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# Nonlinear buckling occurs earlier than linear buckling?

## Nonlinear buckling occurs earlier than linear buckling?

(OP)
When doing a nonlinear buckling analysis (where you have geometric and material nonlinearities), why do nonlinear effects usually happen at lower loads compared to when doing a linear buckling analysis?

Edit: Trying to understand this comment: "Yes, i.e. in plate bending LBA will see a lot of "fake" buckling (since it cannot see membrane state), and it isn't there - so technically the nonlinear analysis in such case would give a better outcome than LBA.
In shells, that would be rare, and I would be careful - it is possible to "miss" buckling and calculate your model in an unstable equilibrium in FEA if you are not sure what to do.
All the best, and good luck!" from this video: https://www.youtube.com/watch?v=EcfswZ0qJws.

### RE: Nonlinear buckling occurs earlier than linear buckling?

Linear buckling represents the theoretical maximum capability (representative of the slightest imperfection). Any imperfection that is greater than "the slightest" (i.e. a typical nonlinear analysis) will result in a lower buckling load (at least for the initial bifurcation - which is not always the same at the maximum load capability).

### RE: Nonlinear buckling occurs earlier than linear buckling?

(OP)
So LBA sees when the slightest imperfection occurs and says: "here we have buckling load", but nonlinear buckling waits for an imperfection which is a bit bigger and says "no here we have buckling load"? I just dont see how bigger imperfection results in lower buckling load, unless bigger imperfection means less stiffness in the structure, hence the lower buckling load in the nonlinear analysis?

### RE: Nonlinear buckling occurs earlier than linear buckling?

Try to drive a nail that has a slight bend in it - It's not all that much less stiff axially, but that bend allows positive feedback to the deformation; in the case of a straight nail that same feedback is present, but with an initial multiplier of zero, reducing the adverse outcome.

I doubt that nonlinear buckling analysis "waits" and it's more likely there is an initial offset that is purposely added to the geometry by the analysis software to force the initial failure.

### RE: Nonlinear buckling occurs earlier than linear buckling?

Are you familiar with a beam-column analysis? If not, look it up. In essence, the the magnitude of the moment in the beam-column is analogous to an increasing imperfection.

### RE: Nonlinear buckling occurs earlier than linear buckling?

> I just dont see how bigger imperfection results in lower buckling load, unless bigger imperfection means less stiffness in the structure, hence the lower buckling load in the nonlinear analysis?

Why is it hard to imagine that? An empty cola can with a perfect shape can carry much more weight than one with a little dent in it.

### RE: Nonlinear buckling occurs earlier than linear buckling?

(OP)
> I doubt that nonlinear buckling analysis "waits" and it's more likely there is an initial offset that is purposely added to the geometry by the analysis software to force the initial failure.

This would make sense, but there are no initial imperfections in my nonlinear model. I am using COMSOL and it does not add geometric imperfections automatically, unless you are talking about something else. Is this the only explanation why nonlinear buckling might occur at lower load than linear buckling? - that nonlinear model has some initial imperfection akin to the empty cola can with a den in it.

To give some context; I am modelling a representative volume element (RVE), which is the smallest volume over which a measurement can be made that will yield a value representative of the whole, say for example a beam. I am using Periodic boundary conditions on the RVE, meaning the boundary conditions are repeating forever. There are no imperfections. I have been told nonlinear effects would occur earlier, but im not sure why.

### RE: Nonlinear buckling occurs earlier than linear buckling?

As you already know, linear buckling analyses generally yield non-conservative results (they overpredict the critical load). The reason for this is because various nonlinear behaviors prevent most structures from achieving this too high critical load predicted by linear buckling analysis. The most significant nonlinearities here are imperfections but also all the other nonlinear behaviors occurring prior to buckling (yielding, large deformations). It’s important to know the assumptions and limitations of linear buckling analyses. Here are some resources that can be helpful for you:
- "Practical Finite Element Analysis for Mechanical Engineers" by D. Madier
- Lukasz Skotny’s materials - the video referenced in your first post is his lecture, this guy is an expert in nonlinear FEA, especially buckling of shells
- Abaqus documentation and training materials - various comparisons of linear and nonlinear buckling simulations

### RE: Nonlinear buckling occurs earlier than linear buckling?

Hi

In my terminology, linear buckling is something comparable to Euler buckling. It is usually performed for a geometrically perfect structure and deformations or material strength are not a parameters in the analysis.
Critical load for a straight column is Pcr = (pi^2 * E * I) / ((buckling length)^2). It is possible to include imperfections in linear buckling but I typically don't.

For nonlinear buckling I include imperfections, deformations and material strength. All those will lower the load capacity compared to the linear buckling approach. I often base the imperfections on the deformations from the lower linear buckling modes. I create an imperfection to initialize the large deformations and ultimately the material fails.

But since this is a FEM forum I would say that the linear buckling analysis often is called something with "buckling" in the software. However, the nonlinear buckling analysis, in my experience, is more likely to be called something with the label "nonlinear" in the solver. Then the user has to handle the imperfections, deformations and material data in the setup to make things work appropriately.

My 2 cents

Thomas

### RE: Nonlinear buckling occurs earlier than linear buckling?

I did my final year project on the softening (ie premature buckling) of trusses due to moments put into the ends of struts by real welded joints rather than pin joints. And that is the sum total of what I remember about that.

Cheers

Greg Locock

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### RE: Nonlinear buckling occurs earlier than linear buckling?

Hi again

My post was written with something fairly simple in mind, like a column or a beam subject to lateral buckling. That makes the theory fairly "clean".

If we consider something more complex or at least use a more nuanced approach.

Just because something buckles it does not necessarily mean collapse (or failure). A simple example is a I-beam with a slender web. Say that the web buckles trough shear and the result is deformation out of plane. That deformation can result in tension in the web, provided that the boundary conditions allow it. And a new situation with equilibrium can occur. The linear buckling analysis will probably find the first part, the nonlinear analysis can track the complete failure.

But this does not necessarily mean a higher load capacity, the nonlinear analysis has different limiting factors. But it can give a very good illustration for the actual failure in a structure. Provided that you can increase the load in steps and then check the results for each increment.

Thomas

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