Simple bending theory – meaning of “plane sections remain plane” In the assumptions of Euler–Bernoulli beam theory, the statement “plane sections before bending remain plane after bending” implies that:

Introduction / Context:Euler–Bernoulli simple bending theory relies on kinematic assumptions to relate curvature, strain, and stress. Understanding “plane sections remain plane” is essential for deriving linear strain distributions.

Given Data / Assumptions:

• Prismatic beam, small deflections.
• Normals to the neutral surface remain straight and rotate but do not warp.
• Shear deformation is neglected (classical beam theory).

Concept / Approach:If a cross-sectional plane remains plane after bending, all points on a line normal to the neutral axis experience the same rotation, leading to a linear variation of longitudinal displacement with distance from the neutral axis, hence a linear strain distribution.

Step-by-Step Reasoning:Assume a radius of curvature R for the neutral surface.A fiber at a distance y from the neutral axis changes length in proportion to y.Thus, axial strain ε_x = y / R, which is linear in y (proportional to distance from the neutral axis).

Verification / Alternative check:Using Hooke’s law σ_x = E ε_x, the stress distribution also becomes linear in y, not uniform. That matches classic linear bending stress σ = M y / I.

Why Other Options Are Wrong:Stress uniformity (a) contradicts bending; (b) strain uniform is incorrect; (e) shear strain is not necessarily zero in reality—it is neglected only in the kinematic assumption for bending response.

Common Pitfalls:Confusing linear strain (correct) with linear stress being the original assumption; in fact linear stress results from linear strain plus Hooke’s law.

Final Answer:Strain is proportional to the distance from the neutral axis