kChapter 15 NORMAN I. NoiuusH AND DUNCAN C. WYLLIE ROCK SLOPE STABILITY ANALYSIS 1. INTRODUC11ON ities of the same set measured normal to the dis- continuity surfaces is called spacing. Together, xcept for the rare case of a completely un- persistence and spacing of discontinuities define E fractured rock unit, the majority of rock the size of blocks. They also influence the dila- masses can be considered as assemblages of intact tancy of the rock mass during shear displace- rock blocks delineated in three dimensions by a ment and determine the extent to. which the system of discontinuities. These discontinuities mechanical properties of the intact rock will can occur as unique randomly oriented features or govern the behavior of the rock mass. as repeating members of a discontinuity set. This Surface properties: The shape and roughness system of structural discontinuities is usually re- of the discontinuity constitute its surface proper- ferred to as the structural fabric of the rock mass ties, which have a direct effect on shear strength. and can consist of bedding surfaces, joints, folia- Infillings:Minerals or other materials that tion, or any other natural break in the rock. In occur between the intact rock walls of disconti- most cases, engineering properties of fractured rock nuities are termed infillings. Their presence can masses, such as strength, permeability, and de- affect the permeability, and shear strength of a formability, are more dependent on the nature of discontinuity. Secondary minerals, such as cal- the structural fabric than on the properties of the cite or quartz, may provide significant cohesion intact rock. For this reason, practitioners in the along discontinuities. However, these thin in- field of rock mechanics have developed the follow- fillings are susceptible to damage by blasting, re- ing parameters to characterize the nature of the sulting in total loss of cohesion. discontinuities that make up the structural fabric: For all but very weak rock materials, the analysis Orientation: The orientation of a discontinuity of rock slope stability is fundamentally a two-part is best defined by two angular parameters: dip process. The first step is to analyze the structural and dip direction. fabric of the site to determine if the orientation of Persistence: Persistence refers to the continuity the discontinuities could result in instability of the or areal extent of a discontinuity and is particu- slope under consideration. This determination is larly important because it defines the potential usually accomplished by means of stereographic volume of the failure mass. Persistence is diffi- analysis of the structural fabric and is often referred cult to quantify; the only reliable means is map- to as kinematic analysis (Piteau and Peckover 1978). ping of bedrock exposures. Once it has been determined that a kinematically Spacing: The distance between two discontinu- possible failure mode is present, the second step re- 392 Landslides: Investigation and Mitigation quires a limit-equilibrium stability analysis to com- Wedge failures involve a failure mass defined by pare the forces resisting failure with the forces two discontinuities with a line of intersection causing failure. The ratio between these two sets of that is inclined out of the slope face [Figure forces is termed the factor of safety, FS. 15-1(b)], For very weak rock where the intact material Toppling failures involve slabs or columns of strength is of the same magnitude as the induced rock defined by discontinuities that dip steeply stresses, the structural geology may not control into the slope face [Figure 15-1(c)], and stability, and classical soil mechanics principles Circular failures occur in rock masses that are for slope stability analysis apply. These procedures either highly fractured or composed of material are discussed in Chapter 13. with low intact strength [Figure 15-1(d)]. 2. TYPES OF ROCK SLOPE FAILURES Recognition of these four categories of failures is essential to the application of appropriate analyt- As shown in Figure 15-1, most rock slope failures ical methods. can be classified into one of four categories depend- ing on the type and degree of structural control: 3. STEREOGRAPHIC ANALYSIS OF STRUCTURAL FABRIC Planar failures are governed by a single disconti- nuity surface dipping out of a slope face [Figure From a rock slope design perspective, the most 15-1(a)], important characteristic of a discontinuity is its (a) Wedge failure on two Planar failure in rock in which a intersecting discontinuities with discontinuity 'daylights" the a line of intersection which slope face "daylights" the slope FIGURE 15-1 (facing pages) Types of rock slope failures: planar failure, wedqe failure, toppling failure, circular failure (diagrams modified from Hoek and Bray 1981). - N.I. NORRIS!-! AND D.C. WYLLIE Rock Slope Stability Analysis 393 orientation, which is best defined by two parame- The rock slope practitioner is referred to the ters: dip and dip direction [Figure 15-2(a)]. The dip work by Hock and Bray (1981), Hock and Brown angle refers to the inclination of the plane below (1980), and Goodman (1976) for in-depth treat- the horizontal and thus ranges from 0 to 90 de- ment of the principles of stereographic analysis. A grees. The dip direction of the plane is the azimuth detailed presentation of the procedures for plot- at which the maximum dip is measured and ranges ting, analyzing, and interpreting data on stereo- from 0 to 360 degrees. The dip direction differs graphic projections is beyond the scope of this from the strike direction by 90 degrees and is the report; however, these techniques are essential to preferred parameter to avoid ambiguity as to the di- rock slope design and to the following discussion. rection of dip. These values are determined by Stereographic presentations remove one dimen- compass measurements on rock outcrops (Chapter sion from consideration so that planes can be rep- 9), oriented drilling techniques, or interpretation resented by lines and lines represented by points. of geologic structural trends (Chapter 8). Stereographic analyses consider only angular re- Interpretation of these geologic structural data lationships between lines, planes, and lines and requires the use of stereographic projections that planes. These analyses do not in any way represent allow the three-dimensional orientation data to he the position or size of the feature. represented and analyzed in two dimensions. The The fundamental concept of stereographic pro- most commonly used projections are the equal- jections consists of a reference sphere that has a area net and the polar net, replications of which fixed orientation of its axis relative to the north are shown in Appendix A. and of its equatorial plane to the horizontal [Figure (c) Toppling failure in hard rock Circular failure in overburden with slabs or columns defined soil, waste rock or heavily by discontinuities that dip fractured rock with no steeply into the slope identifiable structural pattern I 394 Landslides: Investigation and Mitigation FIGURE 15-2 Concepts for northorth stereographic representation of rend linear and planar dip direction features (modified plunge from Hoek and Bray 1981). *dJ Linear Features Planar Features (a) Definition of Terms north north stnke dip direction trend lunge dip point representation of line lower half lower half reference sphere reference sphere great cde representation of a plane Linear Features Planar Features (b) Representation on Reference Sphere 15-2(b)]. Linear features with a specific plunge and the center. Similarly, planar features are positioned trend are positioned in an imaginary sense so that so that the feature passes through the center of the the axis of the feature passes through the center of reference sphere and produces a unique intersec- the reference sphere. The intersection of the linear tion line with the lower half of the reference feature with the lower half of the reference sphere sphere [Figures 15-2(b) and 15-3(b)]. The projec- defines a unique point [Figure 15-3(a)]. Depending tion of this intersection line onto the stereographic on the type of stereographic projection, this point plot results in a unique representation of that plane is rotated down to a unique point on the stereonet. referred to as a great circle. Planes with shallow dips For the purposes of this discussion, only equal-area have great circles that plot near the circumference nets will be considered, although the reader should of the net, and those with steep dips plot near the be aware that equal-angle projections can also be center. Planes are used to represent both'disconti- used. Linear features with shallow plunges plot nuities and slope faces in stereographic analyses. near the circumference of the stereographic pro- A useful alternative method of representing jection, whereas those with steep plunges plot near planes is to use the normal to the plane. This nor- Rock Slope Stability Analysis 395 mal in a stereographic projection will be a unique Zenith point that is referred to as the pole to the plane [Figure 15-3(b)]. Structural mapping data of dis- continuities are often plotted in the pole format Refere1 Spher1e rather than the great-circle format in order to de- tect the presence of preferred orientations, thus defining discontinuity sets, and to determine N mean and extreme values for the orientations of these sets. As shown in Figure 15-4, this process can be facilitated by contouring to accentuate and Equal distinguish the repetitive features from the ran- Area dom or unique features. Computerized graphical Net methods greatly facilitate the analysis of large amounts of structural data. However, when fewer than 50 data points are involved, manual plotting (a) Linear Features and analysis are probably more efficient. The intersection of two planes defines a line in space that is characterized by a trend (0 to 360 degrees) and plunge (0 to 90 degrees).