A Review on Anti-Dip Bedding Rock Slopes Subjected to Flexural Toppling

. Flexural toppling is typically observed in anti-dip bedding rock slopes with a set of steeply dipping parallel joints against the slope face. Tis failure occurs due to the bending of rock strata, similar to cantilever beams. However, the exact failure surface is usually unknown, which leads to an assumption regarding the location and shape of the failure surface that must be made in the theoretical analysis. Tis assumption serves as the basis for stability analysis but may lead to errors if it is not appropriate. Terefore, in this study, we provide a detailed and systematic review of the mechanical model, precondition, failure patterns of rock strata, primary controlling factors, morphological characteristics of slope failure surfaces, and theoretical analysis methods of slope stability. We also introduce two practical application cases to better understand the advantages, disadvantages, and application scope of these methods. Additionally, we discuss the existing issues and potential future research developments in this feld.


Introduction
Toppling refers to the rotation or bending behavior of rock strata in steeply dipping bedding slopes under self-weight and external force [1][2][3]. Tis failure is commonly observed in natural and engineered slopes, particularly in anti-dip bedding rock slopes [4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Although these slopes are typically stable in their natural state, excessive human engineering activities have triggered a large quantity of engineering geological problems associated with such slopes (see Figure 1). Anti-dip bedding rock slopes subjected to toppling failure account for approximately 33% of slope deformation and failure cases studied in recent research [18][19][20][21][22][23]. Tis failure is characterized by a wide distribution, large deformation scale, and complex mechanism, posing signifcant risks to the safety of engineering structures and endangering people's lives and property.
However, slope instabilities induced by toppling failure were rarely reported before 1976, except for a few qualitative descriptions of the phenomena [24]. Te frst recorded case of toppling failure occurred in quarry near the village Elm in Switzerland in 1881, where rock masses composed of clay slates sufered severe damage. Turtle Mountain in Canada was subjected to large-scale toppling and sliding failure in 1903. Talobra frstly described the toppling deformation of anti-dip bedding rock slopes in 1957 [25]. Ten, Muller put forward the defnition of toppling in 1968 [26]. Ashby qualitatively investigated the toppling failure mechanism through model tests in 1971 [27]. However, it was not until 1976 that Goodman and Bray [28] in geotechnical engineering proposed the frst classifcation of toppling failure, i.e., fexural toppling, block toppling, and block-fexure toppling (see Figure 2). In their work, Goodman and Bray [28] provided detailed defnitions and applicable conditions of these diferent failure patterns. Flexural toppling often occurs in thin-bedding slates, phyllites, and schists with a set of steeply dipping parallel joints against the slope face (see Figure 2(a)) [29]. Blocky toppling is prevalent in thick-bedding sedimentary rocks such as limestones, sandstones, and columnar jointed volcanic rocks if such a rock stratum develops another set of approximately orthogonal joints [29]. Geologically, the diference between the above two failure patterns lies in the development of joints in rock masses, while mechanistically, it lies in whether the rock column has bending or tensile resistance. In natural anti-dip bedding rock slopes, mostly toppling failures can be seen as a combination of blocky and fexural toppling patterns, which is generally called as block-fexural toppling (see Figure 2(c)) [30][31][32][33]. Typical rocks susceptible to blockfexure toppling are interbedding sandstone and shale, interbedding chert and shale, and thin-bedding limestone [29]. Te work by Goodman and Bray [28] marked the beginning of investigating the mechanics behind toppling failure. Subsequent research studies have made signifcant progress in understanding the mechanism of blocky toppling failure, the evolution of deformation, the main controlling factors, and triggering factors, leading to the establishment of theoretical analysis methods [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51]. In contrast, fexural toppling failure is more complicated due to the diverse and intricate failure surfaces involved [52][53][54][55]. As the failure surface is often unknown, assumptions regarding the location and shape of the failure surface need to be made during the theoretical analysis of fexural toppling. Tis assumption serves as the basis for stability analysis but may lead to errors if it is not appropriate. Terefore, a comprehensive exploration and summary of all aspects of fexural toppling are necessary to mitigate such errors.
In this study, we aim to provide a detailed and systematic review of the mechanical model, precondition, failure patterns of rock strata, primary controlling factors, morphological characteristics of slope failure surfaces, and theoretical analysis methods of slope stability. By doing so, we hope to enhance our understanding of these aspects and identify any existing issues that need to be addressed. Additionally, we will discuss potential areas for future research in this feld.

Mechanical Model of Anti-Dip Bedding Rock Slope Subjected to Flexural Toppling
Establishing a mechanical model is the initial step in studying the failure mechanism of an anti-dip bedding rock slope subjected to fexural toppling during the theoretical analysis. Goodman and Bray [28] frst put forward a "block limit equilibrium model" by applying the limit equilibrium principle, i.e., GB model (see Figure 3). Te rock strata in this model are considered as rectangular blocks with equal thickness, and they are divided into three zones, i.e., stable zone, toppling zone, and sliding zone, according to static equilibrium conditions. Although this model has been commonly used in stability analysis due to its convenience, some aspects still need to be improved, such as excessive assumptions and limited applicability [56,57]. Chen et al. [56] and Wang et al. [57] enhanced this model in the following ways: (a) the safety factors are calculated using a strength reduction method with cohesion, tensile strength, and frictional angle as reduction parameters to assess the slope stability; (b) the rational failure patterns of rock strata are determined by identifying the minimum safety factor among diferent groups consisting of diferent amounts of sliding and toppling strata; (c) the rock bridge on the base is taken into account and all sides of rock columns are supposed to be fully connected; and (d) the case that the side is not perpendicular to the bottom is also studied. In addition to these enhancements, water pressure and external forces were also considered by other scholars [37][38][39]42] within the framework developed by Goodman and Bray. Considering these additional factors further improved the accuracy and applicability of the stability analysis.
Although the revised GB model is relevant to engineering practice than before, it still does not apply to the stability analysis of numerous anti-dip bedding slopes. For example, Han and Wang [58] found that the most dangerous area of such slopes lies in the top, not the toe, and the limit equilibrium method is not suitable for the slope stability analysis. Furthermore, this model only describes the failure of the blocks on the slope face and does not account for the deformation and failure process. Last but not the least, the stability of such slopes composed of hard strata can be preliminarily analyzed while the mechanical behavior of such slopes composed of soft rock masses cannot be adequately explained.
To address these limitations, Sun and Zhang [59] established a cantilever beam model (see Figure 4) that depicts the complete progression from bending deformation to fracture failure of anti-dip bedding slopes, based on feld investigations and model test results. Subsequent studies have confrmed the validity of this model and have derived the critical conditions for fexural toppling deformation failure [60][61][62][63]. Combined with the maximum tensile stress criterion, Chen and Huang [64] obtained the stress and defection criteria measuring the bending fracture of rock strata. Jiang and Huang [65] derived the critical conditions of the cantilever beam with equal thickness subjected to elastic bucking and bending fracture.
In this cantilever beam model, the rock masses are divided into two types, i.e., the stable zone below the basal plane and the deformed zone above the basal plane. Obviously, this model is suitable for describing the bending behavior of rock masses and can be applied to anti-dip bedding slopes subjected to fexural toppling deformation.
However, feld investigations reveal that the failure of the rock strata at the toe of numerous anti-dip bedding slopes is attributable to the shear damage (see Figure 5) induced by the extrusion force exerted by the upper strata, which cannot be explained by the cantilever beam model. Terefore, diferent mechanical models should be used for simplifed analysis of rock strata in diferent areas.
Te analysis of rock strata in shear failure zone should be conducted using a landslide force model, while the rock strata in bending tension zone should be studied using a superimposed beam model and a cantilever model, as shown in Figure 6. In the shear failure zone, the primary failure mechanism is sliding along the bedding planes or joints. Terefore, a landslide force model is appropriate for analyzing the stability of the rock strata in this zone. Tis model considers the shear strength parameters, such as frictional angle and cohesion, to calculate the safety factors and evaluate the stability of the slope. On the other hand, in the bending tension zone, a number of rock strata experience bending due to the applied external forces and the weight of the overlying strata, while the others undergo bending under self-weight alone. Tese behaviors can be approximated using a superimposed beam model and a cantilever beam model. In this model, the rock strata are treated as superimposed cantilever beams subjected to bending moments. By considering the mechanical properties of the rock, such as the tensile strength and unit weight, the stability of the rock strata in the bending tension zone can be assessed.
By employing these diferent models for the shear failure and bending tension zones, a more comprehensive analysis of the slope stability can be achieved, taking into account the specifc failure mechanisms and behaviors of the rock strata in each zone.

Precondition for Flexural Toppling Failure
From a geometric perspective, the mechanical model of antidip bedding slope subjected to fexural toppling can be categorized as stable. Field investigations reveal that only the steeply dipping bedding slopes are susceptible to toppling failure, while the gently dipping bedding slopes are not. Generally, the geometric morphology of the slope determines the stress distribution, and the characteristics of the structural planes in the rock layers control the strength of rock masses. Largely, they decide the failure process and severity of slope. Sun and Zhang [59] reported that the structural layers can compromise the integrity of rock mass and weaken its stability. Te slope dip angle and the rock stratum dip angle are two crucial factors that refect the geometrical morphology of anti-dip bedding slopes. Goodman and Bray [28] found that the precondition for toppling failure lies in the intense interlayer movement. Te joint frictional angle plays a signifcant role in measuring the resistance to movement between layers. Terefore, there must be a certain relationship between the stratum dip angle, the slope dip angle, and the joint frictional angle for the occurrence of fexural toppling failure in an anti-dip bedding slope. After excavation, when an anti-dip bedding slope is formed, the maximum principal stress σ 1 is nearly parallel to slope face (see Figure 7), while the minimum principal stress σ 3 ignored in most cases is nearly perpendicular to slope face. Te second principle stress σ 2 is assumed to be equal to 0. Consequently, the sliding forces acting on bedding strata are mainly provided by σ 1 , while the sliding resistance forces are also generated by σ 1 . If sliding occurs in rock stratum i, it should satisfy the following relationship: where β is the slope dip angle, η is the rock stratum dip angle, and φ i is the joint frictional angle. Only when inequation (2) is satisfed can fexural toppling occur in a rock stratum. Tis conclusion is supported by An [21], Xie [66], and Qu and Diao [60] through case statistics, numerical simulation, and theoretical analysis. It should be noted that the cohesion of the joints is not considered in equation (1). To address this limitation, Cai [9] proposed a solution by considering a comprehensive joint frictional angle. Tis approach allows for an investigation of the starting conditions for interlayer movement. By adopting a comprehensive joint frictional angle, the analysis can better account for the cohesive forces within the joints.

Advances in Materials Science and Engineering
Tis is particularly relevant in situations where the cohesion of the joints plays a signifcant role in the stability of the slope.

Failure Patterns of Rock Strata in Anti-Dip Bedding Rock Slopes Subjected to Flexural Toppling
Once anti-dip bedding rock slopes reach the precondition for fexural toppling failure, this failure can occur due to the combined efects of self-weight and external forces. Goodman and Bray [28] believed that this failure is due to the exaggerated tensile stress exceeding the strength of rock stratum. Aydan and Kawamoto [63] confrmed this hypothesis through their base friction tests and derived the iterative formula for interlayer thrusts. Adhikary and Dyshin [61,62] further refned this research based on centrifugal tests. However, feld investigations reveal that the failure of the rock strata at the toe of numerous anti-dip bedding slopes is more likely to be subjected to the shear damage [18,67,68]. Qu and Diao [60], Zheng et al. [69][70][71], Qu et al. [72], and Su et al. [73] validated this conclusion through theoretical analysis. Several strata with a small slenderness ratio are unlikely to undergo bending tension failure as they have a strong resistance to toppling. Instead, they are prone to shear failure under self-weight and external forces. Cai et al. [68] concluded that the toppling failure of anti-dip bedding slopes is the result of bending fracture and compression shear. Qu and Diao et al. [60] and Alejano et al. [74], respectively, simulated the bending and shear deformation using UDEC and FEM. Toppling failures in natural rock slopes involve not only the opening and sliding of existing structures but also the tensile and shear failures of rock bridge (see Figure 8) [75][76][77]. More importantly, the deformation and failure of the whole model slope exhibit clear zonal characteristics, i.e., shear deformation occurring at the toe and tensile deformation occurring along the shallow surface of the slope, especially at the edges [78][79][80][81]. Tis phenomenon can be explained through the following analysis: due to the limited deformation space, the deformation of rock masses gradually decreases from the top to the toe of the slope. As a result, the slenderness ratio of deformed rock mass decreases while the fracture resistance increases. Consequently, these rock masses are more likely to undergo shear failure rather than bending fracture failure.
In summary, the failure patterns of rock strata in anti-dip bedding rock slopes subjected to fexural toppling involve bending tension and shear sliding. Rock strata with a small slenderness ratio at the toe of the slope are likely to undergo shear sliding failure, while the rock strata located in the middle and upper portions of the slope are subjected to bending tension failure. demonstrate that there are two kinds of failure surfaces in antidip bedding rock slopes subjected to fexural toppling, i.e., linear-type surface and bilinear-type surface. In general, the anti-dip bedding slopes consisting of soft rock strata often present linear-type failure surfaces (see Figure 9). Aydan and Kawamoto [63] were the frst to propose a linear-type failure surface emanating from the toe of the slope and perpendicular to the joints by performing base friction tests. Subsequent investigations by Adhikary et al. [62] and Adhikary and Dyshin [61] perfected Aydan and Kawamoto's investigation [63], determining that the failure surface forms at an angle of 10°( called failure angle) above the plane perpendicular to the joints. Aydan and Amini [82] found that the failure angle ranges from 0°to 15°. Zheng et al. [71] put forward an analytical method to determine the failure surface and concluded that the failure angle is approximately 13°. Su et al. [73] considered the failure surface as a linear-type plane associated with the minimum safety factor of stability. Te simulated results obtained using the UDEC Trigon approach, as verifed by Zheng et al. [70], further supported the fndings of Zheng et al. [71]. Lian et al. [83] obtained linear-type planes using the distinct lattice spring model, while Qu and Diao [60] developed an optimal method for determining linear-type failure surfaces.

Morphological Characteristics of Failure Surface of Anti-Dip Bedding Rock Slope
On the other hand, anti-dip thin-bedding slopes consisting of hard rock strata often exhibit bilinear-type failure surfaces (see Figure 10). Zuo et al. [84], through several physical model tests, initially identifed the bilinear-type surface. Lu et al. [85] suggested that the bilinear-type surface combines the plane obtained from Adhikary's method [62] with the fracture surface induced by self-weight. Liu [67] and Wu et al. [86] utilized UDEC to obtain bilinear-type failure surfaces of Yangtai landslide. Cai et al. [68] determined the total failure surface of a bedding slope by deriving the fracture depth of each rock stratum using the "reference surface" theory. Zheng et al. [70] arrived at the conclusion that the bilinear-type failure surface consists of two components: the surface of superimposed strata perpendicular to the discontinuities and multiplanar surface of cantilevered strata. Su et al. [87] discovered that the bilinear-type failure surface is made up of three constituents. Te failure surface of rock strata subjected to shear sliding failure is a linear-type plane with an angle above the plane perpendicular to the joints. Te failure surface of superimposed strata subjected to bending tension failure is parallel to the plane perpendicular to the joints. Finally, the failure surface of cantilevered strata subjected to fracture failure under self-weight alone is parallel to the plane perpendicular to the joints.
In summary, anti-dip bedding rock slopes predominantly composed of soft rock strata exhibit linear-type failure surfaces with an angle above the plane perpendicular to the joints. In contrast, anti-dip bedding rock slopes primarily composed of hard rock strata display bilinear-type failure surfaces. In the latter case, the potential failure surface of the strata undergoing shear sliding failure is a linear-type surface with an angle above the plane perpendicular to the joints. Te potential failure surface of a single stratum undergoing bending tension is a linear-type surface parallel to the plane perpendicular to the joints. 6 Advances in Materials Science and Engineering

Primary Controlling Factors Influencing the Stability of Anti-Dip Bedding Rock Slope
Many scholars have investigated the factors that can infuence the stability of anti-dip bedding slopes subjected to fexural toppling. Han and Wang [58] summarized these factors as the initial stress feld, initial slope shape, depth and angle of slope excavation, mechanical parameters of rock strata and joints, structural face interval, and groundwater fuctuation after excavation. Among these factors, the mechanical parameters of rock strata and joints, along with structural face interval, are considered to be the most signifcant. Tis conclusion was supported by Chang et al. [88], who used FEM to confrm the fndings of Han and Wang [58]. Lian et al. [83] found that the failure patterns and deformation of such slopes are signifcantly infuenced by the joint frictional angle and the joint dip angle. Zuo et al. [84] discovered that the strength of discontinuities and the thickness of rock strata have signifcant efects on the stability of anti-dip bedding rock slopes. Lu et al. [89], Zhang et al. [90], and Cheng et al. [91] further expanded on Zuo et al.'s conclusions [84]. Tey found that the deformation time increases as the rock strata dip angle decreases, while the fnal deformation of the slope remains unchanged. Furthermore, they observed that the thinner the back edge of the slope is, the larger the toppling deformation caused by excavation is. Tey also noted that the greater the slope strength, the stronger the integrity of the slope and the more the energy stored by excavation. Additionally, the height of the slope is also a key factor in toppling deformation [9]. Liu [67], Ning et al. [92], Wang et al. [93], Weng et al. [94], and Bowa and Xia [95] have all demonstrated that the slope dip angle and strata dip angle have signifcant efects on the stability of anti-dip rock slopes. Specifcally, deformation worsens as the slope dip angle and strata dip angle increase, particularly when the strata dip angle is between 50°and 70°and the slope dip angle is around 65°. Zheng and Tang [96] as well as Zheng et al. [97] further explored the relationship between the slope dip angle and the depth of toppling deformation. Miao et al. [98], through a stability analysis of the Jiaxi landslide along the Yalong River, identifed heavy rain, valley trenching, excavation conditions, and the development of joints as the main factors infuencing toppling deformation. Zheng et al. [99] frmly believed that lithology (relatively weak rock mass), structure (appropriate thickness and dip angle of rock stratum), and external conditions (valley trenching or slope toe excavation) are the key factors contributing to toppling deformation.
In summary, the rock strata dip angle, the slope dip angle, the joint frictional angle, the joint dip angle, and the external conditions have signifcant efects in determining the fnal failure state of the slope. On the other hand, the thickness of the rock strata, the tensile strength of the intact rock, the joint frictional angle, the joint dip angle, and the weathering degree of rock play signifcant roles on the deformation process.

Theoretical Analysis Methods for Flexural Toppling Stability of Anti-Dip Bedding Slope
Due to the convenience in calculation and application, theoretical methods are widely employed in the stability analysis for anti-dip bedding rock slopes subjected to fexural toppling. Typical methods are Aydan and Kawamoto's limit equilibrium method [63], Majdi and Amini's deformation coordination method [100], and the minimum thrust force method based on Pan Jiazheng's principle [72].

Aydan-Kawamoto Limit Equilibrium Method.
Te fact that the tensile strength is smaller than the maximum tensile stress can be seen as the failure mechanism of anti-dip bedding rock slopes subjected to fexural toppling. Aydan and Kawamoto [63] put forward a limit equilibrium method to study this failure mechanism. Tis method followed four assumptions: (1) each column subjected to self-weight and interlayer force shown in Figure 11 is regarded as a cantilever; (2) every rock stratum is at the limit equilibrium state prior to failure; (3) the tensile stress exerting on each rock stratum along the failure surface is equal to the tensile stress σ t ; and (4) the failure surface is considered to be perpendicular to the joints. Based on the above assumptions, the distribution of axial stress σ x can be written as the following equation: where M is the bending moment, I is the second moment of area, N is the axial force, A is the cross-sectional area, and τ j and τ j−1 are the frictional forces and satisfy Coulomb's friction criterion.
where μ = tan φ j and φ j is the joint frictional angle. According to equations (3) and (4) and the relationship between the forces depicted in Figure 8, the following equation can be obtained: where I j = b 3 j /12, W j = cb j h j , N j = W j sinα, S j = W j cosα, and h j = 0.5 (h j + h j−1 ). F s is the safety factor of the stability, α is the rock stratum dip angle, b j is the thickness of the rock stratum, c is the unit weight, and θ is the angle between the failure surface and the normal vector of the rock stratum. χ ∈ (0, 1) indicates the position of application of the interlayer normal force. Te residual force at the toe of the slope, P 0 , can be calculated using a step-by-step method. If 8 Advances in Materials Science and Engineering P 0 > 0, the slope is unstable; if P 0 = 0, the slope is at the limit equilibrium state; if P 0 < 0, the slope is stable. When Aydan and Kawamoto's method [63] is employed to assess the stability of an anti-dip bedding rock slope, the position of the failure surface must be assumed in advance. Aydan and Kawamoto [63] found that the failure surface is a linear-type plane perpendicular to the rock stratum. Adhikary et al. [62] later discovered that the angle between the failure plane and the plane perpendicular to the joints is approximately 10°, while Adhikary and Dyskin [61] suggested that this angle ranges from 0°to 15°. In fact, this angle is associated with the stress state of a slope, which should be determined by searching the limit equilibrium state of slope.
In general, the methods based on Aydan and Kawamoto's theoretical framework are suitable for searching the failure surface of anti-dip bedding soft slopes and providing preliminary assessments of slope stability. Tese methods ofer valuable insights into the failure mechanisms and can be used as a starting point for further stability analysis and design considerations.

Majdi-Amini Deformation Compatibility Method.
In 2008, Majdi and Amini [100] put forward a theoretical analysis method to study the fexural toppling. Similar to the Aydan-Kawamoto method [63], the rock strata are supposed as cantilevers in this method. However, the interlayer forces are taken into account, and the stability conditions require both equilibrium equation and deformation compatibility equation (see Figure 12). By combining these two types of equations, they proposed an analytical method. Amini et al. [101] and Majdi and Amini [102] further refned this method and established a design formula for the safety factor of the stability.
where b is the thickness of the rock stratum, σ t is the tensile strength of the rock stratum, η is the rock stratum dip angle, c is the unit weight, F s is the safety factor of the stability, and ψ is the equivalent calculated length of a column, which can be obtained from the following equations: where C is a dimensionless parameter, H is the slope height, β is the slope dip angle, θ is the angle between the failure surface and the normal vector of the rock stratum, and q is the distributed load.
It should be noted that the above method neglected the efects of joints and cracks, and that the rock stratum is considered to be a homogenous and isotropic material. Obviously, this method will overestimate the slope stability. To resolve the above problem, they introduced a parameter called joint connectivity indicating the infuence of weak structural plane. Ten, equation (6) can be revised as follows: where k = t 1 /(t 1 + t 2 ) and t 1 and t 2 are, respectively, equal to half the average length of crack and rock bridge. Although the revised method provides better predictions for stability, accurately obtaining these two parameters is challenging. At present, only engineering geological survey or structural plane mapping can be used for estimating these two parameters.
Zhao et al. [52] took the randomly distributed joints into account, and they established a fexural toppling model based on deformation compatibility. In this model, the slope is divided into a free deformation zone and a compatible deformation zone. Te safety factor of each rock stratum is calculated through bending moment, in which the minimum one represents the overall stability of the slope. Tis model ofers an innovative approach to incorporate deformation compatibility and address the challenges associated with randomly distributed joints. Tis contributes to a more accurate evaluation of slope stability, aiding in the design of efective stabilization measures and mitigating potential risks.

Minimum Trust Force Method Based on Pan Jiazheng's
Principle. In 2017, Qu et al. [72] proposed a minimum thrust force method based on Pan Jiazheng's principle to predict the failure surface and the stability of anti-dip bedding rock slopes undergoing fexural toppling. Tey considered that the potentially dangerous failure surface is the surface with the minimum resistance force. In other words, the external force causing the overall failure of the slope is the minimum one. Furthermore, the rock strata are also regarded as cantilevers. Te primary failure patterns of the rock strata are shear sliding and bending tension, which depend on their stress state.
If rock stratum i has a potential of undergoing shear sliding (see Figure 13(a)), the stress follows the Mohr-Coulomb criterion.
where τ is the shear stress, σ is the normal stress, φ is the frictional angle, and c is cohesion. Te contact between layers satisfes the Coulomb friction criterion.
where Q i is the friction force between rock strata i and i + 1, P i is the external force which is needed to cause the shear sliding failure in rock stratum i, Q i−1 is the friction force

10
Advances in Materials Science and Engineering between rock strata i and i − 1, P i−1 is the external force which is needed to cause the shear sliding failure in rock stratum i − 1, and φ i is the joint frictional angle.
By combining equations (9) and (10) and the relationship between the forces shown in Figure 13(a), equation (11) can be derived: where θ is the dip angle of the failure surface, φ is the frictional angle, w i is the weight of rock stratum i, b i is the rock thickness, and θ r is the angle between the failure surface and the plane perpendicular to the joints. If rock stratum i has a potential of undergoing the bending tension failure (see Figure 13(b)), the principle of maximum tensile stress is satisfed.
where M i is the bending moment, N i is the axial force, I is the second moment of area, and σ t is the tensile strength. Te Coulomb friction criterion is satisfed here.
where T i is the external force causing the bending tension failure in rock stratum i and T i−1 is the external force causing the bending tension failure in rock stratum i − 1. By combining equations (12) and (13) and the relationship between the forces shown in Figure 13(b), equation (14) can be obtained: where h i−1 is the contact height between rock strata i and i − 1, h i is the contact height between rock strata i and i + 1, a is the normal angle of rock stratum, and h i is the height equivalent above the failure surface. If P i > T i , it represents that, for stratum i, the external force which is needed to cause shear sliding failure is larger than that needed to cause bending tension failure. In this case, we believe that stratum i is more likely to undergo bending tension failure, and vice versa.
Te failure patterns of the slope can be determined based on the specifc failure mechanisms of rock strata involved. Additionally, the identifcation of the potentially dangerous slope failure surface is performed using the minimum principle developed by Pan Jiazheng [103]. Finally, the external force causing the overall failure of the slope, F, is calculated using a step-by-step method. If F > 0, the slope is stable; if F � 0, the slope is at the limit equilibrium state; if F < 0, the slope is unstable.
Su et al. [87] further refned this method by considering random cross joints. Qu and Diao [103] improved it by modifying the position of application of the interlayer force. Qu et al. [104,105] also investigated the efect of anchor reinforcement and seismic action on the stability of anti-dip bedding slopes. Tese refned methods are particularly suitable for evaluating the stability of hard anti-dip bedding rock slopes.

Other Methods.
In addition to the previously mentioned methods, there are other approaches proposed by experts and scholars, which can be categorized into three types based on their purposes. Tese methods aim to revise the angle between the failure surface and the plane perpendicular to the joints, determine the bilinear-type surface of bedding slopes, or analyze the total failure surface. For revising the angle of the failure surface, several techniques have been developed. Te method developed based on the minimum stability factor theory [73], the optimization method based on limit equilibrium theory [60], the calculation method for the stability factor based on maximum tensile stress principle and the limit equilibrium theory [106], and the calculation method for the stability coefcient based on the Sarma method [107] are examples of such approaches. Other investigations focus on determining the bilinear-type surface. Lu et al. [85] proposed that the bilinear-type surface combines the plane obtained from Adhikary's method [62] with the fracture surface induced by self-weight. By using UDEC, Liu [67] obtained the bilinear-type failure surface of Yangtai slope. Cai et al. [68] determined the total failure surface of a bedding slope by deriving the fracture depth of each rock stratum with the "reference surface" theory. Zheng et al. [108] concluded that the bilinear-type failure surface consists of two parts: the surface of superimposed strata perpendicular to the discontinuities and multiplanar surface of cantilevered strata. Te fnal kinds of research aim to calculate the toppling fracture depth of rock mass. Liu et al. [109] regarded the mechanical process of rock mass undergoing fexural toppling as a plane strain model. By combining it with the maximum tensile stress principle, they derived a critical formula for calculating the toppling Advances in Materials Science and Engineering 11 fracture depth of rock mass. Based on the assumption that the stress between upper and lower cantilever beams presents triangular distribution, Zhu et al. [110] obtained the equation to calculate fracture depth of rock mass. According to the timeliness of development and evolution of toppling deformation, Pang et al. [111] took a Kelvin rheological model combined with the criterion that tensile strain is zero to determine the limit depth of toppling deformation development. Cheng et al. [112] developed a "combined cantilever beam" model that can take soft and hard rocks into account. By applying the maximum tensile stress criterion and piecewise superposition method, they derived the formulas for calculating the fracture depth of hard rock strata and the defection of slope surface in combined rock strata. Zheng et al. [113] derived the rheology constitutive equation for the bending of a cantilever beam based on the results of cantilever beam rheological tests. Using this equation, they obtained the limit depth of the development of toppling deformation. Tese methods proposed by researchers provide alternative ways to analyze the failure behavior of anti-dip bedding rock slopes and ofer valuable insights into predicting failure surfaces and slope stability. By further exploring and evaluating these methods, engineers and geologists can make more accurate assessments and design appropriate measures [114,115] to mitigate the risks associated with fexural toppling failure.

Practical Application Cases
A slate slope in South Anhui in China [67] shown in Figure 14(a) and a rock slope facing the Galandrood Road in Iran [116] shown in Figure 14(b) are taken as practical application cases to demonstrate the advantages and disadvantages of the above methods. Te total failure surfaces of the two slopes are depicted by black lines shown in Figure 14. Table 1 shows the calculation parameters of these slopes [67,116].
Te results presented in Table 2 confrm that the South Anhui slate slope underwent toppling failure while the Galandrood Road slope remained stable, in line with feld investigations. Te failure surfaces of South Anhui slate slope obtained through various methods are depicted in Figure 15, in which the failure surfaces obtained by Qu et al. [72], Su et al. [87], and Qu and Diao [103] are in line with the practical failure surface found in feld investigations. Te stability factors of the Galandrood Road slope obtained with diferent methods shown in Figure 16 range from 1.38 to 2.67, which are in line with the values reported by Majdi and Amini [102]. As a result, these methods can be applied to assess the stability of anti-dip bedding rock slopes subjected to fexural toppling. However, it should be noted that diferent methods will yield varying levels of prediction accuracy.
For the unstable slopes susceptible to toppling, such as the South Anhui slate slope, the overall failure usually occurs   [67,116].

Aydan-Kawamoto
[63] Majdi-Amini [102] Amini et al. [117] Qu et al. [72] Su et al. [87] Qu and Diao [103]  Advances in Materials Science and Engineering before the damage extends to the plane perpendicular to the joints. Consequently, Aydan and Kawamoto's method [63] tends to signifcantly overestimate the stability of such slopes. Majdi and Amini [102] neglected the efects of the shear strength parameters (cohesion and frictional angle of rock strata and joints) on the slope stability, making their method prone to yielding inaccurate stability factors for unstable slopes. Lu et al. [85] underestimated the failure area of the slope, while Zheng et al. [69] overestimated it. Qu et al. [72] did not account for the efects of the cohesion of the joints on the slope stability, leading to an overestimation of the slope stability. Su et al. [87] overestimated the position of application of the interlayer force, resulting in a smaller stability factor than the reasonable estimation. Qu and Diao's method [103] can accurately predict the failure surface and stability of unstable slopes. For the stable slopes, like the Galandrood Road slope, the aforementioned methods can yield reasonable results.

Conclusion and Outlook
In summary, over the past few decades, numerous experts and scholars have extensively studied anti-dip bedding rock slopes subjected to fexural toppling failure, leading to a clear understanding of the failure mechanism. Te theoretical analysis methods, as well as numerical modeling and feld monitoring, can be used to evaluate the stability of such slopes. However, it should be noted that the following conclusions need to be considered in any case: (1) Only when inequation (2) is satisfed can fexural toppling occur in a rock stratum. (2) Rock properties can signifcantly infuence the failure surface morphologies. Anti-dip bedding rock slopes composed of soft rock strata typically exhibit a linear-type failure plane, whereas such slopes consisting of hard rock strata present a bilinear-type failure surface. A searching method should be employed to locate the slope failure surface. (3) Bending tension and shear sliding failure are the primary failure patterns of rock strata. Rock strata with a small slenderness ratio are more prone to shear sliding failure, while the middle and upper rock strata are more likely to undergo bending tension failure. Te shear sliding failure plane is a linear-type plane above the plane perpendicular to the joints, while the bending tension failure plane is parallel to the plane perpendicular to the joints. (4) Diferent mechanical models should be employed to analyze rock strata in diferent areas. Te landslide force model should be used to analyze the rock strata in the shear failure zone, while the superimposed cantilever beam model is more suitable for studying the rock strata in the bending tension zone.
By considering these aspects, the analysis of anti-dip bedding rock slopes subjected to fexural toppling failure can be more comprehensive and accurate. In addition, there are still many problems in the current theoretical studies on the fexural toppling deformation for rock slope, which need to be overcome gradually in the further research.
(1) Te existing theoretical analysis methods only involve penetrating discontinuities and neglect the infuence of microcracks in the rock strata. In fact, the tensile strength is greatly infuenced by microcracks in the original rock. Incorporating these factors into theoretical analysis is challenging but necessary to improve accuracy. (2) Most of the existing studies are based on the natural state of the slope, and few of them involve the efect of external loads. However, external loads such as rainfall, reservoir water fuctuation, excavation unloading, blasting, and earthquake often aggravate the toppling deformation of rock strata and even directly lead to slope instability. Understanding and quantifying the infuence of these external loads on toppling deformation and slope instability are crucial for practical engineering applications. (3) Te theoretical analysis method is simple and convenient but only suitable for simple toppling mechanical model. For example, the limit equilibrium method, which is most commonly used at present, is only applicable in small shallow hard slope, not in large deep toppling deformation. How to establish a theoretical analysis method for large and complex toppling failure deformation to evaluate its stability quickly and accurately, providing reasonable suggestions for practical engineering, is a pressing engineering geological challenge. (4) Single continuous or discontinuous numerical analysis methods are difcult to accurately model the entire process of crack initiation, propagation, block sliding, collision, and movement accumulation in layered rock masses. Additionally, the selection and calibration of simulation parameters is a complex process. It is challenging and of great practical value to develop advanced coupled continuousdiscontinuous analysis methods and determine reasonable calculation parameters to accurately simulate the deformation and failure process of layered rock slopes. (5) Te mechanical model is too simple. Existing studies on toppling failure models are mainly focused on the slopes composed of single lithological rocks. For example, hard rock slopes are considered to be subjected to block toppling, while soft rock slopes are susceptible to fexural toppling. However, the rock mass encountered in practical engineering often has complex internal structure and physical and mechanical properties, and its deformation and failure patterns can present combinations of diferent failure patterns. Terefore, future research should focus on studying the failure patterns of toppling slopes under complex geological conditions, particularly in soft and hard interbedding structures. Understanding the behavior of these slopes will provide valuable insights for engineering practice. (6) Many existing research conclusions are based on 2D slope models, but in actual situations, topping phenomena often occur in ridge-shaped hills with three free faces, and the deformation and sliding direction of rock masses are not entirely consistent with the profle direction. Obviously, 2D slope models cannot consider the infuence of threedimensional terrain features and the distribution of structural planes parallel to the profle direction on the development process of toppling rock mass.
In future research, it would be benefcial to study anti-dip bedding rock slopes by establishing true 3D models to ensure that the research conditions are closer to the real feld situations.
By addressing these issues and expanding our understanding, we can improve the accuracy and reliability of slope stability analysis and contribute to the development of more efective mitigation measures for fexural toppling in anti-dip bedding rock slopes.

Data Availability
All data and materials generated or analyzed in this study are included within the article.

Conflicts of Interest
Te authors declare that they have no conficts of interest.