Bending stress distribution in beams For pure bending of a prismatic member, the direct longitudinal stress σ at a point a distance r from the neutral axis is proportional to which power of r?

Introduction / Context:
Understanding how bending stress varies across the depth of a beam is fundamental for sizing sections and placing reinforcement.

Given Data / Assumptions:

• Prismatic beam under pure bending moment M.
• Linear elastic (Hookean) material; plane sections remain plane.
• Neutral axis passes through the centroid.

Concept / Approach:
The flexure formula states σ = M * y / I, where y is the perpendicular distance from the neutral axis and I is the second moment of area. Hence, σ varies linearly with distance from the neutral axis.

Step-by-Step Solution:

Verification / Alternative check:
At r = 0 (neutral axis), σ = 0; at the extreme fibre (maximum r), σ is maximum—consistent with linear variation.

Why Other Options Are Wrong:

• r^2 or r^3 would imply nonlinear distributions inconsistent with beam theory.
• Independence of r or inverse dependence contradicts observed bending stress profiles.

Common Pitfalls:
Confusing shear stress distribution (parabolic in rectangles) with bending normal stress (linear).

Final Answer:
r