HELD BY,HELD BY,
Department ofDepartment ofMechanical Engineering,Mechanical Engineering,
T.K.I.E.T.,T.K.I.E.T.,Warananagar.Warananagar.
SUBMITTED BY,SUBMITTED BY,
Mr. Digvijay D. Patil.Mr. Digvijay D. Patil.BE Mech.BE Mech.
RIT, SAKHARALE.RIT, SAKHARALE.
Mobile: 9766666871.
AA
PAPER PRESENTATIONON
FINITE ELEMENT METHODFINITE ELEMENT METHODDesign Optimization of Mini Baja Frame
FOR
“EUREKA-JIDNYASA 2K9”“EUREKA-JIDNYASA 2K9”ORGANISED BY,ORGANISED BY,
Tatyasaheb Kore Institute of Engineering and Technology,Tatyasaheb Kore Institute of Engineering and Technology, Warananagar.Warananagar.
1
Abstract:
Think. A Super Computer Analysis Model of a component. There is no prototype
of any component now. This will be possible because of the Miracle made by the
emerging technology and tremendous revolution in the analysis techniques. One of these
techniques is nothing but the Finite Element Method.
Before 1950’s the designs of the components, elements, parts or shapes of a
system were not implicating Finite Element Method. The engineers, scientists were not
able to specify where the design will fail and what will be the exact values of the stress &
strain. They were totally dependent on Trial & Error Method or Update Solve Update
Method. The replacement of any component before its failure or to manufacture a
component, which will sustain given loading conditions, it was required to have a definite
method to get correct solutions of the problems.
But, when in 1950’s the Finite Element Method was introduced in engineering
application, it started giving almost accurate results. The growth of this technique is
attributed to rapid advances in computer technology, particularly over last decade. The
Finite Element Method is becoming an extremely sophisticated tool in engineering
application due to its computerization. This method is widely accepted in many branches
of industries. Due to closeness towards exact solutions, the Finite Element Method is yet
not challenged or paralleled by any other Numerical Method.
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INDEX
3
Sr. No. Title Page No.
Abstract 2
1. Introduction 4
2. Finite Element Method
Setup the Field Variable
Discretization
Approximation Functions
Gradients of the Unknown Quantity and Constitution of Relationships
Elemental Equations
Global Equations
Solve Equations & Solutions
7
3. Computerization of Finite Element Method 10
4. CASE STUDY:
Design Optimization of Mini Baja Frame
(Mini Baja Roll Cage)
13
5. Finite Element Software Analysis of Frame 14
6. Advantages of FEM 17
7. Disadvantages of FEM 17
8. Future with FEM 18
9. Conclusion 18
10. References 19
1. Introduction
What is a Numerical Method?
Numerical Method is the mathematical technique used to enumerate the approximate
results for the given problem. There are several numerical methods. The most important
among them are,
1. Finite Difference Method
2. Finite Element Method
3. Boundary Element Method
4. Finite Volume Method
What is Finite Element Method?
The Finite Element Method is a technique used to derive an approximate solution of any
complex engineering problem that can be reached by subdividing the problem into
smaller, more manageable elements i.e. finite elements. The behavior of the structure can
be easily predicted by solving linear equation’s sets in the form of matrix algebra for
those finite elements.
The finite element method was introduced in 1950’s and has become an engineering tool
applied to various problems which yield approximate results. The problems includes the
complexities dealing with,
1. Geometry 2. Boundary Conditions 3. Loading Conditions
In mechanical problems the elements may be model membranes, beams, plates, solids,
fluids etc. This method contains conversion of the available component data into matrices
& interpolating differential equations and their processing’s.
There are main three types of problems that can be analyzed by FEM.
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Table 1
Sr. No. Problem Engineering Area Application
1.
Steady
State
Problem
Aerospace
Automobiles
Civil
Stress analysis of aircraft frames,
Wings etc.
Stress analysis of camshafts, cylinder
blocks, chassis, connecting rods etc.
Stress analysis of Dams, walls,
bridges, etc.
2. Eigen
Value
Problem
Aerospace
Automobile
Frequency analysis of engine
components, helicopter rotor
blades.etc.
Frequency analysis of gearbox casing
and body shell etc.
3. Transient
Problems
Automobile
Mechanical
Civil
Time dependent analysis of engine
piston
Analysis of impact problems &
dynamic crack propagation
Structural stress waves in rock
structures
Finite element is an approximating numerical method. So, obviously it is having errors.
The magnitude of the error depends on,
1. Type of Model
2. Size of the Model
3. Fitness of the Model
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Why Finite Element Method?
The Finite Element Method is being used in almost every engineering discipline like
automotives, biomedical, electronics, electrical, manufacturing, designing, civil etc. It
also can provide platform to heat mass transfer, dynamics, radiations problems.
Finite element method optimizes new designs, verifies fitness of existing designs,
predicts performance & evaluates new concept. In addition to this, FEM is also taking
applications in accident analysis, reconstruction and forensic investigation.
Why we use Stress Engineering?
The Stress Engineering is a leading provider of finite element analysis service to the
industries. There is lot of difference in engineering knowledge and industrial or practical
experience. The Stress Engineering crashes the band gap between knowledge and
experience by acting as a connecting link in association with Finite Element Method.
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Cross Section of the DAM
Discretization of the DAM
2. Finite Element Method (FEM)
When the Finite Element Method is applied to a problem, following tasks
are performed in step-by-step manner.
1. Setup the Field Variable:
The field variable is the governing variable and deals with the unknown
things which are to be evaluated. E.g. In structural problems, displacement is field
variable and in thermal problem, temperature is the field variable.
2. Discretization:
The points at which the primary
unknowns are required to be
evaluated are called ‘nodes’ or
‘nodal points’, and interfaces
between the elements are called
nodal lines (or nodal/planes/
surfaces). The number of
unknowns at a node is
termed as ‘Nodal degree of
freedom’ (DOF).
The most appropriate
element type is chosen for
the analysis required. The
total number of elements
used and their variation in
size and type within a given
body are primarily matters
of engineering judgment.
One may choose one
dimensional (1-D), two dimensional (2-D), three dimensional (3-D) or an axisymmetric
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element depending upon the physical system under consideration. The axisymmetric
elements are constructed by rotating a 2-D finite element about the axis of symmetry by
360°.
The discretization is the most important step in this method because, the
accuracy of the result depends on the details of the discretization and 90% of the time is
spent in this phase of analysis. It needs sound engineering knowledge and understanding
of physics of the problem to get meaningful results.
3. Approximation Functions:
This step involves choosing a pattern or shape for the distribution of the
unknown quantity within each element. The unknown quantity can be displacement for
stress-strain analysis, temperature in heat flow problems, fluid pressure or velocity for
fluid flow problems. The approximation function is defined within the element using the
nodal values of the element. Linear, quadratic and cubic polynomials are the frequently
used functions. For an n-node element the approximation function can be expressed as,
u = N1u1 + N2u2 + …… + Nnun
Where, u1, u2……un are the nodal unknowns and N1, N2……Nn are the shape functions.
4. Gradients of the Unknown Quantity and constitution of Relationships:
These relationships are necessary for deriving the equations for each finite
element. E.g. In stress analysis problems, gradients and constitutive relationships are
simply the strain displacement and the stress-strain expressions respectively.
Strain/displacement: ε x = du/dx Stress-strain: σx = E ε x (Hooke’s Law)
Where, ε x = Strain in x direction, σx = Stress in x direction, E = Modulus of elasticity.
5. Elemental Equations:
In this step, equations governing the behavior of a generic (typical) finite
element are obtained by invoking available laws and principles. These equations describe
a relationship between the nodal DOF and the nodal forcing parameters for the generic
element. This relationship can be written in compact matrix form.
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[K e] {δ e} = {f e}
[K e] = element property matrix, {δ e} = element vector of unknown DOF,
{f e} = vector of element nodal forcing parameters
6. Global Equations:
Repeated application of the generic element equation, results in the element
equations for other elements. Then the element equations are added together using
method of superposition to obtain global or total equations for the entire body. The
process of superposition is called ‘assembling’. This relationship can be written in
compact matrix form as,
[K] {δ} = {F}
[K] = assembled (global) property matrix, {δ} = assembled (global) vector of nodal
unknown, {F} = assembled (global) vector of element nodal forcing parameters
To evaluate the performance of the body, it is needed to impose boundary conditions on
it. Boundary conditions are the physical constrains or the supports that exist on the body.
7. Solve Equations & Solutions:
The assembled equations are then solved for the δ’s by using gauss elimination or
iterative method. δ’s are called primary unknowns because these are the first quantities
derived by FEM. After deriving primary unknowns are derived. These can be stresses,
strains and forces etc. in structural problems or velocity & discharge in fluid problems.
The derived primary and secondary unknowns are the solutions of the problem. The
results are then interpreted in the tabular form or graphically represented. This simplifies
the understanding of the problems and helps in design decisions.
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3. Computerization of Finite Element Method
The tremendous inventions and advancements in computer science & information
technology, it is assured the effective use of Finite Element Method to solve the
problems. The popular softwares based on FEM are,
Table 2
Software Area of Application
Ansys V 11.0 Structural, Heat, Electrical
NASTRAN Aerospace, Automobile
NISA Structural
ALGOR Stress Analysis
STAD PRO Civil
GT-STUDEL Structural
In these softwares, the finite element method is stored as embedded programs. These
programs can’t be modified or altered. The computer user is able to decide the input
parameters, discretization and nodal numbering. After giving all the required input data
the CPU processes matrix operations as that in FEM to get the solution. The output of the
problem can be displayed on the monitor screen. e.g. the output can be in the form of
nodal displacement, force value or deflection, stress value etc. in case of the structural
problems. This process is done with the help of following important parts of FEM
softwares.
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Preprocessors:
The preprocessor is used to develop the finite element model. There are two methods in
preprocessors to generate element mesh.
User should define individual nodes, elements and build the mesh manually.
A solid model of the component is designed in the CAD software. It is then
imported into the FEM software and the computer generates the mesh
automatically by auto mesh option.
The mesh is nothing but discretization of the component into element continuum. The
preprocessor requires following direct user input data.
1. Coordinate system:
For flexible model generation, it is required to define the coordinate system. Different
coordinate systems like Cartesian, cylindrical, spherical systems are available for global
and local coordinate systems allowing flexible model generation. The local coordinate
system defines the origin in any position with respect to component. E.g. defining an arc
is simpler in spherical and cylindrical system than Cartesian system.
2. Nodes & Elements:
Once the node has been defined it is easy to define a row of nodes by intermeeting the
coordinates. When nodes are generated, they are used to define the elements. The
numbering of the elements starts from the first node. The mesh tool in the FEM software
is so powerful that it can generate 2-D or 3-D meshes very quickly.
3. Geometrical & Material Constants:
This input data is fed to the computer by the user. The geometrical constants include
thickness of model, area, second moment of inertia of the component. The Material
constants are, Young’s Modulus, Poisson’s Ratio.
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4. Loads:
In order to get correct solutions of the problem, it is necessary to define the load on the
model correctly. The loads can be body force, pressure, thermal force etc. The
gravitational force is considered to be absent for easy calculations. The loads are
generally defined at the nodes of the element.
5. Meshing:
When the component model is ready with nodal numberings and loadings, it is meshed
by mesh tool in the software. The mesh size can be controlled manually or it can be auto
meshed. The preprocessor converts the local coordinates into global coordinate system
and also perform necessary analytical calculations of higher order matrices that will take
a lot of time for human approach with the help of solution tool.
Postprocessors:
After deriving the output data it is arranged into tabular form by postprocessor. In case of
structural problems, the nodal displacements & stress values at nodal points are given in
the tabular form. The graphical solutions can also be obtained in the form of graphs by
plot control tool in the FEM software. The output also contains the deformed shapes of
the analyzed model with different colors representing different stress values at different
locations of the components. The maximum stresses are denoted by red color, medium
stresses by yellow or green color and minimum stresses by blue color. Thus, we can
easily come to know where the failure is most likely to occur.
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The Components of Roll Cage (Frame)
Stress Pattern Band
4. Design Optimization of Mini Baja Frame (Roll Cage):
The purpose of the driver’s roll cage in Mini Baja is to prevent the driver who is wearing
a restraint system from being crushed or seriously injured in the event of an impact or a
rollover. The cage must be large enough for the driver’s comfort and should provide
safety supports to the driver.
The main components of the frame are, Rear Roll Hoop (RRH), Roll Hoop Overhead
members (RHO), Lower Frame Side members (LFS), Side Impact members (SIM) &
Front Bracing members (FBM).
5. Finite Element Software Analysis of Frame:
There are a few features of the design that may need some
additional strengthening. For this reason it is deemed that
there should be an analysis of front impact and side impact
(Major Impacts) by loading the frame for those kinds of the
load. However, before these analyses are performed the
loading forces exerted on the vehicle must be completely
defined. With the help of softwares, the stress distribution is
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Front Impact Stress Distribution(Enlarged View)
defined ranging from maximum to minimum values. Different stress values bear different
colors for easy distinguishing. The maximum stress is represented by red color and
minimum by blue color.
1. Front Impact:
Front Impact is assumed to occur when the worst case of collision happens by running
the vehicle into a stationary object. To validate this load the analysis is done using the
maximum speed of the vehicle and the target impact force is used to find the resulting
crash pulse, or deceleration time.
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Front Impact Stress Distribution-After Modification (Enlarged View)
Side Impact Stress Distribution(Enlarged View)
In this case a deceleration of 10 G’s is the assumed loading. This is equivalent to a 7500
lbf load on the vehicle. The analysis figure shows that the high stresses are induced in the
highlighted beam or link element. Then the frame can be predicted to fail at the corners
and centre of the red highlighted element.
Now suppose, if we add two links symmetrically & in diagonal direction, the stress
concentration is greatly reduced. So, this strategy is acceptable.
2. Side Impact:
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Front Bracing Modification
Side Impact Stress Distribution–After Modification (Enlarged View)
Side impact is most likely to occur with the
vehicle being hit by another Mini Baja
vehicle running on it’s neighbouring side. To
validate this load the analysis is done using the target impact force of 5G or 3750 lbf, to
find the resulting stress distribution. To reduce stress concentration we can add two more
links to convert it into structure so that it will sustain the load. This modification
increases the strength of the Lateral Cage of the Mini Baja Frame.
Real World Testing Results:
In real world testing or practical experience, we came to know that, front and side loading
were causing maximum stress region i.e. forecasting the region of failure. When we
modified the designs by addition of component links, the failure regions were almost
eliminated as discussed above.
Case Study Conclusion:
The usage of Finite Element Method (or FEA) is valuable to design the Mini Baja Frame
through Software package. The analysis allowed addition of four important components
to withstand Front and Side impacts. The Finite Element Method gave a very accurate
prediction of where the failure will occur. In practical phenomenon, at the same position,
the Mini Baja Frame was subjected to fail. So, results were almost matching.
6. Advantages of FEM:
● FEM’s biggest advantage is versatility.
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Side Bracing Modification
A variety of problems with complicated geometries, material properties and
boundary conditions can be solved readily by FEM.
● Many General Purpose Finite Element Software Packages (e.g. Ansys) are
available to perform different types of analyses and to reduce the analytical time
for the computation of unknown quantities. These softwares touch several field of
problems like stress analysis, heat transfer, electromagnetic field analyses etc.
● It is relatively easy to control accuracy. Accuracy can be increased by refining the
Finite Element Mesh or choosing more fine Elements or by employing higher
order elements.
7. Disadvantages of FEM:
● Exact solutions can’t be achieved by FEM. The solutions show at least some error
with the exact readings as they are approximately determined.
● Numerical solution is obtained at one time for a specific problem only.
● Experience :
Sound Engineering judgment and some understanding regarding physics of the
problem are required in creating the Finite Element Model.
● Poor selection of the Element type or poor Finite Element Discretization can lead
to disastrous results.
8. Future with FEM:
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The whole world considers Finite Element Method to be the future. Finite Element
Method is important because, it is relatively easy to tackle, relatively easy to computerize
and relatively faster than other methods. It can also offer the financial rewards by
omitting the use of prototype and doing the analysis on the component design on the
computers. It is an approach and science oriented towards applications. Finite Element
Method touches so many aspect of our life. For simplicity we can divide the field into
three major activities: Structural Analysis, Heat Transfer Analysis and Electromagnetic
Analysis. Needless to say they all interact. This will become quite simpler to get along
with FEM. The Finite Element Method is one of the best tools in engineering used for
analysis purpose and it can be surely said that, it will show a rapid revolution and faster
growth of engineering stream.
9. Conclusion:
The Finite Element Method can be considered as the connecting link between Practical
(Industrial) and Theoretical Knowledge. Practical Knowledge is the knowledge gained by
the experiences in the industry and Theoretical Knowledge is the knowledge achieved in
the learning phase.
In Finite Element Method, we are trying partially to predict the natural behavior of the
components. The Finite Element Method being on a large numbers of applications will
carry major role in the Human Development and Better Sanitation.
The miracle of Finite Element Method holds a strong promise of its presence in the field
of every engineering aspect. Not only in mechanical but it is ready to get applied in
almost every field including electronics, instrumentation, mechanical etc.
10. References:
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Books:
01. Finite Element Method : Theory and PracticeBy : Mr. M. J. Fagan.University of Hull.
02. Introduction to Finite Elements in EngineeringBy : Mr. Chandrapatala and Mr. Belegundu.
Journals:
01. Introduction to Finite Element MethodBy : Tai Hun Kwon.
02. Technical Design Report on Mini Baja Vehicle Design Optimization
By : Mr. Jonathan Hastie and Prof. Has hemi.Year : December 2005.Department of Mechanical, Industrial and Manufacturing Engineering &College of Engineering, Northeastern University, Boston.
03. Finite-element methodBy : W. Robert J. Funnel.Department of Biomedical Engineering, McGill University.Year : 2005.
04. SAE International Journals
Internet Sources:
01. www.engineering.uiowa.edu/~uisae
02. www.BajaBuggyBaja450ccOffRoadMiniBajaBuggyS2SPowersports.com.htm
03. www.wolfram.com
04. www.google.co.in (Journals and Books Section)
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