Nonlinear Buckling of Cylinder under External Lateral Pressure 5

[crbeebe]

I am performing nonlinear buckling analysis using COSMOS/M of long thin-walled cylinders under external lateral pressure, using the eigenvalue buckling mode shape as the initial geometric imperfection.

I (perhaps naively) was expecting the resulting load-deflection curve to have a nice sharp knee indicating the buckling load. Instead, I get a nice asymptotic curve that has no discernable slope break point. Does anyone have a recommendation on how I can determine the actual buckling pressure from these results? Or, is there a better way for performing the non-linear analysis?

Many thanks in advance.

 

[TGS4]

What is the magnitude of your initial displacements (for initial imperfrections)? Also, what non-linearities are you modeling (geometric and/or material)? Also, are your loads following the deformed geometry?

If you were to create a simple beam buckling problem, a geometric non-linear solution should closely approximate Euler's buckling load. Just be warned that once you reach a buckling load, the solution will diverge, so you will know the buckling load from the load immediately prior to the divergence.

Cheers!

 

[BATman2]

Crbeebe,

Have you tried a hand calc. from references suchs Roark's Forumla for Stress and Strain? In Rev. 6, check chapter 14, Table 35, Item 19, Thin tube UNIFORM external Lat. pressure. At a minimum you could revise your model to see how the hand calcs match the FEA to give you an idea of where you stand. Note the requirements for r/t and L. Does your case meet this?

It sounds like the tube is going plastic instead of buckling. I would check the Boundary conditions and Mat. Props. again. Is the length correctly modeled?

Not an expert in the non-linear stuff, but just my 2-cents.

Batman2

 

[corus]

Euler's method doesn't give the same answer as a large displacement solution. In fact Euler's method is only an approximation and doesn't give a conservative result.

If you have an asymptotic curve then you could try plotting the relationship between force and displacement and then fitting a line of best fit to the data. For an asymptote you'd expect the curve to be of the form y=b/(x-a) where y tends to infinity at x=a, the buckling load.

corus

 

[bvi]

Regarding the nonlinear analysis, it sounds like you have reached a limit load the cylinder is undergoing plastic collapse. You can consider the load at which the computer fails to converge, or any load less than that, to be the limit load.

As a side note, if you do axisymmetric shell eigenvalue solutions using COSMOS/M, the last time we did this the solutions were unconservative for the lowest couple modes.

 

[crbeebe]

TGS4,

Answers to your questions:
-Largest magnitude of your initial displacements (for initial imperfrections)is 0.03 (R=5, t=.21). So largest init disp is about 14% of thickness?

-Only geometric non-linearity is modelled.

-Loads do not follow deformed geometry...direction and area considered for pressure loading are constant.

This buckling problem is a tube collapse (not an axial column buckling)--picture a toothpaste tube.

Batman2,

I've performed the hand calcs, and they match very well with the eigenvalue buckling solution. So, I'm pretty sure the geometry, BCs, and MPs are okay.

Corus,
You got me thinking, and I ran the analysis further than before to determine the asymptote value of the load factor (I had stopped at about .25 deflection before, thinking that would be adequate). The resulting load factor-deflection plot produces a local maximum load factor at about 320 psi and then the load begins to decrease…acts like a snap-through problem. (I wasn’t expecting that. Does that sound reasonable for this type of problem?) Furthermore, the hand calculation and eigenvalue buckling analyses predicted a buckling pressure of about 450 psi, which is quite different from the 320 psi I get from the NL analysis.

Any suggestions??

 

[crbeebe]

bvi,

Yes, I found the same thing with an axisym model. I'm now running a 1/8 shell and solid models with better results.

Thanks!

 

[btrueblood]

Try varying the relative size of the initial geometric imperfection (i.e. make it smaller, say 5% then 1% then 0.5%), and see whether the "knee" of the curve asymptotes to the same value. What you describe sounds similar to what happens when you analyse (for collapse) a column that has an initial moment or eccentricity in it - the beam doesn't buckle, but has a more "rounded" force-deflection curve.

 

[corus]

crbeebe,
The difference in the predicted buckling load is typical of the difference between Euler's method and NL analysis in that the NL analysis gives a lower load, and as such Euler's method is not conservative.


corus

 

[bvi]

I would think the difference between the nonlinear analysis and the eigenvalue solution is that you have included an initial imperfection. The degree of initial imperfection affects the buckling pressure.

 

[crbeebe]

All,

Thank you for your help. The magnitude of the initial geometric perturbation was the key. I started with a magnitude of 14% of the wall thickness. After your suggestions, I ran with various magnitudes from 14% down to .06%. The knee of resulting load-deflection curves became sharper as the perturbation decreased, which makes sense...the cylinder is getting closer to the perfect circular shape. There was not much change in peak load factor below about 0.5% thickness.

FYI, at 14%, pcr=320 psi; below 0.5%, pcr=360 psi. Big difference due to magnitude of the initial geometric perturbation. Even so, these values are much less than the eigenvalue solution (431 psi).

Again, thanks for the help...I learned a lot.

 

[BATman2]

Crbeebe,

One last thought. Can you conduct a test? It seems you now have a few answers to the same problem.

A had a situation with a large tubular structure that was intended to buckle under a specific load. The NL analysis report one value but that actual test showed a much higher buckling load. The tube had different cross sections along its length, resulting in a buckling initiator at the transition. The NL analysis apparently underestimated the resistance at that transition. The test proved the real strength of the part. (That same test is going to be repeated on the same part from another supplier very. Comparing results should be interesting.)

Batman2

 

[jhardy1]

It is a well-known fact that the actual buckling capacity of thin walled shells is extremely sensitive to physical geometric imperfections, actual boundary conditions, and actual load distribution.

Typical "real" thin-walled shells often have actual buckling captures in the order of 1/2 to 3/4 of the theoretical "perfect" elastic buckling loads. If the shells are manufactured to very tight tolerances, it is possible to achieve actual buckling capacities very close to the theoretical "perfect" buckling capacity.

With respect to support conditions, the buckling capacity of an axially loaded thin-walled cylinder with a "built-in" edge support at the base can be twice that of the same shell with a simply supported edge.

The buckling capacity calculated for a uniformly applied pressure load can be 50% to 100% higher than the calculated buckling load with a slightly varying load with the same overall magnitude. Also, small perturbing forces (e.g. small lateral loads on an axially loaded column) can significantly reduce the calculated collapse load.

The allowable stress limits in design codes for thin-walled shells (e.g. pressure vessels, etc) generally include an allowance for real-world fabrication tolerances etc, so often appear to be very conservative when compared against classical "perfect" buckling theory. If your actual construction tolerances are worse than implied by the respective codes, your actual buckling capacity can be significantly impaired.

When calculating buckling capacities for real-world shells using FEA, you need to make sure your model makes due allowance for geometry imperfection, load variation, perturbing loads, actual support conditions, etc - or you run a very real risk of over-estimating the load capacity.

 

[crbeebe]

Batman2: Thanks for the suggestions. This is an experimental program. These structures will be tested, but no production is planned.

Julian: Thanks for the advice. Good points all.

Now I have closed form solutions (Donnell, Timoshenko, etc.), eigenvalue solutions, and nonlinear solutions. All make sense to me in that the eigenvalue soln's are in line with the theoretical values, and the non-linear soln's predict lower critical buckling values.

HOWEVER, I've just run across another equation for collapse pressure in Roark's (not the Table 35 equations, but an equation found in Section 12.5 of the 6th ed). This equation produces critical buckling values that are much lower than Roark's Table 35 and even the non-linear soln's. Not much said to explain why there is such a large difference from theory...I need to look up the sources. Anybody have experience with this equation? Perhaps just an ultra-conservative bound? I'm not looking for a safety margin...I need to know the actual buckling pressure as accurately as possible.

Thanks!

 

[mtnengr]

Nonlinear analysis of shells of revolution can be a very tricky thing. Using a general FEM program such as Ansys, Nastran or Abquas may or may not give you the correct answer. You have to be very familiar with the characteristics of the program for the loading and geometry being considered.

I have used in the past speciality programs such as Bosor5, Monsap and alike, simply because they were designed for this class of problem. You still have to understand their behavior under the conditions posed.

Imperfection geometry is even more difficult to comprehend. I can supply a host of references in this area. Two that I have authored are: WRC Bull. 313 Apr. 1986 and Pressure Vessel Design PVP Vol 57, pp 49-66. There's more about the subject that I can expand upon, but defer until later.

 

[fkmeyers]

Be careful of codes like BOSOR5. It is an axisymmtric code and should not be used for anything with discontinuities in the theta direction. We had a failure at McDonnell Douglas during a test of a rocket fairing because someone used this instead of a detailed Nastran analysis. The fairing had 3 seperation rails that were assumed continuous but were not. That was a very expensive mistake. But if the structure is truly axisymmetric then BOSOR will be very accurate.

 

[mtnengr]

The originally stated problem was a nonlinear analysis of an axisymmetric cylinder subjected to a lateral pressure. The structure is assumed to be axisymmetric. Of course if a three dimensional structure is being analyzed, then a three dimensional shell analysis program should be used. My choice would be STAGS. This program is an extension of Bosor5. Many of the procedures are based upon the same ideas as in Bosor5. Certainly one could use NASTRAN, but again I must caution for the analyst to understand the behavior of the computer code being used.

The other programs that I had previously mentioned in the WRC Bullten are simple axisymmetric cylindrical shells with stiffeners included. The loadings are assumed to be axisymmetric or torsion. Three separate methods were provided. The purpose of these programs were to demonstarte the effects of modal coupling, ie. responses of harmonics N +M and N -M that could produce lower buckling- imperfection values than a single mode.