This document describes methods of assessing the potential for wedge sliding instability, using stereonet kinematic analysis.

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1 Wedge Failure Kinematic analysis on a stereonet This document describes methods of assessing the potential for wedge sliding instability, using stereonet kinematic analysis. 1. The traditional approach has been to consider the orientation of the intersection line of two planes. 2. An alternative approach considers the normal to the intersection line (wedge pole method). Wedge versus Planar sliding Kinematic wedge sliding and planar sliding analyses using a stereonet are fundamentally very similar. Planar sliding is sometimes considered a special case of wedge sliding, or vice versa. One difference between planar and wedge sliding analyses, is that for wedge sliding analysis, it is not necessary to consider lateral kinematic limits on the sliding direction (i.e. the plus/minus 20 degree limits), as is done with pure planar sliding analysis. In general, as long as the intersection line of a wedge satisfies the daylighting condition, it is not necessary to impose additional lateral restrictions on the sliding direction. Because the joint planes act as release surfaces, it is always possible to remove a wedge with a daylighting intersection line. (In practice, wedges may not be easily removable, e.g. thin wedges can wedge themselves in place, but we do not consider those issues here). Wedges do not always slide along the line of intersection. Depending on the wedge geometry, they may slide on only one of the joint planes. For example, a wedge line of intersection can be nearly parallel to the strike of the slope, and the wedge can still slide out of the slope on one joint plane). This might be called planar wedge sliding, when a wedge slides along one joint plane, rather than the line of intersection. For the purpose of stereonet analysis of wedge sliding, it is assumed: both sliding planes of a wedge have the same friction angle. cohesion of the joint planes equals zero (same as the planar analysis assumption) This friction only analysis assumption allows a rough safety factor to be calculated (tan theta / tan phi).

2 Intersection Line Method The intersection of two arbitrary planes forms a line in 3-dimensional space. Kinematic analysis of wedge stability is simply based on comparing the orientation of the line of intersection of two planes with respect to the slope plane orientation. In order for a wedge to slide, the line of intersection must daylight in the slope face. The most common check for wedge sliding is illustrated in the following figure. If a wedge intersection line falls within the crescent shaped zone defined by the slope plane (great circle) and the friction circle, then wedge sliding can occur. Figure 1: critical wedge sliding zone (slope plane = 65/180, friction angle = 30) Wedge intersection lines which fall within the region indicated in Figure 1, represent daylighting wedges which can slide (i.e. the wedge intersection line dips out of the slope, and more steeply than the friction angle). Because we are analyzing actual sliding directions (rather than poles), the friction circle is defined with respect to the perimeter of the stereonet. This is defined by a cone (small circle) with a vertical axis, and angular radius equal to 90 minus the friction angle. In the above example, the friction angle = 30, so the cone angle = = 60 degrees. The orientation of the slope plane in the above example is 65/180.

3 Wedge Sliding Modes The wedge intersection line must always daylight in order for wedge sliding to be possible, however sliding does not necessarily occur along the line of intersection. For the critical zone shown in Figure 1, there are two possible wedge sliding modes: Sliding along the line of intersection (i.e. sliding on both planes simultaneously) Sliding along one wedge plane The wedge sliding mode can be determined on a stereonet according to Hocking (1976). Assuming an intersection line which daylights in the critical zone shown in Figure 1: If the dip vector of one of the joint planes lies between the slope dip direction, and the line of intersection trend, then the wedge will slide on one plane If the dip vectors of both planes lie outside of this region, then sliding will be along the line of intersection. This is illustrated in the following figures.

4 This could be offered as an optional analysis feature (i.e. show the wedge sliding mode on the stereonet). Another case should be considered: it is possible to have wedges where the line of intersection dips LESS THAN the friction angle, but sliding can still take place on one plane, if the dip vector of the plane falls within the critical zone of Figure 1. This is shown in the following figure. If the wedge intersection line falls in either of the yellow regions, AND The dip vector of one plane falls in the red region. Then wedge sliding can occur on the one plane. Basically, these are wedges where the intersection line has a shallow dip, and strikes sub-parallel to the slope. Figure 2: critical zones for wedge sliding on one plane This wedge failure mode is not usually considered for stereonet analysis (probably due to the difficulties involved in graphically evaluating these wedges). However, it is a valid case, and was noticed while viewing the output from Swedge. This is shown in the following figure, a Combination Analysis with output displayed on the stereonet, and failed wedge intersections highlighted in red. Notice that the extent of the failed wedge intersections corresponds (approximately) to the sum of the yellow and red areas in the above figure, and is larger than the traditional crescent shape considered for wedge sliding on a stereonet. (The output does not correspond exactly, because the actual distribution of poles in the original dataset is not uniformly distributed on the stereonet. If

5 a larger number of uniformly distributed poles were analyzed, the regions would correspond closely.) Figure 3: Swedge combination analysis, stereonet output, failed wedge intersections occupy crescent shaped region (note: failed poles are also highlighted in red). Notice that the crescent shaped region of failed intersections in Figure 3, corresponds to the total crescent shaped region of Figure 2 (red + yellow zones). Failed intersections corresponding to the red zone of Figure 2 may represent sliding on one plane or two planes. Failed intersections corresponding to the yellow regions of Figure 2 can only represent sliding on one plane, because the line of intersection dips less than the friction angle. Note the following special cases: If the dip direction of the intersection line = slope dip direction, then sliding will occur along the line of intersection.

6 Conversely, if the dip vector of one plane has the same dip direction as the slope, then sliding will take place on this plane. In the special case you could have the same dip direction for the intersection line, one plane and the slope, and sliding would be in the common direction of all three. The intersection point could be coincident with the dip vector of one plane, in which case the wedge would slide along the line of intersection and the dip vector of the plane simultaneously.

7 Wedge Pole method of stereonet wedge stability An alternative method of evaluating wedge sliding using stereonets, has been proposed. Rather than plotting the wedge intersection lines, normals to the wedge intersection lines are plotted. This has been called the wedge pole method. The procedure is described in detail in a discussion paper by Laurie Richards (ref). One of the main goals of this alternative method, is to allow the assessment of planar and wedge stability, simultaneously, using the same constructions on the stereonet (i.e. use the same daylight envelope and pole friction cone, and plot poles to planes and wedge poles). Figure : Wedge Pole construction for wedge stability (Note: in Laurie Richards paper, he does not mention the plus/minus strike limits for planar sliding, and implies that planar and wedge sliding both use the full daylight envelope (this is also implied in the original Dips manual). However, the planar strike limits are important, and constitute a practical difference between wedge and planar sliding, which lessens the value of trying to use the same templates for planar and wedge sliding.)

8 Figure : Contouring of wedge poles However, the advantages of the wedge pole method are not clear, when compared to the dip vector method of planar sliding and the intersection line method of wedge sliding. These are also complementary methods, which use the same constructions (slope plane great circle, and plane friction cone measured from the stereonet perimeter). The wedge pole method requires that normals to the wedge intersection lines are generated, a procedure which is not very intuitive and requires an additional step of work. Furthermore, many practitioners and students find the dip vector method of stereonet kinematic analysis, much more intuitive and easier to understand than the pole plot methods. In view of these considerations, the value of the wedge pole method is not clear, although it certainly is a valid alternative.

9 Methods of determining wedge intersection lines We should note that there are various ways of generating the wedge intersection lines. 1. One common method is to first determine the distinct joint sets in your data (usually determined from a contour plot). Then determine the mean orientation plane for each joint set. Then determine all of the possible intersections of the mean planes, and check to see if any of these intersections fall in the critical zone. This is the method presented in the original Dips manual example of wedge sliding analysis. The obvious drawback of this method is that only mean joint planes are considered. This will highlight any obvious wedge stability problem, but does not consider the many individual wedges which can be formed by all possible joint combinations. Figure x: Intersections determined from mean (or major) planes only. 2. Other methods attempt to bracket the range of orientations represented by each joint set, by estimating an upper and lower bound for each set. The intersections of these bounding planes then form curved rectangular regions, which can be checked against the critical wedge sliding zone. A rather inefficient and clumsy attempt to improve upon the mean plane method. This method is suitable as a graphical manual method.

10 Figure x: joint sets bracketed by bounding planes, defines a region of possible intersections. 3. The most comprehensive check, is to generate all possible combinations of two joint planes, and plot all resulting intersection lines on the stereonet. With current computing power, this is now easily done, and allows every possible wedge orientation to be checked. This is available as the Combination Analysis option in Swedge. Other Notes Plotting of upper face plane on stereonet (reduces the number of valid daylighting intersection lines if the upper face has a non-zero dip angle).

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