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 B = R D = 1 2 w L Here, w is the weight per unit length of the rod. 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 ) over the span BC using the formula: M 1 = 1 2 w L ( x − a ) Substitute L 2 for x . M 1 = 1 2 w L ( L 2 − a ) = 1 2 w L ( L − 2 a 2 ) = 1 4 w L ( L − 2 a ) Calculate the ratio of M 1 E I : Ratio = M 1 E I Substitute 1 4 w L ( L − 2 a ) for M 1 . M 1 E I = 1 4 w L ( L − 2 a ) 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 − a ) for b 1 and 1 4 w L ( L − 2 a ) E I for h 1 . A 1 = 1 2 × ( L 2 − a ) × 1 4 w L ( L − 2 a ) E I = 1 2 × ( L − 2 a 2 ) × ( 1 4 w L ( L − 2 a ) ) × 1 E I = 1 16 w L ( L − 2 a ) 2 E I Calculate the moment ( M 2 ) over the span AB using the formula: M 2 = − 1 2 w x 2 Substitute L 2 for x . M 2 = − 1 2 w ( L 2 ) 2 = − 1 2 w ( L 2 4 ) = − 1 8 w L 2 Calculate the ratio of M 2 E I using the relation: Ratio = M 2 E I Substitute − 1 8 w L 2 for M 2 . M 2 E I = − 1 8 w L 2 E I = − 1 8 E I w L 2 Calculate the area ( A 2 ) due to the moment ( M 2 ) using the formula: A 2 = 1 3 b 2 h 2 Here, b 2 is the width of the parabola in area ( A 2 ) and h 2 is the height of the parabola in area ( A 2 ) . Substitute L 2 for b 2 and − 1 8 E I w L 2 for h 2 . A 2 = 1 3 × L 2 × ( − 1 8 E I w L 2 ) = − 1 48 w L 3 E I Place the reference tangent at point C so that the deflection ( y C ) at point C is zero

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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 9.5, Problem 124P

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Find the distance (a) for which the slope at ends A and E is zero.

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Chapter 9.5, Problem 123P

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Chapter 9 Solutions

EBK MECHANICS OF MATERIALS

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Find the distance ( a ) for which the slope at ends A and E is zero.

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

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Find the distance (a) for which the slope at ends A and E is zero.

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