[ Music ]
>> Hello, everyone. Welcome back to Engineering
Mechanics. In earlier sessions, we talked
about particle equilibrium, a very important
topic in engineering statics. If you recall,
the particle is a concentrated mass. We can
ignore the size and shape of an object, making
a particle assumption. Particle assumption
is very useful in solving a wide range of
problems, as we have seen. Having said that,
if you would like to analyze a body closely,
if you want to perform a very detailed analysis,
then rigid body assumption is not adequate,
as we can see. So, what we have to do is to
consider the body as a whole along with dimensions
and size. And when you want to consider a
body in such fashion, then we will have to
perform a rigid body equilibrium analysis,
not a particle equilibrium analysis. So, earlier
we looked at 2-D particle equilibrium. Now
we are going to do a 2-D rigid body equilibrium,
meaning we are going to consider a body with
all its dimensions and size. So, 2-D rigid
body equilibrium. We are still in 2-D, meaning
2 dimensional plane. In particular equilibrium,
in case of 2-D, we have 2 equations because
the particle has no rotation. The particle
is represented by a geometrical unit, point.
A point has no dimension so there was no rotation
associated with a point. So, we have 2 equations
in particle equilibrium. Now, we are moving
onto 2-D rigid body equilibrium and therefore
we will have an additional equation. The additional
equation comes from the fact that the rigid
body can rotate. So, we are going to take
a closer look at that in a minute. But before
we go and do a closer look, let's talk briefly
about rigid body. What is a rigid body? As
the name implies, rigid body means a body
that does not deform. If you want a formal
understanding of rigid body. Say pick up 2
points on the rigid body. Here I have a marker
on my hand. Let's pick up 2 points, maybe
these 2 end points. Draw a line connecting
those 2 endpoints and let's say we subject
this rigid body, body to some load, loading
condition such as the way I am pushing this.
And if you look at it, this body under the
influence of the forces that I am applying,
is not changing its dimension and that would
be a formal definition of rigid body. Rigid
body is a body that does not change its dimension
under the influence of the forces that I apply.
So, with that understanding, we are on number
1. But I do want to give a word of caution
here. All bodies when they are subjected to
loading conditions will have some kind of
a small internal deformations. Those are very
important from an engineering analysis point
of view and you are going to study about that
later in the class called strength of materials.
But in our context, those small internal deformations
do not have any influence on the static equilibrium
study. So, as far as we are concerned in this
class, we will ignore all those small deformations
and we assume rigid body as a body that does
not deform. All right? So, we understand what
is a rigid body assumption. Now, rigid bodies
ought to be supported. All bodies ought to
be supported, right? For example, I am supported
by this floor. If I am not supported by this
floor, I will not be standing here. So, all
bodies ought to be supported and there are
a few popular supports that you are already
aware of. Think about a pin joint or a pin
support. Well, you see this in your life every
day. Every time you open a door or close a
door, you are actually using pins. The doors
are typically supported by hinges which are
actually pin supports. Not just the doors.
Take a scissor like this. The scissor has
actually 3 parts, the upper part, the lower
part and this is connected by a pin. So, these
2 parts are connected by a pin support. All
right? So, there are various supports. Popular
support is a pin support which allows rotation.
All right? There's a roller support. Now,
for example here I have a little metal plate
with a screw in it. It is actually used in
photo industry for mounting cameras but here
I have a slot and I have a, a little screw
that you can see and it can spin, as you can
see. So, it's like a pin support but in addition
to allowing rotation, the slot allows this
body to move along that. Which means this
is not just a pin support. I call it a roller
support which allows the rotation as well
as the movement along this. It doesn't allow
you to move in the vertical direction. So,
this pin support, this roller support will
actually create one reaction. Again, we are
going to take a closer look at this in a minute.
On the other hand, a pin support such as this
allows only rotation. It doesn't allow any
horizontal movement or a vertical movement
meaning the pin connection will give you 2
support reactions. Now, when I say reactions,
reactions are caused when there's a resistance.
Reactions are created when the support restricts
your movement or the support eliminates movement.
So, in this case, in the case of a scissor,
the pin support does not allow any motion
in X direction or any motion in Y direction.
It allows only rotation. That's a pin support.
Again, to repeat the roller support, in this
case the roller allows rotation as well as
horizontal motion but it does not allow a
vertical motion and that would be a support
reaction. So, the objective of 2-D rigid body
equilibrium is to typically calculate the
reactions from the supports. Or sometimes
other external forces. And we are going to
take a closer look at those problems now.
Hello everyone, welcome back. We are going
to take a closer look at the problem. Before
we do that, a quick review. We talked about
rigid body. Rigid body is a body that does
not deform and if you want a, a better way
of explaining, here's is a, an outline of
a rigid body. I pick up 2 points on the rigid
body, connect them by a line, subject this
rigid body to some loading conditions, all
right? And then I look at the line. It does
not change the dimension. So, a rigid body
is a body that does not change its dimensions
under the influence of forces. Any small internal
deformations that take place is not important
from our perspective because it does not have
any bearing on the static equilibrium. So,
that's rigid body, all right? Now, the next
step is we briefly talk about the equilibrium
conditions. Now, I want to draw your attention
to what we talked before. We discussed particle
equilibrium, meaning a system that is represented
as a concentrated mass. And in those cases,
we wrote the equilibrium equations as these.
Sum of the forces allowing X direction equal
to 0 and sum of the forces allowing Y direction
equal to 0. In other words, we had 2 equations
and we could solve for 2 unknowns. And we
are done some examples and you understand
that. Now, we are moving on to rigid body
equilibrium and in rigid body we are concerning
not only the mass but we are also looking
into the dimensions. So, this rigid body in
theory will have motion along X axis, motion
along Y axis, and a rotation about Z axis.
And therefore in order for this body to be
in equilibrium, we have to make sure sum of
all the forces allowing X and Y equal to 0
as well as the moment. So, our equations for
a 2-D rigid body equilibrium will be sum of
the forces allowing X direction equal to 0,
sum of the forces allowing Y direction equal
to 0. I'm writing the scale of equations.
And then sum of all the movement about a point,
for example 0 equal to zero. So, we will have
3 equations in case of 2 dimensional rigid
body equilibrium problem. Meaning we can solve
for 3 unknowns. All right? Lastly, when we
look at the rigid body, these rigid bodies
are not floating in space. In engineering,
in fact a body just floating in space is of
not important. We like to get the bodies interconnected
so they can do something useful. So, all bodies
that we talk about in engineering are somehow
connected. So, in this case, let's take a
look at the supports that we are going to
be considering in this subject, all right?
So, support reactions. All right? Now, I want
to recall your attention to what I talked
before. I showed you a scissor and I showed
how a pin holds these 2 scissors. This is
a pin joint. This pin joint allows only rotation.
It does not allow any translation. So, if
I want to represent that on a diagram, I'm
going write the pin, draw the pin joint like
this. Meaning I can connect something through
a pin here and the pin joint is connected,
represented using this diagram. And as I showed
you, it allows rotation but it does not allow
translation. That means there will be 2 support
reactions. And those 2 reactions are a reaction
along X and a reaction along Y. And I really
do not know the direction but in general,
I'm going to assume the direction is positive
and I do not know. So, this pin joint can
be replaced by 2 support reactions like this.
For example, if I call this joint or this
point as A, I say this is A sub Y and this
as A sub X. So, the pin joint will be represented
by 2 support reactions. The next support reaction
I want to briefly talk about is the roller
support. Roller supports are widely used in
engineering, meaning your body can actually
move along one direction. And the roller support
is represented in many different ways but
one way is to show something like a pin joint
but put them on rollers like this. Meaning
this body allows rotation and also allows
a horizontal motion, as you can see. So, this
can be represented using just one support
reaction such as a Y. Now, if you want a physical
example, I want you to take a closer look
at this body. Here I have this pin which can
move along the horizontal direction but it
cannot move along the vertical direction and
that's why you are getting this vertical support
reaction. All right? So, this is a roller
support. So, let's try this one. This is a
pin support. This is a roller support. And
there's another popular support I want you
to understand is a fixed support. What is
a fixed support? Well, basically you fix something
like this. You fix it like this and you cannot
move in X direction. You cannot move in Y
direction. I cannot rotate above Z. It's fixed.
It's like a tree on the ground which is kind
of uprooted, which is fixed and it cannot
move. And that kind of a joints, connections
are very popular. And sometimes fixed support
is also known as a built-in support. And how
do I represent the fixed support? Well, a
fixed support is represented by a diagram
like this, where a beam is stuck into the
concrete wall, for example. And this obviously
eliminates all motions. Therefore, it'll give
you 3 support reactions. And those are support
reaction along Y axis, support reaction along
X axis and finally it eliminates a rotation
and that rotation can be represented like
M sub A at the moment. It's a couple, couple
moment. All right? So, these are all popular
support reactions that I want to make sure
you understand. So, pin support, roller support,
fixed support. In addition, you will also
hear about a smooth surface. A smooth surface
is like an icy surface and if you stand on
it, obviously the surface supports you because
you're not going down but it doesn't help
you in keeping your motion away from X or
Y direction. In fact, you can move along and
therefore a smooth support which can be represented
simply a floor like this and this is a body.
And if you want to represent a smooth support,
it'll be like a roller support, you'll have
a direction along the vertical axis, all right?
If it is a rough support, you could have 2
support reactions. So, a rough support would
look something like this. So, this is smooth.
This is rough. All right? These are other
supports that I want you to keep it in mind.
There are numerous other supports. Cables
are another support that we discussed a lot.
When you are dealing with a cable, that it's
always tension of the cable, therefore you
know the line of action of the force all the
time. So, these are other popular supports.
Now look, take a look at an example that involves
these supports and we will solve for support
reactions. All right, here is a problem. This
problem is taken from the popular Bear and
Johnson text 10th edition. Simple problem.
And let's take a closer look at the problem.
We have a beam. We have a few applied external
loads and the beam is supported here by a
pin joint. And you can recognize the pin here.
And this is your ground. And then here the
pin is supported by actually a cable. The
cable is tied to something on the ground.
Many students confuse this with a pin joint
but basically a cable is tied here. So, our
interest is only the cable, so we don't even
have to worry about the point C. All right?
So, how do we solve problems like this? Well,
the steps are going to be the same and I want
to make sure you completely understand that
and follow this every time. When we deal with
static equilibrium, equilibrium problems like
this, the first step is to draw a free body
diagram. You know how to draw a free body
diagram and we must do this properly, step
by step. A free body diagram is a self-contained
mathematical model of a physical system. It
has all the information necessary for solving.
So, we must draw a good free body diagram.
You make a mistake in drawing the free body
diagram, your equations will be wrong. And
if your equations are wrong, then obviously
you won't have the correct answer. So, it's
very important you draw a correct free body
diagram. After drawing the free body diagram,
you will have to develop the equations and
then solve it, all right? So, recognize the
supports here, pin support, and a cable support.
And we are going to connect this party or
free this body. I want to show you the body.
This is the body I'm going to free from the
supports. In fact, that's where the word free
body comes from, freeing the body from other
bodies or supports. In this case, there's
only one body that's connected here and we
are going to free from the support. And then
we are going to draw a free body diagram,
all right? So, here is my body again, or the
system again in a smaller diagram. And I'm
going to draw the free body diagram. So, here
is the free body diagram. I want it right,
draw it right here on the side on the right-hand
side. And this is my point A and this is my
point B. I have not drawn this to scale but
it is going to represent the free body of
this B. So, as I told you, pin support here,
therefore it allows 2 support reactions and
those 2 support reactions are going to be
A sub Y along vertical direction and A sub
X along horizontal direction. This is the
2 support reactions from this pin. There is
also support reaction from the cable and you
already know the cable's always on tension.
So, if I disconnect the cable here, the tension
will be on this direction and that's my cable
tension and I'm going to call it T, because
it is my tension. So, these are all the 3
support reactions. And remember. In 2-D rigid
body equilibrium problems, there are 3 equations,
sum of the forces along XY equal to 0 and
movement about a point is equal to 0. So,
there are 3 equations and if you look at it,
we do have 3 unknowns so you should feel good
that we can solve this problem in theory already.
Now, what else is not shown here? We have
shown the free body and we have to now show
all the forces. I have shown the support reactions
which are the forces. That's the first step.
Then I want to show all the external loads
that are shown here. So, I'm going to start
here. This is my 15-pound force. Here is my
20-pound force. Here is my 35-pound force.
Then I have my 20-pound force. And then I
have my 15-pound force. So, I have freed the
body. I have shown the support reactions.
I have shown all the external loads, all the
forces are shown and labeled, and my critical
dimensions are right here. And the only thing
that I have not shown is a coordinate system.
So, I'm going to say my coordinate system
is going to be this, XY. All right? Now that's
a complete free body diagram and with that
information, I can move onto the next step.
Strictly speaking, I should show these dimensions
here, so I'm going to just put that in here,
6 inches, 8 inches, 8 inches, 6 inches. Now,
this is a complete free body diagram, everybody,
understand that. Once I have the free body
diagram, I never have to look back anything
else. A free body diagram must be self-contained
and I believe I have, I have all the information
necessary right here. All right? So, now the
next step is developing the equation. So,
let's start with our traditional approach.
Some of the forces along X direction must
be equal to 0. So if I look up, I have no
force other than A sub X along X and therefore
A sub X equal to 0, that's my first equation
and that is also part of the answer. Because
the question is finding the support reactions.
I got this one. Now, I'm going to sum up the
forces along Y direction equal to 0. So, I
start with this negative 15. Realize these
are all negative direction, negative 15, negative
20, negative 35, negative 20, negative 15,
all these applied loads. Then I'm going to
also have support reaction positive AY and
then negative T equal to 0. I can reorganize
this and if I reorganize this, it is going
to be AY minus T. I'm keeping unknowns on
the left-hand side and the known constant
to the right-hand side. And if you add up
all these, 15, 20, 35, 70, 90, 105. This is
my second equation. It has got 2 unknowns.
So, I cannot solve it. So, I'm going to go
to the next step, all right? The next step
is the easiest thing would be to take a movement
about any of these points and I am going to
take, say, for example, movement above point
B. If I take a movement above point B, I will
not have any movement due to, due to the force
T, tension, because there's no movement, arm.
I'll be able to calculate all other movements
and since there is only one unknown, I'll
be able to find out A sub Y immediately. On
the other hand, some of you say that I'm going
to take movement above point A, that would
be fine as well, in which case you would be
finding the T first. So, let's take movement
about point B, should be equal to 0. And if
I did that, I'm going to start in one end
and keep going so I'm not going to miss any
forces in the process. SO, 15-pound force
gives you a counterclockwise movement. As
you can see, it's going to go counterclockwise.
So, it is 15 pounds times this distance from
here all the way, that'll be this distance
8, 6 plus 8 plus 8 plus 6. That would be 8,
8, 16, 28 inches. All right? Then I'm going
to move onto the next force, 20-pound force.
Again, that is going to give you a counterclockwise
movement. So, plus 20 times this distance,
8, 8, 16, plus 6, 22 plus 35. Which is going
to give me a counterclockwise movement as
well which will be equal to 35 times 14 plus
20 next force times 6. And I'm going to continue
in the next line. I'm done with all these
forces. Now, this 15-pound force won't give
any movement because it is right here. T won't
give any movement. So, the only force that,
and AX obviously won't give any movement.
So, the last unknown support reaction left
in this particular equation would be AY which
is actually going to give you a clockwise
movement. Why? It's directed upwards. So,
it's going to be negative AY times 6 equal
to 0. Now, I'm not going to spend time expanding
right now here. You can do this yourself with
a calculator. If I did that, I'm going to
calculate AY right away and it is going to
be equal to if you did your math, 245 pounds.
So, this equation, movement equation, immediately
gives me the answer for this unknown support
reaction. So, with this information, now I
can go back. This is my third equation actually.
And I can go back and substitute this and
I can find the tension as if I substituted
245 here, my tension will be 140 pounds. And
if you have done this correctly, so I have
my free-body diagram. I have my equations,
and the final answer I will give it like this.
A sub X is equal to 0. A sub Y is equal to
100, sorry, 100, 245 pounds. And the direction
is this. Finally, tension is equal to 140
pounds and it is going to be in this direction
and here is my answer. So, I have solved a
simple beam problem involving 3 support reactions
and, and that's all we have to do about this
problem. And let's move on to the next, next
example.
