Buckling un-braced length of uniformly loaded axial member

[dougseason]

Hi all,

I did a search and didn't find any discussion so I'm starting this new thread. Hope I'm not creating a duplicate topic. For buckling un-braced length of a column with point load at the top, it is clear that the un-braced length equals the length of the column (assuming pin-pin top and bottom). My question is what is the un-braced length when the column is loaded uniformly throughout it's length and not a point load at the top? The axially force & stress diagram would increase linearly. Intuitively, I sense the two loading conditions would affect the buckling behavior of a column differently but I'm not sure how to take the uniform axial load into consideration in an analysis. I understand a conservative assumption to treat it as a point load at the top can be made.

Please chime in.

Thanks

D2

 

[CANPRO]

If you have a copy of "Theory of Elastic Stability" by Timoshenko and Gere, this topic is covered nicely in Chapter 2. If you don't have a copy, I highly recommend getting one.

 

[swearingen]

I second CANPRO's reply. Also, how, exactly, are you getting an axial load applied like that? Just curious...



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[TheDW]

I've checked beams like this before when they're supporting grating panels that wouldn't act as a diaphragm...seismic weight of the grating goes to each joist, each joist applies a seismic axial point load to its girder, check the girder as a beam-column.

 

[SAIL3]

see post dated 15Nov 2010..COLUMN EFFECTIVE LENGTH....

 

[JAE]

Applied loads do not, in themselves, brace things.

However, if the applied load is coming into the column via some structural element that has strength and stiffness capable of bracing the column either nodally or relatively to some other stable structural fixity, then it may indeed brace your column.

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[CANPRO]

JAE, completely agree with your statement but I think the OP was asking a different question. I think what dougseason was getting at is this:

[ul]
[li]Typical buckling load of a column is determined assuming that a point load is applied at the top of the column, and axial stress is uniform through the length of the column (self-weight neglected)[/li]
[li]The end conditions effect the k value, if pin-pin k=1.0[/li]
[li]If the same column is loaded uniformly along its length, the load increases as you approach the base of the column - in this scenario the column can support more load before it buckles[/li]
[li]If you take the total load supported by the column in the 2nd scenario and then back-cacluate for the k value using traditional methods (assuming all the load is applied at the top of the column) - you'll get a k value < 1.0[/li]
[li]The OP was wondering how to calculate the effective k value if the column was uniformly loaded vs point loaded at the top[/li]
[li]Terminology is important here - the unbraced length of the column isn't changing, the loading conditions are changing. If you want to relate this change in capacity to traditional methods and assumptions, you end up with a k value that changes with support conditions and loading conditions. Since we're used to calling the product of k*L the effective length, its easy to mix up the terminology[/li]
[/ul]

 

[JoshPlumSE]

This question is a good one and a bad one at the same time.

It's a good question because it points out how silly the concept of effective length really is. This is a good example why AISC has moved more towards the Direct Analysis Method, which does a better job of capturing the geometric non-linearity of the structure. Now, for use in the Direct Analysis method, I would say that the effective length is really the same for the loaded at the top and the uniformly loaded method.

It's a poor question if you're looking for this effective length to tell you a lot about the structure the way it does for a pin-pin column with a point load at the top. There are, however, ways to calculate the theoretically correct euler buckling load load for a column loaded this way. Asking the difference between this load the euler buckling point load is probably a better question. The AISC design guide on tapered members gives a method in an appendix that works for tapered members under various loading conditions. This is the method of successive approximations. I believe it could be used easily for this example to solve for the euler buckling load for a given column, length load arrangement.

 

[Eagleee]

Ref Teknisk Ståbi p.130, an pin-pin ended axially loaded column with 'alpha x N' axial force at top and 'N' at bottom (varying linearly, alpha < 1) the critical buckling length is 'l x sqrt((1 + 0.88 x alpha) / 1.88)', where 'l' is the length of the column.