Shear Strength Reduction Approach for Slope Stability Analyses

( 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