How do I assemble global matrices for structures with nodes connecting 3 or more beams? (with Finite Element Method)

Hello,

I don't have a code to show because that's what I can't come up with. I have designed a structure made of several beams (to give an idea, it ressembles an American Made Steel Bridge), and I have already obtained the "elemental matrices" for each of them (12x12 Mass and Stiffness matrices)*, all referenced in the same axis.

Basically I am lost when it comes to obtain the global matrices, or the Mass and Sitffness matrices for the whole structure, because of the nodes that connect 3 or 4 beams at once.

I understand the process when it comes to make the assembly of a "unidirectional" element, if you can call it so -like a single large beam that has been divided into multiple finite elements-, because the global matrix simply requires to go finite element to finite elemement following the direction of said single beam. But how can I make the assembly with nodes that connect finite elements in more than one direction?

If it's of any help to unerstand what I'm saying, the model I'm trying to make the assembly of is a simplification of a plane. I've designed the finite elements to match this 3D model I found online (pictures with pink background).

Thank you all in advance!

*The elemental matrices have the same information for rows and columns, and that is X, Y, Z, θx, θy, θz, X, Y, Z, θx, θy, θz

[Mario Malic]

I am not sure if I understand your question, but in global matrix, you have a global coordinate system. For each element, you're supposed to calculate local stiffness matrix and multiply it with the transformation matrix to get its representation in global coordinate system.

I am a bit confused about 12x12 matrices you mention, X, Y, Z, θx, θy, θz, X, Y, Z, θx, θy, θz. How many degrees of freedom has a single node in your model?

[Irene Simó]

Regarding the first you comment, I've obtained already those stiffness and mass matrices for each element (in the global coordinate system, already did the multiplication with the transformation matrix).

Each node has 6 degrees or freedom (X, Y, Z, θx, θy, θz), therefore the 12x12 matrices I have represent the dependence or influence between 2 nodes, hence the 12x12 and not only 6x6. Said 12x12 matrices, given that they convey the information between 2 nodes, are composed by the stiffness and mass of the finite element that connects them.

I feel I'm explaining myself poorly, and it porbably is because I just learned about FEM a few weeks ago and this is my first time working on a project with the method (I need this for uni, I wouldn't have started with a whole plane if I could've chosen myself). What I'm trying to say is that my question is probably quite elemental and it's possible that I make it sound more complicated than it really is, if that makes it easier to understand?

[Mario Malic]

What do you mean by conveying the information between the two nodes? I guess, this is to find the length of the beam.

From the Hooke's law

[K]{x} = {F}

Beam with 1 DOF per node, 3 elements, 4 nodes

% x==== x ==== x ====x   % let's say this is a beam divided into three finite elements
%   1st    2nd   3rd     element
% 1     2      3     4   nodes in global CS, denoted as x

Note on the nodes in the global CS: they don't have to be in this order, when you'd solve them manually, you'd place the global nodes with known boundary conditions on one side of matrix/vector, not sure what's the procedure when you compute this stuff on the computer.

For the purpose of this example, let's leave global nodes as they are

For element 2, one DOF per node, in local CS we have

%  2   3  <-global nodes, in local CS it's 1 and 2
2 [k2 -k2      {u1      {F1
           *         =  
3 -k2  k2]      u2}      F2}

Assembling this into the global CS

%   1   2    3   4   % global nodes
1 [ .   .    .   .          {u1         {F1 
2   .  k2  -k2   .    *      u2      =   F2 
3   . -k2   k2   .           u3          F3   
4   .   .    .   .]          u4}         F4}   

This is only the case where we have one degree of freedom per node.

You can find the lectures (1st or 2nd one) from K. J. Bathe on MIT OCW that explains the assemblage into global stiffness matrix.