Singly reinforced rectangular beam (working stress): A 25 cm wide beam with effective depth 70 cm is provided with 18.75 cm² tension steel. If the modular ratio m = 15, determine the depth of the neutral axis.

Introduction / Context:
Under the working stress method, neutral axis depth in a singly reinforced rectangular beam is obtained using transformed-section analysis. The position of the neutral axis governs the stress distribution in concrete and steel and is a key input for service stress checks and moment capacity calculations within the elastic range.

Given Data / Assumptions:

• Beam width b = 25 cm; effective depth d = 70 cm.
• Area of tension steel Ast = 18.75 cm².
• Modular ratio m = Es/Ec = 15.
• Working stress assumptions: plane sections remain plane; linear elastic behavior; concrete in tension ignored.

Concept / Approach:
For transformed-section analysis, replace steel by an equivalent concrete area m*Ast located at the steel level. The neutral axis depth n (from top) follows by force equilibrium in the transformed section:

m * Ast * (d - n) = (b * n²) / 2

Step-by-Step Solution:
Insert values: 15 * 18.75 * (70 - n) = (25 * n²) / 2Compute constants: 281.25 * (70 - n) = 12.5 * n²Expand: 19687.5 - 281.25 n = 12.5 n²Rearrange: 12.5 n² + 281.25 n - 19687.5 = 0Divide by 12.5: n² + 22.5 n - 1575 = 0Solve: Discriminant = 22.5² + 4 * 1575 = 6806.25; sqrt = 82.5n = (-22.5 + 82.5) / 2 = 30 cm

Verification / Alternative check:
With n = 30 cm, n < d confirming a tension-controlled singly reinforced section in service. Substituting back satisfies the equilibrium equation to numerical accuracy.

Why Other Options Are Wrong:

• 20 cm, 25 cm, 35 cm, 40 cm do not satisfy the transformed equilibrium equation for the given b, d, Ast, and m.

Common Pitfalls:
Using gross area instead of transformed steel area, forgetting to square n in the compression term, or mixing units (cm vs. mm).

Final Answer:
30 cm.