problem_2.m

% Problem 3
clear
clc
setdefaults
% Plug in different values for q to find corresponding deflections.
q = [1,10,20,30,50];
% Young's Modulus for aluminum [Pascals]:
E = 70e9;
% Width of beam [meters]:
b = 0.1;
% Height of beam [meters]:
h = 0.01;
% Second moment of inertia [meters^4]:
I = (b*h.^3)/12;
% Plotting q vs. max deflection:
for i = 1:5
    %----------------------------------------------------------------------
    % Define the 6 segment matrix:
    % Segment length for 6 segments
    h_6 = 1/6;
    A_6 = zeros(5);
    b_6 = ((h_6^4*q(i))/(E*I))*ones(5,1);

    % Diagonal vectors of the A matrix:
    ones_vect_6 = ones(3,1);
    fours_vect_6 = -4*ones(4,1);
    sixes_vect_6 = 6*ones(5,1);
    % Create A matrix by summing the diagonal vectors.
    A_6 = diag(ones_vect_6,-2) + diag(fours_vect_6,-1) + diag(sixes_vect_6) + diag(ones_vect_6,2) + diag(fours_vect_6,1);
    A_6(1,1) = 5;
    A_6(5,5) = 5;
    % Solve for the deflection vector by inv(A) * b
    w_6 = inv(A_6) * b_6;
    w6_max(i) = max(w_6);
    
    %----------------------------------------------------------------------
    % Define the 10 segment matrix:
    % Segment length for 10 segments
    h_10 = 1/10;
    A_10 = zeros(9);
    b_10 = ((h_10^4*q(i))/(E*I))*ones(9,1);

    % Diagonal vectors of the A matrix:
    ones_vect_10 = ones(7,1);
    fours_vect_10 = -4*ones(8,1);
    sixes_vect_10 = 6*ones(9,1);
    % Create A matrix by summing the diagonal vectors.
    A_10 = diag(sixes_vect_10)+diag(fours_vect_10,-1)+diag(fours_vect_10,1)+diag(ones_vect_10,-2)+diag(ones_vect_10,2);
    A_10(1,1) = 5;
    A_10(9,9) = 5;
    % Solve for the deflection vector by inv(A) * b
    w_10 = inv(A_10) * b_10;
    w10_max(i) = max(w_10);
    %----------------------------------------------------------------------
    % Define the 20 segment matrix:
    % Segment length for 20 segments
    h_20=1/20;

    A_20 = zeros(19);
    b_20 = ((h_20^4*q(i))/(E*I))*ones(19,1);

    sixes_vect_20 = 6*ones(1,19);
    fours_vect_20 = -4*ones(1,18);
    ones_vect_20 = ones(1,17);
    % Create A matrix by summing the diagonal vectors.
    A_20 = diag(sixes_vect_20)+diag(fours_vect_20,-1)+diag(fours_vect_20,1)+diag(ones_vect_20,-2)+diag(ones_vect_20,2);
    A_20(1,1) = 5;
    A_20(19,19) = 5;
    % Solve for the deflection vector by inv(A) * b
    w_20 = inv(A_20) * b_20;
    w20_max(i) = max(w_20);
    %----------------------------------------------------------------------
end
% Plot q vs. max deflection for each segment size.  There will be three
% different graphs, each corresponding to a different segment size.
figure
plot(w6_max,q,'-o')
title('Distributed Beam Loading vs. Maximum Deflection for 6 segments');
xlabel('\delta_{x} [m]');
ylabel('q [N/m]');

figure
plot(w10_max,q,'-o');
title('Distributed Beam Loading vs. Maximum Deflection for 10 segments');
xlabel('\delta_{x} [m]');
ylabel('q [N/m]');

figure
plot(w20_max,q,'-o');
title('Distributed Beam Loading vs. Maximum Deflection for 20 segments');
xlabel('\delta_{x} [m]');
ylabel('q [N/m]');