A two-dimensional limit equilibrium computer code for analysis of complex toppling slope failures

Akr Ardestni, Mehdi Amini,*, Kmrn Esmeili

a School of Mining Engineering, College of Engineering, University of Tehran, Tehran, Iran

b Department of Civil and Mineral Engineering, University of Toronto, Toronto, Ontario, Canada

Keywords: Rock slope Toppling failures Analytical solution Windows form application Deterministic analysis Probabilistic analysis

ABSTRACT Evaluation of blocky or layered rock slopes against toppling failures has remained of great concern for engineers in various rock mechanics projects. Several step-by-step analytical solutions have been developed for analyzing these types of slope failures. However, manual application of these analytical solutions for real case studies can be time-consuming,complicated,and in certain cases even impossible.This study will first examine existing methods for toppling failure analyses that are reviewed, modified and generalized to consider the effects of a wide range of external and dead loads on slope stability.Next,based on the generalized presented formulae, a Windows form computer code is programmed using Visual C#for analysis of common types of toppling failures.Input parameters,including slope geometry,joint sets parameters, rock and soil properties, ground water level,dynamic loads, support anchor loads as well as magnitudes and forms of external forces,are first loaded into the code.The input data are then saved and used to graphically draw the slope model. This is followed by automatic identification of the toppling failure mode and a deterministic analysis of the slope stability against this failure mode. The results are presented using a graphical approach.The developed code allows probabilistic introduction of the input parameters via probability distribution functions(PDFs)and thus a probabilistic analysis of the toppling failure modes using Monte-Carlo simulation technique.This allows calculation of the probability of slope failure. Finally, several published case studies and typical examples are analyzed with the developed code. The outcomes are compared with those of the main references to assess the performance and robustness of the developed computer code. The comparisons demonstrate good agreement between the results.

1. Introduction

Toppling failures are among the most common instabilities in natural or man-made rock slopes and often have complicated failure mechanisms. The failures are generally classified into main and secondary modes. The main toppling failures are divided into the flexural, blocky and block-flexure mode, where rock columns generally overturn under their own weights (Fig. 1). However, in secondary toppling failure modes, some external forces motivate the rock columns to become unstable.When a rock mass only has a major discontinuity set, dipping steeply into the slope face, rock columns are prone to flexural failure mode(Fig.1a).In these cases,rock columns may bend or shear and then topple downward. The presence of an additional discontinuity set, perpendicular to the main set in the rock mass,can divide the rock columns into several rock blocks. In this case, the sliding or overturning of the rock blocks may lead to a blocky failure mode(Fig.1b).In many cases,a combination of these two failure mechanisms is observed in toppling failures that are known as the block-flexure failure mode(Fig.1c). A variety of secondary toppling failure modes have been observed and investigated. Fig. 2 shows the schematic diagrams and real case studies of several types of secondary toppling failures.However, slide-toe-toppling and slide-head-toppling failures(Alejano et al., 2010) are the most common secondary toppling failure modes. Since these failures are caused by a combination of sliding failure in the continuous soil or rock mass and main toppling failure in the jointed rock mass,they can be analyzed using the existing analytical models.

Fig.1. Schematic diagrams of main toppling failure modes: (a) flexural, (b) blocky, and (c) blocky-flexural.

Different approaches have been developed for analysis of the main and secondary toppling failures. Most of these approaches stem from the conventional step-by-step approach, in which the stability of all rock columns, rock blocks and soil slices must be sequentially investigated by limit equilibrium analysis. Hence, the method is time-consuming and complicated.The outcomes of this method lead to a unique factor of safety (FoS). Some of the developed solutions allow incorporation of the inherent uncertainty of input parameters for a probabilistic assessment of the slope failure(Tatone and Grasselli,2010).The probabilistic analysis results in the probability of slope failure, which is crucial in rock slope engineering; however, this requires some time-intensive and massive calculations.

In this paper, the existing analytical methods for analyses of the toppling slope failures are first reviewed and some generalized formulations are presented for evaluation of main and secondary toppling failures. Accordingly, these formulations are changed to two computer-learning algorithms which can be used for computer coding. Since these algorithms consist of several iterative processes, they are changed to a new Windows form computer code. Geomechanical and geometrical parameters of a slope are used as input data to evaluate the slope stability against toppling failure, and the results are presented as the final outcomes. To verify the computations of the developed computer code, its outcomes are compared to the existing tools and experiments.

Fig. 2. Goodman and Bray (1976)’s classification of secondary toppling failures.

1.1. A brief literature review

After the Vajont Dam disaster in 1963, Müller (1968) pointed out the possibility of overturning of rock blocks in natural rock slopes.However,“toppling”concept was first introduced by Ashby(1971)who presented a simple criterion for analyzing the stability of a single rock block against blocky toppling failure. This was followed by various studies on toppling failures with the help of some numerical and physical modeling and through several case studies(Aydan and Kawamoto,1992;Adhikary et al.,1997;Aydan and Amini, 2009; Gu and Huang, 2016; Bowa and Xia, 2019). According to Goodman and Bray (1976), toppling failures can be classified into main (blocky, flexural, and block-flexure) and secondary types.In addition,they proposed a step-by-step analytical method for analysis of blocky toppling failure, which has so far been used by various researchers and in many computer codes(Hoek and Bray, 1977; Zanbak, 1983; Choquet and Tanon, 1985;Turner and Schuster,1996; Alejano et al., 2015). Other analytical solutions were investigated and proposed for flexural toppling failure (Aydan and Kawamoto, 1987, 1992; Aydan et al., 1988).Through a combination of Goodman and Bray (1976)’s as well as Aydan and Kawamoto (1987, 1992)’s methods, an analytical method for block-flexure slope failure was proposed(Amini et al.,2012). This method has been recently used for upgrading the Roctopple program by Rocscience. The step-by-step method also was utilized for analysis of some secondary toppling failure modes(Amini et al.,2012,2015,2017;Mohtarami et al.,2014;Amini and Ardestani, 2018). It is noted that both continuum and discontinuum numerical modeling approaches can be used for modeling complex toppling-sliding slope stability analysis(Amini and Ardestani, 2018; Sarfaraz and Amini, 2020). These models allow computing the stability of a slope against toppling-sliding failure as well as slope deformation and displacement. However,analytical and mathematical solutions have their own advantages as they are straightforward,easy to understand,need fewer input parameters, and can be easily coded.

1.2. Discussion

The step-by-step analysis of slopes against toppling failures using the developed analytical solutions requires substantial mathematical calculations. Even for simple slope models, the stability analysis is a time-consuming process. In this regard, various computer codes have been developed, including Roctopple(Rocscience, 2014), Rockpack III (Watts, 2001), and ROCKTOPPLE(Tatone and Grasselli, 2010). However, all of these codes use the traditional step-by-step Goodman and Bray (1976)’s algorithm for slope stability assessments,which is only applicable for analysis of blocky toppling failure mode. Even though block-flexure analysis algorithm recently has been added to the Roctopple, these codes cannot be utilized for evaluating of flexural and secondary toppling modes of failures. Therefore, all existing codes are only useful for assessment of a special mode of toppling failures. Furthermore, in some cases, engineers must analyze a slope against both toppling and circular sliding failures simultaneously and must consider the critical mode of failure.In these circumstances,they can either use two separate codes and analyze their outcomes simultaneously or transfer the results of one code to the next one and evaluate a hybrid failure(Alejano et al.,2010).In this context,a new computer program was developed for evaluating of all main and the most common secondary toppling failure modes as well as circular sliding.

Fig. 3. A rock slope with potential of toppling failure divided into several inclined columns (left); and A free body diagram with one of its rock columns (right).

2. Deterministic analysis of toppling-sliding failures

2.1. Modeling a jointed rock mass prone to toppling failure

Fig. 3 presents a rock slope that is prone to toppling failure and is subjected to water pressure, seismic force, anchorage loading, and external point and distributed loads. A free body diagram of rock column i can be seen in the same figure. The rock slope has a potential of toppling failure and may be overturned under blocky, flexural, or block-flexure toppling mode. According to equations of equilibrium and Mohr-Coulomb failure criterion (Appendix A), interaction forces between rock columns or rock blocks can be found from Eqs. (1)-(4) for toppling, slide, flexural, and shear modes of failure, respectively.Assuming thatfnR(interaction force at the right side of the first rock block, which has the potential of failure) is zero, the failure mode of the block could be determined based on the locations of cross joints; then all interaction forces can be calculated according to these equations using a step-by-step method. Since this process consists of several reiterating functions, it can be easily transformed into an algorithm. Fig. 4 shows the flowchart of an algorithm that is used for analysis of a jointed rock mass prone to toppling failure.

Fig. 4. Flowchart of an algorithm for analysis of a discontinuous rock slope prone to toppling failure.

All coefficients in Eqs. (1)-(4) (ξ) can be calculated from the formulae presented in Appendix A.

2.2. Modeling the continuous domain of a slope prone to secondary toppling failure

In two types of secondary toppling failures(slide-head-toppling and slide-toe-toppling),rock mass has an inclined contact line with a continuous mass (a weak/weathered rock mass or a soil mass)that has a potential of sliding failure. Hence, for analysis of the entire slope against the failures (sliding-toppling), the interaction force between continuous and discontinuous masses must be first determined. The continuous mass is basically divided into several inclined slices (Fig. 5), and based on the equations of equilibrium and Mohr-Coulomb failure criterion (see Appendix A), the magnitude and location of applied inter-slices interaction forces can be calculated by

All coefficients in Eqs.(5)and(6)（ξ）can be also calculated from the formulae presented in Appendix A.

Eq.(5)consists of four unknowns including the magnitude and inclination of inter-slice forces acting on the left and right sides of each slice. Therefore, this equation is statistically indeterminate.According to the existing boundary conditions, the applied right force to the first slice is zero.Moreover,according to Newton’s third law of motion, the magnitudes of interaction forces in which two adjacent slices applied to each other are equal. Mathematically, it means thatThis will reduce the number of unknowns to three. By identifying the angle of the inter-slice forces, the two other unknowns will be eliminated, and the equation will be solved.For this purpose,similar to Morgenstern and Price(1965)’s method, it is assumed that the distribution of the inclination of inter-slice forces follows a mathematical relation. Amini et al.(2017) suggested a semi-sinusoidal relationship for inter-slice forces. However, testing of different semi-sinusoidal relationships shows that reducing the curvature of this function could reduce the slope’s FoS. Therefore, in this study, the following linear relationship was used to determine the direction of the inter-slice forces:

Similar to the discontinuous part of the slope, analysis of the continuous domain is also an iterating process and can be transformed into an algorithm (see Fig. 6).

For analysis of a slope against main toppling failures, the first algorithm (Fig. 4) should be used to calculate the inter-column forces. Meanwhile, a combination of these two algorithms (Figs. 4 and 6) must be applied to a slope that is prone to the secondary toppling failures (toppling-sliding). According to these algorithms,the theoretical normal force that acts on the first slice or block at the toe of slope(f1R)can be found.The sign of this force(P0)can be used to evaluate the overall stability of the slope, where

(1) P0>0→Slope is unstable;

(2) P0<0→Slope is stable;

(3) P0= 0→Slope is at the point of equilibrium.

Moreover, shear parameters of discontinuities, intact rock/soil mass and the sliding surface can be adjusted in a way that the slope could reach to the point of equilibrium and then the obtained shear parameters can be compared with the real values to determine the slope’s FoS.This technique is a common practice for determination of the slope’s FoS(Goodman and Bray,1976).

Fig.5. A section of a hypothetical continuous soil mass/soft rock slope prone to sliding failure and divided into several inclined slices (left);and A free diagram of one of its slices(right).

Fig.6. Flowchart of an algorithm for analysis of continuous domain of a slope prone to sliding failure.

3. TOPPLE2 computer code

The large number of equilibrium computations makes it impossible to use the proposed solution for toppling failure analysis. A computer code is thus required to solve these complex equations in a timely manner and visualize the results in the form of charts and figures that help understanding the problem. In this study, a computer program was developed using Visual C#, which is called TOPPLE2. The program consists of more than 20,000 command lines and can be downloaded(https://topple2.jimdosite.com). The program stores the geometrical, physical, and mechanical characteristics of a slope in a SQL database. An example of an open-pit mine slope created in the program with all geometric details,external loading,and ground water conditions,is shown in Fig. 7. In this fgiure, the blue lines represent the phreatic water level.In order to create a complex slope model,the geometrical and geo-mechanical characteristics of every unit must be introduced to the program separately.

The code combines the units sequentially, creates the whole slope model,and then saves and displays it.This enables creating a complex slope model from the simpler slopes.For example, the slope shown in Fig. 7 consists of eleven different sections. The defined slope is then analyzed deterministically using the equilibrium equations. The modeling results are reported visually to the user. In addition, the program can determine the FoS of the slope using strength reduction method. The developed program can also perform probabilistic slopes stability analysis.The Monte Carlo simulation method is used to determine the distribution function of the FoS and to calculate the probability of failure for the slope.

Fig. 7. A complicated pit slope created in the TOPPLE2 code.

3.1. Defining a slope using the TOPPLE2 program

Each complex slope can be made of some single slopes with different geometrical, mechanical, and physical characteristics.Using the “Material Definition” tab, the physico-mechanical properties of different materials forming the slopes are defined. This includes cohesion, friction angle, unit weight and tensile strength of materials forming the slope. Since the physico-mechanical properties of soil and rock are usually not the exact values,TOPPLE2 allows entering these parameters as various probability distribution functions (PDFs) for a probabilistic analysis. Uniform,normal and log-normal distribution functions can be utilized by the user(see Fig.8).In this section,number of materials which can be defined to the code is unlimited.

The“Joint Definition”tab allows defining the shear properties of all discontinuity sets in the slope(Fig.9).This includes the cohesion and friction angle of each discontinuity set. Similar to “Material Definition” tab, number of defined joint sets is also unlimited.

In the“Slope Definition”tab,singular slopes are defined using pre-defined materials and discontinuity properties (Fig. 10). In order to define these singular slopes, besides selecting the mechanical properties of the slope material and the discontinuities,the geometric characteristics of the slope and joints,as well as the depth and configuration of the groundwater (water level depth from the top of the slope)are defined.To analyze a slope against a special main toppling failure (flexural, blocky, and block-flexure),a new parameter must be defined to the code in this tab by user that is known as “Column Blockiness”. When the value of this parameter is zero, all rock columns are considered to be completely continuous and the slope is analyzed against a pure flexural toppling failure by the code.For a blocky toppling failure,the column blockiness parameter must be selected as 100.In this case, the code assumes a cross joint set present in all rock columns, as well as basal plane and analyzes it against blocky toppling failure.However,if the column blockiness is defined as a special number between 0 and 100, for example 70, the code automatically assigns cross joints only to 70%of the rock columns in the slope model. Hence, this option allows stability analysis of slope against the block-flexure toppling failure. Overall, the code is able to evaluate a slope against all main toppling failures. A general guideline for defining these parameters is presented in Fig.11.

By arranging several single slopes (such as the one shown in Fig.11) back-to-back, a complex slope (such as the one shown in Fig. 7) can be formed. It must be noted that the final slope will be analyzed as a unit slope and defining of several separate singular slopes only allows users to easily create a more complex slope. In the “Model Definition” tab, the dynamic loads acceleration is determined and a model with the desired name is created using the“Create New Model” button. In the “Slope Model Sequences” section, a predetermined singular slope is selected by pressing the“Add a Sequence”button,which allows a singular slope to be added to the model (Fig.12).

Fig. 8. “Material Definition” tab in TOPPLE2 code.

Fig. 9. “Joint Definition” tab in TOPPLE2 code.

Fig.10. “Slope Definition” tab in TOPPLE2 code.

Fig.11. Defining geometrical properties of a slope.

Fig.12. Model definition tab in TOPPLE2.

In this tab, it is also possible to add the reinforcement anchor loads to the slopes.As shown in Fig.11,the anchor angle,anchorage length, activation force, anchorage yield force, and installation location of the anchors on the slope should be determined. The angles of downward installed anchors are considered to be negative. Dead loads (in the forms of point load and distributed force)and their magnitude and location can be added to the slope through the “External Forces” tab (Fig.13).

All predefined models are accessible through the list specified on the right side of the windows form as shown in Fig.14.As it can be seen in this figure,by selecting a defined slope and pressing the“Submit” button, the schematics of the slope are visualized. It is possible to view, edit, and delete data stored in the database through the “Database View” tab.

Fig.13. External forces tab in TOPPLE2.

Fig.14. “Kinematic Analysis” tab: (a) Stereonet projection of a slope with potential of toppling failure; and (b) A graphical presentation of the slope.

3.2. Kinematic and kinetic analysis of a slope against toppling failures using TOPPLE2

By entering the “Kinematic Analysis” tab, the stability of the defined blocky or layered rock slope is examined kinematically against toppling failure and the result is displayed in the form of a stereographic projection. As shown in Fig. 14, the critical zone of toppling failures and great circles referring to stable and unstable rock slopes faces are presented by the code in a stereonet projection.If the pole of the main discontinuity set of rock mass is located in this critical zone, the slope is prone to toppling failure kinematically. Otherwise, the slope can be considered to be stable against the toppling failure.

In the “Analytical Solution” tab, the stability of the slopes is determined analytically using the equations presented in section 3.TOPPLE2 first determines the dominant mode of failure.If the slope only consists of rock columns,the program automatically considers several hypothetical failure surfaces(as shown by the green dashed lines on the rock slope presented in Fig.14b).

Fig.15. Analysis of circular failure in a continuous slope: (a) Several slip surfaces; and (b) Critical slip surface.

Fig.16. Analysis of a secondary toppling failure (slide-toe-toppling): (a) Several possible failure surfaces; and (b) Critical failure surface.

Fig.17. Analysis of a secondary toppling failure (slide-head-toppling): (a) Several possible failure surfaces; and (b) Critical failure surface.

The dip directions of these surfaces are equal to the basal plane angle defined by the user. Several suggestions have been made to determine the critical basal plane angle in rock slopes prone to toppling failure(Aydan and Kawamoto,1992;Adhikary et al.,1997;Aydan and Amini,2009).Thus,this angle can be defined using the suggested theoretical methods or through physical and mathematical modeling and sensitivity analysis.Finally,the stability of all rock columns and blocks is investigated on each of these hypothetical failure surfaces with the step-by-step algorithm, as illustrated in Fig. 4.

Fig.18. The result and output of the analytical solution for the slope model with slide-toe-toppling failure mode.

Fig.19. “Probabilistic/Sensitivity Analysis” tab in TOPPLE2 (the horizontal axis shows the selected parameter value and the vertical axis presents the slope FoS).

For a slope consisting of continuous materials, the computer code examines all possible circular slip surfaces of the slope with a trial-and-error method and determines the critical circular slip surface. Fig.15a shows 150,815 black circular surfaces that are the probable slip surfaces examined by the program to find possible instabilities. Each of these surfaces is an arc of a circle with an arbitrary radius that connects the end of the desired tension crack(red and green points in Fig.15a)to the slope face(green points in Fig.15a).In Fig.15b,the critical failure surface found by the program is presented. The application point of resultant force, applied to each slice, is shown with black dots in Fig.15b.

It should be noted that,in this code, it is assumed that the slip surface passes exactly through the slope toe. According to Spencer, this assumption can only create an error of about 2% in the calculation of the FoS (Spencer, 1967). For stability of slopes prone to secondary toppling(a slope with sliding-toppling failure mode),the computer code can recognize the failure mode and can analyze the slope stability based on the corresponding analytical solution.

Figs. 16 and 17 present a slope with a potential of slide-toetoppling and slide-head-toppling failure, respectively. The stability of these slopes has been analyzed by the TOPPLE2 program,and the critical failure surface for each slope has been plotted in the figures. In a hybrid instability (slide-toe-toppling or slide-headtoppling), potential failure surfaces consist of a stepwise basal plane in toppling zone and a semi-circular shape in sliding zone.The code randomly considers numerous combinations of basal planes for toppling part and semi-circular shape for sliding part,and analyzes all of them and presents the most critical one(Figs.16 and 17). For example, the code analyzes 22,658 potential failure surfaces in a case study presented in Fig.16,and 124,127 potential failure surfaces for the case study are shown in Fig.17, in order to find the most critical failure surface for each slope. The specifications of all rock columns and soil slices of the critical failure surface can be plotted by the code for further assessment.Fig.18 details the stability analysis for the slide-toe-toppling model. For further investigation of the results, the program can store the analysis results in the format of an Excel file. Moreover, the program can generate an AutoCAD output from the slope geometry and the resulting slip surface.

3.3. Probabilistic slope stability analysis using TOPPLE2

The “Probabilistic/Sensitivity Analysis” tab provides options for a sensitivity analysis and a probabilistic slope stability analysis, as shown in Fig.19.

Fig. 20. Probabilistic analysis results produced in TOPPLE2 code.

Fig. 21. The result of the stability analysis of a toppling rock slope: (a) Using TOPPLE2 program; and (b) By Goodman and Bray (1976).

3.3.1. Sensitivity analysis

As shown in the top left panel of Fig.19,there is a special console for sensitivity analysis in this section.Using this part,it is possible to analyze the sensitivity of each pre-defined slope model, against some of the effective input parameters, including friction angle,cohesion coefficient, unit weight, slope height, horizontal and vertical seismic accelerations,thickness of the rock layers,angle of the slope face, angle of the layers, and angle of slip surface. The slope sensitivity analysis is done in a way that after determining the range of a parameter by the user, the software automatically increases the value of that parameter in up to 1000 steps and obtains the FoS for each step model using the strength reduction method.Even though the process requires numerous calculations to fulfill the thresholds, its outcomes can be useful and practical for assessing of real case studies against toppling failures.As shown in Fig.19, the variation of FoS versus the parameter value (slope face angle) is plotted.

Fig. 22. Comparison between the results of slope stability analysis by Goodman and Bray (1976) and TOPPLE2 software.

Fig. 23. A hypothetical soil slope, analyzed by Slide (Rocscience, 2016).

3.3.2. Probabilistic analysis

The program allows some of the input parameters to be defined statistically. These parameters are the friction angle,cohesive coefficient, tensile strength, unit weight, horizontal and vertical seismic accelerations, angle of the slip surface, column blockiness,and angle of the groundwater surface.As shown in the bottom left panel of Fig.19, in the probabilistic analysis console,each of these parameters can be selected arbitrarily to be incorporated in the probabilistic slope stability analysis. The TOPPLE2 uses the Monte Carlo simulation method to perform the probabilistic analysis. In this method, the slope is repeatedly analyzed with random inputs based on the range and the PDFs of the input parameters. At each step, first, according to the mean value,standard deviation, and statistical distribution defined for each parameter,a random value will be generated and used as input to determine the FoS of slope. This process is repeated so that the average calculated FoS values converge.As shown in Fig.20,when the probabilistic slope stability analysis is performed,the stability results can be statistically presented as the lowest,the highest,the mean, and the standard deviation of the FoS, the 95% and 90%confidence limits, the number of stable models and the probability of stability (POS) and the probability of failure (POF). In addition,the statistical distribution of the calculated FoS values is visualized in the form of a histogram.

Fig.24. Analytical result of a hypothetical slope with TOPPLE2 and its comparison with analytical results using different limit-equilibrium methods.

Fig. 25. Analyses of several published case studies prone to slide-toe-toppling failure by TOPPLE2 program.

4. Verification of the developed computer code

In this section, various cases are presented to demonstrate the performance and accuracy of the developed computer code for toppling slope failure. A well-known example of a toppling rock slope stability analysis is shown in Fig.21b(Wyllie and Mah,2014).This typical slope model, which was initially presented and examined by Goodman and Bray (1976), has some geometrical problems and the sizes written on the model do not exactly match.Consequently,a modified version of this slope has been created and analyzed in TOPPLE2 (see Fig. 21a). The results of these two analyses are compared in Fig.22.As can be seen,the predicted failure mechanism and the inter-block forces resulted by both analyses are similar. The slight difference in results is due to the small modification in the geometry of the two models. In addition, the equilibrium equations used in the TOPPLE2 program are much more accurate than the Goodman and Bray (1976)’s calculations. However, the results of the two methods are well matched. This comparison corroborates the proposed computer code and its functionality for rock slope analysis.

Fig. 23 presents a hypothetical slope, modeled by Slide software (Rocscience, 2016). The result of the stability analysis for this slope using TOPPLE2 is shown in Fig. 24 and the critical slip surfaces and the FoS values obtained from various analytical methods (Janbu, Bishop, Spencer, Morgenstern and Price, and ordinary)used in the Slide software are plotted on this figure and compared to the TOPPLE2 results. As can be seen, the critical slip surface and the FoS estimated by the TOPPLE2 program are in good agreement with the results of the conventional methods that was used by other researchers such as Fellenius (1927),Spencer (1967), Morgenstern and Price (1965), Bishop (1956),and Janbu et al. (1956). All of these methods use equations of limit equilibrium for analysis of circular failure. However, since a circular sliding mass is an indeterminate problem, they consider different initial assumptions to solve it. Therefore, the location and shape of critical sliding surfaces proposed by these methods are different.As can be seen,the critical sliding surface predicted by TOPPLE2 is between the well-known methods, thus it suggests that TOPPLE2 is reasonable.

Amini et al. (2017) modeled nine slopes prone to slide-toetoppling failure using physical laboratory tests. These models were developed and analyzed using TOPPLE2. The results of these models and the FoS values calculated by the program are presented in Fig.25.As the geometrical properties of all these physical models are recorded at the point of failure, their estimated FoS values are likely to be around 1.As can be seen,except for two cases(B30 and B25),the estimated FoS values of the modeled slopes are less than 1.It is recognized that the physical models were carried out using a three-dimensional (3D) tilting table; however, TOPPLE2 is a twodimensional (2D) code. The discrepancy between the outcomes of physical and computational models can be due to the shear resistance between the two sides of the physical models and the tilting box. The shear resistance can increase the stability of the physical models. However, in the 2D computational models, the effect of shear resistance is removed.Therefore,the FoS calculated by the 2D computational model is usually lower than that from the 3D physical experiments.

Fig. 26. Analysis of several published case studies prone to slide-head-toppling failure by TOPPLE2 program (Amini et al., 2018).

Recently, Amini et al. (2018) developed several physical slope models in the laboratory and presented a new method for stability analysis of slopes prone to slide-head-toppling failure. In this study, the same slopes were analyzed using the TOPPLE2 program. Fig. 26 presents a comparison between the results of the physical slope analysis and the outputs of the TOPPLE2 program,for three typical slopes.It is clear that the results of the two analyses are almost similar. Nevertheless, the algorithm to determine the critical slip surface in the TOPPLE2 program is much more efficient than the method used by Amini et al.(2018).The program is able to find out the most likely slip surfaces in these rock slopes.

5. Basic assumptions

Several assumptions were considered for the development of TOPPLE2 code. When users are utilizing the code for analysis of a slope, they should pay attention to these assumptions and their corresponding limitations.Primary analysis must approve that the slope has potential of toppling failure. The assumptions are:

(1) Dip direction of main layers (joints) must be into the slope.

(2) In dynamic analysis, the seismic loads are changed into equivalent point load static forces.

(3) All rock blocks are considered to be rigid with sharp edges.

(4) They are only evaluated against toppling and sliding failure modes.

6. Conclusions

A computer code for deterministic and probabilistic analysis of main toppling failure modes and two types of slopes prone to slide-toe-toppling and slide-head-toppling is presented. The program is robust and allows the effect of groundwater, seismic force and other external loads on the stability of slope to be considered. The computer program, TOPPLE2, was developed using Visual C#. The program identifies the possible toppling failure mechanism in a defined slope and examines its stability using appropriate limit equilibrium equations. In addition, the program allows performing sensitivity analysis and probabilistic evaluation of the rock slopes. The accuracy and performance of the developed program were examined using some published case studies. The results demonstrated a good agreement between the analytical and other numerical solutions and the results obtained in TOPPLE2. It can be concluded that the code can be used as a powerful tool for stability analysis of slopes prone to toppling failure.

Declaration of competing interest

The authors wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

List of symbols

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.jrmge.2020.04.006.