Roark's Circular Ring Formulas

[JStephen]

See what you all think about this-
In Roark & Young's Formulas for Stress and Strain, they give stress and deflection for circular rings. In the 5th Edition, that's Table 17, page 220. In the discussion prior to the table, they say, "By superposition, these formulas can be combined so as to cover almost any condition of loading and support likely to occur."

So suppose you have a circular ring with uniform outward radial load, and apply any number of equally spaced radial loads to it. Deflection for the uniform load can be handily calculated, and is simply a uniform growth of the ring. Deflection due to the equally spaced point loads is from Load Case 7. So far, so good. Load situation is shown in the attached sketch, Figure A, and predicted deflection in Figure B.

The catch is that the uniform outward radial load will tend to hold the shape round, but this is not reflected in the formulas for Load Case 7. IE, the ring with the uniform outward load should be much stiffer than predicted by Load Case 7.

Note that the uniform outward load could be replaced by a large number of outward point loads equally spaced and produce similar results.

Two questions, then:
Where exactly is the Roark formulation getting off track? It would seem they are making additional assumptions not stated.
How DO you calculate the deflection in this case?

 

[JStephen]

First question was where Roark's formulation got off track, and I think the answer there is it assumes small deflections AND small hoop stresses, but the small hoop stress limitation is not one of the items listed in the discussion on that section.
Second question was how do you calculate deflection in that case, which remains unanswered, except in a general sense.

 

[rb1957]

i don't think Roark ever got off track. he does not assume small hoop stress, considering ring axial load as hoop stress. he does assume a fairly large section that can develop sizeable internal moments.

where your application of Roark went off track is
1) your stubborn insistence to apply loads to deflected structures, which is not applicable in typical load superposition for rings. it would be quite reasonable to sum the internal loads and the resulting deflections from your individual loadcases.

2) mentioning hoop stress makes me think of thin shell structures rather than heavy rings. If you have a thin shell structure then simple superposition May be inappropriate.

do you have a structure in mind, or is this a thought exercise ?

Quando Omni Flunkus Moritati

 

[prex]

Roark, as most other texts on elasticity, assumes a linear behavior, sometimes together with more assumptions (e.g.thin wall cylinders). Linear behavior requires small deflections (i.e.the loads are applied on the undeformed structure), but in turn small deflections mean also small hoop strain and small hoop stress (the only stress of interest in a ring under uniform pressure). So what?
But also: what is the meaning of 'small hoop stress'? A deflection is small when compared to a characteristic dimension of the structure, a strain is small when it is a fraction of a per cent (at least for steel), and a stress? The only parameter for comparison is the Young's modulus (remember that linear elasticity does not consider yield at all).
And 'Second question was how do you calculate deflection in that case': which one? Assuming you mean 'deflections in a ring under pressure and point loads with large deflections', then there is no simple answer: I guess that a FEM calculation with a non linear code would be the only possible choice.

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[SAIL3]

Padd...I agree that the material properties itself in the sail are not changed...the stiffening affect is a result of the loading condition, tensile membrane stresses in this case...like a typical flag in high wind where the term "starched" is often used. Back to the original question...the term small deflection has been used but never defined. The more I think about it the more I am inclined to zero-in on the magnitude of the final combined stress and if that is less than the max allowable then I am not too concerned about how I got there. This is an extremely complicated theoretical problem when it comes to applying any valid theory to each step in the loading process.Typically the max allowable stress for rings has been conservative(say 15000ksi for A36) in recognition of the many unknowns involved. As far as deflection is concerned I would still use the combined final deflections of each load case. However, all bets are off on trying to figure out the state of stress or deflection in any intermediate step.

 

[JStephen]

Roark assumes that the ring "is of such large radius in comparison with its radial thickness that the deflection theory for straight beams is applicable", so it should be applicable to thin rings, to sections of infinite shells, to circular rubber bands, etc.

The simplest example I can think of to demonstrate the stiffening effect would be to pressurize a hose and then step on it. Does it squish easier (even for small deflections!) when it is pressurized? Cutting a unit slice out of that hose gives you a circular ring which reasonably well matches Roark's assumptions. Stepping on the hose is Load Case 1, pressurizing it is Load Case 7 with a large number of loads. And of course, Roark's formulas in this case predict that the hose is exactly as stiff unpressurized as it is pressurized.

My specific application: I was looking at ways to estimate the deflection in a tank shell due to thermal stresses at the legs. If the shell is more flexible, loads are lower. In this case, the shell is 90" radius, 0.375" thick, and calculated radial deflections were about 3/8".

 

[prex]

In your application you do not have a large deflection condition.
Hence, according to Roark (and linear elasticity), the radial deflection under a radial load is independent of pressure. But of course, if the legs restrain the radial deflection due to pressure, you'll have a load at the legs due to pressure only, that you should add to the thermal load or displacement.

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[RFreund]

Sorry if this was already stated as I tried to skim through this -

Wouldn't the deflection at any point be the deflection outward due to the uniform pressure minus the deflection inward due to the point load?
If you plotted the combined (aka superimposed deflection) around the ring you would get a more 'rounder' cylinder.
Maybe I'm missing something here.

EIT

http://www.HowToEngineer.com

 

[rb1957]

i think the OP's point is applying pressure to the non-circular ring (deformed by the point loads) will give a result different to applying pressure to the circular ring. equally, internal pressure "stiffens" the ring so it's deflection due to the point loads should be different from an unpressurised ring. both these are IMHO large displacement issues, and i think his structure (OP's post, three posts up) is small displacement.

Quando Omni Flunkus Moritati

 

[IDS]

It isn't just a matter of small deflections vs. large deflections. The point that has been missed (unless I missed it) is that in a closed pressurised ring system, such as a car tyre or a section through a closed hose, the application of a point load changes the internal pressure. You can't just add the deflection due to the initial pressure and the bending deflection due to the point load, because the pressure changes.

In the case of a vertical tube filled with a liquid and open at the top the pressure wouldn't change and you could use superimposition to find the deflected shape, provided that the deflections were small.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/