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.
OP said:
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.
OP said:
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.
OP said:
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.
