The Evolution Mechanism of the Slope Toppling Deformation for Early-Warning Analysis

: With massive engineering projects performed in high and steep mountain areas, the evidence 11 of toppling deformation, which has been an important engineering geological problem in construction, 12 has been exposed and observed in quantities. Three key issues in the early warning of toppling slopes are 13 the boundary condition, evolution mechanism, and deformation stability analysis. This paper investigates 14 an evolution mechanism for timely predicting the occurrence of toppling induced slope failure in rock 15 masses, relates boundary formation and progressive development about toppling fracture planes. By 16 describing an instantaneous toppling velocity field and identifying two possible fracture plane geometries 17 (linear and parabolic), the optimal path of toppling fracture plane is searched via critical toppling heights 18 (i.e., minimum loads) calculation using the upper bound theory of limit analysis. It is interesting to find 19 that no matter what the slope structures and mechanical parameters are, the optimal path of toppling 20 fracture plane is straight and most likely oriented perpendicular to the bedding planes. Hereby, 21 considering structural damage will enable progressive toppling deformation instead of systemic failure, 22 the toppling deformation evolution is probably taking place of a loop following the formation of the first 23 fracture plane due to exceeding slope critical height. In the loop, deformation and column inclination 24 updates due to fracture plane formation and fracture plane inclination increase to adjust the changed inclination of columns, as it may take degrees perpendicular to columns. And this progressive formation 26 of ever more inclined fractures plane is what lead to sliding collapse. Altogether we divide the toppling 27 evolution into 5 stages, and define the instability criterion for toppling deformation transform into sliding 28 collapse as the fracture plane inclination being equal to its friction angle. In addition, a PFC2D simulation 29 of the entire slope toppling process is performed to verify this speculative evolution mechanism, and a 30 satisfactory result is acquired. Finally, a deformation calculation model of toppling slopes is proposed 31 for stability analysis in accordance with the instability criterion, which is further applied in a typical 32 toppling case. The findings of this study could lay a foundation for the deformation, stability and early- 33 warning analysis of toppling slopes.


37
As a special deformation mode, toppling slopes are widely distributed in deep-cut canyon areas. 38 The toppling slope is deep-seated, large-scale and occurs over a long timeline. Unlike a landslide, slope 39 toppling does not conform to sliding deformation and does not result in a single fracture plane or an 40 obvious topographic feature. The applicability of analytical models based on the sliding hypothesis in 41 the toppling slope analysis remains to be determined (Huang et al. 2017). Current studies on the stability 42 analysis of toppling slopes are almost always calculated by a static model (Goodman and Bray, 1976;43 Aydan and Kawamoto, 1992; Adhikary et al. 1997;Zhang et al. 2007; Liu et al. 2009; Amini and Majdi, 44 2012). However, a reasonable height-width ratio is key to performing accurate calculations for a thin 45 layered toppling slope (Popov, 1976), and the rationality of using a designated failure plane is debatable. 46 The limit equilibrium method, which ignores the stress-strain relationship of rocks and the evolution 47 of geohazards, cannot be used to predict whether a toppling slope will continuously deform or transform 48 into a large-scale landslide. Regarding the evolution mechanism (Wang et al. 1981; Pritchard and Savigny, 49 1990; Adhikary et al. 1996Adhikary et al. , 2007Goricki and Goodman, 2003;Tosney et al. 2004;Stacey, 2006;Yeung 50 and Wong, 2007; Zheng et al. 2018), there is a lack of consensus. Huang (2008) proposed a 3-stage 51 evolution mode for high rock slopes in 2008 and suggested that stability evaluation is both a strength 52 problem and a deformation problem. Therefore, studying the evolution mechanism of toppling slopes 53 over their long geological history is a novel approach. 54 In the study of the evolution mechanism of toppling slopes, boundary conditions are necessary and 55 essential. Some studies have been conducted on the boundary conditions of toppling slopes, mostly 56 physical or numerical modeling studies. Aydan and Kawamoto (1992) concluded that the fracture plane 57 was perpendicular to the bedding plane from a base friction model test. According to a centrifuge model 58 test, Adhikary et al. (1997) observed that the angle between the fracture plane and the normal to the rock 59 layers was 12-20°. Furthermore, after summarizing some experimental phenomena, Chen et al. (2016) 60 noted that the fracture plane was discontinuous (Fig. 1). In addition, numerical modeling studies are often 61 performed by setting basal discontinuities or cross-joints to seek the boundary conditions (Ishida et al. 62 1987; Bovis and Stewart, 1998;Nichol et al. 2002). nonuniform toppling intensity zoning usually based on the rock quality classification can affect these 69 determinations. Therefore, the main objective of this paper is to search the optimal path of toppling 70 fracture plane and analyze the evolution mechanism as well as prewarning criterion of toppling slope; 71 the 2-dimensional particle flow code PFC2D (Itasca Consulting Group, 2017) is used to verify the study. The limit analysis theory (Chen, 1975) takes the object as an ideal rigid-plastic material to find the 81 ultimate load using upper or lower bound theory. The upper bound theory in the limit analysis derived 82 by the virtual power principle can be described as follows: the calculated failure load is greater than or 83 equal to its practical ultimate load for all kinematically induced collapse mechanisms: 84 where Ti and Fi are the surface force and body force that act on the collapse mechanism, respectively; 86 σij is the internal stress set that reacts to Ti and Fi; ui and εij are the strain sets under Ti, Fi and σij; A and V 87 are the area and volume acted on by Ti, Fi and σij. 88 Based on the upper bound theory, a theoretical method to search for the toppling fracture plane is 89 proposed, and the core assumptions in the analysis are as follows: 90 1. The rock slope is considered an ideal rigid-plastic material and obeys Mohr-Coulomb yield 91 criterion. 92 2. The analysis of the 2-dimensional slope deformation is simplified to a plane strain problem, and 93 the deformation is compatible. 94 3. No primary joint or weak plane is included in the analysis model with the exception of the bedding 95 planes, and the fracture plane initiates at the toe of the slope. 96

Analysis model 97
The analysis model for the toppling system is simplified as a series of superposed cantilever beams 98 or rock columns with equal thicknesses (see Fig. 3). The origin of the rectangular coordinates is set at 99 the toe of the slope. Slope function f(i) in an x-y rectangular coordinate system can be written as follows: 100 where H is the slope height; b is the thickness of the rock columns; α and β are the slope angle and 102 dip angle of the bedding plane, respectively; i is the number of rock columns. For the convenience of the subsequent analysis and calculation, the slope function is rotated 90-β 106 degrees counterclockwise, so that the X-axis is perpendicular to the bedding planes, and the Y-axis is 107 parallel to the bedding planes; thus, slope function f(i)' in the X-Y rectangular coordinate system is: 108 By assuming that the fracture plane function is g(i), the toppling height h(i) of each rock column 110 and its deformation area Si can be represented as follows: 111 where f(i-1)'-g(i) is the relative height or upper section height of the rock columns. 114 If the fracture plane intersects the slope surface, the total number of unstable rock columns in the 115 system can be obtained, and its implicit expression is shown as follows: 116 where n is the total number of unstable rock columns. 118

Instantaneous velocity field description 119
According to the upper bound theory of limit analysis, it is necessary to describe an instantaneous 120 and allowable velocity field for the toppling system virtual power equation. Define that the first rock 121 column freely topples with an instantaneous angular velocity ω1. The interrelationships among the rock 122 columns are considered to be similar at their tops during deformation, which implies that the tangential 123 velocities of the bedding plane vertices are identical, while the interbedded dislocations occur in the 124 radial direction and form imbricates (see Fig. 4). where ω1 and ωi are the instantaneous angular velocities. 131

External power 132
Without considering other external forces (i.e., earthquakes), the body force that acts on the slope 133 system is only due to gravity. External power Wγ of gravity in the described velocity field can be 134 described as follows: 135 where γ is the unit weight of the rock columns. 137

Internal energy consumption 138
In the process of toppling deformation, the internal energy consumption of the system is mainly due 139 to compression along the neutral planes, tension between the neutral planes and resistance caused by the 140 interbedded dislocations, as shown in If the neutral plane coefficient of the rock columns is 1, the compression zone width of each rock 145 column is b/2. Chen (1975) proposed the following internal energy consumption per unit of compressed 146 volume when bending a rock column: 147 where ̇ and ̇ are the internal energy consumption per unit of compressed volume and its 149 maximum compressive strain, respectively. Therefore, the internal energy consumption Wc of the 150 compression zone in the toppling system can be derived as follows: 151 where c and φ are the cohesion and friction angle of the rock material, respectively, and bc is the 153 compression zone width of each rock column. 154 (2) Tension zone 155 Similarly, the tension zone width of each rock column is b/2. Since the pure tensile state, internal 156 energy consumption Wt of the tension zone in the toppling system can be expressed as follows: 157 where σt is the tensile strength of the rock material; bt is the tension zone width of each rock column. 159

(3) Interbedded dislocation 160
When the total number of unstable rock columns in the system is n, the resistances caused by 161 interbedded dislocations are n-1. According to the described velocity field, 3 interbedded dislocation 162 modes can be determined: 163 , ωi＝ωi+1＝ω1, and the interbedded dislocation mode is only dislocation, as 164 shown in Fig. 6(a). The internal energy consumption Wj(a) of this mode can be considered a simple shear 165 power, see Eq. (12). 166 , ωi<ωi+1, and the interbedded dislocation mode is dislocation coupled with 167 tension, as shown in Fig. 6(b). Since the tensile strength of bedding planes is 0, no internal energy 168 consumption occurs during the tension process, but the interbedded resistances will be attenuated by 169 tension. Hence, the internal energy consumption Wj(b) of this mode can be determined with Eq. (13). 170 where cj and φj are the cohesion and friction angle of bedding planes, respectively; tan( + ) is 181 a reduction factor for the attenuated interbedded resistances; θi is the angle between tension velocity 182 and shear velocity , and its tangent value can be proposed by the improved Coulomb model (Chen, 183 1975): 184 Where the ultimate load is expressed as the critical toppling height Hcr, which is implicit in the equation. 189 Therefore, once the fracture plane function g(i) is determined, the critical toppling height can be 190 computed by eliminating the instantaneous angular velocity ωi assumed on both sides of the expression, 191 and different fracture plane geometries corresponds to different critical toppling heights. 192 Thus, when the real slope height is less than the critical toppling height, the slope has not yet 194 undergone toppling deformation. When the real slope height is equal to or even exceeds the critical 195 toppling height, the slope will undergo or has undergone toppling deformation.
where m is the fracture plane function coefficient in the X-Y rectangular coordinate system (Fig. 7). 208 When m＝0, h(i)＝f(i) '-g(i+1), and dislocation occurs only between the bedding planes ( Fig. 6(a)). When 209 , and dislocation coupled with tension occurs between the bedding planes ( where α and β are the slope angle and dip angle of the bedding plane, respectively. 218 For the optimal fracture plane search, either the slope structure or mechanical parameters should be 219 designated in advance to compute and compare the critical toppling heights by changing the straight line 220 inclination. 221 1. Consistent slope structure, changing mechanical parameters: the slope structure is fixed as a 222 Hoek and Bray (1977) classical model, where the slope angle is 56°, and the dip angle is 60°. In addition, 223 it is reasonable to divide the mechanical parameters into 2 categories (i.e., corresponding to the rock 224 mass and bedding plane) to avoid the dimensional disaster, and the designed mechanical parameters are 225 shown in Tables 1 and 2. 226   4 show that regardless of the mechanical parameters or slope structures, the straight 238 line inclination corresponding to the minimum toppling height is 0. Due to the optimal control (Berkovitz, 239 1974), it can be certain that toppling most likely occurs when the linear fracture plane is perpendicular 240 to the bedding plane; alternatively, when toppling occurs, the most favorable linear fracture plane is 241 perpendicular to the bedding plane. 242

Parabola 243
Assuming a parabolic fracture plane, the single-parameter equation for the quadratic coefficient is 244 as follows: 245 where m (m>0) is the fracture plane function coefficient in the X-Y coordinate system. 247 The optimal solution for the quadratic coefficient of the parabolic fracture plane is still performed 248 by the semigraphic method, and the calculation process is not repeated. The calculation results show that 249 the critical height is minimized when the parabolic quadratic coefficient approaches 0, i.e., infinitely near 250 the normal line of the bedding plane, as shown in Fig. 8.

Evolution process speculation 257
Toppling deformation is a dynamic evolution process with an inherent mechanism. Based on the 258 fracture plane initiation characteristics, a toppling evolution mechanism along with an instability 259 (a) Original intact slope 261 Toppling deformation requires a certain external force, so there is a critical height Hcr at which the 262 slope reaches the critical state. When the real slope height is less than this critical height, the slope will 263 not topple and will maintain its intact slope and rock mass structure characteristics, as shown in Fig. 9(a). 264 (b) Initial toppling damage 265 Due to a long-term geological history of valley cutting, the slope height will increase. When the 266 slope height reaches its critical toppling height Hcr, the slope will topple, and the formed initial fracture 267 plane is a straight line perpendicular to the bedding planes (see Fig. 9(b)). However, the gentle fracture 268 plane angle in this stage is usually not sufficiently steep to cause sliding failure, and toppling deformation 269 will slowly progress. 270

(c) Time-dependent deformation 271
Because of the toppling fracture plane must take degrees perpendicular to columns, the fracture 272 plane inclination will increase to adjust the decreasing changed inclination of columns. Thus forms a 273 loop, which corresponds to the time-dependent deformation stage, as shown in Fig. 9(c). Notably, this 274 progressive development of toppling fracture plane is not conducive to maintaining the slope stability. 275

(d) Limit equilibrium state 276
When the fracture plane continues to develop, the slope will be in a limit equilibrium state when the 277 fracture plane inclination is equal to its friction angle (see Fig. 9(d)). This condition indicates that the 278 slope is about to lose its stability and slide. 279 (e) Turning into a landslide 280 After the limit equilibrium state, the slope state will transform from toppling deformation to sliding 281 failure. At this stage, the toppling fracture plane evolves into a sliding surface, which causes the rock 282 masses to slide and forms a landslide, as shown in Fig. 9(e). The speculated evolution process indicates that the prerequisite for slope toppling is gravity load, 292 which is controlled by the slope height. Toppling deformation occurs only when the slope height reaches 293 its critical toppling height. Moreover, the instability criterion of the slope for the transition from toppling 294 deformation to sliding failure is when the inclination angle of the toppling fracture plane is equal to its 295 friction angle; when this criterion is satisfied, the formation of a large-scale landslide is inevitable. 296

Simulation verification 297
To verify the speculative toppling evolution mechanism, the entire slope toppling process is The designed simulation model generated by explosive repulsion is 110 m high and 213 m wide, 306 the effective slope height is 70 m, the slope angle is 70°, the bedding plane dip angle is 65°, and the bed 307 thickness is 2 m. A marker bed and 5 monitor particles are set to observe the entire movement process 308 (Fig. 10).  (Table 5). 314 Before the simulation, the displacement boundary is constrained, and high material parameters are 316 set to achieve the initial equilibrium under gravity (9.81 m/s 2 ). After the initial balance, the material 317 parameters in Table 5 Fig. 11 shows that the initial formed fracture plane presents a straight line almost perpendicular to 328 the bedding plane and does not correspond to an immediate instability. Meanwhile, the simulated 329 evolution process is perfectly consistent with the speculation in section 4.1, where the discontinuities 330 gradually offset to the free state until the slope reaches the limit equilibrium state. When the displacement 331 exceeds the 18-m threshold, the deformation is difficult to control. The colors of the displacement field 332 in the simulation represent multiple stages of fracture deformation. As a matter of fact, Goricki and 333 Goodman (Goricki and Goodman, 2003) found a similar evolution process in their published article. 334

335
For toppling slopes, we believe that the structural damage will result in continuous toppling 336 deformation instead of systemic failure, and the key of early-warning control lies in the deformation 337 stability analysis. According to the toppling evolution process, a deformation calculation model for the 338 prewarning value ymax is proposed, and this prewarning value is applicable from the current state to the 339 limit equilibrium state, as shown in Fig. 12. where H is the slope height, and α is the slope angle. During the actual practice, the monitoring 360 point position could be selected as the analysis object. 361 3. Prewarning value calculation: because the toppling fracture plane must take degrees 362 perpendicular to columns, the rotation angle of the top rock column from the current state to the limit 363 equilibrium state is also θcr. Based on the cosine law, its prewarning value ymax can be calculated: 364  (Table 6), 3 toppling intensities (intense 387 toppling, weak toppling and parent rock zones) are divided by 2 estimated toppling fracture planes, which 388 connect the signs of rock toppling exposed in the adits. Meanwhile, 2 connected straight fracture planes 389 have measured dips of 28.7° and 12.5° and are almost perpendicular to the bedding planes, which is 390 consistent with the above analysis conclusion. Combined with the slope ground fissure distribution, signs 391 and rates of deformation, we preliminarily determine that the slope remains stable. 392 To further analyze the slope deformation stability, the GSI method is applied to quantitatively 394 estimate the current fracture plane mechanical parameters (i.e., intense toppling boundary). The 395 estimated equivalent friction angle is 31.69°, which is also the critical sliding surface angle. If the 396 monitoring point positions are selected as the targets, the prewarning value ymax of the slope can be 397 calculated by Eqs. (20)-(22), and the slope deformation stability analysis can be determined as follows. 398 Table 7 Deformation stability analysis for the toppling slope             (a) Panoramic photo of the investigated toppling slope with rock deformation. (i) A reverse ssure due to toppling in PD01; (ii)~(iv) rock toppling developed in PD07, PD02 and PD03, with V-shaped fracture opening formed in hard rock layers or exible bending formed in soft rock layers. (b) Cross-section pro le of the upstream slope ridge in the maximum deformation direction Overview of the Xingguangsanzu slope Note: The designations employed and the presentation of the material on this map do not imply the expression of any opinion whatsoever on the part of Research Square concerning the legal status of any country, territory, city or area or of its authorities, or concerning the delimitation of its frontiers or boundaries. This map has been provided by the authors.

Figure 14
Deformation stability analysis of the slope