Linear and Non-linear Buckling

[sushi75]

Hello everybody,


I've have to deal with stability which requires a buckling analysis. Howevert, I don't really have a clear idea of the difference between linear and non linear buckling!

What's the difference, and/or how can we decide wether or not it's linear or non linear?

thanks a lot for your help on that topic!!

 

[dhengr]

Tomstress :
You are probably talking about elastic and inelastic buckling. The first works/buckles within the elastic limit of the parts and generally returns to its original shape when unloaded. The second goes beyond the elastic limit of the material, does not return to original shape when unloaded, and quickly gets out of control with little additional load. Look them up in some good Advanced Strength of Materials or Theory of Elasticity text books. There are also a number of good texts on this particular subject, along with many advanced degree theses done on the subject. Dig into them, you’ve got a lot to learn, this is not a subject that lends itself well to a simple formula or a one paragraph forum reply. You should be discussing this with a senior engineer at your workplace.

 

[ESPcomposites]

Linear or nonlinear buckling can be a function of material nonlinearity, as dhengr mentioned. For example, the effect of plasticity significantly affects the buckling load for short columns or intermediate length columns.

However, nonlinear buckling can also be primarily a function of geometry/loading (and essentially independent of material nonlinearity). The classic example is that of a long beam-column (an axially loaded column with a moment), which has a coupled axial/bending nonlinear effect that an eigenvalue (linear) buckling analysis can not properly solve. In such a case, a geometrically nonlinear FE buckling solution is appropriate (or classical analysis). While you can use material nonlinearity, it is not necessary since once the outer fibers stresses reach the yield stress, they increase rapidly with increased load (essentially independent of material plasticity for long columns). Note, as stated above, if the column is short, the effect of material plasticity is significant.

You can perform a geometrically nonlinear buckling analysis to determine the linear buckling solution (should be your first trial). This requires a bit of thought of how the problem works though. If you were to apply just an axially load to a column and perform an nonlinear buckling analysis, you would not observe "buckling". You need to also apply some small amount of "destabilization" to the model. In real life, this is due to imperfections of the structure, minor load eccentricities, etc. When you then apply the axially load (in a nonlinear manner), each step will increase the moment/eccentricity in an nonlinear increasing manner. Eventually, you reach a reach a state where the yield stress is exceeded. At this point, the question of how to address material nonlinear comes up. If the column (or plate, etc.) is "long" then it largely won't matter how much plasticity the material has (and you will have determined the equivalent of the eigenvalue solution). However, if the column (other) is "short" then the effect of plasticity is significant.


Brian

http://www.espcomposites.com

 

[BUGGAR]

This is sort of a side avenue but here goes.

I'm designing a stiffened aluminum plate beam. It's a deep beam where shear controls. The stiffeners increase the shear buckling resistence but I am still designing the beam to function in the buckled range. My stress/strain curve is basically linear up to shear buckling of the web. As the web goes into buckling, the the slope of the curve flattens but still maintains a slight upward slope. Then the web goes into shear field tension (acts like a tension diagonal truss chord between the stiffeners) and the stress/strain curve again increases in slope up to material failure.
This is an example of restrained buckling, vs. the unrestrained buckling of a column.

Ps.: Aircraft wing spars are designed like this.

 

[ESPcomposites]

Right, that is another example of a geometric nonlinearity...followed by a material nonlinearity. Another classic example for plates is for a uniaxially load with all edges simply supported (attached picture).

http://files.engineering.com/getfile.aspx?folder=278e792c-ab80-4ffc-91af-9f383fbe284d&file=plate.jpg

Brian

http://www.espcomposites.com

 

[moon161]

A pic of a buckled shear member (wagner tension field)

http://inrisk.sites.olt.ubc.ca/files/2012/11/Wagner_tension_field.jpg

 

[sushi75]

Thanks a lot for all the replies, very helpful.

Looking at the eigen value problem for buckling, it looks like the one for modal analysis.

Is there any way to make a comparison or analogy between the mass matrix in modal analysis with the geometric stiffness matrix for buckling?

Thanks!