define to be able to get the complete displaced geometry of the structure.
I will look at two different types of structures. One set of structures which are made up of only
actually loaded members if you remember those are trusses. I am going to start off with trusses
because even trusses and beams and frames which are essentially structures made up of members
which can deform flexurally, the treatment is different. Let us look at a truss and again I go back
to the simple truss that I had last time. We want to find out what the Kinematic Indeterminacy of
this truss structure is.
Before that I need to define what a Kinematically determinate remember, it is very easy for you
to know how to get the statically determinate structure. You know when you look at it what a
statically determinate structure looks like. How do you see a Kinematically indeterminate
structure? Let us take this same structure; I want to make this Kinematically indeterminate. How
will I make it Kinematically indeterminate? This is a Kinematically determinate structure. Why
is this Kinematically determinate? Let us look at this structure. Understand what Kinematic
Indeterminacy is.
It is the minimum number of displacement quantities that you need to define, the displaced shape
of a structure. In this structure, how will it displace under loads? How will it deform? Can this
point go anywhere? This point cannot go anywhere because it is restrained both in this direction
as well as this direction; it cannot move anywhere on the plane that means this point cannot go
anywhere. Can this point go anywhere? Neither can this point go anywhere. Can this point go
anywhere?
Again, it cannot, simply because every point has a hinge which restraints the displacements in
the plane of the structure. So what would be the displaced shape of the structure if it was
subjected to loads? Let us put some loads on the structure. These are the possible loads on a truss
structure because a truss structure can only be loaded at the joints. I put all possible loads that
this structure can be subjected to. What will be the displaced shape under these loads?
Remember, all these loads would get directly transferred to the supports and the structure will
not be subjected to any stress. Therefore there is going to be zero strain and since this going to be
zero strain there is going to be no displacement. What is a Kinematically Determinate structure?
It is a structure that does not displace. Is that interesting? Currently, it is not interesting. Later on,
we will see that we need to define a Kinematically Determinate structure before we can use what
is known as the displacement method of analyzing statically indeterminate structures.
Once we have a Kinematically determinate structure, it is very easy to look at this structure and
see what is going to be the Kinematic Indeterminacy or the Degrees of Freedom of this structure.
Let us look at it. Can this point go anywhere? No. Again there is a hinge which is restraining it
from moving anywhere. Can this point go anywhere? What is the restraint? The only restraint is
it cannot move vertically. But is there any restrain for moving it horizontally? No. So, this is a
possible displacement. This node can displace in this direction and therefore this horizontal
displacement is a degree of freedom of the structure. Let us look at this point now. Can this point
go anywhere? Note that this point can go anywhere on this plane and in a plane how many
displacements do you require to define? If you define two orthogonal displacement quantities,
they completely define the displacement of this point anywhere in space.
Now, if you look at this structure, a simple truss has three nodes: one node completely restrained
and cannot go anywhere, this node vertically restraint and can move horizontally, this node is
free to displace in any direction in the plane. We can define two displacement quantities which
will define the motion of this point anywhere in space. Now how many Degrees of Freedom or
what is the Kinematic Indeterminacy of this structure? 1, 2, 3 the Kinematic Indeterminacy (K.I)
of this structure is equal to 3. Therefore if you have a truss, any joint in the truss can move in the
plane and what you need to look at is what the restraints are. Let me redefine this problem. I am
not going to do an algorithmic way of getting the Kinematic Indeterminacy. How many nodes
does this structure have? It has three nodes. Now for the movement of a node or a joint in a plane
is given by two independent displacement quantities. In other words, if I know those two
displacements quantities I know where the joint is in the space. Let us look at it this way. How
many joints? 3. How many Degrees of Freedom per joint? 2. Therefore if this plane truss were to
have no supports what will be the Degrees of Freedom? It would be 2 into the number of joints
so let it be 6. The Degrees of Freedom 2 per joint multiplied by 3 is equal to 6. 6 Degrees of
Freedom since this truss has three joints. This is the unrestrained Degrees of Freedom.
In other words, the truss did not have any support, then the number of degree of freedom would
be 6. However, you do have two supports in the structure. What are the supports? At one joint,
you have a hinge which has two restraints; it stops it from moving vertically and it also stops it
from moving horizontally. It does not allow this joint to go anywhere so it gives two restraints to
this joint. What about this joint? This is a kind of roller joint. How many restraints does it give?
It gives exactly 1 it restrains the vertical motion of this joint. 1 meaning 2, plus 1. The total
number of restraints in this is 3. What is the restrained structure? How many Degrees of Freedom
does it have? 3.
Kinematic Indeterminacy by finding out displacements and also the algorithmic way: Find out
the number of joints in a truss, multiply that by 2. That gives you the unrestrained Degrees of
Freedom. You find out the number of restraints that the body has and then subtract those
restraints and you have the Degrees of Freedom or the Kinematic Indeterminacy of the truss. Let
us look at one another problem. Let us look at the slightly more complicated truss.
....................................................................................
This is the truss; we have to find out the Kinematic Indeterminacy. How would I go about it? We
are going to use only the algorithmic way of getting it, and then I will show you which are those
displacements. How many joints? 4. What is the number of Degrees of Freedom? Note that
Kinematic Indeterminacy and Degrees of Freedom are used interchangeably. What are the
Degrees of Freedom per joint for a planar truss? 2 Degrees of Freedom. What is the total number
of unrestrained Degrees of Freedom? 2 into 4 is equal to 8. We need to define what the restraints
are? Restraints here are one hinge support, two restraints - one roller support and one restraint.
So a total of 3. The total actual Degrees of Freedom is 8 minus 3 is 5. The Kinematic
Indeterminacy is 5, for this structure. Once I have defined the kinematic Degrees of Freedom,
you need to understand what is Kinematic Indeterminacy, it is the minimum number of
displacement quantities, that you need to define, to be able to get the displaced geometry of the
structure. I need to find out and define those displacement Degrees of Freedom. Note that I
should have only 5 independent displacement Degrees of Freedom. Let us look at this. Are there
any Degrees of Freedom at this node? No. Both of them are restrained. Let us go to joint 2;
restraint none. What are the Degrees of Freedom? 1 2. Let us go to 3. Any restraint? No.
What are the Degrees of Freedom 1 2 the vertical displacement and the horizontal displacement.
What about 4? It is restrained in this direction and is not restrained in this direction, so how
many do I have? I have 1 2 3 4 and 5. 5 Degrees of Freedom; 5 Kinematic Indeterminacy. It
should be fairly easy for you now, to get the Kinematic Indeterminacy of a truss type structure.
Planar truss - 2 Degrees of Freedom, space truss - 3 Degrees of Freedom because you need 3
displacement quantities to define the position of a point in space. 3 independent: x displacement,
y displacement and z displacement. What will be the unrestrained Degrees of Freedom for a
space truss? 3 into the number of joints; that will be the unrestrained and then you find out the
number of restraints that you have which are the support conditions or the supports. Subtract the
restraints and you got the Degrees of Freedom for a truss and that defines the Kinematic
Indeterminacy for a truss structure. Let us move on to that for a beam.
This is a structure that you are all familiar with; a simply supported beam. What is the Static
Indeterminacy of the simply supported beam? 0. You know that it is a statically determinate
structure but now we are not looking at Static Indeterminacy we are looking at Kinematic
Indeterminacy. Again there are 2 joints 2 nodes in the structure and you can ask why we do not
use the same method as a truss. For every node to move in the plane, there will be two Degrees
of Freedom. 2 nodes, 2 Degrees of Freedom and three restraints. This is kinematical so if you use
the truss model 2 into 2 so 2 nodes 2 Degrees of Freedom per node that is 4 and there are 3
restraints; 2 here and 1 here one degree of freedom. What is the one degree of freedom this one
(Refer Slide Time: 20:43). Is that correct? Is this single kinematically indeterminate structure
single degree of freedom? No.
Understand that members in a beam and frame behaved differently from a truss; a truss is only
an actually loaded member. All we need to know is where the two joints at the end of a truss
member is and that gives us elongation of the truss and from that we can find out the force in the
truss. Can we do that for a beam? No, because the behaviour of a beam is in the way it is loaded,
it goes like this. The entire concept of displacement, Degrees of Freedom for a beam and a frame
would be different from a truss and that is why we need to look at it differently. How do we look
at it differently? An important point is that a joint that connects beam members or frame
members in addition to the two Degrees of Freedom which define its movement also has another
additional degree of freedom. What does that do?
That degree of freedom actually defines the movement of the members, relative to each other.
What is that? That degree of freedom is the rotation degree of freedom. Whenever you look at a
beam or a frame every joint when it is unrestrained has 3 Degrees of Freedom. So a joint has 3
Degrees of Freedom. What does this degree of freedom do? If I fix this and this, this disappears
because this cannot go up or down.
Remember, that if I have a Kinematically determinate structure I can define the displaced
geometry of the structure completely. If I restrain this and I restrain this, both the displacements
are restrained which means no displacement. But is that true? No. Under this load it displaces in
this manner. Understand that I required additional displacement quantities to define the displaced
shape of the structure and what are those? Look at this rotation and this rotation; these two
rotations gives me the displaced shape of the structure. That is that additional degree of freedom
that I need to define. Whenever you have a beam or a frame you have 3 Degrees of Freedom per
joint. Let us come back to this structure; the simply supported beam. How many Degrees of
Freedom does it have? How many joints? 2. How many unrestrained Degrees of Freedom? 3 per
joint, so a total of 6. Note that this hinge support still restraints only 2 which is this displacement
and this displacement. It has 2 restraints and this one. What was the total number of restraints? 3.
What are the Degrees of Freedom? 3. This simply supported beam has 3 Degrees of Freedom or
its Kinematic Indeterminacy is equal to 3.
What is Kinematic Indeterminacy? Number of unknown displacement quantities. Let us find
those out for this structure just like we did in the truss. Note that it restraints this but allows this,
so this is my first degree of freedom. This is a pin roller; it allows free rotation so this is another
degree of freedom. This joint restraints both the motions but allows it to rotate and that is my
third degree of freedom. In other words, this simply supported beam, if I knew the horizontal
displacement of this point, the rotation at this point and the rotation at this point, I would be able
to give you how this structure has displaced. If I knew the displacement quantities I could draw
the displaced geometry of the structure. This, in essence, defines the Kinematic Indeterminacy
for beams and frames.
I want to introduce one more joint that you are all familiar with; another statically determinate
structure; this is a cantilever beam. What is the Kinematic Indeterminacy of this cantilever
beam? How many joints? 1 and 2 so 2 joints. How many unrestrained Degrees of Freedom? 6.
You need to find out the number of restraints. How many restraints? Let us look at this joint, this
is a fixed support; a fixed support is one which restrains the point from displacing in any
direction and it also restrains the rotation of that point. How will this displace? This will displace
something like this. At this point, the slope would be 0, the displacements both vertically and
horizontally would be 0, so completely restrained. At this point, it is not restrained; it does not
have any support. What does this restrain? It restrains vertical, horizontal and rotation, so there
are three restraints. So the Kinematic Indeterminacy is 3. What are those 3? It is 1, 2, 3. So, for a
beam you should be able to get the Kinematic Indeterminacy or the number of Degrees of
Freedom for the structure. Let us move on to Kinematic Indeterminacy for a frame.
Note I just said that, the Kinematic Indeterminacy for both a beam and a frame are computed in
exactly the same fashion. I can use the same algorithm that I developed for the beam. Look at
this; how many nodes 1, 2, 3, 4 there are four joints in this structure. Following my procedure,
each joint unrestrained has three Degrees of Freedom. The number of unrestrained Degrees of
Freedom equal to 3 into 4 is equal to 12. Unrestrained Degrees of Freedom is 12. Now, we need
to find out what are the restraints in the structure. How many supports? Two supports, 1 here and
1 here. What are they? fixed supports. How many restraints does the fixed support give? 3. How
many fixed support? 2 What are the restraints 6 is equal to 2 into 3. Two supports with 3
restrains each. How many degrees of freedom? 6. Kinematic Indeterminacy, remember, is a same
as Degrees of Freedom 6. I should be able to draw them 1, 2, 3, 4, 5, 6. The vertical, horizontal
and rotation of this joint and the vertical, horizontal and rotation of this joint; 3 per node; total
six Degrees of Freedom. If I know these 6 displacement quantities I can draw the displaced
shape of this frame.
Now, I want to bring in the concept of constraints. Typically what we do, when we consider
beams and frames we tend to neglect, the axial deformations because by and large in a beam and
a frame, the main way that the loads are transferred is through flexural deformations. In other
words, the structure bends to take the loads. While flexural deformations are very large the axial
deformation are not zero, but are very small. We tend to neglect the axial deformations. What
happens then?
Let me go back to a simply supported beam; I neglect the axial deformation of this member,
therefore I am saying that it is axially rigid. Note, in general, for a statically determinate beam,
the forces that you normally find out are the sheer force bending movement not the axial force.
Implicitly, without knowing, you are actually neglecting axial deformation; you never explicitly
mention that. When you are analyzing a frame or a beam the member is axially rigid. What
happens when you have a member which is axially rigid? What happens here is that you are
constraining the structure. It has no effect on the Static Indeterminacy of the structure.
Remember, that it has a tremendous effect on the Kinematic Indeterminacy or the number of
Degrees of Freedom that you get from a structure. You are constraining a structure into one
actually rigid. When I say that this member is actually rigid, this cannot go anywhere. It can
move horizontally. Forget about the rotational Degrees of Freedom; I am looking only at the
axial deformation. It can go this way so this goes this way. What effectively happens to this
member when this goes this way? This joint cannot go anywhere, this joint can move
horizontally. The member is either axially shortened or if the displacement is in this direction it
is elongated. In other words, this degree of freedom actually corresponds to the axial deformation
of the body. If this member is axially rigid, since this point is not going anywhere; it cannot
deform axially. That means this displacement has to be zero.
By making the structure axially rigid I have eliminated a degree of freedom. If I say that this
beam is axially rigid how many Degrees of Freedom does it have? It has 2 rotations. It is a 2
degree of freedom structure. How do I put it into my algorithm? I truly believe that unless you
have an algorithm way of computing Static Indeterminacy and Kinematic Indeterminacy, you are
always going to flounder because you would not know how many indeterminacies there are in
the structure both static and kinematic. Let us go back to algorithm way. How do I include this
effect in the algorithmic way? What we define is two joints unrestrained Degrees of Freedom: 2into 3 is equal to 6, restraints -3. Now, what have I done by making it axially rigid? I am
constraining the structure.
So, a new thing: constraints. Constraints, also reduce the number of Degrees of Freedom in the
structure. In this case how many constraints? It is axially rigid; whatever I have done I have
made this into 0. How many constraints do you think axial rigidity in a member brings in? 1.
Because a single degree of freedom defines the axial deformation of a member and when I make
it axially rigid that is the additional constraint that I am giving; that this body cannot deform
axially. I am making one constraint per member. How many members are there in this structure?
There is 1. How many Degrees of Freedom? 2. I have already shown you the Degrees of
Freedom in the structure. How about the algorithm for getting the Kinematic Indeterminacy of a
beam or a frame? Start off by finding out how many joints there are in the structure. Multiply the
number of joints by 3 that gives you the total number of unrestrained Degrees of Freedom. Then
find out the restraints. What are the restraints that are provided always by supports? You go to
every support and find out what kind of restraint it gives add up all the restraints from all the
supports. Those are the total number of restraints those are to be subtracted from the unrestrained
Degrees of Freedom. In addition to that, you need to look at the constraints. If, a member is
axially rigid how many constraints? Single; it is 1 per member. So, you subtract the total number
of constraints and you got your Degrees of Freedom or the Kinematic Indeterminacy of the
structure.
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