How to explain the limit load by stiffness matrix using finite element analysis?

Maybe you are using Newton-Raphson iterations with load control.  If you have a structure that softens or has the behavior of snap through (and possibly snap back) then load control will not work, it will likely crash at the peak load and the analysis will stop or have trouble converging.

One possible solution is to use displacement control instead.  Or, you may want to use more advanced techniques such as arc length control or generalized displacement control which can trace load versus displacement paths beyond peak points.

Crisfield's book volume 1 talks about some of the reasearch that has been done to monitor certain characteristics of stiffness matrices when they reach peak points.  His book also talks about different control schemes for nonlinear analysis.

 You may like to look at a tutorial I have written which gives load control and displacement control algorithms and also gives references to arc length control and generalized displacement control techniques.  My tutorial may be found at the following location:  http://people.wallawalla.edu/~louie.yaw/Co-rotational_docs/2Dcorot_truss...

You may also like to look at the following location (see chapter 5 in particular and Part III) for nonlinear finite element analysis information:  http://www.tu.kielce.pl/kis/mo/COLORADO_NFEM/colorado/

 I hope this helps,

 Louie

You have good questions.  I have wondered about the answers to your questions also.

 For the case of snap through the answer is somewhat easy.  If for example you load an arch with a point load at midspan then for certain arches you will get snap through.  Usually, such arches are shallow and as the arch is loaded downwards at midspan, there is a load level at which the arch snaps (dynamically) downward so that the arch is now inverted.  Usually, in finite element analysis, this is done by a displacement control scheme under a static incremental iterative procedure.  So the path that is traced has certain portions that are not really static, but in real life would be a path occuring during a dynamic transition from one stable shape of the arch to another stable shape of the arch.  The same thing may happen for shells.  So although we do this as a static analysis, there are certain parts of the analysis that in reality would happen dynamically (unless, of course you tested the structure in a testing machine and loaded it by displacement control very slowly, then the test would mimic a finite element analysis with displacement control).  That is my understanding of snap-through.

The case of snap-back is more difficult.  Physically, I'm not certain if I completely understand why it is happening.  Snap-back may happen for arches and shells just like snap-through.  However, it tends to happen for deeper arches or shell structures and seems to happen also more for structures where the load is offset from midspan of the arch.  For a snap-back analysis in FEM you must use something like arc length control or generalized displacement control.  For an arch, after reaching the peak load the structure may try to snap-through but may also have a temporary dynamic decrease in vertical displacement as the load level also decreases.  The snap back path is just a representation of a possible equilibrium path (that is an unstable equilibrium path).  Again, in reality the snap-back portion of the path would be dynamic rather than static in real life if the structure was free to transition between stable configurations.

It would be good for you to ask others also about this.  I'm still learning about this too.  I have reached some of the above understanding by reading journal articles and books, however, the answer to why snap-back is happening is not clear to me.  The following reference is the best discussion that I have seen so far of snap-through and snap-back

Zdenek Bazant and Luigi Cedolin, STABILITY OF STRUCTURES, Dover, 2003 (pages 278 to 284).

 Also, Crisifield discusses these phenomenon briefly in a journal article and gives reasons why an analyst might bother to determine the equilibrium path for structures that behave this way.  You may want to see the following.

M. A. Crisfield, "A fast incremental/iterative solution procedure that handles snap-through", COMPUTERS AND STRUCTURES, vol. 13,(June 1981), pp. 55-62.

I hope this helps.  I think this is a fascinating subject.  Stability and buckling of structures is very interesting.

Good luck,

Louie

For an analytical determination of the snap-through and snap-back in the localization problem, please take a look at the paper

"On constructing the analytical solutions for localizations

in a slender cylinder composed of an incompressible

hyperelastic material" by Hui-Hui Dai, Yanhong Hao, Zhen Chen

 http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6VJS-4RFSD0R-5&_user=14084&_coverDate=05%2F01%2F2008&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000001598&_version=1&_urlVersion=0&_userid=14084&md5=c6af30754606b249b504a7f660ea49c8

The work may not help you for using FEM to capture the snap-through and snap-back but may be useful for understanding the nature of these phenomena (as Louie mentioned "...why an analyst might bother to determine the equilibrium path for structures that behave this way"

Hui-Hui