Beam of uniform strength: A beam may be designed to have uniform strength (constant maximum fibre stress along its length) by which of the following approaches?

Introduction / Context:
A beam of uniform strength is shaped so that the allowable bending stress is reached everywhere along the span, reducing weight compared to a prismatic beam. This concept is used in cranes, leaf springs, and tapered structural members.

Given Data / Assumptions:

• Bending stress formula: sigma = M / Z where Z is section modulus.
• Allowable stress sigma_allow is constant along the beam.
• Designer may vary cross-sectional dimensions.

Concept / Approach:
For sigma to be constant while M(x) varies with x, the section modulus must vary proportionally: Z(x) = M(x) / sigma_allow. Since Z depends on both width and depth, either parameter (or both) may be varied to match the bending moment diagram.

Step-by-Step Solution:

Verification / Alternative check:
For a cantilever with end load (M ∝ x), a triangular depth variation provides near-constant stress; other profiles can achieve the same with width variation.

Why Other Options Are Wrong:
Options a, b, and c each can be correct individually; limiting the designer to only one would be unnecessarily restrictive.

Common Pitfalls:
Ignoring local stability (buckling of thin webs/flanges) when aggressively tapering members.

Final Answer:
Any one of the above, depending on design convenience