Simple bending (flexure) formula Identify the correct simple bending equation relating bending moment M, section modulus terms, curvature, and stress for a prismatic, linearly elastic beam under pure bending.

Introduction / Context:
The simple bending (flexure) formula links internal bending moment, section geometry, material stiffness, and curvature for beams obeying Euler–Bernoulli assumptions. It is the cornerstone for sizing beams and checking stresses under service and ultimate loads.

Given Data / Assumptions:

• Homogeneous, isotropic material; linear elasticity.
• Plane sections remain plane; no significant shear deformation.
• Pure bending region with constant M and zero shear, or general bending where the relation still holds locally.

Concept / Approach:

The derivation shows neutral axis through the centroid and linear strain distribution ε = y/R. Hooke’s law gives σ = E ε = E y / R. Equilibrium of internal stresses with external moment yields M/I = σ/y = E/R, where I is the second moment of area about the neutral axis, y is distance from the neutral axis, and R is radius of curvature.

Step-by-Step Solution:

Verification / Alternative check:

Specializing to the extreme fiber at y = c gives σ_max = M c / I and section modulus Z = I / c, both standard design quantities, confirming internal consistency.

Why Other Options Are Wrong:

• Options (b), (c), (d) scramble variables or dimensions, violating the derived proportionalities.
• “None of these” is incorrect since the standard identity exists and is well-known.

Common Pitfalls:

• Using section modulus Z in place of I without tracking c correctly.
• Applying the formula outside its assumptions (deep beams with significant shear deformation).

Final Answer:

M / I = σ / y = E / R.