Assumptions of simple (Euler–Bernoulli) bending theory: Which of the following standard assumptions are made for a straight prismatic beam under pure bending?

Introduction / Context:Euler–Bernoulli beam theory is the most widely taught model for flexure. Its assumptions define when the flexure formula σ = M * y / I and the linear strain distribution are valid. Recognizing these assumptions helps engineers assess applicability to real beams and understand deviations (e.g., shear deformation in deep beams).

Given Data / Assumptions (being tested):

• Homogeneous, isotropic material (uniform properties in all directions).
• Linear elasticity (stress proportional to strain within elastic limit).
• Plane sections remain plane and normal to the neutral axis after bending.

Concept / Approach:These assumptions lead to a linear strain profile ε = y / ρ and the flexure relation M / I = E / ρ. They also imply the neutral axis passes through the centroid for homogeneous sections, and that warping and shear effects are neglected in slender beams.

Step-by-Step Solution:

Verification / Alternative check:Timoshenko beam theory relaxes the “sections remain perpendicular” assumption by including shear deformation; differences become significant for deep beams or low shear modulus materials, confirming the role of the assumptions in Euler–Bernoulli theory.

Why Other Options Are Wrong:

• Any subset omits a key pillar; removing one invalidates the simple model.

Common Pitfalls:Applying Euler–Bernoulli to short/deep beams where shear effects are non-negligible; ignoring material anisotropy (e.g., laminated composites).

Final Answer:All of the above