450 ROCK MECHANICS Figure 17 Simplified relations for contours of constant radial stress for a foundation in stratified material resembling a transversely isotropic rock their axial direction parallel to one of the in situ principal stresses Despite these severe restrictions on geometry, the fact that engineers are commonly required rapidly to assess the stability of underground excavations means that these solutions are used extensively Figure 18 shows a cross-section through a circular opening and presents the well-known Kirsch equations for the induced-stress components These equations are used for many applications involving the analysis of stress around tunnels, wellbores, and mine shafts As an example, Figure 18 also shows simplified relations for the specific case of the maximum and minimum values of tangential stress induced at the boundary of an excavation, and how these components vary as the in situ stress ratio varies The figure shows how the maximum and minimum stresses are induced under uniaxial conditions, represented by k ¼ 0, and that for isotropic conditions, represented by k ¼ 1, the induced stress is constant around the boundary For the cases of stress induced around an elliptical excavation, a particularly elegant solution is available for the tangential stress induced at the boundary This is shown in Figure 19 Also, for the particular case of an elliptical opening with a horizontal major axis, the extreme boundary stresses, which are induced in the side walls, roof, and floor, are also given These final two equations can be equated, and doing so shows that the tangential stress is constant around the boundary in the case of an elliptical opening that has an aspect ratio equal to the in situ stress ratio This elegant result is a valuable design guide for engineers wishing to minimize the range of stresses induced around the boundary of an excavation Numerical Analysis With all of the fundamental analyses that are routinely used in rock mechanics and rock engineering, various simplifying assumptions are required in order to make the problem tractable These simplified procedures are perfectly acceptable for the majority of applications, but for applications requiring an accurate understanding of the behaviour of a structure, more complex analyses are necessary These are performed using computer programs that implement sophisticated numerical techniques The techniques most often used in rock mechanics are finite element, boundary element, and finite difference methods for continua, and various discrete element methods