The GATE Grind

GATE 2024 CE (CE2) – Question 39

Solid Mechanics · Column buckling · 2 marks · Multiple choice

A homogeneous, prismatic, linearly elastic steel bar fixed at both the ends has a slenderness ratio ($l/r$) of 105, where $l$ is the bar length and $r$ is the radius of gyration. The coefficient of thermal expansion of steel is $12 \times 10^{-6}$ /°C. Consider the effective length of the steel bar as $0.5l$ and neglect the self-weight of the bar.
The differential increase in temperature (rounded off to the nearest integer) at which the bar buckles is

  1. 298 °C
  2. 85 °C
  3. 400 °C
  4. 250 °C

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) 298 °C

Explanation

A rise in temperature $\Delta T$ of the bar with both ends fixed produces the compressive force $P = EA\alpha\Delta T$. The bar buckles when this reaches the Euler load with the effective length $0.5l$: $\frac{\pi^2EI}{(0.5l)^2} = \frac{4\pi^2EAr^2}{l^2}$. So $\alpha\Delta T = \frac{4\pi^2}{(l/r)^2}$ and $\Delta T = \frac{4\pi^2}{105^2 \times 12 \times 10^{-6}} = \frac{39.478}{0.1323} = 298$ °C.