STABILITY OF SLOPES CURVED IN PLAN – AN EXAMPLE Oldrich Hungr, Geological Engineering Program, UBC, Vancouver Paul Wilson, Thurber Engineering Ltd., Vancouver ABSTRACT The influence of plan curvature of slopes on the Factor of Safety is often described in terms of pre-conceived “rules of thumb”. In this article, the authors describe a systematic parametric study carried out using a three-dimensional Limit Equilibrium program, searching for the critical ellipsoidal sliding surface for a series of slopes straight, concave and convex. It is shown that no simple rule applies and that slope curvature increases the Safety Factor both for convex and concave slopes. The use of three-dimensional analysis removes the need for guessing. The analysis does not consider the influence of slope curvature on the groundwater regime and this may also strongly affect slope stability. RÉSUMÉ L’influence de la courbure du sol sur le Facteur de sécurité est souvent décrite de manière empirique. Dans cet article, les auteurs décrivent une étude paramétrique systématique menée à l’aide d’un programme en 3D basé sur des méthodes d’équilibre limite. Ils ont recherché la surface de rupture ellipsoïdale critique pour différents profils de pente : plan, concave, convexe. Il est démontré dans cet article qu’il n’existe pas de règle simple qui s’applique dans ce cas, et que la courbure de la pente augmente le Facteur de Sécurité pour les pentes convexes comme pour les concaves. Cette analyse ne prend pas en considération l’influence de la courbure sur l’écoulement des eaux souterraines, celle-ci pouvant en réalité affecter de façon importante la stabilité de la pente. L’utilisation de modèle en 3D rend les méthodes empiriques obsolètes. 1. INTRODUCTION Traditionally, Limit Equilibrium slope stability analyses have been carried out in two dimensions, based on the assumption of plane strain conditions. This is an oversimplification, as real slope failures are always three-dimensional. The 3D nature of sliding surfaces in prismatic slopes (straight in plan and uniform in cross-section) results in an increase of the Factor of Safety due to “end effects” (e.g. Baligh and Azzouz, 1975). There is a proof that the Factor of Safety of a given 3D sliding surface must always be greater than the critical 2D Factor of Safety of any slope-cross-section intersecting that surface (Cavounidis, 1987). A common practice is to add a few percent to the 2D Factor of Safety on this account (e.g. Duncan, 1992, Cornforth, 2005). Three-dimensionality is important where slopes are curved in plan, such as in the concave corners of an excavation or convex ridges. Very little has been published on this subject. Piteau and Peckover (1978) reviewed a few studies, showing that concave rock slopes should be more stable and convex ones less stable than straight slopes. The purpose of this article is to examine the effects of plan curvature quantitatively, using a three-dimensional Limit Equilibrium analysis of two contrasting examples. 2. METHOD Extensions of well-known Limit Equilibrium methods of slope stability analysis started appearing in the literature in recent decades. The first methods dealt with purely

cohesive materials (e.g. Baligh and Azzouz, 1975). Later, more general Limit Equilibrium methods of slices have been extended into 3D methods of columns (e.g. the Ordinary Method, Howland, 1977, Bishop and Janbu Simplified Methods, Hungr, 1987, Hungr et al., 1989 the Spencer’s Method, Lam and Fredlund, 1993 and several others). To date, no proven method exists capable of analyzing problems that are strongly asymmetric in geometry or loading (e.g. Hungr, 1994). However, analysis of symmetrical settings is now fairly routinely used, removing the need for arbitrary adjustment of the Factor of Safety for such problems. The technique used for the solution of the present examples is a 3D extension of Simplified Bishop’s Method described in Hungr et al, 1989. The method is based on a direct application of the original Bishop assumptions: 1) the vertical shear stresses mobilized on inter-column surfaces can be neglected and; 2) overall moment equilibrium equation is used to solve for the Factor of Safety. The present implementation includes the modification due to Fredlund and Krahn (1977), incorporating the moment of the normal forces on column bases in the overall moment equilibrium equation. With this modification and the adoption of a well-placed centre of rotation, the method produces reasonable approximations of the Factor of Safety for some non-rotational sliding surfaces, particularly a composite surface consisting of an ellipsoidal scarp combined with a near-horizontal weak plane (Hungr, 1997).

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3. METHOD 3.1 A simple ellipsoidal failure surface. The first example is an extension of a simple circular failure surface in a slope characterized by a friction angle and a cohesion, with no pore-pressure (Figure 1).

D=20m D=-20m

Figure 1. Example 1, ellipsoid: central cross-section. Material parameters: γ=20kN/m3, φ=20˚, c=30kPa. The computer program implementing the method, CLARA-W, can generate curved slope surfaces by rotating the cross-section of the slope (without the sliding surface) around a vertical axis. Three slope configurations were created: a) A straight slope, where the cross-section was not rotated, but expanded into a long prismatic slope representing the plane strain case (Figure 2). b) a concave slope (re-entrant), where the cross-section was rotated around a vertical axis situated at a distance D=20 m in front of the slope toe (Figure 3) and c) a convex slope (ridge) where the vertical axis was -20 m behind the slope crest (Figure 4). The problem was first analysed with the straight slope in plane strain. The program can generate ellipsoidal sliding surfaces symmetrical around a horizontal axis, appearing as circles in the cross-section plane. Each ellipsoid is characterized by the ratio between the half-axis parallel to the axis of rotation and the radius of the central circle, called Aspect Ratio (AR). An AR of 1.0 produces a circle. An AR which is a large number (say, 10000) approximates a cylindrical surface (Figure 2). The program then conducts a conventional grid search trial ellipsoid analysis, for ellipsoids with a single given AR (Figure 5). At each of a grid of trial centers, ellipsoids of a number of radii are analysed to find the “critical ellipsoid” for that Aspect Ratio. Each search looks at approximately 1000 ellipsoids, each with 1000 to 2000 columns. The search then must be extended to different values of the Aspect Ratio. This is not done automatically. A separate grid search, as shown in Figure 5, is carried out and the interim “critical” Factors of Safety are plotted against the AR as shown in Figures 2a, 3a and 4a. The final “critical“ellipsoid is found as a minimum on this plot.

Figure 2. a) F vs. ellipsoid Aspect Ratio, straight slope (D=infinity). The horizontal line at F = 1.587 is the plane strain solution. b) Critical ellipsoid (cylinder), AR=infinity.

Figure 3. a) F vs. ellipsoid Aspect Ratio, concave slope (D=20m). b) Critical ellipsoid, AR=0.9.

Figure 4. a) F vs. ellipsoid Aspect Ratio, convex slope (D=20m). b) Critical ellipsoid, AR=1.5.

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Figure 5 A sample result of the grid search analysis. A search is being conducted for ellipsoids with a given Aspect Ratio (see inset) 3.2 A compound slip surface The second example is a compound failure surface generated by an intersection of an ellipsoid and a horizontal weak plane, daylighting in the toe of the slope. The material properties, applicable on the ellipsoidal scarps, are the same as in Example 1. The weak surface is planar and has zero cohesion and a residual friction angle of 10˚. There is a piezometric surface as shown in Figure 6.

Figure 6. Example 2, compound surface: central cross-section. Material parameters: γ=20kN/m3, φ=20˚, c=30kPa. Weak surface properties: c=0, φ=10˚. The cross- section shown in Figure 6 was again extended into a straight slope (Figure 7), then rotated around vertical axes to create a concave (Figure 8) and convex (Figure 9) slope as described previously. Again, grid searches were conducted for ellipsoids with different Aspect Ratios and plots of AR vs. F to find the Aspect Ratio corresponding to the minimum Factor of Safety. It is interesting that the minimum F for the convex surface was found to correspond to a cylindrical scarp (Figure 9).

Figure 7. a) F vs. AR, straight slope. The horizontal line at F = 1.587 is the plane strain solution. b) Crit. surface.

Figure 8. a) F vs. AR, concave slope. b) critical ellipsoid, AR=1.5.

Figure 9. a) F vs. AR, convex slope. b) critical ellipsoid, AR=1.5.

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4. RESULTS AND CONCLUSIONS The results of the parametric study are summarized in Tables 1 and 2. In each case, the lowest Factor of Safety corresponds to the plane strain case, as predicted by Cavounidis (1987). Increases of the Factor of Safety are given in the last column of each table. For the ellipsoidal surface, they amount to about 10-20%, consistent to frequently used “rules of thumb”. For the compound surface, the concave geometry produces a significant increase in the Factor of Safety (42%). The convex geometry produces significantly larger failure volumes. As shown in Figures 2 and 7, the “end effects” can account for about 20% increase in the Factor of Safety for the two geometries, though more can occur with other configurations (e.g. Hungr et al., 1989). The increases of the Factor of Safety due to three-dimensional effects are potentially significant and difficult to account for by simple approximate rules. They could lead to non-conservative results when stability analysis is used to back-calculate material strength parameters. The assumption that the plane strain case always produces the lowest F is compromised by the fact that most slope failures are limited in lateral extent, due to non-homogeneity of slope cross-sections. There are some geological effects of three-dimensionality that cannot be accounted for by Limit Equilibrium analysis. On one hand, lateral stresses in curved slopes could increase or decrease the confining effect and could also cause strength changes due to loosening (or lack thereof) of the soil or rock mass (e.g. Piteau and Peckover, 1978). On the other hand, convergence of groundwater flow lines could destabilize concave slopes as a result of increased pore pressures. Table 1, Summary, Ellipsoidal Surfaces Slope AR Volume F % Straight ∞ - 1.587 100 Concave 0.9 7560 1.865 117 Convex 1.5 10 100 1.732 109 Table 2, Summary, Compound Surfaces Slope AR Volume F % Straight ∞ - 0.885 100 Concave 0.5 6970 1.260 142 Convex ∞ 17300 0.986 104 ACKNOWLEDGEMENTS The spring, 2001 graduate class of EOSC 529, “Advanced Geotechnique”, in the Geological Engineering Program at the University of British Columbia produced the first version of this parametric study.

REFERENCES Baligh, M.M and Azzouz, A.S., 1975. End effects on the

stability of cohesive slopes. ASCE Journal of the Geotechnical Engineering Division, 101:1105-1117.

Cavounidis, S. 1987. On the Ratio of Factors of Safety in Slope Stability Analyses. Geotechnique, 37 (2), 207-210.

Cornforth, D.H., 2005. Landslides in practice. John Wiley and Sons, Hoboken, N.J.,590 pp.

Duncan, J. M. 1992. State-of-the-Art: Static Stability and Deformation Analysis. In Proc., ASCE Specialty Conference on Stability and Performance of Slopes and Embankments II, Berkeley, California, Geotechnical Special Publication 31, American Society of Civil Engineers, New York.

Fredlund, D.G. and Krahn, J., 1977. Comparison of Slope Stability Methods of Analysis. Canadian Geotechnical Journ., 14:429-439.

Hovland, H.J., 1977. Three-dimensional slope stability analysis method. ASCE Journal of the Geotechnical Engineering Division, 103:971-986.

Hungr, O. 1987. An Extension of Bishop's Simplified Method of Slope Stability Analysis to Three Dimensions. Geotechnique, Vol. 37, No. 1, pp. 113-117.

Hungr, O., 1997. Slope stability analysis. Keynote paper, Procs., 2nd. Panamerican Symposium on Landslides, Rio de Janeiro, Int. Society for Soil Mechanics and Geotechnical Engineering, 3: 123-136.

Hungr, O., E M. Salgado, and P.M. Byrne. 1989. Evaluation of a Three-Dimensional Method of Slope Stability Analysis. Canadian Geotechnical Journal, Vol. 26, pp. 679-686.

Piteau, D.R. and Peckover, F.L., 1978. Engineering of rock slopes. Chapter 9 in Landslides, Analysis and Control. National Academy of Sciences, Nat. Res. Coun., Washington, DC., Special Rep. 176:192-227.

Lam, L. and Fredlund, D.G., 1993. A general limit equilibrium model for three-dimensional slope stability analysis. Canadian Geotechnical Journal, Vol. 30, pp. 905-919.

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