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: 9780100257061

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 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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