Consider two shafts in torsion, each of the same material, length, and cross-sectional area. One
shaft has a solid square cross section and the other shaft has a solid circular section.
(a) Which shaft has the greater maximum shear stress and by what percentage?
(b) Which shaft has the greater angular twist θ and by what percentage?
For a square cross section with side length b and a circular section with diameter d
A_{\text {square }}=A_{\text {circular }} \quad \Rightarrow \quad b^{2}=\frac{\pi}{4} d^{2} \quad \Rightarrow \quad b=\frac{\sqrt{\pi}}{2} d
From Eq. (3-40) with b=c
\left(\tau_{\max }\right)_{\text {square }}=\frac{T}{b c^{2}}\left(3+\frac{1.8}{b / c}\right)=\frac{T}{b^{3}}\left(3+\frac{1.8}{1}\right)=\frac{T}{d^{3}}\left(\frac{2}{\sqrt{\pi}}\right)^{3}(4.8)=6.896 \frac{T}{d^{3}}
For the circular cross section,
\begin{aligned}&\left(\tau_{\max }\right)_{\text {circular }}=\frac{16 T}{\pi d^{3}}=5.093 \frac{T}{d^{3}} \\&\frac{\left(\tau_{\max }\right)_{\text {square }}}{\left(\tau_{\max }\right)_{\text {circular }}}=\frac{6.896 \frac{T}{d^{3}}}{5.093 \frac{T}{d^{3}}}=1.354\end{aligned}
The shear stress in the square cross section is 35.4 \% greater. .
(b) For the square cross section, from the table on p. 102, \beta=0.141 . From Eq. (3-41),
\theta_{\text {square }}=\frac{T l}{\beta b c^{3} G}=\frac{T l}{\beta b^{4} G}=\frac{T l}{0.141\left(\frac{\sqrt{\pi}}{2} d\right)^{4} G}=11.50 \frac{T l}{d^{4} G}
For the circular cross section,
\begin{aligned}&\theta_{r d}=\frac{T l}{G J}=\frac{T l}{G\left(\pi d^{4} / 32\right)}=10.19 \frac{T l}{d^{4} G} \\&\frac{\theta_{s q}}{\theta_{r d}}=\frac{11.50 \frac{T l}{d^{4} G}}{10.19 \frac{T l}{d^{4} G}}=1.129\end{aligned}
The angle of twist in the square cross section is 12.9 \% greater. .
Eq. (3-41):
\theta=\frac{T l}{\beta b c^{3} G}Eq. (3-40):
\tau_{\max }=\frac{T}{\alpha b c^{2}} \doteq \frac{T}{b c^{2}}\left(3+\frac{1.8}{b / c}\right)table on p.102
| b/c | 1 | 1.5 | 1.75 | 2 | 2.5 | 3 | 4 | 6 | 8 | 10 | ∞ |
| α | 0.208 | 0.231 | 0.239 | 0.246 | 0.258 | 0.267 | 0.282 | 0.299 | 0.307 | 0.313 | 0.333 |
| β | 0.141 | 0.196 | 0.214 | 0.228 | 0.249 | 0.263 | 0.281 | 0.299 | 0.307 | 0.313 | 0.333 |