(a) Explanation Given information: The elastic modulus ( E ) is 29 × 10 6 psi . The section of the beam is S6 × 12.5 . Calculation: Refer Appendix C, “Properties of Rolled steel shapes”. The moment of inertia ( I ) for the given section is 22 in . 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 B : ∑ M B = 0 R D × 13 = ( 1.5 × 13 ) + ( 7 × 6.5 ) R D = 65 13 R D = 5 kips Determine the reaction of the support by considering the vertical equilibrium condition: ∑ F y = 0 R B + R D = 1.5 + 7 + 1.5 R B + R D = 10 kips Substitute 5 kips for R D . R B + 5 = 10 R B = 5 kips 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 C using the formula: M 1 = R B × 4.5 Substitute 5 kips for R B . M 1 = 5 × 4.5 = 22.5 kips ⋅ ft Calculate the ratio of M 1 E I : Ratio = M 1 E I Substitute 22.5 kips ⋅ ft for M 1 . M 1 E I = 22.5 kips ⋅ ft 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 4.5 ft for b 1 and 22.5 kips ⋅ ft E I for h 1 . A 1 = 1 2 × 4.5 × 22.5 E I = 50.625 E I kips ⋅ ft 2 Calculate the moment ( M 2 ) by taking moment at C using the formula: M 2 = − 1.5 × ( 2 + 4.5 ) = − 9.75 kips ⋅ ft Calculate the ratio of M 2 E I : Ratio = M 2 E I Substitute − 9.75 kips ⋅ ft for M 2 . M 2 E I = -9 .75kips ⋅ ft E I Calculate the area ( A 2 ) due to the moment ( M 2 ) using the formula: A 2 = 1 2 b 2 h 2 Here, b 2 is the width of the triangle in area ( A 2 ) and h 2 is the height of the triangle in area ( A 2 ) (b) To determine Find the deflection ( y A ) at point A . (c) To determine Find the deflection ( y C ) at point C .

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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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 117P

(a)

To determine

Find the slope (θA) at end A.

(b)

To determine

Find the deflection (yA) at point A.

(c)

To determine

Find the deflection (yC) at point C.

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Chapter 9.5, Problem 116P

Chapter 9.5, Problem 118P

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Find the slope ( θ A ) at end A .

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

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Find the slope (θA) at end A.

Find the deflection (yA) at point A.

Find the deflection (yC) at point C.

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