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We are having an internal company discussion concerning the appropriate application of ACI 318-14 and the design of Sway columns. We are using RAM SS to design the columns, and as a matter, of course, we are conducting a partial second-order analysis. We understand that Bentley's process does not account for the P-δ effect. We are trying to apply section 10.10.6 as called for by 10.10.2.2. The code is not clear on the appropriate K value to use in 10.10.6. A strict reading of the Provisions says use K=1.0, but some here are arguing that since the columns are part of a sway-frame it is not appropriate to use K=1.0. I looked at AISC for some extra understanding and they imply K=1.0 for the calculation of the B1 modifier. I am leery of leaning on this as support since the AISC method is predicated on the use of notional loads. Does anyone have any papers to clarify this or just an opinion regarding the correct K for this case?

Robert Hale, PE

my rational behind using a K=1.0 for this process is you are amplifying the p-Delta moments using the non-sway amplification procedure to capture p-delta imperfections. One very important note is RAM Concrete is hard coded to assume you considered p-Delta in the RAM Frame analysis.

Additionally we set the laterals columns to non-sway to force the right amplification procedure but inflate k to 1.55 to force things to check out slender where kl/r would be less than 22 with a k of 1.

This also only applies to columns defined as lateral elements, Gravity columns should be set to sway or non-sway as appropriate. To further complicate things if you have a two-way slab the k factors calculated by RAM are almost always incorrect.

Open Source Structural Applications: https://github.com/buddyd16/Structural-Engineering

I think you're quoting the code sections from the 2011 version of ACI-318, not the 2014 version.

I think a k=1.0 is appropriate provided you've done the following:
a) Used the 2nd order or P-delta analysis within the program to capture the P-Big Delta effect. This is the vast majority of the 2nd order effects for most reasonable structures. Only when you're dealing with really slender members does p-little delta start becoming more important.
b) the EI value you use for the calculation of the elastic buckling (Pc) takes into account cracking, creep and such. Using their 0.4EI / (1+Beta) approach (ACI 318 - 2011 eqn 10-15) This ends up being something like o.25 EI. For a column that's a really significant stiffness reduction.

#### Quote (RH)

but some here are arguing that since the columns are part of a sway-frame it is not appropriate to use K=1.0.

I believe that K=1.0 and that the sentiment above, expressed by your colleagues, represents a misunderstanding of the fundamentals of the situation. As I see it, the logic chain here works like this:

1) We want to amplify our moments for P-del between the ends of the member.

2) We tackle #1 with a series approximation involving the ratio P/Pcr. ACI presents this as an apparent stiffness reduction which is neither here nor there. Things experiencing compression possess a reduced, apparent flexural stiffness as a result of that compression. The rest is just algebra.

3) Per #3, to get P/Pcr correct, we of course need to use an appropriate effective length for the Euler buckling load. Here's where the rubber meets the road. The moment amplification that we're after is correlated to a buckling mode involving movement between member ends and NOT involving movement at member ends. So the maximum theoretical K value for this situation is 1.0.

For most concrete columns , a effective length factor between 0.5 and 1.0 will more accurately reflect reality. I think that many engineers struggle to wrap their heads around how a location with an applied moment can also be a point of rotational restraint in the same sense as the applied moment. I know that I certainly do.

I never understood this, is it incorrect to utilize the cracked MOI (0.7 for a column and 0.35 for beam etc) in ACI (somewhere in chapter 6 I believe of ACI 318 2014) when calcing Pcr?
Sounds like it is wrong based on what Josh,"ends up being like 0.25 EI for a column" when ACI states that 0.7 can be used for a column.

If so, I have been doing it wrong and need to revise my ways.

Let's start off with recognizing one basic fact. The REAL behavior of concrete is non-linear. Cracking, and creep for material non-linearity. And, then our regular P-Delta and p-little delta for geometric non-linearity.

For analysis, we typically take the material non-linearity out of it by assuming 0.7*Igross for columns and 0.35*Igross for beams. That's the standard way that most engineers account for cracking so that they can ignore (or greatly simplify) material non-linearity. We all know it's a gross over simplification, but that's what we do because it we're engineers not PhD researchers.

Chapter 6.6 Elastic First Order Analysis
When I mentioned the 0.25EI, that was referring to the Chapter 6.6 (of ACI 318-2014). Specifically, the P-little delta amplification factor..... Cm/{1-Pu/(0.75*Pc)}

Where Pc really equals the "euler buckling" load:

Pc = Pe = pi^2 * EI_eff / (KL)^2

In ACI 2014 (and going back a number of code cycles), EI_eff (for the purpose of the p-little delta amplification) can be estimated by section 6.6.4.4.4 (reference from ACI 318-2014).
The first equation given is 0.4*E*Igross / (1+Beta_dns)

Beta_dns is the ratio of maximum sustained factored load to maximum factored load. In the commentary they suggest Beta_dns can be assumed to be 0.6, which makes EI_eff = 0.25 EI gross. But, only for the purpose of this moment amplification equation.

Why do they use such a smaller value for this? To account for creep caused by sustained loads and the effect it could have on very slender columns.

Personally, I'm not a big fan of this type of amplification. This is all based on the assumption that you performed an Elastic - First Order analysis. Meaning, no material nonlinearity (I = 0.7 or 0.35) and no geometric non-linearity (i.e. hand calc methods for P-Big Delta and P-Little Delta).

Chapter 6.7 Elastic Second Order Analysis:
Now, Chapter 6.7 more accurately captures what I really think engineers are doing. The rest of the world (or the academics) would call this GNA (Geometrically non-linear analysis), which means we're accounting for the non-linearity due the change in geometry of the structure. P-Delta and P-little delta.

Rather than mess around with all the funny hand methods for amplification, section 6.7.1.2 (and the commentary) suggest that you capture P-little delta if you merely sub-divide your columns between floor levels. But, that means, their P-little delta amplification is now based on whatever assumptions you used for Icracked.... usually 0.7 Igross.

Hence you have a MAJOR difference between how p-little delta effects are calculated between the two methods of analysis. But, both methods still follow the letter of the code. Now, when is this a big deal?
Only when column slenderness is significant. I don't have a great feel for this for concrete (as most of my stability work has been with steel). But, the KL/r > 22 is a trigger for sway frames and KL/r > 40 is a trigger for non-sway frames.

Personally, I usually start my analysis WITHOUT sub-dividing my columns. When I'm close to a finalized design, then I'll sub-divide the most slender of them. If their moments change by more than 1% then I'd probably sub-divide them all and see how much of a change I get. If it's consistently above 1% difference from the moments before subdivision, then I would take a closer look at the P-little delta effect by hand rather than purely relying on the computer programs P-Delta analysis.

That makes sense Josh, I have been in chapter 6.7 for my designs(elastic second order analysis). This explain why I have been using the 0.7 for columns and 0.35 for beams.

Finally in front on of my ACI book and 6.7 (specifically 6.7.2.1.1) points you to 6.6.3.1 which gives your tradition 0.7/0.35 * Ig ect.

Seems like this is pretty easy to consider with software and less fussy than the hand methods of elastic first order analysis.

Yes, it is much easier for the analysis programs to do their analysis via the provisions of section 6.7. However, I suspect ACI may eventually try to "bridge the gap" between the two methods. Maybe something like a flexural stiffness reduction for columns based on their axial load and slenderness..... something akin to what AISC's Direct Analysis Method does.

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