Derivation of blank thickness

[ghensky]

Does anybody know the derivation of (B31.3, paragraph 304.5.3) the minimum required thickness for the following blank equation: t = d * ((3*P /16*SEW) + c)^0.5, which can be further simplified to (assuming E = 1, W = 1, and no corrosion allowance) t = d(3P/16S)^0.5. So far I managed to get t = PD/4S (not sure if the approach is correct). Thanks in advance.

 

[jte]

That formula was developed by Timoshenko from, if I recall correctly, work done by La Place in solving the equations. It's the same formula used in B16.48. This is a linear approach to a hugely nonlinear problem, and has been demonstrated to be quite conservative.

More details about this is available in two papers, the Journal paper is essentially the same as the conference paper.

http://proceedings.asmedigitalcollection.asme.org/proceeding.aspx?articleID=1570641

http://pressurevesseltech.asmedigitalcollection.asme.org/article.aspx?articleid=1459628

 

[ghensky]

Thanks jte.

 

[Ehzin]

I'm not positive on the methodology they used to derive that equation but I believe this is the approach that was used.

Roark's 7th Edition, Table 11.2, Case 10, Uniformly distributed pressure from ro to a.
The equation provided for special cases is M = KMq(a^2), when r0/a is 0 then Kmra = -0.125
The above factor is assuming that the end plate boundary is fixed. Not necessarily true but it could be considered that the portion exposed to pressure is "fixed" within an outer ring which is the excess blind that sits outside of the gasket diameter.

Plugging this into the mentioned equation then gives a max moment occurring at the gasket interface of M = 1/8*qa^2.
Finding the related stress due to bending is S = 1/8*P*(r)^2 * (t/2) / (b*t^3/12)
Substituting D/2 for r, recognizing b as a unit width so replacing it with 1 you get the following, S = [12*P*(D^2)] / [64*t^2], rearranging for t and reducing gives, t = D[3P/16S]^.5

I don't have a great familiarity with this particular section of Roark's but I believe the above approach makes sense and is what was done to arrive at the B31.3 equation.

Any comments?

Thanks,
Ehzin

 

[jte]

Roark's Formulas for Stress and Strain, or simply Formulas for Stress and Strain by Roark if you have fourth edition or earlier, is a wonderful resource. But it should be recognized that at its heart it is not a text book covering basic principles but a compilation and simplification of results and it is in this that the book is an excellent work.

If you are serious about deriving the formulas, get a copy of Theory of Plates and Shells by Timoshenko and Woinowsky-Kreiger. Advanced Mechanics of Materials by Boresi, Shmidt, and Sidebottom also covers the derivation of formulas for circular plate behavior in part 13.9 of the 5th edition.