Rigid pavement theory: If E is the modulus of elasticity of concrete (kg/cm²), d the slab thickness (cm), μ the Poisson's ratio of concrete, and k the subgrade modulus (kg/cm³), the radius of relative stiffness r (cm) is

Introduction / Context:
The radius of relative stiffness r is a key parameter in Westergaard’s analysis of concrete pavements. It quantifies how a slab spreads a wheel load over the subgrade, controlling interior, edge, and corner stresses and deflections.

Given Data / Assumptions:

• E: modulus of elasticity of concrete (kg/cm²).
• d: slab thickness (cm).
• μ: Poisson’s ratio of concrete (dimensionless).
• k: subgrade reaction modulus (kg/cm³).
• Homogeneous isotropic slab on a Winkler foundation.

Concept / Approach:

The classical closed-form relation for r arises from balancing bending stiffness of the plate (E * d^3 / 12 / (1 - μ^2)) with the elastic spring support k. The dimensional combination that yields length is the fourth-root of stiffness-to-foundation ratio.

Step-by-Step Solution:

Verification / Alternative check:

Check units: D has units kg·cm, dividing by k (kg/cm³) gives cm^4; fourth root yields cm, confirming dimensional consistency.

Why Other Options Are Wrong:

• Option B is the inverse; it would make stiffer slabs produce smaller r, which is non-physical.
• Options C, D, and E use incorrect powers and combinations; they do not produce correct dimensions or behavior trends.

Common Pitfalls:

• Forgetting the (1 − μ^2) term from plate theory.
• Mixing SI and metric units without converting k consistently.

Final Answer:

r = [E * d^3 / (12 * k * (1 - μ^2))]^(1/4).