Project ForgeStrength of Materials
Bending of Beams
GATE 2008
Hard
A simply supported beam has two overhangs of length L each, with a downward point load P at each overhang tip; the supports are 2L apart, directly below each load. The strain energy stored in the beam (flexural rigidity EI) is
- AP²L³/3EI
- B2P²L³/3EI
- 4P²L³/3EI
- D8P²L³/3EI
Explanation
By symmetry each support reaction is P (upward). In each overhang (length L, measuring x from the free tip), M(x)=-Px, so ∫M²dx=P²L³/3 per overhang (two overhangs: 2P²L³/3). In the middle span (length 2L), the moment is constant at -PL, giving ∫M²dx=P²L²·2L=2P²L³. Total ∫M²dx=2P²L³/3+2P²L³=8P²L³/3, so U=(1/2EI)(8P²L³/3)=4P²L³/3EI.
Source: GATE ME 2008 (2008)
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