(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 considering the vertical equilibrium condition: ∑ F y = 0 R A = R B = 1 2 × w 0 × L 2 R A = w 0 L 4 Show the moment ( M E I ) diagram for the reaction of the beam as in Figure 2. Due to symmetric beam with symmetric loading, consider the left side of the beam. Calculate the moment ( M 1 ) using the formula: M 1 = R A × L 2 Substitute w 0 L 4 for R A . M 1 = w 0 L 4 × L 2 = w 0 L 2 8 Calculate the ratio of M 1 E I : Ratio = M 1 E I Substitute w 0 L 2 8 for M 1 . M 1 E I = w 0 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 triangle in area ( A 1 ) and h 1 is the height of the triangle in area ( A 1 ) . Substitute L 2 for b 1 and w 0 L 2 8 E I for h 1 . A 1 = 1 2 × L 2 × w 0 L 2 8 E I = w 0 L 3 32 E I Show the moment ( M E I ) diagram for uniformly varying loading of the beam as in Figure 3. Calculate the moment ( M 2 ) for the beam using the formula: M 2 = 1 2 ( − w 0 L 2 ) × ( 1 3 × L 2 ) = − w 0 L 2 24 E I Calculate the ratio of M 2 E I using the relation: Ratio = M 2 E I Substitute − w 0 L 2 24 E I for M 2 . M 2 E I = − w 0 L 2 24 E I Calculate the area ( A 2 ) due to the moment ( M 2 ) using the formula: A 2 = 1 4 b 2 h 2 Here, b 2 is the width of the triangle in area ( A 2 ) and h 2 is the height of the triangle in area ( A 2 ) (b) To determine Find the deflection ( y C ) at point C .

7th Edition

ISBN: 9780100257061

Author: BEER

Publisher: YUZU

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Slope and Deflection

The slope in a deflected beam is described as an angle in radians made by the tangent of the section with the original axis of the beam. The deflection at any point on the axis of the beam is defined as the lateral displacement of the point before and af…

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Chapter 9.5, Problem 114P

(a)

To determine

Find the slope (θA) at end A.

(b)

To determine

Find the deflection (yC) at point C.

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Chapter 9.5, Problem 113P

Chapter 9.5, Problem 115P

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Find the slope ( θ A ) at end A .

Solution Summary: The author illustrates the moment area method by showing the free body diagram of the beam as in Figure 1.