Stitching together matrices/ indexing matrices

Hi there any help would be appreciated...

I am a mech engineer working with rotor dynamics and am being asked to do FEA on matlab. I have the following formula for the stiffness matrix for each element: K = ( E * A / Le) * [1 -1; -1 1]. What I would like to is compute the values of these matricies for 4 elements, the soln should be 4 K matricies... and my question regarding this is how do I index these 2x2 K matricies to be K1, K2, K3, K4? When I try to do this using a For Loop, I code the following:

%noel is number of elements which is 4

for i = 1: noel

K{i} = ( E * A / Le) * [1 -1; -1 1];

end

-doing this I get an error saying theres an issue with usijng { }, I have tried [] and () and had no luck either.

(I understand it says ask one question per post but I want to include my second one for a little more context to the overall problem)

The next step I need to do is to stitch the matricies together diagonally. What I mean by this is if you consider the 2x2 to be [A B;C D] for each K value, I need to produce one final matrix consisting of adding the first and last elements of each K matrix together. EX: if K1 = [A B; C D] and K2 = [E F; G H] I want to merge them such as K_system = [A B 0; C (d+E) F; 0 G H]. I need to do this for K1-4.

Sorry for the multiple question post, do not feel obliged to answer both, I can post them separately. The dual question is for sake of context.

Thanks,

Kyle

[Asad (Mehrzad) Khoddam]

In FEM, you need to fine local stiffness matrix then assemble it to the global stiffness matrix.

The location of each k_ij in K depends on the location of members and connection between them

[Kyle McLaughlin]

Right, this question is about assembling them into the global matrix, the K matrix is the same for all elements as described above. The problem statement is specifically regarding modeling a free-free beam in axial vibration and that was the equation my prof gave in the slides for "pure axial vibration". I am unaware of how to index these individual K values for the purpose of assembly, in addition to the assembly process of the first and last elements. The logic I imagined to follow was to 1) index K values 2) add the last and first elements of adjacent arrays 3) assemble global matrix 4) repeat for mass matrix 5) perform eig()....

Answer 1

[Asad (Mehrzad) Khoddam]

For each k, you should know the connectivity array, for example if the first element is connected to the following degrees of freedom:

cArray=[1, 2, 5, 6]

Then

K(cArray, cArray) = K(cArray, cArray) + k;

Where k is the local stiffness matrix

[Kyle McLaughlin]

I dont understant, are you saying that I am missing some form of matrix needed to complete this calculation? We did not talk about the requirement of any sort of "connectivity array" in lecture. The problem statement provides 4 elements and nodes = dof = 5

[Kyle McLaughlin]

My understanding of the question is that I have the 2x2 K matrix (defined in the inital question) which represents the stiffness of each element in the free free beam and that I need to stitch together 4 of those

[Asad (Mehrzad) Khoddam]

So the connectivity array of the first element is:

% element 1
%[1, 2, 3, 4]
% element 2
%[3, 4, 5, 6]
% element 3
%[5, 6 , 7, 8]
% and element 4:
%[7, 8, 9, 10]
cArray = [1:4];
K(cArray, cArray) = K(cArray, cArray) + k;
cArray = [3:6];
K(cArray, cArray) = K(cArray, cArray) + k;
cArray = [5:8];
K(cArray, cArray) = K(cArray, cArray) + k;
cArray = [7:10];
K(cArray, cArray) = K(cArray, cArray) + k;

[Kyle McLaughlin]

Thank you for your detailed response, I am still unaware however of what the connectivity array is used for and how it is incoroporated into the calculations. We never discussed this in lecture. What would I do with this code

[Asad (Mehrzad) Khoddam]

You can use this instead:

K(1,1) = k(1,1);
for i=2:9
    K(i,i)=k(1,1)+k(2,2);
    K(i-1,i) = k(1,2);
    K(i,i-1) = k(2,1);
end
K(9,10) = k(1,2);
k(10,9) = k(2,1);
K(10,10) = k(2,2);

[Kyle McLaughlin]

Thank you, this worked to assemble the K matrix and added the values I needed. Much appreciated!!!