Question 12.6: A circular hollow shaft of outer diameter 200 mm and inner d...
A circular hollow shaft of outer diameter 200 mm and inner diameter 160 mm is subjected simultaneously to a torque T = 11.1 kN m and an axial compressive load, P = 362 kN. Calculate: (a) \left(\sigma_T\right)_{\max },(b)\left(\sigma_C\right)_{\max } and (c) \tau_{\max } .
Let us first compute the area properties as follows:
Cross-sectional area,
A=\frac{\pi}{4}\left(d_{ o }^2-d_{ i }^2\right)=\frac{\pi}{4}\left(200^2-160^2\right) mm ^2=11309.73 mm ^2
Polar moment of inertia,
J=\frac{\pi}{32}\left(d_{ o }^4-d_{ i }^4\right)=\left(\frac{\pi}{32}\right)\left(200^4-160^4\right) mm ^4=92.74 \times 10^6 mm ^4
Axial stress \left(\sigma_{x x}\right) due to P is
\sigma_{x x}=\frac{P}{A}=\frac{362\left(10^3\right)}{11309.73} N / mm ^2=32 MPa \text { (compressive) }
or \sigma_{x x}=-32 MPa
and shear stress \left(\tau_{x y}\right) due to T is
\tau_{x y}=\frac{T\left(d_0 / 2\right)}{J}=\frac{11.1\left(10^6\right)(200 / 2)}{92.74\left(10^6\right)} N / mm ^2=11.97 MPa
The stress states are shown in Figure 12.14 on a differential element of the shaft:
Thus, the principal stresses are
\sigma_{1,2}=\frac{\sigma_{x x}}{2} \pm \sqrt{\left\lgroup \frac{\sigma_{x x}}{2} \right\rgroup^2+\tau_{x y}^2}
=-16 \pm \sqrt{(-16)^2+11.97^2} MPa
or \sigma_1=-36 MPa , \quad \sigma_2=4 MPa
and the maximum shear stress is
\tau_{\max }=\sqrt{(-16)^2+11.97^2}=20 MPa
Thus,
\left(\sigma_{ T }\right)_{\max }=4 MPa , \quad\left(\sigma_{ C }\right)_{\max }=36 MPa \text { and } \tau_{\max }=20 MPa
