Flexure theory – Variation of strain with distance from the neutral axis When a straight prismatic beam is subjected to a bending moment within the elastic range, how does the normal strain in any layer vary with its distance from the neutral axis?

Introduction / Context:The linear strain distribution across a beam depth under pure bending is a central result of Bernoulli–Euler beam theory, leading to the classic flexure formula for stress.

Given Data / Assumptions:

• Prismatic, homogeneous beam; small deflections.
• Plane sections remain plane after bending.
• Elastic, isotropic material with Young’s modulus E.

Concept / Approach:If curvature is κ, the strain at a distance y from the neutral axis is ε = κ * y. Hence strain varies linearly from compression at one extreme to tension at the other, and is zero at the neutral axis.

Step-by-Step Explanation:Assume a cross-section rotates to maintain plane sections (Bernoulli hypothesis).Fiber length change is proportional to its radius difference from the neutral surface.Therefore ε(y) = κ * y, indicating direct proportionality to distance from the neutral axis.

Verification / Alternative check:Combining ε = κ y with σ = E ε yields σ = E κ y, the familiar linear stress distribution and the flexure formula σ = M y / I.

Why Other Options Are Wrong:Constant or inverse relationships contradict ε = κ y; independence from distance is incorrect; maximum at the NA is false since ε = 0 at the NA.

Common Pitfalls:Confusing strain with shear stress distribution; assuming non-linear distribution in the elastic range without justification.

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