FREDLUND边坡稳定未来讲义（精品）.ppt

What is the Future for Slope Stability Analysis? (Are We Approaching the Limits of Limit Equilibrium Analyses?),Dr. Delwyn G. Fredlund University of Saskatchewan, Canada Second Symposium and Short Course on Unsaturated Soils and Environmental Geotechnics Budapest, Hungary November 4-5, 2003,Introduction,Limit Equilibrium methods of slices have been “Good” for the geotechnical engineering profession since the methods have produced financial benefit Engineers are often surprised at the results they are able to obtain from Limit Equilibrium methods,So 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 must be found by TRIAL and ERROR,SHAPE and LOCATION are the driving force for a paradigm shift Objectives of this Presentation:,To show the gradual change that is emerging in the way that slope stability analyses can be undertaken To illustrate the benefits associated with improved procedures for the assessment of stresses in a slope,Outline of Presentation,Provide a brief Summary of common Limit Equilibrium methods along with their limitations (2-D & 3-D) Take the FIRST step forward through use of an independent stress analysis Take the SECOND step forward through use of Optimization Techniques,Is a Limit Equilibrium Analysis an Upper Bound or Lower Bound Solution?,Limit Equilibrium Methods primarily satisfy the requirements of an upper bound type of solution Reason: the shape of slip surface is selected by the analyst and thereby a displacement boundary condition is imposed,Limit Equilibrium and Finite Element Based Methods of Analyses,W,W,W,W,W,W,W,W,N,Limit Equilibrium,Method of Analysis,Sm = ta dl,d,l,s,n,d,l,Finite Element Based,Method of Analysis,l,d,ta dl,QUESTION: How can the Normal Stress at the base of a slice be most accurately computed?,Consider the Free Body Diagrams used to calculate the Normal Stress?,Assumption for all Limit Equilibrium Analysis,Soils behave as Mohr-Coulomb materials (i.e., friction, f, and cohesion, c) Factor of safety, Fs, for the cohesive component is equal to the factor of safety for the frictional component Factor of safety is the same for all slices,Summary of Available Equations Associated with a Limit Equilibrium Analysis,Equations (knowns): Quantity Moment equilibrium n Vertical force equilibrium n Horizontal force equilibrium n Mohr-Coulomb failure criterion n 4n,Unknowns: Quantity Total normal force at base of slice n Shear force at the base of slice, Sm n Interslice normal force, E n-1 Interslice shear force, X n-1 Point of application of interslice force, E n-1 Point of application of normal force n Factor of safety, Fs 1 6n-2,Summary of Unknowns Associated with a Limit Equilibrium Analysis,One Fs per sliding mass,Forces Acting on Each Slice,Focus on Sm,b,y,x,S,m,X,R,E,R,E,X,L,Slip,surface,Ground,surface,W,h,R,N,=,s,n,b,b,f,N,L,Phreatic line,Mobilized Shear Force, Sm for Saturated-Unsaturated Soils,(,),(,),s,w,a,s,a,n,s,m,F,u,u,F,u,F,c,S,b,f,b,f,s,b,b,tan,tan,-,+,-,+,=,Only new variable required for solving saturated- unsaturated soils problems is the shear force mobilized,b = Friction angle with respect to matric suction ua = Pore-air pressure uw = Pore-water pressure,Moment equilibrium, Fm: Force equilibrium , Ff:,Pore-air pressures are assumed to be zero gauge,Normal force at base of slice: Limit Equilibrium methods differ in terms of how (XR-XL) is computed and overall statics satisfied Limit Equilibrium problem is indeterminate: Can apply an assumption (Historical solution) Can utilize additional physics (Future solution),(,),F,F,u,F,c,X,X,W,N,b,w,L,R,tan,sin,cos,tan,sin,sin,f,a,a,f,a,b,a,b,+,+,-,-,-,=,s,x,b,a,Area = Interslice,normal force (E),width of slice,b,s,x,t,x,y,s,y,Distance (m),t,x,y,b,a,Area = Interslice,shear force (X),Vertical,slice,Distance (m),=,b,a,xy,dy,X,t,=,b,a,x,dy,E,s,Stresses on the Boundary Between Slices,Morgenstern & Price, 1965,Summary of Limit Equilibrium Methods and Assumptions,Forces Acting on One Slice in Ordinary or Conventional Method,h,W,b,b,a,N = s b,n,N,Sm,Forces Acting on One Slice in Bishops Simplified and Janbus Simplified Methods,h,W,b,b,a,N = s b,n,N,Sm,ER,EL,Summary of Limit Equilibrium Methods and Assumptions,Direction of X and E is the average of the ground surface slope and the slope at the base of a slice,Vertical Horizontal,Lowe and Karafiath,Direction of X and E is parallel to the ground,Vertical Horizontal,Corps of Engineers,Direction of E and X is defined by an arbitrary function. Percent of the function required to satisfy moment and force equilibrium is called ,Vertical Horizontal Moment,Morgenstern-Price, GLE,Assumptions,Equilibrium Satisfied,Method,Forces Acting on One Slice in Spencers, Morgenstern-Price, and GLE Methods,Various Interslice Force Functions Proposed by Morgenstern & Price (1965),Spencers,Wilson and Fredlund (1983) Used a finite element stress analysis (with gravity switched on) to determine a shape for the Interslice Force Function Interslice Force Function for a Deep-seated Slip Surface Through a 1:2 Slope,X = E f(x),Definition of Dimensionless Distance,Generalized Functional Shape,where: K = magnitude of function at mid-slope e = base of natural log C = variable to define inflection point n = variable to define steepness = dimensionless x-position,(,),2,/,),(,n,n,C,Ke,x,f,w,-,=,Wilson and Fredlund, 1983,X = E f(x),Dimensionless Distance,Unique function of “slope angle” for all slip surfaces,“C” coefficient versus slope angle,Unique function of “slope angle” for all slip surfaces,“n” coefficient versus tangent of slope angle,Comparison of Factors of Safety Circular Slip Surface,0,0.2,0.4,0.6,1.80,1.85,1.90,1.95,2.00,2.05,2.10,2.15,2.20,2.25,l,Janbus Generalized,Simplified,Bishop,Spencer,Morgenstern-Price,f(x) = constant,Ordinary = 1.928,Ff,Fm,Fredlund and Krahn 1975,Factor of safety,Moment and Force Limit Equilibrium Factors of Safety For a Circular type slip surface,Moment limit equilibrium analysis,Force limit equilibrium analysis,Fredlund and Krahn, 1975,Lambda, l,Factor of safety,Force and Moment Limit equilibrium Factors of Safety for a planar toe slip surface,Force limit equilibrium analysis,Moment limit equilibrium analysis,Lambda, l,Factor of safety,Krahn 2003,Force and Moment Limit equilibrium Factors of Safety for a composite slip surface,Moment limit equilibrium analysis,Force limit equilibrium analysis,Lambda, l,Factor of safety,Fredlund and Krahn 1975,Force and Moment Limit equilibrium Factors of Safety for a “Sliding Block” type slip surface,Krahn 2003,Extensions of Methods of Slices to Three-dimensional Methods of Columns,Hovland (1977) 3-D of Ordinary Chen and Chameau (1982) 3-D of Spencer Cavounidis (1987) 3-D Fs 2-D Fs Hungr (1987) 3-D of Bishop Simplified Lam and Fredlund (1993) 3-D with f(x) on all 3 planes; 3-D of GLE,Shape and Location Become Even More Difficult to Define in 3-D,Two Perpendicular Sections Through a 3-D Sliding Mass,Section Parallel to Movement,Section Perpendicular to Movement,Free Body Diagram of a Column with All Interslice Forces,Parallel,Perpendicular,Base,Interslice Force Functions for Two of the Directions,X/E,V/P,First Step Forward Question:,Is the Normal Stress at the base of each slice as accurate as can be obtained? Is the Normal Stress only dependent upon the forces on a vertical slice?,Improvement of Normal Stress Computations,Fredlund and Scoular 1999,Limit equilibrium and finite element normal stresses for a toe slip surface,From limit equilibrium analysis,From finite element analysis,Limit equilibrium and finite element normal stresses for a deep-seated slip surface,From finite element analysis,From limit equilibrium analysis,Limit equilibrium and finite element normal stresses for an anchored slope,From finite element analysis,From limit equilibrium analysis,To illustrate procedures for combining a finite element stress analysis with concepts of limiting equilibrium. (i.e., finite element method of slope stability analysis) To compare results of a finite element slope stability analysis and conventional limit equilibrium methods,Using Limit Equilibrium Concepts in a Finite Element Slope Stability Analysis,Objective:,The 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,Hypothesis,Assumption: The stresses computed from “switching-on” gravity are more reasonable than the stresses computed on a vertical slice,Manner of “Importing Stresses” from a Finite Element Analysis into a Limit Equilibrium Analysis,s,n,s,n,tm,Mohr Circle,t,m,IMPORT: Acting Normal Stress Actuating Shear Stress,Limit Equilibrium Analysis,Finite Element Analysis for Stresses,Bishop (1952) - stresses from Limit Equilibrium methods do not agree with actual soil stresses Clough and Woodward (1967) - “meaningful stability analysis can be made only if the stress distribution within the structure can be predicted reliably” Kulhawy (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 Stability,Stress Level Rezendiz 1972,Zienkiewicz,et al,1975,Strength & Stress Level,Adikari and Cummins 1985,Enhanced limit methods,(finite element analysis,with a limit equilibrium,Finite Element Slope Stability Methods,Direct methods,(finite element analysis only),Strength Level,Kulhawy 1969,F,-,-,Z,=,1,3,1,3,D,D,L,L,f,s,s,s,s,F,=,(,c,+,tan,),-,-,c,+,tan,A,1,3,1,3,s,f,s,s,s,s,s,f,D,D,L,L,f,*,F,=,(,c,+,tan,),K,s,f,t,D,D,L,L,Definition of Factor of Safety,Load increase,to failure,analysis),Differences and Similarities Between the Finite Element Slope Stability and Conventional Limit Equilibrium,Differences Solution is determinate Factor of safety equation is linear Similarities Still necessary to assume the shape of the slip surface and search by trial and error to locate the critical slip surface,Why hasnt Finite Element Slope Stability Method been extensively used?,Difficulties and perceptions related to the stress analysis Inability to transfer large amounts of data and find needed information Now: Microcomputer have dramatically changed our ability to combine Finite Element and Limit Equilibrium analyses,Definition of Factor of Safety,Kulhawy (1969) where: Sr = resisting shear strength or Sm = mobilized shear force,=,m,r,FEM,S,S,F,b,f,s,tan,),u,(,c,S,w,n,r,-,+,=,Actuating Shear,Normal Stress,Analysis Study Undertaken by Fredlund and Scoular (1999),Adopted the Kulhawy (1969) procedure Used Sigma/W and Slope/W Poissons ratio range = 0.33 to 0.48 Elastic modulus, E = 20,000 to 200,000 kPa Cohesion, c = 10 to 40 kPa Friction, = 10 to 30 degrees Compared conventional Limit Equilibrium results with Finite Element slope stability results,Location of Center of a Section along the Slip Surface within a Finite Element Analysis,x-Coordinate,y-Coordinate,Slip Surface,Finite Element,(r, s),s,r,Fictitious slice defined with,the Limit Equilibrium analysis,Center of the base of a slice (x, y),Presentation of Finite Element Slope Stability Results,Conditions Analyzed: Dry slope Piezometric line at 3/4 height, exiting at toe Dry slope, partially submerged Piezometric line at 1/2 height and submerged to mid-height,Selected 2:1 Free-Standing Slope with a Piezometric Line Exiting at the Toe of the Slope,20,40,60,80,100,120,20,40,60,80,0,Crest,Toe,2,1,x - Coordinate (m),Note: Dry slope with & without piezometric line,y - Coordinate (m),Selected 2:1 Partially Submerged Slope with a Horizontal Piezometric Line at Mid-Slope,20,40,60,80,100,120,20,40,60,80,0,Crest,Toe,2,1,x - Coordinate (m),Water,y - Coordinate (m),Note: Dry slope with & without piezometric line,0,50,100,150,200,250,300,20,30,40,50,60,70,x-Coordinate (m),Acting and restricting shear stress (kPa),Crest,Toe,Shear Strength,Shear Force,Poisson Ratio,m,= 0.33,Shear Strength and Shear Force for a 2:1 Dry Slope Calculated Using the Finite Element Slope Stability Method,Local and Global Factors of Safety for a 2:1 Dry Slope,0,1,2,3,4,5,6,7,20,25,30,35,40,45,50,55,60,65,70,x-Coordinate,Factor of Safety,Crest,Toe,Local F,(,m,Local F,(m= 0.33),Bishop Method, F,= 2.360,= 2.173,Global Factors of Safety,Bishop 2.360,Janbu,2.173,GLE (F.E. function) 2.356,F,s,(,m,= 0.33,) 2.342,F,s,(,m,= 0.48,) 2.339,Ordinary 2.226,s,Janbu Method, F,s,s,s,Fs = 2.342,Fs = 2.339,= 0.48),Factors of Safety Versus Stability Number for a 2:1 Dry Slope as a Function of c,0.0,0.5,1.0,1.5,2.0,2.5,0,5,10,15,20,25,Factor of Safety,c,= 20kPa,c,= 10kPa,c,= 40kPa,F,s,(,m,= 0.33),F,s,(,m,= 0.48),2:1 Dry Slope,Factor of Safety Versus Stability Coefficient for a 2:1 Dry Slope as a Function of ,0.0,0.5,1.0,1.5,2.0,2.5,0.00,0.02,0.04,0.06,0.08,0.10,0.12,Stability,Coefficient, c,Factor of Safety,f,= 30,f,= 10,f,= 20,2:1 Dry Slope,s,F,s,(,m,= 0.33),F,(,m,= 0.48),F,s,(GLE),s,Factor of Safety Versus Stability Coefficient as a Function of for 2:1 Slope with a Piezometric Line,0.0,0.4,0.8,1.2,1.6,2.0,0.00,0.02,0.04,0.06,0.08,0.10,0.12,Stability Coefficient, c/,g,H,Factor of safety,f,= 30,f,= 20,f,= 10,2:1 Slope with piezometric line,Location of the Critical Slip Surface for a Slope with a Piezometric Line with Soil Properties of c = 40 kPa and f = 30,7,0,1,0,2,0,1,0,0,6,0,5,0,4,0,3,0,9,0,8,0,7,0,1,1,0,5,0,6,0,4,0,1,0,2,0,3,0,8,0,Method,X,Y,R,Factor of safety,G,L,E,(,F,.,E,.,Function),5,8,.,5,5,6,.,0,3,7,.,9,1,.,7,4,1,F,s,(,m,=,0,.,3,3,),5,7,.,5,4,9,.,5,3,4,.,7,1,.,6,2,7,F,s,(,m,=,0,.,4,8,),5,7,.,5,5,3,.,0,3,7,.,8,1,.,6,6,1,Y- Coordinate (m),Location of the Critical Slip Surface for a Slope with a Piezometric Line where the Factor of Safety is Closest to 1.0,7,0,1,0,2,0,1,0,0,6,0,5,0,4,0,3,0,9,0,8,0,7,0,5,0,6,0,4,0,1,0,2,0,3,0,1,1,0,8,0,F,s,(,m,=,0,.,3,3,),s,Method,X,Y,R,Factor of safety,G,L,E,(,F,.,E Function,.,6,3,.,5,5,9,.,0,3,9,.,6,1,.,1,0,2,F,s,(,m,=,0,.,3,3,),6,3,.,0,5,9,.,0,4,1,.,5,1,.,0,7,6,F,(,m,=,0,.,4,8,),6,1,.,5,5,9,.,5,4,2,.,3,1,.,1,0,0,y- Coordinate (m),Factor of Safety Versus Stability Coefficient as a Function of for 2:1 Dry Slope, 1/2 Submerged,0.0,0.5,1.0,1.5,2.0,2.5,3.0,3.5,0.00,0.02,0.04,0.06,0.08,0.10,0.12,Stability,Coefficient, c,/,g,H,Factor of Safety,f,= 20,f,= 10,2:1 Dry slope, one-half submerged,f,= 30,F,s,(,m,= 0.33),F,s,(,m,= 0.48),F,s,(GLE),Factor of Safety Versus Stability Coefficient as a Function of for 2:1 Slope Half Submerged with Piezometric Line,0.0,0.5,1.0,1.5,2.0,2.5,0.00,0.02,0.04,0.06,0.08,0.10,0.12,Stability,Coefficient, c,/,H,Factor of Safety,f,= 30,f,= 10,f,= 20,2:1 Slope, one-half submerged,g,F,s,(,m,= 0.33),F,s,(,m,= 0.48),F,s,(GLE),1,0,2,0,1,0,0,6,0,5,0,4,0,3,0,9,0,8,0,7,0,1,1,0,7,0,5,0,6,0,4,0,1,0,2,0,3,0,8,0,F,s,(,m = 0.33),GLE (F.E. Function),s,Method,X,Y,R,Factor of safety,G,L,E,(,F,.,E,.,Function,5,8,.,0,5,8,.,5,4,0,.,2,2,.,3,0,3,F,s,(,m,=,0,.,3,3,),5,2,.,5,5,0,.,5,3,1,.,8,2,.,2,5,9,F,(,m,=,0,.,