It is usual practice in engineering to design a stable structure by ensuring that the stresses in the elements of the structure are always less than their strength, which may be defined as the yield stress, the ultimate stress, or the fatigue stress, according to the type of the load. Such restrictive design is not practical in the design of underground excavations since the rock surrounding such structures is usually failed, but only occasionally does this give rise to instability of the structure. Thus, while the usual criterion for the design of a stable structure is certainly sufficient, it is not necessary for underground excavations.
A proper understanding of the conditions necessary for stability is vital to the useful application of rock mechanics knowledge to the design of underground structures. Most tests on the properties and strength of brittle rocks are disrupted at or near maximum load by the rapid release into the specimen of energy stored in the resilience of the testing machine. This is in contrast with a phenomenon, which may be observed frequently, where the rock immediately surrounding an excavation is both failed and subject to stress. By postulating a complete stress-strain curve for rock, it can be shown that failure is unstable or stable, depending upon whether the stress-strain curve of the rock lies inside or outside the stress-strain characteristics of the applied stress. Stability is, therefore, a function of the stiffness of the system applying the stress relative to that of the failing rock.
Many workers have established that the strength of rock specimens under a lateral hydrostatic confining stress varies linearly with confining stress, in a way which may be expressed by the equation. Equation (1) (Available in full paper) where Co is the uniaxial compressive strength of a similar specimen; Cr is the strength of the specimen when subjected to a lateral confining stress, s3; and k is a constant.
This is in agreement with the Mohr-Coulomb theory of failure, if the Mohr-Coulomb envelope is assumed to be linear. In this case the theory postulates that a rock specimen under compressive stress will fail by shearing along a plane where the following condition is first satisfied: Equation (2) (Available in full paper) where t and s n,are the shear and normal stresses acting on the plane of failure, µ is a quantity defined as the coefficient of internal friction, and So is known as the cohesion of the material. The relationships between So, µ , Co, and k are Equation (3) (Available in full paper) and Equation (4) (Available in full paper)
Most present theories of rock failure postulate the existence within the rock of small flaws, known as Griffith cracks, the extension of which under sufficiently high stresses causes failure of the rock. Under compressive stresses, flat cracks have their opposite surfaces in contact, and as the applied stress changes, these crack surfaces slide over one another with coefficient of sliding friction µ'.