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

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 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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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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