Using Limit Equilibrium Concepts in Finite Element Slope Stability Analysis

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Proceedings f the nternational ymposium n Slope stability Engineering-IS-shikoku'99,Invited eynote paper, Matsuyama, hikoku, apan. pp.3 47-

November -l , 1999Slop StabilityEngineeing, Yagi,Yamagani&Jiango 1999 Batt<ema, onedam, SBN Osslg O7g

Using imitequilibrium oncepts n finiteelement lope tability nalysis

D.G.Fredlund R.E.G.ScoularUniversity of Saslatchewan, Saskatoon, Sask., Canada

ABSTRACT: This paper reviews thc development of finite element slope stability analyses and proposesthat such a method can orm a practical procedure or solving slope stability problems. Several stope stabititymethods have been proposed hat make use of the finite element merhods; ir"i" "r" summarized n-this paper.The proposed inite.e-lement method s in a fonn th3l-g* be conveniently used n engineering practici. Theprocedure ends tself to Present day numerical modelling techniques. Ihe method tri Ueen ipA"t"a to takeadvantage frecent advances n computer echnology and algorithms.

The combination of a finite element strels analysi: with-a limit equilibrium analysis provides greater cer-tainty and flexibility regarding he intemal distribution of stesses within the soil'mass. The nirmal forc"a-long ny selected lip surface can be calculated rom the sfiess distribution that has been calculateJ *i.rg "linear and non-linear stess analysis. The overall factor of safety or a slope, when the n"it. il"m"nl methodis used, can be defined as the available shear strcngth of the soil dividedbythe resisting slear st roeth. Th.overall actor of safety s a combination f the local factors of safety within the.fop". ile.etuttine-ou"ottfactor of safety retains the basic assumptions nhercnt to the limii equilibrium de'finition oi tt "

-L.to,of

safety.The ocal facton ofsafe-ty att an exprcssion ofthe stability ofthe soil mass at each point along he slip sur-

face' The overall factor of-safety computed using the finite element method shows good "gt""fr"ot with thefacton of safety computed using any one of several imit equilibrium methods. nl nnitJ element methodprovides additional information-regarding the potential performance of a slope; information not availablewhen using raditional limit equilibrium methods. The results ndicate hat it is'important to use he effectiveshear stength characterization f the soil when performing the slope stability anaiysis. The computed actorof safety obtained when using " _toq she{ shength characterizatio; of the soil, rr"y oot agree wiih the factorof safety computed when using he finite element stress analysis method.- K9f wgrds:qlope stability analysis, inite elem€nt, enhanced method direct method strength metho4 stesslevel method, actor ofsafety, Iocal factor ofsafety.

I INTRODUCTION

Limit equilibrium methods of analysis have provento be a widely used and successful method for theassessment f the stability of a slope. Limit equilibrium methods um forces and moments elated o anassumed slip surface passed hrough a soil mass(Fredhmd nd Krabn, 1975; Fredhmd et al, l98l).However, hese methods do not utilize the stress er-sus stain characteristics f the soils involved. It iswell known, and ntuitively rmderstood hat the sta-bility of a slope should be influenced by the stressversus strain characteristics of a soil (Kondner1963). A finite element analysis utilizes a stress ver-sus stain model for the soils involved to calculate

the stresses n the soil mass. These sEesses an sub-sequently be used o compute a factor ofsafety (Fig.1). The complete stress state rom the finite elementanalysis can be "imported" into a limit equilibriumanalysis where the normal stress and the shear stressare computed corresponding o any selected slip sru-face.

The objective of this paper is to demonsEate aprocedure for combining a finite element stressanalysis on a slope with the concepts of a limitingequilibrium method of a'alysis. The final method scalled a "finite element method of slope stabilityanalysis" and the results are compapd to results ob-tained when using conventional limit equilibriummethod of analysis.

31



Finite Element Analysis or Stresses

LimitEquilibrium nalysis

Figue I.Illustration showing sresses that arc "rmported" from a finite element analysis into a limit equilibriurn malysis.

2 BACKGROI.'ND

Bishop (1952) noted that the stresses rom a limitequilibrium method of analysis did not agree withthe achral stresses within an earth stnrcture. Otherresearchers have confirmed this observation bothwith experimental evidence and with numericalmodelling. a Rochelle 1960) estimated he stnessconditions n sGep slopes using photoelastic ests ongelatine models. The results showed that stressesalong a slip surface were over-stressed n the lowerportion of the slip circle. Brown and King (1966)produced critical slip surfaces rom a finite elementstrcss analysis of slopes using a linear elastic soilmodel. The critical slip surfaces were produced byusing he angle of obliqwty, 0, along he slip surface(i.e., dequal to (45o + Q/a)). Each critical slip sur-face represented close approximation o an essen-tially circularshaped lip surface.

Clough and Woodward 1967) rndertook studyto evaluate he effect of incremental loading withsingle step loading as it related to strcsses and de-formations. t was concluded hat: l) stresses nd de-formations n an embankment obtained rom a directapplication of the gravitational body forces on thecomplete structtrre were not completely accurate,and 2) changing Poisson's ratio interferes with therelationship between sEesses nd displacements, e-quiring a new analysis for each case. It was con-cluded that "meaningfttl stability analysis can bemade only d the stress distribution within the struc-ture can be predicted reliably."

Kulhawy (1969) developed compder programto obtain an independent assessment f the nonnaland shear stress distribution along an assumed slip

surface. The normal and shear shesses rom an elastic analysis were used o calculate an overall factoof safety. The formulation of Kulhawy (1969) waclassified as an "Enhanced Limit Strength Method".

A number of finite element slope stability methods have been proposed and the methods can b,categorized as "enhanced limit methods,' or "direcmethods", as shown n Figure 2.

Wright (1969) compared he factors of safety calculated using the "enhanced limit strength" methorwith factors of safety calculated using Bishop's Simplified method 1952). A slip surface was selecterfor comparative purposes hat had a factor ofsafetof 1.0 when using he Bishop's Simplifiedmethod.was concluded hat the factors of safety determinerby the "enhanced limit strength" method (Kulhawf1969)were approximately % higher han hose determined applyrng Bishop's Simplified methodWright et al. (1973), using the "enhanced limirsEength" method, howed hat: l) along one hird orthe slip surface, he local factors of safety are leslthan the overall factor of safety, 2) the factors orsafety calculated by the finite element method usinglinear elastic material properties anged rom 0% tr4.5% higher than those calculated using Bishop'sSimplifiedmethod, and 3) the factors of safety cll-culated by the finite element method using non-linear elastic material properties increased witt.Poisson's atio and arc 2%o o 8% higher han thosecalculated sing he Bishop's Simplifiedmethod.

Rcsindiz (1974) ageed with the concept of usingthe finite element method o calculate he stability ola slope; however, disagreed with points No. 2 andNo.3 of the results f Wright et al. (1973) becausethe factor of safety differences were too smallRes6ndiz had developed a finite element method oi

32



Finile Element Slope StabilityMethods

Direct melhods

(finiteelement analysis only)

Enhanced imitmethods(finile element analysis witha limit equilibriumanalysis)

Load ncreaseto failure

Strength ecreaseto ailure

Definitionof Faclor of

Strength

Kulhaury 969

Stress Level

Zenkiewiaet al 1975Streng[h & Stress Level

Adikari nd Commins 1985

. _I (c ' + dta$')Mn- z ra t

Figure 2. Finite element approaches proposed in comprrting the factor ofsafety in a slope stability analysis.

slope stability analysis defined as an "enhanced limitstress-level" ethod" n 1972 Fig. 2). This methodused he maximum principal stress difference of thesoil at failure to define the factor of safety. Analysesmade using non-linear stress versus strain relation-ships ed to factors of safety which in all cases werehigher (i.e., differences as large as 3Ao/o)han con-ventional acton of safety (e.g., Ordinary method orBishop's implifiedmethod).

Zier/rriewicz et al. (1975) also proposed a finiteelement method of analysis o compute he factor ofsafety by using the principal stess difference n thesoil at failure to define the factor of safety. Themethod s an "enhanced limit stress level method"(Fig. 2). Both the Resdndiz 1972) aurrd,ienktewiczet al. (1975) ormulations re classified s "enhancedlimit stress-level" ethods.Naylor (1982) estdblished wo tJpes of finite ele-ment slope stabilitymethods, "direct,'and an "en-hanced imit" method of analysis. The direct methodused a finite element nodal formulation to define theslip surface and he factor of safety directly from theanalysis. The proposed "direct" slope stabilitymethod defined he factor of safety either as the in-creased oad necessary o cause ailure, or as the re-ciprocal of the reduction in the strength propertiesrequired n order o achieve ailure. These methods

have also been studied by Martins et al. (l9gl) andTan and Donald 1985).

The "enhanced limit" slope stability methods arebased on

stresses alculated using a finite elementanalysis nd combined with a limit equilibriumnoeofanalysis along a prescribed lip surface, o d"fin"the actor ofsafety. The prescribed lip surface s theone defined by the lowest factor of safety and isfound using a trial and error procedure. The stressesalong the slip surface are computed using a finiteelement analysis and can either be usea in ""strength"method or a "stress-level"method. Fariasand Naylor (1996) stated hat when using the "di-rect" finite element method t is, ,'not easv o obtaina safety factor accttrate to within the'confidenceIimits achievable by linit equilibrium methods". Theauthors noted hat: l) a fine mesh s required,Z) acode capable of giving reliable results with ihe

Mohr Coulomb elasto-plastic model for toadingstates close tofailure is needed, and 3) il is tuuallynecessary o carry out a set of analyses with c, andtanQ'progressively reduced by a factor which willbecome he safety actor when ailure is eventuallyreached.'Enhanced imit" methods equire only onefinite element analysis o calculate actors ofsafetyfor a slope with various combinations of c, andtan6'.

33



Adilori and Cumrnins 1985) produccd a finiteelement method hat combine he "sfength" and thenstress-level" methods as defined by Kulhawy(1969) and ZierJriewiczet al. (1975), rEspectively(Fig. 2). The Adikari and Cummins 1985) method

prodtrccd facton of safety that were between thevalues obtained when applyng the Kulhawy (1969)and the Ziern/ri*vicz et al. (1975) methods. It wasnoted hat or near-failure onditions i.e., as definedby Bishop's Sirnplified method, 1955), he value ofthe factor of safety calculated by the Adikari andCummins 1985) method approached .0, while thevalue ofthe factor ofsafety calculated y the Zien-kiewicz et al. (1975) method remained high. Thefactor of safety by the Kulhawy (1969) method alsoapproached nity with the factor ofsafety being de-pendent on the percentage f the sftngth mobiliza-tion in the component materials. The main differ-ence n results appean related o using the stresseson the principal plane (Zienkiewicz et al. 1975)

rather than on the plane. By definition, failure doesnot occur on the plane of principal stress and there-forc, the Zier*iewicz et al. (1975) method or anystressJevel method) s computing a factor of safetythat must be higher than the factors of safety pro-duced by a "strength" method.

Duncan et al. (1996) provided a summary of thelimit equilibrium and finite element methods hathave been proposed or slope tability analyses.

3 SUGCESTED STTIDYFOR COMPARISONBETWEEN TTIE FINITE ELEMENT AND THELMIT EQUILIBRJI.'MMETHODS OF SLOPESTABILITY ANALYSIS

The frnite element slope stability method proposedin this paper s of the "enhancedlimit strength" lpe(Scoular, 1997).The finite element method uses heKulhawy (1969) definition for the factor of safetycombined with a finite element tress nalysis f theslope. Stress nalyses erc done using Poisson's a-tios equal o 0.33 and 0.48. For each stress nalysis,the cohesion nd he angle of internal riction of thesoil were altered as the stability of the slope wascomputed The selected alues or cohesion, c', were10, 20 and 40 kPa, and or the angle of intemal ric-tion, Q',were 10, 20 and 30 degrees.

The finite clement slope stability method pro.duces n overall actor ofsafety hat s an expressionof the sability of the slope based on the calculatedstresses within the slope. Slope stability problemssolved using he finite element method have wo im-portant distinctions from limit equilibrium methods.First, the finite element slope stability equation sdeterminate; herefore, no further assumptions arcrcguircd to complete he calculations. Second, hefactor of safety equation s linear, because he nor-

mal stress at the base'of a slice is known. On theother hand, imit equilibrium methods, sarting withBishop'sSimplifiedmethod 1955), have.uscd nestimated actor of safety when computing the nor-mal force at the base ofa slice. The final factor of

safety s found through an ilerative proc€ss. The fi-nite element method actor of safety s defined usingthe normal and shear stresses omputed using a fi-nite element nalysis.

Finite element numerical stress analyses havebeen available for many years. The finite elementmethod however, has not bccome popular for slopestability studies due to intense computational rc-quirements nd difticulties in assessing he stressversus traincharacteristics fthe soils. n addition,inexpensive nd easy o use imit equilibrium meth-ods have provided factors of safety that appear orepr€sent ailure conditions in the field in mostsituations. Microcomputers now have sufficientcomputational capacity to perform combined stressand imit equilibrium analyses. s a result, t is an-ticipated hat the latter procedure will become morecornmon n engineering ractice.

3.1 Procedure usedfor thefinite element analysisThe enhanced imit (strength) inite element methodproposed by Kulhawy (1969) was selected as themost appropriatemethod or slope stabilityanalysis.The finite element stress-deformation oftware.Sigma/W a proprietary product of Geo-Slope nter-national Ltd., Calgary, Albena, Canada), was modi-fied to utilizes a search algorithm in order to assignand transfer calculated finite element stresses o adesigned point on the slip surface (Bathe, 1982;Krahn et al., 1996). The calculated inite elementcalculated tresses re used o calculate he normaland shear stresses on the slip surface. The latterstrcsses re used o calculate local facton of safetyat the center ofthe base ofeach slice as well as heoverall actor ofsafety for the entire slip surface.

3.2 DeJinition offacor of safety

The overall actor ofsafety is defined n accordancewith the finite element slope stability method dc-scribed y Kulhawy (1969), and expressed s he ra-tio of the sum of the incremental resisting shearstrcngths, S7, o the sum of the mobilized shearforces,Sn', along he slip surface.

- ES .Frrr"r F. (l )

L " f r

The resisting orce for each slice is calculated nterms of the shear strength, a, at the center of a slicemultiplied by the base length of the slice, f. T}rcavailable resisting shear strength for a satu-rated/rursaturated oil (Fredlund and Rahardjo, 1993)can be written as:

34



t s

tII1 - 1 , 1 ) Nr (1 , 1 )

GlobalCoordinates x,y) x - Coordinate

Figurc3. Definitionofthe globaland ocal coordinates or a rectangular inite element.

troo

Eoo;

g, =sp={c'+(oo -u" )tanl'+(u" -u*)tar/b 1p(2)

The mobilized shear orce, S,,, for each slice iscalculated as the mobilized shear stress, 2., at thecenter of a slice multiplied by the base englh, B.S^ = t^f (3)

The ocal factor ofsafety is defined as he ratio ofthe resisting shear orce, ,S7, t a point along the slipsurface divided by the mobilized shear orce, .Sr, atthe same point,

S , r Bt* =L =;7

The resisting shear orce, .S7, and the mobilizedshear force, S,,, tr€ both calculated using thestresses omputed n the finite element analysis. Thenormal stess, on, frd shear stress, rv1; @r b€ 'im-ported" as known values to the limit equilibriumanalysis and the definition of both the overall andlocal actor ofsafety equations are inear.

3.3 Element dentiJication orresponding o thebase of a sliceEach element must be checked o confirm that thcccnter of the base of the slice is located within theelcment under consideration. Then the stresses al-culated by the finite element analysis can be "im-ported" nto the stabilityanalysis. Oncc he elementcmbracing he center of a portion along the slip sur-tace ts located, stress values rom the Gauss pointsof the element can be transferrcd o the nodes bf theclcmentand consequently o the center ofthe base.tne procedure s in accordance with the mcthod de-scribcd y Bathe 19g2).

A common set of coordinates s used o identi$the center of a slice along a slip surface with respectto the surrounding inite element. The global coordi-nates or the center of the base are calculated n or-der to determine the location of the base centerwithin the slope, and to determine which element sassociated ith the center ofthe base. The ocal co-ordinates fthe center ofthe base re hen calculatcdwithin the element that encompasses he center ofthe base Fig. 3).

The global coordinates or the center ofthe baseof a slice are related o the global coordinates f thefinite element nodal poins through use of the shape

fimctions.x = < N > { X } ( 5 )

Where x = global x coordinates or the centerofthe base fa slice;y = globaly coordinates or thecenter ofthe base ofa slice; X} = global x coordi-nates or the element nodal points; {Y}= global ycoordinates or the element nodal points; and <y'{>:matrix of shape irnctions.

The shape unctions <ly'> are defined n terms of .the local coordinates e s). Since the global coordi-nates for the center of the base of a slice and thenodes are known, the local coordinates can be ob-tained by solving Equations (5) and (6), simultane-ously. The shape inctions for a rectangular initeelement with fou nodes are as ollows (Bathe 1982):

N ,= 11 t + r 1 t + s ;

N , =i ( l

- r X , + s )

(4)

(7 )

Local Coordinetes (r,s)

FiniteElement

x - Coordinate

35

(8)



wher€ and s = local coordinates within the element.The ocal coordinates ary between l and +l (Fig.

3). A knowledge f the ocal coordinates s crucial oidentifuing the element overlapping he center of thebase of a slice. By definition, an element surroundsthe center of the base of a slice f the followine con-ditionsare met:For a triangular element,

( 0 < r > l ) a n d ( 0 < s > l ) ( l )For a rectangular element,

( - l < r ) l ) a n d ( - l < s > l ) ( 1 2 )

The center of the base s outside an element f thelocal coordinates are not within the above specifiedranges. The search continues until an element isfound that satisfies hese conditions.

3.4 Transfer of element stresses o the center of thebase ofa slice

Calculated shesses are stored within the computersoftware relative to the Gauss points of an element.SEesses must be transferred rom the Gauss points ofan element o the nodes of the element and then tothe center fthe base ofa slice.

The local coordinates of a point within a finiteelement are defined in relationship o the global co-

ordinates at the nodes of the element by using theshape unctions, as per Equations 5) and (6):

x = 1 N r N 2 N 3 N 4

(14)

where r and y = global coordinate positions withinthe element hat are known as he center of base of aslice (Fig. 4); X and y: global coordinate t the ele-ment nodes; and f7, N2, N and N4 = the shape irnc-tions defined n Equations to 10.

The stresses rom a finite element analysis arestored at the Gauss points. The shape irnctions can

be used o describe he change of a variable within anelement n terms of nodal values. The finite elementslope stability calculations equire hat stresses t thecenter of the base for each slice be within an ele-ment. This is achieved using the following proce-dure:

{ o } , = < N > { F } ( 1 5 )

wherc aln = stresses at the element node; <|y'> =matrix of the shape unctions; md {F} = stress aluesat the Gauss points.

The local Gauss point integration coordinates arc(0.577,0.577), however, when he local Gauss ointintegration coordinates are projected outward to theelement nodes, the local coordinates bccome(1.7320, 1.7320) Gig. 5). This projection s carriedout for each element and the values for the stresscsfrom each contributing element are averaged at eachnode. Accordingly, the values of or, o, and to c,anbe computed at each node of the finite element mesh.The nodal stresses, r, ay, and lryy, f an element aretransferred o the center ofthe base ofa slice alongthe slip surface.

{o}=< N >{o}, (16)

where /ol = stresses at the center of the base of aslice.

The stresses, ,, ay, and rry, can now be computedat he center ofthe base or each slice.

3.5 Thenormal and shear stresses t the center ofaslice

Once the stresses, o\, oy rnd r- are known at thecenter of the base or each slice, the normal stress,on, and he mobilized shear stess, r, , can be cd-culated using Equations 17) and (18), respectively(Higdon etaL 1976):

6 r * o u o r ' o u c o s 2 0On =------+trystnLfl

(r7)o- -o.. sin20

rm = tocos2Q (18)

where o,= total stress n the x-direction at the centcrofthe base;o" = total stress n the y-direction at thecenter ofthe base; r,n = shear stress n the x- and y-

direction at the center of the base; and d = anglemeasured rom the positive x-axis to the line of ap.plicationofthe nonnal stress.

The above steps provide the necessary nformationrequired o calculate he stability of a slope using thefinite element stresses. he calculated values for thenormal stress, o,r, and the mobilized shear stress, -,at the center of the base of a slice are entered ntoEquations 2) and (3) to give the resisting shear orce

N ,= i ( l- r [ l - s )

N .i ( + r X - s )

(e)

(10)

I x , )t ; l- )I xt It - ll x t )

l Y, )t - tl Y, l{ - }t Y , lt - l

lY.

(13)

36



/ Fictitious slice defined withthe Limit EquilibriumanalYsis

x_____+.

x-Coordinate

Figure . Location ofthe center ofthe base long he slip surfacewithin a particular inite element.

u.73m,1.73n1 t1.7320,1.7320)

tr.#o.sro o.s2,6.5r4

(-0.srt, .574 p.1n, 4.iln

(-1.7320, 1.73n) (1.7320,r.7320)

Figure .Gauss oint rojectionso henodes fa finiteclcnent.

(strength) and the mobilized shear force (actuatingshear), espectively.

The local factor ofsafety is computed as the ratioof the resisting shear force to the mobilized shearforce. The overall actor ofsafety is the sum oftheshear orce resistance alues divided by the sum ofthe actuating shear orces along he slip surface.

t

IDo

ooI

O EementNodes

+ Eement C'auss Pcints

4 PARAMETRIC STI.JDIESON A SIMPLE 2:ISLOPE

A slope at 2 horizontal o I vertical is analyzed or 4conditions Scoular, 1997). The first case s a free-standing slope with zero pore-water pressures andthe slope s referred o as a dry slope @ig. 6). Thesecond case is a free-standing slope with a pie-zometric line at thrce quaders of the slope height,and he slope s referred o as a wet slope Fig. 6).

(x, y) known

(r, s) unknown

- Center f the base of a slice x, y)

x - Cmrdinate

37



Piezometric LineI

t l

Eoo

EooI

u 4u60 EO 1OO

x - Coordinate (m)

Figure6. Serected :l free-standing lopcwith a piczomctic rineexiting at the oe ofthe srope.

Pr titd case s a slope partially submerged n wa-r-erTm zero pore_water pressures n the slope (re_{e1-ed o as dry) (Fig. D. The fourth "*" ir'" o*_ltally submerged lopewith a piezometric ine at onehalt of the.slope height (refened o as wet) (Fig.7).lhe partiallysubmerged lope s covercd with w1rcrto one halfofthe slope height, providingsupport orme stope and tncreasing he factors of safety. Thecohesion f the soil was varied rom l0 to 4O kp"and he angle of internal friction was varied

from l0to 30 degrees or each lope 1pe.

4.1 Limit equilibrium analysisThe limit equilibrium analyses are performed using9" F :tt_Limit Equilibrium methoa (CG),(Fredlund & kahn 1977) which provides " ,orn_bined moment and force equilibrium solution. Lempirical finite element interslice force function,911e9 on an independent stess analysis (Fan et al.1986) was ued. The General Limit Equilibriumpethod along with a finite element ntersiice forcefimction provides a method of comparison betureenthe finite element based analysis nd th" limit equi_libriumanalysis.

4.2 Finite element stress nalysisThe finite element shess analysis was performed byswitching-on" gravity for the fr"e-rtaodine ,LD,and for the partially submerged lope. The toaa Lfthe water and the lateral support t provides to the

sl.ope s simulated.by oint oads equal o the weightot.water on the slope. The analyses re performldusrng potsson's ratios of 0.33 and 0.4g, and aIo*g't modulus equal o 20,000 and ZOO,TiOOpa.lne results showed that the stesses change with a"l"rg-g poisson's ratio, but ar" "orrit*t fo,changes n the Young's modulus. his observation sconsrstent ith the observations f Matos 19g2).

s_RESI.JLTS OF TI{E FIMTE ELEMENT SLOPESTABILITYMETHOD

Tl" l*ql factors of safety differs along the overallslip surface (Fig. 8). Ioial factors oi-r"f"r" r"r"*Tpl1t9d for a2:l (dry) slope with a cohesioi quafro 4u t(pa an<lan angle of intemal friction equai to30 degrees. While thi tocal facton-oi;;firafong the slip surface, the overall n rit.- lf"rn"*facton of safeg fall within the .-g" "i t " iir,ritequilibrium factors of safety. The ?itr";;"-;_tween the local factors of safety for poisson;s Ltio,of 0.33 and 0.48, calculated using the fi"il;t";;l1e9 f.*n-p* inFigure.8. " r""toroir"f"tyTTputed by rhe limit equilibrium method and thefinite element."Fd appear o U"u.ry,i.ifrr. fi"resuts appear o be within-_the imits of unccrtaintyassociatcd with slope stability calculations. lc i_nite elemcnt method incorporates he stress-;traincharacteristics f the soil when computins th;;h;strength and actuating shear orce oitne-rol io Gcalculation ofthe factor ofsafety (Fig. 9).

38



Crest

\ l\-

---Water l

I

I roe

60

x-

Coordinatem)

Figure 7. Selected 2: I partially submerged slope with a horizontal piezometric line at mid-slope.

80

EoG

E 4 0oo

I

1(x)0

The factor of safety results computed using thefinite element method (i.e., F3 corresponding o aPoisson's atio of 0.33, F4 conesponding o a Pois-son's atio of 0.48) are compared o the factors ofsafety computed sing he limit equilibriummethod(GLE) and are shown n Tables and 2. To assessthe variations n the factor of safety by each methodofanalysis, he results are grouped according o co-hesion and angle of intemal friction. The factors ofsafety grouped ccording o cohesion, ', are plotted

versus he stabilitynumber,

(7lftan{')/cf,(Janbu'

1954). The factors of safety grouped according othe angle of intemal friction, /', are plotted versus

the stability coefficient, (c/7II) (Taylor, 1937),where 7 s the unit weight of the soil, I/ is the heightof the slope, 'is the angle of intemal riction, and c'is the cohesion.

The factors of safety are grouped according o thesoil parameters nd plotted versus he stability num-ber and the stability coefficient. The greatest differ-ence n factors ofsafety is noticed at high angles ofintemal friction, at low values of cohesion and at themaximum alues of Poisson's atio.

The factors of safety for the (dry) free-standingslope, when grouped according to cohesion andplotted venus the stability number (Fig. l0) show a

S loo

ft zsoh

$ zoo

.EosoO

E 10 0E

F s ooo

X o20 30 40 50

x+oordinate m)

Figure 8. Prescntation fthe local and global faaors ofsafety for a 2:l dry slope.

PoissonRatio,P=0.33

39



Table l. 2:l fiee-standing lope

Soilc 'kPa

20 l0l0 2040 l020 20l0 3040 2020 30

L t 3 l1.260t.3701 .6 t1.794t.892

o.677o.782

r.021t.102I . J J )

WetF3

o.6340.7450.930

t.0771.260

0.6470.7s50.953

Parameters0'degree

GLEFinite

element

DryF3tt= 0.33

F4tt=0.48 l inire p=0.33 p=0.48

element

0.882 0.867 o.8741 .1251.2301.3521.6391.7651.884

l . 5 l1.239

t.6961 .7751 .918

1.239 0.9951.368 l n?r368 0.969 0:988

>', O

rA

oqr

x-coordinate

Figurc9. Shear trength ndshear otce or a 2:l dry slope calculated sing the inireelement method.

Figure 10. Facors ofsafcty versrs stability nurnbcr for a 2:l dry slope as a fimction ofcohesion.

2. 5

2. O.o3 1 . 5o

t r .o6&

0. 5

0. 0

Ical F, 1P= 6.33;lobalFacton ofSafcy

Bi$op 2.3@Janbu 2.173

GLE (F.8. firnctiar) 2.3564 (P =0.33) 23apF, (r = O.+e; 2.339frinary 2226

,mbuMelho4Ft=2.173

+ F I ( G L E )+ Fr (F = 0.33)

- Fs (P =0.48)

40



Ta,ble . 2:l partially submerged lope

SoilckPa

Parameters Dry6' GLEdegree Finiteelement

firnction

WetF3 F4 GLEp= 0.33 p= 0.48 Finite

elementfimction

F3 F4p= 0.33 p= 0.48

l0 t0 0.E4520 l0 1.t49l0 20 1.344

0.843 0.827 0.6491.1 5 1.085 0.8861.425 1.422 1.050

0.635 0.64r0.874 0.8801.046 1.06Et.3t4 1.3431.296 1.316t.505 r.5301.774 t.7951.763 1.786

20 20 1.618 1.586 1.s75 l .3lE40 l0 t.721l0 30 1.86540 20 2.29720 30 2.337

1.722 1.691 I at',

2.081 n.s.&* 1.4822.3E5 2.368 1.E002.268 2.204 1.783

qO 30 I.OOA 2.970 2.899 2.303 2.260 2.274rn.s.a.:no solution chieved

0.00 0.02 0.04 0.06 0.08 0.10 0.12StabilityCoeficent, d( yH)l

t;igure I l. Factor ofsafety versus stability coefficient for a 2:l dry slope as firnaion ofangle ofintemat friction.

StabilityNumber, ( yHtang')/d]ligurc 12. Factor ofsafety versus stability number as a imction ofcohesion for a2'l slope with the piczomeric line at % ofthe.lop hcight.

2.5

) 1 . 5(,(oa

: 1 . 0oEoLL

0. 5

0. 0

2.0

1.6

b* 1 2anoE o.a6

l

0.4

25050

41



2. 0

1 .6

.)og 1 . 2o

€ o.aolr

0.00

slight divergence n the factors of safety when thecohesion approaches 0 kPa and the angle of inter-nal friction approach 0 degrees.

The factors of safety by the finite elementmethod, with a high Poisson's ratio, is greater hanthe General Limit Equilibrium solution. The slightdivergence s evident when he factors ofsafety aregrouped according o the angle of intemal frictionand plotted versus he stability coefficient Fig. I l).It is also evident that at high values of cohesion,(i.e., c'equal to 40 kPa), The factors of safety com-puted when using the General Limit Equilibriummethod are greater than those from the finite ele-ment methods with either Poisson's atio value.

The factors of safety for the (wet) free-standingslope with a piezometric ine at three quarters of theslope height, are grotrped according o the cohesionand plotted versus he stability number (Fig. l2).The results show a slight divergence bet*,een he fi-

nite element actors of safety and the General LimitEquilibrium facton of safety when the cohesion s40 ud20 kPa. The difference between he factors ofsafety by both methods s constant at all values ofcohesion until the angle of intemal friction becomesggual o_30 degrees and cohesion becomes equal tol0kPa(Fig. 3) .

The grouping ofthe factors ofsafety accordins othe angle of intemal friction, ploned venus the ta-bility cocfficient (Fig. l5), shows the sarne pattemas or the (dry) free-standing lope (Fig.l0).

The dif-ferences n the results ar€ more pronounced as thecohesion ecome ess han 0 kPa.The factors of safety for the partiallv submersedslope with a piezometric ine at one haliof the slipeh.eightwere grouped by cohesion and plotted vetsusthe stability number (Fig. 16). The results showclose agreement etween he General Limit Equilibrium method and the finifs sts6sal mettrod. The

0.02 o. ' t2. 1 0Stabitity Coefficent, Ic' (rH ll

Figure 13. Factor of safety vems stability cocfEcient as a filrction of the angle of intemal friction for a 2:l slope with the pie-zometric line at % ofthe slope height.

0 5 1 0 1 5 2 0 2 5

Stability umbcr, (tH an0,)/c,l

Figure 14. Factor of safety ctsus tabilitynumbcr sa fimctionof cohesion or a 2: dry stope , submerged ith water.

3. 5

3. 0

b 2-5o3 2.0o

E r.soGq

i . o

0. 5

0. 0

42



.> 2.5

.g3 2. 0ob i .5tto* 1 . 0

2.5

2.0

8 r r6 ' v

U)

t 1 .06

L

0.00 0.04 0.06 0.08StabifltyCoefficent, e 1H )l

Figure I 5. Factor of safety versus stability coeffcient as a fimction of internal friction for a 2: I dry slope % submerged n water.

Stability Number, I lHtan{')/c'l

Figure 16. Factor on safety versus stability numb€r as a firnction ofcohesion for a 2:t slope halfsubmerged with a horizontal pie-zometric ine.

0 .12.10

0.5

same pattem of divergence s evident as was shownfor the dry soil slope which is partially submerged(Fig. 1a). However, he divergence s not quite asextensive. he same conrments apply to the factor ofsafety versus the stability coefficient as shown inFigure 7.

Plotting he factors of safety for the various slopeconditions, i.e., dry free-standing, et free-standingand dry partially submerged), versus stabilify num-ber on Figure 18, shows he ranking of slopes byfactors of safety. The factor of safety can be esti-mated or a slope hat is similar to one of these casesby calculating he stability number and selecting heappropriate alue of cohesion and angle of intemalfriction.Both the General Limit Equilibrium method and thefinite element method of slope stability producefactors of safety hat are n close agreement. The ad-vantage of the finite element method is that thestress-stain haracteristics fthe soil are used o de-

termine he stress state n the slope. f the limit equi-librium and finite element factors of safety aresimilar for a simple slope than results rom the twomethods can be interpreted n similar manners. Thisshrdy hen sets he stage or using the finite elementmethod for situations where the limit equilibriummethods s known to not yield satisfactory results.The finite element method also produces graphs ofthe local factors of safety hat can be combined withthe shear strenglh-actuating hear orce plots to helpexplain he best support mechanism or the slope.

The close agreement etween he factors of safetywhen using the limit equilibrium method or the fi-nite element method, has historically favored he useof limit equilibrium methods. Examination of thecritical slip surfaces eveals hat while the factors ofsafety values are close, the location of the criticalslip surfaces may be different.

+ FI(GLE)+ Fr (P =0 '33)- F (p = 0.48)

F3 GLE)Fr (P = 0'3 l)Fc F = 0.48)

43



io- q 1 .5

o

t r . o6

lt

0 .5

0 .00.04.0 0 0.08 0 . 1 0 o . 1 2

3.0

2.5

2. 0

1 . 5

1 .0

goU)oo()o

lJ-

Stability Coefficent , Ic.t( rH )l

Figure 17' Factor ofsafety versus sability coefficient as a fimction ofthc angle ofintemal friction for a 2:l slope halfsubmerge.with a horizontal piezometric line.

StabilityNumber, [( yH tang,ltc,lFigure 18' Factor of safety vcrsrs sability nunber as a ftrndion of cohesion fot a 2:l slope, evaluated for dry, piezometic andubmerged conditions.

6 ANALYSIS FOR TI{E LOCATION OF TTIECRITICAL SLIP SI.JRFACE

The ocation of the critical circle changes depend.ingon the sitrution Fiog qotfr"{. The biggesrchangein location of critical slip surface was-experiencJdfor the (wet) free-standing sloae (Figs.

19 and 20)and he (wet) supported lope Figs. 2l and,22\.In general, he finite element method slip surfaces

go deeper han the limit equilibrium slip surfaces orthe (wet) free-standing slope. The partially sub.merged slopes show that the limit equilibrium slipsurfaces go deeper han the finite element methodslip surfaces. For the free- standing slope, he frniteelement method with a Poisson's atio equal o 0.4g

showed he deepest slip surface. For the partialll.ylmerged slope, he finite element method with aPoisson's atio equal o 0.4g, showed .o*ia"r"Utyshallower slip surface.

7 CONCLUSION

The finite element method of slope stabiliw is a vi-able method of analysis hat s now availabie or en_gingenng .practice. The use of the finite "lem.ntmethod yelds morc detailed information on tt.stress trte n the soil than s available rom .onln"n-tional limit equilibrium methods. This informationcan assists.engineers n the design of slopes andslope etaining structures.

--r--- F" (GLE)+ Fr (t t = 0.33)-- Fr (F = 0.4E)

3/4 Piezometnc, C = 10 kpa

4



Cohesion = 40 kPa,Method

GLE (EE. Function)F" (p = 0.3€t)Fg (p = 0.48)

0'= 30", Piezometric litr- 3t4 of the ralay up the slopeX y R FactorofSafety

58.50 56.00 37.88 ,t.74157.50 49.50 34.69 1.62757.50 53.00 37.83 1.661Fs (p = 0.33)

Fs (p = 0.48)

6-? 50.gP? ? 4 0di 3 0

E-? 40o6.gp6 3 0oo

I

10 20 30 40 50 60 70 80 90 loo .t10

x - Coordinate (m)Figure 19. Location of the critical slip surface for a slope with a piczometric line where the soil properties are c, = 40Wa and ('=300.

1 0 2 0 3 0 4 0 5 0 6 0 70 8 0 9 0 1 o o 11 0x - Coordinate (m)

Figure 20' Location of the critical slip surface for a slope with a piezometric line where the factors of safety are closest to I .0.

The value of Poisson's atio can affect he calcu- the finite element method o slope stability prob,lation of the factor of safety as well as the location lems, a u"tt"t *a"ot"nding is required regardingof the slip surface. with an ncreasing application f oe etrect oiFoi.*nt otio and he overall deforma-tion model on the stabilityof slopes.

45

Coh.esion 40 kPa, 0,= 3Oo, iezometric line 3/4 of lhe way up lhe slopeMethod x y R Fjctorofs;feiyGLE (F.E. Function) 63.50 S9.OO 39.56 1.102F, (p = 0.33) 63.00 S9.OO 41.54 1.076F. (p = 0.48) 61.50 S9.SO 42.gg 1.100

Fs (F = 0.48)Fs rr= 0.33)

GLE F.E. unction)



Cohesion = 40 kPa, 0'= 30o, half submeqed slopeMethod /\ Y R FaciorofSafety

GLE (F.E. Function) 58.fi) 58.50 40.20 2.303F, (p = 0.33) 52.50 50.50 31.76 2.259F. (p = 0.48) 51.50 51.50

30.94 2.273

EoGcEoo()I

10 20 30 40 50 60 70 80 90 100 110x - Coordinate (m)

Figure 2l ' Location ofthe critical slip surface or a halfsubmerged slope where he soit properties are c,= 40kpa and, O'= 31,.

. 10 20 30 40 50 60 70 80 90 100 11x -Coord ina te m)

Figute 22. Location of the critical slip srrface or a submerged lopewhere he actors of safety are closesr o 1.0.

The finite element stress.analy-sis rovides input boundary conditions are being used and that a rca-information for the calculation of the stability of a sonable stressdeformation riodel is beine used.slope. Further research must be undertaken on the With this arisurianc€, soil structures can be dner de-stress analysis in order to ensure that the proper sigpedtoaccountforavari€tyofstressconditions.

Eor0.cEooo

I

Cohesion = 40 kPa, 0,= 30o,Method X

GLE (F.E . Func t ion) 62 .50

half submerged stop-

F , ( p = 0 .33) 54 .00F . ( p = 0 .48) 53 .00

Y R Factor of Safety6 3 . 5 0 4 4 . 11 1 . 0 5 05 1 . 5 0 3 1 . 9 7 1 . 0 4 656 .50 34 .69- 1 .068

G L E ( F. E . u n c t i o n )

Fs (tr = 0.48)

F r ( P = 0 . 3 3 )

46



ACKNOWLEDGEMENTS

Thc authors want to acknowledge he initial dis-cussions egarding he potential for using a finiteclemcnt slope stability method rhat were held with

hofl Wong ki Sin of Nanyang Technological Uni-venity, Singapore. These discussions ormed he ba-sis for the study of this topic. The assistance f Dr.Fangshcng Shuai, Ms. Noshin Tadenadeh and Ms.Brigine Boldt-Lrppin n assembling his manuscriptis also acknowledged. he authors are also gnatefulto Gco-Slope ntemational, Calgary, for the modifi-cations made to their software in order thst thisstudycould be readilyperformed.

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Using Limit Equilibrium Concepts in Finite Element Slope Stability Analysis

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