Doubly reinforced beam—role of critical neutral axis: How do maximum stresses develop relative to the position of the actual neutral axis compared to the critical neutral axis?

Introduction / Context:
In flexural design, the position of the neutral axis (NA) relative to a critical (balanced) depth governs the failure mode. For doubly reinforced beams, compression steel supplements the concrete when the section is shallow or moments are high. The comparison of actual NA with the critical NA indicates whether concrete crushes first or steel yields first, guiding ductility and safety considerations.

Given Data / Assumptions:

• Balanced (critical) NA corresponds to simultaneous attainment of limiting concrete strain and steel yield at ultimate.
• Over-reinforced: NA deeper (greater depth) than critical; concrete crushes first.
• Under-reinforced: NA shallower than critical; tensile steel yields first, providing ductile warning.

Concept / Approach:

The distribution of strains is linear in flexure. The strain at the extreme compression fiber and at the steel layers depends on NA depth. Positioning relative to critical depth dictates which material hits its limit first. Doubly reinforced beams allow redistribution: compression steel participates when concrete’s compression block is insufficient.

Step-by-Step Solution:

Verification / Alternative check:

Ultimate limit state strain diagrams and standard code equations for balanced reinforcement confirm these qualitative outcomes independent of specific numerical values.

Why Other Options Are Wrong:

Each of (a), (b), and (c) states a correct condition; hence the combined correct choice is (d) “All of the above statements are correct”.

Common Pitfalls:

Confusing “greater NA depth” with safer behavior; over-reinforced sections are less ductile. Neglecting compression steel contribution in analysis of doubly reinforced sections.

Final Answer:

All of the above statements are correct