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GATE2017 ME-1: 39

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An initially stress-free massless elastic beam of length $L$ and circular cross-section with diameter $d$ $(d << L)$ is held fixed between two walls as shown. The beam material has Young's modulus $E$ and coefficient of thermal expansion $a$.

If the beam is slowly and uniformly heated, the temperature rise required to cause the beam to buckle is proportional to 

  1. $d$
  2. $d^{2}$
  3. $d^{3}$
  4. $d^{4}$
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