Effective Flexural Rigidity (EI) of Composite Cantilever Beam

[lalala1plus1]

Hello,

I am trying to derive the (Euler-Bernoulli) beam equation of a composite beam to determine the vibration and natural frequencies (up to 3 modes) of the beam.

The composite beam can be modeled as in the picture below:

In order to apply the general equation:

(EI)_effw'''' = m'_eff (d²w/dt²)

in which
(EI)_eff : Effective Flexural Rigidity of the composite beam
m'_eff : Effective mass per unit length of the composite beamI know that for pure layered composite beam (with n layers), we can simply add up the flexural rigidity and mass per unit length of each layer: (EI)_eff,layered = Σ E_nI_n and m'_eff,layered = Σ ρ_nb_nt_n

However, for my case, I have 3 different materials laminated in parallel in the middle layer. Is there any way to get the effective EI and m' of this layer, so that I can add them with those from the top and bottom layer to find the total effective flexural rigidity and mass per unit length of the whole beam?

I am urgently looking forward for explanation, and if better, with literature suggestion, which I can read it in detail.

Thank you very much for the help!

 

[lalala1plus1]

To clarify and simplify the problem, let's assume I have a composite cantilevered beam with one material (Material 1) surrounded by another material (Material 2). Both materials are of the same thickness t. The beam is symmetric in the width direction, but not in the length direction (L_a≠L_b). The model can be assumed as in the picture:

How can I find the effective flexural rigidity (EI)_eff and mass per unit length m'_eff of the composite beam in order to apply the beam equation:

(EI)_effw'''' = m'_eff (d²w/dt²)

I am really looking forward for any help! Thanks a lot!

 

[Nidum]

You can't . This problem has to be solved using an analysis method which take into account how mass and stiffness values vary along the length of the beam and possibly across the depth and width as well depending on the proportions of the beam and the level of completeness you want to achieve in your investigation .

 

[Nidum]

We can certainly help you find a practical way of solving for the frequency and mode shapes though .

What is your educational background ? Have you studied the basic theory of this type of problem ?

 

[lalala1plus1]

Hello Nidum,

Thank you very much! I have master in mechanical engineering. I do know the theory of the dynamic of a beam. It is however the first time I am trying to model a composite beam. I have been reading a lot of papers in the past few days, but most of them are just focusing on layered composite beam (each layer consists of only one type of material). And, since I am going to do some parameter studies on the beam, I would like to avoid using FEM-Software.

It will be really great and helpful if you can guide me on how to get the natural frequencies and mode shapes analytically, or provide me some literature related to my problem.

Thank you very much!

 

[Nidum]

If we ignore the variation of properties across the width pro tem and assume that the beam is relatively long compared with it's depth then this problem can be solved in several ways . Possibly the easiest are :

By considering the beam as three separate elements joined on end . You would need to derive the controlling equations from first principles but really this would only be the standard derivation with the added difficulty of taking into account the different properties of the three sections and matching of terms where the sections meet .

By using a lumped property approximation . Whilst more commonly used for numerical solutions these types of problem can be solved analytically as well if not too complex . In this case the elements are just in one chain and the boundary conditions are known so the model definition and solution method should not be too difficult to arrive at .

In both cases the effective mass/ unit length and effective stiffness of the individual sections is derived using standard method for layered beams .

 

[Nidum]

Just aside : I once spent many hours doing vibration calculations for turbine and compressor blades . Later I did a lot of work on high speed shaft stability .

 

[lalala1plus1]

Thanks Nidum,

I guess the method of lumped property approximation will be more appropriate. However I am not quite sure how it can be done. Do you have any literature on it or can you give me some explanation?

It is really lucky to have expert in vibration like you to help me!

 

[lalala1plus1]

Just another question, if I have a sandwich composite beam arranged vertically (in the same direction to the bending deflection, as shown in the picture below), can I still add their EI to determine the effective bending stiffness of the beam?

 

[Nidum]

There is a problem in that your sketches seem to suggest that you are working on short beams with significant width and depth . Analysis of these types of beams is much more complicated than it would be for longer beams with relatively smaller width and depth .

Before doing any more decide on which class of beam you want to analyse ? Can do both if necessary but long and thin is certainly much easier to work on than short and fat when it comes to beam calculations .

 

[lalala1plus1]

Hello Nidum,

The beam is actually very thin (around 2.5% of the length) and considerably slim compared to its length. It was displayed not in ratio in the sketch to show the detail of the materials.

I have been trying to separate the beam into 3 sections and combined them back together. However, the frequency calculated has an offset of ~5-20 Hz from the experimental value.

It would be great if you can give me some advice.

Thank you in advance!

 

[JRMichler]

The beam equation of a beam with constant dimensions and properties integrates to a sinusoidal function with constants of integration determined by the end conditions. Your problem has segmented mass and stiffness, so the mode shapes will not be single frequency sinusoids. To visualize this, think of the effect of making the center middle portion extremely soft/stiff/light/dense on the second and third natural frequency mode shapes. You need an analysis method that will correctly calculate the mode shapes, given that they will not be simple sinusoids.

FEA will do this, but apparently you are not allowed to use it, or it is not available. The math behind FEA should be able to solve this problem. Look up the theory of FEA. Manually model this problem with three elements. FEA natural frequencies are something to do with eigenvalues and eigenvectors and mass and stiffness matrices. The math should be relatively simple for three elements. Each element will use the lumped stiffness and mass that you calculate using the method in your first post. If the results look promising, then divide each element in half, and repeat. Continue until your results are good enough.

 

[Nidum]

@lalala1plus1 : Ok . I'll see what I can do .

 

[Nidum]

Can you explain briefly what method(s)you have tried so far to get a solution and what difficulties you encountered ?

Do you have a specific beam configuration to work on ? If so please give us details .