Protocols for Collecting and Using Traffic Data in Bridge Design (2011)

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8Research Tasks The research effort was organized according to the following nine tasks: • Task 1. Review relevant practice, data, existing speci- fications, and research findings from both foreign and domestic sources on the collection and analysis of truck weight data with particular emphasis on evaluating the stresses and deformations induced in highway bridges. This information shall be assembled from both technical literature and unpublished experiences. Information on the quantity and quality of existing WIM data from studies on or near U.S. Interstate bridges is of particular interest. • Task 2. Describe existing and potential processes to develop and calibrate vehicular loads for superstructure design, fatigue design, deck design, and overload permitting. • Task 3. Develop the vehicular traffic data requirements (type, quantity, and quality) for each of the processes iden- tified in Task 2. The data requirements should be validated by a sensitivity analysis of each data element. • Task 4. Assess the statistical adequacy of existing traffic data to meet the requirements developed in Task 3. Rec- ommend means to eliminate any inadequacies. • Task 5. Using the information from Tasks 1 through 4, rec- ommend candidate protocols for collecting and processing traffic data to calibrate national bridge live-load models. The protocols should include guidance on selecting default values for use when the traffic data do not meet the require- ments developed in Task 3. Prepare an updated, detailed work plan for developing and demonstrating the application of the protocols. • Task 6. Submit an interim report within 6 months of the contract start that documents the results of Tasks 1 through 5 and includes the updated and expanded work plan for developing and demonstrating the protocols for collecting and processing traffic data. The contractor will be expected to meet with NCHRP approximately 1 month later. Work may not proceed on subsequent tasks without NCHRP approval of the work plan. • Task 7. Develop the protocols in accordance with the approved work plan. • Task 8. Demonstrate the application of the protocols using existing national data to develop and calibrate vehicular loads for superstructure design, fatigue design, deck design, and overload permitting. • Task 9. Submit a final report that documents the entire research effort. Overview of Data Collection and Review The first step in the live-load model development process was to assemble and review recent developments and relevant information on practice, specifications, bridge live-load models, WIM systems, WIM data, and studies of truck weights. A purpose of this task was to understand the state of the art and the practice of collecting and utilizing traffic data in bridge design in the United States and in other countries. It included a survey of U.S. state highway agencies. A search of published technical literature in the United States and other countries was conducted for information applicable to this research. Survey A survey questionnaire was e-mailed to the traffic moni- toring divisions of all state DOTs. The purpose of this ques- tionnaire was to obtain detailed information and document practices on issues central to this research, such as types of WIM equipment in use in each state and the locations of WIM sites, WIM equipment calibration procedures, and the C H A P T E R 2 Research Approach

types of traffic data being collected and how they are being used. A copy of the survey questionnaire is contained in Appendix C. The questionnaire consisted of the following five sections: • Section 1—Weigh-in-Motion (WIM) Program • Section 2—WIM Sites • Section 3—WIM Data • Section 4—WIM Data Validation and WIM System Cali- bration • Section 5—WIM Data Analysis and Applications Completed questionnaires were received from the following 27 states: Alaska, Arkansas, California, Connecticut, Florida, Georgia, Hawaii, Idaho, Indiana, Iowa, Kansas, Louisiana, Michigan, Minnesota, Mississippi, Missouri, Nevada, New Jersey, New Mexico, New York, North Dakota, Ohio, Oregon, South Dakota, Virginia, Washington, Wyoming. It is believed that states with significant WIM programs have responded to the questionnaire. State-of-the-Practice Summary State DOT Survey Tabulated responses to all survey questions are contained in Appendix C. Responses to certain key questions are presented in Tables 1 through 4. The responding states have maintained a WIM program over the past 3 to 32 years. The number of high-speed WIM sites varied from 3 to 137, distributed among Interstate and non-Interstate routes. Most states also indicated that they can provide a whole year’s worth of WIM data for statistical analy- ses. This will be an important consideration in the selection of WIM sites for load modeling data because it incorporates any seasonal variability in the traffic data. The types of sensors used at each WIM site, date installed, date last calibrated, number of traffic lanes, and the number of WIM lanes are given in Appen- dix C. These details were valuable when selecting WIM sites for demonstrating the use of the protocols in this project in Task 8. There is great variability in the truck arrival time data recorded by the various systems. The range is from 0.01 second 9 State DOT WIM Program? How Long has the WIM Program been in Operation? Total Number of High-Speed WIM Sites Number of WIM Sites on Interstates Do you have WIM Data Available for a Whole Year? Alaska Yes 10+ years 7 4 Yes Arkansas No Yes California Yes 15 years 137 58 Yes Connecticut Yes 9 years 36 bi- directional +4 LTPP 21 bi- directional +2 LTPP Yes Florida Yes 32 years 40 14 Yes Georgia Yes 10 years 90 30 No Hawaii Yes 18 years 7 2 Yes Idaho Yes 12 years 16 6 Yes. Lots of available WIM data. Indiana Yes 15 years 52 24 Yes Iowa Yes 15 years 28 9 Yes Kansas Yes 14 years 9 perm 70 portable 3 perm 25 portable Yes Louisiana Yes 7 years 3 3 No Michigan Yes 14 years 41 21 Yes Minnesota Yes 22 years 6 2 Yes Mississippi Yes 14 years 15 7 Yes Missouri Yes 10 years 13 7 Yes Nevada Yes 20 years 4 4 Yes New Jersey Yes 13 years 64 14 Yes New Mexico Yes 17 years 18 7 Yes New York Yes 10+ years 21 11 Yes North Dakota Yes 3 years 12 4 Yes. Possibly. Ohio Yes 15 years 44 21 Yes Oregon Yes 8 years 22 18 Yes. In a text format. South Dakota Yes 15 years 14 6 Yes Virginia Yes 3 2 Yes Washington Yes 16 years 37 10 Yes Wyoming Yes 8 years 5 3 Yes Table 1. WIM program details.

to the hour. The fact that many systems record time stamps to 0.01 second accuracy indicates that there is sufficient avail- ability of refined time stamps for estimating multiple-presence probabilities for truck crossings as discussed in the following chapter. The responses show that many states have traffic data of similar quality for a number of years at the same site that may be helpful in estimating trends in truck loadings. It is also interesting to note that of the 27 responding states only Cali- fornia and Oregon have begun using WIM data for bridge design applications. These initiatives appear to be in the early states of implementation. WIM data quality testing and validation is important to ensure that only quality data are made part of the load mod- eling process. Although WIM systems can provide massive amounts of valuable data in a relatively efficient manner, the data must be checked for accuracy. This accuracy check is a WIM user’s quality assurance (QA) program. Most states have implemented QA programs for their WIM systems to check data accuracy. A QA program adds confidence to the validity of the WIM data and alerts the data analyst to prob- lems occurring at the WIM site. The purpose of a QA proce- dure is to help WIM users check data for accuracy and precision. 10 Question 3.4. What is the accuracy of truck arrival time stamps reported in the data set (1 sec, 0.01 sec, etc.)? Are time stamps available for more than one lane at a site? What is the highest resolution possible for truck arrival times? Please explain. State DOT Time Stamp Accuracy Alaska The accuracy of the truck time arrival is .01 for all lanes of data. The time stamp is on each vehicle record (PVR). This was an Alaska requirement to ensure that duplicate records were not loaded to the database. This is the highest resolution possible for truck arrival times. California 0.01 sec. Yes. Connecticut Truck arrival time stamps are reported for each lane at a site and are recorded by 1 second. The IRD software shows the data time stamp recorded to one-hundredth of a second (0.01) in the individual vehicle viewing software. Additional work would be needed to determine the resolution of the data that is reported in the output file formats. Florida Times are recorded to the nearest full second, for all lanes. Georgia Accurate to the .01 second and we weigh in one lane of the roadway. At 10 locations we collect truck traffic in the two outside lanes. Hawaii Time stamps are not checked for minute/second accuracy. We check them for date accuracy, and we check the WIM system clock at least once per month. Observed accuracy for those can range from within 1-2 seconds to 1-2 minutes. Idaho ECM WIM system equipment has a time resolution of one-tenth of a second. The IRD/Diamond WIM systems have a “scientific” mode setting which allows for data collection with a time stamp of one hundred of a second. Indiana The timestamps for the vehicle records are to the 1/100 of a second. The ASCII report, however, will alone show the timestamp to the nearest second. This is a shortcoming that has been identified and will be corrected in future versions of the software. Iowa They are stored by the hour. We can view the info in real time to the second and can be viewed for all lanes at the site. Kansas Accuracy varies because the on-site clock is not externally synchronized. Precision of the arrival time is 1 second, which is the finest resolution available from the equipment. Time stamps are available for each truck, regardless of lane. Michigan The time stamp is down to the second for each lane of travel. So we have the hour, minute, and second the vehicle started to cross the sensor. Minnesota Whole second Mississippi We store the data on an hourly basis in the cardw, but the img file has a time stamp associated with each vehicle. We collect WIM data on all lanes at a permanent site. Missouri Year, month, day, hour Nevada We have never had the need to investigate this but from my experience it is within a second New Jersey Truck arrival time stamps using the “View Vehicle” menu of the IRD office software shows a time stamp of up to 0.01 of a second; processed weight data from the W-record cards only up to one minute. New Mexico Hourly, for all lanes. New York All lanes are monitored and trucks are time stamped to 0.01 seconds. North Dakota 1 second resolution—yes, time stamps available on all lanes at all times Ohio Mettler-Toledo’s time stamp is now sub second at .01 sec. The TMG does not have this resolution and needs to be changed. The time stamp is on each vehicle so it would be by lane. Peek or Pat/IRD do not provide time stamps to the .01 second level. Oregon Time stamp accuracy is within .01 seconds. Time stamps are available for each lane in multi-lane systems. South Dakota Unknown on accuracy of arrival time and are by lane Virginia Time stamps are to the nearest second, and are available for all lanes Washington 12:00:00:00 Wyoming Time stamps are to the second and are by lane. Table 2. Accuracy of truck arrival time stamps.

Literature Review An investigation of published technical literature for the monitoring, collection, and analysis of bridge-related WIM data has been performed utilizing transportation organiza- tion websites including Transportation Research Informa- tion Services (TRIS), National Technical Information Service (NTIS), U.S.DOT, FHWA, and TRB. In addition, transporta- tion engineering websites and databases, and the websites of state departments of transportation and other U.S., Cana- dian, European, Asian, and Australian transportation insti- tutes, were explored for relevant material regarding the use of WIM data. This literature search concentrated on the following WIM research topics concerning the use of WIM to: • Model bridge loads, • Study the growth or trends in truck weights, • Study the multiple presence of trucks, and • Study site-specific bridge loads. The technical literature search resulted in the compilation of a reference list consisting of approximately 250 abstracts, research papers, journal articles, conference papers, and reports with applicability to the project research. The collected material was reviewed for pertinence to the areas of research under consideration. The documents deter- mined to be the most relevant were obtained and a scan of the material was performed. Of the examined material, approxi- mately 70 applicable documents were selected for further evaluation and possible summary preparation. A tabulated summary of approximately 40 documents was prepared from the reviewed material (see Appendix B). Con- tained in each document summary is a brief study descrip- tion, the study findings (if any), and recommendations for further research suggested by the authors. A review of the summarized material (Appendix B) reveals that WIM data have been employed in numerous bridge-related applications in North America and abroad. WIM data have been used to assess current bridge design live loads and to model new design live loads. Several studies 11 State DOT Availability of WIM Data for a Number of Years at the Same Site? Use of WIM Data for Bridge Design Applications? Alaska Yes No Arkansas California Yes Yes. Caltrans has started using WIM data for bridge design recently. Connecticut No. See answer for previous question. No. Based loads permitted through the state. Florida Yes No Georgia No No Hawaii Yes No Idaho Yes Yes. We provide some commercial vehicle weight data and reports to our bridge design people. Indiana Yes No. The data is not provided directly to them. However, through Purdue University or our research section they may be utilizing the data. Iowa Yes No Kansas Yes No Louisiana No No Michigan Yes No. Not to my knowledge. Minnesota Yes No Mississippi No No Missouri Yes No Nevada Yes No New Jersey Yes No New Mexico Yes No New York Yes North Dakota Yes. Possibly. No Ohio Yes Yes. Maumee River crossing design & I think for a few other applications a few years back. Oregon Yes. From “official” state weigh records. These are available from every weigh station location, not just WIM sites. Yes—currently the ODOT Bridge Section is conducting an analysis for bridge redesign, and engineering standards. South Dakota Yes Unknown Virginia No No Washington Yes No, at least not to our knowledge Wyoming Possibly Not to my knowledge Table 3. Multi-year traffic data.

have found that the current load models were insufficient for the actual loading experienced by the bridge population. WIM data have also been applied to the development of new fatigue models and the assessment of existing models. The results of the investigations reveal that fatigue evaluation is highly site specific and that the actual fatigue damage result- ing from the use of WIM data is often underestimated or overestimated by the code-specified fatigue truck. Truck load growth trends have been assessed utilizing WIM data. For instance, a large-scale California study estab- lished that truck volumes have increased over time, however, the gross vehicle weight in the state has remained unchanged. This study also investigated the possibility of applying WIM data that were collected at a bridge site to other nearby bridge locations. The forecasting of truck load spectra as a result of changing truck weight limits also has been investigated by the application of WIM data The examination of truck multiple presence on bridges has employed WIM data to simulate multi-lane, traffic-critical loading events and extreme load effects. The studies differ in bridge span length and number of lanes investigated, how- ever it was generally noted that as the span length increases, the critical loading event is governed by an increasing num- ber of trucks. One study indicated that traffic density should be a deciding factor in the development of multiple presence reduction factors. Numerous studies have investigated WIM-derived site- specific bridge loads for evaluation and design purposes. Generally, it was determined that truck loads are strongly site specific, influenced by factors such as traffic volume, gross vehicle weight, axle weight, local industry, and law enforce- ment effort, and that current load models for design are often not representative of actual site loading. WIM Technology Over the last two decades, highway agencies have recog- nized the advantages of having automated data collection sys- tems that can provide information on truck weights and truck traffic patterns for economic analysis, traffic manage- ment, and various other purposes. The quality and quantity of WIM data has greatly improved in recent years. Due to the development of various WIM technologies, unbiased truck- loads are now being collected at normal highway speeds, in large quantity, and without truck driver knowledge. WIM systems that are utilized to provide high-speed weighing of 12 *Question 4.5 Do you have a Quality Assurance Program in place for your WIM systems to check data accuracy? State DOT Response Alaska No Arkansas NR California No Connecticut Yes. Through the office checks and the LTPP checks conducted by the FHWA regional contractors. Delaware NR Florida Yes Georgia No Hawaii No Idaho Yes. This is a daily ongoing part of our WIM maintenance and processing program. Illinois NR Indiana Yes Iowa Yes Kansas No Louisiana Yes Michigan Yes Minnesota Yes Mississippi Yes Missouri Yes Nevada Yes New Jersey Yes New Mexico Yes New York Yes North Dakota No. Still under development. Ohio Yes. TKO Oregon Yes. A trouble report system. South Dakota Yes Virginia Yes Washington Yes Wyoming Yes Note: No response (NR). Table 4. WIM data quality assurance.

trucks and other traffic are bending plates, load cells, piezo- electric cables, quartz cables, and bridge WIM systems. One major WIM system vendor alone has approximately 500 per- manent WIM sites in operation in 47 states: 246 piezo, 180 bending plate (BP), 42 single load cell (SLC), and 24 Kistler quartz (Table 5). The electronics, software, and storage tech- nologies of WIM data loggers have also advanced in pace with the sensor technology. Weigh-in-motion equipment currently used in the United States can collect data on truck volumes, axle configurations, truck arrival times, and load spectra. They usually at least classify the vehicles into the 13 FHWA classifications (bins). The majority of WIM data collection is done with perma- nently installed weight sensors, although some states may not collect weight data continuously at these sites. Permanent WIM stations provide more extensive datasets at geographi- cally diverse locations over long periods. On a national or regional basis, WIM data is easily obtained from the wide net- work of permanent sites. If more localized site-specific char- acteristics are desired, it may be necessary to utilize portable WIM systems for data collection. Portable devices allow flex- ibility in collecting site-specific traffic data at locations of interest, such as a bridge where significant illegal overloads are suspected. 13 Table 5. WIM sites and lanes (maintained by one vendor).

There are several types of WIM technologies with varying performance and cost considerations (Table 6). Piezoelectric sensor-based systems offer acceptable accuracy (usually ± 15% for gross weights) at such a low cost that their use has become quite widespread for data collection purposes. They can be used as temporary or permanent sites. Strain-based WIM scales and load cell WIM systems provide more accu- racy at a higher cost. Strain- and load cell-based systems are used primarily in permanent applications. New WIM tech- nologies continue to be developed and brought to the mar- ket. Piezoquartz sensors were recently introduced in the United States. They are less sensitive to changes in tempera- ture than the piezo-style sensors, and therefore, are generally more accurate. Use of In-Service Strain Measurements Procedures for using in-service peak strain measurements to directly evaluate the safety (using the LRFR method) of existing bridges have been proposed by some researchers. A 14 Type of Sensor Strengths Concerns Piezoelectric (BL) Easier, faster installation than many other WIM systems. Generally lower cost than most other WIM sensors. Well supported by industry. Can be used for temporary WIM systems. Sensitive to temperature change. Accuracy affected by structural response of roadway. Above average maintenance requirement. Requires multiple sensors per lane. Piezoquartz Easier, faster installation than many other WIM systems. May be more cost-effective (long term) if sensors prove to be long lived. Very accurate sensor. Sensor is not temperature sensitive. Growing support by industry. More expensive that other piezo technologies. Requires multiple sensors per lane. Above average maintenance requirement. Sensor longevity data not available. Accuracy affected by structural response of roadway. Bending Plate Frame separates sensor from pavement structure. Entire tire fits onto sensor. Moderate sensor cost. Sensor is not temperature sensitive. Extensive industry experience with the technology. Longer installation time required than piezo systems. Some systems have experienced premature failure, while others have been very long lived. Load Cell Entire tire fits onto sensor. Frequently considered the “most accurate” of conventional WIM technologies. Some systems have demonstrated very long lifespans. Most expensive WIM system. Requires significant construction effort to install. Cost-effective if constructed and maintained for a long lifespan. Table 6. Sensors commonly used for permanent WIM sites.

considerable part of this effort involves the statistical charac- terization of the live-load effect using an extreme-value the- ory. The strains due to ambient traffic are monitored and recorded to represent the distribution of maximum load effects. The maximum load effect distribution is then projected for longer periods, up to 10 years, for determining the max- imum expected load effect for evaluation. Because it is based on actual bridge response, it eliminates a substantial part of live-load modeling uncertainties, such as those related to dynamic impact and girder distribution factors. Used in com- bination with pavement WIM systems, this method has potential applications in other load modeling applications, particularly for fatigue design and assessment. Combining WIM data with bridge response data could significantly reduce uncertainties inherent in high-speed WIM data. One problem involved in such procedures is that the projection is valid for only the stresses at the point where the strain mea- surements are recorded and the information cannot be gen- eralized for stresses and strain at other locations of the same bridge let alone for application to other bridges. In any case, it is not the aim of this project to develop or recommend new WIM systems. Rather, efforts will concentrate on data now produced by operational WIM networks. LRFD Fatigue Design Fatigue criteria for steel bridges are an important consid- eration for designing for heavy traffic and long expected design lives as well as for assessment of remaining life for existing bridges. As truck weight and volume increase and bridges are maintained in service for increasingly longer peri- ods of time, the fatigue design and assessment issues become even more important. The present AASHTO fatigue truck, which is based on reli- ability analyses, was developed by NCHRP Project 12-28 (Moses et al. 1987). The truck traffic load input was taken for some 30 WIM sites in about 8 states collected in the 1980s. For a suite of bridges, each of over 20,000 trucks was used to calculate the stress ranges and then the fatigue damage aver- aged according to the fatigue damage law (cubic power). The number of cycles to failure depends on the cube of the stress range and comparison with lab test data for various welded details. A variety of random variables was considered to account for material, analysis, and load uncertainties. This information is used to calibrate the fatigue process to a target β for redundant and non-redundant cases. Moses et al. describes the details of this derivation in NCHRP Report 299 (1987). The results were incorporated into two AASHTO guide specifications for fatigue design and fatigue evalua- tion, respectively. Further, the fatigue truck developed for the nominal loading (54-kip, 5-axle truck) in these guide specifications was also incorporated into the AASHTO LRFD specifications and the LRFR manual. Most of the fatigue damage in a bridge is caused by passages of single trucks across the bridge. The total number of truck passages in the 75-year life of a bridge can exceed 100 million. Static strength design must be based on the most severe load effect expected to occur over the life of the bridge. Fatigue design, on the contrary, should be based on typical conditions that occur, because many repetitions are needed to cause a fatigue failure. (Moses et al. 1987). The fatigue load specified in the steel structures section of the LRFD specifications also produces a lower calculated stress range than that of the standard specifications. The fatigue pro- visions of the new specifications are more reflective of the fatigue loads experienced by highway bridges. In the LRFD, a special vehicle is used for fatigue analysis. It consists of one design truck, as specified above, but with the rear (32-kip) axle spacing fixed at 30.0 ft and without an accompanying uniform load (Figure 1). The fatigue load is used to represent the variety of trucks of different types and weights in actual traffic. For the purposes of fatigue design, a truck is defined as any vehicle with more than either two axles or four wheels. The constant axle spacing approximates that 5-axle semi-trailer trucks do most of the fatigue damage to steel bridges. The specified fatigue loading in LRFD produces 15 14’ 30’ 6 k 24 k 24 k Figure 1. LRFD fatigue truck (includes 0.75 load factor).

a lower calculated stress range than produced by the loadings in the standard specifications. This reduction in calculated stress range is offset by an increase in the number of cycles of loading to be considered in the LRFD specifications. The lower stress range and increased number of cycles are more reflective of the actual conditions experienced by bridges. If the maximum stress a detail experiences in its lifetime is less than the constant amplitude fatigue threshold for that detail, the detail is considered to have infinite fatigue life. In the FATIGUE load combination given in the LRFD spec- ifications, a load factor of 0.75 is applied to the fatigue load. The factored fatigue load is equivalent to the AASHTO HS15 load- ing. A revision to Article 3.4.1 was adopted in 2008 to specify two separate fatigue load combinations. For infinite life design under higher traffic volume conditions, the FATIGUE I load combination would be used. In the new FATIGUE I load com- bination, the stress range caused by the fatigue design truck is multiplied by a load factor of 1.50 (or 2.0 * 0.75). For finite life design under lower traffic volume conditions, the FATIGUE II load combination would be used. The FATIGUE II load com- bination retains the load factor of 0.75 applied to the stress range caused by the fatigue design truck. Where the bridge is analyzed using approximate analysis methods, the specified lateral live-load distribution factors for one lane loaded are used in the fatigue check, without the multiple presence factor of 1.2. Where the bridge is analyzed by any refined method, a single fatigue truck is positioned transversely and longitudinally to maximize the stress range at the detail under consideration. A reduced dynamic load allowance of 15% is applied to the fatigue load. Since fatigue is defined in terms of accumulated stress- range cycles over the anticipated service life of the bridge, the fatigue load should be specified along with the frequency of load occurrence/stress cycles. For the purposes of determin- ing the number of stress cycles per truck passage, LRFD Table 6.6.1.2.5-2 may be used (see Table 7). The frequency of the fatigue load is taken as the single-lane average daily truck traffic. The number of cycles to be consid- ered is the number of cycles due to trucks actually anticipated to cross the bridge in the most heavily traveled lane over its design life. The frequency of the fatigue load is taken as the single-lane average daily truck traffic (ADTTSL). In the absence of better information, the single-lane ADTT shall be taken as follows: Where: ADTT = the number of trucks per day in one direction averaged over the design life p = fraction of truck traffic in a single lane. Because of the importance of the lifetime average daily truck volume parameter in fatigue design, the engineer should use whatever site data may be available for making this estimate. It should be made excluding 2-axle trucks consis- tent with the procedure used in calculating the fatigue truck weight. Traffic volume usually grows at an annual rate of about 2% to 5% until reaching a very high limiting value. It is unrealistic to project traffic growth indefinitely into the future. ADT, including all vehicles, is physically limited to about 20,000 vehicles per day per lane. AASHTO Fatigue Guide Specifications (1989, 1990) The Guide Specifications for Fatigue Design of Steel Bridges (AASHTO 1989) and the Guide Specifications for Fatigue Eval- uation of Existing Steel Bridges (AASHTO 1990) specified a 3-axle fatigue tuck having a gross weight of 54 kips to represent the variety of trucks in actual traffic seen in the WIM data col- lected in the early 1980s. The fatigue truck is similar to the LRFD fatigue load once the loads are scaled down using the 0.75 load factor. One key difference was that the guide specification allowed a variable spacing of 14 ft to 30 ft instead of the stan- dard 30-ft main axle spacing. If used, the reduced axle spacing would have resulted in increased fatigue design stresses. To recognize the considerable region-to-region and site- to-site differences in truck weight population, alternatives for determining the gross weight of the fatigue truck were per- mitted. Where the gross-weight histogram for truck traffic (excluding 2-axle trucks) was available, the guide allowed the determination of the gross weight of the fatigue truck from the following: W f Wi i 3 = ( )∑ 1 3 2( ) ADTT p ADTT (1)SL = × 16 Longitudinal Members Span > 40 ft Span 40 ft Simple span girders 1.0 2.0 Continuous span girders near supports 1.5 2.0 Continuous span girders elsewhere 1.0 2.0 Transverse members Spacing > 20 ft Spacing 20 ft 1.0 2.0 Table 7. Stress cycles per truck passage.

Where: fi = fraction of gross weights within an interval i Wi = gross weight at mid-width of interval i The fatigue design was based on the passage of a single fatigue truck across the bridge in the lane under considera- tion. The net effect of closely spaced trucks was considered to be small for normal traffic conditions (Moses et al. 1987). Therefore, the effect of truck superpositions was neglected for span lengths typically covered by the AASHTO standard specifications. If unusual bunching of trucks is expected at a site, a 15% increase of the fatigue truck weight was specified. Fatigue Load Research Research studies have been conducted to establish improved fatigue load models that will cause the same cumulative fatigue damage as the normal truck traffic distribution obtained by WIM measurements in various states. Laman and Nowak (1996) developed a fatigue load model for girder bridges from WIM measurements at five bridge sites in Michigan (Figure 2). Stress cycles also were measured at the midspan of all girders. A new 3-axle fatigue truck was proposed to model vehicles with 3 to 7 axles. The fatigue damage caused by the traffic con- sisting of 3 to 7 axles is equivalent to the damage caused by an equal number of passes of the 3-axle fatigue truck. A 4-axle fatigue truck was proposed for sites with 10- and 11-axle trucks common in Michigan. The AASHTO fatigue truck was not adequate to model the traffic at these sites. It was found that a high percentage of the fatigue damage was dominated by 10- and 11-axle trucks, although they did not dominate the distribution of truck types. It was found that for these sites, two site-specific fatigue trucks could provide relatively accu- rate estimates of fatigue damage accumulation over a range of bridge spans. This illustrates that truck load spectra are strongly site specific. Chotickai and Bowman (2006) developed a new fatigue load model based on WIM data collected from three differ- ent sites in Indiana (Figure 3). The recorded truck traffic was passed over simulated bridge spans to investigate moment range responses in simple and continuous span bridges. Based on Miner’s hypotheses, fatigue damage accumulations were compared with the damage predicted for the 54-kip AASHTO fatigue truck, a modified AASHTO fatigue truck with an equiv- alent effective gross weight, and other fatigue truck models. The simulation results indicate that the use of the 54-kip gross weight in an evaluation of bridge structures can result in a considerable overestimation or underestimation of the extent of the actual fatigue damage. It also was shown that the fatigue trucks given by Laman and Nowak (1996) do not provide an accurate estimate of the fatigue damage accumulation for a wide range of span lengths when compared with fatigue dam- age estimated using the WIM database. Based upon the results of the Indiana WIM study, a new 3-axle fatigue truck and a 4-axle fatigue truck were proposed. The front and rear axle spacing of the new 3-axle fatigue truck are wider than the AASHTO fatigue truck. In addition, a higher percentage of the gross weight is distributed to the front axle compared with the AASHTO fatigue truck. These adjustments were consid- ered to be consistent with statistics of the axle configurations 17 (a) 44.5 - 102 kN 102 - 129 kN 102 - 129 kN 3.5 - 4.0 m 8.8 - 9.8 m (b) 44.5 - 98.0 kN Source: Reprinted with permission from ASCE. 191 - 267 kN 89.0 - 267 kN165 - 267 kN 3.4 - 4.3 m 5.2 - 5.5 m 3.4 - 4.3 m 0.26W (a) 0.41W 0.33W5.5 m 0.13W 0.31W 0.31W 0.25W (b) 3.0 m 5.5 m 8.2 m 10.7 m Source: Reprinted with permission from ASCE. Figure 2. Fatigue trucks (Laman and Nowak 1996). Figure 3. Fatigue trucks (Chotickai and Bowman 2006).

of truck traffic observed. The new 4-axle fatigue truck was most effective when a significant number of 8- to 11-axle trucks pass over a bridge, while the new 3-axle fatigue truck was most effective otherwise. LRFD Deck Design In the LRFD specifications, design load for decks and top slabs of culverts using the transverse strip method is the axle load, represented by a 32-kip axle of the design truck or the design tandem consisting of a pair of 25.0-kip axles spaced 4.0 ft apart, whichever creates the most extreme force effect (Figure 4). The two design vehicles should not be considered in the same load case—consider all trucks of one kind, design truck or design tandem (Figure 5), not a mix of the two. The design truck has the same weights and axle spacing as the HS20 load model, which was adopted in 1944 for bridge design, and has been carried over from the standard specifications. Only the truck load is required for decks and not the truck + distrib- uted loads as required for girders. The slab may be designed by the three following methods: (1.) traditional strip method (2.) empirical design, and (3.) yield-line method. In the traditional strip method, the deck is subdivided into equivalent strips perpendicular to the sup- porting components. The width of equivalent strips is given in LRFD Table 4.6.2.1.3-1. The strips are analyzed as contin- uous beams using one or more loaded lanes to determine maximum live-load moments. The empirical design requires that the designer satisfy a few simple rules regarding deck thick- ness and reinforcement details. An analysis for load effects is not required. Although newer deck design methods were intro- duced, design load effects on bridge decks have remained essentially the same in the LRFD specifications compared with the requirements of the standard specifications. The live-load modeling in LRFD did not specifically address the increasing load effects on bridge decks from the heavier and more com- plex axle configurations of current truck traffic. Axle groups with more than 2 axles are currently not con- sidered for deck design in LRFD. However, LRFD commentary C3.6.1.3.3 states Individual owners may choose to develop other axle weights and configurations to capture the load effects of the actual loads in their jurisdiction based upon local legal load and permitting policies. Triple-axle configurations of single-unit vehicles have been observed to have load effects in excess of the HL-93 tandem axle load. Multi-Axle Specialized Hauling Vehicle To increase load carrying capacity and maximize produc- tivity, in recent years the trucking industry has introduced specialized commercial vehicles with closely spaced multiple axles (Figures 6, 7, and 8). This was the focus of NCHRP Proj- ect 12-63 (Sivakumar et al. 2007). States have adopted a vari- ety of short multi-axle vehicles as posting loads in response to 18 DECK DESIGN – LRFD CODE PROVISIONS 32 k DESIGN AXLE 25 k DESIGN TANDEM25 k 4’ APPROXIMATE STRIP METHOD OF ANALYSIS: +M = 26.0 + 6.6S -M = 48.0 + 3.0S Figure 4. LRFD deck design loads and strip widths. W2 W2 S2 W3 W3 W3 S3 S3 W4 W4 W4 S4 S4 W4 S4 TANDEM TRIDEM QUAD Figure 5. Typical multi-axle loads. Figure 6. Concrete ready-mix truck with quad axle.

their potential for overstressing shorter span bridges and bridge decks (Figures 9 and 10). Additionally, short dump trucks are allowed loads of 60K or more in the rear tri-axle group in many states, as seen in Figure 9. Similar exemptions are granted for tandem axles on concrete mixer trucks and other work trucks (>50K allowed on a tandem). Such axle groups usually include lift axles, which are required to be low- ered when the truck is loaded, but are not always used in this manner. This further exacerbates the overstress problem on decks and short-span bridges. NCHRP 12-63 has compiled a database of over 70 trucks with complex axle configurations currently in use as legal loads in various states. Many states exempt short hauling vehicles from Federal Formula B and axle weight limits under the grandfathered rights granted when the federal weight laws were first enacted. This infor- mation intended for developing new AASHTO load models for evaluation in NCHRP 12-63 can be a valuable resource for developing new axle loads for deck design as the data repre- sents service loads that the new decks will be subjected to on a routine basis. This has implications for the strength and fatigue design provisions for decks in the current LRFD spec- ifications. Current provisions may be grossly underestimat- ing the maximum and repetitive load effects on bridge decks. Furthermore, the multiple presence probabilities for side-by- side axles could be considerably different than truck multiple presence probabilities for trucks. The former influences deck design, which could be much higher than the side-by-side probabilities for trucks as each truck may have several axle groups that could increase multiple-presence. This issue has not been adequately investigated in past calibrations of the bridge code. LRFD Overload Design In addition to routine service loads, bridge owners usually have established procedures that allow the passage of vehicles 19 Figure 7. Multi-axle specialized hauling vehicle (Sivakumar et al. 2007). Figure 8. Truck with tridem and quad axles. 15 15 22.5 22.5 11’ 4’ 4’ GVW = 75 KipsAlabama Tri-Axle 13.5 18 22.5 22.5 9’-7” 4’-1” 4’-6” GVW = 76.5 KipsConnecticut Construction Vehicle 19191916 4.5’4.5’8’ GVW = 73 KipsDelaware DE 4 18.7 18.7 18.713.9 4’-2”4’-2”10’ GVW = 70 KipsFlorida SU4 L = 19’ L = 17’ L = 18’-4” L = 18’-2” Figure 9. State legal loads that exceed federal weight limits (Sivakumar et al. 2007).

above the legally established weight limitations on the high- way system. Depending upon the authorization, these permit vehicles may be allowed to mix with normal traffic or may be required to be escorted in a manner that controls their speed and/or lane position, and the presence of other vehicles on the bridge. The multiple-presence probabilities for permit trucks are significantly different from those used for normal traffic. In the LRFD, the Strength II limit state is specified for checking an owner-specified special design vehicle or permit vehicle during the design process with a reduced load factor of 1.35. No further guidance is given in the LRFD specifica- tions on how this load factor was derived, the level of safety represented by this limit state, site traffic, or how the load fac- tor might be adjusted when design live-loading characteris- tics (such as the gross weight of the permit load, likelihood of trucks exceeding the permitted weight, and multiple presence likelihood) differ significantly from the calibration assump- tions. These kinds of information need to be obtained locally or regionally through WIM measurements and considered in the Strength II design process since the data used for calibra- tion of national codes are unlikely to be representative of all jurisdictions. 20 Figure 10. North Carolina legal loads (Sivakumar et al. 2007).

LRFD Superstructure Design The notional live-load model used in the LRFD specifica- tion for the design of bridges, designated HL93, consists of a combination of the • Design truck or design tandem, and • Design lane load. Each design lane under consideration shall be occupied by either the design truck or tandem, coincident with the lane load, where applicable. The HL93 live-load model was initially developed as a notional representation of shear and moment produced by a group of vehicles routinely permitted on highways of various states under “grandfathered” exclusions to weight laws. The vehicles were based on a 1990 study conducted by TRB that identified 22 representative configurations of vehicles allowed by states as exceptions to weight laws, the smallest and largest of which were a 3-axle, 48-kip single truck, and an 11-axle, 149-kip trailer truck (Kulicki 1994). The notional load model was subsequently compared to the results of truck weight studies, selected WIM data (Ontario), and the 1991 Ontario Highway Bridge Design Code (OHBDC) live-load model. These comparisons showed that the notional load could be scaled by appropriate load factors to be representative of these other load spectra. Comparisons of moments and shears in simple spans and two-span continuous girders ranging in span from 20 to 150 ft produced by HS20 and the envelope of results produced by the 22 representative exclusion vehicles indicated that the HS20 design loading was not representative of vehicles on U.S. high- ways. Five candidate notional loads were identified in the live-load development for the AASHTO LRFD specification (Kulicki 1994). Ratios of force effects for each of these live-load models divided by the corresponding force effect from the envelope of exclusion loads indicated that the load model involving a combination of either a pair of 25-kip tandem axles and the uniform load, or the HS20 and the uniform load, seem to produce the best fit to the exclusion vehicles. The tight clus- tering of force effect ratios for all span lengths, forming bands of data that are essentially horizontal, indicates that the live- load model and load factor can be independent of span length. Thus, the combination of the tandem with uniform load and the HS20 with the uniform load were shown to be an adequate basis for a notional design load in the LRFD specification. The LRFD specification allows site-specific modifications to the design truck, design tandem, and/or the design lane load under the following conditions: • The legal load of a given jurisdiction is significantly greater than typical; • The roadway is expected to carry unusually high percent- ages of truck traffic; • Flow control, such as stop sign, traffic signal, or toll booth, causes trucks to collect on certain areas of a bridge; or • Special industrial loads are common due to the location of the bridge. The dynamic load model was determined to be a function of three major parameters: road surface roughness, bridge dynamics (frequency of vibration), and vehicle dynamics (suspension system). The actual contribution of road rough- ness, bridge dynamics, and vehicle dynamics varies from site to site and is difficult to predict. Therefore, the dynamic load allowance in the LRFD was specified as a constant percentage of live load (Nowak 1993b). Proposed Process to Develop Vehicular Live-Load Models The design and load capacity evaluation of a bridge mem- ber depends on the live-load model and the live-load factor used in the design check equation. The live-load factor γL = 1.75 for Strength I provided in the AASHTO LRFD specifications has been calibrated for use along with the HL93 design load such that bridge members designed with the AASHTO LRFD specifications would achieve a uniform target reliability index β = 3.5. In actuality, conservative rounding up of the originally calibrated live-load factor indicates that the AASHTO LRFD will produce members with reliability index values higher than 3.5. The reliability index calculations use as input a live-load model that estimates the maximum expected live-load effect on a bridge member. (Lmax is the expected lifetime maximum load effect on a bridge.) The model includes the mean value of the maximum expected live-load effect along with the stan- dard deviation and the probability distribution type. The live- load model used during the AASHTO LRFD calibration was obtained from a generic set of truck weight and load effect sta- tistics that are presumed to be valid for any typical bridge site in the United States. Because of its generic nature, the live-load model may not represent the actual loading conditions at a particular bridge site or bridges in a state where the truck weights or traffic conditions do not follow the expected typi- cal model. Under these conditions, site-specific or state- specific live-load models may need to be developed based on actual truck weight and traffic data collected at the site or within the state. Several states are currently using WIM sys- tems to collect vast amounts of truck weight and traffic data that can be used to obtain site-specific and state-specific live- load models for bridge design and load capacity evaluation. This would allow individual states to adjust the AASHTO live- load factors to take into consideration the particular truck 21

traffic conditions throughout a state, a region, or on a partic- ular route. Task 2 of this research project was focused on existing and potential processes to develop and calibrate vehicular loads for superstructure design, fatigue design, deck design, and over- load permitting. The findings are summarized in Appendix D. Several procedures of various levels of complexity exist to esti- mate the maximum expected load effect on a highway bridge. This section describes the recommended procedure that uti- lizes site-specific truck weight and traffic data to obtain esti- mates of the maximum live load for a specified return period. The return period for the design of a new bridge is specified to be 75 years as per the AASHTO LRFD code for Strength I limit state. A 2-year return period has been used for the load capac- ity evaluation of existing bridges during the calibration of the AASHTO LRFR, and a 1-year period has been proposed for estimating the maximum live-load effect for the AASHTO LRFD Strength II limit state. It should be clearly stated that it is not possible to obtain exact values of the maximum expected 75-year load due to the limitations in the available data. In fact, to obtain accurate results, one would need sev- eral cycles of WIM data collected over 75 years for each cycle, which is an impossible task. Even the development of live-load models for 1-year and 2-year return periods would require several cycles of 1-year and 2-year data, which are currently not available due to the relatively recent adoption of WIM technology in the United States. Hence, some form of statisti- cal projection will be needed for any practical load modeling effort as will be described in this section. The approach described in this section uses a normal probability distribution to project the tail end of the collected WIM data histograms. The properties of the normal distribution can then be used to obtain the statistics of the maximum load effects for any return period using extreme value distributions. This section presents the theoretical background for the proposed procedure for modeling the maximum live-load effect on a highway bridge. The proposed approach for obtaining the live-load model requires as input the WIM data collected at a site after being “scrubbed” and processed to remove data outliers as described in other sections of this report. For state-specific load models, data from several repre- sentative sites should be assembled. The statistics of the max- imum live-load effects can then be used to adjust the AASHTO live-load factors so that the Strength I and Strength II limit state designs would achieve safety levels similar to those intended by the AASHTO code writers while simulta- neously accounting for the state-specific loading conditions. 1. Probability Density Function and Frequency Histogram of Truck Load Effects For the single-lane loading of short-span bridges, a truck loading event is defined as the occurrence of a single truck on the bridge. For multiple truck presence in multi-lanes, the loading event may consist of two or more trucks simultane- ously on the bridge. While all currently used WIM systems are capable of providing axle weights and axle spacings for each truck crossing, only WIM systems capable of taking con- tinuous uninterrupted data at normal highway traffic speeds with accurate time stamps are able to identify multi-lane loading events and provide the axle weights, axle spacings, and relative positions of all the trucks involved in each multi- lane loading event. For the cases when uninterrupted multi-lane data with accurate time stamps are not available, multi-lane loading events can be obtained from simulations based on estimates of the number of side-by-side and information on ADTT data. This could be achieved using an approach that will be described in Chapter 3. In the first step, using the WIM data files, the shear force or bending moment effect of each truck loading event in the WIM record is calculated by passing the trucks through the proper influence line. For multi-lane loadings, the combined shear force or moment effect from the trucks that are simul- taneously on the bridge is obtained. The shear or moment for each truck load event is then normalized by dividing the cal- culated value by the shear or moment of the HL93 load model. The shear and moment data for the single-lane loading and the multi-lane loading events are collected into separate per- cent frequency histograms. Each histogram provides a dis- cretized form of the probability density function (pdf) of the shear or moment effects for the site. The histogram is desig- nated as Hx(X) while the pdf is designated as fx(X). The rela- tion between Hx(X) and fx(X) is given by Where: Xl and Xu give the upper and lower bounds of the bin within which X lies. If the bin size is small, then fx(X) can be assumed to be constant within the range of Xl to Xu and Equa- tion (3) becomes Where: ΔX is the bin size. 2. Cumulative Distribution Functions and Cumulative Frequency Histograms Using the histograms and the probability density functions fx1(. . .), fx2(. . .), fxs(. . .), and fz(. . .), the cumulative distribu- tions for the shear or moment effects of single or multi-lane H X f X X f X X Xx x x u l( ) = ( ) = ( ) −( )Δ ( )4 H X f x dxx x X X l u( ) = ( )∫ ( )3 22

loading events can be obtained. The cumulative distribution is assembled from the pdf using the following equation: In essence, Equation 5 assembles all the bins of the histo- gram below a certain value Z into a single bin at Z. This is repeated for all possible values of Z. Thus, Fz(Z) will give the probability that a loading event will produce a load effect less than or equal to Z. 3. Cumulative Probability Function for the Maximum Load Effect over a Return Period of Time, treturn. For the AASHTO Strength I limit state, a bridge structure should be designed to withstand the maximum load effect expected over the design life of the bridge. The AASHTO LRFD code specifies a design life of 75 years. The LRFR bridge load rating also requires checking the capacity to resist the maxi- mum load effects for a 2-year return period. The AASHTO LRFD Strength II limit state implicitly assumes a 1-year return period associated with special permit trucks. It is simply impossible to collect enough data to determine the maximum load effect expected over 75 years of loading. Even getting suf- ficient data for the 1-year and 2-year return periods would require several cycles of 1-year and 2-year data, which are not currently available. Therefore, some form of statistical projec- tion should be performed. The proposed calculation proce- dure uses the cumulative distribution function for individual loading events and then applies a statistical projection to obtain the information required for a 1-year, 2-year, or 75-year return period. To find the cumulative distribution for the maximum loading event in a period of time treturn one has to start by assuming that N loading events occur during this period of time. These events are designated as S1, S2, . . . SN. The maxi- mum of these N events, called Smax,N, is defined as The study team was interested in finding the cumulative probability distribution of Smax,N. This cumulative probability distribution, Fs max N(S), gives the probability that Smax,N is less than or equal to a value S. If Smax,N is less than S, this implies that S1 is less than S, and S2 is less than S, . . . and SN is less than S. Hence, assuming that the loading events are independent, the probability that Smax,N ≤ S can be calculated from F S F S F S F Ss s s smax,N 1 2 N( ) = ( ) ( ) ( )i . . . ( )7 S max S S Smax,N 1 2 N= ( ), , . . . ( )6 F Z f y dyz z Z ( ) = ( ) −∝ ∫ ( )5 If S1, S2, . . . SN are independent random variables that are drawn from the same probability distribution, then and Equation 7 reduces to Note that Equation 9 assumes that the number of events is a known deterministic value. A sensitivity analysis performed as part of this study has however demonstrated that the results of Equation 9 are not highly sensitive to small varia- tions in N as N becomes large. The application of Equation 9 requires high precision in Fs(S). For example, given 3,000 trucks per day over a 75-year return period, the number of single truck events N would be over 82 million. Thus, to obtain the median value of the max- imum event Smax N corresponding to Fsmax N(S) = 0.5 would require Fs(S) to have precision up to the 9th decimal point, which is not possible to obtain without executing some form of statistical projection. To perform the statistical projection, a careful analysis of the tail end of Fs(S) is required. 4. Statistical Fit of Tail End of Probability Distribution of Load Effect of a Single Loading Event The probability distribution of the single loading event does not follow any known probability distribution type. However, careful observations of the tail ends of the WIM data histograms assembled from several sites indicate that the tail ends match the tail ends of normal probability distribu- tions. For example, Figure 11a shows the plot of the data col- lected at the I-81 site in Upstate New York on a normal probability scale. A normal probability plot is executed by F S F Ss s N max N ( ) = ( )⎢⎣ ⎥⎦ ( )9 F S F S F S F Ss s s s1 2 N( ) = ( ) = = ( ) = ( ). . . ( )8 23 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 60-ft simple span moment St an da rd D ev ia te Moment/HL-93 All data Upper 5% Linear fit (upper 5%) y=3.0049x-0.0697 (R2=0.997) Figure 11a. Normal probability plot for moment effect of trucks in drive lane of I-81 NB.

taking the normal inverse of Fs(S) represented by Φ−1[Fs(S)] and plotting versus S. The plot would produce a straight line if S follows a normal distribution. In this case, the mean of S would correspond to the abscissa for which Φ−1[Fs(S)] is zero. The mean plus on standard deviation would correspond to the abscissa for which Φ−1[Fs(S)] is equal to 1.0. The plot of the WIM data shows that the data as a whole does not follow a normal distribution as the curve does not follow a straight line. However, the figure shows that the upper 5% of the data does approach a straight line indicating that the tail end of the data resembles the tail end of a hypo- thetical normal distribution. A linear fit on the normal prob- ability plot of the upper 5% of the data collected at this I-81 site will produce a slope, m, and an intercept, n, which will give the mean of the equivalent normal distribution that best fits the tail end as μevent = −n/m. The standard deviation of the best-fit normal distribution is σevent = 1/m. For the I-81 data, single truck events give μevent = 0.0232 and σevent = 0.333. The regression analysis of the upper 5% of the data produces a regression coefficient R2 = 0.997 indicating a good linear fit. 5. Cumulative Distribution of Maximum Load Effect To verify that the normal fit of the tail end of the data is suf- ficient to obtain good estimates of the maximum load effect for long return periods, the results of Equation 7 are plotted in Figure 11b for two cases. Case 1 uses as input the cumula- tive distribution Fs(S) obtained from the WIM data for single loading events and Case 2 uses the Fs(S) that corresponds to the normal distribution with μevent = 0.0232 and σevent = 0.333. The application of Equation 7 uses different return periods varying from 1 day to 75 years. The number of events for the I-81 site are obtained as Nday = 3200 single truck events. Figure 11b shows good agreement between the projections obtained from the normal fit of the tail and the WIM data for the 1-day, 1-week, and 1-month return periods. The figure also shows that the WIM data are not sufficient to obtain the max- imum load effect for return periods greater than 1 month. However, the use of the normal distribution to model the tail end of WIM data would allow for obtaining the maximum load effect distribution for extended return periods. 6. Extreme Value Distribution of Maximum Load Effect Although the application of Equation 7 can be executed numerically for any parent probability distribution, the fact that the tail end of the WIM data matches that of a normal distribution allows for the application of extreme value the- ory to obtain the statistics of the maximum load effect in closed form. The approach is based on the following known concept as provided in Ang and Tang (2007), which states “if the parent distribution of the initial variable S has a general normal distribution with mean μevent and standard deviation σevent, then the maximum value after N repetitions approaches asymptotically an Extreme Value Type I (Gumbel) distribu- tion” with a dispersion αN given by: 24 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 2 2 . 2 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 M o m e n t / H L - 9 3 F r e q u e n c y 1 day 1 week 1 month from Normal Figure 11b. Cumulative distribution of maximum load effect of single lane events for different return periods.

and a most probable value uN given by: αN and uN can be used to find the mean of the maximum load effect, Lmax, and its standard deviation, σmax, for any return period having N repetitions as and Monte Carlo Simulation An alternative to the statistical projections approach consists of using a Monte Carlo simulation to obtain the maximum load effect. In this approach, results of WIM data α π α max ( )= 6 13 N L uN N max max . ( )= = +μ α 0 577216 12 u N N N event event= + ( ) − ( )( ) + ( )μ σ π2 4 2 2 ln ln ln ln ln N( ) ⎛ ⎝⎜ ⎞ ⎠⎟ ( )11 α σ N event N = ( )2 10 ln ( ) observed over a short period can be used as a basis for projec- tions over longer periods of time. A Monte Carlo simulation requires the performance of an analysis of a large number of times and then assembling the results of the analysis into a histogram that will describe the scatter in the final results. Each iteration is often referred to as a cycle. The process can be executed for the single lane-loading situation or the side- by-side loading. A step-by-step procedure for Monte Carlo simulation is described in the next chapter in Step 12.3 of the draft protocols. Monte Carlo simulation will not be accurate for large pro- jection periods because of the limitations in the originally collected data. Also, the Monte Carlo simulation will be extremely slowed down when the number of repetitions K is very high. Furthermore, one should make sure not to exceed the random number generation limits of the software used, otherwise the generated numbers will not be independent and the final results will be erroneous. Hence, statistical pro- jections must be made to estimate the maximum load effects for long return periods. Alternatively, one can use a smoothed tail end of the WIM histogram by fitting the tail with a known probability distribution function (such as the normal distri- bution) and use that fitted distribution with the Monte Carlo simulation. It should be emphasized however, that the Monte Carlo simulation would still be very inefficient for projections over long return periods. 25