Shear Strength Reduction Approach for Slope Stability Analyses

Shear Strength Reduction Approach for Slope Stability AnalysesMark S. DiederichsGeo-Engineering Centre at Queen’s University and Royal Military College, Kingston, OntarioMatt LatoDepartment of Geological Sciences and Geological Engineering, Queen’s University, Kingston, Ontario.Reginald HammahRocscience Inc. Toronto, OntarioPete QuinnGeo-Engineering Centre at Queen’s University and Royal Military College, Kingston, OntarioABSTRACT: This paper is intended to illustrate the applicability of Shear Strength Reduction (SSR) as ageneral technique for obtaining Factor of Safety estimates for slopes in variable geology, progressive or locallybrittle yield behaviour and with ground-structure interaction. Comparisons are made with Limit Equilibriumsolutions (LEM). Implications of the assumptions required to ensure this correlation are discussed includinguniform stiffness, rigid-plastic behaviour, instantaneous interaction between geological units, andinstantaneous generation of support loads. The paper uses Finite Element Modelling (FEM) as a vehicle fordemonstration although Finite Difference solutions are equally valid. General applicability of the method isdemonstrated and the limitations explored using Discrete Element simulation of multi-block slope failure.1 INTRODUCTIONThe Shear Strength Reduction (SSR) technique (Matsui& San 1992, Dawson et al 1999; Griffiths &Lane 1999; Cala & Flisiak 2003, Hammah et al2005a, 2006) enables finite element (or finite difference)techniques to be used to calculate factors ofsafety for slopes, providing an alternative to limitequilibrium calculations and a potentially more reliableanalysis of slopes with heterogeneous stiffness,strain-softening and passive structure-ground interaction.The methodology is general and can be appliedto other non-linear problems such as multiblockdiscrete element simulations.Geotechnical engineers primarily conduct slopedesign based on calculated factor of safety values.Limit Equilibrium (LEM) techniques that compareresisting forces to driving forces (or moments) areideally suited to the generation of a nominal safetyfactor (Krahn 2003). Instantaneous slide surfacemobilization and consideration of stresses and forcesindependent of pre-failure movement are inherent inthese techniques, and may result in inadequate representationof the system’s actual stability state.Non-linear modelling using the Shear StrengthReduction (SSR) technique can also used to determinefactors of safety. The approach offers a numberof advantages over LEM (Griffiths & Lane, 1999)including the elimination of a priori assumptions onthe shape and location of failure surfaces, the eliminationof assumptions regarding the inclinations andlocations of interslice forces, the capability to modelprogressive failure, the calculation of deformationsand the incorporation of displacement controlledground-structure interaction. The general approachis valid for a wide range of applications.The automated inclusion into specific analysistools mandates the need for some accepted stabilitycriteria. Maximum unbalanced force tolerance,maximum displacement limits or other criteria havebeen incorporated into commercially available codesalthough these are not fundamentally related to themethodology of SSR. Examples of alternative stabilityindicators are proposed in this paper. Stabilityindicators can be tailored for the problem at handthrough manual application of the SSR approach.Some key issues, nevertheless, require furtherresolution including the questions: should factoredshear strength parameters represent the dominantvariables in the definition of factor of safety formore complex failure modes (shear plus toppling,extension, etc); should cohesion and friction be subjectedto the same reduction factors; should all capacityrelated parameters, including support, be includedin the SSR process and should support befactored by the same Strength Reduction Factor asthe soil/rock shear strength? Duncan (1996) providessome insight into the latter issue althoughmore discussion is needed.1.1 Limit Equilibrium AnalysisThe general limit equilibrium approach developedby Fredlund and Krahn (1977) calculates a compositefactor of safety, FS, for moment equilibrium as:Proceedings of the 1st Canada-US Rock Mech. Symposium - Vancouver 2007: Invited Session Keynote Paper

( c'βR+ ( N − uβ) R tanφ')Σ( Wx − Nf ) DdFS Σ= M ±(1)and for force equilibrium as:Σ( c'β cosα+ ( N − uβ) tanφ'cosα)FS M= (2)Σ N sinα− Dcos( ) ωwhere the slice base normal force is:c'β sinα+ uβsinαtanφ'W + ( XR− XL) −N = FS(3)sinαtanφ'cosα+FSwhere u is the pore pressure, W is the slice weight, Dis an applied line load, α is the base inclination andβ, R, x, f, d and ω are geometric parameters.The significant assumptions in this formulationconcern the inter-slice forces (shear and normal).Early techniques (Fellenius 1936) ignored theseforces or made significant simplifying assumptions(Janbu 1954, Bishop 1955). Morgenstern and Price(1965) and Spencer (1967) proposed a solution forthis problem used in the general formulation byFredlund and Krahn (1977). Krahn (2003) summarizesand compares a number of approaches forLEM analysis. These differ primarily in the individualor combined consideration of moment or forceequilibrium and the consideration of inter-slice normaland shear forces and the assumed or calculatedinclination of resultant inter-slice forces as illustratedin Figure 1 and summarized in Table 1.Figure 1: Slice and interslice forces for a composite limit equilibriumapproach to slope stability.The basal stress state in the original LEM formulationsare ultimately a function of gravitationalloading on the individual slices. This does not considerthe influence of topographical and stiffnessbased variations in the underground stress state.A number of authors, including Zou et al. (1995)and Krahn (2003) have proposed the use of linearelastic finite element analysis to provide the initialstress map for a subsequent LEM calculation and theintegration of slice forces. Stianson et al. (2004) investigatethe use of non-linear plastic analysis (forstable problems) to compute the basal stress stateand slice forces. These methods collectively representan important advance over conventional techniques.The use of elastic FEM stress calculationstakes into account the stress variability due to stiffnesscontrasts and topography (as in GeoSlope 2001,for example). The use of plastic analysis to computestresses for an LEM calculation may account forprogressive yield and stress redistribution althoughthe additional complexity of the integrated solutionprocedure is of dubious added value.Another assumption involves the consideration ofline loads in the formulation. Support loading can beconsidered by adjusting resisting forces or drivingforces within the factor of safety equation. One approachto handle this problem is to include so-calledactive support (e.g. pre-tensioned anchors) as a reductionto the driving forces while passive anchors(e.g. grouted dowels) are incorporated as an increasein resisting forces. This selection, while logical, issomewhat arbitrary and this challenge is discussed,without the provision of a unique solution, by Krahn(2003). The displacement dependency of resistingloads provided by some passive support systems arenot considered in this formulation.Finally, LEM techniques must be performed on anassumed continuous slip surface. This surface wasoriginally circular in early formulations but mosttechniques after Fellenius can be applied to noncircularand very general surfaces (Figure 2). Thesurface that generates the minimum factor of safetyis located using a grid or optimized search (centerand radius), block search (relocating vertices of anarbitrary slip polygon) or other statistical techniques.For simple geometries, modern searchmethods yield robust results although for problemsincluding extreme geometries (very thin weak layers,for example), search techniques can often missthe true global minimum (FS) geometry.Table 1: Comparison of equilibrium considerations and interslice force assumptions (Krahn 2003)MethodMomentEquilibriumHoriz. ForceEquilibriumIntersliceNormal ForceIntersliceShear ForceInclination ofInterslice ForceFelleniusBishop SimplifiedJanbu SimplifiedSpencerMorgenstern-PriceCorps of Engin. 1Corps of Engin. 2Lowe-KarafiathYESYESNOYESYESNONONONONOYESYESYESYESYESYESNOYESYESYESYESYESYESYESNONONOYESYESYESYESYESNo ForceHorizontalHorizontalConstantVariable= Crest to Toe Average Dip= Slice Ground Surface Dip= Average of Surface and Base DipProceedings of the 1st Canada-US Rock Mech. Symposium - Vancouver 2007: Invited Session Keynote Paper

einforcement loadings, or shear strain localization(as in Matsui & San 1992) as desired. Figure 4 illustratesthe convergence approach for the same slopein Figure 2.Figure 2: a) LEM grid search using SLIDE (RocScience2005a) for circular slip surface and b) block search (optimizedvertices). FS

In the supported case, however, the results differin terms of the location of the minimum FS ( =SRF), although the Factor of Safety for this newfailure zone falls within the range calculated for thiszone in LEM (FS=1.5 to 1.6). The main difference isthat the LEM still locates the minimum surface(FS=1.37-1.46) through the support as in Figure 11.In addition, the lower right plot in Figure 12 illustratesthat the generation of axial load in the supportsis not constant and does not reach the maximumcapacity. The axial (and shear) load developsas the slope deforms. This is more physically accuratethan the assumption of static full capacity (oreven factored nominal capacity) in LEM analysis.In this analysis, however, the capacity of the boltsare not factored along with soil shear strength in theSSR process. There is, at present, no consensus onwhether they should be (i.e. apply the same SSR tobolt capacity) or how the actual loads in the boltsshould be compared with the bolt capacities to givean FS for the bolts. This is a discussion that isneeded in order to adopt a standard approach forsupported slopes. It is possible to use the SSR approachin a different way and discretely select limitingSRF (for short and long term stability for example)according to accepted practice. This would becompatible with the factored load approach formerging structural and geotechnical engineering.The bolts could then be scaled to provide stabilityfor the strength-reduced slip surface within a structural(steel code) safety margin. This is just one suggestion.It is clear, however, that more discussion iswarranted on the incorporation of support in SSRmodels. Accepting that SSR is mechanically correct,the accounting process for support needs to be furtherdiscussed in the learned community.2.5 Example 5: Stiffness and Strain SofteningThe examples in the previous section were selectedto demonstrate the compatibility of LEM and FEM-SSR techniques for simple slope problems. In eachcase the conditions are fixed and the units are of uniformand high stiffness (the correct approach to obtaindirect comparison between the two methods).The following example explores the effect of multiplematerials, stiffness, strain softening and sensitivitiesto convergence tolerance and mesh density.The layer properties (according to Hoek et al. 2002)for the example in Figure 13 are given in Table 7.Table 7. Strength parameters of the materials in Example 5.UCSMPaUCSresidm i GSI m b s E rmGPaSSt 60 12 21 70 7.2 0.036 16.5Sh1SiltSh215251015251081076065501.92.91.20.0120.0200.0043.09.42.0SSt =Sandstone, Silt =Siltstone, Sh1=shale, Sh2 =weak shaleFigure 13 shows the minimum FS calculated forthree selected LEM solutions using peak and residualstrength parameters for each layer. In this example,the residual strength is specified by reducing thenormalizing UCS. The slope crest is 205m above thelake level.Figure 13: LEM analysis (FS values shown) for a stratifiedrock slope. Left analysis uses peak strength from Table 7;Right analysis uses “resid” strength for the Sandstone.Figure 14 compares the FEM-SSR results formodels with elastic-plastic response (using “peak”parameters from Table 7) using identical elasticproperties and variable (true) elastic properties. Theeffect of stiffness contrasts in this case is not particularlysignificant although there is no reason notto include stiffness variation in an FEM-SSR analysis.The results in either case compare best to thenon-circular Bishop analysis in Figure 13.Figure 14: FEM-SSR analysis of slope in Fig. 13 and Table 7.Shear strain contours and displacement vectors (x25) plotted.Figure 15 shows the SSR results for an analysiswith strain weakening (brittle) behaviour prescribedfor the sandstone. Other lithologies are plastic. SSRis applied to both peak and residual parameters inthis case.Figure 15: FEM-SSR analysis for Example 5 with strain weakeningbehaviour for the sandstone (zero dilation). Water tableis solid line (1). Shear strain (contours) and displacement vectors(x25) are plotted. Note bimodal failure (2 blocks).Proceedings of the 1st Canada-US Rock Mech. Symposium - Vancouver 2007: Invited Session Keynote Paper

Interestingly, the FS (=SRF=1.85) for this analysisis within the range of the LEM analyses for residualparameters. In addition, close examination ofFigure 15 shows that both the non-circular, deepseatedslip surface as well as the shallower circularsurface suggested by alternate LEM analyses (rightside of Figure 13) are both simulated in the FEManalysis (two distinct zones of elevated shear strain).The results shown are for zero-dilation. Failure geometryis similar for dilation (m dil = 0.25m b ) with anelevated FS=SRF=1.92. Table 8 summarizes the impactof mesh density and convergence tolerance.Beyond a reasonable minimum element density, theresults are not very sensitive to mesh configurationalthough poor results are obtained with an elevatedtolerance or the use of 3-noded triangular elements(as opposed to 6-noded linear strain elements).Table 8: Effects of tolerance, maximum iterations per cycleand mesh density for Example 5 with brittle sandstone (Figure15) and zero dilation.Mesh Tolerance Max Iterations FS=SRF(*6 nodes)*Standard 0.001 500 1.85*Standard 0.01 250 2.05*Standard 0.0001 1000 1.82*Coarse (x 0.5) 0.001 500 1.86*Fine (x 2) 0.001 500 1.823 node (stand) 0.001 500 2.033 DISCRETE / DISCONTINUOUS MODELSThe application of SSR is not restricted to continuummodels such as finite element or finite differencetools. This section presents two examples ofapplication in discontinuum analysis.3.1 SSR in Discrete Element AnalysisWhile the equilibrium calculation is based on agroup of slices, limit equilibrium techniques stillconsider the behaviour of a contiguous block of rockor soil in a state of limiting sliding instability. Inmulti-block applications, the development of severalunstable blocks and limited movement does notimmediately indicate failure in an operational sense.The example in Figure 16 illustrates the use of SSRfor a blocky rockmass. The model is created using3DEC and the characterization technique describedby Kalenchuk et al. (2006). A vertical slope is usedfor this simple example. In this model the “actual”joint shear strength properties are specified (cohesoin= 90 kPa, friction angle = 56 deg) although theinitial conditions could be variable and joint specific.The boundary conditions shown are unrealisticand for demonstration only. Failure must be definedand could be based on critical displacement or, as inthis case, a limit for released blocks or cumulativereleased block volume as shown in Figure 17.Figure 16: Schematic 3DEC block models used in Figure 17.Figure 17: Results of SSR analysis for composite model in Figure16. Vertical lines bound the range of FS (=SRF) based ontwo possible failure criteria (see axes). Each point is averagedover several stochastic simulations.3.2 SSR in Discontinuous Deformation AnalysisOne of the fundamental premises of the SSR techniqueis that the factor of safety for any problem canbe related to the factoring of shear strength for materialsand interfaces. Numerous verification examplesillustrate that this gives the same result as an overallforce and moment comparison for sliding problems.It is less clear whether the SSR technique is valid forproblems that do not have a single sliding surfaceand which have internal separation and/or vorticity.The example in Figure 18 demonstrates the applicationof SSR to a simple toppling problem.Figure 18: Example of SSR applied to toppling analysis usingDiscontinuous Deformation Analysis. Critical SRF for interblockfriction and cohesion for this analysis is 1.24. The nominalfriction angle is 30 degrees and the cohesion is 5kPa.Proceedings of the 1st Canada-US Rock Mech. Symposium - Vancouver 2007: Invited Session Keynote Paper

This analysis employs discontinuous deformationanalysis (Sitar et al 2001) assuming that cohesionand friction are the only variables to control stability.The implementation appears simple and reliableenough and reasonable answers can be obtained, althoughit is important to debate the universality ofrelating stability in complex situations (with sliding,extension and rotation) to shear strength alone. Inaddition to the necessary factoring of tensilestrength, crucial in rock mechanics problems, theSSR technique may not adequately capture geometricfactors such as block aspect ratio and rotationalmechanics not related to shear strength. The methodis attractive for non-standard applications but themechanics of each case should be compatible.4 CONCLUSIONSFor problems dominated by sliding, the SSR techniquehas been clearly demonstrated as a valid alternativeto Limit Equilibrium Methods. The techniquecan be used in non-linear finite element or finite differencecontinuum models or in discontinuum analysisand is not unreasonably sensitive to model setup(although mesh density, solution tolerance and shapefunction sensitivity must always be checked).The greatest advantage to the SSR technique isthe lack of a requirement to discretely or iterativelypre-define a sliding surface for consideration althoughmodern LEM techniques have sophisticatedsearch algorithms for non-circular surfaces.In sliding models with no tension crack or with nosupport, the factor of safety can be directly anduniquely related to the factoring of shear strength formaterials and interfaces giving similar results to anoverall force and moment comparison (LEM).If tension is involved then tensile strength mustalso be included in the SSR process for the geomaterial.It may be necessary to debate the validity ofequal factoring of cohesion and friction in the SSRsolution. Cohesion is often specifed with much lessconfidence than frictional properties and yet forshallow problems, dominates over frictional effects.For simple cases the LEM (force and moment)definition of FS coincides with the SSR (FS=criticalSRF) definition. The definitions deviate for morecomplex geologies and material behaviour. Shearstrength may not be the dominant parametric factorin the true factor of safety in many cases.It is also not clear how shear strength reductionshould be standardized in cases where support is involved.More discussion is needed to decide whethersupport strength be fixed during the SSR iterationsor included in the over all strength reduction.REFERENCESCala M, & Flisiak J. 2003. Complex geology slope stabilityanalysis by shear strength reduction. Proc. of the 3rd Int.FLAC Symp., Sudbury. Lisse: Balkema. 99-102.Dawson EM, Roth WH, Dresher A. 1999. Slope stabilityanalysis by strength reduction. Geotechnique, 49(6). 835-840.Duncan, J.M. 1996. 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