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Okay, so as I promised last time, today we will look at some of the examples solved using

the equations that we derived in the last class and that will be done by my students,

Smitty and Prasad. They are working with me as a student. Okay, so you can bill them

as much as you want. Okay, so I hand over to them.

Good morning everyone. Till now we have covered Euler-Bernoulli beam theory, Timoshenko beam

theory and the rod theory, special castor-red rod theory for solving elasticity problems.

And, based on the previous knowledge, we are looking at some, a few examples of rods specifically.

I mean, how do they behave actually? So, the very first example is cantilever beam that

has been covered for each of the cases in the class. The boundary conditions are same

as we have discussed. At the clamped end, there would be no movement, so rotation and

displacements are 0. And, at the free end, there is no movement and a static load is

acting at this free end or constant load is acting at the free end. I think we have solved

the analytical solutions for Euler and Timoshenko beam theory for this case. So, based on those

solutions, I have some plots here. The very first one is load versus tip deflection for

each of the cases. Now, as we can see that for lower loads, all these theories, I mean,

they agree with each other. But, at higher loads, the Timoshenko and Euler beam theories,

they do not, I mean, they are no more reliable actually. As we know that the rod theory incorporates

large deflections, large rotations and everything. So, Timoshenko and Euler beam theory actually

hold for the linear regime only. So, they agree only for smaller loads, but at higher

loads, they deflect from the actual path. So, this is all about tip, load versus tip

deflection. So, any doubts or? So, and next is this. In case of rod theory only, this

is axial position versus tip deflection. This plot we get when we increase the load like

this. The load is being increased from right to left. And we can see that I have taken

length of rod equal to 5 and we can see that as the load increases, there is axial displacement

also and the rod gets shorter and shorter with the tip deflection increasing from 0

to 3. And rod does not get shorter, right?

It displaces. Yeah, it displaces or we can say, I mean,

it will look shorter. I mean, it is not the material.

Yes, with the tip deflection. Initially, it is like this and then it will

be like this. It is simple like this. So, this kind of behavior we do not see in Timoshenko

and Euler-Bernoulli main theories only for rod theory. And then this is that error analysis

that we already covered in the class. I will write down the formula before discussing this

actually. This was displacement called Timoshenko and Euler-Bernoulli divided by this. And it

was I think 1 by 4 a by kappa g and h by L square, right? This we had done already in

the class, this error analysis. Now, for error to be really small, we will put this value

less than 1. So, it will give me L by h is much greater than 1 by 4 a by kappa g. So,

for different values of, here are our material properties and these are the geometric properties

of the rod. So, based on this little formula, we can find out, I mean how much is the error

if we change the geometry of the rod. So, it comes out that for slender and long beams,

I mean for L by h equal to, this is the plot for L by h equal to 20 means a very long rod

or perfectly slender rod. The Timoshenko and Euler-Bernoulli theories, they coincide with

each other. But for shorter beams, I mean that means for L by h equal to 5, the Euler-Bernoulli

theory does not coincide with Timoshenko. So, that is no more reliable now. So, basically

for shorter beams, we should go for Timoshenko beam theory and for longer beams, we can use

Euler-Bernoulli beam theory. So, this was what I mean where to use Euler-Bernoulli beam

theory and where to use Timoshenko based on the geometry of the rod. We can easily find

out that using that little formula.

He can also say that the shorter or longer is a little term and it is related to that

right hand side over there. Can you pull it up? It is possible that there is a scenario