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
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Ar = [E * d^3 / (12 * k * (1 - μ^2))]^(1/4)
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Br = [12 * k * (1 - μ^2) / (E * d^3)]^(1/4)
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Cr = [E * d / (k * (1 - μ))]^(1/2)
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Dr = [E * d^2 / (k * (1 + μ))]^(1/3)
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Er = [E / (k * d)]^(1/2)
Answer
Correct Answer: r = [E * d^3 / (12 * k * (1 - μ^2))]^(1/4)
Explanation
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:
Plate bending stiffness D = E * d^3 / (12 * (1 - μ^2))Relative stiffness r satisfies r^4 = D / kTherefore r = [E * d^3 / (12 * k * (1 - μ^2))]^(1/4)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).