Select the correct set of assumptions in the elementary (Euler–Bernoulli) theory of bending of beams used in structural analysis and design.

Introduction / Context:
The elementary beam theory underpins bending stress and deflection formulae used widely in civil and mechanical engineering. Knowing its assumptions is essential to understand applicability and limitations when dealing with real materials and sections.

Given Data / Assumptions:

• We work within linear elastic theory (Hooke's law).
• Small deflections, plane sections remain plane (implied in classical theory).
• Statements list material homogeneity, isotropy, equal E in tension/compression, and layer behavior in bending.

Concept / Approach:
Euler–Bernoulli bending assumes a linear strain distribution across depth and proportional stress via a single modulus E. Homogeneous and isotropic material behavior ensures uniform response. Taking E the same in tension and compression yields symmetric stress blocks. The phrasing 'each layer acts independently to expand or contract' reflects that layers are free to undergo compatible axial strains dictated by curvature without through-thickness shear slip in the ideal model.

Step-by-Step Solution:
1) Adopt homogeneity → uniform composition/properties.2) Adopt isotropy → properties identical in all directions.3) Use a single E for both tension and compression → linear, symmetric response.4) Permit layers to experience compatible axial strains set by curvature → linear strain profile.5) Conclude the complete set of listed assumptions holds in the elementary theory.

Verification / Alternative check:
Standard texts list: plane sections remain plane, material homogeneous and isotropic, stresses within elastic limit, and E same in tension/compression. The listed statements align with that framework.

Why Other Options Are Wrong:

• Picking only A–D individually omits the completeness of the assumption set.

Common Pitfalls:

• Confusing Euler–Bernoulli with Timoshenko theory (which includes shear deformation).
• Applying elementary bending to composite/anisotropic materials without modification.

Final Answer:
all the above.