*2.2.1 Safety factor for 3D slope stability*

The 3-dimensional model is a refined version of the 2-dimensional by projecting the skid plane into a column and determining the resultant force, as well as the moment based on the x, y, and z directions. The equilibrium force and moments acting on the overall column mass are used to determine the following 3 possible direction of the slip plane:


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

**Figure 4.** *Slope design process [12].*

and implementation. The initial stage of the geotechnical model is determined by 4 parameters, namely the geological model, structural conditions, rock mass and hydrogeological model. At the domains stage, the failure modes are determined by 2 parameters, namely the strength of the material and the condition of the structure. A single slope design arrangement is determined by the regulations or standards used by the company and the capabilities of the equipment used. Determination of the haulage road width and also the overall slope angle is based on mine planning related to the economic aspects of the opening geometry made. Furthermore, the stability analysis of the slope geometry that has been designed refers to the parameters (structure, strength, groundwater, and in-situ stress). After the final design is obtained, a risk assessment is carried out to mitigate the potential for landslides that may occur. In the implementation stage, the functions of dewatering, blasting and monitoring of the progress of the design model and the movement of rock masses

These days the needs and pressure to analyze a slope 3 dimensionally is more sounds. This is because 2D analysis assumes that the width of slope is infinitely wide so then it neglects 3D effect [11]. In most of the cases the width to height ratio is not sufficiently long and varies perpendicular to the slide movement. Therefore, 3D analysis is considered important to be done to produce the representative FoS. Moreover, in 3D analysis the volume of failure can also be estimated while 2D analysis cannot. If the volume can be determinate, it can be useful as one of the

The 3-dimensional model is a refined version of the 2-dimensional by projecting

the skid plane into a column and determining the resultant force, as well as the moment based on the x, y, and z directions. The equilibrium force and moments acting on the overall column mass are used to determine the following 3 possible

considerations in giving failure prevention recommendation.

(**Figure 4**).

**Figure 3.**

*Slope Engineering*

**2.2 Limit equilibrium method 3D**

*Comparison between 3D and 2D single slope analysis [11].*

*2.2.1 Safety factor for 3D slope stability*

1.The column moves in the same direction

2.The column moves towards one another

3.The column moves in the opposite direction

direction of the slip plane:

**80**

For the 3-dimensional analysis, the mass potential of the slip plane is divided into several columns. Ref. [8] give the equation of the Simplified Janbu method deduced from the Morgenstern-Price method to obtain a safety factor value of 3-dimensional analysis (**Figure 5**).

ai is space angle for sliding direction with respect to the projected x – y plane, ax, ay are base inclination along x and y directions measure at the center of each column, Exi, Eyi are inter-column normal forces in x and y directions, respectively, Hxi, Hyi are lateral inter-column shear forces in x and y directions, N'i, Ui are effective normal and base pore watery force, Pvi, Si is vertical external force, and base mobilized shear force, and Xxi, Xyi are vertical inter-column shear force in plane perpendicular to x and y directions.

With the mohr-coulomb collapse criteria, the safety factor is determined using the following equation:

$$\mathbf{F} = \frac{\mathbf{S}\_{\text{fi}}}{\mathbf{S}\_{\text{i}}} = \frac{\mathbf{C}\_{\text{i}} + \mathbf{N}\_{\text{i}}^{\prime} \tan \phi\_{\text{i}}^{\prime}}{\mathbf{S}\_{\text{i}}} \tag{5}$$

where Sfi is ultimate resultant shear force available at the base of column i, N'<sup>i</sup> is the effective base normal force, Ci is (c. Ai) and c and Ai are effective cohesive strength and the base area of the column. The base shear force Si and normal base force Ni are expressed as the components of forces with respect to x, y, and z directions for column i.

$$\mathbf{S}\_{\rm xi} = \mathbf{f}\_1 \mathbf{S}\_{\rm i}, \mathbf{S}\_{\rm yi} = \mathbf{f}\_2 \mathbf{S}\_{\rm i}, \mathbf{S}\_{\rm xi} = \mathbf{f}\_3 \mathbf{S}\_{\rm i} \tag{6}$$

Fz <sup>¼</sup> <sup>0</sup> <sup>¼</sup> Nig3i <sup>þ</sup> Sif3i–ð Þ¼ Wi <sup>þ</sup> Pvi ð Þþ Xxiþ1–Xxi Xyiþ1–Xyi � � (14)

Solving Equation, the base normal and shear forces can be expressed as

Si <sup>¼</sup> Ci <sup>þ</sup> ð Þ Ai � Ui tan <sup>ϕ</sup><sup>0</sup>

Ai <sup>¼</sup> Wi þ ðPvi <sup>þ</sup> <sup>Δ</sup>Exiλ<sup>x</sup> <sup>þ</sup> <sup>Δ</sup>Eyi

Bi ¼ � f3i g3i

<sup>F</sup> <sup>¼</sup> Ci <sup>þ</sup> <sup>N</sup><sup>0</sup>

Considering the overall force equilibrium in x-direction internal force E

<sup>X</sup>Hxi <sup>þ</sup>XNig1i–

<sup>X</sup>Hyi <sup>þ</sup>XNig2i–

Considering overall moment equilibrium in the x-direction

Considering overall force equilibrium in the y-direction

Considering overall moment equilibrium in the y-direction

�

*Three Dimensional Slope Stability Analysis of Open Pit Mine*

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

�

cancels out.

�

�

**Figure 6.**

**83**

*Force equilibrium in columns [8].*

F 1 � Bi tan <sup>ϕ</sup><sup>i</sup> F

g3i

Si

<sup>i</sup> tan ϕ<sup>0</sup> i

<sup>X</sup> Wi <sup>þ</sup> Pvi � Nig3i–Sif3i � �RX <sup>þ</sup><sup>X</sup> Nig1i � Sif 1i � �RZ <sup>¼</sup> 0 (23)

<sup>X</sup> Wi <sup>þ</sup> Pvi � Nig3i–Sif3i � �RY <sup>þ</sup><sup>X</sup> Nig2i � Sif 2i � �RZ <sup>¼</sup> 0 (25)

Fx ¼ 0 ¼ Sif 1i � Nig1i � Hxi þ Hxiþ<sup>1</sup> ¼ Exiþ1–Exi (15) Fy ¼ 0 ¼ Sif 2i � Nig2i � Hyi þ Hyiþ<sup>1</sup> ¼ Eyiþ1–Eyi (16)

Ni ¼ Ai þ BiSi (17)

λyÞ

h i � � (18)

<sup>X</sup>Sif1i <sup>¼</sup> <sup>0</sup> (22)

<sup>X</sup>Sif2i <sup>¼</sup> <sup>0</sup> (24)

(19)

(20)

(21)

i

$$\mathbf{N\_{xi}} = \mathbf{g\_{1}N\_{i}}, \mathbf{N\_{yi}} = \mathbf{g\_{2}N\_{i}}, \mathbf{N\_{zi}} = \mathbf{g\_{3}N\_{i}} \tag{7}$$

where f1, f2, f3 and g1, g2, g3 = unit vectors in the direction of Si and Ni. The projected shear angles a' = same for all columns in the x – y plane in the present formulation, and by using this angle, the space shear angle ai found for each column.

$$\mathbf{a}\_{\mathrm{i}} = \tan^{-1}\left\{\sin\Theta\_{\mathrm{i}}/\left[\cos\Theta\_{\mathrm{i}} + \left(\cos\mathbf{a}\_{\mathrm{ji}}/\tan\mathbf{a}'\cos\mathbf{a}\_{\mathrm{zi}}\right)\right]\right\} \tag{8}$$

$$\theta\_{\mathbf{i}} = \cos^{-1}\left\{\sin \mathbf{a}\_{\mathbf{i}\mathbf{i}} \bullet \sin \mathbf{a}\_{\mathbf{j}\mathbf{i}}\right\} \tag{9}$$

An arbitrary intercolumn shear force function f (x, y) is assumed in the present analysis, and the relationships between the intercolumn shear and normal forces in the x- and y-directions are given as:

$$\mathbf{X}\mathbf{x}\_{i} = \mathbf{E}\mathbf{x}\_{i}\mathbf{f}\begin{pmatrix}\mathbf{x},\mathbf{y}\end{pmatrix}\lambda\_{\mathbf{x}} \tag{10}$$

$$\mathbf{x}\mathbf{y}\_i = \mathbf{E}\mathbf{y}\_i\mathbf{f}(\mathbf{x}, \mathbf{y})\lambda\_\mathbf{y} \tag{11}$$

$$\mathbf{Hx\_i = Ey\_i f(x, y)} \lambda\_{xy} \tag{12}$$

$$\mathbf{H}\mathbf{y}\_i = \mathbf{E}\mathbf{x}\_i \mathbf{f}(\mathbf{x}, \mathbf{y})\lambda\_{\mathbf{y}\mathbf{x}} \tag{13}$$

Where λ<sup>x</sup> and λ<sup>y</sup> are intercolumn shear force mobilization factors in x and y directions, respectively, and λxy and λyx are intercolumn shear force mobilization factors in xz and yz planes. Considering the vertical and horizontal force equilibrium for the ith column in the z, x, and y directions produces the following equations (**Figure 6**):

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

$$\mathbf{F\_z = 0 = N\_i \mathbf{g\_{ji}} + \mathbf{S\_i} f\_{\bar{3}i -} (W\_{\bar{i}} + P\_{\text{vi}}) = (\mathbf{X\_{xi+1}} - \mathbf{X\_{xi}}) + (\mathbf{X\_{yi+1}} - \mathbf{X\_{yi}}) \tag{14}$$

$$\mathbf{F\_x = 0 = S\_i f\_{1i} - N\_i g\_{1i} - H\_{xi} + H\_{xi+1} = E\_{xi+1} - E\_{xi}} \tag{15}$$

$$\mathbf{F\_{y}} = \mathbf{0} = \mathbf{S\_{i}f\_{2i}} - \mathbf{N\_{i}g\_{2i}} - \mathbf{H\_{yi}} + \mathbf{H\_{yi+1}} = \mathbf{E\_{yi+1}} - \mathbf{E\_{yi}} \tag{16}$$

Solving Equation, the base normal and shear forces can be expressed as

$$\mathbf{N}\_{\mathrm{i}} = \mathbf{A}\_{\mathrm{i}} + \mathbf{B}\_{\mathrm{i}} \mathbf{S}\_{\mathrm{i}} \tag{17}$$

$$\mathbf{S}\_{\mathbf{i}} = \frac{\mathbf{C}\_{\mathbf{i}} + (\mathbf{A}\_{\mathbf{i}} - \mathbf{U}\_{\mathbf{i}})\tan\phi'\_{\mathbf{i}}}{\mathbf{F}\left[\mathbf{1} - \left(\frac{\mathbf{B}\_{\mathbf{i}}\tan\phi\_{\mathbf{i}}}{\mathbf{F}}\right)\right]} \tag{18}$$

$$\mathbf{A}\_{i} = \frac{\mathbf{W}\_{i} + (\mathbf{P}\_{\rm vi} + \Delta \mathbf{E} \mathbf{x}\_{i} \boldsymbol{\lambda}\_{\rm x} + \Delta \mathbf{E} \mathbf{y}\_{i} \boldsymbol{\lambda}\_{\rm y})}{\mathbf{g}\_{3i}} \tag{19}$$

$$\mathbf{B}\_{\rm i} = -\frac{\mathbf{f}\_{\rm 3i}}{\mathbf{g}\_{\rm 3i}} \tag{20}$$

$$\mathbf{F} = \frac{\mathbf{C\_i} + \mathbf{N'\_i}\tan\phi'\_i}{\mathbf{S\_i}} \tag{21}$$

Considering the overall force equilibrium in x-direction internal force E cancels out.

$$-\sum \mathbf{H}\_{\mathbf{x}\mathbf{i}} + \sum \mathbf{N}\_{\mathbf{i}} \mathbf{g}\_{\mathbf{i}\mathbf{i}} - \sum \mathbf{S}\_{\mathbf{i}} \mathbf{f}\_{\mathbf{i}\mathbf{i}} = \mathbf{0} \tag{22}$$

Considering overall moment equilibrium in the x-direction

$$-\sum \left(\mathbf{W}\_{\text{i}} + \mathbf{P}\_{\text{vi}} - \mathbf{N}\_{\text{i}} \mathbf{g}\_{\text{3i}} \mathbf{S}\_{\text{i}} \mathbf{f}\_{\text{3i}}\right) \mathbf{R}\mathbf{X} + \sum \left(\mathbf{N}\_{\text{i}} \mathbf{g}\_{\text{1i}} - \mathbf{S}\_{\text{i}} \mathbf{f}\_{\text{1i}}\right) \mathbf{R}\mathbf{Z} = \mathbf{0} \tag{23}$$

Considering overall force equilibrium in the y-direction

$$-\sum \mathbf{H}\_{\rm yi} + \sum \mathbf{N}\_{\rm i} \mathbf{g}\_{2i} - \sum \mathbf{S}\_{\rm i} \mathbf{f}\_{2i} = \mathbf{0} \tag{24}$$

Considering overall moment equilibrium in the y-direction

$$-\sum \left( \mathbf{W}\_{\mathrm{i}} + \mathbf{P}\_{\mathrm{vi}} - \mathbf{N}\_{\mathrm{i}} \mathbf{g}\_{\mathrm{3i}} \mathbf{S}\_{\mathrm{i}} \mathbf{f}\_{\mathrm{3i}} \right) \mathbf{R} \mathbf{Y} + \sum \left( \mathbf{N}\_{\mathrm{i}} \mathbf{g}\_{\mathrm{2i}} - \mathbf{S}\_{\mathrm{i}} \mathbf{f}\_{\mathrm{2i}} \right) \mathbf{R} \mathbf{Z} = \mathbf{0} \tag{25}$$

**Figure 6.** *Force equilibrium in columns [8].*

where Sfi is ultimate resultant shear force available at the base of column i, N'<sup>i</sup> is

Sxi ¼ f 1Si, Syi ¼ f 2Si, Szi ¼ f3Si (6) Nxi ¼ g1Ni, Nyi ¼ g2Ni, Nzi ¼ g3Ni (7)

(8)

cos axi

λ<sup>x</sup> (10)

λ<sup>y</sup> (11)

λxy (12)

λyx (13)

(9)

the effective base normal force, Ci is (c. Ai) and c and Ai are effective cohesive strength and the base area of the column. The base shear force Si and normal base force Ni are expressed as the components of forces with respect to x, y, and z

where f1, f2, f3 and g1, g2, g3 = unit vectors in the direction of Si and Ni. The projected shear angles a' = same for all columns in the x – y plane in the present formulation, and by using this angle, the space shear angle ai found for each

ai <sup>¼</sup> tan �<sup>1</sup> sin <sup>θ</sup>i*<sup>=</sup>* cos <sup>θ</sup><sup>i</sup> <sup>þ</sup> cos ayi*<sup>=</sup>* tan a<sup>0</sup>

<sup>θ</sup><sup>i</sup> <sup>¼</sup> cos �<sup>1</sup> sin axi <sup>∙</sup> sin ayi

Xxi ¼ Exi f x, y

Xyi ¼ Eyi f x, y

Hxi ¼ Eyi f x, y

Hyi ¼ Exi f x, y

Where λ<sup>x</sup> and λ<sup>y</sup> are intercolumn shear force mobilization factors in x and y directions, respectively, and λxy and λyx are intercolumn shear force mobilization factors in xz and yz planes. Considering the vertical and horizontal force equilibrium for the ith column in the z, x, and y directions produces the following equa-

An arbitrary intercolumn shear force function f (x, y) is assumed in the present analysis, and the relationships between the intercolumn shear and normal forces in

directions for column i.

*3 dimensional column [8].*

*Slope Engineering*

the x- and y-directions are given as:

column.

**Figure 5.**

tions (**Figure 6**):

**82**

The directional safety factor Fx and Fy is determined as follows:

$$\mathbf{F\_x} = \frac{\sum [\mathbf{C\_i} + (\mathbf{N\_i} - \mathbf{U\_i})\tan\phi\_i]\mathbf{f\_{1i}}}{\sum \mathbf{N\_i}\mathbf{g\_{1i}} - \sum \mathbf{H\_{xi}}}, \quad (\mathbf{0} < \mathbf{F\_x} < \infty) \tag{26}$$

Fsy ¼

*Three Dimensional Slope Stability Analysis of Open Pit Mine*

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

simplified Janbu's method.

square grid.

**Figure 7.**

**85**

*Illustration of grid point [11].*

*2.2.2 Grid search in determination of slip surface*

points and radius increment can be seen in **Figures 7** and **8**.

the global factor of safety Ff based on force is determined as follows:

<sup>P</sup>Ayi <sup>f</sup> 2i <sup>þ</sup> f3ig2i*=*g3i � � <sup>P</sup> g2i*=*g3i � �ð Þ Wi <sup>þ</sup> Pvi

Ff ¼ Fsx ¼ Fsy (40)

For 3D asymmetric Janbu's method, at the force equilibrium point, the directional factors of safety, Fsx, and Fsy are equal to each other. Under this condition,

The safety factor is also used in vertical and 3D force equilibrium to achieve the

One of the methods that can be used to determine the critical slip surface is the

After assuming the field geomaterial failure, the next step is mass discretization of the sliding mass into a number of columns. Square nets are applied to the sliding mass so that the sliding mass is divided into columns. There are two kinds of columns; the active column where the column is inside the sliding mass boundary line, and the inactive column where these columns are outside the sliding mass boundary line. In the calculation, the inactive columns are ignored so that the discrete sliding mass is determined only from the sum of the active columns. **Figure 9** shows the illustration of the discretization of the sliding mass using a

grid search method [11]. In the grid search method, the first thing to do is to determine the size of the grid box with dimensions x, y, and z. After the grid box is available, the user determines the number of grid points that you want to use in the x, y, and z directions. This point serves as the center of rotation. Each center of rotation can produce a number of circles that are used as slip surfaces. The number of circles produced at each center of rotation from the minimum radius to the maximum radius is called the radius increment. Illustration of the number of grid

(39)

$$\mathbf{F\_{y}} = \frac{\sum [\mathbf{C\_{i}} + (\mathbf{N\_{i-}} \mathbf{U\_{i}}) \tan \phi'\_{i}] \mathbf{f\_{2i}}}{\sum \mathbf{N\_{i}} \mathbf{g\_{2i}} - \sum \mathbf{H\_{yi}}}, \quad \left(\mathbf{0} < \mathbf{F\_{y}} < \infty\right) \tag{27}$$

Formulation 3D Bishop's methods by considering the overall moment equilibrium equations in x or y direction and neglecting the inter-column vertical and horizontal shear forces.

$$\mathbf{F\_{mx}} = \frac{\sum \{ \mathbf{K\_{xi}} [\mathbf{f\_{2i}} \mathbf{R} \mathbf{Z\_i} + \mathbf{f\_{3i}} \mathbf{R} \mathbf{Y\_i}] \}}{\sum (\mathbf{W\_i} + \mathbf{P\_{vi}}) \cdot \mathbf{R} \mathbf{Y\_i} + \sum \mathbf{N\_i} \left( \mathbf{g\_{2i}} \mathbf{R} \mathbf{Z\_i} - \left( \mathbf{g\_{3i}} \mathbf{R} \mathbf{Y\_i} \right) \right)} \tag{28}$$

$$\mathbf{F\_{my}} = \frac{\sum \left\{ \mathbf{K\_{yi}} [\mathbf{f\_{i1}RZ\_i} + \mathbf{f\_{j1}RX\_i}] \right\}}{\sum (\mathbf{W\_i} + \mathbf{P\_{vi}}) \cdot \mathbf{RX\_i} + \sum \mathbf{N\_i} (\mathbf{g\_{i1}RZ\_i} - (\mathbf{g\_{j1}RX\_i}) \mathbf{})} \tag{29}$$

$$\mathbf{K}\_{\rm xi} = \left\{ \frac{\mathbf{C}\_{\rm i} + \left[ (\mathbf{W}\_{\rm i} + \mathbf{P}\_{\rm vi}) / \left( \mathbf{g}\_{\rm 3i} - \mathbf{U}\_{\rm i} \right) \right] \tan \phi\_{\rm i}}{\mathbf{1} + \left( \mathbf{f}\_{\rm 3i} \tan \phi\_{\rm i} / \mathbf{g}\_{\rm 3i} \, \mathbf{F}\_{\rm mx} \right)} \right\} \tag{30}$$

$$\mathbf{K}\_{\rm yi} = \left\{ \frac{\mathbf{C}\_{\rm i} + \left[ (\mathbf{W}\_{\rm i} + \mathbf{P}\_{\rm vi}) / \left( \mathbf{g}\_{\rm 3i} - \mathbf{U}\_{\rm i} \right) \right] \tan \phi\_{\rm i}}{\mathbf{1} + \left( \mathbf{f}\_{\rm 3i} \tan \phi\_{\rm i} / \mathbf{g}\_{\rm 3i} \mathbf{F}\_{\rm my} \right)} \right\} \tag{31}$$

Considering overall moment equilibrium about an axis passing through (x0, y0, z0) and parallel to the z axis gives:

$$
\sum \left( -\mathbf{N\_{i\mathbf{g\_{1i}}} - \mathbf{S\_i f\_{1i}}} \right) \mathbf{R} \mathbf{Y} + \sum \left( \mathbf{N\_{i\mathbf{g\_{2i}}} - \mathbf{S\_i f\_{2i}}} \right) \mathbf{R} \mathbf{X} = \mathbf{0} \tag{32}
$$

$$\mathbf{F\_{mx}} = \frac{\sum [\mathbf{K\_{zi}}(\mathbf{f\_{2i}} \mathbf{R} \mathbf{X\_i} - \mathbf{f\_{3i}} \mathbf{R} \mathbf{Y\_i})]}{\sum \mathbf{N} \left(\mathbf{g\_{2i}} \mathbf{R} \mathbf{X\_i} - \left(\mathbf{g\_{1i}} \mathbf{R} \mathbf{Y\_i}\right)\right)} \tag{33}$$

$$\mathbf{K}\_{\rm zi} = \left\{ \frac{\mathbf{C}\_{\rm i} + \left[ (\mathbf{W}\_{\rm i} + \mathbf{P}\_{\rm vi}) / \left( \mathbf{g}\_{\rm 3i} - \mathbf{U}\_{\rm i} \right) \right] \tan \phi\_{\rm i}}{\mathbf{1} + \left( \mathbf{f}\_{\rm 3i} \tan \phi\_{\rm i} / \mathbf{g}\_{\rm 3i} \, \mathbf{F}\_{\rm mz} \right)} \right\} \tag{34}$$

For the 3D asymmetric Bishop's method, at moment equilibrium point, the directional factors of safety, Fmx, Fmy, and Fmz are equal to each other. Under this condition, the global factor of safety Fm based on moment can be determined as

$$\mathbf{F\_m = F\_{mx} = F\_{my} = F\_{mx}} \tag{35}$$

Formulation 3D simplified Janbu's methods by considering the overall force equilibrium equations and neglecting the inter-column vertical and horizontal shear forces.

$$\mathbf{A}\_{\rm xi} = \frac{\left\{ \mathbf{C}\_{\rm i} + \left[ (\mathbf{W}\_{\rm i-} \mathbf{P}\_{\rm vi}) / \left( \mathbf{g}\_{\rm 3i-} \mathbf{U}\_{\rm i} \right) \right] \tan \phi\_{\rm i} \right\}}{\mathbf{1} + \left( \mathbf{f}\_{\rm 3i} \tan \phi\_{\rm i} / \mathbf{g}\_{\rm 3i} \mathbf{F}\_{\rm xx} \right)} \tag{36}$$

$$\mathbf{A}\_{\rm yi} = \frac{\left\{ \mathbf{C}\_{\rm i} + \left[ (\mathbf{W}\_{\rm i-} \mathbf{P}\_{\rm vi}) / \left( \mathbf{g}\_{\rm 3i-} \mathbf{U}\_{\rm i} \right) \right] \tan \phi\_{\rm i} \right\}}{\mathbf{1} + \left( \mathbf{f}\_{\rm 3i} \tan \phi\_{\rm i} / \mathbf{g}\_{\rm 3i} \mathbf{F}\_{\rm sy} \right)} \tag{37}$$

$$\mathbf{F\_{sx}} = \frac{\sum \mathbf{A\_{xi}} (\mathbf{f\_{1i}} + \mathbf{f\_{3i}} \mathbf{g\_{1i}}/\mathbf{g\_{3i}})}{\sum (\mathbf{g\_{1i}}/\mathbf{g\_{3i}}) (\mathbf{W\_i} + \mathbf{P\_{vi}})} \tag{38}$$

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

The directional safety factor Fx and Fy is determined as follows:

<sup>P</sup> Ci <sup>þ</sup> ð Þ Ni � Ui tan <sup>ϕ</sup><sup>i</sup> ½ �

<sup>P</sup> Ci <sup>þ</sup> ð Þ Ni�Ui tan <sup>ϕ</sup><sup>0</sup> i ½ �

Nig1i � <sup>P</sup>Hxi

Nig2i � <sup>P</sup>Hyi

Kxi <sup>¼</sup> Ci <sup>þ</sup> ð Þ Wi <sup>þ</sup> Pvi *<sup>=</sup>* g3i � Ui

Kyi <sup>¼</sup> Ci <sup>þ</sup> ð Þ Wi <sup>þ</sup> Pvi *<sup>=</sup>* g3i � Ui

� �RY <sup>þ</sup><sup>X</sup> Nig2i � Sif2i

Kzi <sup>¼</sup> Ci <sup>þ</sup> ð Þ Wi <sup>þ</sup> Pvi *<sup>=</sup>* g3i � Ui

Formulation 3D Bishop's methods by considering the overall moment equilibrium equations in x or y direction and neglecting the inter-column vertical and

> <sup>P</sup> Kxi f2iRZi <sup>þ</sup> f3iRYi f g ½ � <sup>P</sup>ð Þ Wi <sup>þ</sup> Pvi RYi <sup>þ</sup> <sup>P</sup>Ni g2iRZi � g3iRYi

> <sup>P</sup> Kyi <sup>f</sup> 1iRZi <sup>þ</sup> f3iRXi ½ � � � <sup>P</sup>ð Þ Wi <sup>þ</sup> Pvi RXi <sup>þ</sup> <sup>P</sup>Ni g1iRZi � g3iRXi

> > 1 þ f3i tan ϕi*=*g3i Fmx � �

> > 1 þ f3i tan ϕi*=*g3i Fmy � �

<sup>P</sup> Kzið Þ <sup>f</sup> 2iRXi � f3iRYi ½ � <sup>P</sup>N g2iRXi � g1iRYi

1 þ f3i tan ϕi*=*g3i Fmz � �

For the 3D asymmetric Bishop's method, at moment equilibrium point, the directional factors of safety, Fmx, Fmy, and Fmz are equal to each other. Under this condition, the global factor of safety Fm based on moment can be determined as

Formulation 3D simplified Janbu's methods by considering the overall force equilibrium equations and neglecting the inter-column vertical and horizontal

Axi <sup>¼</sup> Ci <sup>þ</sup> ð Þ Wi�Pvi *<sup>=</sup>* g3i�Ui

Ayi <sup>¼</sup> Ci <sup>þ</sup> ð Þ Wi�Pvi *<sup>=</sup>* g3i�Ui

<sup>P</sup> g1i*=*g3i

Fsx ¼

� � � � tan <sup>ϕ</sup><sup>i</sup> <sup>g</sup>

� � � � tan <sup>ϕ</sup><sup>i</sup> � � 1 þ f3i tan ϕi*=*g3iFsx

� � � � tan <sup>ϕ</sup><sup>i</sup> � � 1 þ f3i tan ϕi*=*g3iFsy

> <sup>P</sup>Axi f1i <sup>þ</sup> f3ig1i*=*g3i � �

� �ð Þ Wi <sup>þ</sup> Pvi

Considering overall moment equilibrium about an axis passing through (x0, y0,

� � � � tan <sup>ϕ</sup><sup>i</sup> <sup>g</sup>

� � � � tan <sup>ϕ</sup><sup>i</sup> <sup>g</sup>

f 1i

f2i

, 0ð Þ <Fx < ∞ (26)

, 0<Fy < ∞ � � (27)

� � � (28)

� � � (29)

� �RX <sup>¼</sup> 0 (32)

� � � (33)

Fm ¼ Fmx ¼ Fmy ¼ Fmz (35)

� � (36)

� � (37)

(30)

(31)

(34)

(38)

P

P

Fx ¼

Fy ¼

Fmx ¼

Fmy ¼

z0) and parallel to the z axis gives:

shear forces.

**84**

(

(

<sup>X</sup> �Nig1i–Sif 1i

(

Fmz ¼

horizontal shear forces.

*Slope Engineering*

$$\mathbf{F\_{sy}} = \frac{\sum \mathbf{A\_{yi}} (\mathbf{f\_{2i}} + \mathbf{f\_{3i}g\_{2i}}/\mathbf{g\_{3i}})}{\sum (\mathbf{g\_{2i}}/\mathbf{g\_{3i}})(\mathbf{W\_i} + \mathbf{P\_{vi}})} \tag{39}$$

For 3D asymmetric Janbu's method, at the force equilibrium point, the directional factors of safety, Fsx, and Fsy are equal to each other. Under this condition, the global factor of safety Ff based on force is determined as follows:

$$\mathbf{F\_{f}} = \mathbf{F\_{sx}} = \mathbf{F\_{sy}} \tag{40}$$

The safety factor is also used in vertical and 3D force equilibrium to achieve the simplified Janbu's method.

## *2.2.2 Grid search in determination of slip surface*

One of the methods that can be used to determine the critical slip surface is the grid search method [11]. In the grid search method, the first thing to do is to determine the size of the grid box with dimensions x, y, and z. After the grid box is available, the user determines the number of grid points that you want to use in the x, y, and z directions. This point serves as the center of rotation. Each center of rotation can produce a number of circles that are used as slip surfaces. The number of circles produced at each center of rotation from the minimum radius to the maximum radius is called the radius increment. Illustration of the number of grid points and radius increment can be seen in **Figures 7** and **8**.

After assuming the field geomaterial failure, the next step is mass discretization of the sliding mass into a number of columns. Square nets are applied to the sliding mass so that the sliding mass is divided into columns. There are two kinds of columns; the active column where the column is inside the sliding mass boundary line, and the inactive column where these columns are outside the sliding mass boundary line. In the calculation, the inactive columns are ignored so that the discrete sliding mass is determined only from the sum of the active columns. **Figure 9** shows the illustration of the discretization of the sliding mass using a square grid.

**Figure 7.** *Illustration of grid point [11].*

**Figure 8.** *Illustration of the radius increment in the grid search [11].*

**Figure 9.** *Discretization of the sliding mass using a square grid [11].*

**Figure 10.** *3D model for slope stability analysis [11].*

After discretizing the sliding mass, internal and external forces in each column can be calculated based on moment equilibrium, force equilibrium, or both depending on what method of calculation is used (**Figure 10**).

*2.2.3 Cuckoo search in determination of slip surface*

non-circular option is CS method.

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

**Radius increment**

**Table 1.**

**Figure 11.**

**87**

**Number of grid points**

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

**Factor of safety**

*Three Dimensional Slope Stability Analysis of Open Pit Mine*

**Volume (m3 )**

**Location Direction**

20 20 10 0.868 2,245,470 North-HW 246.7 371,070 9,586,190 470

40 40 20 0.873 2,009,660 North-HW 246.5 371,068 9,586,190 443 20 20 10 0.868 2,245,470 North-HW 246.7 371,070 9,586,190 470

40 40 20 0.872 1,934,080 North-HW 246.4 371,069 9,586,200 428 20 20 10 0.868 2,245,470 North-HW 246.7 371,070 9,586,190 470

40 40 20 0.867 1,937,390 North-HW 246.1 371,076 9,586,170 459 20 20 10 0.868 2,245,470 North-HW 246.7 371,070 9,586,190 470

40 40 20 0.872 1,934,080 North-HW 246.4 371,069 9,586,200 428 20 20 10 0.868 2,245,470 North-HW 246.7 371,070 9,586,190 470

40 40 20 0.879 1,968,990 North-HW 246.3 371,078 9,586,190 421

10 30 30 15 0.874 1,998,950 North-HW 246.6 371,069 9,586,190 446

20 30 30 15 0.874 2,028,600 North-HW 246.5 371,069 9,586,200 429

30 30 30 15 0.906 1,628,780 North-HW 246.2 371,092 9,586,210 352

40 30 30 15 0.876 1,756,790 North-HW 246.2 371,076 9,586,200 404

50 30 30 15 0.876 2,068,950 North-HW 246.3 371,077 9,586,180 459

*The influence of number of grid point and radius increment in determining safety factor.*

**of sliding**

**Center of rotation X YZ**

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

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**.


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

#### **Table 1.**

*The influence of number of grid point and radius increment in determining safety factor.*

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