Theory of simple bending – core assumptions Which of the following statements are core assumptions in the classical (Euler–Bernoulli) simple bending theory for beams used in civil and mechanical engineering design?

Introduction / Context:
The Euler–Bernoulli (simple) bending theory forms the backbone of beam analysis in structural engineering. Its assumptions limit the theory's applicability but also make closed-form solutions possible for a wide range of practical problems.

Given Data / Assumptions:

• Linearly elastic response, small deflection theory.
• Prismatic beam unless stated otherwise.
• No significant shear deformation (plane sections remain plane).

Concept / Approach:
Key assumptions include: (1) material linearity and identical modulus in tension and compression, (2) plane sections remain plane and perpendicular to the neutral surface after bending, (3) the material is homogeneous and isotropic, and (4) no net axial force on a pure bending section (resultant of normal stresses over the section is zero), so only bending moment acts.

Step-by-Step Solution:
Assess option A: Same E in tension and compression supports symmetric linear stress–strain behavior.Assess option B: Plane sections remain plane is the cornerstone assumption, eliminating shear strain in bending.Assess option C: Homogeneous and isotropic material ensures uniform properties and a centroidal neutral axis.Assess option D: In pure bending, the algebraic sum of normal stresses over the section is zero, giving zero resultant axial force.Conclusion: All statements align with the simple bending theory.

Verification / Alternative check:
Standard derivations of the flexure formula sigma = M y / I explicitly invoke these assumptions to obtain a linear stress distribution.

Why Other Options Are Wrong:
Each individual statement is correct; therefore selecting any one of A–D alone would be incomplete relative to the complete set of assumptions.

Common Pitfalls:
Applying the simple bending theory to deep beams or materials with different tensile/compressive moduli; such cases require Timoshenko theory or nonlinear material models.

Final Answer:
All the above