Centrifugal pressure, hoop stress, thick-walled cylinder 3

[RyreInc]

Greetings,

I've been tasked with finding the stresses on a cylindrical part, to be rotated at high speed. I don't need a precise answer, but a worst case scenario is a good place to start.

I've found a source (patent 5015940) that gives an equation for centrifugal pressure for a thin-walled cylinder, as well as a few sources that provide hoop stress for a thin-walled rotating cylinder.

However, wall thickness is 35% of radius, and I haven't had luck deducing a formula that combines pressure/force from rotation with a thick-walled body.

Intuition tells me that stresses will be greatest on the outer surface, and the stresses of a thick-walled cylinder should be less than or equal to those of a thin-walled equivalent. Are these assumptions correct? If not, how should I proceed?

Thanks!

 

[flash3780]

Doesn't matter what units you work in, you have to keep everything straight. I prefer to report stress in stones/furlong^2.

Aerospace design is still done (with the exception of NASA) almost entirely in imperial units. The reason is the expense of changing unit systems. You force tons of subcomponents into obsolescence (nuts & bolts, etc). All of your empirical correlations developed from years of engine building have to be converted (with the chance of error, mind you). Not to mention, all of your engineers have a feel for a particular unit system: Experience might tell me the approximate h-value along a turbine blade is in imperial units, for example. For SI units, you have to convert. It makes reviewing designs and analysis difficult for chief engineers who've spent a career learning those sorts of things. You can't exactly stop a design review meeting to run a conversion.

That's not to say that change can't happen. There just has to be a good reason for it. If, for example, foreign sources charged a premium when prints showed up in inches vs mm, engine manufacturers would think long and hard about making the switch.

 

[RyreInc]

Indeed, the real problem was that I had never heard of weight density before!

Back on topic: I'm expanding this problem to a cylinder made from two different materials (steel shaft, surrounded by impregnated nylon). My boundary conditions are As, Bs, An, and Bn (A & B for steel and nylon respectively). I need to define constraints on the system in order to solve for the four unknowns, but so far I can only think of three:

Bs=0, otherwise stresses are infinity at r=0.
radial stress at outside radius = 0.
radial stresses of steel and nylon are equal at boundary.

What am I missing?

 

[rb1957]

"Aerospace design is still done (with the exception of NASA) almost entirely in imperial units." ... on this side of 'the pond'. on the 'ther side they've managed to convert to metric

 

[unclesyd]

Here is the formulas used in a program that covers the whole
gambit, thermal, spin, interference, and axial for shrink fits.
This code was written for a program call Symphony but with a few Minor corrections it can run in Excel.
In your case set the inner diameter to 0.0 or a very, very small number.

http://my.engineering.com/pg/file/unclesyd/read/444170/spin

http://my.engineering.com/pg/file/unclesyd/read/444171/spin2

 

[unclesyd]

Here is graph of the stress from a spinning cylinder.

http://my.engineering.com/pg/file/unclesyd/read/444188/graph-of-stress-from-spin

 

[RyreInc]

unclesyd,

Thank you for these attachments; the form of your equations are more generalized and therefore more useful.

However, after using these formulas I get the same answers as when I treat each cylinder separately. The radial stresses at the middle radius (boundary between materials) are zero, and it seems like this wouldn't be the case...

Also, for F2 of outer cylinder, you continue to use Poisson's ratio for the inner cylinder. A typo I assume?

 

[unclesyd]

It does treat the cylinder separately as there is nothing to impose any stress, as in this case the spin. As long as the concentric cylinder are at zero clearance or a cylinder standing alone the stress are calculated at the inner and outer diameter or can be evaluated at in point between the two boundaries.

I'm trying to get all the essential equations to where you can download them.

 

[RyreInc]

Maybe not quite what you're looking for, but here's your equations as displayed in Mathcad:

http://files.engineering.com/getfil...290dc3dee&file=Rotating_cylinder_stresses.JPG

 

[unclesyd]

Here is a little more information concerning the methodology that was used in these calculations. It's been many moons ago since I looked at this but I think all the necessary nomenclature is there.

http://my.engineering.com/pg/file/unclesyd/read/444582/page-21

http://my.engineering.com/pg/file/unclesyd/read/444583/page-22

http://my.engineering.com/pg/file/unclesyd/read/444584/page-23

http://my.engineering.com/pg/file/unclesyd/read/444585/page-24

http://my.engineering.com/pg/file/unclesyd/read/444586/page-18

http://my.engineering.com/pg/file/unclesyd/read/444587/page-

16

 

[ione]

RyreInc,

Further to the boundary conditions necessary to solve your problem you have to consider that you do realize a shrink fit and a pressure generates at the interface between the two cylinders. The inner cylinder is subjected to an external pressure and the outer cylinder is subjected to an internal pressure. Now the radial displacement comes into play, that is what you lack in your boundary consitions.
You have to consider that the interference ?, necessary to realize the shrink fit, is equal to reduction of the external radius of the inner cylinder plus the increase of the internal radius of the outer cylinder.

 

[RyreInc]

ione,

Thanks, that is indeed the information I wasn't considering. I did think it had something to do with this, but hadn't been able to resolve the particulars. You've defined it quite nicely.

This is where the problem gets tricky, since the nylon was injection molded over the steel, and I don't know the details of this aspect. I might have to call it good enough at this point!

 

[unclesyd]

I would try to use the Nylon properties in the Spin program and look at the results. i would take the boundary conditions as if the Nylon was pushed on withe a very small interference. You might also plug your data into the [a} and [c} equations on page 21 of the essential equations.

http://www.tamshell.com/mds-nylon6-30gf.htm