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# Finding Stress in a Round Flat Plate

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## Finding Stress in a Round Flat Plate

(OP)
I am doing stress calculation on a round flat plate welded to the end of a pipe.  So I looked in "Roark's Formulas for Stress & Strain" 6th Edition.  I went to chapter 10 “Flat Plates” and checked the examples.  In example one (1) they found the largest moment and then plugged that into the stress for a bending moment equation which was STRESS=6M/t^2 where M was M subscript ”c” which I assumed meant the center of the plate; t was the thickness of the plate; ^2 signifies that “t” is squared.  My question is if STRESS=Mc/I how did they derive the equation for this round plate to be STRESS=6M/t^2?

My best guess is they used I=(1/12)bh^3 for a rectangle where b is the base; h is the height and ^3 signifies h is cubed.  Substituting into STRESS=Mc/I and moving 12 from the denominator of the denominator and up to the numerator which gave them the equation STRESS=12Mc/bh^3.  Saying c=(1/2)h  and substituting it into the equation turns the equation into STRESS=12M(1/2h)/bh^3.  Multiplying 12 and ½ gives STRESS=6Mh/bh^3.  Then canceling out the h on top and bottom gives STRESS=6M/bh^2.  Then saying at the center b becomes zero gives you STRESS=6M/bh^2.

If this is not correct let me know. Also, if anyone knows a source to quote for the bending stress on a flat round plate let me know.

### RE: Finding Stress in a Round Flat Plate

If b is a unit width, it takes the value of 1 and falls out of the equation.  Perhaps that's what happened.

### RE: Finding Stress in a Round Flat Plate

Benensky,
You are taking M to be the moment.  It is actually the moment per unit length, which corresponds to (M/b) in your derivation above.  Hence there is no need for any statement such as "at the center b becomes zero" (whatever that might mean).

### RE: Finding Stress in a Round Flat Plate

Also note that b cannot be taken as a unit width (ie, 1" wide) because the stress continuously varies across the plate- it is not the average over a 1" width that is figured, but the moment per width at a point.

### RE: Finding Stress in a Round Flat Plate

Benenensky, follow the example on page 393, example 1.

You need to not only define your circular plate constraints, but also the "plate constant" or D, which figures into deflection, hence maximum moment at the edge of the plate.  The maximum moment is obviously at the edge where the deflection is zero, conversely, maximium deflection at the center where moment is zero.  This gives you the required boundary conditions on a fourth degree ordinary differential equation, hence the means to reduce solve the equation(s) for a closed solution set.

The expression you're concerned about is the result of general theory for plates not covered in the Roarks book.  This is simply a receipe text for engineers skilled in the art.  I have found Timoshenko, Elements of Strengths of Materials, circa 1935, to have a complete argument on solutions sets involving deflections.  There are others.

Anyways, you gave no numbers so it is hard to define your exact uncertainty in this case.  Good luck.

Kenneth J Hueston, PEng
Principal
Sturni-Hueston Engineering Inc

### RE: Finding Stress in a Round Flat Plate

Is your plate a column base or a flat head on a pressure vessel?

### RE: Finding Stress in a Round Flat Plate

(OP)
unclesyd:

flat head on a pressure vessel

-benensky

### RE: Finding Stress in a Round Flat Plate

You will have to design your PV under from the codes or standards recognised by your jurisdictional authorities.  In the US this usually the ASME Section VIII.
Comeback with a little more information as to the design conditions and you will get a little better direction.

### RE: Finding Stress in a Round Flat Plate

Firstly, flat circular plates are uncommon on pressure vessels.

If you are interested in the theory behind the design philosophy, you may delve further than Roark, but in case you only wanted to design a one-off piece, then Megyesy should be the right place to look.

My two cents, though.

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