Wrinkling Stability Criterion Equation 1

[FJCV]

Dear Fellow Engineers,
NACA-TN-3431 shows on page 9 equation 5 the stability criterion for a plate to fail in short wave failure mode (aka. skin wrinkling failure). Mentioned NACA Technical Note refers for this equation to Timoshenko's 1936 book "Theory of Elastic Stability" and states that the skin in the wrinkling mode can be represented by a column on an elastic foundation. In the attachment I tried to derive NACA-TN-3431 eq. 5 starting with Timoshenko's beam equation unsuccessfully. I appreciate it if anybody can review my calculation and recommend a correction. Thx.


Web link to NACA-TN-3431 (Semonian & Peterson 1955 Langley, VA):

https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19930084292.pdf

My derivation as jpg image file:

http://files.engineering.com/getfil...ile=wrinkling_failure_stability_criterion.jpg

My derivation as pdf file:

http://files.engineering.com/getfil...ile=wrinkling_failure_stability_criterion.pdf

 

[Ingenuity]

With both λ and b[sub]s[/sub] in units of length, to be dimensionally correct the following expression is wrong:

I think that the issue is that your Timoshenko reference is using a solid bar column of dimensions b[sub]s[/sub] x t[sub]s[/sub], so the bar flexural stiffness of EI[sub]s[/sub] will NOT be equivalent to your plate flexural stiffness, D[sub]s[/sub] - for a plate - unless you introduce an additional b[sub]s[/sub] term.

 

[FJCV]

Hello Ingenuity,
Thank you for your reply. I appreciate your help.
Can you clarify what should change in the EI = Ds equation.

Are you able to update the derivation so it matches the NACA equation? Thx.

 

[Ingenuity]

FJCV:

You are welcome.

Try this derivation:

PDF copy attached.

CAUTION: I cannot validate that the Timoshenko equation is an appropriate representation of your 'wrinkling' buckling - outside my wheelhouse.

 

[FJCV]

Ingenuity,
Thank you for your help.

 

[Denial]

Not wishing to upset any applecarts here, but I believe that the formula for the buckling load of a column laterally supported in an elastic (Winkler) medium should be
P[sub]cr[/sub] = (pi[sup]2[/sup]EI/L[sup]2[/sup])[m[sup]2[/sup] + kL[sup]4[/sup]/(m[sup]2[/sup]pi[sup]4[/sup]EI)]
The only change being the exponent on the pi in the denominator of the second term in the parentheses should be 4 rather then 2.

 

[Hokie93]

Denial is correct; pi should be raised to the fourth power in the denominator of the second term in the parentheses.

 

[Ingenuity]

Good catch by Denial and Hokie93.

FJCV - you copied Equation 5 from your NASA reference incorrectly in your derivation, Π in the second term denominator for k[sub]m[/sub] is raised to the 4th power, so when you derive for the corrected Timoshenko equation you will indeed arrive at the same k[sub]m[/sub] equation.

For completeness, the reworked derivation is attached.

 

[Denial]

I suspect that the above exponent=2 versus exponent=3 error originated in Roark's "Formulas for Stress and Strain.[ ] Probably a transcription error when it was being copied from Timoshenko (who Roark gives as the source).

Back in 1993 I needed the formula for the buckling load of an infinitely long column laterally supported by a Winkler medium.[ ] I cannot remember the context, but for some reason I decided I needed to derive it from first principles.[ ] When I had finished that [irony]little[/irony] exercise I went to check it by comparing it with Roark's formula, duly simplified for the infinite length case.[ ] Oops.[ ] Something wrong.[ ] Rechecked my workings.[ ] Seemed fine.[ ] Re-rechecked.[ ] Etc.[ ] In desperation I eventually went to the University library to check against Timoshenko.[ ] Bingo.

So I rang McGraw-Hill's Sydney office and reported the error to someone who sounded like he was the right chap, then followed this up with a fax (to the same chap).[ ] But I never checked whether later editions of Roark have corrected the error.[ ] My Roark is still the Fifth Edition, in which the formula in question is Case 4 of Table 34, chapter 14 "Elastic Stability".[ ] I know that the error was also in the Sixth Edition.

Maybe someone on this forum could check later editions?

 

[FJCV]

Hello All,
Thank you all for your responses.

 

[Ingenuity]

Denial:

Roark's 7th Edition (which appears to be re-organized and expanded) has the following:

Little hard to see the expression in the above, so here is a close-up:

So the bracketed second term denominator is π[sup]4[/sup] - thanks to your input. Well done.

 

[Ingenuity]

I suspect that the above exponent=2 versus exponent=3 error

What is this "'exponent=3" you speak of? You mean 4, right?

 

[IDS]

Denial - you will be pleased to know that Roark's 7th Edition has pi^4 (see attached extract).

A good example of how accepting the content of black books without question can be just as bad as accepting the output of black boxes.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[FJCV]

Hello Denial,
I attached a copy of Timoshenko's original book which matches your reference.

Timoshenko:

http://files.engineering.com/getfile.aspx?folder=31b1f9d4-5c02-4356-952c-a34bd992d09a&file=Timo.pdf

 

[Denial]

Thanks all.[ ] Now, two dozen years later, I can sleep a bit easier.
(And, yes, in my 21Jun17@21:14 post above "exponent=3" should have been "exponent=4":[ ] early on a winter's morning I must have been under the influence of cold fingers and/or cold brain.)

 

[dik]

you must be in Denial...