Problem 1 (20 points, Core Course Outcome #6)/MatlabGrader Develop a Matlab function mySimpson 38 that calculates I = fy(x)dr using the composite Simpson's 3/8 method for a set of discrete data points (x, y) that are equally spaced. If the number of subintervals is not divisible by 3 because there is 1 extra subinterval, calculate the integral of the last 4 subintervals using Simpson's 1/3 method. If there are 2 extra subintervals, calculate the integral for the last 2 subintervals using Simpson's 1/3 method. If the number of subintervals is zero (= 1 data point only), the function shall return a value of zero for the integral. If the number of subintervals is one (= 2 data points only), the function shall use the trapezoidal method on the single interval. If the vectors x and y are not of equal length, the function shall execute Matlab's function error("...") with an appropriate error message. As input the function shall take the column vectors x and y that contain the data points. As output the function shall give the calculated integral I. You may call/use any function developed in recitation, homework, and/or exams. Required submission: well commented function source code submitted to Matlab Grader using the Canvas link for Exam 6 - Problem 1 Problem 2 (35 points, Core Course Outcomes 6 & 11) A cantilevered beam of length L = 16 is made of a hybrid material that has a spatially varying product of the area moment of p(x) X El(x) L Figure 1: Sketch of cantilevered beam with varying load p(x) and varying EI(x). inertia and Young's modulus, EI(x,a), where a is a parameter describing the hybrid material composition with 2 ≤ a ≤5. The beam is subjected to a varying load p(x), see Figure 1 (positive values of p are pointing in the positive y direction). Recall that for the beam shown, the shear force F(x) at location r; is the bending moment M(r.) is F(x)= p(x)dx, the beam's slope (z;) as a result of the bending moment is and the beam's deflection y(r.) is 1 M(x)= -[" F(x)dx, 0(xi)= M(z) El(x,a) edz. y(x;) = 0 (x)dr. (1) (2) (3) (4) The load p(x) can be determined by calling the content hidden function loadbeam (x) available in the Canvas assignment. The input argument can be either a scalar value of a single location along the beam, or a column vector containing multiple locations. The output is p(x) at the given input location(s) as either a scalar (for a scalar input ) or a column vector. EI(x, α) can be determined by calling the content hidden function EIbeam (x, alpha) available in the Canvas assignment. Here alpha is the scalar value of a and the input argument x can be either a scalar value of a single location along the beam, or a column vector containing multiple locations. The output is EI(x, a) at the given input location(s) as either a scalar (for a scalar input) or a column vector. The content hidden Matlab function files loadbeam.p and EIbeam.p are available on Canvas in the exam assignment. For a = 3 and N = 32 equally sized intervals along the beam, determine the beam's deflection at it's end, y(L) using only integration methods covered in this class. Graph in separate plots the load p(x), the beam's shear force F(x), bending moment M(x), slope (2), and deflection y(x). Note that all quantities are dimensionless and Eq. (3) does require component-wise division ./ when calculating the integrand using vectors. Use column vectors throughout. State which integration method you have used. On Canvas, you will find a link to a MatlabGrader non-credit test for the script you code. The test let's you check whether your script will calculate the deflection at the beam's end correctly using Newton-Cotes integration techniques covered in class. It is entirely optional and will not earn any credit. Please see the description of the MatlabGrader problem for further details. Required submission to Gradescope: name of integration method used (named in code is not sufficient); beam's deflection at its end, y(L): 5 separate graphs showing p(x), F(x), M(x), 0(x), and y(x) respectively; printout of script and any additional functions not already submitted in prior homework/exams;

Problem 1 (20 points, Core Course Outcome #6)/MatlabGrader Develop a Matlab function mySimpson 38 that calculates I = fy(x)dr using the composite Simpson's 3/8 method for a set of discrete data points (x, y) that are equally spaced. If the number of subintervals is not divisible by 3 because there is 1 extra subinterval, calculate the integral of the last 4 subintervals using Simpson's 1/3 method. If there are 2 extra subintervals, calculate the integral for the last 2 subintervals using Simpson's 1/3 method. If the number of subintervals is zero (= 1 data point only), the function shall return a value of zero for the integral. If the number of subintervals is one (= 2 data points only), the function shall use the trapezoidal method on the single interval. If the vectors x and y are not of equal length, the function shall execute Matlab's function error("...") with an appropriate error message. As input the function shall take the column vectors x and y that contain the data points. As output the function shall give the calculated integral I. You may call/use any function developed in recitation, homework, and/or exams. Required submission: well commented function source code submitted to Matlab Grader using the Canvas link for Exam 6 - Problem 1 Problem 2 (35 points, Core Course Outcomes 6 & 11) A cantilevered beam of length L = 16 is made of a hybrid material that has a spatially varying product of the area moment of p(x) X El(x) L Figure 1: Sketch of cantilevered beam with varying load p(x) and varying EI(x). inertia and Young's modulus, EI(x,a), where a is a parameter describing the hybrid material composition with 2 ≤ a ≤5. The beam is subjected to a varying load p(x), see Figure 1 (positive values of p are pointing in the positive y direction). Recall that for the beam shown, the shear force F(x) at location r; is the bending moment M(r.) is F(x)= p(x)dx, the beam's slope (z;) as a result of the bending moment is and the beam's deflection y(r.) is 1 M(x)= -[" F(x)dx, 0(xi)= M(z) El(x,a) edz. y(x;) = 0 (x)dr. (1) (2) (3) (4) The load p(x) can be determined by calling the content hidden function loadbeam (x) available in the Canvas assignment. The input argument can be either a scalar value of a single location along the beam, or a column vector containing multiple locations. The output is p(x) at the given input location(s) as either a scalar (for a scalar input ) or a column vector. EI(x, α) can be determined by calling the content hidden function EIbeam (x, alpha) available in the Canvas assignment. Here alpha is the scalar value of a and the input argument x can be either a scalar value of a single location along the beam, or a column vector containing multiple locations. The output is EI(x, a) at the given input location(s) as either a scalar (for a scalar input) or a column vector. The content hidden Matlab function files loadbeam.p and EIbeam.p are available on Canvas in the exam assignment. For a = 3 and N = 32 equally sized intervals along the beam, determine the beam's deflection at it's end, y(L) using only integration methods covered in this class. Graph in separate plots the load p(x), the beam's shear force F(x), bending moment M(x), slope (2), and deflection y(x). Note that all quantities are dimensionless and Eq. (3) does require component-wise division ./ when calculating the integrand using vectors. Use column vectors throughout. State which integration method you have used. On Canvas, you will find a link to a MatlabGrader non-credit test for the script you code. The test let's you check whether your script will calculate the deflection at the beam's end correctly using Newton-Cotes integration techniques covered in class. It is entirely optional and will not earn any credit. Please see the description of the MatlabGrader problem for further details. Required submission to Gradescope: name of integration method used (named in code is not sufficient); beam's deflection at its end, y(L): 5 separate graphs showing p(x), F(x), M(x), 0(x), and y(x) respectively; printout of script and any additional functions not already submitted in prior homework/exams;

Operations Research : Applications and Algorithms

4th Edition

ISBN:9780534380588

Author:Wayne L. Winston

Publisher:Wayne L. Winston

Chapter12: Review Of Calculus And Probability

Section12.5: Random Variables, Mean, Variance, And Covariance

Problem 5P

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