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.

patil.digvijay@yahoo.co.in

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.

2

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

5

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.

6

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

13

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:

15

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.

16

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:

17

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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