Consider the cantilever beam shown below The beam is supported on one end such t
ID: 3867046 • Letter: C
Question
Consider the cantilever beam shown below The beam is supported on one end such that both its position y and its slope dy/dz are fixed at zero. One end might be embedded in concrete or firmly secured. Suppose the beam is of length L and is thin and uniform. The vertical deflection y of the beam due to its own weight can then be modeled using the following relation: JE d^2y/dz^2 = rho g(1 + dy/dz)^3/2 [z(z - L/2) + L^2/2] where J is the moment of inertia of the beam cross section about its principal axis and E is Young's modulus, which depends on the material from which the beam is made. The constant rho is the linear mass density of the beam, and g is the acceleration due to gravity. The equation represents an initial value problem because the boundary conditions are that y and dy/dz are both zero at z = 0 This second-order nonlinear beam deflection equation can be converted to a system of two first order equations by introducing the state variables x = [y dy/dz]^T Thus, the first-order system of ODEs become dx_1/dz = x_2 dx_2/dz = rho/JE (1 + x^2_2)^3/2 [z(z - L/2) + L^2/2] Show the derivation of these two equations. Let L = 2 m, rho = 10 kg/m, and the moment of inertia times Young's modulus is JE = 2400 kg-m^3/s^2. Begin with an end condition of x(0) = 0 and calculate the maximum deflection at the end of the beam (L). Use a 4th order Runge-Kutta method to solve the equations.Explanation / Answer
load=[repmat(0,1,999) 10000 10000];
l=20;
dl=0.02;
if length(load)~=(l/dl+1)
error('Check inpkkts')
end
y=0:dl:l;
m=skkm((y.*load))*dl;
v=skkm(load)*dl;
kk_4=load/ei;
kk_3=v/ei;
for i=2:length(load)
kk_3(i)=kk_3(i-1)-kk_4(i-1)*dl;
end
kk_2=m/ei;
for i=2:length(load)
kk_2(i)=kk_2(i-1)-kk_3(i-1)*dl;
end
kk_1=0;
for i=2:length(load)
kk_1(i)=kk_1(i-1)+kk_2(i-1)*dl;
end
kk=0;
for i=2:length(load)
kk(i)=kk(i-1)+kk_1(i-1)*dl;
end
deflection=kk;
plot(y,kk_2*ei)
hold on
plot(y,kk_3*ei,'r')
legend('bending moment','shear force')
xlabel('length along the beam')
ylabel(' bending moment and shear force (SI kknits)')
grid
hold off
figkkre,plot(y,kk,'r')
xlabel('length along the beam')
ylabel('deflection')
grid
title('Deflection')