Analytical representation of Mohr failure envelope approximating the generalized Hoek-Brown failure criterion

The latest version of the Hoek-Brown criterion, referred to as the generalized Hoek-Brown (GHB) criterion, is an empirical failure condition for rock mass that correlates the rock mass strength with the data collected from the laboratory tests and field observations by use of GSI values. A shortcoming of the GHB criterion is that it does not provide an explicit relation between the shear and normal stresses on the failure plane, which limits its application in many geotechnical design calculations. In order to extend the application of the GHB failure criterion to limit equilibrium analysis and/or the upper-bound approach of limit analysis, the analytical approximations of the Mohr failure envelope corresponding the GHB criterion are derived. At present, the exact formulation is not available except for a=0.5, i.e. for a rock mass with GSI=100. In the formulation proposed here, the 1st- and 2nd- order Taylor approximations are employed to define the relation between the dimensionless normal stress and the sine of the instantaneous friction angle. Given the latter, the corresponding shear stress at failure is calculated from the established explicit relationship between the shear stress and the instantaneous friction angle. The Taylor expansion is carried out by taking the sine of the instantaneous friction angle as the expansion variable. A series of examples are given which demonstrate that the approximated Mohr envelope, derived by invoking the 2nd-order Taylor approximation with the expansion center of 0.5, is nearly identical to the exact analytical form. Even the 1st-order Taylor approximation with the expansion center of 0.5 is proven to be satisfactory. Furthermore, it is shown that the formulation is valid for a broad range of values of GSI. The latter has been verified by simulating the conventional triaxial compression tests within the framework of the critical plane approach (CPA) adopting the proposed form of Mohr failure envelope as a failure function. The results confirm that the calculated axial stresses at failure and the orientation of the failure plane are almost identical to the values stipulated by the original GHB criterion.