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1、FREDLUND边坡稳定未来讲义（精品）Still waters run deep.流静水深流静水深,人静心深人静心深 Where there is life,there is hope。有生命必有希望。有生命必有希望Introduction qqLimit Equilibrium methods of slices have been“Good”for the geotechnical engineering profession since the methods have produced financial benefit qqEngineers are often surprised
2、 at the results they are able to obtain from Limit Equilibrium methodsSo Why Change?There are Fundamental Limitations with Limit Equilibrium Methods of Slices?The boundaries for a FREE BODY DIAGRAM are not known-The SHAPE for the slip surface must be assumed-The LOCATION of the critical slip surface
3、 must be found by TRIAL and ERRORSHAPE and LOCATION are the driving force for a paradigm shiftObjectives of this Presentation:qTo show the gradual change that is emerging in the way that slope stability analyses can be undertakenqTo illustrate the benefits associated with improved procedures for the
4、 assessment of stresses in a slopeOutline of PresentationqqProvide a brief Summary of common Limit Equilibrium methods along with their limitations(2-D&3-D)qqTake the FIRST step forward through use of an independent stress analysisqqTake the SECOND step forward through use of Optimization Techniques
5、Is a Limit Equilibrium Analysis an Upper Bound or Lower Bound Solution?qqLimit Equilibrium Methods primarily satisfy the requirements of an upper bound type of solutionReason:the shape of slip surface is selected by the analyst and thereby a displacement boundary condition is imposedLimit Equilibriu
6、m and Finite Element Based Methods of AnalysesWWWWWWWWNLimit EquilibriumMethod of AnalysisSm=t ta dldls sndlFinite Element Based Method of Analysisldt ta dlQUESTION:How can the Normal Stress at the base of a slice be most accurately computed?Consider the Free Body Diagrams used to calculate the Norm
7、al Stress?Assumption for all Limit Equilibrium AnalysisqqSoils behave as Mohr-Coulomb materials(i.e.,friction,(i.e.,friction,and cohesion,c),and cohesion,c)qqFactor of safety,Fs,for the cohesive component is equal to the factor of safety for the frictional componentqqFactor of safety is the same for
8、 all slices()msnsSFuFc=-+b b s sb btanSummary of Available Equations Associated with a Limit Equilibrium AnalysisqqEquations(knowns):Quantityl lMoment equilibriumnl lVertical force equilibriumnl lHorizontal force equilibriumnl lMohr-Coulomb failure criterionn4nqqUnknowns:Unknowns:Quantity Quantityl
9、lTotal normal force at base of slice Total normal force at base of slice n nl lShear force at the base of slice,SShear force at the base of slice,Smmn nl lInterslice normal force,EInterslice normal force,En-1n-1l lInterslice shear force,XInterslice shear force,Xn-1n-1l lPoint of application of inter
10、slice force,EPoint of application of interslice force,E n-1n-1l lPoint of application of normal forcePoint of application of normal forcen nl lFactor of safety,FFactor of safety,Fs s1 1 6n-26n-2Summary of Unknowns Associated with a Limit Equilibrium AnalysisOne FOne Fs s per sliding mass per sliding
11、 massForces Acting on Each SliceFocus on SmbyxSmXREREXLSlip surfaceGround Ground surfacesurfaceWhRN=s snb bb bfNLPhreatic linePhreatic lineMobilized Shear Force,Sm forSaturated-Unsaturated Soils()()swasansmFuuFuFcSb b b b s sb bbtantan-+-+=Only new variable required for solving saturated-unsaturated
12、 soils problems is the shear force mobilized b b =Friction angle with respect to matric suction=Friction angle with respect to matric suctionu ua a=Pore-air pressure=Pore-air pressureu uww =Pore-water pressure=Pore-water pressure qqMoment equilibrium,Fm:qqForce equilibrium,Ff:-+=NfWxtanRtantanuNR cF
13、bwm b bb b -+=a aa a b ba ab bsincostantantancosNuNcFbwfPore-air pressures are assumed to be zero gaugeqqNormal force at base of slice:qqLimit Equilibrium methods differ in terms of how(XR-XL)is computed and overall statics satisfiedqqLimit Equilibrium problem is indeterminate:l lCan apply an assump
14、tion Can apply an assumption(Historical solution)(Historical solution)l lCan utilize additional physics Can utilize additional physics(Future solution)Future solution)()FFuFcXXWNbwLRtansincostansinsin a aa a a ab ba ab b+-=s sxbaArea=Interslice normal force(E)width of slice,b bs sxt txys syDistance(
15、m)Elevation(m)t txybaArea=Interslice shear force(X)Vertical sliceDistance(m)Elevation(m)=baxydyXt t=baxdyEs sStresses on the Boundary Between SlicesMorgenstern&Price,1965Summary of Limit Equilibrium Methods and AssumptionsMethod Equilibrium Satisfied Assumptions Ordinary Moment,to base E and X=0 Bis
16、hops Simplified Vertical,Moment E is horizontal,X=0 Janbus Simplified Vertical,Horizontal E is horizontal,X=0,empirical correction factor,f0,accounts for interslice shear forces Janbus Generalized Vertical,Horizontal E is located by an assumed line of thrust Spencer Vertical,Horizontal,Moment Result
17、ant of E and X are of constant slope Forces Acting on One Slice in Ordinary or Conventional MethodhWbb ba aN=s bN=s b nN NSmForces Acting on One Slice in Bishops Simplified and Janbus Simplified Methods hWbb ba aN=s bN=s b nN NSmERELSummary of Limit Equilibrium Methods and AssumptionsDirection of X
18、and E is the average of the ground surface slope and the slope at the base of a sliceVerticalHorizontalLowe and KarafiathDirection of X and E is parallel to the groundVerticalHorizontalCorps of EngineersDirection of Direction of E E and and X X is defined by is defined by an arbitrary function.Perce
19、nt of an arbitrary function.Percent of the function required to satisfy the function required to satisfy moment and force equilibrium is moment and force equilibrium is called called VerticalHorizontalMomentMorgenstern-Price,GLEAssumptionsAssumptionsEquilibriumEquilibriumSatisfiedSatisfiedMethodMeth
20、od Forces Acting on One Slice in Spencers,Morgenstern-Price,andGLE Methods hWbb ba aN=s bN=s b nN NSmERELXRXLVarious Interslice Force FunctionsProposed byMorgenstern&Price(1965)SpencersWilson and Fredlund(1983)Used a finite element stress analysis(with gravity switched on)to determine a shape for th
21、e Interslice Force FunctionInterslice Force Function Interslice Force Function for a Deep-seated Slip for a Deep-seated Slip Surface Through a 1:2 Surface Through a 1:2 SlopeSlopeX=E f(x)Definition of Dimensionless Distance f(x)is largest at mid-pointInflection points near crest&toeGeneralizedGenera
22、lizedFunctionalFunctionalShapeShapewhere:K=magnitude of function at mid-slopee =base of natural logC=variable to define inflection pointn=variable to define steepness=dimensionless x-position()2/)(nnCKexfw w-=Wilson and Fredlund,1983X=E f(x)Dimensionless DistanceDimensionless DistanceUnique function
23、 of“slope angle”for all slip surfaces“C”coefficient versus slope angleUnique function of“slope angle”for all slip surfaces“n”coefficient versus tangent of slope angleComparison of Factors of Safety Circular Slip Surface00.20.40.61.801.851.901.952.002.052.102.152.202.25l lJanbus GeneralizedSimplified
24、 BishopSpencerMorgenstern-Pricef(x)=constantOrdinary=1.928FfFmFredlund and Krahn1975Factor of safetyMoment and Force Limit Equilibrium Factors of SafetyFor a Circular typeslip surfaceMoment limit equilibrium analysisForce limit equilibrium analysisFredlund and Krahn,1975Lambda,l l Factor of safetyFo
25、rce and MomentLimit equilibriumFactors of Safety for a planar toe slip surfaceForce limit equilibrium analysisMoment limit equilibrium analysisLambda,l l Factor of safetyKrahn 2003Force and MomentLimit equilibrium Factors of Safety for a composite slip surfaceMoment limit equilibrium analysisForce l
26、imit equilibrium analysisLambda,l l Factor of safetyFredlund and Krahn 1975Force and MomentLimit equilibrium Factors of Safety for a“Sliding Block”type slip surfaceMoment limit equilibrium analysisForce limit equilibrium analysisLambda,l l Factor of safetyKrahn 2003Extensions of Methods of Slices to
27、Three-dimensional Methods of ColumnsqqHovland(1977)3-D of OrdinaryqqChen and Chameau(1982)3-D of Spencer qqCavounidis(1987)3-D Fs 2-D FsqqHungr(1987)3-D of Bishop SimplifiedqqLam and Fredlund(1993)3-D with f(x)on all 3 planes;3-D of GLEShape and Location Become Even More Difficult to Define in 3-DTw
28、o Perpendicular Sections Through a 3-D Sliding MassSection Parallel to MovementSection Perpendicular to MovementFree Body Diagram of a Column with All Interslice ForcesParallelPerpendicularBaseInterslice Force Functions for Two of the DirectionsX/EV/PFirst Step ForwardQuestion:qIs the Normal Stress
29、at the base of each slice as accurate as can be obtained?qIs the Normal Stress only dependent upon the forces on a vertical slice?Improvement of Normal Stress ComputationsFredlund and Scoular1999Limit equilibrium and finite element normal stresses for a toe slip surfaceFrom limit equilibrium analysi
30、sFrom finite element analysisLimit equilibrium and finite element normal stresses for a deep-seated slip surfaceFrom finite element analysisFrom limit equilibrium analysisLimit equilibrium and finite element normal stresses for an anchored slopeFrom finite element analysisFrom limit equilibrium anal
31、ysisqqTo illustrate procedures for combining a finite element stress analysis with concepts of limiting equilibrium.(i.e.,finite element method of slope stability analysis)qqTo compare results of a finite element slope stability analysis and conventional limit equilibrium methods Using Limit Equilib
32、rium Concepts in a Finite Element Slope Stability AnalysisObjective:qqThe complete stress state from a finite element analysis can be“imported”into a limit equilibrium framework where the normal stress and the actuating shear stress are computed for any selected slip surface HypothesisAssumption:The
33、 stresses computed from“switching-on”gravity are more reasonable than the stresses computed on a vertical sliceManner of“Importing Stresses”from a Finite Element Analysis into a Limit Equilibrium Analysis s snFinite Element Analysis for StressesLimit Equilibrium Analysiss snt tmMohr Circlet tmIMPORT
34、:Acting Normal StressActuating Shear StressLimit Equilibrium AnalysisFinite Element Analysis for StressesqqBishop(1952)-stresses from Limit Equilibrium methods do not agree with actual soil stresses qqClough and Woodward(1967)-“meaningful stability analysis can be made only if the stress distributio
35、n within the structure can be predicted reliably”qqKulhawy(1969)-used normal and shear stresses from a linear elastic analysis to compute factor of safety“Enhanced Limit Strength Method”Background to Using Stress Analyses in Slope StabilityStress LevelRezendiz 1972Zienkiewicz et al 1975Strength&Stre
36、ss LevelAdikari and Cummins 1985Enhanced limit methods(finite element analysiswith a limit equilibriumFinite Element Slope Stability MethodsDirect methods(finite element analysis only)Strength LevelKulhawy 1969F -Z=1313 D DD DLLfs ss ss ss s F=(c +tan)-c +tanA 1313 s s s ss ss ss ss s D D D DL Lf*F=
37、(c +tan )K s s t tD D D DLLDefinition of Factor of SafetyLoad increaseto failureStrength decreaseto failureanalysis)Differences and Similarities Between the Finite Element Slope Stability and Conventional Limit Equilibrium qqDifferencesl lSolution is determinatel lFactor of safety equation is linear
38、qqSimilaritiesl lStill necessary to assume the shape of the slip surface and search by trial and error to locate the critical slip surfaceWhy hasnt Finite Element Slope Stability Method been extensively used?qqDifficulties and perceptions related to the stress analysisqqInability to transfer large a
39、mounts of data and find needed information Now:Microcomputer have dramatically changed our ability to combine Finite Element and Limit Equilibrium analysesDefinition of Factor of Safety qqKulhawy(1969)qqwhere:l lSr=resisting shear strength or l lSm=mobilized shear force =mrFEMSSFb b s stan)u(cSwnr-+
40、=Actuating ShearActuating ShearNormal StressNormal StressAnalysis Study Undertaken by Fredlund and Scoular(1999)qqAdopted the Adopted the Kulhawy(1969)Kulhawy(1969)procedure procedureqqUsed Sigma/W and Slope/W Used Sigma/W and Slope/W qqPoissons ratio range=0.33 to 0.48Poissons ratio range=0.33 to 0
41、.48qqElastic modulus,E=20,000 to 200,000 kPaElastic modulus,E=20,000 to 200,000 kPaqqCohesion,c=10 to 40 kPaCohesion,c=10 to 40 kPaqqFriction,Friction,=10 to 30 degrees=10 to 30 degreesqqCompared Compared conventional Limit Equilibrium conventional Limit Equilibrium results with Finite Element slope
42、 stability results with Finite Element slope stability results results Location of Center of a Section along the Slip Surface within a Finite Element Analysisxyx-Coordinatey-CoordinateSlip SurfaceFinite Element(r,s)srFictitious slice defined withthe Limit Equilibrium analysisCenter of the base of a
43、slice(x,y)Presentation of Finite Element Slope Stability Results qqConditions Analyzed:l l Dry slopel l Piezometric line at 3/4 height,exiting at toel l Dry slope,partially submergedl l Piezometric line at 1/2 height and submerged to mid-heightSelected 2:1 Free-Standing Slope with a Piezometric Line
44、 Exiting at the Toe of the Slope20406080100120204060800CrestPiezometric LineToe21x-Coordinate (m)Note:Dry slope with&without piezometric liney-Coordinate (m)Selected 2:1 Partially Submerged Slope with a Horizontal Piezometric Line at Mid-Slope20406080100120204060800CrestToe21x -Coordinate (m)WaterPi
45、ezometric Liney -Coordinate (m)Note:Dry slope with&without piezometric line050100150200250300203040506070 x-Coordinate(m)Acting and restricting shear stress(kPa)CrestToeShear StrengthShear ForcePoisson Ratio,m m=0.33Shear Strength and Shear Force for a 2:1 Dry Slope Calculated Using the Finite Eleme
46、nt Slope Stability MethodLocal and Global Factors of Safety for a 2:1 Dry Slope012345672025303540455055606570 x-CoordinateFactor of SafetyCrestToeLocal F(m mLocal F(m m=0.33)Bishop Method,F=2.360=2.173Global Factors of SafetyBishop 2.360Janbu 2.173GLE(F.E.function)2.356Fs(m m=0.33 )2.342Fs(m m=0.48
47、)2.339Ordinary 2.226sJanbu Method,FsssFs=2.342Fs=2.339=0.48)Factors of Safety Versus Stability Number for a 2:1 Dry Slope as a Function of c 0.00.51.01.52.02.50510152025Stability Number,g g Htan /c Factor of Safetyc =20kPac =10kPac=40kPaFs(m m =0.33)Fs(m m =0.48)Fs(GLE)2:1 Dry Slope Factor of Safety
48、 Versus Stability Coefficient for a 2:1 Dry Slope as a Function of 0.00.51.01.52.02.50.000.020.040.060.080.100.12Stability Coefficient,c /g g H Factor of Safety =30 =10 =202:1 Dry SlopesFs(m m =0.33)F(m m =0.48)Fs(GLE)sFactor of Safety Versus Stability Coefficient as a Function of for 2:1 Slope with
49、 a Piezometric Line 0.00.40.81.21.62.00.000.020.040.060.080.100.12Stability Coefficient,c/g g H Factor of safety =30 =20 =102:1 Slope with piezometric lineFs(m m =0.33)Fs(m m =0.48)Fs(GLE)Location of the Critical Slip Surface for a Slope with a Piezometric Line with Soil Properties of c=40 kPa and =
50、3070102010060504030908070110506040102030 x-Coordinate (m)80GLE(F.E.function)Fs(m m=0.33)Fs(m m=0.48)MethodXYRFactor of safetyGLE(F.E.Function)58.556.037.91.741Fs(m m=0.33)57.549.534.71.627Fs(m m=0.48)57.553.037.81.661Y-Coordinate(m)Location of the Critical Slip Surface for a Slope with a Piezometric