Rectangular Pressure Vessels

[jpd2005]

Can anybody tell me where I can find a mathematical proof of the stresses/deflections induced in a rectangular thin wall vessel subjected to an internal pressure?
Many thanks for your help in advance.

 

[jte]

jpd-

Not sure what exactly you mean by "mathematical proof" but you'll find some guidance in ASME VIII-1 App 13 and some in Roark's Formulas for Stress and Strain by Young (you can evaluate the sides as flat plates fixed at the edges).

The problem you'll find is that the mathematical formulations are most likely linear and limited to deflections on the order of 1/2 the thickness of the plate. In reality, it is likely that a thin wall rectangular vessel will have deflections on the order of a few thicknesses of the plate. This puts your situation into the nonlinear world where you will need to iterate a solution which is deformed geometry dependent. Presuming, of course, that you don't go materially nonlinear (exceed yield) first at the corner joints.

jt

 

[JStephen]

Note that the fixed plate edges are really only applicable if the vessel is square. If rectangular, one side is stiffer than the other, and the corner joint should rotate when pressurized.

 

[jpd2005]

Thanks to jte and JStephen for their comments. I was looking to see if anybody had applied castigliano's thin curved bar theory to the problem to a quarter of the tank cross section and had come up with a proof or example I could work with. It's been some years since I done complex integral calculus, which I think is what is needed.
JStephen is right about his observation of applying thin plate theory to the problem, i.e. only deals with square sections.

 

[arto]

"if you have to use calculus, your doing something wrong" go for App. 13

 

[jte]

Regarding square vs rectangular: Naturally, the corner moments are unequal with a rectangular section. So design it as though it was a square with sides equal to the long sides of the rectangular. Any reason that would not give a reasonable, yet conservative result for most common aspect ratios?

arto-

Yup. Or, as one of my former supervisors once said "Whats this big S doing here? Give me some calculations I can understand!"

jt

 

[rmw]

I don't know where you would find it, mine is an old original chemical paper xerox copy but there is a magazine article that gives all the formulae. It is;

Designing Rectangular pressure Vessels (Subtitled; for maximum strength with minimum material)

Charles J. Ladfe (may be labfe-poor copy)
Assistant Mechanical Engineer
Pittsburgh Works
Allis Chalmers Mfg. Co.
Pittsburgh, PA

I can see that it is on page 151 of Machine Design Magazine dated May 08, but I cannot make out the year. If I had to guess based on the poor copy, I would say 1971 or 1941.

It does footnote a 1940 reference for a McGraw Hill publication titled "Theory of Plates and Shells". The authors name is very difficult to read, but might be Timowhanko. (that gets most of the letters that I can read.)

For it to have entered my files, either of 1941 or 1971 would be valid. It would have had to have been pre '73.

Hope this leads you somewhere. Where I worked at the time we had to deal with rectangular pressure vessels, so this information was near and dear to us.

rmw

 

[unclesyd]

rmw,

The name is "Timoshenko" as the author of the book.

 

[rmw]

Thanks Unclesyd,

It was a combination of an old bad copy on old bad paper and old bad eyes.

I couldn't make it out.

You got any guess on the date of the "Machine Design" article?

rmw

 

[JStephen]

"Theory of Plates and Shells" was updated after 1940 (in 1959, I know, possibly later). I bought a new copy in 1985 or so, and still have it. I like Timoshenko's writing. He did a "History of Strength of Materials", a 2- volume Strength of Materials book, "Theory of Elasticity", "Theory of Elastic Stability" and "History of the Theory of Elasticity" among others. Some of the footnotes in "Theory of Plates and Shells" refer to his earlier papers written in Russian.

"Regarding square vs rectangular: Naturally, the corner moments are unequal with a rectangular section. So design it as though it was a square with sides equal to the long sides of the rectangular. Any reason that would not give a reasonable, yet conservative result for most common aspect ratios?"

If it's square, edges are fixed against rotation by symmetry, and maximum bending is at the edge as on a fixed-fixed beam. If edges are free to rotate, maximum stress is 50% higher and at the center (based on beam analogy). If edges are semi-fixed, it should fall somewhere in between those two cases. I think your assumption that this is a conservative design is valid, provided you realize that you don't know where the maximum stress is, and provided you use the same thickness on the short side. Note that the corner rotation on the short side means that on those sides, the plates are actually bent against the load at the ends.