The document provides an overview of finite element analysis (FEA), a numerical method used for solving engineering and mathematical physics problems, particularly those involving complicated geometries and material properties. It outlines the steps involved in FEA, including discretization, the assembly of elements, and the implementation of boundary conditions, alongside its advantages and limitations. The document also discusses various applications of FEA across multiple engineering disciplines and lists available commercial software packages for its implementation.

2. FUNDAMENTALS OF COMPUTER AIDED ENGINEERING
FINITE ELEMENT ANALYSIS : INTRODUCTION
UNIT III – FINITE ELEMENT ANALYSIS

3. By
Prof.(Dr) D Y Dhande
Professor
Department of Mechanical Engineering
AISSMS College Of Engineering, Pune – 411001
Email:dydhande@aissmscoe.com

4. FINITE ELEMENT ANALYSIS : AN INTRODUCTION
CONTENTS
• Need of finite element analysis
• Introduction to approaches used in Finite Element Analysis such as direct
approach and energy approach
• Boundary conditions: Types
• Rayleigh-Ritz Method
• Galerkin Method

5. FINITE ELEMENT ANALYSIS : AN INTRODUCTION
NEED OF FINITE ELEMENT ANALYSIS
• The Finite Element Analysis (FEA) is a numerical method for solving problems of
engineering and mathematical physics.
• Modern Mechanical design involves complicated shapes, sometimes made of different
materials.
• Useful for problems with complicated geometries, loadings, and material properties
where analytical solutions can not be obtained.
• The finite element analysis is based upon the idea of building a complicated object
into small pieces . Applications of the FEM can be found everywhere in day to day life
as well as engineering.

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7. FINITE ELEMENT ANALYSIS : AN INTRODUCTION
FINITE ELEMENT METHOD
• In FEM, the body ( or structure) is divided into finite number of smaller units
known as elements. This process is known as discretization.
• Describe the behaviour of physical quantities on each element.
• Connect (assemble) the elements at the nodes to form an approximate
system of equations for the whole structure.
• Solve the system of equations involving unknown quantities at the nodes.
(e.g. displacements)
• Calculate desired quantities (strains and stresses) at selected elements.

8. FINITE ELEMENT ANALYSIS : AN INTRODUCTION
• Example :

9. FINITE ELEMENT ANALYSIS : AN INTRODUCTION
STAGES OF FINITE ELEMENT METHOD
Basic Steps:
• Discretization
• Selection of approximation of functions
• Formation of elemental stiffness matrix
• Formation of total stiffness matrix
• Formation of element loading matrix
• Formation of total loading matrix
• Formation of overall equilibrium equation
• Implementation of boundary condition
• Calculation of unknown nodal displacements
• Calculation of stresses and strains

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Phases:
• Pre–Processing:
Here a finite element mesh is developed to divide the given geometry into
subdomains for mathematical analysis and the material properties are
applied and also the boundary conditions.
• Solution:
In this phase governing matrix equations are derived and the solution for the
primary quantities is generated.
• Post-Processing:
In the last phase, checking of the validity of the solution generated ,
examination of the values of primary quantities such as displacement and
stresses, errors involved is carried out.

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ADVANTAGES OF FINITE ELEMENT ANALYSIS
• Can readily handle complex geometry.
• Can handle complex analysis types like vibration, heat transfer, fluids etc.
• Can handle complex loading:
i. Node-based loading (point loads).
ii. Element-based loading (pressure, thermal, inertial forces).
iii. Time or frequency dependent loading.
• Can handle complex restraints: Indeterminate structures can be analyzed.
• Can handle bodies comprised of nonhomogeneous materials: Can handle
bodies comprised of non-isotropic materials: Orthotropic & Anisotropic.
• Special material effects are handled such as temperature dependent
properties , plasticity , creep , swelling etc.

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LIMITATIONS OF FINITE ELEMENT ANALYSIS
• A specific numerical result is obtained for a specific problem.
• The FEM is applied to an approximation of the mathematical model of a system (the
source of so-called inherited errors).
• Experience and judgment are needed in order to construct a good finite element
model.
• A powerful computer and reliable FEM software are essential.
• Input and output data may be large and tedious to prepare and interpret.
• Numerical errors such as the limitation of the number of significant digits, rounding
–off occur very often.
• Fluid elements with boundaries at infinity can be computed and treated by using
boundary element method.

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APPLICATIONS OF FINITE ELEMENT ANALYSIS
• Mechanical/Aerospace/Civil/Automotive Engineering
• Structural/Stress Analysis
• Static/Dynamic
• Linear/Nonlinear
• Fluid Flow
• Heat Transfer
• Electromagnetic Fields
• Soil Mechanics
• Acoustics
• Biomechanics

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AVAILABLE COMMERCIAL SOFTWARE PACKAGES FOR FEM
• ANSYS
• NASTRAN
• PATRAN
• NISA / DISPLAY III
• LS DYNA
• HYPERMESH
• CATIA
• Pro-E(CREO)
• SOLID WORKS
• COSMOS
• COMSOL
• DYNA-3D
• ABAQUS

16. FINITE ELEMENT ANALYSIS : AN INTRODUCTION
BOUNDARY CONDITIONS
• Boundary Conditions : The values of variables specified on the boundaries of the
body (or structure) are called as boundary conditions.
• Above figure shows a cantilever beam AB subjected to a uniformly distributed load.
• Let, p =uniformly distributed load acting on the cantilever beam, N/m
y = vertical deflection of the cantilever beam at a distance x, m
= length of the beam , m
E = modulus of elasticity of the beam material, N/mm2
I = moment of inertia of the beam cross section about neutral axis, m4
l

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TYPES OF BOUNDARY CONDITIONS
• There are two types of boundary conditions :
(i) Geometric (Essential) boundary conditions
(ii) Force (Natural ) boundary conditions.
(1) Geometric (Essential) boundary conditions :
• In structural mechanics problems, the geometric or essential boundary conditions include
: specified displacements and slopes. The geometric or essential boundary conditions are
also known as kinematic boundary conditions.
• For above example, the geometric boundary conditions are:
At A (i.e. x = 0 ) are : displacement y = 0 and slope (dy/dx) = 0
(2) Force (Natural ) boundary conditions :
• In structural mechanics problems, the forced or natural boundary conditions include :
specified forces and moments. The force or natural boundary conditions are also known
as specified forces and moments. The force or natural boundary conditions are also

18. FINITE ELEMENT ANALYSIS : AN INTRODUCTION
known as static boundary conditions.
• Example of force boundary conditions: For above example, the force boundary conditions
at B ( i.e. X = l) are:
• Solution to Differential Equations:
• The solution to the differential equation must satisfy the above boundary conditions at A
and B as follows:
(i) the geometric boundary conditions at A (i.e. x = 0 )
(ii) the force boundary conditions at B ( i.e. X = l)
2
2
3
3
: 0
: 0
d y
bending moment EI
dx
d y
shear force EI
dx



19. FINITE ELEMENT ANALYSIS : AN INTRODUCTION
APPROXIMATE NUMERICAL METHODS
• For complex engineering problems, it is extremely difficult and many a times
impossible to obtain the exact analytical solution which satisfies all prescribed
boundary conditions.
• In such cases, approximate numerical methods are used as alternative methods of
finding solutions. These are:
( i) Variational ( Functional Approximation) Methods: Rayleigh- Ritz method and
Galerkin Method
(ii) Finite Element Method
(iii) Finite difference method.

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