Flexure formula relationship at a beam section Let M be the bending moment, I the second moment of area about the neutral axis, R the radius of curvature, E the modulus of elasticity, F the bending stress at a fibre, and Y the distance of that fibre from the neutral axis. Choose the correct fundamental flexure relationship among these quantities.

Introduction / Context:
The Euler–Bernoulli beam theory establishes the flexure formula linking bending moment, curvature, elastic modulus, section stiffness, and fibre stress. Recognizing the exact proportionalities is fundamental to beam design and stress calculations.

Given Data / Assumptions:

• Linear elastic material behavior (Hooke’s law).
• Plane sections remain plane and normal to the neutral axis after bending.
• Small deflections and small strains.

Concept / Approach:
The curvature κ is 1/R and, for small deflection linear theory, κ = M / (E * I). Fibre stress varies linearly with distance Y from the neutral axis: F = (M * Y) / I. Combining these gives the standard identity: M / I = E / R = F / Y.

Step-by-Step Solution:

Verification / Alternative check:
Dimensional consistency: M/I has units of stress per length, equal to E/R (stress per length), and equals F/Y (stress per length), confirming the proportionality.

Why Other Options Are Wrong:

• M / I = R / E = Y / F: Inverts E and R incorrectly.
• M * I = E * R = F * Y: Incorrect products; the theory gives ratios, not products.
• M / I = E * R = F / Y: E * R has wrong dimensions and relation.
• None of these: Incorrect, since option A is the standard relation.

Common Pitfalls:
Confusing curvature definitions, mixing up numerator and denominator, and forgetting that stress varies linearly with Y only under Euler–Bernoulli assumptions.

Final Answer:
M / I = E / R = F / Y