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Home , Shear strength (soil), Slope stability analysis

A New Era in : Reduction Finite Element Technique

This article examines the merits of using finite element analysis versus limit-equilibrium methods for slope and analysis Geomechanics used worldwide by geotechnical

Article prepared for RocNews software tools for and Summer 2004 A New Era in Slope Stability Analysis: Shear Strength Reduction Finite Element Technique

Finite Element (FE) analysis is now widely accepted in routine excavation and support design and analysis. It may also revolutionize routine slope stability analysis.

Practicing engineers and mass along a slip surface. The academics alike are becoming popularity of limit-equilibrium increasingly convinced of the methods is primarily due to their benefits of the Shear Strength relative simplicity, ready ability to Reduction (SSR) technique for evaluate the sensitivity of stability “Things should be determining slope factor of to various input parameters, made as simple as safety. This method is particularly and the experience geotechnical possible, but not any useful when several different engineers have acquired over the simpler.” modes of failure are possible; it years in interpreting calculated — Albert Einstein automatically finds the critical factor of safety values. Limit- mechanism. But has the SSR equilibrium methods require technology matured to the minimal input data. The factor stage where it can be used for of safety values they output routine analysis? In this article, help engineers to guard against we will attempt to answer this uncertainties such as ignorance question, as as offer insights about the reliability of input into today’s slope parameters and loadings, and practices. the possibility that identified failure mechanisms may differ Limit-Equilibrium Slope from actual behaviour. As Stability Analysis well, recommended factor of Limit-equilibrium methods safety values for slopes and are the most commonly used excavations generally ensure approaches for analyzing that deformations are within the stability of slopes. The acceptable range. fundamental assumption at their core is that failure occurs Despite all the benefits, the through sliding of a block or limit-equilibrium approach has some important deficiencies.

2 The technique neglects - literature. They were mostly strain behaviour of and based on an approach that flows rocks. It also makes arbitrary naturally from the definition assumptions (mostly regarding of slope factor of safety, and is inter-slice ) to ensure now commonly referred to as static determinacy. It is awkward the Shear Strength Reduction to use for analyzing stability (SSR) technique. By definition, problems, such as the failure of the factor of safety of a slope cantilever and retaining walls, in is the “ratio of actual soil shear which failure involves deformed strength to the minimum shear wedges. strength required to prevent failure,” or the factor by which The Shear Strength soil shear strength must be Since the finite Reduction Technique reduced to bring a slope to element method Rapid advances in computer the verge of failure (Duncan, was first applied technology and sustained 1996). In the SSR finite element to geotechnical development have pushed the technique elasto-plastic strength engineering in (FEM) is assumed for slope materials. 1966, it has grown and other The material shear strengths tremendously in approaches to the forefront of are progressively reduced until popularity. geotechnical practice. Since it collapse occurs. was first applied to in 1966, the FEM For Mohr-Coulomb material has grown tremendously in shear strength reduced by popularity, primarily due to a factor (of safety) F can be its ability to analyze a very determined from the equation broad range of problems, while yielding realistic results. It can accommodate practically all This equation can be re-written as kinds of , and can model key aspects of material behaviour such as stress paths ( sequence), and In this case, coupled stress-pore pressure variations. and

In the mid 1970s, techniques are reduced Mohr-Coulomb for applying the FEM to shear strength parameters, and slope stability analysis started these values can be input into an appearing in geotechnical FE model and analyzed.

3 Basic Algorithm Advantages For Mohr-Coulomb materials, The elasto-plastic SSR FE the steps for systematically approach offers a number of searching for the critical factor significant advantages over of safety value, F, which brings traditional limit-equilibrium a previously stable slope to the analysis. First, it eliminates the verge of failure, are as follow need for a priori assumptions on failure mechanisms (the type, Step 1: Develop an FE model of a shape, and location of failure slope, using the and surfaces). The SSR technique strength properties established automatically establishes the for the slope materials. critical failure mechanism. It Compute the model and record The SSR FE eliminates artificial separation the maximum total deformation technique minimizes of slope problems into those in the slope. the expertise required involving slip surface failures, in finding critical Step 2: Increase the value of F and those involving failure of failure mechanisms and calculate factored Mohr- deformed wedges. for certain slope Coulomb material parameters Although possibly more problems. as described above. Enter the demanding in other aspects, new strength properties into the the SSR FE technique minimizes slope model and re-compute. the expertise required in finding Record the maximum total critical failure mechanisms deformation. for certain slope problems. Step 3: Repeat Step 2, using At times this goes unnoticed systematic increments of F, until by slope engineers. A typical the FE model does not converge example involves the problem to a solution, i.e. continue to of identifying the critical failure reduce material strength until mechanism beneath a concrete the slope fails. The critical F value on a that just beyond which failure occurs includes a weak layer. The search will be the slope factor of safety. for the critical limit-equilibrium mechanism, using Slide, is (For a slope that is initially described in an AVI movie in unstable, factor of safety values in steps 2 and 3 must be reduced until the FE model converges to a solution.)

4 this issue of RocNews Summer – a combination of block search 2004. Regular non-circular limit- with surface optimization – a equilibrium analysis with the more critical slip surface with a Morgernstern method produced 2.36 factor of safety value was a factor of safety of 3.0. Using located. The images of these advanced features in Slide not results are shown on Figures 1a applied by most modelers and 1b.

Figure 1a. The critical failure surface and factor of safety from conventional Morgernstern limit- equilibrium analysis of a concrete dam. The thin, beige-coloured material beneath the dam is the weak layer.

Figure 1b. The critical failure surface and factor of safety obtained using block search, combined with surface optimization. Notice the differences between this surface and that in Figure 1a.

5 The problem was then modeled solution), and an exaggerated in Phase2, the finite element deformed mesh. Contours of analysis program developed maximum shear strain are shown by Rocscience, using the SSR on Figure 2b. These figures technique. This yielded a factor indicate failure in the weak zone of safety value of 2.35. Figure and a critical failure surface 2a shows the contours of total similar to the non-circular limit- displacement at the modeling equilibrium surface obtained stage preceding total failure from the combination of block (non-convergence of the search and optimization.

Figure 2a. Contours of total displacement calculated for the FE model of the dam, and the (exaggerated) deformed mesh. These contours indicate a critical failure mechanism similar to that obtained from advanced limit- equilibrium analysis.

Figure 2b. Contours of maximum shear strain calculated for the FE model of the dam. Like the previous image, these contours indicate the critical failure mechanism as shearing, passing through the weak zone.

6 Another benefit of the SSR aspect for some embankments approach is its elimination of and excavations. Drawing on arbitrary assumptions regarding the robustness of the FEM, the inclinations and locations the technique performs very of inter-slice forces. As well, well under a wide range of the method can automatically conditions. Lastly, it can be monitor the development of more readily applied to three- failure zones, from localized dimensional slope modeling than areas all the way to total slope limit-equilibrium methods. failure. This is particularly important in the analysis of high Will Use of the SSR Technique Become Widespread? slopes such as those found in large open-pit mines, and the In the past, a number of factors impact of slope excavation on limited application of the SSR nearby structures. Given the technique for routine slope correct deformation properties stability analysis. Compute The FEM is of materials, the SSR method can times for analyses were long, able to compute predict expected deformations computing power adequate for deformations and at the stress levels found in such analysis was expensive, and quantities such as slopes. model preparation and results bending moments, interpretation required extensive allowing the method Due to the FEM’s ability to effort and time due to clumsy, to be used to design compute deformations and difficult-to-use interfaces. The the support elements quantities such as bending above-listed factors combined to in a slope. moments, it can to be used to push SSR costs very high. design the support elements in a slope. For example, in a Most practicing engineers had slope stabilized with piles, the another complaint against the FEM is capable of predicting technique: they felt it was data axial loads, bending moments hungry, requiring material and deformations of the piles, input parameters, which were making it possible for engineers not collected in routine site to select appropriate dimensions investigations, at least not with and materials that ensure reasonable accuracy. They also adequate performance. felt that the reliability of SSR results was unproven. The SSR FE technique can model construction procedures and sequences (i.e. loading paths), a particularly important

7 Cheap, and yet very powerful, that are close to those obtained computers, and tremendous from method-of-slices and other improvements in program limit-equilibrium approaches. interfaces and user-friendliness The references provided at the have radically changed the end of this article will provide situation. Today the typical details for the interested reader. desktop computer can The advances described above perform two-dimensional FE should change perceptions of analysis within minutes. New SSR slope analysis. Most of the commercially available computer issues that hindered application programs, such as Rocscience’s yesterday have been eliminated, Phase2, significantly reduce the or are rapidly being addressed. amount of time required to build A few of the outstanding issues models, to interpret results, and with the SSR technique remain to produce output for reports. and companies like Rocscience Obtaining input material are finding ways of addressing parameters for finite element them. analysis is possible through testing, similar to those required Outstanding Issues with the SSR Technique for traditional settlement or stability analysis. Triaxial To date, all published discussions and direct shear tests can be on the application of the FEM combined with consolidation to slope stability analysis have tests to obtain required input assumed Mohr-Coulomb data. For elasto-plastic analysis strength. This may be primarily involving conventional Mohr- due to the ease with which Coulomb strength, with the reduced Mohr-Coulomb assumption of associative flow strength parameters can be rule, the FEM requires the same calculated for application in the parameters as limit-equilibrium SSR technique. As shown earlier analysis (except for the modulus in the article, reduced of deformation and Poisson’s (c*) and angle (φ*) values ratio). can be explicitly determined from original parameters c and The question of the reliability φ. With nonlinear criteria such as of the SSR technique has been the Generalized Hoek-Brown and comprehensively answered. Power Curve strength models, Several studies have shown that it is impossible to obtain such the technique produces results closed-formed relationships.

8 To get around the difficulty Conclusion described above, Rocscience By examining the merits of engineers have helped develop finite element analysis and limit- an approach that uses a Mohr- equilibrium methods for slope Coulomb approximation of a design and analysis, it is our nonlinear strength criterion. opinion that, for the foreseeable The details of this approach are future, these two approaches will described in a recent paper, coexist. Together, they supply (Hammah et al, 2004). We are engineers with a more complete currently developing a more toolkit for tackling slope stabillity generalized and more accurate problems. The experience, method for performing SSR amassed by the geotechnical analysis for nonlinear material engineering profession over the strength. decades, with limit equilibrium Although the SSR technique is methods is invaluable and highly intuitive and relatively cannot be readily displaced. straightforward to conduct On the other hand, several using any existing finite types of geotechnical problems element program, the manual are not readily analyzed with effort involved in calculating limit equilibrium methods, but reduced material properties for can be handled by the SSR FE multiple factor of safety values approach. Today’s technological can get tedious, therefore advances, combined with the hampering routine use. The reduced costs of computing SSR technique has been built power, enable the SSR technique into a few geotechnical finite to solve these problems easily. In element programs, but aspects addition, the SSR technique can of the problem have not been be used to resolve ambiguities in sufficiently automated to levels limit-equilibrium slope stability found in limit-equilibrium analysis. software packages. Rocscience is Recent improvements to the SSR working on releasing, in the near technique and its applications 2 future, a version of Phase that do indeed have the potential more completely automates SSR to enhance the quality of analysis. slope , and expand our understanding of slope behaviour and interactions between the

9 various factors that influence We invite readers to comment on this stability. While we may not be article or express their thoughts and observations on the issue of limit- ready to abandon tried and true equilibrium versus finite element analysis. approaches all together, we may Contact us at: [email protected]. be on the threshhold of a new era in slope engineering.

Useful References on the SSR Technique

The references below supplied most of the technical background and descriptions of the SSR technique described in this article. Although the list is by no means exhaustive, it will provide the interested reader an excellent starting point for learning more about the SSR technique.

Dawson, E., Motamed, F., Nesarajah, S. and Roth, M., Geotechnical stability analysis by strength reduction, Slope Stability 2000, Proceedings of Sessions of Geo-Denver 2000, ASCE Geotechnical Special Publication no. 101, pp. 99-113. 2000.

Dawson, E.M., Roth, W.H. and Drescher, A., Slope stability analysis by strength reduction, Geotechnique, vol. 49, no. 6, pp. 835-840. 1999.

Duncan, J.M., State of the art: limit equilibrium and finite-element analysis of slopes, Journal of Geotechnical Engineering, vol. 122, no. 7, pp. 577-596. 1996.

Griffiths, D.V. and Lane, P.A., Slope stability analysis by finite elements, Geotechnique, vol. 49, no. 3, pp. 387-403. 1999.

Hammah, R.E., Curran, J.H., Yacoub, T.E. and Corkum, B., Stability analysis of rock slopes using the finite element method. To appear in the Proceedings of Eurock 2004 & 53rd Geomechanics Colloquy. 2004.

Lechman, J.B. and Griffiths, D.V., Analysis of the progression of failure of earth slopes by finite elements, Slope Stability 2000, Proceedings of Sessions of Geo-Denver 2000, ASCE Geotechnical Special Publication no. 101, pp. 250-265. 2000.

Matsui, T. and San, K-C., Finite element slope stability analysis by shear strength reduction technique, Soils and Foundations, vol. 32, no. 1, pp. 59-70. 1992.

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