Axially loaded R.C.C. short column: Given Ac (area of concrete), Asc (area of longitudinal steel), A (gross section area), m (modular ratio), and σc (permissible compressive stress in concrete), which expression for the axial load-carrying strength is correct?

Introduction / Context:
In reinforced concrete (R.C.C.) columns designed by the working stress method, the axial load capacity is obtained by summing the contributions of concrete and steel at their permissible stresses. This question checks whether you recognize the equivalent algebraic forms of the same strength expression when expressed in terms of Ac, Asc, A, and the modular ratio m.

Given Data / Assumptions:

• Ac = area of concrete only.
• Asc = area of longitudinal steel.
• A = gross area = Ac + Asc.
• σc = permissible compressive stress in concrete.
• m = modular ratio (Es/Ec under working stress concepts).
• Short, concentrically loaded column; secondary effects ignored.

Concept / Approach:

Under concentric compression, steel shares the load in proportion to its modulus difference relative to concrete, hence the use of modular ratio m. The strength (permissible axial load) equals the sum of concrete's share plus steel's share converted into an equivalent concrete stress block using m.

Step-by-Step Solution:

Verification / Alternative check:

All three expressions reduce to the same value when you substitute Ac = A − Asc, proving their equivalence algebraically and dimensionally.

Why Other Options Are Wrong:

Options (a), (b), and (c) are each correct individually; the only fully correct choice that captures this is “All of the above.” “None of these” is obviously incorrect.

Common Pitfalls:

Forgetting that A = Ac + Asc; mixing permissible stresses with design strengths; applying the expression to columns with eccentricity without modifications.

Final Answer:

All of the above