Explanation Given information: The flexural rigidity of the beam is E I . The length of the beam ( L ) is 24 ft . Calculation: Show the free body diagram of the beam as in Figure 1. Refer to Figure 1. Calculate the support reactions A and B ( R A and R B ) : Calculate the moment about point A by applying the equation of equilibrium: Sum of moments about point A is equal to 0. ∑ M A = 0 ( R B × 24 ) − 6 − 6 − ( 4 × 24 ) = 0 24 R B − 108 = 0 R B = 4.5 kip Calculate the vertical forces by applying the equation of equilibrium: Sum of vertical forces is equal to 0. ∑ F y = 0 − R A + R B − 4 = 0 − R A + 4.5 = 4 R A = 0.5 kip ( ↓ ) Calculate the moment function with respect of the x distance M ( x ) : M ( x ) = − 0.5 〈 x − 0 〉 − ( − 6 ) 〈 x − 8 〉 0 − ( − 6 ) 〈 x − 16 〉 0 = − 0.5 x + 6 〈 x − 8 〉 0 + 6 〈 x − 16 〉 0 Elastic Curve and slope is as follows: E I d 2 υ d x 2 = M ( x ) Substitute [ − 0.5 x + 6 〈 x − 8 〉 0 + 6 〈 x − 16 〉 0 ] for M ( x ) . E I d 2 υ d x 2 = − 0.5 x + 6 〈 x − 8 〉 0 + 6 〈 x − 16 〉 0 Integrate both sides of the Equation. E I d υ d x = − 0.5 x 2 2 + 6 〈 x − 8 〉 + 6 〈 x − 16 〉 + C 1 = − 0.25 x 2 + 6 〈 x − 8 〉 + 6 〈 x − 16 〉 + C 1 Integrate both sides of the Equation. E I υ = − 0.25 x 3 3 + 6 2 〈 x − 8 〉 2 + 6 2 〈 x − 16 〉 2 + C 1 x + C 2 = − 0

Statics and Mechanics of Materials, Student Value Edition (5th Edition)

5th Edition

ISBN: 9780134382890

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 16.3, Problem 43P

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Statics and Mechanics of Materials, Student Value Edition (5th Edition)

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Find the equation of the elastic curve ( υ ) .

Solution Summary: The author calculates the flexural rigidity of the beam by applying the equation of equilibrium: Sum of moments about point A is equal to 0.

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Chapter 16 Solutions