Investigation by limit analysis on the stability of slopes with cracks

Abstract

A full set of solutions for the stability of homogeneous c, φ slopes with cracks has been obtained by the kinematic method of limit analysis, providing rigorous upper bounds to the true collapse values for any value of engineering interest of φ, the inclination of the slope, and the depth and location of cracks. Previous stability analyses of slopes with cracks are based mainly on limit equilibrium methods, which are not rigorous, and are limited in their capacity for analysis, since they usually require the user to assume a crack depth and location in the slope. Conversely, numerical methods (e.g. finite-element method) struggle to deal with the presence of cracks in the slope, because of the discontinuities introduced in both the static and kinematic fields by the presence of cracks. In this paper, solutions are provided in a general form considering cases of both dry and water-filled cracks. Critical failure mechanisms are determined for cracks of known depth but unspecified location, cracks of known location but unknown depth, and cracks of unspecified location and depth. The upper bounds are achieved by assuming a rigid rotational mechanism (logarithmic spiral failure line). It is also shown that the values obtained provide a significant improvement on the currently available upper bounds based on planar failure mechanisms, providing a reduction in the stability factor of up to 85%. Charts of solutions are presented in dimensionless form for ease of use by practitioners.

An experimental model for slopes subject to weathering

“…The formation of two cracks is observed. Cracks can significantly reduce slope stability [28]. Through the analysis the location of each crack and failure mechanism can be tracked, the deformation of the slope and the velocity of the sliding mass can be calculated.…”

Discussion of “Stability assessment of slopes with cracks using limit analysis”

“…6). The calculation of the external work in this case of a composite partly log-spiral partly linear failure surface requires the calculation of the work done by the fictitious wedge BOCADB minus the work of the fictitious wedge DCAD (Utili 2013). The analytical expressions for the calculation of the external work done by soil masses sliding along composite log-spiral failure surfaces, which requires the use of fictitious wedges bordered by a log-spiral, was first presented in Utili (2005) and Utili and Nova (2008) for the case of slopes with a horizontal upper part subject to a sequence of landslides, and in Utili and Crosta (2011) for the more general case of slopes with an inclined upper part.…”

“…In the search for the failure mechanism of a slope of given inclination and friction angle (␤ and , respectively), the length and location of the crack is free, i.e., the minimization of the function is sought over four independent variables, the angles 0 , h , C (in Utili (2013) the angles , , and are employed with = 0 , = h , and = D ), and ␤= with the additional constraint that "the crack cannot be deeper than the maximum depth of the crack discussed." Concerning the variable ␤=, the discusser has shown that for > 5°, all the failure mechanisms pass through the slope toe, i.e., ␤= = ␤, whatever values of ␤ and are considered (Utili 2013), so that for the drained analyses presented in the paper under discussion with = 10°or greater, the number of variables to be considered in the unconstrained minimization can be reduced to three: 0 , h , and C .…”

“…However, regarding the findings and the formulation of the problem, Utili (2013) and the paper under discussion (Michalowski 2013) are rather different. In this Discussion, the discusser wants to highlight the main complementary and different findings between the two papers and point to some aspects of the paper under discussion that in the discusser's opinion need clarification, especially with regard to the following three topics (each being a section of the present discussion): calculation of the external rate of work for rotational failure mechanisms, failure mechanisms analyzed for pre-existing cracks, and failure mechanisms with crack formation.…”

“…The discusser has recently published a paper in Géotechnique titled "Investigation by limit analysis on the stability of slopes with cracks" (Utili 2013) that includes, for the first time (to the discusser's knowledge), a systematic investigation of the influence of the presence of cracks in uniform slopes for rotational failure mechanisms via the limit analysis upper bound approach. Looking at the discusser's paper and the paper under discussion (Michalowski 2013), a reader may note that the aim of the two papers is the same, namely, to assess quantitatively the effect of the presence of cracks on the stability of slopes employing the upper bound approach of limit analysis.…”