Doubt regarding constants. Elastic Beam Theory.

[AiRAVATA]

Hello. As a good mathematician, I'm having troubles reading some constants for a PDE.

I'm modelling an elastic rod using the equation

$$\rho A U_{tt} - N U_{xx} + E I U_{xxxx} = 0,$$

where "$\rho$ is the beam density, $A$ and $I$ are the area and moment of inertia of the beam cross section respectively, $N$ is the prescribed axial load and $E$ is the Young's modulus." (Nayfeh and Mook)

Now, in the literature, I've found numerical values defined as

$$k_{\parallel} = \frac{A E_{\parallel}}{L},$$

where "$E_{\parallel}$ is the Young (stretching) modulus, $A$ is the cross sectional area of the beam and $L$ is the length of the beam", and

$$k_{\perp} = \frac{3 E_{\perp} I}{L^3}$$

where "$E_{\perp}$ is the Young (bending) modulus, $I$ is the moment of inertia of the beam cross section and $L$ the length of the beam".

My question is the following:

How does the quantities $k_{\parallel}$ and $k_{\perp}$ relate to $N$ and $E I$ on the PDE?

 

[Studiot]

...is the prescribed axial load..."

Beams are structural elements that carry transverse loads. Any axial load is additional and cannot be modeled by simple beam theory.

Please clarify what you mean.

As a note the unusual notation does not help. Why are you using it?

 

[AiRAVATA]

Well, the unusual notation comes from the book of Nayfeh and Mook, and from the literature I've consulted. I used to think that the second derivative term had something to do with some sort of "tension", like in string theory, but now I think I'm wrong. From what I've read, it comes from Timoshenko's beam theory, not Euler-Bernoulli, and indeed, it is because of axial load. So, if I have a simply supported beam, and it has a low aspect ratio, should I drop that term?

 

[Studiot]

Nayfeh and Mook

I know nothing about these authors.

The stuff you posted in post #1 seems all mixed up can we start at the beginning like I asked?
What are you trying to do?
What do you mean axial load?
What about the transverse load?

Post#3 didn't clarify anything.
In particular do you understand my comments about beams? You didn't reply.

If you have a reference from Timoshenko, please state the book (he wrote several) and page.

 

[AiRAVATA]

Sorry for late reply and thanks for your attention. I'm having the flu and that has kept me from thinking about the problem.

First, Nayfeh is a classic author in perturbation theory and nonlinear oscillations, and has several papers on nonlinear beam theory. The reference I give is from his book Nonlinear Oscillations, p. 448. The book can be found online.

Now, what I'm doing is modelling the movement of a beam under some punctual (transversal) forces. The beam aspect ratio is low, meaning that its length is much greater than is width. For now, I'm only interested in linear theory, as the displacement of the beam can be thought to be small. Also, I'm assuming that the constitutive equations are linear, i.e. Hook's law.

Accordingly to Wikipedia (sorry for the poor reference, but my cold has kept me from the library), the term $N U_{xx}$ comes to play when an axial effect is considered, and here lies the source of my confusion.

I used to think that such term arose from considering axial effects, just as it does in the wave equation, and that including it in the beam equation its only meant to make a more "precise" model of the problem. Now, from what I've read, such term is only considered when an external axial force is applied. Why is that? If there is deformation, then the beam must suffer axial effects, and it surely can be approximated by simple equations (even if the phenomenon is more complex). In any case, the numerics will dictate how valid the approximation is.

For the numerics, I've found in the literature the values of the Young modulus for stretching and bending. Clearly, the bending modulus is the one that corresponds to the fourth derivative term, and I have no problem in implementing that value. The problem is relating N with the stretching Young modulus. That is, if it makes sense to include that term.

In any case, what would be helpful by now would be to me to understand what is the difference between the second derivative term in string vs. beam theory and why should I keep it (or not), and if so, what is the relation between the value N and the stretching modulus.

 

[AiRAVATA]

Ok. After a little reading, I now know that N is the axial tension, and in my case, is equal to zero, because the beam is not under the effects of prestress.

 

[Studiot]

I think you are talking about oscillations or vibrations of beams and that is what your differential equation is all about.
Please confirm if this is the case.

I think we have a few problems with terminology to straighten out.

If you have a simply supported beam it means that the beam is unrestrained at the supports and so is free to rotate. As a direct result there can be no horizontal reactions at the supports and no axial forces induced in the beam.
If the beam is subject to axial loads then it is not simply supported.

Apart from the counterbalancing tension and compression introduced by the beam action, axial forces can be introduced by one of four means

1) Prestressing
2) A physical real axial load
3) Loads applied along the beam at angles other than perpendicular to the beam's axis
4) Loads applied eccentrically, due to the shape of the beam

Incidentally we call the axis in the direction of the support reaction the depth of the beam. The other two axis (length and width) are perpendicular to this.
A definition of a beam is that the length >> width.
If this were not the case the element would not be a beam but a slab, plate, shell or membrane.

 

[AiRAVATA]

I think you are talking about oscillations or vibrations of beams and that is what your differential equation is all about.
Please confirm if this is the case.

It thought that was implied given the equation posted in #1.

If you have a simply supported beam it means that the beam is unrestrained at the supports and so is free to rotate. As a direct result there can be no horizontal reactions at the supports and no axial forces induced in the beam.
If the beam is subject to axial loads then it is not simply supported.

Agree. I might have been unclear about this point, but boundary conditions are more complicated than "simply supported", the reason why I didn't elaborated is because they have little to do with the original question.

Apart from the counterbalancing tension and compression introduced by the beam action, axial forces can be introduced by one of four means

1) Prestressing
2) A physical real axial load
3) Loads applied along the beam at angles other than perpendicular to the beam's axis
4) Loads applied eccentrically, due to the shape of the beam

I understand that, but there are none of those in my problem. What got me confused (and still does) is why, contrary to the string, there is no tensile restoring forces given a deformation perpendicular to the axis of the beam.

Incidentally we call the axis in the direction of the support reaction the depth of the beam. The other two axis (length and width) are perpendicular to this.
A definition of a beam is that the length >> width.
If this were not the case the element would not be a beam but a slab, plate, shell or membrane.

Now, you must think I'm dumb.

 

[Studiot]

Now, you must think I'm dumb.

I meant no disrespect, but you did state more than once about the aspect ratio being length to width (which is irrelevant) as opposed to length to depth ratio which is highly significant.

Also neither of your stiffness constants, k, are pertinent to beams, supported in simple or more complex manner.


This one is for cantilevers

$$k = \frac{{3EI}}{{{L^3}}}$$

and this one for axial stretching

$$k = \frac{{AE}}{L}$$

So you can excuse my confusion.

I tried to download the book you referred to, but couldn't as the site won't accept my postcode!

Timoshenko does a chapter on dynamics in his book

'Theory of structures'

There doesn't seem to be anything in his more famous

'Theory of elasticity'

I will post some mathematical derivations over the w/e

Edit

By the way there is always a restoring force, it is just a question of getting the right model.