Indeterminate beam analysis with FEM 3

[technodam]

Hi,
I want to know the analytical procedure with the use of finite element method (i.e. matrix method) to find bending moment, shear force and deflection of an indeterminate (no. of unknowns are more than no. of equations) beam. I know two of the methods viz. force method and three moment method. But these methods can't be generalized for large number of elements and/or forces. so, it will be very useful to have a FE method to solve the mentioned problem.
Thanks!

 

[JAE]

This would take a lot of time and probably more space than an online forum can offer. Sounds like you need to simply take a class in this or simply go buy a good textbook on FE methods.



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[rb1957]

three moment equation can be extended to a large number of supports; how many supports are you dealing with ?

moment distribution can solve a beam with many supports.

to use FE is simply, build a model of the beam. you can build the stiffness matrix of the beam by hand (get a structures text to figure out the stiffness matrix for a beam element, maybe NASTRAN user manual, maybe a post-grad text), and solve the FE problem as a "finite difference" problem.

another day in paradise, or is paradise one day closer ?

 

[ThomasH]

Hi
What do you mean by "no of unknowns are more than no of equations"?
Usually I would say they are equal.

And I agree with JAE, you probably need a good textbook or some class on the subject.

Regards
Thomas

 

[rb1957]

What do you mean by "no of unknowns are more than no of equations"? = an indeterminate problem (if unknowns = equations, like you're used to seeing, then the problem is determinate and can be solved by equations of equilibrium.

another day in paradise, or is paradise one day closer ?

 

[ThomasH]

rb1957

Sorry, my mistake. I did not limit my approach to equations of equilibrium.

Regards

Thomas

 

[technodam]

Thank you JAE and rb1957 for the reply!
But I don't want complete solution to the problem. I just want how to formulate the problem. Let's for example if we have simply supported continuous beam with 3 supports. so, normally we divide the same in two parts and solve with either force or three moment equation. But if I want to use matrix method (stiffness or displacement method) by dividing the beam into say 10 elements, then how to structure a displacement and/or stiffness matrix and write the complete beam equation in the form of [k][x]=[F]?

Can anyone state the method with simple example?

As suggested by rb1957 I did refer the text books for beam stiffness problems. but normally I found determinate problems or at the most eigen value problems. I haven't found indeterminate beam problem solved using FE technique. So, if you find any particular referance for the same, it will be appriciated!

 

[IDS]

A good book with details of programming frame, plate and solid analyses is Programming the Finite Element Method by Smith and Griffiths.

You could also look on my blog where there is a series of posts on programming the stiffness method in Excel:

https://newtonexcelbach.wordpress.com/2014/01/22/frame-analysis-with-excel/

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[technodam]

Thank you IDS!
I have seen your blog. It seems pretty good to have such kind of problems solved with spreadsheet. I have gone through 'Continuous Beam' module and find it good as per my requirements. I will definitly study the macros you wrote and I hope these macros will explain the assembly of stiffness matrix and finally the solution of a problem. But, if possible will you please explain the procedure to calculate stiffness matrix in brief? I think it will be quicker and helpful to those who may not aware of VBA in Excel.

Thanks once again!

 

[IDS]

Technodam - if you work through the first few blog posts, and the associated spreadsheets, that gives you everything you need to know to set up a stiffness matrix on the spreadsheet without any VBA. I don't have time to go through it all here, and a forum really isn't the right format anyway, but if you have any specific questions please ask and I will respond.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[technodam]

Ok IDS. I will go through previous blog posts and try to understand the methodology.
Thanks for the help!

 

[yakpol]

I personally like force method for continuous beams. For unyielding supports and constant EI within a span the solution by force method is much shorter than displacements method (stiffness matrix). Attached is the solution for a concentrated force, which can be exteded to any type of loading.

Hope it helps!

Yakpol