Westergaard Rigid Concrete Slab Formula: Radius of Relative Stiffness l & Corner Stress

Balance concrete flexural rigidity E * h^3 against Winkler subgrade spring k!

Modeling a Portland cement concrete highway pavement or forklift warehouse floor as an elastic plate (Young’s modulus E, thickness h, Poisson’s ratio mu ~= 0.15) resting on a dense liquid Winkler foundation (Modulus of Subgrade Reaction k, in psi/in or MPa/m), Harald M. Westergaard (1926, Public Roads) showed that load-spreading footprint is governed by a single characteristic length — the Radius of Relative Stiffness l = ((E * (h^3)) / (12 * (1 - (mu^2)) * k))^0.25!

Under wheel/post load P acting over circular tire contact radius a, an un-doweled slab CORNER lacks two-way continuity and develops severe top tensile bending stress given by Westergaard’s classic Corner Stress Equation: sigma_c = (3 * P / (h^2)) * (1 - ((a * SQRT(2)) / l)^0.6), which engineers verify against allowable concrete modulus of rupture f_r / FS!

Westergaard Radius of Relative Stiffness l (in or mm)

Evaluates ((E * h^3) / (12 * (1 - mu^2) * k))^0.25 across consistent units: Concrete Modulus E (B2 psi), Slab Thickness h (B3 in), Poisson’s ratio mu (B4 in 0..0.49), and Subgrade Modulus k (B5 psi/in).

=IF(OR(B2 <= 0, B3 <= 0, B4 < 0, B4 >= 0.5, B5 <= 0), "Invalid Concrete E, Thickness h, Poisson mu, or Subgrade k", ROUND(((B2 * (B3 ^ 3)) / (12 * (1 - (B4 ^ 2)) * B5)) ^ 0.25, 2))

Westergaard Slab Corner Tensile Bending Stress sigma_c (psi)

Computes (3 * P / h^2) * (1 - (a * SQRT(2) / l)^0.6) for wheel load P (B6 lb) and contact radius a (B7 in, requiring a * SQRT(2) < l).

=IF(OR(B2 <= 0, B3 <= 0, B4 < 0, B4 >= 0.5, B5 <= 0, B6 <= 0, B7 <= 0), "Invalid Westergaard Slab or Wheel Inputs", ROUND((3 * B6 / (B3 ^ 2)) * (1 - (((B7 * SQRT(2)) / (((B2 * (B3 ^ 3)) / (12 * (1 - (B4 ^ 2)) * B5)) ^ 0.25)) ^ 0.6)), 1))

Further reading

• FHWA — Rigid Pavement Design & Westergaard Plate Theory Reference