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Finite Element Analysis

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What is FEA ? 

The FEA method gives an approximate solution for certain problems

in engineering and science

It is mainly used for problems for which no exact solution is

available

It is a numerical rather than an analytical method

One of the first applications of FEA was to find the stresses and

strains in engineering components under load

FEA requires large computation resources when applied to any

realistic model of an engineering component

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What is FEA ? 

The development of the FE method has depended on the availability

of suitable digital computers for it to run on and graphics technology

to display the results

The FE method can be applied to a wide range of material

properties, linear-elastic (Hookean) behaviour and behaviour

involving deviation from Hooke’s law (plasticity or rubber elasticity)

Many comprehensive general-purpose FE systems are now

available

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What is FEA ? There are more specialised packages for particular applications

(e.g. steel framework, piping)

Depending on the type and complexity of the analysis, large

analyses may benefit from multiple processors

FEA can be applied to one-dimensional (line), 2-D (area) or 3-D

(volume) problems

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What is FEA ? The FE method is now applied to problems involving a wide range

of phenomena

 structural and stress analysis

dynamics and vibrations

heat conduction

fluid mechanics

electrostatic

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What is FEA ? Any continuous problem can be divided up into a number of smaller

areas or volumes which are called finite elements. The process is

called discretisation and the assembly of elements is called the

mesh

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Element types

Elements can be various shapes some examples are

In two dimensions, quadrilateral or triangular

In three-dimensions, brick-shaped, wedge-shaped or tetrahedral

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Linear Elastic Analysis

The displacement is the quantity that is first found in the analysis at

a series of points called nodes

Nodes are placed at the corners of the elements and often at the

midside – depending on the element formulation

Nodes on the boundaries of adjacent elements must be common to

the elements that meet there

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There are a number of incorrect ways of specifying a finite element

mesh

Linear Elastic Analysis

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Linear Elastic Analysis 

The analysis calculates the displacement at the nodes for the

particular loading applied to the FE model

The displacements are found for a finite number of points, namely,

the nodes

The displacement of each particular node is defined in terms of the

displacements of the other nodes of the element

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Linear Elastic Analysis 

The displacement of each node in a 2-D element has two

components:

one parallel to a reference x axis

one parallel to the y axis

these are called degrees of freedom

Each node has two degrees of freedom associated with it in a 2-D

model and three for a 3-D bricked-shaped element

The computer time and the cost of the analysis increases as the

number of degrees of freedom in the model is increased

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Linear Elastic Analysis 

The solution procedure for the displacement finite element method

allows the user to determine a number of unknowns, these are

obtained in the following order:

Firstly calculate the nodal displacements & reactions

the program finds the corresponding strains

from the strains the stresses are computed

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Pre-processing 

Pre-processing is concerned with the creation of the finite element

model and the definition of the way in which it is to be constrained

and loaded

The model can be built up within the pre-processor itself

The success of the analysis depends on:

The simplifications introduced into the model as compared with

the real thing

The choice of the mesh and type of elements to be used

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Pre-processing 

The data for the model must contain information about:

 Appropriate mechanical or physical properties must be allocated

to the elements – e.g. the material of which the structure is

made, plate thickness etc.

Restrictions to the movement of certain nodes (constraints)

The geometry and finite element mesh

Control information for the analysis (type of analysis, etc)

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Pre-processing 

Additional information in the input file specifies the type of analysis,

the way it is to be carried out and the output required

e.g. static analysis

solver to be used

Output required - displacements, stresses at nodes, etc

 With some systems, an input file can be produced without the use

of a pre-processor if the model is particularly simple

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Analysis 

The FE system reads in the data from the input file

It carries out certain checks on this information

If there are no errors in the input file, the analysis is carried out and

output is produced

These files can be examined and the relevant information extracted

The post-processor can be used to present this information in a

user-friendly manner

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Post-processing 

The post-processor takes in the information from the output files and

presents it in a range of different graphical and tabular forms

Colour bands may be used to indicate the value of stress on the

surface of the component

Contour lines of equal stress may be drawn

The model may be rotated and examined from different viewpoints

Point values may be queried

Graphs may be plotted

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The FEA Developer’s View 

The starting point is represented by the differential equations that

govern the phenomena to be modelled

The FEA is then one method of providing numerical solutions to

these equations when analytical solutions are not available

Much of the advanced work in this field is not accessible to the non-

expert

Once the equations for the FE formulation of a problem have been

developed, efficient procedures (algorithms) and programmes have

to be developed