Slender steel column: Linear FEA vs Moment Magnification

[ElliottJames]

Apologies if this seems like an elementary question, but I haven't been able to confirm the answer searching here or elsewhere. Given: a slender steel column with a downward vertical force P at the top and a sideways force P, also at the top. Assume the material is in the elastic region and deflections are not large.

If statics equations are used to find the moment at the base, then AISC code says apply moment magnification equations to adjust the result to account for the P-Delta effect. Those equations are basically a fudge to make first-order analysis a good approximation of what a second-order analysis would achieve. That's fine, no problem there.

Now assume the column is modeled in a _linear_ FEA program, with the column refined into many segments. The result shows the expected deflection curve. Does this result include the P-Delta effect because the FEA calculation automatically considered the P-Delta moment when it solved for the equilibrium solution that balances nodal deflections and strain? Therefore one would not apply moment magnification equations to this linear FEA result? (I understand the best answer would be to use non-linear FEA. What I'm asking is the linear FEA result "good enough" to be used in lieu of the moment magnification approximation).

 

[JoshPlumSE]

It all depends on the program. You said that this was a linear FEA program, right? That would imply that it does NOT consider 2nd order effects, because that is an inherently non-linear effect.

That being said, some programs will use a hand-calc type of moment magnfication method. Others will include an option for P-Delta analysis (though this would make them something more than a pure linear FEA program).

 

[ElliottJames]

@JoshPlum: It's a linear FEA program. And maybe this gets at the heart of my thinking: that linearity (Hooke's Law) holds for each segment. Now as the column is broken up into an increasing number of segments, the accumulated nodal deflections and strains act as a limit function and approach the desired non-linear result, in this particular case anyway.

 

[sponton]

Nope, it most likely doesn't consider second order effects. The FEA code will probably solve equations of equilibrium, compatibility and boundary conditions and display the displacement from the initial position. That's it, it doesn't iterate many times having as the original state the displaced structure for this new iteration, FEA code that takes into acccount second order effects would continue to do this until the change in stress and displacements between positions at tn and tn+1 are basically negligible [t being time]

 

[JoshPlumSE]

Ultimately you have to know your program. There are plenty of "benchmark" problems that you can test against to see if the results are the "correct" non-linear results or not. I'm specifically thinking of the ones in the commentary to AISC 360 (2005 or 2010 version).

So, your best bet is to test it out yourself to be sure. From your description, it is not likely. But, the only way to know for sure is to test it out.

 

[bookowski]

Run one model with a single element and another with divided elements. If it's linear you will get identical results, if it's performing p-d you won't. If you are already sure it's linear then breaking it up does nothing, analysis would be based on a single iteration of the undeformed configuration.

 

[IDS]

The underlying assumption in the standard stiffness method used in frame analysis programs is that deflections are small, and do not affect the stiffness matrix.

It follows that a linear analysis does not include the effect of deflections on the structure geometry, and moment magnification factors must be included.

If the program can do non-linear analysis it is possible to specify linear elastic material properties, but the analysis still needs to iterate to find the deflected shape, taking account of the deflections.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[ElliottJames]

@bookowski wrote: "Run one model with a single element and another with divided elements. If it's linear you will get identical results, if it's performing p-d you won't."

Of course, thank you for the suggestion. Set up a simple model of a 20-foot tall HSS 2.50x0.125 A36 steel tube. 100 lbs of lateral force at the top and 3.1 slugs mass (100 lbs weight) also perched on the top. Results:

[pre]Segs Disp X(in.) Disp Z(in.) Moment @Base(ft-lbs)
1 24.09641 -0.00117 2000
5 24.09641 -0.00117 2000
50 24.09641 -0.00117 2000[/pre]

The only thing that changed was the plot of the tube, it went from straight to a nice curve. Okay, I'm convinced, the P-delta effect is not reflected in the results. Thanks to all for your replies.