Civil, Structural & Geotechnical Engineering
Burmister Two-Layer Pavement & Odemark Equivalent Thickness Formula: Deflection Factor F_2
Calculate two-layer elastic pavement surface deflection w_0 = F_2 * (1.5 * q * a / E_2) and Odemark’s Method of Equivalent Thickness h_e = f * h_1 * (E_1 / E_2)^(1/3) for stiff asphalt or base courses over soft subgrade.
Why Boussinesq fails on layered roads — and how Burmister (1943) & Odemark (1949) quantify load-spreading!
Boussinesq’s 1885 elastic half-space theory assumes uniform soil modulus E from tire to bedrock, predicting flexible circular-plate surface deflection w_Bouss = (2 * (1 - mu_2^2) * q * a) / E_2 (~= 1.5 * q * a / E_2 at Poisson’s ratio mu = 0.5)! In reality, pavements place a stiff upper layer (modulus E_1, thickness h_1) over a much softer subgrade (modulus E_2 << E_1). Solving biharmonic stress functions for a two-layer elastic system, Columbia University professor Donald M. Burmister (1943, Highway Research Board) showed that surface deflection drops by Dimensionless Deflection Factor F_2 < 1: w_0 = F_2 * (1.5 * q * a / E_2)!
Combining Nils Odemark’s (1949) Method of Equivalent Thickness — which transforms upper layer h_1 into an equivalent thickness of subgrade-modulus material h_e = f * h_1 * (((E_1 / E_2) * ((1 - mu_2^2) / (1 - mu_1^2)))^(1/3)) (with correction factor f ~= 0.90 for a two-layer interface) — yields the widely used analytical Burmister-Odemark Deflection Factor: F_2 = (1 / SQRT(1 + (h_e / a)^2)) + (E_2 / E_1) * (1 - (1 / SQRT(1 + (h_e / a)^2)))!
| Two-Layer System (q = 80 psi, a = 6 in, h_1 = 8 in, E_2 = 10,000 psi, mu = 0.35) | Odemark Equivalent h_e (f = 0.9) | Deflection Factor F_2 | Center Deflection w_0 |
|---|---|---|---|
| Unreinforced Subgrade Half-Space (E_1 = E_2 = 10k psi) | 7.20 in | 1.0000 | 0.0720 in (72.0 mils) |
| Granular Base Over Subgrade (E_1 = 40,000 psi, Ratio = 4) | 11.43 in | 0.5986 | 0.0431 in (43.1 mils) |
| Stiff Asphalt Over Subgrade (E_1 = 350,000 psi, Ratio = 35) | 23.55 in | 0.2684 | 0.0193 in (19.3 mils) |
Odemark (1949) Transformed Equivalent Subgrade Thickness h_e (in or mm)
Evaluates 0.9 * h_1 * ((E_1 / E_2)^(1/3)) when Poisson’s ratios match (or full Poisson ratio term) across upper thickness h_1 (B2), upper modulus E_1 (B3 >= E_2), and subgrade modulus E_2 (B4).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B3 < B4), "Requires h_1 > 0 and Stiff Upper Layer E_1 >= Subgrade E_2 > 0", ROUND(0.9 * B2 * ((B3 / B4) ^ (1 / 3)), 2))Burmister-Odemark Two-Layer Dimensionless Deflection Factor F_2 (0 < F_2 <= 1)
Computes (1 / SQRT(1 + (h_e / a)^2)) + (E_2 / E_1) * (1 - 1 / SQRT(1 + (h_e / a)^2)) for tire contact radius a (B5).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B3 < B4, B5 <= 0), "Invalid Two-Layer Inputs", ROUND((1 / SQRT(1 + ((0.9 * B2 * ((B3 / B4) ^ (1 / 3))) / B5) ^ 2)) + (B4 / B3) * (1 - (1 / SQRT(1 + ((0.9 * B2 * ((B3 / B4) ^ (1 / 3))) / B5) ^ 2))), 4))Try the example
Use the original values or change them to test your understanding.