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Buckling Analysis
3

Buckling Analysis

Buckling Analysis

(OP)
I created a simply tube made up of plate elements and ran two buckling cases:

1.  One end is fixed
2.  One end is pinned

Now according to Eulers simple equation for linear buckling, the effetive length changes with the type of constraint.  In Nastran, both cases yield the same eigenvalue.  Am I doing something wrong?  Does Nastran account for the type of constraint like the theoretical formula?

Thanks,
Brian

RE: Buckling Analysis

You are comparing apples and oranges.  The Euler formula is for a column that is either pinned or fixed on the end - the buckling mode for a column involves lateral displacement of the entire column cross-section.  You have modelled a cylinder and the critical buckling mode is likely local displacement of the cylinder wall in a number of waves around the circumference and along the length.  If the cylinder is relatively long the fixity around the circumference at the ends will not make much if any difference.  Plot the buckling mode shape from your FE model to see the difference between the two cases.  Also, NASA SP-8007 is a good reference for cylinder buckling behavior.  It has data to illustrate the known fact that actual buckling loads of cylinders can be significantly lower than the theoretical eigenvalue buckling loads, due to effets of load and geometric imperfections.

RE: Buckling Analysis

(OP)
I see.  That makes more sense now that I am looking at the buckling mode shape.  Now is there a way to incorporate a correlation factor into Nastran so that the analysis it performs is not invalid?  Hand calcs work fine if there is a simple load case, but not for a complex geometry.

RE: Buckling Analysis

You can either factor by hand the eignevalue results using the correlation factors, or you can run a nonlinear analysis which incorporates inital imperfections.  The trick with the latter is selecting the appropriate level of imperfection; often times you ahve to run several cases of different imperfection magnitude and type in order to determine if the response converges.  Some FE codes automate the use of the eigenvector mode shade as the initial imperfection for the nonlinear analysis.  There are a bunch of NASA papers and reports related to buckling of shell structures available on the NASA Langley site, http://techreports.larc.nasa.gov/cgi-bin/NTRS and http://techreports.larc.nasa.gov/ltrs/

RE: Buckling Analysis

2
Reference the NASA document SP-8007.  Look for it on the internet.  I think the NEiNastran website knowledge base has it as well.  This is the Bible for cylindrical shell buckling.

RE: Buckling Analysis

(OP)
Found it.  I have one question so far about the simple isotropic unstiffened cylinder under axial compression.  When you solve for Nx, the result is an axial load per unit width of circumference.  What is this exactly in lamens terms?  

RE: Buckling Analysis

Nx is a line load.  NASA SP-8007 “Buckling of Thin Walled Circular Cylinders” was written mainly for the analysis of rockets and spacecraft which use unstiffened cylindrical shell structures.  It is the Bible here where I work (Delta rocket program).  Note the formula has the recommended 0.6 knockdown factor in it (gamma).  For the basis of comparison to FEA leave this off.  The FEA (using linear buckling analysis) does not account for this.  You would multiply it by what ever result you get from FEA to "knock down" the result to account for actual imperfections which trigger buckling earlier.

Using equation (4) in SP-8007 you have the critical stress.  I would use this and then

Sigma-x = axial stress
E = Young’s Modulus
nu = poisson’s ratio
t = thickness
r = radius

to get the total buckling load, just multiply by area (2*pi*r*t).

When you run an FEA linear buckling analysis, the lowest "positive" mode (eigenvalue) times the total applied load is the buckling load.  Multiple this by your knock down and you have the max allowable load for the structure.  Knock down factors are more critical in cylindrical shell buckling than for other types of structures.  SP-8007 should cover this.  
 

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