Small or large deflection theory in buckling pipe/cylinders? 3

[BuckTU]

Hi fellow Engineer,

this question (two parts) concerns buckling of thick-walled pipe/cylinders under axial compression. An answer will vanish my engineering-blok...

(First)Could someone tell/explain me what the difference is between small- and large-deflection theory Or where I can find it on the internet, because the papers/books I've read are a bit vague about this topic. And (second) have these two theories got anything to do with the incremental and deformation theory?

BuckTU

 

[CESSNA1]

BUCKTU:
Small deflection theory is what most engineers use. it simply means that the deflections are small compared to other dimensions. Small deflections are assumed in the derivation of the elastic equations and engineering theory. For example if the deflection of a loaded beam is less than .01 its length then the deflection is considered to be small. Large deflections are over that. Obviously there is a big gray area where small deflection theory is no longer valid and large deflection theory becomes valid.

Large deflections usually fall under the definition of non-linear analysis.

Buckling theory is separate from these. There are various theories on buckling that are out there.

As a simple example consider a cantiler beam with a load at the free end. As the load increases the beam will deflect more and more. In small deformation theory we assume that the tension and compression on the top and bottom of the beam at the fixed end are equal. We assume that if either one reaches the yield point the beam will "fail". In reality however, what usually happens is the the bottom of the beam (in compression) will buckle and fail initally; below the yield stress of the material.

Regards
Dave

 

[BuckTU]

Thanx Dave (CESSNA1),

But,

has anyone investigated the Incremental and Deformation Theory concerning wrinkling or buckling of thick walled cylinders?? And could someone tell me were I can find material, because most researches mention it, but they do not explain it.

BuckTU

 

[Cockroach]

Yes, we have extensive experience with your concern. Look for Von Mises-Hencky Equation in the literature specific to Pressure Vessel Theory - Thick Wall.

Boresi-SideBottom do a reasonable job of it in Advance Strength of Materials, there are other textbooks available that can be found on various websites.

You need to develop the stress gradient specific to your application. For example, wall loading adds to longitudinal stress beyond the reaction of end caps subjected to internal pressure. This would be a typical example NOT covered in most advance textbooks, but the Von Mises-Hencky Theory addresses it quite well. Also, you can add shear terms into the equation, typically these are neglected when principal loading of hoop, radial and longitudinal pressure vessel equations are applied.

You're getting into some very pretty mathematics. Actually this is one of the more interesting problems engineers are faced with. Good fortune!

Kenneth J Hueston, PEng
Principal
Sturni-Hueston Engineering Inc
Edmonton, Alberta Canada

 

[mtnengr]

Re: BuckTU question

Dr. David Bushnell has written several papers on the subject of Incremenal versus Deformation theory. He used BOSOR5 as his main investigative tool. In producing a PC version of BOSOR5, I have compiled all of Bushnell's papers on a CD. If you need a particular paper, I can send it to you.

Thick cylinders have h/R < 30 (shear stress and normal stresses through the thickness begin to have some influence). At this point, the order of the governing differential equations goes from 8 to 10. It is not until h/R < 5-10 that "thick" shell effects become really important. BOSOR5 uses thin shell theory.

Most FEM 2-D solid programs do not investigate, non-axisymmetric loads and then investigate any non-linear geometric coupling of stresses and rotations. You can use NASTRAN, or similar hog, to produce your own analysis method. This requires you to understand the mechanisms of second order coupling of strains and displacement plus the introduction rotations into the equilibrium equations.

I might suggest you review my paper: "The Inelastic Analysis of Shells of Revolution Subjected to Nonaxisymmetric Loads.", ASME: Recent Advances in Structural Dynamics- PVP Vol. 124, pg 13-25. The assumption is that thin shell theory is being used.

 

[MikalT]

I'm writing a piping utility to solve for pipe span deflection and stress at support points. What are the formulas (equations) to solve for stress and deflection of span pipe with considerations for corrosion allowance, insulation density, etc.? Assume pipe filled with water. I know that there are many variables to consider such as friction forces at support points, etc.

Where can I find a table for moduli of elasticity for various grades of pipe... I have the section moduli and moments of inertia for the various pipe sizes and wall thicknesses.

Thanks!

 

[BuckTU]

Dear MikalT,

Your question will have more effect, if you make your own thread, because it will show-up at NEW threads and not the old ones...Its a Tip.

And what do you mean with "span pipe"?

BuckTU.