(a) Explanation Given information: The section of the cantilever beam ABC is W 130 × 23.8 The young’s modulus ( E ) of steel is 200 GPa . Calculation: Show the free body diagram of the beam as in Figure 1. Calculate the moment due to the reaction at B as below: M 1 = R B × L A B Here, L A B is length of the beam. Substitute 1.8 m for L A B . M 1 = R B × 1.8 = 1.8 R B Calculate the M 1 E I value: Substitute 1.8 R B for M 1 . ( M 1 E I ) = 1.8 R B E I Show the ( M 1 E I ) diagram in below Figure 2. Calculate the area ( A 1 ) using the relation: A 1 = 1 2 ( L A B ) ( M 1 E I ) Substitute 1.8 m for L A B and 1.8 R B E I for ( M 1 E I ) . A 1 = 1 2 ( 1.8 ) ( 1.8 R B E I ) = 1.62 R B E I (1) Calculate the moment ( M 2 ) due the point load: M 2 = − 20 × ( L A B + L B C ) Here, L B C is length of the beam. Substitute 1.8 m for L A B and 1.2 m for L B C . M 2 = − 20 × ( 1.8 + 1.2 ) = − 60 kN ⋅ m Calculate the ( M 2 E I ) value: Substitute − 60 kN ⋅ m for M 2 . ( M 2 E I ) = − 60 E I Show the ( M 2 E I ) diagram in below Figure 3. Calculate the area ( A 2 ) using the relation: A 2 = 1 2 ( L A B + L B C ) ( M 2 E I ) Substitute 1.8 m for L A B , 1.2 m for L B C , and ( − 60 E I ) for ( M 2 E I ) . A 2 = 1 2 ( 1.8 + 1.2 ) ( − 60 E I ) = − 90 E I Refer Appendix C “Properties of rolled steel shape (SI units)” for moment of inertia of section W 130 × 23.8 and the value is 8.91 × 10 6 mm 4 . Calculate the value ( EI ): Substitute 200 GPa for E and 8.91 × 10 6 mm 4 for I . E I = ( 200 GPa × 1 , 000 , 000 kN/m 2 1 GPa ) × ( 8.91 × 10 6 mm 4 × 1 m 4 1 , 000 4 mm 4 ) = 1 , 782 kN ⋅ m 2 Express the tangential deviation equation of C with respect to A : t C / A = A 1 ( ( 2 3 L A B ) + L B C ) + A 2 ( 2 3 ( L A B + L B C ) ) Equate the above equation to zero. 0 = A 1 ( ( 2 3 L A B ) + L B C ) + A 2 ( 2 3 ( L A B + L B C ) ) (2). Substitute 1.62 R B E I for A 1 , − 90 E I for A 2 , 1 , 782 kN ⋅ m 2 for EI , 1 (b) To determine The final value of the reaction ( R B ) at point B .

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Author: BEER

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Chapter 9.6, Problem 153P

(a)

To determine

Find how much point B should be raised to return point C to tits original position.

(b)

To determine

The final value of the reaction (RB) at point B.

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Chapter 9.6, Problem 152P

Chapter 9.6, Problem 154P

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Find how much point B should be raised to return point C to tits original position.

Solution Summary: The author explains how to calculate the moment due to the reaction at B.

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Find how much point B should be raised to return point C to tits original position.

The final value of the reaction (RB) at point B.

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