Bulletin of Engineering Geology and the Environment https://doi.org/10.1007/s10064-018-1386-3

ORIGINAL PAPER

Stability analysis of the Zhangmu multi-layer landslide using the vector sum method in Tibet, China

Mingwei Guo1 & Sujin Liu1,2 & Shunde Yin3 & Shuilin Wang1

Received: 28 February 2018 /Accepted: 10 September 2018 # Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract The Zhangmu landslide is located on the China–Nepal border in Tibet, China, which has recently become a serious threat to the lives and properties of local people. In order to efficiently quantify the stability of the Zhangmu landslide, a new method named the vector sum method (VSM) is proposed. Differing from conventional slope-stability analysis methods, the VSM considers both the magnitude, the direction of force and the strength-reserving definition of the safety factor based on the actual stress field of slope achieved from finite element analysis. Moreover, the global sliding direction of potential landslides was theoretically deduced by the principle of minimum potential energy, while the safety factor can be directly computed by not only the force limit equilibrium of the whole sliding body in the global sliding direction but also the moment limit equilibrium at the moment center. Finally, stability analysis of the Zhangmu landslide was performed by the proposed method, and verified against the rigorous Morgenstern–Price method.

Keywords Vector sum method . Slope stability . Limit equilibrium method . Strength reduction method . Zhangmu landslide

Introduction most residential buildings and public facilities at Zhangmu Port have been built on an ancient rock slide, which is Zhangmu is the only overland and international trading port surrounded by the Boqu River, the Bangcundong Valley, the on the China–Nepal border in southern Tibet, China, located Qiangma Valley and the Zhangmu Valley. Geological survey about 80 km from Kathmandu, the capital of the Kingdom of (Jia et al. 2006;Mao2008;Huetal.2015;Maetal.2017)has Nepal, and 750 km from Lhasa, the capital of Tibet, China revealed that the Zhangmu landslide can be divided into two (Fig. 1). large old debris slides, i.e., the Old Fuliyuan debris slide and According to the latest statistics from the general adminis- the Old Bangcundong debris slide. Furthermore, it has been tration of customs of China in 2017, the total trade volume pointed out that the Zhangmu landslide is a multi-layer land- was about 0.88 billion USD between China and Nepal in slide from the surface to the bedrock comprised of modern 2016, and about 80% of the total trade passed through debris, old debris and rock debris. Therefore, the stability of Zhangmu port, which demonstrates the significant status of this multi-layer landslide has become the key issue in the trade the trade and economic development center of Zhangmu port. and economic development of the region, including the safety However, due to the constraints of its geographic location, of local people’s lives and properties. Since 1993, local landslides have occurred frequently each year. Therefore, Zhangmu port has been investigated * Mingwei Guo many times (Yi and Tang 1996; Xie et al. 2003;Jiaetal. [email protected] 2006;Zhuetal.2010), and preventitive measures have been taken after each geological investigation. However, these 1 State Key Laboratory of Geomechanics and Geotechnical measures have not efficiently prevented local landslides in Engineering, Institute of Rock and Soil Mechanics, Chinese Zhangmu port from occurring repeatedly at different depths Academy of Sciences, Wuhan 430071, Hubei, China or different locations. In 2013, in order to completely solve 2 University of Chinese Academy of Sciences, Beijing 100049, China the stability problem of Zhangmu port, the Regional 3 Department of Civil and Environmental Engineering, University of Collaboration and Innovation Project between the Chinese Waterloo, Waterloo, ON N2L 3G1, Canada Academy of Sciences and Tibet was set up, entitled M. Guo et al.

Fig. 1 Location of Zhangmu town (Ma et al. 2017)

BGeologic exploration, risk assessmentand comprehensive In this study, in order to effectively assess the stability of a prevention and control of Zhangmu Landslide^ (Hu et al. multi-layer landslide, a new approach considering the vector 2015;Maetal.2017). Based on comprehensive investiga- characteristics of force is proposed based on the actual stress tion and analysis of the Zhangmu landslide, the quantitative state of the landslide achieved from the finite element method. stability analysis of the Zhangmu landslide has been studied With this new approach, the multi-layer landslide can be di- and is summarized in this paper. rectly assessed for potential sliding bodies with different Stability analysis of the Zhangmu multi-layer landslide using the vector sum method in Tibet, China depths. The proposed method is demonstrated in the slope Conventional methods for slope stability stability analysis of the Zhangmu multi-layer landslide. analysis

The slope-stability analysis problem involves determining the The Zhangmu multi-layer landslide safety factor and corresponding critical slip surface, and, once the safety factor is defined, the critical slip surface can be Figure 2 shows the full view of Zhangmu port, which covers searched among admissible slip surfaces within the slope. To an area of 1.23 × 106 m2. From the figure, it can be seen that date, many methods have been developed to calculate the most residents live on this large rock slide. According to in- slope stability, such as the conventional Limit Equilibrium tensive geological survey and analysis, the geological struc- Method (LEM) (Bishop 1955; Fredlund and Krahn 1977; ture and landslide boundaries are shown in Fig. 3. Because of Duncan 1996), Strength Reduction Method (SRM) similar formation processes and geological structures for both (Zienkiewicz et al. 1975;GriffithsandLane1999), and the Old Fuliyuan debris slide and the Old Bangcundong debris Limit Analysis Method (LAM) (Chen et al. 2003; Viratjandr slide, unless specified, in the following for the purpose of and Michalowski 2006;Sloan2013; Lim et al. 2017). Among simplification, the Zhangmu multi-layer landslide in this pa- these methods, the definitions of safety factor can be mainly per refers to the Bangcundong multi-layer landslide. divided into two types, i.e. the strength-reserving definition Profile 3 (Fig. 4) is a classical profile of the Bangcundong and the overloading definition (Zheng et al. 2006). landslide, which shows the spatial distribution of the multi- For the strength-reserving definition, the safety factor is layer landslide, for which , from the ground surface to the defined as the number by which the shear strength parameters bedrock, there exists the modern sliding body, the old debris must be factored to bring the slope into the state of limit sliding body, an ancient rock debris deposit and the bedrock. equilibrium or failure. The popular LEM and SRM, which The modern landslide is the local failure of the Old are mainly used to quantify the slope stability, are all based Bangcundong debris slide, which is composed of schist, on this definition. gneiss clasts, sand, and silt. In addition, shear zones composed Considering the sliding body as a rigid body, LEM divides of silt soil with medium gravel have been found at different the sliding body into many slices, but it does not utilize the depths, a deep one can be considered as the shear zone of the stress versus strain characteristics of a slope. In order to obtain Old Bangcundong debris slide, while a shallow one is that of the safety factor and to determine a stability problem, many modern Bangcundong slide. The ancient rock slide is com- assumptions have to be made to establish the force or moment posed of cracked rock with original schist or gneiss between equations based on the limit state of these slices, such as the the bedrock and the debris deposits. Furthermore, the rock magnitude, the direction, and the position of forces acting on slide shear zone has been detected and is a thin layer of gravel the interface of the slices (Bishop 1955;FredlundandKrahn soil. The bedrock is mainly composed of mica schist, quartz 1977;Duncan1996; Zhu et al. 2003; Cheng et al. 2007; Zhou schist, and biotite plagioclase gneiss. and Cheng 2014; Chakraborty and Goswami 2016; Luo et al. For the physical and mechanical properties of the geolog- 2017). Compared with 2D methods, however, 3D limit equi- ical materials, many tests and back analyses were performed librium methods are far from maturity in both theory and to predict the mechanical properties of the geological mate- practice (Lam and Fredlund 1993; Huang and Tsai 2000; rials, and the recommended parameters are given in Table 1. Chen et al. 2006; Cheng and Yip 2007; Zheng and Tham 2009; Zhou and Chen 2013; Lu and Zhu 2016; Basudhar and Lakshminarayana 2017). Despite LEM is still remaining an effective tool in practical problems due to its simplicity and the existing experience in practical engineering, it has some inherent limitations, i.e., indeterminate static conditions, unre- alistic stress distributions, unsatisfied displacement compati- bility or problematic definition of soil–structure interaction. As an alternative approach to LEM, the Finite Element Strength Reduction Method (FE-SRM) was proposed as early as 1975 by Zienkiewicz and has been greatly developed all over the world (Zheng et al. 2005; Cheng et al. 2007; Hamdhan and Schweiger 2013; Isakov and Moryachkov 2014; Shen and Karakus 2014; Tu et al. 2016; Kelesoglu 2016;Tangetal.2017). The FE-SRM possesses some advan- tages over LEM, as it not only automatically locates the Fig. 2 Zhangmu town critical failure surface but also simulates the stress–strain M. Guo et al.

Fig. 3 Geological map of the study area showing landslide boundaries, lithology, and profile lines Stability analysis of the Zhangmu multi-layer landslide using the vector sum method in Tibet, China

Fig. 4 Profile(3–3′) of the Old Bangcundong landslide

behavior and readily monitors the progressive shear failure of Vector sum method the slope under complex geometries and loading conditions. Nevertheless, the judgment on the critical state of the slope, as Considering both the magnitude and the direction of force, the well as the high computational cost, has become the crucial vector sum method (VSM) was first put forward by Ge issues of this method, which affect both the numerical accu- (2008), in which the safety factor is defined as the ratio of racy and the computational efficiency. the total resisting force to the total driving force in the global Besides the strength-reserving definition, for the sliding direction, and, recently, this method has been devel- overloading definition, the safety factor, defined on the oped based on the actual stress of the slope achieved from basis of the strength along the slip surface, is also popu- numerical analysis with some complex conditions (Liu et al. larly recognized (Zou et al. 1995; Kim and Lee 1997; 2017;Fuetal.2017; Zou et al. 2017). However, in previous Pham and Fredlund 2003; Stianson et al. 2011). For a studies of this method, only the force equation of the sliding critical slip surface shaped by a circle or straight line, it body in the sliding direction was considered and the global has a clear physical meaning; however, for a noncircular sliding direction was still determined by an assumption based slip surface, the clarity of this definition was doubted on the sliding failure mechanism (Ge 2010; Guo et al. 2013; because the definition of the safety factor is neither the Wu 2013). vector sums of force nor projections of the algebraic sums In order for the concise stress on the potential slip surface to of force in a particular direction (Ge 2010;Tangetal. be obtained, the complete stress state from the finite element 2016). analysis can be imported into a limit equilibrium analysis,

Table 1 Physical and mechanical properties of geological materials

Position Natural Saturated

Unit Elastic Poisson’s Cohesion Angle of Dilation Cohesion Angle of Dilation weight modulus ratio (kPa) friction angle (°) (kPa) friction angle (°) (kN/m3) (Mpa) (°) (°)

Modern Bangcundong landslide Main body 27.0 20 0.30 27 30 0 21 24 – Shear zone 22.1 15 0.35 20 24 0 18 20 – Old Bangcundong landslide Main body 27.0 30 0.3 29.5 33.5 0 25.0 30.0 0 Shear zone 25.7 15 0.35 26.53 31.83 0 23.0 29.0 0 Ancient rock slide Main body 26.0 300 0.28 3930 31.29 0 3310 29.0 0 Shear zone 22.1 30 0.33 31.12 33.42 0 25.0 28.0 0 Bedrock 26.0 4000 0.25 16,610 45.79 0 6450 30.59 0 M. Guo et al. where the normal and shear stresses are computed with respect to any selected slip surface. Here, the strength-reserving def- inition of the safety factor and the vector characteristics of force are considered simultaneously, and the safety factor can be defined by the force and moment equilibrium based on the strength-reserving definition. Due to the direction of force, all resisting forces along the potential slip surface can be calculated to be a resisting force vector if the vector compo- sition law is used. Consequently, the total driving force vector can be obtained, and the force equilibrium equation can be established in the global sliding direction, when the soil shear Fig. 5 Sketch of simple slope strength is divided by the safety factor to bring the slope to the limit equilibrium state. Similarly, the moment equilibrium equation can also be established at the moment center. 2 3 σ τ τ Therefore, the global safety factor of the slope should be the x xy xz σ ¼ 4 τ σ τ 5 ð Þ smaller of the two obtained from either the force or moment xy y yz 4 τ τ σ equilibrium equations. In the following, the determination of xz yz z the global sliding direction and the establishment of the force and moment equilibrium equations will be described in detail, It can be seen that Eq.1 expresses the static equilibrium based on the strength-reserving definition. state of the potential sliding body. When the slope is about to slide, it can only slide along the Global sliding direction potential slip surface because of the restriction of the bedrock. Assuming the global unit displacement d of the slope occurs Here, the global sliding direction can be theoretically deter- along the potential slip surface, that is, the direction of the mined and given by the principle of minimum potential ener- global displacement d indicates the global sliding direction, gy. The details are as follows. then the local displacement at any location on the potential slip When the static balance of an object is broken, because of surface can be obtained based on the global sliding displace- external loads or a change of material strength, it will try to ment. So, the displacement at the point M on the potential slip reach a state in which the potential energy will be at a relative surface in Fig. 5 canbeconsideredas: minimum. Therefore, when the static state of a slope is bro- ken, the displacements generated in the slope tend to make the ds ¼ d−dn ¼ d−dnn ð5Þ potential energy of the sliding body reach a relative minimum. Thus, the global sliding direction of a slope can be completely dn ¼ d⋅n ð6Þ determined by the principle of minimum potential energy. where dn is the vector form of normal displacement along the normal direction in Eq.6;actually,dn = dn= dnn,inwhichdn The vector expression is the scalar displacement along the normal direction on the point M of the slip surface. Therefore, when the displacement The moment the potential energy of a slope changes into a occurring at point M is ds, the change of potential energy can relative minimum, it will have a stationary value, which offers be given by: the mathematical method to obtain the global sliding direction of the slope. Figure 5 shows a simple slope. The unit weight of Π ¼ Π0 þ ∫SσM ⋅dsdS ð7Þ the slope is γ, the total stress at any point M on the slip surface σ n is M,and is the unit normal direction pointing to the bed- where Π0 is the initial potential energy of the sliding body. rock at the point M. Therefore, the force balance equation for Then, the first-order variation can be written as: the sliding body can be given by: ∂Π ∫V bdV ¼ ∫AσM dA ð1Þ ¼ 0 ð8Þ 2 3 ∂d 0 Equaion 7 canalsobeconsideredas: b ¼ 4 0 5 ð2Þ γ ∂ σ ¼ σ•n ð Þ ∫SσM ⋅dsdS ¼ 0 ð9Þ M 3 ∂θi Stability analysis of the Zhangmu multi-layer landslide using the vector sum method in Tibet, China

where d = d(θi), θi is the independent variation of sliding also be obtained. Therefore, under the critical state of a slope, displacement, and for a 2D problem, i is equal to one, but the total resisting and driving forces should be equal in the for a 3D problem, it is equal to two. global sliding direction, and then the force equilibrium equa- tion can be established in the global sliding direction, as shown in Eq. 15. Scalar expression for 2-D problem ƒ! ƒ!   0 0 ! ƒ! ƒ! ! From Eq. (9), the scalar expression can be deduced as follows ∫l στc þ σnc • − d dl ¼ ∫l στc þ σnc • d dl; ð15Þ for a 2D problem. The global sliding displacement can be simplified as:  ƒ! ƒ! cosθ where σnc and στc are the driving normal and shear stresses, d ¼ ð10Þ sinθ respectively, under critical state at any point M (in black), and, ƒ! ƒ! 0 0 correspondingly, σnc and στc (red color) are the resisting where θ is the sliding angle defined anticlockwise from the X normal and shear stresses, respectively, at point M. axis to the global sliding displacement. Thus, there is only one The driving and resisting forces are further explained in variable for a 2D problem. Then, in Eq. (5), dscan be detail as follows. expressed by:  Âà cosθ cosθ nx Resisting force Based on the strength-reserving definition, as- ds ¼ d−dnn ¼ − nx ny ð11Þ sinθ sinθ ny suming the safety factor by force equilibrium is Ff, if the shear strength of the slope material complies with the Mohr– That is, Coulomb yield criterion, the resisting shear stress at any point  M can be expressed as Eq. 16. 2 1−nx −nxny cosθ ds ¼ d−dnn ¼ 2 ð12Þ −nxny 1−ny sinθ ƒ! ƒ! τ c þ ‖σ 0‖ φ Equation 12 can be substituted into Eq. 8, which can then ‖σ 0‖ ¼ max ¼ nc tan ð Þ τc F F 16 be expressed as: f f  2 where c and φ are the cohesion and the friction angle, respec- nx −1 nxny sinθ ∫l½ŠσMx σMy 2 dl ¼ 0 ð13Þ nxny ny −1 −cosθ tively, of the material. The direction of a resisting shear stress at the critical state is always along the tangential direction where σMxand σMy are the components of σM on the X and Y upward on a slip surface. axes, respectively. Because the decrease of shear strength brings the slope to For the sliding body, under the conditions that the stress sliding failure, the assumption can be made that the normal state of the sliding body and the location of the slip surface are stress stays constant during the evolution process from the known, the only variable is the angle θ, so Eq. 12 can be normal state to the critical state of the slope. Then, the normal further simplified by: stress at the critical state can be considered as:  2 ∫lnxnyσMx þ ny −1 σMydl ƒ! ƒ! σnc ¼ σn ð17Þ tanθ ¼ ÀÁ ð14Þ 2 ∫lnxnyσMy þ nx −1 σMxdl Therefore, all resisting forces along a slip surface in the From Eq. 14, it can be concluded that the global sliding global sliding direction at the critical state can be expressed as: direction can be theoretically determined by the stress state of ƒ! ƒ!  0 0 ! the slope. ∫l στc þ σnc • − d dl ! !  The force equilibrium equation τ max a ƒ! ! ¼ ∫l − σnc • − d dl ð18Þ F f Figure 6 shows the evolution process of a potential sliding ! body from the normal state to the critical state because of the where a stands for the unit direction of resisting shear stress decrease in shear strength of the material. In the VSM, with at point M along a slip surface. the direction of force considered, all resisting forces along a potential slip surface can be determined to be a resisting force Driving force According to the static balance state of a poten- vector, and, correspondingly, the total driving force vector can tial sliding body, Eq. 19 can be obtained based on the M. Guo et al. macroscopic forces and microstresses along the slip surface at Further simplification can be made using Eq. 23, the normal state (see Fig. 6). ! !   τmax a ! ƒ! !  ∫AB • − d dl ¼ ∫ στ • d dl ð23Þ ƒ! ƒ! ƒ! ƒ! F f AB ∫ στ þ σn dl ¼ PH þ PV ð19Þ AB As an invariable along the slip surface, the safety factor can P P where H and V are the total horizontal and vertical forces, be finally expressed as: respectively, acting on the potential sliding body, which be- ! ƒσ! ƒσ! ! long to macroscopic forces. n and τ are, respectively, the ∫AB τ max a • − d dl F ¼  ð Þ normal and shear stresses acting on the bedrock at any point M f ƒ! ! 24 ∫ στ • d dl on the potential slip surface, which belong to microforces of AB the sliding body. Similarly, the macroscopic forces acting on the sliding From Eq. 24, it can be seen that only the shear stress has body can be considered as invariable during the evolution effects on the safety factor when the force equilibrium of a process from the normal state to the critical state of the slope. slope is considered. It is not difficult to obtain the shear stress i.e. the total driving force stays constant during the evolution along the slip surface in Eq. 24 by the finite element method. It process of the slope, and thus the driving force at the critical should be noted that the safety factor here can be directly state can be expressed as: obtained by integrals along the slip surface, while LEM and   SRM require iterative calculations in order to obtain the safety ƒ! ƒ! ! ƒ! ƒ! ! factor. ∫ στ þ σn • d dl ¼ ∫AB στc þ σnc • d dl ð20Þ AB The moment equilibrium equation

Figure 7 shows a simple slope for moment equilibrium, in Safety factorFf With Eqs. 15, 17, 18 and 20, the force equilib- which M is any point on a slip surface, σM (in black) is the rium equation can be simplified by: ƒ! ƒ! 0 0 driving plane stress and σnc , στc (red color) are the resisting ! τ !a ! normal and shear stresses, respectively, at point M. Similarly, ∫ max −ƒσ! • − d d σ ′ ƒσ! ƒσ! AB F n l S (in red) is the resisting plane stress, and nc , τc (in black) f are the driving normal and shear stresses, respectively, under  ƒ! ƒ! ! critical state at point M. ¼ ∫ στ þ σn • d dl ð21Þ AB Here, point O is assumed to be the moment center and the coordinate of the center is shown in pink in Fig. 7a, b. According to the definition of moment, the magnitude of i.e., the moment of the acting force at a certain point is directly ! proportional to the distance from the point to the force. It is !   r τmax a ! ƒ! ! defined as the cross-product of the distance vector and the ∫AB • − d dl þ ∫AB σn • d dl F f acting force vector F, and based on the right-hand rule, and the   moment can be calculated by: ƒ! ! ƒ! ! ¼ ∫ στ • d dl þ ∫ σn • d dl ð22Þ AB AB M ¼ r  F ð25Þ

Fig. 6 The stress state from the normal to the critical state Stability analysis of the Zhangmu multi-layer landslide using the vector sum method in Tibet, China

where the distance vector r is the vector from the moment determine the moment center in section BThe stability analysis point to the acting point of the force. For the simple slope in of a multi-layer Zhangmu landslide^. Thus, for the VSM, the Fig. 7, the total resisting moment along a slip surface can be global sliding direction and moment center should be appro- calculated as: priately determined.   0 0 0 M Mr ¼ ∫l r  σM dl ¼ ∫lr  σnc þ στc dl ð26Þ The stability analysis of the multi-layer Similarly, the total driving moment can also be obtained Zhangmu landslide by: In order to verify the efficiency and accuracy of the VSM for a M Md ¼ ∫lðÞr  σM dl ¼ ∫lr  ðÞσnc þ στc dl ð27Þ multi-layer landslide, and show the calculating process of the For the sliding body, at the critical state of slope caused by proposed method, the multi-layer Bangcundong landslide was shear strength reduction, the magnitude of the total driving chosen to be assessed in this section. moment should be equal to the magnitude of the total resisting In this paper, the finite element method is used to obtain the moment along a slip surface at the moment center, and thus, stress field of slopes composed of elasto-plastic materials. The considering the vector characteristics of force, the moment ideal elasto-plastic constitutive model, the Mohr–Coulomb equilibrium equation can be established as Eq. 28. yield criterion, and the non-associated flow rule are used in the elasto-plastic finite element analysis. Gaussian integration M ¼ −M ð Þ Mr Md 28 points are used to integrate the stiffness matrix in the finite element analysis, and the discontinuous stresses with low ac- Assuming the safety factor of a slope is F by moment m curacy at the boundary of the elements are computed by the equilibrium, and according to Eqs. 17, 18 and 20, Eq. 28 direct calculation of stress integrals. This study adopts the can be simplified as: ! global stress smoothing technique to overcome the low accu- ! racy deficiency. The stress at any position within an element τ max a ∫lr  − dl ¼ ∫lðÞr  στ dl ð29Þ canbecalculatedby: Fm n σ ¼ ∑ N iσi ð31Þ The safety factor Fm can be finally expressed as: i¼1  ! n N ∫lr  −τ max a dl where is the number of nodes of the element, i is the shape Fm ¼ ð30Þ function about a nodal point i,andσi is the corresponding ∫lðÞr  στ dl nodal stress. Based on the mechanical analysis for a potential sliding body in section BThe moment equilibrium equation^, the safe- Calculating model and the conditions ty factor by moment equilibrium can be easily calculated if the moment center is already determined. Based on detailed boundary of the landslide (Fig. 3) and its Similar to the global sliding direction in section BGlobal geological structure profile (3–3′)(Fig.4), the calculating sliding direction^, the moment center is also a key issue for the model of finite element analysis can be established as in Fig. 8. moment Eq. 31. In this paper, we use the points on the slip In this calculation model, the horizontal distance in the X axis surface to define a circle, which is straightforward. And then is 2000 m, and vertical distance in the Y axis is about 1212 m. the circle center can be considered as the center of the mo- Moreover, near the area of the landslide, the density of ele- ment, which is proved to be an efficient technique to ment mesh was remarkably higher than in any other place,

Fig. 7 Simple slope for moment equilibrium

(a) Sliding stress (b) Resisting stress M. Guo et al.

under the saturated state can also be easily obtained. Generally speaking, the elasto-plastic finite element analysis was utilized to obtain the stress state of the slope, and the non-associated flow rules are used during the elasto-plastic finite element analysis, while, in addition, the dilation angle is assumed to be zero during elasto-plastic finite element computing. However, for the modern landslide under saturated condition, using the elasto-plastic finite element technique it is difficult to compute the stress state of the multi-scale landslide because the state of the modern landslide is close to the limit state. Thus, the elastic finite element technique was used for the stress state of the landside. Fig. 8 Finite element analysis model of the Zhangmu landslide The calculated results of the landslide using VSM are listed in Table 2, from which it can be seen that, under natural con- while the total element number is 6597. For boundary condi- dition, it is stable for any potential sliding bodies, and the tions, the bottom is fixed and the lateral boundaries are nor- safety factor is more than 1.10. However, under saturated con- mally restricted. dition, it is stable for old and ancient rock landslides, and Figure 9 shows the local mesh and spatial distribution of unstable for the modern landslide, which reveals that the rain- the multi-layer landslide. Based on the shear zones found in- fall condition greatly reduces the stability of the modern land- side the landslide, in the finite element analysis model, a thin slide. Furthermore, the safety factors by force equilibrium are and weak layer was considered to simulate the shear zone well consistent with those by moment equilibrium in the pro- (Fig. 9), and the potential slip surface was along the shear zone posed method. The global sliding direction is about 25° for from the top to the front edge of the landslide. this multi-layer landslide, which demonstrates the actual to- For the physical and mechanical parameters of the mate- pography of the landslide, which is of help in devising the rials, Table 1 gives the recommended values from detailed treatment of stable piles. investigations and back analysis, from which two different conditions of landslide, natural and saturated conditions, were taken into account, and, because of being deep inside the Comparison analysis bedrock for the hydrostatic level from the data of underground water inside the landslide, the static water was not considered In order to verify the accuracy of the proposed method, the in the stability analysis of this landslide. popular limit equilibrium method (the rigorous Morgenstern– Price method) was also utilized to analyze the stability of the same landslide. According to the geological profile (Fig. 4)of Calculated results and analysis the landslide, the full analysis model of the multi-layer land- slide was established and demonstrated in Fig. 11a, while Based on the computing model and conditions of the slices of the modern landslide, old landslide and ancient rock Zhangmu landslide, the stress state of this landslide can be landslide are shown in Fig. 11b, c and d, respectively. determined using the finite element method, and Fig. 10 dem- Belonging to polygonal-shaped slip surfaces with low-angle onstrates the horizontal S11 and vertical stress S22 contours of dip in this multi-layer landslide, they can be easily analyzed this landslide under normal state. Similarly, the stress contour without convergence problems during iterative calculating for

Fig. 9 Local mesh and spatial distribution of the Zhangmu landslide Stability analysis of the Zhangmu multi-layer landslide using the vector sum method in Tibet, China

Fig. 10 Stress contour of the Zhangmu landslide under normal state

(a) Horizontal stress S11 contour

(b) Vertical stress S22 contour the safety factor. The commercial Geo-slope software was the safety factors by the proposed method are almost the same used to perform the stability analysis of these landslides with as those by the Morgenstern–Price method, and for the mod- the rigorous Morgenstern–Price method. ern landslide, the safety factor with the proposed method is a Table 3 demonstrates the results compared with the pro- slightly larger than that by the Morgenstern–Price method. posed method and the Morgenstern–Price method, from Therefore, for the multi-layer landslide, the VSM can be ap- which it can be seen that the safety factors by the proposed plied to effectively analyze the local or global stability of the method are in good agreement with those by the landslide, and, more importantly, compared with the Morgenstern–Price method. Except for the modern landslide, Morgenstern–Price method, for complex slopes with sliding

Table 2 Calculating results of Bangcundong landslide with Method Modern landslide Old landslide Rock slide VSM Natural Saturated Natural Saturated Natural Saturated

Vector sum Force equilibrium 1.181 0.978 1.212 1.125 1.626 1.310 method Moment equilibrium 1.1941 0.989 1.207 1.120 1.618 1.304 Global sliding angle (°) 24.42 24.43 28.76 28.75 22.68 22.68

The sliding angle is the angle between the sliding direction and the horizontal axis M. Guo et al.

Fig. 11 Full analysis and slices model of the multi-layer landslide

(a) analysis model (b) modern landslide

(c) Old landslide(d) Rock landslide failure mechanisms, this proposed method can effectively de- force vector. More importantly, the global sliding direc- termine its stability without convergence problems. tion was first derived in theory based on the actual stress of the slope, which brings insight into the treatment de- sign for potential landslides. Thus, the proposed method Discussion and conclusions has a clear physical meaning for any slip surface, i.e. the safety factor by force equilibrium stands for the quantified The stability of the multi-layer Zhangmu landslide has been index of the sliding failure mode along the sliding direc- analyzed based on detailed geological survey and analysis. tion; however, the safety factor by moment equilibrium Due to the limitations of conventional slope-stability methods means that of the rotational failure mode around the mo- mentioned in section BConventional methods for slope stabil- ment center. Additionally, this proposed method needs no ity analysis^, for this multi-layer landslide, a new approach assumptions in computing the factor of safety, whereas, in called VSM has been proposed. This method considers the the LEM, many assumptions without physical meaning direction of force and the strength-reserving definition of the about the inter-slice force are needed to compute the fac- safety factor. With this method, the complete stress state of the tor of safety, while the VSM can be easily extended to 3D landslide from finite element analysis is imported into a limit problems based on the actual stress of the slope. In sum- equilibrium analysis where the normal stress and shear stress mary, despite the critical slip surface needing to be can be computed with respect to any selected slip surface. searched, the proposed method is of vast significance to Moreover, the whole sliding body is taken into account as the fundamental theory of slope stability. the research object, and the global safety factor can be deter- 2) According to the characteristics of this method, its merits mined directly by the force and moment equilibrium equation are as follows: one is the clear physical meaning for the of a sliding body. stability of the landslide, that is, both the magnitude and the direction of force are considered, the global sliding 1) Differing from conventional methods, such as LEM and direction of the slope was first put forward and theoreti- FE-SRM, the VSM emphasizes the direction of the force, cally determined, and, naturally, the stability of the slope in which the resisting forces along a potential sliding was performed in the global sliding direction and the cen- body, as well as the driving forces, are respectively deter- ter of the moment. The other merit is computational effi- mined to be the total resisting force vector and the driving ciency. As we know, convergence problems always occur

Table 3 Comparison results of landslide stability Method Modern landslide Old landslide Rock slide

Natural Saturated Natural Saturated Natural Saturated

Vector sum method 1.181 0.978 1.207 1.120 1.618 1.304 Morgenstern–Price method 1.132 0.937 1.236 1.147 1.625 1.309 Stability analysis of the Zhangmu multi-layer landslide using the vector sum method in Tibet, China

during iterative computation for the safety factor in con- Cheng YM, Lansivaara T, Wei WB (2007) Two-dimensional slope sta- ventional methods. However, the VSM has no conver- bility analysis by limit equilibrium and strength reduction methods. Comput Geotech 34:137–150 gence problem, as long as the stress state of the slope Duncan JM (1996) State of the art: limit equilibrium and finite-element can be obtained, the safety factor with this method can analysis of slopes. J Geotech Geoenviron Eng 122(7):577–596 be directly determined by the force and moment equilib- Fredlund DG, Krahn J (1977) Comparison of slope stability methods of – rium equations. The actual stress state of the slope can be analysis. 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The calculated results by the proposed method two-dimensional slope stability analysis. Disast Adv 6(S4):258–269 also demonstrate that both the sliding failure along the Hamdhan IN, Schweiger HF (2013) Finite element method–based anal- sliding direction and the rotational failure around the mo- ysis of an unsaturated soil slope subjected to rainfall infiltration. Int J Geomech 13(5):653–658 ment center control the stability of this landslide. Under Hu RL, Zhang XY,Gao Wet al (2015) Structure and stability of Zhangmu natural condition, it is stable for this multi-layer landslide, deposit in Tibet. 10th Asian Reg Conf IAEG and under saturated condition, it is still stable for the an- Huang CC, Tsai CC (2000) New method for 3D and asymmetricalslope – cient rock landslide (F > 1.30). However, it is unstable for stability analysis. 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