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# Geometric Stiffness Matrix

Sun, 2010-03-21 23:11 - SivaSrinivasKolukula

Hi all

I want to know about Geometric Stiffness Matrix. When we calculate the stiffness of a body when the body is deformed then the stiffness called geometric stiffness. I have refered few books but I could not get the satisfactory details. I want to find the geometric stiffness of a fluid. I dont have any idea about geometric stiffnes. Any reference for this? Any article which explains this?

Sreenivas

Forums:

## Many years ago I was

Many years ago I was at the same point as you, and the reading of the following article helped me a lot.

http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6W6Y-44HXCK2-5...

It gives all the details to implement the geometric stiffness matrix.

Actually there is two methods the Total Lagragian (that recomputes everything with respect to the initial frame, which is presented in the paper) and the Updated Lagrangian (which update the strain-displacement matrices B wich respect to the deformed frame). The theory is well documented in the book Zieckwicz and Taylor for instance, volume 2 chapter 10).

Personally I turned myself on other methods like corotational methods that still computes linear stress-displacement relationship but in a rotated frame, where a rotation is computed from the deformation gradient. This is described in Hauth and al. 'Corotational Simulation of Deformable Solids". This method is fast to compute and sufficiently realistic for my needs.

My blog on research on Hybrid Solvers: http://mechenjoy.blogspot.com/

## Hi tlaverne

Hi tlaverne

Thanks for the references and spontaneous reply. I will go through and if I have any further doubts, I will take your help.

Inspiration and genius--one and the same.

_______________________________________

http://sites.google.com/site/kolukulasivasrinivas/

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Siva Srinivas Kolukula

Junior Research Fellow

## Hi I have refered

Hi

I have refered few other literature as well for geometric stiffness matrix. Geometric stiffness in case of bars, beams, shells etc are well derived and understood. I want to find the geometric stiffness matrix for fluids. How I can find it? I want geometric stiffness for fluids.............any reference for this?

Thanks in advance........

Inspiration and genius--one and the same.

_______________________________________

http://sites.google.com/site/kolukulasivasrinivas/

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Siva Srinivas Kolukula

Junior Research Fellow

## How to make an analogy

Hi Sreenivas,

Geometric Stiffness Matrix is often used in Buckling. It is called Stability Coefficient Matrix or Initial Stress Stiffness Matrix : This Matrix is added to the Conventional Stiffness Matrix in Buckling Analysis. The phenomenon of Buckling is implied by Compressive Forces which generates Bending Stiffness of the Structure and Bending is increased by Reversed Membrane Forces (Tensile) : the corresponding second therm is called Stress Stiffening. Corresponding Matrices are [K] and [Ksigma]. You have to verify if your fluid analysis correspond to a Buckling analysis with the reverse process in order to compute critical loads. The set of equations is ([K]+[Ksigma]) {delta d} = {delta R} with {delta d} : are variations of displacements

Mohammed lamine Moussaoui

## Hi Mohammed lamine,

Hi Mohammed lamine,

I understood the concept of Geometric stiffness in case of solids. I have refered few books which gave me sufficient idea about the geometric stiffness. But, all the books discussed Ksigma or Kg in case of solids. It is used for finding stability at the time of buckling. I want to find geometric stiffness for fluids. Yes, it is the case of buckling analysis. I have a partially filled cylindrical vessel excited verically with fcosΩt. Then i want to attain stability for the plane free surface of fluid. I am assuming cylinder as rigid. At some external excitation frequencies the free surface is unstable. I want to attain the relation between natural frequencies and external frequencies inorder to attain the stability criteria. From the buckling problems I got to know that, in these kind of cases Ksigma has to be evaluated, but I am unable to make it for fluid using analogy of solid structures. Any help or reference?

Thanks in advance.....

Inspiration and genius--one and the same.

_______________________________________

http://sites.google.com/site/kolukulasivasrinivas/

----------------------------------------------------------------------

Siva Srinivas Kolukula

Junior Research Fellow

## Good references for the geometric stiffness matrix

See:

Chapter 2 of http://scholar.lib.vt.edu/theses/available/etd-101198-161441/

Cook, 1995. Finite Element Modeling for Stress Analysis

Zienkiewicz & Taylor 2000, The Finite Element Method, 5th Edition, Vol 2: Solid Mechanics