(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: 9780073398235

Author: Ferdinand P. Beer, E. Russell Johnston Jr., John T. DeWolf, David F. Mazurek

Publisher: McGraw-Hill Education

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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.