Direct Analysis Method K=1 verus Euler Buckling Strength Equations-Consistent logic? 2

[AggieYank]

I understand that Euler buckling load = the load at which a purely axially loaded column becomes unstable, meaning that any lateral load applied to the column would result in huge flexure and flexure deflection. Euler buckling equations use the “effective length” to define the buckling length of a column, where the k factor comes into play for effective length. And the Euler buckling equation is the basis for the non-short column strength equation in AISC. This makes sense.

I also understand that in Direct Analysis Method, that by using factored loads, notional loads, reduced stiffness, and capturing second order effects, that you are directly modelling and understanding forces and moments that will develop in the columns. So, any column buckling which may want to happen will show up in the form of sidesway and bending in the columns in the analysis. This makes sense too.

What I don’t understand is how the two items above interact. DAM directly defines the forces and moments that develop in the columns which is very logical. But then we use the same column strength equations (basically just Euler buckling equation), but we just hold K=1. This doesn’t “click” for me, and it just feels like an approximation that is just “close enough”. Is the real story that we don't care about Euler buckling anymore because we have already modelled the real buckling that could happen and confirmed it can't happen? So then the column strength equation becomes a little more aggressive but not as aggressive as simply using P/A or something similar?

I appreciate any insight or advice you can give.

 

[rb1957]

Euler is a mathematical solution to differential equations (sort of like "spherical chickens in a vaccuum").

The real world effects (eccentricity, ... ovaloid chickens with feathers in an atmosphere ...) that DAM analyzes aren't solvable the same way, and don't arrive at a similar closed form solution.

another day in paradise, or is paradise one day closer ?

 

[JoshPlumSE]

This one is a tough one to answer. There are a number of different aspects to your question and I'm not sure that this can be 100% covered in this type of forum. A couple of quick comments:
1) DA method isn't just trying to capture the type of ELASTCIC buckling that Euler buckling calculations predict. Rather it is trying to also predict the inelastic buckling that really occurs in steel structures. This is an important difference.

2) There are multiple methods of buckling. Flexural buckling in the strong and weak axis, Flexural torsional buckling, torsional buckling et cetera.

More to follow later.

 

[KootK]

Is the real story that we don't care about Euler buckling anymore because we have already modelled the real buckling that could happen and confirmed it can't happen?

Exactly this. Future versions of the direct analysis method won't have even K-values because they won't involve Euler buckling.

If we model all of the factors leading to instability including second order and non-linear effects, there's no need to bother with Euler buckling. And we have the technology to do this.

At present, our codified analysis methods account for many instability sources but not all of them. The K=1 Euler checks attempt to fill that gap. The way in which they do so is generally conservative and, really, not all that consistent with our more modern methods. Apples and oranges.


I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[sponton]

I wouldn't say it's not useful, but then again if you're doing an elastic analysis using FE or any other approximation method and you find out that it buckles or becomes non-linear @ X load, it turns out it is not that load, because in real life things are inelastic in the elastic range because of material imperfections, eccentricities, etc.

Anyways, the direct analysis to overcome this, forces you to analyze the non sway + the sway case with a reduced stiffness in the members adding lateral stiffness, this is somewhat empirical and that's why you can use K=1. Now if you're using some other method [elastic or the magnified moments, you need to calculate the K's for the sway case]. I think the problem of these methods it's the actual reach of what we know or what we pretend to know. I think researchers rather than putting a big safety factor and making an easier equation to calculate the strength of members tend to use what we currently know about analysis and material behaviour and make everything thrice as complicated for a .004$ gain in accuracy.

In a way, whatever analysis may tell you which if you go to the field and compare your models you tend to realize all the crap you take for granted or oversimplify and in no way you will believe the fancy analysis especially when non-linearities are involved.

 

[JoshPlumSE]

3) Real buckling isn't as simple / straight forward as KL/r and Euler buckling formulas. In many cases, you don't have an obvious answer for what the K value should be. Think curved structures or arches or such. Even for straight compression member, what happens when the axial load varies along the length?

4) Then there is overall buckling of the frame / system? That isn't really addressed by simple KL/r assumptions for individual members.

5) If the Direct Analysis Method truly accounted for all forms of stability, then we could take KL = 0 and just use the yield strength of the member as the capacity. And, that option will likely be coming for some future version of the code. However, that is much more dangerous. You're expecting a lot more sophistication out of your analysis (and the engineer who built the analysis model) if you use KL = 0.

So, we have this hybrid method (DA Method) that accounts for stability effects mostly through the direct analysis of the structure. But, the method is simple enough to use that most engineers (and analysis programs) can use it properly with confidence that their results are reasonably conservative.

 

[AggieYank]

Josh and Koot, thanks for reinforcing the idea that the K=1 method is just a simplified conservative way of checking against the Direct Analysis Method Output results, and that i am not simply missing something about trying to understand the K=1.:

It will be interesting to see what future AISC releases say about column strength checks against Direct Analysis Method results.