Structural Number and Physical Pavement Layer Depths

SN represents the structural value of the pavement. SN_f denotes the final designed pavement (after overlay), whereas SN_eff refers to the effective structural number of the pavement as-built (SN_eff,b) or after milling (SN_eff,a).

D represents the physical depth of the pavement layers above the subgrade.

PAVEXpress Design Overlay: NDT Method Illustrated

What is the required design overlay thickness to fulfill structural requirements dictated by existing pavement conditions and the SN_F?

Step 1

Solve for SN (SN_F) using the AASHTO 93 Equation for Flexible Pavements

SN_F is the structural number that satisfies the structural needs of a designed pavement. It is found by iteratively solving for SN in the AASHTO93 equation for flexible pavements using any root-finding method.

In this case, SN = SN_F

This number is needed to compare the remaining structural number of existing pavement (SN_eff,b) and the structural number of the removed pavement (SN_rmv). Ultimately, SN_F allows us to calculate the overlay thickness necessary to fulfill structural requirements implied by existing pavement conditions and geometry.

Details for Solving for SN = SN_F

Define a function F(SN) that is equal to the difference between the left and right sides of the AASHTO 93 equation above. This is a function of SN alone if we assume all other values are held constant.

Using any root finding method, find SN such that F(SN) = 0

(Your answer should have 2 decimals at most; any further accuracy has little physical impact)

Step 2

Solve for E_p using allowed pavement deflection

The effective pavement modulus (E_p), along with the thickness of existing pavement (D_o), is needed to calculate the effective pavement structural number (SN_eff,b) before any removal.

E_p is found by using a root-finding method on the below equation. The results are then used to calculate SN_eff in Step 3.

NOTE: In PAVEXpress, you have the option of calculating E_p (provide other required inputs) or entering its value directly (if known)

Details for solving for E_p

Using a root-finding method, determine E_p such that F(E_p) = 0

NOTE: a (plate radius) should not be confused with a structural coefficient of a pavement layer (a_i) as denoted with a subscript.

Step 3

Calculate the effective pavement structural number after removal (SN_eff,a)

The effective structural number of the remaining pavement after milling (SN_eff,a) is used to determine the structural number that must be supplied by a new AC overlay (SN_OL) to achieve the required SN_F of the designed pavement.

SN_eff,a is found by subtracting the structural number of the removed pavement (SN_rmv) from the structural number of the as-built pavement SN_eff,b.

3.1 – Determine SN_eff,b (using E_p)

3.2 – Determine SN_rmv

3.3 – Determine SN_eff,a

Step 4

Calculate Overlay SN (SN_OL) and Thickness (D_OL)

SN_OL is the minimum structural number of the AC Overlay required to achieve a total pavement structural number of SN_F. The ultimately selected overlay thickness should be the larger of the minimum overlay thickness and the calculated overlay thickness (D_OL).

SN_OL is found by taking the difference between the design structural number (SN_F) and the effective structural number of the remaining pavement after any pavement removal (SN_eff,a).

4.1 – Determine SN_OL

4.2 – Determine D_OL

4.3 – Compare D_OL to minimum overlay thickness

Determine Pavement Structure for Rigid Pavements

PAVEXpress Design uses a modified version of the AASHTO98 methods to determine the required design thickness for a rigid pavement. It is based on a complex systems of dependent empirical equations that are solved iteratively and based on material properties, loading conditions, and mean weather conditions.

In particular, PAVEXpress omits the adjustment step for seasonal temperature changes, which affects the elastic modulus of subgrade support, k’. The solution to AASHTO98R is highly sensitive to the value of k’. In practice, this term is often excluded due to a lack of sufficient data to needed to calculate it. Consequently, PAVEXpress design allows you specify this value directly based on your own adjustments or calculations.

In practice, this means users should adjust their subgrade modulus independently prior to specifying this value in this PAVEXpress module.

Caveats for the use of AASHTO98 for Rigid Pavements

[1] AASHTO98 can be calibrated for location adjustments by including mean annual temperature, precipitation, and wind speeds. The table values provided in AASHTO98 may no longer accurately reflect current conditions. Furthermore, the methods are highly sensitive to weather-based adjustments.

[2] AASHTO98 was designed for pavement thicknesses between 6.0 and 15.0 inches. The model excludes values outside this range, although numerical solutions are mathematically possible. PAVEXpress also follows this rule.

[3] AASHTO98 was developed for pavements that experience between 1.5 and 100 million Design ESALs. It is not intended for pavements that experience design loads outside this range.

[4] AASTO98 provides example solutions for fixed values of temperature differentials. Consequently, values resulting from PAVEXpress based on custom weather conditions will be slightly different than those suggested in the AASHTO98 Guide. Where possible, users are encouraged to compare results from PAVEXpress to those recommended by the tables in AASHTO98.

Differences between AASHTO93 and 98 for Rigid Pavements

In AASHTO93, D is determined iteratively using a single equation that relates loading, structural properties, and performance to the required pavement thickness.

In AASHTO98, D is determined iteratively through a system of dependent equations, rather than a single equation like AASHTO93. That is, D appears in multiple supporting equations that relate effects, including loading conditions, structural properties, performance levels, and now weather conditions.

The AASHTO93 methods include parameters C_d (drainage coefficient) and J (load transfer coefficient that depends on load transfer efficiency). These terms are absent in AASHTO98.

As well, AASHTO98 recommends a different method for determining the value of k and k’ (effective elastic modulus of subgrade support) based on mean local weather information (precipitation, temperature, and wind).

Overview of Solution Methods

AASHTO98 consists of a system of dependent equations that ultimately compare a user’s design conditions to design conditions related to the original AASHTO Road Tests. This is reflected by the set of paired parameters without an apostrophe (e.g. σ_t’ and σ_t, F’ and F, et. al.). In contrast, parameters with an apostrophe denote inputs reference your specific pavement design, rather than the AASHTO Road Test conditions.

It should be noted that k and TD+ are the same for both user conditions and the AASHTO Road Tests. Selected weather condition parameter should reflect the user’s design conditions (rather than the AASHTO Road Test).

The goal is to find the value of D that minimizes the difference between W’ and W as shown in the expression for log₁₀(W’). D is re-calculated until the ratio of W and W’ is as close to unity as possible (or their difference is minimized). In practice, D is typically rounded to the nearest half inch. In PAVEXpress, D is calculated to the nearest 1/10 of an inch so that a user may decide how to interpret and translate their results into a pavement design.

Given that AASHTO limits its results to a range of 6.0 to 15.0 inches, this yields 91 possible practical solutions for D using increments of 0.1 inch.

NOTE: W is sometimes referred to W₁₈ to denote the number of 18-kip [80-kN] ESALs

For each iteration of D, PAVEXpress begins by solving all pre-requisite equations before evaluating the primary equation for log₁₀(W’). This means PAVEXpress evaluates the equations in the following order (color coded):

The table below includes a list of constants from the AASHTO Road Tests. They affect calculations that include parameters with an apostrophe (e.g. σ_t’, F’, et. al.) and are provided in the AASHTO98 Manual.

NOTE: The k value specified in PAVEXpress should reflect your pavement design; this table value reflects the value used in the AASHTO Road Tests

Completely expanded, the map shows the full set of equations required to determine design thickness using AASHTO98 (Rigid) methods

PAVEXpress and Everstress

Introduction

This pavement response calculator can be used to estimate stress, strain or deflection within a layered pavement system due to a static loads. The modulus of elasticity, Poisson’s ratio and thickness must be defined for each layer. Further, the load magnitude, contact pressure and location must be defined for each load considered.

The software which is now contained within PAVEXpress was called Everstress. Everstress was originally developed from the WESLEA layered elastic analysis program (provided by the Waterways Experiment Station, U.S. Army Corps of Engineers). The pavement system is multi-layered elastic using multiple wheel loads (up to 20). The program can analyze a pavement structure containing up to five layers.

Broadly, use of layered elastic analysis along with some type of limiting criteria is an evolving approach to pavement analysis and design; although, it is not new being as it have been available within the pavement community for over 50 years. This process is often referred to as the mechanistic-empirical approach. The mechanistic component is the determination of pavement reactions such as stresses, strains, and deflections within the pavement layers. The empirical portion relates these reactions to the performance of the pavement structure. For instance, it is possible to calculate the amount of deflection at the surface of the pavement. If these deflections are related to the life of the pavement, then an empirical relationship has been established between the mechanistic response of the pavement and its expected performance.

The basic advantages of a mechanistic-empirical pavement analysis and design are:

• A design check against methodologies such as AASHTO 93.
• The assessment of different load magnitudes and configurations.
• The ability to examine how new materials behave in a pavement structure.
• A better definition of the role of construction.
• The accommodation of environmental and aging effects on materials.

The modulus of elasticity (E) and Poisson’s ratio (µ) are used to define each layer in the pavement along with the layer thickness. The calculations estimate the pavement responses of stresses, strains, and deflections. The major assumptions include:

• Materials remain in their elastic range; hence, the use of modulus of elasticity.
• Layers extend infinitely in the horizontal direction and semi-infinitely for the subgrade.
• Tire contact areas are circular.

The layered elastic analysis allows the estimation of critical pavement responses. These critical responses typically considered are shown in the following table:

The figure which follows further illustrates the location of these pavement responses.

Transfer Functions (Failure Criteria)

There are two distress transfer functions discussed here: (1) fatigue cracking due to strains at the bottom of the HMA layer, and (2) rutting due to permanent deformation of the HMA and unbound layers. Estimates of critical strains are commonly used in these models.

General Models

For estimating the loads to failure for a specific type of strain, there are two general models typically used for both estimating fatigue cracking and rutting—one type of model does not contain the modulus of elasticity of the HMA and the other does. These are (after Priest and Timm, 2006):

N_f = k₁ (1 / ε_t)^k2 for fatigue cracking

N_r = k₃ (1 / ε_v)^k4 for rutting

where

• N_f = loads to failure (specifically to a limiting condition).
• ε_t = horizontal tensile strain at the bottom of the HMA.
• ε_v = vertical compressive strain at the top of the subgrade (estimate includes rutting for all pavement layers).

and

N_f = k₁ (1 / ε_t)^k2 (1 / E)^k3 for estimating fatigue cracking

where

• N_f = loads to failure (specifically to a limiting condition).
• ε_t = horizontal tensile strain at the bottom of the HMA.
• E = modulus of elasticity of the HMA.

The coefficients for these models are determined experimentally typically from a combination of laboratory and field test sections.

What follows are specific models that contain actual coefficients.

Fatigue Cracking

Fatigue cracking typically appears on the surface of HMA as a series of interconnected cracks. The transfer functions readily available use the assumption that the tensile strain at the bottom of the HMA layer is related to the fatigue life. However, this is not always the case since it is well documented that often HMA surface cracks start at the top of the pavement (apparently due to aging of the asphalt binder). Numerous views can be found on how the surface initiated cracking process works. The assumption used in PAVEXpress is that tensile strains can be used to estimate fatigue life based on strains at the bottom of the HMA (hence cracks that start at the bottom of the HMA and migrate to the surface).

Suggested values for limiting the horizontal tensile strain at the bottom of the HMA layer and vertical compressive strain at the top of the subgrade to achieve long life are 60 microstrains and 200 microstrains, respectively (Monismith and Long, 1999). But more on this topic shortly.

For fatigue cracking estimates, the model is below has been commonly used (Finn et al, 1977) and was based on laboratory results:

log N_f = 14.82 – 3.291 log (ε_t / 10^-6) – 0.854 log (E_ac / 10³)

where:

• N_f = loads to failure.
• ε_t = horizontal tensile strain at the bottom of the HMA (in/in x 10^-6)
• E_ac = the stiffness of the HMA layer in psi.

Finn et al (1977) applied a shift factor to the laboratory results to estimate field fatigue with the resulting model as follows:

log N_f = 15.947 – 3.291 log (ε_t/10^-6) – 0.854 log (E_ac/10³) for 10% fatigue cracking in the wheelpath

or

log N_f = 16.086 – 3.291 log (ε_t/10-6) – 0.854 log (E_ac/10³) for 45% fatigue cracking in the wheelpath

Fatigue models for HMA have been developed by several agencies and research groups. A limited summary for these models with coefficients follow.

There is variation in the model predictions. Using the first three of the models shown above and two fixed inputs of εt = 200 x 10-6 and E = 500,000 psi:

Fatigue Endurance Limit

An additional fatigue cracking consideration is if strains at the bottom of the HMA layer are kept below a design strain threshold, the initiation of bottom-up fatigue cracking can be prevented. This design threshold is typically called the fatigue endurance limit (FEL) of the asphalt mixture. The FEL represents a level of strain below which there is no cumulative damage over an indefinite number of load cycles (after Tran et al, 2015). Illustration of a fatigue endurance limit is shown below (modified from Prowell et al, 2010):

The value for the endurance limit of the tensile strain at the bottom of the HMA layer has been continually examined. Original work by Monismith and others suggests a value of 60 to 70 microstrains, but currently accepted values range from 70 to 100 microstrains (Thompson and Carpenter, 2004). Research at NCAT suggests higher fatigue endurance limits are possible (Willis et al., 2009, Prowell et al, 2010, Tran et al, 2015). Prowell specifically noted that a range of 75 to 200 microstrains can be supported by currently available data and analyses. The recent analyses done for SHRP R23 applied a 100 microstrain endurance limit for developing long lasting pavement designs with the software PerRoad 3.5.

Rutting

Rutting occurs due to permanent deformation of the asphalt concrete layer and unbound layers. However, as the deformation of asphalt concrete is not well defined, the failure criteria equations are expressed as a function of the vertical compressive strain at the top of the subgrade. The Chevron equation was utilized (Asphalt Institute, 1982). It is shown below:

N_r = 1.077 x 1018 (10^-6/ε_v)^4.4843

where:

• N_r = loads to cause a 0.5 inch rut,
• ε_v = vertical compressive strain (in/in x 10 -6) at the top of the subgrade.

The traffic volume and the performance herein are always expressed in terms of 18,000 lb equivalent single axle loads (ESALs). The major assumption is the HMA mix is well-designed and constructed (which implies good compaction during construction).

Typical Layer Moduli

The table below provides a summary of a few HMA related layer moduli.

HMA Pavement Typical Moduli and Ranges of Moduli

References

Finn, F.N., et al., “The Use of Distress Prediction Subsystems for the Design of Pavement Structures,” Proceedings, 4th International Conference on the Structural Design of Asphalt Pavements, University of Michigan, Ann Arbor, 1977.

Monismith, C.L. and F. Long, 1999a, “Mix Design and Analysis and Structural Section Design for Full Depth Pavement for Interstate Route 710,” Technical Memorandum TM UCB PRC 99-2, Pavement Research Center, Institute for Transportation Studies, University of California, Berkeley.

Monismith, C.L. and F. Long, 1999b, “Overlay Design for Cracked and Seated Portland Cement Concrete (PCC) Pavement – Interstate Route 710,” Technical Memorandum TM UCB PRC 99-3, Pavement Research Center, Institute for Transportation Studies, University of California, Berkeley.

Priest, A.L. and Timm, D.H., 2006, “Methodology and Calibration of Fatigue Transfer Functions for Mechanistic-Empirical Flexible Pavement Design,” NCAT Report 06-03, National Center for Asphalt Technology, Auburn University, December 2006.

Prowell, B.D. et al, 2010, “Validating the Fatigue Endurance Limit for Hot Mix Asphalt,” NCHRP Report 646, Transportation Research Board, 2010.

Thompson, M.R. and S.H. Carpenter, 2004, “Design Principles for Long Lasting HMA Pavements,” Proceedings, Intl. Symp. on Design and Construction of Long Lasting Asphalt Pavements, National Center for Asphalt Technology, Auburn University, Alabama, pp. 365-384.

Timm, D.H., Newcomb, D.E. and Birgisson, B., 1999, “Mechanistic-Empirical Flexible Pavement Thickness Design: The Minnesota Method,” Staff Paper, MN/RC-P99-10, St. Paul: Minnesota Department of Transportation, 1999.

Tran, N., Robbins, M.M., Timm, D.H., Willis, J.R., Rodezno, C. 2015, “Refined Limiting Strain Criteria and Approximate Range of Maximum Thicknesses for Designing Long-Life Asphalt Pavements,” NCAT Report 15-05, National Center for Asphalt Technology, Auburn University, September 2015.

Willis, J. R. and D. H. Timm, 2009, “A Comparison of Laboratory Thresholds to Measured Strains in Full-Scale Pavements,” Proceedings, Intl. Conf. on Perpetual Pavements, Ohio University, Columbus.

The Asphalt Institute, 1982, “Research and Development of The Asphalt Institute’s Thickness Design Manual,” Research Report 82-2, The Asphalt Institute, College Park, Maryland, August 1982.