Bull Eng Geol Environ (2008) 67:471–478 DOI 10.1007/s10064-008-0156-z

ORIGINAL PAPER

Stability analysis of slopes using the finite element method and limiting equilibrium approach

Nezar Atalla Hammouri Æ Abdallah I. Husein Malkawi Æ Mohammad M. A. Yamin

Received: 19 July 2007 / Accepted: 14 April 2008 / Published online: 16 May 2008 Ó Springer-Verlag 2008

Abstract The limit equilibrium method is commonly non draine´es pour des sols argileux et la pre´sence de fentes used for , being relatively simple de traction. Les analyses ont e´te´ re´alise´es avec le logiciel compared with finite element analysis. Both methods were PLAXIS 8.0 (me´thode des e´le´ments finis) et le logiciel used to analyse homogeneous and inhomogeneous slopes, SAS-MCT 4.0 (me´thode des e´quilibres limites). Les fac- taking into account the rapid drawdown condition, the teurs de se´curite´ et les positions des surfaces de rupture undrained and the presence of tension cracks. critiques obtenus par les deux me´thodes sont compare´s. The analyses were carried out using PLAXIS 8.0 (finite element method) and SAS-MCT 4.0 (limit equilibrium Mots cle´s Stabilite´ des pentes Á Equilibres limites Á approach). The safety factor and location of the critical slip Ele´ments finis Á PLAXIS Á SAS-MCT surface obtained from the two methods are compared.

Keywords Stability of slopes Á Limiting equilibrium Á Introduction Finite element Á PLAXIS Á SAS-MCT In recent years there have been rapid developments in the Re´sume´ La me´thode de calcul aux e´quilibres limites est fields of computational methods, design and high- classiquement utilise´e pour les analyses de stabilite´ des speed and low-cost hardware. Of particular relevance to pentes, car relativement simple par rapport aux calculs en slope stability analysis are the limit equilibrium and finite e´le´ments finis. Les deux me´thodes ont e´te´ utilise´es pour element methods. However, when using limiting equilib- e´tudier la stabilite´ de pentes homoge`nes ou he´te´roge`nes, rium methods to analyze slopes, several computational conside´rant des situations de vidange rapide, des conditions difficulties and numerical inconsistencies may occur in locating the critical slip surface (depending on the ) and hence establishing a factor of safety. Despite these inherent limitations, due to its simplicity limiting equilib- N. A. Hammouri (&) rium continues to be the most commonly used approach. GIS and Remote Sensing, Hashemite University, However, as personal computers have become more readily Zerqa, Jordan e-mail: [email protected] available, the finite element method has been increasingly used in slope stability analysis. One of the advantages of A. I. H. Malkawi finite element over limiting equilibrium is that no Geotechnical and Dam Engineering, assumption is needed about the shape or location of the Jordan University of Science and Technology, Irbid 22110, Jordan critical failure surface. In addition, the method can be e-mail: [email protected] easily used with others to calculate stresses, movements, pore pressures in embankments and seepage induced fail- M. M. A. Yamin ure as as for monitoring progressive failure. Civil-, The University of Akron, Akron, OH 44304, USA Duncan (1996) presented a comprehensive review of e-mail: [email protected] both limit equilibrium and finite element analysis of slopes. 123 472 N. A. Hammouri et al.

He compared the results of finite element analyses with analysis technique for maintaining the mass in equilibrium field measurements and found a tendency for the calculated depending on the basic three equilibrium equations. Much deformations to be larger than the measured deformation. research has been carried out in this field since the first Yu et al. (1998) compared limit equilibrium results with attempt by Culmann in 1866 to deal mathematically with rigorous upper and lower bound solutions for the stability the slope stability problem (Yu et al. 1998). of simple earth slopes. They also compared the results The analysis of slopes using the limit equilibrium method using Bishop’s method with those obtained from limit (LEM) has been significantly refined by using various analysis method which takes advantage of the lower and methods of vertical slices. SAS-MCT Version 4 (Malkawi upper bound theorems of plasticity. Kim et al. (1999) and Hassan 2003) uses a newly developed automatic search analysed slopes using both the limit equilibrium method procedure coupled with new Monte Carlo methods of both and limit analysis method and found the results from the random jumping and random walking types for locating the two approaches were generally in good agreement for global critical circular and non-circular slip surface homogeneous slopes, although further work is needed to (Malkawi et al. 2000, 2001a, 2001b). It is based on the most analyze slopes with non-homogeneous soils. Zaki (1999) common five limiting equilibrium methods for determining suggested finite element offers real benefits over limiting the safety factor, i.e., ordinary or Fellenius method (Felle- equilibrium methods. Lane and Griffiths (2000) presented nius 1936), Bishop’s simplified method (Bishop 1955), an assessment of the stability of slopes under drawdown Janbu’s simplified method (Janbu 1973), Spencer’s method conditions using the finite element method to produce (Spencer 1967) and the generalized limiting equilibrium operating charts for circumstances which should be appli- (GLE) method, a discrete version of Morgenstern–Price cable to real structures. Rocscience Inc. (2001) presented a method (Morgenstern and Price 1965). document outlining the capabilities of the finite element method and after comparing the results with those from various limit equilibrium methods, suggested the finite Factors affecting slope stability element method was of more practical use. Kim et al. (2002) analysed several slopes with inhomogeneous In general, the most important factors for slope stability profiles and irregular geometry using both the upper and analysis are: (1) The geometry of slope; (2) The material lower bound analysis and the limit equilibrium methods. properties of the soil; and (3) The forces acting on the Both methods gave a similar factor of safety and location slope. The study considers, three different examples with of the critical slip surface compared with those obtained homogeneous and inhomogeneous slopes taking into using finite element analysis. account the effects of: (a) Rapid drawdown, (b) Undrained clay soils, Brief description of finite element and limit equilibrium (c) Crack location. methods (a) This study will focus only on rapid drawdown as this The finite element method (FEM) is a numerical technique is the more critical case when sudden removal of water can for solving differential equations or boundary value prob- reduce the stability on the edge of a reservoir as there is lems in science and engineering. For further details, readers insufficient time for the pore water pressures to stabilise. are referred to the work of Clough and Woodward (1967), (b) An undrained clay slope is a special case as the Strang and Fix (1973), Hughes (1987), Zienkiewicz and frictional of the soil particles is usually very Taylor (1989). low. Rapid loading of fine-grained soils would allow the PLAXIS Version 8 (Brinkgreve 2002) is a finite element package intended for the two dimensional analysis of Water deformation and stability in geotechnical engineering. The Level program can analyse problems in man-made or natural Water slopes. The safety factor is determined using the //c Level reduction approach where the strength parameters (tan/) and (c) of the soil are successively reduced until failure of H HW the structure occurs. The limit equilibrium method (conventional method) has been widely used in geotechnical engineering problems for many years by applying the perfectly plastic Mohr– Coulomb criterion. The method is a purely numerical static Fig. 1 Application of rapid drawdown condition for example 1 123 Stability analysis using FEM and LEA 473

2.4 F.E.M (0.1) Spencer (Noncircular) (0.1) 2.2 F.E.M (0.15) Spencer (Noncircular) (0.15) 2.0 F.E.M (0.2) Spencer (Noncircular) (0.2) 1.8

1.6

1.4

Safety Factor 1.2

1.0

0.8

0.6 0 102030405060708090100 Drawdown Percent

Fig. 2 Comparison of Spencer and finite element for example 1 (/ = 12°;C/cH = 0.1, 0.15, 0.2)

Fig. 3 Contour of total displacement for example 1 using FEM (C/cH = 0.15; / = 30°). a Drawdown = 20%; b Drawdown = 80%

water in the pore spaces to dissipate. The internal angle of Fig. 4 Critical noncircular slip surface for example 1 using LEM under these conditions is assumed to be zero for (C/cH = 0.15; / = 30°) a Drawdown = 20%; b Drawdown = 80% saturated samples. (c) The existence of tension cracks at the head of a slide is an important indicator of an instability problem. These Effect of rapid drawdown condition cracks may be filled with water (e.g., due to rainfall). In cohesive soils with a low tensile strength, they can become Figure 1 displays the slope geometry for a 2:1 (64°) significant, notably in reducing the length of failure surface homogeneous simple slope with a height of 10 m. The along which the shear strength can be mobilized. slope was analyzed for different values of (C/cH) ratio (0.10, 0.15, and 0.20) and different values of friction angle PLAXIS 8.0 versus SAS-MCT 4.0 (/: 12, 20, 30, and 40°). Lane and Griffiths (2000) con- sidered a range of (C/cH) from 0.01 to 1.0 and friction PLAXIS Version 8 (an FEM package) and SAS-MCT angles (/) from 12 to 40°. An analysis of rapid drawdown Version 4 (an LEM package) were used in this study to was made from 0% (when the reservoir is full and the soil analyze the three examples. is fully saturated) to 100% (when the reservoir is empty but 123 474 N. A. Hammouri et al.

Fig. 5 Slope geometry of example 2 1m

1m 3 4

CU1, φU1, γ1

1 2 6

CU2, φU2, γ2 y

0 x 5

4.5 F.E.M (0.15) Bishop (0.15) 4.0 F.E.M (0.2) Bishop (0.2) 3.5 F.E.M (0.25) Bishop (0.25) 3.0

2.5

2.0

Safety Factor 1.5

1.0

0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 (Cu2/Cu1)

Fig. 6 Comparison of finite element and Bishop results for example 2 for different values of CU2/CU1:CU1/cH = 0.15, 0.2, 0.25 Fig. 8 Contour of total displacement for example 2 using FEM (CU1/cH) = 0.25: a (CU2/CU1) = 1.0; b (CU2/CU1) = 1.5

the soil is fully saturated). Lowering the water level as illustrated in Fig. 1, in the initial situation (H) the water 5.5 F.E.M (Cu2=Cu1) level height is equal to the height of the slope (drawdown 5.0 Bishop (Cu2=Cu1) F.E.M (Cu2>>Cu1) 0%) while in the final situation (Hw) the height of the water Bishop (Cu2>>Cu1) 4.5 level is zero (drawdown = 100%). The drawdown per- centage is given by Hw divided by H 9 100. 4.0 Figure (2) shows a similar general trend for the decrease 3.5 in the safety factor for the different ratios and a friction angle 3.0 of 12° in the low drawdown condition, although the curva- 2.5 ture depends on the method used; Spencer’s method giving Safety Factor

2.0 lower factor of safety values than FEM. With high draw- down percentages, the values for both Spencer’s method and 1.5 FEM are almost the same. This suggests that the value of the 1.0 safety factor is controlled by the weight of the water. Using 0.5 Spencer’s method, which assumes that the inclination of the 0.15 0.20 0.25 0.30 inter-slice forces is the same for all slices, gives a less (Cu1/γH) conservative result than that obtained using FEM. Similar Fig. 7 Comparison of finite element and Bishop results for example results were obtained for all the C/cH ratios (0.10, 0.15, and 2 for CU2 =CU1 and CU2  CU1 0.20) and friction angles (12, 20, 30, and 40°).

123 Stability analysis using FEM and LEA 475

3 both layers is c1 = c2 = 20 kN/m and the undrained friction angle is /U1 = /U2 = 0. The example was analyzed for different ratios of CU1/ cH (0.15, 0.2, 0.25, and 0.3). The for the

layer (CU2) was calculated for different ratios of CU2/CU1 from 0.5 to 3.0 and Bishop’s method used for determining the safety factor for the limiting equilibrium approach, assuming the shape of the critical slip surface is circular.

Figure 6 illustrates the results obtained with a CU1/cH ratio of 0.2 and different CU2/CU1 ratios. The FEM presents an increasing trend in the safety factor as the

CU2/CU1 ratio increases from 0.5 to 3.0 while the LEM analysis indicates the safety factor increases as the CU2/ CU1 ratio increases up to 1.5 after which it becomes constant. Increasing the cohesion for the foundation layer results in increased stability when the critical slip surface passes through that layer, but if the critical slip surface is at the toe, the value of the safety factor will not be changed because of the fixed value of cohesion for the top layer. Therefore, the location of the critical slip surface controls the safety factor. As the FEM presents a deep slip surface (i.e. base circle), the safety factor is higher (Fig. 8) but when the LEM presents a deep slip surface, a toe failure only occurs when the cohesion for the foundation layer is more than 1.5 times the cohesion for the top layer (Fig. 9). In general, the safety factors obtained from the two

methods were very similar (Fig. 7), but when CU2 is much greater than CU1, the angle of the line is greater using FEM than LEM. Again, these results applied for all the CU1/cH ratios used in the analyses.

Effect of crack location

The slope geometry for this example is shown in Fig. (10). The slope consists of four different soil layers with the presence of a water level. The physical properties of the soil layers are summarized in Table 1. Spencer’s method Fig. 9 Critical circular slip surface for example 2 using LEM (CU1/ cH) = 0.25, a (CU2/CU1) = 1.0, b (CU2/CU1) = 1.5 and a non-circular slip surface was used in the LEM analysis. Tension cracks were placed at different locations As the shape of the critical slip surface controls the behind the steep slope (S = 5, 10, 15, 20, 25, 30, and value of the safety factor, the LEM can include a slip 35 m). The depth of the tension crack was 5 m. surface intersecting the slope and passing through the toe As seen in Fig. 11 using LEM there is an increase in of the slope, while with FEM the slip only passes through safety factor with distance from the scarp while the line the toe of the slope (Figs. 3, 4). produced by FEM remained effectively constant. The main reason for this inconsistency is that with FEM the slip Effect of undrained clay soil on slope failure surface is almost the same for all cases of crack location (Fig. 12) while the LEM takes into account the crack The slope geometry of this example is shown in Fig. 5 location as the point where a progressive failure can be where the height of the slope is 6 m, the soil unit weight for initiated (Fig. 13).

123 476 N. A. Hammouri et al.

Fig. 10 Slope geometry and application of cracks location on S slope for example 3 5m 3 4

5m Layer 1

10 Water Level 11 Layer 2

8 9 1 2 Layer 3 7

6 Layer 4 y

0 x 5

Table 1 Physical properties of soil layers for example 3 Layer Friction Cohesion Unit weight angle, / (°) C (kN/m2) c (kN/m3)

1 32 20 18.2 2 30 25 18.0 3 18 40 18.5 4 28 40 18.8

1.29

1.25

1.21

1.17

Safety Factor Fig. 12 Contour of total displacement for example 3 using FEM a 1.13 S = 15 m; b S = 25 m

1.09 FEM of slopes having arbitrary regular and irregular geometry Janbu(Noncircular) were analysed using both methods. Table 2 shows a com- 1.05 0 5 10 15 20 25 30 35 40 parison between obtained safety factor values from both S (m) FEM and LEM; it can be seen that the differences in the values of safety factor achieved were found to be small. Fig. 11 Comparison of finite element and Janbu results for different The critical slip surfaces obtained using both methods crack locations for example 3 compared well. This study supports previous work indicating that there Summary and conclusions are differences between the safety factors obtained using FEM and various LEM methods. In the cases reported here, FEM and LEM were used to study the stability of slopes both methods gave an almost identical shape and location with homogeneous and inhomogeneous soils. The effects for the critical slip surface for a slope, except in the case of of rapid drawdown, undrained clay soils and crack loca- the undrained clay slope where FEM was not able to locate tions were taken in consideration. Three different examples the critical slip surface. 123 Stability analysis using FEM and LEA 477

FEM offers an automated mechanism for searching the critical limit load and its associated lower and upper bound analysis. This is considered a real benefit over LEM which uses either a grid method or random slip surfaces generator procedure as in the SAS-MCT program. With LEM, it is necessary to identify the shape of the critical slip surface (circular or non-circular) before starting the search proce- dure or analyse for both shapes; with FEM this is an automated procedure. In view of the differences in the values of safety factor obtained, it is recommended that the engineer should analyse critical slopes using both FEM and LEM.

References

Bishop W (1955) The use of the slip circle in the stability analysis of slopes. Geotechnique 5(1):7–17 Brinkgreve R (2002) PLAXIS Version 8, finite element code for soil and rock analyses. A. A. Balkema, Netherlands Clough R, Woodward R (1967) Analysis of stresses and deformations. J Soil Mech Found Div ASCE 4:529–549 Duncan J (1996) State of the art: limit equilibrium and finite element analysis of slopes. J Geotech Geoenviron Eng ASCE 122(7):578–584 Fellenius W (1936) Calculation of stability of earth dams. Transac- tions, 2nd congress large dams, Washington, DC, pp 445–462 Hughes R (1987) The finite element method: linear static and dynamic finite element analysis. Prentice-Hall, Englewood Cliffs Husein Malkawi A, Hassan W (2003) SAS-MCT 3.0, stability analysis of slopes using Monte Carlo technique. A Software for 2D and 3D slope stability analysis with dynamic effect, Windows Version Husein Malkawi A, Hassan W, Abdulla F (2000) Uncertainty and reliability analysis applied to slope stability. J Struct Safety 22:161–187 Husein Malkawi A, Hassan W, Sarma S (2001a) An efficient search method for locating circular slip surface using Monte Carlo Fig. 13 Critical noncircular slip surface for example 3 using LEM a technique. Can Geotech J 38(5):1081–1089 S = 15 m; b S = 25 m Husein Malkawi A, Hassan W, Sarma S (2001b) A global search method for locating general slip surface using Monte Carlo Table 2 Comparison between safety factor values obtained from technique. J Geotech Geoenviron Eng ASCE 127(8):688–698 FEM and LEM Morgenstern R, Price V (1965) The analysis of the stability of general slip surfaces. Geotechnique 15(1):79–93 Safety factor LEM Method Safety factor Janbu N (1973) Slope stability computations in embankment-dam difference % engineering. Wiley, New York, pp 47–86 FEM LEM Kim J, Salgado R, Yu H (1999) Limit analysis of soil slopes subjected to pore-water pressures. J Geotech Geoenviron Eng ASCE Example 1 (a) 1.928 1.714 Spencer 12 125(1):49–58 (b) 1.505 1.369 Spencer 10 Kim J, Salgado R, Lee J (2002) Stability analysis of complex soil Example 2 (a) 1.464 1.472 Bishop 1 slopes using limit analysis. J Geotech Geoenviron Eng ASCE (b) 2.193 2.026 Bishop 8 128(7):546–557 Lane P, Griffiths D (2000) Assessment of stability of slopes under Example 3 (a) 1.243 1.152 Janbu 8 drawdown conditions. J Geotech Geoenviron Eng ASCE (b) 1.241 1.218 Janbu 2 126(5):443–450 Rocscience Inc. (2001) Application of the finite element method to slope stability, Toronto Where tension cracks were included at different loca- Spencer E (1967) A method of analysis of the stability of embank- tions, FEM was not able to adequately reflect their ments assuming parallel inter-slice forces. Geotechnique 15:11– significance. 26

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Strang G, Fix J (1973) An analysis of the finite element method. Zaki A (1999) Slope stability analysis overview. University of Prentice-Hall, Englewood Cliffs Toronto Yu H, Salgado R, Sloan W, Kim J (1998) Limit analysis versus Zienkiewicz C, Taylor L (1989) The finite element method, 4th edn, equilibrium for slope stability. J Geotech Geoenviron Eng ASCE 1. McGraw-Hill, New York 124(1):1–11

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