The governing differential equation for the deflection of a simple beam subjected to a uniform (Fig. 1) is given by: d2y 1 dx22 where E is elastic modulus, Iis beam moment of inertia, y is beam deflection, w is the intensity of the uniform load, and x is the distance along the beam measured from the left support (end). The boundary conditions are the deflections y(0) and y(L) are zero where L is beam length. Given L=5 m, l= 0.005 m4, w= 10 KNm, and E-200.000 kPa. Provide the following 1. Solve for the deflection of the beam using the finite difference method. Use an interval size of 1 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: y(x)-24E (-xf + 24x3-13x) 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 the shooting method incorporating MATLAB ode45 and fzero functions. Compare against the exact solution given in item 1. Comment on the results. (For students taking the class at the 7000 level) 4. Find the maximum deflection of the beam using fminbnd. Compare against the exact value of 5wL4/(384E). Provide in plot the results of items 2 and 3 along with the exact solution. Comment on the results 5. Your answers should be provided in the form of a report that includes a description of the project, the resulting system of equations, a printout of your MATLAB files your plots, and your comments on the results Fig. 1. Simple beam subjected to a uniform load (w). The governing differential equation for the deflection of a simple beam subjected to a uniform (Fig. 1) is given by: d2y 1 dx22 where E is elastic modulus, Iis beam moment of inertia, y is beam deflection, w is the intensity of the uniform load, and x is the distance along the beam measured from the left support (end). The boundary conditions are the deflections y(0) and y(L) are zero where L is beam length. Given L=5 m, l= 0.005 m4, w= 10 KNm, and E-200.000 kPa. Provide the following 1. Solve for the deflection of the beam using the finite difference method. Use an interval size of 1 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: y(x)-24E (-xf + 24x3-13x) 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 the shooting method incorporating MATLAB ode45 and fzero functions. Compare against the exact solution given in item 1. Comment on the results. (For students taking the class at the 7000 level) 4. Find the maximum deflection of the beam using fminbnd. Compare against the exact value of 5wL4/(384E). Provide in plot the results of items 2 and 3 along with the exact solution. Comment on the results 5. Your answers should be provided in the form of a report that includes a description of the project, the resulting system of equations, a printout of your MATLAB files your plots, and your comments on the results Fig. 1. Simple beam subjected to a uniform load (w)
