AISC Direct Analysis Method 3

[StrEng007]

I'm looking for any resources that illustrate multiple examples of how to apply the Direct Analysis Method with amplification of first order analysis (B1, B2).
Particularly any example that bridges the gap between applying loads to the first order model and the analysis to determine the internal forces that include the second order effects. Most of the examples I've found either approach the subject from a second order analysis computer method (multiplying loads up by 1.6, dividing results by 1.6), or skip a bunch of steps when it comes to applying reduced stiffness and provide a "table of results".

I'm still trying to understand if the reduced stiffness only play a role in the amplification of forces, and sway ratios to determine application of notional loads, and not the serviceability. It has always baffled me how serviceability checks pertain to deflections within the elastic range of the modulus, while column designs that achieve an inelastic critical load utilize the tangent modulus.

Resources I've already seen:
AISC Design Guide 28
A Comparison of Frame Stability Analysis Methods in ANSI/AISC 360-05, Carter & Geschwinder
Steel Design 5th Edition by Segui

 

[JoshPlumSE]

I'm still trying to understand if the reduced stiffness only play a role in the amplification of forces, and sway ratios to determine application of notional loads, and not the serviceability. It has always baffled me how serviceability checks pertain to deflections within the elastic range of the modulus, while column designs that achieve an inelastic critical load utilize the tangent modulus.

LOL. Yes, this concept can certainly be confusing.

Basic deflection and drift:
First remember that when we're talking about deflections we're usually talking about elastic deflections. Even if we have some P-Delta amplification which accounts for basic GEOMETRIC non-linearity.

Even for seismic, we're almost always talking about an elastic analysis. Maybe with a Cd "amplification" factor based on the presumed amplification based on the ductility demand of the specific seismic resisting system.

Direct Analysis and non-linear effects:
Now, why it it so different when we do the Direct Analysis Method. Well, when we do an analysis that captures P-Big Delta and P-little delta, we are doing a good job of capturing the GEOMETRIC non-linearity of the structure. That actually does a pretty good job of capturing the ELASTIC buckling of steel members and steel frames.

But, steel columns tend to be in a slenderness range where elastic buckling doesn't govern. Most columns will be of a height and slenderness where they fail via IN-ELASTIC buckling. This just means that the P-Big Delta + P-Little Delta analysis that we did is over conservative in terms of predicting column and frame buckling. That is the whole point of the Direct Analysis Method. To get rid of the K factors and solve for amplified moments and forces and such directly.

Of course, we're not doing an analysis that truly accounts for material non-linearity. Instead, we're "fudging" it by softening our columns a bit. An analysis that accounts for true material non-linearity would involve incrementing your load and constantly modifying your stiffness matrix with each "event" where a member's stiffness decreases based on the level of yield in the member. That takes a lot of solutions and computational time. Think Perform-3D or a non-linear push-over analysis. Very challenging.

That's why the Direct Analysis method is an incremental, but still relatively simple improvement over the methods we used in the past (i.e. P-Delta for Geometric nonlinearity and no consideration of material non-linearity in the analysis).

The 1.6 amplification factor and ASD vs LRFD:
What's the point in applying a 1.6 amplification factor to our loads if we're just going to divide our results by 1.6?
Also, why are we using a 1.6 factor instead of 1.5 (which is used elsewhere in AISC specs when comparing ASD vs LRFD loading)?

The first point is that we are trying to make sure that ASD doesn't have an unfair advantage over LRFD. I don't think it's any secret that the steel research academics prefer LRFD as a statistically better derived method. Having separate safety factors related to load variability and totally separate safety factors related to member capacity. Nor is it a secret that the engineering community tends to prefer ASD for the simplicity of not having to factor their loads. Maybe also because deflection and drift are also easier to check with the same load combinations.

So, why do we use 1.6 instead of 1.5? I attended the committee meetings for AISC Chapter C for many years. It's my understanding that the 1.6 was an added safety factor because when the 13th came out (with the DA Method) they hadn't done enough research to be fully confident in the 1.5 factor yet.

I mentioned at a number of meetings how I thought we shouldn't be punishing ASD so much. That the 1.6 factor showed (essentially) some bias in the code against ASD. In my thinking, this this punishment wasn't just a 6.7% increase (1.6 / 1.5), but could be significantly more than that because we were amplifying gravity loads and lateral loads by the same factor.

P*Delta = (1.6*P)*(1.6*delta)

Therefore, this amplification would be more like a (1.6/1.5)^2 increase (i.e. 14%). Potentially more since the effect is so non-linear. Even more if you had to decrease the stiffness of your columns. The committee was totally open to hearing my concerns, but I was the one who was going to have to do the work to demonstrate how significant the 1.6 penalty was compared with 1.5.

Unfortunately, I could never come up with what felt like a realistic example which showed it to be as significant as I thought it would be. So, I eventually dropped the argument.

If anyone has examples that they have which work nicely under LRFD, but fail pretty miserably under an ASD analysis then I'd love to see them. I worked mostly with a heavily loaded portal frame supporting a tank structure. So, a very high P load versus a relatively weak lateral force. But, now I'm thinking I should have looked at a very tall and slender communications tower or something of that nature. Not only are the members relatively slender single angles, but the overall structure is also pretty slender.

 

[Settingsun]

The difference between 1.5 and 1.6 would depend on where you are on the non-linear response. It makes negligible difference for stocky structures because the response is almost linear: multiplying by 1.6 then dividing by 1.6 cancels the same as using 1.5.

Now roughly speaking:
For quite a slender structure where load*1.5 puts you at 50% of elastic buckling capacity, the second order analysis will give about double the response of the linear analysis. Using 1.6*load would instead be at 53% of elastic buckling, and 2.14*response, so 7% penalty as an order of magnitude.

This example is well into the territory where the alternatives to direct analysis aren't permitted because of excessive slenderness.

 

[JoshPlumSE]

Steve -

So, my attempt to demonstrate the problem with 1.6 vs 1.5 was related to finding a structure that worked for the 1.5 application (code check wise and buckling wise) and did NOT work for the 1.6 multiplier (either with code check or buckling). I could not find cases where the difference was really significant for members that worked at the 1.5 amplification. Maybe even comparing them to the LRFD type solution and code checks that would have resulted from that solution.

That's where I was was relatively unsuccessful.

But, I think you have hit the basic problem on the head.... It depends on how close you are to "elastic buckling" either for the compression member or for the frame / structure as a whole. Which is why I was thinking that it might make more sense to do this comparison for a slender communications tower structure.

 

[Settingsun]

Any structure that's at 100% utilisation for 1.5 factor will fail for 1.6 because the nonlinear component of the stress, which doesn't cancel in the multiply-divide process, will be larger. The difference would be small except for structures on the precipice of instability - I wouldn't expect much sympathy from code writers.

The first point is that we are trying to make sure that ASD doesn't have an unfair advantage over LRFD.

It seems clear to me that the intent is that the ASD design process results in the intended margin of safety against overload. When nonlinear response is significant, an apparent 50% margin under working loads means you have significantly less than 50% capacity to increase the load.

 

[StrEng007]

Any good resources for calculation examples besides those listed below?

Resources I've already seen:
AISC Specification and Design Examples
AISC Design Guide 28
A Comparison of Frame Stability Analysis Methods in ANSI/AISC 360-05, Carter & Geschwinder
Steel Design 5th Edition by Segui

 

[JoshPlumSE]

StrEng007 -

Sorry, I don't know of hand calc examples.

I think the implication is that the Direct Analysis Method is primarily intended for automation through computer analysis. This isn't explicitly said, but this is consistent (IMO) with most engineering practitioners.

 

[Settingsun]

StrEng007, perhaps post one of the better examples from your references and ask questions related to that. Sometimes it's easier to discuss specifics rather than general concepts.