ROCK MECHANICS 449 Figure 16 The fj theory for underground excavations (A) The fundamental stress conditions leading to slip, and how these are identified in the fj theory (B) Application to underground excavations in stratified materials with different frictional properties unstable, a further rigorous equilibrium analysis can be undertaken in order to determine their size and hence support requirements These analyses can be time consuming and prone to error if undertaken by hand, and so are usually performed by computer programs Another form of kinematic analysis that is widely used is the so-called fj theory This uses the principle that, if shearing along a fracture is to be avoided, the direction of action of the major principal stress acting on the fracture must lie within an angle fj, the friction angle, of the normal to the fracture Now, close to an underground excavation the major principal stress is tangential to the boundary This means that a geometric analysis can be used to identify those zones on the boundary where the orientation of the major principal stress is such as to induce shearing This is illustrated in Figure 16, for the case of a rock mass that contains one dominant set of fractures, and hence can be assumed to be transversely isotropic Techniques for the analysis of foundations on fractured rock are not as straightforward as for slopes or underground excavations However, with most engineering structures it is possible to use foundations that are sufficiently large as to impose stresses that are small enough to prevent instability from occurring If it can be assumed that the rock beneath the foundation is continuous, then the induced stresses may be determined using analytical solutions developed from the theory of elasticity This approach is widely used in soil mechanics, wherein the assumption of a continuum is often valid, but it is seldom used in rock mechanics for the reason that most near-surface rock masses are fractured However, the concept of ‘bulbs of pressure’, or contours of major principal stress, from these analyses is useful in rock mechanics As the fj theory shows, as the direction of action of the major principal stress is constrained by the presence of a fracture, the shape of the resulting stress contours is controlled by the fracture geometry Figure 17 shows a simplified solution for the radial stress that is applicable to the case of rocks that contain one dominant set of fractures, and so can be assumed to be transversely isotropic The contours of major principal stress are influenced strongly by the fracture geometry, showing that foundations on rock can induce high stresses in zones far removed from the foundation Engineering in Continuous Rock For underground excavations in continuous rock, or in locations where the in situ stresses are sufficiently high as to allow a fractured rock to act as if it were continuous, it is the stress induced around the excavation that leads to instability Simple analytical solutions for the induced stresses are available for only a restricted set of excavation geometries, namely, circular and elliptical openings that are in a state of plane strain This last constraint requires the excavations to be long, of constant cross-section, and with