Sizing a rectangular beam when breadth b is fixed: how does the required depth d vary with the design bending moment M to satisfy a given allowable bending stress?

Introduction / Context:
Preliminary beam sizing commonly uses the bending stress formula M = σ_allow * Z. For rectangular sections with fixed breadth b, designers adjust depth d to meet moment capacity. Understanding the scaling relation between d and M fast-tracks early sizing.

Given Data / Assumptions:

• Rectangular section, breadth b fixed.
• Allowable bending stress σ_allow is fixed by material and code.
• Elastic bending theory applies; serviceability not considered here.

Concept / Approach:

Section modulus for a rectangular section about its strong axis is Z = b d^2 / 6. Required moment capacity is M = σ_allow * Z. Solving for d gives the scaling with M for constant b and σ_allow.

Step-by-Step Solution:

Verification / Alternative check:

Doubling the design moment M requires depth increase by √2, consistent with the square-root relation.

Why Other Options Are Wrong:

Linear, cube-root, or quadratic scaling does not follow from Z = b d^2 / 6 when b is constant. d cannot be independent of M.

Common Pitfalls:

Using I instead of Z directly; forgetting that Z ∝ d^2 for rectangles, leading to the square-root dependence.

Final Answer:

d ∝ M^0.5 (square-root relation)