MATLAB: Euler’s Method and Deflection of Cantilever Beam

eulersmethod

Hi, I'm trying to write the code to calculate the deflection of a beam using euler's method. The deflection of the beam is given as w/(24*EI)*(x^4-4Lx^3+6L^2x^2) and the derivative is w/(24*EI)*(4x^3-12Lx^2+12L^2x). This code works except the answers I am looking for are slightly off. So when x = 1.25, y = 3099, but right now I'm getting x = 2.5, y = 3099. Does anyone know why this is or can help me fix this?
w = 21819;
EI = 106;
L = 5; % length
h=1.25; % step's size
N=5; % number of steps
y(1)=1;
for n=1:N       
y(n+1)= y(n)+h*(w/(24*EI))*(4*x(n).^3 - (12*L*x(n).^2) + (12*L^2*x(n)));
x(n+1)=n*h;          % euler's method, using derivative of given function
end
plot(x,y)

Best Answer

  • Your derivative is incorrect for your specified deflection equation. I think your deflection equation is also not quite right. Simple Euler is never going to give a good result here because of the rapid variation with x, even with a large number of steps. y(1) must be zero to be consistent with the deflection equation. Have a look at the following:
    w = 21819;
    EI = 106;
    L = 5; % length
    N = 100; % number of steps
    h = L/N; % step's size
    ytrue = @(x)w/(24*EI)*(x.^4 - 4*L*x.^3 + 6*L^2*x.^2); % uniformly loaded cantilever beam
    y(1)=0;
    x(1) = 0;
    for n=1:N-1       
    y(n+1)= y(n)+h*w/(24*EI)*(4*x(n).^3 - 12*L*x(n)^.2 + 12*L^2*x(n));
    x(n+1)=n*h;          % euler's method, using derivative of given function
    end
    plot(x,ytrue(x),x,y,'.'),grid
    xlabel('x'),ylabel('displacement')
    legend('true','Euler')