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19 Tire Pressure ( p) Width (W ) Tire Length (L) Figure 13. Tire load applied to pavement surface. Determination of the Effect of Cumulative Axle Load Distribution on Tire Length Because of the difficulty of employing each tire length for axle load intervals to evaluate traffic load effects on propaga- tion of reflection cracking, the effect of the axle load distribu- tion on the tire patch length for each category was used for the evaluation of traffic 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. Figure 12. Comparison of SIF with ANN model The tire patch lengths of corresponding axle load intervals predictions for HMA overlays over a cracked for each category can be calculated using Equation 3 and the asphalt pavement surface layer. characteristics of axle types. Table 11 lists the calculated axle load intervals for all traffic categories, and Table 12 lists the tire patch length increments. in developing the models. These models apply only to the vari- Using the tire patch length and collected traffic data, the able ranges used as input to these models; extrapolation out- cumulative axle load distribution can be determined for each side the range of inference may not produce accurate results. category. Figure 14 illustrates the procedure for determining tire length and the cumulative axle load distribution (CALD) of Traffic Loads and Tire Footprints each category. Such distribution should be produced for all eight traffic categories to account for all types of vehicles and axles. Tire footprints are closer to rectangles than to the com- Figure 15 shows the cumulative axle load distribution of tire monly assumed circular footprints (25). In this project, rec- load for Category 1 of LTPP section 180901 in 2004, which was tangular tire footprints with known tire widths were used; determined using data in Table 12. tire footprint length was calculated from the tire load and the inflation pressure. The length of tire patch was used to eval- uate bending and shearing SIF in asphalt overlays. Also, Modeling of Cumulative Axle because the tire length is proportional to the load, a cumu- Load Distribution lative axle load distribution on tire length for each category Since the frequency distribution of each tire length of a load may be determined, based on collected traffic data such as category is used to evaluate load effects for reflection cracking WIM or AADTT. propagation, the cumulative axle load distribution (CALD) of pavement sections and traffic categories should be developed Tire Patch Length along with the tire length. The CALD of traffic loads or tire lengths follows a sigmoidal curve having a lower asymptote of The tire-load model that assumes a rectangular tire contact zero and a finite upper asymptote as shown in Figure 16. (Details area (as shown in Figure 13) was used to evaluate the effect of of the modeling process are provided in Appendix D). tire load on reflection cracking. After reviewing potential models that describe the statisti- Tire width is assumed to be constant within each traffic cal properties of the cumulative axle load distribution versus category (vehicle class and axle type) even under different tire tire length, the Gompertz model presented in Equation 4, was pressures. Thus, the tire length can be calculated as follows: chosen. Tire Length (in.) = tire load (lb) (3) y = exp [ - exp ( - x )] (4) lb tire pressure 2 tire width (in.) in. where , , and are model parameters.