Beam Finite Element - Deriving the Geometric Stiffness Matrix

Deriving the geometric stiffness matrix for a beam with an axial load. This effect is important when analyzing large, geometrically nonlinear displacements, rotating beam like in the case of turbo machinery as well as for the buckling of columns.
The videos below are prerequisites for the current video. It will be helpful to watch them first.
Transverse Vibration Analysis of an Axially-Loaded Euler-Bernoulli Beam
• Transverse Vibration A...
Column Buckling
• Column Buckling (Conti...

[@Zachthesloth]

Hello sir,
Thank you very much for your work. I'd love to see an official treatment of differential stiffness for any kind of shell or plate element.
- Zach

[@APiccolo]

Man that helps a lot! Keep it up

[@TruongThanhChung]

Thanks

[@devrimmatematik6595]

Thanks for your great video, i understand how to get this geometrical stiffness matrix. One thing i want to say, can you make geometric stiffness matrix in case the beam has uniformly distributed load on it using von karman strains.

[@maurocasadio6244]

Thank you ! really helpfull. One question. Is there any bibliographie wich you recommend that explains more in depth this topic (large deformatino on beam elements)?.

[@Freeball99]

I don't really know of any textbooks that I would recommend for large deformation theory. A lots of this material can be found in articles and research material. Personally, I am a fan of using a co-rotational formulation to handle large displacements. Here is a paper that explains it's use for a beam element. www.dropbox.com/s/y50ohuzbvrzo1a2/Hsiao-A-Corotational-Procedure-That-Handles-Large-Rotations-1987.pdf?dl=0

[@devrimmatematik6595]

Reddy's book has covered that topic i ve mentioned in my previous comment but it is really complicated to understand for me. The book's name is Introduction to Finite Element Method at 4th chapter there is Nonlinear Beam section (pages 87-104). All the formulation exists there but i can't understand it :(.

[@ashwaniyadav4416]

sir ,
please make one video on geometric stiffness matrix for circular disc.

[@Freeball99]

This is likely going to take me a little while to get there, but it's a good suggestion. There is a lot of background material that I will need to cover before that.

[@abdelhakbenkaba1057]

thank you,
if the beam has a linearly variable section, does the geometric matrix change; if yes you can you help me

[@Freeball99]

For this case of an axially-loaded beam with infinitesimal displacements, if it is just the section that changes, then that in itself will have no direct effect on the geometric stiffness matrix so long as P remains constant.

[@abdelhakbenkaba1057]

Now I am relieved, thank you again

[@Freeball99]

@@abdelhakbenkaba1057 Just to be clear...while a variable section does not affect the geometric stiffness matrix, it does affect the regular stiffness matrix due to varying EI.

[@manhtuan2259]

What is "U,x", i don't understand it ? 0:53

[@Freeball99]

This is the derivative of the axial displacement with respect to x.

[@alexandermurysev5949]

Hello! That is a mistake in your calculations - when you put results from eq.2 to eq.1, getting eq.3, your (dw/dx)^2 becomes somehow (d2w/dx2)^2

[@Freeball99]

Yes, you're correct. It's a typo. Should just be the first derivative w,x^2 in equations 3 & 4. I corrected this when I pasted eqn 4 onto the next page.