This document outlines the objectives, course outcomes, and units covered in a finite element analysis course. The objective is to equip students with fundamentals of FEA and introduce the steps involved in discretization, applying boundary conditions, assembling stiffness matrices, and solving problems. The course covers basic FEA concepts, one-dimensional elements, two-dimensional elements, axisymmetric problems, isoparametric elements, and dynamic analysis. Students will learn to formulate and solve structural and heat transfer problems using various finite elements.

1. 17MEC21
FINITE ELEMENT
ANALYSIS
NANDHA ENGINEERING
COLLEGE,ERODE-52

2. OBJECTIVE:
•To equip the students with the finite element analysis fundamentals
•To enable the students to formulate the design problems using Finite
Element Analysis
•To introduce the steps involved in discretization, application of
boundary conditions, assembly of stiffness matrix and solution
COURSE OUTCOMES:
At the end of the course, the students will be able to
CO1: Formulate the mathematical model for solution of engineering
design problems
CO2: Solve heat transfer and structural problems using 2D
elements
CO3: Explain the stages in solving engineering problems under
axisymmetric condition
CO4: Analyze and solve the real time problems using isoparametric
elements
CO5: Determine the solution for real time 1D structural problems
using structural dynamic analysis

3. UNIT I : BASIC CONCEPTS AND 1D
ELEMENTS
Basic concepts - general procedure for
FEA - discretization - weak form - weighted
residual method - Ritz method- applications
- finite element modeling - coordinates -
shape functions - stiffness matrix and
assembly - boundary conditions - solution of
equations - mechanical loads, stresses and
thermal effects - bar and beam elements

4. UNIT II : 2D ELEMENTS (6+6)
Finite element modeling - Poisson
equation - Laplace equation -
plane stress, plane strain - CST
element -element equations, load
vectors and boundary conditions –
Pascal’s triangles - assembly -
application in two dimensional heat
transfer problems

5. UNIT III : AXISYMMETRIC PROBLEMS (6+6)
Vector variable problems - elasticity
equations - axisymmetric problems -
formulation - element matrices -assembly
- boundary conditions and solutions
UNIT IV : ISOPARAMETRIC ELEMENTS (6+6)
Isoparametric elements - four node
quadrilateral element - shape functions -
Jacobian matrix - element stiffness matrix
and force vector - numerical integration -
stiffness integration - displacement and
stress calculations

6. UNIT V : DYNAMIC ANALYSIS
(6+6)
Types of dynamic analysis - general
dynamic equation of motion, point and
distributed mass - lumped and
consistent mass - mass matrices
formulation of bar and beam element -
undamped - free vibration - eigen
value and eigen vectors problems

7. TEXT BOOKS:
1. S S Rao, “The Finite Element Method in Engineering”, 5th
ed., Elsevier, 2012
2. Chandrupatla.T.R and Belegundu.A.D, “Introduction to Finite
Elements in Engineering”, 4th ed., Pearson Education, New
Delhi, 2015
REFERENCES:
1. Seshu. P, “A Text book on Finite Element Analysis”, 1st
ed., PHI Learning Pvt. Ltd., New Delhi, 2009
2. David V Hutton, “Fundamentals of Finite Element Analysis”,
1st ed., Tata McGraw Hill International Edition, 2005
3. Jalaludeen.S.Md “Finite Element Analysis in
Engineering”, 5th ed., anuradha publications, 2013
4. Reddy J.N, “An Introduction to Finite Element Method”, 3rd
ed., McGraw Hill International Edition, 2005.
5. Zienkiewicz. O.C and Taylor, R.L, “The Finite Element
Method: Its basis and fundamentals”, 7th ed., Elsevier, 2013

8. History of Finite Element Methods
• 1941 – Hrenikoff proposed framework method
• 1943 – Courant used principle of stationary potential energy
and piecewise function approximation
• 1953 – Stiffness equations were written and solved using digital
computers.
• 1960 – Clough made up the name “finite element method”
• 1970s – FEA carried on “mainframe” computers
• 1980s – FEM code run on PCs
• 2000s – Parallel implementation of FEM (large-scale analysis,
virtual design)
Courant Clough

9. ⚫ To reduce the amount of prototype
testing.
⚫ Computer Simulation allows multiple
“what if “scenarios to be tested quickly
and effectively.
⚫ To simulate designs those are not
suitable for prototype testing. E.g.
Surgical Implants such as an artificial
knee.
Why is FEA needed?

10. THREE STAGES OF
FEA/ANSYS
⚫ Preprocessing: defining the problem; the
major steps in preprocessing are given
below:
⚫ Define key points/lines/areas/volumes
⚫ Define element type and
material/geometric properties
⚫ Mesh lines/areas/volumes as required
⚫ The amount of detail required will depend
on the dimensionality of the analysis (i.e.
1D, 2D, axi-symmetric, 3D).

11. ⚫ Solution: assigning loads, constraints and
solving; here we specify the loads (point or
pressure), constraints (translational and
rotational) and finally solve the resulting set of
equations.
⚫ Post processing: further processing and
viewing of the results; in this stage one may
wish to see:
⚫ Lists of nodal displacements
⚫ Element forces and moments
⚫ Deflection plots
⚫ Stress contour diagrams

12. INTRODUCTION
⚫ A fundamental premise of using the finite
element procedure is that the body is
sub-divided up into small discrete regions
known as finite elements.
⚫ These elements defined by nodes and
interpolation functions. Governing
equations are written for each element and
these elements are assembled into a global
matrix. Loads and constraints are applied
and the solution is then determined.

14. ELEMENTS
1.small portion of a system 2.Definite shape 3. Should have min two nodes
4.Loads act only at the nodes
One-Dimensional Elements
Line
Rods, Beams, Trusses,
Frames
Two-Dimensional Elements
Triangular, Quadrilateral
Plates, Shells, 2-D
Continua
Three-Dimensional Elements
Tetrahedral, Rectangular Prism
(Brick)
3-D Continua

15. ELEMENT TYPES

16. Discretization Concepts

17. Development of Finite Element Equation
• The Finite Element Equation Must Incorporate the Appropriate Physics of the
Problem
• For Problems in Structural Solid Mechanics, the Appropriate Physics Comes from
Either Strength of Materials or Theory of Elasticity
• FEM Equations are Commonly Developed Using Direct, Variational-Virtual Work or
Weighted Residual Methods
Variational-Virtual Work Method
Based on the concept of virtual displacements, leads to relations between internal
and external virtual work and to minimization of system potential energy for
equilibrium
Weighted Residual Method
Starting with the governing differential equation, special mathematical operations
develop the “weak form” that can be incorporated into a FEM equation. This method
is particularly suited for problems that have no variational statement. For stress
analysis problems, a Ritz-Galerkin WRM will yield a result identical to that found by
variational methods.
Direct Method
Based on physical reasoning and limited to simple cases, this method is worth
studying because it enhances physical understanding of the process

18. Basic Steps in the Finite Element Method
Time Independent Problems
- Domain Discretization
- Select Element Type (Shape and Approximation)
- Derive Element Equations (Variational and Energy Methods)
- Assemble Element Equations to Form Global System
[K]{U} = {F}
[K] = Stiffness or Property Matrix
{U} = Nodal Displacement Vector
{F} = Nodal Force Vector
- Incorporate Boundary and Initial Conditions
- Solve Assembled System of Equations for Unknown Nodal
Displacements and Secondary Unknowns of Stress and
Strain Values

19. TOPICS COVERED
⚫ GENERIC FORM OF FINITE
ELEMENT EQUATIONS
1.RAYLEIGH RITZ METHOD
2. WEIGHTED RESIDUAL
METHOD
3.BAR ELEMENT.

21. One Dimensional Examples
Static Case
1 2
u1
u2
Bar Element
Uniaxial Deformation of Bars
Using Strength of Materials
Theory
Beam Element
Deflection of Elastic Beams
Using Euler-Bernouli Theory
1 2
w1
w2
θ2
θ1

22. GENERAL STEPS OF THE FINITE ELEMENT
ANALYSIS
⚫ Discretization of structure > Numbering of Nodes
and Elements > Selection of Displacement
function or interpolation function > Define the
material behavior by using Strain – Displacement
and Stress – Strain relationships > Derivation of
element stiffness matrix and equations >
Assemble the element equations to obtain the
global or total equations > Applying boundary
conditions > Solution for the unknown
displacements > computation of the element
strains and stresses from the nodal displacements

23. Advantages of Finite Element Method
⚫ 1. FEM can handle irregular geometry in a
convenient manner.
⚫ 2. Handles general load conditions without
difficulty
⚫ 3. Non – homogeneous materials can be handled
easily.
⚫ 4. Higher order elements may be implemented.
Disadvantages of Finite Element Method
⚫ 1. It requires a digital computer and fairly extensive
⚫ 2. It requires longer execution time compared with

24. APPLICATIONS OF FINITE ELEMENT ANALYSIS
⚫ Structural Problems:
⚫ 1. Stress analysis including truss and frame analysis
⚫ 2. Stress concentration problems typically associated
with holes, fillets or other changes in geometry in a
body.
⚫ 3. Buckling Analysis: Example: Connecting rod
subjected to axial compression.
⚫ 4. Vibration Analysis: Example: A beam subjected to
different types of loading.
⚫ Non - Structural Problems:
⚫ 1. Heat Transfer analysis: Example: Steady state thermal
analysis on composite cylinder.
⚫ 2. Fluid flow analysis: Example: Fluid flow through

25. BASED ON FEA
What is ANSYS?
•General purpose finite element modeling package for numerically solving a
wide variety of mechanical problems.
What is meant by finite element?
•A small units having definite shape of geometry and nodes
What is the basic of finite element method?
•Discretization
State the three phases of finite element method.
• Preprocessing ,Analysis ,Post Processing
State the methods of engineering analysis?
•Experimental ,Analytical methods
What are the h versions of finite element method?
•The order of polynomial approximation for all elements and numbers of
elements are kept increased
What is Discretization?
The art of dividing a structure in to a convenient number of smaller components
Defining the Job name
Utility Menu>File>Change Job name

26. BASED ON FEA AND ANSYS
What is the effect of size and number of elements on the solution by
FEM.?
•Smaller size and more no. elements more FEA accuracy
How to improve accuracy of solution by FEM?
•By increasing no. of elements , by fine meshing, by choosing higher order
polynomial function
What is DOF?
•It is a variable that describes the behavior of a node in an element.
What is meshing?
•The minimum number of elements that give you a converged solution.
In how many methods ANSYS can be used?
•Graphical User Interface or GUI and Command files
What kind of hardware do need to run a ANSYS?
•A PC with a sufficiently fast processor, at least 2GB RAM, and at least 500 GB
of hard disk
Name any FEA software
1.ANSYS 2.NASTRAN 3.COSMOS 4.NISA 5.ASKA 6.DYNA 6.I-DEAS

27. •Finite element analysis.
Finite element method is a numerical method for of engineering, mathematical,
physics. In the finite element method, instead of solving the problem for the entire
body in one operation, we formulate the equations for each finite element and
combine them to obtain the solution of the whole body
•Finite element
A small unit having definite shape of geometry and nodes is called finite element.
•State the methods of engineering analysis.
There are three methods of engineering analysis. 1) Experimental method.2) Analytical
method.3) Numerical method or Approximate method.
•Types of boundary conditions
There are two types of boundary conditions; they are Primary boundary condition
.Secondary boundary condition.
•Structural and Non-structural problem
Structural problem: In structural problems, displacement at each nodal point is
obtained. By using these displacement solutions, stress and strain in each element can
be calculated.
Non Structural problem: In non structural problem, temperatures or fluid pressure at
each nodal point is obtained. By using these values, Properties such as heat flow, fluid
flow, etc for each element can be calculated.

28. •Name the weighted residual methods.
1.Point collocation method.2. Sub domain collocation method.3.Least square method
4. Gale kin’s method
•Rayleigh Ritz method.
Rayleigh Ritz method is a integral approach method which is useful for solving complex
structural problems, encountered in finite element analysis
•Total potential energy.
The total potential energy π of an elastic body, is defined as the sum of total strain energy
U and potential energy of the external forces,(W).
Total potential energy, π = Strain energy (U) - Potential energy of the external forces (W).
•Compare essential boundary conditions and natural boundary conditions.
There are two types of boundary conditions. They are:
1.Primary boundary condition (or) Essential boundary condition the boundary condition,
which in terms of field variable, is known as primary boundary condition.
2.Secondary boundary condition or natural boundary conditions: The boundary
conditions, which are in the differential form of field variables, are known as secondary
boundary condition.
•Compare boundary value problem and initial value problem
The solution of differential equation is obtained for physical problems, which satisfies some
specified conditions known as boundary conditions. The differential equation together with
these boundary conditions, subjected to a boundary value problem. The differential
equation together with initial conditions subjected to an initial value problem.

29. •Give examples for essential (forced or geometric) and
non-essential (natural) boundary conditions.
•geometric boundary conditions are displacement, slopes, etc.
•natural boundary conditions are bending moment, shear force, etc.
•Node or Joint
Each kind of finite element has a specific structural shape and is
interconnected with the adjacent elements by nodal points or nodes. At the
nodes, degrees of freedom are located. The forces will act only at nodes
and not at any other place in the element
What is meant by DOF?
When the force or reaction acts at nodal point, node is deformation. The
deformation includes displacement, rotations, and/or strains. These are
collectively known as degrees of freedom (DOF).
What is aspect ratio?
Aspect ratio is defined as the ratio of the largest dimension of to the
smallest dimension. In many cases, as the aspect ratio increases, the
inaccuracy of the solution increases. The conclusion of many researches is
that the aspect ratio should be close to unity as possible.

30. What are h and p versions of finite element method?
h version and p versions are used to improve the accuracy of the finite element
method.
In h versions, the order of polynomial approximation for all elements is kept constant
and the number of elements is increased.
In p version, the number of elements is maintained constant and the order of
polynomial approximation of element is increased. During Discretization,
Mention the places where it is necessary to place a node.
The following places are necessary to place a node during discretization process.
1. Concentrated load-acting point.
2. Cross section changing point
3. Different material inters junction point
4. Sudden change in load point.
•Steps in FEM.
• Discretization
• Selection of the displacement models
• Deriving element stiffness matrices
• Assembly of overall equations/ matrices
• Solution for unknown displacements
• Computations for the strains/stresses