Question 3.3: A hollow shaft and a solid shaft constructed of the same mat...
A hollow shaft and a solid shaft constructed of the same material have the same length and the same outer radius R (Fig. 3-13). The inner radius of the hollow shaft is 0.6R.
(a) Assuming that both shafts are subjected to the same torque, compare their shear stresses, angles of twist, and weights.
(b) Determine the strength-to-weight ratios for both shafts.
(a) Comparison or shear stresses. The maximum shear stresses, given by the torsion formula [Eq. (3-13) \tau_{\max }=\frac{T r}{I_{P}} ], are proportional to 1/I_{P} inasmuch as the torques and radii are the same. For the hollow shaft, we get
I_{P}=\frac{\pi R^{4}}{2}-\frac{\pi(0.6 R)^{4}}{2}=0.4352 \pi R^{4}and for the solid shaft,
I_{P}=\frac{\pi R^{4}}{2}=0.5 \pi R^{4}Therefore, the ratio \beta_{1} of the maximum shear stress in the hollow shaft to that in the solid shaft is
\beta_{1}=\frac{\tau_{H}}{\tau_{S}}=\frac{0.5 \pi R^{4}}{0.4352 \pi R^{4}}=1.15where the subscripts H and S refer to the hollow shaft and the solid shaft, respectively.
Comparison of angles of twist. The angles of twist [Eq. (3-17) \phi=\frac{T L}{G I_{P}} ] are also proportional to 1/I_{P}, because the torques T, lengths L, and moduli of elasticity G are the same for both shafts. Therefore, their ratio is the same as for the shear stresses:
Comparison of weights. The weights of the shafts are proportional to their cross-sectional areas; consequently, the weight of the solid shaft is proportional to \pi \mathrm{R}^{2} and the weight of the hollow shaft is pro-portional to
\pi R^{2}-\pi(0.6 R)^{2}=0.64 \pi R^{2}Therefore, the ratio of the weight of the hollow shaft to the weight of the solid shaft is
\beta_{3}=\frac{W_{H}}{W_{s}}=\frac{0.64 \pi R^{2}}{\pi R^{2}}=0.64From the preceding ratios we again see the inherent advantage of hollow shafts. In this example, the hollow shaft has 15% greater stress and 15% greater angle of rotation than the solid shaft but 36% less weight.
(b) Strength-to-weight ratios. The relative efficiency of a structure is sometimes measured by its strength-to-weight ratio, which is defined for a bar in torsion as the allowable torque divided by the weight. The allowable torque for the hollow shaft of Fig. 3-13a (from the torsion formula) is
and for the solid shaft is
T_{S}=\frac{\tau_{\max } I_{P}}{R}=\frac{\tau_{\max }\left(0.5 \pi R^{4}\right)}{R}=0.5 \pi R^{3} \tau_{\max }The weights of the shafts are equal to the cross-sectional areas times the length L times the weight density γ of the material:
W_{H}=0.64 \pi R^{2} L \gamma \quad W_{s}=\pi R^{2} L \gammaThus, the strength-to-weight ratios S_{H} and S_{S} for the hollow and solid bars, respectively, are
S_{H}=\frac{T_{H}}{W_{H}}=0.68 \frac{\tau_{\max } R}{\gamma L} \quad S_{S}=\frac{T_{S}}{W_{S}}=0.5 \frac{\tau_{\max } R}{\gamma L}In this example, the strength-to-weight ratio of the hollow shaft is 36% greater than the strength-to-weight ratio for the solid shaft, demonstrating once again the relative efficiency of hollow shafts. For a thinner shaft, the percentage will increase; for a thicker shaft, it will decrease.