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This is a copy of the final published version of a paper published via gold open access in International Journal of Rock Mechanics and Mining Sciences.
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Babiker, A.F.A., Smith, C.C., Gilbert, M. and Ashby, J.P. (2014) Non-associative limit analysis of the toppling-sliding failure of rock slopes. International Journal of Rock Mechanics and Mining Sciences, 71. 1 - 11. Doi: 10.1016/j.ijrmms.2014.06.008
Non-associative limit analysis of the toppling-sliding failureof rock slopes
Ahmed Faysal A. Babiker a, Colin C. Smith a,n, Matthew Gilbert a, John P. Ashby b
a Department of Civil & Structural Engineering, University of Sheffield, Mappin Street, Sheffield S1 3JD, UKb Ashby Consultants Limited, 1/26 Bassett Road, Auckland 1050, New Zealand
a r t i c l e i n f o
Article history:Received 23 February 2013Received in revised form4 March 2014Accepted 22 June 2014
Keywords:Rock slope stabilityToppling-slidingNon-associative frictionPlasticityLimit analysis
a b s t r a c t
Limit analysis is a powerful procedure which is widely used in geotechnical engineering for the analysisof collapse states. However, when applied to toppling-sliding failures in rock slopes, overestimates ofstability can arise, and hence limit equilibrium and DEM approaches have proved more popular. In thispaper it is shown that limit analysis tends to overestimate stability due to the presence of potentiallyunrealistic dilatancy at the joints. To address this, a modified plastic limit analysis procedure incorpo-rating a non-associative, low dilation, friction model is proposed. Originally developed to assess thestability of masonry walls, the procedure is here extended to allow an envelope of potential solutions tobe obtained. The numerical results obtained are found to closely match both analytical and experimentalresults from the literature, demonstrating the significant promise of the procedure.& 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license
The influence of geological discontinuities on the stability ofrock slopes and in particular on their modes of failure had beenrecognised by a number of workers by the 1960s, as documentedin the literature, e.g. Hoek and Bray . Examples of rock slopefailure modes are shown in Fig. 1, and include planar 2-dimensional wedge, 3-dimensional wedge and circular failuremodes. Plane failures occur by sliding on a single discontinuity,whereas wedge failures require two or more intersecting disconti-nuities. Although circular arc failures were originally recognised insoil slopes, they can occur in large scale rock slopes where thediscontinuities are closely spaced, or where the intervening rock issufficiently weak to fail internally.
Toppling failure involves rotation of jointed columns aboutpivot points. Pure toppling failure only occurs in very steep oroverhanging slopes. On the other hand, toppling-sliding failure canoccur in flatter slopes or in larger scale slopes, where topplingcolumns force the rock near the toe of the slope to fail by slidingor shearing through intact material. The resulting failure surfaceis often approximately circular in shape. Studies of toppling-sliding failure were initiated in the late 1960s and early 1970susing physical models, e.g. Ashby . Subsequently various workers
have identified the toppling-sliding mechanism in failures observedin the field .
1.2. Toppling-sliding stability analysis
Various numerical and analytical procedures have been pro-posed to solve problems involving toppling-sliding failure of rockslopes. Considering numerical procedures first, the distinct ele-ment method (DEM) originally developed by Cundall  has beenapplied by workers such as Ashby , Ishida et al.  and Lanaroet al. . With DEM large displacements and separation of blockscan be modelled by incrementally solving equations of motion,using an explicit finite-difference method that requires small timesteps. Corners of blocks are rounded to model crushing, therebyeliminating singularities. In order to balance the kinetic energy inthe system a damping coefficient is introduced, tailored accordingto block size.
Considering analytical procedures, analysis of rock slope stabi-lity is usually undertaken by a combination of kinematic analysisand limit equilibrium analysis. Kinematic analysis involves inter-preting the geometry of the geological discontinuities and theslope to determine whether failure by a given mechanism isfeasible. The rock slope stability analysis is then carried out usinga limit equilibrium analysis of the given geometry as shown bye.g. Goodman and Bray  and Hoek and Bray . However in alimit equilibrium analysis the goal is to ensure global equilibriumis satisfied, regardless of kinematic considerations, i.e. the failuremechanism identified need not actually be kinematically admis-sible. Since then, in the absence of other readily available tools,
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journal homepage: www.elsevier.com/locate/ijrmms
International Journal ofRock Mechanics & Mining Sciences
http://dx.doi.org/10.1016/j.ijrmms.2014.06.0081365-1609/& 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/).
n Corresponding author. Tel.: 44 114 2225717.E-mail address: firstname.lastname@example.org (C.C. Smith).
International Journal of Rock Mechanics & Mining Sciences 71 (2014) 111
limit equilibrium procedures have been favoured by most workersin the field, e.g. Wyllie , Zanbak , Aydan and Kawamoto ,Adhikary et al. , and Sagaseta et al. .
In recent years, research on rock slope stability has continued.Many researchers have continued to focus on toppling-slidingfailures, e.g. Liu et al. , Tatone and Grasselli , and Liu et al., with limit equilibrium methods generally used to obtainsolutions. Additionally, some researchers have considered flexuraltoppling scenarios, in which flexural cracking of blocks is con-sidered (e.g., ) or block-flexure toppling, in which over-turning of some blocks is accompanied by flexural cracking ofothers (e.g. ). In this paper, the focus will be on modellingtoppling-sliding failures, without flexural cracking, though nowutilising a numerical approach based on the principles of limitanalysis rather than limit equilibrium, which has been the pre-dominant methodology used in this field to date.
1.3. Limit analysis
In limit analysis (see e.g. ) both global equilibrium andkinematic admissibility are enforced, potentially leading to morerigorous solutions when compared to limit equilibrium methodswhich generally consider equilibrium only and often requireadditional assumptions about forces acting in the system. Theonly issue is that in conventional limit analysis the relationshipbetween inter-block forces and displacements is governed by theassociative flow rule, which means that sliding along a joint will beaccompanied by separation, or dilatancy . Thus a tangentialdisplacement t along a joint will be accompanied by a normaldisplacement n, according to the following equation:
n t tan ; 1where is the angle of dilatancy. This will equal the angle offriction when using a conventional limit analysis approach dueto the normality principle, leading to what is termed associativefriction. This theoretical principle makes the analysis of theproblem relatively straightforward by eliminating a source of staticindeterminacy and allows the application of the powerful upperand lower bound theorems of plasticity. However, little or nodilation is typically observed in practice (i.e. Fig. 2b vs. 2a). Whilethis does not tend to affect analysis of sliding failures due to thesimple mechanics and kinematics involved, analysis of toppling-sliding failure is complicated by the indeterminacy that affects thelocation of the points of contact between blocks and the forces atthose points. This can lead to a conventional limit analysis over-estimating the actual stability of rock slopes, given that thesegenerally comprise tightly packed assemblages of blocks.
However, since the essential features of a limit analysis approachare otherwise attractive, here a method of modelling problems witha reduced, more realistic, level of dilation within a limit analysisframework is explored, leading