Steel Column & Beam Unbraced Lengths & the Direct Analysis Method 2

[jochav5280]

Good evening:

I would appreciate help in understanding how to specify the unbraced lengths for steel columns and beams. With the implementation of AISC's Direct Analysis Method, I'd also like to understand how using a K-factor = 1 impacts definition of these unbraced lengths.

For instance, with cantilever beam-columns, am I to define the unbraced column and beam lengths as the cantilever dimension or should they be doubled since cantilevered members are effectively unbraced for twice their actual length.

Would someone also help me understand how the Direct Analysis Method is able to set K=1?

Thank you for your help!

Best regards,

jochav5280

 

[BAretired]

I have not used the Direct Analysis Method, but the attached article may be of some help:

https://engineering.purdue.edu/~jliu/courses/CE470/PPT_PDF/CE470DAM_S14.pdf

BA

 

[RPMG]

To specify L for all cases, it is generally to where beams/girder connect, and not the total height length of the column.

To specify K for first or second order analysis, see the commentary in the AISC manual. I don't have it on hand to give an exact reference. A cantilevered column would have a recommended K as 2.1.

To specify K for direct analysis, K = 1.0. For the cantilevered column (K=1.0), destabilizing effects will be accounted for by:

Notional Loads or Directly modeling the column as slanted​

And reducing the stiffness of the column for the effects of residual stress. For instance, EI might be replaced by 0.80*EI​

Note that axial force itself destabilizes the column.​

The idea to switch from effective length to direct method, in my opinion, is for analysis software which has difficulty computing K.

 

[KootK]

I was really hoping that this thread would garner more discussion. I think that this issue is massively important and, in general, very poorly understood. If I understand your question correctly, OP, how to go about applying DDM provisions is not your primary concern. There's lots of information out there for that. Rather, I think that you're interested in acquiring a fundamental understanding of the DDM philosophy in general, and how it is that K can can be set to unity in particular. Me too.

I have yet to see a solid explanation for this in print anywhere, including the AISC manual commentary and the AISC design guide on structural stability. Perhaps the folks who compose those documents are so stability smart that they can't fathom how this stuff could be anything less than obvious to practitioners. In what follows, I will do my best to explain what I know. Be mindful of the fact that I'm not at all confident in my explanation. If others jump in and Eng-Spank me on my errors, that will be just fine by me.

First, the easy stuff.

I would appreciate help in understanding how to specify the unbraced lengths for steel columns and beams.

K=1 for in plane bending and buckling. All members, all the time. K should be set to traditional, effective length factors for out of plane buckling, torsional buckling, and lateral torsional buckling.

For instance, with cantilever beam-columns, am I to define the unbraced column and beam lengths as the cantilever dimension or should they be doubled since cantilevered members are effectively unbraced for twice their actual length.

Yup, K=1.0. Just like RPMG described. I dug out the graph below which compares the traditional and DDM (K=1.0) methods for precisely the case that you've described. Reassuringly, the answers are pretty close. But, then, why are the answers close? And will they always be close? Those are the tricky questions.

And now the tough part.

Would someone also help me understand how the Direct Analysis Method is able to set K=1?

Some General Stability Background

First off, DDM is all about the stability design of sway frames. For non-sway frames, the method still works but is of little consequence. DDM limits your K to unity and, in a non-sway frame, most of your columns would be K <= 1.0 for traditional K-factor design anyhow.

Fundamentally, buckling is a subset of general instability. And instability, in the general sense, is reduction of stiffness to zero with regard to a particular degree of freedom. For conventional steel framing, this usually takes the form of an axial load at which flexural stiffness reduces to zero.

There are at least two distinct ways to come at stability problems. The traditional method for assessing instability is bifurcation analysis which includes K-factor design and Euler buckling. Basically, there is no movement at all until the load hits the bifurcation point / Euler buckling load. Then BAM! Infinite displacement. This is the ELM curve shown below.

The second common approach for assessing stability is moment magnification. As applied loads increase, and system stiffness is taxed, P-big-delta effects generate secondary moments which must be designed for. If these magnified moments are self limiting and can be accommodated, the system is stable. If the magnified moments grow without bound, the system is unstable. This is the DDM curve shown below.

Of the two approaches -- bifurcation and moment magnification -- moment magnification is the more "real". Things that buckle in real life don't just sit still and then suddenly go bonkers. Rather, instability develops more gradually as load is increased, approaching the critical load value asymptotically. This is a result of numerous forms of "imperfections" that function as perturbations to an otherwise stable system. It isn't the case that second order analysis (DDM) needs to match bifurcation results (K-factor). Rather, it is a credit to the K-factor method that, for simple scenarios, it is able to do such a good job of mimicking more accurate second order results.

Comparing DDM and K-factor Methods

Using traditional, K-factor sway frame design, you could come up with some pretty big K values (1.5, 3.0, 10.0). It's important to realize that those values never did reflect the buckled shape of a single, floor to floor column. Rather, they reflected the sway buckling mode shape of a frame assemblage that might include three stories worth of columns and as many as four intersecting girders. The critical load of that assemblage, divided by the Euler buckling load for the individual column under consideration (K=1), is the classic alignment chart K-Factor.

A second way that the K-factor method addressed stability affects was through the application of the B2 factor in beam-column design. This amplified first order sway moments to account for P-big-delta effects and assumed that all columns within a story would sway buckle simultaneously.

So how do we address sway frame instability now that K=1.0? We do it through moment magnification (the paragraph above in red). And we get those magnified moments from a second order analysis that includes the stiffness modifications and imperfection modelling that RPMG mentioned (notional loads, 0.8EI, etc). If the second order analysis is done on a computer, it will likely include algorithms like load stepping and geometric stiffness matrices to account for both P-big-delta and P-little-delta.

How do we know that DDM will provide comparable answers to the K-factor method? We don't. In fact, for all but the simplest of cases -- like your cantilevered column - the two methods will often give markedly different results. And the DDM results are the "righter" ones, by far. A lot of serious simplifications went into the development of the K-factor method. It was, of course, a brilliant solution to a very difficult problem, formulated back when computational power was limited. It's somewhat ironic that, nowadays, our comfort with K-factor seems to be muddling our understanding of the more fundamental stability principles embodied in DDM.

Pretty Picture

The greatest trick that bond stress ever pulled was convincing the world it didn't exist.

 

[JoshPlumSE]

I won't disagree with anything that KootK said, though I will point out a reference that I think explains in a way that makes it easier to understand.... at least for me.

I'm referring to an AISC webinar by Shankar Nair on the subject of stability, "A New Approach to Design for Stability". See link below.

http://www.aisc.org/content.aspx?id=4498

He starts with the basics... "What is Stability?" and defines what we mean when we say buckling. Specifically pointing out that bifurcation buckling doesn't happen in reality. Then he talks about amplification of deflection due to the presence of axial load. It goes on from there, of course. That's part 1.

Part two talks more about what the conventional K factor method is trying to do and where there are problems with that method.

Part three talks more about the Direct Analysis Method and how it addresses all the issues that were discussed in part 2.

 

[humanengr]

I think I understand OP's original question, but it seems to me the replies did not address the root of that question.

Its clear how the direct analysis method is used to calculate the demand on, for example, a column.
I believe the original question, was once you know that demand, you must use AISC Chapter E to determine the strength of the column. Those equations depend on the laterally unbraced length, L multiplied by K.
We know that K is 1.0, but does that mean L is the member length without any further "adjustment".

For example; A multistory building moment frame (unbraced) has a total length of 100 ft. from the base of the column to the top of the building. Each story height is 10 ft. So, is the column unbraced length in the plance of the frame = 10 ft.?

Prior to D.A.M., the unbraced length would have been 10 ft. X a factor ( I don't want to mention the "K" word), so what is it now, Specifically for determining the strength per ASIC Chapter E?

It seems hard to grasp that the column laterally unbraced length, L in the plane of the frame is only 10 ft. The "buckling length" of the column would be expected to be more than 10 ft, therefore shouldn't the unbraced length (L, in Chapter E) be greater than 10 ft?

 

[JoshPlumSE]

HumanEngr -

Watch those videos from Shankar Nair on AISC's website. I think it should make more sense after seeing them.

A quick summary, there is no such think as a buckling failure in a column. Rather it is a flexural failure brought on by the presence of axial force which amplifies the flexural forces and stresses in the member.

The direct analysis method attempts to capture this amplification on the demand side. Thereby turning what used to be be fictitiously referred to as an axial failure into more of the true bending failure that it is.

If the method were a bit more sophisticated you could actually run your analysis with KL = 0. That's the concept. The K = 1.0 is more of a transition value that can be used to bridge the gap between the old (really unsophisticated) analyses that we used to do and the future (extremely sophisticated) analyses that would allow KL = 0.

I believe the 2015 AISC code when it is released will have some provisions suggestion that you can use KL = 0 when you perform a more advanced analysis that meets some additional restrictions above and beyond what the DA method has.

 

[KootK]

Those equations depend on the laterally unbraced length, L multiplied by K. We know that K is 1.0, but does that mean L is the member length without any further "adjustment".

You're exactly right of course. What KL is, is the question. And a discussion of what K is, without a tandem discussion of what L is incomplete at best. And your moment frame example is very salient. It's been a frustrating example for me because, until very recently, I haven't been able to provide a satisfactory explanation for why I believe that L, and therefore KL in your example ought to be 10'

Just last week, I wrapped up AISC's 12 hr night school program on stability. Above all else, I wanted to better understand the KL=1 business. As we've discussed above, DDM is fundamentally a KL=0 method as it's not based on bifurcation. In the mountain of presentation slides there's a lone slide that deals with this. It says that K=1:

To capture potential non-sway buckling of
light members subjected to large axial load

So there you have it. L=10' in your example because the purpose is to check that inter-story buckling doesn't occur prior to whole frame sway buckling which captured using advanced analysis techniques.

I watched the Shankar Nair stuff that Josh recommended. Josh is right: it was very enlightening and a worthwhile investment of time.

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.