assemble global stiffness matrix

%Hello, I want to assemble global stiffness matrix of 3 element connected each other to get stiffness matrix of whole structure, but could not figure it out how to write a function. Can you please help me out.

%Igonoring shear deformation, the element stiffness matrix in local

%coordination for element 1

clc;

clear;

close;

%syms A;

%syms R;

syms E

syms I

syms R

syms phi real;

E = 1;

I_z = 1;

E_Iz = E*I_z;

L = R; %R

%A = 1;

A = 100000000000;

K1 = A*E/L;

K2 = 12*E*I_z/L^3;

K3= 6*E*I_z/L^2;

K4 = 4*E*I_z/L;

K5 = 2*E*I_z/L;

% for element 1

Ke_1_local = [K1 0 0 -K1 0 0;

0 K2 K3 0 -K2 K3;

0 K3 K4 0 -K3 K5;

-K1 0 0 K1 0 0;

0 -K2 -K3 0 K2 -K3;

0 K3 K5 0 -K3 K4 ];

% transformation matrix Gamma

Phi = 90;

gamma_1 = [cosd(Phi) sind(Phi) 0 0 0 0;

-sind(Phi) cosd(Phi) 0 0 0 0;

0 0 1 0 0 0 ;

0 0 0 cosd(Phi) sind(Phi) 0;

0 0 0 -sind(Phi) cosd(Phi) 0;

0 0 0 0 0 1];

%element stiffness matrix in global coordinate

Ke_1_global = gamma_1'.*Ke_1_local.*gamma_1;

%For arch

%Element 2

phi_1 = 0;

phi_2 = pi/2;

%equillibrium matrix

phi2 = [-1 0 0;

0 -1 0;

-R*(sin(phi_2)-sin(phi_1)) R*(cos(phi_1)-cos(phi_2)) -1];

%flexibility matrix

Q_b = @(phi) [-R*(sin(phi)-sin(phi_1)) R*(cos(phi_1)-cos(phi)) -1];

d2 = @(phi) (Q_b(phi)'.*Q_b(phi))./E_Iz;

d2_phi = d2(Phi);

d2_int = int(d2_phi, 1e-8, pi/2);

d_3 = vpa(d2_int);

%Siffness matrix

Kff_3 = inv(d_3);

Kfs_3 = inv(d_3)*phi2';

Ksf_3 = phi2*inv(d_3);

Kss_3 = phi2*inv(d_3)*phi2';

K_2_global = round([Kff_3 Kfs_3; Ksf_3 Kss_3], 2);

%Element3

phi_1 = pi/2;

phi_2 = 3*pi/4;

%equillibrium matrix

phi2 = [-1 0 0;

0 -1 0;

-R*(sin(phi_2)-sin(phi_1)) R*(cos(phi_1)-cos(phi_2)) -1];

%flexibility matrix

Q_b = @(phi) [-R*(sin(phi)-sin(phi_1)) R*(cos(phi_1)-cos(phi)) -1];

d3 = @(phi) (Q_b(phi)'.*Q_b(phi))./E_Iz;

d3_phi = d3(Phi);

d3_int = int(d3_phi, phi_1, phi_2);

d_3 = vpa(d3_int);

%Siffness matrix

Kff_3 = inv(d_3);

Kfs_3 = inv(d_3)*phi2';

Ksf_3 = phi2*inv(d_3);

Kss_3= phi2*inv(d_3)*phi2';

K_3_global = round([Kff_3 Kfs_3; Ksf_3 Kss_3], 2);

%assemble local stiffness matrix

connect = {[1 2], [2,3]; [3,4]};

nele = 3; %number of element

ndof = 3; %number of dof perelement

Answer 1

[Ayush]

Dear Milan,

I understand that your intent is to construct the global stiffness matrix for a structure comprising of three interconnected elements, aiming to obtain the stiffness matrix for the entire structure by implementing a function.

I attempted to execute the provided code on my system, but encountered an error that states:

```

Error using vertcat

Dimensions of arrays being concatenated are not consistent.

Error in MLAnswers6_w3 (line 80)

connect = {[1 2], [2,3]; [3,4]};

```

However, to assemble the global stiffness matrix for the structure with three interconnected elements, the following steps can be followed:

1. Initialize the global stiffness matrix, denoted as “K_global”, as a matrix of zeros with dimensions “(ndof*nele) x (ndof*nele)”.

2. For each element, compute its global stiffness matrix, denoted as “K_element_global”, by utilizing the transformation matrix gamma and the element stiffness matrix “Ke_local”.

3. Transpose the gamma matrix and multiply it with “Ke_local”.

4. Multiply the resulting matrix with “gamma”.

5. Assemble the local stiffness matrices into the global stiffness matrix.

6. For each element, determine the corresponding degrees of freedom (DOFs) in the global stiffness matrix.

7. Add the local stiffness matrix to the appropriate locations within the global stiffness matrix.

Example code to achieve the described steps:

function K_global = assembleGlobalStiffnessMatrix(nele, ndof, Ke_local, gamma)
K_global = zeros(ndof*nele);
for element = 1:nele
    % Calculate the global stiffness matrix for the current element
    Ke_element_global = gamma' * Ke_local * gamma;
    % Identify the corresponding DOFs in the global stiffness matrix    
    dofs = (element-1)*ndof + 1 : element*ndof;
    % Assemble the local stiffness matrix into the global stiffness matrix    
    K_global(dofs, dofs) = K_global(dofs, dofs) + Ke_element_global;   
end
end
% Call the function to assemble the global stiffness matrix
K_global = assembleGlobalStiffnessMatrix(nele, ndof, Ke_1_local, gamma_1);

% Add the stiffness matrices of the remaining elements
K_global = K_global + K_2_global + K_3_global;

Hope this helps!