Question

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

• A. \(\frac{w H^2}{8 E}\)
• B. \(\frac{w H^2}{6 E}\)
• C. \(\frac{w R H}{8 E}\)
• D. \(\frac{w R H}{6 E}\)

Hint:

When calculating deflection due to self-weight, remember that the deflection depends on the material's specific weight, the height of the structure, and its elastic modulus. Use the deflection formula for axial-loaded structures.

Answer 1

The Correct Option is A

The problem asks for the vertical deflection at the mid-height of a solid circular cone under self-weight. The self-weight of the cone generates a stress distribution along its height, and this stress causes deflection at different points.
To calculate the deflection due to self-weight, we can use the formula for the deflection of a structural element under its own weight. The deflection at the mid-height of a cone (a tapered structure) due to its weight can be derived using the general equation for deflection in a column under axial load: \[ \delta = \frac{w H^2}{8 E} \] Where:
- \(w\) is the specific weight of the material,
- \(H\) is the height of the cone,
- \(E\) is the elastic modulus of the material.
This formula is derived from the elastic theory of materials considering the geometry of the cone and the self-weight distribution. Thus, the correct answer is (A).