BUCKLING OF A TAPERED PLATE 2

[StressMan2506]

Fellow stress/structural engineers:

I am processing a concession in which a tool mark has been blended out of a flange of an I-section beam. The result is that, over a limited length the thickness varies from nominal at the toe of the flange/web fillet to approx. 2/3 nominal at the free edge. Analysis using Table 15.2, case 1e of Roark's Formulas for Stress & Strain, 7th edition, yields a failure if I consider just the minimum thickness; it passes using the mean thickness. I'm wondering if I have underestimated the tendency of the free edge to buckle. My question is this:

Does anyone know of an accepted method/formulation for a plate tapered perpendicular to the direction of load? I hope the attached sketch shows my meaning.

Thanks in anticipation.

 

[rb1957]

the two easy ways to represent the tapered thickness would be use the minimum or use the average; but then i guess you've already thought of those.

if this is blending out a tool mark, it won't be very long (out-of-page) ...

Quando Omni Flunkus Moritati

 

[edbgtr]

'Morning StressMan2506,
Find attached a chapter from my personal set of stress notes for a stepped flange of 2 thicknesses across the width of the flange. By carefully estimating the "equivalent" flange to the one you have, using the 2 thickness data, you should get a better estimation of the local buckling of your discrepant flange.
Ed.

 

[StressMan2506]

Thanks, guys.

You're right, rb1957, I did consider min. & mean and I agree that blending out a tool mark generally affects a small length, but, in this case, the blend was not as well executed as it could have been. If the thin run had passed, the job would have been done. The mean run left me feeling that I couldn't confidently state that the edge had been enveloped...

I will be having a shot with the method you sent, edbgtr.

Cheers,
StressMan2506

 

[StressMan2506]

Hi edbgtr:

I've taken a look at your attachment. With reference to Roark, 7th edn, Table 15.2, case 1d, it appears that the work was based upon a long,narrow plate. The following statement appears at the top of Roark's table:

"...the smaller width should be greater than 10 times the thickness unless otherwise specified."​

No similar statement appears in the attachment, but given my first statement, I infer that there is a similar limitation in the applicability of the stepped plate formulation. Is t[sub]v[/sub]+t[sub]o[/sub] limited to some proportion of b?

Thanks in anticipation.

 

[edbgtr]

Hi Again Stressman2506,

For a typical 2000 series Al alloy, you'll find that the yield stress b/t buckling ratio for a one-edge-free compression panel is close to 10:1. If you apply Needham crippling, this works out at around 8 to 8.5. I.e. it is often used as a guideline for designing one-edge-free flanges for I or C beams. The method from my notes indeed uses a flange aspect ratio of >5 where the k[sub]c[/sub] value is 0.43 for a classical solution and 0.388 for the Poisson Ratio reduced value, ν = 0.3. If the clamping effect of the flange at the web is larger than for an SS boundary condition, then a higher coefficient is sometimes used by some aircraft companies. No doubt they have justified, via test, the use of the higher k[sub]c[/sub] value.

So to answer your question, yes it is desirable to design your flange not to buckle or cripple under design loads.

The method given in my notes gives the parameter ratios for which it is valid. It is not advisable to use said curves (functions) outside of their analysed parameter ratio range. The interpolation carpet plot functions have been "forced", mathematically, to be valid within the specified validity ranges.

Being involved in MRB work myself, I may do some additional work in the near future to produce a "method" for determining the buckling coefficient for tapered compression flange elements. Currently, I do not know of one in any of the commonly available company stress manuals.

Cheers,

Ed.

 

[StressMan2506]

Thanks, again Ed. A couple of points:

I take it that the stepped plate notes are your own work. If so, I would not be able to cite the method in my concession justification.​

My flange has b/t < 5, so Roark's table doesn't apply. Assuming a uniform flange of thickness = post-blend minimum, stress = 494 N/mm[sup]2[/sup]. Setting b/t = 10 gives critical stress = 459 N/mm[sup]2[/sup]. Yield is 500 N/mm[sup]2[/sup]. On this basis, engineering judgement tells me that the blended flange will not buckle and is stressed below yield, i.e. it is safe. My difficulty is that our customer requests calculations for everything. I have found a thick plate formulation. (Buckling of Thick Rectangular Plates, Srinivas & Rao, AIAA Journal, Aug 1969) but I can't get it to yield sensible results. I'll have to revisit it.​

Louis

 

[edbgtr]

Hi StressMan2506,

The work is my own, based on some raw results data (the tabulated coefficients) I obtained and calibrated with my own classical and FE analysis. It is important that such FE based data agrees with classical solutions at the points where it can be tested. The transition of the data curves between known classical solutions must also be smooth and free of short period waves and spikes.

If I’m interpreting your numbers correctly, you should be in a good position to clear the discrepancy you have.

From the b/t < 5 ratio you mention for the flanges, you could have what is called a compact section as defined by our Civil Engineering cousins, i.e. you don’t have any instabilities on the compression side of your beam, including the web and hopefully a correctly dimensioned and stable web/flange joint. You also seem to be in the fortunate position of knowing the critical loading applied to your beam. This means you can make a decision based on your calculations, i.e. producing a substantiation/justification based on the numbers; your calculations. Is the discrepancy at the point of maximum load (bending, shear, axial, torsion) on the beam? If not, then that section may not be stressed to its allowable capacity.

Your material, I assume is of aircraft quality from an approved supplier, so your material data is reliable and in most cases has characteristics a few percent larger than the allowables from the MMPDS. Even better if you’re using OEM material allowable data, as that often has a bit of additional conservatism built into the values relative to the MMPDS.

If you have blended out whatever damage you had on the flange in accordance with an accepted depth to blend-out length ratio, thereby reducing the K[sub]t[/sub] factor to somewhere around 1.1, then you should be in a position to make a good assessment regarding the continued fatigue life of the component.

It is difficult for engineers on this forum to gain a complete picture of your problem, as we are not in your shoes, but from what you have said in your recent post, it sounds like you may still have a recoverable member that you can substantiate by calculation.

I hope that our discussions have brought you closer to a solution to your problem.

Ed.

 

[StressMan2506]

Thanks once again, Ed. I studied civil engineering and worked for some years in the offshore oil & gas industry; the term "compact section" rings a bell.

I'm close to writing a justification based on numbers and reasoning, but I may have overlooked lateral loading; I need to look further into the dossier.

Louis

 

[edbgtr]

StressMan2506,
I have found a reference to tapered flange buckling within a report produced by Y Kim and Teoman Peköz at Cornell University. This work looks pretty well researched and the residing Professor Peköz is well known in the Civil Engineering discipline.
To save you the trouble of searching for the item, I have attached a copy of the report from their website.
Regards,
Ed.

 

[StressMan2506]

Thank you very much!

 

[edbgtr]

StressMan2506,
You got me going on this one, and I have done some additional work on the subject, which I think you and others here may find interesting and useful. The work, as you will see, has been bench-marked against published papers and my own FEA. The work I have done is contained in the attached pdf.
I welcome discussion and suggestions so we can all improve our skills and create new tools to use in our substantiation work.
Ed.

 

[Sparweb]

Edbgtr,
Glad to see you're still cracking clever problems like these.
To confirm that I'm starting out on the right foot reading your paper, I have a question.
Does " h[sub]b[/sub] / h[sub]o[/sub] " refer to the ratio of high/low applied stress on the unit plate on the simply supported sides?
I believe that I see a relationship but I'm not sure about it yet.

Is the analysis making a comparison between a tapering stress applied to a constant flange thickness, with a constant load applied to a tapering thickness? This would cause a tapered stress distribution, so it seems like that is similar to the illustrated example. I want to be sure I an working with the right mental picture before making assumptions as I go further. Making the next step to say that these two different situations can give rise to similar buckling configurations would be a very interesting way to sneak up on a complex problem.


STF

 

[edbgtr]

'Morning STF,

Thanks for your response and good questions.

I applied a uniform stress across the tapered b-edge of the plate. This tied up with the uniform stress assumption one usually makes with this type of flange. If the stress distribution is non-uniform then the buckling coefficient changes to one for a non-uniform stress distribution per McComb's Notes and Bulson. The tapered loading shown in the Kobayashi chart is the load/unit length across the face of the b-edge. The h[sub]b[/sub]/h[sub]o[/sub] represents the thickness ratio t[sub]r[/sub]/t[sub]o[/sub] I use as my preferred symbols.

Throughout the FE analysis performed I ensured that the applied compressive stress was uniform across the tapered section. You'll see this as the σ[sub]ave[/sub] values given in the last set of mode shapes at the end of the article. If the stress had not been uniform, I would not have found agreement with Mizusawa's work. The load I applied at one end of the model and balanced by the constraints at the other, was 25000N. The average stress is the load given here divided by the total area of the tapered section.

As far as the sneaking up on another problem is concerned, I checked these results against the McCombs Supplement curves for tapered stress distribution (compression + bend buckling) over a SS-FS flange, and yes there are similarities in the values at a/b=0.5, but the buckling coefficients with taper ratios > 1 are higher at the higher a/b ratios. Well spotted.

Regards,
Ed.

 

[Sparweb]

Thanks for the clarifications Ed. Good thing I asked, or I would have been thinking about combined bending+compression when it wasn't applicable to this work.
My McCombs book is at work so I don't have it to refer to over the weekend.

STF