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Finite element method (FEM)is a numerical technique for solving boundary value problems in which a large domain is divided into smaller pieces or elements. The solution is determined by asuuming certain ploynomials. The small pieces are called finite element and the polynomials are called shape functions.

? Since the properties of each element are evaluated separately differnt material properties can be incorporated for each element.

? There is no restriction in the shape of the medium.

? Any type of boundary condition can be adopted.

?The computational cost is high.

?The solution is approximate and several checks are required.

? Local or element coordinates

? Natural coodinates

? Simple natural coodinates

? Area coordiantesor Triangular coordiantes

? Generalised coordinates

? Discretization of the structure

? Selection of suitable displacement fuction

? Finding the element properties

? Assembling the element properties

? Applying the boundary conditions

? Solving the system of equations

? Computing additional results

Discretization is the process of subdividing the given body into a number of elements which results in a system of equivalent finite elements.

?The geometry of the body

?The number of independent space coordinates

?The nature of stress variation expected

Displcement function is defined as simple functions which are assumed to approximate the displacements for each element. They may assumed in the form of poynomials, or trignometrical functions.

**The various terms used in FEM are explained below:**

**? Finite element - **Small elements used for subdividing the given domain tobe analysed are called finite elements. The seelements may be 1D, 2D or 3D elements depend in on the type of structure.

**? Nodes and nodal points - **The intersection of the differnt sides of elements are called nodes. Nodes are of two types - external nodes and internal nodes.

**O External nodes -** The nodal point connecting adjacent elements.

**O Internal nodes-** The extra nodes used to increase the accuracy of solution.

**? Nodal lines -** The interface between elements are called nodal lines.

**? Continuum -** The domain in which matter exists at every point is called a continuum. It can be assumed as having infinite number of connected particles.

**? Primary unknowns -** The main unknowns involved in the formulation of the element properties are known as primary unknowns.

**? Secondary unknowns -** These unknowns are derived from primary unknowns are known as secondary unknowns. In displacement formulations, displacements are treated as primary unknowns and stress, strain, moments and shear force are treated as secondary unknowns.

**The various elements used in FEM are classified as:**

? One dimensional elements(1D elements)

? Two dimensional elements(2D elements)

? Three dimensional elements(3D elements)

3-D elements are used for modeling solid bodies and the various 3-Delements are tetrahedron, hexa hedron, and curved rectangular solid.

Axisymmetric elements are obtained by rotatinga1-D line about an axis. Axisymmetric elements are shown in the figure below.

Shape function is also called an approximate function or an interpolation function whose value is equal to unity at the node considered and zeros at all other nodes. Shape function is represented by Ni where i =nodeno.

**The properties of shape functions are:**

? Theno of shape functions will be equal to theno of nodes present in the element.

? Shape function will have a unit value at the node considered and zero value at other nodes.

? The sum of all the shape function is equal to 1. i. e. SNi =1

Element aspect ratio is defined as the ratio of the largest dimension of the element to its smallest dimension.

**The possible locations for nodes are:**

?Point of application of concentrated load.

?Location where there is a change in intensity of loads

?Locations where there are discontinuities in the geometry of the structure

?Interfaces between materials of different properties.

**Displacement functions should have the following characteristics:**

The displacement field should be continuous.

? The displacement function should be compatible between adjacent elements

? The displacement field must represent constant strain states of elements

? The displacement function must represent rigid body displacements of an element.

Plane strain is a state of strain in which normal strain and shear strain directed perpendicular to the plane of body is assumed to be zero.