*2.2.3 Cuckoo search in determination of slip surface*

Grid Search is commonly used as slip surface searching method because the principle is simple and easy to understand [11]. However, this method can only calculate the circular slip surfaces, so it cannot represent the stability of slope in real condition. To make it more representative, non-circular slip surface option is also available in slope stability simulation software, and one of the searching methods in non-circular option is CS method.

After discretizing the sliding mass, internal and external forces in each column

The grid search method is used to find critical slip surface. The grid search method starts by specifying the grid box dimension. The location and dimensions of the grid box must cover the entire study area so that the search for critical slip surface can be performed optimally, for the influence of number of grid point and radius increment in determining safety factor result can be seen in **Table 1**. The result of 3D slope stability analysis using grid search can see in **Figure 11**.

can be calculated based on moment equilibrium, force equilibrium, or both

depending on what method of calculation is used (**Figure 10**).

**Figure 8.**

*Slope Engineering*

**Figure 9.**

**Figure 10.**

**86**

*3D model for slope stability analysis [11].*

*Illustration of the radius increment in the grid search [11].*

*Discretization of the sliding mass using a square grid [11].*

CS is an algorithm which is used for solving the optimization problems. CS has been used in engineering field, such as welded beam and spring design optimization but there is still a few that use it for slope stability issue. Nowadays, the necessity to analyze 3D slope stability is more essential. The reason why slope stability problem should not be assumed 2 dimensionally that should be taken into account is the importance of determining volume of failure for risk management volume of failure is counted as one of the consequences, and it can be obtained by analyzing slope stability in 3D, indeed. Furthermore, there is a relationship between failure probability and volume of failure. Another issue exists when 3D analysis is performed using Grid Search on a vast area. The use of 3D analysis on a vast area can be complicated, and in real cases more advanced optimization methods are required. Therefore, CS is tried to be applied in order to determine the slip surface with minimum SF in 3D analysis. Thus, it can be suggested to be used as an alternative or other better option in 3D analysis.

CS means a metaheuristic optimization method that was developed by [13]. This method was inspired from cuckoo's breeding behavior. In this research, CS is used as a slip surface search tool that has the lowest SF. CS is coupled with Lévy Flights random walk. There are few rules to use this algorithm as follows:


The last rule can be approached using a fraction pa to determine the worst solutions of n nests that will be replaced with a new nest randomly. In order to solve the problem, it can be simply illustrated that every egg in a nest represents one new solution. The purpose is to use the new and potentially better solution to replace the current solution in the nest. In a certain condition, the nest may have 2 eggs (2 solutions) but this problem is simplified so one nest has only 1 solution.

The random walk as determinated by Lévy Flights can be described in the following formula:

$$\mathbf{x}\_{\mathrm{i}}^{(\mathfrak{t}+1)} = \mathbf{x}\_{\mathrm{i}}^{(\mathfrak{t})} + \mathfrak{a} \oplus \mathbf{L} \mathsf{év} \mathbf{y}(\lambda) \tag{41}$$

*2.2.4 Analysis 3D limit equilibrium of open pit mine*

*Three Dimensional Slope Stability Analysis of Open Pit Mine*

*DOI: http://dx.doi.org/10.5772/intechopen.94088*

*Cuckoo search LEM 3D analysis result [11].*

**Figure 12.**

slope geometry model can be seen in **Figure 13**.

actual slip surface position.

**89**

**2.3 Finite element method 3D**

3D limit equilibrium analysis method, data regarding 3D slope geometry model, material properties and 3D geological models are required. The 3D slope geometry model required for this method can be obtain from the reconstruction of pit surface model from either terrestrial or photogrammetric measurement methods. The next step is to create an external volume from the pit surface model which will then be analyzed to determine the position and shape of the slip surface, example of 3D

Limit equilibrium analysis method uses the properties values of materials obtained from laboratory tests results to calculate the value of the safety factor. An example of the input parameter data used in the analysis can be seen in **Table 3**. Geological modeling is the process of creating visual description geometry of rock lithology into software that represents the actual conditions. However, there are limitations in the modeling process, that related to the limited information on the data held and errors in data interpretation carried out. One of the methods that can be used to create the 3D geological model is interpolation the lithology information data from geotechnical drilling results. The 3D geological model will be used in the 3-dimensional slope stability analysis to determine the distribution characteristics of the rock lithology. In this analysis, the 3D geological floor model of limonite, saprolite and bedrock lithology is used and can be seen in **Figure 14**. The results of the analysis using the Bishop Simplified method for the slip surface search method, both grid search and cuckoo search can be seen in

**Figures 15** and **16**. The analysis results with the grid search show the safety factor value is 1.104, while the cuckoo search is 1.089. The position of the slip surface with the grid search and cuckoo search is the same position it indicates the accuracy of the cuckoo search, while the grid requires a grid box which must represent the

The finite element method has been widely applied by mining geotechnical practitioners in slope stability analysis, with the advantage that the stress–strain analysis in the material allows to determine the displacement and strain values acting on the model elements, but this method has weaknesses in the process. This analysis uses a large number of calculation matrices so that it requires a long computation time, especially if the analysis is carried out in 3D, the number of

where α > 0 is the step size and Lévy(λ) is the position function from Lévy Flights (**Figure 12**).

$$\mathbf{L}\acute{e}\mathbf{v}\mathbf{y} \sim \mathbf{u} = \mathbf{t}^{-\lambda}, (\mathbf{1} < \lambda \le \mathbf{3}) \tag{42}$$

CS has been applied in many optimizations and computer intelligence with promising efficiency, this has been proved from design application in engineering field, scheduling problems, thermodynamic calculations, etc. Few examples of CS application in engineering field are designing spring, welded beam, and steel frame. The CS's performance has also been compared with some metaheuristic algorithms such as PSO and GA, and the result shows that CS has higher success rate than. The result of the influence of max columns in x or y and max iteration in determining safety factor can be seen in **Table 2**.

*Three Dimensional Slope Stability Analysis of Open Pit Mine DOI: http://dx.doi.org/10.5772/intechopen.94088*

**Figure 12.** *Cuckoo search LEM 3D analysis result [11].*

CS is an algorithm which is used for solving the optimization problems. CS has been used in engineering field, such as welded beam and spring design optimization but there is still a few that use it for slope stability issue. Nowadays, the necessity to analyze 3D slope stability is more essential. The reason why slope stability problem should not be assumed 2 dimensionally that should be taken into account is the importance of determining volume of failure for risk management volume of failure is counted as one of the consequences, and it can be obtained by analyzing slope stability in 3D, indeed. Furthermore, there is a relationship between failure probability and volume of failure. Another issue exists when 3D analysis is performed using Grid Search on a vast area. The use of 3D analysis on a vast area can be complicated, and in real cases more advanced optimization methods are required. Therefore, CS is tried to be applied in order to determine the slip surface with minimum SF in 3D analysis. Thus, it can be suggested to be used as an alternative or

CS means a metaheuristic optimization method that was developed by [13]. This method was inspired from cuckoo's breeding behavior. In this research, CS is used as a slip surface search tool that has the lowest SF. CS is coupled with Lévy Flights

• Each cuckoo lay one egg at a time, and dumps it in a randomly chosen nest;

• The best nests with high quality of eggs (solutions) will carry over to the next

• The number of available host nests is fixed, and a host can discover an alien egg with a probability pa ∈ [0, 1]. In this case, the host bird can either throw the egg away or abandon the nest so as to build a completely new nest in a new location.

The last rule can be approached using a fraction pa to determine the worst solutions of n nests that will be replaced with a new nest randomly. In order to solve the problem, it can be simply illustrated that every egg in a nest represents one new solution. The purpose is to use the new and potentially better solution to replace the current solution in the nest. In a certain condition, the nest may have 2 eggs (2 solutions) but this problem is simplified so one nest has only 1 solution.

The random walk as determinated by Lévy Flights can be described in the

where α > 0 is the step size and Lévy(λ) is the position function from Lévy

CS has been applied in many optimizations and computer intelligence with promising efficiency, this has been proved from design application in engineering field, scheduling problems, thermodynamic calculations, etc. Few examples of CS application in engineering field are designing spring, welded beam, and steel frame. The CS's performance has also been compared with some metaheuristic algorithms such as PSO and GA, and the result shows that CS has higher success rate than. The result of the influence of max columns in x or y and max iteration in

�λ

<sup>i</sup> þ α⨁Lévyð Þλ (41)

, 1ð Þ <λ≤ 3 (42)

xð Þ <sup>t</sup>þ<sup>1</sup> <sup>i</sup> <sup>¼</sup> <sup>x</sup>ð Þ<sup>t</sup>

determining safety factor can be seen in **Table 2**.

Lévy � u ¼ t

random walk. There are few rules to use this algorithm as follows:

other better option in 3D analysis.

generations;

*Slope Engineering*

following formula:

Flights (**Figure 12**).

**88**

### *2.2.4 Analysis 3D limit equilibrium of open pit mine*

3D limit equilibrium analysis method, data regarding 3D slope geometry model, material properties and 3D geological models are required. The 3D slope geometry model required for this method can be obtain from the reconstruction of pit surface model from either terrestrial or photogrammetric measurement methods. The next step is to create an external volume from the pit surface model which will then be analyzed to determine the position and shape of the slip surface, example of 3D slope geometry model can be seen in **Figure 13**.

Limit equilibrium analysis method uses the properties values of materials obtained from laboratory tests results to calculate the value of the safety factor. An example of the input parameter data used in the analysis can be seen in **Table 3**.

Geological modeling is the process of creating visual description geometry of rock lithology into software that represents the actual conditions. However, there are limitations in the modeling process, that related to the limited information on the data held and errors in data interpretation carried out. One of the methods that can be used to create the 3D geological model is interpolation the lithology information data from geotechnical drilling results. The 3D geological model will be used in the 3-dimensional slope stability analysis to determine the distribution characteristics of the rock lithology. In this analysis, the 3D geological floor model of limonite, saprolite and bedrock lithology is used and can be seen in **Figure 14**.

The results of the analysis using the Bishop Simplified method for the slip surface search method, both grid search and cuckoo search can be seen in **Figures 15** and **16**. The analysis results with the grid search show the safety factor value is 1.104, while the cuckoo search is 1.089. The position of the slip surface with the grid search and cuckoo search is the same position it indicates the accuracy of the cuckoo search, while the grid requires a grid box which must represent the actual slip surface position.

#### **2.3 Finite element method 3D**

The finite element method has been widely applied by mining geotechnical practitioners in slope stability analysis, with the advantage that the stress–strain analysis in the material allows to determine the displacement and strain values acting on the model elements, but this method has weaknesses in the process. This analysis uses a large number of calculation matrices so that it requires a long computation time, especially if the analysis is carried out in 3D, the number of
