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Chapter 5: Problem 2

The solid shaft of radius \(r\) is subjected to a torque \(\mathbf{T}\). Determine the radius \(r^{\prime}\) of the inner core of the shaft that resists one- quarter of the applied torque \((T / 4) .\) Solve the problem two ways: (a) by using the torsion formula, (b) by finding the resultant of the shear-stress distribution.

Short Answer

Expert verified
Using the torsion formula, the radius of the inner core that resists one-fourth of the applied torque is \( r' = 0.8409r \). Using the resultant of the shear-stress distribution, that radius is \( r' = 0.5774r \)

Step by step solution

01

Way (A): Using the Torsion Formula

From the equation of torsion \( T / J = \tau / r \), the elementary torque \( dT \) is given by \( dT = r dA \tau = r^2 dA (T / J) \). Integrating from \( r' \) to \( r \) gives \( T / 4 = \int_{r'}^r r^2 dA (T / J) \) which results in \( T / 4 = (T / J) \int_{r'}^r r^2 dA \). This gives \[ r' = ( \sqrt[4] {1/2} )r \] So the radius of the inner core that resists one-quarter of the applied torque is be \( r' = 0.8409r \)
02

Way (B): Finding the Resultant of the Shear-Stress Distribution

For a given shear stress function \( \tau = (T \cdot r) / J \) of the radial position \( r \), the shear stress \( \tau \) is zero at the center and increases linearly to maximum at the outer surface. Thus, the equivalent resultant shear stress that gives the same torque \( T / 4 \) is located at \( 2r / 3 \), Therefore \( r' = 2 / 3 * r \) which results in \( r' = 0.5774r \) So, the radius of the inner core that resists one-quarter of the applied torque, using the resultant of the shear-stress distribution, will be \( r' = 0.5774r \)

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Most popular questions from this chapter

A torque is applied to the shaft having a radius of \(80 \mathrm{mm} .\) If the material obeys a shear stress-strain relation of \(\tau=500 \gamma^{1 / 4}\) MPa, determine the torque that must be applied to the shaft so that the maximum shear strain becomes 0.008 rad.

The tubular drive shaft for the propeller of a hovercraft is \(6 \mathrm{m}\) long. If the motor delivers \(4 \mathrm{MW}\) of power to the shaft when the propellers rotate at 25 rad \(/\) s, determine the required inner diameter of the shaft if the outer diameter is \(250 \mathrm{mm}\). What is the angle of twist of the shaft when it is operating? Take \(\tau_{\text {allow }}=90 \mathrm{MPa}\) and \(G=75 \mathrm{GPa}\)

The propellers of a ship are connected to an A-36 steel shaft that is \(60 \mathrm{m}\) long and has an outer diameter of \(340 \mathrm{mm}\) and inner diameter of \(260 \mathrm{mm}\). If the power output is \(4.5 \mathrm{MW}\) when the shaft rotates at \(20 \mathrm{rad} / \mathrm{s},\) determine the maximum torsional stress in the shaft and its angle of twist.

The 60-mm-diameter shaft is made of \(6061-\) T 6 aluminum. If the allowable shear stress is \(\tau_{\text {allow }}=80 \mathrm{MPa}\) and the angle of twist of disk \(A\) relative to disk \(C\) is limited so that it does not exceed 0.06 rad, determine the maximum allowable torque \(\mathbf{T}\)

A shaft is made of an aluminum alloy having an allowable shear stress of \(\tau_{\text {allow }}=100\) MPa. If the diameter of the shaft is \(100 \mathrm{mm}\), determine the maximum torque \(\mathbf{T}\) that can be transmitted. What would be the maximum torque \(\mathbf{T}^{\prime}\) if a 75 -mm-diameter hole were bored through the shaft? Sketch the shear-stress distribution along a radial line in each case.
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