Evaluation of Deep-Seated Slope Stability of Embankments over Deep Mixed Foundations

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1 Abstract Evaluation of Deep-Seated Slope Stability of Embankments over Deep Mixed Foundations Jie Han 1, Jin-Chun Chai 2, Dov Leshchinsky 3, and Shui-Long Shen 4, When embankments are constructed over soft foundations, deep-seated slope stability often becomes one of the controlling factors in design. Deep mixing methods have been commonly used as an alternative to solve the deep-seated slope stability problem. Bishop s modified method is a commonly adopted approach for analyzing the slope stability of embankments on deep mixed foundations. Bishop s modified method assumed slopes fail along a circular slip surface and the soils along this slip surface provide shear resistance. However, experimental studies have showed that deep mixed columns under a combination of vertical and horizontal forces could fail due to shearing or bending. The possible failure modes depend on the combination of the forces, the strengths of soft soils and deep mixed columns, dimensions and arrangements of deep mixed columns. Since deep mixed columns are formed by mixing a certain amount of admixture (cement or lime or a combination) with soil, they can have a wide range of strengths. A numerical method was used in this study to evaluate the factors of safety varying with the strengths of deep mixed columns and their arrangements with three rows of columns having two different thickness. Mohr- Coulomb failure criterion was used for embankment fill, foundation soil, and deep mixed columns. A row of deep mixed columns was modeled as a wall in 2-D for simplicity of analysis. The numerical analysis indicated that the critical slip surface of the deep-seated slope failure was not circular when the deep mixed columns were used. The factors of safety obtained using the numerical method were compared with those using Bishop s modified method and Spencer s three-part wedge method. The comparisons indicated that Bishop s modified method yielded significantly higher factors of safety than the numerical method, especially when the deep mixed columns had higher strengths. The Spencer s three-part wedge method yielded lower factors of safety than the numerical method. 1 Ph.D., PE, Assistant Professor, Dept. of Civil Engineering, Widener University, One University Place, PA 19013, USA, Tel.: , Fax: , jxh0305@mail.yahoo.com 2 Ph.D., Associate Professor, Institute of Lowland Technology, Saga University, 1 Honjo, Saga , JAPAN, Tel. and Fax: , chai@cc.saga-u.ac.jp 3Professor, Dept. of Civil and Environmental Engineering, University of Delaware, Newark, DE 19716, USA 4 Ph.D., Associate Professor, School of Civil Engineering and Mechanics, Shanghai Jiao Tong University, 1954 Hua Shan Road, Shanghai , CHINA, suiryu_shen@yahoo.com.cn

2 Introduction When designing embankments over soft foundations, geotechnical engineers may face a number of challenges, which include bearing capacity failure, excessive total and differential settlements, and slope instability, etc. The slope instability of embankments may develop locally, near the facing, inside the embankment, or through the foundation soil as local failure, surficial failure, general slope failure, or deep-seated failure as shown in Figure 1. The deep-seated slope failure is also referred as a global slope failure, mainly induced by a soft foundation existing under the embankment. A number of techniques have been successfully adopted to prevent deep-seated slope failure, such as ground improvement techniques and use of geosynthetics or piles. As one of ground improvement techniques, deep mixed (DM) columns have been commonly used as an alternative to solve deep-seated slope stability problems. Terashi (2002) indicated that nearly 60% of on-land application in Japan and perhaps roughly 85% of Nordic applications are for the settlement reduction and improvement of stability of embankment by means of group of treated soil columns. This paper focuses on the evaluation of deep-seated slope stability of embankments over deep mixed foundations using limit equilibrium methods and a numerical solution. General slope failure Deep seated failure Local failure Surficial failure Figure 1. Potential Slope Stability Failures Limit equilibrium methods have been commonly adopted for analyzing the deepseated slope stability of embankments over deep mixed foundations. Bishop s modified method with a circular slip surface is probably the most commonly used limit equilibrium method. In the analysis of DM foundations, the DM columns and the soil are either treated as individual components or as a composite ground to resist the shear stresses. In this study, the DM columns are treated as individual DM walls in 2-D analyses. Limit equilibrium methods assume that the shear strengths of the columns are always fully mobilized if a slip surface cuts through any part of the columns. In reality, the resistance of the columns depends on the intersected location by the slip surface. As shown in Figure 2, the columns may only provide very limited resistance at the locations A and C because the soil around the columns may fail prior

3 to the failure of the columns or the slip surfaces may go around the columns to create noncircular slip surface or the columns may behave as piles punching into the upper or lower soil layer as described by Broms and Wong (1985). In addition, the centrifuge model tests done by Kitazume and Terashi (1991) indicated that the DM columns failed by bending under a combination of vertical and horizontal forces. Broms (1999) also indicated that horizontal forces would reduce the bearing capacity of DM columns. Since the bending strengths of the DM columns are much lower than their shear strengths, Kitazume et al. (1997) was concerned about the possibility of overestimation using Bishop s slip circle analysis in the current design. A B C Figure 2. Potential slip locations through DM columns In recent years, numerical methods have been increasingly used for analyzing slope stability including the computation of its factor of safety. San et al. (1994) indicated that finite element and limit equilibrium methods could consistently determine the locations of critical slip surfaces and required tensile strength of reinforcement in geosynthetic-reinforced slopes. Dawson et al. (1999) concluded that the factors of safety of unreinforced slopes obtained using a finite difference method (FLAC - Fast Lagrangian Analysis of Continua) were in good agreement with those using the limit equilibrium method with a log-spiral slip surface. Han et al. (2002) used the same finite difference software (FLAC) to obtain the identical corresponding factors of safety of unreinforced and geosynthetic-reinforced slopes as the Bishop s modified method (limit equilibrium method). The technique used for computing the factor of safety of slope stability in the numerical method is discussed in the following section. As compared with limit equilibrium methods, numerical methods have the following advantages in solving the factor of safety of slope stability (Cundall, 2002): (1) no pre-defined slip surface is needed; (2) the slip surface can be any shapes; (3) no assumptions are needed for functions of inter-slice force angles; (4) multiple failure surfaces are possible; (5) structures (such as footings, tunnels, etc.) and/or structural elements (such as beams, cables, etc.) and interfaces can be included; and (6) kinematics is satisfied. However, numerical methods generally require the user to have more knowledge and experience in order to properly use it. A complicated and large size problem may require significant computation time. The inclusion of structural elements and interfaces may create numerical instability. It is difficult to

4 perform a specific search when it is sometimes needed (for example, surficial slope stability needs to be prevented in order to study the deep-seated slope stability). The comparisons of numerical solutions and limit equilibrium methods are presented in Table 1. Considering the complexity of the failure mechanisms, the finite difference method (FLAC) was adopted in this study to evaluate the deep-seated slope stability of embankments over DM foundations. The computed factors of safety by the numerical method were compared with those obtained using Bishop s circular slip surface method and the Spencer s three-part wedge method. Table 1. Comparisons of Numerical Solutions and Limit Equilibrium Methods (Cundall, 2002) Numerical solution Limit equilibrium Equilibrium Satisfied everywhere Satisfied only for specific objects (slices) Stresses Computed everywhere using field equations Computed approximately on certain surfaces Deformation Part of the solution Not considered Failure Yield condition satisfied everywhere; failure surfaces develop automatically as condition dictate Failure allowed only on certain pre-defined surfaces; no check on yield condition elsewhere Kinematics The mechanisms that develop satisfy kinematic constraints Kinematics are not considered mechanisms may not be feasible Numerical Method The finite difference program (FLAC 2D Version 4.0) developed by the Itasca Consulting Group, Inc. was adopted in this study for numerical analyses of slope stability of embankments over DM foundations. A shear strength reduction technique was adopted in this program to solve for a factor of safety of slope stability. Dawson et al. (1999) exhibited the use of the shear strength reduction technique in this finite difference program and verified numerical results with limit equilibrium results for simple slopes. In this technique, a series of trial factors of safety are used to adjust the cohesion, c and the friction angle, φ, of soil as follows: c φ trial trial 1 = FS trial c 1 = arctan FS trial tan φ Adjusted cohesion and friction angle of soil layers are re-inputted in the model for equilibrium analysis. The factor of safety is sought when the specific adjusted

5 cohesion and friction angle make the slope become instability from a verge stable condition or verge stable from an unstable condition. Modeling The geometry and material properties of the models used in this study are shown in Figure 3. Deep mixed columns were modeled as continuous walls. Mohr-Coulomb failure criteria were used for embankment fill, soft soil, firm soil, and deep mixed walls. The properties of embankment fill, soft soil, and firm soil were kept constant. The unit weight of all the soil layers including DM walls was 18kN/m 3. In this study, DM walls are installed at the toe, the shoulder, and the mid-point between the toe and the shoulder with a thickness of 1.0m or 2.0m although they can be installed under the whole embankment in practice. For a parametric study, the DM walls have an undrained shear strength varying from 100kPa to 1000kPa and friction angle equal to zero. The same models were used in numerical and limit equilibrium analyses. 20m 10m 30m Soil-cement column c=100 to 1000kPa, φ= Embankment fill 5m c=0, φ=34 0 1m or 2m Soft soil c=10kpa, φ=0 0 10m Firm soil c=100kpa, φ=0 0 5m Figure 3. Numerical Analysis Model Analysis of Results Numerical Analyses An embankment over untreated ground was selected as a baseline case. The shearstrain rate contours of this case were shown in Figure 4. It is shown that high shear strain rates developed in the embankment and the soft soil, which created a critical slip zone. The shape of this critical slip zone is circular, which is consistent with the slip surface assumed in Bishop s method. It is apparent that the critical slip zone was bottomed out by the firm soil underneath the soft soil. Figure 5 presents shear-strain rate contours for an embankment over a DM foundation. There is no continuous shear-strain rate zone. It is shown that the shear-strain rate contours intercepted by soil-cement walls. The high shear-strain rate zones developed in the embankment and soft soil in front and behind the DM walls. In front of the column at the toe, the

6 high shear-strain rate zone was caused by the rotation of the column towards the soft soil. ( )x10-8 ( )x10-8 ( )x10-8 (7.5-10)x10-8 ( )x10-8 (7.5-10)x10-8 ( )x10-8 (5-7.5)x10-8 Soft soil Firm soil Figure 4. Shear Strain Rates Developed in the Embankment and Untreated Ground (6-8)x10-8 (4-6)x10-8 >10x10-8 (8-10)x10-8 (6-8)x10-8 (4-6)x10-8 < 4x10-8 (2-4)x10-8 Column Column Column (2-4)x10-8 Soft soil Firm soil Figure 5. Shear Strain Rates Developed in the Embankment and thedm Foundation (wall thickness = 1.0m; undrained shear strength of DM column, c u = 100kPa) Since the critical slip surface was not very obvious based on the shear strain rate contours in the embankment and the DM foundation, the velocity vectors obtained by the numerical method were plotted in Figure 6 to assist in evaluating the critical slip surface. The velocity vectors shown in Figure 6 represent the movement of the slope and the soft soil with DM columns. The shape of the critical slip surface can be approximately considered as a three-part wedge type. This critical slip surface was simulated in the slope stability analysis using Spencer s three-part wedge method in the next section.

7 Figure 6. Velocity Vectors Developed in the Embankment and the DM Foundation (wall thickness = 1.0m; undrained shear strength of DM column, c u = 100kPa) Limit Equilibrium Analyses The limit equilibrium program, ReSSA2.0, developed by ADAMA Engineering, Inc., was used in this study to conduct slope stability analyses using Bishop s modified method and Spencer s three-part wedge method. The tangent to critical circles at their points of exit for Bishop s method is limited in ReSSA to a maximum of 50 degrees. This limitation is to avoid numerical errors leading to misleading values of factor of safety when the circles emerge too steeply (Whitman and Bailey, 1967). The critical slip surface based on Bishop s modified method for the embankment over DM columns is shown in Figure 7. Since the slip surface cut through three rows of DM walls, all DM walls mobilized their full strengths. Figure 7. Bishop s Circular Slip Surface Analysis (wall thickness = 1.0m; undrained shear strength of DM column, c u = 100kPa) The critical slip surface based on Spencer s three-part wedge method for the embankment over DM columns is shown in Figure 8. As compared with the shape of critical slip surface based on the velocity vectors, the three-part wedge slip surface

8 has a good representation of the critical slip surface obtained by the numerical method. Factor of safety Figure 8. Critical Slip Surface in Three-Part Wedge Analysis (undrained shear strength of DM column, c u = 100kPa) The computed factors of safety against deep-seated slope failure using the FLAC numerical method, Bishop s modified method, and Spencer s three-part wedge method are plotted in Figure 9. When the undrained shear strength of DM walls is equal to 10kPa, it represents a case with an untreated ground. It is shown that all three methods computed almost identical factors of safety. With an increase of the undrained shear strength of DM walls, the factors of safety computed by these three methods become different. Bishop s modified method yielded significantly higher factors of safety than those obtained by the numerical method when the undrained shear strength of the DM walls is higher. The difference may result from the misrepresentation of the circular slip surface and fully mobilized strength of DM walls assumed in Bishop s method for this specific problem. However, the three-part wedge method yielded lower or conservative factors of safety as compared with the numerical method. Spencer s method assumed that the interslice force angles from the left and right vertical sides of each slice are equal. The inclusion of vertical strong elements (DM walls) may change functions of interslice force angles between the left and right vertical sides of slice. Conclusions Numerical analyses indicated that the critical slip surface of the embankment over a deep mixed foundation for this specific problem was not circular. Bishop s modified method computed significantly higher factors of safety than the numerical method when the undrained shear strength of the deep mixed walls. Spencer s three-part wedge method yielded lower factors of safety than the numerical method.

9 2.5 2 Three-row soil-cement walls Wall thickness = 1m Factor of Safety, FOS Bishop's modified method Numerical method (FLAC) Three-part wedge method Undrained Shear Strength of DM Walls, c u (kpa) (a) Three-row soil-cement walls Wall thickness = 2m Factor of Safety, FOS Bishop's modified method Numerical method (FLAC) Three-part wedge method Undrained Shear Strength of DM Walls, c u (kpa) (b) Figure 9. Computed Factors of Safety using Limit Equilibrium and Numerical Methods (wall thickness = 2.0m)

10 Acknowledgement This research work was conducted under the research fellowship provided by the Japan Society for Promotion of Science (JSPS) for the first author as a visiting associate professor at Saga University, Japan. This support is greatly appreciated. The authors are thankful for valuable discussions with Prof. N. Miura at Saga University on this specific research topic. References Bishop, A.W. (1955). The use of the slip circle in the stability analysis of slopes. Geotechnique, 5, Broms, B.B. and Wong, I.H. (1985). Embankment piles. Third International Geotechnical Seminar Soil Improvement Methods, Singapore, November. Broms, B.B. (1999). Can lime/cement columns be used in Singapore and Southeast Asia? 3 rd GRC Lecture, Nov. 19, Nanyang Technological University and NTU- PWD Geotechnical research Centre, 214p. Cundall, P.A. (2002). The replacement of limit equilibrium methods in design with numerical solutions for factor of safety. Powerpoint presentation, Itasca Consulting Group, Inc. Dawson, E.M., Roth, W.H., and Drescher, A. (1999). Slope stability analysis by strength reduction. Geotechnique 49(6), Han, J., Leshchinsky, D., and Shao, Y. (2002). Influence of tensile stiffness of geosynthetic reinforcements on performance of reinforced slopes. Proceedings of Geosynthetics 7 th ICG, Delmas, Gourc & Girard (eds), Swets & Zeitlinger, Lisse, Itasca Consulting Group, Inc. (2002). FLAC/Slope User s Guide, 1 st Edition, 82p. Kitazume, M. and Terashi, M. (1991). Effect of local soil improvement on the behavior of revetment. Proc. Geo-Coast 91, 1, Kitazume, M., Omine, K., Miyake, M., and Fujisawa, H. (1997). JGS TC Report: Japanese design procedures and recent activities of DMM. Grouting and Deep Mixing, Yonekura, Terashi, and Shibazaki (eds), Balkema, Rotterdam, San, K.C., Leshchinsky, D., and Matsui, T. (1994). Geosynthetic reinforced slopes: limit equilibrium and finite element analyses. Soils and Foundations, 34(2), Terashi, M. (2002). The state of practice in deep mixing methods. ASCE Geotechnical Special Publication No. 120, L.F. Johnsen, D.A. Bruce, and M. J. Byle (eds.), Vol. 1, Whitman and Bailey (1967). Use of computers for slope stability analysis. Journal of Soil Mechanics and Foundation Engineering, ASCE, 93(SM4),