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

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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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