Elastic elongation under self-weight – Conical bar versus prismatic bar A conical bar and a prismatic (uniform) bar of the same material and length L hang under their own weight. The elongation of the conical bar is how much of the elongation of the prismatic bar?

Introduction / Context:
Bars elongate under their self-weight because the material below each cross-section supports the weight of material beneath. The distribution of cross-sectional area changes how stress accumulates along the length, making tapering an effective way to reduce elongation.

Given Data / Assumptions:

• Length L, same material (Young’s modulus E and unit weight γ) for both bars.
• Prismatic bar has constant area A; conical bar tapers linearly to the tip.
• Small deformations and linear elasticity.

Concept / Approach:
Axial strain at a section equals axial stress / E. Stress at a height x from the top equals weight below that section divided by the local area. Integrating strain along the length gives total elongation. For self-weight: δ_prismatic = (γ L^2) / (2 E). For a conical bar: δ_conical = (γ L^2) / (3 E).

Step-by-Step Solution:
For the prismatic bar: δ_pr = (γ L^2) / (2 E).For the conical bar: δ_co = (γ L^2) / (3 E) (due to area increasing toward the top, lowering stress).Ratio: δ_co / δ_pr = (1/3) / (1/2) = 2/3.Hence the conical bar elongates by two-thirds of the prismatic bar’s elongation.

Verification / Alternative check:
A taper reduces stress near the top by providing more area where the force is larger, so a smaller overall extension compared to a uniform bar is physically reasonable.

Why Other Options Are Wrong:
“Equal to” ignores area variation; “half” and “one-third” underestimate; “three-fourths” overestimates relative to the established 2/3.

Common Pitfalls:
Neglecting to integrate strain properly; using average stress instead of the varying stress distribution; mixing up factors 1/2 and 1/3.

Final Answer:
two-third