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- 1. 1 TITLE: SLOPE STABILITY (FINITE SLOPES) 1.0 INTRODUCTION Slope stability analysis is an important area in geotechnical engineering. Most textbooks on soil mechanics include several methods of slope stability analysis. Before the calculation of slope stability in these methods, some assumptions, for example, the side forces and their directions, have to be given out artificially in order to build the equations of equilibrium. With the development of cheaper personal computer, finite element method has been increasingly used in slope stability analysis. The advantage of a finite element approach in the analysis of slope stability problems over traditional limit equilibrium methods is that no assumption needs to be made in advance about the shape or location of the failure surface, slice side forces and their directions. Various calculations carried out illustrate perfectly benefits that can be gained from modeling the behavior by the finite elements method. The method can be extended to account for seepage induced failures, brittle soil behaviors, random field soil properties, and engineering interventions such as geo-textiles, soil nailing, drains and retaining walls. Generally, there are two approaches to analyze slope stability using finite element method. One approach is to increase the gravity load and the second approach is to reduce the strength characteristics of the soil mass. 2.0 APPLICATION SLOPE STABILITY EVALUATION FINITE ELEMENT METHODS The various limit equilibrium methods are based on the arbitrary choosing a series of slip surfaces and of defining that which gives the minimal value of the safety factor. Nowadays, we attend an intensive use of numerical analysis methods giving access to the constraints and deformations within the formations constituting the subsoil. For that purpose, it is necessary to know the behavior law of the considered formations; then, the volume of ground is divided into simple geometric elements, each element being subjected to the action of the close elements. The calculation will consist in determining stress fields and displacements compatible with the mechanic equations and the behavior law adopted. Many works were done in the finite elements field and we could cite works of ZIENKIEWICZ [5] or DHATT [6]. The finite element method makes it possible to calculate stresses and deformations state in a rock mass, subjected to its self weight with the assumption of the behavior law adopted. In our calculations, a model with internal friction without work hardening (perfect elastoplastic Model: Mohr-Coulomb) is used, which corresponds to the basic assumptions of the analytical methods. In our work, we will use the method of reduction of soil resistance properties, known as the “c-φ reduction” method. The c-φ method is based on the reduction of the shear strength (c) and the tangent of the friction angle (tanφ) of the soil. The parameters are reduced in steps until the soil mass fails. Plaxis uses a factor to relate the reduction in the parameters during the calculation at any stage with the input parameters according to the following equation: Where Msf is the reduction factor at any stage during calculations, tanφinput and cinput are the input parameters of the soil, tanφ reduced and creduced are the reduced parameters calculated during the analysis [9]. The characteristics of the interfaces, if there is, are reduced in same time. On the other hand, the characteristics of the elements of structure like the plates and the anchoring are not influenced by Phi-C reduction. The total multiplier Msf is used to define the value of the soil strength parameters at a given stage in the analysis at the failure stage of the slope, the total safety factor is given as follows:
- 2. 2 The safety factor found using the method of c-φ reduction according to the criterion of Mohr Coulomb remains comparable with those found by the analytical methods in both cases with or without presence of water. The difference noted is the fact that for the analytical methods ,safety factors are assumed constants along the failure surface. Moreover, finite element methods that provide access to stresses and strains within the soil, offer the possibility of a detailed operating calculations as curves: displacements, the evolution of the safety factor according to displacement , the localization of deformations and plastic zones. The taking into account of the behavior law in the codes with the finite elements makes it possible to better determine the stress and strain state in various points. The total displacements figure highlights the limit between the zone where there is no displacement (zero value) and the zones where displacements occur (non null values) We note the circular form of this limit which points out the slip surface adopted by the analytical methods. These displacements are important at the slope and the highest value is in mid-slope. The horizontal component of displacements exceeds the vertical’s. The rupture curve identification in Plaxis is based on the localization of the deformations on the slope we once again, find the circular form of slip surfaces. 3.0 APPLICATION BY 3D ELASTO-PLASTIC FINITE-ELEMENT METHOD This application works for the stabilities of the abutments of Houhe gravity-arch dam using elasto-plastic ﬁnite element analyses and block theory. The gravity-arch dam is constructed by expanding the existing gravity dam. Due to the complex geologic conditions, the natural abutments do not meet the stability requirements. Engineering measures including concrete plugs and concrete backﬁlls are thus designed to increase the safety of both abutments. The effectiveness of these measures is evaluated in this study. The paper performs three-dimensional elasto-plastic ﬁnite element analyses to obtain the stress and deformation characteristics of the abutments and the dam. The stabilities of sliding blocks on the abutments are then evaluated by incorporating the results of ﬁnite-element method analyses. It is found that the use of concrete plugs signiﬁcantly increases the factors of safety against sliding for both abutments. Concrete backﬁlls not only provide the sliding resistance, but also are effective in reducing the deformation of the abutments and improve the stress conditions along the base of the dam. The dam works under satisfactory conditions as the results of applying these engineering measures. It may work if it is applied in the slope stability process. A major advantage of FEM lies in its ﬂexibility in modeling complex geometries and geologic conditions. Current FEM codes can simulate a variety of material behaviors as well as incorporate the inﬂuence of construction procedures. Construction of the global ﬁnite element model of abutments and dam can be performed with ease using advanced preprocessing modules. The analyses provide more realistic distribution of thrust forces acting on the abutments for analyses of their stability. For approaches such as the limit equilibrium analyses or block theory thrust forces are generally assumed to be constant and are estimated by simpliﬁed beam or arch models. This can be quite different from real situations. Analyses by FEM also provide comprehensive information on the stress and deformation characteristics of both the dam and the abutments. By incorporating the appropriate plastic material models, FEM is theoretically capable of automatically identifying the location and places of the failure surface and indicating the extent of progressive failure. Grifﬁths and Lane, for example, applied iterative FEM for automatic analyses of slope stability problems by assuming that failure occurs when the results are non-convergent after a speciﬁc number of iterations. However, identifying failure planes by FEM alone is computational extensive, especially when there involves complex materials behaviors and complex potential sliding wedges. Besides, there are no universally accepted failure criteria to implement for the FEM analysis. FEM model and analyses are performed to provide more accurate stress distributions acting on the surfaces of those blocks. This results in a more accurate evaluation of the stability conditions of the abutments. In addition to this, FEM analyses also provide important information such as stress and deformation characteristics.
- 3. 3 4.0 APLICATION FINITE ELEMNT ANALYSIS FOR EVALUATION OF SLOPE STABILITYINDUCED BY CUTTING This paper concerned an evaluation of stability condition of slope. A slope stability induced by cutting was evaluated by a finite element analysis. The finite element analysis employed a constitutive model in which non- associated strain hardening-softening elasto-plastic material was assumed. In-site investigation was done by an inclinometer for the boreholes. For defining the soil stratum, a new data processing system was applied in this paper to generate a soil ground model, using many boring data. Soil samples were taken and subjected to geotechnical laboratory tests. A triaxial compression test (CU ) was performed to determine the shear strength. The numerical analysis did not consider the pore pressure, because no ground water was appeared in that area. The deformation obtained by the numerical analysis was close to the results of inclinometer of borehole observed by in-site investigation. The finite element analysis was able to predict the estimation of the slope stability induced by cutting. 4.1 LABORATORY TEST Undisturbed sample lay around the sliding surface was taken from the cutting slope surface. The physical properties of the sample are summarized in Table 1. The grain size distribution curve is shown in Fig.8. In order to investigate the relationship between peak strength of the material, triaxial compression test (CU ) was conducted. The internal friction angle and cohesion were 25 degrees and 12.5kN/m2, respectively. Yatabe (2004) reported that a residual friction angle of clay of crystalline schist of Sambagawa metamorphic belt was about 15 degrees. 5.0 APPLICATION OF THE FINITE ELEMENT METHOD TO SLOPE STABILITY This document outlines the capabilities of the finite element method in the analysis of slope stability problems. The manuscript describes the constitutive laws of material behaviour such as the Mohr-Coulomb failure criterion, and material properties input parameters, required to adequately model slope failure. It also discusses advanced topics such as strength reduction techniques and the definition of slope collapse. Several slopes are analyzed with the finite element method, and the results compared with outcomes from various limit equilibrium methods. Conclusions for the practical use of the finite element method are also given. Slope stability analysis is an important area in geotechnical engineering. Most textbooks on soil mechanics include several methods of slope stability analysis. A detailed review of equilibrium methods of slope stability analysis is presented by Duncan (Duncan, 1996). These methods include the ordinary method of slices, Bishop’s modified method, force equilibrium methods, Janbu’s generalized procedure of Slices, Morgenstern and Price’s method and Spencer’s method. These methods, in general, require the soil mass to be divided into slices. The directions of the forces acting on each slice in the slope are assumed. This assumption is a key role in distinguishing one limit equilibrium method from another. Limit equilibrium methods require a continuous surface passes the soil mass. This surface is essential in calculating the minimum factor of safety (FOS) against sliding or shear failure. Before the calculation of slope stability in these methods, some assumptions, for example, the side forces and their directions, have to be given out artificially in order to build the equations of equilibrium. With the development of cheaper personal computer, finite element method has been increasingly used in slope stability analysis. The advantage of a finite element approach in the analysis of slope stability problems over traditional limit equilibrium methods is that no assumption needs to be made in advance about the shape or location of the failure surface, slice side forces and their directions. The method can be applied with complex slope configurations and soil deposits in two or three dimensions to model virtually all types of mechanisms. General soil material models that include Mohr-Coulomb and numerous others can be employed. The equilibrium stresses, strains, and the associated shear strengths in the soil mass can be computed very accurately. The critical failure mechanism developed can be extremely general and need not be simple circular or logarithmic spiral arcs. The method can be extended to account for seepage induced failures, brittle soil behaviors, random field soil properties, and engineering interventions such as geo-textiles, soil nailing, drains and retaining walls (Swan et al, 1999). This method can give information about the deformations at working stress levels and is able to monitor progressive failure including overall shear failure (Griffiths, 1999).
- 4. 4 6.0 APPLICATION FINITE ELEMENT REALIBALITY ANALYSIS OF SLOPE STABILITY The method of nonlinear finite element reliability analysis (FERA) of slope stability using the technique of slip surface stress analysis (SSA) is studied. The limit state function that can consider the direction of slip surface is given, and the formulations of FERA based on incremental tangent stiffness method and modified Aitken accelerating algorithm are developed. The limited step length iteration method (LSLIM) is adopted to calculate the reliability index. The nonlinear FERA code using the SSA technique is developed and the main flow chart is illustrated. Numerical examples are used to demonstrate the efficiency and robustness of this method. It is found that the accelerating convergence algorithm proposed in this study proves to be very efficient for it can reduce the iteration number greatly, and LSLIM is also efficient for it can assure the convergence of the iteration of the reliability index. In structural reliability analysis using the FORM, the most popular iterative searching algorithm of the design point is perhaps the so-called first order second moment method (FOSM), which was originally developed by Hasofer and Lind (1974), and later extended to non-normal random variables by Rackwitz and Fiessler (1978). However, it has been demonstrated that if the limit state function is highly nonlinear, the iterative computation of the design point and reliability index will probably be divergent in the FOSM algorithm. Since limit state function of the FERA of slope stability is usually a highly nonlinear implicit function of soil parameters, it is necessary to study some new iterative algorithms to assure the successful running of the nonlinear FERA code. In this study, a new iterative algorithm— LSLIM, is presented to carry out the reliability analysis. LSLIM is another searching algorithm of FORM. The iterative procedure in standard normal space has been listed by Gong (2003). Since the FERA for the calculation of stress increment is carried out in the original space, the iteration of design point and reliability index should also be performed in the original space. When basic variables are independent normally distributed variables, 7.0 APPLICATION STRENGHT REDUCTION AND STEP-LOADING FINITE ELEMENT APPROACHES IN GEOTECHNICAL ENGINEERING The finite element limit analysis method has the advantages of both numerical and traditional limit equilibrium techniques and it is particularly useful to geotechnical engineering. This method has been developed in China, following well-accepted international procedures, to enhance understanding of stability issues in a number of geotechnical settings. Great advancements have been made in basic theory, the improvement of computational precision, and the broadening of practical applications. These applications are evidence of the design improvements and benefits made possible in geotechnical engineering by finite element modeling. As the stability analysis is related to the analysis of force and strength instead of displacement, a perfect elastorplastic constitutive model is sufficient for accurate finite element calculations without the consideration of hardening and softening rocks or soils. The yield criterion is very important in the finite element limit analysis method as it has great effects on the computational results. When the unit weight of the soil γ = 0 , the ultimate bearing capacity of rigid and smooth strip foundations has closed form solutions in Prandtl. A finite element limit analysis model can be used to analyze rigid and smooth strip foundations. The finite element model The load-displacement curve of the central point of the foundation obtained from the finite element limit analysis method . It can be interpreted that the displacement of the central point of the foundation increases as the load increases, and increases suddenly and sharply when the foundation fails. 8.0 APPLICATION OF SLOPE STABILITY ANALYSIS BY FINITE ELEMENTS The majority of slope stability analyses performed in practice still use traditional limit equilibrium approaches involving methods of slices that have remained essentially unchanged for decades. This was not the outcome envisaged when Whitman and Bailey (1967) set criteria for the then emerging methods to become readily accessible to all engineers. The finite element method represents a powerful alternative approach for slope stability analysis which is accurate, versatile and requires fewer a priori assumptions, especially regarding the failure mechanism.
- 5. 5 Slope failure in the finite element model occurs “naturally” through the zones in which the shear strength of the soil is insufficient to resist the shear stresses. This paper describes several examples of finite element slope stability analysis with comparison against other solution methods, including the influence of layering and s free surface on slope and dam stability. Graphical output is included to illustrate deformations and mechanisms of failure. It is argued that the finite element method of slope stability analysis is more powerful alternative to traditional limit equilibrium methods and its widespread use should now be standard in geotechnical practice. Slope stability represents an area of geotechnical analysis in which a nonlinear finite element approach offers real benefits over existing methods. Duncan’s review of finite element analysis of slopes concentrated mainly on deformation rather than stability analysis of slopes, however attention was drawn to some important early papers in which elasto-plastic soil models were used to assess stability. Smith and Hobbs (19740) reported results and obtained reasonable agreement with Taylor’s (1937) charts. Zienkiewicz et al (1975) considered a c’.ø’slope and obtained good agreement with slip circle solutions. Subsequent use of the finite element method in slope stability analysis has added further confidence in the method. 9.0 CASE STUDY: RAILWAY SLOPE The case study relates to a railway slope in Moroccan Prérif, between Tangier city and Tangier-Med port. The geological formations are consisted of sandstone and marls alternations (Figure 2). The important rains caused a landslide on a ravine which damaged locally the railway. Stratigraphic column of sedimentary formations constituting the embankment.
- 6. 6 Finite element mesh of slope profile. 10.0 CONCLUSION The analysis and design of failing slopes and highways embankment requires an in-depth understanding of the failure mechanism in order to choose the right slope stability analysis method. The present study made it possible to compare on a real geometrical model the computation results of the safety factor (defining the state of the slope stability compared to the limit equilibrium) by various methods: limit equi-librium and finite elements methods. The behavior law stress-strain which is lacking to the limit equilibrium meth- ods is integrated into the finite elements methods. The results obtained with slices methods and FEM are similar. However, the results obtained using the finite ele- ments are nearest those obtained by Bishop’s method than Fellenius’ method. If we compare the sliding surfaces obtained with the slices methods with representations of the total displace- ment increments obtained with FEM, it is possible to see that the failure mechanism was very well simulated by FEM. In the analyzed case it is possible to see the circu- lar shape of the sliding surfaces in the graphics of the total displacements increments.The determination of the safety factor is insufficient to identify problems of slope stability, the various calcula- tions performed illustrate perfectly the benefits that can be gained from modeling the behavior by FEM: 1) the calculation of displacements obtained by FEM allows to estimate the actual settlement and optimize ways of re-inforcement; 2) the prediction of failure mechanism; 3) the use of the results of field tests to better approximate the real behaviour of structures.
- 7. 7 11.0 REFERENCES J. M. Duncan, “State of the Art: Limit Equilibrium and Finite-Element Analysis of Slopes,” Journal of Geotech- nical Engineering, Vol. 122, No. 7, 1996, pp. 577-596. doi:10.1061/(ASCE)0733-9410(1996)122:7(577) A. W. Bishop, “The Use of the Slip Circle in the Stability Analysis of Slopes,” Géotechnique, Vol. 5, No. 1, 1955, pp. 7-17. doi:10.1680/geot.1955.5.1.7 J. M. Duncan, A. L. Buchignani and M. De Wet, “An Engineering Manual for Slope Stability Studies,” Virginia Polytechnic Institute, Blacksburg, 1987. O. C. Zienkiewicz and R. L. Taylor, “The Finite Element Method Solid Mechanics,” Butterworth-Heinemann, Ox-ford, Vol. 2, 2000, p. 459. Goodman RE, Powell C. Investigations of blocks in foundations and abutments of concrete dams. J Geotech Geoenviron Eng 2003;129(2):105–16. Fairhurst C, Lorig L. Improved design in rock and soil engineering with numerical modeling. In: Sharma VM, Saxena KR, Woods RD, editors. Distinct element modeling in geomechanics. Rotterdam, The Netherlands: Balkema; 1999. p. 27–46. Griffiths DW, Lane PA. Slope stabilityanaly sis by finite elements. Geotechnique 1999;49(3):387–403. Copen MD, Lindholm EA, Tarbox GS. Design of concrete dams. In: Golze AR, editor. Handbook of dam engineering. New York: Van Nostrand Reinhold; 1977. Zienkiewicz O C, Humpheson C, Lewis R W. Associated and non-associated visco-plasticity in soil mechanics. Geotechnique, 1975, 25 (4): 671–689. Matsui T, San K C. Finite element slope stability analysis by shear strength reduction technique. Soils and Foundations, 1992, 32 (1): 59–70. Manzari, M.T., Nour, M.A., 2000. Significance of soil dilatancy in slope stability analysis. Journal of Geotechnical and Geoenvironmental Engineering, 126(1):75-80. [doi:10.1061/(ASCE)1090-0241(2000)126:1(75)] Mellah, R., Auvinet, G., Masrouri, F., 2000. Stochastic finite element method applied to non-linear analysis of embankments. Probabilistic Engineering Mechanics, 15(3):251-259. [doi:10.1016/S0266-8920(99)00024-7] Arai, K., and Tagyo, K. (1985), “Determination of noncircular slip surface giving the minimum factor of safety in slope stability analysis.” Soils and Foundations. Vol.25, No.1, pp.43-51. Malkawi, A.I.H.,Hassan, W.F., and Sarma, S.K. (2001), “Global search method for locating general slip surfaces using monte carlo techniques.” Journal of Geotechnical and Geoenvironmental Engineering. Vol.127, No.8, August, pp. 688-698. Greco, V.R. (1996), “Efficient Monte Carlo technique for locating critical slip surface.” Journal of Geotechnical Engineering. Vol.122, No.7, July, pp. 517-525. Kim, J., Salgado, R., Lee, J. (2002), “Stability analysis of complex soil slopes using limit analysis.” Journal of Geotechnical and Geoenvironmental Engineering. Vol.128, No.7, July, pp. 546-557. Yamagami, T. and Ueta, Y. (1988), “Search noncircular slip surfaces by the Morgenstern-Price method.” Proc. 6th Int. Conf. Numerical Methods in Geomechanics, pp. 1335-1340. M.K. Kockar, H. Akgun (2003), “Methodology for tunnel and portal support design in mixed limestone, schist and phyllite conditions: a case study in Turkey.” Int. J. Rock Mech. and Min Sci, Vol. 40, pp. 173–196.

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