Deflection of circular plate with hole 1

[Timoko]

Hallo,

i would like to know an approach to this problem. i am stuck and can not find how to relate the angele in the solution because it is circular.

- please chek Photo

Best Regards
Timo

 

[JStephen]

iI don't really understand your diagram. But regardless- with circular plates loaded and supported axisymetrically, see the circular plate formulas in Roark's Formulas for Stress and Strain. For non-symmetric load cases or supports, you can often make some approximations (yield line analysis, etc.) to show adequacy, but getting a reasonably accurate estimate of deflection would be likely be a finite element problem.

 

[Timoko]

Hallo Stephan,

i guess this is advanced Strength of material problem or as you said can be as 1d Element (FEM) represented. However, Roakers formulas are simple and can not be approximated to this problem because:

- a/b Ratio is too small
- non-uniform cross-section

i can solve this problem when it is beam. However this is circular corss section and the second moment of inertia should be integrated with respect to Theta. i am lost!

Thanks in Advance

 

[SAIL3]

FEA would be an appropriate design tool for this problem...if that is not an option, I would try to approximate the solution using an upper and lower-bound approach..
I am assuming there is only one conc load on the inner lip from the OP's sketch....
1. LOADING.
assume area of influence of the conc load is approx 1/4 of the circle....so I would find an equivalent
uniform load by dividing the conc load by the length of the inner arc of 90 degrees and apply this
equivalent uniform load to the whole inner lip....and use Roarks formulas.
I might even try a 45 degree area of influence and divide the conc load by that inner arc length to get
the equivalent uniform load....and then take the average of both approaches..
2. MATERIAL.
Case 1. Assume "d" is the thickeness of both the length "a" & "b"...solve..
Case 2. Assume "c" is the thickeness of both the length "a" & "b"....solve..
add 75% of the difference in results between case 1 and case 2 and add it to case 1....one can vary the 75% depending on the length of a and b.....
material cost should not be a factor in this unless one is making a 1000 of them so I would tend to be conservative in my assumptions...engineering judgement will be involved ,as it should be in every engineering problem...

 

[racookpe1978]

On that small a edge plate that thick, with that large a internal diameter, you'll be very close to make a linear assumption to start with.

 

[rb1957]

without any dim'ns we're just guessing ... you might approximate the problem to a beam1 unit wide, cantilever at one end ... but that's clearly a conservative assumption.

you could consider the smaller ring (the one with the load) to be supported by the larger ring ... ract the applied point load around an arc ... +-30deg ? +-45deg ? +-90deg ... uniform reaction, cos(theta) varying reaction ... reacting the applied shear, and a bending distribution reacting the moment. Then guess some more with the larger ring ...

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

 

[racookpe1978]

Are members (rings ?) "a" and "b" welded together then welded to "c", or are they all made out of one piece with the "fixed "c" ring?

 

[Timoko]

Hallo,

@ SAIl3:
-your approach is correct for certain accuracy. Roakers formulas are useless for an Engineer who want to understand the origin of these formulas. i dont want to use blind approach without analytical derivation. and in the internet i couldnt see derivations of roakers formulas!
-I am not talking about cost of material and production. this problem pop in my mind and i wanted to know how to solve it step by step !!ANALYTICALLY!!; it is not a projekt or ...
However, Thanks in Advance!

@racookpe1978

As explained i am not talking about dimensions i need a solution as function of (Roh, E, I, a,b) i am not taking values seriously i need to formulate or derive to get the solutions which may look like: deflection=EA/3I*(a/b).... and so on. but thanks at least you have tried.

@rb1957:

good Engineer dont need dimension to derive strength of material problem. These are variables a, b,....so i need solution as function of(Roh, E, I, a,b). but thanks at least you have tried.


@: No they are all made of one piece.

 

[Timoko]

Picture update is here

 

[shaneelliss]

I think you'll find that most people on this forum are a lot more practical in how they would approach a hard problem then a professor at a university might be. Most of us wouldn't bother to derive the exact formula when an approximate analysis that is a little conservative will get us by in the real world. I am not saying there is nobody here that could derive this formula for you, but I would be surprised if it happened, because even though it is kind of an interesting problem, it isn't interesting enough for most people to spend a couple hours or days deriving a formula for a stranger on the internet, especially when that stranger has said it has no practical purpose for him/her.

I would model it as a cantilever beam with two different I's, using an average width for each part of the stepped beam, with a proportionally wider cross section for the deeper part of the beam. It won't be an exact answer but I am a practicing engineer, not a professor.

 

[Timoko]

Hallo shaneelliss,

i think we miss the definition of Enginnering. if you are Practicing Engineer who copy formulas and apply them to find approximate solution with reallity and you sure thats work. then good luck but please be sure to do test your component befor it is used by humans. i wish that you will not heart anybody with your (blind) approach without real understanding as Practicing Engineer which in my opinien not Engineerig rather than copying.

regarding derivation, i am sure that no one will spend a day driving it for me but this was not my question. i need to see approaches of different people!.

Best luck with your Practicing Engineering. I hope you are not working in R and D field becuase this is really dangerous.

 

[shaneelliss]

No, I am not in R&D, so you don't have to worry about anything blowing up because of my "dangerous" methods.

However, if you think that simplifying hard to analyze problems into easier to analyze problems that give you approximate but safe answers is not engineering, you are wrong. Almost everything engineers do is finding solutions that will approximate reality.

I gave you the approach I would take and with a spreadsheet and small enough increments, it will give you an answer that will be close enough to the exact solution that it won't matter that it is approximate.

 

[Timoko]

that is good to hear, well this is another part of the story, offcourse we as Engineers want to make things simple as Possible, but Not simpler as Einstein said. This means lets go deep in theory understand it then if we have the key of theoretical knowledge then we can open any door without complexibility. so never be as an Oil drooplet in glass of water. you will be floating in the top without having deep understanding. Sometimes you will face Problems in life where there is no formula to apply to find the approximate solution so you have to develop your own!.

 

[pperlich]

Timoko,
I think you need to relax a little bit. It is often not practical for practicing engineers to find exact solutions to every problem. There is nothing wrong with making reasonable AND conservative assumptions. This is called engineering judgment. There is nothing unsafe about it, if it is done properly. It is a noble endeavor to try to find exact solutions but so, so many real life problems are too complex to ever get an exact solution.

Best of luck.

 

[racookpe1978]

i think we miss the definition of Enginnering. if you are Practicing Engineer who copy formulas and apply them to find approximate solution with reallity and you sure thats work. then good luck but please be sure to do test your component befor it is used by humans. i wish that you will not heart anybody with your (blind) approach without real understanding as Practicing Engineer which in my opinien not Engineerig rather than copying.

To the contrary. Explicitly the to the contrary.

EVERY analysis in "physics" begins with extreme simplifications and approximations. In the 'real world" of real metals and alloys and composites and rocks and dirt and sand, NOTHING behaves like it does in the pretty FEA models. True, the latest models begin to approximate realty if pure metals are involved and pure reaction forces and perfect crystals and "average properties" are indeed "average" over the whole structure which has been perfectly modeled in all details.

The "by the book" designers use the book because it is (usually) safest and (usually) has been written and tested in practice by experienced engineers who have both seen the failures in real life and the successful designs in real life. It is the FEA modeled designs and CAD prototypes and grad school "projects" and ivory tower research-granted that fail much more often. True, a "by the book" "read the book" designers are not always the best. They can be wrong in some circumstances - and are seldom the least expensive nor the fastest, most efficient. But I would trust them over an untried mouth-of-the-CAD machine model. NASA has failed too many times recently dong just that. We used to lose rockets and missiles and satellites because we were testing them, trying to find out what could go wrong. Now? We find out things can go wrong when they do fail in Mar's atmosphere, when their programs use the wrong altitude, when the missile is rounding Venus, on the way to Jupiter, etc.

Is the little simplistic ring above correct? No. because no real world force actually acts on a single circle as you show with the green arrows all perfectly uniform around the circumference but with zero surface area that compresses the metal at its points of contact. Will a 1000 (unit) IS and 1100 OD double ring that is 10 units high behave differently than a 10 unit ID and 11 unit OD that is 10 units high? Will a 1.0 unit ID and 1.1 unit OD double ring 1 unit high behave differently than a 10 unit OD that is 100 units high? Will a foam rubber double ring behave differently than a marble double ring? Will that marble double ring with all of its internal stresses and flaws behave differently than a pure glass or ceramic double ring?

 

[fel3]

EVERY analysis in "physics" begins with extreme simplifications and approximations: spherical chickens.

==========
"Is it the only lesson of history that mankind is unteachable?"
--Winston S. Churchill

 

[fel3]

"EVERY analysis in "physics" begins with extreme simplifications and approximations." (racookpe1978)

To wit: spherical chickens.

==========
"Is it the only lesson of history that mankind is unteachable?"
--Winston S. Churchill

 

[Timoko]

@ pperlich: nice advice!. no worries i am quite relax). do you think that one can reach Exact solution?? well offcourse NO. The plate it self is simplified so at least i need an exact solution for this simplification. if i simplify the simplified solution i will get rubish. Rather than that FEM anyway is an Approximation of a problem since Elemnet dimension can not be Zero!. so Please read all what i write then you will get my point better. Thank you for joining the discussion.

@ racookpe1978: nice advice. you have written something which is obvious for most of Engineers. However as you said the Circular Plate is simplified and exact analytical solution is possible to get. simplifying the simplified solution gives rubbish. so no need for wasting time in producing rubbish. rather than than very simplified solution of the plate i have it and its not enough for the moment i need to go deeper. But story you have written i totally agree with it.

@fel3: thats true but spherical chickens analysis is completed. Time to go deeper.

so lets all remember that Engineers must make things as simple as Possible, but Not SIMPLER as Einstein said.

 

[seaaggie]

@fel3,
I thought it was spherical cows?

Thanks,
Mark

 

[fel3]

Timoke…

I went to Fresno State, which isn't exactly known for its physics program. All they could afford were the chickens.

Fred

==========
"Is it the only lesson of history that mankind is unteachable?"
--Winston S. Churchill