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# Nonlinear buckling doing Newton-Raphson method instead of Arc-length method

## Nonlinear buckling doing Newton-Raphson method instead of Arc-length method

(OP)
I am trying to do what is mentioned in this paper, except using arc-length method, I need to do it with Newton-Raphson method:

They use Lagrangian approach in this paper, which I will also do:

And arc-length method, but I need to do it with Newton-Raphson method:

Are there any papers out there that talk about nonlinear analysis (preferably buckling) using Lagrangian approach and Arc-length method? The paper mentioned above which uses arc-length method is named: "Numerical investigation of stiffness and buckling response of simple and optimized infill structures (Fengwen, Sigmund)"

### RE: Nonlinear buckling doing Newton-Raphson method instead of Arc-length method

What are you looking for - theory or examples ? If you need the former, just study how the N-R method works - its principles are the same whether the equilibrium path involves buckling or not. And if you need some examples then there are several benchmark problems that can be used to compare the performance of N-R vs Riks method. Abaqus documentation features some of them. They usually include snap-through and consist of various frames, trusses (e.g. Mises truss) and arcs. But imperfection sensitive cylindrical shell would be a nice case to study as well.

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