Question

# A thin tubular shaft of circular cross section with inside diameter 100 mm is subjected to a torque of $5000\ \mathrm{N} \cdot \mathrm{m}$. If the allowable shear stress is 42 MPa, determine the required wall thickness t by using (a) the approximate theory for a thin-walled tube, and (b) the exact torsion theory for a circular bar.

Solution

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a) To determine the required thickness of the wall using the approximate theory, we will use the expression for maximum shear stress in the circular cross section 3-83:

\begin{aligned} &\tau_{max}=\frac{T}{2\pi r^2 t} \end{aligned}

Where we will set the shear stress in the shaft to its maximum allowable value of $\tau_{max}=42$ MPa, $T=5000$ N$\cdot$m$=5000\times 10^3$ N$\cdot$mm is the torque acting on the shaft, $t$ is the thickness of the tube. $r$ is the median radius of the shaft. We can express the median radius as:

\begin{aligned} &r=\frac{100}{2}+\frac{t}{2}\\\\ &r=50+\frac{t}{2} \end{aligned}

Substituting the set expressions:

\begin{aligned} &42=\frac{5000\times 10^3}{2\pi \Big(50+\dfrac{t}{2}\Big)^2 t} \end{aligned}

Solving numerically for $t$ gives us solution $t=6.662$ mm.

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