Simple bending in beams: How does the normal bending stress vary with distance from the neutral axis in a prismatic beam under pure (simple) bending?

Introduction / Context:The flexure (bending) formula provides the stress distribution across a beam's section under pure bending. Understanding how stress varies with depth is essential for sizing sections and placing reinforcement.

Given Data / Assumptions:

• Beam is prismatic and behaves elastically.
• Plane sections remain plane (Bernoulli-Euler assumption).
• No shear (pure bending) at the section considered.

Concept / Approach:The bending stress is given by σ = M * y / I, where M is the bending moment about the neutral axis, y is the distance from the neutral axis, and I is the second moment of area. Hence, σ varies linearly with y: zero at the neutral axis and maximum at the extreme fibers.

Step-by-Step Solution:Write flexure formula: σ = M * y / I.For fixed M and I, σ ∝ y.Thus, stress is directly proportional to the distance from the neutral axis.

Verification / Alternative check:Stress diagram is triangular over the depth, changing sign across the neutral axis for sagging vs hogging moments.

Why Other Options Are Wrong:Inversely proportional: contradicts σ = M y / I.Curvilinear/constant/zero at extreme fibers: all inconsistent with linear distribution and boundary conditions.

Common Pitfalls:Confusing bending stress with shear stress, which has a parabolic distribution in rectangular sections.

Final Answer:Directly proportional to the distance from the neutral axis.