Project ForgeStrength of Materials
Bending of Beams
GATE 2016
Expert
A simply supported beam of length 2L is subjected to a moment M at the mid-point x=0. The slope at the center x=0 is
- AML/(2EI)
- BML/(3EI)
- ML/(6EI)
- DML/(12EI)
Explanation
By Castigliano's theorem, the rotation at the point of application of moment M equals ∂U/∂M, where U = ∫M(x)²/(2EI)dx over the full span. With reactions ±M/(2L), the bending moment magnitude on each half-span is (M/2L)(L-|x|). Integrating gives U=M²L/(12EI) (both halves contribute equally), so θ(0)=∂U/∂M = ML/(6EI).
Source: GATE ME 2016 Set-2 (2016)
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