Explanation Given information: The elastic modulus ( E ) is 200 GPa . The section of the beam is W310 × 60 . Calculation: Refer Appendix C, “Properties of Rolled steel shapes”. The moment of inertia ( I ) for the given section is 128 × 10 6 mm 4 or 128 × 10 − 6 m 4 . 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 about A : ∑ M A = 0 R B × 10 = ( w × 10 × 10 2 ) + 40 − 40 R B = 50 w 10 R B = 5 w Determine the reaction of the support by considering the vertical equilibrium condition: ∑ F y = 0 R A + R B = ( w × 10 ) R A + R B = 10 w Substitute 5 w for R B . R A + 5 w = 10 w R A = 5 w 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 ) by considering the reaction of the support using the formula: M 1 = R A × x Substitute 5 w for R A and 5 m for x . M 1 = 5 w × 5 = 25 w Calculate the ratio of M 1 E I : Ratio = M 1 E I Substitute 25 w for M 1 , 200 GPa for E , and 128 × 10 − 6 m 4 for I . M 1 E I = 25 w 200 × 10 6 × 128 × 10 − 6 = ( 9.76563 × 10 − 4 ) w 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 5m for b 1 and ( 9.76563 × 10 − 4 ) w for h 1 . A 1 = 1 2 × 5 × 9.76563 × 10 − 4 w = 2.4414 w × 10 − 3 kN ⋅ m 2 Calculate the moment ( M 2 ) by considering moment acting on the beam using the formula: M 2 = − 40 kN ⋅ m Here, negative sign indicates that the moment is acting anticlockwise direction. Calculate the ratio of M 2 E I : Ratio = M 2 E I Substitute − 40 kN ⋅ m for M 2 , 200 GPa for E and 128 × 10 − 6 m 4 for I . M 2 E I = − 40 kN ⋅ m 200 × 10 6 × 128 × 10 − 6 = − 1.5625 × 10 − 3 Calculate the area ( A 2 ) due to the moment ( M 2 ) using the formula: A 2 = b 2 h 2 Here, b 2 is the width of the rectangle in area ( A 2 ) and h 2 is the height of the rectangle in area ( A 2 )

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ISBN: 8220100257063

Author: BEER

Publisher: YUZU

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Chapter 9.5, Problem 122P

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Find the load (w) for which the deflection is zero at midpoint C of the beam.

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Chapter 9.5, Problem 121P

Chapter 9.5, Problem 123P

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Find the load ( w ) for which the deflection is zero at midpoint C of the beam.

Solution Summary: The author calculates the moment of inertia (I) for the given section of the beam by considering the vertical equilibrium condition.

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Find the load (w) for which the deflection is zero at midpoint C of the beam.

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