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20 Table 11. Typical characteristics for axle types (24). Axle Tire Width Tire Pressure Category Tires Axle Load Interval (lb) Type (in.) (PSI) 1 Single 7.874 40 (< 6,000 lb) 3,000 ~ 40,000 lb Single 2 Dual 8.740 120 (> 6,000 lb) at 1,000 lb intervals 3 Single 7.874 120 6,000 ~ 80,000 lb Tandem 4 Dual 8.740 120 at 2,000 lb intervals 5 Single 7.874 120 12,000 ~ 102,000 lb Tridem 6 Dual 8.740 120 at 3,000 lb intervals 7 Single 7.874 120 12,000 ~ 102,000 lb Quadrem 8 Dual 8.740 120 at 3,000 lb intervals The Gompertz model is appropriate because it has a clear The distribution factor of C1 represents the minimum axle physical boundary condition which shows asymptotes at y = load (tire length) to be considered for load related distress. 0 and y = and is asymmetric about its inflection point which The lower limits of axle load and tire length are presented in occurs at / (26). The parameter in the model indicates the Table 15. C2 is the factor at which the cumulative distribution upper asymptote which is equal to 1.00 (100%) for the cumu- reaches 100 percent first. L1 and L2 are the tire lengths corre- lative axle load distribution curve. The parameter describes sponding to C1 and C2, respectively. The model parameters how wide the rising portion of the curve is. In addition, the and plots of calibrated cumulative axle load distribution versus parameter indicates the slope of the cumulative axle load tire lengths for all categories of Section 180901 are provided in distribution curve. Figure 17 illustrates a typical curve of the Appendix A. Gompertz model. For Level 1 data inputs, the model parameters for the The parameter should be equal to 1.00 because the cumulative axle load distribution can be computed using cumulative axle load distribution curve has a physical bound- WIM data for each category, while the default values for Level ary condition ranging from 0 to 1.00 (i.e., 0 to 100 percent). 3 input are provided. The default model parameters, shown Therefore, the modified model for cumulative axle load dis- in Table 14, were prepared using traffic data from the LTPP tribution is: database. Also, Table 16 presents the default CALD values which were determined based on the model parameter default C ( Li ) j = exp [ - exp ( - Lij )] (5) values. where Determination of Hourly Number Lij = ith tire length in tire patch length increment in of Axles traffic category j; In order to analyze reflection cracking propagation C(Li)j = cumulative axle load distribution factor at Li caused by bending or shearing, the hourly number of axles within traffic category j; and should be considered in each of the tire length increments , = model parameters describing the curve width and within each traffic category. The number of axles can slope, respectively. be calculated from the probability density which is deter- The collected traffic data from WIM or AADTT for a given mined based on the cumulative axle load distribution section were used to develop the model parameters and in of tire lengths in each category (details of the process of the modified Gompertz model of Equation 5. The results pro- determining the hourly traffic distribution are provided in vided a good fit of the data along with relatively high signifi- Appendix E). cance. Table 13 lists the developed model parameters and for the traffic category 1 of LTPP Section 180901. Typical Probability Density on Tire Patch Length model parameters for each traffic category are presented in Table 14. The probability density of the tire patch length is the fre- Figure 18 shows a plot of the calibrated model data and the quency distribution of each tire length in a category, which corresponding measured traffic data for the LTPP section. is required to determine the number of traffic loads during

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21 Table 12. Tire patch length increment for traffic categories (inches). Tire Traffic Category Patch Length 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 - - - - each hour of each day. The number of traffic loads for each where P (Lj) is the probability density function within traffic 1-hour time period in each day for eight traffic categories category j and C (Lj) is the cumulative probability within traf- and tire length increments is used to calculate the bending fic category j. or shearing stress intensity factor. The probability density of For instance, the probability density function for the Cat- tire patch lengths for each traffic category can be deter- egory 1 of LTPP Section 180901 can be determined, based on mined from the cumulative axle load distribution function the cumulative axle load distribution of the section provided as follows: in Figure 18; results are shown in Figure 19. The probability density for all categories of the LTPP sections is provided in dC ( L j ) Appendix E. The default probability density for Level 3 data P (Lj ) = (6) dL j input is presented in Table 17.

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22 Tire Length Cumulative Axle Load Distribution Collecting the axle load interval Collecting the number of axle loads for each category for each category from WIM or AADTT Axle load distribution factor Axle load (lb) Tire load (lb) = Number of tires No. of alxe loads for each tire length = Total No. of axle loads Tire length (in.) tire load (lb) = tire pressure (lb/in.2) tire width (in.) Cumulative axle load distribution Cumulative Axle Load Distribution on Tire Length Figure 14. Determination of cumulative axle load distribution on tire patch length. 1.00 Cumulative Axle Load Distribution 0.80 0.60 0.40 0.20 0.00 0 3 6 9 12 15 18 21 Tire Length (in.) Figure 15. Cumulative axle load distribution versus tire length (Category 1 of LTPP section 180901 in 2004). Cumulative Axle Load Distribution Pi = f (Li) Maximum load P2 = 1 Y (=1.0 for CALD curve) Minimum load to be considered P1 e-1 L1 L2 Tire Length 1 2 X Figure 16. Typical tire length versus cumulative axle load distribution. Figure 17. Gompertz model curve.

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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 C1 0.071 L1 3.704 0.967 C2 1.000 L2 16.933 R2 0.982 Table 14. CALD model parameter default values determined based on LTPP data. Parameters Traffic Category R2 1 3.44056 0.73836 0.980 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 1.00 P2 Cumulative Axle Load Distribution Measured 0.80 Model 0.60 0.40 0.20 P1 L2 0.00 0 3 L1 6 9 12 15 18 21 Tire Length (in.) Figure 18. Cumulative axle load distribution versus tire length (Category 1 of LTPP Section 180901 in 2004).

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Table 15. Minimum values to be considered for load related distress. Minimum Values Traffic Category Axle Type Axle load (lb) Tire Length (in.) 1 3.704 Single 3,000 2 1.669 3 1.588 Tandem 6,000 4 0.715 5 2.117 Tridem 12,000 6 0.953 7 1.588 Quad 12,000 8 0.715 Table 16. Default cumulative axle load distribution for each traffic category. Category No.* 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.

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0.40 0.35 Probability Density 0.30 0.25 0.20 0.15 0.10 0.05 0.00 00 23 29 35 41 47 53 8 4 0 6 2 8 3 9 5 1 7 .5 .6 .7 .7 .8 .8 .9 .9 .0 .1 .1 0. 4. 5. 6. 7. 8. 9. 10 11 12 13 14 15 16 17 19 20 21 Tire Length (in.) Figure 19. Probability density function of tire length (Category 1, LTPP Section 180901). Table 17. Default probability density for each traffic category. Traffic Category No.* 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 0.0551 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 0.0010 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.