Slope stability analysis by ®nite elements

Slope stability analysis by ®nite elements

D. V. GRIFFITHS� and P. A. LANE{

The majority of slope stability analyses per-formed in practice still use traditional limitequilibrium approaches involving methods ofslices that have remained essentially unchangedfor decades. This was not the outcome envisagedwhen Whitman & Bailey (1967) set criteria forthe then emerging methods to become readilyaccessible to all engineers. The ®nite elementmethod represents a powerful alternative ap-proach for slope stability analysis which is accu-rate, versatile and requires fewer a prioriassumptions, especially, regarding the failuremechanism. Slope failure in the ®nite elementmodel occurs `naturally' through the zones inwhich the shear strength of the soil is insuf®-cient to resist the shear stresses. The paperdescribes several examples of ®nite elementslope stability analysis with comparison againstother solution methods, including the in¯uenceof a free surface on slope and dam stability.Graphical output is included to illustrate defor-mations and mechanisms of failure. It is arguedthat the ®nite element method of slope stabilityanalysis is a more powerful alternative to tradi-tional limit equilibrium methods and its wide-spread use should now be standard ingeotechnical practice.

KEYWORDS: dams; limit equilibrium methods; num-erical modelling; plasticity; slopes.

En grande majoriteÂ, les analyses de stabiliteÂ depente meneÂes dans la pratique continuent aÁutiliser les meÂthodes traditionnelles d'eÂquilibrelimite et des systeÁmes de tranches qui n'ontpratiquement pas changeÂ depuis des dizainesd'anneÂes. Ce n'eÂtait pas le reÂsultat envisageÂquand Whitman et Bailey (1967) ont eÂtabli descriteÁres pour que ces meÂthodes alors eÂmer-geantes puissent devenir facilement accessibles aÁtous les ingeÂnieurs. La meÂthode d'eÂleÂments ®nisqui repreÂsente une alternative puissante pour lesanalyses de stabiliteÂ de pente, est exacte, poly-valente et demande moins d'hypotheÁses àpriori', surtout en ce qui concerne les meÂca-nismes de rupture. La rupture de pente dans lemodeÁle aÁ eÂleÂments ®nis se produit `naturelle-ment' aÁ travers des zones dans lesquelles lareÂsistance au cisaillement du sol est insuf®santepour reÂsister aux contraintes tangentielles. CetexposeÂ deÂcrit plusieurs exemples d'analyses destabiliteÂ de pente utilisant les eÂleÂments ®nis eteÂtablit des comparaisons avec d'autres meÂth-odes, comme l'in¯uence d'une surface libre surla stabiliteÂ d'une pente et d'une digue. Nousjoignons une repreÂsentation graphique pour il-lustrer les deÂformations et meÂcanismes de rup-ture. Nous avancËons que la meÂthode d'eÂleÂments®nis pour analyser la stabiliteÂ des pentes consti-tue une alternative plus puissante aux meÂthodestraditionnelles d'eÂquilibre limite et que son utili-sation devrait maintenant devenir une pratiquestandard en geÂotechnique.

INTRODUCTION

Elasto-plastic analysis of geotechnical problemsusing the ®nite element (FE) method has beenwidely accepted in the research arena for manyyears; however, its routine use in geotechnicalpractice for slope stability analysis still remainslimited. The reason for this lack of acceptance isnot entirely clear; however, advocates of FE tech-niques in academe must take some responsibility.

Practising engineers are often sceptical of the needfor such complexity, especially in view of the poorquality of soil property data often available fromroutine site investigations. Although this scepticismis often warranted, there are certain types ofgeotechnical problem for which the FE approachoffers real bene®ts. The challenge for an experi-enced engineer is to know which kind of problemwould bene®t from a FE treatment and whichwould not.

In general, linear problems such as the predic-tion of settlements and deformations, the calcula-tion of ¯ow quantities due to steady seepage or thestudy of transient effects due to consolidation areall highly amenable to solution by ®nite elements.Traditional approaches involving charts, tables or

Grif®ths, D. V. & Lane, P. A. (1999). GeÂotechnique 49, No. 3, 387±403

387

Manuscript received 28 May 1998; revised manuscriptaccepted 8 December 1998. Discussion on this papercloses 3 September 1999; for further details see p. ii.� Colorado School of Mines, Golden.{ UMIST, Manchester.

graphical methods will often be adequate for rou-tine problems but the FE approach may be valuableif awkward geometries or material variations areencountered which are not covered by traditionalchart solutions.

The use of nonlinear analysis in routine geotech-nical practice is harder to justify, because there isusually a signi®cant increase in complexity whichis more likely to require the help of a modellingspecialist. Nonlinear analyses are inherently itera-tive in nature, because the material properties and/or the geometry of the problem are themselves afunction of the `solution'. Objections to nonlinearanalyses on the grounds that they require excessivecomputational power, however, have been largelyovertaken by developments in, and falling costs of,computer hardware. A desktop computer with astandard processor is now capable of performingnonlinear analyses such as those described in thispaper in a reasonable time spanÐminutes ratherthan hours or days.

Slope stability represents an area of geotechni-cal analysis in which a nonlinear FE approachoffers real bene®ts over existing methods. As thispaper will show, slope stability analysis by elasto-plastic ®nite elements is accurate, robust andsimple enough for routine use by practising en-gineers. Perception of the FE method as complexand potentially misleading is unwarranted andignores the real possibility that misleading resultscan be obtained with conventional `slip circle'approaches. The graphical capabilities of FE pro-grams also allow better understanding of themechanisms of failure, simplifying the outputfrom reams of paper to manageable graphs andplots of displacements.

TRADITIONAL METHODS OF SLOPE STABILITY

ANALYSIS

Most textbooks on soil mechanics or geotech-nical engineering will include reference to severalalternative methods of slope stability analysis. In asurvey of equilibrium methods of slope stabilityanalysis reported by Duncan (1996), the character-istics of a large number of methods were sum-marized, including the ordinary method of slices(Fellenius, 1936), Bishop's Modi®ed Method(Bishop, 1955), force equilibrium methods (e.g.Lowe & Kara®ath, 1960), Janbu's generalized pro-cedure of slices (Janbu, 1968), Morgenstern andPrice's method (Morgenstern & Price, 1965) andSpencer's method (Spencer, 1967).

Although there seems to be some consensus thatSpencer's method is one of the most reliable, text-books continue to describe the others in somedetail, and the wide selection of available methodsis at best confusing to the potential user. Forexample, the controversy was recently revisited by

Lambe & Silva (1995), who maintained that theordinary method of slices had an undeservedly badreputation.

A dif®culty with all the equilibrium methods isthat they are based on the assumption that thefailing soil mass can be divided into slices. This inturn necessitates futher assumptions relating to sideforce directions between slices, with consequentimplications for equilibrium. The assumption madeabout the side forces is one of the main character-istics that distinguishes one limit equilibrium meth-od from another, and yet is itself an entirelyarti®cial distinction.

FINITE ELEMENT METHOD FOR SLOPE STABILITY

ANALYSIS

Duncan's review of FE analysis of slopes con-centrated mainly on deformation rather than stabi-lity analysis of slopes; however, attention wasdrawn to some important early papers in whichelasto-plastic soil models were used to assess stabi-lity. Smith & Hobbs (1974) reported results oföu � 0 slopes and obtained reasonable agreementwith Taylor's (1937) charts. Zienkiewicz et al.(1975) considered a c9, ö9 slope and obtained goodagreement with slip circle solutions. Grif®ths(1980) extended this work to show reliable slopestability results over a wide range of soil propertiesand geometries as compared with charts of Bishop& Morgenstern (1960). Subsequent use of the FEmethod in slope stability analysis has added furthercon®dence in the method (e.g. Grif®ths, 1989;Potts et al., 1990; Matsui & San, 1992). Duncanmentions the potential for improved graphical re-sults and reporting utilizing FE, but cautionsagainst arti®cial accuracy being assumed when theinput parameters themselves are so variable.

Wong (1984) gives a useful summary of poten-tial sources of error in the FE modelling of slopestability, although recent results, including thosepresented in this paper, indicate that better accu-racy is now possible.

Advantages of the ®nite element methodThe advantages of a FE approach to slope

stability analysis over traditional limit equilibriummethods can be summarized as follows:

(a) No assumption needs to be made in advanceabout the shape or location of the failuresurface. Failure occurs `naturally' through thezones within the soil mass in which the soilshear strength is unable to sustain the appliedshear stresses.

(b) Since there is no concept of slices in the FEapproach, there is no need for assumptionsabout slice side forces. The FE method

388 GRIFFITHS AND LANE

preserves global equilibrium until `failure' isreached.

(c) If realistic soil compressibility data are avail-able, the FE solutions will give informationabout deformations at working stress levels.

(d) The FE method is able to monitor progressivefailure up to and including overall shearfailure.

Brief description of the ®nite element modelThe programs used in this paper are based

closely on Program 6.2 in the text by Smith &Grif®ths (1998), the main difference being theability to model more general geometries and soilproperty variations, including variable water levelsand pore pressures. Further graphical output cap-abilities have been added. The programs are fortwo-dimensional plane strain analysis of elastic±perfectly plastic soils with a Mohr±Coulomb failurecriterion utilizing eight-node quadrilateral elementswith reduced integration (four Gauss points perelement) in the gravity loads generation, the stiff-ness matrix generation and the stress redistributionphases of the algorithm. The soil is initially as-sumed to be elastic and the model generates normaland shear stresses at all Gauss points within themesh. These stresses are then compared with theMohr±Coulomb failure criterion. If the stresses at aparticular Gauss point lie within the Mohr±Cou-lomb failure envelope, then that location is assumedto remain elastic. If the stresses lie on or outsidethe failure envelope, then that location is assumedto be yielding. Yielding stresses are redistributedthroughout the mesh utilizing the visco-plastic algo-rithm (Perzyna, 1966; Zienkiewicz & Cormeau,1974). Overall shear failure occurs when a suf®-cient number of Gauss points have yielded to allowa mechanism to develop.

The analyses presented in this paper do notattempt to model tension cracks. Although `no-tension' criteria can be incorporated into elasto-plastic FE analyses (e.g. Naylor & Pande, 1981),this additional constraint on stress levels compli-cates the algorithm, and, in addition, there is stillsome debate as to how `tension' should properlybe de®ned. Further research in this area is war-ranted.

Soil modelThe soil model used in this study consists of six

parameters, as shown in Table 1.The dilation angle ø affects the volume change

of the soil during yielding. It is well known thatthe actual volume change exhibited by a soil dur-ing yielding is quite variable. For example, amedium±dense material during shearing mightinitially exhibit some volume decrease (ø, 0),

followed by a dilative phase (ø. 0), leading even-tually to yield under constant volume conditions(ø � 0). Clearly, this type of detailed volumetricmodelling is beyond the scope of the elastic±perfectly plastic models used in this study, where aconstant dilation angle is implied.

The question then arises as to what value of øto use. If ø � ö, then the plasticity ¯ow rule isàssociated' and direct comparisons with theoremsfrom classical plasticity can be made. It is also thecase that when the ¯ow rule is associated, thestress and velocity characteristics coincide, so clo-ser agreement can be expected between failuremechanisms predicted by ®nite elements and criti-cal failure surfaces generated by limit equilibriummethods.

In spite of these potential advantages of usingan associated ¯ow rule, it is also well known thatassociated ¯ow rules with frictional soil modelspredict far greater dilation than is ever observed inreality. This in turn leads to increased failure loadprediction, especially in `con®ned' problems suchas bearing capacity (Grif®ths, 1982). This short-coming has led some of the most successful con-stitutive soil models to incorporate non-associatedplasticity (e.g. Molenkamp, 1981; Hicks & Bough-rarou, 1998).

Slope stability analysis is relatively uncon®ned,so the choice of dilation angle is less important.As the main objective of the current study is theaccurate prediction of slope factors of safety, acompromise value of ø � 0, corresponding to anon-associated ¯ow rule with zero volume changeduring yield, has been used throughout this paper.It will be shown that this value of ø enables themodel to give reliable factors of safety and areasonable indication of the location and shape ofthe potential failure surfaces.

The parameters c9 and ö9 refer to the effectivecohesion and friction angle of the soil. Althougha number of failure criteria have been suggestedfor modelling the strength of soil (e.g. Grif®ths,1990), the Mohr±Coulomb criterion remains theone most widely used in geotechnical practice andhas been used throughout this paper. In terms ofprincipal stresses and assuming a compression±negative sign convention, the criterion can bewritten as follows:

Table 1. Six-parameter soil model

ö9 Friction anglec9 Cohesionø Dilation angleE9 Young's modulusí9 Poisson's ratioã Unit weight

SLOPE STABILITY ANALYSIS BY FINITE ELEMENTS 389

F � ó 91 � ó 932

sinö9ÿ ó 91 ÿ ó 932

ÿ c9 cosö9 (1)

where ó 91 and ó 93 are the major and minor princi-pal effective stresses.

The failure function F can be interpreted asfollows:

F , 0 stresses inside failure envelope (elastic)F � 0 stresses on failure envelope (yielding)F . 0 stresses outside failure envelope (yielding

and must be redistributed)

The elastic parameters E9 and í9 refer toYoung's modulus and Poissons's ratio of the soil. Ifa value of Poisson's ratio is assumed (typicaldrained values lie in the range 0:2 , í9 , 0:3), thevalue of Young's modulus can be related to thecompressibility of the soil as measured in a one-dimensional oedometer (e.g. Lambe & Whitman,1969):

E9 � (1� í9)(1ÿ 2í9)

mv(1ÿ í9)(2)

where mv is the coef®cient of volume compressi-bility.

Although the actual values given to the elasticparameters have a profound in¯uence on thecomputed deformations prior to failure, they havelittle in¯uence on the predicted factor of safetyin slope stability analysis. Thus, in the absenceof meaningful data for E9 and í9, they can begiven nominal values (e.g. E9 � 105 kN=m2 andí9 � 0:3).

The total unit weight ã assigned to the soil isproportional to the nodal self-weight loads gener-ated by gravity.

In summary, the most important parameters in aFE slope stability analysis are the same as theywould be in a traditional approach, namely, thetotal unit weight ã, the shear strength parametersc9 and ö9, and the geometry of the problem.

Gravity loadingThe forces generated by the self-weight of the

soil are computed using a standard gravity `turn-on' procedure involving integrals over each ele-ment of the form:

p(e) � ã�

V e

NT d(vol) (3)

where N values are the shape functions of theelement and the superscript e refers to the elementnumber. This integral evaluates the area of eachelement, multiplies by the total unit weight of thesoil and distributes the net vertical force consis-

tently to all the nodes. These element forces areassembled into a global gravity force vector that isapplied to the FE mesh in order to generate theinitial stress state of the problem.

The present work applies gravity in a singleincrement to an initially stress-free slope. Othershave shown that under elastic conditions, sequentialloading in the form of incremental gravity applica-tion or embanking affects deformations but notstresses (e.g. Clough & Woodward, 1967). In non-linear analyses, it is recognized that the stress pathsfollowed due to sequential excavation may be quitedifferent to those followed under a gravity `turn-on' procedure; however, the factor of safety ap-pears unaffected when using simple elasto-plasticmodels (e.g. Borja et al. 1989; Smith & Grif®ths,1998).

In comparing our results with limit equilibriumsolutions which generally take no account of load-ing sequence, our experience has shown that thepredicted factor of safety is insensitive to the formof gravity application when using elastic±perfectlyplastic Mohr±Coulomb models. An example of thisinsensitivity is demonstrated later in the paper.

The factor of safety may be sensitive to load-ing sequence when implementing more complexconstitutive laws, such as those that attempt toreproduce volumetric changes accurately in anundrained or partially drained environment. Forexample, Hicks & Wong (1988) showed that theeffective stress path could have a big in¯uenceon the factor of safety of an undrained slope.

Determination of the factor of safetyThe factor of safety (FOS) of a soil slope is

de®ned here as the number by which the originalshear strength parameters must be divided in orderto bring the slope to the point of failure. (Thisde®nition of the factor of safety is exactly thesame as that used in traditional limit equilibriummethods, namely the ratio of restoring to drivingmoments.) The factored shear strength parametersc9f and ö9f , are therefore given by:

c9f � c9=FOS (4)

ö9f � arctantanö9

FOS

� �(5)

This method has been referred to as the `shearstrength reduction technique' (e.g. Matsui & San,1992) and allows for the interesting option ofapplying different factors of safety to the c9 andtanö9 terms. In this paper, however, the samefactor is always applied to both terms. To ®nd the`true' FOS, it is necessary to initiate a systematicsearch for the value of FOS that will just cause theslope to fail. This is achieved by the program


solving the problem repeatedly using a sequence ofuser-speci®ed FOS values.

De®nition of failureThere are several possible de®nitions of failure,

e.g. some test of bulging of the slope pro®le(Snitbhan & Chen, 1976), limiting of the shearstresses on the potential failure surface (Duncan &Dunlop, 1969) or non-convergence of the solution(Zienkiewicz & Taylor, 1989). These are discussedin Abramson et al. (1995) from the original paperby Wong (1984) but without resolution. In theexamples studied here, the non-convergence optionis taken as being a suitable indicator of failure.

When the algorithm cannot converge within auser-speci®ed maximum number of iterations, theimplication is that no stress distribution can befound that is simultaneously able to satisfy boththe Mohr±Coulomb failure criterion and globalequilibrium. If the algorithm is unable to satisfythese criteria, `failure' is said to have occurred.Slope failure and numerical non-convergence occursimultaneously, and are accompanied by a dramaticincrease in the nodal displacements within themesh. Most of the results shown in this paper usedan iteration ceiling of 1000 and are presented inthe form of a graph of FOS versus E9ämax=ãH2 (adimensionless displacement), where ämax is themaximum nodal displacement at convergence andH is the height of the slope. This graph may beused alongside the displaced mesh and vector plotsto indicate both the factor of safety and the natureof the failure mechanism.

SLOPE STABILITY EXAMPLES AND VALIDATION

Several examples of FE slope stability analysisare now presented with validation against tradi-

tional stability analyses where possible. Initial con-sideration will be given to slopes containing nopore pressures in which total and effective stressesare equal. This is followed by examples of inhomo-geneous undrained clay slopes. Finally, submergedand partially submerged slopes are considered inwhich pore pressures are taken into account.

Example 1: Homogeneous slope with no foundationlayer ( D � 1)

(See Fig. 5 for the de®nition of D.) The homo-geneous slope shown in Fig. 1 has the followingsoil properties:

ö9 � 208

c9=ãH � 0:05

The slope is inclined at an angle of 26´578 (2:1)to the horizontal, and the boundary conditions aregiven as vertical rollers on the left boundary andfull ®xity at the base. Gravity loads were appliedto the mesh and the trial factor of safety (FOS)gradually increased until convergence could not beachieved within the iteration limit as shown inTable 2.

The table indicates that six trial factors of safetywere attempted, ranging from 0´8 to 1´4. Eachfactor of safety represented a completely indepen-dent analysis in which the soil strength parameterswere scaled by FOS as indicated in equations (4)and (5). Some ef®ciencies are possible in that thegravity loads and global stiffness matrix are thesame in each analysis and are therefore generatedonce only.

The Ìterations' column indicates the number ofiterations for convergence corresponding to eachFOS value. The algorithm has to work harder toachieve convergence as the `true' FOS is approached.

H

1.2H 2H

Rol

lers

Fixed

Fig. 1. Example 1: Mesh for a homogeneous slope with a slope angle of 26´578 (2:1), ö9 � 208, c9=ãH � 0´05


When FOS � 1:4, there is a sudden increase in thedimensionless displacement E9ämax=ãH2, and thealgorithm is unable to converge within the iterationlimit. Fig. 2 shows a plot of the data from Table 2, andindicates close agreement between the FE result andthe factor of safety given for the same problem by thecharts of Bishop & Morgenstern (1960).

Figure 3 shows the in¯uence of gravity loadingincrement size on displacements in example 1.With a `failure' factor of safety of FOS � 1:4applied to the soil properties, the four graphscorrespond to the maximum displacement obtainedwhen gravity was applied in a single increment ascompared with that obtained with two, three or ®veequal increments. The ®gure demonstrates that thedisplacement obtained with full gravity loading isbarely affected by the increment size.

Figures 4(a) and 4(b) give the nodal displace-ment vectors and the deformed mesh corresponding

to the unconverged situation with FOS � 1:4. Thedeformed mesh corresponding to this unconvergedsolution gives a rather diffuse indication of thefailure mechanism. This is due to the relativelycrude FE mesh, which must remain continuouseven at `failure'. Conventional FE analysis is un-able to model gross discontinuities along potentialfailure surfaces, although techniques have beendescribed for enhancing the visualization of thefailure surfaces (e.g. Grif®ths & Kidger, 1995).More advanced FE methods for modelling shearbands in conjunction with adaptive mesh re®ne-ment techniques have been described by Loret &Prevost (1991) and Zienkiewicz et al. (1995).

Example 2: Homogeneous slope with a foundationlayer ( D � 1:5)

Figure 5 shows that a foundation layer of thick-ness H=2 has now been added to the base of theslope of example 1, with all other properties andgeometry remaining the same.

The initial mesh and the deformed mesh atfailure are shown in Figs 5(a) and 5(b) respectively.It is clear from Fig. 5(b) that a mechanism of the`toe failure' type has been obtained. Fig. 2 indi-cates that the critical factor of safety is essentiallyunchanged from example 1 at FOS � 1:4, althoughthe displacements are increased due to the greatervolume of compressible soil.

This FE result con®rms that the addition of thefoundation layer has not led to any perceptiblechange in the factor of safety of the slope. Bishop& Morgenstern (1960) give FOS � 1:752 as one

Table 2. Results from example 1

FOS E9ämax=ãH2 Iterations

0´80 0´379 21´00 0´381 101´20 0´422 201´30 0´453 411´35 0´544 7921´40 1´476 1000

Bishop & Morgenstern (1960)FOS 5 1.380

0.25

0.5

0.75

11.3

1.51.8

2

2.32.5

2.8

33.3

3.5

3.8

4

E′δ

max

/γH

2

0.8 1 1.2 1.4 1.6

FOS

D 5 1.0 (Example 1)D 5 1.5 (Example 2)

Fig. 2. Examples 1 and 2: FOS versus dimensionlessdisplacement. The rapid increase in displacement andthe lack of convergence when FOS � 1´4 indicatesslope failure

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.60 0.2 0.4 0.6 0.8 1 1.2

E′δ

max

/γH

2

Gravity factor

1 increment2 equal increments3 equal increments5 equal increments

Fig. 3. In¯uence of gravity increment size on maxi-mum displacement at failure (FOS � 1´4) from exam-ple 1


Fig. 4. Example 1: Deformed mesh plots corresponding to the unconvergedsolution with FOS � 1´4: (a) nodal displacement vectors; (b) deformedmesh

(a)

(b)

Fig. 5. Example 2: Homogeneous slope with a foundation layer. Slope angle 26´578 (2:1), ö9 � 208, c9=ãH � 0´05,D � 1´5: (a) undeformed mesh; (b) mesh corresponding to unconverged solution with FOS � 1´4

DH

H

(a)

(b)


possible solution for this example (D � 1:5, c9=(ãH) � 0:05, ö9 � 208, 2:1 slope), although it isimportant to check the alternative solution corre-sponding to D � 1:0 to verify which gives thelower FOS. The charts of Cousins (1978) essen-tially agree with the FE result and indicate that,with a foundation layer, the critical circular me-chanism at its lowest point passes fractionallybelow the base of the slope and gives a slightlylower factor of safety than when there is nofoundation layer present.

Solving this example using a proprietary slipcircle program also found the possible `result' ofFOS � 1:7 when a failure circle tangent to thebase of the foundation was assumed. It was neces-sary to force the slip circle to pass through the toeto obtain the `correct' FOS of 1´4.

This example demonstrates one of the mainadvantages of FE slope stability analysis over con-ventional methods. The FE approach requires no apriori assumption of the location or shape of thecritical surface. Failure occurs `naturally' withinthe zones of the soil mass where the shear strengthof the soil is insuf®cient to resist the shear stresses.The use of a limit equilibirum method requires, atthe very least, some experience and care on thepart of the user in order to initiate appropriatesearch procedures which avoid the possibility ofhoming in on the wrong `critical' circle.

Example 3: An undrained clay slope with a thinweak layer

The next example demonstrates a stability analy-sis of a slope of undrained clay. Effective stressanalysis of an undrained slope of this type could

be performed with respect to effective stresses byadding a large apparent ¯uid bulk modulus to thesoil constitutive matrix (Naylor, 1974). In this case,however, a total stress analysis using a Trescafailure criterion (öu � 0) is presented.

Figure 6 shows a slope on a foundation layer(D � 2) of undrained clay. The slope includes athin layer of weaker material which initially runsparallel to the slope, then horizontally in thefoundation and ®nally outcrops at an angle of458 beyond the toe. Although this example mayseem contrived, it is not unlike the situation of athin, weak liner within a land®ll system. Thefactor of safety of the slope was estimated by FEanalysis for a range of values of the undrainedshear strength of the thin layer (cu2) while main-taining the strength of the surrounding soil atcu1=ãH � 0:25.

The FE results shown in Fig. 7 give the com-puted factor of safety expressed to the nearest0´05. For a homogeneous slope (cu2=cu1 � 1), thecomputed factor of safety was close to the Taylorsolution (Taylor, 1937) of FOS � 1:47 and gavethe expected circular base failure mechanism. Asthe strength of the thin layer was gradually re-duced, a distinct change in the nature of the resultswas observed when cu2=cu1 � 0:6.

Also shown on this ®gure are limit equilibriumsolutions obtained using Janbu's method assumingboth circular (base failure) and three-line wedgemechanisms following the path of the weak layer.The discontinuity when cu2=cu1 � 0:6 clearly re-presents the transition between the circular me-chanism and the non-circular mechanism governedby the weak layer. For cu2=cu1 . 0:6, the (circular)base failure mechanism governs the behaviour, and

H

H

2H 2H

0.6H 1.2H

cu2

cu1

cu1

2

1

2

1

2H

1.2H

1

0.6H

1

0.4H

0.4H

cu2 , cu1

φu 5 0

Fig. 6. Example 3: Undrained clay slope with a foundation layer including a thin weak layer (D � 2,cu1=ãH � 0´25)


the factor of safety is essentially unaffected by thestrength of the weaker thin layer. For cu2=cu1 , 0:6,the (non-circular) thin layer mechanism takes overand the factor of safety falls linearly.

This behaviour is explained more clearly inFig. 8, which shows the deformed mesh at failurefor three different values of the ratio cu2=cu1. Fig.8(a), corresponding to the homogeneous case(cu2=cu1 � 1), indicates an essentially circular fail-ure mechanism tangent to the ®rm base as pre-dicted by Taylor. Fig. 8(c), in which the strengthof the thin layer is only 20% of that of thesurrounding soil (cu2=cu1 � 0:2), indicates a highlyconcentrated non-circular mechanism closely fol-lowing the path of the thin weak layer. Fig. 8(b),in which the strength of the thin layer is 60% ofthat of the surrounding soil (cu2=cu1 � 0:6), indi-cates considerable complexity and ambiguity. Atleast two con¯icting mechanisms are apparent.First, there is a base failure mechanism merging

2

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

FO

S

0.20 0.4 0.6 0.8 1cu2/cu1

Finite elements

cu1/γH 5 0.25 Taylor (1937)FOS 5 1.47

Slope program—wedgesSlope program—circles

Fig. 7. Example 3: Computed factor of safety (FOS)for different values of cu2=cu1

Fig. 8. Example 3: Deformed meshes at failure corresponding to the un-converged solution for three different values of cu2=cu1 (a) cu2=cu1 � 1´0;(b) cu2=cu1 � 0´6; (c) cu2=cu1 � 0´2

(a)

(b)

(c)


with the weak layer beyond the toe of the slope,and second, there is a mechanism running alongthe weak layer parallel to the slope and outcrop-ping at the toe.

Without prior knowledge of the two alternativemechanisms, a traditional limit equilibrium searchcould seriously overestimate the factor of safety.This is illustrated in Fig. 7 where, for example, acircular mechanism with cu2=cu1 � 0:2 would in-dicate FOS � 1:3, when the correct factor of safetyis closer to 0´6.

Example 4: An undrained clay slope with a weakfoundation layer

In this case, the same slope geometry and ®niteelement mesh as in the previous example has beenused but with a different type of inhomogeneity, asshown in Fig. 9. The shear strength of the slopematerial has been maintained at a constant value ofcu1=ãH � 0:25, while the shear strength of thefoundation layer has been varied. The relative sizeof the two shear strengths has again been expressedas the ratio cu2=cu1. Fig. 10 shows the computedfactor of safety for a range of cu2=cu1 values,together with classical solutions of Taylor for thetwo cases when cu2 � cu1 and cu2 � cu1. There isclearly a change of behaviour occurring atcu2=cu1 � 1:5, as indicated by the ¯attening out ofthe curve. Also shown on this ®gure are limitequilibrium solutions for both toe and base circlemechanisms. The discontinuity corresponding tocu2=cu1 � 1:5 obviously represents the transitionbetween these two fundamental mechanisms.

This transition is clearly demonstrated by the FEfailure mechanisms shown in Fig. 11. Whencu2 � cu1 (Fig. 11(a)), a deep-seated base mechan-ism is observed (Fig. 11(a)), whereas a shallow`toe' mechanism is seen when cu2 � cu1 (Fig.11(c)). The result corresponding to the approximatetransition point at cu2 � 1:5cu1 (Fig. 11(b)) shows

an ambiguous situation in which both mechanismsare trying to form at the same time. It is interest-ing to note that the lower soil must be approxi-mately 50% stronger than the upper soil before thetoe mechanism becomes the most critical.

The previous two examples have shown thateven in quite simple cases, complex interactionscan occur between con¯icting mechanisms withinheterogeneous slopes which can be detected bythe FE approach. For more complicated stabilityproblems involving several soil property groupssuch as a zoned earth embankment, the FEapproach is arguably the only rational method thatwill generate the correct factor of safety andindicate the location and shape of the criticalmechanism.

2H 2H

H

H

cu1

cu2

2H

21

φu 5 0

Fig. 9. Example 4: Undrained clay slope with a weak foundation layer(D � 2, cu1=ãH � 0´25)

Finite element analysisBase circle analysisToe circle anlaysis

Taylor(cu2 .. cu1)FOS 5 2.10

Taylor(cu2 5 cu1)FOS 5 1.47

cu1/γH 5 0.25

0 0.5 1.5 2.5 3.51 2 3 4cu2/cu1

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

FOS

Fig. 10. Example 4: Computed factor of safety (FOS)for different values of cu2=cu1


INFLUENCE OF FREE SURFACE AND RESERVOIR

LOADING ON SLOPE STABILITY

We now consider the in¯uence of a free surfacewithin an earth slope and reservoir loading on theoutside of a slope as shown in Fig. 12.

Regarding the role of the free surface, a rigor-ous approach would ®rst involve obtaining agood-quality ¯ow net for free surface ¯ow throughthe slope, enabling pore pressures to be accuratelyestimated at any point within the ¯ow region. Forthe purposes of slope stability analysis, however,it is usually considered suf®ciently accurate andconservative to estimate pore pressure at a pointas the product of the unit weight of water (ãw)and the vertical distance of the point beneath thefree surface. In Fig. 12 the pore pressures at twolocations, A and B, have been calculated usingthis assumption.

In the context of FE analysis, the pore pressuresare computed at all submerged (Gauss) points asdescribed above, and subtracted from the total nor-mal stresses computed at the same locations follow-ing the application of surface and gravity loads. Theresulting effective stresses are then used in the

remaining parts of the algorithm relating to theassessment of Mohr±Coulomb yield and elasto-plas-tic stress redistribution. Note that the gravity loadsare computed using total unit weights of the soil.

The external loading due to the reservoir ismodelled by applying a normal stress to the faceof the slope equal to the water pressure. Thus, asshown in Fig. 13, the applied stress increaseslinearly with water depth and remains constantalong the horizontal foundation level. These stres-ses are converted into equivalent nodal loads onthe FE mesh (e.g. Smith & Grif®ths, 1998) andadded to the initial gravity loading.

Example 5: Homogeneous slope with horizontalfree surface

Figure 14 shows a similar slope to that analysedin example 1, but with a horizontal free surface at adepth L below the crest. Using the method describedabove, the factor of safety of the slope has beencomputed for several different values of the draw-down ratio (L=H), which has been varied from ÿ0:2(slope completely submerged with the water level

Fig. 11. Example 4: Deformed meshes at failure corresponding to theunconverged solution for three different values of cu2=cu1: (a) cu2=cu1 � 0´6;(b) cu2=cu1 � 1´5; (c) cu2=cu1 � 2´0

(a)

(b)

(c)


0:2H above the crest), to 1´0 (water level at the baseof the slope). The problem could be interpreted as a`slow' drawdown problem in which a reservoir, ini-tially above the crest of the slope, is graduallylowered to the base, with the water level within theslope maintaining the same level. A constant totalunit weight has been assigned to the entire slope,both above and below the water level.

The interesting result shown in Fig. 15 indicatesthat the factor of safety reaches a minimum ofFOS � 1:3 when L=H � 0:7. A limit equilibriumsolution shown on the same ®gure indicates asimilar trend (e.g. Cousins, 1978). The specialcases corresponding to L=H � 0 and L=H � 1agree well with chart solutions given, respectively,by Morgenstern (1963) (F � 1:85), and Bishop &Morgenstern (1960) (FOS � 1:4). The fully sub-merged slope (L=H < 0) is more stable than the`dry' slope (L=H > 1), as indicated by a higherfactor of safety.

An explanation of the observed minimum is dueto the cohesive strength of the slope (which isunaffected by buoyancy) and the trade-off betweensoil weight and soil shear strength as the drawdownlevel is varied. In the initial stages of drawdown(L=H , 0:7), the increased weight of the slope has aproportionally greater destabilizing effect than theincreased frictional strength, and the factor of safetyfalls. At higher drawdown levels (L=H . 0:7), how-

Free surface

Embankment

A

hA

uA 5 hAγW

uB 5 hBγW

hB

hW

B

Reservoir Level

Fig. 12. Slope with free surface and reservoir loading

Free surface

Embankment

Reservoir Level

Linearly increasingnormal stress fromzero to hWγW hW

Constant normalstress 5 hWγW

Fig. 13. Detail of submerged area of slope beneath free-standing reservoirwater showing stresses to be applied to the surface of the mesh asequivalent nodal loads

H

L 5 0L (negative)

L (positive)

Fig. 14. Example 5: `Slow' drawdown problem. Homo-geneous slope with a horizontal free surface. Slopeangle 26´578 (2:1), ö9 � 208, c9=ãH � 0´05 (above andbelow free surface)


ever, the increased frictional strength starts to have agreater in¯uence than the increased weight, and thefactor of safety rises. Other results of this type havebeen reported by Lane & Grif®ths (1997) for a slopewhich was stable (FOS . 1) when `dry' or fullysubmerged, but became unstable (FOS , 1) at acritical value of the drawdown ratio L=H . It shouldalso be pointed out from the horizontal part of thegraph in Fig. 15, corresponding to L=H < 0, thatthe factor of safety for a fully submerged slope isunaffected by the depth of water above the crest.

Excellent agreement with Morgenstern (1963)

for `rapid drawdown' problems has also beendemonstrated for a range of slopes using a similarapproach (Lane & Grif®ths, 1997).

Example 6: Two-sided earth embankmentThe example given in Fig. 16 is of an actual

earth dam cross-section including a free surfacewhich slopes from the reservoir level to foundationlevel on the downstream side (Torres & Coffman,1997). For the purposes of this example, the materi-al properties have been made homogeneous. Fig. 17

Finite elementsLimit equilibrium

FO

S

2

1.8

1.6

1.4

1.2

Morgenstern(1963)

FOS 5 1.85

Bishop &Morgenstern(1960)

FOS 5 1.4

L/H0 0.120.120.2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Fig. 15. Example 5: Factor of safety in `slow' drawdown problem fordifferent values of the drawdown ratio L=H

Fig. 17. Example 6: Finite element mesh

Reservoir level

17.1

33.5

7.3

18° 23°

124.4 33.5

H 5 21.3

7.3

Dimensions in metres

Free surface

Fig. 16. Example 6: Two-sided earth embankment with a sloping free surface. Homogeneous dam, ö9 � 378,c9 � 13´8 kN=m2, ã � 18´2 kN=m3 (above and below WT)


shows the FE model used for the slope stabilityanalysis (Paice, 1997). The boundary conditionsconsist of vertical rollers on the faces at the left andright ends of the foundation layer with full ®xity atthe base. It should be noted that the downstreamslope of the embankment is slightly steeper than theupstream slope.

A second analysis was also performed with nofree surface corresponding to the embankment be-fore the reservoir was ®lled. FE slope stabilityanalysis led to the results shown in Fig. 18. Bothcases were also solved using a conventional limitequilibrium approach which gave FOS � 1:90 witha free surface and FOS � 2:42 without a free sur-face. The limit equilibrium and FE factors of safetyvalues were in close agreement in both cases.

Regarding the critical mechanisms of failure,Figs 19 and 20 show the deformed mesh cor-responding to the unconverged FE solution ascompared with the slip circle that gave thelowest factors of safety from the limit equilibriumapproach. As expected, the lowest factor of safetyoccurs on the steeper, downstream side of theembankment in both cases. It should also be notedthat both the FE and limit equilibrium resultsindicate a toe failure for the case with no freesurface (Fig. 19), and a deeper mechanism extend-ing into the foundation layer for the case with a

free surface (Fig. 20). Fig. 21 shows the corre-sponding displacement vectors from the FE solu-tions. Reasonably good agreement between thelocations of the failure mechanisms by both typesof analysis is indicated.

Fig. 19. Example 6 with no free surface: (a) deformed mesh corresponding to the unconverged solution by®nite elements; (b) the critical slip circle by limit equilibrium. Both methods give FOS � 2´4

(a)

(b)

R 5 62.7m

5.4m

62.4m

5

10

15

20

25

30

351.2

E′δ

max

/γH

2

1 1.4 1.6 1.8 2 2.2 2.4 2.6

FOS

FOS 5 1.90 FOS 5 2.42

Limitequilibriumsolutions

No free surfaceWith a free surface

Fig. 18. Example 6: FOS versus dimensionless dis-placement


THE CRITERIA FOR COMPUTER-AIDED ANALYSIS

Whitman & Bailey (1967) looked forward to thefuture of computer-aided analysis for engineers andset criteria by which it could be judged. Theircomments were originally addressed to the automa-tion of limit equilibrium methods, but they alsocommented on the then emerging numerical analy-sis techniques.

They judged that the system must be suf®cientlyaccurate for con®dence in its use and appropriate

for the parameters being input. FE analysis meetsthese criteria with a degree of accuracy decided bythe engineer in designing the model.

It should be possible, in a realistic timescale, todo suf®cient trials to examine all the key modes ofbehaviour; to consider different times in the life ofthe structure and to vary parameters during designto test options for cost and ef®ciency. All this isnow possible with FE methods.

Finally, the method of human±machine commu-

Fig. 20. Example 6 with a free surface: (a) deformed mesh corresponding to the unconverged solution by®nite elements; (b) the critical slip circle by limit equilibrium. Both methods give FOS � 1´9

(a)

(b)

R 547.8m

11.5m

43.9m

Fig. 21. Example 6: Displacement vectors corresponding to the unconverged solution by ®nite elements:(a) no free surface; (b) with a free surface. Only those displacement vectors that have a magnitude . 10% ofthe maximum are shown

(b)

FOS 5 1.9

(a)

FOS 5 2.4


nication must be user-friendly and readily accessi-ble. This is partly a matter of program design buteasily achieved. Graphical output greatly enhancesthe process of design and analysis over and abovethat from the numerical results.

Similarly, Chowdhury (1981), in his discussionof Sarma (1979), commented on the perceivedreluctance to develop alternatives to limit equili-brium methods for practice when the tools to do sowere already available. Since then, numerous appli-cations and experience have veri®ed the possibili-ties offered by ®nite elements.

The key issues of cost and turnaround time havebeen overtaken by the falling cost of powerfulhardware and processor speeds which now makethe FE method available to engineers at less thanthe cost of their CAD systems. What remains isthe concern of powerful tools used wrongly. Thatis no more true of ®nite elements after years ofapplication than of limit equilibrium methods,which can themselves produce seriously misleadingresults. Engineering judgement is still essential,whichever method is being used.

CONCLUDING REMARKS

The FE method in conjunction with an elastic±perfectly plastic (Mohr±Coulomb) stress±strainmethod has been shown to be a reliable and robustmethod for assessing the factor of safety of slopes.One of the main advantages of the FE approach isthat the factor of safety emerges naturally from theanalysis without the user having to commit to anyparticular form of the mechanism a priori.

The FE approach for determining the factor ofsafety of slopes has satis®ed the criteria for effec-tive computer-aided analysis. The widespread useof this method should now be seriously consideredby geotechnical practitioners as a more powerfulalternative to traditional limit equilibrium methods.

ACKNOWLEDGEMENTS

The writers acknowledge the support of NSFGrant No. CMS 97-13442. Dr Lane was supportedby the Peter Allen Scholarship Fund of UMIST.

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Slope stability analysis by ®nite elements