(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 × P R A = P 2 Determine the maximum moment ( M max ) by taking moment at A : M max = R A × 2 a Substitute P 2 for R A . M max = P 2 × 2 a = P a Show the moment ( M E I ) diagram for the given loading 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 = P a 2 Calculate the ratio of M 1 E I : Ratio = M 1 E I Substitute P a 2 for M 1 . M 1 E I = P a 2 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 a for b 1 and P a 2 E I for h 1 . A 1 = 1 2 × a × P a 2 E I = P a 2 4 E I Calculate the moment ( M 2 ) by extending the load below the beam using the formula: M 2 = P a 2 Calculate the ratio of M 2 E I : Ratio = M 2 E I Substitute P a 2 for M 2 and 2 EI for EI . M 2 E I = P a 2 ( 2 E I ) = P a 4 E I Calculate the area ( A 2 ) due to the moment ( M 2 ) using the formula: A 2 = 1 2 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 ) . Substitute a for b 2 and P a 4 E I for h 2 . A 2 = 1 2 × a × P a 4 E I = P a 2 8 E I Calculate the moment ( M 3 ) using the formula: M 3 = P a 2 Calculate the ratio of M 3 E I using the relation: Ratio = M 3 E I Substitute P a 2 for M 3 (b) To determine Find the deflection ( y C ) at point C .

7th Edition

ISBN: 8220100257063

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 115P

(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 114P

Chapter 9.5, Problem 116P

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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.