Reply to the discussion by Utili on “Stability assessment of slopes with cracks using limit analysis”

Abstract

The author would like to thank the discusser for his interest in the paper. The discusser provided multiple comments, and suggested that clarifications be offered in regard to three areas: (i) calculations of the work rate of the weight of the soil, (ii) failure mechanism with a pre-existing crack, and (iii) collapse mechanism with crack formation. The author’s response comments are given in that order. It was indicated in the paper under discussion (Michalowski 2013) that the equations for calculating the work rate of the soil weight were omitted, because they can be found in multiple papers, including the original paper by Chen et al. (1969). The discusser does not seem to agree with this statement as the shape of the moving mass of a slope with a crack (or cracks) is different from that considered by Chen and others. However, proper application of the original equations makes new derivations entirely unnecessary. The work rate of the soil weight was considered in the paper by first calculating the work rate of the moving “uncracked” block ACOBDA (Fig. 6 of the paper under discussion), and then calculating the work rate for a vertical slope of height CD to the right of the crack (ACDA). Subtracting the latter from the former yields the required work rate. Calculating the two rates does not require any equations beyond those in the paper of Chen et al. (1969). The author did not find it necessary to repeat these equations in his paper. The second issue raised by the discusser relates to the mechanism with an existing crack(s). This part of the discusser’s enquiry is more of a commentary, with an emphasis on some of his own contributions in a recently published paper (Utili 2013). Regarding below-toe failures, indeed, the critical mechanisms for slopes with > 5° (with no pore-water pressure) are all toe mechanisms, but the credit for this should go to Chen and Giger who indicated this clearly in 1971 (Chen and Giger 1971). However, the range where below-toe failure is found critical is wider for “shallow” slopes in the presence of pore-water pressure (Michalowski 1995). The search for the best upper bound to the dimensionless critical height H/c was formulated as a constrained optimization problem (the depth of an existing vertical crack cannot be larger than the critical height of a vertical slope in the same soil). However, in most cases the crack depth constraint did not alter the solution, i.e., the best solution was found away from the constraint. The exceptions were found among slopes subjected to pore-water pressure, which the discusser did not consider. Further comments of the discusser are related to his formulation of the problem (Utili 2013), and they reveal a different, perhaps questionable, philosophy of approach from that of the author. It may be helpful to bring up some of the fundamental aspects of limit analysis. The kinematic theorem of limit analysis only indicates that the integral of the rate of internal (dissipated) work is not smaller than the integral of the work rates of true external forces in any kinematically admissible mechanism. Consequently, the theorem can be used to find an upper bound to an active limit load or, as in the paper under discussion, an upper bound to dimensionless group H/c. However, the theorem cannot be used to find the location (or depth) of an existing crack. The location of an existing crack must be a part of the geometry definition of the problem, just as the inclination of the slope is, and it must be known a priori. However, in his cited paper (Utili 2013), the discusser formulated three different problems: one with an “unspecified location” of the crack, one with unspecified crack depth, and one with both being unknown. The method can only provide information as to where the most adverse location of the crack would be in the kinematically admissible mechanism considered in the analysis. This location was calculated in both Michalowski (2013) and Utili (2013) by making the location of the crack one of the variables in the optimization procedure. This, however, cannot be considered as finding the true location of the crack, just as the mechanism used in the analysis cannot be considered a true failure mechanism. Therefore, in the paper under discussion, the problem was formulated differently: the slopes considered had multiple cracks and, among those pre-existing cracks, there was always one in very close proximity to the most critical location determined by the mechanism used in the solution. This is a subtle difference in formulation from that of the discusser, but it allows one to arrive at the solution without suggesting that the crack location so determined is the true location of a pre-existing crack.