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1、What is the Future for Slope Stability Analysis? (Are We Approaching the Limits of Limit Equilibrium Analyses?),Dr. Delwyn G. Fredlund University of Saskatchewan, CanadaSecond Symposium and Short Course on Unsaturated Soils and Environmental GeotechnicsBudapest, HungaryNovember 4-5, 2003,Introductio
2、n,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 wi
3、th 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
4、Presentation:,To show the gradual change that is emerging in the way that slope stability analyses can be undertakenTo 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 met
5、hods 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 sati
6、sfy 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 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 d
7、l,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 usedto calculate the Normal Stress?,Assumption for all Limit Equilibrium Analysis,Soils behave as Mohr-Coulomb mater
8、ials (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 componentFactor of safety is the same for all slices,Summary of Available Equations Associated with a Limit Equilibrium Analysis,Equations (knowns): Quantity
9、 Moment equilibrium n Vertical force equilibrium n Horizontal force equilibrium n Mohr-Coulomb failure criterion n4n,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 in
10、terslice force, E n-1 Point of application of normal force n Factor of safety, Fs 16n-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,Phreati
11、c 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 = Po
12、re-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 indete
13、rminate: 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
14、,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 O
15、ne 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,
