Does FEM analysis correctly model buckling? 1

[JAE]

For finite element models, say RISA or some other program, where you have a structure under various loadings, does the PDelta analysis performed by these programs properly estimate buckling?

We've been having this disussion in our office and one view is that the PDelta certainly estimates the second order effects, but that this is not the same as Euler buckling.

The other view is that Euler buckling is simply a derivation of second order effects that uses an assumed out-of-plane initial distortion to get second order effects started. So with a finite element model with forces in two or three global directions you will, by nature, have the second order effects started and the buckling load will be at least approximated.

What do you think?

 

[Lion06]

JAE-
I don't think that the FEM programs are intended to capture buckling in that manner. The Euler buckling load is for a perfectly straight column.

I tried, with an FEM program - RAM Elements (Advanse, at the time) - to capture this behavior and I just couldn't mimic it.

Since the Euler buckling load is for a perfectly straight column such that any disturbance will cause buckling upon reaching the critical load, I had two choices in my mind.

1) I could model a straight column with a small lateral load at mid-height, or
2) I could model a column with a slight curve.

I chose option 2, and modeled a column with a half-sine wave shape. I made it like 20' long and broke it into 1' segments. I selected a column shape that should buckle elastically. I then loaded the column near the buckling load to see what kind of behavior I would get. I expected to see lateral displacements that would begin to really increase significantly as I approached the buckling load. What I found was that there wasn't a sudden increase in the lateral displacement at (or near the buckling load), but rather a more gradual increase. It wasn't linear, but it wasn't as sudden as I thought it would be.

What really threw me for a loop was when I got above the buckling load, it actually displaced in the opposite direction. I still haven't figured that one out.

At no time, however, did it fail to converge, which was also a little confusing. I didn't run this test on a lot of different column sections, heights, etc, so it's possible I could have got different results under different conditions.

To get back to the actual question......... I'm not sure I understand the question exactly. We typically use the buckling load on the design side, right? The analysis kind of is what it is. You get moments and axial forces and compare that to the capacities (which account for the possibility of buckling).

 

[JoshPlumSE]

JAE -

The answer to your question is yes and no.

Yes, in the sense that an elastic P-Delta analysis (if done properly) will generally capture the member's buckling or the frame's buckling.

No, in the sense that there are limitations related to modeling and results.

1) First, as Lion6 points out, there has to be some form of initial displacemnt. Either a built in out of straightness or some form of lateral load applied to your model.

2) Second, in order to capture individual member buckling you need to capture the P-Little Delta effect in addition to the P-Big Delta effect. This will often require that the member be sub-divided with additional nodes along the length of the column.

3) The results from an elastice FEM program (like RISA) can really only capture the ELASTIC buckling effects. The good news is that AISC's Direct Analysis Method (which is a pseudo-elastic method) contains adjustments to the elastic analysis which will allow it to approximately capture the inelastic member buckling.

Below is a link to a download from the RISA website. In the link is a set of "benchmark" problems from the AISC commentary which are used to determine if you analysis can adequately capture the elastic buckling effects (i.e. p-little delta). As you can see in the presentation, the RISA results match very well... but only if you sub-divide the column.

http://www.risatech.com/documents/pdfs/RISADirectAnalysis.zip

Hope that helps!!

 

[ishvaaag]

I have made an attempt to aproximate some biarticulated member 10 m long, square tube 100 mm outer face side, 6 mm thick, Fy 275 MPa in RISA 3D version 9.1. In this attempt I have divided the length in 4 parts, and imparted a parabolic initial imperfection resulting in the central node being 1 cm from theoretical axis and the 2 following nodes, 0.75 cm. Within this frame, you make the load bigger and RISA follows the amplification of the deformation till that the center is displaced -in the same side than the initial imperfection- over an extra 24 cm, at which moment the unity bending check, set in this model as per LRFD 1999 attains a value 1. At this moment, the standing limit load as per the check is 7.15 tonnes, where the Euler buckling load is 6.91 tonnes.

Seeing that we have used 4 elements only, we have used an initial imperfection, and RISA acknowledges other deformations, particularly axial deformation, uses K=1 on segment lengths of 2.5 m for the check (quarters) it seems not initially a bad match.

For the elastic buckling, if you substitute the 0.877 by one, and do not reduce the axial limit, you also get 6.91 as the limiting axial load. So again, under that concept of the coefficients, a close match.

Yet I am not entirely satisfied since both the 0.877 in the formulation should figure instead of 1 and a reduction factor 0.85 to give the limit for the check, i.e., one would expect a lower limit load for the 1999 LRFD check, and hence a lower limit load in my model.

I upload an image then zipped model.

 

[ishvaaag]

A RISA 3D 9.1 model.

 

[JoshPlumSE]

Isvaag -

Keep in mind that a bending code check of 1.0 does not correspond to buckling.... When you approach the buckling limit your code check should become "infinite".

Also, in order to nicely match the "theoretical" value you would need to turn off shear deformation on the Global Parameters.

You do both of these things and you should get very close to the buckling values. Though the program will always start to "diverge" on the P-Delta analysis slightly before the actual Euler buckling value.

 

[JAE]

Thanks for the replies.

Lion06 - Euler buckling would require an intential out-of-plumbness so that was basically my question - with an out-of-plumbness in the model already, would buckling occur.

Josh - I did the benchmark problems (found in the back of the AISC manual) on RISA some years ago and found that (as you state) they do indeed capture the little delta effect.

My particular project involves a stainless steel sculpture about 17 feet tall - a somewhat irregular shape. So we modeled it in RISA.

Attached is a stress plot of what we were working with.

Since it has all sorts of planes and curves at odd angles to xyz we felt that there was already an intential out-of-plumbness of sorts to begin with.

I see what you are saying about the inelastic effects in the Euler buckling. RISA would assume elastic. (so when will RISA get some inelastic analysis developed? Hurry up!! )

 

[ishvaaag]

I have found a hint of answer for that in that the member is as slender that exceeds one valid good for compression, hence no check "per the code". The division in segments overcomes that and the program gains the ability to check, but at the shorter segments at which the slender equation may even not govern. I may elaborate this more later.

 

[ishvaaag]

By whatever the reason I have re-posted the same RISA model file. DIsregard it in the last post.

 

[BAretired]

JAE,

That sculpture looks like a parachutist's nightmare.

BA

 

[ishvaaag]

A small work claryfying what happens for one example with the P-Delta treatment within RISA-3D for 2 codes, AISC LRFD 2nd 1999 and AISC 360-05.

It is the same slender member than above but with 6 m length.

Both the code and RISA 3D show substantial consistence in the results, even if the statement of what to do may be done better.

 

[JoshPlumSE]

The key here is that ths is done entirely with plate elements. The P-Delta effect is not currently considered for plate elemetnts in RISA. RISA does it for linear elements and for wall panels. But, it doesn't do it for plate elements. Therefore, you cannot capture elastic buckling for these types of plate models.

 

[Ron]

Josh probably knows more about this than any of us, but I'll throw my $0.02 in...

In my experience, FEA only captures the precipice of buckling, not the process. Once it starts, all bets are off.

 

[JAE]

JoshPlum,
I thought that the PDelta was simply a routine in the matrix solution that cycles through independantly of the type of element (i.e. two node "beam" vs. multi-node plate).

How does RISA differentiate between doing PDelta in a model that has both plates and beams? That doesn't make sense. The matrix solution is dealing with joints and their mathematical relationship between themselves (in terms of stiffness against relative movement).

 

[ishvaaag]

The confirmation of what JoshPlum asserts by example.

RISA 3D does not account for P-Delta in plate elements even when the option P-Delta is marked.

How to ... surely is a matter of what enters equilibrium and not in the FEM setup. FEM is all mathematical manipulation, that's part of the force/nature of the method.

 

[SAIL3]

From what I know(or think I know), FEA only indicates which members that have reached a stress level that would indicate buckling according to the criteria of whatever code you are working with.
More importantly, it does not redistribute loads after a member
has buckled...one has to manually remove that member from the model(make it inactive).
FEA is a very good number-cruncher and rule-checker but in no way
is it a substitute for engineering judgement and experience.

 

[ishvaaag]

SAIL3, what you say is the normal scope of programs targeted to practical design the structures. I have commented on RISA because it is one of those that I have in such intent, and the question was somewhat targeted as well on if were the structural design programs calculating the thing well. If they perform column checks they must be doing so in the codified way, otherwise are defective on that matter.

There are however FEM packages very well adapted to find the buckling load of some structures in accord with the constitutive laws of the materials, restraints and other conditions that may apply, irrespective of some code formulation.

 

[JAE]

SAIL3 - FEM analysis in programs that have PDelta-built-in functions will solve multiple matrix solutions - iterating by relocating the joint coordinates to new, deflected positions (based on the solution preceeding it) and then re-running the analysis with the structure starting in the newly deformed position.

The question, I think, is: does the buckling occur prior to a portion of the model reaching an inelastic state, or can PDelta "buckling" occur - a diverging of the model - prior to inelasticity.

I thought the initial divergence, under Euler buckling, wasn't inelastic as JoshPlumb above suggests. I agree that once it begins to buckle, it eventually will reach and exceed yield stress. Just not sure if Fy must be exceeded prior to model divergence.

 

[ishvaaag]

As I think on the matter, I think there are instances where the stiffness reduction can come from members buckling elastically, so for these cases JAE is right. You can have an elastically buckled column returning to shape retiring the load, still, it has diverged before.

 

[SAIL3]

JAE-I do not think that elastic buckling or inelastic buckling are concepts that a FEA program can understand. These are a result of engineering observations and understanding. The program
interprets the results based on a set of rules dictated by the applicable code. It can determine wheather it fails or is still ok
based on this criteria.
Beam/Col.....the program can easily handle the P-little Delta and after applying the loads calculate the deflections. Then it can
iterate the loads on this deflected shape thus covering P-Big Delta and check if the members are still ok.
That's as far as it goes...if the structre was initially stable, i do not believe that it will say it is now unstable because a certain member failed.....the engineer has to address the failed member himself. There is no post-buckling analysis.
Plate elements....an entirely different animal altogether.
Most of the rules pertaining to bm/cols dictated by the codes do not apply here, especially concerning stability.
So the engineer has to get really involved in interpretating the results to determine if they make sense.
Using RISA's plate elements in your model may give you a general idea of the resulting stress levels.
I always check the resulting stress levels and try to understand how they got there. Your model shows one area of peak stress which in reality may not exist as plate can easily redistribute loads as one area begins to yield.
The biggest pitfall in plate strucures are in-plane compressive stresses. So to that end I would add the following comments on your structure:
With the following assumptions....no interior steel skeleton and plate probably made out of gage ss.
Loads will gravitate towards the stiffest areas of the structure...so all the verical corners will carry the majority of the loads.
This leaves one with a skeleton of corners with a fill-in of plate that is just delivering the load to the stiff areas.
Determine axial loads in corner based on this assumption.
Check corner as an angle compression member..how big a leg?...use b/t limitations from aisc code...are these angles(corners) braced by the remaining pl?...hmmmmm..probably not.
Anyway, that is a general approach that I would look at.
JoshPlum has alot more credability and in-depth knowledge of the RISA program than I do, so I would tend to pay more attention to his remarks than my long-winded comments.