Bridges for Service Life Beyond 100 Years: Service Limit State Design (2014)

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94 C h a p t e r 5 5.1 Development of Live Load Models for Service Limit States 5.1.1 Introduction The consideration of limit states, both ultimate (strength) and serviceability, requires the knowledge of loads. The objective of this task is to determine the statistical parameters of live load for the limit states considered in AASHTO LRFD (2012). For strength limit states, the live load statistics were deter- mined in NCHRP Project 12-33 and documented in NCHRP Report 368 (Nowak 1999). The emphasis was placed on pre- diction of the extreme expected live load effects in the 75-year lifetime of a bridge. The database at that time was a truck sur- vey carried out by the Ontario Ministry of Transportation in Canada. The basic statistical parameters of the maximum 75-year live load effect (moment and shear force) were deter- mined by extrapolating the truck survey data. It was assumed that the survey represented 2 weeks of heavy traffic. The pro- cedure is described in NCHRP Report 368 (Nowak 1999). The serviceability limit states require additional statistical parameters, not only the maximum values, but also load spectra (i.e., frequency of occurrence of loads). The maxi- mum values are needed for shorter time periods, such as a day, week, month, or year. At present, a considerable amount of WIM (weigh in motion) truck data is available and the research team had access to two sources: NCHRP Project 12-76 data (Sivakumar et al. 2011) and Federal Highway Administration (FHWA) files. This chapter provides docu- mentation on the development of the statistical parameters of live load for service limit states (SLSs) and fatigue. The analysis includes consideration of the WIM database from NCHRP Project 12-76 and FHWA. The obtained data included over 65 million vehicles. Of that number, about 10 million were deleted or filtered because of obvious errors, leaving about 55 million. Data from New York (about 7.8 mil- lion records) and Indiana other than site SPS-6 (about 13 million records) were also removed. The New York data were not considered because they included a considerable number of extremely heavy vehicles. It was decided that these data would have a strong effect on the statistical parameters, which would cause the remaining states to be unnecessarily penalized. Indiana data could not be considered because the format was not compatible with the other states. The consid- ered database included about 35 million vehicles. The obtained WIM data include the following information for each location and each recorded vehicle: number of axles, spacing between axles, axle loads, gross vehicle weight (GVW), vehicle speed, and exact time of measurement. Sta- tistical parameters are determined for the GVW and moment caused by the vehicles, including a cumulative distribution function (CDF); a bias factor (l) that is equal to the mean-to- nominal ratio (i.e., the ratio of the mean value and the nomi- nal, or design, value); and the coefficient of variation (CV), V, which is equal to the ratio of the standard deviation (s) to the mean (µ). The CDFs for the WIM data for each site were plotted on normal probability paper, which is described in Chapter 3, Section 3.2.1. 5.1.2 WIM Database The truck survey includes WIM truck measurements from 52 sites obtained from NCHRP Project 12-76 and FHWA. The data obtained from FHWA, which are summarized below, included trucks recorded from special pavement studies (SPSs); each SPS is followed by a number that identi- fies the study’s location (e.g., SPS-1 is Special Pavement Study, Location 1): • Arizona (SPS-1)—Data recorded continuously from Janu- ary 2008 until December 2008; • Arizona (SPS-2)—Data recorded continuously from Janu- ary 2008 until December 2008; Live Load for Calibration

95 • Arkansas (SPS-2)—Data recorded continuously from January 2008 until December 2008; • Colorado (SPS-2)—Data recorded continuously from January 2008 until December 2008; • Delaware (SPS-1)—Data recorded continuously from January 2008 until December 2008; • Illinois (SPS-6)—Data recorded continuously from Janu- ary 2008 until December 2008; • Indiana (SPS-6)—Data recorded continuously from July 2008 until December 2008; • Kansas (SPS-2)—Data recorded continuously from Janu- ary 2008 until December 2008; • Louisiana (SPS-1)—Data recorded continuously from January 2008 until December 2008; • Maine (SPS-5)—Data recorded continuously from Janu- ary 2008 until December 2008; • Maryland (SPS-5)—Data recorded continuously from January 2008 until December 2008; • Minnesota (SPS-5)—Data recorded continuously from January 2008 until December 2008; • New Mexico (SPS-1)—Data recorded continuously from May 2008 until December 2008; • New Mexico (SPS-5)—Data recorded continuously from May 2008 until December 2008; • Pennsylvania (SPS-6)—Data recorded continuously from January 2008 until December 2008; • Tennessee (SPS-6)—Data recorded continuously from January 2008 until December 2008; • Virginia (SPS-1)—Data recorded continuously from Janu- ary 2008 until December 2008; and • Wisconsin (SPS-1)—Data recorded continuously from January 2008 until December 2008. Data obtained from NCHRP projects are also summarized here, and include trucks recorded from California • Lodi (Site 003)—Data recorded continuously from June 2006 until March 2007; • Antelope Eastbound (Site 003)—Data recorded almost continuously from April 2006 until March 2007 (107 days missing); • Antelope Westbound (Site 003)—Data recorded almost continuously from April 2006 until March 2007 (109 days missing); • LA 710 Southbound (Site 059)—Data recorded continu- ously from April 2006 until March 2007; • LA 710 Northbound (Site 060)—Data recorded almost continuously from April 2006 until March 2007 (32 days missing); and • Bowman (Site 072)—Data recorded almost continuously from April 2006 until February 2007 (139 days missing). Florida • US-29 (Site 9916)—Data recorded continuously from January 2005 until December 2005 (11 days missing); • I-95 (Site 9919)—Data recorded continuously from Janu- ary 2005 until December 2005 (16 days missing); • I-75 (Site 9926)—Data recorded almost continuously from January 2005 until December 2005 (100 days missing); • I-10 (Site 9936)—Data recorded almost continuously from January 2005 until December 2005 (100 days missing); and • State Route (Site 9927)—Data recorded almost continuously from January 2004 until December 2004 (5 days missing). Indiana • Site 9511—Data recorded continuously from January 2006 until December 2006; • Site 9512—Data recorded continuously from January 2006 until December 2006; • Site 9532—Data recorded continuously from January 2006 until December 2006; • Site 9534—Data recorded continuously from January 2006 until December 2006; and • Site 9552—Data recorded continuously from January 2006 until December 2006. Mississippi • I-10 (Site 3015)—Data recorded almost continuously from January 2006 until December 2006 (28 days missing); • I-55 (Site 2606)—Data recorded almost continuously from January 2006 until December 2006 (16 days missing); • I-55 (Site 4506)—Data recorded almost continuously from March 2006 until December 2006 (39 days missing); • US-49 (Site 6104)—Data recorded almost continuously from January 2006 until December 2006 (5 days missing); and • US-61 (Site 7900)—Data recorded almost continuously from January 2006 until December 2006 (49 days missing). New York • I-95 Northbound (Site 0199)—Data recorded continu- ously from March 2006 until December 2006; • I-95 Southbound (Site 0199)—Data recorded continu- ously from July 2006 until November 2006; • I-495 Westbound (Site 0580)—Data recorded continu- ously from January 2006 until December 2006; • I-495 Eastbound (Site 0580)—Data recorded continuously from January 2006 until December 2006; • Highway 12 (Site 2680)—Data recorded continuously from January 2005 until December 2005; • I-84 Eastbound and Westbound (Site 8280)—Data recorded continuously from January 2006 until December 2006; • I-84 Eastbound and Westbound (Site 8382)—Data recorded continuously from January 2005 until December 2005;

96 • I-81 Northbound and Southbound (Site 9121)—Data recorded continuously from January 2005 until December 2005; and • Highway 17 Eastbound and Westbound (Site 9631)—Data recorded continuously from February 2006 until Decem- ber 2006. 5.1.3 WIM Data Filtering The WIM data both from NCHRP Project 12-76 and FHWA include vehicle records that appear to be incorrect. There are various reasons for questioning the data (e.g., GVW is too low, unrealistic geometry). The data were filtered to eliminate questionable vehicles by using the following criteria: • Weight per axle less than 2 kips or greater than 70 kips, based on NCHRP 12-76; • Record in which the first axle spacing was less than 5 ft, based on NCHRP 12-76; • Record in which any axle spacing was less than 3.4 ft, based on NCHRP 12-76; • Record in which GVW varied from the sum of the axle weights by more than 10%, based on NCHRP 12-76; • Record in which the length of the truck varied from the sum of the axle spacings by more than 1 ft, based on NCHRP 12-76; • Record that had a GVW less than a threshold; at various times the threshold was 10 or 12 kips; • Record in which the steering axle was less than 6 kips, based on NCHRP 12-76; • Record in which the sum of the axle spacing lengths was less than 7 ft, based on Pelphrey et al. (2008); • Class of the vehicle according to FHWA, from Class 3 to 14, to filter out cars, motorcycles, and so on; and • Speed ranges from 10 to 100 mph, based on NCHRP 12-76. The filtering process is illustrated in the flowchart in Fig- ure 5.1. Because a heavy vehicle meeting all the conditional filters involving GVW would pass the filters, the research team reviewed exceptionally heavy vehicles to check if their configuration resembled permit vehicles, such as cranes and garbage trucks. The data were divided into two sets. The first set contained regular truck traffic. These data were used for the live load model for SLSs. The remaining set of data included permit vehicles and illegally overloaded vehicles, which occurred rela- tively infrequently. The latter data were used along with the Figure 5.1. Flowchart of the filtering process.

97 regular truck traffic for live load for SLS II. The GVW criteria of 20 kips in Step 3 is a traditional, albeit arbitrary, cutoff used in virtually all previous fatigue studies to reduce the calculation effort by not considering light traffic, which does not contribute significantly to cumulative damage. The CDFs of GVWs were plotted on probability paper; examples are shown in Figure 5.2 to Figure 5.5. The live load model based on the Ontario truck survey data that were used in calibration for strength limit states is also shown. The rela- tive position of the Ontario curve is a result of the intentional selection of seemingly heavy vehicles, albeit based solely on the appearance of the vehicles. Figure 5.2 represents the CDF of the GVW of trucks from FHWA sites plotted on probability paper. Data collected from 14 sites represent 1 year of traffic, data from the Indiana site represent 6 months of traffic, and data from the New Mexico sites represent 8 months of traffic. The maximum truck GVW was 220 kips. Mean values ranged from 20 to 65 kips. Figure 5.3 to Figure 5.5 represent CDFs of the GVWs for Ontario and the following states: Oregon and Florida (Fig- ure 5.3), Indiana and Mississippi (Figure 5.4), and California and New York (Figure 5.5) (i.e., the NCHRP 12-76 data). The corresponding traffic data from these figures are given in Table 5.1. Figure 5.2. CDF of GVW FHWA and Ontario data. GVW [kips] St an da rd N or m al V ar ia bl e Figure 5.3. CDFs of GVW for Oregon, Florida, and Ontario. St an da rd N or m al V ar ia bl e GVW [kips] St an da rd N or m al V ar ia bl e GVW [kips] NCHRP Data - Oregon NCHRP Data - Florida

98 Figure 5.4. CDFs of GVW for Indiana, Mississippi, and Ontario. St an da rd N or m al V ar ia bl e GVW [kips] GVW [kips] St an da rd N or m al V ar ia bl e NCHRP Data - Indiana NCHRP Data - Mississippi Figure 5.5. CDFs of GVW for California, New York, and Ontario. GVW [kips] St an da rd N or m al V ar ia bl e GVW [kips] NCHRP Data - California NCHRP Data - New York St an da rd N or m al V ar ia bl e Table 5.1. Summary of State Sites and Their Traffic Data for Figures 5.3 to 5.5 Figure State No. of Sites No. of Months of Data Maximum GVW (kips) Mean Value Range (kips) Figure 5.3 Oregon 4 4 200 43–52 Florida 5 12 250 20–50 Figure 5.4 Indiana 5 12 250 25–57 Mississippi 5 12 260 38–57 Figure 5.5 California 2 8.7 250 40–50 1 7 New York 7 12 380 35–50

99 Table 5.2. WIM Locations and Number of Recorded Vehicles Site No. of Days in Data Total No. of Truck Records Lane ADTT Arizona (SPS-1) 365 35,572 97 Arizona (SPS-2) 365 1,430,461 3,919 Arkansas (SPS-2) 365 1,675,349 4,590 Colorado (SPS-2) 365 343,603 941 Delaware (SPS-1) 365 201,677 553 Illinois (SPS-6) 365 854,075 2,340 Indiana (SPS-6) 214 185,267 508 Kansas (SPS-2) 365 477,922 1,309 Louisiana (SPS-1) 365 85,702 235 Maine (SPS-5) 365 183,576 503 Maryland (SPS-5) 365 164,389 450 Minnesota (SPS-5) 365 55,572 152 New Mexico (SPS-1) 245 117,102 321 New Mexico (SPS-5) 245 608,280 1,667 Pennsylvania (SPS-6) 365 1,495,741 4,098 Tennessee (SPS-6) 365 1,622,320 4,445 Virginia (SPS-1) 365 259,190 710 Wisconsin (SPS-1) 365 226,943 622 California Antelope EB 258 837,667 2,192a California Antelope WB 256 943,147 2,258a California Bowman 134 651,090 2,018a California LA-710 NB 333 4,092,484 6,380a California LA-710 SB 365 4,661,287 8,366a California Lodi 304 3,298,499 5,186a Florida I-10 354 1,641,480 2,207a Florida I-95 349 2,112,518 2,558a Florida US-29 354 389,164 606a Mississippi I-10 337 1,965,022 2,967a Mississippi I-55UI 268 1,232,223 2,054a Mississippi I-55R 349 1,333,268 1,790a Mississippi US-49 359 1,225,138 1,475a Mississippi US-61 319 159,299 254a Total 35,856,898 Note: EB = eastbound; WB = westbound; NB = northbound; SB = southbound. a NCHRP data are for multilane cases; the lane with maximum ADTT is listed. As an initial observation, the data shown in Figure 5.2 to Figure 5.5 are generally consistent for the majority of the sites (consistent refers to the similarity of the general shape of the curves, i.e., the CDFs). Exceptions are the following heavily loaded sites from New York: • Site 9121 on I-81 by Whitney Point; • Site 8382 on I-84 by Port Jervis; • Site 8280 on I-84 by Fishkill; and • Site 0580 on I-495 in Queens in New York City. Because these sites were so exceptional, it was decided not to include the New York WIM data in developing a national, notional SLS live load. In addition, several sites for which the recording format differed or had considerably less than one tier of data were eliminated from consideration. A summary of the remaining 32 sites and filtered data, including the WIM locations, number of records, and average daily truck traffic (ADTT), is shown in Table 5.2. Approximately 35 million records are represented by these sites. A copy of the raw WIM data and of the filtered WIM data is available at http://www.trb.org/Main/Blurbs/170201.aspx. A sample of the filtered WIM data is included in Appendix F. The CDFs of GVWs and moment are plotted as separate curves for each location. The legend for all CDFs is shown in Figure 5.6. 5.2 Initial Data analysis 5.2.1 Gross Vehicle Weight The CDFs for the GVWs from the remaining FHWA and NCHRP sites are plotted on probability paper in Figure 5.7. Each of the 32 curves represents a different location. The result- ing curves indicate that the distribution of GVW is not normal. Irregularity of the CDFs is a result of different types of vehicles (such as long and short, fully loaded and empty, or loaded by volume only) in the WIM data. For the considered locations, the mean GVWs are between 25 and 65 kips. The upper tails of the CDF curves show a similar trend, but there is a considerable spread of the maximum values, from 150 to over 250 kips. 5.2.2 Moments from WIM Data The distribution of simple-span moments due to WIM trucks was obtained by calculating the maximum bending moment for each vehicle in the database. Each vehicle was run over influence lines to determine the maximum moment by using a specially developed computer program. The calculations were carried out for spans from 30 to 200 ft. For easier inter- pretation and comparison of results, the calculated WIM data moments were then divided by the corresponding HL-93 moment. Normalizing the data to a common reference makes

100 Figure 5.6. Legend for all graphs. FHWA Data NCHRP Data the data easier to interpret. HL-93 was a convenient reference and ties this work to the original strength limit state calibra- tion and associated published information. The CDFs for the ratio of the WIM truck moment and HL-93 moment are plotted on normal probability paper in Figure 5.8 to Figure 5.12; the shape of the CDF curves is simi- lar to that of GVW. The mean WIM moments were between 0.2 and 0.4 of the HL-93 moments for all span lengths consid- ered. The probability of a WIM moment exceeding 0.4 to 0.5 of the HL-93 moment was about 0.15. The maximum values of the WIM moment were between 1.0 and 1.4 of HL-93 moment in most cases. The obtained results served as the basis for determining the statistical parameters of live load needed for the reliability analysis of the serviceability limit states. 5.2.3 Filtering of Presumed Illegal Overloads and Special Permit Loads The goal of this analysis was to observe the change in the very top tail of the distribution after removing the heaviest vehicles from the database. These extremely heavy vehicles seemed to be either permit vehicles that should be included in the design pro- cess (as some states do) or vehicles reviewed for permit issuance by using the Strength II limit state load combination; otherwise, they are illegal overloads. An example of the heaviest truck in the WIM data is presented in Figure 5.13. This truck was recorded at Site 8382 near Port Jervis, New York. The total length of the truck was 100.6 ft. The GVW was 391.4 kips. The position of the 12 axles, their weight, and the vehicle’s length suggest that it should be categorized as a permit vehicle. WIM equipment cap- tures each vehicle, including permit vehicles, as a string of axles, and an FHWA designation is given based on the best FHWA category that fits the detected configuration. Heavy vehicles are assumed to be permit vehicles or illegally loaded vehicles. The initial study indicated that the removal of a very small number of the heaviest vehicles drastically changed the upper tail of the CDF of moments and shears. It was decided to explore this by investigating the number of vehicles that exceeded an upper value of 1.35 times HL-93, which corresponds to the max- imum bias ratio obtained from the Ontario measurements. The results of the analysis for sites from New York and Mis- sissippi were plotted on probability paper and are shown in Figures 5.14 to 5.16. It can be observed that, as expected, the very upper tail of the distribution changed drastically by removing only a very small percentage of vehicles. For example, for 90-ft spans at New York Site 8382 (Fig- ure 5.15), the bias changes from about 2.35 to about 1.65—but only when considering the six largest moment ratios (corre- sponding to the six heaviest trucks, including the 391-kip vehi- cle shown in Figure 5.13) out of the 1.55 million data records remaining after application of the additional filter to remove moments less than 15% of the corresponding HL-93 moment. Even for the WIM sites that demonstrated very extreme tails, these extreme trucks constituted only the upper 0.01% to 0.22% of the truck population. For most of the locations reviewed, the percentage was lower (see Table 5.3). The heavi- est loads may have an important impact on calibration of the ultimate or strength limit states; however, in the case of SLSs, the upper tail of the CDF of the live load is not important, as it is the main body of the CDF that affects SLS performance. Therefore, for SLS calibration, it was decided to ignore the upper tip of the CDF of live load. 5.2.4 Multiple Presence Analysis Multiple presence was investigated by a correlation analysis of the WIM data sets. The objective of the correlation analysis was to select two trucks that were simultaneously positioned on the bridge as shown in Figure 5.17 and that satisfied the following requirements: • Both trucks had the same number of axles. • GVWs of the trucks were within ±5%. • All corresponding spacings between axles were within ±10%. The maximum load effect is often caused by the simultane- ous presence of two or more trucks on a bridge. The statistical parameters of these effects are influenced by the degree of correlation. In calibration for the strength limit states, certain probabilities of occurrence of correlated trucks were assumed on the basis of engineering judgment applied to limited obser- vations of the presence of multiple trucks of unknown weight. The available WIM data allowed for verification of these assumptions. (text continues on page 108)

101 Figure 5.7. CDFs of GVWs.

102 Figure 5.8. CDFs of WIM moment and HL-93 moment ratio, span 5 30 ft.

103 Figure 5.9. CDFs of WIM moment and HL-93 moment ratio, span 5 60 ft.

104 Figure 5.10. CDFs of WIM moment and HL-93 moment ratio, span 5 90 ft.

105 Figure 5.11. CDFs of WIM moment and HL-93 moment ratio, span 5 120 ft.

106 Figure 5.12. CDFs of WIM moment and HL-93 moment ratio, span 5 200 ft.

107 Truck Moment / HL93 Moment St an da rd N or m al V ar ia bl e Truck Moment / HL93 Moment St an da rd N or m al V ar ia bl e New York 8280 Span 90 ft New York 8382 Span 90 ft Figure 5.15. Data removal from New York Sites 8280 and 8382. Truck Moment / HL93 Moment St an da rd N or m al V ar ia bl e Truck Moment / HL93 Moment New York 0580 Span 90 ft New York 2680 Span 90 ft St an da rd N or m al V ar ia bl e Figure 5.14. Data removal from New York Sites 0580 and 2680. Figure 5.13. Configuration of extremely loaded truck.

108 Table 5.3. Removal of Heaviest Vehicles (90-ft Span) Figure State Site No. of Trucks Before Filtering No. of Trucks After Filtering No. of Removed Trucks Removed Trucks (%) Figure 5.14 New York 0580 2,474,407 2,468,952 5,455 0.22 Figure 5.14 New York 2680 89,286 89,250 36 0.04 Figure 5.15 New York 8280 1,717,972 1,717,428 544 0.03 Figure 5.15 New York 8382 1,551,454 1,550,914 540 0.03 Figure 5.16 New York 9121 1,235,963 1,235,886 77 0.01 Figure 5.16 Mississippi I-10 2,103,302 2,103,300 2 0.00 Truck Moment / HL93 Moment St an da rd N or m a l V a ria bl e Truck Moment / HL93 Moment St an da rd N or m a l V a ria bl e New York 9121 Span 90 ft Mississippi - I10 Span 90 ft Figure 5.16. Data removal from New York Site 9121 and Mississippi I-10 locations. The selected trucks were plotted on probability paper and compared with all recorded vehicles. The GVW of both cor- related trucks were added together and divided by two to obtain the average GVW. (Note that the correlation criteria ensure that the average is similar to the two selected trucks in each pair.) The comparison of the mean correlated GVW of the trucks recorded in adjacent lanes with the GVW of the whole population from Florida and New York is shown in Figure 5.19. Two Trucks: one AfTer The oTher Filtering the data resulted in the selection of 8,380 fully cor- related trucks in one lane in Florida and 9,868 fully correlated trucks in one lane in New York. Histograms of these trucks are shown in Figure 5.20. The comparison of the mean cor- related GVW of the trucks recorded in one lane with the GVW of the whole data set from Florida and New York is shown in Figure 5.21. A special program was developed to filter the data by using the time of a record and the speed of the truck to find instances when either of the events shown in Figure 5.17 occurred involving similar trucks. The filter resulted in select- ing the observed cases of two trucks with a headway distance less than 200 ft in either the same lane or two adjacent lanes. Two Trucks: Side by Side The analysis of the degree of correlation was performed for Site 9936 in Florida along I-10 and Site 8382 in New York with 1,654,004 and 1,594,674 site-specific total records, respectively. Filtering the data resulted in the selection of 2,518 fully correlated trucks in adjacent lanes in Florida and 3,748 fully correlated trucks in adjacent lanes in New York. Histograms of the GVWs of these fully correlated side-by- side trucks are shown in Figure 5.18. (continued from page 100)

109 Figure 5.17. Two cases of the simultaneous presence of two trucks with headway distance less than 200 ft. T1 T2 Headway Distance max 200 ft T1 T2 Headway Distance max 200 ft Figure 5.18. Histograms of trucks side by side (a) on Florida I-10 and (b) at New York Site 8382. (a) (b) Florida I-10 New York Site 8382 Figure 5.19. Comparison of mean GVW and GVW of the whole population for (a) Florida and (b) New York. (a) (b) Florida I-10 New York Site 8382

110 Figure 5.20. Histogram of trucks one after another (a) on Florida I-10 and (b) at New York Site 8382. Florida I-10 New York Site 8382 (a) (b) Figure 5.21. Comparison of mean GVW and GVW of the whole population for (a) Florida and (b) New York. (a) (b) Florida I-10 New York Site 8382 ImplIcATIons for specIfIcATIon DevelopmenT The study of multiple presence based on WIM data indicated that, for SLSs, the vehicles representing the extreme tails of the CDF need not be considered as being simultaneously present in multiple lanes. The implication is that only a single-lane live load model needs to be considered on the load side (Q) of limit state functions. The resistance side (R) of limit state functions should represent the requirements of the applica- ble design requirement, even if that is a multiple-lane loading situation. The issue of multiple load lanes was considered in the development of HL-93 for AASHTO LRFD strength limit states, and the conclusion was that extreme truck load does not occur simultaneously with another fully correlated extreme truck, but was considered to occur simultaneously with a truck about 15% to 20% lighter. This two-lane loading was correlated to the design loading of two lanes of HL-93 with a load factor of 1.75 and a multiple presence factor of 1.0. (The multiple presence factor for a single-lane loading is 1.20 to account for the occasional truck that creates more force effect than the family of configurations used to develop the HL-93 load configuration.) 5.2.5 Project Guidelines Regarding Live load The following guidelines are based on live load bias factors and CVs determined from the preliminary analysis of WIM measurements and previous work by the research team (Nowak 1999): • The use of dynamic load as 10% of live load, with CV = 80%, is recommended.

111 • Generally use a single loaded lane (no multiple loaded lanes). • The national load (i.e., notional load) should not try to encompass all WIM records. Some of the extremely heavy vehicles are permit loads and some are illegal overloads. A relatively small number of loads were excluded for most of the SLS studies, but they were included for the overload limit state. • It is likely that different probabilities of exceedance will be used for various limit states based on consequences. • Some jurisdictions may need exceptions based on their legal loads and extent of enforcement. • The basic HL-93 load model, scaled by calibrated load fac- tors, is appropriate for SLS. With these recommendations, the evaluation of numerical live load models continued. The processes used and results obtained are summarized here. Further details and extensive graphical presentations are contained in Rakoczy (2011). 5.3 Statistical parameters for Service Limit States Other than Fatigue 5.3.1 Maximum Moments for Different Time Periods The maximum moment is a random variable. It depends on the period of time, ADTT, and distribution of traffic (e.g., CDF of WIM moments). For a given CDF of WIM moments [F(x)], period of time (T), and ADTT, the mean value of the maxi- mum moment can be determined as follows. The total number of vehicles (N) expected during the considered time period T (in days) is T × ADTT. The expected or mean value of the max- imum moment for time T [Mmax(T)] is equal to the moment corresponding to probability {1 - F[1/N(T)]}, where F(x) is the CDF of WIM moments, which is F-1[1 - 1/N(T)], where F-1 is the inverse of CDF. The objective is to determine the mean maximum moment for different time periods (i.e., 1 day, 2 weeks, 1 month, 2 months, 6 months, 1 year, 5 years, 50 years, 75 years, and 100 years). The number of recorded vehicles for each location is given in Table 5.2. The data were collected over different time periods, in most cases about 1 year, but the number of vehicles varies because ADTT varies. Each CDF in Figure 5.8 to Figure 5.12 includes the number of data points equal to the corresponding number of vehicles (N). For each CDF, the vertical coordinate of the maximum moment (Zmax) is given by Equation 5.1: 1 (5.1)max 1z N( )= − Φ− where -F-1 is the inverse standard normal distribution func- tion. For example, if N = 1,000,000, then Zmax = 4.75. In further analysis, five ADTTs were considered: 250, 1,000, 2,500, 5,000, and 10,000. The calculations were performed separately for each ADTT. To determine the mean maximum moments corresponding to the considered time periods, the vertical coordinates were found first. Starting with ADTT = 250, the vertical coordinate of the mean maximum 1-day moment z is given by Equation 5.2: 1 250 2.65 (5.2)1z ( )= − Φ =− because the number of trucks per 1 day is 250. The mean maximum 2-week moment z is given by Equa- tion 5.3: 1 3500 3.44 (5.3)1z ( )= − Φ =− because the number of trucks per 2 weeks is (250 trucks) (14 days) = 3,500 trucks. Finally, the mean maximum 100-year moment z is given by Equation 5.4: 1 9,125,000 5.18 (5.4)1z ( )= − Φ =− because the number of trucks per 100 years is (250 trucks) (365 days)(100 years) = 9,125,000 trucks. Similarly, for ADTT = 1,000, the vertical coordinate of the mean maximum 1-day moment z is given by Equation 5.5: 1 1000 3.09 (5.5)1z ( )= − Φ =− because the number of trucks per 1 day is 1,000. The mean maximum 2-week moment z is given by Equa- tion 5.6: 1 14,000 3.8 (5.6)1z ( )= − Φ =− because the number of trucks per 2 weeks is (1,000 trucks) (14 days) = 14,000 trucks. Finally, the mean maximum 100-year moment z is given by Equation 5.7: 1 36,500,000 5.67 (5.7)1z ( )= − Φ =− because the number of trucks per 100 years is (1,000 trucks) (365 days)(100 years) = 36,500,000 trucks. Values of z for the considered ADTTs and time periods from 1 day to 100 years are summarized in Table 5.4. For example, for the WIM moments in Figure 5.11 (span = 120 ft), the vertical coordinates corresponding to different time periods are shown in Figure 5.22 for ADTT = 1,000. There were 32 WIM locations and, therefore, 32 curves rep- resenting CDFs of WIM moments in each of Figures 5.8 to 5.12. The mean maximum moment can be obtained directly

112 Table 5.4. Vertical Coordinates for the Mean Maximum Moment Time Period ADTT 250 1,000 2,500 5,000 10,000 1 Day 2.65 3.09 3.35 3.54 3.72 2 Weeks 3.44 3.08 4.02 4.18 4.33 1 Month 3.65 4.00 4.20 4.35 4.50 2 Months 3.82 4.15 4.35 4.50 4.65 6 Months 4.09 4.39 4.59 4.73 4.87 1 Year 4.24 4.55 4.73 4.87 5.01 5 Years 4.59 4.87 5.05 5.18 5.31 50 Years 5.05 5.31 5.47 5.60 5.72 75 Years 5.13 5.38 5.55 5.67 5.78 100 Years 5.18 5.44 5.60 5.72 5.83 from the graph by reading the moment ratio (horizontal axis) corresponding to the vertical coordinate representing the con- sidered time period. For example, from Figure 5.22, the mean maximum 1-day moment ratio for Florida US-29 is 0.95, and the mean maximum 1-year moment ratio is 1.39. Values for lon- ger time periods were projected or interpolated as appropriate. For each ADTT and span length, there are 32 values of the mean maximum 1-day moment, 32 values of the mean maxi- mum 2-week moment, and so on. For an easier review and comparison, CDFs of these 32 values obtained from Fig- ure 5.22 were plotted on normal probability paper and are shown in Figure 5.23. There is one CDF for 1-day values, one for 2 weeks, and so on. These are CDFs of extreme variables, as each of the 32 values is the maximum moment for a WIM loca- tion. The obtained CDFs are almost parallel; in particular, this applies to the upper part. Because of regularity, it is easier to determine the statistical parameters. Each data point repre- sents the mean of the maximum value for one of 32 WIM loca- tions, which means that the CDFs in Figure 5.23 are extreme value distributions rather than hypothetical curves. 5.3.2 Statistical Parameters of Live Load It was assumed that the 32 WIM locations considered are rep- resentative for the truck traffic in the United States. The statisti- cal parameters (the mean maximum and CV of the maximum live load) were determined for each WIM location. The CDFs of the mean maximum values were plotted on probability paper. This is an extreme value distribution. The mean of these mean maximum values can be considered as the mean maxi- mum national live load. The standard deviation of the mean maximum values can be determined from the graphs (slope of the CDF). However, the WIM locations were not selected randomly; rather, the selection was based on the availability of WIM stations with truck data and the credibility of the measured data (truck records). If the considered WIM loca- tions are biased (i.e., nonrepresentative), then the processed database can underestimate or overestimate the statistical parameters of the national live load. Therefore, for the pur- pose of further reliability analysis, it is conservatively assumed that the calculated mean maximum live load is increased by 1.5 standard deviations. The probability of exceeding this value (mean plus 1.5 standard deviations) is about 5%, so that it will be exceeded by 5% of 32 WIM locations (i.e., in one or two WIM locations). As the upper parts of the CDFs are almost straight lines, the fitting by normal distributions is justified. The mean values can be read directly from the graph as the intersection of CDFs (represented by straight lines) and the horizontal axis at zero on the vertical scale. This process is depicted in Figure 5.24. The visual comparison of how the actual CDF fits a straight line is much better than any curve-fitting formula because the research team was mostly interested in only some parts of the CDF. Different curves can have different slopes, which are reflected in the standard deviations. Calculations were carried out for all considered cases of ADTT and span length. The results, which were extrapolated to 100 years and span length of 300 ft, are summarized in Table 5.5 to Table 5.9. Statistical parameters were calculated for a variety of ADTTs (500, 1,000, 2,500, 5,000, and 10,000); however, the AASHTO LRFD is based on 5,000 (consistent with strength limit states). Live load data for values of ADTT other than 5,000 were tabulated so owners can repeat the cali- bration process with other data. For a given bridge, use of a lower ADTT should lead to a higher reliability index. Bias factors vary depending on ADTT for shorter time periods; however, for longer time periods, the bias factor is about 1.4. 5.3.3 Reactions Tables of statistics for reactions of simply supported spans were developed for the same spans, time periods, and ADTTs as presented for bending moments by using a methodology analogous to the one presented in Section 5.3.2. The results are shown in Table 5.10 to Table 5.14. Graphical representa- tions are presented in Rakoczy (2011). 5.3.4 Axle Loads Statistical parameters for various time periods and ADTTs are developed using a methodology analogous to that pre- sented in Section 5.3.2 applied to axle loads instead of moments. The results are presented in Table 5.15. (text continues on page 121)

113 Figure 5.22. Vertical coordinates for different time periods for ADTT 5 1,000 and span 5 120 ft.

114 Figure 5.23. CDFs of mean maximum moment ratios for ADTT 5 1,000 and span length 5 120 ft.

115 Figure 5.24. Determination of mean values at 1.5 s.

116 Table 5.5. Statistical Parameters of Live Load Moments for ADTT 250, l 5   1.5s Time Period Span 30 ft 60 ft 90 ft 120 ft 200 ft 300 ft l µ CV l µ CV l µ CV l µ CV l µ CV l µ CV 1 Day 0.92 0.65 0.28 0.82 0.64 0.23 0.80 0.66 0.17 0.79 0.65 0.15 0.71 0.56 0.18 0.61 0.48 0.18 2 Weeks 1.06 0.80 0.21 1.05 0.80 0.16 1.01 0.80 0.18 1.02 0.80 0.16 0.93 0.73 0.16 0.84 0.67 0.16 1 Month 1.12 0.85 0.21 1.09 0.85 0.19 1.08 0.85 0.18 1.08 0.85 0.17 1.01 0.78 0.19 0.90 0.73 0.16 2 Months 1.14 0.90 0.18 1.15 0.91 0.17 1.14 0.90 0.18 1.14 0.90 0.17 1.05 0.85 0.15 0.95 0.77 0.15 6 Months 1.19 0.95 0.17 1.23 0.96 0.19 1.20 0.97 0.15 1.19 0.98 0.14 1.12 0.91 0.15 1.04 0.85 0.15 1 Year 1.23 1.00 0.15 1.27 0.98 0.19 1.24 1.00 0.16 1.22 1.04 0.12 1.15 0.94 0.15 1.08 0.88 0.15 5 Years 1.31 1.07 0.15 1.35 1.09 0.16 1.31 1.13 0.11 1.31 1.14 0.10 1.25 1.02 0.15 1.18 0.97 0.15 50 Years 1.37 1.17 0.11 1.39 1.16 0.13 1.39 1.25 0.07 1.37 1.19 0.10 1.32 1.06 0.16 1.25 1.02 0.15 75 Years 1.38 1.20 0.10 1.40 1.19 0.12 1.41 1.27 0.07 1.39 1.21 0.10 1.34 1.08 0.16 1.27 1.04 0.15 100 Years 1.39 1.22 0.09 1.43 1.21 0.12 1.42 1.28 0.07 1.41 1.22 0.10 1.35 1.09 0.16 1.29 1.05 0.15 Table 5.6. Statistical Parameters of Live Load Moments for ADTT 1,000, l 5   1.5s Time Period Span 30 ft 60 ft 90 ft 120 ft 200 ft 300 ft l µ CV l µ CV l µ CV l µ CV l µ CV l µ CV 1 Day 0.99 0.72 0.28 0.89 0.71 0.20 0.90 0.72 0.17 0.89 0.71 0.17 0.81 0.63 0.19 0.71 0.55 0.19 2 Weeks 1.14 0.87 0.21 1.13 0.90 0.16 1.13 0.89 0.18 1.14 0.91 0.16 1.06 0.85 0.16 0.97 0.77 0.16 1 Month 1.18 0.95 0.16 1.19 0.95 0.16 1.19 0.95 0.17 1.19 0.96 0.16 1.11 0.91 0.14 1.01 0.83 0.14 2 Months 1.23 0.99 0.16 1.26 0.99 0.18 1.26 1.00 0.17 1.23 1.03 0.13 1.16 0.96 0.14 1.07 0.89 0.14 6 Months 1.27 1.04 0.14 1.31 1.05 0.16 1.30 1.10 0.12 1.27 1.09 0.11 1.22 0.99 0.15 1.15 0.93 0.15 1 Year 1.33 1.07 0.16 1.34 1.08 0.16 1.32 1.15 0.10 1.31 1.14 0.10 1.25 1.01 0.16 1.18 0.95 0.16 5 Years 1.37 1.11 0.15 1.37 1.14 0.13 1.36 1.21 0.08 1.35 1.17 0.10 1.30 1.06 0.15 1.24 1.01 0.15 50 Years 1.38 1.24 0.07 1.42 1.21 0.12 1.41 1.26 0.08 1.41 1.21 0.11 1.35 1.11 0.14 1.28 1.05 0.14 75 Years 1.40 1.26 0.07 1.42 1.23 0.11 1.42 1.28 0.07 1.41 1.23 0.10 1.36 1.13 0.13 1.29 1.07 0.13 100 Years 1.40 1.27 0.07 1.44 1.24 0.11 1.43 1.29 0.07 1.43 1.24 0.10 1.37 1.14 0.13 1.30 1.09 0.13

117 Table 5.7. Statistical Parameters of Live Load Moments for ADTT 2,500, l 5   1.5s Time Period Span 30 ft 60 ft 90 ft 120 ft 200 ft 300 ft l µ CV l µ CV l µ CV l µ CV l µ CV l µ CV 1 Day 1.03 0.80 0.19 0.97 0.79 0.18 0.97 0.77 0.17 0.98 0.78 0.17 0.90 0.70 0.19 0.80 0.62 0.19 2 Weeks 1.20 0.93 0.19 1.20 0.96 0.17 1.20 0.96 0.17 1.20 0.97 0.15 1.12 0.92 0.14 1.02 0.84 0.14 1 Month 1.23 0.99 0.16 1.25 0.99 0.17 1.26 1.00 0.17 1.22 1.04 0.12 1.16 0.95 0.15 1.09 0.89 0.15 2 Months 1.28 1.04 0.15 1.31 1.04 0.17 1.29 1.11 0.11 1.27 1.12 0.09 1.21 0.98 0.15 1.12 0.91 0.15 6 Months 1.31 1.07 0.15 1.34 1.07 0.17 1.32 1.15 0.10 1.31 1.14 0.10 1.25 1.01 0.16 1.18 0.95 0.16 1 Year 1.34 1.11 0.14 1.35 1.11 0.14 1.36 1.19 0.09 1.34 1.17 0.09 1.28 1.04 0.15 1.21 0.98 0.15 5 Years 1.36 1.15 0.12 1.39 1.18 0.12 1.39 1.24 0.08 1.38 1.20 0.10 1.33 1.07 0.16 1.26 1.01 0.16 50 Years 1.40 1.25 0.08 1.42 1.22 0.11 1.43 1.29 0.07 1.43 1.23 0.11 1.37 1.11 0.15 1.29 1.05 0.15 75 Years 1.40 1.26 0.07 1.43 1.24 0.10 1.43 1.30 0.07 1.44 1.24 0.10 1.37 1.13 0.14 1.29 1.06 0.14 100 Years 1.40 1.27 0.07 1.44 1.25 0.10 1.44 1.31 0.07 1.44 1.25 0.10 1.39 1.14 0.14 1.32 1.09 0.14 Table 5.8. Statistical Parameters of Live Load Moments for ADTT 5,000, l 5   1.5s Time Period Span 30 ft 60 ft 90 ft 120 ft 200 ft 300 ft l µ CV l µ CV l µ CV l µ CV l µ CV l µ CV 1 Day 1.08 0.85 0.18 1.02 0.82 0.17 1.03 0.82 0.17 1.03 0.82 0.17 0.95 0.75 0.17 0.84 0.67 0.17 2 Weeks 1.24 0.98 0.17 1.26 1.00 0.17 1.24 1.00 0.16 1.24 1.04 0.13 1.16 0.96 0.14 1.06 0.88 0.14 1 Month 1.28 1.04 0.15 1.32 1.03 0.18 1.30 1.12 0.11 1.26 1.11 0.09 1.20 0.99 0.14 1.13 0.93 0.14 2 Months 1.31 1.07 0.15 1.34 1.07 0.17 1.32 1.15 0.10 1.31 1.14 0.10 1.23 1.02 0.14 1.16 0.96 0.14 6 Months 1.34 1.11 0.14 1.35 1.11 0.14 1.34 1.19 0.08 1.32 1.17 0.09 1.28 1.04 0.15 1.23 1.00 0.15 1 Year 1.35 1.14 0.12 1.38 1.14 0.14 1.38 1.21 0.09 1.36 1.19 0.09 1.31 1.07 0.15 1.25 1.02 0.15 5 Years 1.39 1.16 0.13 1.40 1.19 0.12 1.40 1.25 0.08 1.41 1.21 0.11 1.34 1.10 0.15 1.28 1.05 0.15 50 Years 1.41 1.21 0.11 1.44 1.24 0.10 1.44 1.27 0.09 1.46 1.23 0.12 1.39 1.13 0.15 1.30 1.06 0.15 75 Years 1.42 1.22 0.11 1.45 1.25 0.10 1.45 1.29 0.08 1.46 1.25 0.11 1.40 1.14 0.15 1.31 1.07 0.15 100 Years 1.42 1.23 0.11 1.45 1.26 0.10 1.47 1.30 0.08 1.47 1.26 0.11 1.40 1.15 0.15 1.33 1.08 0.15

118 Table 5.9. Statistical Parameters of Live Load Moments for ADTT 10,000, l 5   1.5s Time Period Span 30 ft 60 ft 90 ft 120 ft 200 ft 300 ft l µ CV l µ CV l µ CV l µ CV l µ CV l µ CV 1 Day 1.17 0.88 0.22 1.09 0.89 0.16 1.11 0.87 0.18 1.13 0.87 0.20 1.02 0.81 0.17 0.91 0.75 0.17 2 Weeks 1.29 1.02 0.18 1.31 1.04 0.17 1.29 1.11 0.11 1.27 1.12 0.09 1.22 0.98 0.16 1.16 0.93 0.16 1 Month 1.32 1.06 0.16 1.34 1.08 0.16 1.32 1.15 0.10 1.29 1.14 0.09 1.25 1.01 0.16 1.20 0.97 0.16 2 Months 1.35 1.09 0.16 1.35 1.11 0.14 1.35 1.18 0.09 1.32 1.17 0.09 1.28 1.04 0.15 1.23 1.00 0.15 6 Months 1.35 1.12 0.13 1.37 1.14 0.13 1.37 1.20 0.09 1.34 1.19 0.08 1.30 1.06 0.15 1.25 1.02 0.15 1 Year 1.37 1.17 0.11 1.39 1.16 0.13 1.39 1.24 0.08 1.38 1.20 0.10 1.32 1.08 0.15 1.27 1.04 0.15 5 Years 1.39 1.24 0.08 1.41 1.21 0.11 1.42 1.27 0.08 1.42 1.22 0.11 1.37 1.11 0.15 1.30 1.06 0.15 50 Years 1.40 1.28 0.06 1.45 1.24 0.11 1.45 1.30 0.08 1.46 1.25 0.11 1.40 1.14 0.15 1.31 1.07 0.15 75 Years 1.41 1.29 0.06 1.46 1.26 0.10 1.47 1.32 0.08 1.47 1.26 0.11 1.40 1.16 0.14 1.32 1.09 0.14 100 Years 1.42 1.30 0.06 1.47 1.27 0.10 1.49 1.33 0.08 1.48 1.27 0.11 1.42 1.17 0.14 1.33 1.10 0.14 Table 5.10. Statistical Parameters of Live Load Reactions for ADTT 250, l 5   1.5s Time Period Span 30 ft 60 ft 90 ft 120 ft 200 ft 300 ft µ  1.5s µ CV µ  1.5s µ CV µ  1.5s µ CV µ  1.5s µ CV µ  1.5s µ CV µ  1.5s µ CV 1 Day 1.02 0.85 0.13 0.88 0.74 0.12 0.88 0.74 0.12 0.86 0.72 0.13 0.73 0.61 0.13 0.57 0.48 0.13 2 Weeks 1.22 1.02 0.13 1.08 0.91 0.12 1.11 0.94 0.12 1.08 0.90 0.13 0.97 0.80 0.14 0.82 0.68 0.14 1 Month 1.28 1.07 0.13 1.14 0.96 0.13 1.17 0.99 0.12 1.15 0.97 0.12 1.06 0.88 0.14 0.93 0.77 0.14 2 Months 1.32 1.11 0.13 1.19 1.01 0.12 1.22 1.04 0.12 1.20 1.02 0.12 1.12 0.92 0.14 0.98 0.81 0.14 6 Months 1.37 1.16 0.12 1.27 1.07 0.12 1.32 1.11 0.13 1.30 1.10 0.12 1.18 0.97 0.14 1.08 0.89 0.14 1 Year 1.41 1.20 0.12 1.31 1.10 0.13 1.37 1.14 0.13 1.35 1.12 0.13 1.22 1.01 0.14 1.12 0.93 0.14 5 Years 1.49 1.26 0.12 1.38 1.15 0.13 1.46 1.22 0.13 1.44 1.20 0.13 1.35 1.11 0.14 1.24 1.02 0.14 50 Years 1.54 1.30 0.12 1.49 1.23 0.14 1.52 1.28 0.13 1.52 1.28 0.13 1.45 1.18 0.15 1.36 1.11 0.15 75 Years 1.55 1.31 0.12 1.50 1.24 0.14 1.55 1.29 0.13 1.55 1.29 0.13 1.46 1.19 0.15 1.37 1.12 0.15 100 Years 1.56 1.32 0.12 1.50 1.25 0.14 1.55 1.30 0.13 1.55 1.30 0.13 1.47 1.20 0.15 1.38 1.12 0.15

119 Table 5.11. Statistical Parameters of Live Load Reactions for ADTT 1,000, l 5   1.5s Time Period Span 30 ft 60 ft 90 ft 120 ft 200 ft 300 ft µ  1.5s µ CV µ  1.5s µ CV µ  1.5s µ CV µ  1.5s µ CV µ  1.5s µ CV µ  1.5s µ CV 1 Day 1.14 0.94 0.14 0.95 0.80 0.13 0.94 0.80 0.11 0.91 0.79 0.10 0.84 0.70 0.13 0.74 0.62 0.13 2 Weeks 1.31 1.10 0.13 1.17 0.99 0.12 1.19 1.02 0.11 1.19 1.02 0.11 1.09 0.91 0.13 0.97 0.81 0.13 1 Month 1.35 1.15 0.12 1.23 1.03 0.13 1.26 1.08 0.11 1.25 1.07 0.11 1.17 0.97 0.13 1.06 0.88 0.13 2 Months 1.38 1.18 0.11 1.26 1.08 0.11 1.31 1.11 0.12 1.31 1.11 0.12 1.22 1.01 0.14 1.11 0.92 0.14 6 Months 1.42 1.22 0.11 1.29 1.11 0.11 1.38 1.15 0.13 1.37 1.16 0.12 1.28 1.05 0.14 1.18 0.97 0.14 1 Year 1.45 1.25 0.11 1.32 1.14 0.11 1.40 1.19 0.12 1.40 1.19 0.12 1.32 1.09 0.14 1.21 1.00 0.14 5 Years 1.50 1.29 0.11 1.40 1.20 0.11 1.49 1.26 0.12 1.50 1.26 0.13 1.38 1.14 0.14 1.28 1.06 0.14 50 Years 1.56 1.33 0.11 1.46 1.25 0.11 1.56 1.30 0.13 1.57 1.30 0.14 1.47 1.20 0.15 1.35 1.10 0.15 75 Years 1.57 1.34 0.11 1.47 1.26 0.11 1.57 1.31 0.13 1.58 1.31 0.14 1.48 1.21 0.15 1.36 1.11 0.15 100 Years 1.57 1.35 0.11 1.48 1.27 0.11 1.57 1.32 0.13 1.59 1.32 0.14 1.49 1.22 0.15 1.36 1.12 0.15 Table 5.12. Statistical Parameters of Live Load Reactions for ADTT 2,500, l 5   1.5s Time Period Span 30 ft 60 ft 90 ft 120 ft 200 ft 300 ft µ  1.5s µ CV µ  1.5s µ CV µ  1.5s µ CV µ  1.5s µ CV µ  1.5s µ CV µ  1.5s µ CV 1 Day 1.18 1.00 0.12 1.02 0.88 0.10 1.07 0.90 0.12 1.04 0.89 0.11 0.93 0.78 0.13 0.79 0.66 0.13 2 Weeks 1.35 1.14 0.12 1.23 1.05 0.11 1.29 1.09 0.12 1.29 1.09 0.12 1.19 0.99 0.13 1.06 0.89 0.13 1 Month 1.38 1.17 0.12 1.26 1.08 0.11 1.35 1.14 0.12 1.34 1.13 0.12 1.23 1.02 0.14 1.12 0.93 0.14 2 Months 1.41 1.20 0.12 1.29 1.11 0.11 1.40 1.17 0.13 1.38 1.17 0.12 1.29 1.06 0.14 1.17 0.96 0.14 6 Months 1.47 1.24 0.12 1.34 1.14 0.11 1.44 1.20 0.13 1.44 1.20 0.13 1.33 1.09 0.15 1.22 1.00 0.15 1 Year 1.49 1.25 0.13 1.36 1.16 0.11 1.47 1.23 0.13 1.48 1.24 0.13 1.38 1.12 0.15 1.25 1.02 0.15 5 Years 1.55 1.29 0.13 1.44 1.21 0.12 1.55 1.29 0.13 1.54 1.28 0.13 1.43 1.17 0.15 1.31 1.08 0.15 50 Years 1.59 1.33 0.13 1.53 1.27 0.13 1.58 1.32 0.13 1.59 1.32 0.14 1.50 1.21 0.16 1.38 1.11 0.16 75 Years 1.60 1.34 0.13 1.54 1.28 0.13 1.59 1.33 0.13 1.60 1.33 0.14 1.51 1.22 0.16 1.39 1.12 0.16 100 Years 1.60 1.35 0.13 1.54 1.29 0.13 1.59 1.34 0.13 1.61 1.34 0.14 1.51 1.23 0.16 1.40 1.13 0.16

120 Table 5.13. Statistical Parameters of Live Load Reactions for ADTT 5,000, l 5   1.5s Time Period Span 30 ft 60 ft 90 ft 120 ft 200 ft 300 ft µ  1.5s µ CV µ  1.5s µ CV µ  1.5s µ CV µ  1.5s µ CV µ  1.5s µ CV µ  1.5s µ CV 1 Day 1.25 1.05 0.12 1.09 0.94 0.11 1.14 0.96 0.13 1.12 0.94 0.13 1.02 0.84 0.14 0.90 0.74 0.14 2 Weeks 1.42 1.19 0.13 1.30 1.10 0.12 1.36 1.13 0.13 1.36 1.13 0.13 1.26 1.03 0.15 1.13 0.93 0.15 1 Month 1.46 1.22 0.13 1.34 1.13 0.12 1.39 1.16 0.13 1.40 1.17 0.13 1.30 1.06 0.15 1.18 0.96 0.15 2 Months 1.48 1.24 0.13 1.36 1.15 0.12 1.43 1.20 0.13 1.44 1.20 0.13 1.33 1.09 0.15 1.21 0.99 0.15 6 Months 1.51 1.27 0.13 1.39 1.18 0.12 1.47 1.23 0.13 1.48 1.24 0.13 1.39 1.13 0.15 1.27 1.03 0.15 1 Year 1.54 1.28 0.13 1.41 1.20 0.12 1.50 1.26 0.13 1.51 1.27 0.13 1.41 1.15 0.15 1.29 1.06 0.15 5 Years 1.58 1.32 0.13 1.48 1.25 0.12 1.54 1.30 0.12 1.56 1.30 0.13 1.46 1.19 0.15 1.34 1.09 0.15 50 Years 1.62 1.36 0.13 1.53 1.29 0.12 1.59 1.35 0.12 1.61 1.35 0.13 1.52 1.23 0.15 1.40 1.14 0.15 75 Years 1.63 1.37 0.12 1.54 1.30 0.12 1.60 1.36 0.12 1.62 1.36 0.13 1.53 1.24 0.15 1.41 1.15 0.15 100 Years 1.63 1.38 0.12 1.55 1.31 0.12 1.61 1.37 0.12 1.62 1.37 0.13 1.53 1.25 0.15 1.42 1.15 0.15 Table 5.14. Statistical Parameters of Live Load Reactions for ADTT 10,000, l 5   1.5s Time Period Span 30 ft 60 ft 90 ft 120 ft 200 ft 300 ft µ  1.5s µ CV µ  1.5s µ CV µ  1.5s µ CV µ  1.5s µ CV µ  1.5s µ CV µ  1.5s µ CV 1 Day 1.31 1.10 0.13 1.20 1.00 0.13 1.23 1.03 0.13 1.21 1.01 0.13 1.11 0.91 0.14 0.98 0.81 0.14 2 Weeks 1.45 1.21 0.13 1.35 1.12 0.13 1.40 1.17 0.13 1.41 1.18 0.13 1.31 1.07 0.15 1.19 0.97 0.15 1 Month 1.48 1.24 0.13 1.39 1.16 0.13 1.43 1.20 0.13 1.45 1.21 0.13 1.36 1.10 0.15 1.24 1.00 0.15 2 Months 1.50 1.26 0.13 1.42 1.19 0.13 1.46 1.23 0.12 1.48 1.24 0.13 1.39 1.13 0.15 1.27 1.03 0.15 6 Months 1.52 1.28 0.13 1.45 1.21 0.13 1.48 1.25 0.12 1.52 1.26 0.13 1.41 1.15 0.15 1.31 1.07 0.15 1 Year 1.55 1.29 0.13 1.46 1.22 0.13 1.51 1.28 0.12 1.54 1.28 0.13 1.44 1.17 0.15 1.33 1.08 0.15 5 Years 1.60 1.34 0.13 1.50 1.26 0.13 1.55 1.31 0.12 1.59 1.33 0.13 1.49 1.22 0.15 1.37 1.12 0.15 50 Years 1.64 1.37 0.13 1.56 1.30 0.13 1.62 1.36 0.13 1.62 1.35 0.13 1.54 1.25 0.15 1.43 1.16 0.15 75 Years 1.65 1.38 0.13 1.57 1.31 0.13 1.63 1.37 0.12 1.63 1.36 0.13 1.55 1.26 0.15 1.44 1.17 0.15 100 Years 1.66 1.39 0.13 1.57 1.32 0.13 1.63 1.38 0.12 1.64 1.37 0.13 1.55 1.27 0.15 1.45 1.18 0.15

121 Table 5.15. Statistical Parameters for Axle Loads, l 5   1.5s Time Period ADTT 250 1,000 2,500 5,000 10,000 l CV (%) l CV (%) l CV (%) l CV (%) l CV (%) 1 Day 0.91 0.17 1.00 0.17 1.07 0.16 1.11 0.16 1.15 0.16 2 Weeks 1.09 0.16 1.17 0.16 1.24 0.15 1.29 0.15 1.32 0.15 1 Month 1.14 0.16 1.23 0.15 1.28 0.15 1.32 0.14 1.36 0.14 2 Months 1.18 0.15 1.27 0.15 1.32 0.14 1.36 0.14 1.38 0.14 6 Months 1.24 0.15 1.32 0.14 1.37 0.14 1.40 0.14 1.42 0.13 1 Year 1.30 0.14 1.37 0.14 1.41 0.13 1.42 0.13 1.45 0.13 5 Years 1.38 0.14 1.43 0.13 1.46 0.13 1.47 0.13 1.49 0.13 50 Years 1.45 0.13 1.48 0.13 1.50 0.13 1.51 0.13 1.53 0.12 75 Years 1.45 0.13 1.48 0.12 1.50 0.12 1.51 0.12 1.53 0.12 100 Years 1.46 0.13 1.49 0.12 1.51 0.12 1.52 0.12 1.53 0.12 failure. However, knowledge about the real fatigue stress caused by current truck traffic, which was based on research done in the 1980s, was limited and outdated. The current AASHTO LRFD (2012) has two fatigue limit states. Fatigue Limit State I is related to infinite load-induced fatigue life. The fatigue load in this limit state reflects the load levels found to be representative of the maximum stress range of the truck population for infinite fatigue life design. Fatigue Limit State II is related to finite load-induced fatigue life. The fatigue load in this limit state is intended to reflect a load level found to be representative of the effective stress range of the truck population with respect to the induced number of load cycles and their cumulative damage effects on the bridge components. Only Fatigue I applies to fatigue of concrete and the considered types of reinforcement. The focus of this section is to develop statistical models of fatigue load based on the WIM truck survey data. The fatigue load is intended to be used in calibration of the design provi- sions in the AASHTO LRFD (2012). The WIM measurements provide an unbiased data set. The 15 WIM sites provided by FHWA are considered as representative for the United States for this analysis. Only sites with one full year of constant reading were used for fatigue analysis. Three cases are considered: midspan moment for a simply supported bridge, moment at the interior support of a two- span continuous bridge, and moment at 0.4 of the span length of a continuous bridge. The surveyed vehicles were run over influence lines as traffic streams to determine the number and magnitude of moment cycles for a wide range of span lengths for each case. The fatigue load time history was then devel- oped for the bending moment. The Fatigue II (finite life) load Figure 5.25. Fatigue failure on S-N curve. 5.4 Development of Statistical parameters of Fatigue Load 5.4.1 Objective Fatigue is one of the major causes of distress in steel highway bridges. Cracking or rupture of components and connections calls for costly repairs or replacements. The durability of affected structures can be enhanced by applying reliability theory to this limit state. The limit state of fatigue is reached when accumu- lated load spectra exceed the fatigue resistance of material. A rational approach to the evaluation of existing bridges and design for new bridges requires knowledge of the load-carrying capacity and accumulated loads, as shown on Figure 5.25. A considerable effort was directed toward tests of materials under cyclic loading to establish the so-called S-N curves, where S is the applied stress, and N is the number of load applications to (continued from page 112)

122 was calculated as an equivalent moment by using the linear damage rule first proposed by Palmgren (1924) and later pop- ularized by Miner (1945) as the Palmgren–Miner rule. The Fatigue I (infinite life) load for each location was determined by finding the highest 0.01% of the load cycles and using the smallest of them as the fatigue load for the considered loca- tion. The obtained results combined with fatigue resistance models served as the basis for the development of calibrated criteria for SLS in the AASHTO LRFD. 5.4.2 WIM Data Used for Fatigue Calculation To be consistent with research done by Fisher (1977), in addi- tion to the two filters used for live load, a third filter was used to remove light trucks with GVW under 20 kips because light vehicles cause relatively low fatigue damage. A summary of the data used for fatigue analysis, including WIM locations, number of records, and ADTT, is shown in Table 5.16. 5.4.3 Truck Traffic Simulation and Calculation of Bending Moment Time History Live load on bridges is caused mainly by moving trucks. Longer bridges often experience more than one vehicle in one span at the same time. Multiple vehicles in one span produce a larger load effect than a single truck. For fatigue load calculations, it Table 5.16. WIM Locations and Number of Vehicles Used for Fatigue Analysis Site No. of Days in Data Total No. of Truck Records Single-Lane ADTT Arizona (SPS-1) 365 26,501 97 Arizona (SPS-2) 365 1,391,098 3,919 Arkansas (SPS-2) 365 1,642,334 4,590 Colorado (SPS-2) 365 326,017 941 Delaware (SPS-1) 365 175,889 553 Illinois (SPS-6) 365 821,809 2,340 Kansas (SPS-2) 365 456,881 1,309 Louisiana (SPS-1) 365 70,831 235 Maine (SPS-5) 365 172,333 503 Maryland (SPS-5) 365 124,474 450 Minnesota (SPS-5) 365 47,794 152 Pennsylvania (SPS-6) 365 1,458,818 4,098 Tennessee (SPS-6) 365 1,583,151 4,445 Virginia (SPS-1) 365 237,804 710 Wisconsin (SPS-1) 365 209,239 622 is very important to find the largest load cycles, because they cause the major fatigue damage. Experimental studies showed that there is a linear relationship between the magnitude of load cycle and fatigue damage. S-N curves for fatigue load tests show a log-log relationship between the cycle amplitude and the number of cycles to failure. This relationship is reflected in the Palmgren–Miner formula for equivalent load, shown as Equations 5.13 and 5.14. Recent WIM data provide much more complex and more accurate information about measured trucks. The WIM data include not only axle loads and spacing between axles, but also truck speed and time of measurement with an accuracy of 1 s. Using these data, the team simulated truck traffic on a bridge for a 1-year period, and the time history of the bend- ing moment was recorded. This allowed calculation of the load effect due to the presence of multiple trucks. Calcula- tions were carried out for span lengths from 30 to 200 ft. The considered continuous bridges had two equal-length spans. Examples of moment time histories for a single truck passage are shown in Figure 5.26, Figure 5.27, and Figure 5.28. 5.4.4 Rainflow Cycle-Counting Method The development of fatigue load models requires a collection of the actual load time histories. The collected time histories must be processed to obtain a usable form. In general, load histories may be considered as either narrow-band or wide- band processes, as shown in Figure 5.29. Narrow-band time histories are characterized by an approximately constant period. Wide-band time histories are characterized by a variable fre- quency and random amplitude. For fatigue calculations, the stress range is determined (i.e., the difference between peak and valley). Bending moment histories due to truck passages are wide band. The cycles are irregular with variable frequencies and amplitudes. Wide-band histories do not allow for simple cycle counting. The Palmgren–Miner rule is applicable only when the individual events are isolated, (i.e., narrow-band time histories). Different counting procedures have been pro- posed and used, all of which were studied and compared to select the most efficient approach for this study. Only two counting algorithms seemed to provide accurate results: rain- flow and range pair (Dowling 1972). Rainflow counting was used in this study. A rainflow cycle-counting procedure was proposed for the first time by Matsuishi and Endo in 1968. This method counts the number of full reversal cycles, as well as partial cycles, and their range amplitude for a given load time history. A full reversal cycle occurs when the cycle range goes up to its peak and back to the starting position. A partial cycle goes in only one direction, from the valley to the peak or from the peak to the valley.

123 Figure 5.27. Bending moment time history for a single truck passage on continuous bridges at middle support. Figure 5.26. Bending moment time history for a single truck passage on simple-supported bridges at middle of the span.

124 Figure 5.29. Wide-band versus narrow-band history. Figure 5.28. Bending moment time history for a single truck passage on continuous bridges at 0.4 of the span length. The summary of the steps in rainflow cycle counting are as follows: 1. Reduce the time history to a sequence of (tensile) peaks and (compressive) troughs. 2. Imagine that the time history is a template for a rigid sheet (pagoda roof). 3. Turn the sheet clockwise 90° (earliest time to the top). 4. Each tensile peak is imagined as a source of water that “drips” down the pagoda. 5. Count the number of half-cycles by looking for termina- tions in the flow occurring when • It reaches the end of the time history (Figure 5.30, Path 3-4-end or Path 4-5-7-9-11-end); • It merges with a flow that started at an earlier tensile peak; or • It flows opposite a tensile peak of greater magnitude (Figure 5.30, Path 5-6, 6-6′, 8-8′, or 10-10′). 6. Repeat Step 5 for compressive troughs. 7. Assign a magnitude to each half-cycle equal to the stress difference between its start and termination (Table 5.17). 8. Pair up half-cycles of identical magnitude to count the number of complete cycles (Table 5.18). Typically, there are some residual half-cycles (Downing and Socie 1982). The moment time histories obtained from the truck traffic simulation for each WIM site, span length, and case were pro- cessed using the rainflow counting method. Total number of cycles was divided by number of trucks in the database to get an average number of load cycles per truck passage. The results for the simple-span case are summarized in Table 5.19, for the negative moment over the support in continuous spans in Table 5.20, and for positive moment at 0.4 of the span length in continuous bridges in Table 5.21. For simply supported bridges, the number of cycles at the midspan was 2 to 2.5 cycles per truck passage for short spans; this value dropped linearly to 1 cycle for a span length of about 100 ft. Similarly, for continuous bridges at 0.4 of the span length, the

125 Figure 5.30. Rainflow counting diagram. Table 5.17. Half-Cycles After Rainflow Counting Positive Direction Negative Direction Range Amplitude Range Amplitude 1–2 2 2–3 3 3–4–end 4 4–5–7–9–11–end 6 5–6 1 6–6′ 1 7–8 1 8–8′ 1 9–10 4 10–10′ 4 11–12 5 12–13 4 13–14 2 — — Note: For range values, see Figure 5.30; — = no further negative direction values. Table 5.18. Load Cycles After Rainflow Counting Amplitude No. of Cycles 1 2 2 1 3 0.5 4 2 5 0.5 6 0.5 Table 5.19. Total Number of Load Cycles and Average Number of Load Cycles per Truck Passage for Simply Supported Bridges at the Midspan Site No. of Vehicles No. of Cycles No. of Cycles per Truck 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft Arizona (SPS-1) 26,501 59,427.5 36,397 27,321 26,505 26,501 2.24 1.37 1.03 1.00 1.00 Arizona (SPS-2) 1,391,098 3,667,719.5 2,632,482.5 1,650,818.0 1,407,468.0 1,397,629.5 2.64 1.89 1.19 1.01 1.00 Arkansas (SPS-2) 1,642,334 4,216,668.5 3,108,866.5 1,983,249.5 1,667,856.0 1,640,182.5 2.57 1.89 1.21 1.02 1.00 Colorado (SPS-2) 326,017 824,366.5 591,565.5 377,138.0 328,271.0 327,680.5 2.53 1.81 1.16 1.01 1.01 Delaware (SPS-1) 175,889 391,173.0 272,989.0 184,061.0 176,696.5 175,664.5 2.22 1.55 1.05 1.00 1.00 Illinois (SPS-6) 821,809 2,104,493.5 1,552,007.5 990,256.0 831,086.0 823,435.0 2.56 1.89 1.20 1.01 1.00 Kansas (SPS-2) 456,881 1,182,596.0 839,726.0 542,967.5 460,973.5 459,671.5 2.59 1.84 1.19 1.01 1.01 Louisiana (SPS-1) 70,831 162,679.5 113,121.5 74,619.5 70,947.0 70,838.0 2.30 1.60 1.05 1.00 1.00 Maine (SPS-5) 172,333 417,837.5 294,010.5 185,121.0 173,174.0 172,727.0 2.42 1.71 1.07 1.00 1.00 (continued on next page)

126 Table 5.19. Total Number of Load Cycles and Average Number of Load Cycles per Truck Passage for Simply Supported Bridges at the Midspan (continued) Site No. of Vehicles No. of Cycles No. of Cycles per Truck 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft Maryland (SPS-5) 124,474 271,233.5 186,120.0 129,968.0 124,930.5 124,482.0 2.18 1.50 1.04 1.00 1.00 Minnesota (SPS-5) 47,794 96,065.0 68,750.0 48,829.0 47,798.0 47,752.0 2.01 1.44 1.02 1.00 1.00 Pennsylvania (SPS-6) 1,458,818 3,669,978.0 2,667,443.0 1,676,101.0 1,477,196.0 1,459,284.0 2.52 1.83 1.15 1.01 1.00 Tennessee (SPS-6) 1,583,151 3,492,829.0 2,816,652.0 1,673,936.0 1,600,563.0 1,583,300.0 2.21 1.78 1.06 1.01 1.00 Virginia (SPS-1) 237,804 563,467.5 416,252.5 260,806.0 239,251.0 238,315.0 2.37 1.75 1.10 1.01 1.00 Wisconsin (SPS-1) 209,239 483,546.0 366,955.0 225,109.0 210,644.0 210,164.5 2.31 1.75 1.08 1.01 1.00 Table 5.20. Total Number of Load Cycles and Average Number of Load Cycles per Truck Passage for Continuous Bridges at the Middle Support Site No. of Vehicles No. of Cycles No. of Cycles per Truck 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft Arizona (SPS-1) 26,501 65,563.5 64,115.5 69,703.5 65,402 58,905 2.47 2.42 2.63 2.47 2.22 Arizona (SPS-2) 1,391,098 4,584,915.0 4,804,207.0 4,971,600.0 4,220,277.5 3,423,766.0 3.30 3.45 3.57 3.03 2.46 Arkansas (SPS-2) 1,642,334 5,437,711.0 5,654,802.0 5,774,335.5 4,949,930.5 3,902,161.0 3.31 3.44 3.52 3.01 2.38 Colorado (SPS-2) 326,017 1,020,374.5 989,200.0 1,100,728.5 983,802.0 767,937.0 3.13 3.03 3.38 3.02 2.36 Delaware (SPS-1) 175,889 543,754.5 502,112.5 527,143.0 484,787.5 419,294.5 3.09 2.85 3.00 2.76 2.38 Illinois (SPS-6) 821,809 2,716,902.0 2,768,327.0 2,836,337.0 2,489,643.5 1,987,891.5 3.31 3.37 3.45 3.03 2.42 Kansas (SPS-2) 456,881 1,505,890.5 1,507,880.5 1,608,769.0 1,387,383.0 1,116,965.5 3.30 3.30 3.52 3.04 2.44 Louisiana (SPS-1) 70,831 217,990.0 199,088.0 215,738.0 200,995.5 166,450.5 3.08 2.81 3.05 2.84 2.35 Maine (SPS-5) 172,333 518,377.5 502,246.5 558,181.0 508,993.0 383,351.5 3.01 2.91 3.24 2.95 2.22 Maryland (SPS-5) 124,474 397,197.5 346,614.5 376,056.5 342,106.5 290,348.0 3.19 2.78 3.02 2.75 2.33 Minnesota (SPS-5) 47,794 135,741.0 131,289.0 139,940.0 123,124.0 107,837.0 2.84 2.75 2.93 2.58 2.26 Pennsylvania (SPS-6) 1,458,818 3,896,713.0 3,604,125.0 4,019,137.0 3,955,368.0 3,174,582.0 2.67 2.47 2.76 2.71 2.18 Tennessee (SPS-6) 1,583,151 4,298,789.0 3,889,255.0 4,468,069.0 4,346,233.0 3,427,878.0 2.72 2.46 2.82 2.75 2.17 Virginia (SPS-1) 237,804 743,162.0 716,559.5 770,125.5 700,670.5 561,742.5 3.13 3.01 3.24 2.95 2.36 Wisconsin (SPS-1) 209,239 646,250.5 633,403.0 657,828.5 608,381.0 492,283.5 3.09 3.03 3.14 2.91 2.35

127 Table 5.21. Total Number of Load Cycles and Average Number of Load Cycles per Truck Passage for Continuous Bridges at 0.4 of the Span Length Site No. of Vehicles No. of Cycles No. of Cycles per Truck 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft Arizona (SPS-1) 26,501 68,688.0 39,328.0 29,363.0 27,695.0 26,509.0 2.59 1.48 1.11 1.05 1.00 Arizona (SPS-2) 1,391,098 4,032,130.0 2,699,800.5 2,281,797.0 2,017,321.5 1,767,920.0 2.90 1.94 1.64 1.45 1.27 Arkansas (SPS-2) 1,642,334 5,610,372.0 4,069,843.0 3,532,308.0 3,132,234.0 2,872,888.0 3.42 2.48 2.15 1.91 1.75 Colorado (SPS-2) 326,017 885,651.0 617,440.5 458,136.5 410,761.5 385,205.5 2.72 1.89 1.41 1.26 1.18 Delaware (SPS-1) 175,889 410,830.0 293,946.0 223,028.5 210,104.0 199,350.0 2.34 1.67 1.27 1.19 1.13 Illinois (SPS-6) 821,809 2,304,196.0 1,579,655.0 1,313,036.5 1,118,188.0 1,037,709.5 2.80 1.92 1.60 1.36 1.26 Kansas (SPS-2) 456,881 1,292,694.0 872,400.0 702,959.5 616,645.0 554,203.0 2.83 1.91 1.54 1.35 1.21 Louisiana (SPS-1) 70,831 171,703.5 120,584.5 91,168.0 85,553.5 80,458.0 2.42 1.70 1.29 1.21 1.14 Maine (SPS-5) 172,333 433,793.5 313,517.0 231,617.0 204,775.5 190,443.0 2.52 1.82 1.34 1.19 1.11 Maryland (SPS-5) 124,474 279,856.5 200,955.5 155,882.5 143,168.0 138,347.5 2.25 1.61 1.25 1.15 1.11 Minnesota (SPS-5) 47,794 123,298.0 70,383.5 59,891.5 52,727.5 48,541.5 2.58 1.47 1.25 1.10 1.02 Pennsylvania (SPS-6) 1,458,818 3,992,907.0 2,784,565.0 2,243,835.5 1,943,551.0 1,756,756.0 2.74 1.91 1.54 1.33 1.20 Tennessee (SPS-6) 1,583,151 4,590,126.0 2,929,061.5 2,273,958.5 1,888,805.5 1,651,117.5 2.90 1.85 1.44 1.19 1.04 Virginia (SPS-1) 237,804 599,977.0 434,778.0 338,100.0 299,309.0 278,883.5 2.52 1.83 1.42 1.26 1.17 Wisconsin (SPS-1) 209,239 516,843.0 376,098.5 298,936.5 267,981.0 246,176.0 2.47 1.80 1.43 1.28 1.18 number of cycles per truck was 2.3 to 3.5 for short spans, which dropped to 1 to 1.5 cycles for a span length of about 100 ft. The results for negative moment over the support in continuous bridges were 2.5 to 3.5 for short spans and about 2.5 for longer spans. More load cycles for short spans is caused by groups of axles rather than whole trucks due to relatively short spans compared with the vehicle length. 5.4.5 Fatigue Damage Accumulation and Equivalent Fatigue Load Because bridge structures are subjected to loads of different magnitude and frequency occurring at different times, the load can be considered as a randomly varying amplitude load. The effect of such a loading can be accounted for by applying a cumulative damage rule. Many rules have been proposed. According to the Palmgren–Miner rule, which seems to provide a reasonable means of accounting for ran- dom variable loading, fatigue damage due to variable ampli- tude loading is expressed by Equation 5.8: 1 (5.8)1 1 2 2 3 3 n N n N n N n N n N n n i i … ∑+ + + + = = where ni/Ni is the incremental damage that results from the stress range cycles with magnitude Si that occurs ni times (Figure 5.31), and Ni is the number of cycles to failure with a constant amplitude equal to Si (Figure 5.32). Failure occurs when the sum of the incremental damage equals or exceeds 1. The tests of welded details (Fisher et al. 1983; Schilling et al.

128 Time St re ss ra ng e, S S3 S2 S1 n1 n2 n3 Figure 5.31. Number of cycles ni for stress range Si. St re ss ra ng e, S Number of load cycles to failure, N S1 S3 S2 N3 N1 N2 Figure 5.32. Number of load cycles to failure Ni for stress range Si. 1977) and Barsom’s crack growth studies (Rolfe and Barsom 1977) showed a good correlation with the Palmgren–Miner rule assumptions. Schilling et al. (1977) showed that the Palmgren–Miner rule can be used to develop an equivalent constant amplitude cyclic loading that produces the same fatigue damage as a variable amplitude load for the same number of load cycles. This theory is based on the exponential model of the stress range–life relationship as given by Equation 5.9 (Fisher 1977): (5.9)N AS n= − where N = number of cycles to failure; S = nominal stress range; A = a constant for a given detail; and n = slope constant. The concept of fatigue design based on stress range alone was adopted by AASHO in 1974 (Fisher et al. 1970, 1974). Equa- tion 5.10 is obtained by substituting Equation 5.9 into Equa- tion 5.8: 1 (5.10) n AS i i n∑ =− Substituting Equation 5.11 into Equation 5.10 yields (5.11)n p Ni i T= 1 (5.12) p N AS p S AS i T i n i e n i n∑∑ = =− − − or (5.13) S p S S p S e n i i n e i i nn ∑ ∑ = = The exponent n for most structural metal details is about 3. Equation 5.13 is often referred to as a root mean cube of the stress distribution. The equivalent stress is a convenient con- cept to be used for comparison of stress histograms obtained using the rainflow counting method. Because fatigue crack nucleation and further propagation occur mostly at tensile stress conditions that are related to bending moment, it is convenient to use the bending moment formulation instead of the stress formulation for an equiva- lent load. The bending moment formulation of Equation 5.13 is given by Equation 5.14: (5.14)eqM p Mi i nn ∑= where Meq = equivalent moment cycle load; Mi = incremental moment cycle; and pi = probability of occurrence of Mi. Calculation of the equivalent moment requires the prob- ability of occurrence for each incremental moment Mi. The corresponding probability distribution functions (PDFs) of the moment cycles for each site were calculated for spans from 30 to 200 ft. As an example, the PDFs for moments cor- responding to the FHWA WIM data from Arkansas (SPS-1) are shown in Figure 5.33. The area under the curve represent- ing the PDF for each span length is equal to 1. The equivalent moment was calculated from moment cycles obtained using rainflow counting. The equivalent load was calculated for all considered WIM sites, a wide range of span lengths between 30 and 200 ft, and three bridge configu- rations. Next, the calculated equivalent moments were divided by moment due to the AASHTO LRFD fatigue truck. Results are summarized in Table 5.22 to Table 5.24. The results show that the moment ratio is smaller for short spans. 5.4.6 Fatigue Limit State II: Fatigue Damage Ratio Finite fatigue life depends on the number of load cycles dur- ing the service life and their magnitude. According to the AASHTO LRFD (2012) provisions, the number of load cycles

129 30 ft. To compare fatigue damage due to design fatigue load and actual fatigue load, it is convenient to remove the resistance part from limit state Equations 5.16, 5.17, and 5.18 [AASHTO LRFD (2012, Equations 6.6.1.2.5-1 and 6.6.1.2.5-2)]: (5.16)f F n( ) ( )γ ∆ ≤ ∆ (5.17)3F A Nn ( )∆ = (5.18)3f A N ( )γ ∆ ≤ where g = load factor; D f = force effect (i.e., live load stress range due to the pas- sage of a fatigue truck); A = resistance constant that depends on the class of the structural detail; and N = number of load cycles during the service life calcu- lated according to Equation 5.15. Stress due to truck passage is calculated according to Equa- tion 5.19: (5.19)f M S∆ = where S is section modulus and M is moment due to truck passage. Figure 5.33. PDFs of WIM moments for data from Arkansas (SPS-1). Table 5.22. Equivalent Moments for Simply Supported Bridges at the Midspan Site No. of Vehicles Equivalent Moment (kip-ft) Equivalent Moment/HL-93 Fatigue Moment 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft Arizona (SPS-1) 26,501 151.63 426.18 889.67 1,362.17 2,593.90 0.62 0.78 0.82 0.84 0.85 Arizona (SPS-2) 1,391,098 145.82 357.81 790.59 1,316.93 2,601.01 0.60 0.66 0.73 0.81 0.85 Arkansas (SPS-2) 1,642,334 146.25 354.83 770.41 1,290.54 2,554.63 0.60 0.65 0.71 0.79 0.83 Colorado (SPS-2) 326,017 132.61 325.49 713.45 1,173.08 2,311.31 0.54 0.60 0.66 0.72 0.75 Delaware (SPS-1) 175,889 155.16 400.92 831.01 1,270.55 2,424.36 0.64 0.74 0.77 0.78 0.79 Illinois (SPS-6) 821,809 146.48 354.91 762.76 1,279.33 2,532.79 0.60 0.65 0.70 0.79 0.83 Kansas (SPS-2) 456,881 141.00 355.18 767.58 1,277.57 2,524.67 0.58 0.65 0.71 0.79 0.82 Louisiana (SPS-1) 70,831 142.42 363.30 775.00 1,202.37 2,318.98 0.58 0.67 0.72 0.74 0.76 Maine (SPS-5) 172,333 129.72 328.38 707.39 1,126.24 2,206.36 0.53 0.60 0.65 0.69 0.72 Maryland (SPS-5) 124,474 132.44 335.88 675.87 1,033.81 1,982.63 0.54 0.62 0.62 0.64 0.65 Minnesota (SPS-5) 47,794 142.39 353.48 731.81 1,138.96 2,219.99 0.58 0.65 0.68 0.70 0.72 Pennsylvania (SPS-6) 1,458,818 151.46 363.23 777.74 1,259.78 2,468.70 0.62 0.67 0.72 0.78 0.81 Tennessee (SPS-6) 1,583,151 153.14 351.05 772.72 1,227.46 2,417.64 0.63 0.65 0.71 0.76 0.79 Virginia (SPS-1) 237,804 140.35 344.56 749.93 1,202.76 2,356.27 0.58 0.63 0.69 0.74 0.77 Wisconsin (SPS-1) 209,239 142.47 360.19 772.69 1,213.03 2,349.64 0.58 0.66 0.71 0.75 0.77 during bridge service life (N) is calculated using Equa- tion 6.6.1.2.5-3, shown here as Equation 5.15: 365 75 (5.15)SLN n ADTT )() )( (= where (ADTT)SL is a single lane of ADTT, and n is the number of load cycles per truck taken from Table 5.25 [AASHTO LRFD (2012, Table 6.6.1.2.5-2)]. The magnitude of load cycles is calculated as a stress due to the HL-93 fatigue truck with the second axle spacing equal to

130 Table 5.23. Equivalent Moments for Continuous Bridges at the Middle Support Site No. of Vehicles Equivalent Moment (kip-ft) Equivalent Moment/HL-93 Fatigue Moment 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft Arizona (SPS-1) 26,501 -96.91 -212.79 -314.01 -483.56 -960.50 0.53 0.59 0.59 0.63 0.72 Arizona (SPS-2) 1,391,098 -89.91 -221.06 -296.65 -454.32 -975.62 0.49 0.61 0.56 0.60 0.73 Arkansas (SPS-2) 1,642,334 -87.98 -219.14 -294.68 -450.44 -998.99 0.48 0.61 0.56 0.59 0.74 Colorado (SPS-2) 326,017 -82.94 -203.52 -268.50 -407.76 -844.78 0.45 0.56 0.51 0.54 0.63 Delaware (SPS-1) 175,889 -90.38 -214.99 -299.91 -451.62 -896.29 0.49 0.60 0.57 0.59 0.67 Illinois (SPS-6) 821,809 -87.55 -219.79 -295.45 -444.61 -964.62 0.48 0.61 0.56 0.58 0.72 Kansas (SPS-2) 456,881 -85.97 -216.73 -290.84 -439.49 -916.36 0.47 0.60 0.55 0.58 0.68 Louisiana (SPS-1) 70,831 -86.45 -205.76 -280.85 -423.51 -858.73 0.47 0.57 0.53 0.56 0.64 Maine (SPS-5) 172,333 -79.39 -198.30 -262.39 -393.39 -825.92 0.43 0.55 0.50 0.52 0.62 Maryland (SPS-5) 124,474 -79.35 -192.49 -263.24 -403.19 -814.86 0.43 0.53 0.50 0.53 0.61 Minnesota (SPS-5) 47,794 -79.86 -201.32 -270.79 -405.61 -814.03 0.43 0.56 0.51 0.53 0.61 Pennsylvania (SPS-6) 1,458,818 -90.89 -235.11 -310.77 -449.53 -974.43 0.49 0.65 0.59 0.59 0.73 Tennessee (SPS-6) 1,583,151 -87.39 -231.37 -300.99 -436.22 -961.13 0.48 0.64 0.57 0.57 0.72 Virginia (SPS-1) 237,804 -84.56 -208.61 -278.84 -418.94 -868.36 0.46 0.58 0.53 0.55 0.65 Wisconsin (SPS-1) 209,239 -83.68 -206.92 -285.18 -422.87 -860.95 0.45 0.57 0.54 0.56 0.64 Table 5.24. Equivalent Moments for Continuous Bridges at 0.4 of the Span Length Site No. of Vehicles Equivalent Moment (kip-ft) Equivalent Moment/HL-93 Fatigue Moment 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft Arizona (SPS-1) 26,501 134.25 413.51 838.11 1,291.21 2,503.65 0.55 0.71 0.76 0.80 0.83 Arizona (SPS-2) 1,391,098 133.46 349.66 663.89 1,096.11 2,282.14 0.54 0.60 0.60 0.68 0.75 Arkansas (SPS-2) 1,642,334 122.64 272.45 540.68 899.92 1,881.91 0.50 0.47 0.49 0.55 0.62 Colorado (SPS-2) 326,017 121.69 317.11 634.34 1,032.34 2,101.21 0.49 0.54 0.58 0.64 0.69 Delaware (SPS-1) 175,889 144.84 386.50 743.25 1,143.10 2,230.39 0.59 0.66 0.68 0.70 0.74 Illinois (SPS-6) 821,809 135.47 345.78 657.19 1,091.43 2,222.58 0.55 0.59 0.60 0.67 0.73 Kansas (SPS-2) 456,881 129.86 342.26 665.25 1,095.22 2,272.29 0.53 0.58 0.61 0.68 0.75 Louisiana (SPS-1) 70,831 131.42 353.55 691.10 1,076.33 2,130.97 0.53 0.60 0.63 0.66 0.70 Maine (SPS-5) 172,333 121.26 312.87 618.65 1,008.24 2,050.51 0.49 0.53 0.56 0.62 0.68 Maryland (SPS-5) 124,474 126.68 339.42 654.83 1,023.41 1,994.36 0.52 0.58 0.60 0.63 0.66 Minnesota (SPS-5) 47,794 120.71 344.34 655.63 1,054.97 2,132.90 0.49 0.59 0.60 0.65 0.70 Pennsylvania (SPS-6) 1,458,818 135.74 352.48 668.88 1,087.55 2,204.49 0.55 0.60 0.61 0.67 0.73 Tennessee (SPS-6) 1,583,151 128.44 339.17 665.81 1,104.94 2,275.40 0.52 0.58 0.61 0.68 0.75 Virginia (SPS-1) 237,804 130.01 334.69 649.10 1,055.60 2,142.89 0.53 0.57 0.59 0.65 0.71 Wisconsin (SPS-1) 209,239 133.10 349.47 666.43 1,061.17 2,138.88 0.54 0.60 0.61 0.65 0.71

131 To calculate the ratio of fatigue damage caused by the actual fatigue load and design fatigue load, the load factor has to be removed from Equation 5.18. From Equations 5.18 and 5.19, it is possible to calculate the ratio of fatigue damage due to the actual load and fatigue damage due to design load by using Equation 5.20: Boundary of Actual Fatigue Damage eq 3S M S A NR = where Meq = equivalent moment from Miner’s Rule; A = resistance constant; and NR = actual number of cycles. p1 3 eq A N S MR = Boundary of Code Fatigue Damage 3S M S A N = where M = moment due to fatigue design truck; A = resistance constant; and N = number of cycles (from Equation 5.15). 1 3 A N S M = p (5.20) 3 eq 3 3 eq 3 eq A N S M A N S M A N N A SM SM N N M M R R R = λ = λ = p p p p p l is the ratio of the fatigue damage due to the actual fatigue load to the fatigue damage due to the design fatigue load. Because resistance was removed from Equation 5.20, the fatigue damage ratio is the same regardless of the bridge com- ponent or detail class. The fatigue damage ratio was calculated according to the current AASHTO LRFD provisions for each WIM site, span length, and case. Results are summarized in Table 5.27 to Table 5.29 in the column labeled Fatigue Damage Ratio (cur- rent). The fatigue damage ratio is smaller for shorter spans. The difference between short and longer spans is due to dif- ferent code provisions for short spans with a given number of load cycles per truck passage (see Table 5.25). For short spans, a truck causes more load cycles than for longer spans. How- ever, it is balanced by a smaller moment ratio (equivalent moment/HL-93 fatigue truck moment) for short spans. If the number of load cycles due to a truck passage were equal for all spans, as shown in Table 5.26, then the resulting fatigue damage ratio would be more uniform. The fatigue damage ratio for the proposed fatigue design was calculated for each WIM site, span length, and case. The results are summarized in Table 5.27 to Table 5.29 in the col- umn labeled Fatigue Damage Ratio (proposed). For simply supported bridges at midspan and continuous bridges at 0.4 of the span length, the results are very uniform for all span lengths. At the middle support of continuous bridges, the difference between short and longer spans is reduced by about 10%. Because fatigue resistance depends on structural detail and material characteristics but not on span length, the variation in fatigue load due to span length produces a variation in reli- ability indices. The design parameters proposed in Section 5.4.9 eliminate this problem. 5.4.7 Fatigue Limit State I: Maximum Moment Range Ratio Fatigue Limit State I is related to an infinite load-induced fatigue life. The fatigue load in this limit state reflects the load levels found to be representative of the maximum stress range of the truck population for an infinite fatigue life design (AASHTO LRFD 2012). In other words, if the majority of stress cycles are below a threshold magnitude [(D F)TH], then failure will require so many load cycles that the considered detail will have an infinite fatigue life. (D F)TH is a boundary between the finite and infinite fatigue life, as shown in Figure 5.34. Table 5.25. Number of Cycles per Truck Passage (n) for AASHTO Fatigue Design Longitudinal Members n Span Length >40 ft Span Length <–40 ft Simple-span girders 1.0 2.0 Continuous girders Near interior support 1.5 2.0 Elsewhere 1.0 2.0 Table 5.26. Number of Cycles per Truck Passage (n) for Proposed Fatigue Design Longitudinal Members n Simple-span girders 1.0 Continuous girders Near interior support 1.5 Elsewhere 1.0

132 Table 5.27. Fatigue Damage Ratios for Simply Supported Bridges at the Midspan Site No. of Vehicles Fatigue Damage Ratio (current) Fatigue Damage Ratio (proposed) 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft Arizona (SPS-1) 26,501 0.65 0.87 0.83 0.84 0.85 0.81 0.87 0.83 0.84 0.85 Arizona (SPS-2) 1,391,098 0.66 0.81 0.77 0.81 0.85 0.83 0.81 0.77 0.81 0.85 Arkansas (SPS-2) 1,642,334 0.65 0.81 0.76 0.80 0.83 0.82 0.81 0.76 0.80 0.83 Colorado (SPS-2) 326,017 0.59 0.73 0.69 0.72 0.76 0.74 0.73 0.69 0.72 0.76 Delaware (SPS-1) 175,889 0.66 0.85 0.78 0.78 0.79 0.83 0.85 0.78 0.78 0.79 Illinois (SPS-6) 821,809 0.65 0.81 0.75 0.79 0.83 0.82 0.81 0.75 0.79 0.83 Kansas (SPS-2) 456,881 0.63 0.80 0.75 0.79 0.83 0.79 0.80 0.75 0.79 0.83 Louisiana (SPS-1) 70,831 0.61 0.78 0.73 0.74 0.76 0.77 0.78 0.73 0.74 0.76 Maine (SPS-5) 172,333 0.57 0.72 0.67 0.69 0.72 0.71 0.72 0.67 0.69 0.72 Maryland (SPS-5) 124,474 0.56 0.71 0.63 0.64 0.65 0.70 0.71 0.63 0.64 0.65 Minnesota (SPS-5) 47,794 0.58 0.73 0.68 0.70 0.72 0.74 0.73 0.68 0.70 0.72 Pennsylvania (SPS-6) 1,458,818 0.67 0.82 0.75 0.78 0.81 0.84 0.82 0.75 0.78 0.81 Tennessee (SPS-6) 1,583,151 0.65 0.78 0.73 0.76 0.79 0.82 0.78 0.73 0.76 0.79 Virginia (SPS-1) 237,804 0.61 0.76 0.71 0.74 0.77 0.77 0.76 0.71 0.74 0.77 Wisconsin (SPS-1) 209,239 0.61 0.80 0.73 0.75 0.77 0.77 0.80 0.73 0.75 0.77 Table 5.28. Fatigue Damage Ratios for Continuous Bridges at the Middle Support Site No. of Vehicles Fatigue Damage Ratio (current) Fatigue Damage Ratio (proposed) 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft Arizona (SPS-1) 26,501 0.57 0.69 0.71 0.75 0.82 0.62 0.69 0.71 0.75 0.82 Arizona (SPS-2) 1,391,098 0.58 0.81 0.75 0.75 0.86 0.64 0.81 0.75 0.75 0.86 Arkansas (SPS-2) 1,642,334 0.57 0.80 0.74 0.75 0.87 0.62 0.80 0.74 0.75 0.87 Colorado (SPS-2) 326,017 0.52 0.71 0.66 0.68 0.73 0.58 0.71 0.66 0.68 0.73 Delaware (SPS-1) 175,889 0.57 0.74 0.71 0.73 0.78 0.63 0.74 0.71 0.73 0.78 Illinois (SPS-6) 821,809 0.56 0.80 0.74 0.74 0.84 0.62 0.80 0.74 0.74 0.84 Kansas (SPS-2) 456,881 0.55 0.78 0.73 0.73 0.80 0.61 0.78 0.73 0.73 0.80 Louisiana (SPS-1) 70,831 0.54 0.70 0.67 0.69 0.74 0.60 0.70 0.67 0.69 0.74 Maine (SPS-5) 172,333 0.49 0.69 0.64 0.65 0.70 0.54 0.69 0.64 0.65 0.70 Maryland (SPS-5) 124,474 0.50 0.66 0.63 0.65 0.70 0.55 0.66 0.63 0.65 0.70 Minnesota (SPS-5) 47,794 0.49 0.68 0.64 0.64 0.70 0.54 0.68 0.64 0.64 0.70 Pennsylvania (SPS-6) 1,458,818 0.54 0.77 0.72 0.72 0.82 0.60 0.77 0.72 0.72 0.82 Tennessee (SPS-6) 1,583,151 0.53 0.76 0.70 0.70 0.81 0.58 0.76 0.70 0.70 0.81 Virginia (SPS-1) 237,804 0.53 0.73 0.68 0.69 0.75 0.59 0.73 0.68 0.69 0.75 Wisconsin (SPS-1) 209,239 0.53 0.73 0.69 0.69 0.75 0.58 0.73 0.69 0.69 0.75

133 Table 5.29. Fatigue Damage Ratios for Continuous Bridges at 0.4 of the Span Length Site No. of Vehicles Fatigue Damage Ratio (current) Fatigue Damage Ratio (proposed) 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft Arizona (SPS-1) 26,501 0.60 0.81 0.79 0.81 0.83 0.75 0.81 0.79 0.81 0.83 Arizona (SPS-2) 1,391,098 0.61 0.75 0.71 0.76 0.82 0.77 0.75 0.71 0.76 0.82 Arkansas (SPS-2) 1,642,334 0.60 0.63 0.64 0.69 0.75 0.75 0.63 0.64 0.69 0.75 Colorado (SPS-2) 326,017 0.55 0.67 0.65 0.69 0.73 0.69 0.67 0.65 0.69 0.73 Delaware (SPS-1) 175,889 0.62 0.78 0.73 0.75 0.77 0.78 0.78 0.73 0.75 0.77 Illinois (SPS-6) 821,809 0.62 0.73 0.70 0.75 0.79 0.78 0.73 0.70 0.75 0.79 Kansas (SPS-2) 456,881 0.59 0.73 0.70 0.75 0.80 0.75 0.73 0.70 0.75 0.80 Louisiana (SPS-1) 70,831 0.57 0.72 0.68 0.71 0.73 0.72 0.72 0.68 0.71 0.73 Maine (SPS-5) 172,333 0.53 0.65 0.62 0.66 0.70 0.67 0.65 0.62 0.66 0.70 Maryland (SPS-5) 124,474 0.54 0.68 0.64 0.66 0.68 0.67 0.68 0.64 0.66 0.68 Minnesota (SPS-5) 47,794 0.53 0.67 0.64 0.67 0.71 0.67 0.67 0.64 0.67 0.71 Pennsylvania (SPS-6) 1,458,818 0.61 0.75 0.70 0.74 0.77 0.77 0.75 0.70 0.74 0.77 Tennessee (SPS-6) 1,583,151 0.59 0.71 0.68 0.72 0.76 0.74 0.71 0.68 0.72 0.76 Virginia (SPS-1) 237,804 0.57 0.70 0.66 0.70 0.75 0.72 0.70 0.66 0.70 0.75 Wisconsin (SPS-1) 209,239 0.58 0.73 0.68 0.71 0.75 0.73 0.73 0.68 0.71 0.75 Number of load cycles to failure (log scale) Resistance, R Finite Fatigue Life Infinite Fatigue Life (F)TH St re ss ra ng e (lo g s ca le) Figure 5.34. The threshold stress (DF)TH on an S-N curve. Fatigue Limit State I refers to the stress value that has 1/10,000 probability of being exceeded. It is assumed that the distribution of stress has the same CDF shape as that of the corresponding moments. Thus, the fatigue load analysis is performed using the developed CDFs for moments for vari- ous considered sites, cases, and spans from 30 to 200 ft. The moment corresponding to the upper 0.01% is determined as a percentile corresponding to the probability of 0.9999, or 3.8 on the vertical axis in Figure 5.35. This moment represents the maximum stress range corresponding to an unlimited fatigue life. For example, for the WIM data from Arkansas (SPS-1), the moment for span of 120 ft corresponding to the upper 0.01% is 2,505.5 kip-ft (Figure 5.35). The calculations were performed for the considered loca- tions, cases, and span lengths. The obtained values of moment were divided by the corresponding AASHTO fatigue truck moment. The results are summarized in Table 5.30 to Table 5.32. 5.4.8 Statistical Parameters of Fatigue Live Load The objective was to determine the statistical parameters of fatigue load that can be considered as representative for the national load. The statistical parameters will be different for the maximum and equivalent fatigue load specified for Fatigue Limit States I and II, respectively. The ratios of the 1/10,000 moment to the HL-93 fatigue moment were plotted on normal probability paper and are shown in Figure 5.36 to Figure 5.38, and the proposed fatigue damage ratios are shown in Figure 5.39 to Figure 5.41. Each point on the graphs represents one of 15 sites considered. To determine the statistical parameters from the graphs, a straight line was fitted for each distribution. A straight line cor- responds to a normal distribution on the normal probability paper. The intersection of the straight line with the horizontal axis is at the mean value. The standard deviation is determined from the slope of the straight line. The statistical parameters of fatigue load (i.e., mean, µ, and CV), based on data from 15 con- sidered sites, were calculated as the ratio of standard deviation (s) and the mean and are listed in Table 5.33 and Table 5.34. (text continues on page 142)

134 Figure 5.35. Moment corresponding to the upper 0.01%, span 5 120 ft. Table 5.30. Maximum Moment Range for Simply Supported Bridges at the Midspan Site No. of Vehicles 1/10,000 Moment Cycle 1/10,000 Moment/HL-93 Fatigue Moment 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft Arizona (SPS-1) 26,501 424 1,003 1,761 2,754 5,640 1.74 1.84 1.63 1.70 1.84 Arizona (SPS-2) 1,391,098 308 765 1,416 2,246 4,711 1.26 1.41 1.31 1.38 1.54 Arkansas (SPS-2) 1,642,334 352 860 1,526 2,460 5,066 1.44 1.58 1.41 1.52 1.65 Colorado (SPS-2) 326,017 336 814 1,497 2,409 4,854 1.38 1.50 1.38 1.48 1.58 Delaware (SPS-1) 175,889 454 1,257 2,302 3,212 5,735 1.86 2.31 2.12 1.98 1.87 Illinois (SPS-6) 821,809 350 844 1,480 2,408 5,033 1.43 1.55 1.37 1.48 1.64 Kansas (SPS-2) 456,881 411 1,018 1,989 3,112 6,083 1.69 1.87 1.84 1.92 1.99 Louisiana (SPS-1) 70,831 460 1,237 2,126 3,332 6,616 1.89 2.27 1.96 2.05 2.16 Maine (SPS-5) 172,333 397 964 1,722 2,726 5,549 1.63 1.77 1.59 1.68 1.81 Maryland (SPS-5) 124,474 412 1,038 1,802 2,599 5,061 1.69 1.91 1.66 1.60 1.65 Minnesota (SPS-5) 47,794 392 1,111 2,220 3,316 6,225 1.61 2.04 2.05 2.04 2.03 Pennsylvania (SPS-6) 1,458,818 402 1,003 1,730 2,623 5,291 1.65 1.84 1.60 1.62 1.73 Tennessee (SPS-6) 1,583,151 419 1,020 1,652 2,387 4,906 1.72 1.88 1.52 1.47 1.60 Virginia (SPS-1) 237,804 369 946 1,709 2,562 5,055 1.51 1.74 1.58 1.58 1.65 Wisconsin (SPS-1) 209,239 393 968 1,712 2,717 5,396 1.61 1.78 1.58 1.67 1.76

135 Table 5.32. Maximum Moment Range for Continuous Bridges at 0.4 of the Span Length Site No. of Vehicles 1/10,000 Moment Cycle 1/10,000 Moment/HL-93 Fatigue Moment 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft Arizona (SPS-1) 26,501 399 976 1,764 2,769 5,542 1.62 1.67 1.61 1.71 1.83 Arizona (SPS-2) 1,391,098 293 761 1,431 2,228 4,636 1.19 1.30 1.30 1.37 1.53 Arkansas (SPS-2) 1,642,334 338 849 1,527 2,416 4,914 1.37 1.45 1.39 1.49 1.62 Colorado (SPS-2) 326,017 319 805 1,528 2,428 4,857 1.30 1.38 1.39 1.50 1.60 Delaware (SPS-1) 175,889 439 1,279 2,243 3,141 5,635 1.78 2.19 2.04 1.94 1.86 Illinois (SPS-6) 821,809 334 814 1,508 2,399 4,893 1.36 1.39 1.37 1.48 1.61 Kansas (SPS-2) 456,881 394 1,049 1,983 3,088 5,988 1.60 1.79 1.81 1.90 1.98 Louisiana (SPS-1) 70,831 458 1,126 2,174 3,349 6,486 1.86 1.92 1.98 2.06 2.14 Maine (SPS-5) 172,333 377 937 1,811 2,768 5,525 1.53 1.60 1.65 1.71 1.82 Maryland (SPS-5) 124,474 406 1,036 1,817 2,618 4,941 1.65 1.77 1.65 1.61 1.63 Minnesota (SPS-5) 47,794 382 1,142 2,134 3,223 6,065 1.55 1.95 1.94 1.99 2.00 Pennsylvania (SPS-6) 1,458,818 395 1,020 1,726 2,608 5,243 1.61 1.74 1.57 1.61 1.73 Tennessee (SPS-6) 1,583,151 416 1,012 1,636 2,379 4,868 1.69 1.73 1.49 1.47 1.61 Virginia (SPS-1) 237,804 356 955 1,704 2,509 4,947 1.45 1.63 1.55 1.55 1.63 Wisconsin (SPS-1) 209,239 375 958 1,705 2,662 5,326 1.53 1.64 1.55 1.64 1.76 Table 5.31. Maximum Moment Range for Continuous Bridges at the Middle Support Site No. of Vehicles 1/10,000 Moment Cycle 1/10,000 Moment/HL-93 Fatigue Moment 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft Arizona (SPS-1) 26,501 -266 -701 -1,026 -1,608 -3,089 1.45 1.95 1.94 2.11 2.30 Arizona (SPS-2) 1,391,098 -211 -549 -968 -1,526 -3,019 1.15 1.52 1.83 2.00 2.25 Arkansas (SPS-2) 1,642,334 -213 -643 -995 -1,522 -3,187 1.16 1.78 1.88 2.00 2.38 Colorado (SPS-2) 326,017 -231 -579 -877 -1,312 -2,813 1.25 1.61 1.66 1.72 2.10 Delaware (SPS-1) 175,889 -248 -650 -1,173 -1,643 -3,303 1.35 1.80 2.21 2.16 2.46 Illinois (SPS-6) 821,809 -207 -640 -1,005 -1,506 -3,093 1.13 1.78 1.90 1.98 2.31 Kansas (SPS-2) 456,881 -294 -755 -1,015 -1,469 -2,937 1.60 2.10 1.92 1.93 2.19 Louisiana (SPS-1) 70,831 -278 -815 -1,128 -1,539 -3,255 1.51 2.26 2.13 2.02 2.43 Maine (SPS-5) 172,333 -251 -694 -970 -1,418 -2,967 1.37 1.93 1.83 1.86 2.21 Maryland (SPS-5) 124,474 -240 -592 -1,049 -1,564 -3,281 1.31 1.64 1.98 2.05 2.45 Minnesota (SPS-5) 47,794 -292 -695 -1,034 -1,487 -2,753 1.59 1.93 1.95 1.95 2.05 Pennsylvania (SPS-6) 1,458,818 -245 -638 -1,067 -1,588 -3,131 1.33 1.77 2.01 2.09 2.33 Tennessee (SPS-6) 1,583,151 -222 -628 -1,025 -1,559 -2,977 1.21 1.74 1.93 2.05 2.22 Virginia (SPS-1) 237,804 -223 -603 -973 -1,477 -3,010 1.21 1.67 1.84 1.94 2.24 Wisconsin (SPS-1) 209,239 -250 -671 -953 -1,394 -2,892 1.36 1.86 1.80 1.83 2.16

136 Figure 5.36. Maximum moment range ratio (Fatigue Limit State I) for simple-supported bridges at the midspan.

137 Figure 5.37. Maximum moment range ratio (Fatigue Limit State I) for continuous bridges at the middle support. Ratio: “1/10000 Moment” / HL-93 Fatigue Moment

138 Figure 5.38. Maximum moment range ratio (Fatigue Limit State I) for continuous bridges at 0.4 of the span length. Ratio: “1/10000 Moment” / HL-93 Fatigue

139 Figure 5.39. Fatigue damage ratio for proposed change (Fatigue Limit State II) for simple-supported bridges at the midspan.

140 Figure 5.40. Fatigue damage ratio for proposed change (Fatigue Limit State II) for continuous bridges at the middle support.

141 Figure 5.41. Fatigue damage ratio for proposed change (Fatigue Limit State II) for continuous bridges at 0.4 of the span length.

142 Table 5.33. Maximum Moment Range Ratio for Fatigue Limit State I Bridge Type Span (ft) Mean Mean  1.5s CV Simple-supported midspan 30 1.60 1.90 0.13 60 1.83 2.24 0.15 90 1.60 1.96 0.15 120 1.64 1.88 0.10 200 1.70 2.15 0.18 Continuous middle support 30 1.35 1.61 0.13 60 1.81 2.13 0.12 90 1.92 2.18 0.09 120 1.97 2.17 0.07 200 2.27 2.47 0.06 Continuous 0.4 of the span length 30 1.54 1.86 0.14 60 1.67 2.06 0.16 90 1.60 1.92 0.13 120 1.65 1.97 0.13 200 1.72 2.11 0.15 Table 5.34. Proposed Fatigue Damage Ratio for Fatigue Limit State II Bridge Type Span (ft) Mean Mean  1.5s CV Simple-supported midspan 30 0.79 0.87 0.07 60 0.78 0.86 0.06 90 0.73 0.81 0.07 120 0.76 0.84 0.07 200 0.78 0.86 0.07 Continuous middle support 30 0.59 0.65 0.07 60 0.74 0.82 0.07 90 0.69 0.77 0.07 120 0.71 0.78 0.06 200 0.79 0.87 0.07 Continuous 0.4 of the span length 30 0.73 0.81 0.07 60 0.72 0.80 0.07 90 0.68 0.75 0.07 120 0.72 0.79 0.06 200 0.76 0.84 0.07 It is assumed that the considered 15 WIM locations are rep- resentative of the truck traffic in the United States. For the pur- pose of further reliability analysis, it is recommended to assume that the mean fatigue load is equal to the mean for the 15 WIM locations plus 1.5 standard deviations (1.5 s). The probability of exceeding this value is about 5%, and as Figure 5.42 shows, 95% of sites in the United States are below this value. The moment ratios corresponding to the mean plus 1.5 standard deviations are also listed in Table 5.33 and Table 5.34. The statistical parameters were calculated for all consid- ered cases and span length. 5.4.9 Recommendations Use of the proposed number of cycles of stress range per truck shown in Table 5.26 resulted in the relatively tightly clus- tered moment range ratios shown in Table 5.33 and Table 5.34 for the Fatigue II and Fatigue I limit states, respectively. As with other live load recommendations in this report, the val- ues to be considered for calibration are the moment ratios at the “mean plus 1.5 standard deviations” and the CVs. For sim- plicity, the recommended values for the calibration of the fatigue limit states are further simplified into single values independent of span length as follows: • For Fatigue I, use stress ranges (loads) based on 2.0 HL-93 and a CV = 0.12. • For Fatigue II, use stress ranges (loads) based on 0.80 HL-93 and a CV = 0.07. The corresponding load factors are determined from Monte Carlo simulation using the statistics of resistance based on past laboratory testing, as summarized in Keating and Fisher (1986). The development of the load factors for steel and concrete components and details is explained in Chapter 6. Figure 5.42. Probability density function of the national fatigue load. Pr ob ab ili ty Fatigue Load Ratio 95% of the population Probability Density Function Mean Mean + 1.5σ (continued from page 133)

143 5.5 Development of Overload (Service II) parameters WIM data also forms the basis for estimating how often a given design moment (or shear) is exceeded. Table 5.35 shows the number of times the live load moment exceeded 100%, 110%, 120%, and 130% of HL-93 for the 32 WIM sites. One of the sites, Florida Route 29, clearly has a unique traffic pat- tern. The Florida Department of Transportation explained that truck traffic from several other highways was being directed onto this road, which undoubtedly accounted for the relatively large number of times the HL-93 was exceeded for the various percentages indicated. The total number of times the various ratios of HL-93 were exceeded, excluding Florida Route 29, is shown in the Table 5.35, as well as the average number per site. Most of the data were collected for a year, so that the lowest row in the table indicates the average number of times each of the criteria was exceeded on an average site during a year. This information was used to assess the signifi- cance of the Service II limit states in Chapter 6. Table 5.35. Number of Times WIM Moments Exceeded Factored HL-93 Loadings Site Moment Ratio Truck/HL-93 >–1.1 Ratio Truck/HL-93 >–1.2 Ratio Truck/HL-93 >–1.3 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft Arizona (SPS-1) 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Arizona (SPS-2) 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 Arkansas (SPS-2) 2 7 3 0 0 0 3 0 0 0 0 0 0 0 0 Colorado (SPS-2) 0 2 5 4 0 0 0 2 0 0 0 0 0 0 0 Delaware (SPS-1) 36 33 22 11 0 10 22 10 1 0 1 11 1 0 0 Illinois (SPS-6) 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 Indiana (SPS-6) 3 11 11 10 2 2 4 5 4 0 0 0 1 0 0 Kansas (SPS-2) 16 33 35 31 2 7 16 17 7 0 6 7 6 0 0 Louisiana (SPS-1) 44 6 12 14 7 26 6 7 7 0 6 6 5 4 0 Maine (SPS-5) 4 4 5 2 0 0 4 2 0 0 0 2 0 0 0 Maryland (SPS-5) 5 6 2 2 0 0 1 1 0 0 0 1 0 0 0 Minnesota (SPS-5) 7 5 6 5 0 4 2 2 1 0 2 1 1 0 0 New Mexico (SPS-1) 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 New Mexico (SPS-5) 3 1 1 2 0 2 0 0 0 0 0 0 0 0 0 Pennsylvania (SPS-6) 32 22 17 14 1 13 17 13 1 0 3 13 2 0 0 Tennessee (SPS-6) 53 4 4 0 0 5 1 0 0 0 1 0 0 0 0 (continued on next page)

144 Table 5.35. Number of Times WIM Moments Exceeded Factored HL-93 Loadings (continued) Site Moment Ratio Truck/HL-93 >–1.1 Ratio Truck/HL-93 >–1.2 Ratio Truck/HL-93 >–1.3 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft 30 ft 60 ft 90 ft 120 ft 200 ft Virginia (SPS-1) 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 Wisconsin (SPS-1) 1 0 3 3 1 0 0 1 1 0 0 0 0 0 0 California Antelope EB 0 1 0 0 5 0 0 0 0 0 0 0 0 0 0 California Antelope WB 0 5 4 13 28 0 0 0 1 9 0 0 0 0 1 California Bowman 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 California LA-710 NB 1 31 50 51 15 0 6 24 19 0 0 0 4 1 0 California LA-710 SB 1 17 45 48 14 0 3 18 19 0 0 0 1 1 0 California Lodi 0 4 16 46 140 0 0 1 2 32 0 0 0 0 2 Florida I-10 79 40 46 75 37 22 16 14 17 5 10 5 4 5 2 Florida I-95 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Florida US-29 653 495 322 245 106 360 266 174 119 51 177 160 82 59 21 Mississippi I-10 24 22 31 33 22 7 2 10 19 2 2 2 2 2 1 Mississippi I-55UI 0 0 0 1 2 0 0 0 0 0 0 0 0 0 0 Mississippi I-55R 19 30 48 58 32 7 8 16 21 19 2 3 5 8 9 Mississippi US-49 0 0 2 1 0 0 0 0 0 0 0 0 0 0 0 Mississippi US-61 0 0 1 2 1 0 0 1 1 0 0 0 0 0 0 Total ( without Florida US-29) 331 285 373 430 310 105 111 144 121 68 33 51 32 21 15 Average per site per year 10.7 9.2 12.0 13.9 10.0 3.4 3.6 4.6 3.9 2.2 1.1 1.6 1.0 0.7 0.5