The bi-axial stress test is a two-dimensional (2-D) field test in which horizontal and vertical stresses are applied to a circular β€œhole in the plate” problem. The diametric closures are measured at three uniformly spaced locations. Using the Kirsch solution, and assuming an isotropic elastic medium, the magnitude of the in-plane far-field principal stresses (𝜎1, 𝜎2) and their orientation (𝛼) can be uniquely determined. However, the Kirsch Solution has limitations, since the two-dimensionality is applicable only for conditions of plane strain or plane stress; and in reality, the out-of-plane stress condition is somewhat in between. The major outcome of this paper is the development of a methodology to determine the three-dimensional principal stresses (𝜎1, 𝜎2,𝜎3) and their orientations (𝛼,𝛽,𝛾) using the two-dimensional Kirsch Solution in three mutually perpendicular planes.


The Sanford Underground Research Facility (SURF, formerly known as DUSEL) is a dedicated underground scientific research facility located in Lead, South Dakota, at the former site of the Homestake gold mine. The SURF mission includes the construction of large underground cavern openings at depths of about 1500 meters to house large-scale physics experiments. A primary rock mechanics concern is the stability of these proposed large-diameter caverns (up to 165 meters) in a host rock which has been characterized to be orthotropic in the following conditions: (1) geometry, (2) in situ stress, (3) elasticity, and (4) strength. Of these four conditions of orthotropy, the second dimension (in situ stress) will be examined in this paper. The objective of this research is to present the theoretical development and implementation of the equations that define two- and three-dimensional orthotropic in situ stress. The equations are developed based on an interpretation of the biaxial stress test in two dimensions and extrapolation of multiple orthotropic field test results in three dimensions. The two-dimensional results are presented in the next section. These two-dimensional equations are used in the development of the three-dimensional equations for determination of the principal stress magnitudes and orientation, which are given in the subsequent section. The method is validated with a numerical example and will be applied to a data set of test results performed in situ to determine the orthotropic stress state (both magnitudes and orientations) in an underground mining location. The results of this mathematical development are applied to an orthotropic biaxial stress field test in Section 4, followed by a number of important conclusions in Section 5. The paper is concluded with a list of applicable references and several supporting figures and tables of results.

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