Introduction to Finite Element Method

Dr. R. Krishnakumar
Department of Mechanical Engineering
Indian Institute of Technology, Madras
Lecture - 2
In the last class we saw certain examples on the applications of finite element analysis
for real life problems. This was done for companies, famous companies as probably
you would have seen in our slides before. We are going to continue on these
examples. The whole idea here is to give you an overview of where all finite element
can be applied in the industry. This will be very useful for us to later conceive a
problem, workout a model and look at the results. Let us now see further examples of
the application of finite element analysis. This list is not complete, there are lot more
applications we are going to talk, but that we are going to do later in the course. But
before that let us continue further and look at certain more complex examples rather,
that we can do using finite element analysis. Let us start with the first example in the
second class.

(Refer Slide Time: 2:12)

This example is for the application of finite element analysis in the design of what are
called as piston rings. Piston rings though looks very nice and simple and small are
not very easy to design due to various aspects. These guys, these piston rings are one
of the components which are abused to very great extent in an engine and hence it is

important that we understand how the piston ring behaves and what will be the
stresses and how the stress analysis can help us to design better rings and so on.
Especially the piston ring becomes important because of the fact that we are now
talking about Euro I norm, Euro II norm and other very stringent pollution control
norms and hence it is important that the ring performs quite well in the engine.

We are going to only have a birds eye view of what can be done. Lot more can be
done, right now we do not have the time or the expertise to go into various aspects.
But at least it is important to understand how this technique can be used? Now let us
look at how that can be used for the design of piston rings? Let us now look at how
we develop the force for the piston ring in our first slide. Let us now look at that slide.

(Refer Slide Time: 3:46)

This slide actually gives you an idea as to how forces in a ring can be calculated. You
can see that the complete analysis of the dynamics of the engine has been carried out
using what is called as multibody dynamics. As the ring sits on the piston and as it
goes up and down, the rings actually tilt; they start hitting the sides. The piston starts
tilting and along with that there is some sort of a tilt and a flutter for the ring as well.
It is important for us to understand the forces that go into the rings. For that we do the
multibody dynamic analysis.

We will not talk about multibody dynamic analysis in this course but nevertheless
multibody dynamic analysis becomes important for certain components to get the
forces. In these multibody components or multibody analysis, most of the time we
treat the body as rigid. It is also possible to treat them as flexible but most of the time
we can treat it as rigid and get the forces that act on the component. So, we can find
out for this particular ring what would be the forces that would act?

Let us look at the next slide and see what it tells us as far as the ring is concerned.

(Refer Slide Time: 5:18)

But actually this slide gives us what is called as the finite element model. We are
going to discuss more and more about this model later in the course but let us now
accept this as a finite element model. We will discuss those lines and what are called
as elements which happen to be there towards the end of the class, but nevertheless
you see there are two regions. As you see, there are two regions; one region is some
sort of a white region and the other there, is a red region. These rings, these piston
rings consist of what are called as molybdenum coating. The red region in the ring
depicts the molybdenum that has been coated on a steel ring. Ring can be steel, cast
iron or whatever it is, but in this case molybdenum is coated onto a steel. This
particular design is called as inlay design.

You can also see that or you can understand immediately that it is possible in finite
element analysis to get what is called as the combination of materials. Like it is
possible to say that we will have steel in one part and we have what is called as
molybdenum in the other part and do an analysis together.

Let us look at the next slide and we will see how the results are?

(Refer Slide Time: 6:44)

You can see that the molybdenum coating has an effect on the stresses and you can
see that there are regions where the stresses are high and there are regions where the
stresses are low as well. Let us not worry about what is the stress value, what is the
allowable stress value and so on, but it so happens that in this particular design the
stresses are high. Let us look at another design and that is depicted in the next slide.

(Refer Slide Time: 7:20)

You can see a second design here and now you see again there is a red region and
there is a white region. This red region gives the molybdenum part and the white
region gives the steel part. This design is what I would call as a sandwich design
where the molybdenum is coated throughout the thickness of the ring. Remember that
we are looking at a cross section of a ring and we call this problem as an axisymmetric problem. Let us look at the results for this problem or for this design.

(Refer Slide Time: 7:55)

That is the stresses; though you may not be able to clearly see what the stresses are,
but I can assure you that stresses happen to be slightly lower than the previous design.
That does not mean that this design is good, but it means that for these stresses or for
the forces that act on this component which has been derived or arrived at from our
previous model of multibody dynamics, this design happens to be better than the
previous design. There are lot more issues to rings, but we will stop this example at
this stage and may be if time permits we will continue it at a later part of the course.

Let us look at our next problem which is a very, very interesting problem.

(Refer Slide Time: 8:43)

The problem is the design of a connecting rod. Please remember that in the previous
problem, the material which we considered was steel as well as molybdenum; a part
of the component was steel, a part of the component was molybdenum. Now we are
going to consider a material which is porous, which has porosity, whose density is not
complete. For example we are going to consider a P/M component, powder
metallurgy component, where the normal densities are not attained; not necessarily
attained but only about 90% of the density is attained. This is a connecting rod for a
two wheeler and now let us look at how we develop this particular component using
finite element analysis.

(Refer Slide Time: 9:39)

The next slide gives the solid model of an existing connecting rod for the same
engine. It is a very nice picture. You can visualize that it has an I- section and that this
connecting rod is performing or it performs very nicely and you can ask me why do
you want to change this connecting rod. All the time we always strive to achieve
excellence; we want to reduce the weight, we want to reduce the cost and so on. So,
from that angle it is now realized that if this forged connecting rod, which is existing
connecting rod is replaced by means of a powder metallurgy connecting rod with
certain special processes, the cost can be considerably reduced. From that point of
view we are redesigning the connecting rod.

The next question that you may ask me is why not I use straight away this connecting
rod with a new material or the new process? That may not be possible because the
stresses that may result when I use this kind of technique are not conducive for the
new material. In other words the stresses may not be withstood by the new material
which is a P/M material. Number 2, the process which I am going to have, which I am
going to use in order to manufacture this connecting rod also may not be very
palatable or may not be very friendly for the existing design. Hence I have to change
the design.

The next question you may ask is why do you want to go through finite element
analysis, why not straight away do it by a small hand calculation and start developing

certain prototypes and then arrive at an optimum design. There is a problem to it

because if you want to arrive at an optimum design, you have to do a number of trials;
maybe you may have to do 4 or 5 trials in order to achieve at the correct result.
Because certain cases hand calculations may not give you, analytical solutions may
not exist, the results that you are looking for and the stresses may not be predicted
correctly. So, if you have to do about 5, say examples of 5 prototypes, then you are
going to spend a lot of money in order to do this. Hence what we do is we simulate
the actual condition or determine the stresses in the actual condition using finite
element analysis and then say that look these are the 2 or 3 designs. These are what
are called as suboptimal design and I will use now these designs, so reduce my trials
to 1 or 2 in order that I can save lot of time as well as money.

Let us look at the next slide and see how exactly we are going to do this.

(Refer Slide Time: 12:36)

This is the solid model of a new connecting rod; connecting rod which has been
designed after a number of trials in the computer, which does not cost me much and
the time also required in order to do it, is very small or in other words I can do the
whole thing in a matter of about 4 days. Let us now look at the results. The next slide
gives me the results of the existing connecting rod.

(Refer Slide Time: 12:59)

This is the stress levels; very nice connecting rod with performing very well, there is
no doubt about it. Let us look at the next slide and see the stress analysis of the P/M
connecting rod. The stresses have been so adjusted that the maximum stress that exist
when the connecting rod is put in operation, put in an engine is less than what could
be withstood by this particular material, by this particular P/M material; that has been
taken care of. The thicknesses have been optimized and hence it is ready for
manufacturing. The manufacturing itself can be continued after this stage, a finite
element analysis as shown in the next slide.

(Refer Slide Time: 13:52)

You can see that it is possible to integrate completely CAD and CAM with finite
element analysis sitting in the middle. We started with a solid model, we went ahead,
we did finite element analysis and we optimized the shape. Now it is possible for me
to push this geometry to CAM and then get the dye assembly for this P/M connecting
rod, get the cut location data and machine this and so on. Today the technology is
available as an integrated piece to start a design from solid model, visualize it, do
analysis and then completely prepare for the manufacturing process. So, that is an
interesting example. This piece is under trial.

Let us look at the next, again an interesting example from the field of machine tools.

(Refer Slide Time: 14:47)

All of you know that a chuck is a very important component of a lathe. Today, we are
looking at very high speeds at which this chuck operate or in other words these are
high speed chucks and there are lot of problems of wear of these chucks. When there
is wear of this chuck, all of you know that it will not bite or hold the component
properly and there are problems of quality of the component and so on and I need not
explain all these things to you; you are mechanical engineers you know these things.
So, the whole idea here is, is it possible to enhance the performance of a high speed
chuck by completely redesigning this chuck.

Let us look at the first slide in the series.

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(Refer Slide Time: 15:36)

That slide shows you a solid model of an existing lathe chuck. This problem is very
interesting and very complicated because the chuck is in contact at number of points
to the body of or this chuck jar rather is in contact at number of points to the body of
the chuck. This is what is called as the jaw of this chuck and hence my interest here is
to find out what are the contact stresses that exist in this jaw?

(Refer Slide Time: 16:11)

The next slide shows you the contact stresses of this chuck jaw as it operate in a
chuck. Now, it is possible to look at this stresses and tell whether these stresses are

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nice enough for you to not to wear or is it going to be subjected to wear. There are lot
more issues of wear but nevertheless this gives me a first hand figure whether stresses
are going to be critical. It so happens that the stresses happen to be critical and we
redesign this chuck and the redesigned chuck looks something like this, which you
can see in the next slide.

(Refer Slide Time: 16:55)

So, that is the re-modified chuck or the redesigned chuck and the stresses were lower
for this particular chuck. So, performance enhancement by modifying a machine tool
component is possible using finite element analysis.

Let us now look at the next example.

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(Refer Slide Time: 17:14)

The next example is from the world of manufacturing. In other words next example
talks about process modeling of what is called as a P/M gear rolling process. The next
slide tells you or shows the model of this, for this process.

(Refer Slide Time: 17:34)

What is that we are doing in this process? All of you know that most of the gears are
forged gears and few of them may be cast as well depending upon the component or
rather the material. In this particular piece the gear, which is the bottom part is made
up of powder metallurgy, it is a P/M gear. Today world over what they do is to look at

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a P/M component and see whether it is possible to selectively densify those areas
where the stresses are higher. For example in this particular gear is it possible to say
where the stresses would be higher and is it possible to densify only those areas? For
example the flank and the root of the gear may be subjected to high stresses and is it
possible to now compact only those places and leaving out the center of the gear? If
you do that, that is good enough for us to use a P/M gear for many of the load bearing
applications which again means that we can save lot of money.

Let us go back and look at that model. You can see from the model that there are two
pieces. That model will now show you that there is a bottom piece which is actually
the gear and the top piece which is what is called as the dye. The gear actually starts
rotating under the influence of the dye and the dye starts compacting. The next slide
will show you what happens because of this process? The results are quite clear from
the next slide.

Now, what is that we are looking at in this slide? We are going to look at what is the
compaction or how efficient has there been as the compaction takes place? In other
words what we are going to do is to look at what is called as a relative density of the
gear.

(Refer Slide Time: 19:54)

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As you can see that in the gear this is the root and that it is important that we have
higher densities only in this region and that the densities need not be that very high in
this region. So, as the dye rolls, it is, it is a regular rolling process, gear rolling
process. As the dye rolls in one direction and then in the other, the material gets
densified on the surface to that extent that it can withstand all the loads or all the
stresses that would come during the operation or it can withstand the operational
loads. It is possible now to design the process using finite element analysis.

(Refer Slide Time: 20:56)

Let us now look at or let us stop for a moment and look at this quite closely. You can
see the equivalent plastic strain during rolling. As I was just explaining to you, the top
one is the dye and which is considered to be rigid, the bottom is the component. You
can see the plastic strains developing which gives rise to what is called as compaction.

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(Refer Slide Time: 21:24)

Let us look at the next slide and that gives us a better picture of the relative densities
or compaction efficiency of the process which we have been talking about. You can
see that as it rolls, the flank slowly gets compacted. Maybe 4 or 5 different cycles are
required in order to completely compact the surface. What is that we are trying to do
using finite element here? We are trying to say or predict how this compaction takes
place in the rolling process? Why is that we are doing, because again here we can
reduce number of trials that are required.

This is a very expensive process or the trials may be quite expensive. So, we can
reduce both the money as well as the time by doing this kind of simulation and then
arriving at the optimum design of both the gear. After all what I have to do is I have to
add some material or I have to redesign this gear so that when it gets compacted my
profile stays good. It should be good to perform or the profile should be such that the
performance of the gear is not affected. So, that is one thing. The next one is how is
that I am going to design the dye? Is there modifications that are required to the dye
or can it be a regular rolling dye itself.

I can do all those kind of simulations, arrive at what should be the shape and also
arrive at what would be the speeds at which I have to rotate and also look at how
many number of rollings are required and so, all things can be done beforehand. After
all we may follow two processes; one process to just compact only material at the

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surface and the other what is called as root rolling process where we may be
interested to compact what is in the root. In other words we may require special dyes
or special shapes of the dye in order to compact material which is existing at this root.
So, this kind of geometry optimization in a process is also possible by using finite
element analysis.

Let us look at the next slide and see what we get.

(Refer Slide Time: 23:43)

The process is complete and the distribution after two rounds, we can keep on seeing
2 rounds, 3 rounds and so on; the distribution of this density is complete and that is
shown in this particular slide.

Let us go to the next slide and see the next problem.

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(Refer Slide Time: 24:04)

Now, it is not that we are going to or is it possible to simulate only this gear? It is
possible to simulate a whole range of manufacturing processes. We are going to study
these things in this course as to how to simulate manufacturing processes. In IIT
Madras we have developed a non-linear finite element code to simulate forging
processes and this is operational in a company and you can see the results of such a
code in the next slide.

(Refer Slide Time: 24:41)

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May be slide is not very good but nevertheless it will become clear in the next slide as
to what we are trying to achieve.

(Refer Slide Time: 24:50)

We are trying to achieve or we are trying to see whether the filling of a dye is
complete and the next slide clearly shows the type of other results that can be
obtained from this kind of work.

(Refer Slide Time: 25:04)

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From this slide it is very clear that it is possible to predict plastic strains using finite
element analysis. You can see that actually a cup has been formed and the process as
you may recognize is what is called as a forward backward extrusion. During this
process the slide shows what would be the maximum plastic strains? If you have the
data to find out whether maximum plastic strains that are achieved in this process
could be withstood by the material, if you have that data, then you can see or you can
say whether this process is good; that is number one. In other words, you can also
look at the process beforehand, simulate the process beforehand and say whether I am
going to achieve this process or not? If not what is to be done?

Number 2 - it may be that the dyes and the punch that are used in the complete
manufacture of the dyes are not good enough to withstand the stresses that may be
generated during the process of manufacture. It is possible for us to find out what
would be the stresses during manufacturing as well. So, it is possible to say whether a
punch will break. If it is going to break, then what would be the modifications that are
required and whether these modifications will successfully alleviate the problems or
remove the problems? All those things can be determined beforehand using finite
element analysis.

Let us now look at the next example and see what is the lesson it teaches?

(Refer Slide Time: 26:51)

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Not only are forgings possible, but whole set of manufacturing processes can be
simulated. We are not going to list them now because it is quite a long list. But just to
give a variety it is also possible to look at compaction, powder compaction process
and see what the density distribution is in a typical component. We will come back to
these examples later. We will first have a birds eye view of where all this process can
be achieved. Let us look at the next slide and see what it teaches us?

(Refer Slide Time: 27:27)

This is another manufacturing process. The manufacturing process that is involved

here is the study of welding of thin sheets. All of you know about distortions in
welding. What are the things that are involved in welding? There are two things that
are, two problems that are important in order to study welding. One is that there is
heat transfer; as you weld, there is a weld pool and heat transfer takes place. First of
all, you have to do a temperature distribution study or we have to determine the
temperatures.

What is the next step? Using this temperature is it possible to solve a mechanics
problem and find out what would be the distortions? We are interested in distortions
ultimately. These are coupled problems; a heat transfer problem followed by
mechanics problem, again a heat transfer problem, mechanics problem and so on. We
are going to study this kind of things in this course but nevertheless it is important to

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understand that finite element can also be applied to study heat transfer problems and
coupled problems as well.

Let us now look at the result of the study.

(Refer Slide Time: 28:45)

You can see the temperature distribution, first of all, during welding; as the torch
moves how the temperature gets distributed. You can see very well that it is quite
concentrated zone in the zone which is very close to the welding zone. Let us look at
the next slide and see what it teaches us.

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(Refer Slide Time: 29:06)

This gives the result of distortion after welding or in other words this is the complete,
this is the result of the complete study, both heat transfer as well as the mechanics
problem; you can see that the sheet is completely distorted. What are we going to do
with the sheet? You remember that in the last class we had talked about undulations
on the surface of a coach. This coach is made up of this kind of such kind of welded
sheets. They are 2 mm sheets very, very thin sheets and hence there are problems of
such kind of distortions.

This is what manifests as undulations; apart from the design or the assembly, the
coach, it is the fundamental process of manufacturing the coach itself which happens
to be welding which gives rise to distortions. How are we going to avoid this or is it
possible now to see to it that we can get a straight sheet or straightening as the
process, is it possible? Yes, that is possible. World over people whoever manufactures
it; whether it is manufactured in Japan or Germany people straighten out this piece.
How do they straighten it? That is what is called as a magnetic forming process.

Let us not worry about this process because it is quite involved but electromagnets are
used to straighten this piece, make this piece go to a plastic deformation or make this
piece go to plastic range and they make it straight. There is a small problem here.
These equipments are expensive. So, if I have to get an equipment in order to do this
kind of work, then I have to spend quite a lot of money. If you look at this kind of

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magnetic forming process equipment, there are about five varieties of equipments that
exist

What is the hitch here? If I have the first variety say variety one and get it and use it
for this kind of straightening process then, if the sheet does not become straight then I
lose lot of money. We have to import; in India, we have to import these pieces so, we
have to lose lot of money. The question here is can I beforehand find out what is the
type of equipment that I have to use in order to straighten? What should be the
specification of the equipment in order that I can straighten this piece? Yes, it is
possible.

Let us look at the next slide.

(Refer Slide Time: 31:42)

The next slide is the result of such a simulation, finite element simulation and you can
see that it is possible to even simulate a magnetic forming process and then look at a
straight piece and arrive at what is called as the optimum configuration of the
equipment that is used in order to make this kind of flat sheets. Let us pause for a
moment; let us get back and see the things that we have seen so far or in other words
let us now look at what are the applications of finite element analysis?

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Let us get some answers from you. Let us see where all we have applied? Summarize,
what are the applications. Let us start from the first example and see what are the
areas in which we have applied finite element analysis? Let us get some answers from
you.

Yes, the first one is rotor. LP rotor; LP loader is the first example but what is that we
got? Yes, stress analysis; in other words it is possible to look at a new design; so for
stress analysis for design.

(Refer Slide Time: 33:12)

Quite a few examples; in fact whether it is repair or whether it is a original design

whatever it is we can do stress analysis for design. What is the next type of examples
that we saw? Non-linear analysis; we did contact analysis. For example for the wheel
we did contact analysis to predict failures. It may not be that when we do stress
analysis for design as a first step we may be interested in a very complicated problem.
Many times we do a simpler analysis but sometimes when we have to investigate
failures then we have to do a much more complex and thorough analysis. We have to
predict where we have to do much more complex and thorough analysis. So, nonlinear analysis to predict, I would say performance; yes, it is that you may argue that
why not we do it here itself? Yes, it is possible that you do this kind of analysis right
in the beginning. But as a tradition most people do not do a very detailed analysis
when we start the design a performance because 80% to 90% of the components

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go through with the simple linear stress analysis. But that may not be sufficient when
the problem becomes complex and we have trouble in the component hence we have
to do a thorough non-linear analysis. This may involve difficult things like contact
with what is called as material non-linearities and geometric non-linearities and so on.
All these things are possible by using finite element analysis.

What is the third example?

(Refer Slide Time: 35:09)

It is even possible to look at assembly processes. We used it for assembly processes

and look at deformation during assembly; so, stress and now deformation assembly
process deformation. What is the next issue or what is that we got as the next issue?

Contact analysis we have already covered here. When you look at say for example
piston ring, then automatically you saw that there are two materials. So, material is
not a constraint. That is the lesson that we learnt. One part of the component can be of
one material and the other can be of another material. So, material is not a constraint.
What is the next lesson? Yes

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(Refer Slide Time: 36:17)

Process model; so, process modeling using FEM that is the reality. This process
modeling consists of both heat transfer sometimes, heat transfer analysis and a
mechanics problem or stress analysis and it can be coupled as well. What is the next
thing we saw? Is there any other lesson you learnt? Equipment selection; lastly it is
possible to use this for equipment selection. This list is no way complete; you know it
is not that the list is complete. In fact today finite element analysis is used in
biomechanics, extensively used in biomechanics. Unfortunately we will not have time
to cover all the applications in this course, but biomechanics is one of the areas where
finite element has really taken off.

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(Refer Slide Time: 37:55)

From I think 87 to 97, there have been 1000 papers published on finite element
analysis to human bodies, different parts of the human body. There are nearly I think
350 papers; 300 to 350 papers on heart alone, human heart and other things that are
important to us or cardiovascular system if I can call it, in that alone there are about
300 to 350 papers. There are extensive finite element studies for dental applications.
So, biomechanics is a very important area today for the application of finite element
analysis. There are lot of things that can be determined using finite element analysis
in biomechanics and if there is time we will see one or two examples, maybe towards
the end of the course. But let me warn you that many of these things are very difficult.
They are not very straightforward and easy because of the way nature has designed
us. The behavior of each and every part in our body is much, much more complex
than the material which we use in order to do things.

There is no comparison between how our tissues behaves and say metal behaves; they
are totally different and hence the problems that we get in using finite element for
biomechanics are much more complex. There are other areas as well. For example in
electrical engineering there are lot of applications in electrical engineering. Here again
the applications are quite recent and again we are not going to talk more about these
things in this course. We will restrict ourselves to design and manufacturing. We will
look at only thermal problems, fluid mechanics problem. For example there has been

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lot of applications of finite element analysis in fluid mechanics; that again will not be
covered in this course.

As I told you, finite element is a very, very vast subject. There has been tens and
thousands of papers published, so we will restrict ourselves to the fundamental finite
element for design and manufacturing. There are some applications of fluid
mechanics in manufacturing but because of lack of time in this particular course I will
not cover all those aspects. With this background saying that these are the applications
of finite element analysis, let us now look at what is involved in finite element
analysis.

Let us look at the last slide in this particular series.

(Refer Slide Time: 40:50)

Have a close look at that. What does it indicate? What are the things that you see?
You see lot of lines or what I would call as network. They intersect at points which
are called as nodes. What is that you see or what is that you get or what is the
information you get out of seeing that picture?

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(Refer Slide Time: 41:19)

You see that there are lot of lines like this or in some other figure you would have
seen lines like this. These lines meet at certain points and so on; these lines meet at
certain points or the bodies which we considered for analysis have been discretized,
have been broken down into what are called as elements. These are what are called as
elements and this is an element and they are bounded by what are called as nodes
which sit at the intersection of these elements. Say for example that is a node, that is a
node, that is a node; that is a node and so on.

So, nodes and elements are the ones which are important in finite element analysis or
what is that we have done philosophically? A complex component has been broken
down into what are called as elements. What are these elements? How does it help us
and what are the advantages of using this kind of approach is what we are going to see
in this course. Essentially what is that we are looking for in a design problem?

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(Refer Slide Time: 43:17)

We have a force and we have a body and we are looking at what are the
deformations? So, I want a relationship between force and deformation. Let us call
this deformation as simple displacement. There is a difference between displacement
and deformation, . one, let us say that the body gets deformed and we will call this
deformed quantity as displacement. If I have a bar and if I apply a load say P the end
of the bar gets displaced by a certain amount say u. The body here in this case is the
bar. The force that happens to be there on this body is P and what is that you are
looking for? You are looking for this displacement, right.

If this is the L or the length of the bar we can easily find out what is the strain what is
the stress and so on. What is that we are interested in? We are interested to find out
what is this deformation or displacement? As mechanical engineers whenever I say
that there is a force on one hand and displacement on the other hand what is it that
comes to your mind? Stiffness; beautiful.

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(Refer Slide Time: 44:56)

Suppose I have a spring for example and I apply a force here and I ask you what is the
displacement? The first question you would ask me is what is the stiffness of the
spring or in other words if this is the force and this is the displacement of the end of
the spring then F is equal to ku. So, all mechanical engineers or all engineers
generally know that this is a very fundamental equation. What is the difficulty in
applying this straight away to a much more complex problem like what we have or
what we had in these particular examples.

For example can we apply this to a railway wheel or to a side wall and so on? It is not
possible, it is not very straightforward. Why? Because here I could immediately say
that the spring displacement is measured or is characterized by the displacement of
the end of the spring. When I say displacement of the spring immediately you know
that I mean how much it is going to displace, this tip, this guy sitting here. On the
other hand if you look at a three dimensional body, which we encounter in the actual
practice I cannot say by one number that this is the displacement. What does it mean?
It means that I just cannot come and tell you that this is the maximum displacement;
your interest is just not in one number.

Your interest is in the complete displacement of the component at every point. You
would come and tell me the displacement at this point, this point, this point, this point
and so on.

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(Refer Slide Time: 46:54)

If I take the complete sheet, you would ask me what the displacement of this complete
sheet is. Maybe to make it more realistic let me put a small window there. It is a
complete sheet. Not only you will ask me the displacement of complete sheet but you
would also be interested in strain distribution throughout the sheet. As we saw in one
of our slides that the plastic strains of the forging, for example, was different at
different places. It is not that you are only interested in displacement; it is only one
small part of the problem. You are interested in strains and strain may not be constant
whereas this problem, it is not so and stress in this problem is constant throughout this
bar, but here the stresses may not be constant. Most components are such that stresses
will not be constant throughout.

How do you now tackle that problem? Yes, this is a nice answer that you have spring
and you have a relationship with force and displacement. How do you extend this?
Can we develop a concept called stiffness and if we develop a concept called stiffness
is it going to be one number? That is the question we are going to answer in the next
two classes.

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(Refer Slide Time: 48:24)

The next class we look at what is stiffness from a finite element point of view. We
will answer that in the next class

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