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5Introduction This report addresses the issues associated with reflec- tion cracking and reports the development of mechanics- based models for use in mechanistic-empirical procedures for the analysis and design of HMA overlays. These mechanics- based models are capable of predicting reflection crack- ing in HMA overlays of flexible and rigid pavements and incorporate associated computational software for use in mechanistic-empirical procedures for overlay design and analysis. Reflection CrackingâDefinition and Mechanisms Reï¬ection cracking can be deï¬ned as the cracking of a resurfacing or overlay above underlying cracks or joints, with movement of some form in the underlying pavement as its probable cause. As illustrated in Figure 3, this cracking can result from trafï¬c and environmentally induced causes. The existing joints or cracks can affect reï¬ection cracking in two forms (7): 1. The existing joints or cracks provide stress concentration at the bottom of an asphalt overlay, which will lead to the continued crack growth into the asphalt overlay layers. 2. If the stress-concentrating effect of the existing joints or cracks has been nulliï¬ed by some means, a secondary effect of the existing joints or cracks is that a maximum deï¬ection of the pavement under a wheel load will occur at the crack. Thus, the maximum stresses in the overlay will occur at this point, making it the most likely location for crack growth to initiate as an indirect effect of the exist- ing crack. Stress concentration, which is the ï¬rst effect of existing joints or cracks, plays the dominant role in reï¬ection crack- ing due to the movement in the existing pavements in the vicinity of joints or cracks. This movement may be induced by bending or shearing action resulting from trafï¬c loads or daily and seasonal temperature changes. In general, reï¬ection cracking is caused by the combination of these three mecha- nisms. Every pass of a trafï¬c load will induce two shearing and one bending effect on the HMA overlay that are affected by the daily temperature (see Figure 4). Thus the combina- tion of all three mechanisms is required to successfully model reï¬ection cracking. In addition, crack initiation and propa- gation is also inï¬uenced by the existing pavement structure (see Figure 5) and conditions, reï¬ection cracking counter- measures (e.g., reinforcing, interlayer), HMA mixture proper- ties, the degree of load transfer at joints and cracks, and others. Therefore, these three mechanisms and these inï¬uencing factors must be taken into account in developing a reï¬ection cracking model. Available Reflection Cracking Models Concern about reflection cracking of asphalt overlays over existing pavements dates back to 1932, when Gary and Martin (10) studied this problem. Subsequently, many studies have been conducted to address this problem and many models have been developed to analyze or predict reï¬ec- tion cracking. In general, these models can be categorized as follows: ⢠Empirical model; ⢠Extended multi-layer linear elastic model; ⢠Equilibrium equations based models; ⢠Finite element plus traditional fatigue equation model; ⢠Finite element plus fracture mechanics model; ⢠Crack band theory based model; ⢠Cohesive zone cracking model; and ⢠Non-local continuum damage mechanics based model. C H A P T E R 2 Findings
6Selection of a Reflection Cracking Model A comprehensive review of available models and their capa- bilities, advantages, and disadvantages is provided in Appen- dix R. The review revealed that the ï¬nite element plus fracture mechanics model provides the most desirable attributes. The major requirements of the selected model for compatibility with the MEPDG include the following: ⢠The model must use the same input data as the MEPDG, including the trafï¬c, weather, material properties, and pave- ment layer geometry. ⢠The model must be capable of using the same level of com- putational capability as the present MEPDG (e.g., it should not require the use of a supercomputer). ⢠The model must be capable of accepting the additional information that distinguishes the reï¬ection cracking mech- anisms, i.e., load transfer and crack or joint spacing in the underlying old pavement surface layer. ⢠The model must be capable of accepting the additional layer geometry associated with the more common overlay conï¬gurations and be capable of being enhanced in the future as computational capabilities increase. ⢠The model must be user friendly, accepting overlay design input data in a graphical format. ⢠The model must be able to calculate the crack growth potential directly from the input data, and it should permit the consideration of commonly used overlay ap- proaches including interlayers, fabrics, and reinforcing layers. ⢠The computational time required by the use of the model must be short enough to allow effective use of the time in evaluating alternative overlay designs. Surface initiated cracking Thermal contraction Warping Crack growth Temperature gradient giving greater contraction at surface Thermally induced fatigue Crack growth Thermal expansion and contractionExisting layer Sub-base HMA overlay Traffic movement Traffic induced fatigue Crack growth Figure 3. Mechanisms of reflection cracking (8). Old surface course Position of wheel load Tip of the crack Overlay A B C Void St re ss es a t t he ti p o f t he c ra ck Sh ea rin g st re ss Be nd in g st re ss Figure 4. Bending and shear mechanisms (9). Overlay Interlayer Level-up Existing pavement Base/Subbase Figure 5. Illustration of an asphalt overlay system.
7These compatibility requirements automatically eliminate a number of approaches such as the empirical model, extended multi-layer linear elastic program, equilibrium equations, and advanced models requiring use of supercomputers. While this use may be desirable in the future when super- computer use becomes more widespread and the MEPDG has been converted to such use, it is more desirable at the present to develop a model that uses the level of computa- tional power that is commonly available for state design offices. After considering the different model features and compatibility requirements, the ï¬nite element plus fracture mechanics model was selected. However, the running time of a ï¬nite element program proved not to provide a practi- cal tool for design so it was decided to incorporate the results of multiple runs into a faster algorithm. Nearly 100,000 runs of a ï¬nite element program were made in order to obtain a broad range of computed fracture mechanics results. These numerical results were then incorporated into a total of 18 separate ANN algorithms. These algorithms provide accept- able computational speeds and are the core of the overlay reï¬ection cracking model developed in this project. The key concept in fracture mechanics is Paris and Erdoganâs Law (11) for modeling crack propagation, particularly for fracture-micromechanics applications. Expressed in Equa- tion 1, Paris and Erdoganâs Law has been successfully applied to HMA by many researchers, for the analysis of experimental test data and prediction of reï¬ection- and low temperature- cracking (4, 12). where c = the crack length; N = the number of loading cycles; A and n = fracture properties of the asphalt mixture; and ÎK = the SIF amplitude (depends on the stress level, the geometry of the pavement structure, the fracture mode, and crack length). There are three fracture modes: tensile, shearing, and tearing. The number of loading cycles, Nf, needed to propagate a crack (initial length, c0) through the pavement layer thick- ness, h, can be estimated by numerical integration in the form of Equation 2. Because the SIF is one of the key parameters in Paris and Erdoganâs Law, the speed and accuracy of computing SIF val- ues is a very critical aspect of crack propagation analysis and the development of a reï¬ection cracking model. N dc A K f n C h = ( )â« Î0 2( ) dc dN A K n = ( ) Î ( )1 Process of Constructing a Calibrated Reflection Cracking Model The process of constructing a calibrated reï¬ection crack- ing model (Figure 6) involves several steps: ⢠Select a sufï¬cient number of overlay sections to provide a good likelihood of having a sufï¬cient amount of good qual- ity data (including sequential distress measurements; pave- ment structure and materials property data; and trafï¬c and weather data) to permit development of a set of calibrated reï¬ection cracking model coefï¬cients. As a rule of thumb, at least 20 such sections are required to develop a complete set of calibration coefï¬cients. ⢠Collect pavement structure data (including layer thickness, construction dates, and nondestructive testing data on each pavement section) and the mixture design data for the overlay. ⢠Collect pavement distress data (including the total length of cracking in the old pavement surface prior to overlay and the lengths and levels of severity of reflection crack- ing) for at least three (preferably more) sets of sequential observations. ⢠Collect trafï¬c data on each pavement section including the input data used in the MEPDG program (i.e., the trafï¬c load spectrum rather than the total 18-kip equivalent sin- gle axle loads). ⢠Collect climatic data on each of the candidate sections of overlay, including the data needed to accurately calculate the overlay temperature with depth below the surface (i.e., hourly air temperature, solar radiation, and surface reï¬ectance). ⢠Develop a ï¬nite element mechanistic method for calculating the SIF in overlays for thermal, bending, and shearing traf- ï¬c stresses as a crack grows up through different thicknesses of overlay. ⢠Develop a numerically accurate and computationally efï¬- cient method of predicting the SIF computed by the ï¬nite element method. ⢠Develop a method of dealing with different trafï¬c loads and tire footprints in calculating the SIF. ⢠Analyze the ï¬eld distress data into a standard form to rep- resent the total length of reï¬ected cracks that appear with time at the different levels of severity (i.e., low, medium and high). At this point, the number of test sections with actual observed reï¬ection cracking data can be determined (in some cases, not all levels of severity have been observed). ⢠Develop a method for accurately calculating the hourly and dailytemperaturesinanoverlay at the current tip of the crack. ⢠Write a program to calculate the stiffness, tensile strength, compliance, and fracture coefï¬cients of the overlay mix- ture using the mixture properties of volumetric contents of the mixture components, aggregate gradation, and binder master curve characteristics. These properties must be
8computed for the loading times for thermal stresses and the passage of trafï¬c loads representing single, tandem, tridem, and quadrem axles. ⢠Develop a computational technique for calculating the total crack growth caused by single, tandem, tridem, and quadrem axles passing over a growing crack, and include the healing shift factor that increases with the length of time between trafï¬c loads. ⢠Develop a program to calculate the viscoelastic thermal stresses in the overlay. ⢠Develop a computational technique to calculate the growth of a crack caused by daily thermal stresses. ⢠Write a supervisory program that will combine the compu- tations of crack growth by each of the separate mechanisms (thermal, bending, and shearing). ⢠Write an interface program to permit the user to input data in the same format used in the MEPDG. ⢠Calculate the number of days required for each of the three fracture mechanisms to cause a reï¬ection crack to grow through the thickness of the overlay. ⢠Develop sets of computation-to-ï¬eld calibration coefï¬cients for each type of pavement-overlay structure and climatic zone. The work performed in each of these steps to develop the hot-mix overlay reï¬ection cracking model and its calibration coefï¬cients is summarized in the following sections. Overlay Sections with Sufficient Data for Model Development After a thorough review of the data available in the LTPP database and for test sections in New York and Texas, the researchers concluded that there is sufficient high quality data to develop sets of calibration coefï¬cients for the reï¬ec- tion cracking model for each major climatic zone in the United States. Data were collected for a total of 11 pavement- structure-overlay-climatic zone sets representing 411 over- lay sections, as shown in Table 1. The LTPP sections provided the bulk of the data that were used for modeling reï¬ection cracking in the different types of pavement structure. The distribution of these sections within the different climatic zones is given in Table 2. The sections were well distributed throughout the United States and Canada, in 37 states, the District of Columbia, and six Cana- dian provinces as listed in Table 3. Collection of Pavement Structure Data The pavement data used were the layer thickness of each pavement layer and the Falling Weight Deï¬ectometer (FWD) data for each test section prior to the placement of the over- lay including the temperature at which the FWD data were obtained. The layer moduli of the old pavement were back- Figure 6. Flow chart of the process of constructing a calibrated reflection cracking model.
9*The abbreviations are listed in order from the old pavement surface layer upward to the overlay. AC = existing hot mix asphalt surface layer, JCP = jointed concrete pavement, CRC = continuously reinforced concrete surface layer, Mill = old surface layer was milled before the overlay was placed. SAMI = (Strain Absorbing Membrane Interlayer) indicates that a compliant interlayer was placed between the old surface layer and the hot mix overlay. A reinforcing interlayer was placed between a leveling course and the hot mix overlay. AC OL = hot mix asphalt overlay. + WF, DF, WNF, and DNF designate Wet-Freeze; Dry Freeze; Wet-No Freeze; and Dry-No Free, Data Set Pavement Structure* No. of Test Sections Climatic Zone+ 1 AC/mill/AC OL 62 WF 2 AC/mill/AC OL 47 WNF 3 JCP/AC OL 69 WF 4 AC/AC OL 59 WF 5 AC/AC OL 33 WNF 6 AC/SAMI/AC OL 26 WF 7 CRC/AC OL 21 WF 8 AC/AC OL 16 DF 9 AC/mill/AC OL 16 DNF 10 AC/SAMI/AC OL 12 WNF 11 AC/Grid/AC OL 50 NY, Texas Totals 411 respectively Table 1. Overlay sections for model development. Pavement-Overlay Structure Description Total Test Sections No. of Test Sections at Each Climatic Zone WF DF WNF DNF AC/AC OL AC, then AC overlay 108 59 16 33 - AC/Mill/AC OL AC, then Mill+AC overlay 125 62 - 47 16 CRC/AC OL CRC, then AC overlay 21 21 - - - JRC/AC + JPC/AC OL JRC or JPC, then AC overlay 69 69 - - - AC/SC or FC/AC OL AC, then seal coat or friction course +AC overlay 38 26 - 12 - Total 361 237 16 92 16 Table 2. LTPP test sections used for calibration. calculated using the program MODULUS (13). The LTPP data included the deï¬ections measured at many equally spaced locations within each test section. The mean of the backcalculated moduli for each layer was used as the modu- lus of that layer for the entire test section. In addition to the layer thickness and the backcalculated moduli of the old pavement, the mixture design data of the overlay was available, including the volumetric composition of the asphalt mixture, the gradation of the aggregate, and some indication of the grade of the asphalt binder. The grade was used to determine the six characteristics of the master curve of the binder according to the CAM model (the glassy shear mod- ulus, Gg, the crossover frequency, ÏR, the rheological index, R, the deï¬ning temperature, Td, and the two time-temperature
10 shift coefï¬cients, C1 and C2). The method of making this con- version is explained in Appendix G. These six properties of the master curve of extracted binders were measured and reported in SHRP studies (4) and tabulated in Appendix G. This infor- mation was used together with the calculated temperature to determine the input to the ANN models of Witczakâs 1999 (2) and 2006 (3) models of the complex modulus. The overlay test sections in Texas and New York City con- tributed high quality data and the unique feature of having the overlays reinforced by geosynthetic interlayers. Pavement Distress Data Collection The pavement distress data included the total length and severity of the cracks in the old pavement surface prior to the placement of the overlay and the length and severity of the cracks reï¬ected through the overlay. Only transverse cracks were considered as reï¬ection cracks in each test section. In order to have reliable Ï and β values for the S-shaped curves that were ï¬tted to the distress data, at least three separate and sequential observations of distress were required. In some cases, no distress data were recorded on the old pavement surface prior to overlay and a mathematical method had to be devised to estimate the original amount of cracking which was subject to reï¬ection. The mathematical method used was the Systems Identiï¬cation method which is described in detail subsequently in this chapter and also in Appendix L. Traffic Data Collection Trafï¬c data is a key element for the design and analysis of a HMA overlay structure or a new pavement structure. For compatibility with the MEPDG, trafï¬c was described by the actual load distribution (spectrum) for each axle type (single, tandem, tridem, or quadrem axle) for each vehicle (truck) class or number of tires (single or dual). State Climate Zone No. of Sections State Climate Zone No. of Sections State Climate Zone No. of Sections Alabama WNF 14 Massachusetts WF - South Dakota WF 3 Alaska WF 4 Michigan WF 12 Tennessee WF/WNF 4/1 Arizona DNF 11 Minnesota WF 21 Texas DF/ WNF 2/28 Arkansas WNF 2 Mississippi WNF 13 Utah DF - California DNF/WNF 5/3 Missouri WF 23 Vermont WF 4 Colorado DF 9 Montana DF - Virginia WF/WNF 1/3 Connecticut WF 6 Nebraska DF/ WF - Washington DF 1 Delaware WF 1 Nevada DF/DNF - West Virginia WF - D.C. WF 1 New Hampshire WF 1 Wisconsin WF 20 Florida WNF 2 New Jersey WF 21 Wyoming DF - Georgia WNF 2 New Mexico DF/DNF - Alberta WF 4 Idaho DF/WF - New York WF 4 British Columbia DF - Illinois WF 13 North Carolina WF/WNF 2/11 Manitoba WF 9 Indiana WF 14 Ohio WF 3 New Brunswick WF 1 Iowa WF 10 Oklahoma WF/WNF 1/11 Nova Scotia WF - Kansas WF 13 Oregon WNF 1 Ontario WF 3 Kentucky WF 2 Pennsylvania WF 7 Quebec WF 9 Maine WF 6 Rhode Island WF - Saskatchewan DF 4 Maryland WF 14 South Carolina WNF 1 - - - Table 3. Distribution of LTPP test sections by states.
11 The daily trafï¬c distribution data was determined based on the trafï¬c data collected in the ï¬eld over the years. However, it was found that some sections did not have enough ï¬eld data to determine the trafï¬c characteristics, while others had complete historical trafï¬c data. In order to consider the level of collected trafï¬c data, a hierarchical approach was adopted in the MEPDG and also is used in this project. The three lev- els were deï¬ned based on the availability of collected trafï¬c data and Weigh-In-Motion (WIM) data which is used to determine the normalized axle load distribution for each axle and vehicle types (14): ⢠Level 1: Very good knowledge of past and future trafï¬c characteristics and site/segment speciï¬c WIM data; ⢠Level 2: Modest knowledge of past and future trafï¬c char- acteristics and regional default summaries WIM data; and ⢠Level 3: Poor knowledge of past and future trafï¬c charac- teristics and national default summaries WIM data or only Average Annual Daily Truck Trafï¬c (AADTT) available. Categorization of Traffic Loads In order to analyze trafï¬c load effects for reï¬ection crack- ing, the annual number of axle loads for each vehicle class and axle type were entered in the analysis process. The number of axle loads was determined using the trafï¬c load categorized based on the FHWA vehicle class, the axle type, and the num- ber of tires (details of this process are found in Appendix C). Classification of Vehicles FHWA deï¬nes vehicles into 13 classes, as shown in Table 4, depending on whether they carry passengers or commodities. Nonpassenger vehicles (class 4 to class 13) are divided by the number of axles and the trailer units (15). Although a bus (vehicle class 4) is a passenger vehicle, the term truck trafï¬c applies to the load level and includes both trucks and buses since the proportion of buses in the trafï¬c ï¬ow is relatively small (16). Because the light axle load groups, such as vehicle classes 1 to 3, do not have signiï¬cant effects on load related distresses, the trafï¬c analysis in this study considered only the heavier load groups (i.e., classes 4 to 13). Axle Load Distribution Factor The axle load distribution is deï¬ned as the classiï¬cation of trafï¬c loading in terms of the number of load applications by each axle type (single, tandem, tridem, or quadrem) within a given range of axle load. The axle load distribution factor is the percentage of the total axle applications in each load inter- val by an axle type for a speciï¬c vehicle class (classes 4 to 13) (14, 17). The load ranges and intervals for each axle type are listed in Table 5. The determination of the axle load distribution requires WIM data, which is the number of axles measured within each axle load range by axle types of each vehicle class. LTPP guidelines require that the vehicle axle weights should be col- lected using a WIM sensor by vehicle classes, type of axle, and axle load intervals. Using measured WIM data, the distribu- tion is calculated by averaging the number of axles measured within each load interval of an axle type for a vehicle class divided by the total number of axles for all load intervals for a given vehicle class. The normalized axle load distribution factors should total 100 for each axle type within each truck class. Table 6 presents an example of FHWA W-4 Truck Vehicle Class Schema Description 4 Buses 5 Two-axle, single-unit trucks 6 Three-axle, single-unit trucks 7 Four-axle or more than four-axle single-unit trucks 8 Four-axle or less than four-axle single trailer trucks 9 Five-axle single trailer trucks 10 Six-axle or more than six-axle single trailer trucks 11 Five-axle or less than five-axle multi-trailer trucks 12 Six-axle multi-trailer trucks 13 â Seven-axle or more than seven-axle multi-trailer trucks Table 4. FHWA vehicle classification.
12 Axle Type Axle Load Interval Single Axles 3,000 ~ 40,000 lb at 1,000 lb intervals Tandem Axles 6,000 ~ 80,000 lb at 2,000 lb intervals Tridem Axles 12,000 ~ 102,000 lb at 3,000 lb intervals Quad Axles Table 5. Load intervals for each axle type. Axle Load (lb) Vehicle Class 4 5 6 7 3,000 0 53,818 183 11 4,000 10 54,606 558 52 5,000 42 39,113 993 139 6,000 175 20,289 1,099 168 7,000 988 24,555 2,426 252 8,000 10,687 22,491 5,617 298 9,000 9,713 13,719 8,154 365 10,000 10,156 12,839 12,423 879 11,000 6,011 7,127 8,945 1,516 12,000 5,875 6,413 7,725 2,913 13,000 3,409 3,511 3,257 2,464 14,000 2,947 3,128 2,289 2,710 15,000 1,640 1,756 975 1,740 16,000 1,239 1,513 725 1,419 17,000 679 834 285 664 18,000 446 800 235 423 19,000 212 424 104 159 20,000 181 360 73 111 21,000 106 261 44 70 22,000 51 131 22 46 23,000 41 135 6 26 24,000 21 85 4 9 25,000 24 90 3 12 26,000 11 43 1 7 27,000 4 33 1 2 28,000 1 12 3 1 29,000 4 25 0 1 30,000 3 13 0 0 31,000 1 16 2 0 32,000 2 8 0 0 33,000 0 5 0 0 34,000 0 2 1 0 35,000 0 0 0 0 36,000 0 0 0 0 37,000 0 2 0 0 38,000 0 0 0 0 39,000 0 0 0 0 40,000 0 0 0 0 Total 54,679 268,157 56,153 16,457 Table 6. Number of single axle loads for vehicle class 4 to 7 (LTPP Section 180901 in 2004).
13 Weight Tables in which WIM data are typically reported for vehicle classes 4, 5, 6, and 7 for LTPP test section 180901 in 2004. Figure 7 shows the annual normalized single axle load distribution calculated using the data in Table 6. Categorizing Traffic Load In this study, the trafï¬c loads were categorized based on the vehicle class, axle type, and number of tires to facilitate the analysis of load effects for reï¬ection cracking. Every truck in each vehicle class has single, tandem, tridem, or/and quadrem axles, and each axle has single or dual tires. Table 7 lists the number of axles for each axle type and vehicle class. All axles of vehicle classes 4 and 5 and single axles of class 6 and 7 vehicles have single tires while the others have dual tires. Thus the matrix of vehicle class and axle types can be categorized according to the number of tires. When the steer- ing and non-steering axles are put together in the single axle type, the matrix can be characterized into eight categories based on the vehicle class, the axle type, and the number of tires. The total number of axle loads for each category was used to determine the axle load distribution factor for the 0 5 10 15 20 25 0 10000 20000 30000 40000 50000 Axle Load (lb) A xl e Lo ad D is tri bu tio n (% ) Vehicle class 4 Vehicle class 5 Vehicle class 6 Vehicle class 7 Figure 7. Annual normalized single axle load distribution for vehicle class 4 to 7 (LTPP Section 180901 in 2004). Vehicle Class Number of Axles Single* Tandem Tridem Quadrem 4 1 1 5 2 (1) 6 1 1 7 1 1 8 3 (2) 9 1 2 10 1 1 1 11 5 (4) 12 4 (3) 1 13 3 (2) 2 The number of nonsteering single axles are shown in parentheses. Shaded areas are vehicle classes that use single tires; unshaded areas are vehicle classes that use dual tires. Table 7. Number of axles for each vehicle class.
14 Table 8. Vehicle class related to axle and tire categories. Vehicle Class Single Axle Tandem Axle Tridem Axle Quad Axle 4 1 3 5 7 5 6 4 6 8 7 8 2 9 10 11 12 13 14 Shaded areas are vehicle classes that use single tires; unshaded areas are vehicle classes that use dual tires. Single Tire Dual Tire analysis of the trafï¬c load effect on reï¬ection cracking. Table 8 shows the categorization of the axles of each vehicle class based on tire conï¬guration. Categories 1, 3, 5, and 7 have single tires and the categories 2, 4, 6, and 8 have dual tires. These eight categories are used in calculating the trafï¬c stresses which are partly the cause of reï¬ection cracking. Climatic Data Collection The climatic data were collected from two principal sources in addition to the LTPP database. The hourly solar radiation and the daily air temperature and wind speed were needed to make accurate estimates of the temperature in the overlay. In addition to these data, the temperature model requires the albedo of the pavement surface, its thermal conductivity, and emissivity and absorption coefï¬cients. The solar radiation data can be obtained from the internet at METSTAT Model (MeteorologicalâStatistical Solar Model) and the SUNY Model for the State University of New York at Albany (http:// rredc.nrel.gov/solar/old data/nsrdb/). The daily climate data on air temperature and wind speed can be found at http:// www.ltpp-products.com/DataPave/. It was necessary to deve- lop a different temperature model than the one contained in the Enhanced Integrated Climatic Model (EICM) in order to cal- culate the temperatures to a higher degree of accuracy (more detail is provided in Appendix B). Although temperatures predicted with the EICM model sat- isfy pavement design needs in general, there have been some large differences when compared to measured pavement tem- perature (18). These differences are most likely caused by the assumption that heat ï¬uxes at the pavement surface are exactly balanced by conduction into the ground well below the surface; inaccuracy of climatic data (especially calculated solar radia- tion); and the assumptions of the constant temperature bound- ary condition and site-independent model parameter values. Recently, signiï¬cant improvement over the EICM model has been achieved by several groups using a similar one- dimensional heat transfer model, but with an unsteady-state surface heat ï¬ux boundary condition, measured model input data, and site-speciï¬c model parameters that were optimized based on measured pavement temperatures (19, 20, 21). Figure 8 presents a comparison of the temperatures mea- sured at different depths below the pavement surface with those calculated with the EICM model (18). Figure 9 shows a comparison between the measured tem- peratures and those calculated with the new model used in this project (the model is described later in this chapter: details are provided in Appendix B). The one dimensional heat transfer model employs an unsteady-state heat ï¬ux boundary condition at the pavement surface, a depth-independent heat ï¬ux 3 m below the surface, and the ability to estimate site-speciï¬c model parameters using known measured pavement temperatures. Finite Element Method for Calculating SIF In this project, it was found that the computational time to calculate new stress intensity factors using the ï¬nite element method at the daily location of the tip of the crack was too
15 long. Therefore a method was adopted to calculate the SIF for a wide variety of conditions, pavement structures, and crack lengths using a ï¬nite element method and then to model the computed results with the computationally efï¬cient ANN algorithm (the method used to generate these sets of SIF is presented in detail in Appendix Q). The ï¬nite element method is a two-dimensional method which uses Fourier Series to represent the effects of loads that act at some distance from the two-dimensional plane where the calculations are made (details are provided in Appendix Q). A comparison of the results obtained using this method with the results obtained using a true three-dimensional ï¬nite element program revealed differences of 2 to 5 percent. Use of a two- dimensional ï¬nite element program was ruled out because of the long computational time it required (the time required for a three-dimensional program is of course much greater). The following six basic pavement structures were used in the computations: ⢠Asphalt overlay over cracked asphalt surface; ⢠Asphalt overlay over compliant interlayer (SAMI) over cracked asphalt surface; ⢠Asphalt overlay with reinforcing geosynthetic layer over cracked asphalt surface; ⢠Asphalt overlay over jointed concrete surface; ⢠Asphalt overlay with reinforcing geosynthetic layer over jointed concrete surface; and ⢠Asphalt overlay over cracked continuously reinforced con- crete surface. The following three different loading conditions were used in the ï¬nite element calculations of the SIF at the tip of the crack: ⢠Thermal stress; ⢠Bending stress due to trafï¬c; and ⢠Shearing stress due to trafï¬c. Figure 8. Typical daily pavement temperature prediction using EICM model (18). Figure 9. Typical daily pavement temperature prediction using improved model.
16 The following two tire and axles configurations were included in the trafï¬c stress ï¬nite element computations: ⢠Single axle, single tire ⢠Single axle, dual tire Collateral studies with multiple axles showed that the SIF beneath one axle is affected by the loads of other axles at stan- dard axle spacings by no more than about 10 percent. This effect was included in the computations of the SIF for tan- dem, tridem, and quadrem axles. Table 9 lists the pavement structures and the number of computer runs performed for developing the SIF. The total number of computer runs was 94,500. The number of bending SIF computations was reduced because the bending stresses become compressive only a short distance into the overlay. The variables included in the ï¬nite element computational runs were the layer thickness, modulus of overlay, surface layer, and base course and the crack or joint spacing. In the thermal stress cases, different levels of thermal expansion coefï¬cient were used. With the jointed concrete pavement structures, dif- ferent levels of load transfer efï¬ciency were used. For those cases where a compliant interlayer (SAMI) was used, the thick- ness and modulus of that layer were also varied. In those pave- ment structures in which reinforcing geosynthetics were used, the thickness and the grid stiffness were used. Because there are no uniform industry standards for specifying the properties of these commercially available products, three levels of geosyn- thetic stiffness (high, medium, and low) were used in the com- puter runs. The appropriate level can be chosen by the user by referring to the graph in Figure 10. The user will calculate the reinforcing product stiffness (in MN-mm/m2 units) and enter the value on the graph in Figure 10 at the corresponding rein- forcing product thickness (in mm). The curved line that is clos- est to this point provides the stiffness level that should be used as input to the design program. For a geo-grid, the stiffness is computed from its geometric and material properties as Ea/s where E is the material modulus (MN/m2); a is the rib cross-sectional area, (mm2); s is the rib spacing (mm); and t is the vertical rib thickness (mm). If the reinforcing material is a sheet instead of a grid, then the overlay reinforcing stiffness in MN-mm/m2 is calculated as Et. Successful use of geo-grids as reinforcing interlayers depends upon embedding the grid within the overlay so that there is Table 9. Number of computer runs of SIF. Pavement Structures Number of Test Sections Computer Runs of Stress Intensity Factors with Varying Crack Lengths Thermal Shear Bending AC/AC OL 233 1,620 25,920 4,320 JCP/AC OL 69 14,580 25,920 4,320 AC/SAMI/AC OL 38 6,480 - - AC/GRID/AC OL 50 9,720 - - CRC/AC OL 21 1,620 - - Figure 10. Overlay reinforcing stiffness versus reinforcing thickness.
17 aggregate interpenetration through the grid from above and below to lock the grid into place. The grid will provide no rein- forcing function if there is slippage between the grid and the overlay material. For this reason, in all ï¬nite element runs the reinforcing interlayer was placed within the overlay on top of a leveling course. Method of Predicting SIF The method chosen to model the computed results from the ï¬nite element runs was the ANN algorithm. The formu- lation of an accurate and computationally efï¬cient algorithm using the ANN approach is described in the literature (22, 23, 24). The ï¬nal model is a multi-layered equation which can achieve very good ï¬ts to the original data. A total of 18 differ- ent ANN models were constructed for use in this program (see Table 10). Details of the ANN models of SIF for the dif- ferent pavement structure and thermal and trafï¬c cases is provided in Appendix F, including graphs of the computed SIF versus those predicted by the ANN algorithms. The coef- ï¬cient of determination (R2) of all of these models was above 0.99 except for one case of pure bending in an asphalt over- lay over a jointed concrete pavement. For this model, the R2 value was 0.83 because tensile stresses rarely occur in asphalt overlays due to pure bending and as a consequence there were very few data points. As noted in Table 10, ANN Models 1, 2, and 3 provide dif- ferent degrees of interlayer slip between the overlay and the underlying old asphalt surface for the thermal stress case. ANN Model 4 provides a compliant interlayer (SAMI) between the overlay and the old asphalt surface layer for the thermal stress case. Model 5 provides a no-slip condition between the overlay and the underlying old asphalt surface for the thermal stress case. Model 6 provides a no-slip condition between the overlay and an underlying jointed concrete pavement for the thermal stress case. ANN Models 7 through 12 provide the Bending SIF for a tire located directly above the crack or joint in the old pavement surface; dual and single tire models are provided. ANN Models 13 through 18 provide the Shearing and Bending SIF that occur when the leading or trailing edge of the tire is above the crack or joint in the old pavement sur- face layer; both dual and single tire models are provided. The ANN Models 8, 10, and 12 are designated by the term âOnly Positiveâ (i.e., only tensile SIF are modeled). This condition occurs when the crack is still small and in the bottom part of the overlay. Negative SIF can be calculated and occur in com- pressive stress areas where cracks will not grow. Figure 11 shows the different pavement structures and trafï¬c or thermal stress cases that apply to each ANN model. An example of the ï¬t of an ANN model to the SIF data that was calculated by the ï¬nite element program is given in Fig- ure 12 for the case of thermal stress in a HMA overlay over an old cracked asphalt pavement surface for which the R2 was 0.9982. Appendix F provides similar graphs for all of ANN models and describes the variations and ranges of overlay pave- ment structure and the material and interface properties used Model No Model Name Load Case 1 AC_over_AC_Interlayer_slip_L Thermal 2 AC_over_AC_Interlayer_slip_M Thermal 3 AC_over_AC_Interlayer_slip_H Thermal 4 AC_SC_AC Thermal 5 AC_over_AC Thermal 6 AC_over_PCC Thermal 7 Pure_Bending_AC_over_AC_Dual_Tire_Together Traffic 8 Pure_Bending_AC_over_AC_Dual_Tire_Together_Only_Positive Traffic 9 Pure_Bending_AC_over_AC_Single_Tire_Together Traffic 10 Pure_Bending_AC_over_AC_Single_Tire_Together_Only_Positive Traffic 11 Pure_Bending_AC_over_PCC_Single_Tire_Together Traffic 12 Pure_Bending_AC_over_PCC_Single_Tire_Together_Only_Positive Traffic 13 AC_Over_AC_Shearing_Bend_Part_Dual_Tire Traffic 14 AC_Over_AC_Shearing_Shear_Part_Dual_Tire Traffic 15 AC_Over_PCC_Shearing_Shear_Part_Dual_Tire Traffic 16 AC_Over_AC_Shearing_Bend_Part_Single_Tire Traffic 17 AC_Over_AC_Shearing_Shear_Part_Single_Tire Traffic 18 AC_Over_PCC_Shearing_Shear_Part_Single_Tire Traffic Table 10. Artificial neural network models for stress intensity factors.
18 Pavement Model Thermal Cases Traffic Load Cases AC Inter Layer LevelingCourse AC AC Over PCC AC Over AC and AC Mill AC AC InterLayer LevelingCourse AC-H AC InterLayer LevelingCourse AC-M AC InterLayer LevelingCourse AC-L AC Over SC Over AC Shearing Load Cases Pure Bending Load Cases Pure_Bending_ACoverAC_DualTire_Together Pure_Bending_ACoverAC_DualTire_Together (Only Positive) Pure_Bending_ACoverAC_SingleTire_Together Pure_Bending_ACoverAC_SingleTire_Together (Only Positive) Pure_Bending_ACPCC_SingleTire_Together Pure_Bending_ACPCC_SingleTire_Together (Only Positive) AC_AC_Shearing_BendPart_dualTire AC_AC_Shearing_ShearPart_dualTire AC_PCC_Shearing_ShearPart_dualTire AC_AC_Shearing_BendPart_singleTire AC_AC_Shearing_ShearPart_singleTire AC_PCC_Shearing_ShearPart_singleTire Figure 11. ANN models applications to overlaid pavement structures.
19 in developing the models. These models apply only to the vari- able ranges used as input to these models; extrapolation out- side the range of inference may not produce accurate results. Traffic Loads and Tire Footprints Tire footprints are closer to rectangles than to the com- monly assumed circular footprints (25). In this project, rec- tangular tire footprints with known tire widths were used; tire footprint length was calculated from the tire load and the inï¬ation pressure. The length of tire patch was used to eval- uate bending and shearing SIF in asphalt overlays. Also, because the tire length is proportional to the load, a cumu- lative axle load distribution on tire length for each category may be determined, based on collected traffic data such as WIM or AADTT. Tire Patch Length The tire-load model that assumes a rectangular tire contact area (as shown in Figure 13) was used to evaluate the effect of tire load on reï¬ection cracking. Tire width is assumed to be constant within each trafï¬c category (vehicle class and axle type) even under different tire pressures. Thus, the tire length can be calculated as follows: Tire Length (in.) tire load (lb) tire pressu = re lb in. tire width (in.) 2 âââ ââ â à ( )3 Determination of the Effect of Cumulative Axle Load Distribution on Tire Length Because of the difï¬culty of employing each tire length for axle load intervals to evaluate trafï¬c load effects on propaga- tion of reï¬ection cracking, the effect of the axle load distribu- tion on the tire patch length for each category was used for the evaluation of trafï¬c load. The axle load distribution inter- vals can be converted into tire length intervals using the char- acteristics of each axle type presented in Table 11. The tire patch lengths of corresponding axle load intervals for each category can be calculated using Equation 3 and the characteristics of axle types. Table 11 lists the calculated axle load intervals for all trafï¬c categories, and Table 12 lists the tire patch length increments. Using the tire patch length and collected trafï¬c data, the cumulative axle load distribution can be determined for each category. Figure 14 illustrates the procedure for determining tire length and the cumulative axle load distribution (CALD) of each category. Such distribution should be produced for all eight trafï¬c categories to account for all types of vehicles and axles. Figure 15 shows the cumulative axle load distribution of tire load for Category 1 of LTPP section 180901 in 2004, which was determined using data in Table 12. Modeling of Cumulative Axle Load Distribution Since the frequency distribution of each tire length of a load category is used to evaluate load effects for reï¬ection cracking propagation, the cumulative axle load distribution (CALD) of pavement sections and trafï¬c categories should be developed along with the tire length. The CALD of trafï¬c loads or tire lengths follows a sigmoidal curve having a lower asymptote of zero and a ï¬nite upper asymptote as shown in Figure 16. (Details of the modeling process are provided in Appendix D). After reviewing potential models that describe the statisti- cal properties of the cumulative axle load distribution versus tire length, the Gompertz model presented in Equation 4, was chosen. where α, β, and γ are model parameters. y x= â â( )[ ]α β γexp exp ( )4 Figure 12. Comparison of SIF with ANN model predictions for HMA overlays over a cracked asphalt pavement surface layer. Tire Length (L) Width (W ) Tire Pressure (p) Figure 13. Tire load applied to pavement surface.
20 The Gompertz model is appropriate because it has a clear physical boundary condition which shows asymptotes at y = 0 and y = α and is asymmetric about its inï¬ection point which occurs at β/γ (26). The parameter α in the model indicates the upper asymptote which is equal to 1.00 (100%) for the cumu- lative axle load distribution curve. The parameter β describes how wide the rising portion of the curve is. In addition, the parameter γ indicates the slope of the cumulative axle load distribution curve. Figure 17 illustrates a typical curve of the Gompertz model. The parameter α should be equal to 1.00 because the cumulative axle load distribution curve has a physical bound- ary condition ranging from 0 to 1.00 (i.e., 0 to 100 percent). Therefore, the modiï¬ed model for cumulative axle load dis- tribution is: where Lij = ith tire length in tire patch length increment in trafï¬c category j; C(Li)j = cumulative axle load distribution factor at Li within trafï¬c category j; and β, γ = model parameters describing the curve width and slope, respectively. The collected trafï¬c data from WIM or AADTT for a given section were used to develop the model parameters β and γ in the modiï¬ed Gompertz model of Equation 5. The results pro- vided a good ï¬t of the data along with relatively high signiï¬- cance. Table 13 lists the developed model parameters β and γ for the trafï¬c category 1 of LTPP Section 180901. Typical model parameters for each trafï¬c category are presented in Table 14. Figure 18 shows a plot of the calibrated model data and the corresponding measured trafï¬c data for the LTPP section. C L Li j ij( ) = â â( )[ ]exp exp ( )β γ 5 The distribution factor of C1 represents the minimum axle load (tire length) to be considered for load related distress. The lower limits of axle load and tire length are presented in Table 15. C2 is the factor at which the cumulative distribution reaches 100 percent ï¬rst. L1 and L2 are the tire lengths corre- sponding to C1 and C2, respectively. The model parameters and plots of calibrated cumulative axle load distribution versus tire lengths for all categories of Section 180901 are provided in Appendix A. For Level 1 data inputs, the model parameters for the cumulative axle load distribution can be computed using WIM data for each category, while the default values for Level 3 input are provided. The default model parameters, shown in Table 14, were prepared using trafï¬c data from the LTPP database. Also, Table 16 presents the default CALD values which were determined based on the model parameter default values. Determination of Hourly Number of Axles In order to analyze reflection cracking propagation caused by bending or shearing, the hourly number of axles should be considered in each of the tire length increments within each traffic category. The number of axles can be calculated from the probability density which is deter- mined based on the cumulative axle load distribution of tire lengths in each category (details of the process of determining the hourly traffic distribution are provided in Appendix E). Probability Density on Tire Patch Length The probability density of the tire patch length is the fre- quency distribution of each tire length in a category, which is required to determine the number of traffic loads during Category Axle Type Tires Tire Width (in.) Tire Pressure (PSI) Axle Load Interval (lb) 1 Single Single 7.874 40 (< 6,000 lb) 120 (> 6,000 lb) 3,000 ~ 40,000 lb at 1,000 lb intervals 2 Dual 8.740 3 Tandem Single 7.874 120 6,000 ~ 80,000 lb at 2,000 lb intervals 4 Dual 8.740 120 5 Tridem Single 7.874 120 12,000 ~ 102,000 lb at 3,000 lb intervals 6 Dual 8.740 120 7 Quadrem Single 7.874 120 12,000 ~ 102,000 lb at 3,000 lb intervals 8 Dual 8.740 120 Table 11. Typical characteristics for axle types (24).
21 Tire Patch Length Traffic Category 1 2 3 4 5 6 7 8 1 3.704 1.669 1.588 0.715 2.117 0.953 1.588 0.715 2 4.233 1.907 2.117 0.953 2.646 1.192 1.984 0.894 3 4.763 2.145 2.646 1.192 3.175 1.430 2.381 1.073 4 5.292 2.384 3.175 1.430 3.704 1.669 2.778 1.251 5 5.821 2.622 3.704 1.669 4.233 1.907 3.175 1.430 6 6.350 2.860 4.233 1.907 4.763 2.145 3.572 1.609 7 6.879 3.099 4.763 2.145 5.292 2.384 3.969 1.788 8 7.408 3.337 5.292 2.384 5.821 2.622 4.366 1.967 9 7.938 3.576 5.821 2.622 6.350 2.860 4.763 2.145 10 8.467 3.814 6.350 2.860 6.879 3.099 5.159 2.324 11 8.996 4.052 6.879 3.099 7.408 3.337 5.556 2.503 12 9.525 4.291 7.408 3.337 7.938 3.576 5.953 2.682 13 10.054 4.529 7.938 3.576 8.467 3.814 6.350 2.860 14 10.583 4.767 8.467 3.814 8.996 4.052 6.747 3.039 15 11.113 5.006 8.996 4.052 9.525 4.291 7.144 3.218 16 11.642 5.244 9.525 4.291 10.054 4.529 7.541 3.397 17 12.171 5.482 10.054 4.529 10.583 4.767 7.938 3.576 18 12.700 5.721 10.583 4.767 11.113 5.006 8.334 3.754 19 13.229 5.959 11.113 5.006 11.642 5.244 8.731 3.933 20 13.758 6.198 11.642 5.244 12.171 5.482 9.128 4.112 21 14.288 6.436 12.171 5.482 12.700 5.721 9.525 4.291 22 14.817 6.674 12.700 5.721 13.229 5.959 9.922 4.469 23 15.346 6.913 13.229 5.959 13.758 6.198 10.319 4.648 24 15.875 7.151 13.758 6.198 14.288 6.436 10.716 4.827 25 16.404 7.389 14.288 6.436 14.817 6.674 11.113 5.006 26 16.933 7.628 14.817 6.674 15.346 6.913 11.509 5.184 27 17.463 7.866 15.346 6.913 15.875 7.151 11.906 5.363 28 17.992 8.105 15.875 7.151 16.404 7.389 12.303 5.542 29 18.521 8.343 16.404 7.389 16.933 7.628 12.700 5.721 30 19.050 8.581 16.933 7.628 17.463 7.866 13.097 5.900 31 19.579 8.820 17.463 7.866 17.992 8.105 13.494 6.078 32 20.108 9.058 17.992 8.105 18.521 8.343 13.891 6.257 33 20.638 9.296 18.521 8.343 19.050 8.581 14.288 6.436 34 21.167 9.535 19.050 8.581 19.579 8.820 14.684 6.615 35 - - 19.579 8.820 20.108 9.058 15.081 6.793 36 - - 20.108 9.058 20.638 9.296 15.478 6.972 37 - - 20.638 9.296 21.167 9.535 15.875 7.151 38 - - 21.167 9.535 - - - - Table 12. Tire patch length increment for traffic categories (inches). each hour of each day. The number of traffic loads for each 1-hour time period in each day for eight traffic categories and tire length increments is used to calculate the bending or shearing stress intensity factor. The probability density of tire patch lengths for each traffic category can be deter- mined from the cumulative axle load distribution function as follows: P L dC L dL j j j ( ) = ( ) ( )6 where P (Lj) is the probability density function within trafï¬c category j and C (Lj) is the cumulative probability within traf- ï¬c category j. For instance, the probability density function for the Cat- egory 1 of LTPP Section 180901 can be determined, based on the cumulative axle load distribution of the section provided in Figure 18; results are shown in Figure 19. The probability density for all categories of the LTPP sections is provided in Appendix E. The default probability density for Level 3 data input is presented in Table 17.
22 Collecting the axle load interval for each category Axle load (lb) Number of tires Tire length (in.) tire load (lb) tire pressure (lb/in.2) à tire width (in.) Collecting the number of axle loads for each category from WIM or AADTT Axle load distribution factor No. of alxe loads for each tire length Total No. of axle loads Cumulative axle load distribution Tire load (lb) = Cumulative Axle Load Distribution on Tire Length Tire Length Cumulative Axle Load Distribution = = Figure 14. Determination of cumulative axle load distribution on tire patch length. 0.00 0.20 0.40 0.60 0.80 1.00 0 3 6 9 12 15 18 21 Tire Length (in.) Cu m ul at iv e Ax le L oa d Di st rib ut io n Figure 15. Cumulative axle load distribution versus tire length (Category 1 of LTPP section 180901 in 2004). P2 = 1 P1 L1 L2 Pi = f (Li) Minimum load to be considered Maximum load Tire Length Cu m ul at iv e A xl e Lo ad D ist rib ut io n Figure 16. Typical tire length versus cumulative axle load distribution. X α (=1.0 for CALD curve) Y e-1 β1 γ γ β2 Figure 17. Gompertz model curve.
23 Table 13. Model parameters and CALD on tire length (Category 1 of LTPP section 180901 in 2004). Parameter Values CALD Value Tire Length (in.) 4.301 0.071 3.704 0.967 1.000 C1 C2 L1 L2 16.933 0.982 β γ R2 Traffic Category Parameters 1 3.44056 0.73836 β γ 0.980 R2 2 3.58353 1.61999 0.999 3 1.62387 0.48959 0.972 4 2.03042 1.04234 0.990 5 1.72904 1.10906 0.906 6 1.92533 1.02297 0.982 7 1.47412 0.98443 0.969 8 2.70840 1.48446 0.956 Table 14. CALD model parameter default values determined based on LTPP data. 0.00 0.20 0.40 0.60 0.80 1.00 0 3 6 9 12 15 18 21 Tire Length (in.) Cu m ul at iv e Ax le L oa d Di st rib ut io n Measured Model P2 P1 L1 L2 Figure 18. Cumulative axle load distribution versus tire length (Category 1 of LTPP Section 180901 in 2004).
Traffic Category Axle Type Minimum Values Axle load (lb) Tire Length (in.) 1 Single 3,000 3.704 2 1.669 3 Tandem 6,000 1.588 4 0.715 5 Tridem 12,000 2.117 6 0.953 7 Quad 12,000 1.588 8 0.715 Table 15. Minimum values to be considered for load related distress. No.* Category 1 2 3 4 5 6 7 8 1 0.1320 0.0896 0.0971 0.0269 0.5835 0.0754 0.4005 0.0056 2 0.2541 0.1941 0.1654 0.0596 0.7411 0.1318 0.5384 0.0187 3 0.3958 0.3282 0.2494 0.1109 0.8465 0.2044 0.6578 0.0472 4 0.5341 0.4689 0.3424 0.1799 0.9115 0.2882 0.7532 0.0962 5 0.6542 0.5977 0.4373 0.2624 0.9498 0.3772 0.8255 0.1660 6 0.7505 0.7048 0.5281 0.3522 0.9718 0.4658 0.8783 0.2523 7 0.8235 0.7884 0.6110 0.4431 0.9842 0.5496 0.9160 0.3478 8 0.8769 0.8508 0.6837 0.5300 0.9912 0.6256 0.9423 0.4449 9 0.9149 0.8960 0.7457 0.6094 0.9951 0.6924 0.9606 0.5373 10 0.9416 0.9281 0.7973 0.6796 0.9973 0.7497 0.9732 0.6210 11 0.9601 0.9505 0.8396 0.7398 0.9985 0.7979 0.9818 0.6940 12 0.9728 0.9661 0.8738 0.7905 0.9992 0.8379 0.9876 0.7557 13 0.9815 0.9768 0.9011 0.8325 0.9995 0.8706 0.9916 0.8067 14 0.9875 0.9842 0.9228 0.8668 0.9997 0.8971 0.9943 0.8481 15 0.9915 0.9892 0.9399 0.8945 0.9999 0.9184 0.9962 0.8813 16 0.9942 0.9927 0.9533 0.9167 0.9999 0.9355 0.9974 0.9076 17 0.9961 0.9950 0.9637 0.9344 1.0000 0.9491 0.9982 0.9284 18 0.9974 0.9966 0.9719 0.9484 1.0000 0.9599 0.9988 0.9446 19 0.9982 0.9977 0.9782 0.9596 1.0000 0.9684 0.9992 0.9572 20 0.9988 0.9984 0.9832 0.9683 1.0000 0.9752 0.9995 0.9670 21 0.9992 0.9989 0.9870 0.9752 1.0000 0.9805 0.9996 0.9746 22 0.9994 0.9993 0.9899 0.9806 1.0000 0.9847 0.9997 0.9805 23 0.9996 0.9995 0.9922 0.9848 1.0000 0.9880 0.9998 0.9850 24 0.9997 0.9997 0.9940 0.9882 1.0000 0.9906 0.9999 0.9885 25 0.9998 0.9998 0.9954 0.9907 1.0000 0.9926 0.9999 0.9911 26 0.9999 0.9998 0.9964 0.9928 1.0000 0.9942 0.9999 0.9932 27 0.9999 0.9999 0.9972 0.9944 1.0000 0.9954 1.0000 0.9948 28 0.9999 0.9999 0.9979 0.9956 1.0000 0.9964 1.0000 0.9960 29 1.0000 1.0000 0.9984 0.9966 1.0000 0.9972 1.0000 0.9969 30 1.0000 1.0000 0.9987 0.9973 1.0000 0.9978 1.0000 0.9976 31 1.0000 1.0000 0.9990 0.9979 1.0000 0.9983 1.0000 0.9982 32 1.0000 1.0000 0.9992 0.9984 1.0000 0.9987 1.0000 0.9986 33 1.0000 1.0000 0.9994 0.9987 1.0000 0.9989 1.0000 0.9989 34 1.0000 1.0000 0.9995 0.9990 1.0000 0.9992 1.0000 0.9992 35 - - 0.9997 0.9992 1.0000 0.9994 1.0000 0.9994 36 - - 0.9997 0.9994 1.0000 0.9995 1.0000 0.9995 37 - - 0.9998 0.9995 1.0000 1.0000 1.0000 1.0000 38 - - 1.0000 1.0000 - - - - * Number represents the tire patch length increment listed in Table 12. Table 16. Default cumulative axle load distribution for each traffic category.
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.0 0 4.2 3 5.2 9 6.3 5 7.4 1 8.4 7 9.5 3 10 .58 11 .64 12 .70 13 . 76 14 . 82 15 .88 16 .93 17 .99 19 .05 20 .11 21 .17 Tire Length (in.) Pr ob ab ili ty D en si ty Figure 19. Probability density function of tire length (Category 1, LTPP Section 180901). No.* Traffic Category 1 2 3 4 5 6 7 8 1 0.1974 0.3502 0.1109 0.1014 0.3486 0.1993 0.3608 0.0429 2 0.2570 0.5155 0.1457 0.1753 0.2462 0.2733 0.3282 0.1103 3 0.2709 0.5924 0.1696 0.2542 0.1564 0.3320 0.2713 0.2140 4 0.2473 0.5753 0.1797 0.3216 0.0936 0.3668 0.2102 0.3344 5 0.2050 0.4984 0.1771 0.3659 0.0543 0.3762 0.1558 0.4426 6 0.1591 0.3994 0.1651 0.3831 0.0309 0.3640 0.1122 0.5158 7 0.1181 0.3037 0.1474 0.3759 0.0174 0.3365 0.0792 0.5453 8 0.0851 0.2227 0.1273 0.3508 0.0097 0.3002 0.055 1 0.5349 9 0.0601 0.1594 0.1071 0.3146 0.0054 0.2604 0.0380 0.4954 10 0.0418 0.1122 0.0884 0.2736 0.0030 0.2209 0.0260 0.4392 11 0.0289 0.0781 0.0719 0.2324 0.0017 0.1842 0.0178 0.3763 12 0.0198 0.0540 0.0577 0.1937 0.0009 0.1516 0.0121 0.3143 13 0.0135 0.0371 0.0459 0.1591 0.0005 0.1234 0.0082 0.2573 14 0.0092 0.0254 0.0363 0.1292 0.0003 0.0997 0.0056 0.2074 15 0.0062 0.0174 0.0285 0.1040 0.0002 0.0800 0.0038 0.1653 16 0.0042 0.0118 0.0223 0.0831 0.0001 0.0638 0.0026 0.1306 17 0.0029 0.0081 0.0174 0.0661 0.0000 0.0507 0.0017 0.1024 18 0.0019 0.0055 0.0136 0.0523 0.0000 0.0402 0.0012 0.0799 19 0.0013 0.0037 0.0105 0.0413 0.0000 0.0318 0.0008 0.0621 20 0.0009 0.0025 0.0082 0.0325 0.0000 0.0251 0.0005 0.0481 21 0.0006 0.0017 0.0063 0.0255 0.0000 0 .0198 0.0004 0.0372 22 0.0004 0.0012 0.0049 0.0200 0.0000 0.0156 0.0002 0.0287 23 0.0003 0.0008 0.0038 0.0157 0.0000 0.0122 0.0002 0.0221 24 0.0002 0.0005 0.0029 0.0123 0.0000 0.0096 0.0001 0.0170 25 0.0001 0.0004 0.0023 0.0096 0.0000 0.0075 0.0001 0.0131 26 0.0001 0.0003 0.0018 0.0075 0.0000 0.0059 0.0001 0.0101 27 0.0001 0.0002 0.0014 0.0059 0.0000 0.0046 0.0000 0.0077 28 0.0000 0.0001 0.0010 0.0046 0.0000 0.0036 0.0000 0.0059 29 0.0000 0.0001 0.0008 0.0036 0.0000 0.0029 0.0000 0.0046 30 0.0000 0 .0001 0.0006 0.0028 0.0000 0.0022 0.0000 0.0035 31 0.0000 0.0000 0.0005 0.0022 0.0000 0.0018 0.0000 0.0027 32 0.0000 0.0000 0.0004 0.0017 0.0000 0.0014 0.0000 0.0021 33 0.0000 0.0000 0.0003 0.0013 0.0000 0.0011 0.0000 0.0016 34 0.0000 0.0000 0.0002 0.0 010 0.0000 0.0008 0.0000 0.0012 35 0.0002 0.0008 0.0000 0.0007 0.0000 0.0008 36 0.0001 0.0005 0.0000 0.0004 0.0000 0.0005 37 0.0001 0.0004 0.0000 0.0000 0.0000 0.0000 38 0.0000 0.0000 * Number represents the tire patch length increment listed in Table 12. Table 17. Default probability density for each traffic category.
26 Reflection Cracking Amount and Severity Model The development of reï¬ection crack amount and severity versus time or number of load repetitions follows a sigmoidal curve having a ï¬nite upper asymptote for three different sever- ity levels, as shown in Figure 20. Jayawickrama and Lytton (12) proposed an s-shaped empir- ical model to describe the amount and severity development of reï¬ection cracking on an asphalt overlay. This model describes the reï¬ection cracking versus the number of load repetition or time relationship as follows: where RFAS = reï¬ection cracking amount and severity, ranging from 0 to 100%; DTotal = total number of days since the overlay construc- tion was completed; and Ï and β = calibration parameters for each severity level. The parameter Ï is the scale factor for reï¬ection cracking amount and severity. A large Ï value indicates that much accu- mulated damage must occur to reach a given level of reï¬ection cracking amount; that is, the parameter Ï describes the spread of the rising portion of the curve, the parameter Ï is always equal to the total number of days to reach 36.8 percent (= 1/e) of the total amount of expected reï¬ection cracking. The param- eter β is the shape factor that describes how steep the rising por- tion of the curve is as shown in Figure 21 (more details on this amount and severity model are provided in Appendix K). RFAS e DTotal= â âââ ââ â100 7i Ï Î² ( ) Because the reï¬ection cracking severity and amount model has a clear physical boundary condition (0 to 100 percent) and the parameters, Ï and β, have physical meanings, the model was selected for the calibration with different ï¬eld data sets. However, because the database used for model calibra- tion consisted of the length of observed transverse cracks when the survey was performed before and after overlay, the variables of the original reï¬ection cracking model were mod- iï¬ed as follows: where D(Ni) = percent of reï¬ection crack length of maximum crack length at N; i = ith crack observation; Ni = number of days after overlay. The percent of reflection crack length, D(Ni), at each observation was calculated by dividing the observed length of transverse crack, after overlay construction, by the total length of transverse crack on an existing pavement surface just before overlay construction. The total crack length on an existing surface can be described as the likelihood of maximum reflection cracking length on an overlay surface. The number of days after overlay, Ni, is determined by counting the days after overlay construction when a given set of observations were made. Based on the field data obtained from test sections, the parameters Ï and β were calibrated for different severity levels. Three sets of param- eters were calibrated when all three severity levels were available: D N ei Ni( )( ) = â âââ ââ â% ( ) Ï Î² 8 0 10 20 30 40 50 60 70 80 90 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Time (Days) Cr ac k Le ng th o r A re a (% ) Low+Medium+High (L+M+H) Severity Medium+High (M+H) Severity High (H) Severity Figure 20. Typical development of reflection crack by severity levels.
27 0 20 40 60 80 100 - 2,000 4,000 6,000 8,000 10,000 No. of Days % C ra ck L en gt h β1 < 1.0 e -1 = 36.8% Ï1 Ï2 Ï3 β2 = 1.0 β3 > 1.0 ⢠ÏH and βH for high severity level, ⢠ÏH and βH for medium and high severity levels, and ⢠ÏLMH and βLMH for low, medium, and high severity levels. The reï¬ection cracking models were calibrated using ï¬eld data for overlay test sections which have sufï¬cient crack measurements. Figure 22 shows the crack amounts expressed as percentages of the original crack length observed in LTPP section 270506, at three severity conditions: high; high or medium; and high, medium, or low severity. The same type of data was obtained for the test sections in New York City (27); example is shown in Figure 23. These sections included different types of geosynthetic materials reinforcing the overlays. Another set of reï¬ection cracking data was found in the geosynthetically reinforced test sections in Texas, an example of which is shown in Figure 24 (28). These sections included HMA overlays over cracked asphalt surface layers in Amarillo and HMA overlays over jointed concrete pavements in Mar- lin. Geosynthetic materials were used to reinforce all of the overlays in both of these sites. Reï¬ection cracking data for the LTPP sites, New York test sections, and Texas test sections are presented in Appendix J. Calibration of Field Reflection Cracking Model One objective of the study was to calibrate the reflection cracking amount and severity model with reflection crack data observed in the field. The calibration refers to the mathematical process through which the total error or dif- ference between observed and predicted values of distress 0 10 20 30 40 50 60 70 80 90 100 0 1000 2000 3000 4000 5000 No. of Days % C ar ck L en gt h L+M+H M+H H Severity Level Figure 22. Reflection cracking amount versus number of days for LTPP section 270506. Figure 21. Parameters in reflection cracking severity model.
is minimized. The process used to achieve the calibration, which determines Ï and β in the reflection cracking model, was conducted using available field reflection cracking data and an iterative method of the System Identification process (details of the calibration process are presented in Appendix L). System Identification Process The reflection cracking amount and severity model at a given severity level was considered to have been calibrated when the error between observed and predicted crack lengths was minimized. Since the predicted crack length is calcu- lated by the calibrated model at each test section, a solution method was required to determine the parameters Ï and β in the model. In this study, the system identification process was used. The purpose of the system identiï¬cation process is to develop a mathematical model which describes the behavior of a sys- tem (real physical process). The actual system and the mathe- matical model are identiï¬ed when the error between them is minimized or satisï¬es the error criteria; otherwise, the model 28 % R ef le ct iv e C ra ck L en gt h Figure 23. Reflection cracking amount data from test sections in New York City. 0 10 20 30 40 50 60 70 80 90 100 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Age, days % R ef le ct iv e C ra ck L en gt h Control GlassGrid Pavetrack Hatelit Petrogrid Pavedry Stargrid Add 1"HMA HotinPlace PFC w/ LevelUp PFC w/ SealCoat Figure 24. Low severity reflection cracking amount data in Amarillo, Texas.
should be adjusted until the error is reduced sufï¬ciently (29). The system identiï¬cation process considers three different error minimization models depending on the choice of residuals combined with the model: forward model, inverse model, and generalized model shown in Figure 25. The forward approach minimizes the output error between the model and the system while using the same input. In the inverse approach, the input error is minimized based on the same output. The generalized model is a combination of the forward and inverse approach when the model is invertible (29). When the system output is ï¬xed because it is observed or obtained from an actual system, the output from the model must be reï¬ned to calibrate the mathematical model includ- ing its parameters. In this project, the reï¬ection cracking amount and severity model (mathematical model) was cali- brated based on observed reï¬ection crack data (actual system output) to produce the predicted crack data (model output) which is close to the observed crack data. An optimal model for the physical system is obtained when the output error between the system and the model is small enough to meet an error criterion. However, if the error does not meet the criterion, the parameters in the mathematical model should be corrected by a parameter adjustment and adaptation algorithm. The correction process is performed iteratively until the error becomes small enough to meet the error criterion. Figure 26 depicts use of the forward model in the system identiï¬cation process including the parameter adjustment and adaptation algorithm for the reï¬ection crack- ing model calibration. Parameter Adjustment and Adaption Algorithm A parameter adjustment and adaptation algorithm was developed based on the Taylor series expansion as follows (30) where [Fki] = sensitivity matrix = m, n = number of output data and model parameters, respectively; fk = mathematical model; pi = model parameters; {αi} = change vector (relative change of parameters) = [α1, α2, . . . , αn]T; and {rk} = residual vector (error between system and model outputs) = [r1, r2, . . . , rm]T. â â Ã( )== ââ f p p f m nk i i ki n k m 11 matrix ; F rki i k[ ]{ } = { }α ( )9 29 SYSTEM MODEL: M OUTPUT ERROR + â SYSTEM INVERSE MODEL: M-1 INPUT ERROR + â SYSTEM +âM1 M2-1 GENERALIZED ERROR (a) Forward Model (b) Inverse Model (c) Generalized Model Figure 25. Methods for system identification process (29). Fields Reflection Cracking Model Parameter Adjustment and Adaption Algorithm Output from System Output from Model Output error Parameters of Model (Ï, β)Good No Good System Model (Observed Cracks) (Predicted Cracks) Figure 26. Scheme of system identification process.
The minimization of error contained within the residual vector {rk} is analogous to the reduction of error employed in least squared error analysis. The squared error between actual output and predicted output is calculated by using a mathematical model to determine the sensitivity of the weighting parameters for allocating the squared error. It is possible to adjust the model parameters until there is no squared error remaining; however, because of the pres- ence of random error, the values in the residual matrix {rk} should not be forced to zero (31). The elements in the residual vector {rk} which represents errors between the actual and model outputs are determined based on model parameters pi assuming at each iteration in the process, that they are known values. The sensitivity matrix [Fki], which reflects the sensitivity of the output from mathematical model fk to the assumed parameters pi, is also a known value. Therefore, the unknown change vector {αi} presents the relative changes of the model parameters and is the tar- get vector to be determined in the process. Equation 9 can be rewritten as: The change vector {αi} obtained using an initial assumption of parameters, can then be used to determine the following new set of parameters. Where j is the iteration count. Solutions for the parameters in the model are found by min- imizing the change vector {αi}. Therefore, the iteration process using Equation 10 is continued until there is no squared error remaining or the desired convergence is reached. Thus, the iteration was repeated until the elements in the change vector {αi} are less than 0.01. Calibrating Reflection Cracking Model of Test Sections Based on the system identiï¬cation and the parameter adjustment algorithm, the reï¬ection cracking models were calibrated using the data obtained from LTPP, New York City, and Texas asphalt overlay test sections. The process was used to ï¬t the predicted crack length to the measured crack length by iteration. Thus, the parameter adjustment algo- rithm of Equation 9 can be expressed as follows: F r[ ]{ } = { }α ( )12 p pi j i j i + = +( )1 1 0 6 11. ( )α αi kiT ki ki T kF F F r{ } = [ ] [ ] { }â1 10( ) where D â (Ni) = crack length at Ni, calculated using Ïj and βj, and D(Ni) = measured crack length at Ni. The parameters Ï and β in the model were determined by iteration. The convergence criterion was set to 1.0 percent (i.e., the iteration was repeated until the elements in the change vector were less than 0.01). The percent crack length at each of the pavement ages was used to develop the model parameters Ï and β in the reflection cracking model along with the system identifica- tion process. Table 18 presents an example of the percent of reflection cracking development of all severity levels of four LTPP sections, which were calculated based on reflec- tion crack data in Table 18. Table 19 shows the developed model parameters, and Figures 27 through 30 present plots of the compiled and measured data for four LTPP sections. The calibrated parameters for all asphalt overlay test sec- tions in the LTPP, New York City, and Texas are listed in Appendix M. The model parameters Ï and β for the three levels of distress are the âfield dataâ which will be calibrated to the number of days for a crack to propagate through the overlay computed with the reflection cracking model. The coefficients by which the different modes of crack propa- gation relate to these field derived model parameters are the âcalibration coefficientsâ which define a particular application (pavement structure, climatic zone, region) of the reflection cracking model. The Ï and β calibration coef- ficients for all of the overlaid sections are tabulated in Appendix M. â ( ) â ( ) â ( ) â ( ) â ( ) â D N D N D N D N D N j j j j j j 1 1 1 1 2 Ï Ï Î² β Ï Ï D N D N D N D N D N D N j j i j j i 2 2 2( ) â ( ) â ( ) â ( ) â ( ) â β β Ï Ï i j j i j D N ( ) â ( ) ⡠⣠â¢â¢â¢â¢â¢â¢â¢â¢â¢â¢ ⤠⦠â¥â¥â¥â¥â¥â¥â¥â¥â¥â¥ â + β β Ï 1 Ï Ï Î² β β j j j j j D N D N D N + â ⡠⣠â¢â¢â¢â¢ ⤠⦠â¥â¥â¥â¥ = ( )â ( ) ( ) 1 1 1 1 D N D N D N D N D N D N i i i 2 2 2 ( )â ( ) ( ) ( )â ( ) ( ) ⡠⣠â¢â¢â¢â¢â¢â¢â¢ â¢â¢â¢ ⤠⦠â¥â¥â¥â¥â¥â¥â¥â¥â¥â¥ ( )13 30
31 Table 18. Reflection cracking development of L+M+H for LTPP test sections. Section No. Overlay Type Number of Days after Overlay % Crack Length 340503 AC/AC OL 0 0 1,200 0 1,492 0 2,283 9.86 2,640 8.28 3,004 20.41 3,367 28.12 3,757 29.59 4,121 45.92 4,247 55.78 270506 AC/Mill/AC OL 0 0 52 0 641 28.76 1,110 30.84 1,803 46.60 2,595 64.55 3,183 67.02 3,600 66 .46 3,992 66.07 240563 AC/FC/AC OL 0 0 1,226 20.60 1,799 65.76 2,590 70.47 2,990 64.76 3,375 81.89 3,802 88.59 4,012 92.06 4,395 93.30 55B901 JRC/AC OL 0 0 114 3.91 358 37.32 869 75.44 2,499 93.86 3,760 95.14 4,410 99 .12 Table 19. Calibrated model parameters of LTPP sections. LTPP Section No. Overlay Type Model Parameters (L+M+H) 340503 AC/AC 2.365 3,617.12 β Ï 270506 AC/Mill/AC 0.702 1,004.85 240563 AC/FC/AC 2.276 1461.25 55B901 JRC/AC 1.159 329.42
32 0 10 20 30 40 50 60 70 80 90 100 0 1000 2000 3000 4000 5000 No. of Days % C ra ck L en gt h Measured Data Model Figure 27. Computed and measured reflection crack for LTPP section 340503. Measured Data Model 0 10 20 30 40 50 60 70 80 90 100 0 1000 2000 3000 4000 5000 No. of Days % C ra ck L en gt h Figure 28. Computed and measured reflection crack for LTPP section 270506. Measured Data Model 0 10 20 30 40 50 60 70 80 90 100 0 1000 2000 3000 4000 5000 No. of Days % C ra ck L en gt h Figure 29. Computed and measured reflection crack for LTPP section 240563.
Prediction of Temperature in a HMA Overlay A new temperature model was developed to better predict temperature variations with depth within the overlay. The model differs somewhat from the model in the EICM (32, 33, 34) that is used in the MEPDG. A comparison of the temper- atures in a pavement surface as measured and as calculated by the EICM model is shown in Figure 8. Figure 9 shows a com- parison between the measured temperatures and as calcu- lated with the new model. The ï¬gures illustrate that the new model matches the measured temperatures more closely than the EICM model. The new one-dimensional model was devel- oped based on radiation and conduction energy balance fun- damentals (details of the model are presented in Appendix B). The heat transfer process is depicted in Figure 31. Sources of heat transfer at the pavement surface are solar radiation and reï¬ection of the solar radiation at the surface by a fraction αË, the albedo; absorption of atmospheric down-welling long- wave radiation by the pavement surface; emission by long- wave radiation to the atmosphere; convective heat transfer between pavement surface; and the air close to the surface, which is enhanced by wind. Below the surface and within the pavement and ground beneath it, heat is transferred by con- duction. Not included in this model is heat transfer enhance- ment by precipitation. Mathematical details of this model follow. Heat Transfer in Pavement Heat transfer in the pavement is governed by the classical thermal diffusion equation where T = the pavement temperature as a function of time and depth below the surface (x); â â = â â T t T x α 2 2 14( ) 33 Measured Data Model 0 10 20 30 40 50 60 70 80 90 100 0 1000 2000 3000 4000 5000 No. of Days % C ra ck L en gt h Figure 30. Computed and measured reflection crack for LTPP section 55B901. Atmospheric downwelling long-wave radiation (Qa) Outgoing long-wave radiation (Qr) Heat convection by wind (Qc) Solar radiation (Qs) Pavement Heat conduction Figure 31. Schematic presentation of heat transfer model of pavement.
α = the thermal diffusivity; α = k/ÏC where k is the thermal conductivity; Ï = the density; and C = the pavement heat capacity. Together with this equation, a ï¬ux boundary condition at the pavement surface and a second ï¬ux condition at 3 m below the surface are considered. The Surface Boundary Condition Considering a differential element of the pavement surface, its thermal energy (temperature) will change to the extent that the ï¬uxes from above and from below do not balance. The various ï¬uxes shown in Figure 31 lead to the following sur- face condition: where ÏC = volumetric heat capacity of the pavement; T5 = pavement surface temperature; x = the depth below the pavement surface; = the (differential) pavement thickness for the energy balance; Qs = heat ï¬ux due to solar radiation; Î±Ë = albedo of pavement surface, the fraction of reï¬ected solar radiation; Qa = down-welling long-wave radiation heat ï¬ux from the atmosphere; Qr = outgoing long-wave radiation heat ï¬ux from the pavement surface; Qc = the convective heat ï¬ux between the surface and the air; and Qf = the heat ï¬ux within the pavement at the pavement surface. The incoming and outgoing long-wave radiation (in W m-2) are calculated by: where a = absorption coefï¬cient of pavement; = emission coefï¬cient of pavement; Ts = pavement surface temperature, k; Ta = air temperature, k; and Ï = 5.68 à 10â8W mâ2Kâ4 is Stefan-Boltzman constant Q Tr s= Ï 4 17( ) Q Ta a a= Ï 4 16( ) Îx 2 Ï Î±C x T t Q Q Q Q Q Qs s s a r c f Î 2 15 â â = â + â â â i ( ) The convective heat ï¬ux (in W m-2) is calculated as: where hc is the heat transfer coefï¬cient from the empirical equation (33). where U is the hourly wind speed (i m sâ1) and a and d are two-dimensionless empirical parameters. The heat ï¬ux within the pavement at the surface is expressed by Fourierâs equation: where Ts is pavement surface temperature, and k is thermal conductivity of asphalt concrete (in W mâ1 kâ1). Combining these results, the following equation serves as the surface boundary condition: The Bottom Boundary Condition Commonly, a constant-temperature boundary condition at some distance below the surface is reported. For example, Hermansson (19) used the annual mean temperature 5 m below the surface as a bottom boundary condition and Gui (20) used a measured temperature of 33.5°C at a depth of 3 m as the boundary condition. In the EICM model, temper- atures were measured from water wells across the country at a depth of 10 to 18 m, from which an isothermal map was constructed. Such a constant-temperature boundary condi- tion has the advantage of simplicity. In this project, a different approach was used. Measured data in the LTPP database showed that temperatures at a depth beyond 2 m tend to vary approximately linearly with depth. Therefore, a ï¬ux boundary condition was used at a depth of 3 m. That is This boundary condition is independent of location and does not require a speciï¬c value. In addition, it is easy to implement this boundary condition in the ï¬nite difference â â = T x 3 22m independent of depth ( ) Ï Î± Ï ÏC x T t Q Q T T h T T s s s a a s c s a Î 2 4 4 â â = â + â â â( )+ i k T x sâ â ( )21 Q k T x f s = â â ( )20 h a T T c s a = +âââ ââ ââââ ââ â698 24 0 00144 2 0 3 . . . i i abs U T T d s a+ +( )( ) ⡠⣠â¢â¢â¢ ⤠⦠â¥â¥â¥0 00097 19 0 3 . ( ) . abs Q h T Tc c s a= â( ) ( )18 34
calculation procedure. However, this thermal gradient bound- ary condition is not accurate if extrapolated to too great a depth. Numerical Solution of the Model This model was solved numerically using a ï¬nite difference approximation method, together with required input data (i.e., hourly solar radiation, air temperature, wind speed, and model parameter values). In the numerical solution, the pavement thickness was divided into cells, which are thinner near the surface and thicker at deeper levels. Each cell is given a temperature (equal to air temperature) at the start of the calculation as an initial condition. The model then calculates a new temperature for each cell (several times for each simu- lated hour) at each time step. Obtaining Hourly Climatic Input Data For any pavement site, model calculation requires accurate site-speciï¬c hourly climatic data and model parameters, including hourly solar radiation, hourly air temperature, and daily average wind speed data in an hourly format. Hourly solar radiation can be collected from the National Solar Radiation Database (NSRDB). Hourly solar radiation data are modeled using State University of New York at Albany (SUNY) or Meteorological-Statistical (METSTAT) models based on satellite images, covering nearly all parts of the coun- try from 1990 to 2005. Daily average wind speed can be collected directly from the Virtual Weather Station program in the LTPP. Additionally, daily wind speed can be obtained directly from the National Climatic Data Center (NCDC) or the meteorological network in each state. Hourly wind speed is preferred, but such data are difï¬cult to obtain and more vulnerable to environmental conditions, adding difï¬culty in the interpolation. However, the model is not overly sensitive to the wind speed such that daily values are adequate. Hourly air temperature data are not commonly available, yet reasonable estimates of these hourly temperatures are needed for accurate temperature calculations. In order to estimate hourly wind speed data, a method was developed to estimate hourly air temperatures from daily maximum and minimum air temperatures. Recorded daily maximum and minimum air temperatures can be obtained easily from the Virtual Weather Stations in the LTPP database or NCDC. A conventional method to impute hourly air temperatures ï¬ts a sinusoidal function to daily maximum and minimum air temperatures (e.g., 32, 33, 34). However, the daily proï¬le of air temperature is not exactly sinusoidal. Typically, air tem- perature rises from the daily minimum temperature to the daily maximum temperature in about 9 hours, and decreases from the daily maximum temperature to the daily minimum in about 15 hours. A more accurate air temperature prediction method would incorporate this non-sinusoidal pattern. In order to obtain a more representative pattern of daily air temperatures, data over an entire year were obtained from the Automatic Weather Station (AWS) in the LTPP database and analyzed using a seasonal trend decomposition time series analysis. The trend trace is a moving average of the measured data, which represents the daily average temperature through- out the year. The âseasonalâ trace is obtained by subtracting the trend line from the measured data and ï¬nding a local poly- nomial which best ï¬ts the result. This trace represents the reg- ular pattern of daily air temperature, which is used instead of a sinusoidal function. With a known daily pattern of air temperature, the hourly air temperatures can be reconstructed from daily maximum and minimum measured data. First, the daily average air tem- perature data are taken from the trend trace. Then, the trend and the seasonal traces obtained from the time series analysis are added together. Finally, the result is linearly transformed to ï¬t the measured data, day by day. The non-sinusoidal pre- dicted daily temperature patterns at six different sites are shown in Figure 32. More details on this temperature model are presented in Appendix B, including comparisons of the predicted pave- ment temperatures at different depths with the measured temperatures. Also presented are maps of the geographic dis- tributions of the temperature variables of albedo, emissivity, and absorptivity, all of which were shown by a sensitivity analysis to be important input variables. Stiffness, Tensile Strength, Compliance, and Fracture Properties of Mixtures The properties of a HMA mixture in an overlay must be estimated both accurately and with computational efï¬ciency to achieve an overlay design resistant to reï¬ection cracking. The stiffness and compliance of the mixture must be calcu- lated at widely different temperatures and loading rates (ther- mal and trafï¬c). The tensile strength must also be calculated over the same wide ranges of temperature and loading rates. The fracture properties (i.e., Paris and Erdoganâs Law coefï¬- cients) must be calculated. These coefï¬cients are also sensi- tive to temperature and loading rates. For these reasons, ANN algorithms which reproduce Witczakâs 1999 (2) and 2006 (3) Complex Modulus models were developed to form the basis for calculating the overlay stiffness under trafï¬c loads and computing the viscoelastic thermal stress for thermal reï¬ec- tion cracking. The method used to construct ANN algorithms is described in the literature (22). The accuracy with which these algorithms reproduce the Witczak Complex Modulus 35
models is described. The tensile strength was determined by Schapery (36, 37) to be an important variable in making real- istic estimates of the Paris and Erdoganâs Law fracture coefï¬- cient, A. Earlier studies reported tensile strengths obtained of ï¬eld cores taken from pavement sections well distributed around the United States and Canada (4, 38) and were con- sidered to be representative of HMA mixtures. The calibra- tion coefï¬cients from these studies (4) could be used to predict both thermal and trafï¬c related reï¬ection cracking and healing between trafï¬c loads. Artificial Neural Network Algorithms for Witczakâs Complex Modulus Models The binder input data required for the 1999 Witczak model (2) are the viscosity, η, and frequency of loading in Hertz, as shown in Table 20. Figure 33 shows a graphic comparison between the data obtained using Witczakâs 1999 model and those developed using the ANN algorithm (35) (statistical comparisons are provided in Appendix G). The R2 values are 0.68 and 0.98 for the 1999 Witczak model and the ANN algo- rithm, respectively, when using the source input data The binder input data for the 2006 Witczak model is the magnitude of the shear modulus, G*, of the binder and its phase angle, δc, in degrees, as shown in Table 21. Also shown in that table are the other required input variables for the model, which include the same gradation and volumetric composition as in the 1999 model. The output is the magnitude of the Complex Modulus. A comparison of the behavior of the 2006 Witczak model (3) and the corresponding ANN algorithm is shown in Figure 34. The R2 values are 0.77 and 0.96 for the 2006 Witczak model and the corresponding ANN algorithm, respectively. The user input to the ANN mixture modulus model is the gradation and volumetric parts of the mix design and the binder data. In keeping with the MEPDG format, the binder data can be input at any of three levels. The binder data are the six properties of the CAM model (i.e., Gg, the glassy mod- ulus in Gpa; R, the Rheological Index; Ïrm, the cross-over fre- quency in rad/sec; Td, the deï¬ning temperature in °C; and the two time-temperature shift parameters, C1 and C2) (39). The 36 0 2 4 6 8 10 12 14 16 18 20 22 24 26 -8 -6 -4 -2 0 2 4 6 8 Te m pe ra tu re F lu tu at io n Pa tte rn A ro un d D ai ly Av er ag e Te m pe ra tu re (D eg ree C ) Hours Texas (48-1000) Texas (48-0800) Texas (48-A800) South Dakota (46-0800) Utah (49-0800) Nevada (32-0100) Figure 32. Predicted daily air temperatures at six different LTPP test sites. Input Output R2 Gradation Volumetric Binder Data Witczak ANN 3/4 (%) 3/8 (%) #4 (%) #200 (%) Va (%) Vbeff (%) η 106 poise fc Hz E* psi 0.68 0.98 Table 20. Data input for 1999 Witczak model (viscosity, , and frequency of loading in Hertz).
37 Figure 33. Comparison of Witczak 1999 model with ANN algorithm. Input Output R2 Gradation Volumetric Binder Data Witczak ANN 3/4 (%) 3/8 (%) #4 (%) #200 (%) Va (%) Vbeff (%) Log|G*| 106 psi δc deg E* psi 0.77 0.96 Table 21. Input data for the 2006 Witczak model (magnitude of the shear modulus of the binder and its phase angle in degrees). Figure 34. Comparison of Witczak 2006 model with ANN algorithm.
user may input these six properties with Level 1 input. In Level 2 input, the user may specify the Performance Grade (PG) of the binder and the climatic region in which the over- lay is to be placed and the program will internally calculate the six CAM parameters that correspond to the PG speciï¬ed. In Level 3 input, the user only needs to specify the climatic region where the overlay will be built. These simpliï¬cations can be made because the mean values of the six CAM param- eters for each of the four climatic regions in North America are stored. A total of 48 sets of CAM parameters were mea- sured on binders extracted from cores (4); the mean values for each of the climatic regions are listed in Table 22. The use of the ANN algorithms permits the calculation of the magnitude and phase angle of a mixture modulus at any temperature and loading rate. The input of mixture moduli at different loading times to the viscoelastic thermal stress subprogram is also generated by use of the ANN algorithm. Models of Tensile Strength of Mixtures The tensile strength of a mixture depends upon the tem- perature and loading rate application. In order to have a real- istic estimate of the tensile strength of a mixture in the ï¬eld, the relationships between the mixture modulus and tensile strength that were measured in earlier research (4) were used and converted from the U.S. Customary units to the Interna- tional System (SI) of units. The relationship developed for the slowest rate (0.005 in./min) was used for the thermal fracture properties (Equation 23) and that developed for the most rapid rate (0.5 in./min) was used for the trafï¬c fracture prop- erties (Equation 24). Where E(t,T) designates the mixture modulus in MPa and the Ït designates tensile strength in kPa. Ï t E t T = ( ) à à ⡠â£â¢ ⤠â¦â¥6 895 1000 6 895 45 5 24 1 1 56 . , . . ( . ) Ï t E t T = ( ) à à ⡠â£â¢ ⤠â¦â¥6 895 1000 6 895 21 3 23 1 1 95 . , . . ( . ) Models of Paris and Erdoganâs Law Fracture Coefficients A and n Earlier studies (4) provided formulas for the Paris and Erdo- ganâs Law fracture coefï¬cients A and n which were found to work well for predicting reï¬ection cracking without being altered. The formulas presented in these studies had been cal- ibrated to ï¬eld fatigue cracking data in each of the four climatic zones. The form of the equations for both A and n were taken from viscoelastic crack growth theory by Schapery (36, 37) (some details on developing these formulas using a Systems Identiï¬cation method are presented in Appendix J). where mmix = the log-log slope of the mixture modulus versus loading time graph for the current temperature and loading rate; E (t, T) = the mixture relaxation modulus (in MPa) at load- ing time, t (in sec.), and temperature, T (in °C); D1 = the coefï¬cient of the mixture creep compliance expressed in a power law form (in kPaâ1); and Ït = tensile strength (in kPa). The magnitudes of the Complex Modulus at three differ- ent loading times, t, which are set at one log cycle apart and at the current temperature, were determined by the selected ANN Witczak model and ï¬t by linear regression to produce the values of E1 and mmix that are used in deï¬ning the current value of the Creep Compliance coefï¬cient, D1. The values of D1 and mmix and the tensile strength, Ït, which is computed from the current value of E(t, T) are then used in the Schapery log , log log ( )mixE t T E m t( )( ) = ( )â ( )1 28 D mix mix 1 1 27= ( )sin ( ) m E m Ï Ï log log log ( )A = g g D g2 3 mix 4+ + m t1 26Ï n g g mmix = +0 1 25( ) 38 Climatic Region Ïrm, (rad/sec) R Td (°C) C1 C2 Gg (Gpa) Wet-Freeze 0.01516 1.935 -5.8 31.57 199.2 0.861 Wet-No Freeze 7.06E-05 2.261 -6.41 42.49 259.3 0.906 Dry-Freeze 0.001397 2.286 -6.22 38.77 239.0 1.571 Dry-No Freeze 0.000845 2.032 -6.07 41.55 266.9 0.532 Table 22. Mean CAM model parameters for the four climatic regions.
equation for the logarithm of the Paris and Erdoganâs Law coefï¬cient, A. Healing Coefficients In addition to the fracture coefï¬cients, the healing coefï¬- cients obtained in earlier studies (4) are used to account for the healing shift function that occurs between the trafï¬c loads on the overlay. The healing shift function is The rest period in seconds between load applications is cal- culated as the number of seconds in a day (86,400) divided by the average daily trafï¬c in vehicles per day. Values for the coef- ï¬cients g0 through g6 were determined for each of the four cli- matic zones; these are listed in Table 23 (4). These coefï¬cients were used without alteration and the fracture coefï¬cients g0 through g4wereappliedwithout modiï¬cation to determine both the thermal and trafï¬c fracture properties. The healing coefï¬- cients were used only with the trafï¬c crack growth equations. Stress Wave Pattern Correction for Viscoelastic Crack Growth Schaperyâs theory of crack growth in viscoelastic materials takes into account the loading time and the shape of the stress pulse during the time that the material is being loaded (30, 31). The normalized wave shape, w(t), has a peak value of 1.0. The wave shape rises to 1.0 and falls back to zero in a length of time, Ît. The correction term for viscoelastic crack growth ak is given by the following equation. The exponent, n, is the Paris and Erdoganâs Law exponent which is given in Equation 25 and is typically between 2 and 6. If the applied load is a square wave, the integral is equal a w t dtk nt = ( )â«0 30Î ( ) SF g thealing rest g = + ( )1 295 6Î ( ) to 1.0. If the stress wave is a rising and falling shape as is commonly the case with trafï¬c and thermal stresses, the value of ak is usually considerably less than 1.0. Appendix F shows the patterns of the rise and fall of the stress waves caused by the pas- sage of single, tandem, tridem, and quadrem axles. These pat- terns were used in determining the effect during each day of each set of axle groupings on the growth of reï¬ection cracks. Computational Method for Crack Growth Due to Traffic Although the SIF for bending and shear occur at the same time under trafï¬c loads, the crack growth technique adopted in this project calculates the growth of cracks due to each of the two stresses separately. Thus, Paris and Erdoganâs Law for bending and shearing are provided by Equations 31 and 32, respectively. The wave patterns for akâthe viscoelastic factorâare shown in Appendix F for each of the types of trafï¬c loading: bending and shearing and each of the four axle groupings. With shearing stresses, there is a peak shearing stress as the leading edge of the tire approaches the reï¬ection crack and then another peak shearing stress of a different sign as the trailing edge of the tire leaves the location of the reflection crack. Thus there are two peak shearing SIF with the passage of a single tire. Examples of these patterns are shown in Fig- ures 35 and 36 for bending and shearing, respectively. The time increment, Ît, for the tridem axle group to pass over a given point on a pavement is given in Equation 33. Ît L V j sec ( )( ) = + âââ â â â 18 33 ft ft sec dc dN A 2K shearing a shearing11 n k= ( )[ ] ( )[ ] ( )32 dc dN A K bending a bending1 n k= ( )[ ] ( )[ ] ( )31 39 Coefficient Climatic Zone Wet-Freeze Wet-No Freeze Dry-Freeze Dry-No Freeze g0 -2.09 -1.429 -2.121 -2.024 g1 1.952 1.971 1.677 1.952 g2 -6.108 -6.174 -5.937 -6.107 g3 0.154 0.19 0.192 1.53 g4 -2.111 -2.079 -2.048 -2.113 g5 0.037 0.128 0.071 0.057 g6 0.261 1.075 0.762 0.492 Table 23. Climatic zone variations of fracture and healing coefficients for HMA.
The incremental crack growth each day is calculated from the accumulated effects of all of the trafï¬c that have passed over the reï¬ection crack during that day as follows: Crack length, cn on the nth day of this crack growth process is the sum of all of the n incremental crack growth increments: When the sum of the bending crack increments reaches the point in the overlay where the bending stresses become com- pressive, that deï¬nes what is termed âPosition 1.â At crack c cn i i i n = = = âÎ 1 36( ) dc A 2K a dN SF IIi n ki i i=1 i=n healing = ( ) ( ) âââ â â ââ 1 for shearing ( )35 dc A K a dN SF foIi n ki i healing = ( ) ( ) âââ â â â= = â i i n 1 1 r bending and ( )34 lengths above this point, bending stresses no longer contribute to the growth of cracks and crack growth is due only to ther- mal and shearing stresses. The number of days that are required for cracks caused by each type of stress to reach Position 1 are recorded. Then the number of days required to grow a crack from Position 1 to the surface of the overlay because of shear 40 Crack or Joint 4.0 ft4.0 ft Overlay Old Surface Lj Lj Lj Figure 35a. Bending loading pattern for a tridem axle. W(t) Load Wave Shape [W(t)]n 0.0 0.72 Lj Lj Lj 0.095 0.84 0.76 0.76 0.095 5.0 ft 5.0 ft 4.0 ft 4.0 ft 0.92 0.92 0.72 0.0 0.84 (18 + Lj) ft Î t 0.84 0.0 0.0 (0.72)n (0.92)n (0.76)n (0.92)n (0.095)n (0.84)n (0.84)n (0.095)n (0.76)n (0.72)n (0.84)n Figure 35b. Normalized SIF and ak wave patterns for tridem bending loading. Crack or Joint Overlay Lj Lj Lj Lj Lj Lj Lj Lj 4.0 ft Old Surface Figure 36a. Shearing loading pattern for a tandem axle.
and thermal stresses is computed. This gives a total of ï¬ve numbers of days that are computed and recorded in the process of growing a reï¬ection crack up through an overlay. All ï¬ve numbers of days are used in the calibration equations to estimate the values of Ï (the scale parameter) and β (the shape parameter) of the observed ï¬eld cracking data. Computational Method for Viscoelastic Thermal Stresses In determining the thermal viscoelastic modulus of an in- service mixture for use in calibrating the program, Falling Weight Deï¬ectometer (FWD) data was used to determine the modulus of the overlay and the mixture temperature at the time of the measurement. The overlay asphalt mixture was used in the 2006 Witczak modulus model to calculate the modulus of the mixture. Discrepancy between the FWD modulus and that predicted by the Witczak 2006 model (3) at the same temperature was ascribed to the rubbery stiffness of the mixture, Eâ. If the stiffness of the Witczak 2006 model was greater than the FWD back calculated modulus, the value of Eâ was set to zero. Otherwise, the value of Eâ was used in calculating the master curve of the relaxation moduli at 11 different loading times to E t T E E t T FWD , , ( )( ) = + ( )â 2006 37 make use of the viscoelastic thermal stress computation algo- rithm (38). The details of the viscoelastic thermal stress compu- tation are given in Appendix I. It makes use of Dirichlet series (Equation 38) to represent the master curve of the relaxation modulus of the asphalt mixture at any loading time, t. where tr is the reduced time of the master curve of the relax- ation modulus of the asphalt mixture. The Dirichlet series coefï¬cients, Ei, and relaxation times, Ti, are determined with a collocation process using the Witczak 2006 mixture model (3) and the Eâ is determined from the analysis of the FWD ï¬eld data. Within the program, the temperature and relaxation modulus of the overlay at the current tip of the growing reï¬ec- tion crack is calculated for each day, and then used to calcu- late the viscoelastic thermal stress at the tip of the crack, the thermal SIF, and the incremental growth of the crack that occurs that day. The actual thermal loading time is converted to a reduced loading time using the Williams-Landel-Ferry (WLF) time-temperature shift (39). As with the growth of cracks due to trafï¬c stresses, the incremental crack growth each day is accumulated until Position 1 is reached. The num- ber of days required to reach Position 1 is recorded as is the number of days required for the crack to grow up the rest of the way to the overlay surface. These two numbers of days are used in the calibration equations to estimate the value of Ï, the scale parameter of the observed ï¬eld cracking data. The viscoelastic thermal stress is calculated using a Boltz- man Superposition Integral in numerical form (details of this calculation are given in Appendix I). The general form of the thermal stress integration is The initial strain is calculated relative to the deï¬ning tem- perature of the master curve Td, and the remaining thermal strain value αÎT is the hourly change of thermal strain. It is the rate of change of the difference between and αÎT that accumulates the thermal stress at the tip of the growing ther- mal reï¬ection crack. The thermal stress at the tip of the crack is calculated for every hour of each day and the highest calcu- lated stress is used to calculate the thermal SIF and the incre- mental growth of the reï¬ection crack for that day. Paris and Erdoganâs Law coefï¬cients were calculated using the modu- lus for the critical time and temperature each day to calculate the D1, mmix, and Ït values. The Ît was set at one 1 and the ak was set at 1.0 for the thermal case. The values for the coefï¬cient of thermal expansion (con- traction) α can be input at Level 1 and Level 3. The Level 3 input is the mean value of the thermal coefï¬cients that were Ï Ï Î± Ï Ï Ï Ï t E t T dr r tr( ) = â( ) â â( )â = = â« Î 0 39( ) E t E E er i tr Ti i n( ) = +â â = â 1 38( ) 41 5.0 ft Lj Lj 4.0 ft 5.0 ft (14 + Lj) ft. 1.11 1.11 1.11 1.11 (1.11)n Î t (1.11)n (1.11)n (1.11)n W(t) Load Wave Shape [W(t)]n Figure 36b. Normalized SIF and ak wave patterns for tandem shearing loading.
used in the earlier research for each of the four climatic zones (4). In addition, the absorption, emissivity, and albedo coef- ï¬cients for the same climatic zones may be input at Level 1 and Level 3. Table 24 gives the mean values for these coefï¬- cients. Absorption, emissivity, and albedo coefï¬cients vary throughout the United States and with the seasons (detailed values are given in Appendix B). Supervisory Program to Compute Crack Growth The supervisory program for the separate analysis of the growing reï¬ection cracks by each of the three mechanisms (bending, shearing, and thermal stress) takes into account all of the identiï¬ed details and produces ï¬ve numbers of days to reach a deï¬ning point in the growth of a reï¬ection crack up through an overlay. One deï¬ning point is Position 1, at which point bending produces no additional crack growth. The other deï¬ning point is the surface of the overlay. The ï¬ve numbers of days are the number of days to reach Position 1 by bending, shearing, and thermal stress and the number of days for the crack to grow from Position 1 to the surface of the overlay because of shearing and thermal stress. The supervisory program handles each of the crack growth processes separately and combines them after the calculations for each of the three separate mechanisms have been completed. Figures 37, 38, and 39 illustrate the calculation processes for thermal stress, bending, and shearing, respectively. In all cases, the initial crack length was taken to be the depth of the old pave- ment surface layer. In all crack growth calculations, the initial crack size, co, is assumed to be the thickness of the overlaid pave- ment surface layer. This value enters into the calculation of the SIF at the tip of the crack each day. User Interface Program for Input and Output Data The input and output user interface for this program has been designed to have the same appearance as the user interface for the MEPDG software. An example of an input screen show- ing the input of a pavement layered structure is given in Figure 40. The user output is a graphic plot of the three severity level distress curves for reï¬ection cracking, if the original ï¬eld data had a sufï¬cient amount of observed data to include high, medium and low levels of severity (details of the various options for input and output are given in Appendix O). Computation-to-Field Calibration Coefficients At the end of the computations, there are ï¬ve calculated numbers of days for a crack to reach a designated point within an overlay (Position I) where the bending stresses become compressive and no longer cause crack growth (Posi- tion II, the surface of the overlay). These ï¬ve numbers of days are illustrated in Figure 41. These five numbers of days can be combined in several ways to model the value of Ï, the scale parameter of the amount, and severity of the observed reflection cracking distress. One way of modeling the Ï-value is to assume that the principal cause of reflection cracking is bending stress and another way is to assume that shearing stress is the prin- cipal cause of reflection cracking. In both cases, it becomes necessary to find how many days of each of the other types of cracking are the equivalent of the number of days of the principal cause of the distress. This concept is illustrated in Figure 42 which shows all three Ï-values (i.e., ÏLMH, ÏMH, and ÏH). The linear regression form of the model for the ÏLMH- value assuming that bending stress is the principal cause of the reflection cracking up to Position I and shearing is the principal cracking mechanism from Position I up to the sur- face of the overlay, is presented in Equations 40 through 42. The thermal calibration model for the low+medium+high distress curve is given by Equation 40. The thermal calibration models for the Medium + High and High distress curves are given by Equations 41 and 42, respectively. Ï Î± α αH fB f B f T f B f S f N N N N N N = â â â ââ â â â + 1 10 11 1 1 12 1 1 T f T f S N N 2 13 14 2 2 42α αâ â ââ â â â ( ) Calibration Coefficients: α α α α α β10 11 12 13 14, , , , , H Ï Î± α αMH fB f B f T f B f S f T N N N N N N = â â â ââ â â â + 1 5 6 1 1 7 1 1 2 α α8 9 2 2 41â â ââ â â â N N f T f S ( ) Calibration Coefficients: α α α α α β5 6 7 8 9, , , , , MH Ï Î± α αLMH fB f B f T f B f S f T N N N N N N = â â â ââ â â â + 1 0 1 1 1 2 1 1 2 3 4 2 2 40α αâ â ââ â â â N N f T f S ( ) Calibration Coefficients: α α α α α β0 1 2 3 4, , , , , LMH 42 Climate Zone Thermal Coeff. α Absorption Emissivity Albedo Wet-Freeze 0.0000248 0.7468 0.6628 0.3027 Wet-No Freeze 2.287E-05 0.7381 0.6571 0.1988 Dry-Freeze 0.00002298 0.7357 0.6071 0.2429 Dry-No Freeze 2.468E-05 0.7000 0.5750 0.2000 Table 24. Level 3 thermal coefficients.
43 Hourly Solar Radiation Hourly Wind Speed Daily Wind Speed Emissivity Coefficient Absorption Coefficient Albedo a d Daily Air Temperature Hourly Air Temperature Pavement Temperature (ÎT) Viscoelastic Thermal Stress Ï(T) Thermal Stress Intensity Factor (SIF) (Artificial Neural Network Model) Crack Growth ÎC=A[J]n ÎN Relaxation Modulus at Crack Tip (Artificial Neural Network Model) 1999 Model? 2006 Model? Gradation Volumetric Composition Frequency (fc) Viscosity (η) Gradation Volumetric Composition Phase Angle (δb) Shear Modulus of Asphalt (G*) Binder Properties 2006 Model1999 Model Is ΣÎC⥠Overlay Thickness No. of Days NfT1, NfT2 Yes No Fracture Properties A,n Weather Data Collocation E inverse Figure 37. Flow chart of the thermal crack growth computations. A similar set of coefï¬cients can be derived by linear regres- sion analysis, assuming that bending is the principal mode of reï¬ection cracking until it reaches Position I and then shear- ing stress is the principal mode of reï¬ection cracking from Position I to the surface of the overlay. An example of this assumed calibration form is shown in Equation 43. Ï Î± α αLMH fB f B f T f B f S f S N N N N N N = â â â ââ â â â + 1 0 1 1 1 2 1 1 2 3 4 2 2 43α αâ â ââ â â â N N f S f T ( ) The calibration coefï¬cients are, as in the ï¬rst form of this model, α0, α1, α2, α3 and α4. Similar models are assumed for the scale parameters ÏMH and ÏH; similar linear regression models were used to model the shape parameter, β. In performing the calibration analysis, the thermal, bend- ing, and shearing forms of equation were tried and the one which proved to have the highest coefficient of determina- tion, R2, was selected. In general, the model with bending as the principal cracking mechanism up to Position I and ther- mal stress as the principal cracking mechanism from there to the surface of the overlay had the highest R2-value with all
overlay types except one. The exception was the AC overlay with reinforcing over AC in the Dry-Freeze Zone. Bending was the principal crack growth mode up to Position I in this model. Shearing was the principal cracking mode from Posi- tion I to the overlay. An example of this latter model is in Equation 43. The complete set of calibration coefï¬cients for each of the types of pavement structures and overlays and the statistical measures of their ï¬t to the observed data is provided in Appendix N. In some cases, no distress was observed at the high severity level and in other cases, only low severity distress was observed. In such cases, the calibration coefï¬cients for the reï¬ection cracking model were limited to those sets which were actually observed. A separate calibration program was assembled to assist in the development of other regional computational model-to-ï¬eld data calibration coefï¬cients (a Userâs Guide to this Calibration Program is presented in Appendix P). Validation of the Calibration Coefficients In reviewing the detailed data for each of the test sections, it was determined that there were only 150 overlay sections with unique data. In some cases multiple sections were located 44 Hourly Solar Radiation Hourly Wind Speed Daily Wind Speed Emissivity Coefficient Absorption Coefficient Albedo a d Daily Air Temperature Hourly Air Temperature Pavement Temperature (ÎT) Each Vehicle Class (Axle and Tire Loads) Bending Stress Intensity Factor (SIF) (Artificial Neural Network Model) Crack Growth ÎC=A[J]n ÎN Layer Relaxation Modulus (Artificial Neural Network Model) 1999 Model? 2006 Model? Gradation Volumetric Composition Frequency (fc) Viscosity (η) Gradation Volumetric Composition Phase Angle (δb) Shear Modulus of Asphalt (G*) Binder Properties 2006 Model1999 Model Is ΣÎC⥠Overlay Thickness? No. of Days NfB Yes No Fracture Properties A,n Daily Traffic Vehicle Class Distribution Weather Data Figure 38. Flow chart of the bending crack growth computations.
on the same highway (i.e., each with the same trafï¬c, climate, and pavement structure). Therefore, the calibration and vali- dation process was conducted in a different way than was ini- tially envisioned. It was planned to separate the overlay sections of each type into two groups: one group would be used to develop the calibration coefï¬cients and the other group would be used to verify that their distress accumulation could be sat- isfactorily predicted. The small number of unique pavement- overlay sections did not permit splitting the sections into two such groups. Instead, it was necessary to use the observed data of all of the sections in both developing the calibration coefï¬- cients by regression analysis and comparing the set of coefï¬- cients with the original ï¬eld data. If the set of calibration coef- ï¬cients reproduced the original observed scales, trends, and patterns of Ï and β values with a high coefï¬cient of determi- nation (R2) and with an acceptable scatter pattern around the line of equality, the set of coefï¬cients were accepted as valid. If they did not produce an acceptable ï¬t to all of the original observed Ï and β values, then the calibration coefï¬cients were revised by trying a new model with bending or shearing as the principal distress mode. The model with the highest coefï¬cient of determination (R2) was selected. No separate validation 45 Hourly Solar Radiation Hourly Wind Speed Daily Wind Speed EmissivityCoefficient Absorption Coefficient Albedo a d Daily Air Temperature Hourly Air Temperature Pavement Temperature (ÎT) Each Vehicle Class (Axle and Tire Loads) Shear Stress Intensity Factor (SIF) (Artificial Neural Network Model) Crack Growth ÎC=A[J]n ÎN Layer Relaxation Modulus (Artificial Neural Network Model) 1999 Model? 2006 Model? Gradation Volumetric Composition Frequency (fc) Viscosity (η) Gradation Volumetric Composition Phase Angle (δb) Shear Modulus of Asphalt (G*) Binder Properties 2006 Model1999 Model Is ΣÎC⥠Overlay Thickness? No. of Days NfS1, NfS2 Yes No Fracture Properties A,n Daily Traffic Vehicle Class Distribution Weather Data Figure 39. Flow chart of shearing crack growth computations.
46 Figure 40. Example user input screen for the reflection cracking program. Overlay C Position I Position II NfB1 Bending Stress NfS1 NfS2 Shearing Stress NfT1 NfT2 Thermal Stress Figure 41. Definition of the number of days of crack growth. ⢠NfB1 = Number of days for crack growth due to bending to reach Position I. ⢠NfT1 = Number of days for thermal crack growth to reach Position I. ⢠NfS1 = Number of days for crack growth due to shearing stress to reach Position I. ⢠NfT2 = Number of days for thermal crack growth to go from Position I to Position II. ⢠NfS2 = Number of days for crack growth due to shearing stress to go from Position I to Position II.
process was pursued with the limited number of overlay test sections that was available. The purpose of validation is to check if the equations derived by regression analysis correctly ï¬t the observed scales, trends, and patterns of the ï¬eld data. Validation is required because the regression analysis assumes that all of the error is in the observed dependent values and not in the independent variables. In this case, the independent variables were pre- dicted using fracture mechanics concepts and material prop- erties which are based on tabulated asphalt mixture, pavement structure, trafï¬c and climatic variables. Thus, there are likely to be errors in the independent variables as well as in the dependent variables (i.e., the values of Ï and β which were ï¬t- ted to the observed distress). A total of 33 possible sets of calibration coefï¬cients could have been developed if low, medium, and high levels of crack- ing severity had been observed in all pavement types. Because some of these severity levels were missing, a total of 24 sets of calibration coefï¬cients were developed (all of the sets of cali- bration coefï¬cients are found in Appendix N). After arriving at a ï¬nal set of calibration coefï¬cients, a fur- ther quality control step was taken by graphically plotting the distress patterns for all of the test sections to make certain that the predicted patterns of distress accumulation were both reasonable and realistic. Logical tests were programmed into the design program to make certain that the predicted distress patterns will be correctly ordered from low to medium to high levels of distress. Examples of the ï¬nal predicted values of Ï and β plotted against the observed ï¬eld values are pro- vided in the Chapter 3 (full set of such plots is provided in Appendix N). 47 0 100 % T o ta l L e n gt h o f C ra ck s No. of Days Medium+High ÏLMH ÏMH ÏH Low+Medium+High High severity 36.8% Figure 42. Illustration of amount and severity of reflection cracking distress curves.