Question

The governing differential equation for the deflection of a cantilever beam subjected to a point load at its free end (Fig. 1) is given by: 2 dx2 where E is elastic modulus, Izz is beam moment of inertia, y 1s beam deflection, P is the point load, and x is the distance along the beam measured from the free end. The boundary conditions are the deflection y(L) is zero and the slope (dy/dx) at x-L is zero where L is beam length. Given L-2 m, /-0.0005 m4, Provide the following P- 10 kN, and E- 200,000 MPa. 1. Solve for the deflection of the beam using the finite difference method. Use arn interval size of 0.5 m for the finite difference approximation. Write the resulting system o equations in matrix form and solve it using MATLAB. Provide a plot comparing your results against the analytical solution given by: (x)- 6EIzz 2. 3. Redo item 1 using an interval equal to 0.01 m. Comment on the results. Solve for the deflection of the beam using a method incorporating MATLAB ode45. Compare against the exact solution given in item 1. Comment on the results. Provide in plot the results of items 2 and 3 along with the exact solution. Comment on the results 4.

SOLVE USING MATLAB PLEASE THANKS!

Answer 1

From the given question, the moment at any section location at a distance 'x' from free and due to 'p' is M x = -P x We know that EI _2z d^2/dy^2 = M x = -P x Integrating the above equation, with respect to 'x' So, EI _z z dy /dy = -Px^2 /2 + C_1 Equation 1 First boundary condition at x = L and slope, dy /dx = 0 So, 0 = -Px^2 /2 + C_1 rightdoublewardsarrow C_1 = Px^2/2 Now we need to take integration again for above equation 1 So, EI_zz y = -P x^3/6 + C_1 x + C_2 Equation 2 Second boundary condition at x = L and y = 0 Now substitute the C_1 to above equation, 0 = -P x^3/6 + Px^2/2 x + C_2 0 = -P L^3/6 + PL^2/2 + C_2 0 = -P L^3/6 + P L^3/2 + C_2 0 = -P L^3 + 3P L^3/6 + C_2 0 = 2PL^3/6 + C_2
Copyable Code:

x = [0:0.01:2];
y = 1*10.^(-5)*(-(x.^3)/6+(2*x)-8/3);
yn = 1*10^(-5)/6*((2-x).^2).*(4+x);
plot(x,yn,x,y)