Concept of a beam of uniform strength: is it true that, in such a beam, the bending stress at every cross-section is constant and equal to the allowable stress?

Introduction / Context:The idea of a beam of uniform strength is a classic design concept where the cross-section is varied along the length to keep the bending stress constant and equal to the permissible value everywhere, thus using material efficiently.

Given Data / Assumptions:

• Beam material follows linear elasticity.
• Sectional properties (e.g., depth or width) can vary with x along the span.
• Allowable bending stress is a prescribed limit for the material.

Concept / Approach:For bending, sigma = M / Z, where Z is the section modulus. In a beam of uniform strength, Z(x) is tailored so that sigma(x) = sigma_allow for the actual bending moment distribution M(x). This requires the cross-section to be smaller where moments are small and larger where moments are high.

Step-by-Step Solution:

Verification / Alternative check:Check a candidate variable section by computing sigma(x) across the span; if constant and equal to sigma_allow, the design meets the definition.

Why Other Options Are Wrong:Stipulations about section type, span length, or varying E are not part of the definition; the key is tailoring Z(x) to M(x).

Common Pitfalls:Assuming “uniform strength” means constant cross-section; it means constant stress by varying sectional properties to match the bending moment diagram.

Final Answer:

True