Project ForgeStrength of Materials
Bending of Beams
GATE 2017
Hard
An initially stress-free massless elastic beam of length L and circular cross-section with diameter d (d ≪ L) is held fixed between two rigid walls. The beam material has Young's modulus E and coefficient of thermal expansion α. If the beam is slowly and uniformly heated, the temperature rise required to cause the beam to buckle is proportional to
- Ad
- d²
- Cd³
- Dd⁴
Explanation
Euler buckling load Pcr = π²EI/L², with I = πd⁴/64, so Pcr ∝ d⁴. Restrained thermal force = EAαΔT, with A = πd²/4, so the force ∝ d²·ΔT. Setting the thermal force equal to Pcr: d²·ΔT ∝ d⁴, giving ΔT ∝ d².
Source: GATE ME 2017 Session 1 (ME-1) (2017)
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