Lever arm in a singly reinforced RCC beam: Given permissible stresses σ_st = 1400 kg/cm^2 (steel), σ_cbc = 80 kg/cm^2 (concrete), modular ratio m = 18, beam width b = 30 cm, and effective depth d = 46 cm, determine the lever arm z of the section (working-stress, balanced condition).

Introduction / Context:
The lever arm z in a singly reinforced beam (working-stress method) is the distance between the compressive resultant in concrete and the tensile force in steel. At the balanced condition, both concrete and steel reach their permissible stresses simultaneously, enabling a direct geometric solution for the neutral axis and lever arm.

Given Data / Assumptions:

• σ_st = 1400 kg/cm^2, σ_cbc = 80 kg/cm^2, modular ratio m = 18.
• Rectangular section with effective depth d = 46 cm (tension steel depth from top fibre).
• Working-stress theory, linear stress distribution.

Concept / Approach:
For a balanced section in WSM: σ_st / σ_cbc = m * (d − x) / x, where x is the neutral axis depth from the top. Once x is known, the lever arm z is taken as the distance from the compressive resultant (at x/3 from the top for a triangular stress block) to the tension steel at depth d, i.e., z = d − x/3.

Step-by-Step Solution:
Compute ratio: σ_st / (m * σ_cbc) = 1400 / (18 * 80) = 1400 / 1440 ≈ 0.9722.Solve for neutral axis: (d − x) / x = 0.9722 ⇒ x = d / (1 + 0.9722) = 46 / 1.9722 ≈ 23.33 cm.Lever arm: z = d − x/3 = 46 − 23.33/3 ≈ 46 − 7.78 ≈ 38.22 cm.Rounded practical value ≈ 38 cm.

Verification / Alternative check:
Using the same inputs with more precise arithmetic gives z within a few millimetres of 38 cm; design rounding to whole centimetres is customary at preliminary stages.

Why Other Options Are Wrong:
37 cm / 39 cm / 40 cm / 36 cm: Do not match the computed balanced-section lever arm with given stresses and modular ratio.

Common Pitfalls:

• Using z = j * d from reinforced concrete constants without confirming balanced condition.
• Forgetting that the concrete resultant for a triangular stress block is at x/3 from the top.

Final Answer:
38 cm