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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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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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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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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.
