Geometric Nonlinear Analysis - Misc. Questions 4

[Celt83]

I have a few areas where I am hitting a mental block with Geometric Nonlinear analysis, I am long out of school and trying to expand my knowledge on these analysis methods and also dabble in programming a solver which I've found to be the best way to get intimate knowledge with these things.

Questions:
1. For loads between nodes how are these incorporated into the typical Newton-Raphson solution process? Is the linear elastic solution done as a first step used to determine the nodal forces to then enter into the nonlinear phase or are the fixed end forces carried into the nonlinear steps?

2. For building structures where deformations are expected to be "small" should the elastic stiffness matrix be updated in each nonlinear iteration utilizing the deformed geometry or only the geometric stiffness matrix?

3. Are there any references out there that provide a full example of the solution and force recovery process or just any other books that cover the subject? (I have Matrix Structural Analysis 2nd Edition by McGuire Gallagher Ziemian and recently found Analysis of Geometrically Nonlinear Structures by Levy, however neither of these seem to address loads between nodes)

 

[FEA way]

I think that you should start with "Non-linear Finite Element Analysis of Solids and Structures" by Crisfield. It's one of the best books on this topic.

 

[IceBreakerSours]

I like Klaus Jurgen Bathe's book. Years after reading his book, I found out that MIT threw an excellent couple of lectures series by him up on YouTube. I found those really valuable.

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[Celt83]

FEA way: Thank you for the book suggestion will look into that one.

IceBreakerSours: Thank you, I didn't think to look in Bathe's book. I purchased that one a bit ago but have not begun an in-depth study of it yet as I wanted to build up a good knowledge of the direct stiffness method for frames before jumping into the more general finite element methods, seems I'm getting into topics now where I should probably make the leap into building up a foundation on the FEM side which I imagine will also help me with some other items related to energy/work, approx. functions, etc.

 

[IDS]

I like:
Programming the Finite Element Method
Author(s):I. M. Smith, D. V. Griffiths, L. Margetts

https://onlinelibrary.wiley.com/doi/book/10.1002/9781119189237

Very practical, with detailed code that works, and just enough theory to understand what is going on, and also good coverage of modelling soils, which is useful for those involved with structures where soil-structure interaction is important.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[Celt83]

Thanks IDS I got a great deal on the Smith and Griffiths book along with their numerical methods book about a year ago and just recently stumbled onto Dr. Griffiths website which has the full fortran code for both books: [URL unfurl="true"]https://inside.mines.edu/~vgriffit/[/url]

I don’t recall them having a section on geometric nonlinear analysis but I only skimmed the book at the time will need to take a more thorough read through it. I also purchased a fortran book recently to try and get a better handle on the code presented in some of my older references.

 

[IDS]

Celt83 - from a quick look at my paper copy (3rd edition), program 4.6 is "Stability Analysis of Frames" and says:
This is achieved by modifying the element stiffness matrices to account for axial loading using the "geometric" matrix as described in Chapter 2.

Sounds like it could be what you are looking for.

On the coding side, I did some work with F2py a few years ago, but had some problems with compatibility with the latest Python at the time.

It might be worth looking at the current version though.


Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[IDS]

My latest blog posts on calling Fortran from Python (2016) don't use f2py:

https://newtonexcelbach.com/2016/07/17/linking-excel-to-fortran-via-xlwings-python-and-ctypes/

https://newtonexcelbach.com/2016/07/27/linking-excel-to-fortran-with-ctypes-update/

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[Celt83]

Thank you IDS, looks like they changed that program in the 4th version of the text and it is now a beam buckling example with a single point load at the end node.

So far everything I’m finding that addresses stability does not speak to how the fixed end force vector is incorporated when there is loading between nodes.

Much appreciated for the links on calling Fortran from Python.

 

[centondollar]

"Questions:
1. For loads between nodes how are these incorporated into the typical Newton-Raphson solution process? Is the linear elastic solution done as a first step used to determine the nodal forces to then enter into the nonlinear phase or are the fixed end forces carried into the nonlinear steps?"
If your load is not dependent on deformation, this does not matter, and the procedure is as for a normal element. Add point forces or moments to nodes, or calculate equivalent nodal forces and moments from force/moment distributed in the element, and add these to the force vector.

"
2. For building structures where deformations are expected to be "small" should the elastic stiffness matrix be updated in each nonlinear iteration utilizing the deformed geometry or only the geometric stiffness matrix?
"
The basic geometric non-linearity with moderate rotation and infinitesimal strain is based on the total Lagrangian approach - the reference frame is the original geometry, and the "deformation" is accommodated by updating the tangent stiffness (axial-bending coupling), which causes axial and bending displacement to change at each load step and iteration. The end result, after all load steps, is axial deformation and bending deformation and rotation with respect to the original geometry.

The stiffness is the tangent stiffness. You may use only one elastic stiffness (same for all load increments and iterations, unusual), the tangent stiffness updated only at the beginning of each load step (modified Newton-Raphson, common), or the tangent stiffness updated at each iteration in each load step (Newton-Raphson, common, but computationally more expensive than modified N-R).

"
3. Are there any references out there that provide a full example of the solution and force recovery process or just any other books that cover the subject? (I have Matrix Structural Analysis 2nd Edition by McGuire Gallagher Ziemian and recently found Analysis of Geometrically Nonlinear Structures by Levy, however neither of these seem to address loads between nodes)
"
The non-linear FE book by Reddy comes to mind. It shows the geometrically non-linear formulation for beams (Euler, Timoshenko) and shells, with detailed descriptions of the system of equations, tangent stiffness, locking phenomena etc.

 

[Celt83]

Thank you for the information Centondollar.