Nonlinear Column Buckling

[spacecase]

Greetings all,

I need to tighten up my buckling analysis and would like to head into the nonlinear realm. I am basically dealing with an eccentrically loaded column. The compressive loads applied to the column are not enough to yield the material, so I should be in the elastic range. Running NASTRAN linear buckling with fixed free conditions yields a Pcr of 54lbs. Using the Secant formula to take into account eccentricity, I have a value of 12lbs.

How would you all go about running this in a nonlinear analysis? I know there is a nonlinear eigenvalue buckling solution that can be done. I’ve also heard that you can run a nonlinear static run and see where it diverges.

I tried the nonlinear static run up to 60lbs and did not see a divergence. I incremented for every 0.5lbs in that run.

I am a newbie analyst and really love the knowledge sharing that goes on here! Thank you all.

 

[GBor]

I'm a little confused, but must admit that I am not a buckling expert. If, however, your load is insufficient to yield the material, non-linear analysis really shouldn't result in any difference from a linear static analysis. Non-linear analysis, depending on your material model, will follow the stress-strain curve, but this curve is linear prior to the yield point...am I missing something?

 

[TGS4]

GBor, spacecase is referring to a geometric non-linear buckling analysis. The best piece of advise that I can give to you is that you need to introduce a small (the fun part is that you have to determine what constitutes small) perturbation into the geometry in order to initiate the buckling failure. You are, however, on the right track that you will basically run the analysis until it fails to converge. My trick for figuring out what sort of perturbation to use is to base it on the linear (eigenvalue) buckling solution.

Feel free to come back with more questions, although I am pretty sure that this topic has been discussed before in the forums.

 

[rb1957]

like GBor, I'm confused with your post.

"fixed free" end conditions ... 1 end fixed & 1 end free, or pinned; thw rod "free" could be redundant if you mean both ends fixed.

does the NASTRAN 54 lbs relate to the secant calc'd 12 lbs in any way? possibly you mean that the euler buckling load is 54 lbs and adding the offset to the loading conditions reduced this to 12 lbs. The offset loaded column isn't a traditional euler failure, in that there is a moment in the column from the start, which justs increases to a critical value; thus it is easier to analyze. but the problem is still non-linear (geometrically as well as materially).

if you tried a non-linear NASTRAn run and got good results for a load of 60 lbs, when you were expecting failure at 12 lbs, then something is very much amiss. one thought would be to model a simple column with a very simple material, something you can hand calc, to make sure you're modelling what you think you are.

good luck

 

[crisb]

What sort of elements are you using, beam, shell, 3D? You need a fine enough mesh to describe the buckling shape. I have seen people try to do a buckling analysis on beam elemnts without any subdivisions!

 

[rb1957]

i disagree with TGS4, for what that's worth !

i don't think you need to introduce a flaw into your strcuture, since with eccentric loading your column is starting to deflect from the get-go (unlike an euler column).

 

[spacecase]

crisb,
The model is made up of a fine mesh of tets w/ midside nodes. It is a flexure that has some bulk added to the base for stress control.

rb1957,
Fixed-Free boundary conditions refers to fixing the base of the column and leaving the top of the column unconstrained. This is the most conservative boundary condition for buckling. The NASTRAN 54lbs runs a long the same lines as the Euler buckling solution, and is in the ball park of the hand calc. The Secant formula is a nonlinear hand calc that takes into account eccentric loading. I do not believe you should rely on Eulers once eccentricy comes into play. So right now I am relying this 12lb value for the critical value, but its causing my margins of safety to border on the negative side of things. And of course there is not a desire to alter the design.

TGS4,
I remember reading some of the past posts that mentioned adding a perturbation, but I guess that it wasn't clear to me what that was exactly.

I am assuming this a small transverse force applied at the loading point? Is it common practice to apply this force or is it something for only small eccentricities? What kind rule of thumb percentage of the linear critical load do you look for?

As far as application in the nonlinear run, do I apply this force in the first load step, then remove it in the second load step? Big Thanks!!!

 

[rb1957]

this is a site i pulled up for eccentrically loaded columns ...

http://www.stanford.edu/class/me111/handouts/lecture_14.pdf#search='columns eccentric load'

i'm sure you know all this ...
be carefull applying some of the results in your specific application. the "secant column formula" which is the last result derived assumes a linear stress field. if you wanted to use this you'd need to calculate an equivalent cozzone stress to account for material non-linearity (if that's what you're after, remembering that this problem is geometrically non-linear too). this site gives predictions for the deflected shape of the column which should be applicable to your case, but i wonder if these are applicable in a partially plastic column (maybe not).

still i'm confused by the differnet laods you quoted in your original post, and how they relate to one another.

 

[spacecase]

rb1957,

If this situation had to take in account yielding of the material, the elastic modulus used in the secant formula should be replaced with a tangent modulus. Fortunately, I can avoid this because everything should still be in the elastic range.

The NASTRAN 54lb load ignores the eccentricity all together and assumes the load is at the center of the column, then applies Euler theory. That is why it is much higher than when eccentricity is taken into account. I beleive it is typical to first get the Euler, linear buckling load before heading into a nonlinear analysis.

 

[rb1957]

then the NASTRAN 54 lbs is the euler buckling load, which you have very sensibly checked with with a hand calc.

one end fixed, the other free is a pretty conservative model, much like a flag pole.

The Euler load is still relevent to the secant solution, you can reformulate the expression to Mmax = Pe*sec(pi/2*sqrt(P/Pcr)) and follow this through to the stress expression. I think the advantage of this expression is that it incorpates the effects of end conditions; in your case Leff = 2*L.

still don't understand how your non-linear NASTRAN didn't detect failure closer to your hand calc of 12 lbs, but then that was your original question.

i'd stick to the hand calc, and if you're getting -ve MSs then you'll have to change something !

 

[corus]

You probably need to include large displacement in your analysis to include non-linear effects without yielding.

corus

 

[spacecase]

Corus,

I do have large displacement included. I am thinking pertubing the structure is will do the trick. However, from the sounds of things, this may be something of an art.

 

[rb1957]

is your model recalculating geometry to account for displacement (geometric non-linearity) ?

"large displacement" (as far as i know) changes the stress formulation of the elements ('cause samll displacement assumption no longer holds) but doesn't change the shape of the structure (which you have to in your case).

i really don't think you need to perturbate (is that a verb?) your model, since you're not looking for the euler load.

 

[Kwan]

Is your shape subjected to crippling or is this a solid bar? What is the shape of your cross section?

 

[spacecase]

It is a rectangular cross section: 10mm x 1mm. The length is a little over a 1.25" With the the fixed-free end conditions, the slenderness ratio is 224, should be well into the Euler buckling range. The material is Invar.

The nonlinear elements are given large displacement effects which include updated element coordinates and follower forces. I can't say I am too familiar yet with what the MSC Nastran code is doing behind the curtain. Maybe someone can chime in on that one.

 

[corus]

I'm surprised that you are able to arrive at a fixed Pcr with large displacement as the solution is asymptotic, ie. with an increasing load the displacements will increase to some large, but unknown value for a load. In that sense it's not eigenvalue buckling but a simple load versus diplacement plot which you can only estimate the buckling load. In that regard I don't think euler's method works with large displacement solutions. Euler's method is not conservative and will give a lower estimate of the buckling load. A large displacement analysis with increasing load will give a better estimate.

corus

 

[GregLocock]

rb1957, I think that depends which non linear code you end up using. For instance our crash models (and for that matter my stabiliser bar models) do change the shape of the structure as they load up.



Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[spacecase]

Corus,

The fixed Pcr I stated was from a linear buckling run, which would be Eulers. You are correct, the nonlinear run provides only load and displacement data. I agree that Eulers will not be conservative in this case. That is why I turned to the Secant formula and am trying to get some kind of verification through the nonlinear FEA. Unfortunately, it seems everyone in my office seems to avoid nonlinear like the plague, so newbie support is limited in this area.

 

[rb1957]

boy this is going on, and on ...

i'm surprised that corus thinks that euler is unconservative, i'm assuming that the OP either extracted the eigen value for his column, centrally loaded. with his very high L'/rho (200+) this is pretty certainly an euler column. then using the secant formulae (for his eccentrically loaded column) he's calculated the maximum load that can be applied (given a whole bunch of assumptions).

then he tried to model the thing (how does the split enz song ... "that was my mistake" ?), and doesn't get the same answer. i'd put my money on the hand calc. btw, which direction did you apply the load in the NASTRAN model ... your euler load assumes buckling on the weak axis, and your strong axis is about 100 times stronger.

 

[spacecase]

A compressive load that causes the structure to bend about the weak axis. In a seperate analysis, I've applied a lateral load in the strong axis direction to look at lateral-torsional buckling.