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Could someone provide me help to solve this Euler-Bernoulli beam equation by using finite difference method and Newton Raphson please.

With boundary value of y(0) = 0 and dy/ds(L) = 0

I am continually getting answers that are nowhere near the results from the bvp4c command.

darova
on 2 Dec 2019

Here is my attempt

clc,clear

N = 10; % 10 Set parameters

L = 16*0.3048; %meter

b = 19.625 * 0.0254; d = 1.625 * 0.0254;

p = 26.5851; I = (b*(d^3))/12; h = L/N; F = 0.7436; E = 68.9e6;

alpha = (h^2)/(E*I);

w = p*b*d;

fode = @(s,f) [f(2); -1/E/I*(w*(L-s)+F)*cos(f(1))];

fbound= @(ya,yb) [ya(1)-0; yb(2)-0];

ss = linspace(0,5);

finit = [0 0];

solinit = bvpinit(ss,finit);

sol = bvp4c(fode,fbound,solinit);

plot(sol.x,sol.y(1,:))

Thiago Henrique Gomes Lobato
on 1 Dec 2019

I didn't check exactly your FEM implementations but one thing that I quickly noticed is that the Newton-Rapson is an iteractive approach, so maybe you get different results simply because your result did not converged. When I iterate about your code I get a very different result that converges actually fast (3 iterations):

% substitute by your images

N = 20 % Set parameters

L = 16*0.3048; %meter

b = 19.625 * 0.0254; d = 1.625 * 0.0254;

p = 26.5851; I = (b*d^3)/12; h = L/N; F = 0.7436; E = 68.9e6;

alpha = h^2/E*I;

w = p*b*d;

S = [h:h:L] ;

y = ones(N,1);

e = ones(N,1);

A = spdiags([e -2*e e],[-1 0 1],N,N);

A(N-1,N) = -2; % fictitious boundaries method

Iterations = 1000;

tol = 1e-20;

for idx =1:Iterations

function_1 = zeros(N,1);

for n = 1:N

function_1(n) = alpha*(w*(L-S(n))+F)*cos(y(n));

end

Fun = A*y + function_1;

% To create the Jacobian of F(y)

Dfdia = zeros(N,1);

for n = 1:N

Dfdia(n) = alpha*(w*(L-S(n))+F)*sin(y(n));

end

J = diag(Dfdia);

step = inv(A+J)*Fun;

y = (y-step);

if norm(step)<tol

Iter = idx

break

end

end

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