Reinforced concrete beam (b = 25 cm, effective depth d = 70 cm): The extreme-fiber stresses are 62.5 kg/cm² in concrete (compression) and 250 kg/cm² in steel (tension). Using modular ratio m = 15 and elastic theory, determine the depth of the neutral axis from the top compression face.

Introduction / Context:Neutral axis depth in a singly reinforced concrete beam under working-stress (elastic) theory is obtained from strain compatibility between concrete and steel using the modular ratio m. This problem tests the direct use of the steel-to-concrete stress ratio to find the neutral axis without needing the steel area.

Given Data / Assumptions:

• Beam breadth b = 25 cm, effective depth d = 70 cm.
• Concrete stress at top σc = 62.5 kg/cm² (compression).
• Steel stress σs = 250 kg/cm² (tension).
• Modular ratio m = 15.
• Elastic theory (linear strain distribution) and concentric bending; concrete in tension ignored.

Concept / Approach:

From similar triangles in the strain diagram and Hooke’s law: σs/σc = m * ( (d − x) / x ), where x is the neutral-axis (NA) depth from the top compression face. Rearranging gives x in terms of m, d, and the stress ratio σs/σc.

Step-by-Step Solution:

Verification / Alternative check:

You can also write x = m d / ( (σs/σc) + m ) = 15 * 70 / (4 + 15) = 1050 / 19 ≈ 55.3 cm, confirming the result.

Why Other Options Are Wrong:

25–40 cm and 30–35 cm are far smaller than the computed 55.3 cm using the given stresses and m; they contradict the compatibility equation.

Common Pitfalls:

Mixing effective depth with overall depth; forgetting to apply m; using force equilibrium that requires As (not provided) instead of direct strain compatibility.

Final Answer:

55 cm (approximately)