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?
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Ad ∝ M^0.5 (square-root relation)
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Bd ∝ M (linear relation)
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Cd ∝ M^(1/3) (cube-root relation)
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Dd ∝ M^2 (quadratic relation)
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Ed is independent of M
Answer
Correct Answer: d ∝ M^0.5 (square-root relation)
Explanation
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:
Z = b d^2 / 6.M = σ_allow * Z = σ_allow * (b d^2 / 6).Rearrange: d^2 = (6 M) / (σ_allow b) ⇒ d = √[ (6 M) / (σ_allow b) ].Therefore, d ∝ √M for fixed b and σ_allow.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)