Some questions on AISC beam/column stability 3

[StrEng007]

*I'm disclaiming my questions with the fact that I don't often design full structural systems using steel. A majority of my work utilizes shear walls for lateral stability. I'm interested in consulting on more braced frame/moment frames, thus I have some questions.

After a read through AISC 360-16 (Spec.) Chapter C and subsequent appendices 7 & 8, I've got the general idea of what AISC wants in regards to stability analysis. To be honest, I cannot say that I fully understood all of these requirements. To get started, here is a quick read on my understanding thus far.

1. When it comes to stability, AISC wants our analysis to include the effects of a bunch of stuff (member deformations, 2nd order effects local/global, geometric imperfections, stiffness reductions... the list goes on.)
2. The Spec. offers (3) ways to go about capturing all of these intricacies in our demand and capacity analysis. They are:

#1 Direct Analysis Method​

•Second Order Analysis (Chapter C)​

•Approximate Second-Order Analysis (Appendix 8)​

#2 Effective Length Method (Appendix 7)​

#3 First-Order Analysis Method (Appendix 7)​

3. The Direct Analysis method is the preferred choice. It eliminates the need for K factors and takes into account a lot of the imperfections that exist in actual fabrication/construction.
4. Both the effective length method and first-order analysis method require a maximum 2nd order to 1st order 'drift check' to be applicable.

Again, most of my steel design work involves utilizing beams and columns as single components, with the occasional beam-column requirements. Very rarely are these members part of any sway frame type action that lends itself to the effects of P-D.

My questions are (bare with me on some of the painfully obvious ones):
1. When AISC speaks to applying a 'rigorous' 2nd order analysis, they are explicitly talking about modeling software, correct?

1.1. If that's the case, does selecting the type of 2nd order analysis in your modeling software allow you to satisfy this criteria. Simply, if I tell the program to perform a 2nd order direct analysis method, are the required strengths (ie member forces) ready to be compared directly to the available strength of each member? My understanding is that the capacity of members rely on K=1.0.​

1.2. Is there anything else I need to do to these results to make them compatible with the requirement of the Spec.? I've heard of multiplying by a factor, running your analysis, then dividing by the same factor. I have a hard time believing that our software isn't doing that for us.​

1.3. While it's easy to specify this in a program, if we're being honest with ourselves, how many out there know what is actually happening behind the scenes on the load demand analysis within the program?​

1.4 Is the approximate 2nd order analysis under the direct analysis method ever provided via software calculations? This is the method I've used in the past to check a couple beam-column scenarios, but I recall it falling under the provisions of amplified first order effects... not the direct analysis method.


2. Please confirm the effective length method is a second order analysis that relies on software. There is no first order magnification possible with this method.

2.1. It seems like something changed here from the first time I learned this using the AISC 360-05 specification. Back then, the direct analysis method was in the appendix and Chapter C allowed moment magnification with the use of calculated K factors for beam-columns that were part of sway frames.
​

3. How do you honor the 2nd order drift check for the First-Order Analysis method, without first calculating the 2nd order analysis?

I'll probably have more questions once some of the steel heavy hitters start coming out of the wood work. Thanks in advance.

 

[milkshakelake]

While it's easy to specify this in a program, if we're being honest with ourselves, how many out there know what is actually happening behind the scenes on the load demand analysis within the program?

For ETABS, there's a simple option to add P-delta effects using a load combination. I just use that. It's based on the practice of a large firm around here that does high rises. That's pretty much it. Sorry I can't go more in-depth about what it actually does. But if it's good enough for the big boys, it's good enough for me. Never had any building department or peer reviewer bring this up when I present results of a model.

 

[lexpatrie]

Oh boy, a numbered list! Yeah, except it's not working the way I want it to. Okay, bullet list it is!

[ul]
[li]1.1 - The K=1 bit isn't really correct, as I understand it you don't calculate an explicit K because it's accommodated inside the analysis. If you desperately wanted to, you could "extract" a K factor, but stability is handled with more detailed calculations rather than the simplification of K factors.[/li]
[li]1.2 - The "multiply then divide" sounds like factoring loads. This is perhaps only required for seismic where drift is supposed to be based on Limit States. It would depend on the rigor of your software, which may vary that it handles this or not. Tech support.[/li]
[li]1.3 - Probably nobody, except the one guy at the software company that did it all.[/li]
[li]1.4 - If you mean the "old" Mlt, Mnt (B1 B2?) factors from way back, probably no software did that, unless it was custom by some guy doing it for a research project. If I remember this right, the approximate is more painful if you're doing it in a 2D frame program because you have to do a lot of manual tinkering, two models, that kind of thing. It involved a No translation and a lateral translation model and then you combined the results. If I recall it's still around, but there's limits to keep it from getting "really plastic" or having excessive non-linearity.[/li]
[li]2 - Doubtful. Effective length is just another term for K factor.[/li]
[li]2.1 - At some point it left the appendix and joined the real code, and the K factor went into the appendix. So yeah, it's much harder to design a portal frame without software. I have dealt with it this way, however.[/li]
[li]3 - I thought there was a ratio. You do a (manual) first order analysis, then do some extra stuff to account for stability, then compare the "final" to the first order and if they are below a certain ratio it's okay. If I remember it right, if it fails you have to either resize or do a full on analysis that you were trying to avoid.[/li]
[/ul]

At one point in my graduate studies, I took a class on this as it was emerging with Jerry Hajjar. I learned a lot more than I ever wanted, especially about GWBasic. I'm speaking from recollection as of 2017 or so, not directly from that class which was a good deal before that.

I'll add that the way I checked my personal program in grad school was to use a single element, lateral load and vertical load on a vertical cantilever. There's a closed form exact solution for this, so if you are below plastic limits, the program should be able to reproduce that fairly well. The problem with my program was somewhere in the guts one of my matrix resets on the local stiffness matrix cleared a single precision variable, rather than the double precision variable of the same name. Cost me a lot of time.

Regards,
Brian

Comments above pertain to (I suppose 11th to ) 14th edition which is the last one I looked at, don't have a 15th or 16th yet. The background reading on this subject gets really involved, but Bill McGuire, Ron Ziemien's grad thesis, and William LeMessurier were all there batting research back and forth when it first developed. This was around the time Intel Pentium had issues with their processors not doing math.

I'm not sure the difference between direct analysis and advanced analysis, what they were calling it in grad school was advanced analysis.

More recent (14th edition) AISC Seminar slideshow -

https://www.aisc.org/globalassets/c...z-handouts/direct-analysis-method-handout.pdf

https://www.aisc.org/education/cont...ct-analysis-method-applications-and-examples/

 

[Luceid]

Been discussing this exact section in the past couple of days - will be following this discussion as well

 

[RattlinBog]

I don't pretend to be an expert on the various stability methods, but I've had some practice in grad school recently. If I'm wrong somewhere, my bad. There's a lot to sift through. My reply will only consider AISC 360-16 (15th Ed manual spec), not anything older or newer. Spec references in brackets [#] below.

I have some practice with:
Ch. C - Design for Stability (namely the Direct Analysis Method)
App. 7 - Effective Length Method
App. 7 - First-Order Analysis Method
App. 8 - Approx. Second-Order Analysis

I have no practice with:
App. 1 - Design by Advanced Analysis

Your questions:
1. Not sure--probably. I suppose you could do an iterative 2nd-order analysis by hand, but...

1.1/1.2: Direct Analysis Method requires:
[C2.1(b)] 2nd-order analysis including P-big delta (turn it on if using software) and P-little delta effects. For P-little delta, you need to divide axially-loaded members with multiple nodes. (If not using software, you can run through App. 8 instead.)
[C2.2] Consider initial system imperfections by either directly modeling displacements or, more commonly, applying notional lateral loads and including in your load combos. (Notional loads entered manually.)
[C2.3(a)] Stiffness adjustment. All members that contribute to the stability of the structure must have their stiffness factored by 0.8. User note says better to apply stiffness reduction to all members. (Risa-3D automatically does this; can be changed under settings. Not sure about other programs.)
[C2.3(b)] Another stiffness adjustment. Members whose flexural stiffness contribute to the stability of the structure shall have their flexural stiffness factored by "tau", the value of which depends on compressive strength ratios.
Edit: the stiffness adjustments above only apply to determining required strength, not available strength. No stiffness reductions for actual capacity (ASD or LRFD).
[C3] Yes, K = 1.0, typically. "The effective length (KL) for flexural buckling of all members shall be taken as the unbraced length unless a smaller value is justified by rational analysis."

1.2: Your question about multiplying by a factor then dividing by it is probably in reference to section C2.1(d): "...For design by ASD the second-order analysis shall be carried out under 1.6 times the ASD load combinations, and the results shall be divided by 1.6 to obtain the required strengths of components." It's an extra step if using ASD. Not required with LRFD.

1.3: See 1.1. I use Risa-3D, and I feel like you have a decent amount of control with notional loads, stiffness reduction, etc. It's not all automatic and hidden. I wouldn't be able to replicate the iterative analysis, though.

1.4: I don't believe so. I've run through App. 8 a couple times and just built a little Excel sheet for it.

2: Hmm...I'll check with my professor. Not sure about the 1st order/2nd order question. I don't have experience with older AISC codes, but I believe Effective Length Method has changed some. Now it's in App. 7, and it requires applying notional loads per C2.2b to determine required strength. However, stiffness reduction is not required. For available strength, you find effective length KL, using the alignment charts in the App. 7 commentary to find K. K <= 1.0 for non-sway and K >= 1.0 for sway frames.

3: Not sure. User note in App. 7 says you can take the 2nd-order to 1st-order drift ratio as the B2 multiplier from App. 8. The ratio limit of 1.5 applies to both the First-Order Analysis Method and Effective Length Method. For First-Order Analysis Method, don't forget to also apply notional loads per 7.3.2 (larger than Direct Analysis Method...). No stiffness reduction and K = 1.0.

App. 8 note: If using App. 8 Approx. 2nd-order Method, you need to include notional loads in lateral translation case. The stiffness reductions per C2.3 would also apply if you're doing the Approx. 2nd-order method under the umbrella of Direct Analysis Method (Ch. C).

 

[StrEng007]

Wow, a lot to decipher here. Thank you for the replies, I'm hoping we can keep this conversation moving.

Isn't it funny how you can get so used to doing something one way and then get completely thrown off base when you crack open AISC 360-16 (new, but not new since AISC is already pushing the gold manual), and notice that slenderness ratios for flexural buckling limit states don't have the familiar K in the KL/r equation. Wait, what?!

As a point of reference, I learned and began practicing from 360-05. Bought 360-10 a while later and shelved the 13th edition. I've had the 360-16 PDF for several years and reluctantly look at it. Project budgets don't allow for all these iterations and continually cracking the code, and so you'll find yourself on this blog.

I used to criticize but now I understand why my predecessors had such a tight grip on their trusty AISC 9th Edition Manuals.

 

[RattlinBog]

After comparing the different methods (AISC 360-16) in grad school, my takeaway has been to use the Direct Analysis Method in your software of choice if you can. In Risa-3D (can't speak for other programs), the only "extra" steps are to apply notional loads (0.2% of gravity) and split axial members into multiple nodes. It's not bad at all. Risa handles stiffness reduction and P-Delta iteration.

I'm pretty sure the effective length method used to be simpler back in the day. Now the current code requires notional loads be applied too. If you have software that can handle it, I'm not sure why you would do anything other than Direct Analysis Method. If you just have hand calcs, better have some good spreadsheets too. Heck, I would probably stick with First-Order Analysis Method over the current Effective Length Method. No need to deal with K and alignment charts. I think AISC is trying to push K out the door. KL became Lc = KL. They're not being too subtle

 

[lexpatrie]

K factor and the alignment charts can only take you so far, analytically speaking. While I'm unaware of any failures that are directly attributed to the alignment charts, there's potential there in some fairly infrequent (obviously, I suppose) circumstances.

9th is kind of dangerously obsolete (no flames please!). I came into it with the first LRFD and boy did the workforce one place just stare daggers that I had it, the eyerolls and the contemptuous glares. I got a free publication from AISC with the company membership and I got the third LRFD versus the ninth and then they really got pissed. pissed. It was sad, lame, and unprofessional, but hey, that was the workplace. Good riddance.

Anyway - Primarily when it comes to HSS for vertical loads (0.93 thickness for the older HSS specifications). If you bother with Cb there's a better equation "now" since around, what, 2005? Actually it's the Zoruba article on it from 2005, the changed equation was in the 1999 LRFD spec. So 11th by the new numbering scheme. Which is nearly fifteen years old, right? 2005 is 13th? It's a great reference to have around, as there are still a large body of valid technical articles on the subjects that come up there, and it's easier to trace them through newer codes if you know where they were "back when". I've been on a bit of a scanning rampage lately and re-reading some older articles, so it's fairly fresh.

The codes change and get more detailed because we keep trying to do more and more with less and less, or do strange stuff nonstop.

Back to the regularly scheduled on-topic conversation, now.

This may be of some use, although old. Wow, 2009 is now OLD.

Software and the Direct Analysis Method, Gebremeskel, P.E., MSC, Dec 2009

On an unrelated note, I think that's the ONE article in Modern Steel Constructions history that's not some cutesy click-bait title that's hard to figure out what it was next time you want to go back and read it. Hurray for an article title that reflects the content. (does this beam make my building look heavy, indeed)

 

[montbIanc]

This is a good explanation of the different methods, from the engineer who led the introduction of direct analysis to the code:

https://www.youtube.com/watch?v=iOH6kaXX3Mk&list=PLvJE8_LjpVk35oVNRd9OQj6tg0vbl-34b&index=9

Q1. Software is the easiest way of doing the direct analysis in this day and age, but it can be done by hand. B1 & B2 multipliers are acceptable within their limitations without iterative calculations.

Q1.1 Depends on the software but you'd hope selecting an option called direct analysis does direct analysis properly. As always, checking the computer is doing what you think it's doing is critically important.

Q1.2 This is almost certainly the way to use direct analysis with allowable strength design. The point of this is that you need to do the non-linear analysis at the ultimate limit state, so the method is fundamentally incompatible with ASD. This is the workaround. They pick a single, approximate load factor and use that for the non-linear analysis.

Q1.3 Again depends on the software. If it's just a general-purpose, stiffness matrix analysis, it's fairly straightforward. Matrices are used to solve a bunch of simultaneous equations. The programs that integrate code requirements vary more from program to program.

Q1.4 The approximate second-order analysis is a hand method that's an alternative to iterative second-order calculations (generally by software). Second-order analysis is a part of direct analysis, but there are other parts involved (so second-order analysis on its own is not the same thing as direct analysis). As such, approximate second-order analysis can be used for that part of direct analysis, subject to the limitations placed on approximate analysis. (Software could do this approximate analysis, but general purpose analysis software would likely do a more rigorous second-order analysis.)

Q2 Effective length methods use first-order analyses. [Edited. I'm out of date and it appears that second order analysis is needed alongside the effective length factor, which introduces some overlap/double counting of stability effects.] The youtube video I linked to provides a basic example that shows how this works, and also its shortcomings (basically that it covers member design but not connection design or how second-order effects in beam-columns affect the connected members).

Q2.1 They've swapped places between main part of the code and appendix. The reason is that direct analysis is permitted for all structures whereas the effective length method is now restricted to structures with low-moderate second-order effects.

Q3 Again, software is easiest but the B1/B2 multipliers can be used to check this requirement.

 

[RobertHale]

Here are my thoughts;
[ol 1]
[li]Effectively, they require the use of software for the analysis for all but the most simple of structural systems (think single cantilevered columns)[/li]
[ol 1]
[li]Most software will require a little more than just selecting one button. Bentley's RAM product, for instance, will require you to specify a using a "P-Delta" analysis (including the load factors that apply - a quirk of their geometric stiffness approach), specifying reduced stiffness for the lateral members (tau) (and this may need to be revised based on the axial demand of some members,) specifying the use of the B1 factor to capture/approximate along member second-order moments, and specifying notional loadings to generate initial geometric imperfections. You really have to dig into the software you are using and make sure you have checked all the boxes[/li]
[li]See the comments in the list above; the big checkboxes are both types of P-Delta captured using the appropriate load (e.g., live load contributes), are the potential initial imperfections in geometry captures, are the member stiffnesses reduced appropriately to capture the expected softening from partial yielding?[/li]
[li]Do I have a concept of what they are trying to accomplish? Yes. Could I do it by hand? Maybe if I had a year to work on a single frame. I have the benefit of taking a steel class from Don White in graduate school. The original 2005 spec referred to this as "Second Order Analysis by Amplified First-Order Elastic Analysis." but you also have to look at the way compression capacity was calculated. 2005 still had K factors in the capacity equations. You will not find any K factors in Chapter E anymore.
[li]Yes, you can get most software to produce these calculations; after all, it was the method we designed steel before 2005.[/li]
[/ol][/li]
[li]The effective length method can be a second-order method. Choosing between direct analysis and effective length is a choice between comparing amplified "nominal" loading with a nominal capacity vs comparing your "nominal" loads to a reduced load capacity[/li]
[ol 1]
[li]I think AISC has changed a little bit of terminology and they are clearly pushing more rigorous analysis methods to the forefront. Your description sounds right, but since 2005 all the methods of analysis and design have been bouncing around in the spec.[/li]
[/ol]
[li]Unless I am just completely misunderstanding the question, can't you just calculate B2?[/li][/ol]

 

[canwesteng]

For what its worth, the most demanding computational parts of direct analysis can be replaced by bumping up notional loads. I think if you were able to design a simple portal frame by hand before, you could still do so now, since the K method has required second order analysis for a while.

 

[sticksandtriangles]

I am attaching a paper I put together for a young engineer lunch learn about 5 years ago. It goes through how to do a RAM elements and ETABs Direct analysis procedure and shows some of the pitfalls with our software.

My personal opinion, I wish AISC would get rid of this term " 'rigorous' 2nd order analysis". What the heck does 2nd order analysis even mean? Let's talk in terms of non-linearity.

I think this should be re-written as for DAM "you must perform a geometric non-linear analysis, considering both P Big delta and P little delta" or something along those lines.

For ETABS, there's a simple option to add P-delta effects using a load combination.

I have used this option as well, but prefer to make a few test load cases where the P-delta effects are explicitly considered with our lateral cases. If I feel comfortable the "rigourous" geometric nonlinearity matches the "softening" of the stiffness matrix method, I will switch over to the less computationally intense method. Lots of times there are big enough detlas I stick with the explicity non-linear geometric analysis, like moment deltas in the range 10% on moment frames/shearwalls.

For what its worth, the most demanding computational parts of direct analysis can be replaced by bumping up notional loads. I think if you were able to design a simple portal frame by hand before, you could still do so now, since the K method has required second order analysis for a while.

Completely agree, I typically default to 0.003 on the notional loads and move on.

S&T - www.re-tug.com

 

[JoshPlumSE]

I think this should be re-written as for DAM "you must perform a geometric non-linear analysis, considering both P Big delta and P little delta" or something along those lines.

I tend to agree. When I was attending the AISC committee meetings on stability, there was some hand-wringing over this terminology.

There are many who would prefer to say "geometric non-linearity" rather than "2nd order analysis". I think that's a more precise and descriptive term.

However, in the US, we adopted this term "2nd order" terminology related to P-Delta type effects. That was done a long time ago. And, the codes have used that term for so long, that's it's tough to just change it immediately. They feel like that could cause some confusion.

To me, it feels like we're talking about math and differential equations when we say 2nd order effects. Whereas when we say "geometric non-linearity" it's pretty clear that we're talking about how the geometry of the structure leads to non-linear deformation.

 

[JoshPlumSE]

1.2. Is there anything else I need to do to these results to make them compatible with the requirement of the Spec.? I've heard of multiplying by a factor, running your analysis, then dividing by the same factor. I have a hard time believing that our software isn't doing that for us.

First, a quick explanation WHY we are asked to do this. If we didn't, then LRFD would be penalized compared to ASD design as far as 2nd order effects are concerned. If you're using LRFD, of course, this isn't necessary.

A while ago, I proposed that the 1.6 was too biased against ASD, that AISC should be using a value more like 1.5 which is the number we used to account for differences between LRFD and ASD loads in other places. I tried to put together examples demonstrating that it penalized ASD design too much.... I never really came up with an example that showed as much of a penalty as I thought there would be.

I still don't like that we use 1.5 sometimes and 1.6 other times. It seems counter-intuitive to me. But, I couldn't demonstrate any true ASD penalty, so I gave up.


Finally, I want to agree with you and point out that most software that is geared towards AISC design will probably do this (the 1.6 load multiplier followed by a dividing your results by 1.6). However, that wasn't the case when the AISC 13th edition first came out. Even now, you would be wise to dig into your program documentation and / or run some tests to determine if this is really happening or now.

 

[montbIanc]

First, a quick explanation WHY we are asked to do this. If we didn't, then LRFD would be penalized compared to ASD design as far as 2nd order effects are concerned.

I would state it somewhat the other way around. Analysis at allowable load level doesn't demonstrate that the target resistance to overload has been achieved.