An initially stress-free massless elastic beam has length $L$ and circular diameter $d\ll L$, with both ends fixed between rigid walls. Material constants are Young’s modulus E and thermal-expansion coefficient $\alpha$. When uniformly heated slowly, its buckling temperature rise is proportional to
Restrained expansion causes compression $N=EA\alpha\Delta T$. Fixed-fixed Euler buckling occurs at $N_{cr}=4\pi^2EI/L^2$. Equating them yields $\Delta T_{cr}=4\pi^2I/(A\alpha L^2)$. For a circular section, $I/A=d^2/16$, giving $\Delta T_{cr}=\pi^2d^2/(4\alpha L^2)$. With other parameters fixed, this is proportional to $d^2$, B.
More Euler theory of columns questions
GATE 2022 ME (ME1) Q16– A uniform light slender beam AB of section modulus EI is pinned by a frictionless joint A…
GATE 2021 ME (ME2) Q34– A column with one end fixed and one end free has a critical buckling load of 100 N. For…
GATE 2020 ME (ME1) Q40– The truss has four members of length l and flexural rigidity EI, and one member of length…
GATE 2020 ME (ME2) Q48– The uniaxial compression test graph is for a 1 m long solid metal bar. Response is linear…
GATE 2019 ME (ME1) Q50– The prismatic beam shown is pinned at both ends, length π m, square side 6 mm, E = 200 GPa…
GATE 2018 ME (ME1) Q29– A 15 mm × 10 mm rectangular steel column is 1.5 m long and simply supported at both ends.…
