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# non linear analysis

I am trying to learn non linear FEA ( Newton Raphson method)and this is what my understanding is. Kindly advise if you think my understanding is correct.

1) At the beginning of the analysis, I am using a small load step and using the current stiffess matrix ( which I am designating as initial tangent stiffness) I am solving for the initial displacement step for this load.

Here my question is: How this initial tangent stiffness is calculated?

The way I look at it:

- This is the stiffness considering the elastic properties , the geometry and the " assumed" condition of the contact in case of contact analysis.

- This analysis can be done using either non linear geometry consideration or linear geometry condition although it is better to use non linear geometry.

- During course of this analysis if the "assumed " state of contact has changed and if the computed stress is post yield point, the stiffness matrix has to be recalculated based on the plasticity concepts and/or the load step may have to changed to "allow" for the old state of assumed contact or consider the new state of contact ( in this case load step need not change)and further run the solution till the time the state of either elasticy/plasticty and or the assumed state of contact has not converged.

2) One step (1) is completed, the internal forces are computed based on the solution of the computed displacement step ( to do this, the current state of dislacement is multiplied with the "current" stiffness matrix) and the difference between the internal and external forces are computed to find the residual.

3) With the residual and the new stiffness matrix ( this is the "final" stiffness matrix from the last step which was used to cpmute the internal forces) again start the step(1) above till the time the residual does not become within a tolerance limit ie close to zero

Kindly advise if my understanding is correct.

Regards

## nonlinear analysis

I think you generally have the right idea. There may be a few things that could be stated better in your explanation, but generally it sounds about right.

One thing I would recommend is the following. You say that you calculate the internal forces by mutiplying the current state of displacement times the current stiffness matrix. This does work. However, some day in the course of making computer algorithms it may be better to calculate your internal forces elementwise based on the integral of (transpose(B)σ) dV, which is how you will see most FE books say to do it. I have done it the way you mentioned, but there were sometimes circumstances when it was better to do it with the integral method for each element, transform the resulting forces to the global level, and then assemble the global internal force vector in order to calculate the residual.

You might like to see a tutorial that I have on my homepage which describes the 2D co-rotational analysis of trusses. There are some nonlinear analysis algorithms in the paper. It is for the case of geometric nonlinearities but material nonlinearities can be inserted at the relevant locations as well. By carefully following my algorithm you might see what I'm talking about. I also list some references that are good for different nonlinear analysis methods such as load control, displacement control, arc length control and generalized displacement control.

Here is the link to the paper I'm am referring to - http://people.wallawalla.edu/~louie.yaw/Co-rotational_docs/2Dcorot_truss...

The best way to learn this is to program it yourself and see how it works. The first few chapters of Crisifield's book vol 1 was quite helpful when I was starting to learn nonlinear analysis with NR iterations.

good luck,

Louie