Working-stress design in reinforced concrete: If the permissible compressive stress in bending for concrete is C (in kg/cm²), what is the appropriate modular ratio m to be used for transformed-section analysis?

Introduction / Context:

The modular ratio m is a cornerstone of the working-stress method in reinforced concrete design. It converts steel area into an equivalent concrete area for transformed-section analysis under service loads. Classical codes and handbooks define m in terms of a constant divided by the permissible compressive stress in bending for concrete, reflecting creep and long-term effects.

Given Data / Assumptions:

• Permissible compressive stress in bending for concrete = C (kg/cm²).
• Working-stress (service) design framework.
• Standard long-term modular ratio expression used in many exam problems.

Concept / Approach:

The long-term (effective) modular ratio is taken as m = 2800/(3C) when C is in kg/cm². The 2800 factor and the divisor 3 represent empirical adjustments for creep and sustained loading, leading to a reduced effective stiffness of concrete under service conditions compared to its instantaneous modulus.

Step-by-Step Solution:

Verification / Alternative check:

Some references list an instantaneous m ≈ Ec/Es, but for working-stress design, the long-term expression with 2800/(3C) is traditionally used for service calculations and matches the exam-style answer set.

Why Other Options Are Wrong:

• 2800/C: Omits the divisor 3; too large for long-term effective m.
• 2300/2C and 700/C: Do not reflect the standard constant or divisor.
• 2800/C²: Incorrect dimensional form; m must be dimensionless, not vary with 1/C².

Common Pitfalls:

• Confusing instantaneous modulus ratio with the long-term modular ratio used in WSM.
• Mixing units (MPa vs kg/cm²). Keep C in kg/cm² to use the 2800 constant correctly.

Final Answer:

2800/3C