448 ROCK MECHANICS Figure 14 Example of kinematic assessment overlay for rock slope instability due to plane and wedge sliding the excavation, and therefore may be unstable Again, for no external forces and fractures possessing only frictional strength, the kinematic criteria relate to orientations only, with the result that, once more, the hemispherical projection can be used rapidly to assess stability Figure 14 shows an overlay that can be used for assessing instability due to plane and wedge sliding in a rock mass for which the friction angle is 30 To use this overlay, the fracture data (both poles and great circles) are plotted on a sheet of tracing paper and placed on a hemispherical projection net, with a second sheet of tracing paper on which the assessment overlay is drawn placed over the top The overlay is then rotated to identify those slope orientations that give rise to either poles or intersections falling within the unstable zones Because the overlay also shows various slope angles, it can quickly be used to select a slope angle that avoids the kinematic conditions leading to instability A similar analysis can be undertaken for underground excavations Figure 15 shows three hemispherical projections illustrating the unstable kinematic conditions that apply to a tetrahedral block formed by three fractures and the horizontal roof to a tunnel or cavern These blocks are identified on the projection as areas bounded by three great circles If the vertical direction (i.e., the centre of the projection) lies within the projection of the block, this indicates that the block is able to fall In the case of sliding instability, the direction of sliding is the steepest of either a line of maximum dip or the intersection of two planes For blocks that are identified as being Figure 15 Identification of falling, sliding, and stable blocks in the roof of an underground excavation (A) Block 345 falls verti cally (B) Block 234 slides on the intersection of planes and (C) Block 245 slides on the line of maximum dip of plane