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 Structural Mechanics Constraints 

Hello friends welcome to the lecture today's topic is constraints. We will cover them in detail like we'll discuss what are constraints one of the type of constraints and how they behave under loading conditions okay so,

Structural Mechanics Constraints

What is constraint ?

Well constraint word itself means limitations or restrictions so these are the structural support conditions that are mathematically expressible as constraints on individual degree of freedom so what we can say is that constraints are supports that are used to restrain structure against relative rigid body motion.

So for any component of structure we have six degree of freedom and six degree of freedom means that there are six rigid body motions so now you will think like what is rigid body motion well if you'll go by the physics definition a rigid body is a solid body in which deformation is zero or you can see that it is so small that it can be neglected so we can say that the distance between any two given points on a rigid body remains constant in time regardless of external forces exerted on it now for an example let's take a body which is having some length and breadth so this is a five oi it is having specific length in prayer.

 

 so what I am doing I'm just translating the body in X direction and then in y direction and this at the same time I am providing rotation to it so even after displacing from its original position the fiber is same its length and breadth is same like original length only so we can say that this motion is a rigid body motion okay. 

Rigid body motions Structural Mechanics ?

Then for a planar body planar body means any body in a plane like a 2d structure we have three rigid body motions how so this is a this is a body in XY plane so what are the possible motions what are the possible rigid body motions it can have so it can translate in X direction it can translate in y direction and it can rotate in XY plane that is it can rotate about z axis so these three are the possible motions is body can have so what we can say is that for a planner body we had three rigid body motions out of which two are translation and one is rotation.

3d body

Okay then for a 3d body.

We have six rigid body motions out of which three are translation and 3r rotation for an example this is a block. Now you can translate this block in three directions either in X Y or either in Z similarly you can rotate this block about x axis about Y axis or about z axis. Okay so these were this was all about the rigid body motion now you know that structural system transfer there loading through a series of elements to the ground each connection is designed so that it can transfer or support a specific type of load or a loading conditions so that is that this is our next topic and that means we are going to discuss. 

What are the different type of suppose ?

1~Pin Joint

So first one is pin joint 

a pin support can resist both vertical and horizontal forces but not a moment they will allow the structure member to rotate but not to translate in any direction okay but it is also true that a pin connection could only rotate in could only could only allow rotation only in one direction okay so what you will get as a reaction forces you will get one horizontal force and one vertical force or you can say that you will get one reaction force at some angle.

So this is the representation of the pin joint Rx is our horizontal reaction force in X direction and our Y is the horizontal reaction horizontal so a vertical reaction force in y direction so their resultant is are at an angle of theta now from simulation yeah you can see this so this is the how pin joint behaves and then under any loading conditions and from this what you can conclude is like a single pin connection is not sufficient to make a structure stable so another support must be provided at some point to prevent the rotation of the structure.

2~Hinge Joint

So next joint is  hinge joint is also called as revolute joint they carry shear and axial forces but not moments so they allow the jointed members to have different but same displacement now see what is prevented is relative displacement of membranes and what is allowed there is rotation horizontal and vertical displacements so this is a representation of hinge joint so these two bars as you can see they are joined by a hint support there's a hinge joint in between so when you when you will make a Freebody diagram or you will provide reactions you will get our eggs that are equivalent opposite directed horizontal and the vertical forces our X and our Y and this is the simulation this is how our hinge joint is going to behave under loading conditions.

3~Roller Joint 

Ok so next one is roller joint but roller joints are used between two surfaces okay which can move relative between each other and that surface can be horizontal vertical or sloped at any angle the resultant reaction force will be a single force whose direction will be either perpendicular will be perpendicular to the surface either it is two words or away that totally depends upon the what kind of loading what about what are the loading conditions okay so rotation axis rotation axis is not fixed in the roller joint so it is so it lets the surface to create a linear motion it can only carry vertical loads and so we can say that they prevent normal translation but it is capable of tangential translation so this is the representation of roller joints what we will get is one reaction that that is for vertical load together as you know it K it can carry only vertical loads and this is how it is going to behave under loading conditions next is 

 

4~Fixed Joint

fixed fixed connections are very common and they demand greater at engineering construction because they are they are often the source of building failure so thick supports can resist vertical and horizontal forces as well as movement since they restrain both rotation and translation they are also known as rigid supports this means that a structure only needs one pick support in order to be stable so the representation of fixed supports always includes two forces one will be the horizontal force one will the vertical force and then a moment in this the red bar is the fix it is a fixed one so it then you can say that it's a fixed wing now the next one is, 

5~Link Joint

in link joint joint bodies are connected to manage force and movement and in this translation is allowed only perpendicular to the link and restricted in the direction of link so what is prevented is translation in the direction of link and what is allowed is translation perpendicular to the link and the rotation see here this linkage is making an angle theta with the horizontal so when you will represent this will get a force resultant force at an angle theta and what are the possible motions see it is it can move perpendicular to the link but it is not allowed to move in the direction of link so this is our link joint so this was all about drawings now

Avoid the rigid body motion Of Planer Structure.

in order to find the Stresses and Strain one need to avoid the rigid body motion why see from the basic definition of strain.

What what you know is that strain is defined as the change in length of fiber divided by the original length of fiber but in case of rigid body motion. there will be no change in length of fiber so there will be no deformation which means no stress and strain. so in order to avoid the rigid body motion one needs to apply constraint all possible rigid body motions can be avoided. okay so let's take an example for this case yeah we can see that there is a random structure which is not subjected to any constraints.

 

 okay so if they apply load at this time it is a simple dynamics problem which will result in a rigid body translation and rotation body will rotate or translate what whatever Lu whatever type of load you will apply it will accordingly translate or rotate okay now as there are two translations and one rigid body motion now what I am doing is I am providing a pin constraint to this okay so now from basic definition of pin joint that we have covered what you know is what we can say is that the two translation are constrained but still there is one motion that is left which one is that is rotation still it can rotate it can rotate about that point right pin point so in order to avoid that rotation let's apply the roller joint so now what we'll do I will apply a roller joint whose orientation is offset from the axis shown

So Y offset from the X's will know more about the reason in the detail soon so let's take a simple rectangular box which is subjected to pin and roller joint such that all motions are constrained now can you please pause for a moment and think why it's rigid motion is constrained what now what I'm going to do is its change the orientation of roller such that the roller is perpendicular to the axis okay if I say that the infinitely Small body rotation is not constrained what will be your expression really is that same like this girl don't worry let's see why it happens so if we go back to the basic definition of a rigid body the fiber length must remain unchanged okay so let's investigate that does it remain constant or changes now we will do the mathematics the question that one need to ask first is what are the approximations we use while solving the solution the first approximation that we are going to use is it is a approximation of linear analysis we watch what it says is for a very small angle sine theta is equal is approximate to theta and cos Delta dies approximate to 1 that means now if we rotate the structure about pin joint this is a pin joint and now I'm going to rotate it about it's some small del theta a new simple geometry.

What you'll see is that the mathematically the length remains constant so even after computing analytically of performing simulation it will not fetch good result cause of linear approximation see initially the length was L and after reflecting it by an angle Delta Theta it's it's length will change to L cos theta right so what will be the final length it will be L plus del n so del L is equal to what is the increment in length initial length minus initial length minus the final length when you'll do this and you will apply the approximation that we have used scores feta will reduce to 1 and 1 minus 1 will give you 0 which will ultimately give you Delta L is equals to 0 ok so I hope now it's clear to you so we can handle this type of problem by using nonlinear analysis and this is something that we will cover in our upcoming lectures in details so thank you and if you liked the video please subscribe for more interesting Lectures

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