Explanation Formula used: The differential equation of deflection of beam is as E I d 4 y d x 4 = w Calculation: A cantilever beam is clamped at the end x =0 and is free at the end x = L . So, the differential equation of deflection of beam is as follows: E I d 4 y p d x 4 = w Where EI is a measure of stiffness of beam and w is the weight distribution along the beam. 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 4 . Differentiate the equation y p = A x 4 for the first order is as follows: y ' p = 4 A x 3 Differentiate the equation y ' p = 4 A x 3 for the second order is as follows: y " p = 12 A x 2 Again, differentiate the equation y " p = 12 A x 2 for the third order is as follows: y " ' p = 24 A x Again, differentiate the equation y " ' p = 24 A x for the fourth order is as follows: y i v p = 24 A Substitute the value of y i v p = 24 A in the differential equation d 4 y p d x 4 = w . 24 A E I = w A = w 24 E I 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 + w 24 E I x 4 Substitute the boundary conditions y = 0 when x = 0 in y = c 1 + c 2 x + c 3 x 2 + c 4 x 3 + w 24 E I x 4 . 0 = c 1 + c 2 × 0 + c 3 × 0 2 + c 4 × 0 3 + w 24 E I 0 4 c 1 = 0 Differentiate the equation y = c 1 + c 2 x + c 3 x 2 + c 4 x 3 + w 24 E I x 4 is as follows: y ' = c 2 + 2 c 3 x + 3 c 4 x 2 + 4 w 24 E I x 3 Substitute the value of x = 0 and y ' = 0 in y ' = c 2 + 2 c 3 x + 3 c 4 x 2 + 4 w 24 E I x 3 0 = c 2

11th Edition

ISBN: 9780134508290

Author: Evans

Publisher: PEARSON CUSTOM PUB.(CONSIGNMENT)

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Chapter 31.10, Problem 27E

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The equation of the deflection y in terms of x.

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Chapter 31.10, Problem 26E

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Chapter 31 Solutions

EBK BASIC TECHNICAL MATHEMATICS

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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