(a) Explanation Calculation: Use moment area method: Show the free body diagram of the beam as in Figure 1. Determine the reaction of the support by taking moment at point B . ∑ M B = 0 − ( R A × L ) + ( w × 3 L 2 ) × ( L 2 2 ) = 0 R A = 3 8 w L Determine the reaction of the support by considering the vertical equilibrium condition: ∑ F y = 0 R A + R B = w ( L + L 2 ) R A + R B = 3 w L 2 Substitute 3 8 w L for R A . 3 8 w L + R B = 3 w L 2 R B = 9 w L 8 Show the moment ( M E I ) diagram for the given loading of the beam as in Figure 2. Calculate the moment ( M 1 ) using the formula: M 1 = R A × L Substitute 3 w L 8 for R A . M 1 = 3 w L 8 × L = 3 w L 2 8 Calculate the ratio of M 1 E I : Ratio = M 1 E I Substitute 3 w L 2 8 for M 1 . M 1 E I = 3 w L 2 8 E I = 3 w L 2 8 E I Calculate the area ( A 1 ) due to the moment ( M 1 ) using the formula: A 1 = 1 2 b 1 h 1 Here, b 1 is the width of the rectangle in area ( A 1 ) and h 1 is the height of the rectangle in area ( A 1 ) . Substitute L for b 1 and 3 w L 2 8 for h 1 . A 1 = 1 2 × L × ( 3 w L 2 8 ) = 3 w L 3 16 E I Calculate the moment ( M 2 ) using the formula: M 2 = − ( w L ) × L 2 = − w L 2 2 Calculate the ratio of M 2 E I using the relation: Ratio = M 2 E I Substitute − w L 2 2 for M 2 . M 2 E I = − w L 2 2 E I Calculate the area ( A 2 ) due to the moment ( M 2 ) using the formula: A 2 = 1 3 b 2 h 2 Here, b 2 is the width of the parabola in area ( A 2 ) and h 2 is the height of the parabola in area ( A 2 ) . Substitute L for b 2 and − w L 2 2 E I for h 2 (b) To determine Find the deflection ( y D ) at point D .

7th Edition

ISBN: 9780100257061

Author: BEER

Publisher: YUZU

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Chapter 9.6, Problem 134P

(a)

To determine

Find the slope (θA) at point A.

(b)

To determine

Find the deflection (yD) at point D.

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Find the slope ( θ A ) at point A .

Solution Summary: The author explains how to determine the reaction of the support by considering the vertical equilibrium condition.

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Find the slope (θA) at point A.

Find the deflection (yD) at point D.

Chapter 9 Solutions