Assemble Global Stiffness Matrix

%% INPUTs:
A = 2300;                 % Cross-Sectional Area of member (in mm^2)
E = 2*10^5;               % Elastic Modulus (in N/mm/mm)
L(:,1) = 4000;            % Length of Member-1 (in mm)
L(:,2) = 6000;            % Length of Member-2 (in mm)
t(:,1) = 0;               % Theta-1           
t(:,2) = 90;              % Theta-2 
t(:,3) = rad2deg(atan(L(:,2)/L(:,1)));   % Theta-3
N = numel(t)
%% OUTPUTs:
L(:,3) = sqrt(L(:,1)^2 + L(:,2)^2);         %Length of Member-3 (in mm)
for i = 1:N
    k{i} = [A*E/L(:,i) 0 -A*E/L(:,i) 0;0 0 0 0;-A*E/L(:,i) 0 A*E/L(:,i) 0;0 0 0 0];
    C{1,i} = [cosd(t(:,i)) -sind(t(:,i)) 0 0;sind(t(:,i)) cosd(t(:,i)) 0 0;0 0 cosd(t(:,i)) -sind(t(:,i));0 0 sind(t(:,i)) cosd(t(:,i))];
    D{1,i} = transpose(C{1,i});
    K{1,i} = C{1,i}*k{1,i}*D{1,i};
end

From this code, I'm able to get (4*4) size [K] matrices such that K{1,1} = K1; K{1,2} = K2 & K{1,3} = K3. Now, I want them to assemble in such a way that it ends up into a global matrices of K (6*6) matrix.

K{1,1} - U1 V1 U2 V2

K{1,2} - U2 V2 U3 V3

K{1,3} - U1 V1 U3 V3

Need global assembly matrix such that: [K] based on U1 V1 U2 V2 U3 V3. For more clarification, I'm attaching image.

Answer 1

[Voss]

%% INPUTs:
A = 2300;                 % Cross-Sectional Area of member (in mm^2)
E = 2*10^5;               % Elastic Modulus (in N/mm/mm)
L(:,1) = 4000;            % Length of Member-1 (in mm)
L(:,2) = 6000;            % Length of Member-2 (in mm)
t(:,1) = 0;               % Theta-1           
t(:,2) = 90;              % Theta-2 
t(:,3) = rad2deg(atan(L(:,2)/L(:,1)));   % Theta-3
N = numel(t)

N = 3

%% OUTPUTs:
L(:,3) = sqrt(L(:,1)^2 + L(:,2)^2);         %Length of Member-3 (in mm)
for i = 1:N
    k{i} = [A*E/L(:,i) 0 -A*E/L(:,i) 0;0 0 0 0;-A*E/L(:,i) 0 A*E/L(:,i) 0;0 0 0 0];
    C{1,i} = [cosd(t(:,i)) -sind(t(:,i)) 0 0;sind(t(:,i)) cosd(t(:,i)) 0 0;0 0 cosd(t(:,i)) -sind(t(:,i));0 0 sind(t(:,i)) cosd(t(:,i))];
    D{1,i} = transpose(C{1,i});
    K{1,i} = C{1,i}*k{1,i}*D{1,i};
end

%% assemble global K:
idx = {1:4, 3:6, [1 2 5 6]};
K_global = zeros(6);
for ii = 1:numel(idx)
    K_global(idx{ii},idx{ii}) = K_global(idx{ii},idx{ii})+K{ii};
end

%% display global K:
format short g
disp(K_global)

   1.3463e+05        29442    -1.15e+05            0       -19628       -29442
        29442        44163            0            0       -29442       -44163
    -1.15e+05            0     1.15e+05            0            0            0
            0            0            0        76667            0       -76667
       -19628       -29442            0            0        19628        29442
       -29442       -44163            0       -76667        29442   1.2083e+05

[YQ]

Thanks a lot!!

[Voss]

You're welcome!

[YQ]

I've one more doubt associated with the same task:

U(1,1) = 0; % Displacement in x direction at Node-1 (mm)

U(2,1) = 0; % Displacement in y direction at Node-1 (mm)

U(3,1) = u2; % Displacement in x direction at Node-2 (mm)

U(4,1) = 0; % Displacement in y direction at Node-2 (mm)

U(5,1) = u3; % Displacement in x direction at Node-3 (mm)

U(6,1) = v3; % Displacement in y direction at Node-3 (mm)

F(1,1) = f1; % Force in x direction at Node-1 (N)

F(2,1) = f2; % Force in y direction at Node-1 (N)

F(3,1) = 0; % Force in x direction at Node-2 (N)

F(4,1) = f4; % Force in x direction at Node-2 (N)

F(5,1) = 12000; % Force in x direction at Node-3 (N)

F(6,1) = 0; % Force in y direction at Node-3 (N)

These are the values of matrix U & F both are 6*1 matrix and both contains some unknowns.

How to find unknown with the help of this equation:

[K_global]*[U] = [F] ?

[Voss]

%% INPUTs:
A = 2300;                 % Cross-Sectional Area of member (in mm^2)
E = 2*10^5;               % Elastic Modulus (in N/mm/mm)
L(:,1) = 4000;            % Length of Member-1 (in mm)
L(:,2) = 6000;            % Length of Member-2 (in mm)
t(:,1) = 0;               % Theta-1           
t(:,2) = 90;              % Theta-2 
t(:,3) = rad2deg(atan(L(:,2)/L(:,1)));   % Theta-3
N = numel(t)

N = 3

%% OUTPUTs:
L(:,3) = sqrt(L(:,1)^2 + L(:,2)^2);         %Length of Member-3 (in mm)
for i = 1:N
    k{i} = [A*E/L(:,i) 0 -A*E/L(:,i) 0;0 0 0 0;-A*E/L(:,i) 0 A*E/L(:,i) 0;0 0 0 0];
    C{1,i} = [cosd(t(:,i)) -sind(t(:,i)) 0 0;sind(t(:,i)) cosd(t(:,i)) 0 0;0 0 cosd(t(:,i)) -sind(t(:,i));0 0 sind(t(:,i)) cosd(t(:,i))];
    D{1,i} = transpose(C{1,i});
    K{1,i} = C{1,i}*k{1,i}*D{1,i};
end

%% assemble global K:
idx = {1:4, 3:6, [1 2 5 6]};
K_global = zeros(6);
for ii = 1:numel(idx)
    K_global(idx{ii},idx{ii}) = K_global(idx{ii},idx{ii})+K{ii};
end

%% Define U and F:
syms('f1','f2','f4','u2','u3','v3')
U = [0;0;u2;0;u3;v3];
F = [f1;f2;0;f4;12000;0];

%% Solve
soln = solve(K_global*U==F)

soln = struct with fields:
    f1: -12000
    f2: -758663023165439976000/42147945731413333
    f4: 758663023165440048000/42147945731413333
    u2: 0
    u3: 109555278165705605241441878016000/113699629798874301526546156126207
    v3: -9895604649984000/42147945731413333

Requires Symbolic Math Toolbox.

[YQ]

I already did this but I need to store the results in matrix form. Is this possible ?

[Torsten]

A(1,1) = double(soln.f1)
A(2,1) = double(soln.f2)

and so on.

[YQ]

thanks

Answer 2

[Aquatris]

I think this is what you are looking for. You can extend the logic in a better way if you have more than 3 nodes.

% create random 4x4 matrices with different magnitude to see the results
% clearly
K1 = rand(4);
K2 = rand(4)+10;
K3 = rand(4)+100;

K = zeros(6); % initialize global K

idx_1 = [1 2]; % column and row number corresponding to U1 V1
idx_2 = [3 4]; % column and row number corresponding to U2 V2
idx_3 = [5 6]; % column and row number corresponding to U3 V3

% Assemble the K matrix
K([idx_1 idx_2],[idx_1 idx_2]) = K([idx_1 idx_2],[idx_1 idx_2]) + K1; % U1 V1 U2 V2 part
K([idx_2 idx_3],[idx_2 idx_3]) = K([idx_2 idx_3],[idx_2 idx_3]) + K2; % U2 V2 U3 V3 part
K([idx_1 idx_3],[idx_1 idx_3]) = K([idx_1 idx_3],[idx_1 idx_3]) + K3; % U1 V1 U3 V3 part

disp(K)

  100.5298  100.5934    0.4374    0.6968  100.4007  100.8383
  101.4022  101.0052    0.0152    0.0705  100.2386  100.2537
    0.5424    0.3041   11.4423   11.5954   10.3730   10.4325
    0.3082    0.5288   11.3503   11.0871   10.8858   10.6074
  100.9454  100.6493   10.0834   10.2776  111.6072  111.5421
  100.3796  100.1210   10.0190   10.5509  111.3972  111.4889

[YQ]

Thanks a lot.