(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 + P L 2 − P L 4 = 0 R A L = 2 P L − P L 4 R A = P L 4 L R A = P 4 Determine the reaction of the support by taking moment at point A . ∑ M A = 0 R B L + P ( 3 L ) 4 − P L 2 = 0 R B L = − 3 P L + 2 P L 4 R B = − P L 4 L R B = − P 4 Show the shear force diagram ( V ) of the loading beam as in Figure 2. Due to symmetric beam with symmetric loading, consider the left side of the beam. Show the moment ( M E I ) diagram for the left side of the given loading beam as in Figure 2. Calculate the moment ( M 1 ) using the formula: M 1 = R A ( L 2 ) Substitute P 4 for R A . M 1 = ( P 4 ) ( L 2 ) = P L 8 Calculate the ratio of M 1 E I using the relation: Ratio = M 1 E I Substitute P L 8 for M 1 . M 1 E I = P L 8 E I = P L 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 and P L 8 E I for h. A = 1 2 × L 2 × ( P L 8 E I ) = P L 2 32 E I Calculate the moment ( M 2 ) using the formula: M 2 = R A ( L 2 ) Substitute P 4 for R A . M 2 = ( P 4 ) ( L 2 ) = P L 8 Calculate the ratio of M 2 E I : Ratio = M 2 E I Substitute P L 8 for M 2 . M 2 E I = P L 8 E I = P L 8 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 L 6 for b 2 and P L 8 E I for h 2 . A 2 = 1 2 × L 6 × ( P L 8 E I ) = P L 2 96 E I Calculate the moment ( M 3 ) using the formula: M 3 = R B ( L 4 ) Substitute − P 4 for R B . M 3 = − ( P 4 ) ( L 4 ) = − P L 16 Calculate the ratio of M 3 E I : Ratio = M 3 E I Substitute − P L 16 for M 3 . M 3 E I = − P L 16 E I = − P L 16 E I Calculate the area ( A 3 ) due to the moment ( M 3 ) using the formula: A 3 = 1 2 b 3 h 3 Here, b 3 is the width of the triangle in area ( A 3 ) and h 3 is the height of the triangle in area ( A 3 ) . Substitute L 12 for b 3 and − P L 16 E I for h 3 . A 3 = 1 2 × L 12 × ( − P L 16 E I ) = − P L 2 384 E I Calculate the moment ( M 4 ) using the formula: M 4 = R B ( L 4 ) Substitute − P 4 for R B . M 4 = − ( P 4 ) ( L 4 ) = − P L 16 Calculate the ratio of M 4 E I : Ratio = M 4 E I Substitute − P L 16 for M 4 (b) To determine Find the slope ( y A ) at point A .

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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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 9.6, Problem 126P

(a)

To determine

Find the deflection (yD) at point D.

(b)

To determine

Find the slope (yA) at point A.

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Chapter 9.6, Problem 125P

Chapter 9.6, Problem 127P

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Find the deflection ( y D ) at point D .

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