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GATE 2022 CE (CE2) – Question 51

Structural Analysis · Deflections of determinate beams, frames and trusses · 2 marks · Numerical answer

Two cantilevers UV and WX, each length 2a, have fixed ends U and X. Their rigidities are EI and 2EI. A vertical spring joins free ends V,W. P=100 kN acts at V and Q at the midpoint of WX. With k=20 kN/m, EI=10⁵ kNm², a=5 m, find Q for zero spring force (nearest kN).

Oppositely fixed cantilevers UV and WX of length 2a, joined at their free ends by a vertical spring. See the attached source image.

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Correct answer: 640

Explanation

Zero spring force requires equal downward displacement of V and W. The first cantilever has length 2a and tip load P, giving $\delta_V=8Pa^3/(3EI)$.

For the second cantilever of rigidity 2EI, a load Q at distance a from its fixed end gives free-end deflection $\delta_W=Qa^2(6a-a)/(12EI)=5Qa^3/(12EI)$. Equating gives $Q=32P/5=\mathbf{640}$ kN.