A scaled finite element model 2

[BeFEA]

Hello! I have a question that I would like to share with you. Although there is little chance to encounter this problem in practice, I think it is good for a structural engineer to pause and ponder about it for a few seconds. For me, it is actually a real-life problem. Here it goes:

- Let's suppose we have a simple reinforced concrete portal frame subjected to a ground motion record (one story, one bay). There is a lumped mass M at the beam level. Let's assume the story height is 3 m and the bay width is 6 m.

Now,
for some reason, I want to numerically model a scaled version of the frame, which has a story height 2 m, bay width 4 m and cross sections down-scaled accordingly. How should I model the mass and the ground motion record? Will the mass be increased/decreased? Will the ground motion remain the same, or should I scale the time and/or acceleration? Let's say that the same material properties are used in both the model and the real portal frame.

The purpose is to have the deformations scaled with the same scale factor applied to the geometry (i.e., the resulting deformations in the model should be scaled by 2/3).

I don't know the answer, that's why I am asking you. Nevertheless, I am writing here some of my ideas that could lead to an answer:
1) I think this is somehow analogous to laboratory testing of scale specimens (for example in a shaking table).
2) I think we should apply some "similitude law". For example, the Cauchy-Froude law.
Any idea how to actually do this?

 

[ChadV]

This is way back in the sands of time for me, but I think the first step would be to list all input and response parameters for the full-sized problem and determine a reduced set of non-dimensional parameters from Buckingham's Pi Theorem (

https://en.wikipedia.org/wiki/Buckingham_π_theorem).

Then for the reduced model, you would set some parameters based on your reduced scale and adjust others (like the added mass and ground motion intensity) to have the non-dimensional parameters the same in both full-size and reduced.

I think you would get some interesting parameters for this... ratio of natural frequencies in terms of density, lumped mass, A, I... plus the fact that your response isn't a single value but a time-history. All this to say... have fun!

 

[RPMG]

for scaling, keep the building frequency the same = sqrt (k/m), and scale mass. k is the frame stiffness, and I/L^3 should be proportional.

 

[BeFEA]

Thank you for your answers. I don't know why, but think I had never heard of the Buckingham's Pi Theorem. I didn't go through it in detail but it seems complicated. I keep believing that there is a simpler solution. For instance, if someone has experience with the application of Similitude laws (such as the Cauchy-Froude law), it would be very helpful. Keeping the frequencies the same violates the Cauchy-Froude similitude law (if I am not mistaken). The law says that in a scaled model, the time is scaled by the square root of the scale factor. Now, I don't know what that means. Is it the period of the structure that is scaled? Is it the time in the input accelerogram? Any idea?

 

[Denial]

I've probably got more sands and more time working against me than has ChadV, but FWIW his suggestion of Buckingham's Pi Theorem is the same approach I would take.[ ] I'll even go so far as to say that most similitude laws reflect Buckingham, directly or indirectly.