6,Chapter 13 J. MICHAEL DUNCAN SOIL SLOPE STABILITY ANALYSIS nalyses of slopes can be divided into two cat- ities, are described in Chapter 15. Techniques for Aegories: those used to evaluate the stability of evaluating rock strength are discussed in Chapter slopes and those used to estimate slope movement. 14, and those for evaluating soil strength are dis- Although stability and movement are closely re- cussed in Chapter.12. lated, two different and distinct types of analyses Limit equilibrium and finite-element analyses are almost always used to evaluate them. are described in subsequent sections of this chap- ter. Methods that are useful for practical purposes are emphasized, and their uses are illustrated by 1. INTRODUCTION examples. Stability of slopes is usually analyzed by methods of limit equilibrium. These analyses require infor- 2.MECHANICS OF LIMIT EQUILIBRIUM mation about the strength of the soil, but do not ANALYSES require information about its stress-strain behav- ior. They provide no information about the In limit equilibrium techniques, slope stability is magnitude of movements of the slope. analyzed by first computing the factor of safety. Movements of slopes are usually analyzed by the This value must be determined for the surface that finite-element method. Understanding the stress- is most likely to fail by sliding, the so-called criti- strain behavior of the soils is required for these cal slip surface. Iterative procedures are used, each analyses, and information regarding the strengths of involving the selection of a potential sliding mass, the soils may also be required. Although these subdivision of this mass into a series of slices, and methods define movements and stresses through- consideration of the equilibrium of each of these out the slope, they do not provide a direct measure slices by one of several possible computational of stability, such as the factor of safety calculated by methods. These methods have varying degrees of limit equilibrium analyses. computational accuracy depending on the suit- The focus in this chapter is on soil slopes and ability of the underlying simplifying assumptions mechanisms involving shear failure within the soil for the situation being analyzed. mass. The methods described are applicable to landslides in weak rocks, where the location of the 2.1 Factor of Safety rupture surface is not controlled by existing discon- tinuities within the mass (see also Chapter 21). The factor of safety is defined as the ratio of the Analyses of slopes in strong rock, where instability shear strength divided by the shear stress required mechanisms are controlled by existing discontinu- for equilibrium of the slope: 337 338 Landslides: Investigation and Mitigation shear strength in a different value of F. The smaller the value of F = shear stress required for equilibrium (13.1) (the smaller the ratio of shear strength to shear stress required for equilibrium), the more likely which can be expressed as failure is to occur by sliding along that surface. Of all possible slip surfaces, the one where sliding is (13.2) most likely to occur is the one with the minimum teq value of F. This is the critical slip surface. in which To evaluate the stability of a slope by limit equi- librium methods, it is necessary to perform calcula- F = factor of safety, tions for a considerable number of possible slip c = cohesion intercept on Mohr-Coulomb surfaces in order to determine the location of the strength diagram, critical slip surface and the corresponding mini- 4) = angle of internal friction of soil, mum value of F. This process is described as a = normal stress on slip surface, and searching for the critical slip surface. It is an essen- shear stress required for equilibrium. tial part of slope stability analyses. The cohesive and frictional components of 2.2 Equations and Unknowns strength (c and 4)) are defined in Chapter 12. The factor of safety defined by Equations 13.1 All of the practically useftil methods of analysis are and 13.2 is an overall measure of the amount by methods of slices, so called because they subdivide which the strength of the soil would have to fall the potential sliding mass into slices for purposes short of the values described by c and 4) in order for of analysis. Usually the slice boundaries are verti- the slope to fail. This strength-related definition of cal, as shown in Figure 13-1. Two useful simplifi- F is well suited for practical purposes because soil catioris are achieved by subdividing the mass into strength is usually the parameter that is most diffi- slices: cult to evaluate, and it usually involves consider- able uncertainty. The base of each slice passes through only one In discussing equilibrium methods of analysis, type of material, and it is sometimes said that the factor of safety is The slices are narrow enough that the segments assumed to have the same value at all points on the of the slip surface at the base of each slice can slip surface. It is unlikely that this would ever be be accurately represented by a straight line. true for a real slope. Because the assumption is unlikely to be true, describing this assumption as The equilibrium conditions can be considered one of the fundamental premises of limit equilib- slice by slice. If a condition of equilibrium is sat- rium methods seems to cast doubt on their prac- ticality and reliability. It is therefore important to understand that Equation 13.1 defines F, and it does not involve the assumption that F is the same at all points along the slip surface for a slope not at failure. Rather, F is the numerical answer to the Iar Slip Suace question, By what factor would the strength have to be reduced to bring the slope to failure by slid- ing along a particular potential slip surface? The answer to this question can be determined reliably FIGURE 13-1 by limit equilibrium methods and is of consider- Division of potential able practical value. sliding masses into To calculate a value of F as described above, a rcuIar Slip Suace slices (slip surface is idealization of potential slip surface must be described. The slip surface of rupture; surface is a mechanical idealization of the surface see Chapter 3, of rupture (see Chapter 3, Section 3.1 and Table Section 3.1). 3-3). In general, each potential slip surface results Soil Slope Stability Analysis 339 isfied for each and every slice, it is also satisfied for FIGURE 13-2 the entire mass. The forces on a slice are shown in Forces on slice i. Figure 13-2. The number of equations of equilibrium avail- able depends on the number of slices (N) and the number of equilibrium conditions that are used. As shown in Table 134, the number of equations available is 2N if only force equilibrium is to be sat- isfied and 3N if both force and moment equilibria are to be satisfied. If only force equilibrium is sat- isfied, the number of unknowns is 3N - 1. If both force and moment equilibria are satisfied, the num- ber of unknowns is 5N - 2. In the special case in which N = 1, the problem is statically determinate, and the number of equilibrium equations is equal to the number of unknowns. To represent a slip surface with realistic shapes, it is usually necessary to use 10 to 40 slices, and the number of unknowns therefore exceeds the number of equations. The excess of unknowns over equations is N - 1 for force equilibrium analyses and 2N - 2 for analyses that satisfy all conditions of equilibrium. Thus, such problems are statically indeterminate, and Different methods use different assumptions to assumptions are required to make up the imbal- make up the balance between equations and ance between equations and unknowns. unknowns, and Some methods, such as the ordinary method of slices (Fellenius 1927) and Bishop's modified 2.3 Characteristics of Methods method (Bishop 1955), do not satisfy all condi- The various methods of limit equilibrium analysis tions of equilibrium or even the conditions of differ from each other in two regards: force equilibrium. Table 13-1 Equations and Unknowns in Limit Equilibrium Analysis of Slope Stability EQUATIONS UNKNOWNS Methods That Satisfy Only Force Equilibrium N = horizontal equilibrium N = normal forces on bases of slices N = vertical equilibrium N - 1 = side forces N - 1 = side force angles, 9 1 = factor of safety 2N total equations 3N - 1 total unknowns Methods That Satisfy Both Force and Moment Equilibria N = horizontal equilibrium N = normal forces on bases of slices N = vertical equilibrium N = locations of normal forces on bases N = moment equilibrium N - 1 = side forces N - 1 = side force angles, 9 N - 1 = locations of side forces on slices I= factor of safety 3N total equations 5N - 2 total unknowns 340 Landslides: Investigation and Mitigation These methods therefore involve fewer equations solve for the factor of safety without using and unknowns than those shown in Table 13-1. assumptions. The computational accuracy involves The characteristics of various practically used only the accuracy with which the shear stress methods with regard to the conditions of equilib- required for equilibrium (t) and the normal stress rium that they satisfy, the assumptions they involve, can be evaluated. Computational accuracy is and their computational accuracies are summa- thus distinct from overall accuracy, which in- rized in Table 13-2. volves also the accuracy with which site condi- tions and shear strengths can be evaluated. By separating issues that determine computational 2.4 Computational Accuracy accuracy from the other