So welcome to our first tutorial today and you will find the collection of

these exercises together with the sample solutions on Stutt-Onn. Did you find them?

Yeah, okay, perfect. So the first exercise is to consider a truss which is discretized by five

linear finite elements and consists of six nodes with the global coordinates as given below.

And the first question is to formulate the general definition of the global shape function phi i of

x and yeah this is the first part here and to this end let me just draw.

So one one.

So this is our diagram here x yeah and how would you sketch the shape function?

So first step and then we will give the mathematical formulation.

Linear, yeah.

How?

Yes, like a hat and the maximum is at one. The maximum is one at which?

Which x? Yeah, xi exactly. So maybe I can just

sketch such an element. So this would be the global coordinates here xi minus one,

xi and xi plus one. So and then we have a shape function that looks like this.

It is one at the node it is assigned to and it is zero at all the other nodes.

So this is function phi i of x and we have here one.

Okay then let me introduce the length of these two adjacent elements.

So this is lei minus one. This is lei.

And yeah now the question is how to formulate this mathematically.

How would you do that? Knowing these coordinates, length and so on.

Yes, so we have to distinguish several cases. So it's zero for which x?

Less than. Yes, second part.

One.

One. That would mean that it would be constant.

So here in this part for, let me just move that a little bit.

So for yeah what would be the appropriate interval here?

Yes.

What would be the appropriate formulation here?

Yes.

From minus.

Yeah.

Xi.

Which length?

Yes.

Yes.

I'm not entirely happy with that.

This formulation it's another constant.

Because Xi is constant, Xi minus one is constant and the length is constant.

Yes, yes that's it.

This is x.

Yeah and the slope is positive because we have this ascending part.

So this is x minus Xi minus one divided by the length of this part.

Okay, so then we have another part here.

Yeah, what would be the appropriate formulation here?

Yes, Xi plus one minus x.

Yes.

So and then we have another zero here for x larger than Xi plus one.

So this is the piecewise linear definition here.

Okay.