**6. Finite element method (FEM)**

This method of design was developed by Clough et al., (1950). Due to wide application of this method in mining engineering especially tunneling, it get more attention for solving mining problems and popularity in this field [19]. The FEM divide problem into small parts and connect these parts at a point/nodes at the apexes and at the boundaries of meshing/discretization. The FEM has many applications in modeling in rock engineering design due to dealing with nonlinearity, boundary conditions and heterogeneity problems [26, 27].

The unidentified function over each element in FEM estimated through test function having its nodal values of anonymous system (in polynomial form). This practice is the fundamental supposition of FEM. For experimental function, it is mandatory to satisfy the principal of PDFs. In this research the FEM based software Phase2 was used for analysis of stresses and total displacement around tunnel. For experimental function it must be satisfied the principal of PDFs, which is given in Eq. (7).

$$\mathbf{u}\_i^\epsilon = \sum\_{j=1}^M N\_{ij} \mathbf{u}\_i^\epsilon \tag{7}$$

Where,

*Nij* is the shape function or interpolation function; this must be defined into inherent coordinates for use of Gaussian quadratic integration, *M* is the element order.

Using shape function the problem original PDFs can be substituted by the arithmetical equation as given below.

$$\sum\_{j=1}^{N} \left[ K\_{\vec{\eta}}^{\epsilon} \right] \left\{ \mathbf{u}\_{i}^{\epsilon} \right\} = \sum\_{j=1}^{N} f\_{i}^{\epsilon} \text{ or } Ku = F \tag{8}$$

1.Finite Difference Method (FDM)

*Division of numerical models and methods [24, 25].*

3.Boundary Element Method (BEM)

*Finite Difference Method (FDM).*

*Finite Element Method (FEM).*

*Boundary Element Method (BEM).*

The Finite difference method (FDM) is the direct calculation of PDEs and transmitted the creative PDEs in term of unknown at grid point into a system of algebraic equations by interchange the fractional derivatives with difference at irregular or regular grid forced over problem areas. This system is solved due to establishing the required initial and boundary condition. This method is old but widely applied in the numerical modeling in rock mechanics. This method is based

The Finite element method (FEM) splits the problem into sub-elements of smaller sizes and shapes with fitting the number of nodes at the vertices and at the side of discretization. FEM is mostly used to estimate the behavior of PDEs at elemental level and for signifying the behavior of elements; it produces the local algebraic equation. After creating the local equation the FEM gathered it according to topographic relation of node and elements and further put it into worldwide system of algebraic equation for receiving the required information after

The Boundary element method is the precise method then FEM and FDM because of its easiness. This method involves the discretization of solution areas at boundary and thus decreases the problem dimension by simplifying the design

for explicit approach of discreet element method (DEM) [26].

establishing the definite initial and boundary situations.

2.Finite Element Method (FEM)

**Figure 7.**

*Slope Engineering*

**68**

#### Where,

*Keij* is the coefficient matrix, ue <sup>i</sup> vector is the nodal value vector having unidentified variables, *f e <sup>i</sup>* is consist of body force contribution and initial boundary condition, *K* is the global stiffness matrix.

*Keij* is also called the element stiffness matrix in term of elasticity problem which is given by Eq. (9).

$$\int\_{\Omega} \mathbf{K}\_{ij}^{\epsilon} = \int\_{\Omega \dot{i}} ([B\_i][\mathbf{N}\_i])^T [D\_i][B\_i] d\Omega \tag{9}$$

*<sup>x</sup>* <sup>¼</sup> <sup>X</sup>*<sup>n</sup> i*�1

*<sup>y</sup>* <sup>¼</sup> <sup>X</sup>*<sup>n</sup> i*�1

*<sup>z</sup>* <sup>¼</sup> <sup>X</sup>*<sup>n</sup> i*�1

For two dimensional element domain the relation between strain and

*εx εy γxy εz*

Δ*σ* is the vector of stress components, Δ*ε* represents corresponding components

It is formed when added the stiffness matrices of all elements. The equation for

of strains and DT is a square matrix that is constant in the elastic case.

*ε* ¼

**6.4 Relation between strain and displacement**

*Some element forms and node position used in two dimensional [28].*

*Design Techniques in Rock and Soil Engineering DOI: http://dx.doi.org/10.5772/intechopen.90195*

displacement represent by Eq. (12) [28].

**6.5 Relation between stress and strain**

It may express as:

**6.6 Global stiffness matrix**

global stiffness is given as:

Where,

**71**

**Figure 8.**

*Nixi*

*Niyi*

(11)

(12)

*Nizi*

*u*1 *v*1 *u*2 *v*2 … *un vn*

Δ*σ* ¼ *DT*Δ*ε* (13)

Where,

*Di* is the elasticity matrix; *Bi* is the geometry matrix which is determined from the relation between displacement and strain.

In FEM the material properties of different materials can easily feed into FEM by assigning different properties to different elements distinctly.

### **6.1 Finite elements**

The element may be in numerous forms i.e. one dimensional, two dimensional and three dimensional elements. One dimensional element having cross-sectional area and usually denoted by line sections or segment. Two dimensional element fields consist of triangle and quadrilateral. Three dimensional element field described by tetrahedron and parallelepiped. Some element shapes and node position used in two dimensional element fields [28] (**Figure 8**).

#### **6.2 Shape function**

It is the displacement within the element at any point when related to the displacement of the nodes. For instance the displacement of u and v within the quadrilateral element at any point represented by Eq. (10).

$$
\begin{bmatrix} u \\ v \\ v \end{bmatrix} = \begin{bmatrix} N\mathbf{1} & \mathbf{0} & N\mathbf{2} & \mathbf{0} & N\mathbf{3} & \mathbf{0} & N\mathbf{4} & \mathbf{0} \\\\ \mathbf{0} & N\mathbf{1} & \mathbf{0} & N\mathbf{2} & \mathbf{0} & N\mathbf{3} & \mathbf{0} & N\mathbf{4} \end{bmatrix} \begin{bmatrix} u\mathbf{1} \\ v\mathbf{i} \\ u\mathbf{2} \\ v\mathbf{2} \\ u\mathbf{3} \\ v\mathbf{3} \\ u\mathbf{4} \\ u\mathbf{4} \\ v\mathbf{4} \end{bmatrix} \tag{10}
$$

Where,

*u1, v1 … .u4, v4* are nodal displacement and *N1-N4* are shape function and that are connected with the nodes 1–4 correspondingly.

#### **6.3 Coordinate transformation**

The shape function is additionally used for coordinate's alteration of element in order to simplify the integration for calculation of stiffness matrix of some quantities for element. The coordinates (*x, y, z*), within the element of a point represented by Eq. (11) [28].

*Design Techniques in Rock and Soil Engineering DOI: http://dx.doi.org/10.5772/intechopen.90195*

Where,

*Slope Engineering*

unidentified variables, *f*

is given by Eq. (9).

Where,

**6.1 Finite elements**

**6.2 Shape function**

*u v* � � � �

Where,

by Eq. (11) [28].

**70**

� � �

**6.3 Coordinate transformation**

� <sup>¼</sup> *<sup>N</sup>*1 0 *<sup>N</sup>*<sup>2</sup> 0 *N*1 0

are connected with the nodes 1–4 correspondingly.

*Keij* is the coefficient matrix, ue

condition, *K* is the global stiffness matrix.

the relation between displacement and strain.

*e*

*Ke ij* ¼ ð Ω*i*

assigning different properties to different elements distinctly.

tion used in two dimensional element fields [28] (**Figure 8**).

quadrilateral element at any point represented by Eq. (10).

�

<sup>i</sup> vector is the nodal value vector having

*<sup>i</sup>* is consist of body force contribution and initial boundary

(9)

*Keij* is also called the element stiffness matrix in term of elasticity problem which

*Di* is the elasticity matrix; *Bi* is the geometry matrix which is determined from

In FEM the material properties of different materials can easily feed into FEM by

The element may be in numerous forms i.e. one dimensional, two dimensional and three dimensional elements. One dimensional element having cross-sectional area and usually denoted by line sections or segment. Two dimensional element fields consist of triangle and quadrilateral. Three dimensional element field described by tetrahedron and parallelepiped. Some element shapes and node posi-

It is the displacement within the element at any point when related to the displacement of the nodes. For instance the displacement of u and v within the

" #

*u1, v1 … .u4, v4* are nodal displacement and *N1-N4* are shape function and that

The shape function is additionally used for coordinate's alteration of element in order to simplify the integration for calculation of stiffness matrix of some quantities for element. The coordinates (*x, y, z*), within the element of a point represented

0 *N*3 0 *N*4 0 *N*2 0 *N*3 0 *N*4

*u*1 *vi u*2 *v*2 *u*3 *v*3 *u*4 *v*4

(10)

*Bi* ½ � *Ni* ð Þ ½ � *<sup>T</sup> Di* ½ � *Bi* ½ �*d*<sup>Ω</sup>

**Figure 8.** *Some element forms and node position used in two dimensional [28].*

$$\begin{aligned} \mathbf{x} &= \sum\_{i=1}^{n} N\_i \mathbf{x}\_i \\ \mathbf{y} &= \sum\_{i=1}^{n} N\_i \mathbf{y}\_i \\ \mathbf{z} &= \sum\_{i=1}^{n} N\_i \mathbf{z}\_i \end{aligned} \tag{11}$$

#### **6.4 Relation between strain and displacement**

For two dimensional element domain the relation between strain and displacement represent by Eq. (12) [28].

$$e = \begin{bmatrix} \varepsilon\_x \\ \varepsilon\_y \\ \varepsilon\_y \\ \gamma\_{xy} \\ \varepsilon\_x \end{bmatrix} = B \begin{bmatrix} u1 \\ v1 \\ u2 \\ v2 \\ \cdots \\ un \\ vm \end{bmatrix} \tag{12}$$

#### **6.5 Relation between stress and strain**

It may express as:

$$
\Delta \sigma = D\_T \Delta \varepsilon \tag{13}
$$

Where,

Δ*σ* is the vector of stress components, Δ*ε* represents corresponding components of strains and DT is a square matrix that is constant in the elastic case.

#### **6.6 Global stiffness matrix**

It is formed when added the stiffness matrices of all elements. The equation for global stiffness is given as:

$$K\Delta\delta = \Delta R\tag{14}$$

**Conflict of interest**

**Author details**

Zahid Ur Rehman<sup>1</sup>

and Bushra Nawaz<sup>1</sup>

Peshawar, Pakistan

Peshawar, Pakistan

**73**

provided the original work is properly cited.

\*, Sajjad Hussain<sup>1</sup>

, Noor Mohammad<sup>1</sup>

1 Department of Mining Engineering, University of Engineering and Technology

2 Department of Civil Engineering, University of Engineering and Technology

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: engr.zahid@uetpeshawar.edu.pk

, Akhtar Gul<sup>2</sup>

We have no conflict of interest.

*Design Techniques in Rock and Soil Engineering DOI: http://dx.doi.org/10.5772/intechopen.90195*

**Notes/thanks/other declarations**

Thanks and warm regards.

Where.

Δ*δ* is unknown vector having increments of nodal displacement due to increment force Δ*R*.

For material linear elastic material behavior the equation may by write as (Scheldt, 2002).

$$K\delta = R \tag{15}$$
