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Comparison of slopestability methodsof analysisrD. G. FnpoluND ANDJ. KnesN

(Jnit'ersityo. fSaskatchev'an, Saskatoon, Sask., Canada 57N 0W0Received October 12 , 19 7

Accepted April 4,1977The papercompare six methodsof s l ice commonly used or slopes tab i l i tyanalysis.The factor

of safety equations are written in the same form, recognizing whether moment and (or) forceequil ibr ium s explic i t ly sat isf ied.The normal force equation s of the same orm for al l methodswith the exceptionofthe ordinary method. Th e method ofhandling th e nters l ice orcesdif feren-t iates he normal force equations.A new derivat ion for the Morgenstern-hice method is presentedan d is cal led the'best-f i tregression'solut ion. It involves the independentsolut ion of the force and moment equil ibr ium_faitors of safety fo r various val ues of tr. The be st-fit regression solution gives th e same actor ofsafety as the'Newton-Raphson'solut ion. The best-f i t regression solut ion is readily com-prehencled, iv ing a completeunderstanding fthe variat ion ofthe factor of safetywi th t r .

L 'art ic le pr6senteune comparaisonde s six m6thodesde tranches uti l is6escouramment poul 'I 'analysede la stabi l i t6de s pentes. Les 6quationsde s facteurs de s6curit6 sont 6cr i tes selon am6me forme, en montrant s i les condit ions d'dquil ibre de moment ou de force sont satisfa itesexplic i tement. L 'dquation de la force normale est de la m6me forme pour toutes es mdthodes rI 'exceptionde la m6thodedes ranchesordinaires.La fagonde tra iter es orces ntertranches stce qu i d if f6rencie es 6quationsde force normale.Un e nouvelle d6rivat ion appel6e solut ion "best-f i t regression" de la m6thode de Morgens-tern-Price es t pr6sent6e. El le consiste r d6terminer ind6pendemment es tacteuls tl e secu-r i t sat is fa isant ux 6quil ibres de force et de moment pour dif f6rentes valeurs de t r . La solu-t ion "best-f i t regression" donne es memes acteursde s6curit6de la solut ionde Newton-Raph-son. l-a solut ion "bes t - f i t regression" es t p lus aci le r comprendreet donne un e imaged6tai l l6ede la variat ion du facteur de s6curit6avec I" .c a n .Ge o te c h .. . r 4 .4 2 g ( |g 1 7 \ l r r a d u i t p a r la re v u e l

IntroductionThe geotechnical engineer frequently useslimit equilibrium methods of analysis whenstudyingslope stability problems. The methods

of slices have become the most commonmethods due to their ability to accommodatecomplex geometrics and variable soil andwater pressureconditions (Terzaghi and Peck1967). During the past three decades ap-proximately one dozen methods of sliceshavebeen developed (Wright 1969). They difierin (i ) the statics employed in deriving thefactor of safety equation and (ii) the assump-tion used to render the problem determinate(Fredlund 1975).This paper is primarily concerned with sixof the most commonly used methods:(i ) Ordinary or Fellenius method (some-times referred to as the Swedish circle methodor the conventional method)(ii) Simplified Bishop method

'Presented at rh e 29th Canadian GeotechnicalConference, Vancouver, B.C., October 13-15, 1976.

(i i i) Spencer'smethod(iv) Janbu'ssimplif iedmethod(v ) Janbu's igorousmethod(vi) Morgenstern-Price methodThe objectivesof this paper are:( 1) to compare the various methods ofslices in terms of consistent procedures forderiving the factor of safety equations. Al lequations are extended to the case of a com-posite failure surfaceand also consider partialsubmergence, line loadings, and earthquakeloadings.(2) to present a new derivation for theMorgenstern-Price method. The proposedderivation is more consistent with that usedfor the other methods of analysis but utilizesthe elements of statics and the assumptionproposed by Morgenstern and Price (1965).The Newton-Raphson numerical technique isnot used to compute the factor of safetyand )..(3) to compare the factors of safety ob-tained by each of the methods for severalexample problems. The University of Sas-


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CAN. GEOTECH.J. VOL. 14,1977.-?,0WATER

WATER,A p\ - - BEDROCK

Frc. 1. Forces acting for the method of slicesapplied to a composite sliding surface.katchewan SLOPE computer program wasused for all computer analyses (Fredlund197 ) .(4 ) to compare the relat ive computationalcosts involved in usine the various methodsof analysis.

Definition of ProblemFigure I shows the forces that must be de-fined for a general slope stability problem.The variables associated with each slice aredefined as follows:

W : total weight of the slice of width 6 andheight ftP : total normal force on the baseof the sliceover a length /S- : shear force mobilized on the base of theslice. It is a percentage of the shearstrength as definedby the Mohr-Coulombequation.That is , S- : / {c ' + lPll - ultan $'|lF where c' : effective cohesionparameter,0' : effectiveangle of internalfriction, -F: factor of safety, and u :porewater pressureR : radius or the moment arm associatedwiththe mobilized shear orce S.

f : perpendicular offset of the normal forcefrom the centerofrotationx : horizontal distance from the slice to thecenter of rotationc( : anglebetween he tangentto the center ofthe baseofeach sliceand the horizontal

: horizontal interslice orces: subscriptdesignatingeft side: subscript esignatingight side: vertical nterslice orces: Se lSmlCsmic coefficient o accountor a dynam-ic horizontal force

e : vertical distance rom the centroid ofeachslice o the centerof rotationA uniform load on the surfacecan be takeninto account as a soil layer of suitable unitweight and density.The following variablesarerequired o definea line oad:L : line load (force per unit width)u) : angle of the line load from the horizontald : perpendicular distance rom the line loadto thecenter frotationThe effectof partial submergence f the slopeor tensioncracks n water requires he definitionof addi t ional ar iablesA : resultantwater forcesa : perpendicular distance rom the resultantwater force to the centerofrotationDerivations for Factor of Safety

The elements of statics that can be used toderive the factor of safety are summations offorces in two directions and the summationof moments. These, along with the failurecriteria, are insufficient to make the problemdeterminate. More information must be knownabout either the normal force distribution orthe interslice force distribution. Either addi-

ELRXt-

timmugFdeOstftcspfcffaip

rr,1*fffififiiiilffi|#ffiffi#l$,


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FREDL UND AND KRAHN 431

O=RESULTANTNTERSLICEFORCE

NOTE:Q^ . IS NOT EQUALANO OPPOSITETOo , ^

LO

PoFtc. 2 Interslice orces or the ordinary method.

tional elementsof physicsor an assumptionmust be invoked o render he problemdeter-minate.All methodsconsideredn this paperuse the latter procedure, each assumptiongiving rise to a different methodof analysis.For comparisonpurposes,each equation isderivedusing a consistentutilization of theequations f statics.Ordinaryor FelleniusMethodThe ordinary method is considered hesimplestof the methodsof slicessince t isthe only procedure hat results in a linearfactor of safetyequation. t is generally tatedthat the interslice orcescan be neglected e-cause hey are parallel to the base of eachslice (Fellenius 1936). However, Newton'sprincipleof 'actionequals eaction's not satis-fiedbetween lices Fig. 2). The ndiscriminatechange n directionof the resultant ntersliceforce from one slice to the next results nfactor of safety errors that may be as muchas 60Vo(Whitmanand Bailey1967).The normal force on the baseof each sliceis derived either from summationof forcesperpendicularo the baseor from the sum-mation of forces n the verticaland horizontaldirections.t l l l r u : o

W - P c o s d - S - s i n a : 01 2 ) F ' : o

S - c o s a - P s i n a - k W : 0Substituting2] into [1] and solving or thenormal force givest 3 l P : W c o s o . - k W s i n a

The factor of safety is derived from thesummation of moments about a common point(i.e. either a fictitious or real center of rotationfor the entire mass).l4l l,vro : o

L w * - I s - R - L p f + l k w e + A a+ L d - - o

Introducinghe failure criteriaand thenormal force from [3] and solving for thefactor of safety givesf{c'lR + (P - ul)R an$'}r


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cAN. GEOTECH. J. VOL. 14 , 1977

Spencer's MethodSpencer'smethod assumes here is a con-stani relationship between he magnitude of theinterslice shear and normal forces (Spencer1967 .l 7 l t an + :5EL ERwhere d = angle of the resultant intersliceforce from the horizontal.Spencer (1967) summed forces perpendic-ular to the interslice forces to derive thenormal force. The same result can be ob-tained by summing forces in a vertical an dhorizontal direction.t8l lru : o

W - ( X * - X t ) - P c o s o - S - s i n a : 0te l IF ' : o

- (E* - EL) + P s i na - S-cos ' z * kW : OThe normal force can be derived from [8 ]an d then the horizontal interslice orce is ob-tained from [9].

il01c'l sin a ul tan 6' si n q' l ,- F -+ -----7-)rm"

Spencer 1967) derived wo factor of safetyequations.On e is based on the summation ofmoments about a common point and the otheron the summation of forces in a directionparallel to the interslice forces. The momentequation is the same as for the ordinary an dthe simplif iedBishop methods (i .e. [4]). Th efactor of safetyequation s the same as [5].The factor of safety equation based onforce equilibrium can also be derived bysumming orces n a horizontal direction.nl l IFn : o

L(EL - ER ) + IP si n a - ls - co so+ L k W + A - I - c o s < , . ' : 0

The intersliceforces (Et, - -En) must cancelout and the factor of safety equation withrespect to force equilibrium reduces toF{c'l cos a -t (P - uI ) tan $' co sa}L , L J a t - l f s i na + l k W + A - L c o s < , ,

Spencer's method yields two factors ofsafely for each angle of side forces. However,at some angle of the interslice forces, the twofactors of safety are equal (Fig. 3) and bothmoment and force equilibrium are satisfied.The Corps of Engineersmethod, sometimesreferred to as Taylor's modified Swedishmethod, is equivalent to the force equilibriumportion of Spencer'smethod in which the direc-iion of the interslice orces s assumed,generallyat an angle equal to the averagesurfaceslope'Janbu's Simplified MethodJanbu's simplified method uses a correctionfactor lo to account for the effect of the inter-slice shear forces. The correction factor isrelated to cohesion, angle of internal friction,and the shape of the failure surface (Janbuet at . 1956). The normal force is derived fromth e summation of vertical forces ([8]) ' withth e intersliceshear orces gnored.r r3 l l r -+ :+ 1lm"

The horizontal force equilibrium equationis used to derive the factor of safety (i'e'tl I I ) . The sum of the interslice forces mustcancel and the factor of safety equationbecomes

f {c ' l cosoc (P - ul ) tan { 'cos a}U4 l Fo :Fn is used to designate he factor of safetyuncorrected for the interslice shear forces'

Frc. 3 Variation of the factor of safety withrespect to moment and force equilibrium- vs '- theangleof the side orces.Soil properties-: '/1h =^0'02;V' : lO"t r" = 0.5. Geometry: slope = 26'5';he ight 100 t (30 m).

sinTul tan

p : I w - ( E * - E r ) t a n o

FL ,< lLoo rs

sr0E ro 15 20FORCE NGLE, (DEGREES)


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FREDL UND AND KRAHN 433The corrected factor of safety isl15l F = loFolanbu's Rigorous MethodJanbu's rigorous method assumes hat thepoint at which the interslice forces act canbe defined by a 'line of thrust'. New termsused are defined as follows (see Fig. 4) :tr, tn = vertical distancefrom the base of theslice to the line of thrust on the left and rightsides of the slice, respectively; dt = anglebetween the line of thrust on the right side ofa slice and the horizontal.The normal force on the base of the sliceis derived from the summation of verticalforces.

t-t l 6 l P : l w - ( X ^ - X , )Lc' l s ina ul tan6 ' sinal,- F-- +- i--- l t^"

The factor of safety equation is derivedfrom the summation of horizontal forces (i.e.[1]). Janbu's rigorous analysisdif fers fromthe simplified analysis n that the shear forcesare kept in the derivation of the normal force.The factor of safety equation is the same asSpencer's quationbasedon fo rce equil ibrium(i .e. [12] ) .In order to solve the factor of safety equa-tion, the interslice shear forces must beevaluated. For the first iteration. th e shears

Frc. 4. Forcesrigorous method.

are set to zero. For subsequent iterations,the interslice forces are computed from thesum of the moments about the center of thebase of each slice.u7l LM" 0

XLbl2 + XRbl2 - Er l t , + (b l2) tanul+ ER /L + (bl2) ta n a - b ta n a, f - kwhl2 :0After rearranging I 7], several ermsbecomenegligibleas the width b of the slice s reducedto a width dr. These erms are (Xr, - XL) b/2,(E', - Er.) (b/2) Ian a and (Er - EL)btan a1.Eliminating these terms and dividing byth e slicewidth, the shear orce on the right sideof a slice s

i l8 l Xn : En tana, - (ER - E)tRlb+ (kw /b ) (h /2 )

The horizontal interslice forces, requiredfor solving [18], ar e obtained by combiningthe summation of vertical and horizontalforces on each slice.t19l (E * - Er ) : lW - (X * - Xy)l ta n a

- S . / c o s a * k WThe horizontal nters lice orces are obtainedby integration from left to right across theslope. The magnitude of the interslice shearforces in [19] lag by one iteration. Eachiteration gives a new set of shearforces. Thevertical and horizontal components of l ine

loads must also be taken into account whenthey are encountered.M o gens ern-P r c e M ethodThe Morgenstern-Pricemethod assumes narbitrary mathematical function to describethe directionof the interslice orces.t20l xf x) = Ylswhere ). = a constant to be evaluated insolving for the factor of safety, an d l(x) =functional variation with respect to x. Figure5 shows typical functions (i.e. l(x) ). For aconstant function, the Morgenstern-Pricemethod is the same as the Spencer method.Figure 6 shows how the half sine functionand ), are used to designatethe direction ofthe interslice forces.

acting on each slice fo r Janbu's Morgenstern an d Price (1965) based theirsolution on the summation of tangential and


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FREDT-UND ND KRAHN

8 0 loo t20 t40 'FIc. 8. Exampleproblem.

iteration. The moment and force equilibriumfactors of safety are solved for a range of )"values and a specifled side force function.These factors of safety areplotted in a mannersimilar to that used for Spencer's method(Fig. 7). The factors of safetyvs . ), are f it bya secondorder polynomial regression nd thepoint of intersection satisfies both force andmoment equilibrium.Comparison ol Methods of AnalysisAII methods of slices satisfying overallmoment equilibrium can be written in thesame form.122) F_: fc'lR f ItP - ul)R tan $'^ - lwx - LPf + lkwe * Aa-r Ld

All methods atisfying verall orce equilib-rium have he followins orm for the factorofsafetyequation:^ Lr ' l cosa * IfP - ul) tan 'cosar12-l E - --L ' J J ' t - l r s i n a + l k w + A - L c o s < ' ,

The factor of safety equations can bevisualized s consisting f the following com-ponents:

From a theoretical standpoint, the derivedfactor of safety equations differ in (i ) theequations of statics satisfiedexplicitly for theoverall slope and (ii) the assumption o makethe problem determinate.Th e assumption sedchanges he evaluation of the interslice forcesin the normal force equation (Table l). Allmethods, with the exception of the ordinarymethod, have the same form of equation forth e normal force.

Xt)l 2 4 l : I w - ( x s . -c'l sino- - +F

u l t a n 6 ' s i n a ,- - - - - - | l nF ) ' " ' "whete mo = cos cY ( sin a tan 6' ) / F .

It is possible to view the analytical aspectsof slope stabil ity in terms of one factor ofsafety equation satisfying overall momentequilibrium and another satisfying overallforce equilibrium. Then each method becomesa special case of the 'best-fit regression'solu-tion to the Morgenstern-Price method.Tasrn . Comparisonf factorof safety quations

Momentequilibrium ForceequilibriumFactor of safety basedon

Moment Force Normalequili equili- forcebrium brium equationohesion Zc'lRFriction Z(P-ul)R tan $'Weight 2l4txNormal LPfEarthquake ZkltePartialsubmergence AaLine oading Ld

2c'l cosu2(P-ul)tan$'cos c!P sin a>ktv

MethodOrdinary or FelleniusSimplified BishopSpencer'sJanbu's simplifiedJanbu's rigorousMorgenstern-Price

xxxt3 1t6 lt10lt13 lt16lI24lAI cos


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FREDLUND ND KRAHN 437Tnslp 3. Comparison of two solutions to the Morgenstern-Price method**

University ofAlberta programside force functionUniversity of SaskatchewanSLOPE programside force function

Constant Half sine Constant Half sine Clippedsine*Caseno. ExampleProblem

I Simple :1 slope,40t (1 2m)high, ' :20",c' : 600'sf(29 kPa)2 Sameas I with a thin, weaklayerwith $' : l0', c' : O3 Same s exceptwith r" : 0.254 Sameas2 exceptwith r" : 0.25for both materials5 Same s exceptwith a Piezo-metric line6 Same s2 excePt ith a Piezo-metric ine for both materials

2.085 0.257 2.085 0.314 2.O760.254 2.076 0.318 2.083 0.3901 . 3940 . 182 1 . 3860 . 218 1 . 3780 . 159 1 . 3700 . 187 1 . 3640 . 2031 . 7720 . 351 1 . 7700 . 432 1 . 7650 . 244 1 . 7640 . 304 1 . 7790 . 4171 . 1 3 70 . 3 3 4 . l r 7 0 . 4 4 1 . 1 2 4 0 . 1 1 61 . 1 1 80 . 1 3 01 . 1 1 30 . 1 3 81 . 8380 . 270 1 . 8370 . 331 1 . 8330 . 234 1 . 8320 . 290 1 . 8320 . 3001.265 0.159Notconverging.250 0.097 1.245 0' 0 l 1.242 0.104

*coordinates x : 0, J : 0.5, and t : 1.0' y : o,21.- - ,**ioG.ince on both -Morgenstern-Pricesolutions s 0.001.

FI r + sU)$ r + otrog

r .65r . 60r 5 5

t . 50

t . 301 . 25t.?o

+ - S P E N C E R. - MORGENSTERN-RICEf (x )= CONSTANT

ORDINARYo.2 \

Frc.10. Comparisonf factors f safetyor case .safety by the various methods remains similarwhether the failure surface is circular or com-posite. For example, the simplified Bishopmethod gives factors of safety that are alwaysvery similar in magnitude to the Spencer andMorgenstern-Price method. This is due to thesmall influencethat the side force function hason the moment equilibrium factor of safety

equation. In the six example cases, he aver-age difference in the factor of safety wasapproximately .1%.Comparison of Two Solutions o theMorgenstern-PriceMethod

Morgenstern an d Price ( 1965 originallysolved heir methodusing he Newton-Raphsonnumerical echnique.This paper has presentedan alternate procedure that has been referredto as the 'best-f it regression'method. The twomethodsof solution were comparedusing theUniversity of Alberta computer program(Krahn et al . 1971) fo r the original methodand the University of Saskatchewancomputerprogram (Fredlund 1974) for the alternatesolution. n addit ion, t is possible o comparethe above solutionswith Spencer'smethod.Table 3 shows a comparison of the twosolutions or the example problems (Fig. 8) .Figures 1 and 12 graphically display thecomparisons.Although the computer programsuse difierent methods for the input of thegeometry and side force function and differenttechniques for solving the equations, thefactors of safety are essentiallythe same.The examplecases how that when the sideforce function is either a constant or a halfsine, the average factor of safety from theUniversity of Alberta computer programdifters from the Universitv of Saskatchewan

SIMPL IF IE

./l, /-I

o . 6. 4


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43 8 CAN. GEOTECH. J. VOL, 14. 1977o. t4

t 6 0r 5 5

o o o 6\Frc. I l E ffect of side force function on factor ofsafety or case3.

t . 4 5

r .40t . 3 5

o.2Frc. 12. Effect of side force function on factor ofsafety or case4.

computerprogram by less than 0.7%. Usingthe Universityof Saskatchewanrogram, heSpencermethod and the Morgenstern-Pricemethod (for a constantside force function)differ by less han 0.2%. The average. valuescomputedby the two programsdiffer by ap-

o t 2o. ro

l o o +I

o02o oROINARY SIMPLTFIEO NEUS JANBU,S SPENCERSMORGENSTERNBISHOP SIMPLIFIEO IGOROUS -PRICE

METHOD FANALYSISFrc. 13. Time per stabil i ty analysis r ial for allmethodsof analysis.

proximately 9Vo. However, as shown above,this difference does not significantly affect thefinal factor of safety.

Comparison of Computing CostsA simple2: I slopewa s selected o comparethe computercosts (i .e. CP U time) associatedwith the various methods of analysis. Th eslope was 440 ft long and was divided into5-ft slices.The resultsshown in Fig. 13 wereobtained using the Univeristy of SaskatchewanSLOPE program run on an IBM 370 model158 computer.The simplified Bishop method required0.012 min for each stabil ity analysis. Th eordinary method requiredapproximately6O %as much time. The factor of safety by Spencer'smethod was computed using four side forceangles. The calculations associatedwith eachside orce angle equired0.024 min. Th e factorof safety by the Morgenstern-Price method wascomputed using six ). values.Each trial required0.021 min. At least hree estimates f the sideforce angle or ), value are required to obtainthe factor of safety. Therefore, the Spenceror

Morgenstern-Price methods are at least sixtimes as costly to run as the simplified Bishopmethod. The above relative costs are slightlyafiectedby the width of slice and the toleranceused in solving the nonlinear factor of safetyequations.

UJF:)z:g(tsFUo-IJJ=F

o.o8o 0 6

r 8 5-trJtL I e.)atLO r z

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