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

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. Sporadycz-nie 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. Przeprowadzo-no 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, 30-059 KRAKOW, POLAND

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konieczność wykonania przestrzennych analiz ze względu na geometrię zbocza oraz budowę geologicz-ną. 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 przeprowa-dzenie obliczeń z wykorzystaniem SSR dla identyfi kacji 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 rozpa-trywanych 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 fl at 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 con-sidered 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 (fi nite difference method) to model rock slope landslides in Austria. They calibrated a 3D model with surface displacement meas-urements. 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 fi t to in situ measurements.

Suarez and Gonzalez (2003) used shear strength reduction technique (SSR) for three-dimensional slope stability analysis of open pit gold mine in Boinas East (Spain). The

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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 de-veloped 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 2.5×10–6 m/s. This tool allowed very successful slope stability engineer-ing 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 simplifi ed solution for long, infi nite 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 rea-sonable only for specifi c 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/m3) estimated FS = 1.371. Appli-cation of Bishop method for stability analysis of the same slope produces FS = 1.001.

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

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Singhroy (2004). SAR framework was applied in Poland to identify mass movements due to underground mining excavation (Krawczyk & Perski, 2000). Accuracy of verti-cal displacement measurements with SAR may even range a few millimetres. Unfortu-nately, 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 observa-tion 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 verifi ed 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.

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

R

R

a) b)

Piteau and Jennings (1970) studied the infl uence 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

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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 confi rmed 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 infl uence 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)

FS as a function of radius (R) for slope inclination α = 45°

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

FS = 1.3R = 100 m R = 250 m R = 500 m R → ∞α = 75° α = 65° α = 55° α = 45°

These results fully confi rmed recommendations from Hoek & Bray (1981). It must be however noted, the designers are reluctant to take advantage of the benefi cial 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 confi rms 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 three-dimensional 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 (Ita-sca 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/m3). 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

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of radii are presented on fi g. 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.

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

Concave slope

Convex slope

PS slope

Radius, m

Facto

rofsafe

ty(F

S)

1 10 100 1000 �

1

1.1

1.2

1.3

1.4

1.5

1.05

1.15

1.25

1.35

1.45

The results of axisymmetric modelling were confi rmed by three-dimensional numeri-cal calculations. The failure surface identifi ed (through the shear strain distribution) for concave slope (radius equal 10 m) is presented on fi g. 3. Three-dimensional numerical calculations produced FS = 1.19. 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 = 1.06. And again, this value is in good agreement with FS = 1.07 obtained from axisymmetric model.

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Fig. 3. Failure surface identifi ed for concave slope, FS = 1.19

Rys. 3. Powierzchnia poślizgu zidentyfi kowana dla zbocza wklęsłego, FS = 1.19

Fig. 4. Failure surface identifi ed for convex slope, FS = 1.06

Rys. 4. Powierzchnia poślizgu zidentyfi kowana dla zbocza wypukłego, FS = 1.06

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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 1.48. These analyses were discussed later in this paper.

For further verifi cation 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 (fi g. 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 confi rmed by Brząkała (2003). He performed a series of 2D numerical calculations for concave and convex slopes. He concluded that FS for con-cave and convex slopes increased about 10-15% comparing with plane strain condition.

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That confi rmed 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 63.43° (the mechanical properties of the soil: c = 38 kPa, φ = 45°, γ = 25.7 kN/m3). The calculations were performed for 3D models with shear strength reduction technique. The SSR calculations for axisymmetric models were carried out for verifi cation with Zettler et al. (1999). The results of calculations for verifi cation were presented in table 2.

TABLE 2

The results of calculations for verifi cation with Zettler et al. (1999)TABLICA 2

Wyniki obliczeń weryfi kujących pracę Zettlera et al. (1999)

SlopeFactor of safety – FS

Zettler et al. (1999) SSR concave 1.83 1.71convex 1.37 1.32

PS 1.41 1.24

The results of calculations performed for verifi cation 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 signifi cant. It was defi nitely 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 con-cave 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/m3). 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 fi g. 6.

It’s quite clear that FS values obtained from Bishop and Janbu were signifi cantly 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

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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.

Fig. 6. The results of SSR calculation for verifi cation with Cheng et al. (2005) – concave slope

Rys. 6. Porównanie wartości FS z pracy Cheng et al. (2005) i otrzymanych z SSR – zbocze wklęsłe

�

Bishop

Janbu

SSR

PS

Radius, m

Facto

rofsafe

ty(F

S)

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

0 20 40 60 80 100

Fig. 7. The results of SSR calculation for verifi cation with Cheng et al. (2005) – convex slope

Rys. 7. Porównanie wartości FS z pracy Cheng et al. (2005) i otrzymanych z SSR – zbocze wypukłe

Bishop

Janbu

SSR

PS

1.4

1.5

1.6

1.7

�

Radius, m

0 20 40 60 80 100

Facto

rofsafe

ty(F

S)

Considering the results of 3D calculations, one can easily fi nd that FS values obtained from SSR method are lower than estimated from LEM. Similar case was analysed by

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Xing (1988). He presented the results of 3D calculations for concave slopes with Felle-nius method extended to three dimensions. Xing (1988) considered a 400 m high slope (inclination 35o) for different radii (the mechanical properties of the soil: c = 660 kPa, φ = 20°, γ = 27 kN/m3). The comparison of Xing (1988) calculation results with SSR (axisymmetric and plane strain models) is presented on fi g. 8.

Fig. 8. The results of SSR calculation for verifi cation with Xing (1988)

Rys. 8. Porównanie wartości FS z pracy Xinga (1988) i otrzymanych z SSR

LEM

LEM - 2D

SSR

SSR - 2D

1 2 3 4 5 6

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Facto

rofsafe

ty(F

S)

Radius/Slope height

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 verifi cation with the application of shear strength reduction technique.

The spatial slope stability analyses with simplifi ed 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 = 17.66 kN/m3.

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.

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TABLE 3

The results of calculations for verifi cation with Jiang et al. (2003)TABLICA 3

Wyniki obliczeń weryfi kujących pracę Jianga et al. (2003)

SlopeFactor of safety – FS

Jiang et al. (2003) SSR 3Dconcave 1.87 1.86convex 1.70 1.76

PS 1.48 1.61

The failure surfaces identifi ed with SSR for convex and concave slopes are showed on fi g.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 = 1.605. 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 identifi ed with SSR for conclave and convex slope

Rys. 9. Powierzchnie poślizgu zidentyfi kowane za pomocą SSR dla zbocza wklęsłego i wypukłego

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3. Summary

There is a widespread opinion that two-dimensional slope stability analysis (in dif-ferent 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 over-simplifi cation 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 confi rm that it’s impossible to perform correct slope stability analysis with 2D methods for certain cases. It’s necessary to build three-di-mensional models. Only these models allow a precise reconstruction of slope geology. Numerical calculations with SSR allowing identifi cation 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 fi nally calculations, are quite time consuming. It may be however stated, three-dimen-sional numerical modelling would be widely used in the near future, especially in case of complex geology slopes.

4. REFERENCES

A n t o n e l l o G., C a s a g l i N., F a r i n a P., L e v e D., N i c o G., S i e b e r A.J., T a r c h i D., 2004. Ground-based SAR interferometry for monitoring mass movements. Landslides. Vol.1, pp. 21-28.

Ay a l e v L., Ya q m a g i s h i H., U g a v a N., 2004. Landslide susceptibility mapping using GIS-based weighted linear combination, the case in Tsugawa area of Agano River, Niigata Prefecture, Japan. Landslides. Vol.1, p. 73-81.

B a b u G.L.S., M u k e s h M.D., 2002. Landslide analysis in Geographic Information Systems. www.gisdevelopment.net/application/natural_hazards/landslides.

B e r a r d i n o P., C o s t a n t i n i M., F r a n c e s c h e t t i G., I o d i c e A., P i e t r a n e r a L., R i z z o V., 2003. Use of differential SAR interferometry in monitoring and modelling large slope instability at Maratea (Basilicata, Italy). Engineering Geology. Vol. 68, p. 31-51.

B r z ą k a ł a W., 2003. About the stability of curved embankments. (in polish: O stateczności obwałowań zakrzywionych). Inżynieria Morska i Geotechnika (Maritime Engineering & Geotechnics). Vol. 23.

C a ł a M., F l i s i a k J., 2000. Slope stability in the light of LEM and numerical calculations. (in polish: Analiza statecz-ności 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. 27-37.

C a ł a M., F l i s i a k J., 2001. 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. 111-114.

C a ł a M., F l i s i a k J., 2002. The infl uence 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. 83-92.

C a ł a M., F l i s i a k 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. 99-102.

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C a ł a M., F l i s i a k J., 2003b. Slope stability analysis with numerical and limit equilibrium methods. In Burczynski, Fedelinski & Majchrzak (eds.) Computer Methods in Mechanics; CMM-2003.

C h e n g Y.M., L i u H.T., We i W.B., A u S.K., 2005. Location of critical three-dimensional non-spherical failure surface by NURBS functions and ellipsoid with applications to highway slopes. Computers and Geotechnics. Vol. 32, p. 387-399.

C l e r i c i A., P e r e g o S., T e l l i n i C., Ve s c o v i P., 2006. A GIS-based automated procedure for landslide suscep-tibility mapping by the Conditional Analysis method: the Baganza valley case study (Italian northern Apennines). Environmental Geology. Vol. 50, p. 941-961.

C o e J.A., E l l i s W.L., G o d t J.W., S a v a g e W.Z., S a v a g e J.E., M i c h a e l J.A., K i b l e r J.D., P o w e r s P.S., L i d k e D.J., D e b r a y S., 2003. Seasonal movement of the Slumgullion landslide determined from Global Posi-tioning System surveys and fi eld instrumentation, July 1998-March 2002. Engineering Geology. Vol. 68, p. 67-101.

C o l e s a n t i C., Wa s o w s k i J., 2004. Satellite SAR interferometry for wide-area slope hazard detection and site-specifi c monitoring of slow landslides. Landslides: Evaluation and Stabilisation (edited by Lacerda, Erlich, Fontura & Sayao) Taylor & Francis Group, London. p. 795-802.

C o m m e n d S., G e i s e r F., T a c h e r L., 2004. 3D numerical modeling of a landslide in Switzerland. Numerical Models in Geomechanics – NUMOG IX. Pande & Pietruszczak (eds.). Taylor & Francis Group, London, p. 595-601.

C o t e c c h i a V., 2006. The second Hans Cloos Lecture. Experience drawn from the great Ancona landslide of 1982. Bull. Eng. Geol. Env. Vol 65, p. 1-41.

D o n a t i L., T u r r i n i M.C., 2002. An objective method to rank the importance of the factors predisposing to landslides with the GIS methodology : application to an area of the Apennines (Valnerina; Perugia, Italy). Engineering Geology. Vol. 63, p. 277-289.

D u m a n T.Y., C a s n T., E m r e O., K e c e r M., D o g a n A., A t e s S., D u r m a z S., 2005. Landslide inventory of northwestern Anatolia, Turkey. Engineering Geology. Vol. 77, p. 99-114.

F a l l R., A z z a m R., N o u b a c t e p C., 2006. A multi-method approach to study the stability of natural slopes and landslide susceptibility mapping. Engineering Geology. Vol. 82, p. 241-263.

F e r r e r o A.M., G i a n f r a n c o F., Vo y a t I.H., 2004. Rock slope stability analysis based on photogrammetric surveys. Landslides: Evaluation and Stabilisation (edited by Lacerda, Erlich, Fontura & Sayao) Taylor & Francis Group, London. p. 789-794.

H o e k E., B r a y J.W., 1981. Rock slope engineering. 3rd edition. London. The Institute of Mining & Metallurgy. H o e k E., R i p p e r e K.H., S t a c e y P.F., 2000. Large scale slope designs – A review of the state of the art. Slope

stability in surface mining (edited by Hustrulid, McCarter & Van Zyl). Society for Mining, Metallurgy and Explora-tion Inc. Littleton, p. 115-124.

Itasca Consulting Group, Inc. 2002. FLAC3D – Fast Lagrangian Analysis of Continua in 3 Dimensions, Ver. 2.1 User’s Guide. Minneapolis: Itasca.

Itasca Consulting Group, Inc. 2005. FLAC – Fast Lagrangian Analysis of Continua, Ver. 5.0 User’s Guide. Minneapolis: Itasca.

J i a n g J.C., B a k e r R., Ya m a g a m i T., 2003. The effect of strength envelope nonlinearity on slope. Canadian Geotechnical Journal. Vol. 40, p. 308-325.

J i b s o n R.W., H a r p E.L., M i c h a e l J.A., 2000. A method for producing digital probabilistic seismic landslide hazard map. Engineering Geology. Vol. 58, pp. 271-289.

K r a h n J., 2004. Stability Modelling with SLOPE/W. Geo-SLOPE Int. Ltd. K r a w c z y k A., P e r s k i Z., 2000. Application of satellite radar interferometry on the areas of underground exploita-

tion of copper ore in LGOM – Poland. Proc. Of 11th Congress of the International Society for Mine Surveying. Vol. 2, p. 209-218.

L e s h c h i n s k y D., H u a n g Ch-Ch., 1992. Generalized slope stability analysis: interpretation, modifi cation and comparison. Journal of Geotechnical Engineering. Vol. 118, p. 1559-1576.

L e d e s m a A., H u r l i m a n n M., P r a t P., S c a v i a C., A m a t r u d a G., P o i s e l R., R o t h W., P r e h A., 2002. D8-Guidelines of the use of numerical methods for the prediction of failure. Impact of large landslides in the mountain environment: identifi cation and mitigation of risk.

89

L o r i g L., 1999. Lessons learned from slope stability studies. FLAC and numerical modeling in geomechanics (Detournay & Hart). A.A. Balkema. Rotterdam. pp. 17-21.

L o r i g L., Va r o n a P., 2000. Practical slope stability analysis using fi nite-difference codes. Slope stability in surface mining (edited by Hustrulid, McCarter & Van Zyl). Society for Mining, Metallurgy and Exploration Inc. Littleton, p. 115-124.

M o r a P., B a l d i P., C a s u l a G., F a b r i s M., G h i r o t t i M., M a z z i n i E., P e s c i A., 2003. Global Positioning Systems and digital photogrammetry for the monitoring of mass movements: application to the monitoring of mass movements: application to the Ca’ di Malta landslide (northern Appenines, Italy). Engineering Geology. Vol. 68, p. 103-121.

P a s c u l l i A., C a l i s t a M., M a n g i f i e s t a M., 2006. The effects of spatial variability of mechanical parameters on a 3D landslide study. 4th International FLAC Symposium on Numerical Methods in Geomechanics – Hart & Varona (eds.). p. 27-34.

P i t e a u D.R., J e n n i n g s J.E., 1970. The effects of plan geometry on the stability of natural slopes in rock in the Kimberley area of South Africa. Proc. Of the Second Congress of the International Society of Rock Mechanics. Belgrade. Vol. 3, paper 7-4.

P o i s e l R., R o t h W., P r e h A., A n g e r e r H., 2001. The Eiblschrofen rock falls – interpretation of monitoring results based on FLAC3D investigations. FLAC and Numerical Modeling in Geomechanics. Edited by BillauxD, Rachez X., Detournay Ch. & Hart R.). Swets & Zeitlinger. A.A. Balkema Publishers. p. 157-161.

P o i s e l R., R o t h W., P r e h A., T e n t s c h e r t E., A n g e r e r H., 2002. The Eiblschrofern rock falls – interpretation of monitoring results. Landslides (ed. by Rybar, Stemberk & Wagner). Swets & Zeitlinger, Lisse, p. 449-454.

P e y b e r n e s C., 2003. Analysis of hydraulic fracture risk in a yoned dam with FLAC3D. In Brummer, Andrieux, De-tournay & Hart (eds.) FLAC and Numerical Modelling in Geomechanics: A.A. Balkema Publishers. p. 103-108.

S a h a A.K., G u p t a R.P., S a r k a r I., A r o r a M.K., C s a p l o v i c s E., 2005. An approach for GIS-based statistical landslide susceptibility zonation – with a case study in the Himalayas. Landslides. Vol. 2, p. 61-69.

S i n g h r o y V., 2004. Remote sensing of landslides. Landslide hazard and risk. Edited by Glade T., Anderson M. & Crozier M. J. Wiley & Sons Ltd. p. 469-492.

S q u a r z o n i C., D e l a c o u r t C., A l l e m a n d P., 2005. Differential single-frequency GPS monitoring of the La Valette landslide (French Alps). Engineering Geology. Vol. 79, p. 215-229.

S u a r e z A.V., G o n z a l e z L.I.A., 2003. 3D slope stability analysis At Boinas East gold mine. In Brummer, An-drieux, Detournay & Hart (eds.) FLAC and Numerical Modelling in Geomechanics: A.A. Balkema Publishers. p. 117-123.

T a p i a A., G o m e z P., 2006. Three-dimensional calibration and stability analyses for the west wall at Radomiro Tomic (RT) Mine. 4th International FLAC Symposium on Numerical Methods in Geomechanics – Hart & Varona (eds.). p. 21-26.

Va l d i v i a C., L o r i g L., 2000. Slope stability at Escondida Mine. Slope stability in surface mining (edited by Hustrulid, McCarter & Van Zyl). Society for Mining, Metallurgy and Exploration Inc. Littleton, p. 153-162.

Westen Van. 2004. Geo-Information tools for landslide risk assessment: an overview of recent developments. Landsli-des: Evaluation and Stabilisation (edited by Lacerda, Erlich, Fontura & Sayao) Taylor & Francis Group, London. p. 39-56.

X i n g Z., 1988. Three-dimensional stability analysis of concave slopes in plan view. Journal of Geotechnical Engine-ering. Vol. 114, p. 658-671.

Z e t t l e r A.H., P o i s e l R., R o t h W., P r e h A., 1999: Slope stability analysis based on the shear reduction technique in 3D. FLAC and numerical modeling in geomechanics (Detournay & Hart). A.A. Balkema. Rotterdam. p. 11-16.

Received: 05 January 2007