Column Buckling - Half in compression, Half in tension 1

[rongerabbit]

Hello everyone,

I've been trying to solve this problem for a while but can't seem to find a solution.
I went through Galambos and Timoshenko but can't seem to find a case that would apply.

I'm trying to figure out which effective length to use for a column that is in tension for half the column and in compression for the other half.
This effectively means that a load upwards of P would be applied at top, and a load downwards of 2P at the middle with no bracing at mid-height.

If anyone has any references that they know of that could be of help, that would be much appreciated.

Thanks in advance.

 

[rb1957]

btw, we're simplifying the real world in that i can't see the real world applying the load along the axis of the beam ... theoretical solution seems to be the consider the beam as two 1/2 length columns (ie effective leneth = 0.5L), sure there's some stiffness from the upper column, but that doesn't play into the euler solution. offset the 2P load, mis-align the 2P load and all bets are off !

Quando Omni Flunkus Moritati

 

[Denial]

BAretired,

You are correct.[ ] As expected (because of the small-displacement nature of a linear buckling solution) the model you defined gives identical results to my setup-2 above.

 

[IDS]

You are correct. As expected (because of the small-displacement nature of a linear buckling solution) the model you defined gives identical results to my setup-2 above.

But in a non-linear analysis (and reality) having the top node restrained in the Y direction will make a big difference, it will prevent buckling. As the lower part of the column approaches the buckling load a greater proportion of the applied load will be transferred to the top support, and the top half of the column, which is in tension, will restrain the lower half. The behaviour will be non-linear, but there will be no buckling until the top column reaches its yield stress.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[SAIL3]

I think IDS has identified the problem with restraining the top node in the Y dir..if restrained in Y dir it no longer behaves as a col in the euler-sense and will not fail until upper portion reaches yield...and if so, I would not really call it a col......getting back to the problem of applying a conc load at mid-height with no bracing at that location....this is a red flag for me...I may not be able to accurately describe my caution on this but here are some nagging questions that will not go away....col buckling is a dynamic event and not a static one....having a conc load at the very point of potential max lateral deflection has an additional destabilising affect on the col that I have no way of calculating....using Newmark's approximation proceedure does not address this and, ofcourse, neither does computer modeling....

 

[rb1957]

if the column had a pin joint in the middle, and the load was carefully applied to both pieces, then this'd look much like an euler column, no? i don't think euler requires there to be lateral stability at the column ends (tho' we know the real world does !)

the extreme model would be a column (like a flag pole) loaded at the free end (fixed at the base), which is a known solution. the real problem is better than this as there is some lateral stiffness and some moment continuity (allowing the "free" end to react with the rest-of-the-world).


Quando Omni Flunkus Moritati

 

[paddingtongreen]

rongerabbit, This looks dangerous to me:

"An example for a design application would be an unbraced beam in a chevron braced bay.
The beam would be braced along its strong axis at mid-point but not along its weak axis. Half of the beam could also hypothetically be in tension and half in compression."

This is a system where the force in the beam is applied by a pair of braces, one in tension, one in compression. You can see continuity between the compression brace and the compression half of the beam. The system needs lateral restraint unless a conservative effective length factor is used.

However, the beam is often in compression, carrying load to the bracing, not the other way around.

Michael.
"Science adjusts its views based on what's observed. Faith is the denial of observation so that belief can be preserved." ~ Tim Minchin

 

[rongerabbit]

Thanks again for all the input.

I would definitely be exercising caution in such a design case but here is an example (allowed by code) where this scenario would occur.

If you're designing a tubular EBF (ie. totally unrestrained along its length except at the ends and at the braces along the y-axis). The code demands a minimum Iy/Ix ratio for design but does not give guidance as to how to design the beam portion outside of the link.
Again, in this instance, there would be a portion of the beam in tension and a portion of the beam in compression (very similar to chevron braces).
In my opinion, it would be too conservative to consider the typical pinned-pinned Euler weak axis buckling load for the entire length but it is hard to justify any other case.

It sounds like designers should not tempt fate in such a design scenario and just design the beam-column by considering the entire laterally unbraced length along the weak axis.