Explanation Given information: The length of the shaft is 3 ft. The diameter of the shafts is 1.5 in. The torque applied at the top of the gear is 600 lb ⋅ ft . Calculation: Let the torque at the fixed end A of the shaft be T A and the torque at the fixed end B of the shaft be T B . Sketch the free body diagram of the shafts as shown in Figure 1. Apply Equilibrium equation to the shaft with 4 in. gear radius. Sum of moments at x is equal to 0. Σ M x = 0 T A + F ( 4 in . ) − 600 lb ⋅ ft = 0 T A + F ( 4 in . × 1 ft 12 in . ) − 600 lb ⋅ ft = 0 F = 1 , 800 − 3 T A (1) Apply Equilibrium equation to the shaft with 2 in. gear radius. Sum of moments at x is equal to 0. Σ M x = 0 T B − F ( 2 in . ) = 0 T B − F ( 2 in . × 1 ft 12 in . ) = 0 F = 6 T B (2) Equate Equation (1) and Equation (2) as shown below. 1 , 800 − 3 T A = 6 T B 3 T A + 6 T B = 1 , 800 T A + 2 T B = 600 (3) The angle of twist ( ϕ ) can be expressed as, ϕ = T L J G Here, T is the torque applied, L is the length of the shaft, J is the polar moment of inertia, and G is the shear modulus of elasticity of the material. Apply compatibility condition. The angle of twist of E is same as the angle of twist of gear F . 4 ( ϕ E ) = 2 ( ϕ F ) 4 ( T A L J G ) = 2 ( T B L J G ) 4 T A = 2 T B T A = 0.5 T B (4) Substitute 0.5 T B for T A in Equation (3). 0.5 T B + 2 T B = 600 T B = 240 lb ⋅ ft Substitute 240 lb ⋅ ft for T B in Equation (4). T A = 0.5 ( 240 lb ⋅ ft ) = 120 lb ⋅ ft The radius of the shaft BD is 0.75 in. The polar moment of inertia for a solid shaft of radius ( r ) is π 2 r 4 . The torsion formula for maximum shear stress in the solid shaft BD ( τ B D ) max is shown below: ( τ B D ) max = T B c J Here, the polar moment of inertia is J and the outer radius of the shaft is c

Statics and Mechanics of Materials (5th Edition)

5th Edition

ISBN: 9780134382593

Author: Russell C. Hibbeler

Publisher: PEARSON

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Design of Shafts and Shaft Components

The shaft is basically defined as the machine element that rotates, is in circular cross-section, it is majorly used to transfer the power from one part of the system to another, or from a power generating machine to a power absorbing machine. The shaft …

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Chapter 10.5, Problem 75P

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Find the maximum shear stress in the shaft BD and AC.

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Chapter 10.5, Problem 74P

Chapter 10.5, Problem 76P

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Find the maximum shear stress in the shaft BD and AC .

Solution Summary: The author calculates the maximum shear stress in the shaft BD and AC. The torque applied at the top of the gear is 600lbcdot

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Find the maximum shear stress in the shaft BD and AC.

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