Working-stress (elastic) theory for a cracked rectangular beam: If m is the modular ratio, r = As/(b d) is the steel ratio, and d is the overall effective depth, the neutral axis depth x from the top is given by which expression?

Introduction / Context:
For a cracked elastic section (concrete in tension neglected), the neutral axis of a singly reinforced beam is obtained using the transformed-section method. The neutral axis passes through the centroid of the transformed section (compression concrete + transformed steel area).

Given Data / Assumptions:

• Modular ratio m = Es/Ec.
• Steel ratio r = As/(b d).
• Rectangular section; concrete tension ignored, linear elasticity.

Concept / Approach:

Locate the centroid (neutral axis) of the transformed section by equating moments of areas about the neutral axis: (b * x) * (x/2) = m As (d − x). This yields a quadratic in k = x/d.

Step-by-Step Solution:

Verification / Alternative check:

Check limiting cases: as r → 0, k → 0; as r increases, k grows sublinearly, consistent with mechanics.

Why Other Options Are Wrong:

Options (b), (c), (d), and (e) do not satisfy the derived quadratic or have incorrect functional forms.

Common Pitfalls:

Using area equilibrium instead of first moment of area; forgetting to neglect concrete in tension when cracked; algebraic sign errors.

Final Answer:

x = d * [ - m r + sqrt( m^2 r^2 + 2 m r ) ]