Beam of uniform strength with constant depth – how should the width vary? For a prismatic beam designed for uniform strength while keeping its depth constant, the required width along the span must vary in proportion to which quantity?

Introduction / Context:“Uniform strength” beams are shaped so that the extreme-fibre stress remains approximately constant along the span. When depth is held constant, width must be adjusted according to how internal bending moment varies with position.

Given Data / Assumptions:

• Rectangular section with constant depth d.
• Allowable bending stress σ_allow is constant along the span.
• Small deflection theory; material remains elastic.

Concept / Approach:For a rectangular beam, section modulus Z = b d^2 / 6. Bending stress is σ = M / Z = 6M / (b d^2). To keep σ = σ_allow, the width b must be adjusted so that b ∝ M.

Step-by-Step Solution:σ_allow = 6M / (b d^2)Rearrange → b = 6M / (σ_allow d^2)Hence, b varies directly with M(x), the bending moment diagram.

Verification / Alternative check:For a simply supported beam with a central point load, M is triangular from support to midspan, and so is the required b(x) when d is held constant.

Why Other Options Are Wrong:

• Shear force primarily governs web/web-like regions; it does not determine bending stress shape for uniform strength.
• M^2 leads to unnecessary over-variation.
• Deflection depends on EI and load distribution; it is not the basis for stress-uniformity shaping.

Common Pitfalls:Confusing uniform strength with uniform stiffness; the former targets constant stress, the latter targets constant EI.

Final Answer:Bending moment M(x)