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Geometric non-linearity:Incremental (Euler) solution

 Hi, 

Geometric non-linearity:Incremental solution

1.I've been reading the book by Crisfield on Non Linear finite element analysis and have just started with the first chapter wherein the author introduces geometric non-linearity.

I have some questions on the incremental (Euler) approach here:

I want to know whether my inetrpretations below are correct:

1)The incremental approach makes use of the tangent stiffness which relates small changes in load to small changes in displacement.

2)The "LOAD" is incremented in steps.

3)At each step we calculate the incremented displacement which is Kt^-1 * del W

where:

Kt -> is the tangent stiffness matrix

del W -> load incremented in each step

3)The tangent stiffness matrix for each step is taken as that one obtained from the previous step / iteration

4)The procedure is continued till all the load is applied

5)Here:the difference in graph between the exact and the incremented solution brings about (reflects) the difference in equilibrium.

Questions>>

A)Are all my interpretations(1,2,3,4) above correct?

B)I am not clear about 5- I've mentioned that the "the difference in graph between the exact and the incremented solution brings about (reflects) the difference in equilibrium"---which mathematical step reflects this point. I do not follow how does this difference reflected in the mathematical equations?

Any comemnts will be appreciated!

kajal  

 

 

Peyman Khosravi's picture

Hi Kajal;

Your interpretations are correct. However I should add some points. Tangent stiffeness matrix is not only calculated based on the current shape of the structure, it also contains the effect of curent internal forces. i.e. a tangent stiffness matrix is the sumation of the stress stiffness and structural (or material) striffness. In simple words, if an element has compressive internal force, its stiffness matrix reduces, until it finally buckles.

In the method you mentioned, no itaration is used, so little by little you diverge from the actual equilibrium path. the reason is that for each load step (or increment) you should repeat the process using e.g. Newton-Raphson method until you fall to the real path, and then apply the next load step. If you are not following what I say, consider this. If linear analysis is not exact for a problem, why should we get an exact answer for each increment if we are useing again only one linear analysis for each load increment? So you see that a small error happens at each load increment which then adds up and finally little by little you diverge from the real path.

Regards

Peyman Khosravi

Thanks Peyman for the reply.

However, my question is- in linear analysis the resultant equations so obtained always  always satisfy equilibrium-in incremental procdure as above-what is the reason that we have diverted from equilibrium-surely-incremental procedure is more superior than linear analysis.

Peyman Khosravi's picture

I think if you look at figures 1-3 and 1-4 (pages 6 and 7) of the vol.1 of the Crisfield you will get your answer, however I am not sure if I have understood your question correctly. Please explain more so I can help you.

Thanks

Peyman Khosravi

Thank you Peymann--i get the point

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