(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 A . ∑ M B = 0 − R B L + − M 0 = 0 R B = − M 0 L Determine the reaction of the support by considering the vertical equilibrium condition: ∑ F y = 0 R A + R B = 0 Substitute − M 0 L for R B . R A − M 0 L = 0 R A = M 0 L Show the moment ( M E I ) diagram of the given loading beam as in Figure 2. Calculate the moment ( M 1 ) using the formula: M 1 = R A ( L 3 ) Substitute M 0 L for R A . M 1 = ( M 0 L ) ( L 3 ) = M 0 3 Calculate the ratio of M 1 E I : Ratio = M 1 E I Substitute M 0 3 for M 1 . M 1 E I = M 0 3 E I = M 0 3 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 3 for b 1 and M 0 3 for h 1 . A = 1 2 × L 3 × ( M 0 3 E I ) = M 0 L 18 E I Calculate the moment ( M 2 ) using the formula: M 2 = R B ( 2 L 3 ) Substitute − M 0 L for R B . M 2 = ( − M 0 L ) ( 2 L 3 ) = − 2 M 0 3 Calculate the ratio of M 2 E I : Ratio = M 2 E I Substitute − 2 M 0 3 for M 2 . M 2 E I = − 2 M 0 3 E I = − 2 M 0 3 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 2 L 3 for b 2 and − 2 M 0 3 E I for h 2 . A 2 = 1 2 × 2 L 3 × ( − 2 M 0 3 ) = − 2 M 0 L 9 E I Place the reference tangent at point A . Show the tangent for deflection related to reference tangent as in Figure 3. Calculate the deflection at the end A related to the point B ( t B / A ) using the formula: t B / A = [ A 1 × ( 2 L 3 + ( 1 3 × L 3 ) ) ] + [ A 2 × ( 2 3 × 2 L 3 ) ] Substitute M 0 L 18 E I for A 1 and − 2 M 0 L 9 E I for A 2 (b) To determine Find the slope ( y A ) at point A .

7th Edition

ISBN: 8220102804487

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 9.6, Problem 127P

(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 126P

Chapter 9.6, Problem 128P

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EBK MECHANICS OF MATERIALS

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Find the deflection ( y D ) at point D .

Solution Summary: The author calculates the moment (y_D) using the formula.