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WIM-based assessment of load effects on bridges due to various classes of heavy trucks
Engineering Structures 140 (2017) 189–198
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WIM-based assessment of load effects on bridges due to various classes of heavy trucks Habib Tabatabai ⇑, Hani Titi, Jian Zhao Department of Civil and Environmental Engineering, University of Wisconsin, Milwaukee, WI 53211, USA
a r t i c l e
i n f o
Article history: Received 14 November 2016 Revised 24 January 2017 Accepted 23 February 2017
Keywords: Weigh-in-motion Truck loads Statistical analysis Copulas WIM Bridge loads Monte Carlo simulations Permit vehicles
a b s t r a c t This study presents statistical evaluations of extreme load effects on bridges due to different truck classes. Using the developed information, a set of procedures are proposed to conduct probabilistic assessments of the relative severity of any heavy truck loading on simply-supported bridges. The truck information used in the analyses was recorded in one entire year by weigh-in-motion stations located throughout the State of Wisconsin. Data on the heaviest five percent of trucks in each truck class-axle group were extracted for analyses. Best-fit unimodal and multimodal statistical distributions for all axle loads and axle spacings (in each truck class-axle group) were determined. Standard and empirical copulas were generated to allow consideration of interdependencies between various marginal distributions. The accuracy of the developed marginal distributions and empirical copulas were verified using multivariate Monte Carlo simulations. Simulations were also used to determine magnitudes (and percentiles) of moments and shears on simply-supported bridges of various span lengths of up to 240 ft (73.2 m) for different truck class-axle groups. Procedures are provided for assessing the relative magnitude of moments/ shears due to any heavy truck as percentile of each truck class-axle group, or as percentile of the entire truck population. This could allow the process of issuing truck permits to be based on a chosen probability of exceedance. Results indicate that the Class 9 truck statistics can be used to represent the extreme load effects associated with all truck classes/groups combined. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction There are 17 Weigh-In-Motion (WIM) stations in the State of Wisconsin that record truck weight information as vehicles pass over their sensors at normal speeds. The WIM data include all legal and illegal trucks that may cross the WIM sensors, and thus provide a reasonably complete picture of truck loads. WIM information include individual truck class designation, number of axles, axle loads, distances (spacings) between adjacent axles, and total truck weight. Understanding the statistical variability of axle weights and spacings for different classes of truck loads is important with respect to probabilistic evaluation of overweight truck effects on bridges and pavements. Several previous works have analyzed WIM data [1–7]. However, this study focused on the distribution of heavy truck loads (total weights exceeding the 95th percentile level or the heaviest 5% -- H5P) in each truck class-axle grouping, as these heavier loads are most relevant to strength-based limit states. Loads associated ⇑ Corresponding author. E-mail addresses: [email protected] (H. Tabatabai), [email protected] (H. Titi), [email protected] (J. Zhao). http://dx.doi.org/10.1016/j.engstruct.2017.02.060 0141-0296/Ó 2017 Elsevier Ltd. All rights reserved.
with the fatigue limit states are not addressed in this study. The choice of the 95th percentile level was based on the fact that other researchers [1,2] have used this level to represent the tail end of load effects. This 5% range is believed to provide a reasonable balance between the desire to properly capture the extreme load effects relevant to design while accurately representing the statistical distribution of the tail. This study utilized distributions of all individual axle loads and axle spacings instead of overall truck weights or generic axle load distributions. The moment and shear effects on bridges are greatly influenced by axle spacings (as well as axle loads). For example, moments due to axle loads that are spaced far apart can be significantly lower than the corresponding moments associated with closely-spaced axles. It is reasonable to consider that the axle spacing information for trucks that are designed for heavy loads would be different from typical axle spacings in other lighter-capacity trucks (because of truck frame design and other considerations). The Federal Highway Administration’s (FHWA) ‘‘Truck Characteristics Analysis Report” [8] points to a wide range of axle spacings for different truck classes. The Road Safety Authority of Ireland [9] suggests relationships between maximum weights and axle spacings or truck lengths. Therefore, in this study, understanding and
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utilizing reasonably accurate axle spacing information is considered crucial for assessing the moment and shear effects of heavy trucks on bridges. The axle spacing and axle load variables should not be considered independent of each other when simulating truck loads. By separating and analyzing the H5P data, the accuracy of predictions of the effects of heavy loads is expected to improve significantly. Fitting a statistical distribution to the H5P data would provide a more accurate representation of heavy (or extreme) loads compared to analyzing the tail of a distribution that is fitted to the entire dataset, as the tail data are most relevant to failure. Researchers such as Ditlevsen [10], Orien et al. [11], and Diebolt et al. [12] have discussed various issues with the extrapolation to distribution tails as well as the importance of finding distributions that fit the tail data properly. Sivakumar et al. [2] state that ‘‘the most important parameters for load modeling are those that describe the shape of the tail end of the truck load effects histogram”. Sivakumar et al. used an equivalent normal probability distribution to project the upper tail of their WIM data even though they correctly noted that the entire dataset was not normally distributed [2]. Sivakumar et al. [2] state that the tail end of the single load event ‘‘does not follow any known probability distribution type.” This observation illustrates the complexity of fitting a distribution to the tail of data from all truck classes combined. In this study, the tail distributions of different truck classes are determined separately. This allows consideration of the effects of individual truck classes. On the other hand, if needed, distribution of various truck classes can be combined together (based on the fraction of each truck class) to arrive at the desired overall tail distribution. However, when such distributions are combined, the particular axle load - spacing relationship would not be maintained properly. Therefore, the analyses in this study were done separately for each truck class-axle group. Srinivas et al. [13] describe an approach for determining multivariate statistical distributions of truck axle weights and spacing using copulas. Their proposed approach was used in this research to find the relevant distributions with corresponding copulas. It is believed that considering axle weight and axle spacing as independent variables would not be as accurate since any interdependencies among various axle loads and spacings would not be retained. Also, conducting multivariate analyses using linear correlation coefficients would not describe the dependence accurately in such cases (Srinivas et al. [13] and Joe [14]). Therefore, multivariate analyses and simulations using copulas were adopted in this study. The overall objectives of this research were to probabilistically assess the load effects of the heaviest 5% of trucks belonging to various truck classes on simply supported bridges, and to develop procedures that could be used to quantify the risk of exceedance of moments and shears by the actual truck population compared to those generated by specific trucks. To do so, statistical tools were developed to simulate heavy truck load effects on bridges and pavements, and to generate estimates of moments and shears (as well as corresponding percentile levels) associated with each truck group (truck class-number of axles). Such percentile data would be useful in statistically quantifying the relative effects of any specific heavy truck on simply supported bridges with various span lengths. For example, using this information, the effects of a truck subject to issuance of permits can be quantified as percentiles of moments and shears generated by a specific truck class-axle group or all truck groups combined. Bridge owners would then have a choice to limit the percentile level to an acceptable level across different span lengths. Also, moment and shear values associated with a particular span length and percentile level (say 99th percentile) could be determined for bridge rating purposes.
To achieve these objectives, a research study was conducted within the scope of the following tasks: Collect, synthesize, and analyze WIM truck data from all stations in Wisconsin for the entire year of 2007. Analyze the WIM data to obtain axle weight and axle spacing information for heavy trucks in various truck classes. Determine unimodal or multimodal (when applicable) statistical distributions for all axle loads and axle spacings for the heaviest five percent of all trucks in each truck group. Determine multivariate ‘‘copulas” that map relationships between the different marginal distributions. Conduct multivariate Monte Carlo simulations to compare simulated and actual histograms and validate the approach. Conduct multivariate Monte-Carlo simulation studies on simply-supported bridge spans based on the developed marginal distributions and copulas. Evaluate the relative moment/shear effects (and percentiles) of each truck class/axle group on simply supported bridges with spans ranging from 20 to 240 ft (6.1–73.2 m). Develop procedures to estimate overall (across all truck classes) shear and moment percentile levels using the corresponding percentile information from Class 9 trucks. Assess moments and shears due to the Wisconsin Standard Permit Vehicle (Wis-SPV) and the AASHTO design truck (HL-93) relative to individual truck classes and all truck groups combined. The software programs used in the data analysis phase of this study included Crystal BallÒ, a forecasting and simulation program, and ModelRiskÒ, a quantitative risk analysis program. Both programs run within the Microsoft ExcelÒ platform, and both can fit statistical distributions to a given dataset. ModelRiskÒ can also fit standard or empirical copulas. Crystal BallÒ and ModelRiskÒ were run together to perform Monte Carlo simulations involving marginal distributions and copulas.
2. WIM data FHWA has established various truck classifications as defined by the Traffic Monitoring Guide (TMG) [15]. TMG identifies thirteen different vehicle classes. Partial descriptions of these classes are shown in Table 1. In addition to the thirteen FHWA vehicle classes, the WIM data also include what are referred to as truck classes 14 and 15. Class 14 includes truck-trailer combinations and class 15 is allocated to unclassified trucks and/or system errors. Approximately six million truck records (FHWA truck classes 5 through 15) were obtained. Data from Wisconsin WIM stations were exported into spreadsheets for analyses. Data were then sorted based on truck class. For each truck class, a dataset containing the H5P truck data (trucks that weigh more than the 95th percentile level) was created. Truck data were first tested for validity, and invalid truck records were discarded. Three validity tests were performed for each truck record: 1. Is the total weight reported for each truck within 5% of the sum of all axle weights reported? 2. Does the number of axles reported match the number of axle weights reported? 3. Are all axle spacings reported reasonable? For example, records that showed axle spacing of less than 20 in. (508 mm) were discarded.
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H. Tabatabai et al. / Engineering Structures 140 (2017) 189–198 Table 1 FHWA Vehicle Classes (TMG, 2001). Class
Vehicle type
Description
1 2
Motorcycles Passenger Cars
3
Other Two-Axle, Four-Tire Single Unit Vehicles
4
Buses
5
Two-Axle, Six-Tire, Single-Unit Trucks
6
Three-Axle Single-Unit Trucks
7 8
Four or More Axle Single-Unit Trucks Four or Fewer Axle Single-Trailer Trucks
9 10
Five-Axle Single-Trailer Trucks Six or More Axle Single-Trailer Trucks
11
Five or fewer Axle Multi-Trailer Trucks
12 13
Six-Axle Multi-Trailer Trucks Seven or More Axle Multi-Trailer Trucks
All two or three-wheeled motorized vehicles All sedans, coupes, and station wagons manufactured primarily for the purpose of carrying passengers and including those passenger cars pulling recreational or other light trailers All two-axle, four-tire, vehicles, other than passenger cars. Included in this classification are pickups, panels, vans, and other vehicles such as campers, motor homes, ambulances, hearses, carryalls, and minibuses All vehicles manufactured as traditional passenger-carrying buses with two axles and six tires or three or more axles. All vehicles on a single frame including trucks, camping and recreational vehicles, motor homes, etc., with two axles and dual rear wheels All vehicles on a single frame including trucks, camping and recreational vehicles, motor homes, etc., with three axles All trucks on a single frame with four or more axles All vehicles with four or fewer axles consisting of two units, one of which is a tractor or straight truck power unit All five-axle vehicles consisting of two units, one of which is a tractor or straight truck power unit All vehicles with six or more axles consisting of two units, one of which is a tractor or straight truck power unit All vehicles with five or fewer axles consisting of three or more units, one of which is a tractor or straight truck power unit All six-axle vehicles consisting of three or more units, one of which is a tractor or straight truck power unit All vehicles with seven or more axles consisting of three or more units, one of which is a tractor or straight truck power unit
Of the total of 5,761,802 unfiltered records for classes 5 through 15, only 4,352 records (or 0.08%) were discarded based on the above three criteria. Some truck classes may have multiple number of axles. For example, class 7 trucks could have either 4 or 5 axles (designated here as truck class-axle groups or truck groups 07–04 and 07–05), while class 8 trucks could have 3 or 4 axles (truck groups 08–03 and 08–04). Table 2 breaks down the number of trucks based on truck groups. WIM data for classes 13 and 15 trucks include a large number of axle variations within the same class. Data associated with the same number of axles within each class were separated before calculating the distributions. For example, two sets of distributions were determined for class 7 trucks (i.e. 07–04 and 07–05). Table 2 shows the number of filtered trucks in each class (and axle) category as a percentage of trucks in that class as well as percentage of all filtered trucks. Class 9 trucks made up over 61.7% of all WIM trucks. There were over 3.5 million class 9 vehicles in the 2007 data. The second and third most common trucks are classes 8 and 5 at 14.7% and 13.0%, respectively. The maximum and 95 percentile values for the total truck weight in each truck class are also shown in Table 2. For example, the maximum recorded total weight for a class 9 was 1079 kN and the 95 percentile weight was 466.8 kN. Truck classes 9 through 13 and some class 15 truck groups have the highest 95 percentile weight and the heaviest overall maximum weights (Table 2). In the following discussions, truck axles are labeled alphabetically with Axle A being the front axle. Axle spacing labels indicate the two axles at the ends of each spacing. For example, axle spacing BC refers to the distance between axles B and C (Fig. 1). 3. Data analysis The H5P data for all class-axle groups were used to generate best-fit statistical distributions using the ModelRiskÒ software. Data from all stations were combined. Limited Analyses of Variance (ANOVA) showed that truck weights in different WIM stations did not belong to the same distribution. Best fit distributions were determined for each axle weight, axle spacing and the total weight in each truck class-axle category using the maximum likelihood estimation. The total weight distributions were obtained for information only, and were not used in moment and shear assessments. The fitting options within the software could not directly accommodate bimodal (‘‘double hump”) or multi-modal statistical
distributions. However, some H5P axle spacing distributions were in fact multi-modal. Therefore, when data warranted such considerations, the following semi-manual approach was used to determine multi-modal best fit distributions: A histogram of the data was generated. The best fit single-mode distributions were determined. If the histogram indicated multi-modal (‘‘multi-hump”) behavior, then the data was manually separated into grouping around each peak. Best fit single-mode distributions for each group were determined. The number of data points within each grouping divided by the total number of data points is the probability (p) associated with the distribution in that grouping. For example, a tri-modal distribution can be represented as follows: Multi-Modal Distribution ¼ p1 Distribution1 þ p2 Distribution2 þ p3 Distribution3
The resulting multi-modal distribution was plotted and compared with the histogram to make sure that the data agreed with the distribution. Single-mode distribution parameters for all class-axle groups are reported in a research report by Tabatabai et al. [16]. Other related studies addressed a number of aspects of analyses of Wisconsin WIM data [17,18]. In this paper, details of the statistical distributions are not presented for brevity. Additional data are provided as supplementary materials accompanying this paper. Selected histogram and distribution plot for Class 9 trucks are compared in Fig. 2. The motivation behind determining statistical distributions for each parameter in a truck class/axle group was to run Monte Carlo simulations using those marginal distributions. One could perform such simulations assuming that the various axle loads and spacings are independent of each other. However, if such parameters were considered independent of each other, then the relationships between different axle loads and spacings, if any, would be ignored. Srinivas et al. [13] suggest that copulas should be used to model the interdependence of truck load information. Copulas have been widely used in financial and insurance industries to assess financial risk in instruments such as the derivatives [13,19]. Copulas were first introduced by Sklar in 1959 [19]. Copula functions can describe the dependence between the variables involved. The multivariate functions can be estimated by linking the marginal distributions with the copula function [13].
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Table 2 Detailed information on different truck class-axle groups. Truck Class
Truck Group (Class – Axles)
Un-filtered Trucks
No of Filtered Trucks
% of Total
% of Truck Class
Class 5 Class 6 Class 7
05–02 06–03 Total 07–04 07–05 Total 08–03 08–04 09–05 Total 10–06 10–07 11–05 12–06 Total 13–07 13–08 13–09 13–10 13–11 13–12 13–13 14–05 Total 15–02 15–03 15–04 15–05 15–06 15–07 15–08 15–09 15–10 15–11 15–12 15–13 15–14
749,409 251,914 73,172
748,658 251,795 73,138 25,753 47,385 848,483 634,745 213,738 3,553,613 77,185 72,939 4246 94,572 30,576 10,595 9738 680 75 65 10 8 19 1128 67,708 3071 13,617 10,013 9057 19,507 4164 4781 1264 727 489 384 341 293
13.00% 4.37% 1.27% 0.45% 0.82% 14.74% 11.02% 3.71% 61.72% 1.34% 1.27% 0.07% 1.64% 0.53% 0.18% 0.17% 0.01% 0.00% 0.00% 0.00% 0.00% 0.00% 0.02% 1.18% 0.05% 0.24% 0.17% 0.16% 0.34% 0.07% 0.08% 0.02% 0.01% 0.01% 0.01% 0.01% 0.01%
100.00% 100.00%
Class 8
Class 9 Class 10
Class 11 Class 12 Class 13
Class 14 Class 15
849,072
3,554,700 77,764
94,578 30,577 10,907
1131 68,578
35.21% 64.79% 74.81% 25.19% 100.00% 94.50% 5.50% 100.00% 100.00% 91.91% 6.42% 0.71% 0.61% 0.09% 0.08% 0.18% 100.00% 4.54% 20.11% 14.79% 13.38% 28.81% 6.15% 7.06% 1.87% 1.07% 0.72% 0.57% 0.50% 0.43%
H5P 95th percentile (KN)
Max weight (KN)
109.8 237.3 398.1 355.0 414.8 205.9 115.7 304.0 466.8 518.7 509.9 638.4 519.7 577.6 579.5 572.7 543.3 724.7 724.7 755.1 930.6 1095.4 316.7 435.4 153.9 119.7 254.0 328.5 413.8 445.2 723.7 811.0 899.2 743.3 410.9 272.6 235.3
349.1 533.4 832.5 832.5 764.9 680.5 646.2 680.5 1078.7 1188.5 1188.5 1048.3 805.1 913.0 1459.2 1459.2 715.9 788.4 892.4 755.1 930.6 1095.4 450.1 1885.7 220.6 246.2 413.8 465.8 604.0 582.5 1184.6 1538.6 1885.7 1066.9 1600.3 1432.7 584.5
Fig. 1. Axle load and spacing designations.
There are many types of functions that can serve as copulas. Two prominent groups of copulas are Elliptical Copulas and Archimedean Copulas. The Gaussian and Student’s T copulas belong in the Elliptical group while Clayton, Gumbel and Frank copulas belong in the Archimedean group [13]. On the other hand, empirical copulas are based on actual data and are not fit to particular mathematical functions [20]. Best-fit standard copulas were determined by the software based on data entered into a spreadsheet. The empirical copulas were also determined by the software based on its proprietary approach. In this study, both approaches (standard and empirical
copulas) were examined. Additional information on the copulas are provided as supplementary materials and in Ref. [16]. As expected, the empirical copulas (which utilize actual data each time) were best able to simulate total truck weight distributions when such simulations were compared with corresponding histograms. Therefore, empirical copulas were used for the Monte Carlo simulations. The Crystal BallÒ software was used for Monte Carlo simulations. Two types of Monte Carlo simulations were performed. First, analyses were performed to demonstrate that the simulation results were similar to the histograms of data. Fig. 3 shows such
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Axle B Weight 0.025
0.02
0.02
Probability
Probability
Axle A Weight 0.025
0.015 0.01 0.005 0
0.015 0.01 0.005
0
40
80
120
160
0
200
0
40
80
100 kg
0.02
0.02
0.015 0.01 0.005
0.01 0.005
0
50
100
150 100 kg
200
250
0
300
0
50
100
BC Spacing
150 100 kg
200
250
300
CD Spacing 0.07
0.6
0.06
0.5
0.05
Probability
Probability
200
0.015
0.7
0.4 0.3 0.2 0.1 0
160
Axle E Weight 0.025
Probability
Probability
Axle D Weight 0.025
0
120 100 kg
0.04 0.03 0.02 0.01
0
20
40
60
80
0
100
0
20
40
60
100 mm
80 100 100 mm
120
140
160
Fig. 2. Representative H5P Histograms and distributions for Class 9 trucks.
Simulation Result
WIM Data 0.07
Probability
0.06 0.05 0.04 0.03 0.02 0.01 0
100
120
140
160
180
200
Kips Fig. 3. Total truck weight - Comparison of Class 9 truck simulation results with H5P WIM data.
a comparison for Class 9 trucks. In this case, individual axle load simulations (considering copulas) were made for all five axles and the individual axle loads were added together to estimate the total weight for each simulation run. The resulting simulated and actual total weight histograms are shown in Fig. 3. Overall, the comparisons between simulated and actual histograms were favorable for all truck groups. For the second type of analyses, a spreadsheet was setup to calculate bending moment and shear envelopes for any moving truck
arrangements (up to 10 axles) using influence lines. A simple-span bridge condition with spans ranging from 20 ft to 240 ft (6.1– 73.2 m) was considered. Each simulated truck was ‘‘marched” across the span in increments and the moment and shear envelopes were calculated. For simulations, the distributions and copulas were applied to all axle loads and axle spacings. Statistical information on moments and shears due to H5P simulations of truck class/axle groups were then developed for simplespan bridges. In addition, the 845 kN Wis-SPV was analyzed, and
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the maximum moments and shears due to Wis-SPV were compared with the simulation results to determine the effect of WisSPV (as a percentile of the simulation results) for each truck class–axle group. The Wisconsin Department of Transportation requires that the Wis-SPV vehicle be checked during design, rehabilitation, or rating of all Wisconsin bridges [21]. This vehicle is evaluated in conjunction with lane loading, with the dynamic allowance and single-lane-loading option utilized. Fig. 4 shows axle loads and spacings for the Wis-SPV truck [21]. The summary of Monte Carlo simulation results for Class 9 trucks are shown in Tables 3 and 4. Moment and shear values cor-
responding to various overall percentile levels are provided. Since H5P data have the heaviest overall weights in that class, it is reasonable to assume that the effects due to H5P vehicles will result in higher moments and shears compared to the remaining 95% of trucks in that class. Therefore, the overall percentile is estimated through the following relationship:
Overall percentile ¼ 95 þ ðH5P percentileÞ 0:05 It should be noted that there is nonlinear behavior between the 99th and 100th percentile results. Therefore, to improve accuracy between 99th and 100th percentiles, analyses were run at 0.05
63 ft (19.2 m) 19.0
26.6
26.6
26.6
22.8
22.8
22.8
22.8
190 kips
84.5
118.3
118.3
118.3
101.4
101.4
101.4
101.4
845 kN
Fig. 4. The 845 kN Wisconsin Standard Permit Vehicle (Wis-SPV).
Table 3 Summary of moments - Monte Carlo simulation results for H5P Class 9 trucks. Max Moment
Percentiles
Span
95th
ft
m
Moment (kN-m)
20 40 60 80 100 120 140 160 180 200 220 240
6.1 12.2 18.3 24.4 30.5 36.6 42.7 48.8 54.9 61.0 67.1 73.2
180.9 505.1 929.2 1361.9 1996.7 2749.0 3510.5 4259.2 5013.7 5758.3 6520.6 7236.1
96th
97th
98th
99th
99.5th
281.0 676.3 1138.6 1720.2 2515.3 3326.0 4134.2 4943.0 5751.5 6560.0 7369.7 8180.0
307.7 732.3 1235.5 1864.4 2718.3 3586.5 4455.5 5325.7 6198.3 7075.2 7941.7 8813.5
331.1 784.4 1326.0 2004.3 2925.5 3866.1 4803.6 5750.4 6691.5 7634.2 8573.4 9515.0
359.1 845.7 1431.1 2171.5 3170.0 4183.8 5199.1 6213.4 7235.0 8253.5 9264.0 10279.0
379.7 891.2 1513.1 2296.0 3344.5 4416.1 5486.6 6568.4 7643.4 8713.4 9793.3 10859.4
99.9th
99.95th
100th
413.3 968.9 1645.9 2508.2 3638.4 4791.0 5956.8 7122.8 8288.4 9453.6 10635.7 11785.9
424.8 1004.0 1699.9 2597.4 3768.2 4957.3 6169.5 7371.2 8568.6 9765.7 10968.8 12153.5
503.6 1303.1 2506.4 4068.5 5650.0 7159.6 8788.8 10294.8 11865.8 13436.1 14986.4 16499.1
Max Moment
Percentiles
Span
99.6th
fet
m
20 40 60 80 100 120 140 160 180 200 220 240
6.1 12.2 18.3 24.4 30.5 36.6 42.7 48.8 54.9 61.0 67.1 73.2
99.7th
99.8th Moment (kN-m)
385.5 904.5 1533.9 2330.5 3400.1 4485.6 5572.5 6665.5 7755.3 8839.5 9925.7 11016.0
392.2 921.7 1560.8 2367.8 3453.4 4550.5 5647.5 6754.7 7861.2 8968.7 10074.3 11174.5
400.2 939.7 1593.9 2418.3 3525.7 4643.7 5768.1 6893.3 8017.1 9145.1 10282.6 11406.3
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H. Tabatabai et al. / Engineering Structures 140 (2017) 189–198 Table 4 Summary of shears - Monte Carlo simulation results for H5P Class 9 trucks. Max Shear
Percentiles
Span
95th
96th
97th
98th
99th
99.5th
123.7 193.5 230.9 268.2 306.9 331.8 349.2 364.8 372.3 382.1 388.8 395.9
194.4 238.4 279.8 328.7 362.1 384.3 400.3 412.4 421.7 428.8 435.0 439.9
210.8 258.4 300.7 353.6 389.7 413.7 431.0 443.9 454.2 462.2 468.8 474.6
226.4 277.6 322.9 381.7 421.2 447.5 466.2 480.4 491.1 499.5 507.1 512.9
244.2 300.3 349.2 413.2 455.5 484.0 504.0 519.1 531.1 540.9 548.5 554.2
257.8 316.8 369.5 436.9 480.9 510.4 531.5 548.2 560.9 570.6 578.9 586.1
99.6th
99.7th
99.8th
99.9th
99.95th
100th
279.7 344.8 402.6 476.5 522.6 554.3 576.9 595.5 610.7 619.9 629.1 635.7
289.8 356.2 416.8 491.7 539.8 573.6 596.7 614.6 628.3 639.2 648.9 655.3
348.3 470.6 617.9 703.7 754.4 785.6 816.7 830.5 842.0 857.6 867.4 871.4
ft
m
Shear (kN)
20 40 60 80 100 120 140 160 180 200 220 240
6.1 12.2 18.3 24.4 30.5 36.6 42.7 48.8 54.9 61.0 67.1 73.2 Max Shear
Percentiles
Span ft
m
20 40 60 80 100 120 140 160 180 200 220 240
6.1 12.2 18.3 24.4 30.5 36.6 42.7 48.8 54.9 61.0 67.1 73.2
Shear (kN) 261.6 321.1 374.3 443.0 487.3 517.5 539.1 555.8 568.3 578.5 586.5 593.4
265.8 326.6 381.1 450.5 495.6 526.2 547.8 564.5 577.3 587.5 595.6 602.8
271.3 333.2 389.4 460.3 506.8 537.4 559.5 576.6 589.3 599.5 607.8 615.3
Table 5 Simulation moments corresponding to 80th percentile of H5P data or 99th percentile for selected class-axle groups. Span
Truck Class - Number of Axles 05–02
ft
m
Moment (kN-m)
20 40 60 80 100 120 140 160 180 200 220 240
6.1 12.2 18.3 24.4 30.5 36.6 42.7 48.8 54.9 61.0 67.1 73.2
156 347 596 846 1097 1348 1599 1851 2102 2352 2602 2855
07–05
08–04
09–05
10–06
13–07
HL-93 Truck
456 1179 2033 2908 3782 4655 5528 6411 7279 8164 9048 9929
280 666 1118 1657 2263 2882 3508 4147 4785 5429 6068 6712
359 845 1430 2171 3168 4182 5197 6210 7232 8250 9260 10,274
405 978 1863 2880 3942 5016 6095 7179 8258 9342 10,424 11,519
406 1050 1812 2729 3848 5044 6276 7531 8811 10,089 11,379 12,639
217 610 1094 1580 2067 2554 3042 3530 4017 4505 4993 5481
percentile increments, so that interpolations can be made more accurately. The 95th through 99th percentile values are shown at 1 percentile increments in Tables 3 and 4. In addition, the 99.5th, 99.6th, 99.7th, 99.8th, 99.9th and the 99.95th percentile level are also shown in those tables to improve accuracy. Tables 5 and 6 show the 99th percentile moments and shears for different span lengths corresponding to selected truck class/ axle groupings. Information on all other truck groups are presented as supplementary materials included with this paper. The 99th percentile values are shown because Moses [1] suggests this level for characterizing the expected maximum live load. Sivakumar et al. [2] also used the 99th percentile level as ‘‘the maximum anticipated load for design”. It should be noted that Moses [1] and Sivakumar et al. [2] were referring to the 99th percentile truck weight, and not the 99th percentile
moments and shears due to individual truck class-axle groups used in this study. Tables 5 and 6 also show moments and shears due to the HL-93 design truck, respectively. The HL-93 truck values shown are for one design lane without dynamic allowance. The effect of lane loading is not included in the HL93 results shown. As stated earlier, the Class 9 truck group is the most prevalent truck class on Wisconsin roads. Therefore, it is reasonable to focus on the effects of that truck class. An effort was made to assess whether the results of Class 9 truck simulations can be used to represent the entire truck population with respect to the maximum anticipated load effects for design or rating purposes. The 99th percentile moment and shear values associated with the Class 9 truck group would, of course, not necessarily coincide with the 99th percentile levels across all trucks.
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Table 6 Simulation shears corresponding to 80th percentile of H5P data or 99th percentile for selected class-axle groups. Span
Truck Class - Number of Axles 05–02
ft
m
Shear (kN)
20 40 60 80 100 120 140 160 180 200 220 240
6.1 12.2 18.3 24.4 30.5 36.6 42.7 48.8 54.9 61.0 67.1 73.2
102.3 125.4 133.4 137.4 140.1 141.4 142.8 143.7 144.6 145.0 145.4 145.9
07–05
08–04
09–05
10–06
13–07
HL-93 Truck
314.5 408.3 449.2 469.3 481.7 489.3 495.1 499.5 502.2 504.8 507.5 509.3
189.5 230.4 255.3 279.3 300.2 314.5 324.7 331.8 338.0 342.9 346.9 350.1
244.2 300.2 349.2 413.2 455.5 483.9 504.0 519.1 531.1 540.9 548.4 554.2
274.0 382.1 462.1 509.3 538.2 558.2 572.0 583.1 592.0 598.3 604.5 608.5
281.6 360.6 427.7 498.0 550.0 587.7 616.4 637.9 654.1 667.8 679.9 688.1
185.0 245.5 270.4 282.9 290.5 295.3 298.9 301.6 303.8 305.1 306.5 307.8
Table 7 Comparison of 99th and 98.59th percentile moments for Class 9 trucks with HL-93 moments. Span
Moments
Ratio Class 9 (98.59th)/HL-93
Class 9 - 99th Percentile moments
Class 9–98.59th Percentile moments
HL-93
ft
m
kN-m
Percentile of All Trucks
kN-m
Percentile of All Trucks
kN-m
20 40 60 80 100 120 140 160 180 200 220 240
6.1 12.2 18.3 24.4 30.5 36.6 42.7 48.8 54.9 61.0 67.1 73.2
358.9 845.3 1430.4 2170.5 3168.4 4181.9 5196.7 6210.5 7231.5 8249.6 9259.6 10274.1
99.28 99.26 99.23 99.22 99.26 99.28 99.29 99.30 99.30 99.30 99.31 99.31
346.5 818.2 1383.3 2091.1 3054.8 4033.8 5015.5 5999.4 6984.5 7969.5 8953.2 9931.0
99.00 98.99 98.95 98.93 98.96 98.99 99.00 99.01 99.01 99.02 99.02 99.02
217.0 610.5 1094.3 1580.1 2067.0 2554.3 3041.8 3529.5 4017.3 4505.2 4993.1 5481.0
1.60 1.34 1.26 1.32 1.48 1.58 1.65 1.70 1.74 1.77 1.79 1.81
Average Percentile
99.28
Average Percentile
98.99
Average Ratio
1.60
The truck groups with higher 95th percentile total weights than the Class 9 group represent a small fraction of trucks. These include Class 10 trucks (representing 1.34% of all trucks), Class 11 trucks (1.64% of all trucks), Class 12 trucks (0.53% of all trucks), and Class 13 trucks (0.18% of all trucks). To determine where the Class 9 moment and shear percentile values fall with respect to the 99th percentile of moment and shear values for all trucks, the following procedures were devised and implemented. First, the Class 9 moment and shear values were compared against the moment and shear simulation results for all other truck groups to determine percentile levels associated with those values at each span length. Second, knowing the percentile levels and the total number of trucks in each group, the number of trucks exceeding the target moment/shear values in each truck group can be determined. Third, the sum of these truck numbers across all truck groups was determined and divided by the number of all trucks to determine the percent of trucks exceeding the target moment/shear values across all truck groups. Finally, subtracting this percentage from 100 would provide the desired percentile level. Tables 7 and 8 show percentile levels for moment and shears, respectively, for various span lengths. Again, because of space limitations, complete data for all truck groups are presented as supplementary materials. The average overall (all truck classes combined) percentile levels were 99.28 and 99.30 for moment and shear, respectively. Therefore, the overall percentile levels are slightly above the target of 99th percentile level. After a number of trials, the Class 9 percentile levels that would result in the overall 99th percentile levels were determined. These were Class
9 percentiles of 98.59 and 98.55 (roughly 98.6th percentile) for moment and shear, respectively. As shown in Tables 7 and 8, these moment and shear levels would result in an average overall percentile levels of 99 for both moment and shear. These moment and shear values represent the maximum anticipated moment and shear values for design based on this set of Wisconsin WIM data. The ratios of these 99th percentile moments and shears to the HL-93 values are shown in Tables 7 and 8. The average ratios across all span lengths is approximately 1.60. However, these ratios can vary for different span lengths. Moments and shears due to any specific design load and/or permit vehicle (such as Wis-SPV) can be compared with the percentiles in Tables 3 and 4. For example, the moment and shear effects due to Wis-SPV are compared with simulation results in Tables 9 and 10. The moments due to Wis-SPV for various span lengths were, on average, at the 99.96th percentile of Class 9 results, and the 99.95th percentile of all trucks. The corresponding percentile levels for shear were 99.96 for Class 9 and 99.97 for all trucks. Therefore, the overall average probability of exceedance is 5 in 10,000 for moment and 3 in 10,000 for shear. 4. Summary and conclusions This study involved statistical evaluation of heavy truck loads that were recorded using WIM stations located throughout the State of Wisconsin. Nearly 6 million vehicle records were collected for the year 2007. Information on trucks Classes 5 through 15 were retained (i.e. non-truck classes were eliminated).
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H. Tabatabai et al. / Engineering Structures 140 (2017) 189–198 Table 8 Comparison of 99th and 98.55th percentile shears for Class 9 trucks with HL-93 shears. Span
Shears
Ratio Class 9 (98.55th)/HL-93
Class 9 - 99th Percentile shears
Class 9–98.55th Percentile shears
HL-93 Loading
ft
m
kN
Percentile of All Trucks
kN
Percentile of All Trucks
kN
20 40 60 80 100 120 140 160 180 200 220 240
6.1 12.2 18.3 24.4 30.5 36.6 42.7 48.8 54.9 61.0 67.1 73.2
244.2 300.3 349.2 413.2 455.5 484.0 504.0 519.1 531.1 540.9 548.5 554.2
99.28 99.25 99.25 99.29 99.31 99.32 99.32 99.32 99.32 99.33 99.33 99.33
235.6 289.2 336.2 398.1 438.6 466.0 485.8 500.7 512.1 521.1 528.5 534.8
98.98 98.95 98.94 98.99 99.01 99.01 99.02 99.02 99.02 99.02 99.02 99.02
185.0 245.5 270.4 282.9 290.5 295.3 298.9 301.6 303.8 305.1 306.5 307.8
1.27 1.18 1.24 1.41 1.51 1.58 1.63 1.66 1.69 1.71 1.72 1.74
Average Percentile
99.30
Average Percentile
99.00
Average Ratio
1.54
Table 9 Wis-SPV moments compared with simulation results for various truck groups. Span
Maximum Moment
Truck Class-Axle Group 05–02
ft
m
kN-m
Percentiles
20 40 60 80 100 120 140 160 180 200 220 240
6.1 12.2 18.3 24.4 30.5 36.6 42.7 48.8 54.9 61.0 67.1 73.2
428.2 1074.9 1782.8 2665.4 3962.7 5322.1 6690.9 8083.7 9448.1 10787.4 12184.5 13587.2
100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
07–05
08–04
09–05
10–06
13–07
ALL
98.33 98.02 97.52 98.18 99.35 99.80 99.92 99.96 99.96 99.97 99.98 99.98
99.95 99.97 99.96 99.96 99.98 100.00 100.00 100.00 100.00 100.00 100.00 100.00
99.95 99.96 99.96 99.95 99.96 99.96 99.96 99.96 99.96 99.96 99.97 99.97
99.37 99.61 98.50 98.10 99.04 99.45 99.65 99.77 99.82 99.84 99.87 99.89
99.29 99.13 98.85 98.80 99.16 99.28 99.34 99.38 99.39 99.40 99.43 99.46
99.94 99.95 99.93 99.93 99.95 99.96 99.97 99.97 99.97 99.97 99.98 99.98
Table 10 Wis-SPV shears compared with simulation results for various truck groups. Span
Maximum Shear
Truck Class-Axle Group 05–02
ft
m
kN
Percentiles
20 40 60 80 100 120 140 160 180 200 220 240
6.1 12.2 18.3 24.4 30.5 36.6 42.7 48.8 54.9 61.0 67.1 73.2
284 359 450 546 604 644 677 693 707 724 736 743
100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
07–05
08–04
09–05
10–06
13–07
ALL
97.85 97.56 99.02 99.85 99.95 99.97 99.98 99.99 99.99 100.00 100.00 100.00
99.95 99.96 99.99 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
99.92 99.95 99.96 99.96 99.97 99.97 99.97 99.97 99.97 99.97 99.97 99.97
99.26 98.29 98.73 99.51 99.75 99.83 99.88 99.90 99.91 99.92 99.92 99.92
99.05 98.97 99.26 99.47 99.52 99.52 99.56 99.52 99.51 99.53 99.5 99.52
99.92 99.92 99.95 99.97 99.97 99.98 99.98 99.98 99.98 99.98 99.98 99.98
If a truck class contained multiple numbers of axles, that class was further sub-divided such that each sub-group contained only one particular number of axles. Data in each class-axle groupings were sorted based on gross vehicle weight, and the heaviest 5 percent of truck records in each group were separated and analyzed. Statistical analyses were performed on the H5P data. Using the H5P data, best-fit unimodal and/or multimodal marginal distributions were determined for each axle weight and spacing in each truck class-axle group. Furthermore, copulas were determined to allow multivariate Monte Carlo simulations.
Multivariate Monte Carlo simulations on H5P data in each classaxle group were conducted using software programs running within a spreadsheet. The following observations are made: (1) Truck simulations for each class-axle grouping were performed and successfully tested for validity against the histogram of data. (2) Some H5P axle spacing distributions are multimodal, and multimodal marginal distributions must be used in such cases for proper simulations.
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(3) Empirical copulas provide more accurate simulations when compared to standard copula functions determined by data fitting. All simulation results reported here are based on empirical copulas determined using the software. (4) The marginal distributions and copulas determined here can be used to statistically assess heavy truck impact on bridges and pavements based any load-dependent metric. (5) The percentile results derived can be used to assess the relative impact of any truck arrangement compared to simulation results. Moments and shears due to the Wisconsin Standard Permit Vehicle were, on average, approximately equivalent to the 99.8th percentile level for all trucks. (6) The percentiles of simulated shear and moment for various truck class-axle groups can be used to estimate percentiles for all trucks (all truck groups combined) using the procedures discussed here. Such data can then be used to estimate the 99th percentile shear and moment values (for different span lengths). These results can then be used for bridge load ratings based on each state’s own WIM data. (7) The Class 9 truck statistics can be used to represent the entire truck population for the purpose of estimating WIMbased design loads, or for issuance of permits. Class 9 percentile level of 98.6 and 98.55 are equivalent to the 99th percentile level associated with the entire truck population for moments and shears, respectively. As the most common truck class, the Class 9 data with its available statistical information can provide a relatively simple tool for assessing the probabilistic structural impact of all trucks. (8) Using the proposed probabilistic approach and the results of this research, bridge owners can set criteria (percentile thresholds) for issuance of single-trip or annual overweight truck permits.
Acknowledgements This research was completed with the partial support and partnership of the National Center for Freight and Infrastructure Research and Education (CFIRE), a Tier 1 University Transportation Center (UTC) funded by the U.S. Department of Transportation (USDOT) Research and Innovative Technology Administration (RITA). Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.engstruct.2017. 02.060.
References [1] Moses F. Calibration of load factors for LRFR bridge evaluation. NCHRP Report 454. Washington DC: Transportation Research Board; 2001. [2] Sivakumar B, Ghosn M, Moses F. Protocols for collecting and using traffic data in bridge design. NCHRP Report 683. Washington DC: National Research Council; 2011. [3] Fiorillo G, Ghosn M. Procedure for statistical categorization of overweight vehicles in a WIM database. J Transport Eng, ASCE 2014. ISSN 0733-947X/ 04014011(11). [4] Ghosn M, Sivakumar B, Miao F. Development of state-specific load and resistance factor rating method. J Bridge Eng, ASCE 2013;18(5):351–61. [5] Lu Q, Harvey J, Le T, Lea J, Quinley R, Redo D, et al. Truck traffic analysis using weigh-in-motion (WIM) data in California. Report by Pavement Research Center, University of California, Berkeley, available online at:; June 2002. [6] Obrien EJ, Enright B. Using weigh-in-motion data to determine aggressiveness of traffic for bridge loading. J Bridge Eng, ASCE 2013;18(3):232–9. [7] Haider SW, Harichandran RS. The impact of weigh-in-motion measurement error on mechanistic-empirical pavement design guide reliability. In: Proceedings of the transportation and development institute congress. p. 548–57. [8] FHWA. Truck characteristics analysis. Federal Highway Administration; July 1999. [9] RSA. Guidelines on maximum weights and dimensions of mechanically propelled vehicles and trailers, including maneuverability criteria. Road Safety Authority, Ireland; February 2013. [10] Ditlevsen O. Distribution arbitrariness in structural reliability. In: Shinozuka Schuller, Yao, editors, Structural safety and reliability, proceedings of ICOSSAR’ 93: 6th international conference on structural safety and reliability. Rotterdam: Balkema; 1994. p. 1241–47. [11] Obrien EJ, Enright B, Getachew A. Importance of the tail in truck weight modeling for bridge assessment. J Bridge Eng, ASCE 2010;15(2):210–3. [12] Diebold J, Garrido M, Girard S. A goodness-of-fit test for the distribution tail. Topics in extreme values, Nove Science, NY; 2007. p. 95–109, 9781600217142. [13] Srinivas S, Menon D, Prasad AM. Multivariate simulation and multimodal dependence modeling of vehicle axle weights and copulas. J Transport Eng, ASCE 2006;132(12):945–55. [14] Joe H. Multivariate models and dependence concepts. London: Chapman and Hall Publishers; 1997. [15] TMG. Traffic monitoring guide. Federal Highway Administration, available online at ; 2001. [16] Tabatabai H, Zhao J, Lee C-W. Statistical analysis of heavy truck loads using Wisconsin weigh-in-motion data. Report No. 01–02, National Center for Freight & Infrastructure Research & Education (CFIRE), University of Wisconsin-Madison; 2009. 197 p. [17] Zhao J, Tabatabai H. Analysis of permit vehicle loads in Wisconsin. Wisconsin Highway Research Program, SPR # 0092-08-15, Report No. WHRP 09-03, September 2009. 183 p. [18] Zhao J, Tabatabai H. Evaluation of a permit vehicle model using weigh-inmotion truck records. J Bridge Eng, ASCE 2012;17(2):389–92. [19] Sklar A. Fonctions de repa’rticion a’ n dimensions et leurs marges, vol. 8. Paris: Publications de l’Institut de Statistiue del’Universit de Paris; 1959. p. 229–31. [20] ModelRisk. By Vose Software. ; 2012. [21] WisDOT Bridge Manual. Chapter 45 – Bridge Rating. Wisconsin Department of Transportation, available at: ; November 2016.
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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
WIM-based assessment of load effects on bridges due to various classes of heavy trucks Habib Tabatabai ⇑, Hani Titi, Jian Zhao Department of Civil and Environmental Engineering, University of Wisconsin, Milwaukee, WI 53211, USA
a r t i c l e
i n f o
Article history: Received 14 November 2016 Revised 24 January 2017 Accepted 23 February 2017
Keywords: Weigh-in-motion Truck loads Statistical analysis Copulas WIM Bridge loads Monte Carlo simulations Permit vehicles
a b s t r a c t This study presents statistical evaluations of extreme load effects on bridges due to different truck classes. Using the developed information, a set of procedures are proposed to conduct probabilistic assessments of the relative severity of any heavy truck loading on simply-supported bridges. The truck information used in the analyses was recorded in one entire year by weigh-in-motion stations located throughout the State of Wisconsin. Data on the heaviest five percent of trucks in each truck class-axle group were extracted for analyses. Best-fit unimodal and multimodal statistical distributions for all axle loads and axle spacings (in each truck class-axle group) were determined. Standard and empirical copulas were generated to allow consideration of interdependencies between various marginal distributions. The accuracy of the developed marginal distributions and empirical copulas were verified using multivariate Monte Carlo simulations. Simulations were also used to determine magnitudes (and percentiles) of moments and shears on simply-supported bridges of various span lengths of up to 240 ft (73.2 m) for different truck class-axle groups. Procedures are provided for assessing the relative magnitude of moments/ shears due to any heavy truck as percentile of each truck class-axle group, or as percentile of the entire truck population. This could allow the process of issuing truck permits to be based on a chosen probability of exceedance. Results indicate that the Class 9 truck statistics can be used to represent the extreme load effects associated with all truck classes/groups combined. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction There are 17 Weigh-In-Motion (WIM) stations in the State of Wisconsin that record truck weight information as vehicles pass over their sensors at normal speeds. The WIM data include all legal and illegal trucks that may cross the WIM sensors, and thus provide a reasonably complete picture of truck loads. WIM information include individual truck class designation, number of axles, axle loads, distances (spacings) between adjacent axles, and total truck weight. Understanding the statistical variability of axle weights and spacings for different classes of truck loads is important with respect to probabilistic evaluation of overweight truck effects on bridges and pavements. Several previous works have analyzed WIM data [1–7]. However, this study focused on the distribution of heavy truck loads (total weights exceeding the 95th percentile level or the heaviest 5% -- H5P) in each truck class-axle grouping, as these heavier loads are most relevant to strength-based limit states. Loads associated ⇑ Corresponding author. E-mail addresses: [email protected] (H. Tabatabai), [email protected] (H. Titi), [email protected] (J. Zhao). http://dx.doi.org/10.1016/j.engstruct.2017.02.060 0141-0296/Ó 2017 Elsevier Ltd. All rights reserved.
with the fatigue limit states are not addressed in this study. The choice of the 95th percentile level was based on the fact that other researchers [1,2] have used this level to represent the tail end of load effects. This 5% range is believed to provide a reasonable balance between the desire to properly capture the extreme load effects relevant to design while accurately representing the statistical distribution of the tail. This study utilized distributions of all individual axle loads and axle spacings instead of overall truck weights or generic axle load distributions. The moment and shear effects on bridges are greatly influenced by axle spacings (as well as axle loads). For example, moments due to axle loads that are spaced far apart can be significantly lower than the corresponding moments associated with closely-spaced axles. It is reasonable to consider that the axle spacing information for trucks that are designed for heavy loads would be different from typical axle spacings in other lighter-capacity trucks (because of truck frame design and other considerations). The Federal Highway Administration’s (FHWA) ‘‘Truck Characteristics Analysis Report” [8] points to a wide range of axle spacings for different truck classes. The Road Safety Authority of Ireland [9] suggests relationships between maximum weights and axle spacings or truck lengths. Therefore, in this study, understanding and
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utilizing reasonably accurate axle spacing information is considered crucial for assessing the moment and shear effects of heavy trucks on bridges. The axle spacing and axle load variables should not be considered independent of each other when simulating truck loads. By separating and analyzing the H5P data, the accuracy of predictions of the effects of heavy loads is expected to improve significantly. Fitting a statistical distribution to the H5P data would provide a more accurate representation of heavy (or extreme) loads compared to analyzing the tail of a distribution that is fitted to the entire dataset, as the tail data are most relevant to failure. Researchers such as Ditlevsen [10], Orien et al. [11], and Diebolt et al. [12] have discussed various issues with the extrapolation to distribution tails as well as the importance of finding distributions that fit the tail data properly. Sivakumar et al. [2] state that ‘‘the most important parameters for load modeling are those that describe the shape of the tail end of the truck load effects histogram”. Sivakumar et al. used an equivalent normal probability distribution to project the upper tail of their WIM data even though they correctly noted that the entire dataset was not normally distributed [2]. Sivakumar et al. [2] state that the tail end of the single load event ‘‘does not follow any known probability distribution type.” This observation illustrates the complexity of fitting a distribution to the tail of data from all truck classes combined. In this study, the tail distributions of different truck classes are determined separately. This allows consideration of the effects of individual truck classes. On the other hand, if needed, distribution of various truck classes can be combined together (based on the fraction of each truck class) to arrive at the desired overall tail distribution. However, when such distributions are combined, the particular axle load - spacing relationship would not be maintained properly. Therefore, the analyses in this study were done separately for each truck class-axle group. Srinivas et al. [13] describe an approach for determining multivariate statistical distributions of truck axle weights and spacing using copulas. Their proposed approach was used in this research to find the relevant distributions with corresponding copulas. It is believed that considering axle weight and axle spacing as independent variables would not be as accurate since any interdependencies among various axle loads and spacings would not be retained. Also, conducting multivariate analyses using linear correlation coefficients would not describe the dependence accurately in such cases (Srinivas et al. [13] and Joe [14]). Therefore, multivariate analyses and simulations using copulas were adopted in this study. The overall objectives of this research were to probabilistically assess the load effects of the heaviest 5% of trucks belonging to various truck classes on simply supported bridges, and to develop procedures that could be used to quantify the risk of exceedance of moments and shears by the actual truck population compared to those generated by specific trucks. To do so, statistical tools were developed to simulate heavy truck load effects on bridges and pavements, and to generate estimates of moments and shears (as well as corresponding percentile levels) associated with each truck group (truck class-number of axles). Such percentile data would be useful in statistically quantifying the relative effects of any specific heavy truck on simply supported bridges with various span lengths. For example, using this information, the effects of a truck subject to issuance of permits can be quantified as percentiles of moments and shears generated by a specific truck class-axle group or all truck groups combined. Bridge owners would then have a choice to limit the percentile level to an acceptable level across different span lengths. Also, moment and shear values associated with a particular span length and percentile level (say 99th percentile) could be determined for bridge rating purposes.
To achieve these objectives, a research study was conducted within the scope of the following tasks: Collect, synthesize, and analyze WIM truck data from all stations in Wisconsin for the entire year of 2007. Analyze the WIM data to obtain axle weight and axle spacing information for heavy trucks in various truck classes. Determine unimodal or multimodal (when applicable) statistical distributions for all axle loads and axle spacings for the heaviest five percent of all trucks in each truck group. Determine multivariate ‘‘copulas” that map relationships between the different marginal distributions. Conduct multivariate Monte Carlo simulations to compare simulated and actual histograms and validate the approach. Conduct multivariate Monte-Carlo simulation studies on simply-supported bridge spans based on the developed marginal distributions and copulas. Evaluate the relative moment/shear effects (and percentiles) of each truck class/axle group on simply supported bridges with spans ranging from 20 to 240 ft (6.1–73.2 m). Develop procedures to estimate overall (across all truck classes) shear and moment percentile levels using the corresponding percentile information from Class 9 trucks. Assess moments and shears due to the Wisconsin Standard Permit Vehicle (Wis-SPV) and the AASHTO design truck (HL-93) relative to individual truck classes and all truck groups combined. The software programs used in the data analysis phase of this study included Crystal BallÒ, a forecasting and simulation program, and ModelRiskÒ, a quantitative risk analysis program. Both programs run within the Microsoft ExcelÒ platform, and both can fit statistical distributions to a given dataset. ModelRiskÒ can also fit standard or empirical copulas. Crystal BallÒ and ModelRiskÒ were run together to perform Monte Carlo simulations involving marginal distributions and copulas.
2. WIM data FHWA has established various truck classifications as defined by the Traffic Monitoring Guide (TMG) [15]. TMG identifies thirteen different vehicle classes. Partial descriptions of these classes are shown in Table 1. In addition to the thirteen FHWA vehicle classes, the WIM data also include what are referred to as truck classes 14 and 15. Class 14 includes truck-trailer combinations and class 15 is allocated to unclassified trucks and/or system errors. Approximately six million truck records (FHWA truck classes 5 through 15) were obtained. Data from Wisconsin WIM stations were exported into spreadsheets for analyses. Data were then sorted based on truck class. For each truck class, a dataset containing the H5P truck data (trucks that weigh more than the 95th percentile level) was created. Truck data were first tested for validity, and invalid truck records were discarded. Three validity tests were performed for each truck record: 1. Is the total weight reported for each truck within 5% of the sum of all axle weights reported? 2. Does the number of axles reported match the number of axle weights reported? 3. Are all axle spacings reported reasonable? For example, records that showed axle spacing of less than 20 in. (508 mm) were discarded.
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H. Tabatabai et al. / Engineering Structures 140 (2017) 189–198 Table 1 FHWA Vehicle Classes (TMG, 2001). Class
Vehicle type
Description
1 2
Motorcycles Passenger Cars
3
Other Two-Axle, Four-Tire Single Unit Vehicles
4
Buses
5
Two-Axle, Six-Tire, Single-Unit Trucks
6
Three-Axle Single-Unit Trucks
7 8
Four or More Axle Single-Unit Trucks Four or Fewer Axle Single-Trailer Trucks
9 10
Five-Axle Single-Trailer Trucks Six or More Axle Single-Trailer Trucks
11
Five or fewer Axle Multi-Trailer Trucks
12 13
Six-Axle Multi-Trailer Trucks Seven or More Axle Multi-Trailer Trucks
All two or three-wheeled motorized vehicles All sedans, coupes, and station wagons manufactured primarily for the purpose of carrying passengers and including those passenger cars pulling recreational or other light trailers All two-axle, four-tire, vehicles, other than passenger cars. Included in this classification are pickups, panels, vans, and other vehicles such as campers, motor homes, ambulances, hearses, carryalls, and minibuses All vehicles manufactured as traditional passenger-carrying buses with two axles and six tires or three or more axles. All vehicles on a single frame including trucks, camping and recreational vehicles, motor homes, etc., with two axles and dual rear wheels All vehicles on a single frame including trucks, camping and recreational vehicles, motor homes, etc., with three axles All trucks on a single frame with four or more axles All vehicles with four or fewer axles consisting of two units, one of which is a tractor or straight truck power unit All five-axle vehicles consisting of two units, one of which is a tractor or straight truck power unit All vehicles with six or more axles consisting of two units, one of which is a tractor or straight truck power unit All vehicles with five or fewer axles consisting of three or more units, one of which is a tractor or straight truck power unit All six-axle vehicles consisting of three or more units, one of which is a tractor or straight truck power unit All vehicles with seven or more axles consisting of three or more units, one of which is a tractor or straight truck power unit
Of the total of 5,761,802 unfiltered records for classes 5 through 15, only 4,352 records (or 0.08%) were discarded based on the above three criteria. Some truck classes may have multiple number of axles. For example, class 7 trucks could have either 4 or 5 axles (designated here as truck class-axle groups or truck groups 07–04 and 07–05), while class 8 trucks could have 3 or 4 axles (truck groups 08–03 and 08–04). Table 2 breaks down the number of trucks based on truck groups. WIM data for classes 13 and 15 trucks include a large number of axle variations within the same class. Data associated with the same number of axles within each class were separated before calculating the distributions. For example, two sets of distributions were determined for class 7 trucks (i.e. 07–04 and 07–05). Table 2 shows the number of filtered trucks in each class (and axle) category as a percentage of trucks in that class as well as percentage of all filtered trucks. Class 9 trucks made up over 61.7% of all WIM trucks. There were over 3.5 million class 9 vehicles in the 2007 data. The second and third most common trucks are classes 8 and 5 at 14.7% and 13.0%, respectively. The maximum and 95 percentile values for the total truck weight in each truck class are also shown in Table 2. For example, the maximum recorded total weight for a class 9 was 1079 kN and the 95 percentile weight was 466.8 kN. Truck classes 9 through 13 and some class 15 truck groups have the highest 95 percentile weight and the heaviest overall maximum weights (Table 2). In the following discussions, truck axles are labeled alphabetically with Axle A being the front axle. Axle spacing labels indicate the two axles at the ends of each spacing. For example, axle spacing BC refers to the distance between axles B and C (Fig. 1). 3. Data analysis The H5P data for all class-axle groups were used to generate best-fit statistical distributions using the ModelRiskÒ software. Data from all stations were combined. Limited Analyses of Variance (ANOVA) showed that truck weights in different WIM stations did not belong to the same distribution. Best fit distributions were determined for each axle weight, axle spacing and the total weight in each truck class-axle category using the maximum likelihood estimation. The total weight distributions were obtained for information only, and were not used in moment and shear assessments. The fitting options within the software could not directly accommodate bimodal (‘‘double hump”) or multi-modal statistical
distributions. However, some H5P axle spacing distributions were in fact multi-modal. Therefore, when data warranted such considerations, the following semi-manual approach was used to determine multi-modal best fit distributions: A histogram of the data was generated. The best fit single-mode distributions were determined. If the histogram indicated multi-modal (‘‘multi-hump”) behavior, then the data was manually separated into grouping around each peak. Best fit single-mode distributions for each group were determined. The number of data points within each grouping divided by the total number of data points is the probability (p) associated with the distribution in that grouping. For example, a tri-modal distribution can be represented as follows: Multi-Modal Distribution ¼ p1 Distribution1 þ p2 Distribution2 þ p3 Distribution3
The resulting multi-modal distribution was plotted and compared with the histogram to make sure that the data agreed with the distribution. Single-mode distribution parameters for all class-axle groups are reported in a research report by Tabatabai et al. [16]. Other related studies addressed a number of aspects of analyses of Wisconsin WIM data [17,18]. In this paper, details of the statistical distributions are not presented for brevity. Additional data are provided as supplementary materials accompanying this paper. Selected histogram and distribution plot for Class 9 trucks are compared in Fig. 2. The motivation behind determining statistical distributions for each parameter in a truck class/axle group was to run Monte Carlo simulations using those marginal distributions. One could perform such simulations assuming that the various axle loads and spacings are independent of each other. However, if such parameters were considered independent of each other, then the relationships between different axle loads and spacings, if any, would be ignored. Srinivas et al. [13] suggest that copulas should be used to model the interdependence of truck load information. Copulas have been widely used in financial and insurance industries to assess financial risk in instruments such as the derivatives [13,19]. Copulas were first introduced by Sklar in 1959 [19]. Copula functions can describe the dependence between the variables involved. The multivariate functions can be estimated by linking the marginal distributions with the copula function [13].
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Table 2 Detailed information on different truck class-axle groups. Truck Class
Truck Group (Class – Axles)
Un-filtered Trucks
No of Filtered Trucks
% of Total
% of Truck Class
Class 5 Class 6 Class 7
05–02 06–03 Total 07–04 07–05 Total 08–03 08–04 09–05 Total 10–06 10–07 11–05 12–06 Total 13–07 13–08 13–09 13–10 13–11 13–12 13–13 14–05 Total 15–02 15–03 15–04 15–05 15–06 15–07 15–08 15–09 15–10 15–11 15–12 15–13 15–14
749,409 251,914 73,172
748,658 251,795 73,138 25,753 47,385 848,483 634,745 213,738 3,553,613 77,185 72,939 4246 94,572 30,576 10,595 9738 680 75 65 10 8 19 1128 67,708 3071 13,617 10,013 9057 19,507 4164 4781 1264 727 489 384 341 293
13.00% 4.37% 1.27% 0.45% 0.82% 14.74% 11.02% 3.71% 61.72% 1.34% 1.27% 0.07% 1.64% 0.53% 0.18% 0.17% 0.01% 0.00% 0.00% 0.00% 0.00% 0.00% 0.02% 1.18% 0.05% 0.24% 0.17% 0.16% 0.34% 0.07% 0.08% 0.02% 0.01% 0.01% 0.01% 0.01% 0.01%
100.00% 100.00%
Class 8
Class 9 Class 10
Class 11 Class 12 Class 13
Class 14 Class 15
849,072
3,554,700 77,764
94,578 30,577 10,907
1131 68,578
35.21% 64.79% 74.81% 25.19% 100.00% 94.50% 5.50% 100.00% 100.00% 91.91% 6.42% 0.71% 0.61% 0.09% 0.08% 0.18% 100.00% 4.54% 20.11% 14.79% 13.38% 28.81% 6.15% 7.06% 1.87% 1.07% 0.72% 0.57% 0.50% 0.43%
H5P 95th percentile (KN)
Max weight (KN)
109.8 237.3 398.1 355.0 414.8 205.9 115.7 304.0 466.8 518.7 509.9 638.4 519.7 577.6 579.5 572.7 543.3 724.7 724.7 755.1 930.6 1095.4 316.7 435.4 153.9 119.7 254.0 328.5 413.8 445.2 723.7 811.0 899.2 743.3 410.9 272.6 235.3
349.1 533.4 832.5 832.5 764.9 680.5 646.2 680.5 1078.7 1188.5 1188.5 1048.3 805.1 913.0 1459.2 1459.2 715.9 788.4 892.4 755.1 930.6 1095.4 450.1 1885.7 220.6 246.2 413.8 465.8 604.0 582.5 1184.6 1538.6 1885.7 1066.9 1600.3 1432.7 584.5
Fig. 1. Axle load and spacing designations.
There are many types of functions that can serve as copulas. Two prominent groups of copulas are Elliptical Copulas and Archimedean Copulas. The Gaussian and Student’s T copulas belong in the Elliptical group while Clayton, Gumbel and Frank copulas belong in the Archimedean group [13]. On the other hand, empirical copulas are based on actual data and are not fit to particular mathematical functions [20]. Best-fit standard copulas were determined by the software based on data entered into a spreadsheet. The empirical copulas were also determined by the software based on its proprietary approach. In this study, both approaches (standard and empirical
copulas) were examined. Additional information on the copulas are provided as supplementary materials and in Ref. [16]. As expected, the empirical copulas (which utilize actual data each time) were best able to simulate total truck weight distributions when such simulations were compared with corresponding histograms. Therefore, empirical copulas were used for the Monte Carlo simulations. The Crystal BallÒ software was used for Monte Carlo simulations. Two types of Monte Carlo simulations were performed. First, analyses were performed to demonstrate that the simulation results were similar to the histograms of data. Fig. 3 shows such
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Axle B Weight 0.025
0.02
0.02
Probability
Probability
Axle A Weight 0.025
0.015 0.01 0.005 0
0.015 0.01 0.005
0
40
80
120
160
0
200
0
40
80
100 kg
0.02
0.02
0.015 0.01 0.005
0.01 0.005
0
50
100
150 100 kg
200
250
0
300
0
50
100
BC Spacing
150 100 kg
200
250
300
CD Spacing 0.07
0.6
0.06
0.5
0.05
Probability
Probability
200
0.015
0.7
0.4 0.3 0.2 0.1 0
160
Axle E Weight 0.025
Probability
Probability
Axle D Weight 0.025
0
120 100 kg
0.04 0.03 0.02 0.01
0
20
40
60
80
0
100
0
20
40
60
100 mm
80 100 100 mm
120
140
160
Fig. 2. Representative H5P Histograms and distributions for Class 9 trucks.
Simulation Result
WIM Data 0.07
Probability
0.06 0.05 0.04 0.03 0.02 0.01 0
100
120
140
160
180
200
Kips Fig. 3. Total truck weight - Comparison of Class 9 truck simulation results with H5P WIM data.
a comparison for Class 9 trucks. In this case, individual axle load simulations (considering copulas) were made for all five axles and the individual axle loads were added together to estimate the total weight for each simulation run. The resulting simulated and actual total weight histograms are shown in Fig. 3. Overall, the comparisons between simulated and actual histograms were favorable for all truck groups. For the second type of analyses, a spreadsheet was setup to calculate bending moment and shear envelopes for any moving truck
arrangements (up to 10 axles) using influence lines. A simple-span bridge condition with spans ranging from 20 ft to 240 ft (6.1– 73.2 m) was considered. Each simulated truck was ‘‘marched” across the span in increments and the moment and shear envelopes were calculated. For simulations, the distributions and copulas were applied to all axle loads and axle spacings. Statistical information on moments and shears due to H5P simulations of truck class/axle groups were then developed for simplespan bridges. In addition, the 845 kN Wis-SPV was analyzed, and
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the maximum moments and shears due to Wis-SPV were compared with the simulation results to determine the effect of WisSPV (as a percentile of the simulation results) for each truck class–axle group. The Wisconsin Department of Transportation requires that the Wis-SPV vehicle be checked during design, rehabilitation, or rating of all Wisconsin bridges [21]. This vehicle is evaluated in conjunction with lane loading, with the dynamic allowance and single-lane-loading option utilized. Fig. 4 shows axle loads and spacings for the Wis-SPV truck [21]. The summary of Monte Carlo simulation results for Class 9 trucks are shown in Tables 3 and 4. Moment and shear values cor-
responding to various overall percentile levels are provided. Since H5P data have the heaviest overall weights in that class, it is reasonable to assume that the effects due to H5P vehicles will result in higher moments and shears compared to the remaining 95% of trucks in that class. Therefore, the overall percentile is estimated through the following relationship:
Overall percentile ¼ 95 þ ðH5P percentileÞ 0:05 It should be noted that there is nonlinear behavior between the 99th and 100th percentile results. Therefore, to improve accuracy between 99th and 100th percentiles, analyses were run at 0.05
63 ft (19.2 m) 19.0
26.6
26.6
26.6
22.8
22.8
22.8
22.8
190 kips
84.5
118.3
118.3
118.3
101.4
101.4
101.4
101.4
845 kN
Fig. 4. The 845 kN Wisconsin Standard Permit Vehicle (Wis-SPV).
Table 3 Summary of moments - Monte Carlo simulation results for H5P Class 9 trucks. Max Moment
Percentiles
Span
95th
ft
m
Moment (kN-m)
20 40 60 80 100 120 140 160 180 200 220 240
6.1 12.2 18.3 24.4 30.5 36.6 42.7 48.8 54.9 61.0 67.1 73.2
180.9 505.1 929.2 1361.9 1996.7 2749.0 3510.5 4259.2 5013.7 5758.3 6520.6 7236.1
96th
97th
98th
99th
99.5th
281.0 676.3 1138.6 1720.2 2515.3 3326.0 4134.2 4943.0 5751.5 6560.0 7369.7 8180.0
307.7 732.3 1235.5 1864.4 2718.3 3586.5 4455.5 5325.7 6198.3 7075.2 7941.7 8813.5
331.1 784.4 1326.0 2004.3 2925.5 3866.1 4803.6 5750.4 6691.5 7634.2 8573.4 9515.0
359.1 845.7 1431.1 2171.5 3170.0 4183.8 5199.1 6213.4 7235.0 8253.5 9264.0 10279.0
379.7 891.2 1513.1 2296.0 3344.5 4416.1 5486.6 6568.4 7643.4 8713.4 9793.3 10859.4
99.9th
99.95th
100th
413.3 968.9 1645.9 2508.2 3638.4 4791.0 5956.8 7122.8 8288.4 9453.6 10635.7 11785.9
424.8 1004.0 1699.9 2597.4 3768.2 4957.3 6169.5 7371.2 8568.6 9765.7 10968.8 12153.5
503.6 1303.1 2506.4 4068.5 5650.0 7159.6 8788.8 10294.8 11865.8 13436.1 14986.4 16499.1
Max Moment
Percentiles
Span
99.6th
fet
m
20 40 60 80 100 120 140 160 180 200 220 240
6.1 12.2 18.3 24.4 30.5 36.6 42.7 48.8 54.9 61.0 67.1 73.2
99.7th
99.8th Moment (kN-m)
385.5 904.5 1533.9 2330.5 3400.1 4485.6 5572.5 6665.5 7755.3 8839.5 9925.7 11016.0
392.2 921.7 1560.8 2367.8 3453.4 4550.5 5647.5 6754.7 7861.2 8968.7 10074.3 11174.5
400.2 939.7 1593.9 2418.3 3525.7 4643.7 5768.1 6893.3 8017.1 9145.1 10282.6 11406.3
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H. Tabatabai et al. / Engineering Structures 140 (2017) 189–198 Table 4 Summary of shears - Monte Carlo simulation results for H5P Class 9 trucks. Max Shear
Percentiles
Span
95th
96th
97th
98th
99th
99.5th
123.7 193.5 230.9 268.2 306.9 331.8 349.2 364.8 372.3 382.1 388.8 395.9
194.4 238.4 279.8 328.7 362.1 384.3 400.3 412.4 421.7 428.8 435.0 439.9
210.8 258.4 300.7 353.6 389.7 413.7 431.0 443.9 454.2 462.2 468.8 474.6
226.4 277.6 322.9 381.7 421.2 447.5 466.2 480.4 491.1 499.5 507.1 512.9
244.2 300.3 349.2 413.2 455.5 484.0 504.0 519.1 531.1 540.9 548.5 554.2
257.8 316.8 369.5 436.9 480.9 510.4 531.5 548.2 560.9 570.6 578.9 586.1
99.6th
99.7th
99.8th
99.9th
99.95th
100th
279.7 344.8 402.6 476.5 522.6 554.3 576.9 595.5 610.7 619.9 629.1 635.7
289.8 356.2 416.8 491.7 539.8 573.6 596.7 614.6 628.3 639.2 648.9 655.3
348.3 470.6 617.9 703.7 754.4 785.6 816.7 830.5 842.0 857.6 867.4 871.4
ft
m
Shear (kN)
20 40 60 80 100 120 140 160 180 200 220 240
6.1 12.2 18.3 24.4 30.5 36.6 42.7 48.8 54.9 61.0 67.1 73.2 Max Shear
Percentiles
Span ft
m
20 40 60 80 100 120 140 160 180 200 220 240
6.1 12.2 18.3 24.4 30.5 36.6 42.7 48.8 54.9 61.0 67.1 73.2
Shear (kN) 261.6 321.1 374.3 443.0 487.3 517.5 539.1 555.8 568.3 578.5 586.5 593.4
265.8 326.6 381.1 450.5 495.6 526.2 547.8 564.5 577.3 587.5 595.6 602.8
271.3 333.2 389.4 460.3 506.8 537.4 559.5 576.6 589.3 599.5 607.8 615.3
Table 5 Simulation moments corresponding to 80th percentile of H5P data or 99th percentile for selected class-axle groups. Span
Truck Class - Number of Axles 05–02
ft
m
Moment (kN-m)
20 40 60 80 100 120 140 160 180 200 220 240
6.1 12.2 18.3 24.4 30.5 36.6 42.7 48.8 54.9 61.0 67.1 73.2
156 347 596 846 1097 1348 1599 1851 2102 2352 2602 2855
07–05
08–04
09–05
10–06
13–07
HL-93 Truck
456 1179 2033 2908 3782 4655 5528 6411 7279 8164 9048 9929
280 666 1118 1657 2263 2882 3508 4147 4785 5429 6068 6712
359 845 1430 2171 3168 4182 5197 6210 7232 8250 9260 10,274
405 978 1863 2880 3942 5016 6095 7179 8258 9342 10,424 11,519
406 1050 1812 2729 3848 5044 6276 7531 8811 10,089 11,379 12,639
217 610 1094 1580 2067 2554 3042 3530 4017 4505 4993 5481
percentile increments, so that interpolations can be made more accurately. The 95th through 99th percentile values are shown at 1 percentile increments in Tables 3 and 4. In addition, the 99.5th, 99.6th, 99.7th, 99.8th, 99.9th and the 99.95th percentile level are also shown in those tables to improve accuracy. Tables 5 and 6 show the 99th percentile moments and shears for different span lengths corresponding to selected truck class/ axle groupings. Information on all other truck groups are presented as supplementary materials included with this paper. The 99th percentile values are shown because Moses [1] suggests this level for characterizing the expected maximum live load. Sivakumar et al. [2] also used the 99th percentile level as ‘‘the maximum anticipated load for design”. It should be noted that Moses [1] and Sivakumar et al. [2] were referring to the 99th percentile truck weight, and not the 99th percentile
moments and shears due to individual truck class-axle groups used in this study. Tables 5 and 6 also show moments and shears due to the HL-93 design truck, respectively. The HL-93 truck values shown are for one design lane without dynamic allowance. The effect of lane loading is not included in the HL93 results shown. As stated earlier, the Class 9 truck group is the most prevalent truck class on Wisconsin roads. Therefore, it is reasonable to focus on the effects of that truck class. An effort was made to assess whether the results of Class 9 truck simulations can be used to represent the entire truck population with respect to the maximum anticipated load effects for design or rating purposes. The 99th percentile moment and shear values associated with the Class 9 truck group would, of course, not necessarily coincide with the 99th percentile levels across all trucks.
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Table 6 Simulation shears corresponding to 80th percentile of H5P data or 99th percentile for selected class-axle groups. Span
Truck Class - Number of Axles 05–02
ft
m
Shear (kN)
20 40 60 80 100 120 140 160 180 200 220 240
6.1 12.2 18.3 24.4 30.5 36.6 42.7 48.8 54.9 61.0 67.1 73.2
102.3 125.4 133.4 137.4 140.1 141.4 142.8 143.7 144.6 145.0 145.4 145.9
07–05
08–04
09–05
10–06
13–07
HL-93 Truck
314.5 408.3 449.2 469.3 481.7 489.3 495.1 499.5 502.2 504.8 507.5 509.3
189.5 230.4 255.3 279.3 300.2 314.5 324.7 331.8 338.0 342.9 346.9 350.1
244.2 300.2 349.2 413.2 455.5 483.9 504.0 519.1 531.1 540.9 548.4 554.2
274.0 382.1 462.1 509.3 538.2 558.2 572.0 583.1 592.0 598.3 604.5 608.5
281.6 360.6 427.7 498.0 550.0 587.7 616.4 637.9 654.1 667.8 679.9 688.1
185.0 245.5 270.4 282.9 290.5 295.3 298.9 301.6 303.8 305.1 306.5 307.8
Table 7 Comparison of 99th and 98.59th percentile moments for Class 9 trucks with HL-93 moments. Span
Moments
Ratio Class 9 (98.59th)/HL-93
Class 9 - 99th Percentile moments
Class 9–98.59th Percentile moments
HL-93
ft
m
kN-m
Percentile of All Trucks
kN-m
Percentile of All Trucks
kN-m
20 40 60 80 100 120 140 160 180 200 220 240
6.1 12.2 18.3 24.4 30.5 36.6 42.7 48.8 54.9 61.0 67.1 73.2
358.9 845.3 1430.4 2170.5 3168.4 4181.9 5196.7 6210.5 7231.5 8249.6 9259.6 10274.1
99.28 99.26 99.23 99.22 99.26 99.28 99.29 99.30 99.30 99.30 99.31 99.31
346.5 818.2 1383.3 2091.1 3054.8 4033.8 5015.5 5999.4 6984.5 7969.5 8953.2 9931.0
99.00 98.99 98.95 98.93 98.96 98.99 99.00 99.01 99.01 99.02 99.02 99.02
217.0 610.5 1094.3 1580.1 2067.0 2554.3 3041.8 3529.5 4017.3 4505.2 4993.1 5481.0
1.60 1.34 1.26 1.32 1.48 1.58 1.65 1.70 1.74 1.77 1.79 1.81
Average Percentile
99.28
Average Percentile
98.99
Average Ratio
1.60
The truck groups with higher 95th percentile total weights than the Class 9 group represent a small fraction of trucks. These include Class 10 trucks (representing 1.34% of all trucks), Class 11 trucks (1.64% of all trucks), Class 12 trucks (0.53% of all trucks), and Class 13 trucks (0.18% of all trucks). To determine where the Class 9 moment and shear percentile values fall with respect to the 99th percentile of moment and shear values for all trucks, the following procedures were devised and implemented. First, the Class 9 moment and shear values were compared against the moment and shear simulation results for all other truck groups to determine percentile levels associated with those values at each span length. Second, knowing the percentile levels and the total number of trucks in each group, the number of trucks exceeding the target moment/shear values in each truck group can be determined. Third, the sum of these truck numbers across all truck groups was determined and divided by the number of all trucks to determine the percent of trucks exceeding the target moment/shear values across all truck groups. Finally, subtracting this percentage from 100 would provide the desired percentile level. Tables 7 and 8 show percentile levels for moment and shears, respectively, for various span lengths. Again, because of space limitations, complete data for all truck groups are presented as supplementary materials. The average overall (all truck classes combined) percentile levels were 99.28 and 99.30 for moment and shear, respectively. Therefore, the overall percentile levels are slightly above the target of 99th percentile level. After a number of trials, the Class 9 percentile levels that would result in the overall 99th percentile levels were determined. These were Class
9 percentiles of 98.59 and 98.55 (roughly 98.6th percentile) for moment and shear, respectively. As shown in Tables 7 and 8, these moment and shear levels would result in an average overall percentile levels of 99 for both moment and shear. These moment and shear values represent the maximum anticipated moment and shear values for design based on this set of Wisconsin WIM data. The ratios of these 99th percentile moments and shears to the HL-93 values are shown in Tables 7 and 8. The average ratios across all span lengths is approximately 1.60. However, these ratios can vary for different span lengths. Moments and shears due to any specific design load and/or permit vehicle (such as Wis-SPV) can be compared with the percentiles in Tables 3 and 4. For example, the moment and shear effects due to Wis-SPV are compared with simulation results in Tables 9 and 10. The moments due to Wis-SPV for various span lengths were, on average, at the 99.96th percentile of Class 9 results, and the 99.95th percentile of all trucks. The corresponding percentile levels for shear were 99.96 for Class 9 and 99.97 for all trucks. Therefore, the overall average probability of exceedance is 5 in 10,000 for moment and 3 in 10,000 for shear. 4. Summary and conclusions This study involved statistical evaluation of heavy truck loads that were recorded using WIM stations located throughout the State of Wisconsin. Nearly 6 million vehicle records were collected for the year 2007. Information on trucks Classes 5 through 15 were retained (i.e. non-truck classes were eliminated).
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H. Tabatabai et al. / Engineering Structures 140 (2017) 189–198 Table 8 Comparison of 99th and 98.55th percentile shears for Class 9 trucks with HL-93 shears. Span
Shears
Ratio Class 9 (98.55th)/HL-93
Class 9 - 99th Percentile shears
Class 9–98.55th Percentile shears
HL-93 Loading
ft
m
kN
Percentile of All Trucks
kN
Percentile of All Trucks
kN
20 40 60 80 100 120 140 160 180 200 220 240
6.1 12.2 18.3 24.4 30.5 36.6 42.7 48.8 54.9 61.0 67.1 73.2
244.2 300.3 349.2 413.2 455.5 484.0 504.0 519.1 531.1 540.9 548.5 554.2
99.28 99.25 99.25 99.29 99.31 99.32 99.32 99.32 99.32 99.33 99.33 99.33
235.6 289.2 336.2 398.1 438.6 466.0 485.8 500.7 512.1 521.1 528.5 534.8
98.98 98.95 98.94 98.99 99.01 99.01 99.02 99.02 99.02 99.02 99.02 99.02
185.0 245.5 270.4 282.9 290.5 295.3 298.9 301.6 303.8 305.1 306.5 307.8
1.27 1.18 1.24 1.41 1.51 1.58 1.63 1.66 1.69 1.71 1.72 1.74
Average Percentile
99.30
Average Percentile
99.00
Average Ratio
1.54
Table 9 Wis-SPV moments compared with simulation results for various truck groups. Span
Maximum Moment
Truck Class-Axle Group 05–02
ft
m
kN-m
Percentiles
20 40 60 80 100 120 140 160 180 200 220 240
6.1 12.2 18.3 24.4 30.5 36.6 42.7 48.8 54.9 61.0 67.1 73.2
428.2 1074.9 1782.8 2665.4 3962.7 5322.1 6690.9 8083.7 9448.1 10787.4 12184.5 13587.2
100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
07–05
08–04
09–05
10–06
13–07
ALL
98.33 98.02 97.52 98.18 99.35 99.80 99.92 99.96 99.96 99.97 99.98 99.98
99.95 99.97 99.96 99.96 99.98 100.00 100.00 100.00 100.00 100.00 100.00 100.00
99.95 99.96 99.96 99.95 99.96 99.96 99.96 99.96 99.96 99.96 99.97 99.97
99.37 99.61 98.50 98.10 99.04 99.45 99.65 99.77 99.82 99.84 99.87 99.89
99.29 99.13 98.85 98.80 99.16 99.28 99.34 99.38 99.39 99.40 99.43 99.46
99.94 99.95 99.93 99.93 99.95 99.96 99.97 99.97 99.97 99.97 99.98 99.98
Table 10 Wis-SPV shears compared with simulation results for various truck groups. Span
Maximum Shear
Truck Class-Axle Group 05–02
ft
m
kN
Percentiles
20 40 60 80 100 120 140 160 180 200 220 240
6.1 12.2 18.3 24.4 30.5 36.6 42.7 48.8 54.9 61.0 67.1 73.2
284 359 450 546 604 644 677 693 707 724 736 743
100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
07–05
08–04
09–05
10–06
13–07
ALL
97.85 97.56 99.02 99.85 99.95 99.97 99.98 99.99 99.99 100.00 100.00 100.00
99.95 99.96 99.99 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
99.92 99.95 99.96 99.96 99.97 99.97 99.97 99.97 99.97 99.97 99.97 99.97
99.26 98.29 98.73 99.51 99.75 99.83 99.88 99.90 99.91 99.92 99.92 99.92
99.05 98.97 99.26 99.47 99.52 99.52 99.56 99.52 99.51 99.53 99.5 99.52
99.92 99.92 99.95 99.97 99.97 99.98 99.98 99.98 99.98 99.98 99.98 99.98
If a truck class contained multiple numbers of axles, that class was further sub-divided such that each sub-group contained only one particular number of axles. Data in each class-axle groupings were sorted based on gross vehicle weight, and the heaviest 5 percent of truck records in each group were separated and analyzed. Statistical analyses were performed on the H5P data. Using the H5P data, best-fit unimodal and/or multimodal marginal distributions were determined for each axle weight and spacing in each truck class-axle group. Furthermore, copulas were determined to allow multivariate Monte Carlo simulations.
Multivariate Monte Carlo simulations on H5P data in each classaxle group were conducted using software programs running within a spreadsheet. The following observations are made: (1) Truck simulations for each class-axle grouping were performed and successfully tested for validity against the histogram of data. (2) Some H5P axle spacing distributions are multimodal, and multimodal marginal distributions must be used in such cases for proper simulations.
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(3) Empirical copulas provide more accurate simulations when compared to standard copula functions determined by data fitting. All simulation results reported here are based on empirical copulas determined using the software. (4) The marginal distributions and copulas determined here can be used to statistically assess heavy truck impact on bridges and pavements based any load-dependent metric. (5) The percentile results derived can be used to assess the relative impact of any truck arrangement compared to simulation results. Moments and shears due to the Wisconsin Standard Permit Vehicle were, on average, approximately equivalent to the 99.8th percentile level for all trucks. (6) The percentiles of simulated shear and moment for various truck class-axle groups can be used to estimate percentiles for all trucks (all truck groups combined) using the procedures discussed here. Such data can then be used to estimate the 99th percentile shear and moment values (for different span lengths). These results can then be used for bridge load ratings based on each state’s own WIM data. (7) The Class 9 truck statistics can be used to represent the entire truck population for the purpose of estimating WIMbased design loads, or for issuance of permits. Class 9 percentile level of 98.6 and 98.55 are equivalent to the 99th percentile level associated with the entire truck population for moments and shears, respectively. As the most common truck class, the Class 9 data with its available statistical information can provide a relatively simple tool for assessing the probabilistic structural impact of all trucks. (8) Using the proposed probabilistic approach and the results of this research, bridge owners can set criteria (percentile thresholds) for issuance of single-trip or annual overweight truck permits.
Acknowledgements This research was completed with the partial support and partnership of the National Center for Freight and Infrastructure Research and Education (CFIRE), a Tier 1 University Transportation Center (UTC) funded by the U.S. Department of Transportation (USDOT) Research and Innovative Technology Administration (RITA). Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.engstruct.2017. 02.060.
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