Flexure theory — consequence of “plane sections remain plane” (What strain/stress distribution follows from the Bernoulli–Euler assumption?)

Introduction / Context:
The classical bending theory assumes that plane sections normal to the neutral axis before bending remain plane and normal after bending. This foundational hypothesis underpins linear strain distributions used in both steel and concrete flexure design.

Given Data / Assumptions:

• Small deflections and linear geometry changes.
• Bernoulli–Euler kinematics (no warping of cross sections).
• Material behavior can later be linked via constitutive laws (e.g., Hooke’s law).

Concept / Approach:
If plane cross sections remain plane, longitudinal fibers separated by a distance y from the neutral axis must change length proportionally to y when the member bends to a curvature kappa. Therefore, axial strain varies linearly with y: epsilon(y) = kappa * y. Stress distribution is then obtained by applying the material law (elastic or nonlinear), but linear strain is the immediate kinematic consequence.

Step-by-Step Solution:
1) Assume planar cross section after bending → no shear distortion across depth.2) Impose compatibility with curvature → fiber extension/compression ∝ distance from NA.3) Conclude epsilon(y) varies linearly across the depth.4) Apply stress–strain law separately to get stress distribution (elastic case → linear stress).

Verification / Alternative check:
Beam bending formula M/I = sigma/y = E/R derives from linear strain and Hooke’s law; experiments confirm linear strain in elastic range.

Why Other Options Are Wrong:
(a) Uniform strain contradicts curvature; (b) uniform stress would imply zero bending; (d) stress proportional to strain is a material law, not a direct result of the plane-sections assumption.

Common Pitfalls:
Conflating kinematic assumption with constitutive law; assuming linear stress in plasticity where material nonlinearity alters stress despite linear strain.

Final Answer:
linearly varying strain over the cross section