In this article, we will mainly use the two-dimensional slope stability analysis software called RocScience (Slide module version 6.020). Slide is a 2D software that consists of performing a slope stability analysis using the limit equilibrium method ( Harabinova & Panulinova, 2017 ), for non-circular and circular surfaces in soils and rocks. Deterministic or probabilistic methods are taken into account by Slide.

Table 5 shows the data provided to the Slide software and the results obtained by the Slide software to perform a stability analysis.

Along with other software, similar examples have been used in the soil and rock engineering literature to illustrate the above-mentioned approaches to rock slope design ( Nilsen, 2000 ; Bedi & Harisson, 2013 ; Nomikos & Sofianos, 2014 ; Abderrahmane & Abdelmadjid, 2014a ; 2014b ; Nilsen, 2017 ; Lahmili et al., 2018 ; Mathe & Ferentinou, 2021 ; Harish Kumar et al., 2020 ; Mezaini et al., 2021 ; Shannon, 2021 ).

This study will use a selected profile on the wall of a flank in the case of an open-pit mine, to illustrate the contribution of RocSicence (Slide module version 6.020), the deterministic approach (limit equilibrium method) and the probabilistic approach through a simulation example. The behavior of the soil was studied with the Mohr Coulomb model which is a criterion accepted in most geotechnical models. It is expressed by the following equation:

$\tau =C^{\prime }+\left(\sigma -\mathcal{U}\right)\tan {\phi }^{\prime }$ Équation (1)

With, 𝜏: Shear stress; $C^{\prime }$: Effective cohesion; 𝜎: Total stress; 𝒰: Interstitial pressure; ${\phi }^{\prime }$ Effective angle of internal friction.

For stability analysis, the choice can be made between limit equilibrium methods, discretization methods and probabilistic methods ( Samir, 2008 ; Harabinova & Panulinova, 2017 ). Melouka (2003) gave more details on these methods. Limit equilibrium methods can be quick and effective, in some cases, using software, charts and tables from Taylor, Bishop, Morgenstern, Spencer, etc. Table 6 shows the different stability assessment methods.

In this paper, we will mainly use the deterministic limit equilibrium method (Bishop Simplified, Janbu Simplified, Spencer and GLE/Morgenstern-Price) to make this sensitivity analysis of the geometric parameters of the bench and the mechanical parameters of the layers in order to determine the most favorable conditions for stability, whose factor of safety is greater than or equal to 1.50. The conditions corresponding to the geometric and mechanical parameters at the sliding moment have a direct influence on the factor of safety in static and pseudo-static conditions ( Fredj et al., 2017 ). In our case, all analyses were made using non-circular surfaces with the Automatic Search technique of the failure surface “Auto Refine Search”.

In our simulation example, we initially considered the configuration of a stack of five benches of different thicknesses and natures. The bedrock is a granite ranging from fine to coarse grains intruded by pegmatites topped by limestones and sandstones that constitute the rocks of the mining wall. For the case studied, the slope design of these five benches is characterized by benches 20 m of height and 5 m of width, with a slope angle of 76˚. The total height of the excavated area is 80 m with an overall edge angle of 66˚. The profile below shows the lithological detail of the rock mass, limited with a window of 125 meters and 315 meters long ( Figure 1 ). Table 7 shows the physical-mechanical characteristics of the different layers in the case study.

For each model tested, the slopes stability was verified under static and pseudo-static conditions relating to felling work. In our case, we took into account horizontal and vertical seismicity, the values of which are respectively 0.5 for horizontal acceleration and 0.25 for vertical acceleration.

Table 8 shows the results analysis carried out before the excavation work. According to these results, the excavation area did not have any stability problems before because all the factors of safety are greater than 1.5. This indicates a long-term stability state of the mine slopes and edges in both static and pseudo-static conditions. It is found that the smallest factor of safety is provided by the Janbu Simplified method ( Harabinova & Panulinova, 2017 ; Aryanti et al., 2018 ).

Table 9 and Table 10 show the results horizontal displacements on the factor of safety. The variation in geometric parameters concerned the slope angle (α), the bench height (HG) and the overall edge angle (β) as well as horizontal displacements.

According to the results obtained in Table 9, the most stable model under different conditions is the last model, whose bench height is 10 m with a slope angle of 63˚ and an overall edge angle of 43˚. According to Read and Stacey (2009) , it is observed that with this model, all factors of safety are greater than 1.5 in static conditions, which shows a state of long-term stability of the mine slopes and edges ( Harabinova & Panulinova, 2017 ). Table 9 shows that under these conditions, pseudo-static stability is maintained because all safety factors are greater than 1. The Janbu Simplified method always gives the smallest factor of safety based on the results with different models and under different conditions.

Figure 2 shows that with the different numerical calculation methods a horizontal displacement on the X axis does not cause instability of the slopes and edges of the mine because all the factors of safety obtained are greater than 1. From X equal to 47 m to 149 m, there is a slight decrease in the factor of safety. Table 10 and Figure 2 show that from X equal to 164 m, the factor of safety increases before falling to X equal to 180 m and then resuming its increase at X equal to 188 m.

The influence of mechanical parameters (C and φ) on the factor of safety was

studied first by varying the internal friction angle of the grains while fixing the value of cohesion and vice versa. Each layer was analyzed independently of the other and at the end, we made coupling of variation of cohesion and the internal friction angle of the grains. The results obtained are indicated in the tables and diagrams below.

According to the results indicated in Table 11 and Table 12 , the variation of the mechanical parameters (C and φ), does not cause instability in the sandstone layer under its own weight (static conditions). Figure 3(b) shows that on the other hand, in Pseudo-static conditions an internal friction angle of the grains (φ) less than 40˚ while maintaining the value of the internal cohesion of the grains at 73 KPa causes instability of the slopes and edges of the mine because the calculated factors of safety are less than 1. Similarly, Figure 4(b) shows that in Pseudo-static conditions, an internal cohesion of the grains lower than 65 KPa causes instability of the slopes and edges of the mine.

Table 13 and Table 14 show that in static and Pseudo-static conditions, the factor of safety increases with increase of the mechanical parameters (C and φ), in limestone. According to Figure 5(b) , it should be noted that instabilities are inevitable below a friction angle of less than 45˚. Similarly, Figure 6(b) shows that instabilities are possible below an internal cohesion of grains lower than 120 KPa.

Table 15 and Table 16 show that in the pegmatite layer, the variation of mechanical parameters does not have enough effect on the factor of safety under static conditions. Figure 7(b) shows that under Pseudo-static conditions, this layer is stable within a range of values of the internal friction angle of the grains from 30˚ to 35˚, outside of which an increase or decrease in this parameter will cause instabilities. Similarly, Figure 8(b) shows that outside an interval of internal cohesion of grains values ranging from 250 to 300 KPa, instabilities are inevitable with an increase or decrease in the parameter.

Figure 9 and Figure 10 in the fine-grained granite layers, and Figure 11 and Figure 12 in the coarse-grained granite layers show that variation in mechanical parameters has an insignificant effect on the factor of safety under static and pseudo-static conditions. Table 17 and Table 18 show that with an internal grain friction angle of less than 40˚, instabilities are unavoidable in the fine-grained granite. These results clearly show that the stability of these last two layers is

higher than the stability of the first three layers. Table 19 , Table 20 , Figure 11 and Figure 12 show that the variation of the mechanical parameters in the coarse-grained granite layer has no effect on the factor of safety, therefore, the slopes and the edges of the mine.

The influence of coupling mechanical parameters (C and φ) on the factor of safety was studied by varying both the internal friction angle and the internal cohesion of grains in each layer. Each layer was analyzed independently of the other in static and pseudo-static conditions. Tables 21-25 and Figures 13-17 show the results obtained. According to Table 21 and Figure 13 , and Table 22 and Figure 14 , in the sandstone and limestone layers, respectively, the results of safety factor calculations show that below a certain value of mechanical parameters, instabilities are unavoidable under the influence of a coupled variation in pseudo-static conditions.

Table 23 and Figure 15 show that the pegmatite layer shows very high instability under the influence of coupled variation of mechanical parameters. Table 24 and Figure 16 that the effect of variation of mechanical parameters on the factor of safety is very limited in the fine-grained granite layer. On the other hand, Table 25 and Figure 17 show that the variation of the mechanical parameters has no effect on the factor of safety in the coarse-grained granite layer.