Searching main source of coefficients in Roark's stress deflection formula for rectangular plate ? 3

[Mech research]

I am studying a case related to rectangular plates with uniformly distributed load, fixed at all four sides.
For stress, deflection calculation at the center of the plate by Roark's formula. Roark uses the coefficient according to a/b aspect ratio.
I tried to search for the references as called in Roark's book to get the main source of these coefficients (alfa, beta, beta1 for stress and deflection )but did not find any.
Could anyone share the document or screenshot, if someone has it?
Thanks in advance.

 

[JedClampett]

I believe those are based on Timoshenko "Theory of Plates and Shells." Alas and alack, I have a copy, but it's at work and I'm quarantined. They might be from the Bureau of Reclamation "Moment and Reactions on Flat Plates" but I doubt it.

 

[Mech research]

Thank you so much, JedClampett. I downloaded it from the internet. I did not find any table with same values as in Roark's book.

 

[CarlB]

Perhaps this Table 35 from Timoshenko? Provides moments vs stresses, and deflection is in terms of "D", flexural rigidity.
[wasn't able to upload the pdf, posting an image)

 

[JedClampett]

The data from Roark is usually heavily edited (three cases of a/b instead of fifteen)from Timoshenko in the interest of saving space. There's also a couple of typos that affect the results, although I'm not sure if it's from this case.
Even if someone gave you the screenshot, the explanation is important too.

 

[CarlB]

Trying again with attachments. Maybe that wasn't an allowable edit to a post.

 

[r13]

B.O.R Engineering Monograph No. 27. See column on the left, and illustration graph at the bottom left for the aspect ratio "a" and "b".

BTW, if you are a student, you shouldn't use the Roark's Formulas, or this Monograph. Instead you shall read "Kirchhoff-Love plate/shell theory", or "Mindlin-Reissner theory" (a refinement from the former). If you are a junior engineer, you shall be able to use FEM to verify the accuracy of results from Roarks, and from this monograph.

 

[IDS]

Looking for the documents referenced in Roark I found:

https://archive.org/details/TheoryOfPlatesAndShells/page/n1/mode/thumb

which is the complete Theory of Plates and Shells, and can be viewed on-line or downloaded.

I have done a quick comparison of the Roark and Timoshenko tables with the B.O.R Engineering Monograph No. 27 (Moody) and W.D. Pilkey's FORMULAS FOR STRESS, STRAIN, AND STRUCTURAL MATRICES, compared with a plate/shell model in Strand7.

I looked at a rectangular steel plate, 10 mm thick, with a = 1.0 or 2.0 m and b = 1.0 m, with uniform load = 1 kPa, E = 200,000 MPa and Poisson's Ratio = 0.3

The factors from the four sources are summarised below:

Note that:
The Pilkey factors are given in the form of a cubic of alpha (a/b)
The first Pilkey factor for C5 (shown bold) is shown as 0.4247 in the text, but this appears to be an error
The Roark factors for deflection incorporate the parameter D, which is applied separately by Timoshenko.
The Roark factors Beta1 and Beta2 are to calculate stress, rather than bending moment.

The deflections and bending moments derived from these factors are shown below for a/b = 2, compared with the Strand7 results:

The results are:
Deflection
MX midspan
MY midspan
MX mid-edge
MY mid-edge

The results are reasonably consistent for deflection and MX values, but there are significant differences in MY, both between Timoshenko and Pilkey, and compared with the Strand7 results. Note that the Moody results were prepared for concrete structures, and assume a Poisson's Ratio of 0.2.

I have attached my spreadsheet, with more details of the results, and images of the relevant tables from the documents, including a readable copy of the Moody table.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[IDS]

Coincidentally I got this link from Academia today:

https://www.academia.edu/39462258/T...te_Analysis_Numerical_and_Engineering_Methods

Theories and Applications of Plate Analysis Classical, Numerical and Engineering Methods Rudolph Szilard, Dr.-Ing., P.E. Professor Emeritus of Structural Mechanics University of Hawaii, United States Retired Chairman, Department of Structural Mechanics University of Dortmund, Germany



Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[IDS]

If you are a junior engineer, you shall be able to use FEM to verify the accuracy of results from Roarks, and from this monograph.

I'd say you should do that if you are a senior engineer as well, or anything in between, although I'd do it the other way round (use the tables to check the FEM results).

Typos in these books are not uncommon (see the example from Pilkey being out by a factor of 10 in my post above).

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[rb1957]

why in Pikey equation is the alpha^2 term negative ("-E26") ? I don't know the source, just looks "odd".

another day in paradise, or is paradise one day closer ?

 

[r13]

A structural engineer needs to know his tools. In early encounter with a analysis software, it should be tested and calibrated with known solutions to confirm its capability and accuracy, also find the restrains/limitations on modelling technics. In reverse, a senior engineer should be able to spot problems in a non-sense analysis output, and in most cases, be able to verify the judgement by simple hand calculation.

 

[IDS]

why in Pikey equation is the alpha^2 term negative ("-E26") ? I don't know the source, just looks "odd".

It looks like the Pilkey numbers are based on fitting a cubic polynomial to someone's data, but not Timoshenko's. I have fitted a cubic to the Timoshenko numbers for the central Mx, and get the results shown below:

The Timoshenko polynomial factors are very different from the Pilkey values, although they all have the same sign, and the resulting moment factors are all within +-3%.

Also note that if the polynomial factors are used outside the specified "alpha" range (1.0 to 2.0), the results are nowhere near the correct value.



Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[rb1957]

I imagine whoever fitted the cubic to timshenko's data picked the four points where they agree and said "near enough" for the others (at least the cubic fit is conservative).

another day in paradise, or is paradise one day closer ?

 

[JStephen]

A lot of Timoshenko's work is based on earlier sources (quite often in German or Russian), and he lists extensive references as he goes.
A lot of Timoshenko's solutions consist of solving partial differential equations with a double series as the end result, in which case, you have good bit of work ahead of you to get actual numbers out of the solution. I don't know to what extent Roark might have done this.
I think in a lot of cases, Roark completely rearranged equations, coordinates, and variable names to try to maintain consistency, so finding the solution in his reference, it might be presented in a somewhat different form. IE, maybe the reference gives you moments, and Roark gives you stresses and stuff like that, or the long edge is A in one reference and B in the other and L in the third, etc.

 

[IDS]

maybe the reference gives you moments, and Roark gives you stresses and stuff like that, or the long edge is A in one reference and B in the other and L in the third, etc.

Yes, both of those apply in the examples I looked at, which is why the Timoshenko and Roark moments agree (to 3 S.F.), even though the tabulated factors are very different.

Roark's deflection results are different though, which suggests that his source for the deflection factors was different.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[IDS]

My results posted above, together with download links are now on-line at:

https://newtonexcelbach.com/2020/08/09/deflections-and-moments-in-rectangular-plates/

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/