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.

 

[BAretired]

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

 

[SAIL3]

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....

 

[BAretired]

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

BA

 

[SAIL3]

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....

 

[BAretired]

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

 

[msquared48]

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

 

[BAretired]

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

 

[dhengr]

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

 

[gobsmacked]

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.

 

[rongerabbit]

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.

 

[Denial]

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?

 

[WillisV]

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

 

[Denial]

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.

 

[Denial]

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").

 

[dhengr]

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).

 

[GregLocock]

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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[Denial]

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[ ][ ]).

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

 

[BAretired]

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

 

[JoshPlumSE]

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.