Pavement Design: Method

Pavement Design

This page describes what the tool computes: the design traffic in equivalent single axle loads, the AASHTO 1993 empirical design of a flexible pavement (structural number and layer thicknesses) and of a rigid pavement (slab thickness), the traffic life of an entered section, and a cross-check by the Corps of Engineers CBR equation. The design equations are regressions of the AASHO Road Test of 1958 to 1960, published in the AASHTO Guide for Design of Pavement Structures (1993). The Guide is cited only: the tool uses its equations as mathematics, in the form printed in the Federal Highway Administration reference manual FHWA-NHI-05-037 (Geotechnical Aspects of Pavements, 2006), and every coefficient the Guide tabulates (layer, drainage and load-transfer coefficients, reliability levels, loss of support) is an entered input. No table or chart of the Guide is reproduced.

Units

The equations are empirical and hold only in their native units: thickness in inches, moduli and strengths in psi, the modulus of subgrade reaction in pci (lbf/in³), axle loads in kips, traffic in 18-kip equivalent single axle loads. The tool converts the inputs to these units through the site's unit dictionary, evaluates the equations there, and converts the thicknesses back. Layer coefficients are per inch of thickness in both unit systems (a coefficient of 0.44 per inch is 0.0173 per millimetre). Traffic, reliability, serviceability, CBR and R-value are plain numbers; seasonal moduli are typed in psi (MPa in SI).

Design traffic

The design traffic W₁₈ is the number of 18-kip (80 kN) equivalent single axle loads (ESALs) in the design lane over the design period. From a vehicle mix, each class contributes its share of the average daily traffic times its truck factor, the ESALs per vehicle:

W₁₈ = ADT · Σ(share_i · T_f,i) · 365 · D_D · D_L · G, with G = ((1 + g)ⁿ − 1) / g

D_D and D_L are the directional and lane distribution, G is the sum of n years of traffic growing at the rate g (G = n without growth). The truck factor of a row is entered (from weigh-station data) or computed as the sum of the load-equivalency factors of its axle groups.

The load-equivalency factor of an axle group of load L_x kips with L₂ axles (1 single, 2 tandem, 3 tridem) is the ratio W₁₈ / W_x of the Road Test equations. For flexible pavements:

log(W_x / W₁₈) = 4.79 log(18 + 1) − 4.79 log(L_x + L₂) + 4.33 log L₂ + G_t / β_x − G_t / β₁₈, with G_t = log[(4.2 − p_t) / (4.2 − 1.5)] and β_x = 0.40 + 0.081 (L_x + L₂)^3.23 / [(SN + 1)^5.19 L₂^3.23]

and for rigid pavements:

log(W_x / W₁₈) = 4.62 log(18 + 1) − 4.62 log(L_x + L₂) + 3.28 log L₂ + G_t / β_x − G_t / β₁₈, with G_t = log[(4.5 − p_t) / (4.5 − 1.5)] and β_x = 1.00 + 3.63 (L_x + L₂)^5.20 / [(D + 1)^8.46 L₂^3.52]

Both equations are ratios of the Road Test performance equations as the Highway Research Board printed them (Special Report 61E, 1962): log W = log ρ + G_t / β, with β = 0.4 + 0.081 (L₁ + L₂)^3.23 / [(SN + 1)^5.19 L₂^3.23] and ρ = 10^5.93 (SN + 1)^9.36 L₂^4.33 / (L₁ + L₂)^4.79 for flexible pavements (its equations 13, 17 and 18) and β = 1 + 3.63 (L₁ + L₂)^5.20 / [(D + 1)^8.46 L₂^3.52], ρ = 10^5.85 (D + 1)^7.35 L₂^3.28 / (L₁ + L₂)^4.62 for rigid pavements (equations 62, 65 and 66); dividing the 18-kip single axle by the axle group cancels the pavement term of ρ. The Road Test had single and tandem axles only: the tridem (L₂ = 3) is the Guide's extension of the same form. The equations also reproduce 169 factors of the Guide's tables as reprinted in an FHWA document (flexible SN 5 and rigid D 9 in, p_t 2.5, tridems included) to the printed digits.

Factors at the design. The factors depend on the pavement they act on (SN or D), and the pavement depends on the ESALs they give. By default the tool iterates: the factors are evaluated at an assumed SN (or D), the design equation is solved for the required structural number above the roadbed (or the required slab thickness) at the resulting ESALs, before rounding, and that value is assumed in the next step, until the required and assumed values agree within 0.001 (0.001 in). The entered SN or slab for factors (5 and 9 in by default) is the starting value, and the steps are tabulated. The dependence is weak, so three or four steps suffice; the fixed point does not depend on the starting value. The other option evaluates the factors once at the entered SN or slab, which is the usual hand procedure. Rows with an entered truck factor do not change in the iteration. The fourth-power approximation (L / L_ref)⁴ is printed beside, with L_ref the load at which the equation gives 1 for that axle group (18 kip for a single axle; about 33 kip for a tandem and 48 kip for a tridem on a flexible pavement of SN 5).

Reliability and serviceability

Reliability enters as the term Z_R S₀ added to log W₁₈: Z_R is the standard normal deviate of the design reliability R (the normal quantile of 1 − R, negative above 50 %) and S₀ the overall standard deviation of the traffic and performance predictions. Designing for R = 85 % with S₀ = 0.45 is designing for 2.9 times the predicted traffic at 50 %. Mean values, not conservative ones, belong in every other input, because the conservatism is in this term. The tool computes Z_R exactly; published tables differ from it in the third decimal, and an entered deviate may replace it.

Performance is measured by the Present Serviceability Index. The design loss for traffic is ΔPSI = p₀ − p_t less any loss assigned to swelling or frost heave, with p₀ the initial and p_t the terminal index. FHWA-NHI-05-037 quotes p₀ = 4.2 for flexible and 4.4 for rigid pavements and p_t of 2.5 or more for major highways.

Roadbed soil

The flexible equation takes the resilient modulus M_R of the roadbed soil. It is entered, or estimated from the California Bearing Ratio, M_R = 1,500 CBR psi (stated for CBR below 10), or from the stabilometer R-value, M_R = 1,000 + 555 R psi. Where the modulus changes through the year, each season's value is given the relative damage u_f = 1.18 × 10⁸ M_R^−2.32; the effective modulus is the one whose damage equals the mean of the seasons. Because the weakest seasons dominate the mean damage, the effective modulus lies well below the average modulus.

Flexible pavement

log W₁₈ = Z_R S₀ + 9.36 log(SN + 1) − 0.20 + log[ΔPSI / (4.2 − 1.5)] / [0.40 + 1,094 / (SN + 1)^5.19] + 2.32 log M_R − 8.07

The equation is solved for the structural number SN by bisection. SN is an abstract thickness: SN = a₁ D₁ + a₂ m₂ D₂ + a₃ m₃ D₃, with D the layer thicknesses in inches, a the structural layer coefficients and m the drainage coefficients of the unbound layers. For untreated granular materials the coefficients follow from the layer modulus, a₂ = 0.249 log E_BS − 0.977 for a base and a₃ = 0.227 log E_SB − 0.839 for a subbase (E in psi); the tool uses these when the coefficient is left blank. Coefficients of asphalt and treated layers are entered.

The layered procedure sets the thicknesses so that every layer is protected by enough structure above it. The structural number needed above the base, SN₁, is the solution of the design equation with the base modulus in place of M_R; the surface is D₁ ≥ SN₁ / a₁, rounded, and supplies SN₁* = a₁ D₁. With the subbase modulus the equation gives SN₂, the number needed above the subbase, and D₂ ≥ (SN₂ − SN₁*) / (a₂ m₂). The lowest layer makes up the rest of the total: D₃ ≥ (SN − SN₁* − SN₂*) / (a₃ m₃). Thicknesses are rounded to entered increments (half an inch for the surface and one inch for unbound layers are the usual ones) and are not taken below entered minimums. Many other combinations give the same SN; cost, drainage, frost and construction decide between them.

Rigid pavement

log W₁₈ = Z_R S₀ + 7.35 log(D + 1) − 0.06 + log[ΔPSI / (4.5 − 1.5)] / [1 + 1.624 × 10⁷ / (D + 1)^8.46] + (4.22 − 0.32 p_t) log{ S_c′ C_d (D^0.75 − 1.132) / [215.63 J (D^0.75 − 18.42 / (E_c / k)^0.25)] }

D is the slab thickness, S_c′ the mean modulus of rupture and E_c the modulus of elasticity of the concrete, C_d the drainage coefficient, J the joint load-transfer coefficient and k the effective modulus of subgrade reaction. The equation is solved for D by bisection on its rising branch and the result is rounded up to the entered increment. The slab thickness is sensitive to the traffic, the concrete strength, J and C_d, and only slightly to k.

The design k is the Guide's composite value after its corrections. For a granular subbase of thickness D_SB and modulus E_SB on a deep roadbed the regression behind the Guide's chart is

ln k_∞ = −2.807 + 0.1253 (ln D_SB)² + 1.062 ln M_R + 0.1282 ln D_SB ln E_SB − 0.4114 ln D_SB − 0.0581 ln E_SB − 0.1317 ln D_SB ln M_R

which the tool prints for comparison when the subbase is entered. The corrections for shallow rock, for the seasons and for loss of support are chart readings of the Guide and are not computed: the design k is entered. Loss of support is a large correction (423 pci becomes 38 pci in the FHWA baseline example), so the composite value must not be entered as the design value.

Entered section, traffic life and remaining life

For an entered flexible section or slab the same equation gives the ESALs the section carries to the terminal serviceability at the design reliability. The tool compares that life with the design traffic, converts it to years with the first-year traffic and the growth rate, and, when the ESALs carried to date are entered, prints the fraction of the traffic life that remains, 1 − N_p / N. This is a traffic ratio only; the Guide's overlay design also uses the measured condition of the pavement.

CBR cross-check

The Corps of Engineers CBR equation gives the thickness of pavement needed above a soil of a given CBR to protect it from a wheel load:

t = α √( P / (8.1 CBR) − A / π ), with A = P / p

t is in inches, P the single (or equivalent single) wheel load in lb, p the tire contact pressure in psi and α the load-repetition factor read from the Tri-Service chart for the number of coverages (1.0 leaves the basic equation). The tool evaluates it for one entered wheel and compares it with the total thickness of the AASHTO section. The two methods rest on different tests and criteria and need not agree; the comparison is a plausibility check.

Limits

The equations describe the pavements, materials, loads and climate of one road test, extended by judgement, and predict serviceability loss only. They do not check rutting, fatigue cracking, faulting or frost action. Airfield pavements, low-volume and aggregate-surfaced roads, frost-protection thickness, overlays, joint and reinforcement design and mechanistic-empirical design are not part of this tool.

Checks

The iteration of the factors is tested against independent scipy root-finding of the same fixed point for the default mix and sixty seeded mixes. The library is tested against the worked examples of FHWA-NHI-05-037 (flexible SN 4.61, 5.07 and 5.76 with their layer thicknesses; rigid slabs of 10.4, 10.5, 11.5 and 12.3 in; composite k of 423 pci), against all 12,000 required structural numbers tabulated in the Florida DOT Flexible Pavement Design Manual (2008), against load-equivalency factors reprinted by FHWA, and against independent scipy solutions of seeded cases. The FHWA flexible layer examples are reproduced with a base modulus of 45,000 psi; the 40,000 psi stated in that manual gives a surface structural number of 2.46, not the printed 2.35.