The effect of internal pressure on a hollow cylindrical column. 1

[MechSwampEng]

I have a drill string that has an internal pressure. Along with this internal pressure they are going to compressing this string by pulling from the top. I know how to calc the critical load, and critical stress for 0 psi. Does anybody know of a method for determining the effect of internal pressure on the critical load of the column ? It is a fixed free, short column. The cross-section of the string is hollow circular(pipe).

 

[prex]

Didn't find a reference to your problem in my favourite books.
Trying to guess, I would reason as follows.
The calculation of the critical load is based onto a moment equation, the moment being that of the load with respect to column axis when deflected. Now this moment is unaffected by the pressure inside the pipe, so the conclusion is: the critical load is unaffected.
Another phenomenon to be considered is a possible change of the internal volume of the pipe due to the post buckling deflection. If the pipe is closed (not a piston), this change in volume would cause a change in pressure (depending also on the compressibility of the fluid) that would influence the critical load.
Just by gut feeling, I would say there is no change in volume, so no effect of pressure. However for a relatively thin pipe, the deflection would cause some ovalization, and cannot exclude that this would influence the critical load a bit.

prex

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[MechSwampEng]

Thank you for your response. However, when I do a minimum thickness calculation for pipe, in finding the membrane stress intensity you must calculate the principal stresses. One of these principal stresses is axial load. Of course when it comes to comparing the largest and the smallest stress, the axial is usually in the middle and therefore not considered. In my case, the axial load is over 100,000lbs with an internal pressure of 10,000psi. I have found documentation stating that in the case of a tensile load an internal pressure will actually fight the deformation of the pipe. But, I can not find anything regarding a compressive axial load. Tensile loads do not have the "Buckling" effect to worry about.

 

[MechSwampEng]

It seems to me that internal pressure, or the radial stress component, is trying to expand the pipe. When you compress a pipe, let's say a short one with no buckling, the OD increases. When you find a critical buckling load there is also a critical stress, if one were to determine this critical stress level it seems like you should be able to work backwards with the stress value created by pressure and determine the load that would bring the stress level up to critical ?

 

[prex]

Thinking a bit more to your problem, I seem to recall now that in the fabrication of rocket sections, with say 5 m diameter and thickness of may be 5 mm, they stabilize them during fabrication by a small gas pressure. However this has nothing to do with column instability, it is used to avoid local instabilities due to local loads caused by handling.
Column buckling is due to the following phenomenon. When the column starts to buckle, the deflection will have two effects: one is the elastic energy due to bending stored in the bar, the other one is the shortening of the column again due solely to the bending deformation, that will need a work to be done by the load.
When the elastic energy equals the work done, then the instabilty occurs.
As you see the axial stress in the column, contrary to what one would expect, has no role to play, that's why I think that the buckling load is independent of the internal pressure (or possibly with a very low dependency due to secondary phenomena).
If the buckling could be caused or initiated by a local instability, then the internal pressure would help in avoiding that. However this would be true particularly for thin cylinders, but I don't think you have a very thin one with that pressure.
However all the above is true for the so called elastic instability. When the plastic one is controlling (and I recall now that you mentioned a short cylinder), then you could be on the right path, as the plasticization of the section under the axial compressive load will not start, until the axial stress due to pressure has been overcome (provided your pipe is end capped).
However I'm not able at the moment to provide definite design suggestions for this. Of course you should account for a loss of pressure when an axial load is still acting. Also you should check that local yielding does not occur where the stress due to pressure is not present (cylinder ends).

prex

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[MechSwampEng]

Thank you for your reply Prex. The points you brought up are good ones and further add to my vexation. The lack of published information on this subject is frustrating.

 

[jistre]

Internal pressures in a capped tube will act to stiffen the tube and increase buckling resistance. High internal pressures in a capped system will push against the end caps and try to elongate the column, putting it into tension logitudinally. Since buckling requires longitudinal compressive stresses to occur, the logitudinal tension preloads have to be overcome before any portion of the column can be put into compression and approach the instability regions.

Here is a link to the test of a U.S. Patent awarded for a structural member made to reduce buckling using precisely this approach. As a bonus, at the end, he goes through the calculations showing the effect of the internal pressures in increasing the critical load of the column. Hope this helps, and good luck.

http://www.patentstorm.us/patents/5555678-fulltext.html

 

[jistre]

One caveat, this is for a typical slender column. I read 'drill string' and pictured precisely that and missed the "short" part. If the column isn't slender, then my statement and the patent linked do not necessarily apply.

 

[MechSwampEng]

Thank You jistre. This information is almost exactly what I needed. The column length and slenderness ratio will vary. I am simply attempting to create a method to deal with this situation. I told them to reduce the length in order to make it a short column until I had a better idea of how to go about this.

 

[prex]

jistre,
sorry, but can't agree at all with what you state.
Will you please revise my arguments above, where I state, in essence, that, for buckling to occur, an external load tending to shorten the column is required indeed, but the presence of compressive stresses in the column section is not required.
That patent is, in my opinion, simply phantasy and cannot work (like perhaps many other patents granted all around the world).
Would it be possible for you to indicate a bibliographic reference for what you state?
This problem is an interesting one.
By the way, MechSwampEng, sorry, I'm not so sure of what I stated above concerning the plastic buckling, as the phenomenon of instability remains the same, simply the formulae are no more the same, as Hooke's law is no more valid. Plastic buckling doesn't mean at all, as I implied above, full yielding of the cross section under the axial load.

prex

http://www.xcalcs.com

: Online tools for structural design

http://www.megamag.it

: Magnetic brakes for fun rides

http://www.levitans.com

: Air bearing pads

 

[MechSwampEng]

What I see in the patent makes some sense to me. I have dealt with some high torque-high pressure tapered thread connections that somewhat demonstrate this concept. These connections are torqued up to almost 90% of their yield strength so they have an immense amount of force holding them together. When a bending moment is applied, the moment has to overcome the preload of the joint before the two parts will seperate. My interpretation of Critical buckling loads and stresses, incorrect as it may be, is that once the stress levels in the part have reached this boundary there is insufficient support or strength retained in the part to resist lateral movement. I think that without extensive testing and formula verification that we could go back and forth on this subject.

 

[MechSwampEng]

For example: Say there is a lift eye mounted at the top of our column. A crane is applying a 100,000lb tensile load. There is also a weight hanging from the lift eye. (of course the weight would have to be centered and everything, I'm thinking in perfect world terms). Until the hanging weight reaches 100,000lbs there will be no effect on the pipe. Say that the critical load for that pipe was 20,000lbs. Would having to apply a 120,000lb weight seem to cause the pipe to buckle?

Of course this situation is completely theoretical and for the sake of discussion and further confusion. If this sounds like a problem that someone is having, please don't site me as a viable reference.....

 

[jistre]

MechSwampEng, I've thought more about this, and I believe that I mistakenly blurred the lines between two failure modes and have to retract my argument. My argument about prestresses applies to a situation when the material will fail due to yielding. Then, a prestress has to be overcome for the part to fail. The example is the reduced load bearing capabilities of an open soda can versus a closed one. The can ultimately fails due to yielding of its shell.

However, buckling failure is not because the material yields. The column under load fails in buckling because it becomes unstable and can no longer hold the load without a small lateral deflection. The load stays constant, but the column's bearing capabilities decrease with deflection, so there is a feedback loop and the deflection rapidly becomes larger and larger. The column can end up yielding in a buckling situation, but it can also buckle and not yield. Just squeeze a thin rod of metal slightly axially. You can make it buckle, then release it and it returns to normal. The two failure modes aren't the same mechanism.

prex makes a valid point in that the Euler buckling equation pretty much states that the critical load is a function only of the geometry of the part and its stiffness. The only way I can see for pressure to affect the critical load without changing geometry is to somehow change the effective stiffness of the system. However, I don't know if that's possible and since my argument was not based on that, it is invalid for this discussion.

prex is also correct about looking at that patent application very critically. It seems as if he may have made the same mistake I did because he talks both of buckling and of superposition of loads.