Consider the stress conditions in a homogeneous soil w ith no preexisting failure planes. Near the surface in flat-lying terrain ( Figure 16. 3 a) ，the direction of maximum principal stress σ1( due to the weight of overlying material) is vertical，and the direction of minimum principal stress σ3is horizontal. In the vicinity of a slope，on the other hand，the stress distribution becomes skew ed，in the manner show n in Figure 16. 3 b. As show n there，one consequence of this stress pattern is that the planes of incipient failure，oriented at α = 45° － φ /2 from the σ3 direction，are curved. In soil mechanics these possible planes of failure are called slip circles or slip surfaces. The limit-equilibrium analysis is carried out using the Mohr-Coulomb failure criteria， and a factor of safety，Fs，defined as the ratio of shearing strength on the slip surface to shearing stress on the slip surface，is calculated. If Fs＞ 1，the slope is considered to be stable with respect to that slip surface. The slip surface with the lowest value of Fsis regarded as the incipient failure plane. If Fs≤1 on the critical surface，failure is imminent.
Figure 16. 3 Orientation of principal stresses
Consider a homogeneous，isotropic clay soil for which the angle of internal friction approaches zero. In such cases，the shear strength of the soil is derived solely from its cohesion c，and the Mohr- Coulomb failure law ［Equation ( 16. 7) ］becomes simply Sr= c. For such soils，the slip surface can be closely approximated by a circle ( Figure 16. 4 a) . The factor of safety will be given by the ratio of the resisting moment to the disturbing moment about the point O. The disturbing force is simply the weight W of the potential slide，and the resisting force is that of the cohesive strength c acting along the length l between points A and B. For this simple case，
Figure 16. 4 Slope stability analysis by ( a) circular arc and
For more complex situations a more sophisticated method of analysis is needed and this is provided by the conventional method of slices. It can be applied to slip surfaces of irregular geometry and to cases where c and φ ( or c' and φ ') vary along the slip surface. This method also invokes the effective stress principle by considering the reduction in soil strength along the slip surface due to the fluid pressures ( or pore pressures，as they are commonly called in the slope stability literature) that exist there on saturated slopes. For the conventional method，the slide is divided into a series of vertical slices. Figure 16. 4 b shows the geometry of an individual slice， and Figure 16. 4 c indicates the conditions of force equilibrium and stress equilibrium that exist at point C on the slip surface at the base of the slice. At C，the shearing stress is ( W sin θ) / l and the shearing strength Sris given，as before，by
For σ = ( W cos θ) / l ，Equation ( 16. 10) becomes
@2and the factor of safety is given by
The conventional method of slices was improved by Bishop in 1955，who recognized the need to take into account the horizontal and vertical stresses produced along the slice boundaries due to the interactions between one slice and another. The resulting equation for Fsis somewhat more complicated than Equation ( 16. 12) ，but it is of the same from. Bishop and Morgenstern in 1960 produced sets of charts and graphs that simplify the application of the Bishop method of slices. Morgenstern and Price in 1965 generalized the Bishop approach even further，and their technique for irregular slopes and general slip surfaces in nonhomogeneous media has been widely computerized. Computer packages for the routine analysis of complex slope stability problems are now in wide use.
To apply the limit equilibrium method to a given slope，whether by computer or by hand，the basic approach is to measure c' and φ ' for the slope material，calculate W，l，θ and p for the various slices，and calculate Fsfor the various slip surface under analysis.
Of all the required data，probably the most sensitive is the pore pressure p along the potential sliding planes. If economics permit，it may be possible to install piezometers in the slope at the depth of the anticipated failure plane. The measured hydraulic heads，h，can then be converted to pore pressures by means of the usual relationship:
p = ρg ( h － z) ( 16. 13) where z is the elevation of the piezometer intake. In many cases，however，field instrumentation is not feasible，and it behooves us to reexamine the hydrogeology of slopes in light of the needs of slope stability analysis.
( Sources: Freeze et al. ，1979)
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