## AASHTO Equations

PaveXpress has been designed and built using widely accepted pavement design practices and methodologies. The following page provides detail for what can be found “under the hood” of PaveXpress as you explore your own pavement designs.

## Flexible

#### 1993 AASHTO Flexible Pavement Structural Design

Empirical equations are used to relate observed or measurable phenomena (pavement characteristics) with outcomes (pavement performance). This article presents the 1993 AASHTO Guide basic design equation for flexible pavements. This empirical equation is widely used and has the following form: ###### Where:
W18
predicted number of 80 kN (18,000 lb.) ESALs
ZR
standard normal deviate
So
combined standard error of the traffic prediction and performance prediction
SN
Structural Number (an index that is indicative of the total pavement thickness required)
a1D1 + a2D2m2 + a3D3m3+…ai = ithlayer coefficientDi = ithlayer thickness (inches)mi = ith layer drainage coefficient
ΔPSI
difference between the initial design serviceability index, po, and the design terminal serviceability index, pt
MR

This equation is not the only empirical equation available but it does give a good sense of what an empirical equation looks like, what factors it considers and how empirical observations are incorporated into an empirical equation. The rest of this section will discuss the specific assumptions, inputs and outputs associated with the 1993 AASHTO Guide flexible pavement empirical design equation. The following subsections discuss:

• Assumptions
• Inputs
• Outputs
##### Assumptions

From the AASHO Road Test, equations were developed which related loss in serviceability, traffic, and pavement thickness. Because they were developed for the specific conditions of the AASHO Road Test, these equations have some significant limitations:

• The equations were developed based on the specific pavement materials and roadbed soil present at the AASHO Road Test.
• The equations were developed based on the environment at the AASHO Road Test only.
• The equations are based on an accelerated two-year testing period rather than a longer, more typical 20+ year pavement life. Therefore, environmental factors were difficult if not impossible to extrapolate out to a longer period.
• The loads used to develop the equations were operating vehicles with identical axle loads and configurations, as opposed to mixed traffic.

In order to apply the equations developed as a result of the AASHO Road Test, some basic assumptions are needed:

• The characterization of subgrade support may be extended to other subgrade soils by an abstract soil support scale.
• Loading can be applied to mixed traffic by use of ESALs.
• Material characterizations may be applied to other surfaces, bases, and subbases by assigning appropriate layer coefficients.
• The accelerated testing done at the AASHO Road Test (2-year period) can be extended to a longer design period.

When using the 1993 AASHTO Guide empirical equation or any other empirical equation, it is extremely important to know the equation’s limitations and basic assumptions. Otherwise, it is quite easy to use an equation with conditions and materials for which it was never intended. This can lead to invalid results at the least and incorrect results at the worst.

##### Inputs

The 1993 AASHTO Guide equation requires a number of inputs related to loads, pavement structure and subgrade support. These inputs are:

• Reliability. The reliability of the pavement design-performance process is the probability that a pavement section designed using the process will perform satisfactorily over the traffic and environmental conditions for the design period (AASHTO, 1993). In other words, there must be some assurance that a pavement will perform as intended given variability in such things as construction, environment and materials. The ZR and So variables account for reliability.
• Pavement structure. The pavement structure is characterized by the Structural Number (SN). The Structural Number is an abstract number expressing the structural strength of a pavement required for given combinations of soil support (MR), total traffic expressed in ESALs, terminal serviceability and environment. The Structural Number is converted to actual layer thicknesses (e.g., 150 mm (6 inches) of HMA) using a layer coefficient (a) that represents the relative strength of the construction materials in that layer. Additionally, all layers below the HMA layer are assigned a drainage coefficient (m) that represents the relative loss of strength in a layer due to its drainage characteristics and the total time it is exposed to near-saturation moisture conditions. Generally, quick-draining layers that almost never become saturated can have coefficients as high as 1.4 while slow-draining layers that are often saturated can have drainage coefficients as low as 0.40. Keep in mind that a drainage coefficient is basically a way of making a specific layer thicker. If a fundamental drainage problem is suspected, thicker layers may only be of marginal benefit – a better solution is to address the actual drainage problem by using very dense layers (to minimize water infiltration) or designing a drainage system. Because of the peril associated with its use, often times the drainage coefficient is neglected (i.e., set as m = 1.0).
• Serviceable life. The difference in present serviceability index (PSI) between construction and end-of-life is the serviceability life. The equation compares this to default values of 4.2 for the immediately-after-construction value and 1.5 for end-of-life (terminal serviceability). Typical values used now are:

Post-construction: 4.0 – 5.0 depending upon construction quality, smoothness, etc.

End-of-life (called “terminal serviceability”): 1.5 – 3.0 depending upon road use (e.g., interstate highway, urban arterial, residential)

• Subgrade support. Subgrade support is characterized by the subgrade’s resilient modulus(MR). Intuitively, the amount of structural support offered by the subgrade should be a large factor in determining the required pavement structure.
##### Outputs

The 1993 AASHTO Guide equation can be solved for any one of the variables as long as all the others are supplied. Typically, the output is either total ESALs or the required Structural Number (or the associated pavement layer depths). To be most accurate, the flexible pavement equation described in this chapter should be solved simultaneously with the flexible pavement ESAL equation. This solution method is an iterative process that solves for ESALs in both equations by varying the Structural Number. It is iterative because the Structural Number (SN) has two key influences:

1. The Structural Number determines the total number of ESALs that a particular pavement can support. This is evident in the flexible pavement design equation presented in this section.
2. The Structural Number also determines what the 80 kN (18,000 lb.) ESAL is for a given load.

Therefore, the Structural Number is required to determine the number of ESALs to design for before the pavement is ever designed. The iterative design process usually proceeds as follows:

1. Determine and gather flexible pavement design inputs (ZR, So, ΔPSI and MR).
2. Determine and gather flexible pavement ESAL equation inputs (Lx, L2x, G).
3. Assume a Structural Number (SN).
4. Determine the equivalency factor for each load type by solving the ESAL equation using the assumed SN for each load type.
5. Estimate the traffic count for each load type for the entire design life of the pavement and multiply it by the calculated ESAL to obtain the total number of ESALs expected over the design life of the pavement.
6. Insert the assumed SN into the design equation and calculate the total number of ESALs that the pavement will support over its design life.
7. Compare the ESAL values in #5 and #6. If they are reasonably close (say within 5 percent) use the assumed SN. If they are not reasonably close, assume a different SN, go to step #4 and repeat the process.
##### Example Calculation

To view an example calculation using the AASHTO 93 equation visit http://www.pavementinteractive.org/article/flexible-pavement-empirical-design-example.

## Rigid

#### 1993 AASHTO Rigid Pavement Structural Design

Empirical equations are used to relate observed or measurable phenomena with outcomes. There are many different types of empirical equations available today but this section will present the 1993 AASHTO Guide basic design equation for rigid pavements as an example. This equation is widely used and has the following form (see Figure 6.5 for the nomograph form) ###### Where:
W18
predicted number of 80 kN (18,000 lb.) ESALs
ZR
standard normal deviate
So
combined standard error of the traffic prediction and performance prediction
D
slab depth (inches)
pt
terminal serviceability index
ΔPSI
difference between the initial design serviceability index, po, and the design terminal serviceability index, pt
Sow
modulus of rupture of PCC (flexural strength)
Cd
drainage coefficient
J
Ec
Elastic modulus of PCC
k

This equation is not the only empirical equation available but it does give a good sense of what an empirical equation looks like, what factors it considers and how empirical observations are incorporated into an equation. The rest of this section will discuss the specific assumptions, inputs and outputs associated with the 1993 AASHTO Guide flexible pavement empirical design equation. The following subsections discuss:

• Assumptions
• Inputs
• Outputs
##### Assumptions

From the AASHO Road Test, equations were developed which related loss in serviceability, traffic, and pavement thickness. These equations were developed for the specific conditions of the AASHO Road Test and therefore involved some significant limitations:

• The equations were developed based on the specific pavement materials and roadbed soil present at the AASHO Road Test.
• The equations were developed based on the environment at the AASHO Road Test only.
• The equations are based on an accelerated two-year testing period rather than a longer, more typical 20+ year pavement life. Therefore, environmental factors were difficult if not impossible to extrapolate out to a longer period.
• The loads used to develop the equations were operating vehicles with identical axle loads and configurations, as opposed to mixed traffic.
• For JPCP and JRCP, all transverse joints were the same spacing. JPCP was 4.6 m (15 ft) and JRCP was 12.2 m (40 ft). All
transverse joints used dowel bars.
• All PCC was of the same mix design and used the same aggregate and portland cement.

In order to apply the equations developed as a result of the AASHO Road Test, some basic assumptions are needed:

• The characterization of subgrade support may be extended to other subgrade soils by an abstract soil support scale.
• Loading can be applied to mixed traffic by use of ESALs.
• Material characterizations may be applied to other surfaces, bases, and subbases by assigning appropriate values.
• The accelerated testing done at the AASHO Road Test (2-year period) can be extended to a longer design period.

When using the 1993 AASHTO Guide empirical equation or any other empirical equation, it is extremely important to know the equation’s limitations and basic assumptions. Otherwise, it is quite easy to use an equation with conditions and materials for which it was never intended. This can lead to invalid results at the least and incorrect results at the worst.

##### Inputs

The 1993 AASHTO Guide equation requires a number of inputs related to loads, pavement structure and subgrade support. These inputs are:

• Reliability. The reliability of the pavement design-performance process is the probability that a pavement section designed using the process will perform satisfactorily over the traffic and environmental conditions for the design period (AASHTO, 1993). In other words, there must be some assurance that a pavement will perform as intended given variability in such things as construction, environment and materials. The ZR and So variables account for reliability.
• PCC elastic modulus. If no value is known, the PCC elastic modulus (Ec) can be estimated from relationships such as the following: ###### Where:
Ec
PCC elastic modulus
fct
PCC compressive strength

If no compressive strength data are available (or cannot be assumed), assume Ec = 27,500 MPa (4,000,000 psi), which corresponds to a compressive strength of 34.5 MPa (5000 psi).

• PCC modulus of rupture (flexural strength). The modulus of rupture (S’c) is typically obtained from a flexural strength test.
• Slab depth. The pavement structure is best characterized by slab depth (D). The number of ESALs a rigid pavement can carry over its lifetime is very sensitive to slab depth. As a general rule, beyond about 200 mm (8 inches) the load carrying capacity of a rigid pavement doubles for each additional 25 mm (1 inch) of slab thickness.
• Drainage coefficient. Rigid pavement is assigned a drainage coefficient (Cd) that represents the relative loss of strength due to its drainage characteristics and the total time it is exposed to near-saturation moisture conditions. Generally, quick-draining layers that almost never become saturated can have coefficients as high as 1.2 while slow-draining layers that are often saturated can have drainage coefficients as low as 0.80. If subsurface drainage is expected to be a problem, positive drainage measures should be taken. In general, the use of drainage coefficients to overcome poor drainage conditions is not recommended (i.e. more slab thickness does not necessarily solve water-related problems). Because of the peril associated with its use, often times the drainage coefficient is neglected (i.e., set as Cd = 1.0).
• Serviceable life. The difference in present serviceability index (PSI)between construction and end-of-life is the serviceability life. The equation compares this to default values of 4.2 for the immediately-after-construction value and 1.5 for end-of-life (terminal serviceability). Typical values used now are:
• Post-construction: 4.0 – 5.0 depending upon construction quality, smoothness, etc.
• End-of-life (called “terminal serviceability” and designated “pt“): 1.5 – 3.0 depending upon road use (e.g., interstate highway, urban arterial, residential)
• Load transfer coefficient (J Factor). This accounts for load transfer efficiency. Essentially, the lower the J Factor the better the load transfer. The J Factor for the AASHO Road Test was estimated to be 3.2. Typical J factor values are as shown below.
Condition
J Factor
Undoweled PCC on crushed aggregate surfacing
3.8
Doweled PCC on crushed aggregate surfacing
3.2
Doweled PCC on HMA (without widened outside lane) and tied PCC shoulders
2.7
CRCP with HMA shoulders
2.9 – 3.2
CRCP with tied PCC shoulders
2.3 – 2.9
• Modulus of subgrade reaction. The modulus of subgrade reaction (k) is used to estimate the “support” of the PCC slab by the layers below. Usually, an “effective” k (keff) is calculated which reflects base, subbase and subgrade contributions as well as the loss of support that occurs over time due to erosion and stripping of the base, subbase and subgrade. Typically, large changes in keff have only a modest impact on PCC slab thickness.
##### Outputs

The 1993 AASHTO Guide equation can be solved for any one of the variables as long as all the others are supplied. Typically, the output is either total ESALs or the required slab depth (D). In design, the rigid pavement equation described in this chapter is typically solved simultaneously with the rigid pavement ESAL equation. The solution is an iterative process that solves for ESALs in both equations by varying the slab depth (D). The solution is iterative because the slab depth (D) has two key influences:

1. The slab depth (D) determines the total number of ESALs that a particular pavement can support. This is evident in the rigid pavement design equation presented in this section.
2. The slab depth also determines what the equivalent 80 kN (18,000 lb.) single axle load is for a given load.

Therefore, the slab depth (D) is required to determine the number of ESALs to design for before the pavement is ever designed. The iterative design process usually proceeds as follows:

1. Determine and gather rigid pavement design inputs (ZR, So, DPSI, pt, Ec, S’c, J, Cd and keff).
2. Determine and gather rigid pavement ESAL equation inputs (Lx, L2x, G)
3. Assume a slab depth (D).
4. Determine the equivalency factor for each load type by solving the ESAL equation using the assumed slab depth (D) for each load type.
5. Estimate the traffic count for each load type for the entire design life of the pavement and multiply it by the calculated ESAL to obtain the total number of ESALs expected over the design life of the pavement.
6. Insert the assumed slab depth (D) into the design equation and calculate the total number of ESALs that the pavement will support over its design life.
7. Compare the ESAL values in #5 and #6. If they are reasonably close (say within 5 percent) use the assumed slab depth (D). If they are not reasonably close, assume a different slab depth (D), go to step #4 and repeat the process.
##### Example Calculation

To view an example calculation using the AASHTO 93 equation visit http://www.pavementinteractive.org/article/rigid-pavement-empirical-design-example.

## Layered Elastic Analysis

#### 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:

Location
Response
Pavement surface (1)
Vertical deflection (good estimate of overall pavement strength)
Bottom of HMA layer (2)
Horizontal tensile strain (prediction of fatigue life)
Top of intermediate layer (3) (base or subbase)
Vertical compressive strain (prediction of rutting)
Vertical compressive strain (prediction of rutting) 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):

Nf = k1 (1 / εt)k2 for fatigue cracking

Nr = k3 (1 / εv)k4 for rutting

where

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

and

Nf = k1 (1 / εt)k2 (1 / E)k3 for estimating fatigue cracking

where

• Nf = loads to failure (specifically to a limiting condition).
• εt = 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 Nf = 14.82 – 3.291 log (εt / 10-6) – 0.854 log (Eac / 103)

where:

• Nf = loads to failure.
• εt = horizontal tensile strain at the bottom of the HMA (in/in x 10-6)
• Eac = 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 Nf = 15.947 – 3.291 log (εt/10-6) – 0.854 log (Eac/103) for 10% fatigue cracking in the wheelpath

or

log Nf = 16.086 – 3.291 log (εt/10-6) – 0.854 log (Eac/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.

Minnesota DOT Timm et al (1999) Nf = 2.83 x 10-6 (1 / εt)3.206 with εt input with units of in/in x 10-6
PerRoad 3.5 Road 3.5 Nf = 2.83 x 10-6 (1 / εt)3.148 with εt input with units of in/in x 10-6
NCHRP and AASHTO Road Test Data (45% fatigue cracking in the wheelpath) Finn et al (1977) Nf = (18.4)[(0.0042)(1/εt)3.291 (1 / E)0.854] with εt input with units of in/in x 10-6 and E is in psi.
NCAT for HMA thicknesses of 5 in or less Priest and Timm (2006) Nf = 0.4875 (1 / εt)3.031 (1 / E)0.6529 with εt input with units of in/in x 10-6 and E is in psi.
NCAT for HMA thicknesses of 7 to 9 in. Priest and Timm (2006) Nf = 0.4831 (1 / εt)3.063 (1 / E)0.5992 with εt input with units of in/in x 10-6 and E is in psi.

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:

Sponsor Fatigue Model εt = 200 x 10-6 and E = 500 x 103 psi
Minnesota DOT Nf = 2.83 x 10-6 (1 / εt)3.206 Nf = 2.0 million
PerRoad 3.5 Nf = 2.83 x 10-6 (1 / εt)3.148 Nf = 1.2 million
NCHRP and AASHTO Road Test Data (45% fatigue cracking in the wheelpath) Nf = (18.4)[(0.0042)(1 / εt)3.291 (1 / E)0.854] Nf = 1.6 million

The results are within a range of 1.2 to 2.0 million loads to a fatigue cracking failure (although the amount of fatigue cracking does vary among these models).

##### 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:

Nr = 1.077 x 1018 (10-6v)4.4843

where:

• Nr = loads to cause a 0.5 inch rut,
• εv = 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

Material Modulus Range (psi)
HMA (temperature dependent) 50,000 to 4,000,000
A commonly used default value is 500,000 psi for uncracked HMA
Cracked HMA Range 50,000 to 500,000
Cracked HMA (10% of wheelpath—slight to moderate fatigue cracks) 100,000 to 250,000
Pulverized HMA 40,000
Granular Base (unstabilized) 20,000 to 50,000
Sandy soils 5,000 to 30,000
Silty soils 5,000 to 20,000
Clayey soils 5,000 to 15,000
##### 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.