**Figure 13.**

*3D slope geometry model.*


#### **Table 3.**

*Material properties.*

elements and nodes used will also be high so that the computation time can run for days depending on the number of nodes and types on mesh used. The computation time is closely related to the maximum number of iterations in the calculation of the

force error or load imbalance (solid tolerance) in determining the convergence of model, the higher the maximum number of iterations, the calculation in determining the convergence level in the analysis result model will be more accurate, but

Slope stability analysis using the finite element method takes into account the stress–strain analysis that works on each element of the model and focuses more on the analysis of the value of the deformation that occurs rather than the level of slope stability [2]. The advantages of analysis using the finite element method when

• No need to assume the position of the slip surface, failure occur in zones where the shear strength of the material cannot maintain stability, due to the shear

• It does not require the concept of slice or columns, so it does not require an

• Analyze the stress–strain so that it can see the deformation and the effective

• The finite element method can monitor the movement of rock masses towards

The value of the safety factor (SF) in the finite element method is defined as the ratio of the actual material shear strength to the shear strength of the material when the model failure. This concept is similar to the limit equilibrium method, where the shear strength ratio of a material to its driving force [1]. The strength of the material

cf <sup>¼</sup> <sup>c</sup>

With strength reduction factor (SRF) is the value of the factor for the decrease in the strength of the material, cf is the cohesion value when the model failure and ϕ<sup>f</sup> is the value of the internal friction angle when the model failure. In determining the strength of the material at failure, the technique of shear strength reduction (SSR) is used, where the actual material strength parameter is decreased step by step until the model become failure condition (non-convergent) [14]. The value of the material strength reduction factor for the Mohr-Coulomb criteria can be

SRF þ

The systematic stage in the model analysis to determine the critical value of the

tan ϕ

tan ϕ

ϕ<sup>f</sup> ¼ arctan

τ SRF <sup>¼</sup> <sup>c</sup>

τ

material shear strength reduction factor (Critical SRF) is as follows:

SRF (43)

SRF (44)

SRF (45)

SRF <sup>¼</sup> cf <sup>þ</sup> tan <sup>ϕ</sup><sup>f</sup> (46)

unfortunately it will take a long time in the analysis.

*Three Dimensional Slope Stability Analysis of Open Pit Mine*

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

compared to the limit equilibrium method are as follows:

approach of forces acting on global equilibrium.

*2.3.1 Strength reduction factor for 3D slope stability*

at failure can be written in the following equation:

stress that works due to gravity.

stress.

failure.

determined as follows:

**93**

**Figure 14.** *3D geological floor model.*

**Figure 15.** *Result 3D analysis using grid search.*

**Figure 16.** *Result 3D analysis using cuckoo search.*

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

force error or load imbalance (solid tolerance) in determining the convergence of model, the higher the maximum number of iterations, the calculation in determining the convergence level in the analysis result model will be more accurate, but unfortunately it will take a long time in the analysis.

Slope stability analysis using the finite element method takes into account the stress–strain analysis that works on each element of the model and focuses more on the analysis of the value of the deformation that occurs rather than the level of slope stability [2]. The advantages of analysis using the finite element method when compared to the limit equilibrium method are as follows:


#### *2.3.1 Strength reduction factor for 3D slope stability*

The value of the safety factor (SF) in the finite element method is defined as the ratio of the actual material shear strength to the shear strength of the material when the model failure. This concept is similar to the limit equilibrium method, where the shear strength ratio of a material to its driving force [1]. The strength of the material at failure can be written in the following equation:

$$\mathbf{c}\_{\mathbf{f}} = \frac{\mathbf{c}}{\mathbf{SRF}} \tag{43}$$

$$\phi\_{\rm f} = \arctan \frac{\tan \phi}{\text{SRF}} \tag{44}$$

With strength reduction factor (SRF) is the value of the factor for the decrease in the strength of the material, cf is the cohesion value when the model failure and ϕ<sup>f</sup> is the value of the internal friction angle when the model failure. In determining the strength of the material at failure, the technique of shear strength reduction (SSR) is used, where the actual material strength parameter is decreased step by step until the model become failure condition (non-convergent) [14]. The value of the material strength reduction factor for the Mohr-Coulomb criteria can be determined as follows:

$$\frac{\pi}{\text{SRF}} = \frac{\text{c}}{\text{SRF}} + \frac{\tan \phi}{\text{SRF}} \tag{45}$$

$$\frac{\pi}{\text{SRF}} = \mathbf{c}\_{\text{f}} + \tan \phi\_{\text{f}} \tag{46}$$

The systematic stage in the model analysis to determine the critical value of the material shear strength reduction factor (Critical SRF) is as follows:

**Figure 14.**

**Figure 15.**

**Figure 16.**

**92**

*Result 3D analysis using cuckoo search.*

*Result 3D analysis using grid search.*

*3D geological floor model.*

*Slope Engineering*


these elements is called meshing. Mesh type will affect the analysis result. Greater number of elements will lead to a more accurate analysis result. However, this will result a more time-consuming computation. There are various mesh types in 3-dimensional analysis, there are 4-nodes and 10-nodes with uniform and graded shape available to this method. The example of elements and nodes utilization in

To recapitulate the results of the analysis can be seen in **Figure 18**. From this figure provides information that there is an effect of using the mesh type on the srf value of the analysis results obtained, this is because the number of elements and

In model analysis using the finite element method, the determination of the convergence criteria is limited by the analysis calculation (maximum iteration), the more iterations allowed, the more accurate the analysis results will be, but it also needs to be considered in terms of the slope model being analyzed, in the use of the maximum number. Optimal iteration related to the level of efficiency in the analysis

At first, the model will be analyzed the stresses that work on each element due to the applied load, it will get the principal major stress and minor for each element,

FEM are given in **Figure 17**.

**Figure 17.**

*4-node graded and uniform.*

nodes used is also different.

so as not to waste too long.

**Figure 18.** *SRF VS mesh type.*

**95**

*2.3.3 Maximum iteration optimization*

*Three Dimensional Slope Stability Analysis of Open Pit Mine*

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

3D model analysis is an extension of 2D analysis, but in modeling complex geometries 2D analysis cannot simulate them properly. The 2D analysis assumes that the slope width is infinitely wide [11]. However, in many cases the 2D analysis is considered more conservative because it results in a lower safety factor value compared to the 3-dimensional analysis. The weakness of the analysis which is carried out in 3 dimensions is that it takes a lot of time and money, which makes practitioners not want to switch. The results of the 3D analysis can be conservative if they are validated with a 2D analysis in the most critical areas [2]. For conservative results, the 2D and 3D analyzes should not be significantly different, however in many cases the 3D analysis provides a higher safety factor value. The advantage given if the analysis is carried out in 3D is that it can represent the actual slope conditions in real terms and can determine the critical position. The value of the safety factor in the finite element method analysis with the 3-dimensional model is determined by the use of the material shear strength reduction factor (SRF) technique. in order to obtain the true safety factor, the srf is gradually increased until the model become failure (non-convergent). When this critical value is found, the safety factor of the slope model is equal to the reduction in material strength (SF) ≈ (SRF).

The determination of the critical value of srf is determined by increasing and decreasing the SRF value step by step until the highest SRF value is obtained in the model to be able to achieve the convergence criteria, see **Table 4**, the slope model enters a non-convergent (failure) at srf 1.05, so that the critical value of srf is 1.04 and highlighted in bold color.

### *2.3.2 Mesh type*


Finite element analysis method employs the utilization of elements and nodes to perform stress–strain analysis acting on each element. The process of establishing

**Table 4.** *Critical SRF determination.* *Three Dimensional Slope Stability Analysis of Open Pit Mine DOI: http://dx.doi.org/10.5772/intechopen.94088*

**Figure 17.** *4-node graded and uniform.*

• The first stage is to prepare the model for analysis using the finite element method, determine characteristics of strength and deformation properties of the material.

• Repeat the second stage using a systematic increase in the value of the material strength reduction factor, until the model becomes a non-convergent condition (failure). The critical SRF is determined at the highest srf value in the model

3D model analysis is an extension of 2D analysis, but in modeling complex geometries 2D analysis cannot simulate them properly. The 2D analysis assumes that the slope width is infinitely wide [11]. However, in many cases the 2D analysis is considered more conservative because it results in a lower safety factor value compared to the 3-dimensional analysis. The weakness of the analysis which is carried out in 3 dimensions is that it takes a lot of time and money, which makes practitioners not want to switch. The results of the 3D analysis can be conservative if they are validated with a 2D analysis in the most critical areas [2]. For conservative results, the 2D and 3D analyzes should not be significantly different, however in many cases the 3D analysis provides a higher safety factor value. The advantage given if the analysis is carried out in 3D is that it can represent the actual slope conditions in real terms and can determine the critical position. The value of the safety factor in the finite element method analysis with the 3-dimensional model is determined by the use of the material shear strength reduction factor (SRF) technique. in order to obtain the true safety factor, the srf is gradually increased until the model become failure (non-convergent). When this critical value is found, the safety factor of the slope

• The second stage is to increase the value of the material strength reduction factor (SRF) and calculate the material parameters using the Mohr-Coulomb criteria, then input the new material properties data into the model and

recalculate and record the maximum total deformation value.

model is equal to the reduction in material strength (SF) ≈ (SRF).

and highlighted in bold color.

*2.3.2 Mesh type*

*Slope Engineering*

**Table 4.**

**94**

*Critical SRF determination.*

The determination of the critical value of srf is determined by increasing and decreasing the SRF value step by step until the highest SRF value is obtained in the model to be able to achieve the convergence criteria, see **Table 4**, the slope model enters a non-convergent (failure) at srf 1.05, so that the critical value of srf is 1.04

Finite element analysis method employs the utilization of elements and nodes to perform stress–strain analysis acting on each element. The process of establishing

**Step SRF Solid Tolerance Convergence** 1 0.0005 Yes 1.3 0.2813 No 1.14 0.0691 No 1.06 0.0152 No 1.02 0.0008 Yes **1.04 0.0009 Yes** 1.05 0.0078 No

to achieve the convergent condition.

these elements is called meshing. Mesh type will affect the analysis result. Greater number of elements will lead to a more accurate analysis result. However, this will result a more time-consuming computation. There are various mesh types in 3-dimensional analysis, there are 4-nodes and 10-nodes with uniform and graded shape available to this method. The example of elements and nodes utilization in FEM are given in **Figure 17**.

To recapitulate the results of the analysis can be seen in **Figure 18**. From this figure provides information that there is an effect of using the mesh type on the srf value of the analysis results obtained, this is because the number of elements and nodes used is also different.

## *2.3.3 Maximum iteration optimization*

In model analysis using the finite element method, the determination of the convergence criteria is limited by the analysis calculation (maximum iteration), the more iterations allowed, the more accurate the analysis results will be, but it also needs to be considered in terms of the slope model being analyzed, in the use of the maximum number. Optimal iteration related to the level of efficiency in the analysis so as not to waste too long.

At first, the model will be analyzed the stresses that work on each element due to the applied load, it will get the principal major stress and minor for each element,

**Figure 18.** *SRF VS mesh type.*

then enter the material properties for each material and start entering the material strength reduction factor (SRF) value for each element, at this stage the material properties value can be increased or decreased depending on the error value (solid tolerance) obtained. Furthermore, an Elasto-plastic analysis was performed using the Mohr-Coulomb failure criterion to obtain a new error value (solid tolerance). If the error value is still above the maximum value within the allowable iteration calculation limit, the SRF value will be lowered until the error value is below the maximum limit. The recapitulate the analysis results can be seen in **Table 5.**

geometry and 3D geological models. Young and poisson ratio are input parameters in this analysis. If the boundary equilibrium analysis requires the position and shape of the slip plane, the analysis does not require these assumptions but uses the elements and nodes that are attached to the model. For an example of a 3D mesh model, see **Figure 19**. For this example, case a 4-node element type with a graded

*Three Dimensional Slope Stability Analysis of Open Pit Mine*

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

gradient is used.

**Figure 19.** *3D meshing model.*

**Figure 20.** *3D restraint model.*

**Figure 21.**

**97**

*3D FEM analysis result.*

The analysis shows that the higher the maximum number of iterations, the SRF value also increases **Table 4**. As the SRF value increases, in the maximum number of iterations of 400 the SRF value constant at a value of 1.04, so this provides information that at a maximum of 400 iterations is the maximum optimal number of iterations. If seen from the effect of the number of iterations on the total displacement, the results will fluctuate, this is closely related to the srf value because the higher the srf value, the total displacement will also increase, but at 1.04 the srf value does not change/is consistent but the total displacement fluctuates because it is solid tolerance, the resulting solid tolerance also varies due to the use of different maximum iterations, this is because the slope conditions remain non-convergent (energy equilibrium is not achieved) above srf 1.04 in the number of iterations that have been set, however, solid tolerance will get closer to the maximum value, while for the computation time it is very clear that the higher the maximum number of iterations, the computation time will also increase, because it will take more time to search for convergence with the maximum iteration limit given, although the results will remain the same, that non-converging above 1.04 and the last converging value is 1.04.
