Column Unbraced Length Question

[gregeckel]

I have a quick theoretical question.

Let’s assume we have an unbraced column 100 feet long. The top of the column is a roller (translation fixed, vertical free) and bottom of the column is a pin. There is a large point load concentric on the column close to the base (let’s assume about 15’ up).

How do I determine my K and L.

I don’t need an actual solution to the problem just a code reference or the theory is fine. The numbers I gave are just to scale up the magnitude of the idea I’m getting at.

Thanks,
Greg

 

[JoshPlumSE]

I did a quick test using RISA-3D. Assuming that the P-Delta analysis can adequately predict the elastic buckling behavior of the system.

I used a HSS10x10x1/4, 100ft long, simply supported at both ends and divided into 40 smaller pieces (so that I can capture the P-little delta effect).

Then I added in a load of 1/500 of the vertical load at the joint with the applied vertical load. Similar to the notional loads of AISC's direct analysis method which is used to approximate elastic buckling directly in an analysis.

The assumption is that when the P-Delta analysis starts to diverge then the column has buckled. This should be an easy test to perform for anyone with a program that does a good P-Delta analysis.

I found that the elastic buckling of the column occured somewhere between 115.3 kips and 115.2 kips. That would correspond to a KL value of 591 inches or 49 feet.

If I move the notional load up to the mid-point of the column (rather than apply it at the 15ft mark) then the buckling still occurs at the same load level.

 

[miecz]

JoshPlum

I must be missing something. My model diverges at 79 kips, and from that I get a KL of 43 ft. Could you post your .r3d file?

 

[JoshPlumSE]

Here is the RISA-3D file. Note a couple of things:
1) Shear deformation is turned off.

2) I'm not using the stiffness reduction associated with the Direct Analysis Method (that is done to help an elastice analysis program mimic the inelastic buckling effects).

3) I'm not using the 13th edition ASD code (which requires that loads be multiplied by 1.6 before accounting for the P-Delta effects).

Josh

 

[JLNJ]

Josh has a pin at the top instead of a vertical roller.

I get 78.5 kips as well.

 

[miecz]

JoshPlum

Thanks for posting the file. I can't see much difference with my model (attached), except that I had the boundary condition at the top free in the y direction, and I split the column into 40 members. I'm getting divergence at 79 kips.

 

[JoshPlumSE]

That makes sense... It should be a roller rather than a full pin. If I had reviewed the axial force in the column I would have known to change it. Therefore, it appears that the KL value from the RISA analysis really ends up being closer to 59 feet. That's almost exactly what the theoretically derived values were from the other folks.

Note that the formula for euler buckling (without safety factors) as I understand it is Pcrit = pi^2*E*I/(KL)^2. That's the formula I used to convert Pcrit = 79 kips to an equivalent KL of 59 feet.

Josh