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Column Buckling - Half in compression, Half in tension

Column Buckling - Half in compression, Half in tension

Column Buckling - Half in compression, Half in tension

(OP)
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.

RE: Column Buckling - Half in compression, Half in tension

That problem should be soluble using successive approximations (Newmark's Numerical Procedures together with the Conjugate Beam Method). Assume a deflected shape of the buckled member and keep correcting it until it has satisfactory agreement at several points along the member. Timoshenko has an example in "Theory of Elastic Stability" starting at p. 120.

The effective length is the column length assuming it is pinned top and bottom.

BA

RE: Column Buckling - Half in compression, Half in tension

if no bracing @ mid-height, I would use the full length as the effective length and P as the compression load applied at the top....

RE: Column Buckling - Half in compression, Half in tension

SAIL3, that would be very conservative but not very accurate.

BA

RE: Column Buckling - Half in compression, Half in tension

BA...I am not aware of any exact solution to this problem.....granted, this is my own engineering judgement and not a rigorous theoretical analysis...here is my reasoning..using the full length as the effective length would give me a conservative value as far as effective length is concerned.....now the real problem arises when one applies a conc load at mid-height verses applying it at the top of the column..
1.when the load is applied at the top of col it's final location is fixed/known after the assumed buckling deflection of the column because of the col being braced at that location..

2.when applied at mid-height the location of the load is dependent on the assumed lateral deflection of the col at mid-height and is a so-called "following load" which in itself adds to to the destabilizing affect..so by assuming the load is applied at the top of the col would be unconservative and consequently reduce the conservatism of assuming the effective length as the full length...

I would consider the final result to be more accurate than you claim...ofcourse, you may prove me totally misguided on this....

RE: Column Buckling - Half in compression, Half in tension

SAIL3,
In a practical design situation, I would likely do exactly the same as you suggested, but if I wanted to predict the result more precisely, I would use the method I suggested above. While it may not be an exact solution to the problem, it can be as exact as you wish by simply repeating the cycle as often as required to find the buckled shape of the column.

BA

RE: Column Buckling - Half in compression, Half in tension

In reality, though, what situation would allow you to apply a downward load at the center of the column and yet have no lateral bracing?

Mike McCann
MMC Engineering

RE: Column Buckling - Half in compression, Half in tension

It is a theoretical question. If a member is hinged at both ends and an axial load is applied at mid point, one half of the member feels tension while the other half feels compression. If there is no bracing at the load point, the buckling load can be determined to any degree of precision required.

Whether or not this has practical value is another matter, but there may be some application where it is of value.

BA

RE: Column Buckling - Half in compression, Half in tension

I agree with BA. I think the best way to analyze this problem is with Newmark’s Method.

RE: Column Buckling - Half in compression, Half in tension

A practical application could be a portion of a slender beam web where load is applied upwards to the top flange, and a separate downwards load at the mid-height of the web.

RE: Column Buckling - Half in compression, Half in tension

(OP)
Thank you all for your input.
The question is a theoretical one although I am trying to potentially use it for design applications.

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.

There's very detailed solutions for point loads at mid point where designers can use a buckling length of ~0.8L but none for one portion of the column in tension and the other in compression.
The problem I was having is defining the buckling shape for such a case to arrive at a solution.

Thanks again for all your input; I'll definitely look at Timoshenko's book more closely this week.


RE: Column Buckling - Half in compression, Half in tension

Interesting. I just ran a pair of numerical examples through the Strand7 software. I modelled a standard pin-ended column, with arbitrarily chosen values for length, bending stiffness and Young's modulus.

Setup 1. Downwards load of 1 applied at top. Predicted buckling load factor 15.3 (which is exactly what Prof Euler would have told me).

Setup 2. Upwards load of 1 applied at top, and downwards load of 2 applied at midpoint. Predicted buckling load factor 61.2 (which is exactly four times the prediction for setup 1).

Something in my bones tells me that this is not a coincidence, and that there is some simple theoretical truth underpinning what is going on. Maybe a few lines of algebra will shine a light on it?

RE: Column Buckling - Half in compression, Half in tension

Algebra: 1/2 length now in compression. Buckling based on 1/(L=.5)^2 = 4

RE: Column Buckling - Half in compression, Half in tension

Yes, so it appears.

But your half-length column is not a pin-ended column as per Euler.  It is partially free to sway at its top, and that freedom to sway is partially limited by the tension in the half-length member above it.  What I want the algebra to show is why these two partial effects exactly cancel each other out.

RE: Column Buckling - Half in compression, Half in tension

I just ran my über-simplistic Strand7 model again.

Setup 3. No load applied at top, and downwards load of 1 applied at midpoint.  Predicted buckling load factor 28.9 (which probably doesn't help us all that much other than to rule out the unlikely possibility of it being 30.6, which would have been "a coincidence too far").

RE: Column Buckling - Half in compression, Half in tension

What boundary conditions did you actually apply at the two ends of the column, other than pinned ends? I think the L/2 of the bottom half of the column comes into play as WillisV suggests. And, at the mid height where the 2P load is applied downward, the column is partially restrained from moving vertically, by the tension P/A above it, this should increase the buckling load. At the mid height the column can only move down or laterally to the extent that (P/AE)(L/2) = δ will allow. And, until the upper half actually starts to yield, so δ can really grow, the lower half will probably not buckle. Even then, the upper half of the column will offer some moment restraint at the mid height as a function of EI/(L/2).

RE: Column Buckling - Half in compression, Half in tension

LPS for dhengr, it is a zen like question - what happens if a column buckles but nobody can tell? The upper column still supports the load in tension. If you were to plot deflection vs load applied there would be a step change by a factor of 2, or something approximating it, as the lower part buckles.

In practice the system would also become laterally soft at the same time, which in real trellis type frames is the issue.

Cheers

Greg Locock


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RE: Column Buckling - Half in compression, Half in tension

An elaboration on my über-simple model.  Two dimensional frame analysis.  X horizontal and Y vertical.  Node at very bottom restrained against X and Y translations, but fully free to rotate.  Node at very top restrained against X translation only, and also fully free to rotate.  Node at midpoint unrestrained in any way.  Beam elements (2x8 of them) running continuously from the very bottom node to the very top node (or it might have been from the top to the bottom smile ).

In other words, to mis-quote Gilbert & Sullivan, "it was the very model of a modern Euler column".

RE: Column Buckling - Half in compression, Half in tension

Denial,
Another way to model it which should produce the same result: Node at very bottom restrained against X and Y translations, but fully free to rotate. Node at very top restrained against X and Y translations and also fully free to rotate. Node at midpoint unrestrained in any way. Downward load of 2P applied at midpoint. No other load applied.

Beam elements as before.

BA

RE: Column Buckling - Half in compression, Half in tension

I'd go with the method of successive approximations described by BA. There is an example given in AISC's design guide on tapered members. It works well for stepped columns or columns with discontinuous loading or such. Therefore, I would imagine that it would work for your case.

RE: Column Buckling - Half in compression, Half in tension

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

RE: Column Buckling - Half in compression, Half in tension

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.

RE: Column Buckling - Half in compression, Half in tension

Quote:

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/

RE: Column Buckling - Half in compression, Half in tension

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....

RE: Column Buckling - Half in compression, Half in tension

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

RE: Column Buckling - Half in compression, Half in tension

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

RE: Column Buckling - Half in compression, Half in tension

(OP)
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.



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