Archives of Mining Sciences 52, Issue 1 (2007) 75 89

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1 Archives of Mining Sciences 52, Issue 1 (2007) MAREK CAŁA* CONVEX AND CONCAVE SLOPE STABILITY ANALYSES WITH NUMERICAL METHODS ANALIZA STATECZNOŚCI ZBOCZY ZAKRZYWIONYCH Z ZASTOSOWANIEM METOD NUMERYCZNYCH This paper deals with the stability of convex and concave slopes. These types of slopes can be often found in the open pit mines. Two dimensional limit equilibrium methods are usually applied for slope stability analysis. Limit equilibrium methods extended to three dimensions are used occasionally. The stability of spatial columns (instead of slices) is analysed in these cases. This paper shows the possibility of application of three and two-dimensional numerical calculations for stability analysis of concave and convex slopes. The shear strength reduction method was used to calculate the value of safety factor. The results of calculations with shear strength reduction method were compared with ones obtained from limit equilibrium methods. The considerations presented below allow formulating the conclusion, that proper two dimensional slope stability analyses are impossible for many cases. It s necessary to perform a three dimensional numerical calculations, which allow to model spatial geometry and complex geology of any slope. Keywords: slope stability, numerical calculations W niniejszej pracy zajęto się analizą stateczności zboczy zakrzywionych. Z tego rodzaju zboczami (wklęsłymi lub wypukłymi) mamy często do czynienia w kopalniach odkrywkowych. Z reguły do analiz stateczności zboczy wykorzystuje się metody równowagi granicznej w płaskich przekrojach. Sporadycznie na świecie stosuje się metody równowagi granicznej rozszerzone do trzech wymiarów. Rozpatruje się wówczas stateczność nie płaskich bloków lecz przestrzennych kolumn. W poniższej pracy pokazano możliwości zastosowania przestrzennych i płaskich obliczeń numerycznych do analiz stateczności zboczy zakrzywionych. Wykorzystano w tym celu metodę redukcji wytrzymałości na ścinanie (SSR). Pozwala ona na określenie wartości minimalnego wskaźnika stateczności dla dowolnego zbocza. Przeprowadzono krytyczną analizę porównawczą obliczeń z zastosowaniem metod równowagi granicznej i metod numerycznych. Na ogół uważa się, że prowadzenie obliczeń w różnych przekrojach płaskich daje w efekcie rozsądne wyniki analiz sytuacji przestrzennych. Należy jednakże zauważyć, że w pewnych przypadkach zachodzi * AGH UNIVERSITY OF SCIENCE AND TECHNOLOGY, DEPARTMENT OF GEOMECHANICS, CIVIL ENGINEERING & GEOTECHNICS; AL. MICKIEWICZA 30, KRAKOW, POLAND

2 76 konieczność wykonania przestrzennych analiz ze względu na geometrię zbocza oraz budowę geologiczną. Analizy stateczności w płaskich przekrojach często prowadzą do zbytniego upraszczania problemu. Widać także, że wartości wskaźników stateczności (FS) uzyskane za pomocą przestrzennych (3D) metod numerycznych są znacznie niższe od wartości uzyskanych z metod równowagi granicznej (LEM). Przedstawione poniżej rozważania potwierdzają fakt, że w wielu przypadkach przeprowadzenie prawidłowej analizy stateczności zboczy na drodze klasycznych obliczeń dwuwymiarowych (2D) jest niemożliwe. Konieczna jest budowa przestrzennych modeli numerycznych, które pozwalają na w miarę wierne odtworzenie budowy geologicznej rozpatrywanego zbocza. Następnym krokiem jest przeprowadzenie obliczeń z wykorzystaniem SSR dla identyfikacji możliwych powierzchni poślizgu. Jak widać z przytoczonych przykładów, wyniki obliczeń numerycznych mogą się znacząco różnić od wyników uzyskanych z metod równowagi granicznej zaadoptowanych w 3D. Wyniki obliczeń numerycznych wskazują jednoznacznie, iż wartość wskaźnika stateczności zboczy wklęsłych i wypukłych jest większa niż w przypadku zboczy analizowanych w płaskim stanie odkształcenia. Dla analizowanych zboczy wklęsłych wartość FS jest istotnie wyższa (o 0.1) od przypadku w płaskim stanie odkształcenia (PS) dla promienia około 5-krotnie większego od wysokości zbocza. Dla rozpatrywanych zboczy wypukłych istotny wzrost FS występuje tylko przy małych wartościach promienia oscylujących około 0.6 H. Można zatem stwierdzić że w przypadku zboczy wypukłych wzrost ten jest stosunkowo niewielki. Słowa kluczowe: stateczność skarp i zboczy, obliczenia numeryczne 1. Introduction Using several cross-sections may sometimes provide a reasonable assessment of the 3D effect. However, in some cases 3D calculations are necessary in order to take the complexity of geology under consideration. Stability of spatial columns (not flat slices) is analysed in three-dimensional (3D) extensions of limit equilibrium methods (LEM). In spite of serious number of proposed 3D LEM methods, such analyses are not very often used for practical purposes. Application of LEM to solve 3D problems is also rather limited due to several simplifying assumptions. In must be noted, however, that an increasing number of investigators use 3D numerical calculations for estimating slope stability. Zettler et al. (1999) showed the stability analysis of three-dimensional slope. They considered concave and convex slopes and estimated its factors of safety (FS) as a function of geometry. Lorig and Varona (2000) applied shear strength reduction technique (detailed description of that method was presented in Cała & Flisiak, 2001, 2002 and 2003a, b) for 3D slope stability analysis for open pit gold mine in Bonias East (Spain). Poisel et al. (2001) used a FLAC3D code (finite difference method) to model rock slope landslides in Austria. They calibrated a 3D model with surface displacement measurements. Ledesma et al. (2002) presented a few examples of 3D stability analyses with FLAC3D and DRAC (Finite Element Method) codes obtaining a good agreement of the results. Peybernes (2003) applied FLAC3D to analyse the stability of earth dam because only the results of 3D calculations fit to in situ measurements. Suarez and Gonzalez (2003) used shear strength reduction technique (SSR) for threedimensional slope stability analysis of open pit gold mine in Boinas East (Spain). The

3 model of landslide progress as a function of time (considering also the changes of water table) was presented by Commend et al. (2004). They calibrated and validated the model using displacement measurements. Than, using calibrated model, they simulated several ways of slope stabilisation. Pasculli et al. (2006) presented a three dimensional stability analysis of the slope located close to Roccamontepiano (Italy) with SSR implemented to FLAC3D. They carried out about 53 FLAC3D runs of the same model of an actual landslide with different sets of selected numerical parameters. It produced the scatter of FS results relatively from 1.25 to 1.6. Tapia and Gomez (2006) described a three dimensional calibration and stability analyses for the western wall at Radomiro Tomic mine (Chile) with FLAC3D. The developed numerical model was successful in reproducing the instabilities that occurred in years 2000 and 2004 and was also applied for the assessment of alternative future design. They also formulated a failure criterion: increase of a displacement velocity of approximately m/s. This tool allowed very successful slope stability engineering at this particular open pit mine. Slope stability analyses with distinct element method (DEM) were not very often used. Such calculations were carried out for slopes dominated by several joint sets. Such example was presented by Valdivia and Lorig (2000), where three-dimensional model of open pit Escondida (Chile) was produced with 3DEC code. The same code was also utilised by Ferrero et al. (2004) to analyse the rock slope stability close to Arnad (Italy). 3DEC was used for to analyse complex geology rock slope stability by Poisel et al. (2002). Geographic information systems (GIS) are often used to process and analyse data that are relevant in evaluating natural hazards. GIS technology is applied to construct a three-dimensional surface map including some mechanical properties of soil or rock (Jibson et al., 2000; Donati i Turrini, 2002; Coe et al., 2003; Mora et al., 2003; Ayalev et al., 2004; Duman et al., 2005; Saha et al., 2005; Fall et al., 2006; Clerici et al., 2006). A very good overview considering these problems was presented by Van Westen (2004). It must be however noted that stability analyses are performed assuming very simple failure modes. The simplified solution for long, infinite slope with (or without seepage) is often applied. Babu and Mukesh (2002) performed stability analysis using the method of Culman (1875). Assuming such a method of factor of safety estimation may be reasonable only for specific geological conditions (for example for rock mass with oriented joint set). Otherwise, it may lead to serious mistakes. For example, using the Cullman method for stability analysis of the 20 m high (slope angle 40 ) dry slope (mechanical properties of the soil: c = 20 kpa, φ = 20, γ = 20 kn/m 3 ) estimated FS = Application of Bishop method for stability analysis of the same slope produces FS = The research framework SAR (synthetic aperture radar) interferometry was intensely applied in the last years for the monitoring of mass movements (Berardino et al., 2003, Antonello et al., 2004, Colesanti & Wasowski, 2004; Singhroy, 2004; Squarzoni et al., 2005; Cotecchia, 2006). A very good overview on these problems was presented by 77

4 78 Singhroy (2004). SAR framework was applied in Poland to identify mass movements due to underground mining excavation (Krawczyk & Perski, 2000). Accuracy of vertical displacement measurements with SAR may even range a few millimetres. Unfortunately, frequency of measurements is a serious disadvantage. SAR images for our part of Europe may be obtained from ESR-1 (only archive pictures) and ESR-2 satellites. The circulation period of ESR-2 satellite is equal 35 days. Such a frequency of observation may be useful only for very big slopes. It may be applied for outer damps in case of open pit mines. 2. Concave and convex slope stability It s well known fact that convex and concave slopes have different value of factor of safety. It was verified on a serious number of cases (Hoek & Bray, 1981; Hoek et al., 2000). Fig. 1 shows one quarter of the model for each slope due to the double symmetry of the model. R R a) b) Fig. 1. Convex and concave slope: a) concave slope, b) convex slope Rys. 1. Zakrzywione zbocza wklęsłe i wypukłe: a) zbocze wklęsłe, b) zbocze wypukłe Piteau and Jennings (1970) studied the influence of plan curvature on the stability of slopes in diamond mines in South Africa. The average slope height was 100 m. They found that the average slope angle for slopes with radii of 60 m was 39.5 as compared to 27.3 for slopes with radii of 300 m. Hoek and Bray (1981) summarized their experience with the stabilising effect of slope curvature as follows. When the radius of curvature of a concave slope is less than the height of the slope, the slope angle can be 10 steeper than the angle suggested by

5 conventional stability analysis. As the radius increases to the value greater than the slope height, the correction should be decreased. For the radii in excess of twice the slope height, the slope angle given by a conventional stability analysis should be used. These recommendations were confirmed through numerical calculations utilising SSR with FLAC by Lorig (1999) and Lorig & Varona (2000). They studied the stability of 500 m high concave slope (inclination 45 ). They check the influence of the increase of the radii on the FS value. Than, for given FS values they estimated the allowable slope inclination (assuming FS = 1.3) for different radii. The results of these calculations are presented in table 1. TABLE 1 The results of numerical calculations performed by Lorig (1999) and Lorig & Varona (2000) TABLICA 1 Wyniki obliczeń numerycznych z prac Loriga (1999) oraz Loriga i Varona (2000) 79 α = 45 FS = 1.3 FS as a function of radius (R) for slope inclination α = 45 R = 100 m R = 250 m R = 500 m R FS = 1.75 FS = 1.65 FS = 1.55 FS = 1.3 Slope inclination as a function of radius (R) for FS = 1.3 = const R = 100 m R = 250 m R = 500 m R α = 75 α = 65 α = 55 α = 45 These results fully confirmed recommendations from Hoek & Bray (1981). It must be however noted, the designers are reluctant to take advantage of the beneficial effects of the slope curvature, because of the presence of discontinuities can often negate the effects. Suarez and Gonzalez (2003) also pointed to an increase of FS for concave slope. They used the shear strength reduction technique to analyse the slope stability and found out that FS increase may range 0.45 (what clearly confirms results of calculations performed by Lorig and Varona (2000). The convex and concave slope stability analysis may be performed utilising 2D and 3D numerical modelling. The two dimensional, axisymmetric calculations may be used only assuming circular curvature. If the shape of curvature is different than only threedimensional calculations may be used. A series of numerical analysis were performed to better quantify the effects of slope curvature on its stability. The computer codes, FLAC (Itasca 2005) and FLAC3D (Itasca 2002) were used for numerical calculations and SLOPE/W (Krahn 2004) for LEM analysis. All analysis assumed a 10 m high (slope angle 45 ) dry slope consisting of isotropic homogeneous material (mechanical properties of the soil: c = 9 kpa, φ = 25, γ = 20 kn/m 3 ). The radius for convex and concave slopes was changed from 0 m to 5000 m. The calculations were performed for axisymmetric models. The factors of safety as a function

6 80 of radii are presented on fig. 2. The radii values were presented in logarithmic scale to clarify the results (that s why FS values for R = 0 were missed; they are 1.48 and 1.22 for concave and convex slope respectively). Fig. 2 clearly shows that both concave and convex slopes have higher FS than slope analysed in plane strain (PS) condition. Finally, for R, factors of safety for both cases tend to the value obtained from plane strain calculations, what is 1.0 in that case Concave slope Convex slope PS slope 1.35 Factor of safety (FS) Radius, m Fig. 2. FS as a function of radius R for convex, concave and PS slopes Rys. 2. FS w funkcji promienia krzywizny dla zboczy wklęsłych, wypukłych oraz PSO The results of axisymmetric modelling were confirmed by three-dimensional numerical calculations. The failure surface identified (through the shear strain distribution) for concave slope (radius equal 10 m) is presented on fig. 3. Three-dimensional numerical calculations produced FS = This value is in good agreement with FS = 1.2 obtained from axisymmetric model. Fig. 4 shows the failure surface for convex slope of 10 m radius. A 3D numerical calculations with SSR produced FS = And again, this value is in good agreement with FS = 1.07 obtained from axisymmetric model.

7 81 Fig. 3. Failure surface identified for concave slope, FS = 1.19 Rys. 3. Powierzchnia poślizgu zidentyfikowana dla zbocza wklęsłego, FS = 1.19 Fig. 4. Failure surface identified for convex slope, FS = 1.06 Rys. 4. Powierzchnia poślizgu zidentyfikowana dla zbocza wypukłego, FS = 1.06

8 82 It may be concluded that, for analysed concave slopes, FS value is considerably higher (over 0.1) than for plain slopes till radii values equal 5 times slope height. For convex slopes the increase of factor of safety is visible only for radii values of 0.6 H. This is not a very big increase, but quite nice for the inhabitant of Kraków due to the number of mounds located near the city. The similar conclusions were presented by Jiang et al. (2003). They analysed concave, convex and plain slopes and obtained the following values of FS: 1.87, 1.7 and These analyses were discussed later in this paper. For further verification of convex slope case, numerical calculations were performed for a model composed of convex slope and an embankment. A 60 m long embankment (soil mechanical properties were assumed precisely as in previous analysis) ended with a half of convex slope was considered. Only a quarter of the model was analysed due to double symmetry. The calculations resulted in factor of safety FS = 1.03 and failure surface ranging the whole embankment and a part of convex slope (fig. 5). Probably for the longer embankment FS values would tend to 1.0 (a FS for plane strain condition). This is a proof that the embankment (not its convex end) is the weakest element of the slope. Fig. 5. Failure surface ranging for embankment and partially its convex end Rys. 5. Powierzchnia poślizgu obejmująca wał i częściowo jego zakrzywione zakończenie These conclusions were also confirmed by Brząkała (2003). He performed a series of 2D numerical calculations for concave and convex slopes. He concluded that FS for concave and convex slopes increased about 10-15% comparing with plane strain condition.

9 That confirmed that is s incorrect to claim that FS for convex slope is lower than for (geometrically similar) slope in plane strain condition. That kind of error was committed by Zettler et al. (1999). They analysed the stability of homogeneous slope, height 25 m and inclination of (the mechanical properties of the soil: c = 38 kpa, φ = 45, γ = 25.7 kn/m 3 ). The calculations were performed for 3D models with shear strength reduction technique. The SSR calculations for axisymmetric models were carried out for verification with Zettler et al. (1999). The results of calculations for verification were presented in table The results of calculations for verification with Zettler et al. (1999) Wyniki obliczeń weryfikujących pracę Zettlera et al. (1999) TABLE 2 TABLICA 2 Slope Factor of safety FS Zettler et al. (1999) SSR concave convex PS The results of calculations performed for verification showed considerable differences in FS values. These differences in FS values were 0.12 and 0.05 for concave and convex slopes respectively. That may be assumed as acceptable numbers. But the difference of 0.17 between factors of safety for plane strain condition (PS) is significant. It was definitely not a correct result and misinformed about higher FS of PS slope comparing to concave one. The doubtful results of convex and concave slope stability analysis were showed by Cheng et al. (2005). They applied Bishop and Janbu methods extended to 3D for concave and convex slope stability analyses. They analysed a 10 m high (inclination 45 ) slope stability for different radii values (the mechanical properties of the soil were as follows: c = 10 kpa, φ = 36, γ = 19.5 kn/m 3 ). The numerical calculations with SSR method (utilising axial symmetry of the problem) were performed for the same slopes. The results for concave slopes are presented on fig. 6. It s quite clear that FS values obtained from Bishop and Janbu were significantly higher than estimated from numerical calculations. For decreasing values of radii, factors of safety evaluated with SSR were considerably increased. Unfortunately, in cannot be compared with the results from LEM, because Cheng et al. (2005) did not show it in their paper. Fig. 7 shows the comparison of calculation results for convex slopes. Factors of safety evaluated with Bishop and Janbu methods were considerably higher (about 0.1) than FS obtained with the application of numerical methods. It must be however noted, FS values produced by LEM increased with the increase of radii. That is not a reasonable

10 84 trend. The results from SSR method showed just the opposite trend. The decrease of radius produces small increase of safety factor and this is intuitively correct. 2.1 Factor of safety (FS) Bishop Janbu SSR PS Radius, m Fig. 6. The results of SSR calculation for verification with Cheng et al. (2005) concave slope Rys. 6. Porównanie wartości FS z pracy Cheng et al. (2005) i otrzymanych z SSR zbocze wklęsłe 1.7 Factor of safety (FS) Bishop Janbu SSR PS Radius, m Fig. 7. The results of SSR calculation for verification with Cheng et al. (2005) convex slope Rys. 7. Porównanie wartości FS z pracy Cheng et al. (2005) i otrzymanych z SSR zbocze wypukłe Considering the results of 3D calculations, one can easily find that FS values obtained from SSR method are lower than estimated from LEM. Similar case was analysed by

11 Xing (1988). He presented the results of 3D calculations for concave slopes with Fellenius method extended to three dimensions. Xing (1988) considered a 400 m high slope (inclination 35 o ) for different radii (the mechanical properties of the soil: c = 660 kpa, φ = 20, γ = 27 kn/m 3 ). The comparison of Xing (1988) calculation results with SSR (axisymmetric and plane strain models) is presented on fig Factor of safety (FS) LEM SSR LEM - 2D SSR-2D Radius/Slope height Fig. 8. The results of SSR calculation for verification with Xing (1988) Rys. 8. Porównanie wartości FS z pracy Xinga (1988) i otrzymanych z SSR Due to the considerable height of the slope, FS values were presented as a function of ratio of radius to slope height (R/H). It s quite clear that FS obtained from SSR are lower than FS estimated from limit equilibrium methods. Initially, (for small radius values) these difference range even to 0.3, but later it decrease to 0.1 for plane strain conditions. An interesting case of convex slope (precisely: conical) stability analysis with LEM in 3D was described by Leshchinski & Huang (1992). Unfortunately, poor description of the problem excludes its verification with the application of shear strength reduction technique. The spatial slope stability analyses with simplified Janbu method are presented by Jiang et al. (2003). They estimated the stability of homogeneous slopes (height 8 m, slope 33.7, radius 18 m) in 2D and 3D. It was assumed that soil had the following mechanical properties: c = 11.7 kpa, φ = 24.7, γ = kn/m 3. 2D and 3D numerical calculations with SSR were performed for to verify the results of Jiang et al. (2003). The results of the calculations are presented in table 3.

12 86 The results of calculations for verification with Jiang et al. (2003) Wyniki obliczeń weryfikujących pracę Jianga et al. (2003) TABLE 3 TABLICA 3 Slope Factor of safety FS Jiang et al. (2003) SSR 3D concave convex PS The failure surfaces identified with SSR for convex and concave slopes are showed on fig.9. Up to now it s the only one case when SSR analysis resulted in higher FS value then FS obtained from LEM analysis in 3D. The results of 2D LEM analysis are also quite interesting. Calculations with different methods produced the following results: Bishop: FS = 1.608, Fellenius: FS = 1.53, Janbu: FS = 1.493, Spencer: FS = However, the difference between results for convex slope is only equal 0.06 and is permissible. FS values for concave slopes are practically equal (the difference of 0.01 is very small). Fig. 9. Failure surfaces identified with SSR for conclave and convex slope Rys. 9. Powierzchnie poślizgu zidentyfikowane za pomocą SSR dla zbocza wklęsłego i wypukłego

13 87 3. Summary There is a widespread opinion that two-dimensional slope stability analysis (in different sections) may produce reasonable results for spatial situation. But sometimes, it s even necessary to perform a three dimensional analyses due to a spatial geometry and geology of the slope. The two dimensional slope stability analyses often lead to oversimplification of the problem. The factors of safety estimated from numerical methods are considerably lower than FS evaluated with limit equilibrium methods in 3D. The examples presented above confirm that it s impossible to perform correct slope stability analysis with 2D methods for certain cases. It s necessary to build three-dimensional models. Only these models allow a precise reconstruction of slope geology. Numerical calculations with SSR allowing identification of failure surface should be the next step. The results of 3D numerical calculations with SSR may be considerably different than obtained from limit equilibrium methods adopted for 3D. The results of three dimensional numerical slope stability analyses are promising. This is not an easy task because building of the spatial model geometry, validation and finally calculations, are quite time consuming. It may be however stated, three-dimensional numerical modelling would be widely used in the near future, especially in case of complex geology slopes. 4. REFERENCES Antonello G., Casagli N., F arina P., Leve D., N ico G., S ieber A.J., Tarchi D., Ground-based SAR interferometry for monitoring mass movements. Landslides. Vol.1, pp Ayalev L., Yaqmagishi H., Ugava N., Landslide susceptibility mapping using GIS-based weighted linear combination, the case in Tsugawa area of Agano River, Niigata Prefecture, Japan. Landslides. Vol.1, p Babu G.L.S., Mukesh M.D., Landslide analysis in Geographic Information Systems. application/natural_hazards/landslides. Berardino P., Costantini M., Franceschetti G., Iodice A., Pietranera L., Rizzo V., Use of differential SAR interferometry in monitoring and modelling large slope instability at Maratea (Basilicata, Italy). Engineering Geology. Vol. 68, p Brząkała W., About the stability of curved embankments. (in polish: O stateczności obwałowań zakrzywionych). Inżynieria Morska i Geotechnika (Maritime Engineering & Geotechnics). Vol. 23. Cała M., Flisiak J., Slope stability in the light of LEM and numerical calculations. (in polish: Analiza stateczności skarp i zboczy w świetle obliczeń analitycznych i numerycznych). XXIII Winter School of Rock Mechanics (XXIII Zimowa Szkoła Mechaniki Górotworu). Edited by KGBiG. Kraków, p Cała M., Flisiak J., Slope stability analysis with FLAC and limit equilibrium methods. FLAC and Numerical Modeling in Geomechanics (edited by Bilaux, Rachez, Detournay & Hart). A.A. Balkema Publishers, p Cała M., Flisiak J., The influence of weak layer on slope stability (in polish: Analiza wpływu słabej warstwy na stateczność skarp). XXV Winter School of Rock Mechanics (XXV Zimowa Szkoła Mechaniki Górotworu). Edited by KGBiG, Kraków, p Cała M., Flisiak J., 2003a. Complex geology slope stability analysis by shear strength reduction. In Brummer, Andrieux, Detournay & Hart (eds.) FLAC and Numerical Modelling in Geomechanics: A.A. Balkema Publishers. p

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