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Introduction of Finite Element Analysis
1. ME8692--FINITE ELEMENT ANALYSIS
UNIT I - INTRODUCTION
Historical Background – Mathematical Modeling of field
problems in Engineering – Governing Equations – Discrete and
continuous models – Boundary, Initial and Eigen Value
problems– Weighted Residual Methods – Variational
Formulation of Boundary Value Problems – Ritz Technique –
Basic concepts of the Finite Element Method.
4. Analytical methods
Problems are expressed by mathematical differential
equations.
It is used only for simple geometries and loading
conditions.
5. Numerical methods (or)
Approximate methods
Problems involving complex material properties and
boundary conditions.
The following three methods are coming under
numerical solutions.
Functional Approximation
Finite Difference Method (FDM)
Finite Element Method (FEA)
6. Finite Element Method (FEA)
Finite element method is a numerical method for solving
problems of engineering and mathematical physics.
In this method, a body or structure in which the analysis to
be carried out is subdivided into smaller elements of finite
dimensions called finite elements. Then the body is
considered as an assembly of these elements connected at a
finite number of joints called Nodes.
The properties of each type of finite element is obtained
and assembled together and solved as whole to get
solution.
This method extensively used in the field of structural
mechanics, fluid mechanics , heat transfer, mass transfer,
electric and magnetic fields problems
7. Based on application, finite element problems are
classified as follows
Structural Problems
Displacement,Stress and Strain in each element can be
calculated
Non-structural Problems
Temperature (or) Fluid pressure at each nodal point is
obtained
8. General Steps of the Finite Element
Analysis
The following two general methods are associated with
the FEA.
Force Method : Internal forces are considered as the
unknowns of the problem
Displacement or stiffness method : Displacement of the
nodes are considered as the unknowns of the problem.
9. Step 1 : Discretization of structure
The art of subdividing a structure into a convenient
number of smaller elements is known as discretization.
Smaller elements are classified as follows
One dimensional elements
Two dimensional elements
Three dimensional elements
Axis symmetric elements
10. One dimensional elements
A bar and beam elements are considered as one
dimensional elements.
The simplest line element also known as linear element
has two nodes, one at each end as shown in figure.
13. Axis symmetric elements
The axis symmetric element is developed by rotating a
triangle or quadrilateral about a fixed axis located in
the plane of the element through 360°.
14. Step 2 : Numbering of Nodes and
Elements
The nodes and elements should be numbered after
discretization process. The numbering process is most
important since it decide the size of the stiffness matrix
and it leads the reduction of memory requirement.
15.
16.
17.
18.
19. Step 3 : Selection of a Displacement
Function or Interpolation Function
20.
21. Step 4 : Define the material behaviour by using
Strain-Displacement and Stress-Strain
Relationships
22. Step 5 : Derivation of element stiffness
matrix and equations
23.
24. Step 6 : Assemble the element equations
to obtain the global or total equations
25. Step 7 : Applying Boundary Conditions
From the above equation, Global stiffness matrix is a
singular matrix. Boundary conditions are applied in
that matrix.
26. Step 8 : Solution for the unknown
displacements
The unknown displacements {u} are calculated by
using Gaussian elimination method or Gauss Seidel
method.
27. Step 9 : Computation of the element strains
and stresses from the nodal displacements
28.
29. Step 10 : Interpret the results
Analysis and evaluation of the solution results is
referred to as post processing. Post processor
computer programs help the user to interpret the
results by displaying them in graphical form.
32. Discretization
The art of subdividing a structure into a convenient
number of smaller components is known as
Discretization. These smaller components are then put
together.
The process of uniting the various elements together is
called Assemblage. The assemblage of such elements
then represents the original body.
Discretization can be classified as follows
Natural
Artificial (Continuum)
33. Natural Discretization
In structural analysis, a truss is considered as a natural
system. The various members of the truss constitute
the elements. These elements are connected at various
joints is known as nodes