Flexure Formula — Bending Equation Which relation correctly represents the bending equation linking bending moment, curvature, and stress distribution for pure bending of a prismatic beam?

Introduction:
The classic flexure formula connects internal bending moment with the linear stress distribution and beam curvature under pure bending for linearly elastic materials.

Given Data / Assumptions:

• Prismatic beam, homogeneous and isotropic.
• Pure bending region (no shear), small deflection theory.
• Plane sections remain plane, linear elastic behavior.

Concept / Approach:
The bending equation is derived from strain compatibility and Hooke's law. It states that normal stress varies linearly with distance y from the neutral axis, and curvature is proportional to bending moment through E and I.

Step-by-Step Solution:
Strain at a fiber: epsilon = y / RStress: sigma = E * epsilon = E * (y / R)Resultant moment: M = integral(sigma * y dA) = (E/R) * integral(y^2 dA) = (E/R) * IRearrange: M / I = sigma / y = E / R

Verification / Alternative check:
Units: M/I has units of stress/curvature; sigma/y equals E/R, consistent dimensionally.

Why Other Options Are Wrong:
M / y = sigma / I = R / E: inverts terms incorrectly.M / I = y / sigma = R / E: places y and sigma in the wrong ratio.M * I = sigma * y = E * R: incorrect dimensional relation.

Common Pitfalls:
Using y with sign mistakes and misidentifying the neutral axis where sigma = 0.

Final Answer:
M / I = sigma / y = E / R