Beams of uniform strength vs. uniform section – where is the economy? Beams designed for uniform strength (depth or width varied so that allowable stress is just reached everywhere) are generally more economical than prismatic beams (uniform section) particularly for:

Introduction / Context:
A beam of uniform strength has its cross-section tailored along the span so that the permissible stress is fully utilized everywhere. This contrasts with a prismatic (uniform) beam where much of the material near the supports may be under-stressed when the maximum bending moment occurs near midspan.

Given Data / Assumptions:

• Allowable stress in bending is the design limit.
• Self-weight may be relevant on long spans.
• Elastic behavior and small deflection theory.

Concept / Approach:
Because bending moment M(x) varies along the beam, section modulus Z(x) = M(x) / σ_allow should ideally vary in the same proportion to keep σ ≈ σ_allow everywhere. This redistribution of material (greater depth or width where M is larger) saves weight and cost while maintaining strength.

Step-by-Step Solution:
For a simply supported beam with UDL or midspan load, M(x) peaks near midspan and diminishes toward supports.Uniform strength design makes Z(x) largest where M(x) is largest and vice versa, reducing excess material near supports.Economy is greatest when M(x) varies strongly along the span—typically long spans with distributed loads.

Verification / Alternative check:
Weight savings are often reported by comparing volume integrals of variable vs. uniform sections for the same strength limit; the more variable M(x) is, the larger the saving.

Why Other Options Are Wrong:

• (b) Short spans with heavy loads do not benefit much because the region of high M is small and fabrication complexity may dominate.
• (c) Load magnitude alone does not decide economy.
• (d) Near-uniform M(x) gives little advantage to varying the section.
• (e) Axial compression members are not bending-dominated beams.

Common Pitfalls:
Ignoring deflection criteria; varying depth can also improve stiffness distribution but must satisfy serviceability limits.

Final Answer:
Large spans where bending moment varies significantly