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GATE 2021 ME (ME1) – Question 41

Mechanics of Materials · Stress, strain and elastic constants · 2 marks · Multiple choice

A right solid circular cone standing on its base on a horizontal surface is of height $H$ and base radius $R$. The cone is made of a material with specific weight $w$ and elastic modulus $E$. The vertical deflection at the mid-height of the cone due to self-weight is given by

  1. $wH^2/(8E)$
  2. $wH^2/(6E)$
  3. $wRH/(8E)$
  4. $wRH/(6E)$

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Show answer and explanation

Correct answer: (A) $wH^2/(8E)$

Explanation

At height z above the base, the cone above the cut has height H−z and volume A(z)(H−z)/3. Compressive stress is therefore $w(H-z)/3$.

Integrate strain from the fixed base to H/2: $\delta=\int_0^{H/2}w(H-z)/(3E)dz=wH^2/(8E)$. **Answer: A**. Radius cancels from weight divided by area.