So welcome to our tutorial today. Today we have exercise 12 which is the second exercise

focusing on beam elements and here we have given a beam element of length le which is subjected to a constant distributed load q0 with a global and local coordinates xyz and xi1, xi2, xi3 and Hermite polynomials of third order are used as shape functions.

These are exactly the polynomials we introduced in the lecture.

So let us start here. First of all we should compute the element stiffness matrix and to this end we just look up the relations in the lecture notes or in our summary.

Let me just open this.

So in the recopulation you will find

here the relation for the element stiffness matrix.

Let me copy that

and furthermore the B matrix, sorry the B vector

which is given for the Hermite polynomials here.

If you would be asked in an examination to compute the BE vector then you would have to start from the shape functions and put that together.

And you know BE contains the second derivatives in essence.

Okay, so with that we can proceed to set up the equation for the stiffness matrix.

So it's 12, 1

and I don't want to do that in full detail just to give you an idea.

We take only a part of the entries here.

So BE transpose times BE would mean

that you have first a column vector and then a row vector.

I just want to compute the first row here.

So I put here

3 by 2

xi

and then of course I have here a JE to the power of minus 2

and from the other vector

we get also JE to the power of minus 2.

So in total we have JE to the power of minus 4.

Then we have 3 by 2 xi

JE 3 by 2 xi minus 1 by 2

minus 3 by 2 xi and then a little bit too few space.

Then the last entry is JE

3 by 2 xi plus 1 by 2.

And I skip here the other entries since this is quite time consuming.

So this gets a 4 by 4 matrix as we said in the morning.

So first entry is 9 by 4 xi squared.

Then we should read JE here.

Second entry is JE 9 by 4 xi squared minus 3 by 4 xi.

Then we have minus 9 by 4 xi squared.

And last entry JE 9 by 4 xi squared plus 3 by 4 xi.

And then the entire matrix by JE to the power of minus 4.

And then these are the entries I don't want to write down in detail here.

So and now we can just select some entries we want to compute.

I just want to compute KE 1111.

KE 1211.

And I just want to know from you which are these entries actually.

So what could be KE 1111?

Maybe to this end just to remind you how the stiffness matrix is set up.

How would you write this block matrix?

KE 1111.

The first?

Yes.