So welcome to our lecture today. Today I would like to continue with the concept of beam elements

within the finite element method framework. And to this end I just want to recapitulate what we did last time.

We derived the weak form and based on the weak form we continued then first with the introduction of shape functions.

These are so called Hermit polynomials and they are given here.

And as you see we have two types of shape functions with a superscript zero.

This refers to the approximation of the deflection and with a superscript one it refers to the slopes or the first derivative of the deflection.

And you see the shape functions are one at the node they are assigned to.

So the red one is one here and the blue one is one here.

And you see their slopes are zero at all the nodes and vice versa.

The values the shape functions take, the shape functions associated with the slopes, the values they take are zero.

But their slopes are one at the node they are assigned to.

So the slope of the red curve is one at the left hand node and the slope of the blue one is one at the right hand node.

After that we approximated the deflection which is the solution in our case.

And we introduced also a handy notation here.

Here we have the vector of shape functions but in contrast to our considerations before we now have these two types of shape functions together with the Jacobian.

And the vector of W here contains both the nodal deflections and the nodal slopes.

So for a two node element it has four entries here because each node has two degrees of freedom.

Namely the deflection and the slope or the translation or the rotation if you want it like that.

And the same happens for the test functions.

So also we get here this vector of shape functions N times the vector of test functions.

After that we consider the derivatives of the deflection and of the test function.

And when we have a look back to the weak form here you see we need both the second derivative of the deflection and the second derivative of the test function.

And in this integral we just need the test function.

So we derived the second derivatives of W and V and we ended up with this setup here.

We have a vector B which contains the second derivatives of the shape functions and also partly the Jacobian.

And we end up with this setup.

If you take the derivative with respect to the local coordinates Xi here then we get this specific B vector.

And here I would like to continue today and for the second derivative of W we have the second derivative of W.

First of all I would like to ask whether there is anything to be discussed. Yes.

One question I have chosen and why is the Jacobian only multiplied with the shape function of the slope?

Why is the Jacobian only multiplied with the shape function of the slope?

To this end let us have a look to the derivation of the shape functions.

So we had here the four conditions.

The approximations of the shape functions, sorry, the approximations of the slope and of the deflection should fulfill.

Namely they should take the nodal values of the deflection and the derivative should take the nodal values of the derivatives.

So when we take the derivative we get terms which contain here the inverse of the Jacobian or one by the Jacobian.

And you see since this is here associated with the deflection this appears in these two equations related to the nodal slopes.

And when you solve now for these coefficients a, b, c and d and you sort then you see that the j,

since this comes here in the equations for the slopes, this again appears here as a factor in the terms for the slopes.

Is this clear?

Yeah, so this is the origin. If you would compute it by hand on your own then you would see that this explicitly appears only in the slope terms.

Okay, anything else?

If this is not the case I would like to come back to the weak form.

And...

Now we want to introduce in the weak form the approximations of the shape, sorry, the approximations of the shape of the slope.

Approximations of the shape, sorry, the test function and the second derivatives of test function and deflection.

And my question is if we introduce that and do the same procedure as we did in chapters 3 and 4 and also 5,

what do we end up with if we now introduce the approximation, what could be the title of the following subsection?

Approximation of shape and denominator.

Yes, but we get, actually in all our considerations we get two typical quantities.