(a) Explanation Given information: The elastic modulus ( E ) is 200 GPa . The section of the beam is W460 × 74 . Calculation: Refer Appendix C, “Properties of Rolled steel shapes”. The moment of inertia ( I ) for the given section is 333 × 10 6 mm 4 or 333 × 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 E × 5 = ( 150 × 3 ) + ( 150 × 2 ) + 60 − 60 R E = 750 5 R E = 150 kN Determine the reaction of the support by considering the vertical equilibrium condition: ∑ F y = 0 R A + R E = 150 + 150 R A + R E = 300 kN Substitute 150kN for R E . R A + 150 = 300 R A = 150 kN Show the moment ( M E I ) diagram for the left side of the given loading beam as in Figure 2. Due to symmetric beam with symmetric loading, consider the left side of the beam. Calculate the moment ( M 1 ) by taking moment at A using the formula: M 1 = R A × 2 Substitute 150 kips for R A . M 1 = 150 × 2 = 300 kN ⋅ m Calculate the ratio of M 1 E I : Ratio = M 1 E I Substitute 300 kN ⋅ m for M 1 , 200 GPa for E , and 333 × 10 − 6 m 4 for I . M 1 E I = 300kN ⋅ m 200 × 10 6 × 333 × 10 − 6 = 4.505 × 10 − 3 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 2 m for b 1 and 4.505 × 10 − 3 for h 1 . A 1 = 1 2 × 2 × 4.505 × 10 − 3 = 4.505 × 10 − 3 kN ⋅ m 2 Calculate the moment ( M 2 ) using the formula: M 2 = 300 kN ⋅ m Calculate the ratio of M 2 E I : Ratio = M 2 E I Substitute 300 kN ⋅ m for M 2 , 200 GPa for E , and 333 × 10 − 6 m 4 for I . M 2 E I = 300 kN ⋅ m 200 × 10 6 × 333 × 10 − 6 = 4.505 × 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 ) (b) To determine Find the deflection ( y C ) at point C .

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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 119P

(a)

To determine

Find the slope (θA) at end A.

(b)

To determine

Find the deflection (yC) at point C.

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Chapter 9.5, Problem 118P

Chapter 9.5, Problem 120P

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Find the slope ( θ A ) at end A .

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