Explanation Write the expression for the moment of inertia of the shaft. I = π d 4 64 (I) Here, the diameter of the shaft is d and the moment of inertia is I . Draw the free body diagram of the beam. Figure (1) The free body diagram of the beam in the direction of y- axis is shown in figure (1). Write the expression for the deflection along y -axis for beam 6 using Table A-9. y A = F 1 b 1 x 6 E I l ( x 2 + b 1 2 − l 2 ) (II) Here, the deflection of the beam is y A , the net force at point A is F 1 , the location of point A from right end is b 1 , the elongation distance of any given point from left end is x , the total length of the beam is l , the Young modulus of the material is E and moment of inertia of the beam is I . Write the expression for the force component at point A along y -axis. F 1 y = F 1 sin 45 ° − F 1 cos 45 ° (III) Here, the force component at A along y-axis is F 1 y and the force at point A is F 1 The free body diagram of the beam in the direction of z- axis is shown below. Figure (2) Write the deflection equation along z -axis for beam 6 using Table A-9. z A = F 1 z b 1 x 6 E I l ( x 2 + b 1 2 − l 2 ) + F 2 b 2 x 6 E I l ( x 2 + b 2 2 − l 2 ) (IV) Here, the deflection of the beam is z A , the net force at point B is F 2 and the location of point B from right end is b 2 . Write the force component at point A along z -axis. F 1 z = F 1 cos 45 ° − F 1 sin 45 ° (V) Here, the force component at A along z-axis is F 1 y and the force at point A is F 1 . Write the expression for net deflection at point A . δ = y A 2 + z A 2 (VI) Here, the net deflection at point A is δ . Write the expression for slope along z- axis at point A . ( θ A ) z = ( d y d x ) = { d d x [ F 1 b 1 x 6 E I l ( x 2 + b 1 2 − l 2 ) ] } = F 1 b 1 6 E I l ( 3 x 2 + b 1 2 − l 2 ) (VII) Here, the slope along z- axis at point is ( θ A ) z . Write expression for the slope along y- axis at point A . ( θ A ) y = − ( d z d x ) = − { d d x [ F 1 z b 1 x 6 E I l ( x 2 + b 1 2 − l 2 ) + F 2 b 2 x 6 E I l ( x 2 + b 2 2 − l 2 ) ] } = − F 1 z b 1 6 E I l ( 3 x 2 + b 1 2 − l 2 ) − F 2 b 2 6 E I l ( 3 x 2 + b 2 2 − l 2 ) (VIII) Here, the slope along z- axis at point is ( θ A ) y . Write the expression for the net slope at point A . Θ A = ( θ A ) z 2 + ( θ A ) y 2 (IX) Here, the net slope is Θ A . Write the magnitude of net force at point A along y -axis. F 1 = T 1 + T 2 (X) Here, the net force at point A along y -axis is F 1 , the tension at point A is T 1 and the tension at point A is T 2 . Write the magnitude of net force at point B along z -axis. F 2 = T 1 B + T 2 B . (XI) Here, the net force at point B along y -axis is F 2 , the tension at point B is T 1 B and the tension at point B is T 2 B . Conclusion: Substitute 20 in for d in the Equation (I). I = π ( 20 mm ) 4 64 = 502654.82 64 mm 4 ≈ 7854 mm 4 Substitute − 300 N for F 1 and − 45 N for F 1 in Equation (III). F 1 y = ( − 300 N ) sin 45 ° + ( − 45 N ) sin 45 ° = − 243.95 N Substitute − 300 N for F 2 and 45 N for F 2 in Equation (V). F 1 z = 300 N cos 45 ° + 45 N cos 45 ° = 243.95 N Substitute − 250 N for T 1 B and − 37.5 N for T 2 B in Equation (XI). F 2 = − 250 N + ( − 37.5 N ) = − 287.5 N Substitute − 243.95 N for F 1 y , 300 mm for x , 550 mm for b 1 , 850 mm for l , 207 × 10 3 MPa for E and 7854 mm 4 for I in the equation (II). y A = { ( − 243.95 N ) ( 550 mm ) ( 300 mm ) 6 ( 207 × 10 3 MPa ) ( 7854 mm 4 ) ( 850 mm ) × ( ( 300 mm ) 2 + ( 550 mm ) 2 − ( 850 mm ) 2 ) } = ( 4.8545 × 10 − 6 ) ( − 330000 ) ( N / MPa ⋅ mm ) ( 1 MPa 1 N / mm 2 ) = 1.60 mm Thus, the net displacement of the shaft at point A along y -axis is 1.60 mm . Substitute 243.95 N for F 1 z , 300 mm for x , 550 mm for b 1 , 150 mm for b 2 , 850 mm for l , 207 × 10 3 MPa for E and 7854 mm 4 for I in the equation (IV)

Shigley's Mechanical Engineering Design (McGraw-Hill Series in Mechanical Engineering)

10th Edition

ISBN: 9780073398204

Author: Richard G Budynas, Keith J Nisbett

Publisher: McGraw-Hill Education

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Chapter 4, Problem 26P

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The slope of the shaft at point A.

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The deflection of the shaft at point A . The slope of the shaft at point A .

Solution Summary: The author explains the deflection of the shaft at point A, the moment of inertia, and the elongation distance of any given point from left end.

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The deflection of the shaft at point A.

The slope of the shaft at point A.

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The deflection of the shaft at point A . The slope of the shaft at point A .

Solution Summary: The author explains the deflection of the shaft at point A, the moment of inertia, and the elongation distance of any given point from left end.

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The deflection of the shaft at point A. The slope of the shaft at point A.

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Chapter 4 Solutions

The deflection of the shaft at point A.

The slope of the shaft at point A.