Theoretical and numerical study on the block-flexure toppling failure of rock slopes

Journal Pre-proof Theoretical and numerical study on the block-flexure toppling failure of rock slopes Yun Zheng, Congxin Chen, Tingting Liu, Haina Zhang, Chaoyi Sun

PII:

S0013-7952(18)32233-6

DOI:

https://doi.org/10.1016/j.enggeo.2019.105309

Reference:

ENGEO 105309

To appear in: Received Date:

24 December 2018

Revised Date:

8 July 2019

Accepted Date:

24 September 2019

Please cite this article as: Zheng Y, Chen C, Liu T, Zhang H, Sun C, Theoretical and numerical study on the block-flexure toppling failure of rock slopes, Engineering Geology (2019), doi: https://doi.org/10.1016/j.enggeo.2019.105309

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Theoretical and numerical study on the block-flexure toppling failure of rock slopes Yun Zheng1, Congxin Chen1, Tingting Liu1, 2*, Haina Zhang1, Chaoyi Sun1*

1

State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil

Mechanics, Chinese Academy of Sciences, Wuhan, Hubei 430071, China 2

Key Laboratory of Roadway Bridge and Structure Engineering, Wuhan University of Technology,

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Wuhan, Hubei 430070, China

Corresponding author (Tingting Liu). Tel./fax:+86 13667131170. Email: [email protected]

*

Corresponding author (Chaoyi Sun). Tel./fax:+86 13628608480. Email: [email protected]

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*

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Highlights

Two theoretical models for block-flexure toppling failure were proposed.



Detailed damage information of the joints during block-flexure was studied.



Effects of cross joint spacing on block-flexure were discussed.

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Abstract

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

Block-flexure toppling failure is frequently encountered in rock slopes. In this paper, we first

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proposed two theoretical models to calculate the factor of safety against block-flexure toppling, namely, step by step analysis model (LEM-SSAM) and overall analysis model (LEM-OAM). Then, we developed two special MATLAB codes for a quick stability estimation based on the proposed methods, respectively. Next, the universal distinct element code (UDEC) was used to study the failure mechanisms of block-flexure toppling. In order to realistically simulate the excavation of slope and

obtain the detailed damage information of joints, we developed two FISH functions in UDEC. Finally, we conducted a comparative analysis of the limit equilibrium method and the numerical simulations through twelve theoretical models. The results show that many aspects of block-flexure toppling failure have been captured using UDEC combined with the developed FISH functions. The LEM-OAM and UDEC agree well while a larger factor of safety is obtained using the LEM-SSAM. It is found that the dip angles and friction angles of the joints dipping steeply into face have a great influence on

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block-flexure toppling failure while the influence of the cross joint spacing is less significant.

Keywords: Rock slopes; block-flexure toppling; limit equilibrium method; numerical simulation;

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strength reduction method.

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1 Introduction

Toppling failure is one main instability mode of rock slopes, which has caused serious economic

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losses and casualties in engineering activities (Müller 1968; Wyllie, 1980; Alejano et al., 2010; Li et al., 2015; Liu et al., 2016). Goodman and Bray (1976) has described various types of toppling failures that

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may be encountered in the field. They classified toppling failures into three distinct modes, i.e., block

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toppling, flexural toppling and block-flexure toppling. Block toppling occurs in rock slopes with two sets of orthogonal joints. One set of the joints dips steeply into face and the other set is gently inclined cross joint controlling the height of blocks. If the set of cross joints is removed from the aforementioned rock slopes, then they will be subjected to flexural toppling failure. Generally, the failure surface of a flexural topple is not as well defined as a block topple (Wyllie et al., 2004). Block-flexure topple is a hybrid of block topple and flexural topple, which is characterized by block

toppling of blocky columns and flexural toppling of continuous columns (Goodman and Bray, 1976). Based on the limit equilibrium theory, Goodman and Bray (1976) first proposed the stepwise analysis method for block toppling, which provided a theoretical basis for later researches. The method was based on the following assumptions: (a) a state of limit friction equilibrium exists simultaneously along the interface between rock blocks; (b) the normal side forces act on the top of blocks. In the past few decades, many scholars have developed the stepwise analysis method from different aspects. To

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achieve a rapid analysis of block toppling, Liu et al. (2009) proposed a transfer coefficient method for

this type of failure. For rock slopes with infinitesimal thickness of blocks, general analytical solutions were provided for quantitatively assessing this type of failure (Bobet, 1999; Sagaseta et al., 2001; Liu

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et al., 2008). Risk evaluation methods for rock slopes against block toppling were also developed using

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the reliability theory (Scavia et al, 1990; Chai et al, 2015). Yagoda and Hatzor (2013) and Jaimes and

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Candia (2018) have deduced the analytical equations of block toppling in the cases of seismic loading. Alejano et al. (2015, 2018) extended the stepwise analysis method to analyze rock blocks with rounded

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edges due to spheroidal weathering. Alejano et al. (2010), Manera et al. (2014), and Amini et al. (2018) proposed the methods for analyzing complex toppling-circular slope failures using the limit

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equilibrium equations.

Assuming cohesionless behavior of joints and regarding rock columns as superimposed beams,

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Aydan and Kawamoto (1992) first proposed a theoretical method for flexural toppling failure. Amini et al. (2009) extended this method to consider the deformation of rock columns and established the equations to identify the acting point of the total side forces. Based on the Aydan's method, Zheng et al. (2018b) proposed a new method which is able to consider the cohesion between rock columns. By dividing the failure zone into different subzones, Zheng et al. (2018a) and Zhang et al. (2018) proposed

new methods to determine the subzone boundaries, the failure surface and the factor of safety. Using solid mechanics and fracture mechanics approaches, the flexural toppling problems of rock slopes with geo-structural defects were investigated by Majdi and Amini (2011). For the combination of flexural toppling failure and other failure types (circular sliding failure, plane sliding failure and wedge sliding failure), some studies have also been conducted (Mohtarami et al., 2014; Amini et al., 2017, 2018). Numerical models are also useful for studying the mechanism of toppling failure in rock slopes

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and have the advantage that they can be used regardless of the geometrical complexity of the study area. For example, the universal distinct element code (UDEC) developed by Itasca (2004) can simulate the

mechanical behavior of jointed rock masses very well. Thus, it has become one of the most commonly

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used software packages used in geotechnical engineering. The application of UDEC to studying

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toppling failure was first validated by Barla et al. (1995). Since then, a large number of studies on

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toppling failure have been carried out using UDEC. Alejano and Alonso (2005) extended the shear strength reduction technique proposed by Dawson et al. (1999) to the computing of factors of safety

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against toppling. Thereafter, Alejano et al. (2010) successfully employed UDEC to back analyze a complex toppling-circular slope failure. Liu et al. (2016) conducted a numerical study on deep-seated

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large-scale flexural toppling failure in metamorphic rocks using UDEC. Zheng et al. (2019) used UDEC combined with the strain-softening model (the strength parameter may be lower significantly to

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reflect a lower shear strength and null tensile strength as expected after failure) to investigate the flexural toppling failure of rock slopes before and after being reinforced with rock bolts. In addition to UDEC, many other numerical methods have been employed to study toppling failure in rock slopes, e.g. equivalent continuum model, numerical manifold method, and distinct lattice spring model (Adhikary and Dyskin, 2007; Zhang et al., 2010; Lian et al., 2017).

As described above, an enormous amount of theoretical analyses and numerical simulations have been conducted on toppling failure of rock slopes. However, current studies are mainly focused on block toppling and flexural toppling. Studies of block-flexure failure are much rarer. Since block-flexure is the most common type of topple in rock slopes, its failure mechanisms and analytical methods deserve the most attention (Amini et al., 2012). In this paper, the stability of rock slopes against block-flexural toppling was studied using the

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limit equilibrium and strength reduction numerical methods. The limit equilibrium method consists of

two models, i.e. the step by step analysis model (LEM-SSAM) and overall analysis model (LEM-OAM). The evolutionary process of block-flexural toppling failure was revealed using UDEC

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combined with the new developed FISH functions. Then, further parameter studies were conducted

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friction angle on block-flexural toppling failure.

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using these two methods to investigate the effect of the cross joint spacing, joint dip angle, and joint

2 Formation conditions of block-flexure toppling failure

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Rock slopes with the potential to undergo block-flexure toppling failure mainly consist of two sets of joints (Fig. 1). One of them is well developed, dipping steeply into face (usually bedding planes),

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and the other is a set of randomly distributed, pseudo-continuous cross joints (Goodman and Bray,

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1976). When failure of the slope occurs, some rock columns are tensile fractured due to bending stresses, and some overturn under the combined action of their own weights and interlayer forces. Because of the large number of small movements in block-flexure toppling, there are fewer tension cracks than in flexural toppling, and fewer edge-to-face contacts and voids than in block toppling (Wyllie et al., 2004). Block-flexure toppling failure is common in syncline and anticline flanks or steeply dipping

monoclinic stratum areas due to their structural features (Gu and Huang, 2016; Huang and Gu, 2017). The lithologic conditions of such slopes are dominated by one or two interbedded layered rocks, such as sandstone, shale, limestone, etc (Amini et al., 2012; Alejano et al., 2010; Chai et al., 2015; Li et al., 2015; Gu and Huang, 2016; Huang and Gu, 2017; Zhang et al., 2018). Many factors can induce such a failure, including earthquakes, rainfall, fluctuation of reservoir water level, excavation and so on (Amini et al., 2012; Alejano et al., 2010; Chai et al., 2015; Li et al., 2015; Gu and Huang, 2016; Zhang

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et al., 2016; Huang and Gu, 2017; Hu et al., 2019).

In addition, block-flexure toppling failure often occurs within the weathering zone. This is

because weathering not only degrades the strength of intact rock, but also results in remarkable

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decrease of interlayer strength, especially under the combined action of rainfall infiltration and

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earthquake (Huang et al., 2019). In other words, two parameters that control the stability of rock slopes

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against block-flexure toppling failure are both reduced by weathering, i.e. the tensile strength of intact rock and shear strength of joints dipping steeply into face.

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3 The limit equilibrium method

Since the real block-flexure toppling failure cases are heavily affected by the complex material

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properties and geological conditions, this study starts with the simplest cases to understand

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fundamental mechanisms. The theoretical model is assumed to be composited of two sets of joints (Fig. 2). The set of joints dipping steeply into face is persistent and equally spaced. The set of cross joints perpendicular to the former set is pseudo-continuous and widely spaced with equal spacing. The connectivity rate of the cross joints is 0.5 and the length of rock-bridge is equal to the width of rock column. It can be seen that two types of rock columns exist in the slope, i.e. continuous columns and blocky columns with cross joints. Therefore, in order to assess the stability of such a slope, mechanical

equations should be established for these two types of rock columns, respectively. In the following theoretical analysis, the failure surface of block-flexure toppling is assumed to develop along the cross joint across the toe of slope.

3.1 Mechanical analysis of continuous columns

Continuous columns may be subjected to flexural toppling or shearing failures and extensive

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studies on their failure mechanisms have been done (Amini et al., 2012; Mohtarami et al., 2014, and Zheng et al., 2018a, 2018b). Due to the forces to which the column is submitted, it can be stable or unstable against a failure mechanism that can be toppling or shearing. The presented calculations

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would indicate which mechanism is more prone to occur and the force needed to control this instability

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mechanism for the particular column under scrutiny.

(a) Column i is subjected to flexural toppling failure but is stable against shearing failure (Fig.

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3(a)). In this scenario, the following conditions are satisfied (Aydan and Kawamoto, 1992):

Qi  Pi tan d  Qi 1  Pi 1 tan d

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where

(1)

Qi 1 and Qi are the shear forces acting on the left and right sides of column i , respectively;

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Pi 1 and Pi are the normal forces acting on the left and right sides of column i , respectively;  d

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is the friction angle of the joints dipping steeply into face. Then, the value of

Pi 1,t =

where

Pi 1,t can be calculated using the following force equation:

1  ti tan d  i 1hi 1  2 Fs 

 ti tan d   i hi  2 Fs    

  t wi cos   2 I i hi     wi sin     Pi     F t t 2  s i   i   

(2)

hi 1 and hi are the height of rock columns i  1 and i , respectively; i 1 and  i are

non-dimensional parameters defining the acting points of normal side forces;

ti is the thickness of

column i ;

I i is the moment of inertia of column i , with the form of Ii  ti 3 /12 ;  t is the

tensile strength of intact rock;

Fs is the factor of safety; wi is the weight of column i ;  b is dip

angle of the cross joints. (b) Column i is subjected to shearing failure but is stable against flexural toppling failure (Fig. 3(b)). In this scenario, it is also assumed that (Zheng et al., 2018a):

Qi  Pi tan d  Qi 1  Pi 1 tan d

 tan   ct wi cos    tan    i  Fs  Fs Pi 1, s =Pi  tan d tan  1 Fs 2 where c and

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Pi 1, s can be calculated using the following force equation:

(4)

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Then, the value of

(3)

 are the cohesion and friction angle of intact rock, respectively.

(5)

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Pi 1  max  Pi 1,t , Pi 1, s ,0 

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determined using the following equation:

Pi 1 acting on the right side of column i  1 is

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It should be noted that the real value of

3.2 Mechanical analysis of blocky columns with cross joints

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Blocky columns (these with cross joints) have the potential for block toppling failure or sliding

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failure. Due to the existence of cross joints, there may be several rock blocks above the failure surface (Fig. 2). This implies that the stability analysis of a blocky column involves multiple rock blocks, which is more complex compared to that of continuous columns. In this study, the following two models are considered when slope instability occurs: (1) Step by step analysis model (LEM-SSAM): each rock block above the failure surface is assumed to be in the limit equilibrium state with the occurrence of block toppling failure (Fig. 4a) or

sliding failure (Fig. 4b). (2) Overall analysis model (LEM-OAM): the blocky column with cross joints is assumed to behave as an equivalent monolithic column, and the whole is in a state of limit equilibrium subjected to block toppling failure (Fig. 5a) or sliding failure (Fig. 5b). In order to facilitate the analysis, a two-dimensional matrix ( i ,

j ) was used to characterize the

location of rock block under consideration, where i denotes the column number counted from the toe

j denotes the block number counted from the top of the column under consideration

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of the slope and (Fig. 4).

j ) may be subjected to block toppling failure or sliding

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For the LEM-SSAM, since block ( i ,

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3.2.1 Step by step analysis model (LEM-SSAM)

failure, two categories are included in the analysis:

j ) is subjected to block toppling failure but is stable against sliding failure (Fig.

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(a) Block ( i ,

4a). In this scenario, the following conditions are satisfied (Goodman and Bray, 1976):

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Qi , j  Pi , j tan d  Qi 1, j  Pi 1, j tan d

(6)

(i,

j ),

respectively; Qi 1, j and Qi , j are the shear forces acting on the left and right sides of block

(i ,

j ),

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where Pi 1, j and Pi , j are the normal forces acting on the left and right sides of block

respectively.

Then, the value of

Pi 1, j ,t can be calculated using the following force equation:

tan d 1  hi , jTi , j -1    ei , j 1  ti Ni , j -1 wi , j cos  Fs 2  Pi , j  ti  hi , j tan   +  i 1, j hi , j 2 i 1, j hi , j i 1, j hi , j

i , j hi , j  ti Pi 1, j ,t 

where hi , j is the height of block ( i ,

(7)

j ); i 1, j and  i , j are non-dimensional parameters defining

the side force distribution relationship; wi , j is the weight of block ( i ,

j ); ei , j 1 is a

non-dimensional parameter defining the acting point of normal force on the base of block ( i ,

j  1 ),

which satisfies the following relations:

1  ei , j 1  2  ei , j 1  0 

when block (i, j  1) has the potential for block toplling failure (8)

when block (i, j  1) has the potential for sliding failure

(b) Block ( i ,

j ) is subjected to sliding failure but is stable against block toppling failure (Fig.

Qi , j  Pi , j tan d  Qi 1, j  Pi 1, j tan d

(9)

Pi 1, j , s can be calculated using the following force equation:

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Then, the value of

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4b). In this scenario, it is also assumed that (Goodman and Bray, 1976):

where

b

(10)

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Pi -1, j , s

 tan b  tan b wi , j cos    tan    Ni , j 1  Ti , j 1 Fs Fs    Pi , j tan b tan d 1 Fs 2 is the friction angle of the cross joints.

It should be noted that the real value of

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Pi 1, j  max  Pi 1, j , s , Pi 1, j ,t ,0

Pi 1, j is determined using the following equation: (11)

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Then, the value of normal force acting on the base of block ( i ,

j  1 ), Ni , j , can be calculated

using the following force equation:

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Ni , j  wi , j cos    Pi , j  Pi 1, j  tan d +Ni , j 1

(12)

The calculation of shear force acting on the base of block ( i ,

j  1 ), Ti , j , should be divided into

the following two cases: (i) When block ( i , using Eq. (13):

j ) is subjected to block toppling failure, the value of Ti , j can be calculated

Ti , j  wi , j sin   Pi , j  Pi 1, j  Ti , j -1 (ii) When block ( i ,

(13)

j ) is subjected to sliding failure, the value of Ti , j can be calculated using

Eq. (14):

Ti , j  Ni , j tan d

(14)

For Eqs. (7) and (10), the value of Pi , j exerted by continuous column can be obtained by the value of

i 1

on block ( i ,

j)

Pi calculated using Eqs. (2), (4) and (5), and its distribution.

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In the case of deformable rock columns or blocks, not all the normal side stresses are transmitted in one point. On the contrary, they are distributed along with the interlayers. Aydan and Kawamoto

(1992) suggested that the distribution of normal side stresses ranges from a triangular to a point

max  n  i, j  1 ,0   max  n  i, j  ,0  2

hi , j

(15a)

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Pi , j 

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in Fig. 6. Then, the value of Pi , j can be computed as:

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distribution. In this study, the normal side stresses are also assumed to distribute linearly, as illustrated

(15b)

2 Pi   A  i   1 h , i  1 / 3 i  n  i,0    0,   1 / 3  i

(15c)

 3 i  3  1 ,0   i  1 / 3  i  n  i, mi   3 i  2 Ai  = ,1 / 3   i  2 / 3  n  i,0   1  3 i  3 i  2 , 2 / 3  i  1   3 i  3

(15d)

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  m j     hi ,m   n  i,0  1  m 1 A  i   1 ,  i  1 / 3  hi        n  i, j     m j   hi ,m  m 1 Pi ,  i  1 / 3  hi 2

where

 n  i,0 and  n  i, mi  are the normal side stresses on the top and base of column i ,  n  i, j  1 and  n  i, j  are normal side stresses on the top and base of block

respectively,

 i, j  ,

respectively, and

mi is the number of blocks contained in the blocky column under

consideration. In addition, when

i =0

and

i =1 ,

the distribution of normal side stresses follows a point

distribution (Fig. 6 (a) and (h)). Then, we have

It should be born in mind that the normal force

Pi 1 exerted by column i on column i  1 is

j  mi

P

(17)

i 1, j

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3.2.2 Overall analysis model (LEM-OAM)

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j 1

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calculated using the following equation:

Pi 1 

(16)

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Pi ,1 =Pi ,2 =...=Pi ,mi 1  0  Pi ,1 =Pi   if i  0 and P =P =...=P  0  if i  1 Pi ,mi  Pi i ,2 i ,3 i , mi  

For the LEM-OAM, since blocks ( i , 1) to ( i , mi ) are assumed to behave as a long rock column

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which is recorded as column i (Fig. 5), the analysis is divided into the following two categories:

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(a) Column i is subjected to block toppling failure but is stable against sliding failure (Fig. 5a). In this scenario, the following conditions are satisfied (Goodman and Bray, 1976):

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Qi  Pi tan d  Qi 1  Pi 1 tan d

Then, the value of

Pi 1,t can be calculated using the following force equation:

tan d  t  h tan   w cos  Fs Pi  i i i i 1hi 2 i 1hi

i hi  ti Pi 1,t 

(18)

(19)

(b) Column i is subjected to sliding failure but is stable against block toppling failure (Fig. 5b).

In this scenario, it is also assumed that (Goodman and Bray, 1976):

Qi  Pi tan d  Qi 1  Pi 1 tan d Then, the value of

Pi 1, j , s can be calculated using the following force equation:

 tan b  wi cos    tan    Fs   Pi  tan b tan d 1 Fs 2

It should be noted that the real value of

(21)

Pi 1 acting on the right side of column i  1 without

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Pi -1, s

(20)

cross joints is determined using the following equation:

Pi 1  max  Pi 1,t , Pi 1, s ,0 

(22)

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The LEM-OAM is potentially more realistic compared to the LEM-SSAM. This is because the

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latter forcing all blocks to limit equilibrium may over-estimate the stability of the slope. Moreover,

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model tests of rock slopes with two orthogonal discontinuity sets also indicated that the blocky column with the cross joints will behave as an equivalent continuous column (Aydan et al., 1989). A

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comparison of the results predicted for the factors of safety between the LEM-SSAM and LEM-OAM approaches is carried out in section 5.

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3.3 Analysis of the acting point of total normal side forces

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The acting point of total normal side forces has a significant effect on the calculated value of the moment a rock block or column bears. Therefore, it is one of the most important parameters in the analysis of toppling failure. Generally, it is assumed to be the same for all rock blocks or columns in previous studies (e.g., Goodman and Bray, 1976; Aydan and Kawamoto, 1992; Adhikary et al., 1997; Liu et al., 2009; Amini et al., 2012; Zhang et al., 2018). Goodman and Bray (1976) suggested that the

acting point of total normal side forces is on the top of each rock block for block toppling, i.e.

 =1.

Flexural toppling is more complex compared to that in block toppling, and the acting point may vary from the bottom to the top of the rock column. In other words, the value of

 varies from 0 to 1

depending on the mechanical properties of the slope, as pointed out by Aydan and Kawamoto (1992). However, when toppling failure occurs, tension cracks are expected to appear along the joints dipping steeply into face (e.g., Alzo'ubi, 2009; Zheng et al., 2018a; Amini et al., 2018). Wyllie et al.

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(2004) suggested that this condition is very useful in the field identification of topples. Moreover, the depth of tension cracks is gradually deepened from the toe to the crest of the slope and eventually extends to the failure surface at the uppermost crack (Fig. 1). This indicates that the value of



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gradually decreases from the toe to the crest of the slope instead of being a constant value. Therefore, it

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can be concluded that in the analysis of block-flexure toppling failure illustrated in Fig. 2, the value of

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 at the last rock column or block unstable behind the slope head is zero while that of rock column 1 (the one located at the toe of the slope) is nonzero.

To solve this problem, we assume a primary function with the form of

  xi    F  xi  for

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predicting the acting point of total normal side forces, where F  xi  is a function ranging from 0 to 1;

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xi is the distance from the upper boundary of rock column 1 (the one located at the toe of the slope) to that of the rock column or block under scrutiny, and it is used to characterize the location of rock

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columns or blocks (Fig. 2);



is a positive constant determined according to F  xi  and boundary

conditions. F  xi  can be considered as a monotonically decreasing function and it satisfies

F  xup  =0 on the upper boundary, where xup is the abscissa of the upper boundary of the last rock

column or block unstable behind the slope head. Many commonly used functions can meet the requirements, such as linear function, quarter-sine

function, negative exponential function, etc. (Fig. 7). Here, a simple linear function is used to describe

F  xi  with the form as:

F  xi   

(23)

L is the length of the failure surface.

Then,

  xi 

can be expressed as:

 xi   1 L  

  xi     



The value of

(24)

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where

xi 1 L

can be determined according to the lower boundary conditions related to the

specific slope. For example for column 1 (the one located at the toe of the slope), if the acting point of

 =1), the value of  =1 then can be calculated by

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total normal side forces is located on its top (i.e. submitting

x1 =0 into Eq. (24). Therefore, using Eq. (24), we have

  xi   

xi 1 L

x1

into Eq. (25).

The value of follows (Fig. 6):

 i , j can be calculated according to the distribution of normal side stresses as

xi  0

(i.e. Fig. 6(a)), we have

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(i) When

for every rock column or block can be determined by submitting its x

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coordinate

  xi 

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Then, the value of

re

(25)

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xi , j  0

(ii) When

0  xi  1/ 3

(26a) (i.e. Fig. 6(b)), we have

  0   i, j  xi , j   hi , j  3  i , j 1   i , j   2 i , j 1   i , j h  3  i , j 1   i , j  i , j 

2 / 3  xi  1

 i . j 1  0,  i . j  0

(i.e. Fig. 6(g)), we have

  0   2 i , j -1 +  i , j xi , j   hi , j  3  i , j -1   i , j   2 i , j 1   i , j h  3  i , j 1   i , j  i , j 



 i . j 1  0,  i . j  0  i . j 1  0,  i . j  0



1 xi , j   0

(i.e. Fig. 6(h)), we have

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xi =1

(26c)

 i . j 1  0,  i . j  0

j 1 j 1

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(v) When

(26b)

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(iii) When

 i . j 1  0,  i . j  0



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

 i . j 1  0,  i . j  0

(26d)

(vi) For the other scenarios (i.e. Fig. 6(c) to (f)), we have

2 i , j 1   i , j

3  i , j 1   i , j 

hi , j

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xi , j 

(26f)

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3.4 Stability criterion

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Under limit equilibrium conditions,

Fs  1 , the external force, P0 , needed at the toe of slope for

keeping it stable can be computed by using the above expressions in a step-wise manner. Since there is no such a force at the toe of slope, the value of

P0  0 , unstable P0  0 , at limiting equilibrium

P0 can be used to evaluate the stability as follows:

P0  0 , stable Assuming

P0  0 , the factor of safety of rock slopes against block-flexure toppling failure, Fs ,

can be calculated using the dichotomy method. It can be seen that the proposed approaches require extensive calculations, which are time-consuming and error-prone if the calculations are carried out manually. Therefore, on the basis of the proposed methods, two special MATLAB codes for a quick stability estimation of rock slopes have

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been developed, respectively. The side forces for all rock columns or blocks and the factor of safety can be quickly computed using these two programs.

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4 The numerical modelling approach

UDEC is commonly used to study the failure mechanisms of jointed slopes and its application to

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toppling failures has also been validated (Alejano et al., 2005; Alzo'ubi, 2009; Zheng et al., 2018a,

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2018b). UDEC solve the contact problem between blocks very well, which can simulate the rotation and sliding of blocks at the same time. Furthermore, constitutive models provided in UDEC can be

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used to simulate the failure of intact rock, such as Mohr-Coulomb model, strain-softening model, Hoek-Brown model, etc. (Itasca, 2004). Due to these features, UDEC owns a significant advantage in

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simulating block-flexure toppling, which involves a combined failure of intact rock and joints.

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Therefore, it was employed in this work.

4.1 Model configuration

Many gneiss anti-dip slopes with steep angles have been formed during the process of mining granite mines in China's Shanxi province (Chen et al., 2015). Therefore, this paper takes this kind of slope as an example to investigate the mechanism of block-flexure toppling failure. Using UDEC, a

model is created to simulate block-flexure toppling failure in a 40-m slope dipping 70° (  c =70°), as shown in Fig. 8. The set of joints dipping 70° into face has a spacing of 3 m and the set of cross joints dipping 20° has a spacing of 9 m. The model is divided into 106940 triangular zones with a maximum edge length of 1.0 m. This slope model is called model B0. To simulate nonlinear material softening behavior of intact rock, strain-softening model is used in the numerical simulation. The commonly used joint constitutive model, coulomb slip model, is used to describe the behavior of joints.

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The parameters for the numerical simulations are summarized in Tables 1 and 2. It should be noted that the properties of the intact rock and joints (e.g. cohesion, friction angle, tensile strength,

elastic modulus, Poisson's ratio) were obtained via laboratory tests (Chen et al., 2015). The cohesion

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and friction angle of the intact rock are irrelevant for the flexural response. However, they control the

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stability of the slope against sliding failure.

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The normal stiffness of joints was estimated using the following equations suggested by Cho et al. (2007):

 K   4 3 G  kn  n   , 1  n  10  zmin 

G

E 2 1   

K

and

(27b) (27c)

G are the bulk and shear moduli of the intact rock, respectively, E is the elastic

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where

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E 3 1  2  

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K

(27a)

modulus of the intact rock, and



is the Poisson's ratio of the intact rock.

Since joints are more likely to displace along the tangential direction compared to that along the

normal direction, smaller shear stiffness (0.2 to 0.8 times the normal stiffness value) is widely used to describe the behavior of the joints in numerical models (Alzo'ubi, 2009; Gao et al., 2014; Li et al., 2015; Du and Huang, 2016; Huang and Gu, 2017; Huang et al., 2018; Yan et al., 2019). In this study, the

shear stiffness value was taken half of the normal stiffness value, as shown in Table 2.

4.2 Block-flexure toppling failure simulation

In the numerical simulation, the model with the initial geometry was first brought to the initial stress equilibrium under gravity loading. Thereafter, the blocks located above the cut slope were deleted to simulate slope excavation. Since rock masses are elasto-plastic materials, a sudden

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excavation of the slope may result in an unstable model response (Gao et al., 2014). To simulate a more realistic slope excavation and minimize the influence of transients on material failure, a FISH function

“SLOPE_EXCAVATION.FIS” was developed to detect excavation boundaries and gradually relax the

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forces on them. The forces applies on the slope surface were relaxed in a total of 10 stages, with a

applied on the slope surface are relaxed).

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release factor R increasing from 0 to 1 in an increment of 0.1 (R=0.1 denotes 10 % of the forces

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To obtain the detailed damage information of the two sets of joints during the failure process, a FISH function “JOINT-DAMAGE-TRACKING.FIS” was also developed to track the length of joints

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where shear or tension failure occurs. Only the joints in the area of interest are tracked. Damage factor,

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D = 0 to 1, is defined as the ratio of the length of failed joints to the total length of the set of joints under consideration, where

Ds1 , Dt1 , and Dst1 denote shear, tension, and total damage factor of the

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joints dipping steeply into face, respectively;

Ds 2 , Dt 2 , and Dst 2 denote shear, tension, and total

damage factor of the cross joints, respectively. The results of block-flexure toppling simulations are illustrated in Figs. 9-13. Figs. 9 and 10 show

the development of shear and tension failure of joints, respectively. Fig. 11 shows the distribution of side normal stresses corresponding to Fig. 10(f) (i.e. R=1.0). Fig. 12(a) and (b) show the progressive

damage of the two sets of joints during block-flexure toppling failure, respectively. Fig. 13 shows the failure surface of the slope. According to the results, the following interpretation can be summarized: (1) When the excavation of slope begins, shear failure first occurs along the set of joints dipping steeply into face and gradually extends to the deep part with the increase of release factor. This is because the stress redistribution within the slope causes an outward movement of each rock column, producing an interlayer slip. Due to the constraining effect of the continuous columns on both sides,

 b ), few cross

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and the internal friction angle of cross joints is larger than its inclination angle ( b joints are subjected to shear failure (Fig. 9).

(2) In the initial stage of unloading, neither of the two joint sets is subjected to tension failure (Fig.

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10(a)). However, with the increase of the release factor, tension cracks first occur along the set of joints

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dipping steeply into face in the upper part of the slope (Fig. 10(b)). Then, the tension cracks extends to

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the deep and the rear part (Fig. 10(c) and (d)), and finally the joints below the crest of the slope are also subjected to tension failure (Fig. 10(e) and (f)). During the process, the failure of cross joints gradually

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develops from the shallow part within the slope to the deep part (Fig. 10(c) to (f)). The failure of the cross joint across the toe of the slope is the most serious and is basically in the open state (Fig. 10(f)).

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Furthermore, it can be seen from Fig. 11 that the normal stresses acting on the side of a column or block is approximately a linear distribution. This suggests that our assumption that the normal side

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stresses are distributed linearly is a reasonable one. (3) Shear failure of the set of joints dipping steeply into face plays a critical role in the early stages

of slope failure. Other damages, such as tension failure of the joints dipping steeply into face, shear and tension failure of the cross joints, do not occur until shear damage factor

Ds1 exceeds 0.8 (Fig. 12).

This is because the prerequisite for toppling failure of rock slopes is interlayer slip, which will result in

the shear failure. When the

Ds1 value reaches 0.9, the total damage Dst1 approximately reaches 1.0.

Thereafter, the tension damage begins to increase associated with the reduction of shear damage (Fig. 12(a)). This indicates that a transformation of joint shear failure to tension failure is involved in the failure process. When slope instability occurs, tension failure dominates over shear failure with a damage factor

Ds1 of around 0.6. This is not surprising as many tension cracks have been initiated

and developed along the joints. For the set of cross joints, in the early stages of unloading ( R  0.6 ),

Ds 2 is greater than the tension damage factor Dt 2 . Then the Ds 2 value

basically remains unchanged while the

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the shear damage factor

Ds 2 value increases nonlinearly with the increase of R .

Finally, tension failure dominates over shear failure by a ratio of approximately 2.5:1 (Fig. 12(b)).

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(4) Tension failure of intact rock indicates that flexural toppling of continuous columns has

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occurred (Fig. 13). Meanwhile, it can be found that the cross joint across the toe of the slope has been

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subjected to be a through tension failure (Fig. 10(f)). This indicates that block toppling failure of blocky columns has occurred. Thus, failure characteristics of block-toppling failure in rock slopes can

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be correctly captured using the UDEC model combined with the developed FISH functions. The result predicts a failure surface that initiates, develops and propagates along the cross joint across the toe of

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the slope (Fig. 10(f) and Fig.13). This validates the rationality of the assumption about the failure

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surface used in the theoretical analysis.

5 Comparison of theoretical and numerical results To deepen our understanding of the mechanism of block-flexure toppling failure, and highlight the

advantages and disadvantages of these two methods, we performed the limit equilibrium and numerical simulation analysis on twelve theoretical models. The calculation parameters are shown in Tables 1 and 2. Model B0 is the basic model, and the other models are obtained by changing one or two parameters

on the basis of model B0. For a better comparison with the limit equilibrium method, strength reduction method (SRM) proposed by Dawson et al. (1999) is used to calculate the factor of safety of the slope, Fs , in the numerical simulation (henceforth, UDEC-SRM). To determine the value of



in Eq. (24), the user-defined FISH function (C_RESULT)

developed by the authors (Zheng et al., 2018b) was used to obtain the distribution of side forces acting

x =0). The results show that, when x =0, the value of   x  0 

between 0.40 and 0.45 for all the twelve theoretical models. Thus, of the parameter

is

 =0.40 to 0.45. The average value

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on the lower boundary (Fig. 2,

 = 0.425 was used in the following limit equilibrium analysis.

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5.1 Influence of cross joint spacing

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We analyzed five slopes with different cross joint spacing (models B0, and S1-S4). Model S3 contains only one pseudo-continuous cross joint (Fig. 14 (c)) and Model S4 is an ideal anti-inclined

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rock slope without cross joints (Fig. 14 (d)).

The results are shown in Table 3. It can be seen that the set of cross joints has significant influence

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on the stability of rock slopes with such a structure. If the slope only contains the set of joints dipping

Fs values obtained

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steeply into face, it will be subjected to a pure flexural toppling failure and the

using the UDEC-SRM and limit equilibrium method proposed by Adhikary et al. (1997) are 1.35 and

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1.47, respectively. If the set of cross joints is added to the aforementioned slope, then its instability mode turns into block-flexure toppling and the than 25%. However, the

Fs value decreases to around 1, reducing by more

Fs value does not change substantially with the cross joint spacing. This is

not surprising as in this case blocky columns with cross joints behave as equivalent monolithic columns (Figs. 13 and 14 (a) to (c)). This phenomenon can be further verified by the good general agreement

between the safety factors obtained using the LEM-OAM and UDEC-SRM, as shown in Table 3. In addition, it can be seen that the factors of safety obtained using the LEM-SSAM are generally larger than those computed by the LEM-OAM and UDEC-SRM, and the difference increases as the joint spacing decreases. This is because forcing all blocks of a blocky column to limit equilibrium will yield a smaller stabilization force. In other words, the LEM-SSAM over-estimates the stability of the slope. The failure surfaces under different cross joint spacing are illustrated in Fig. 14. It is clear that, if

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there is a set of cross joints even only one cross joint in the slope, the failure surface basically remains

the same as the joint spacing changes (Figs. 13 and 14 (a) to (c)). In all the cases, the failure surface has been found to be perpendicular to the joints dipping steeply into face and distributed along the cross

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joint across the toe of the slope. However, the situation changes immediately if the cross joints are

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removed from the slope. In this case, the failure surface is found to orient at an angle of around 8°

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upward from the normal to the joints dipping steeply into face (Fig. 14(d)). Therefore, it can be confirmed that the cross joint across the toe of the slope plays a critical role in determining the location

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of the failure surface.

In order to see more clearly the failure development, we ran the UDEC models much more steps

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(over 5 million) after the formation of failure surfaces. In Fig. 15, we plot the collapse conditions of slopes with different cross joint spacing. It can be seen clearly that all four slope models have

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undergone toppling failure. It is also noticeable that a portion of the upper surface of each plane is exposed in a series of back facing, or obsequent scarps, such as those presented in Fig. 15(b). This is a typical feature of toppling failure, as pointed out by Wyllie et al. (2004). Further inspection of Fig. 15 suggests that, there are several differences between the block-flexure toppling mechanism identified and standard flexural toppling without cross joints. For the former type

of toppling failure, it involves not only the bending deformation of continuous columns, but also the overturning of blocky columns, especially when the slope contains multiple cross joints (Fig. 15 (a) and (b)). For the latter, only pure bending deformation of rock columns can be observed (Fig. 15(d)). Once again, it implies that the set of pseudo-continuous cross joints plays an important role in the failure mode of such slopes. Moreover, it can be seen that almost all cross joints above the one across the toe of the slope have undergone tension failure (in red). Typical edge-to-face contacts between upper and

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lower rock blocks are formed due to rotation of blocks about the fixed base and even some voids can be clearly observed. This indicates that the blocky columns would disintegrate into multiple small blocks

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after the occurrence of block-flexure toppling failure.

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5.2 Influence of joint dip angle

Four models with different dip angles of the joints dipping steeply into face are analyzed in this

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section (models B0, D1, D2, and D3), with the value increasing from 60° to 75° in an increment of 5°. The set of cross joints remain perpendicular to the set of joints dipping steeply into face. The results of

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model calculation are illustrated in Figs. 16-18.

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It can be seen that, for the four examples, typical block-flexure toppling failures have occurred and the failure surfaces are distributed along the cross joint across the toe of the slope, not changing

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with the dip angles of the joints (Figs. 16 and 17). The LEM-OAM and UDEC-SRM compare well as the

Fs value obtained using these methods differ by no more than 5%. However, the Fs obtained

using the LEM-SSAM are greater that the LEM-OAM and UDEC-SRM values (Fig. 18). This is because the LEM-SSAM assumes that toppling or sliding failure occurs in every rock block above the failure surface, which overestimates their resistance. In fact, blocky columns with cross joints tend to

behave as an equivalent continuous column instead of discrete blocks, as pointed out by Aydan et al. (1989).

5.3 Influence of the friction angle of the joints dipping steeply into face

Fig. 19 shows the

Fs value obtained using the LEM-OAM, LEM-SSAM and UDEC-SRM,

respectively, at different friction angles of the joints dipping steeply into face,

 d , (models B0, F1-F4).

LEM-SSAM are significantly larger. The value of

Fs obtained using

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It can be seen that the LEM-OAM and UDEC-SRM compare well while the

Fs increases almost linearly as  d increases.

This is because interlayer slip becomes more difficult as

d

increases and then a larger reduction

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factor is required for interlayer slip to occur (Zheng et al., 2018a). For example, the

Fs obtained

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using the UDEC-SRM at 35°, high joint friction angle, is about 1.12, which is about 1.2 times of the value at 25° of joint friction angle. Thus, when reinforcing rock slopes subjected to block-flexure

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toppling failures, increasing the shear strength of the joints dipping steeply into face will be an effective method.

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Fig. 20 shows the simulated failure surfaces at different

d

values. No significant changes have

value is varied from 25° to 35°. In all the cases, the failure surfaces are

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been observed when the

d

found to initiate, develop and propagate along the cross joint across the toe of the slope.

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6 Discussion

Due to the randomness and discontinuity of joints, pure toppling failures (block and flexural) are

rare in nature, while block-flexure toppling is the most common type (Amini et al., 2012). This kind of toppling failure involves a combination of fracture of long rock columns and rotation of short columns (Goodman and Bray, 1976). Our results show that many aspects of block-flexure toppling can be

correctly captured by the UDEC model combined with the developed FISH functions (Figs. 13 to 17). The numerical response indicates that the set of pseudo-continuous cross joints not only controls the failure mode of the slope but also has a significant influence on its stability. However, it can be seen from the simulated results that not all cross joints suffer from shear or tensile failure. The failure occurs mainly along the cross joint across the toe of the slope (Figs. 9, 10, 13, 14, 16, and 17). In other words, these blocky columns with cross joints behave as equivalent monolithic

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columns and the whole overturn around the bottom when toppling failure occurs. This is consistent

with the failure process witnessed in the laboratory and onsite (Aydan et al., 1989; Amini et al., 2017; Huang and Gu, 2017; Zhang et al., 2018), as shown in Fig. 21.

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The reason why the LEM-OAM and UDEC-SRM compare well while the factors of safety

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obtained using the LEM-SSAM are significantly larger is also due to the different assumptions about

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the instability mechanism of blocky columns. The LEM-SSAM assuming every rock block located above the failure surface undergoes tension or shear failure (i.e. forcing all blocks to limit equilibrium)

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will yield a smaller stabilization force, resulting in a larger factor of safety. Therefore, the LEM-SSAM does not seem to be a good approach for the analysis of block-flexure toppling.

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7 Conclusions

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Block-flexure is the most common toppling type in natural and artificial rock slopes. In this paper, the mechanisms of this type of failure are studied using limit equilibrium and numerical models, respectively. The limit equilibrium method consists of two models, namely, step by step analysis model (LEM-SSAM) and overall analysis model (LEM-OAM). Two FISH functions are developed in UDEC, namely, “SLOPE_EXCAVATION.FIS” and “JOINT-DAMAGE-TRACKING.FIS”. The former is used to simulate a more realistic slope excavation and minimize the influence of transients on material

failure, and the latter is used to obtain the detailed damage information of the two sets of joints during the failure process. Based on the results of theoretical analysis and numerical simulations, the following conclusions can be drawn: (1) Using the universal distinct element code (UDEC) and the developed FISH functions, block-flexure toppling failure characterized by block toppling of blocky columns and flexural toppling of continuous columns is explicitly captured. The failure surface is found to initiate, develop and

(2) The value of

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propagate along the cross joint across the toe of slope.

Fs obtained using the limit equilibrium method with overall analysis model

(LEM-OAM) is in good agreement with the results calculated by UDEC with strength reduction

Fs value obtained using the step by step analysis model (LEM-SSAM) is

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method (UDEC-SRM). The

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risky because it overestimates the resistance of blocky columns with cross joints. Assuming results are

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reliable, the LEM-OAM is more favored in slope design than UDEC-SRM, as it is convenient for parameter studies, back analysis, reliability calculations, etc.

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(3) Shear failure of the joints dipping steeply into face plays a critical role in the initiation of block-flexure toppling. As block-flexure toppling failure develops in rock slopes, shear failure

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gradually transforms into tension failure. When slope instability occurs, tension failure dominates over shear failure by a ratio of approximately 1.5:1.

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(4) When slope instability occurs, for the set of pseudo-continuous cross joints, tension failure also

dominates over shear failure by a ratio of approximately 2.5:1. The tension failure is primarily distributed on the cross joint across the toe of the slope, which implies that blocky columns with cross joints behave as equivalent monolithic columns. (5) The dip angles and friction angles of the joints dipping steeply into face have a great influence

on block-flexure topping stability. The influence of the cross joint spacing is less significant. However, the stability and failure mode of the slope depends on whether it contains one cross joint across the toe of slope. If the slope only contains a set of joints dipping steeply into face, it will be subjected to a pure flexural toppling failure. Then, if a pseudo-continuous cross joint across the toe is added to the slope, its failure mode will turns into block-flexure toppling failure and the stability is drastically reduced. It should be noted that the slope model used in this work is selected from the perspective of

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simplified analysis. The regularity needed to carry out theoretical models is not usually met in practice. In future work, therefore, theoretical models considering the random distribution of the cross joints will

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be studied.

Acknowledgments

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The research was financially supported by National Natural Science Foundation of China (Grant

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Nos. 11602284, 11472293, and 41807280), Natural Science Foundation of Hubei province, China (Grant Nos. 2017CFB508 and 2018CFB450) and the Open Research Fund of the State Key Laboratory

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of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese

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Academy of Sciences (Grant No. Z015005).

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toppling rock slopes. In: Proc Int 7th Congress of the ISRM. Tokyo. Japan, pp. 417-421

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Bobet A., 1999. Analytical solutions for toppling failure. Int. J. Rock Mech. Mining Sci. 36, 971-80.

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Bryan S.A., Tatone, Giovanni G., 2010. Rocktopple: a spreadsheet-based program for probabilistic block-toppling analysis. Comput. Geotech. 36(1), 98-114.

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Chai, B., Tang, Z., Zhang, A., Du, J., Su, H., Yi, H., 2015. An uncertainty method for probabilistic analysis of buildings impacted by rockfall in a limestone quarry in fengshan, southwestern china. Rock

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Mech. Rock Eng. 48(5), 1981-1996.

Chen, C.X., Zheng, Y., Zhang W., 2015. Report on the Stability Evaluation of Huabei Granite Mine

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Slope. Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan, China (in Chinese).

Cho, N., Martin, C.D., Sego, D.C., 2007. A clumped particle model for rock. Int. J. Rock Mech. Min. Sci. 44 (7), 997–1010. Dawson, E.M., Roth, W.H., Drescher, A., 1999. Slope stability analysis by strength reduction.

Géotechnique. 49(6), 835-840. Gao, F.Q., Stead, D., Kang, H.P., 2014. Simulation of roof shear failure in coal mine roadways using an innovative UDEC trigon approach. Comput. Geotech. 61(3), 33-41. Goodman, R.E., Bray, J.W., 1976. Toppling of rock slopes. In: Rock Engineering for Foundations & Slopes. ASCE, pp. 201–234. Gu, D., Huang, D., 2016. A complex rock topple-rock slide failure of an anaclinal rock slope in the wu

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gorge, yangtze river, china. Eng. Geol. 208, 165-180.

Jaimes, M.A., Candia, G., 2108 Toppling of rigid electric equipment during earthquakes. Eng. Struct. 168, 229-242.

-p

Hu J., Liu H.L., Li L.P., Zhou S., Zhang Q., Sun S.Q., 2019. Stability Analysis of Dangerous Rockmass

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Considering Rainfall and Seismic Activity with a Case Study in China’s Three Gorges Area. Pol. J.

lP

Environ. Stud. 28, 631-645.

Huang, D., Gu, D.M., 2017. Influence of filling-drawdown cycles of the Three Gorges reservoir on

489-506.

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deformation and failure behaviors of anaclinal rock slopes in the Wu Gorge. Geomorphology 295,

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Huang, D., Gu, D.M., Song, Y. X., Cen, D.F., Zeng, B., 2018. Towards a complete understanding of the triggering mechanism of a large reactivated landslide in the three gorges reservoir. Eng. Geol. 238,

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36-51.

Huang, D., Li, Y.Q., Song, Y.X., Xu, Q., Pei, X.J., 2019. Insights into the catastrophic Xinmo rock avalanche in Maoxian county, China: Combined effects of historical earthquakes and landslide amplification. Eng. Geol. 258, 105158. https://doi.org/10.1016/j.enggeo.2019.105158 Itasca. User manual for UDEC, Version4.1. Minneapolis: Itasca; 2004.

Li, Z., Wang, J., Li, L., Wang, L., Liang, R.Y., 2015. A case study integrating numerical simulation and GB-InSAR monitoring to analyze flexural toppling of an anti-dip slope in fushun open pit. Eng. Geol. 197, 20-32. Lian, J.J., Li, Q., Deng, X.F., Zhao, G.F., & Chen, Z.Y., 2018. A numerical study on toppling failure of a jointed rock slope by using the distinct lattice spring model. Rock Mech. Rock Eng. 51(2), 513-530.

of rock slopes. Int. J. Rock Mech. Mining Sci. 45(8), 1361-1372.

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Liu C.H., Jaksa M.B., Meyers A.G., 2008. Improved analytical solution for toppling stability analysis

Liu, C.H., Jaksa, M.B., Meyers, A.G., 2009. A transfer coefficient method for rock slope toppling. Can Geotech J. 46(1), 1-9.

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Liu, M., Liu, F.Z., Huang, R.Q., Pei, X.J., 2016. Deep-seated large-scale toppling failure in

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metamorphic rocks: a case study of the erguxi slope in southwest china. J. Mt. Sci. 13 (12), 2094-2110.

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Majdi, A., Amini, M., 2011. Analysis of geo-structural defects in flexural toppling failure. Int. J. Rock Mech. Mining Sci. 48(2), 175-186.

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Manera Bassa, C., Oyanguren, P.R., Philippon, R.G., Fernández-Pello, Lois M., 2014 The toppling of large blocks on the northeast slope of the Meirama mine. In: Alejano, L.R., Perucho.

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Mohtarami, E., Jafari, A., Amini, M., 2014 Stability analysis of slopes against combined circular– toppling failure. Int. J. Rock Mech. Mining Sci. 67(2), 43-56.

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Müller, L., 1968. New considerations on the vaiont slide. Rock Mech. Eng. Geol. 6 (1-2), 1–91 Wyllie, D.C., 1980. Toppling rock slope failures examples of analysis and stabilization. Rock Mech. 13 (2), 89-98. Wyllie, D.C., Mah, C.W., Hoek, E., Bray, J.J., 2004. Rock Slope Engineering: Civil and Mining. Scavia, C., Barla, G., Bernaudo, V., 1990. Probabilistic stability analysis of block toppling failure in

rock slopes. Int. J. Rock Mech. Mining Sci. 27(6), 465-478. Sagaseta, C., Sanchez, J.M. Canizal, J., 2001. A general analytical solution for the required anchor force in rock slopes with toppling failure. Int. J. Rock Mech. Mining Sci. 38(3), 421-435. Yagoda, B.G., Hatzor, Y.H., 2013. A new failure mode chart for toppling and sliding with consideration of earthquake inertia force. Int. J. Rock Mech. Mining Sci. 64(6), 122-131.

element method. Acta Geotech. 14(2), 403-416.

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Yan C.Z., Jiao Y.Y., Yang S., 2019. A 2D coupled hydro-thermal model for the combined finite-discrete

Zhang, G., Zhao, Y., Peng, X., 2010. Simulation of toppling failure of rock slope by numerical manifold method. Int. J. Comp. Meth-Sing., 07(01), 167-189.

-p

Zhang, Z.L., Wang, T., Wu, S.R., Tang, H.M., 2016. Rock toppling failure mode influenced by local

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response to earthquakes. Bull. Eng. Geol. Environ. 75, 1361-1375.

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Zhang, G.C., Wang F., Zhang H., Tang H.M., Li, X.H., Zhong, Y., 2018. New stability calculation method for rock slopes subject to flexural toppling failure. Int. J. Rock. Mech. Min. Sci. 106, 319-328.

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Zheng Y., Chen C.X., Liu T.T., Song D.R., Meng F., 2019. Stability analysis of anti-dip bedding rock slopes locally reinforced by rock bolts. Eng. Geol. 251, 228-240.

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Zheng, Y., Chen, C., Liu, T.T., Xia, K.Z., Liu, X.M., 2018a. Stability analysis of rock slopes against sliding or flexural-toppling failure. Bull. Eng. Geol. Environ. 77, 1383-1403.

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Zheng, Y., Chen, C.X., Liu, T.T., Zhang, H.N., Xia, K.Z., Liu, F., 2018b. Study on the mechanisms of flexural toppling failure in anti-inclined rock slopes using numerical and limit equilibrium models. Eng. Geol. 237, 116-128.

Tension cracks

X

0 c

b

Blocky columns

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Continuous column

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The failure surface

d

H Y

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Fig.1 Schematic diagram of block-flexure toppling failure modified from Wyllie et al. (2004).

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Fig.2 Theoretical slope model, where block-flexure toppling failures are likely to occur.

(a)

(b)

Fig.3 Mechanical analysis of continuous columns: (a) the column has the potential for flexural toppling

ro of

failure, (b) the column has the potential for shearing failure.

Ni , j 1

 i, j 

Pi 1, j

i -1, j 1hi , j

Qi , j

Pi , j

i , j hi , j

Si , j 1 Qi , j Pi , j

 i, j 

Pi 1, j

Qi 1, j

Qi 1, j

Si , j

ei , j

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Si , j wi , j

Ni , j

ur

wi , j

(a)

Ni , j 1

Si , j 1

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ei , j 1

lP

re

-p

 i, j 

Ni , j

(b)

Fig.4 Mechanical analysis of blocky columns using the step by step analysis model (LEM-SSAM): (a) the

block has the potential for block toppling failure, (b) the block has the potential for sliding failure.

ro of -p

(b)

re

(a)

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Fig.5 Mechanical analysis of blocky columns using the overall analysis model (LEM-OAM): (a) the

blocky column has the potential for block toppling failure, (b) the blocky column has the potential for

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sliding failure.

…

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…

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Fig.6 Illustration of the point where the total inter-column normal force Pi acts on a column for

various distributions of normal side stresses: (a)

i =0 ,

(b) 0 

i  1 / 3 ,

(c)

i =1 / 3 ,

(d)

1 / 3  i  1 / 2 , (e) 1 / 2  i  2 / 3 , (f) i =2 / 3 , and (g) 2 / 3  i  1 , (h) i =1 .

ro of

Function F(xi)

(a) 1

F(xi)= Linear Function

0

-p

x coordinate

na

lP

F(xi)= Quarter-Sine Function

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Function F(xi)

(b) 1

0

x coordinate

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Function F(xi)

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(c) 1

F(xi)= Negative Exponential Function

0 x coordinate

Fig.7 Typical functions for determination of the acting point of normal side forces in the analysis of

TLE : .

(*10^2)

toppling failure: (a) a linear function, (b) a quarter-sine function, and (c) a negative exponential

C (Version 6.00)

function.

LEGEND

2018 9:48:51 6840 2.683E-01 sec

1.200

54.56

145.44 0.800

40

The cut slope ° 70 ° 20 70

°

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40

0.400

0.000

200

-0.400

-p

Fig.8 Model geometry and boundary conditions of the slope model using UDEC (Length unit: m).

0.600

1.400

1.800

Ⅱ (R=0.2)

ur

na

Ⅰ(R=0.1)

1.000 (*10^2)

lP

0.200

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nsulting Group, Inc. is, Minnesota USA

Ⅳ (R=0.6)

Ⅴ (R=0.8)

Ⅵ (R=1.0)

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Ⅲ (R=0.4)

Fig.9 Development of shear failure in the joints during the process of block-flexure toppling failure.

b (R=0.2)

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a (R=0.1)

d (R=0.6)

-p

c (R=0.4)

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e (R=0.8)

f (R=1.0)

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Fig.10 Development of tension failure in the joints during the process of block-flexure toppling failure.

Joint 32

Joint 24

Joint 15

Joint 8

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Joint 1

Fig.11 Plots showing the distribution of normal side stresses along joints 1, 2, 15, 24, and 32

corresponding to Fig 10(f) (i.e. R=1.0).

1.0 Total damage III 0.8

IV

Damage factor

II

f V

0.6

0.4

Tension damage

e

Shear damage

d 0.2 I

VI

c

b 0.0

0a

20

40

60

80

Calculation time ( s )

(a)

ro of

1.0

0.8

0.6 Tension damage

f

IV

0.2

III

II I b 0.0 a 0

e V

Shear damage

VI

d

c 20

40

60

Calculation time ( s )

lP

(b)

80

-p

0.4

re

Damage factor

Total damage

Fig.12 Progressive damage of the two sets of joints during block-flexure toppling failure: (a)

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progressive damage of the joints dipping steeply into face, (b) progressive damage of the cross joints.

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Refer to Figs. 8 and 9 for the stages in each plot.

The failure surface

Fig.13 Failure surface plot for the slope with the potential for block-flexure toppling failure.

The failure surface

The failure surface

(a)

(b) 8° The normal to the joints dipping into face

The failure surface

The failure surface (d)

ro of

(c)

Fig.14 Failure surface plots for the models with different cross joint spacing

s2 : (a) s2 =6 m, (b) s2

=12 m, (c) the slope contains only one pseudo-continuous cross joint across the toe of slope, and (d)

JOB TITLE : .

-p

the model is an ideal anti-inclined slope without cross joints. It should be noted that the failure

UDEC (Version 6.00)

surfaces are obtained using strength reduction method, referring to Table 3 for the JOB TITLE : .

JOB TITLE : .

UDEC (Version 6.00)

UDEC (Version 6.00)

each plot.

LEGEND

Voids UDEC (Version 6.00)

na

UDEC (Version 6.00) LEGEND

Edge-to-face contacts

1-Jul-2019 6:36:29 cycle 8698658 time = 3.429E+02 sec block plot joints with FN or SN = 0.0

LEGEND

(*10^2)

1.200

Jo 0.200

0.600

0.800

0.800

(*10^2)

(*10^2)

0.400

0.400

Obsequent scarps

1.200

9-Oct-2018 4:33:05 cycle 2391690 time = 9.484E+01 sec block plot joints with FN or SN = 0.0

1.200

0.000

0.000

(b)

ur

(a)

Itasca Consulting Group, Inc. Minneapolis, Minnesota USA

values in

lP

JOB TITLE : .

Fs

1.200

LEGEND

29-Jun-2019 14:05:25 29-Jun-2019 14:05:25 cycle 5791418 cycle 5791418 time = 2.386E+02 sec block plot time = 2.386E+02 sec joints with FN or SN = 0.0 block plot JOB TITLE : . joints with FN or SN = 0.0

4-Jul-2019 10:50:12 cycle 10989996 time = 4.296E+02 sec block plot joints with FN or SN = 0.0

re

(*10^2)

LEGEND

0.800

0.800

-0.400

-0.400

0.400

0.400

Itasca Consulting Group, Inc. Minneapolis, Minnesota USA

1.000 (*10^2)

1.400

1.800

0.200

0.600

1.000 (*10^2)

1.400

1.800

0.000

0.000

(c)

(d) -0.400

-0.400

Itasca Consulting Group, Inc. Minneapolis, Minnesota USA

Fig.15 The UDEC models with much more steps (over 5 million) after the formation of failure surfaces Itasca Consulting Group, Inc. Minneapolis, Minnesota USA 0.200

0.600

1.000 (*10^2)

at different cross joint spacing

1.400

s2 :

1.800

(a)

s2 =6

0.200

m, (b)

0.600

s2 =12

1.000 (*10^2)

1.400

1.800

m, (c) the slope contains only one

Itasca Consulting Group, Inc. Minneapolis, Minnesota USA 0.200

0.600

1.000 (*10^2)

1.400

pseudo-continuous cross joint across the toe of the slope, and (d) the model is an ideal anti-inclined

slope without cross joints (see Fig. 14 for the corresponding failure surfaces of each model). The red

ro of

line represents tension failure of joints.

The failure surface

The failure surface

(b)

-p

(a)

re

The failure surface

(d)

lP

(c)

The failure surface

Fig.16 Failure surface plots for the models with different dip angles of the set of joints dipping steeply

 d : (a)  d =60 °, (b)  d =65 °, (c)  d =70 °, and (d)  d =75 °. It should be noted that

na

into face

the failure surfaces are obtained using strength reduction method, referring to Fig. 18 for the JOB TITLE : .

UDEC (Version 6.00)

Fs

(*10^2)

(*10^2)

1.200

1.200

0.800

0.800

0.400

0.400

0.000

0.000

UDEC (Version 6.00)

values in each plot.

LEGEND

3-Jul-2019 15:19:10 cycle 5811786 time = 2.275E+02 sec block plot joints with FN or SN = 0.0

Jo

LEGEND 29-Jun-2019 8:31:18 cycle 4600882 time = 1.824E+02 sec block plot joints with FN or SN = 0.0

ur

JOB TITLE : .

(a)

(b) -0.400

Itasca Consulting Group, Inc. Minneapolis, Minnesota USA

-0.400

Itasca Consulting Group, Inc. Minneapolis, Minnesota USA 0.200

0.600

1.000 (*10^2)

1.400

1.800

0.200

0.600

1.000 (*10^2)

1.400

1.800

JOB TITLE : .

JOB TITLE : .

UDEC (Version 6.00)

UDEC (Version 6.00) LEGEND

LEGEND

4-Jul-2019 14:29:53 cycle 14757212 time = 5.699E+02 sec block plot joints with FN or SN = 0.0

30-Jun-2019 8:26:29 cycle 12390665 time = 4.838E+02 sec block plot joints with FN or SN = 0.0

(*10^2)

(*10^2)

1.200

1.200

0.800

0.800

0.400

0.400

0.000

0.000

(c)

(d) -0.400

-0.400

Itasca Consulting Group, Inc. Minneapolis, Minnesota USA

Fig.17 The UDEC models withItasca much more steps (over 5 million) after the formation of failure surfaces Consulting Group, Inc. Minneapolis, Minnesota USA 0.200

0.600

1.000 (*10^2)

1.400

0.200

1.800

0.600

at different dip angles of the set of joints dipping steeply into face

1.000 (*10^2)

1.400

1.800

 d : (a)  d =60 °, (b)  d =65 °, (c)

ro of

 d =70 °, and (d)  d =75 °(see Fig. 16 for the corresponding failure surfaces of each model).. The red

re

-p

line represents tension failure of joints.

1.6

23%

1.2

1.0

65

ur

60

na

Factor of safety

1.4

lP

UDEC-SSRT LEM-OAM LEM-SSAM

4%

70

75 o

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Inclination angle of the joints dipping steeply into face ( )

Fig.18 Relationship between the factor of safety and the dip angle of the set of joints dipping steeply

into face. UDEC-SRM: strength reduction method, LEM-OAM: overall analysis model, and LEM-SSAM:

step by step analysis model.

1.4 UDEC-SSRT LEM-OAM LEM-SSAM

ro of

Factor of safety

20%

1.2

1.0 5%

26

28

30

32

34

36

-p

24

o

re

Friction angle of the joints ( )

Fig.19 Relationship between the factor of safety and the friction angle of the set of joints dipping

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steeply into face. UDEC-SRM: strength reduction method, LEM-OAM: overall analysis model, and

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LEM-SSAM: step by step analysis model.

The failure surface

The failure surface

(a)

(b)

The failure surface

The failure surface (d)

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(c)

Fig.20 Failure surface plots for the models with different friction angles of the set of joints dipping steeply into face,  d : (a)

 d =25 °, (b)  d =28 °, (c)  d =32 °, and (d)  d =35 °. It should be noted

re

values in each plot.

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lP

Fs

-p

that the failure surfaces are obtained using strength reduction method, referring to Fig. 15 for the

(b)

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(c)

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-p

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(a)

(d)

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Fig.21 Photographs of toppling failures reported in literatures: (a) to (d) are modified from Aydan et al.

(1989), Amini et al. (2017), Huang and Gu (2017), and Zhang et al. (2018), respectively. It can be seen

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from these photographs that blocky columns with cross joints behave as equivalent monolithic

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columns and the whole overturn around the bottom when toppling failure occurs.

Table 1 Basic parameters of rock slopes adopted in the theoretical and numerical models Parameters

Value

H (m)

Dip angle of the cut slope,

40

 c (°)  (°)

Friction angle of the intact rock , Cohesion of the intact rock,

70

Unit weight of the intact rock ,



 t (MPa)

2.0 (01*)

(kN/m3)

Elastic modulus of the intact rock ,

2500

E (GPa)

45



0.25

Residual values for friction angle, cohesion and tension

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re

-p

1*

3.0 (01*)

c (MPa)

Tensile strength of the intact rock ,

Poisson’s ratio,

45 (451*)

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Height of the slope,

Table 2 Parameters of the two sets of joints adopted in the theoretical and numerical models Value Parameters S1

S2

S3

S4

D1

D2

D3

F1

F2

F3

F4

Dip angle of the joints dipping steeply into face,  d (°)

70

70

70

70

70

60

65

75

70

70

70

70

Spacing of the joints dipping steeply into face , s1 (m)

3

3

3

3

3

3

3

3

3

3

3

3

Friction angle of the joints dipping steeply into face ,  d (°)

30

30

30

30

30

30

30

30

25

28

32

35

Cohesion of the joints dipping steeply into face , c j d (MPa)

0

0

0

0

0

0

0

Tensile strength of the joints dipping steeply into face ,  tj d (MPa)

0

0

0

0

0

0

0

Normal stiffness of the joints dipping steeply into face, k n1 (GPa/m)

270

270

270

270

270

Shear stiffness of the joints dipping steeply into face, k s1 (GPa/m)

135

135

135

135

135

135

Dip angle of the cross joints,  b (°)

20

20

20

20

0

0

0

0

0

0

0

0

0

270

270

270

270

270

135

re

135

135

135

135

135

20

30

25

15

20

20

20

20

-p

0

lP

na

ro of

B0

270

270

9

6

12

-1*

-2*

9

9

9

9

9

9

9

Cohesion of the cross joints , c jb (MPa)

0

0

0

0

0

0

0

0

0

0

0

0

Tensile strength of the cross joints ,  tjb (MPa)

0

0

0

0

0

0

0

0

0

0

0

0

Friction angle of the cross joints , b (°)

30

30

30

30

30

30

30

30

30

30

30

30

Normal stiffness of the kn 2 cross joints, (GPa/m)

270

270

270

270

270

270

270

270

270

270

270

270

Shear stiffness of the ks 2 cross joints, (GPa/m)

135

135

135

135

135

135

135

135

135

135

135

135

Jo

ur

Spacing of the cross joints , s2 (m)

Model S3 contains only one pseudo-continuous cross joint across the toe of slope.

2*

Model S4 is an ideal anti-inclined slope without cross joints.

Jo

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na

lP

re

-p

ro of

1*

Table 3 Factors of safety for models B0 and S1-S4

s1 / s2 1*

UDEC-SSRT2*

LEM-OAM2*

LEM-SSAM2*

S1

2

0.98

1.01

1.16

B0

3

1.00

1.01

1.18

S2

4

1.02

1.01

1.09

S3

-3*

1.03

1.01

1.01

S4

-4*

1.35

1.475*

1.475*

Model

1*

Ratios of the spacing of joints dipping steeply into face to the cross joint spacing.

2*

ro of

UDEC-SSRT: strength reduction method; LEM-OAM: overall analysis model; LEM-SSAM: step by step analysis model. 3*

Model S3 contains only one pseudo-continuous cross joint across the toe of slope.

4*

Model S4 is an ideal anti-inclined slope without cross joints.

5*

Jo

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na

lP

re

-p

the factors of safety against pure flexural toppling failure, which is obtained using the limit equilibrium method proposed by Adhikary et al. (1997).