CBR → Resilient Modulus Conversion

When modulus data is unavailable, CBR can be converted to resilient modulus using an empirical correlation. The most valid across a wide range of CBR values is that from the MEPDG / AASHTOWare Pavement ME 2000:

For subgrades with CBR above 20%, direct laboratory resilient modulus testing per ASTM D3999 is recommended in place of CBR-based estimation.

Modulus of Subgrade Reaction

The k-value is defined as the ratio of applied contact pressure to the resulting surface deflection:

Elastic Multilayer System

The subgrade support is modeled as a system of n elastic layers per Burmister’s 1945 theory. Each layer is defined by its modulus, Poisson’s ratio, and thickness, with the bottom layer (natural subgrade) extending to infinite depth. The following assumptions apply:

• Linear (Hookean) relationship between stress and deformation.
• Each layer is homogeneous and isotropic — same properties throughout.
• Full friction (bonded interfaces) between adjacent layers.
• All layers are infinite in the horizontal direction.
• A uniform circular load of radius a = 15 in. (38.1 cm) is applied at the surface.

Equivalent Modulus (Ê)

The n−1 layers above the subgrade are collapsed into a single equivalent modulus Ê using a thickness-weighted geometric mean. This is the key step that converts the multi-layer system into a form usable by the Palmer-Barber 1940 deflection formula:

Each layer is weighted by its thickness and its stiffness relative to the top layer. When only a subgrade exists with no layers above it, Ê = E_n and the formula reduces to the single half-space case.

Surface Deflection

Following Palmer and Barber 1940, the total vertical deflection at the center of a uniformly loaded flexible circular plate on the multilayer system is:

With q in psi, a in inches, and E_n in psi, the deflection Δ₀ is in inches. Substituting into the k-value definition and canceling q gives a k-value that is load-independent — a critical property of the method.

Rigid-Plate Correction

The Palmer-Barber deflection above assumes a flexible plate with uniform pressure distribution. A physical plate load test uses a rigid plate, under which the pressure distribution is non-uniform. Ullidtz 1987 expressed the required pressure for a rigid plate as:

Integrating this expression across the plate area gives a total load equivalent to 78.53% of the uniformly distributed load. Since all deflection expressions are directly proportional to applied pressure, the rigid-plate deflection is also 78.53% of the flexible-plate deflection at center. A lower deflection implies greater structural capacity, so the k-value must be increased:

Final Formula

Combining the rigid-plate correction with the Palmer-Barber deflection and substituting the fixed plate radius a = 15 in. yields the final closed-form expression, as shown in Cañas Silva 2010. The constant 0.04244 = 1.2732 / (2 × 15 in.):

Cañas Silva 2010 developed an Excel spreadsheet and software that calibrated this equation to results from MEPDG and then compared the results of the calibrated model to published composite k-value tables from PCA and FAA. Because the nature of the calibration included using two different plate radii, the following adjustment developed by Robert Rodden and Juan Pablo Covarrubias in 2026 to match the calibration is used instead:

And the dynamic k-value is computed as:

The static k-value is then: