Let us move on to our next chapter on Beams.
You know we have already seen this animation, we will; we have already seen it in the context
of trusses. Now we will refocus our attention on the beams. First striking feature what
you get is the cross-sectional dimensions are much smaller than the length of the member.
And they are also called as rafters, and you should notice that they are supported only
at the joints supported at the ends. They are the simplest beams that you can analyze.
And what is the kind of load that this experiences; essentially transverse loading. You have the
beams used in a roof truss; they are also used in a bridge truss. I have what are known
as floor beams, they are again supported only at the ends. They are very simple to analyze.
There are also beams that are supported in between, we will see a classification of them
in a little while later. Here again, the loading is transverse to the
member. What you have to notice is, a beam supports transverse load. The cross-sectional
dimensions are much smaller compared to the length.
And you have various classification of the beams. Right now, we should be able to identify,
what is the support that I have shown on this. You can identify that this as a fixed support,
and I have a slender member like this. I have an end load, and this has a particular name:
this is called a cantilever. You have many examples of a cantilever, that
you come across in your surroundings. One simple example is your sunshade. Many of the
sunshades in the building are essentially cantilever. And look at the branch of a tree;
the branch of a tree is again a cantilever. And there is also something very interesting
in the case of a branch. How does the cross-section of the branch varies along its length; very
intelligent. God has understood engineering mechanics much better than you, and I learn.
You have that tapered so that, where you need a higher cross-section to withstand the bending
moment, you have a higher cross-section. It is not like a constant cross-section that
I have shown here. So, nature is very, very intelligent. And you call this beam as simply
supported. You should be able to identify what is the kind of end support I have used;
please note that this is only a symbolism. You should know how to interpret the symbols.
What is the striking difference between this support and the support shown here? The difference
is in the rollers. There are many ways books represent these supports.
So, I have tried to show different forms of them, so that you can interpret the symbolism
whenever you come across a problem from any book. Such a beam is very simple to analyze,
and you also call that as simply supported. And I have a slight difference between the
beam shown here, and the beam shown below. What is done here is, the support on the right
is moved inside, and you have the beam protruding out of the support. Have you come across any
such practical example in your life in your surroundings? Because I have always been emphasizing
that you are trying to analyze systems around you; that is the purpose of this course.
How many of you are swimmers? No, we are all very studious people, we never go near sports.
That is very very unfortunate, and you know if you have gone there; swimming pool, if
you find the person is stepping on to the spring board to dive. The support may not
be this far away; this may be closer to this. So, you have a practical example where divers
when they jump into a swimming pool. You have an overhung beam. Either an overhung beam
or it could be a cantilever depending on the construction that they have made in that place.
And this is called an overhung beam. So, there are different names given to these
beams based on the way supports are located, and how the supports are made: I have a fixed
support here, I have a hinged support here, or a pinned connection, I have a roller support
here. I have in this beam: I have a pinned connection here, I have roller support shown
in the rest of the beam. And you have to give a name to this, what is the kind of name the
people have coined; I have many supports, and people call this as a continuous beam.
See for every type of beam we come across; you can always find the practical structure
around you. Only then you learn engineering. Can you identify a practical structure which
everybody has to use that convenience when you go home? When you travel by a train, please
also note the rails: how the rails are supported, and you have this support at periodic intervals.
You have not three; you have thousands of supports, and not only that, people also go
and qualify those supports as a beam on an elastic foundation. So, people can go very
close to reality when you go for higher level courses.
In this level of the course, we say everything is rigid. We say the supports are again rigid
in the sense they do not sink. Whereas in practical rails, the floor is not rigid; you
also accommodate the elastic behaviour of the soil beneath you. I have another example
here: what is the difference between a cantilever, and what I see here? The end is not free,
but the end is supported. Can you tell me a practical example that you normally come
across? You have a big bungalow, and you have a big portico. Many of the big porticos they
will have a support at the end. And I have another type of beam, I have shown
this as a fixed support. I also have another fixed support; you call this as a fixed beam.
See, I have the luxury of putting the beam as a nice rectangular piece, you can simply
write it as a line. If you can closely look at the way I have listed the beams, do you
think that you can learn something more from the way they are classified from whatever
you have learned earlier? See, in this course, we assume everything
as rigid. And I use equilibrium equations, and this is a planar situation. I can have
only three equations written; so, I can determine only three unknowns. So, one of the basic
exercises that you will do for all problems is whether the problem is solvable from equations
of statics, and you call it by the name statically determinate. I have given you the clue. Can
you apply your mind and find out how the beams are listed in this, how they can be classified?
What do you have in this segment, and what do you have in this segment? Even if you take
one example on the left, one example on the right, you can find out the classification.
When I have; let us take a simply supported beam. How many unknowns I have here? I have
two, and I have one. I can write three equations; I can solve for all of them. On the other
hand, when I go here how many unknowns I have; I will not be able to solve it from equations
of statics, I have to bring in the deformation behaviour. So that, I can write one more equation.
I need as many numbers of unknowns; I should have as many numbers of equations. So, the
way the beams are listed here. This set can be classified as statically determinate, and
this side can be classified as statically indeterminate. And I have shown a variety of members here,
and you have to tell me which of these are beams. In the previous slide, we saw everything
as horizontal members. I have a horizontal member like this, but I have a vertical member
here. Can you call this as a beam? Yeah, 50 percent of you say, you can call it as a beam.
And 50 percent of you say, you cannot call it as a beam.
Because we have seen in the previous slide, all beams are horizontal. But I asked you
to notice one important fact when we looked at the roof truss and the bridge truss, which
also had rafters and floor beams; the loading was transverse to the beam. See in English,
if you go, say what is a “beam”. I can consider that as a light beam, or I can also
say somebody with a beaming smile; the word has multiple meanings.
So, in the context of engineering mechanics, a member that supports transverse load is
called the beam. The member can be vertical like this, but still, the load is applied
transverse to this. The member could be inclined like this, but the load is transverse to this.
One of the common confusion students have is, whenever they look at a member which is
horizontal, they do not think; they jump on to an immediate conclusion that it is a beam.
I have clearly shown that this is supporting only an axial load. You do not call that from
an engineer mechanics point of view as a beam. And you know recently, Kerala is in spate;
and all dams are overflowing they are sending out water. Can you extend your knowledge of
whatever you have discussed as a beam to what you see in a dam? I can see the dam cross-section
like this, and I would have a water column applying a force here. And this you know very
well the pressure increases as you go down, you have basically a triangular loading acting
on the dam. So, this is one practical example, where you
have a distributed loading acting. And I said, when I do an analysis of any civil engineering
construction, the self-weight is so important cannot be ignored. How do I model that? That
is what is shown here. I have a beam; I have a uniformly distributed load; this is nothing
but accommodating the weight of the beam as a load acting on the beam. See the books,
in order to train you to handle distributed load gives different types of distributed
loading, to get a mathematical practice. But if you look at from a practical standpoint,
you will come across either uniformly distributed; in many cases, representing the self-weight
of the beam or you will have a loading, which is triangular indicating, the load that could
come from a liquid column. So, we should be comfortable in handling uniformly distributed
load and triangle loading. You cannot escape out of it; they are very, very common. And
you should be comfortable in handling this. And you have many other examples: you have
a crane. This is acting like a beam. And you should also be surprised to learn that there
are springs, which are used in automobiles; if you see a huge truck, if you look at the
rear wheel, it is supported on a structure like this. See, normally you have seen a spring
which has coiled like this. That is a normal string that you come across; this also behaves
like a spring, but you have to analyze the members subjected to bending. They are called
leaf springs; they are practical examples. Then we move on to some fun facts we have
seen that Romans were good in building aqueducts, and we recently visited this place: it is
called a Mathoor hanging trough. This is the longest trough bridge in Asia at a height
of 115 feet, and I have this side-view of the bridge. You could see very tall columns
supporting the bridge, and essentially an aqueduct. See, this is taken at a time when
there are no floods in Kerala; so, there is no water in the bridge that we had seen. That
gives you an idea where you are at about 115 feet from the floor, and this is the beam
of 115 feet long. And there is also another interesting bridge;
this bridge is made of roots. Roots of trees around 500 years old are grown for the purpose
of making a bridge in Cherrapunji, Meghalaya, which is known for the heaviest rainfall in
India. And the locals were very clever. They had made this, and you can understand from
the solid mechanics point of view, engineering mechanics point of view; whenever I have a
transverse loading, the structure behaves like a beam.
And what is the tree is, it is actually an Indian rubber tree “Ficus Elastica”; grows
secondary roots to climb boulders in this mountainous region. This property was exploited
by the locals to make bridges to cross streams, which receive heavy rainfall. It is a local
answer to a human need. And mind you these bridges stay for 200 years, whereas man-made
bridges life span is 75 to 100 years. The difference is here only humans can walk;
that is how the bridge is constructed, and I was also very surprised, they had multilayer
bridges. So, this is the bridge in Umshiang, and it is a double-decker root bridge. So,
when there is heavy flow of water, they will use the top bridge and the other one in the
bottom bridge. And there is also another very interesting
bridge here; this is at Dwarka. This is the Dwarka temple. You have this Gomti river.
This is a suspension bridge; what is interesting is very similar to the roots that you have
seen. You have an array of cables; they are needed to support this hanging bridge.
We read the beam separately. We have seen the roof as well as the bridge, a combination
of truss and floor beams. And here you have a combination of a beam and cables, and we
will also see a combination of trusses, floor beams, and cables. All of these are needed
to support transverse loading. See, this bridge is a short bridge, maybe about 500 meters
on Gomti river, supported by strings as well as your bridge you have the pier.
And one of the very famous bridge, which people would like to go and see when they want to
tour the world, is the Golden gate bridge at San Francisco. And what is interesting
to note here is, this is a very long bridge of 3 kilometers: close to 3 kilometers and
from a distance-can you perceive this as a slender member? This also gives you some ideas
about idealizations. If I want to find out the natural frequency of this the bridge,
I could consider this as a slender beam and get first level. My results will be far away
from the final result but may be acceptable as a starting figure for you to aid your thinking.
Suppose I take a closer picture of it, I do see that - what I see as the slender member
is actually a truss; I have this truss which you have seen, and this is suspended by cables.
And, one of the records shows, they have used 1 lakh 30 thousand kilometers of wire to support
this bridge - mind-boggling number, fine. And it is a massive structure; you will feel
it only when you travel on this. So, you have the feeling of travelling on
the bridge. You can see these cables supporting this, thus a massive pier which is supporting
this suspension. And what I want you to notice is, when we see here the board, the board
would say there is a maximum speed limit of 45 miles per hour. See the idea here is, when
you design bridges, you will also have to load them carefully.
See, after the bridge was established in the 1960s, they had a 50th-year celebration, and
they had invited people around to come and visit the bridge. Without their anticipation,
about a lakh of people landed on the bridge; the bridge was not designed for this, it is
a suspension bridge. So, you develop some structural problem at the mid-span of the
bridge. So, they have now decided for the 75th year they are not going to invite the
public, because loading is very important, fine. And this is on the Pacific Ocean; that
is separating the San Francisco bay and this, and they have designed the bridge to withstand
wind speeds of 160 kilometers per hour. See, the scientific community learnt a lesson
from Tacoma Narrows Bridge; that collapsed because of wind forces. And it appears it
can accommodate a swing of about 8 meters! Imagine, if there is going to be a swing like
this, you should not get on to the bridge and then walk. See, now Kerala is in spate;
all the bridges where you have; they all have overflowing water beneath, one tendency for
human beings to go and watch. And it is having a heavy current of water beneath. And imagine
thousands of people stand on the bridge; you will also have to go with the water. So, do
not try out all of these stunts. And I would also like to show you another
interesting application where the technology is going, you all feel that glass is brittle
and this is a 70-foot cantilever bridge at Grand Canyon. See scientific development is
so good that we have mastered how to make bridges out of glass. They are very strong;
you can temper them, and you use some kind of a coating and fibers and so on, and so
forth, you have this. And what you will have to look at is you see
the Canyon beneath you very clearly, and they have also said that it has good friction so
that you can easily walk. You can also see people walking on top of it. Of course, this
is not meant for your car to travel on this; human beings can walk over it. To that extent,
you know this is achieved, and you find that glass is used. Glass is also attractive because
it is a green material. If glass is broken, I can melt it, use it for some other purpose.
So, that is how all civil engineering constructions are now heading towards to. You go to any
mall; you will find railings are replaced by glass unless you understand glass from
mechanics point of view, you cannot employ them there.
Let us now get onto, our understanding of a slender member. You know, I have shown a
small portion of the member; imagine that this member is very long, the cross-sectional
dimensions are much smaller compared to its length. And I have also taken a member which
has a plane of symmetry. I have also shown the axis. I have the x-axis
along the axis of the member: y and z axis. Suppose, I pass an imaginary plane through
this slender member, what are the forces that a slender member can transmit? We have to
understand that, and there is also a symbolism that is used in higher-level studies. They
use the symbolism like this. I could have a force acting on the surface of the member.
This is denoted by two subscripts: first subscripts denote the plane on which it is acting. So,
you can very well see that this surface is actually the x-plane, plane is indicated by
the outward normal. And the force is in the direction of y. So, that is why this is put
as F xy. This is the notation that you will come across in your second level of course.
But in this course, this is simply represented as capital V with a subscript y indicating
in the y-direction. There are multiple symbols used in textbooks.
My interest is to expose you to these symbolisms. You have to know that for you to interpret
when you come across. I would like you to make a neat sketch and put these forces. I
could also have a moment like this; when I have a moment like this, the member can bend
in this plane. I have a slender member; this moment will bend it. See only in this course,
you make a distinction between; what is the moment that causes bending of the member,
what is the moment that causes twisting. I think in your earlier learning; you have
always associated moment as a twisting moment; you have never even bothered about whether
a moment can produce bending. If I have the moment acting on the axis, the member will
twist. So, I could have a moment like this; this is denoted as M xz. So, z is the direction
this is the positive direction that I have here, this will cause bending in xy plane.
I have this member; I have bending in this plane, or bending in this plane, ok. When
I apply the bending moment anticlockwise, it will bend like this; when I apply the bending
moment clockwise, it will bend like this. I could also have a bending moment, denoted
as M xy. So, when I have this member, it was initially bending in this plane. Now I can
also bend like this, or like this in the horizontal plane. Then finally, I can also have a force
in the x-direction, and a twisting moment. The moment M xx is also a moment. Has the
same units as Newton-meters. Because it does the action of twisting the member, you qualify
that as a twisting moment; you also call this as a torque. I think in your study of physics,
you have worried about torque, and you have always associated moment to a torque; you
are never associated the moment to bending like this or bending like this.
In higher studies, you will have two subscripts: one of the first subscript denotes the plane,
and the second subscript denotes the direction. So, a slender member - in fact, can support
three forces; it can transmit three forces, as well as three moments. If I have to design
a slender member, I need to know what are the forces acting on it. And I would like
to find out the critical location so that I can design the cross-section appropriately.
So, I essentially need to know the variation of these forces, not at one section, how does
it vary along the length of the member. That is what you get from diagrams like; an
axial force diagram. We had seen Fxx is acting along the axis of the member. You may want
to see how does it change from, section to section along the length of the beam. We call
that as an axial force diagram. I could also have a diagram, which indicates the variation
of shear force: it could be F xy or F xz either of the two. Then I would also be interested
in knowing the variation of the bending moment, along the length of the member; I call such
a diagram as a bending moment diagram. And I may also be interested how does the twisting
moment varies from cross-section to cross-section. So, the idea of this chapter is to learn how
to plot; of the four, two of the diagrams, shear force diagram and bending moment diagram.
And in this what is important, I have a beam like this. Suppose, the beam is subjected
to a bending moment like this. You could see, I am applying a clockwise bending moment,
fine. And, how do you find what happens to the top fiber? This is obviously, not rigid.
It is a very flexible member. I am not a superman, but still, I can show the deflection so easily
here. With small forces, this is bending like this.
Can you see something happening to the top fiber physically, and something happening
to the bottom fiber physically? What happens to the top fiber? It is stretched, and the
bottom fiber is compressed. This is what I had shown in your earlier discussion. When
we are comparing a truss and a beam, in the case of a beam, their forces the stresses
developed; vary in a triangular fashion. One half of it experiences- tension, another half
of it experiences- compression. I have shown the bending moment like this
clockwise. Suppose I have the bending moment anticlockwise, reverse of this would happen.
So, from your design point of view, more than the sign of the bending moment, you are worried
about the magnitude of the bending moment. That is very, very important. The magnitude
is very important for your design purpose, that is what you are really looking at. Nevertheless,
when I want to draw the diagram along the length of the beam. I should have both the
magnitude and direction of that beam bending moment properly denoted; that requires a systematic
training. We will see the nitty-gritty details now.
Let me take a simple beam. I have a beam supported on a pin-joint and a roller-support. Here
also given the dimensions, you could replace this bar, as a simple line for your notes.
And, one of the first steps in analyzing this problem is, to determine the reactions. It
is also given; the distances a and b are such, a plus b equal to the total length L of the
beam. Let us write the free body diagram of the simply supported beam. You should put
the coordinate axis and replace the supports by the forces, which we have already learned;
how do I replace a pin-joint, I do not know the direction of the resistance. So, I denote
this by a horizontal component and a vertical component.
See earlier, we have put this as a reaction, and put it as R A x, R A y, and so on. It
will be too cumbersome to write like this. I have labeled this as A. So, you can also
simply write it as A suffix x and A suffix y. And, how do I replace the support at the;
it is a roller support. So, I have only one reaction By.
So, I can use the equations of statics. When I say a sigma F x equal to 0, that gives me
A x equal to 0. I say sigma F y equal to 0, that gives means A y plus B y equal to P.
And I can take moment about any point. I take it about point A. When I take it about point
A, I have the force P, and I have the reaction B y.
And what is the way force P gives the bending moment; it is clockwise. I would like you
to visualize that, it gives the clockwise bending moment. And the reaction B y gives
the anticlockwise bending moment, when I look from point A. So, I get this as minus P a
plus B y into L equal to 0. I get this reaction at B as P into a by L. It is not Newton meters;
it is Newtons, ok. You correct this. And I have A y equal to P b by L Newtons.
In fact, once you are experienced, you should be able to determine this reaction, just by
inspection. When you are learning it in the initial stages, you will write sigma F x equal
to 0, sigma F y equal to 0, sigma M equal to 0. After solving 10 problems, you should
be able to write them by inspection, because this is very, very simple, ok.
I have to find out, what is the bending moment. So, I take an arbitrary section, through the
beam between 0 and a; I have taken the origin here. So, between 0 and a, I will take a section.
So, that will tell me what happens in this section. If I take another cut between a and
a plus b, I would be able to find out what is the general variation in the section.
Now, let me make an imaginary cut at a distance x from the point A. I call this as section
1 and section 2, and I draw the free body diagram. When I say the free body diagram,
I should put what force the slender member is transmitting. We have already seen in general
it can transmit: three forces and three moments. Here the problem is simple. So, it will be
essentially transmitting a shear force and a bending moment.
And in your earlier discussion, we have said this is an unknown; I have the reference axis
like this, I could represent the unknowns as positive quantities, to start with. I can
put the shear force V like this, and we have followed a convention: anticlockwise moment
is positive and clockwise moment is negative. I do not know what is the magnitude of this,
what are the directions. To start with, I put this as, in the positive direction. My
mathematics will tell me whether the assumption is right or wrong; when I analyze the free
body. But I just follow the principle, fine. So, when I do this, I get sigma F y equal
to 0, I get V equal to minus Pb by L, I determine the moment, that gives me M equal to Pbx by
L. Is there any difficulty at this stage? There is no difficulty at all. Now, let me
solve the problem not by taking the section 1, let me solve the problem by taking section
2. I have the section 2; I can start afresh.
When I start afresh, if I do not know the direction, assume it in the positive direction,
and then proceed with it. So, let me put the shear force acting in the vertical direction
like this, and your bending moment is anticlockwise. Let me solve this free body diagram. Let me
put F y equal to 0. I get V equal to Pb by L. I have M equal to 0. I get this as finally
when I do the simplification. I get this as minus Pbx divided by L.
Individual free body diagrams are systemically analyzed, absolutely no problem. Have we missed
anything? In a free body diagram, if you do not know anything, you assume it in any direction;
your mathematics will give me the final direction, ok. You will not say that whenever I solve
the problem, I solve from left to right. We would solve a problem by taking a section
out of the two sections, which is simpler to solve, we will take it and then determine
the unknowns. Ultimately, I want to find out what happens
at section xx; that is my interest. If my interest is only that, it is simpler to handle
the section 1 and get the answers; section 2 is not good for finding out for this section,
at a distance x. On the other hand, if I have to find out what is the variation in this
segment of the beam. I can make a cut here; then, this one will be simpler to analyze
than what you have it here. So, I want to set a stage. We may have to
solve a given problem, starting from left to right or right to left, which one is simpler
in handling mathematics. We had one convention. If you do not know the unknown forces put
them as positive, and then find out the values - it is not sufficient. Do you see the need
for developing a sign convention? If you look at the answers that you have got in this,
and the answers that you have got earlier. I have got the answer for this. When I summarize,
I get this as V equal to minus Pb by L, M equal to Pbx by L. When I analyze the segment
2, I get V as Pb by L, M as minus Pb x by L. When I finally, go and translate this on
this, when I change them; both the results will give me the same answer. But the question
is, when I want to plot a bending moment diagram or a shear force diagram, how will I write
this as positive or negative? Individually, what I have solved in this free body diagram
is correct; individually, what I have solved this free body diagram is correct. But collectively,
we have missed a very important point. You know I would like to spend sufficient
time on the sign convention; the idea is each book follows a different sign convention.
See, after learning beams under me, you should be able solve problem from any book and verify
your answer is correct or not with his sign convention. Because people rush through the
sign convention. You should not rush through a sign convention; you should get your fundamentals
clear. I want you to go back and ponder today: what
are the ways that we could rationalize our steps. So, in this class we have looked at:
what is a beam, what are the different classifications of the beams. We have looked at: cantilever,
simply supported, overhung, continuous beam, propped cantilever, and also a fixed-fixed
beam. And we have also classified them as statically determinate and statically indeterminate.
Then we looked at several examples of, what is a beam in things around you. And I said
of the distributed loadings; you should be comfortable in handling, uniformly distributed
load, and triangular loading. Because these type of loadings you come across in many of
the practical applications, you cannot miss them.
Then we have also looked at beam is essentially a slender member, and a slender member can
transmit in general three forces and three moments. And in this chapter, we are confining
our attention to the slender member behaving like a beam. So, we take a simpler beam; for
analysis, the loads are in the same plane as the beam is drawn. You essentially have
only a shear force and a bending moment. And we have looked at then, difficulty in
associating the sign for a given bending moment or shear force; when we solve the problem
from left to right or right to left, you need to have a sign convention. We have not done
any mistake individually. But collectively, when I want to solve a complicated problem
with several loadings, I may have multiple sections, I should have a convention for me
to analyze from left to right or from right to left. So that, I could identify a strategy
how to solve the problem in the easiest manner possible with less mathematical computation
-Yet, I get a result that can be interpreted without any doubt.
Thank you.
