Beams of uniform strength: If the beam depth is kept constant along the span, how should the width b vary with the local bending moment M to maintain constant extreme-fibre stress?

Introduction / Context:
A beam of uniform strength maintains the same allowable bending stress at every section under the applied loading. If depth cannot be varied (architectural constraint), designers can tailor the width b to keep bending stress constant.

Given Data / Assumptions:

• Beam depth d is constant along the span.
• Allowable bending stress σ_allow is fixed.
• Material remains linear-elastic.

Concept / Approach:
Bending stress at the extreme fibre is σ = M / Z. For a rectangular section, section modulus Z = b d^2 / 6. Holding σ = σ_allow constant gives M / (b d^2 / 6) = σ_allow, thus b ∝ M when d is constant.

Step-by-Step Solution:
Start with σ = M / Z.Use Z_rect = b d^2 / 6.Set σ = constant ⇒ M / (b d^2 / 6) = constant.Rearrange ⇒ b ∝ M, since d is constant.

Verification / Alternative check:
If M doubles near midspan compared to ends, maintaining the same σ requires doubling b at midspan when depth is fixed, matching intuitive expectations for a uniform-strength design.

Why Other Options Are Wrong:

• b ∝ 1/M or √M or M^2: these do not satisfy σ = constant with fixed depth.
• b constant: would cause σ to track M, defeating uniform strength.

Common Pitfalls:

• Using moment of inertia I instead of section modulus Z directly; Z is the correct geometric property for bending stress at the extreme fibre.

Final Answer:
b ∝ M.