Self-weight scaling of a bar: If all linear dimensions of a prismatic bar are scaled up by a factor n (n > 1), in what ratio does the maximum axial stress produced by its own weight increase?

Introduction / Context:
Dimensional scaling arguments help predict how stresses change when structures are built larger or smaller but with similar proportions. For vertical members carrying only their own weight, the governing parameter is length, not cross-sectional area.

Given Data / Assumptions:

• Geometrically similar scaling: length, breadth, and depth multiply by n.
• Material properties (density ρ and modulus) unchanged.
• Bar is vertical; stress of interest is the maximum axial stress due to self-weight.

Concept / Approach:
For a vertical prismatic bar of constant area A and length L, the maximum axial stress due to its own weight is σ_max = ρ * g * L (at the top). This expression is independent of A because weight increases with A but so does the load-carrying area, canceling out.

Step-by-Step Solution:
Original: σ_1 = ρ g L.After scaling: L_2 = n L; area scales as A_2 = n^2 A; weight per unit length scales as ρ g A_2.However, stress at the top remains σ_2 = ρ g L_2 = ρ g (n L) = n (ρ g L) = n σ_1.Hence, the stress increases in the ratio n : 1.

Verification / Alternative check:
Dimensional analysis confirms σ scales with length only for self-weight in a prismatic bar; finite-element or calculus derivations of σ(y) = ρ g (L − y) lead to the same conclusion.

Why Other Options Are Wrong:
1 : n or 1 : n^2 imply stress decreases with size, which is incorrect for self-weight in prismatic bars.n^2 : 1 exaggerates the scaling (stress does not depend on area).Remains unchanged contradicts the direct proportionality to length.

Common Pitfalls:
Assuming stress scales with total weight without accounting for simultaneous scaling of area, or overlooking that σ_max occurs at the top and equals ρ g L.

Final Answer:
n : 1.