1. Under a uniform load, small deflections y of a simply supported beam are given by...

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1. Under a uniform load, small deflections y of a simply supported beam are given by d'y EI 1¹²y = qr(x - L)/2; 0< x < L; dx² y(0) = y(L) = 0 (1) = where L = 10 is the length of the beam, EI= 1900 is modulus of elasticity times moment of inertia, and q = -0.6 is load distribution. The beam extends from z = 0 to x = L. The goals are to find y(x) at points separated by Az = 0.1 step, plot y(x) versus z, and verify the solution. (a) Using appropriate finite differencing scheme, formulate the finite difference approxima- tion. To illustrate your work, use 6 total grid points (2 of them are on the boundaries, y = 0 and y = L). Write down finite-difference approximations for the interior points. Then write your algebraic equations in the matrix-vector form with unknown vector on the left hand side and all known quantities on the right hand side. Clearly define your notations. (b) Write a computer program to solve the above equation fora general uniform rid spacing of Ar. Provide your program as an attachment (.m or .py). Using Ar= 0.1, plot y(x) versus z. Note that the solution must satisfy boundary conditions. Provide your plot. (c) To verify your solution, obtain an analytical solution. Show all your steps. Plot the analytical solution on the same figure above, use different linestyle and clearly label your figures with proper legend. Provide your plot. 1. Under a uniform load, small deflections y of a simply supported beam are given by d'y EI 1¹²y = qr(x - L)/2; 0< x < L; dx² y(0) = y(L) = 0 (1) = where L = 10 is the length of the beam, EI= 1900 is modulus of elasticity times moment of inertia, and q = -0.6 is load distribution. The beam extends from z = 0 to x = L. The goals are to find y(x) at points separated by Az = 0.1 step, plot y(x) versus z, and verify the solution. (a) Using appropriate finite differencing scheme, formulate the finite difference approxima- tion. To illustrate your work, use 6 total grid points (2 of them are on the boundaries, y = 0 and y = L). Write down finite-difference approximations for the interior points. Then write your algebraic equations in the matrix-vector form with unknown vector on the left hand side and all known quantities on the right hand side. Clearly define your notations. (b) Write a computer program to solve the above equation fora general uniform rid spacing of Ar. Provide your program as an attachment (.m or .py). Using Ar= 0.1, plot y(x) versus z. Note that the solution must satisfy boundary conditions. Provide your plot. (c) To verify your solution, obtain an analytical solution. Show all your steps. Plot the analytical solution on the same figure above, use different linestyle and clearly label your figures with proper legend. Provide your plot.