References...............................................................................................................................21

ListofFiguresFigure1:SauMauPingSlopeacrossfromendangeredapartmentblocks........................................4

Figure2:GoldSimModelLayout........................................................................................................7

Figure3:SauMauPingSlopeDiagram.............................................................................................9

Figure4:ProbabilisticDistributionoftheTensionCrack....................................................................9

Figure5:FrictionAngle-BetaandTruncatedNormalDistributions.................................................10

Figure6:Cohesion-BetaandTruncatedNormalDistributions........................................................10

Figure7:Relationshipbetweenfrictionanglesandcohesivestrengthsmobilisedatfailureofslopes

invariousmaterials..........................................................................................................................11

Figure8:ProbabilisticDistributionoftheEarthquake.......................................................................12

Figure9:ProbabilisticDistributionofWaterLevelintheTensionCrack...........................................13Figure10:CDFofFSforBaseCaseScenario(BetaDistribution)....................................................16

Figure11:CDFofFSforBaseCaseScenario(TruncatedNormalDistribution)...............................16

Figure12:Changeinp(FS


3/25

3/21

1 SUMMARY

AprobabilisticstabilityanalysiswascarriedoutusingGoldSimonthecasestudyAslope

stabilityprobleminHongKong(Hoek,2007),toassesstheshort-termandlong-termstability

oftheSauMauPingslope.Fourremedialmeasures(reductionofslopeheight,reductionof

slope angle, reinforcement and drainage) were considered for improving the long-term

stability of the slope; short-termstability wasnotconsidered an issueunder themodelled

conditions.Costswere estimatedfor each of the remedialmeasures. Reducing the slope

anglewas selectedas the preferredalternativedue toa technical andeconomicpointof

view.

2 OBJECTIVE

Theobjectiveofthisprojectistocarryoutaprobabilisticanalysisonawidespreadproblem

thatthreatensthesafetyofpeopleandtheirhomes,slopestability.Aprobabilisticapproach

aided by GoldSim, a Monte Carlo simulation software, will be used to model the slope

stability problem, generate a distribution for the factor of safety of the slope and finally,

generateadistributionfortheprobabilityoffailureandprobabilityofacceptanceoftheslope.

Allofthesedistributions,alongwithestimatedcosts,willbeusedinmakingadecisiononthe

mostappropriateremedialmeasure that shouldbe takenin orderto ensure the long term

stability of the slope. The problem chosen for this analysis will be introduced in the next

section.

3 INTRODUCTION

Aliteraturereviewforslopestabilityproblemsleadtoaveryinterestingcasestudywrittenby

EveretHoektitledAslopestabilityprobleminHongKong.Inthisarticle,thecaseofarock

slopelocatedalongtheSauMauPingRoadinKowloonisdiscussed.Thisrockslopeposed

potentialdangertotheapartmentblockslocatedacrossaroadwhichhousedapproximately

10,000people.Thecriticalconcernforengineerswasthatamajorrockslidecouldoccurdue

toexceptionallyheavy rainsand crossthe roadcausingdamage to theapartmentblocks.

Beforemakingadecisiononwhethertoevacuateresidents,somecrucialquestionsneeded

tobeanswered.Thesequestionswerethefollowing:

Whatwasthefactorofsafetyoftheslopeunderearthquakeandheavyrainconditions

Whatwastheacceptablefactorofsafetyforshorttermandlongtermconditions

Whatstepswouldberequiredtoachievethesefactorsofsafety

Figure1showstheSauMauPingSlopeandtheapartmentblockslocatedacrosstheroad.

Theoriginalsolutiontothiscasetookatraditionaldeterministicapproach.However,forthe

purposesofthisproject,aprobabilisticapproachwillbetakentoevaluateandcomparethe


4/25

4/21

originalengineeringdesigndecisiontothedecisiontakenwhentheweightofinfluencefrom

theproblemvariablesaretakenintoaccount.

AsecondcasestudybyE.Hoek,Factorofsafetyandprobabilityoffailure,describesthe

recommendedmethodologyforcarryingoutasensitivityanalysisoftheprobabilityoffailure.

Theprobabilitydistributionsusedinthisapproach(i.e.truncatednormalandexponential),as

wellastheprobability functionsthatwerechosen byourgroup for experimental purposes(i.e. betafunction),will bedescribed in thestatisticssection thatfollows.TheMonteCarlo

methodofsimulationwillbeused inorder togenerateafinaldistributionofresultsforthe

factorofsafetycalculationandtheprobabilityoffailureandacceptance.

Figure1:SauMauPingSlopeacrossfromendangeredapartmentblocks

(Source:PracticalRockEngineering,E.Hoek)

4 STATISTICS

4.1 NormalDistribution

Thenormaldistributionisacontinuousprobabilitydistributionwithtwoparameters,mean

andvariance2,denotedbyN(,2).Theshapeofthenormaldistributionislikeabellwitha

singlepeak at themeanofdata.Thespread ofa normaldistribution iscontrolled by thestandarddeviation.Thesmallerthestandarddeviation,themoreconcentratedthedata.

Thenormaldensityfunctionis

{ = #$ e{%

.


5/25

5/21

It is symmetric around the mean value, and inflection points occur at . Normal

distributions are one of the most common distributions used for probabilistic analysis in

geotechnicalengineeringduetoitssimplicityandwideapplicability.Giventhatitsrangefrom

to cancauseproblemswhenusedinconjunctionwithaMonteCarloanalysis,usually

theyare truncated by minimum andmaximumvalues. In this particular case,a truncated

normaldistributionwasusedtodefinethefollowingvariables:

Frictionangle

Cohesion

Tensioncrackdepth

4.2 BetaDistribution

The beta distribution is a continuous probability distributions having two positive shape

parameters,denotedbyand.TheBetadistribution,initsstandardform,rangesfromzero

toone,andtakesawiderangeofshapes.Thebetadensityfunctionis

{;, = [#{1 \# u[#{1 u\#du#"

The beta distribution is often used to describe the uncertainty or random variation of a

probabilityvalue.Itcanrescaleandshifttocreatedistributionswithawiderangeofshapes

andover any finite range. Betadistributions are very flexible andcan beused to replace

manyothercommondistributions.Asecondmodelwithbetadistributionsassignedtofriction

angleandcohesionwasalsodevelopedforcomparisonpurposesforthisanalysis.

4.3 ExponentialDistribution

The exponential distribution is a continuous probability distribution. It describes the time

between events inaPoisson process.For a givenPoisson process, the timeT between

consecutiveoccurrencesofeventshasanexponentialdistributionwiththefollowingdensity

function:

{ = exp{ J 00 otherwise

Exponentialdistributionswereusedtomodeltheeffectofearthquakeandheavyrainonthe

slope. Given the need of creating truncated exponential distributions for modelling the

earthquakeand storm (seeSection5.1.3formoreon this), and considering thatGoldSimcannottruncateanexponentialdistribution,atruncatedgammadistributionwasusedinstead.

This was possible due to the fact that an exponential distribution is a special case of a

gammadistributionwiththeshapeparameterk=1.

Sinceagammadistributionisdefinedby:

Mean=k*t


6/25

6/21

Standarddeviation= k t$

Bysubstitutingk=1boththemeanandstandarddeviationwillbecomeequaltot.Therefore,

anexponentialdistributionwithmeanequaltoXisequivalenttoagammadistributionwith

meanequaltothestandarddeviation(whichisequaltoX).

5 GOLDSIMMODEL

Asmentionedabove,GoldSim(Version10.11(SP4),AcademicLicense)wasusedtoassess

theprobabilityofacceptanceoftheSauMauPingslope.TheMonteCarloanalysiswasrun

considering 5,000 realizations in every case. Figure 2 illustrates the layout of the model

createdtoaccomplishthis.


7/25

7/21

F

igure2:GoldSimM

odelLayout


8/25

8/21

5.1 BaseCase

The Factors ofSafety for theBaseCasewerecalculated basedon the sameprocedures

followedonthecasestudyAslopestabilityprobleminHongKong(Hoek,2007),whichare

summarizedbythefollowingequation:

Where,

Symbol Parameter

c Cohesivestrengthalongslidingsurface

A Baseareaofwedge

W Weightofslidingmass

p Angleoffailuresurface,measuredfromHorizontal

Horizontalearthquakeacceleration

U UpliftforceduetowaterpressureonfailureSurfaceV Horizontalforceduetowaterintensioncrack

Frictionangleofslidingsurface

For more information regarding the equations used in the Factor of Safety calculations,

pleaserefertoAppendixA.

5.1.1 GeometryoftheSlope

TheSauMauPingroadwascutinamassofunweatheredgranite,withsheetjointsparallel

totheexposedfaceofthecutslope.Thesesheetjointsarethemostprobablesurfaceof

failureoftheslope,andwereestimatedwithadipof35.

Accordingtotheinvestigationcarriedout,therockslopecanberepresentedbythediagram

showninFigure3.Asillustratedthere,the60mhighgraniteslopeisdividedinthree20m

high bencheswith inclinationsof70 to the horizontal,and anoverall slopeangleof50.

Therewerealsotensioncracksatsomeplacesbehindthecrestoftheslope,withvariable

depths. The GoldSimmodel developed for this project assessed only the stability of the

overallslopebyincludingthetensioncrack,andnotacasewithoutit.Individualevaluationof

thestabilityofeachbenchwasnotcarriedout.


9/25

9/21

Figure3:SauMauPingSlopeDiagram


The depth of the tension crack was modelled as a stochastic variable with a normal

distribution, following the recommendation on Factor of safety and probability of failure

(Hoek, 2007).Figure4 illustrates the dataused tocreate theprobabilistic distribution.For

more detailed information regarding the equations used in estimation of the standard

deviation,minimumandmaximumdepthofthetensioncrackpleaserefertoAppendixA.The

meandepthofthetensioncrackwasestimatedashalfthemaximumdepth.

Figure4:ProbabilisticDistributionoftheTensionCrack

5.1.2 MaterialProperties

Due to the lack of shear strength information available on the SauMauPing slope, unit

weight, frictionangle and cohesionof the rockslopehadtobeassumed tocomplete the


10/25

10/21

stabilityanalysis.Theseassumptionswerebasedonpublishedinformationforsimilarrocks,

asdescribedinthecasestudyAslopestabilityprobleminHongKong(Hoek,2007).

Table1showstheparametersselectedfortheprobabilisticanalysiscarriedoutaspartofthis

project.Figures5and6illustratethestochasticdistributionsusedtomodelfrictionangleand

cohesion.

Table1:MaterialProperties

Mean CoV Std.dev Min Max Distribution

Rockfrictionangle 40 10% 3.75 30 45 Beta&Normal

Rockcohesion 125kN/m2 30% 37.5kN/m2 50kN/m2 200kN/m2Beta&Normal

Unitweight 25.5kN/m3 - - - - Deterministic

Figure5:FrictionAngle-BetaandTruncatedNormalDistributions

(betadistribution)

(truncatednormaldistribution)

Figure6:Cohesion-BetaandTruncatedNormalDistributions

(betadistribution)

(truncatednormaldistribution)

Theunitweightwasestimatedasadeterministicvaluebecausenoinformationwasfound

with regard toprobabilistic distributions that couldbeused toestimate this value.Friction

angleandcohesionwhereestimatedfortheoriginalanalysisusingtheplotonFigure7,from

thecasestudyAslopestabilityprobleminHongKong(Hoek,2007).Theplotalsoshows


11/25

11/21

the rangeofestimated shear strength values for the sheet joints inunweathered granite,

developedfortheoriginalanalysis.

Figure7:Relationshipbetweenfrictionanglesandcohesivestrengthsmobilisedatfailureofslopesinvariousmaterials


TheoriginalanalysisusedtheenvelopeofshearstrengthshownonFigure7toobtainmean

valuesforthefrictionangleandcohesion.Consideringthatthesestrengthparameterswereestimatedfromtheavailableliterature,andthatthereportedparametersarelikelyreductions

ofavarietyoffieldandlabtestsaswellasobservationsfromthefield,amoreappropriate

approachtoassessthestabilityoftheSauMauPingslopewouldbetoassumetheshear

strengthasarandomvariableamongagivenrange,ratherthanafixedone.Followingthis

line of thought, we used the same envelope on Figure 7 to generate our probabilistic

distributions, setting up the minimums, maximums and means after it. The standard

deviationsusedforouranalysiswerecalculatedwithatypicalcoefficientofvariationforeach


12/25

12/21

parameter.Thesecoefficientsofvariationwereselectedafterrevisingsomeliterature(Park,

1999;ParkandWest,2001;Rththi,1988).

5.1.3 EarthquakeandWaterinTensionCrack

Besides geometry and inherent material parameters, other factors could also affect the

stability of the slope, such as earthquakes (inducing external forces to the system) and

storms(changingthesaturationconditionsofthesystem).Inthisspecificcase,earthquakes

werenot perceived asamayor threatgiventhat the region of studywasnot deemed as

highlyseismic.Afterdiscussingthesubjectwithlocalexperts, thedevelopersoftheoriginal

stabilityanalysis(Hoek,2007)consideredappropriatetoincludeaminoracceleration(inthe

formofapseudo-staticforce)duetoearthquakeloadingintothesystem.Thisacceleration

wasestimatedtobe0.08g.

Asapseudo-staticforce,theearthquakewasinputintothemodelasafractionofthegravity

acceleration,appliedtotheweightoftheslidingmass.Thepseudo-staticforcewastreated

asaprobabilisticvariable,inthiscasewithatruncatedgammadistribution(seesection4.3)

withameanandstandarddeviationof0.08gandamaximumvalueof0.16g,asshowninFigure8.

Figure8:ProbabilisticDistributionoftheEarthquake

Typhoons,ontheotherhand,areverycommoninHongKongandoneofthesestormscould

easily fill the tension cracks with water and saturate the whole slope. Storm events are

usually represented by exponential distributions, but given that we needed tomodel the

amount of water in the tension crack, and the tension crack has amaximum depth, theexponential distribution representing the water level in the tension crack should also be

truncatedwithamaximumvalueequaltothewholedepthofthetensioncrack.TheGamma

distributionshownonFigure9wasusedtomodelthewaterlevelinthetensioncrack,where

Zisthedepthofthetensioncrack(modelledasanormaldistribution,asexplainedabove).


13/25

13/21

Figure9:ProbabilisticDistributionofWaterLevelintheTensionCrack

5.2 RemedialMeasures

AstheprobabilityofacceptanceoftheBaseCaseexceededour acceptancecriteria(see

Section7),fourdifferentandindependentremedialmeasureswereassessedtoimprovethe

stabilityoftheSauMauPingslope.Thesemeasureswere:

Reductionoftheslopeheight

Reductionoftheslopeangle

Reinforcementoftheslope(bycablebolts)

Drainageoftheslope

5.2.1 ReductionoftheSlopeHeight

Inordertoassessthereductionoftheslopeheight,changesintheheightoftheslopewereevaluatedusingtheGoldSimmodel.Reductionsof5mwereassesseduptoatotalheightof

30 m, changing also the tension crack depth distribution with a new maximum (with

consistentmeanandstandarddeviation)whilekeepingtherestofthemodelinputsthesame

aswiththeBaseCase.ResultsarediscussedinSection7.Forfurtherdetailedinformation

aboutthecalculationsofthenewtensioncrackdepthparameterspleaserefertoAppendixA.

5.2.2 ReductionoftheSlopeAngle

Similarasinthepreviouscase,theoverallangleoftheslopewaschangedbyintervalsof

first5andthen1uptoadipof40.Thetensioncrackdepthdistributionwasalsomodified

toreflectthechangeintheslopeangle.TherestoftheGoldSimmodelparameterswerekept

thesameasintheBaseCase.PleaseseeSection7fordiscussionoftheresults.Formore

information regarding the calculations for thenew tension crackdepthparameters please

refertoAppendixA.


14/25

14/21

5.2.3 ReinforcementoftheSlope(byCablebolts)

Cableboltsof25mlongwereincludedinthemodelasanewforceintheFactorofSafety

equationshownonSection5.1.Thenewequationisasfollows:

WhereTistheforceappliedbythecableboltandistheangleofthecableboltwith

respect to the horizontal. Four different T values, from 2,800 kN to 4,000 kN, were

assessed, all of them for a equal to 35, and four factors of safety were calculated

accordingly.ResultsarediscussedinSection7.

5.2.4 DrainageoftheSlope

Toassesstheeffectofdrainingtheslope,nowaterpressurewasincludedintheFactorof

Safety calculations for this remedialmeasure.Please see Section 7 fordiscussion of the

results.

6 COSTS

Costswereestimatedbasedonliteraturereviewandprofessionalengineeringjudgementof

the authors of this report. The costspresentedhere are just anapproximation basedon

severalassumptions,giventhattheinformationavailableintheoriginalcasestudydoesnot

focusoncostingaspects.Thecostsusedforthisprojectareusedsimplyforcomparisonof

differentalternatives.Theseshouldnotbeusedforadifferentpurpose.

Excavationvolumes, numberofboltsand lengthofthedrainagestructuresalsohad tobe

estimatedinordertoobtaincomparablecostsbetweendifferentalternatives.Thecostshave

beencalculatedforasectionoftheslopeof1.0mwidth.

Cost of material excavation was estimated as 50.00 $CAD/m3 of material. This amount

includestransportationofthecutmaterialtoawastedump5kmawayfromthesiteandis

basedonpreviousexperienceandontheSMEMiningEngineeringHandbook(SMEwebsite,

online source). The excavation volume estimated for this alternative was calculated by

assumingexcavationsupto50mupstreamofthecrestoftheslope.

CostofcableboltinstallationwasobtainedfromthebookCableboltinginUndergroundMines

(HutchinsonandDiederichs,1996),which suggesteda valueof $CAD30.00 permeterof

cablebolt.Thelengthofthecableboltswasestimatedasthelengthfromthefaceoftheslope

uptotheslipsurface(i.e.sheetjoints),whichwas20m,plusanadditional5mforgrouting

and proper anchoring of the bolts. The number of cable bolts required was calculated

assumingayieldstressof400kNperbolt(GIAindustriabwebpage,onlinesource).


15/25


16/25

16/21

Figure10:CDFofFSforBaseCaseScenario(BetaDistribution)

Figure11:CDFofFSforBaseCaseScenario(TruncatedNormalDistribution)

Firstandforemost,Figures10and11showthattheselectionofabetaortruncatednormal

distribution doesnotsignificantlyaffect theoutcomeof thestability analysis.The resulting

probabilities of acceptance for the beta distribution as well as the truncated normal

distributiondiffer invalueby1 3%. The betadistribution results are higher (i.e.more

conservative). This is negligible for a stability evaluation like the one carried out in this

project,alongwithmanyotherformsofgeotechnicalanalysis,wheremanyassumptionsand


17/25

17/21

simplificationsaremade.Itisworthnotingthatgiventheflexibilityofthebetadistribution,it

can take different shapes.Therefore, the similarity in the results of this specific analysis

betweenthebetaandtruncatednormaldistributionsisduetotheshapeofthecurrentbeta

distribution (which issimilar to the truncatednormal),since it isrelated to the parameters

(mean,standarddeviation,minandmax)usedtodefineit.

Figures10and11alsoshowthattheprobabilityoffailureforbothdistributionsisabout1%,whichdoesnotimplyahighrisk;theprobabilityofacceptancefortheshorttermscenariois

8.71%forthebetadistributionand6.77%forthetruncatednormal.Evenifthep(FS


18/25

18/21



19/25

19/21



20/25

20/21

Figure15:CombinedPlotShowingp(FS


21/25

21/21

betweenreducingslopeheightandslopeangleisduetotheexcavationvolumesrequiredfor

eachofthem,beingthefirstonealmosttwiceasmuch.Moreover,Figure10showsthatto

reachap(FS


22/25

APPENDIXA:

-CALCULATIONS-


23/25

1 FACTOROFSAFETYCALCULATIONS

FactorofSafetycalculationsweredonefollowing themethodshownonthecasestudyA

slopestabilityprobleminHongKong(Hoek,2007).FigureA-1showsthediagramofforces

usedtoelaboratetheFSequation,basedonlimitequilibriumconcepts.Itisworthnotingthat

only forceswereconsideredwhencalculatingFS(notmomentums),and thatalltheforces

pass through the center ofgravityof the slidingmass.This isa simplification,but itwasrequiredtouseGoldsimtosolvetheproblem.Furthermore,giventheassumptionsmadeto

modelthecharacteristicsoftheslope,itwillnotbeveryinfluentialinthefinalresults.

FigureA-1

Where,

Symbol Parameter

c Cohesivestrengthalongslidingsurface

A Baseareaofwedge


f Angleoftheslope,measuredfromHorizontal

p Angleoffailuresurface,measuredfromHorizontal

Horizontalearthquakeacceleration

U UpliftforceduetowaterpressureonfailureSurfaceb Distancebehindtensioncrackandcrest

z Tensioncrackdepth

zw Waterlevelontensioncrackandfailuresurface


T Forcerepresentingthecableboltreinforcement

Inclinationofreinforcement

V Horizontalforceduetowaterintensioncrack

Frictionangleofslidingsurface


24/25

EquationsusedtocompletetheFScalculationsweretakenfromAslopestabilityproblemin

HongKong(Hoek,2007),andareshownonFigureA2.

FigureA-2

Equationforcalculatingzwasobtainedbyminimizationofthefollowingequation(hoekand

Bray,1974):

Thisminimizationwasdoneconsideringdryconditionsontheslope,whichisasimplification

butisacceptableforthescopeofthisanalysis(Hoek,2007).

2 TENSIONCRACKDEPTHDISTRIBUTION

Atruncatednormaldistributionwasselectedtomodeltheuncertaintiesonthedepthof the

tension crack, following the recommendation on the case study Factor of safety and

probabilityoffailure(Hoek,2007).Themaximumdepthofthetensioncrackoccurswhenthe

crackislocatedatthecrestoftheslope(Hoek,2007),andwasestimatedby:


25/25

z=H(1tanp/tanf)=24.75m

Themeanforthetensioncrackdistributionwasselectedashalfofthemaximumcrackdepth,

while an arbitrary standard deviation of 3 m was estimated, again following the

recommendationonthecasestudyFactorofsafetyandprobabilityoffailure(Hoek,2007).

Aminimumdepthof0.01mwasselectedforthetensioncracktoavoidnumericalissues.

TheabovedescribedparameterswereusedfortheBaseCasescenario.Whenchangingthe

modeltoassessthereductioninheightandslopeangleremedialmeasures,themaximum,

mean, and standard deviation of the tension crack distribution were changed as well, to

reflect the modifications made on the slope. The new values for max, mean and std.

deviation were estimated as a fraction of the original ones, keeping the same ratio the

changeinheightorslopeangle,asshownintablesA-1andA-2.

TableA-1

H(m) Maxcrackdepth(m) Std.deviation(m)

60 24.75 3.0055 22.69 2.75

50 20.62 2.50

45 18.56 2.25

40 16.50 2.00

35 14.44 1.75

30 12.37 1.50

TableA-2

Slopeangle() Maxcrackdepth(m) Std.deviation(m)

50 24.75 3.00

45 17.99 2.18

44 16.49 2.00

43 14.95 1.81

42 13.34 1.62

41 11.67 1.41

Stability Assessment of a Critical Rock Slope

Documents

Transcript of Stability Assessment of a Critical Rock Slope