Flagpole Steel Column - K=1 for all cases? 1

[AggieYank]

Say a steel pipe has rigid (moment) base, and cantilevers up 10 feet. A 5 kip weight is vertically attached at the top.

For all load cases, is K=1 as long as you use Direct Analysis Method to determine the forces?

In other words:
For "Dead load only", "Dead + Wind", etc: Because you are including the notional load, reduced stiffness, you can use K=1?

I can't quite wrap my head around how you can avoid using K=2.0 (or higher sometimes) for an axial load of a cantilever, based on the buckling shape.

Thanks an advance.

 

[JoshPlumSE]

Aggie -

Keep in mind, this is only valid if you use the OTHER provisions of the direct analysis method as well. Flexural stiffness reduction, 2nd order analysis and notional load (or direct modeling of out-of-plumbness).

The notional load gives you some initial displacement or moment. So, it is not longer a pure axial situation.

The 2nd order analysis captures the P-Delta (and little delta) effects. Meaning the axial loads tendency to increase the bending moment is directly accounted for in the analysis. This does a good job of capturing ELASTIC buckling of your flagpole.... provided you have the initial moment or displacement to amplify.

The stiffness reduction is an approximate way to capture the IN-elastic buckling that would really occur in a typical steel structures. You're still using a nominally elastic analysis, but you're reducing the stiffness in order to make the 2nd order analysis buckle sooner.

 

[TehMightyEngineer]

Because direct analysis should include second order effects of P-Δ (displacements relative to member ends) and P-δ (local deformation relative to the element chord between end nodes). These effects are approximated by the effective length factor K and the column buckling stress Fe (which uses K). Thus, using K = 2.0 for direct analysis would be double penalizing the column strength and wouldn't make sense.

Note that you aren't required to use direct analysis (except in certain cases) so you could use K = 2.0 and employ the Amplified First-Order Elastic Analysis in section C2.1b of the AISC (or similar if you're under a different code) which is the more traditional method of designing what you're talking about.

Maine EIT, Civil/Structural.

 

[AggieYank]

Thanks for the replies.

Relating euler buckling stress to second order effects inducing small moments is where I get lost. I understand that Euler buckling is different than having axial force on a shape with a small bend.

Euler buckling is related to what the shape would look like in the buckled position. So, for example, a cantilever column would have a buckling shape twice the length. Having k=2 makes sense.

For our example column, is the logic that the effects of notional loads, reduced stiffness, 2nd order effects will simply end up with similar sizes? Or, is the logic that it is a more detailed calculation of Euler buckling somehow?

 

[WillisV]

The logic is that you get about the same column size (the interaction ratio goes to 1.0 at about the same axial load level) - with the older effective length method you calculated a reduced axial capacity to reach an itneraction ratio of 1.0, with the direct analysis method a higher axial capacity is calculated, but the applied moment side of the interaction equation begins to control and you end up with about the same thing.

 

[Bagman2524]

I'm with AggieYank on this one. A cantilever's deflected shape is half of a simple beam's deflected shape. Thus and effective length k factor of 2.0 is used. What's the use of using a K factor with the Direct Analysis Method if it's not really an effective length factor.

"Look for 3 things in a person intelligence, energy and integrity. If they don't have the last one, don't even bother with the first 2. W. Buffet

 

[WillisV]

@Bagman - K is always 1.0 with the direct analysis method. Always.

 

[TehMightyEngineer]

Bagman: My understanding is they have K = 1 simply so that you can utilize the same equations. They could just have easily said "remove K from all equations" but saying K = 1 seems much simpler to me.

Maine EIT, Civil/Structural.

 

[WillisV]

It is actually permitted to use K less than 1 in direct analysis, but that is not typically done.

 

[Bagman2524]

Table 2-1 has a comparison of the 3 methods available for stability analysis; Direct Analysis Method, Effective Length Method & First Order Analysis Method. Per the table K = 1 for all frames in DAM. Why even consider the k factor for DAM?

"Look for 3 things in a person intelligence, energy and integrity. If they don't have the last one, don't even bother with the first 2. W. Buffet

 

[WillisV]

As TehMightyEngineer says - it is to leave the strength equations the same - just insert K=1 when using DAM, or K= whatever when using other methods. If it bothers you, just white out the legs of the K so it looks like a one.

 

[KootK]

This is tricky. I struggle with it myself. And this is an excellent example to test our "faith" as it were.

The only reason that we use the "K" value with tradition Euler buckling calculations is that we're too lazy to do separate calculations for all of the different permutations (pin-fix, cantilever-sway, et.).

For normal column buckling, instability occurs when P-delta moments cause unbounded displacement somewhere along the member. This is true for both traditional effective length methods as well as for the direct analysis method. This is obviously just one of many ways to define instability.

With effective length methods, we find the compressive load at which the load-displacement curve stops being vertical and dog legs to the left (or right). This is the classic Euler bifurcation load. Of course, it's not a perfect model. Nature would never do anything this cleanly. Instead, the load-displacement curve become non-linear earlier in the loading history and the horizontal and vertical lines of the load-displacement curve are connected by a fillet curve of sorts at the knuckle. In this context, we need "K" to serve as an indexing value to get us from our real buckling mode back to the classic Euler mode (pin-pin, K=1) which we find expedient to calculate.

With the direct analysis method, we use non-linear analysis techniques (load stepping, geometric stiffess matrices, etc.) to essentially trace out a more realistic load-displacement curve. Here instability is noted computationally by identifying instances where displacement growth is no longer proportional to load increase. Mathematically, it's an eigenvalue problem if you roll that way.




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

 

[KootK]

This is tricky. I struggle with it myself.

The only reason that we use the "K" value with traditional Euler buckling calculations is that we're too lazy to do separate calculations for all of the different boundary condition permutations (pin-fix, cantilever-sway, et.). We could -- and some folks have -- generated Euler buckling equations for all of the common boundary conditions that do not involve the "K-factor". In fact, I was forced to do this in a graduate stability class. The reason that we use the "K" factor is simply expedience. It's a convenient scaling factor that we can use to transform the classic pin-pin, K=1 Euler solution to suit our real boundary conditions.

When we use the direct analysis method, we employ second order computational techniques that mathematically test for instability using the real structure and the real boundary conditions (as input at least). This includes techniques like load stepping, eigenvalue analysis, and the use of geometric stiffness matrices. Since the mathematical solution matches the structural model in this scenario, there is no need to scale the mathematical solution with a "K" factor.

Computational efficiency aside, I believe the the direct analysis method is the more rational method. It can readily account for interesting phenomena like lean-on column bracing etc. For better or worse, code development organizations are clearly assuming that we're computer jockeys.




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

 

[KootK]

Hmm. Please ignore my first post. I'd intended for my second post to replace it, not add to it.

In response AggieYank's comment: "Or, is the logic that it is a more detailed calculation of Euler buckling somehow?"

Yes, that is exactly right. If you looked at purely elastic buckling of a pin-pin column and used DAM without the residual stress modifiers, you'd get nearly the same answer as the Euler equation. And, if you subdivided the member into infinitely many sub members for your second order analysis, you'd get exactly the same answer.

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