Question

Two shafts, one of solid section and the other of hollow section, of same material and weight having same length are subjected to equal torsional force. What is the torsional stiffness of hollow shaft?

• A. Equal to that of the solid shaft
• B. Less than that of the solid shaft
• C. More than that of the solid shaft
• D. Exactly half of that of the solid shaft

Answer 1

The correct option is C More than that of the solid shaft

Let the diameter of solid shaft be $d_s$. Let the external and internal diameter of the hollow shaft be $d_{o}and{d}_i$ respectively. It is given that both the shafts are made of same material, have same weight and length and are subjected to equal torsional force.

$\therefore$ Weight of solid shaft = Weight of hollow shaft

$\Rightarrow \gamma \times \frac{\pi }{4}{d}_s^{2}L=\gamma \times \frac{\pi }{4}(d_o^2-d_i^2)L$

$\Rightarrow d_s^2=d_o^2-d_i^2.....\left(i\right)$

Now, we know that, torsional stiffness is given as

$k=\frac{T}{\theta }=\frac{G{I}_P}{L}$

For same value of G and L

$k=\frac{T}{\theta }\propto I_p$

$\Rightarrow \frac{k_h}{k_s}=\frac{\frac{\pi }{32}(d_o^4-d_i^4)}{\frac{\pi }{32}\left(d_s^4\right)}$

$\Rightarrow \frac{k_h}{k_s}=\frac{(d_o^2{)}^2-(d_i^2{)}^2}{(d_s^2{)}^2}$

$\Rightarrow \frac{k_h}{k_s}=\frac{(d_o^2-d_i^2)(d_o^2+d_i^2)}{(d_s^2{)}^2}\left[from\right(i\left)\right]$

$\Rightarrow \frac{k_h}{k_s}=\frac{d_0^2+d_i^2}{d_o^2-d_i^2}$

$Now,thequantity\frac{d_o^2+d_i^2}{d_o^2-d_i^2}isalwaysgreaterthan1$

$\therefore k_h>k_s$