Solution to Problem 325 Torsion

Problem 325
The two steel shaft shown in Fig. P-325, each with one end built into a rigid support have flanges rigidly attached to their free ends. The shafts are to be bolted together at their flanges. However, initially there is a 6° mismatch in the location of the bolt holes as shown in the figure. Determine the maximum shearing stress in each shaft after the shafts are bolted together. Use G = 12 × 10⁶ psi and neglect deformations of the bolts and flanges.

Solution 325

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$\theta_{\text{of 6.5' shaft}} + \theta_{\text{of 3.25' shaft}} = 6^\circ$

$\left( \dfrac{TL}{JG} \right)_{\text{of 6.5' shaft}} + \left( \dfrac{TL}{JG} \right)_{\text{of 3.25' shaft}} = 6^\circ \left( \dfrac{\pi}{180^\circ} \right)$

$\dfrac{T(6.5)(12^2)}{\frac{1}{32}\pi (2^4)(12 \times 10^6)} + \dfrac{T(3.25)(12^2)}{\frac{1}{32}\pi (1.5^4)(12 \times 10^6)} = \dfrac{\pi}{30}$

$T = 817.32 \, \text{lb}\cdot\text{ft}$

$\tau_{max} = \dfrac{16T}{\pi D^3}$

$\tau_{\text{of 6.5' shaft}} = \dfrac{16(817.32)(12)}{\pi (2^3)} = 6243.86 \, \text{ psi}$ answer

$\tau_{\text{of 3.25' shaft}} = \dfrac{16(817.32)(12)}{\pi (1.5^3)} = 14\,800.27 \, \text{ psi}$ answer

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