GATE 2023 AE – Question 48
The thickness of a uniform hollow circular shaft is equal to the difference between the outer radius and the inner radius. The ratio of the inner diameter to outer diameter of the shaft is 0.5. For the shaft reacting to an applied torque, the ratio of the maximum shear stress τ to the maximum shear stress thin wall τ − obtained using the thin-wall approximation is . (round off to one decimal place)
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Correct answer: 1.1 to 1.3
Explanation
**Geometry.** The inner-to-outer diameter ratio is 0.5, so $r_i=0.5\,r_o$. The thickness is $t=r_o-r_i=0.5\,r_o$, and the mean radius is $r_m=\dfrac{r_o+r_i}{2}=0.75\,r_o$.
**Exact maximum shear stress** (at the outer surface):
$$\tau=\frac{T\,r_o}{J},\qquad J=\frac{\pi}{2}\left(r_o^4-r_i^4\right).$$
**Thin-wall approximation:**
$$\tau_{thin}=\frac{T}{2\pi r_m^2\,t}.$$
**Ratio:**
$$\frac{\tau}{\tau_{thin}}=\frac{T r_o\big/\left[\frac\pi2(r_o^4-r_i^4)\right]}{T\big/\left(2\pi r_m^2t\right)}=\frac{4\,r_o\,r_m^2\,t}{r_o^4-r_i^4}.$$
Substituting (with $r_o=1$): numerator $4(1)(0.5625)(0.5)=1.125$ and denominator $1-0.0625=0.9375$:
$$\frac{\tau}{\tau_{thin}}=\frac{1.125}{0.9375}=\mathbf{1.2}.$$
Official GATE 2023 answer key: https://gate2026.iitg.ac.in/doc/download/Answer_keys2023/AE_ANS_GATE2023.pdf