Bending stress–moment relationship in beams Fill the blank: The bending stress in a beam section is __________ the bending moment at that section (other factors constant).

Introduction / Context:The flexure formula relates bending stress to internal bending moment and section properties. Understanding proportionality guides selection of sections with appropriate moment of inertia to limit stresses under a given loading.

Given Data / Assumptions:

• Bernoulli–Euler theory: plane sections remain plane.
• Linear elastic material; small deflections.
• Stress evaluated at a given fiber distance y from the neutral axis.

Concept / Approach:The flexure formula is sigma = M * y / I. For a fixed section (I and y fixed at a particular fiber), sigma varies directly with M. Therefore, as bending moment increases, bending stress increases proportionally at that location.

Step-by-Step Solution:Start with sigma = M * y / I.Hold y and I constant for the chosen fiber and section.Conclude sigma ∝ M (direct proportionality).

Verification / Alternative check:Dimensional analysis and linearity of constitutive law corroborate the proportional relationship in elastic bending.

Why Other Options Are Wrong:Equal/less/more than are not functional relationships; “inversely proportional” contradicts the formula.

Common Pitfalls:Comparing stresses at different fibers (varying y); forgetting that changing section (I) alters the proportionality constant.

Final Answer:directly proportional to