Closure to “Lower-Bound Finite Elements Limit Analysis for Hoek-Brown Materials Using Semidefinite Programming” by Jyant Kumar and Debasis Mohapatra

The writers are thankful to the discussers for their constructive discussion on the work. Rather than keeping 𝛼=0.5 for all geological strength index (GSI) values, the discussers presented a method to account for the actual value of 𝛼 in the generalized Hoek-Brown (GHB) criterion by using the reduced values of the material input parameters to solve the problem by using the modified Hoek-Brown (MHB) criterion; for the definitions of GSI and 𝛼, refer to the original paper, and the MHB and GHB criteria are defined exactly the same way as introduced by the discussers. In the original paper, it was pointed out that due to an assumption of using 𝛼=0.5, a significant overestimation of the collapse load occurs for GSI≤30. The discussers provided a detailed investigation on exploring how the choice of 𝛼 affects the solutions for values of GSI≤30. Figs. 2–4 plotted by the discussers provide the difference between the MHB and GHB criteria for different values of GSI, stress range, and 𝑚𝑖. These figures explain the reason for the difference between the obtained solution and that reported in the literature by Merifield et al. (2006) and Clausen (2013). The main contribution of the discussers is that a new method has been proposed to take into account the true value of 𝛼 by choosing the reduced values of the input strength parameters (𝜎𝑐⁢𝑖, 𝑚𝑏, and 𝑠) to be used in the MHB criterion based on the least-square optimization so that the yield envelope of the MHB criterion fits closely with the GHB criterion. The method introduced by the discussers is simple and straightforward and can be easily implemented. By using two worked examples, the discussers indicated that the reduced material input parameters can significantly reduce the strength overestimation of the MHB criterion for lower values of GSI, although no comparison of the final solution with the GHB was provided in the discussion. The major disadvantage, however, associated with the recommended approach of the discussers is the choice of appropriate range of 𝜎1 in order to compute the reduced values of the input strength parameters to be used in the MHB criterion. It is, however, going to be a difficult exercise to select the correct range of 𝜎1. If the selected range of 𝜎1 is kept very large, it is not going to be an easy task to find the perfect match between the two yield envelopes using the GHB and the MHB criteria.