Buckling capacity of twisted plate strut 1

[bugbus]

I saw one of these brackets holding up a kitchen benchtop and it got me thinking about how the buckling capacity of this twisted strut would compare to the equivalent untwisted version?


I suppose it basically boils down to something similar to below (ignoring the twist itself):


Does anyone know if there is an exact solution to the buckling load for a given n? Or even as n -> infinity?

I thought this was an interesting/fun problem, might play around with this over the weekend.

 

[bugbus]

OK, it turns out this could be more straightforward than I first thought

I was playing around with this in a FE software. As an assumption, I modelled the stiffer parts as fully rigid elements. I don't think that would be unreasonable, e.g., for a typical plate with aspect ratio of 5, the relative flexural rigidity is 25x more in the stronger direction.

It turns out that the buckling load can just be calculated based on an effective length equal to the sum of the lengths of the more flexible parts (in this case Lef = 0.5L because the segments are roughly equal). The buckled shape is interesting, the red parts are rigid whereas the blue parts are not.

I suppose if (EI)₁ and (EI)₂ are a little more similar in magnitude, it would be somewhat more of a complicated situation. Might investigate further if I get bored.



Intuitively it kind of makes sense. The rigid red elements preserve the same slope at the ends of the blue elements, so that when all the blue elements are pieced together, it is equivalent to a strut with length L/2.

 

[GregLocock]

You could probably model it as a series of linear flexible joints joined by rigid struts and do an energy balance. that is minimise F in F*d=sum(1/2k*theta_n²)

 

[Smoulder]

Feels like will end up like springs. 1/resultant = 1/stiff + 1/flexible.

 

[bugbus]

@GregLocock, will definitely look at that as an alternative approach, thanks
@Smoulder, agree, it seems like the solution would end up being similar to that

As far as finding a closed solution to the critical buckling load, I'm basically stuck at this point.

My thinking is that the variation of flexural rigidity along the length of the column is a square wave, which could be represented as a Fourier series expansion:

This is kind of neat, because it boils down to a series of sin() terms which seems to suit the differential equation for buckling load quite nicely.

This would then require the following differential equation to be solved for the critical buckling load Pcr:

I'll keep trying to solve this thing, but I suspect that the limits of my maths knowledge has been reached.

 

[GregLocock]

I'll just whack my suggested approach into Grok and see what the result looks like.

 

[human909]

An interesting topic.

It has been examined in various sources:

https://ascelibrary.org/doi/abs/10.1061/(ASCE)EM.1943-7889.0000144

https://www.sciencedirect.com/science/article/pii/S0141029623012026

 

[JoshPlumSE]

Does anyone know if there is an exact solution to the buckling load for a given n? Or even as n -> infinity?

Sort of. If you look at the AISC design guide for tapered members (DG - 25), there is a method given in appendix A.2 (Method of Successive Approximations) that is actually pretty easy to implement in a spreadsheet. I did it at one point the example in Appendix C (C.3).

I can't seem to find that spreadsheet though. I'll look again later, when I have time. If I find it, I will post it here.

FWIW, an FEM program that does eigen buckling would be just as good, of course.

 

[bugbus]

Very interesting suggestions, thanks all

 

[GregLocock]

Here's a script, generated by grok, not too sure I believe it!

 

[rb1957]

from your first analysis (2nd post) it seemed that the flat pieces were critical and the twisted pieces were effectively rigid for buckling about the weak axis.

so if n = 5, 3 pieces will be aligned, and effective L is 60% (or (n+1)/2n if n is odd); if n is even then Leff = 50%

 

[Stress_Eng]

There are a number of methods you can use to produce a beam column buckling template. One, as suggested above, is by successive approximations, another is by a strain energy method, where you assume the displaced shape by using a trigonometric series. I've created one where I've solved the 2nd order differential equation and applied it to numerous segments (solved by equating boundary conditions between the segments). The following pictures are examples for your case. There's a section view of the modeled beam in compression and a couple of the buckling wave-forms. Hope you find a suitable method.