Explanation Given information: Rods AB and AC have a diameter of 20 mm and are made of 6061-T6 aluminum alloy. They are connected to the rigid collar A which slides freely along the vertical guide rod. The 50-kg block D is dropped from height h above the collar. Consideration: For 6061-T6 aluminum alloy, The modulus of elasticity is E = 68.9 × 10 9 N/m 2 . The acceleration due to gravity is g = 9.81 m/sec 2 . The allowable normal stress is σ Y = 255 MPa . Joint A : Show the free-body diagram of the joint A as in Figure 1. Along the horizontal direction: Determine the force in the member AB in terms of member AC by resolving the horizontal component of forces. ∑ F x = 0 F A B sin 30 ° − F A C sin 30 ° = 0 F A B = F A C = F Along the vertical direction: Determine the force in the segment AD in terms of F by resolving the vertical component of forces. ∑ F y = 0 F A B cos 30 ° + F A C cos 30 ° − P A = 0 0.866 F + 0.866 F − P A = 0 P A = 1.732 F Area of the rods: Determine the area of the rods ( A ) using the formula. A = π d 2 4 (1) Here, d is diameter of therods. Substitute 20 mm for d in Equation (1). A = π ( 20 mm × 1 m 1,000 mm ) 2 4 = 3.142 × 10 − 4 m 2 Compatibility condition: Show the free-body diagram of the geometry to find the deflection as in Figure 2. Refer to Figure 2. δ F = Δ A cos 30 ° Determine the deflection at point A using the basic formula. δ = P L A E (2) Here, P is force in the rods, L is length of the rods, and E is Young’s modulus of the materials. Substitute Δ A cos 30 ° for δ F , F for P, 400 mm for L , 3.142 × 10 − 4 m 2 for A , and 68.9 × 10 9 N/m 2 for G in Equation (2). Δ A cos 30 ° = F × 400 mm × 1 m 1,000 mm 3.142 × 10 − 4 × 68.9 × 10 9 Δ A = ( 2.13383 × 10 − 8 ) F Stiffness of the spring: Determine the equivalent stiffness for the system using the relation. k = P A Δ A (3) Substitute 1.732 F for P A and ( 2.13383 × 10 − 8 ) F for Δ A in Equation (3). k = 1.732 F ( 2.13383 × 10 − 8 ) F = ( 81.17 × 10 6 ) N/m Maximum normal stress: Determine the maximum force acting in the rods AB and AC using the relation. ( σ max ) A B = ( σ max ) A C = σ Y σ Y = F max A (4) Substitute 255 MPa for σ Y and 3

9th Edition

ISBN: 9780133254426

Author: Russell C. Hibbeler

Publisher: Prentice Hall

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Chapter 14.4, Problem 14.52P

To determine

the maximum height h from which the 50-kg block D can be dropped without causing yielding in the rods when the block strikes the collar.

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Chapter 14.4, Problem 14.51P

Chapter 14.4, Problem 14.53P

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