02 Oct 2019
— Edited @
03 Nov 2021
We are using VABS to model section properties of composite circular section beams (golf shafts).
We noticed some counter intuitive results in the following situation. Your advice would be more than welcome.
As you could see in the attached file we are modeling one multi layer section with one orthotropic material (unidirectional UD carbon fiber prepreg).
The case 1 is the base-line laminate made of 32 plies with orientations at 0,+45 and -45 layers.
In case 2 we have the same 32 layers but the orientations of the 6 inner most layers were changed to 90 (hoop direction). Inner and outer diameters remain the same.
We were expecting the section bending stiffness (first term of the Timoshenko stiffness matrix) to decrease from case 1 to case 2. But it increased.
I created case 3 and 4 with even more 90° plies in the inside of the tube. The bending stiffness decreases.
In case 5 the first 6 plies are at +-45°. As expected, the bending stiffness decreases.
Could you please let me know if you see some modeling problems in the files or if you can explain such behavior.
The attached is actually a zip file. Change the extention to .zip to open it. Inside the zip you will find all the related .dat files and an Excel spreadsheet summarizing the results.
Thanks in advance for your advice
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07 Oct 2019
Two other things I would like you to try before we know how to advice you: 1) compute the results at the shear center. 2) compare the flexibility matrix at the shear center.
07 Oct 2019
As the section are circular the shaer centers are at 0,0 as you can see in the attached .dat.k files.
does it answer your question?
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07 Oct 2019
Are all layers of equal thickness ? If so can you provide the thickness of each layers ? And could you provide the radius of the central reference line ?
08 Oct 2019
Could you find attached a new version of the spreadsheet. In this version you could see the inner and outer diameter of the tube as well as the thickness of each layer.
Thanks in advance for your help.
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08 Oct 2019
Please also provide corresponding comparison of the flexibility matrix.
08 Oct 2019
The flexibility matrices are in the dat.K file I sent in my first message. The attached .txt file is actually a zip file. Change the extension to be able to unzip it (for some reason this forum does not accept zip file as attachement)
Let me know if it's not what you meant.
08 Oct 2019
I wanted to clarify that you are interested in the anomaly which presents itself in "EA" which is the axial stiffness of the cross-section and not "EI" which is the bending stiffness of the cross-section.
In the post you have mentioned that its the bending stiffness and the excel file it is the axial stiffness which you have highlighted.
09 Oct 2019
You are right, I made a confusion in my spreadsheet. We are interested in the bending stiffness of the section and we are getting it from the term 44 of the classical stiffness matrix. Updated spreadseet attached.
According to the VABS documentation, EI2 EI3 are the terms 33 and 44 of the diagonalized stiffness matrix for isotropic materials. Our material is not isotropic. As the non diagonal terms are quite low I do not diagonalize the matrix. I do not really know how bad are these assumptions.
Would you have a more formal method to extract the EI from VABS results for non isotropic section?
However it does not change the problem as we notice the drop in stiffness (calculated by Gebt through natural frequency) when we introduce hoop fibers. This last points illustrates that the section with hoop fiber "stiffer" in bending than the section without. Which is counter intuitive.
I look forward to continuing this investigation.
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10 Oct 2019
For the case of laminated circular tubes Prof. Yu and I have developed an analytical form of the beam stiffness matrix using the basic principles of VABS. I have attached a pdf of the formulas obtained.
These formulas use the values obtained by representing the laminated section in the form of plate and getting its corresponding plate stiffness matrices. In the formulas "Aij, Bij and Dij" represent the corresponding plate stiffness values.
I have attached an excel file which shows results obtained from the formulas for the different cases you have put up and the results are close, though not exact because of the thin-walled assumptions used. These formulas do provide a useful insight to the anomaly which has presented itself.
It can be seen that the axial stiffness does indeed increase as we go from case one to case two, which can be attributed to hoop direction stiffness as well as the Poisson's ratio increasing because of the unique layup configuration. This can also be extended to the case of 33 value, with the increase in the effective Young's modulus of the section in the form of "EA", we can say that "EI" will also show the same trend.
A sensitivity analysis can be conducted using the formulas but coming to a conclusion for the 33 value from the formula itself is difficult as the value depends on a number of quantities as shown.
I hope this clarifies your doubt and I look forward to your views on it.
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11 Oct 2019
Thanks for these explanation.
I understand the stiffness could be increased by the addition of 90° plies (less ovalisation under load...) but I am surprised VABS could capture this phenomenum explicaitaly in the stiffness matrixces without non linear approach.
Could you confirm it does. I would not want to use an artifact to make our design.
Thanks for your help.
14 Oct 2019
Thomas, I believe this is not a nonlinear effect. It is purely due to interaction of the complex 3D stress state which produces an effect counter-intuitive to our intuition obtained from isotropic homogeneous reasoning.
17 Oct 2019
Dear Prof Yu,
Thanks for your explanation. This is now clear this counter intuitive effect is not an artifact of the model and is captured by VABS for non isotropic material.
It's great such behavior could be seen at section level.
It's very nice to be able to capture these effects without the need for a complete shell model.
Thanks for your valuable help.