So, welcome to our tutorial today.

Today we have exercise number 10, which is considering a mechanical system as it is sketched below.

And yes, the extent of such an exercise is very much comparable to that what you might be asked for in an examination.

And with that we have also another overview of the relevant, or most relevant topics of the consideration of the mechanical problem in two and three dimensions.

So, the system that is given is such a structure. It is clamped at either side. It has a height of L and a length of 2L.

And it is subjected to two point loads, which is F here at the top and F at the bottom.

The discretization of the system is here done by four triangular elements. The node numbers are given, the element numbers are given, and a part of the connectivity matrix is given as well.

Furthermore, it is assumed the state of plane stress. So, there is no stress in the third direction, just in the x1, x2 direction.

Then the Young's modulus is given, the force is given, the length is given, and this is also a simplification here.

The Ponsons ratio is set to zero. This facilitates the calculation a bit.

In the first question we are asked to complete the connectivity matrix under assumption of a counterclockwise numbering of the local element nodes.

So, let us do this as a first step here.

Please feel free to start filling in the connectivity matrix.

And I will just copy that a second time here.

This is the connectivity matrix.

And maybe take, let's say, two minutes at most to complete this and then we discuss the result here.

I start completing this.

Okay.

Okay.

Okay.

Okay.

So, who needs more time to complete this?

Nobody? Okay. Yeah, this should be the result. You can check on your own whether it is correct or not.

And with that I would like to move on with 10.2. Here we should compute the element stiffness matrix per unit thickness for element one.

And we also should use Gauss quadrature with a single Gauss point.

So, let me just recall what is the basic relation we need to compute the stiffness matrix.

And to this end it makes quite much sense to have a look to our recapitulations.

We have given here the formulation for the element stiffness matrix.

And we see a couple of ingredients here.

What do we need specifically to compute step by step to finally end up with the quantities we need here for the stiffness matrix?

Yes. So, what is hidden in small n?

The shape functions. We need the shape functions. The derivatives.

The Jacobian. Yeah, let us just put that together as a list here.

So, of course we need the shape functions. We need the Jacobian, the determinant of the Jacobian, the inverse of the Jacobian.

Then we need derivatives of the shape functions.

Here for the Jacobian we already used derivatives. Which derivatives did we use for the Jacobian with respect to what variable?

Exactly. So, this is the next step actually. We need to compute the derivatives with respect to the global coordinates.

Which then leads to BEU, this matrix of derivatives.

And of course we have to set up the elasticity matrix here in folk notation, C tilde.

And then we have to numerically integrate.

Okay, so let us start with the shape functions. We have a triangular element. What are our shape functions?

Yes, that is it. That is it. So, then the Jacobian. I would like to ask you as a little exercise to compute the Jacobian.

If you are not sure how to do that, either we discuss it here or you look it up in our previous either examples or exercises.

I will do that here in parallel. And if you want to check whether you are able to do that, then please don't have a look here.

Otherwise, if you are unsure, of course you are welcome to see what I am doing here.

Okay.

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