A few basic theory questions about buckling

[jrw501]

I'm sure these will be trivial to many of you, but...

1. Will buckling occur in slender columns (steel is probably easier to think about for me as an example, but I guess infinitely well confined concrete or something may work too) even if the column itself is ideal (no imperfections, is plumb, etc...) and the load is applied ideally through the centroid? Or will it just shorten elastically then plastically undergoing poisson effects?

The second one relates more toward visualizing postbuckling strength (but also buckled sections in general).

2. If a plate girder's web buckles, is the stress that was being applied before buckling still acting on the web, or is it assumed that all/some of it is shed to the compression flange increasing the stress there? I guess specifically one thing I'm thinking about is in relation to the Dc term in AASHTO and what the depth of the web in compression would be if we assumed it buckled.

Sorry if these are terrible or somewhat nonsensical questions, these aren't for any projects or anything I'm working on as an EIT, just stuff I've been thinking about that I've never heard discussed or felt fully confident about. If you think it may help me to ask tangential questions that may improve my understanding, that'd be great too. Thanks.

 

[Lion06]

1. This is a purely academic question since no column can exist without some imperfection in load application or manufacturing. If, however, you could make those conditions real then yes, it would only shorten elastically until it reaches yield. If the section is completely homogenous and all fibers reach yield at the same point then no post-elastic buckling would occur either. The point you're talking about is called the bifurcation point. It's the buckling load where it can buckle either direction (that direction is determined by imperfections or applied lateral load) or the academic solution that you're inquiring about here.

2. I'd be very interested to hear others' opinions on post-buckling behavior and strength. It would make sense to me that a column that buckles elastically has greater post buckling strength than a column that buckles inelastically simply because the E hasn't begun to reduce yet as a result of softening from yielding of the section and therefore is more stiff (flexurally).

 

[jrw501]

Thanks for affirming my thoughts for the first one, I realize in practice there's no way to simulate these conditions, but I have never been 100% sure that buckling wasn't a phenomenon.

Every time I've seen the buckling equation derived it's been with an assumed eccentricity which in turn I've always assumed was due to practical material or geometric/fabrication imperfections, but I don't think the cause (perhaps -- a cause) for the eccentricity was ever mentioned in classes I've taken. It was more or less just, "let's find the axial load required to hold this column at some deflection y."

 

[IDS]

To add to the first one, it's not just a matter of manufacturing flaws. Any finite deviation from perfect coincidence of the the centroid of the section and the centroid of the applied load would be sufficient to instigate buckling. Thermal vibration of the molecules of the column would be enough, and if you reduced the temperature of your section to absolute zero it would still buckle because of Heisenberg's Uncertainty Principle!

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[dik]

Euler buckling occurs without imperfections. You assume a slight deformation, but at the end of the day, it does not become part of the formula. For really slender columns, the buckling load is almost part of the exact Euler solution.

Dik

 

[GregLocock]

All Euler states is that a slender column can support the critical load in equilibrium in one of two ways, either in axial compression, or by bending. That's where the maths stops.

In practice it is found that a sufficiently good slender column will buckle at a load similar to that critical load.



Cheers

Greg Locock


New here? Try reading these, they might help FAQ731-376

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[rb1957]

Euler was a mathematician and didn't deal with the real world (but an ideal mathematical one). as such he showed that an ideal column will buckle.

for 2) i'd say the web is stabilizing the compression flange ... if the web buckles 1st, i'd expect the compression flange would probably collapse (or cripple). of all things are possible, say a very thick compression flange and a very thin web ... the web could buckle (read up on on-set of elastic buckling) and the additional load would be carried by the heacy stable flange (the web would continue it buckling load)

 

[IDS]

Anyone interested in the theoretical side of it might enjoy my article linking Euler's column buckling equation with the period of travel through a hole through the middle of the Earth:

http://newtonexcelbach.wordpress.co...and-the-hole-through-the-middle-of-the-earth/

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[paddingtongreen]

A column that buckles elastically is unchanged and will return to it's original shape when the load is removed.

"If a plate girder's web buckles, is the stress that was being applied before buckling still acting on the web" Is it buckled under a load or at the support, I don't think there is a generic answer.

Michael.
Timing has a lot to do with the outcome of a rain dance.

 

[dik]

Sorry Greg, my statement's still correct... not too shabby for a 1700's dude... He was also the initiator of the mathematics currently called topology. To reiterate... for slender columns, his buckling equation is nearly 'bang on'. The scatter at the intermediate column critical loading is mostly due to imperfections, internal stresses and that sort of ilk.

His mathematical solutions to problems of the real world are 'real'...

Dik

 

[Lion06]

I don't know that I agree that Euler buckling occurs without imperfections. Without some initial distortion (imperfection) there is zero bending. The Euler equation is the load at which a perfectly straight column will buckle if any kind of imperfection is introduced. You can satisfy the solution to the differential equation with
kL = Npi (which is they typical buckling behavior), but it can also be satisfied with delta = 0 (this is a trivial case, obviously, but it is the perfectly ideal case as questioned above with no imperfection whatsoever in member or load application). The delta = 0 solution is in unstable equilibrium, but equilibrium nonetheless.

 

[Lion06]

The scatter in non-slender columns is due in large part to residual stresses and material softening (reduction of E) due to material yielding.

 

[jrw501]

"If a plate girder's web buckles, is the stress that was being applied before buckling still acting on the web"

Is it buckled under a load or at the support, I don't think there is a generic answer.

I guess I was thinking about the web buckling under a load and about the ability of the section to still carry load if the web buckles.

I also agree with StrEIT's assessment of my first question and after spending some downtime today reading about the historical evolution of buckling calculations it definitely seems like one of the assumptions required is that at some point an unideal factor is introduced (if anyone is interested the paper was an ASCE J. Struct. Engr. paper called "Column Buckling Theory: Historic Highlights" by Bruce Johnston -- I'm not sure on the rules with respect to posting it).

As a side note I just wanted to say thank you to everyone that posts here (including many that don't appear in this thread like JAE, BAretired, and others). I have been sporadically reading this forum for a few months now and as a young engineer who just started working within the last month or so I find many of these topics (even though I'm presently working for a bridge design firm) really interesting / thought provoking.

 

[rb1957]

geez, we don't want to go around provoking thoughts,
now do we ... i mean they can bite ya

 

[BAretired]

1. Will buckling occur in slender columns (steel is probably easier to think about for me as an example, but I guess infinitely well confined concrete or something may work too) even if the column itself is ideal (no imperfections, is plumb, etc...) and the load is applied ideally through the centroid? Or will it just shorten elastically then plastically undergoing poisson effects?

Like any unstable structure, a slender column will fail when gravity load reaches the critical buckling load. Even if the column is presumed ideal, the environment is not. A slight air movement is enough to cause lateral deflection. So is the pull of the moon or other heavenly bodies.

2. If a plate girder's web buckles, is the stress that was being applied before buckling still acting on the web, or is it assumed that all/some of it is shed to the compression flange increasing the stress there? I guess specifically one thing I'm thinking about is in relation to the Dc term in AASHTO and what the depth of the web in compression would be if we assumed it buckled.

Can't answer briefly. Check these links for more detail:

http://www.google.co.uk/search?q=te...=en&sourceid=gd&rlz=1Q1IRFA_enCA356CA356&aq=t

BA