I briefly glanced this article. Seems interesting but I haven't read it yet - hopefully I'll still be able to help.
First, lets say a few words regarding the non-linear buckling analysis. It is important to realize that contrary to linear analysis, it takes into account the following:
- Geometrical non linearity - the model shape changes according to the
applied loads (deformation) so the load distribution through the
model is updated accordingly. Step by step
- Material non linearity - As the stresses reached the yield point,
the YOUNG modulus changes and so is the model stiffness matrix.
Yielded zones will not carry excessive loads anymore and the load
distribution will be updated
- Imperfections - small deflections that make a difference between a
"clean" FEA model to a real life case
In the sense of it, buckling modes are the shapes a body will tend to wear while being subjected to the a specific load pattern. There are shapes for bending loads, torsional loads and so on. For each mode there is critical load the model could not further withstand and that will eventually cause it to buckle.
While running a buckling analysis, we apply loads on a model and examine its behavior. We would be able to obtain the deflection it gets and to check whether buckling occurs under these loads.
If we run a non-linear analysis, it is a good advice to add a small imperfection, in the shape of the relevant mode (i.e, a shape the will contribute to the deflection under the specific case). Do not forget that physically and mathematically, the model will first begin to deform when linearity governs (small deflections, constant stiffness) - so it will try to deform into one of the modes that can be acquired with linear buckling analysis.
Hope this helps...