Linear Buckling 3

[IngNicola]

SHORTLY (and not accurate) THEORY: a buckling load factor:

= 1 means that the applied load is exact the load for buckling the structure.
< 1 means that the applied load buckle the structure
> 1 means the structure will not buckle

I omit negative values.

But if you have to design a structure which is your own limit buckling factor?
Someone thinks 2.5 is a good safe factor to represent the geometric and loading imperfections. what do yout think?

 

[SWComposites]

The answer depends on many things.
What type of structure?
What is the geometry? (for instanct, thin walled cylinders are very sensitive to imperfections)
How is it loaded?
What is the application?
Are you using a linear eigenvalue analysis or a non-linear analysis?

 

[mtnengr]

All of the comments by SWComposites are valid. But, to jump to some idea of what knockdown factors can be experienced, one should look at the work of Koiter. His thesis was first published in 1946 (in Dutch) and later translated by the Air Force to English in 1960's. This work is applicable to any structure.

Many (>1000) papers have been written on imperfection sensitivity of shell structures, particular cylindrical thin shells subject to axial loads or external pressures. One of my papers (WRC Bulletin 313: "Computer Programs for Imperfection Sensitivity Analysis of Stiffened Cylindrical Shells", Apr 1986, R.L. Citerley) provides four computer programs that examines the characteristics of the specific "Koiter knockdown factor."

If interested, the publication can be obtained from WRC or be sent via PDF format from this author.

For the above cases, the asymptote can be as low as 0.2 (ie, an 80% reduction factor of the classical linear critical buckling for modest imperfection geometries).

 

[corus]

There is a german DIN structural standard that gives recommended safety factors for buckling, taking into account imperfections. If the german standard isn't suitable then look at the latest design standard which will be applicable for your application. It's better to use a referenced and approved source than relying on what people 'think'.

corus

 

[mtnengr]

Corus suggests* that you use a "standard" that gives safety factors for buckling. Fine, if you throughly understand the basis of the standard and whether it is applicable to your case. Sometimes the "standard" is flat out wrong, as in the case of earlier ASME standards on imperfection sensitivity for pressure vessels. See my discussion in Chapter 2.3 "Imperfection Sensitivity and Post-Buckling Behavior of Shells", Pressure Vessels and Piping Design Technology-1982- A Decade of Progress, ASME.

An engineer should understand just how a standard is developed. The basis of it's material being used and just who were the members of the consortium that developed the standard. It has taken the ASME over eighty years of scholarly and industrial input to arrive at a "Buckling/Imperfection Standard." There is still a need for additional thought in applying these standards.

All of the references used by this author will allow the engineer to 'THINK' about the problem and rely upon his own education to arrive at a defendable position. To be sure, "standards" are one source, but not necessarily the only source. You should not use a "standard" just because it is a "standard". Even if it is a contractual obligtion to use a specific standard, one should observe the conservativeness of the standard and should act accordingly.

* No mention as to what specific DIN standard should be used.

 

[gfbotha]

Whilst understanding the need for standards & codes, I really like the comments made by Mtnengr (thought I’m alone). All the years I preferred making up my own mind about a particular design rather than to simply follow some standard (maybe I’m just stubborn!!). Even standards differ and get amended… If allowed, one should use them as a guide only.

Getting back to the original question: The most important issues for me are: likely manufacturing imperfections, loading effects/misalignment, and linear buckling vs non-linear analysis.

One needs to dig in quite deep to say whether a factor 2.5 is sufficient or not.

 

[corus]

I wonder if in some years time people will say, mtnengr's paper was just flat out wrong?

To say that you don't use a design standard* because in your opinion you think it is 'flat out wrong' is beyond belief. I look forward to reading about future liability cases for a failed design where this defence has been put forward.

*DIN 18800 part 1

corus

 

[gfbotha]

Well, yes, I would push for a more cautious approach; and agree that Mtnengr came over a bit too strong regarding that particular phrase. There are issues like different assumptions, conditions and supporting technologies of the particular era.

Also, if one does not fully understand the underlying factors, you better do not deviate from the standard! My understanding is that some industries are more bound by standards than others. In that sense I consider myself lucky…

Regards
Gert

 

[IngNicola]

thank you!

first of all:

> corus, if I ask it's to improve my knowledge and not to be an inaccurate engineer,

Then
"If a more accurate estimate of the buckling load is required, it is recommended that a nonlinear analysis be carried out so that the effect of pre-buckling deformation can be included and the post buckling capacity predicted"

I'm modelling a bridge beam (only one with length 60 m and doubleT steel section) with a linear eigenvalue analysis. The beam is loaded with its own weight (no other loads). The first analisys give 0.9 as buckling factor, with some trasversal beam I arrive about 7.8 as buckling factor.

thanks

 

[cbrn]

Hi,
IngNicola, it's not my field, but I seem to know that in Italy the latest Technical Norms for civil constructions impose to use non-linear buckling. The reason is the same already pointed out: by definition, linear buckling doesn't consider second-order effects, which can cause the "limit load" to drop by very huge factors. The same concept is inherent in the EN 13445 for Pressure Vessels.
So, I can understand the point underlying Mtnengr's position, but I (like others, it seems...) prefer applying a Norm (*) rather than relying on "personal taste".

IMO the discussion in this thread could be developed in order to better understand:
- how it can be that a code is "flat out wrong" (it shouldn't be, due to the process a Norm whithstands before being issued...)
- in this case, which actions have been taken in the engineering practice when it was discovered that the code was wrong (wasn't the code used at all? Was it used everywhere and did some structure collapse because of the code's application? etc...)
- if buckling is - nowadays - taken into account by the Codes with sufficient accuracy and/or completeness

*: "applying" a Norm means first of all having understood it.

Regards

 

[mtnengr]

My "flat out wrong" comment had to do with the earlier versions of the ASME Code suggesting that critical buckling factors could be increased by including imperfections rather than a decrease.

A discussion of the above comment was presented, as well as, the suggestion of performing a non-linear analysis was proposed in a paper: "The Use of Manufacturing Tolerances in Imperfection Analysis of Shells", PVP Vol 57, pp 49-66. A five step process was proposed.

One step relies upon a valid non-linear analysis. This is a major hurdle. An exprienced analyst must determine if the results are correct. Numerical instabilities can cause the results from a dynamic analysis to appear to be a decrease (or possible increase) in load carrying capacity and not be the true representation. Comparison of results from different non-linear tools have been shown to vary widely (more so in dynamic analysis case than static). One should not accept the results from a computer program blindly. If you do then, Corus's comment about being proven incorrect is possible.

Papers like the above can stimulate an engineering society to act and include the ideas into a "standard." Yes, only time will tell if a proposed approach is the "correct approach."

 

[GregLocock]

When i was at uni I did a final year paper on structural design. It was interesting to see how many mistakes were codified into the UK standards at the time, concerning the design of steel frameworks.

Sticking to a code may be an acceptable legal defence, but not checking weird result from that code is very poor engineering.

Cheers

Greg Locock

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

 

[jte]

Sticking to a code may be an acceptable legal defence, but not checking weird result from that code is very poor engineering.

I'll second that. I often think people should read the foreword to their codes/standards a bit more often. In my case, I like the foreword from ASME VIII-1 which includes the following statement: The Code is not a handbook and cannot replace education, experience, and the use of engineering judgement.

You need to follow the code, but a good engineer won't follow it blindly.

jt

 

[cbrn]

heartfully agree with the last three comments.

Thanks, regards

 

[torpen]

Hello,

you can check the sensitivity of the structure to imperfections by :
1 - performing a linear eigenvalue buckling analysis
2 - taking as new geometry the first buckling mode (or a combination of few modes) with an amplitude equals the amplitude imperfections for instance
3 - running a non-linear analysis with the new geometry
4 - performing a linear eingenvalue on the deformed structure
The buckling load factor is equal to :
(load * last but one "time") + (linear load factor * last time step).

You can omit steps 1 and 2 and use the initial geometry for the step 3.

Regards

Torpen.

 

[mtnengr]

I think that there is a need for an expansion of Torpen’s comments.

The selection process for an imperfection shape must consider the specific shell theory being used. Most shell theories include the effects of an imperfection through the derivative of the imperfection shape. The strain-displacement relations include the square of the rotations and the equilibrium equations are modified by the coupling terms of the membrane components and the rotations.

If Sander’s non-linear shell theory is considered, the twist of the normal is included as possible coupling. The derivative of the imperfection in each orthogonal direction is included as an added part of these rotations (including twist) and not the imperfection itself.

The question that requires consideration is: How is the shell to be analyzed? There are essentially two possibilities: Fourier decomposition or three-dimensional shell modeling.

Fourier decomposition considers all components to be expressed in terms of both sine and cosine components. When these conditions are considered, terms of responses include cos(m±n)sin(m±n), etc. Since the main imposed forces (accelerations) are generally from the Fourier functions n = 0 and ± 1, the responding components would be the direct coupling of n = 0 with the critical shape function (k) and the adjacent modes k+1, k-1 and i+j = k and/or i-j = k , where i and j are adjacent secondary critical modes.
If real life imperfection geometries are included, then the Fourier representation of the imperfection geometry should be added to the analysis set. The most critical shapes for cylinders are diamond shapes and for spheres are hexagon shapes.

Three-dimensional shell analysis requires more ingenuity. Since the observed critical buckling behavior follows the Fourier representations, one again has to consider imperfection shapes described as a Fourier series in both orthogonal directions. If a non-linear state is considered at a specific time, then repeated eigenvalue analysis procedures will provide an estimate of the critical load condition from that state (don’t forget to include body forces). If direct load (time dependent) analysis is considered, one has to define what suggests buckling.

Increasing incremental load factors can be considered for either the direct load or Fourier decomposition method. A growth of 1000 fold of the initial imperfection is considered to be an indication that buckling has occurred. For the Fourier method, one can observe the time dependent response. A continuing growth in the critical buckling load will be observed. The energy is transferred from one or multiple modes to the single critical buckling mode over a finite time (usually associated with multiple times of the inverse of the fundamental dynamic natural frequency).

Although Torpen’s comments are essentially correct, a great deal more of engineering thought should be considered.