## Buckling Factor without imperfection < Buckling Factor with imperfection (?!)

## Buckling Factor without imperfection < Buckling Factor with imperfection (?!)

(OP)

Hi gents,

As per figure below, I have performed simple test for beam in Critical Buckling Load linear analysis.

First model is beam without any imperfection introduced. Buckling Load Multiplier (BLM) is 65.

Second model is based on scaled unitless deformation from the first model, hence imperfection is introduced.

Similar Critical Buckling Load linear analysis is performed, now BLM is 165. Loading and supports the same.

What is going on, do you have any suggestions? One shall expect that with imperfection BLM should be less, but not higher than without imperfection modelled...

As per figure below, I have performed simple test for beam in Critical Buckling Load linear analysis.

First model is beam without any imperfection introduced. Buckling Load Multiplier (BLM) is 65.

Second model is based on scaled unitless deformation from the first model, hence imperfection is introduced.

Similar Critical Buckling Load linear analysis is performed, now BLM is 165. Loading and supports the same.

What is going on, do you have any suggestions? One shall expect that with imperfection BLM should be less, but not higher than without imperfection modelled...

## RE: Buckling Factor without imperfection < Buckling Factor with imperfection (?!)

After writing the below, I think I've (partially) solved your problem. It would be helpful if you described you "first" and "second" analyses in more detail (i.e. are they eigenvalue buckling analyses, geometrically nonlinear/large displacement analyses etc). As a general rule, imperfections have very little effect on eigenvalue (or "linearly elastic") buckling analyses. For GMNL anlayses, obviously you need an imperfection to get the thing going.

Once you define your "first" and "second" analyses, maybe we can explore the problem further.

Other things that would be helpful....

1. Geometry (preferably a sketch)

2. Element Formulation/Mesh Props

3. Support Conditions

4. Material Properties

5. What type of buckling mode you expect (flexural, LTB etc etc).

5. Loading

5. How the longitudinal stiffener is formed (or is it just a single "beam" cross-section).

6. An approximate solution i.e. using Ncr = pi^2*EI/Lcr^2

gl, ukbridge.

## RE: Buckling Factor without imperfection < Buckling Factor with imperfection (?!)

Trying to address your questions:

1. It is a simple I-beam with longitudinal stiffener for benchmark purposes.

2. Plate elements used.

3. Translations at end edges of profile contours are fixed.

4. Linear steel.

5. LTB.

6. Uniformly distributed load acting at top flange/web edge along all length of the beam.

7. As per item 1.

8. Not provided.

Both analyses are (linear) eigenvalue buckling analyses, once difference is, the second is with imperfection included.

## RE: Buckling Factor without imperfection < Buckling Factor with imperfection (?!)

Thanks for following up with the helpful summary.

Yes, geometrical imperfections will/should have very little effect on an Eigenvalue Buckling Analysis. This is because an eigenvalue buckling analysis is an inherently first-order, linearly elastic analysis.

A detailed explanation is outside the scope of this post, but a really good introduction to this topic can be found in this relatively simple paper for practicing bridge engineers. http://www.atkinsglobal.co.uk/~/media/Files/A/Atki...

Simply put, to study the effect of imperfections (if any) you will have to conduct a geometrically nonlinear, or large displacement analysis. The above paper will go into much better detail than I ever could on this. This is also a good paper/text by the same author https://www.scribd.com/document/206882241/EN1993-P... (A leading UK Bridge Engineer). Again, this is a very good read which will help you understand the differences http://www.steelconstruction.info/File:SCI_ED008.p....

That still doesn't quite answer the question on why you're getting very different load factors. Maybe its worth making another copy of your original model and defining the imperfection to be doubly sure. I've also made some comments on your model description:

1. OK,

2. I think strictly speaking you should be using shell elements (plate elements are typically for 2D applications like slabs). I'm unfamiliar with your software though.

3. By this, do you mean you have assigned translational restraints in all principal axes, to the lines representing the bottom and top flanges at the ends? Although its not a "real" application, and for testing purposes, be careful with how you idealise your supports. Knowledge of "what" the problem is obviously matters a bunch, but are you sure you want the edges of the beams to be torsionally restrained - I suppose without any bracing or a another beam to give it some warping resistance its a mechanism otherwise i.e. there is no torsional restraint.

4. Ok.

5. Yes, now I know a bit about the problem LTB will obviously govern here.

6. Fine, although I might try and further simplify the problem as a unit UDL applied to the TF where it meets the web panels.

7. Ok.

9. There are formulae available for simple cases in your local building code or textbook. You will need to calculate Mcr, the elastic critical buckling moment. This is the theoretical bending moment at which the beam will theoretically buckling (not accounting for imperfections). This is a helpful guide; http://eurocodes.jrc.ec.europa.eu/doc/WS2008/SN003.... For bench-marking purposes and to make your validation as simple as possible, ensure that your model is a doubly-symmetric I-beam (i.e. bbf=btf and ttf = tbf) - the formulae are much simpler to use. To get the Mcr from your model, work out the BM in the beam in a static analysis (i.e. wl^2/8) and multiple it by the buckling load factor. Then simply compare etc.

Once you've gone through all that and you're still having trouble, feel free to shout. gl!

## RE: Buckling Factor without imperfection < Buckling Factor with imperfection (?!)

I will close this matter for the moment as it is not critical and lesson learnt will be there is no point of using imperfection at eigenvalue analysis, e.g. see EN 1993-1-6.

Thanks !

## RE: Buckling Factor without imperfection < Buckling Factor with imperfection (?!)

Unless its an academic matter i.e. for a journal paper, or (extremely) desperate its often not worth the effort of accounting for second-order effects in your actual analysis. In exceptional cases, e.g. for an extremely large pier which will require an enormous amount of reinforcement, it'd be economical to do so.

It is far easier to simply do an elastic buckling analysis (like you've done), and account for geometric imperfections (second-order effects) using the EN 1993-1-1 buckling curves. Something thats often forgotten, if your Buckling Eigenvalue < 10 you must allow for additional second-order effects using one of the clauses in section 5. Think its something like M_enh = M(1st_order)*LF/(1-1/LF), make sure you look it up.

gl

## RE: Buckling Factor without imperfection < Buckling Factor with imperfection (?!)

It seems you've fully fixed both ends. That makes all your stresses tensile. As you introduce the imperfection, you are creating more tensile strain at the start of then analysis and those raise the buckling factor of safety.

Think about a beam that isn't simply supported, but instead fully fixed at both ends, where is the compression in it? Without compression it won't buckle. Your buckling mode appears more complex than this, but the concept still applies.