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The relationship between the slope angle and the landslide size derived from limit equilibrium simulations
GEOMOR-05088; No of Pages 4 Geomorphology xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
Geomorphology journal homepage: www.elsevier.com/locate/geomorph
The relationship between the slope angle and the landslide size derived from limit equilibrium simulations Xiao-Li Chen a,⁎, Chun-Guo Liu b, Zu-Feng Chang c, Qing Zhou a a b c
Key Lab of Active Tectonics and Volcano, Institute of Geology, China Earthquake Administration, Beijing 100029, China China Earthquake Networks Center, Beijing 100045, China Earthquake Administration of Yunnan Province, Kunming 650041, China
a r t i c l e
i n f o
Article history: Received 5 December 2014 Received in revised form 28 January 2015 Accepted 29 January 2015 Available online xxxx Keywords: Slope stability General limit equilibrium (GLE) Factor of safety (Fs) Landslide volume
a b s t r a c t Katz et al. (2014) carried out a study of controls on the size and geometry of landslides using two-dimensional discrete element numerical simulations. One of their conclusions is that in addition to the peak strength of the slope material, the initial slope angle is another factor that controls the amount of material available for landslides, thus the size, of the resultant landslide. It means that in steeper slopes, more material disintegrates for a given material strength, and consequently the produced landslide is larger. However, in our studies based on limit equilibrium simulations, the sliding mass volume decreases with the increasing slope angle for a given material strength, just contrary to the result of Katz et al. One possible explanation is that when the slope angle in our model increases, the geometry of the potential critical slip surface changes, leading to a decrease of the amount of material available for potential sliding that compensates the increasing gravity effect owing to the enlargement of the slope angle. It suggests that there exist different controls of the slope angle on the landslide size for given material strength. © 2015 Elsevier B.V. All rights reserved.
1. Introduction
2. Methods
Katz et al. (2014) carried out a study of controls on the size and geometry of landslides through two-dimensional discrete element numerical simulations. They hypothesized that the observed global characteristic landslide size is a result of the limited thickness of disintegrated and weathered material that exists on hillslopes. They also suggested that the primary controls on the landslide geometry could be proved to be the residual friction angle of the slope material and the original slope angle. Moreover, they observed that the initial slope angle is an additional factor that controls the amount of material available for landsliding, thus the size of the resultant landslide. In steeper slopes, more material disintegrates for a given material strength, and thus the resultant landslide size is larger. However, in our studies based on limit equilibrium simulations, the sliding mass volume decreases with the increasing slope angle, just contrary to the result of Katz et al. (2014). In this brief comment paper, we present our work on this subject and attempt to explain the difference between our results and that of Katz et al. (2014).
We performed two-dimensional slope stability calculations using the general limit equilibrium (GLE) method incorporated in the software SLOPE/W of GeoStudio for stability analysis of slopes. The initial code of this software was developed by D.G. Fredlund at the University of Saskatchewan and has been on the market since 1977 (Fredlund, 1974; Fredlund and Krahn, 1977; GEO-SLOPE International Ltd., 2010). The GLE formulation satisfies two equations of factor of safety (Fs) (Fs is the ratio between the total available shear strength along the slip surface and the summation of the gravitational driving forces) and allows for a range of interslice shear-normal force assumptions, showing its advantages in the limit equilibrium analysis (Fellenius, 1936; Janbu, 1954; Bishop, 1955; Morgenstern and Price, 1965; Fredlund, 1974; Fredlund and Krahn, 1977; GEO-SLOPE International Ltd., 2010). The GLE method can be applied to any kinematically admissible slip surface shape (GEO-SLOPE International Ltd., 2010), which makes it more flexible and practical. The SLOPE/W software provides some options to locate the position of the critical slip surface within the slope body. Finding the critical slip surface involves a trial procedure (Janbu, 1954;Bishop, 1955; Morgenstern and Price, 1965). This is repeated for many possible slip surfaces, and at the end, the trial slip surface with the lowest Fs is considered as the governing or critical slip surface. From previous work,
DOI of original article: http://dx.doi.org/10.1016/j.geomorph.2014.05.021. ⁎ Corresponding author. Tel.: +86 10 62009056. E-mail address: [email protected] (X.-L. Chen).
http://dx.doi.org/10.1016/j.geomorph.2015.01.036 0169-555X/© 2015 Elsevier B.V. All rights reserved.
Please cite this article as: Chen, X.-L., et al., The relationship between the slope angle and the landslide size derived from limit equilibrium simulations, Geomorphology (2015), http://dx.doi.org/10.1016/j.geomorph.2015.01.036
2
X.-L. Chen et al. / Geomorphology xxx (2015) xxx–xxx
the option Auto-Locate in this software can lead to a reasonable result (GEO-SLOPE International Ltd., 2010). So we used the Auto-Locate option to search for the critical slip surface instead of specifying a slip surface. This is more reasonable because the slip surfaces are usually undetected in many cases in reality. To study the relationship between the slope angle and the sliding mass volume, we constructed a two-dimensional homogeneous slope model (Fig. 1). In this model, the slope height (H), slope angle (α) and slope base length (L) are the parameters to determine the geometry of the slope. The height of the slope was set to be 80 m, which was a constant value. Meanwhile, the slope angle α was adjusted by changing the horizontal distance L (i.e., tan α = H/L). In the GLE factors of safety equations, the material strength parameters are important for the equilibrium, which are characterized by material cohesion (c) and internal friction angle (φ). In addition, material unit weight (γ) here is used to determine the sliding mass weight or gravitational force. The slope material parameters for the simulation are listed in Table 1, which are close to the Katz et al. (2014) model. During simulations, the slope height H was kept the same, while the right end of the slope base was moved from 210 to 60 m by an interval of 10 m for each step, which means that the length of the slope base as well as the slope angle were reduced step by step. With these conditions, slope stability calculations were performed by analyzing, for each slope base length, 2000 slip surfaces to determine the one with the minimum Fs. Thus the potential slip surface within the slope body, which was initially a curved shape, was determined, and the corresponding sliding mass volume for each step was calculated. 3. Results Totally, we obtained 17 simulation results. Limited by the space, here we only present five representative results for slope angles 23°, 33°, 45°, 53°, and 65° (Fig. 2). They indicate that the potential slip mass volume decreases with the increasing slope angle (Fig. 3). The potential sliding volume reaches the maximum value of 9097 m3 when the slope has the minimum angle of 23° (corresponding to the right end of the slope base which is at the horizontal axis of 210 m, Fig. 2A), whereas the potential sliding mass volume decreases to the minimum value of 474 m3 when the slope angle approaches 65° (the right end of the slope base is at the horizontal axis of around 58 m) (Fig. 2E). It seems that such a relationship is associated with the changes of geometry of the critical slip surface. For example, when the slope angle is 23°, the
Table 1 Slope material strength parameters for simulation.a c (MPa)
φ (°)
γ (kN m−3)
1.0
30
25.0
a
c (MPa): cohesion; φ (°): internal friction angle; γ (kN m−3): material unit weight.
critical slip surface is a big curved plane (Fig. 2A). As the slope angle increases, the curvature and the length of this slip surface become smaller, thus leading to a decrease in the thickness as well as the volume of potential sliding material. When the slope angle is around 65°, the slip surface turns into a straight plane with a very small length, correspondingly the least volume of potential sliding material (Fig. 2E). 4. Discussion The limit equilibrium method of slices in SLOPE/W is based purely on the principle of statics, which says nothing about strains and displacements. The lack of a stress–strain constitutive relationship to ensure displacement compatibility creates many of the difficulties to obtain a converged solution under certain conditions (GEO-SLOPE International Ltd., 2010). For example, a steep slip surface makes it difficult to obtain a converged solution, which accounts for why the greatest slope angle is less than 70° in this simulation. However, the limitations do not necessarily prevent using the method in practice; it has been widely applied in slope stability analysis, and the outcomes are versatile (Wilson and Keefer, 1985; Lam and Fredlund, 1993; Jibson et al., 2000; Miles and Keefer, 2001; Jibson and Michael, 2009; Frattini and Crosta, 2013; Chen et al., 2014). Comparing the simulation work of slope failure conducted by Katz et al. (2014) and this study, except for the different model sizes, we found that the obvious difference is the principles of simulation methods. Although it is difficult to explain in physical mechanism the reasons for such differences at present, we speculate that there may exist various controls of the slope angle on landslide size for given material strength. One possible explanation is that when the slope angle in our model increases, the amount of material for potential sliding also decreases, which compensates the increasing gravity effect due to the enlargement of the slope angle. The purpose of our simulation was to find the possible critical slip surface within the slope body. So the processes leading to slope failure and the position where the failure starts are not the concern. In Katz et al.'s (2014) work, they indicated that the slope failure initiated at
Fig. 1. Sketch of the model geometry. H: slope height. L: slope base length, which is the difference between the left end and the right end of the slope base. α: slope angle.
Please cite this article as: Chen, X.-L., et al., The relationship between the slope angle and the landslide size derived from limit equilibrium simulations, Geomorphology (2015), http://dx.doi.org/10.1016/j.geomorph.2015.01.036
X.-L. Chen et al. / Geomorphology xxx (2015) xxx–xxx
3
Fig. 2. Potential sliding surfaces and Fs values versus different slope angles as a result of the leftward motion of the right end of the slope base. Blue shaded regions represent potential slip mass. The horizontal axis is the same as that in Fig. 1.
the foot of the slope and propagated upward along the slope. However, in our study, as the slope base length becomes shorter (i.e., the slope becomes steeper), the geometry of the potential slip surface in the cross section changes from a circular shape into a planar shape; meanwhile the landslide is progressively restricted to the upper part of the slope body, resulting in a decrease of landslide size (Figs. 2 and 3). Our results are similar to those of Frattini and Crosta (2013), which indicate that small landslides need high slope gradient to be unstable. Indeed, circular and planar slip surfaces – which are automatically located by the software in our simulations – are common rupture types for landslides in nature as observed (Wilson and Keefer, 1983; Keefer, 1984; Wen et al., 2004; Kieffer et al., 2006; Qi et al., 2009, 2010, 2011; Huang et al., 2010; Chen et al., 2012; Xu et al., 2013). Rotational slides usually occur on curved rupture surfaces, while
translational slides occur on planar ruptures (Varnes, 1978; Cruden and Varnes, 1996; Tusnaki, 2002; Highland and Bobrowsky, 2008). What Katz et al. (2014) derived from their numerical simulations may be able to explain some phenomena, while our modeling can also account for some observations. In our simulations, these changes in slip surface geometry suggest that slope failure mechanics or types are probably subjected to the initial slope angle to some degree. 5. Summary This work focuses on the relationship between landslide size and slope angle based on limit equilibrium simulations for given material strength. Our results show that as the slope angle increases, the sliding mass volume or the potential slide size decreases, just contrary to the
Please cite this article as: Chen, X.-L., et al., The relationship between the slope angle and the landslide size derived from limit equilibrium simulations, Geomorphology (2015), http://dx.doi.org/10.1016/j.geomorph.2015.01.036
4
X.-L. Chen et al. / Geomorphology xxx (2015) xxx–xxx
10000
Sliding mass volume (m3)
9000 8000 7000 6000 5000 4000 3000 2000 1000 0 0
50
100
150
200
Distance (m) Fig. 3. Slip mass volumes versus the varying position of the right end of the slope base that represents the length of the slope base. The horizontal axis is the same as that in Fig. 1. As the right end of the slope base moves toward the left, the slope base length as well as the slope angle are reduced.
results of Katz et al. (2014) from numerical simulations. One possible explanation is that when the slope angle in our model increases, the geometry of the potential critical slip surface changes, leading to a decrease in the amount of material for potential sliding that compensates for the increasing gravity effect owing to the enlargement of the slope angle. This suggests that there exist various controls of the slope angle on the size of landslides for given material strength. Acknowledgments This research was supported by the National Key Basic Research Program of China (Grant No. 2013CB733205) and the Project of Special Fund for Earthquake Scientific Research (201408002). The authors would like to express their thanks to anonymous referees and Dr. Richard A. Marston for their helpful comments. References Bishop, A.W., 1955. The use of the slip circle in the stability analysis of slopes. Geotechnique 5, 7–17. Chen, X.L., Ran, H.L., Yang, W.T., 2012. Evaluation of factors controlling large earthquakeinduced landslides by the Wenchuan earthquake. Nat. Hazards Earth Syst. Sci. 12, 3645–3657. Chen, X.L., Liu, C.G., Yu, L., Lin, C.X., 2014. Critical acceleration as a criterion in seismic landslide susceptibility assessment. Geomorphology 217, 15–22. Cruden, D.M., Varnes, D.J., 1996. Landslide types and processes. In: Turner, A.K., Schuster, R.L. (Eds.), Landslides — Investigation and Mitigation. National Research Council, Transportation Research Board. National Academy Press, Washington, D.C., pp. 36–75. Fellenius, W., 1936. Calculation of the stability of earth dams. Proceedings of the Second Congress of Large Dams. 4, pp. 445–463 (Washington D.C.). Frattini, P., Crosta, G.B., 2013. The role of material properties and landscape morphology on landslide size distributions. Earth Planet. Sci. Lett. 361, 310–319. Fredlund, D.G., 1974. Slope Stability Analysis. User's Manual CD-4. Department of Civil Engineering, University of Saskatchewan, Saskatoon, Canada. Fredlund, D.G., Krahn, J., 1977. Comparison of slope stability methods of analysis. Can. Geotech. J. 14, 429–439. GEO-SLOPE International Ltd., 2010. Stability Modeling with SLOPE/W 2007 Version. Printed in Canada. Highland, L.M., Bobrowsky, P., 2008. The Landslide Handbook — A Guide to Understanding Landslides. 1325. Geological Survey Circular, Reston, Virginia, U.S. (129 pp.). Huang, R.Q., Pei, X.J., Zhang, W.F., Li, S.G., Li, B.L., 2010. Further examination on characteristics and formation mechanism of Daguangbao landslide. J. Eng. Geol. 17 (6), 725–736 (in Chinese, with English Abstr.).
Janbu, N., 1954. Applications of composite slip surfaces for stability analysis. Proceedings of the European Conference on the Stability of Earth Slopes, Stockholm. 3, pp. 39–43. Jibson, R.W., Michael, J.A., 2009. Maps showing seismic landslide hazards in Anchorage, Alaska. U.S. Geological Survey Scientific Investigations Map 3077, scale 1:25,000, 11-p. pamphlet. Jibson, R.W., Harp, E.L., Michael, J.A., 2000. A method for producing digital probabilistic seismic landslide. Eng. Geol. 58, 271–289. Katz, O., Morgan, J.K., Aharonov, E., Dugan, B., 2014. Controls on the size and geometry of landslides: insights from discrete element numerical simulations. Geomorphology 220, 104–113. Keefer, D.K., 1984. Landslides caused by earthquakes. Geol. Soc. Am. Bull. 95, 406–421. Kieffer, D.S., Jibson, R.W., Rathje, E.M., Keith, K., 2006. Landslides triggered by the 2004 Niigata Ken Chuetsu, Japan, earthquake. Earthquake Spectra 22, S47–S73. Lam, L., Fredlund, D.G., 1993. A general limit equilibrium model for three-dimensional slope stability analysis. Can. Geotech. J. 30, 905–919. Miles, S.B., Keefer, D.K., 2001. Seismic Landslide Hazard for the Cities of Oakland and Piedmont, California. http://pubs.usgs.gov/mf/2001/2379/oakpamph.pdf. Morgenstern, N.R., Price, V.E., 1965. The analysis of the stability of general slip surfaces. Geotechnique 15, 79–93. Qi, S.W., Xu, Q., Liu, C.L., Zhang, B., Liang, N., Tong, L.Q., 2009. Slope instabilities in the severest disaster areas of 5.12 Wenchuan earthquake. J. Eng. Geol. 17 (1), 39–49 (in Chinese, with English Abstr.). Qi, S.W., Xu, Q., Lan, H.X., Zhang, B., Liu, J.Y., 2010. Spatial distribution analysis of landslides triggered by 2008.5.12 Wenchuan earthquake, China. Eng. Geol. 116, 95–108. Qi, S.W., Xu, Q., Zhang, B., Zhou, Y.D., Lan, H.X., Li, L.H., 2011. Source characteristics of long runout rock avaklanches triggered by the 2008 Wenchuan earthquake, China. J. Asia Earth Sci. 40, 896–906. Tusnaki, R. (Ed.), 2002. Landslide in Japan. Published by Japan Landslide Society & National Conference of Landslide Control. Varnes, D.J., 1978. Slope movement types and processes. In: Schuster, R.L., Krizek, R.J. (Eds.), Landslides — Analysis and Control. National Research Council, Transportation Research Board, Washington, D.C., pp. 11–23. Wen, B.P., Wang, S.J., Wang, E.Z., Zhang, J.M., 2004. Characteristics of rapid giant landslides in China. Landslides 4, 247–261. Wilson, R.C., Keefer, D.K., 1983. Dynamic analysis of a slope failure from the 6 August 1979 Coyote Lake, California, earthquake. Bull. Seismol. Soc. Am. 73, 863–877. Wilson, R.C., Keefer, D.K., 1985. Predicting areal limits of earthquake-induced landsliding. In: Ziony, J.I. (Ed.), Evaluating Earthquake Hazards in Los Angeles Region — An EarthScience Perspective. U.S. Geological Survey, Professional Paper 1360, pp. 317–345. Xu, C., Xu, X.W., Zheng, W.J., Wei, Z.Y., Tan, X.B., Han, Z.J., Li, C.Y., Liang, M.J., Li, Z.Q., Wang, H., Wang, M.M., Ren, J.J., Zhang, S.M., He, Z.T., 2013. Landslides triggered by the April 20, 2013 Lushan, Sichuan Province Ms7.0 strong earthquake of China. Seismol. Geodyn. 35 (3), 641–660 (in Chinese, with English Abstr.).
Please cite this article as: Chen, X.-L., et al., The relationship between the slope angle and the landslide size derived from limit equilibrium simulations, Geomorphology (2015), http://dx.doi.org/10.1016/j.geomorph.2015.01.036
Contents lists available at ScienceDirect
Geomorphology journal homepage: www.elsevier.com/locate/geomorph
The relationship between the slope angle and the landslide size derived from limit equilibrium simulations Xiao-Li Chen a,⁎, Chun-Guo Liu b, Zu-Feng Chang c, Qing Zhou a a b c
Key Lab of Active Tectonics and Volcano, Institute of Geology, China Earthquake Administration, Beijing 100029, China China Earthquake Networks Center, Beijing 100045, China Earthquake Administration of Yunnan Province, Kunming 650041, China
a r t i c l e
i n f o
Article history: Received 5 December 2014 Received in revised form 28 January 2015 Accepted 29 January 2015 Available online xxxx Keywords: Slope stability General limit equilibrium (GLE) Factor of safety (Fs) Landslide volume
a b s t r a c t Katz et al. (2014) carried out a study of controls on the size and geometry of landslides using two-dimensional discrete element numerical simulations. One of their conclusions is that in addition to the peak strength of the slope material, the initial slope angle is another factor that controls the amount of material available for landslides, thus the size, of the resultant landslide. It means that in steeper slopes, more material disintegrates for a given material strength, and consequently the produced landslide is larger. However, in our studies based on limit equilibrium simulations, the sliding mass volume decreases with the increasing slope angle for a given material strength, just contrary to the result of Katz et al. One possible explanation is that when the slope angle in our model increases, the geometry of the potential critical slip surface changes, leading to a decrease of the amount of material available for potential sliding that compensates the increasing gravity effect owing to the enlargement of the slope angle. It suggests that there exist different controls of the slope angle on the landslide size for given material strength. © 2015 Elsevier B.V. All rights reserved.
1. Introduction
2. Methods
Katz et al. (2014) carried out a study of controls on the size and geometry of landslides through two-dimensional discrete element numerical simulations. They hypothesized that the observed global characteristic landslide size is a result of the limited thickness of disintegrated and weathered material that exists on hillslopes. They also suggested that the primary controls on the landslide geometry could be proved to be the residual friction angle of the slope material and the original slope angle. Moreover, they observed that the initial slope angle is an additional factor that controls the amount of material available for landsliding, thus the size of the resultant landslide. In steeper slopes, more material disintegrates for a given material strength, and thus the resultant landslide size is larger. However, in our studies based on limit equilibrium simulations, the sliding mass volume decreases with the increasing slope angle, just contrary to the result of Katz et al. (2014). In this brief comment paper, we present our work on this subject and attempt to explain the difference between our results and that of Katz et al. (2014).
We performed two-dimensional slope stability calculations using the general limit equilibrium (GLE) method incorporated in the software SLOPE/W of GeoStudio for stability analysis of slopes. The initial code of this software was developed by D.G. Fredlund at the University of Saskatchewan and has been on the market since 1977 (Fredlund, 1974; Fredlund and Krahn, 1977; GEO-SLOPE International Ltd., 2010). The GLE formulation satisfies two equations of factor of safety (Fs) (Fs is the ratio between the total available shear strength along the slip surface and the summation of the gravitational driving forces) and allows for a range of interslice shear-normal force assumptions, showing its advantages in the limit equilibrium analysis (Fellenius, 1936; Janbu, 1954; Bishop, 1955; Morgenstern and Price, 1965; Fredlund, 1974; Fredlund and Krahn, 1977; GEO-SLOPE International Ltd., 2010). The GLE method can be applied to any kinematically admissible slip surface shape (GEO-SLOPE International Ltd., 2010), which makes it more flexible and practical. The SLOPE/W software provides some options to locate the position of the critical slip surface within the slope body. Finding the critical slip surface involves a trial procedure (Janbu, 1954;Bishop, 1955; Morgenstern and Price, 1965). This is repeated for many possible slip surfaces, and at the end, the trial slip surface with the lowest Fs is considered as the governing or critical slip surface. From previous work,
DOI of original article: http://dx.doi.org/10.1016/j.geomorph.2014.05.021. ⁎ Corresponding author. Tel.: +86 10 62009056. E-mail address: [email protected] (X.-L. Chen).
http://dx.doi.org/10.1016/j.geomorph.2015.01.036 0169-555X/© 2015 Elsevier B.V. All rights reserved.
Please cite this article as: Chen, X.-L., et al., The relationship between the slope angle and the landslide size derived from limit equilibrium simulations, Geomorphology (2015), http://dx.doi.org/10.1016/j.geomorph.2015.01.036
2
X.-L. Chen et al. / Geomorphology xxx (2015) xxx–xxx
the option Auto-Locate in this software can lead to a reasonable result (GEO-SLOPE International Ltd., 2010). So we used the Auto-Locate option to search for the critical slip surface instead of specifying a slip surface. This is more reasonable because the slip surfaces are usually undetected in many cases in reality. To study the relationship between the slope angle and the sliding mass volume, we constructed a two-dimensional homogeneous slope model (Fig. 1). In this model, the slope height (H), slope angle (α) and slope base length (L) are the parameters to determine the geometry of the slope. The height of the slope was set to be 80 m, which was a constant value. Meanwhile, the slope angle α was adjusted by changing the horizontal distance L (i.e., tan α = H/L). In the GLE factors of safety equations, the material strength parameters are important for the equilibrium, which are characterized by material cohesion (c) and internal friction angle (φ). In addition, material unit weight (γ) here is used to determine the sliding mass weight or gravitational force. The slope material parameters for the simulation are listed in Table 1, which are close to the Katz et al. (2014) model. During simulations, the slope height H was kept the same, while the right end of the slope base was moved from 210 to 60 m by an interval of 10 m for each step, which means that the length of the slope base as well as the slope angle were reduced step by step. With these conditions, slope stability calculations were performed by analyzing, for each slope base length, 2000 slip surfaces to determine the one with the minimum Fs. Thus the potential slip surface within the slope body, which was initially a curved shape, was determined, and the corresponding sliding mass volume for each step was calculated. 3. Results Totally, we obtained 17 simulation results. Limited by the space, here we only present five representative results for slope angles 23°, 33°, 45°, 53°, and 65° (Fig. 2). They indicate that the potential slip mass volume decreases with the increasing slope angle (Fig. 3). The potential sliding volume reaches the maximum value of 9097 m3 when the slope has the minimum angle of 23° (corresponding to the right end of the slope base which is at the horizontal axis of 210 m, Fig. 2A), whereas the potential sliding mass volume decreases to the minimum value of 474 m3 when the slope angle approaches 65° (the right end of the slope base is at the horizontal axis of around 58 m) (Fig. 2E). It seems that such a relationship is associated with the changes of geometry of the critical slip surface. For example, when the slope angle is 23°, the
Table 1 Slope material strength parameters for simulation.a c (MPa)
φ (°)
γ (kN m−3)
1.0
30
25.0
a
c (MPa): cohesion; φ (°): internal friction angle; γ (kN m−3): material unit weight.
critical slip surface is a big curved plane (Fig. 2A). As the slope angle increases, the curvature and the length of this slip surface become smaller, thus leading to a decrease in the thickness as well as the volume of potential sliding material. When the slope angle is around 65°, the slip surface turns into a straight plane with a very small length, correspondingly the least volume of potential sliding material (Fig. 2E). 4. Discussion The limit equilibrium method of slices in SLOPE/W is based purely on the principle of statics, which says nothing about strains and displacements. The lack of a stress–strain constitutive relationship to ensure displacement compatibility creates many of the difficulties to obtain a converged solution under certain conditions (GEO-SLOPE International Ltd., 2010). For example, a steep slip surface makes it difficult to obtain a converged solution, which accounts for why the greatest slope angle is less than 70° in this simulation. However, the limitations do not necessarily prevent using the method in practice; it has been widely applied in slope stability analysis, and the outcomes are versatile (Wilson and Keefer, 1985; Lam and Fredlund, 1993; Jibson et al., 2000; Miles and Keefer, 2001; Jibson and Michael, 2009; Frattini and Crosta, 2013; Chen et al., 2014). Comparing the simulation work of slope failure conducted by Katz et al. (2014) and this study, except for the different model sizes, we found that the obvious difference is the principles of simulation methods. Although it is difficult to explain in physical mechanism the reasons for such differences at present, we speculate that there may exist various controls of the slope angle on landslide size for given material strength. One possible explanation is that when the slope angle in our model increases, the amount of material for potential sliding also decreases, which compensates the increasing gravity effect due to the enlargement of the slope angle. The purpose of our simulation was to find the possible critical slip surface within the slope body. So the processes leading to slope failure and the position where the failure starts are not the concern. In Katz et al.'s (2014) work, they indicated that the slope failure initiated at
Fig. 1. Sketch of the model geometry. H: slope height. L: slope base length, which is the difference between the left end and the right end of the slope base. α: slope angle.
Please cite this article as: Chen, X.-L., et al., The relationship between the slope angle and the landslide size derived from limit equilibrium simulations, Geomorphology (2015), http://dx.doi.org/10.1016/j.geomorph.2015.01.036
X.-L. Chen et al. / Geomorphology xxx (2015) xxx–xxx
3
Fig. 2. Potential sliding surfaces and Fs values versus different slope angles as a result of the leftward motion of the right end of the slope base. Blue shaded regions represent potential slip mass. The horizontal axis is the same as that in Fig. 1.
the foot of the slope and propagated upward along the slope. However, in our study, as the slope base length becomes shorter (i.e., the slope becomes steeper), the geometry of the potential slip surface in the cross section changes from a circular shape into a planar shape; meanwhile the landslide is progressively restricted to the upper part of the slope body, resulting in a decrease of landslide size (Figs. 2 and 3). Our results are similar to those of Frattini and Crosta (2013), which indicate that small landslides need high slope gradient to be unstable. Indeed, circular and planar slip surfaces – which are automatically located by the software in our simulations – are common rupture types for landslides in nature as observed (Wilson and Keefer, 1983; Keefer, 1984; Wen et al., 2004; Kieffer et al., 2006; Qi et al., 2009, 2010, 2011; Huang et al., 2010; Chen et al., 2012; Xu et al., 2013). Rotational slides usually occur on curved rupture surfaces, while
translational slides occur on planar ruptures (Varnes, 1978; Cruden and Varnes, 1996; Tusnaki, 2002; Highland and Bobrowsky, 2008). What Katz et al. (2014) derived from their numerical simulations may be able to explain some phenomena, while our modeling can also account for some observations. In our simulations, these changes in slip surface geometry suggest that slope failure mechanics or types are probably subjected to the initial slope angle to some degree. 5. Summary This work focuses on the relationship between landslide size and slope angle based on limit equilibrium simulations for given material strength. Our results show that as the slope angle increases, the sliding mass volume or the potential slide size decreases, just contrary to the
Please cite this article as: Chen, X.-L., et al., The relationship between the slope angle and the landslide size derived from limit equilibrium simulations, Geomorphology (2015), http://dx.doi.org/10.1016/j.geomorph.2015.01.036
4
X.-L. Chen et al. / Geomorphology xxx (2015) xxx–xxx
10000
Sliding mass volume (m3)
9000 8000 7000 6000 5000 4000 3000 2000 1000 0 0
50
100
150
200
Distance (m) Fig. 3. Slip mass volumes versus the varying position of the right end of the slope base that represents the length of the slope base. The horizontal axis is the same as that in Fig. 1. As the right end of the slope base moves toward the left, the slope base length as well as the slope angle are reduced.
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Please cite this article as: Chen, X.-L., et al., The relationship between the slope angle and the landslide size derived from limit equilibrium simulations, Geomorphology (2015), http://dx.doi.org/10.1016/j.geomorph.2015.01.036