Dear Song Mengli,

The post-buckling behavior is derived from a branching problem (2nd order theory). At a branching problem the balance is ambiguous. In the branching point leaves the system the primary path and goes into a secondary branch. The transition is performed by infinitesimal loads. A distinction is made in benign post-buckling behavior and malicious post-buckling behavior. To my knowledge, it has not yet succeeded to find an general proof of a minimal post-buckling load.

Good luck

Michael Lersow

Literature:

Henry L. Langhaar

tower, ASCE J. Eng. Mech., 96, 1970, 753-779 mit Boresi: Buckling and post-buckling behavior of a cylindrical shell subjected to external pressure, TAM

2 KB (157 Wörter) - 19:50, 4. Apr. 2015

This blog post might help you understand the possibilities of a post-buckling problem. http://www.comsol.com/blogs/buckling-structures-suddenly-collapse/

I think the main confusion is because, in most cases, when you perform a post-buckling analysis you introduce some type of imperfection (geometry, mesh, small force, numerical, etc) which creates a smooth transition curve with no sharp location to identify as the "buckling point". The smaller the imperfection, the sharper and smaller the transition zone.  Depending on the geometry and material hardening properties, the post buckling can be stable, unstable or marginally stable.

For an stable condition, you could consider post buckling when your force is larger the the critical force obtained from a linear buckling analysis (of course the safety factor comes into play to consider real life imperfections).

For an unstable condition you could use the same criteria as before or if you are sure your imperfections are quite small, you can define post buckling as the point past your maximum force.

Dear Song Mengli,

due to the response from Luis Alonso Quirós Fonseca, I would like to add something. Luis Alonso Quirós Fonseca has particular explains the influence of imperfection of the structure. But there are still other influences on the results of the buckling behaviors, among others of the simulation process, because, it is a branching problem , see below.

If this satisfied your question? I don`t know!

Supplements: At your question: "What is the criteria to define exactly what is post-buckling"?  first one should ask an another question: "of which structure"? and "how is sensitivity of imperfection of the structure"? First, I have enclosed a publishing of Jakob Weber, but in German, which answered these questions. Weber explained therein: The sensitivity of imperfection of a structure is depends from its post-buckling behavior. Therefore, there are three basic forms of the post buckling behavior: "symmetric stable", "symmetrical unstable" and "asymmetrical", see on left side of publishing.

The simulation of the buckling behavior of structures is one of the most difficult tasks of applied mechanics.There have prevailed 2 species of simulation of buckling:

Eigenvalue buckling and Updated-Lagrange-Methode (Geometrienichtlinearität)

With the Updated-Lagrange-Methode , see also Weber,  the reality can be represented more accurately. I have compiled this in one image. The Updated-Lagrange-Methode follows the curve on the right.

As can be seen, it is quite reasonable to refrain the definition of a safety margin.

I have attached a second publishing, of Herbert Schmidt, but also in German. On page 81 ff, right column, he describes both the "benign - local buckling" as well as the "malicious - global buckling" post-buckling behavior from experience with typhoons or hurricanes.  Therein, Schmidt also describes how imperfection of load-bearing structure is exploited by structural engineers. The "benign" post-buckling behavior is include in some guidelines thereby,  because for hurricane-like wind-bearing a strict buckling safety proof is not required, also in the international much applied guidelines for tank structures, such as of the American Petroleum Institute.

I hope that also this comment is for you helpful. All the best and Good luck!

Michael Lersow

Hello Ariadne,

You advise is very helpful. The second paper is used also by KIT (Weber). The three basic forms of the post buckling behavior: "symmetric stable", "symmetrical unstable" and "asymmetrical" are therein described and you can find there the felicitous phrase: “A division of elastic stability theory and its applications into separate categories of buckling and postbuckling is not entirely rational, but the creation of such an artificial distinction is almost essential to bring either into manageable proportions.”

Have a good day.

Michael Lersow

Dear Michael, thanks very much for your detailed explanation and supplements, it really helps! Dear Luis, thanks for your information as well! Dear Ariadna, thanks for your useful attachment, they all very valuable for me!

I will spend a little bit more time to focus on all your suggestion, and find more based on your recommendations, and get back to you if anything I find is still unclear.

Thanks very much!

Mengli

hi all

what is the different between ultimate compressive vs post-buckling?

1. Every material behaves distinctly, based on the nature of the load. The nature of the load can be tensile, compressive, shear, bending and torsional mode. A simple plate, for instance, behaves differently in tension (takes much higher load, before failure) to that of in compression (buckles at much smaller load), and so their allowables.

2. Ultimate compressive meaning the maximum load that a material can withstand before failure, in a compression environment. whereas, the Post-buckling is the concept of study of the response of a compression member, beyond the critical buckling load. They are highly non-linear in nature, which is evident from the P-Delta curve , quoted few shares above by Dr. Michael Lersow .