A hollow circular section shaft 2 m long is firmly supported at each end and has an outside diameter of 80 mm. The shaft is subjected to a torque of 12 kNm applied at a point 1.5 m from one end. If the shear stress in the shaft is limited to 150 N/mm^2 and the angle of twist to 1.5°, calculate the

Question

A hollow circular section shaft 2 m long is firmly supported at each end and has an outside diameter of 80 mm. The shaft is subjected to a torque of 12 kNm applied at a point 1.5 m from one end. If the shear stress in the shaft is limited to 150 N/mm² and the angle of twist to 1.5°, calculate the maximum allowable internal diameter. The shear modulus G = 80000 N/mm².

Accepted Answer

Referring to Fig. S.11.2 and using Eq. (11.8)

[latex] T_{ A }=\frac{L_{ BC }}{L_{ AB }+L_{ BC }} T [/latex] (11.8)

[latex] T_{ A }=\left(\frac{0.5}{2.0} ight) imes 12=3 kNm [/latex]

Therefore, from equilibrium

[latex] T_{C}=9 kNm [/latex]

Maximum shear stress condition:
From Eq. (11.4)

[latex] \frac{T}{J}=\frac{ au}{r}=G \frac{ heta}{L} [/latex] (11.4)

[latex] \frac{T_{\max }}{J}=\frac{ au_{\max }}{\left(\frac{D}{2} ight)} [/latex] (i)

where D is the external diameter of the shaft and [latex] J=\pi\left(D^{4}-d^{4} ight) / 32 [/latex] in which d is the internal diameter of the shaft. Then, substituting in Eq. (i)

[latex] \frac{9 imes 10^{6}}{\left[\frac{\pi\left(80^{4}-d^{4} ight)}{32} ight]}=\frac{150}{\left(\frac{80}{2} ight)} [/latex]

which gives

[latex] d=63.7 mm [/latex]

Maximum angle of twist condition:
The maximum angle of twist will occur at B; then either the portion AB or the portion BC may be considered. Considering AB, from Eq. (11.4)

[latex] heta_{ B }=\frac{T_{ AB } L_{ AB }}{G J} [/latex]

i.e.

[latex] 1.5 imes \frac{\pi}{180}=\frac{3 imes 10^{6} imes 1.5 imes 10^{3}}{80000 imes \pi\left[\frac{\left(80^{4}-d^{4} ight)}{32} ight]} [/latex]

from which

[latex] d=66.1 mm [/latex]

The maximum allowable internal diameter is therefore 63.7 mm.

Question 11.2: A hollow circular section shaft 2 m long is firmly supported...

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