To determine The torque T causes the inner surface of the shaft to yield. To sketch: The residual shear-stress distribution in the shaft. Explanation Given information: Shaft is subjected to a torque T . Calculation: The outer radius of the shaft is 3 in. The inner radius of the shaft is 1.5 in. Refer the shear stress-strain diagram: The value of yield shear stress τ Y = 10 ksi . The value of yield shear strain γ Y = 0.004 rad . The expression to find the torque ( T ) in the shaft with inner radius ( c i ) and outer radius ( c 0 ) is shown below: T = 2 π ∫ c i c 0 τ ρ 2 d ρ (1) Substitute τ Y for τ , 1.5 in . for c i , and 3 in . for c 0 in Equation (1). T = 2 π τ Y ∫ 1.5 in . 3 in . ρ 2 d ρ = 2 π τ Y [ ρ 3 3 ] 1.5 in . 3 in . = 2 π ( 10 ksi ) ( ( 3 in . ) 3 3 − ( 1.5 in . ) 3 3 ) = 494.8008 kip ⋅ in . T = 494.8008 kip ⋅ in . ( 1 ft 12 in . ) = 41.2334 kip ⋅ ft ≈ 41.2 kip ⋅ ft Therefore, the torque T causes the inner surface of the shaft to yield is 41.2 kip ⋅ ft _ Find the angle of twist ( ϕ ) of the shaft using the relation: ϕ = γ Y ρ Y L (2) Here, L is the length of the shaft. Substitute 0.004 rad for γ Y , 1.5 in. for ρ Y , and 3 ft for L in Equation (2). ϕ = 0.004 rad 1.5 in . ( 3 ft ) = 0.004 rad 1.5 in . ( 3 ft × 12 in . 1 ft ) = 0.096 rad Find the residual shear stress when the reverse torque of T ' = 20 kN ⋅ m is applied to the shaft. Find the shear stress ( τ ' ) at ρ = c 0 using the relation shown below: ( τ ' ) at ρ = c 0 = T ' c 0 J (3) Find the polar moment of inertia of outer radius ( c 0 ) and inner radius ( c i ) as shown below: J = π 2 ( c 0 4 − c i 4 ) = π 2 ( ( 3 in

9th Edition

ISBN: 9780133254426

Author: Russell C. Hibbeler

Publisher: Prentice Hall

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Any functioning component, such as machine parts, structural members, pressure vessels, and so on, are all under the influence of more than one force. When anybody is under the influence of more than one force such as shear forces, bending moments, axial…

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Chapter 5.10, Problem 5.138P

To determine

The torque T causes the inner surface of the shaft to yield.

To sketch:

The residual shear-stress distribution in the shaft.

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Chapter 5.10, Problem 5.137P

Chapter 5.10, Problem 5.139P

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