Explanation Formula used: The differential equation of deflection of beam is as E I d 4 y d x 4 = w Calculation: A beam of length(L)=10 is hinged at both the ends and has a variable load distribution of w = k E I x where the value of k = 7.2 × 10 − 4 / m and x is the distance from on end So, the differential equation of deflection of beam is as follows: E I d 4 y d x 4 = w Where EI is a measure of stiffness of beam and w is the weight distribution along the beam. Substitute the value of w = k E I x in E I d 4 y d x 4 = w E I d 4 y d x 4 = k E I x d 4 y d x 4 = k x The auxiliary equation is: m 4 = 0 m = 0 , 0 , 0 , 0 So, the general solution to the above differential equation is y c = c 1 + c 2 x + c 3 x 2 + c 4 x 3 To solve the nonhomogeneous solution of the form, let y p = A x 5 Differentiate the equation y p = A x 5 for the first order is as follows: y ' p = 5 A x 4 Differentiate the equation y ' p = 5 A x 4 for the second order is as follows: y " p = 20 A x 3 Again, differentiate the equation y " p = 20 A x 3 for the third order is as follows: y " ' p = 60 A x 2 Again, differentiate the equation y " ' p = 60 A x 2 for the fourth order is as follows: y i v p = 120 A x Substitute the value of y i v p = 120 A x in the differential equation d 4 y d x 4 = k x . 120 A x = k x A = k 120 Thus, the complete solution of the differential equation is as follows: y = c 1 + c 2 x + c 3 x 2 + c 4 x 3 + k 120 x 5 Substitute the boundary conditions y = 0 when x = 0 in y = c 1 + c 2 x + c 3 x 2 + c 4 x 3 + k 120 x 5 . 0 = c 1 + c 2 × 0 + c 3 × 0 2 + c 4 × 0 3 + k 120 × 0 5 c 1 = 0 Differentiate the equation y c = c 1 + c 2 x + c 3 x 2 + c 4 x 3 + k 120 x 5 is as follows: y ' = c 2 + 2 c 3 x + 3 c 4 x 2 + k 24 x 4 Again, differentiate the equation y ' = c 2 + 2 c 3 x + 3 c 4 x 2 + k 24 x 4 is as follows: y " = 2 c 3 + 6 c 4 x + k 6 x 3 Substitute the second boundary condition y = 0 and x = L in y = c 1 + c 2 x + c 3 x 2 + c 4 x 3 + k 120 x 5

Pearson eText for Basic Technical Mathematics with Calculus -- Instant Access (Pearson+)

11th Edition

ISBN: 9780137554843

Author: Allyn Washington, Richard Evans

Publisher: PEARSON+

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Chapter 31.10, Problem 28E

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The equation of the deflection y in terms of x.

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Chapter 31.10, Problem 27E

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Chapter 31 Solutions

Pearson eText for Basic Technical Mathematics with Calculus -- Instant Access (Pearson+)

Ch. 31.1 - Show that is a solution of . Is it the general...Ch. 31.1 - Prob. 1E Ch. 31.1 - In Exercises 1 and 2, show that the indicated...Ch. 31.1 - In Exercises 3–6, determine whether the given...Ch. 31.1 - Prob. 4E Ch. 31.1 - In Exercises 3–6, determine whether the given...Ch. 31.1 - Prob. 6E Ch. 31.1 - In Exercises 7–10, show that each function y =...Ch. 31.1 - Prob. 8E Ch. 31.1 - In Exercises 7–10, show that each function y =...

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