Hi,

I have two local stiffness matrices (6x6) calculated in SMath. Now I want to assemble them into a single global stiffness matrix (8x8):



Red numbers represent degrees of freedom. The assemblage is done in such a way that terms from these two local matrices which correspond to the same degree of freedom have to be added. So for example I take the term 5-5 (column numbered as 5 and row numbered as 5) from k1 and add it to the term 5-5 from k2. The problem is that these terms are located in different parts of the matrices. For example aforementioned term 5-5 is actually in the 3rd column and 3rd row of the first matrix, in the 5th column and 5th row of the second matrix and goes to the same place (5th column, 5th row) of the global matrix. Of course I can do it manually but it's a lot of work for larger matrices (even here it takes some time). Is there an automated way to do it in SMath ?

Thanks in advance for your help.

Thanks for reply. I've seen this sample. It's really great but I'm not sure how to proceed in my case. I attached the .sm file.

Stiffness matrix.sm (40 KiB) downloaded 173 time(s).

Wrote

Thanks for reply. I've seen this sample. It's really great but I'm not sure how to proceed in my case. I attached the .sm file.

Stiffness matrix.sm (40 KiB) downloaded 173 time(s).

There should be two ways for three methods I could think of.
I won't add algorithm style here, these are enough I guess.
You can also combine method1 and method2 but I wouldn't call this a new way.

Regards

Stiffness matrix_rev.sm (58 KiB) downloaded 96 time(s).

Hi. As Overlord says, there are an algorithmic way. You can try it also.

Stiffness matrix.sm (33 KiB) downloaded 134 time(s).

Best regards.
Alvaro.

Thank you very much. After the assembly, selected rows and columns of the global stiffness matrix have to be removed to account for boundary conditions. For example I would like to delete columns 1-4 and rows 1-4. So far I was just manually rewriting remaining terms to new matrix (K_new) but again it's quite tedious work. What's the best way to do it automatically in SMath ? Column and row numbers for deletion are not always ordered like that. For example it could be 2 an 5-8 as well.

Hi. Check alg(#,#,#) and minor(#,#,#), both are in the matrix toolbar.

Best regards.
Alvaro.

Wrote

Thank you very much. After the assembly, selected rows and columns of the global stiffness matrix have to be removed to account for boundary conditions. For example I would like to delete columns 1-4 and rows 1-4. So far I was just manually rewriting remaining terms to new matrix (K_new) but again it's quite tedious work. What's the best way to do it automatically in SMath ? Column and row numbers for deletion are not always ordered like that. For example it could be 2 an 5-8 as well.

Is this what you are wishing to accomplish?
Specified rows and columns automatically removed from matrix.
There are also matrix extension commands to remove columns and rows.
But I wanted to write those functions without extension.

Regards

PS: If you want to see the results without removal,
assigning rowsdel=[9] and colsdel=[9] shall be enough.
The rowsdel and colsdel should always be a vector.

Stiffness matrix_rev2.sm (45 KiB) downloaded 82 time(s).

Wrote

Check alg(#,#,#) and minor(#,#,#), both are in the matrix toolbar.

They are both resident since my first SS 5346
duplicate of vminor from f(x) menu and duplicate of panel Minor,
as exemplified in disambiguation from Samples document.

Better removal for columns and rows added, algorithm rewritten.
And much more conventional way for not removal implemented.
Entering zero for not remove wanted row or col is easier.

Regards

Stiffness matrix_rev3.sm (53 KiB) downloaded 89 time(s).

Hi EngMath! I have some documents that I used for my Finite Element Method course. I automatized that assembly task. Attached here:

01 - 1-element bar: 01-BARRAS DE 2 ELEMENTOS.sm (137 KiB) downloaded 117 time(s).

02 - 2-element bar: 02-BARRAS DE 3 ELEMENTOS.sm (54 KiB) downloaded 111 time(s).

03 - trusses: 03-ARMADURAS.sm (124 KiB) downloaded 99 time(s).

04 - 1-element frame: 04-PORTICO DE 1 BARRA.sm (58 KiB) downloaded 99 time(s).

05 - 2-element frame: 05-PORTICO DE 2 BARRAS.sm (60 KiB) downloaded 101 time(s).

06 - 3-element frame: 06-PORTICO DE 3 BARRAS.sm (65 KiB) downloaded 111 time(s).

07 - plates: 07-PLACAS.sm (132 KiB) downloaded 107 time(s).

08 - axisymmetry: 08-AXISIMETRIA.sm (393 KiB) downloaded 98 time(s).

In all of them you will find my algorithm for the assembly task. You will find also the reduced stiffness matrix, I created functions to delete specific rows and columns.

Documents {01,02}, {04,05,06} are the same, only with different number of rows (I created those this way to use it quickly in the SS Android version).

I combined this with previous code for matrix assembly and it works great.

Thanks again, you guys are awesome, you really helped me a lot. And thank you Alvaro for sharing your FEM implementations in SMath. The 7th one (plates) is just what I'm working on now.

Great! Glad it helped! I made some quick translations to my plate worksheet.

07-PLATES.sm (135 KiB) downloaded 100 time(s).

07-PLATES.pdf (502 KiB) downloaded 115 time(s).

07-PLATES (with code).pdf (540 KiB) downloaded 125 time(s).

Great, thanks Alvaro :-) Are these examples from some book ?

You're welcome EngMath! Yes, they are from Introduction to Finite Elements in Engineering by Chandrupatla and Belegundu.
You can download it here:
https://1drv.ms/b/s!Ap2rcjsf-xok1woTCf8p26Qpw5Hn?e=SkhCnB

This example shows the use of an index vector (f, contains the unconstrained dofs) for reduction of the global system.

The input is via dof and connectivity tables.

Each bar element has 4 dofs defined by a line in the connectivity matrix
Each global dof is a line in the dof matrix with each containing

- initial position
- external nodal load
- constraint flag (1-constrained, 0-free)
- prescribed displacement (only relevant if dof is constrained, can be nonzero)


Fachwerk UE2.sm (80 KiB) downloaded 109 time(s).

Wrote

Red numbers represent degrees of freedom. The assemblage is done in such a way that terms from these two local matrices which correspond to the same degree of freedom have to be added. So for example I take the term 5-5 (column numbered as 5 and row numbered as 5) from k1 and add it to the term 5-5 from k2. The problem is that these terms are located in different parts of the matrices. For example aforementioned term 5-5 is actually in the 3rd column and 3rd row of the first matrix, in the 5th column and 5th row of the second matrix and goes to the same place (5th column, 5th row) of the global matrix. Of course I can do it manually but it's a lot of work for larger matrices (even here it takes some time). Is there an automated way to do it in SMath ?

Stifness matrix is not my discipline.
From square/rectangular matrix ...
You can extract paired symmetric [5,5] or asymmetric [2,5] col/row.
Extract...single col, single row.
You can delete pair wise single col/row or any cols/rows in any combination.
You can delete only col or only row.
Once in hand the new matrix, you can insert valued col or row
or col & row anywhere in the new matrix.
That step is one by one at this point of my proposal, might be automated.
All that from minimalist handling matrix.
Your 'K' matrices don't speak to me ... more visible example would help.
Cheers ... Jean

Wrote

might be automated.

... ½ automated left for clic/clic/DONE gurus.

Any ideas on how to integrate internal releases into the 2D Frame stiffness matrices?



Seems to be a few 2D truss projects, and some 2D Frame projects, but I'm curious about internal releases for frame structures.
Maybe someone can save me a deep dive into my old text books.
Thanks

Wrote

Any ideas on how to integrate internal releases into the 2D Frame stiffness matrices?

Here I attach the other elementary stiffness matrices. I didn't tested these in FE programs (always worked with rigid-rigid ends in FE), but when I was student of indeterminate structures (displacement method) I did some demonstrations of stiffness factors for each case, and these are well verified (except for the last page, that I wrote a few minutes ago just to complete the case of unitary rotation about the hinge (not needed for displacement method)).

Having in mind that the k(i,j) element of an elementary stiffness matrix is the reaction value in the j-displacement direction caused by an i-unitary displacement, you could trace each of the values found in my demonstrations.

Hope it works for you and please comment if it did once you have implemented it into your code.

I attach those demonstrations too (in spanish, but structural mechanics is a universal language :d )

Regards!

Alvaro

File not found.File not found.

Wrote

Here I attach the other elementary stiffness matrices. I didn't tested these in FE programs (always worked with rigid-rigid ends in FE), but when I was student of indeterminate structures (displacement method) I did some demonstrations of stiffness factors for each case, and these are well verified (except for the last page, that I wrote a few minutes ago just to complete the case of unitary rotation about the hinge (not needed for displacement method)).

Having in mind that the k(i,j) element of an elementary stiffness matrix is the reaction value in the j-displacement direction caused by an i-unitary displacement, you could trace each of the values found in my demonstrations.

Hope it works for you and please comment if it did once you have implemented it into your code.

I attach those demonstrations too (in spanish, but structural mechanics is a universal language :d )

Regards!

Alvaro

This worked perfectly, thanks Alvaro! For now I just added an extra vector that set a condition for each member and had the stiffness and transformation matrix builder you have in your example cycle through an if statement to decide which stiffness matrix was used for each member as the total stiffness matrix is built. Attached is the modified version of your 3-beam frame example, which technically could be used for any number of beams.
I verified the results in the attached file with some 2D structural analysis software and it appears to be correct correct. Take a look and let me know. I'm sure there are more systematic ways of accomplishing this, but as a proof of concept, I'm happy with the results.

Working Stiffness Matrix Idea - Internal Releases.sm (100 KiB) downloaded 126 time(s).

Now I just need to blend some of the graphical and data input ideas from other users and make this a more functional mini 2Dstructural analysis tool. Would be nice to get working standalone exe via SMath Viewer.