Project ForgeA right solid circular cone standing on its base on a horizontal surface has height H and base radius R, specific weight w, and elastic modulus E. The vertical deflection at mid-height of the cone due to self-weight is given by
- wH²/8E
- BwH²/6E
- CwRH/8E
- DwRH/6E
Explanation
Measuring height y from the base, the axial stress at y is σ(y) = w×(volume of cone above y)/(area at y) = w×[(1/3)πr(y)²(H-y)]/[πr(y)²] = w(H-y)/3 — remarkably independent of the radius, since the taper cancels exactly. Strain ε(y)=σ(y)/E. Displacement at y (relative to the fixed base) = ∫₀^y ε dy' = (w/3E)[Hy-y²/2]. At y=H/2: u=(w/3E)[H²/2-H²/8]=(w/3E)(3H²/8)=wH²/(8E).
Source: GATE ME 2021 Session 1 (ME-1) (2021)
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