Effect of Truck Weight on Bridge Network Costs (2003)

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25 CHAPTER 3 CONCEPT OF RECOMMENDED METHODOLOGY FOR ESTIMATING BRIDGE NETWORK COSTS DUE TO TRUCK WEIGHT LIMIT CHANGES This chapter presents the concept of the recommended methodology, whose procedure is given in Appendix A. Before applying this methodology to a specific scenario of truck weight limit change, a planning period, PP, in years needs to be determined by the user. This period defines the time span during which the cost impact will be considered effective. This period is recommended to be consistent with the agency’s planning period, so that parameters for projecting to the future would be readily available. These parameters may include discount rate, traffic growth rate, and expected funding lev- els. A 20-year period may be used as the default value for PP, if more specific information is not available to help select a more realistic period. As mentioned above, four cost-impact categories are cov- ered in the methodology: 1. Fatigue of existing steel bridges, 2. Fatigue of existing RC decks, 3. Deficiency due to overstress for existing bridges, and 4. Deficiency due to overstress for new bridges. It should be noted that there are other categories that con- tribute to the cost impact as a result of truck weight limit changes. One example may be fatigue failure of steel expan- sion joints. However, significant work would be needed to develop quantitative methods for estimating the costs that can cover a variety of situations over the nation. This amount of work has been deemed to be beyond the scope of this project. Further, these costs are believed to be relatively less significant than those covered here. Nevertheless, agencies that have spe- cific data and models for cost impact categories other than those covered here should be encouraged to include them in application of the recommended methodology. Accordingly, the four prioritized cost-impact categories are addressed respectively in the following sections. The con- cept presented is intended to be consistent with the typical practice of state transportation agencies, largely guided by the AASHTO specifications. For those areas where no clearly defined guidelines exist or current guidelines can be improved based on the latest research results, recommendations are made based on the latest knowledge in the relevant areas. An example of such a situation is the calculation of the life- averaged daily truck traffic, Ta, needed for fatigue assess- ment for steel bridge components. It is discussed in detail in Section 3.3. Before these four cost-impact categories are discussed below, two general concepts need to be presented and dis- cussed first. They are (1) the recommended data require- ments for the methodology and (2) the recommended method for predicting truck-load spectra due to truck-weight-limit changes. These two concepts are relevant to all four cost- impact categories. The data requirements are very important to the implementation of the recommended methodology. They should be set to accommodate a variety of situations over the country with respect to data availability. The pre- diction method for truck weight spectra is part of the proce- dure for each cost impact category. This is because the load spectrum is the cause of cost impact. When applying the recommended methodology in Appen- dix A, an upper and a lower level of data requirements are recognized, referred to as Level II and Level I respectively. These levels refer to individual cost-impact categories. Level I represents the minimum requirement, which is expected to be reachable by all state highway agencies. Level II is a high- est requirement, which may be met by only a few agencies at this time but not simultaneously for all cost impact cate- gories. With foreseeable advancement in data availability (e.g., through Virtis becoming loaded and operational), more agencies will be able to meet this level of requirement in the future. Note that these two levels also have implications to the amount of analysis effort and the level of accuracy for the result. In general, the higher level is expected to produce a more accurate result. 3.1 DATA REQUIREMENTS FOR APPLYING THE RECOMMENDED METHODOLOGY It was recognized, prior to starting the development of the recommended methodology, that the availability of required data could be a critical issue for successful implementation of the research product. This availability varies significantly among state agencies and even further among local agencies that may be interested in using the recommended method- ology as well. It is also acknowledged that this may have a

significant implication on the reliability of the application result, because more detailed data generally would permit bet- ter fidelity to the analysis and therefore more accurate results. Based on the survey results discussed in Section 2.2 related to the variability of available data, two levels of data require- ment are designed for the recommended methodology. This is to assure that the methodology at least can be used by all state highway agencies in the country. The lower data require- ment level is referred to as Level I, where a set of default data has been prepared for application, in case relevant informa- tion is not readily available. The higher level is referred to as Level II, which represents a situation where data are avail- able for all specific sites (bridges). In the software developed for the recommended methodology, selecting which level to perform analysis at is allowed for each cost impact category and for any bridge. Completely satisfying the Level II data requirement for all cost impact categories and for every bridge in the network is not realistic for an agency. On the other hand, the agency may be able to satisfy the Level II requirement for an individual cost impact category. Further, the Level I analy- sis requires the minimum data that all state agencies are expected to be able to provide. The software module for the recommended methodology is also flexible enough to permit an agency to perform analy- sis at a level between Levels I and II. For example, for cer- tain bridge sites, the agency may have site-specific data (such as WIM data and bridge structure details) but, for other sites, such data are not readily available. In reality, a large number of state agencies may perform analyses at such a “hybrid” level. This flexibility accommodates virtually all situations in terms of data availability. 3.2 PREDICTING CHANGES IN TWHS AND WHEEL WEIGHT HISTOGRAMS This section presents the concept of the recommended method for predicting truck load spectra as a result of truck- weight-limit changes. This includes two load types: TWHs and wheel weight histograms (WWHs). The former is rele- vant to Cost-Impact Categories 1, 3, and 4 for estimating costs related to steel fatigue of existing bridges, deficiency of exist- ing bridges, and deficiency of new bridges. The latter repre- sents the load causing RC bridge deck fatigue. The recom- mended prediction method for the latter case is based on the result for the former. Thus, it is discussed later. 3.2.1 Related Definitions In this section and thereafter, several terms are used with specific definitions. They are given here to facilitate further presentation. They are also used in the appendixes when pre- senting the recommended methodology and other relevant information. 26 Base Case refers to conditions without the proposed changes in truck weight limits. Alternative Scenario refers to conditions with the proposed changes in truck weight limits. Practical maximum gross vehicle weight (PMGVW) is the assumed maximum weight at which a given vehicle can oper- ate under a given set of truck weight limits. PMGVW cap- tures the net effect on gross weight of the various types of weight and dimension limits (e.g., gross weight limit, axle weight limits, bridge formula, length limits, etc.), as well as practical considerations such as maximum weights on steer- ing axles. Tare weight is the weight of a truck when it is carrying no freight. Payload is the weight of the freight carried on a truck. Operating weight is the total gross weight of a truck (tare plus payload), which is interchangeable with GVW. Payload ton-miles is a measure of the amount of freight car- ried by trucks. It is calculated as the product of payload (in tons) and VMT. For example, 1,000 VMT by a truck with a tare weight of 20,000 pounds and an operating weight of 50,000 pounds represents 15,000 payload ton-miles (1,000 VMT and a payload of 15 tons). Using the SI system, this quantity is measured by kN-kilometers. Empty/loaded ratio (rE/L) specifies the amount of empty VMT associated with repositioning trucks after they have delivered their payloads. For example for construction trucks hauling dirt from one site to another and then returning to the first site empty, the empty/loaded ratio is typically 1.0. The empty/loaded ratio for trucks carrying intercity freight is much less since such trucks often carry freight in both directions. Weight-limited traffic is VMT by trucks whose loading is assumed to be directly affected by the weight limit scenario under consideration. Much of weight-limited traffic oper- ates at weights close to the PMGVW. However, they may not be exactly at the PMGVW because of imprecision in truck loading and weighing practices. Also, some vehicles may start out from their home base fully loaded and distribute a portion of their payload at each of several locations. Further, some vehicles, such as garbage trucks, may start out empty and increase their payloads over the course of their trip and return to their home base at the PMGVW. In defining weight- limited traffic, the researchers exclude vehicles operating under special permits that exempt them from the PMGVW. For example, all states grant exceptions to weight-limits for non-divisible loads. The weights of these vehicles will not be affected by changes in the PMGVW. They also exclude trucks

whose PMGVWs are determined by limits in other states. When a truck travels through several states with different truck weight limits, its PMGVW is the most restrictive PMGVW in all of the states in which it operates. Load shifts occur when operators load trucks heavier or lighter in response to changes in the PMGVW of these truck types. Truck type shifts occur when operators shift freight from one type of truck to another because of a change in their rel- ative PMGVWs. Exogenous shifts occur when payloads change. Examples of such shifts are (1) payload shifts from trucks to rail or vice versa and (2) economic growth. 3.2.2 Recommended Method for Predicting TWHs A new method is presented in this section to offer improve- ments over the proposed methods commented in Section 2.4, to better deal with current proposed changes in truck weight limits. Note that this new method is able to deal with both cases of increase and decrease in truck weight limits for plan- ning purposes, although virtually all observed weight-limit changes have been increases. It needs to be emphasized that, for low-density commodi- ties, the permissible payload is usually limited by its cubic capacity, controlled by truck size limits (including those on width, height, trailer length, and number of trailers). For high- density commodities with divisible loads (e.g., sand, coal, beets, and hay), shippers will try to load their trucks as close as possible to the PMGVW. Increases or decreases in PMGVWs for these vehicles (due to changes in truck weight regulations) will directly affect TWHs for these vehicles. However, observ- ing and estimating how weight-limit changes affect TWHs is complicated because these changes also influence the follow- ing factors. (1) Total travel distances of loaded trucks (VMT) required to transport a given amount of payload from one location to another. (2) Travel distances by empty trucks returning to their home base or repositioning to pick up their next load. (Single-unit trucks such as dump trucks and con- crete mixers typically operate up to half of their mileage empty. For combinations, the percentage of empty mileage is less, typically 10 to 30 percent (TRB 1990b). (3) The type(s) of truck configuration used to carry freight. (4) Competition for freight between trucks and other modes (most impor- tantly rail). (5) The total amount of freight shipped. These fac- tors have not been adequately covered in the earlier efforts reviewed above. Changes in TWHs due to truck-weight-limit changes may be classified into the following three types of freight shifting: (1) Load shifts without changing truck types (truck configu- rations), referred to as truckload shift hereafter. (2) Load shifts 27 with changing of truck configuration, referred to as truck type shift below. (3) Exogenous shifts, such as economy growth and mode shift (e.g., from and to rail) due to competition. The new method presented next specifically deals with these shifts. Testing for the proposed method is also presented using measured truck weight data spanning weight limit change in the states of Arkansas and Idaho. This recommended method of predicting changes in TWHs assumes that a TWH for the Base Case is available for each type of vehicle (as listed in Data Set A-5.2.1 of Appendix A) except for automobiles and 4-tire light trucks. Theses two types of vehicles are not relevant to bridge structures in strength and fatigue-related issues. These assumed TWHs for the Base Case may be obtained using WIM data, possibly available with highway agencies. The FHWA VMT data for Year 2000 may be used as the default data set. A sample of it for a functional class of roads in the State of Minnesota is shown as Data Set A-5.2.1 in Appendix A. Note that this sample needs to be normalized to be a TWH. In other words, each term in that table needs to be divided by the sum of all these terms. Then the sum of the resulting terms will be 1, which qualifies the data to be a TWH. This default data set is available for 12 functional classes in each of the 50 states. 3.2.2.1 Truck Load Shift In truckload shifting as a result of truck-weight-limit change, trucks of a given type can be loaded heavier (or lighter). This is because the Alternative Scenario’s PMGVW is higher (or lower) than the Base Case PMGVW. This type of change in TWHs is expected to occur when the Alterna- tive Scenario does not require trucks to change their config- uration for carrying the new allowable loads. A typical exam- ple of truckload shift in the United States is the increase of legal GVW limit from 320 or 326 to 356 kN (72 or 73.28 to 80 kips) in the 1970s and 80s. Virtually only 5-axle (3-S2) trucks reacted to this weight-limit change and increased their payloads. Accordingly, load-shifting will be limited within the type of vehicle. In other words, only the TWH for that type of vehicle will be subject to change (shifting). This shifting should be performed only for weight-limit-dependent truck traffic. This amount of traffic is identified using a window shown in Fig. 3.1 over the Bases Case TWH (for the im- pacted type of trucks) assumed to be available. Namely, the traffic that is within this window may be subject to shifting. The window is defined by five parameters: a1, a2, b1, b2, and c, which are discussed below. These parameters are referred to as window parameters. Parameters b1 and b2 define a neighborhood of weight-limit- sensitive traffic, with reference to the Base Case’s PMGVW. When GVWBC /PMGVWBC is close to 1 between 1 − a1 and 1 + a2, the level of weight-limit-dependence is described by c. It indicates the percentage of the traffic that is to be changed

under the Alternative Scenario. Beyond this small range to the left, the level of weight-limit-dependence is assumed to vary linearly from c at 1 − a1 to zero at 1 − b1 being the lower boundary of the neighborhood. To the right from 1 + a2 a sim- ilar behavior is assumed of weight-limit-dependence up to 1 + b2. Fig. 3.1 can also be expressed analytically as follows (3.2.2.1) where TTGVWk stands for truck traffic at weight within the kth GVW interval in the TWH, and TT ′GVWk is the amount of traf- fic that is to be shifted (to be replaced by another amount of traffic) to a different GVW. The subscript BC refers to Base Case. For practical application, when the Base Case TWH is not expressed in traffic amount but in frequency, all TTs are replaced by corresponding frequencies. After the weight-limit-dependent traffic TT ′GVWk,BC is iden- tified as in Eq. 3.2.2.1, the following equations will be thereto applied in modifying the TWH, as a response to the consid- ered changes in truck weight limits: cTT b a for: 1 b GVW PMGVW a cTT TT for: 1 a GVW PMGVW a cTT b a for: 1 + a GVW PMGVW GVWk,BC 1 1 1 k,BC BC GVWk,BC GVWk,BC 1 k,BC BC GVWk,BC 2 k,BC BC GVW PMGVW b b GVWPMGVW k,BC BC k,BC BC − +  + −  − − < < − ′ = < < + − + − < < 1 1 1 2 1 1 1 1 2 2 2 ( ) ( ) b Otherwise 2 0 28 GVWAS = GVWk,BC (PMGVWAS /PMGVWk,BC) (3.2.2.2a) TTGVW, AS = TT ′GVWk,BC(GVWk,BC − TAREBC)/(GVWAS −TAREAS) (3.2.2.2b) where the subscripts BC and AS refer to the Base Case and Alternative Scenario, respectively. TARE is the empty weight of truck. TTGVW,AS is the truck traffic at weight GVWAS under the Alternative Scenario. Eq. 3.2.2.2a indicates change in operating weight. It occurs only within the window defined in Fig. 3.1 (Eq. 3.2.2.1). Eq. 3.2.2.2b enforces the condition that the total payload travel (in kN-km) is conserved during load-shifting since the total amount of freight carried remains constant, that is, TTGVW,AS(GVWAS − TAREAS) = TT ′GVWk,BC(GVWk,BC − TAREBC) (3.2.2.3) Note that when PMGVWAS is greater than PMGVWBC repre- senting an increase in weight limit, the total amount of truck traffic will decrease since fewer trips will be required to transport the same amount of freight (payload). It also should be noted that possible payload changes are covered in Sec- tion 3.2.2.3 addressing external factors, such as economy- growth-dependent payload increase and competition-induced payload shift from or to rail. In applying these equations, GVWBC is taken at the mid- point of a weight interval falling in the window defined in Eq. 3.2.2.1 (Fig. 3.1). Consequently, the value of GVWAS accord- ing to Eq. 3.2.2.2a generally will not match the midpoint of a weight interval. It is then appropriate to distribute TTGVW,AS between two neighboring weight intervals to achieve the desired value of GVWAS, which are designated as the i th and the i + 1 th intervals, respectively. The distribution ratios pi and pi + 1 for the i th and the i + 1 th weight intervals are required to satisfy the following equations: pi + pi + 1 = 1 (3.2.2.4) pi GVWi,AS + pi + 1 GVWi + 1,AS = GVWAS (3.2.2.5) Then the truck traffic equal to piTTGVW,AS is to be moved to the i th GVW interval and pi + 1 TTGVW,AS to the i + 1 th interval. For example, assume that 25 kN-increments are used for defining weight intervals. Use PMGVWAS = 356 kN (80 kips) and PMGVWBC = 326kN (73.28 kips) to express the legal weight limit change for some states in the 1970s and 80s. Typically under this GVW limit change, 3-S2 trucks could increase their weights to the new limit of 356 kN without changing their configurations. For GVWk,BC = 312.5 kN repre- senting a weight range from 300 to 325 kN, GVWAS is equal to 341.3 kN according to Eq. 3.2.2.2a. TT340.6,AS computed using Eq.3.2.2.2b will then be distributed between the weight inter- vals 325 to 350 kN (with midpoint GVWi,AS equal to 337.5 kN) 0 100% 100% GVWBC/PMGVW BC Percent of Traffic To Be Shifted a1 a2 b1 b2 c Figure 3.1. Window for truck traffic shifting.

and 350 to 375 kN (with midpoint GVWi + 1,AS equal to 362.5 kN). The distribution ratios pi and pi + 1 respectively are 85 per- cent and 15 percent, by satisfying Eqs. 3.2.2.4 and 3.2.2.5. The following assumptions have been used for the pro- posed method. (A) Not all truck traffic is weight limited. For many commodities (e.g., potato chips), the cubic capacity of the truck is the limiting factor. (B) Heavier trucks exces- sively above PMGVWBC and operating under special permits may not react to weight-limit changes if other factors (e.g., the permit fee charge system) do not change. (C) The total payload traveled (in kN-km) remains the same before and after the weight limit change, i.e., Payload (in kN) × Distance of Travel (in km) = Constant (3.2.2.6) Eq. 3.2.2.6 has been expressed in Eq. 3.2.2.2b, and the distri- bution of this traffic over the truck-weight intervals is altered because of shifting. It is also important to note that, truck trans- portation is influenced by many factors. Therefore, selecting these parameters a1, a2, b1, b2, and c for the proposed method may require measured data and appropriate engineering judg- ment. For example, for trucks operating in multiple states, the PMGVW is generally controlled by limits in the most restric- tive state. It may be different from PMGVW for trucks oper- ating in a limited area where truck weight limits are uniform, which could dictate the selection of these parameters. 3.2.2.2 Truck-Type Shift The same equations as Eqs. 3.2.2.1 and 3.2.2.2 used for truck load shifting are recommended to be used for truck- type shifting. However, TTGVWk,BC, TT ′GVWk,BC, PMGVWBC and TAREBC now refer to the truck type from which traffic is shifted, and TTGVW,AS, PMGVWAS, and TAREAS refer to the truck type to which traffic is shifted. For example, assume again 25-kN (5.6-kips) increments for weight intervals. Consider the scenario of GVW weight- limit increase from 356 kN (80 kips) as PMGVWBC to 431 kN (97 kips) as PMGVWAS. The 3-S2 trucks controlled by the current weight limit 356 kN (80 kips) would need to change to 3-S3 configurations to add weight and also satisfy other requirements, such as the wheel weight limits with consider- ation to pavement fatigue. This would cause truck-type shift as a result of the proposed truck-weight-limit change. For GVWk,BC = 362.5 kN representing a weight interval between 350 and 375 kN (on 5-axles), GVWAS is found to be 438.9 kN according to Eq. 3.2.2.2a and it will have to be carried by 6 axles. TT439.5,AS computed using Eq. 3.2.2.2b will then be distributed between two weight intervals: (1) 425 to 450 kN (with midpoint GVWi,AS equal to 437.5 kN) and (2) 450 to 475 kN (with midpoint GVWi + 1,AS equal to 462.5 kN). The distribution ratios pi and pi + 1, respectively, are 94.5 per- cent for the former and 5.5 percent for the latter, by satisfying Eqs. 3.2.2.4 and 3.2.2.5. Note again that the both intervals refer to 6-axle trucks now. 29 3.2.2.3 Exogenous Shift Exogenous shifts here refer to those changes to TWHs due to external factors, instead of those between weight intervals (truck load shifts) and between different truck types or con- figurations (truck type shifts). The influencing factors may be, for example, economic growth or competitiveness with other transportation modes (e.g., rail). Cambridge Systematics et al. (1997) and USDOT (1999) provide detailed discussions on transportation modal shifts for freight demand predictions. The guidelines presented there help in understanding relevant issues and in estimating the amount of truck traffic change. The first step of accounting for these effects is to identify the traffic in the TWHs that is subject to exogenous shift. This can be approached in the same way as it was in Section 3.2.2.1 using a window, although the window needs to be specifically defined according to the situation. For the case of overall eco- nomic growth as a likely example, all traffic should be sub- ject to change, unless otherwise objected. This may be read- ily taken into account by using a growth factor to be applied to all traffic. Equivalently, this can be done to the total traffic for bridge related analyses: ADTTAS = g ADTTBC (3.2.2.7) where g is the growth factor, which could be estimated based on data at the network level. For the case of transportation modal change due to truck- weight-limit changes, it would be reasonable to use the same window in Fig. 3.1 for identifying the impacted traffic. In addition, a multiplier r can be applied to the affected traffic at weight GVW: TTGVW,AS = rGVWk TT ′GVWk,BC (GVWk,BC − TAREBC)/(GVWAS − TAREAS) (3.2.2.8) As indicated, rGVWk can be a function of operating weight GVW at the kth interval. The multiplier is higher than 1.0 for traffic increase and less than 1.0 for decrease. Note that this case of exogenous shift may be likely accompanied by the other two kinds of load-shift. Thus, Eqs. 3.2.2.1 to 3.2.2.5 will be simultaneously applicable. Further, Eq. 3.2.2.7 can be viewed as a special case of Eq. 3.2.2.8 if we set g = r = con- stant and understand that all traffic is subject to this change. 3.2.2.4 Adjustment of Empty Truck Traffic Empty truck traffic here refers to the traffic of trucks with no or little payload. Theoretically, this amount of traffic needs to be adjusted for each affected truck type, depending on how much empty truck traffic will be changed as a result of the above three types of load shift. Specifically, an empty-to- loaded ratio rE/L can be used for identifying this amount of

traffic subject to adjustment. These changes should be made to the intervals surrounding the tare weight, as follows ∆TTTARE − GVW,BC = −rE/L TT ′GVWk,BC (3.2.2.9) ∆TTTARE − GVW,AS = +rE/L TTGVWi,AS (3.2.2.10) where TT ′GVWk,BC and TTGVWi,AS are, respectively, the reduced and increased traffic amounts. GVW for these two traffic amounts is different as indicated because of the weight-limit change modeled by Eq. 3.2.2.2a. Accordingly, ∆TTTARE− GVW,BC and ∆TTTARE − GVW,AS are the decreased and increased empty truck traffic amounts at the tare weights corresponding to the respective GVWs. When these traffic amounts are identi- fied, their distribution to neighboring weight intervals can be done in the same way as in Eqs. 3.2.2.4 and 3.2.2.5. It is of interest to note that adjustment for empty truck traffic may become insignificant with respect to analyses for bridge strength and fatigue, depending on the magnitude of tare weight because this adjustment is concerned with the lower end of the TWH and bridge strength and fatigue are more rel- evant to the higher end of the TWH. 3.2.3 Testing Examples for the Recommended Method Example 1: Effect of Legal Weight-Limit Change on TWH The legal GVW limit in Arkansas was increased from 326 kN (73.28 kips) to 356 kN (80 kips) in 1983, when a num- 30 ber of states did the same to be in accordance with each other. Truck weight data for 1981 and 1986 are used here respec- tively as before (Base Case) and after (Alternative Scenario) situations, to be sure that the effects of this weight-limit change had fully developed, because it may take some time for truckers to be prepared for a significant change to their operation. These data were acquired at weigh stations over the State, provided by the Arkansas Highway and Trans- portation Department. Such data are considered reliable for weighed axles (and thus GVW as the sum of axle weights). Compared with WIM data available now but not available at the time, weigh station data may miss some overloaded trucks. Nevertheless, for the purpose of testing and illustrat- ing application of the proposed method for predicting TWHs, these data are judged to be adequate. Only 5-axle (mainly 3-S2) trucks are considered in this test because of the following reasons: (1) The available data include a statistically significant number of 5-axle trucks weighed, but only a few trucks of other types. (2) A vast major- ity of trucks are of this type, traveling in the State’s (and in the entire country’s) highway system. (3) This truck-group’s behavior was expected to change because of the weight-limit change. Fig. 3.2 shows two TWHs based on the measured data in 1981 and 1986. Note that the peak of heavy weights shifted from the 300–325 kN (67–73 kips) interval in 1981 to the 325–350 kN (73–79 kips) interval in 1986. It also should be noted that the 125–150 kN (28–34 kips) interval has a notice- ably higher frequency in 1981 than in 1986. It appears that this was caused by the Motor Carrier Act of 1980 (TRB 1990a). Figure 3.2. Comparison of TWHs for Arkansas: 1981 and 1986 measured (Example 1).

The Act significantly altered the trucking industry by allow- ing carriers to increase their service territories and the types of commodities they could transport. This deregulation greatly decreased the amount of empty backhaul traffic by allowing trucks to obtain loads for the backhaul portion of their trip, instead of returning to their operation base empty. As a result, fewer empty trips were shown in the 1986 TWH in the 125–150 kN (28-34 kips) interval. Fig. 3.3 shows the predicted TWH as a result of the weight-limit change, by applying the proposed method using a1 = a2 = 10%, b1 = b2 = 20%, c = 95%, and rE/L = 0.2 in Eqs. 3.2.2.1, 3.2.2.9, and 3.2.2.10. These parameters are selected based on review of previously reported data in the literature and more recent data studied in the FHWA’s truck size and weight study (USDOT 1998). Only truck-load-shift is con- sidered because only 5-axle (mainly 3-S2) vehicles were expected to react to the limit change. The 1986 measured data are also shown in the same figure for comparison. It is seen that the heavy-truck peak’s shift is clearly captured by the proposed method—from the 300–325 kN (73-79 kips) inter- val to the 325–350 kN (67-73 kips) interval. It is apparent that the light-truck peaks at the 100–125 kN interval in the two TWHs are noticeably different, not due to the truck-weight- limit change as commented on above. 31 For steel bridge fatigue evaluation, TWHs are used as a load spectrum. According to the AASHTO procedure (1990), an equivalent truck weight is calculated based on equivalence in fatigue damage (Moses et al. 1987): Weqv = (Σ f i GVW i3)1/3 (3.2.3.1) where GVWi is the GVW for interval i in the histogram (taken as the midpoint), and f i is its frequency. For this test example, Weqv is calculated using the predicted and measured TWHs for comparison. They agree with each other fairly well, as shown in Table 3.1. Example 2: Effect of Overweight-Permit Weight-Limit Change on TWH In July 1998, Idaho Transportation Department (ITD) launched a pilot project of lifting its 467-kN (105-kips) limit for annual overweight permits to 574 kN (129 kips). The vehi- cle configurations are still restricted according to the Bridge Formula. Under this change, these overloads are permissible only on two specific routes in the State. For all possibly impacted truck types, the new permit weight limit requires some changes to the vehicle configuration based on the Bridge Figure 3.3. Comparison of TWHs for Arkansas: predicted vs. measured (Example 1).

Formula. It should be noted that typically permit weight- limit changes might affect only a small fraction of the total truck traffic. WIM data were obtained before the weight limit change in 1997 and after the change in 1998 and 1999. The 1997 data are used as the Base Case for predicting the TWH under the new permit weight limit. Based on the registered permits and conversations with the registered trucking companies, it is assumed that only those trucks with more than 5 axles may respond to this limit change. Further, the PMGVWAS for 6-, 7-, 8-, and 9-axle vehicles are set respectively at 512 kN (115 kips), 529 kN (119 kips), 552 kN (124 kips), and 574 kN (129 kips), according to the Bridge Formula. The recom- mended default window parameters are used: a1 = a2 = 10%; b1 = b2 = 20%, c = 95%, and rE/L = 0.2. Table 3.2 shows com- parison of the resulting equivalent truck weight for steel bridge fatigue evaluation according to the AASHTO specifications (1990). The results based on the predicted and measured TWHs are very close to each other. Note that this example includes truck-type shifts. Exogenous shifts have been con- sidered to be negligible if any, because the new permit limit is only applicable to limited routes. 3.2.4 An Illustrative Example of Predicting TWH An example of lifting the legal GVW limit of 80 kips to 97 kips is used here for illustration. Under this scenario, the axle weight limits will not change (i.e., single-axle weight up to 20 kips and tandem axle weight up to 34 kips). Note that this is one of the realistic scenarios that have been debated and investigated in previous studies. In those scenarios, change to axle weight limits has not been an option. 32 As a first step for this example, the types (configurations) of truck that will be affected need to be identified. This iden- tification requires knowledge of truck weight and size regu- lations. Based on this knowledge, it is then possible to approx- imate the trend of change in trucking behavior. For this example, the up limit weight of 80 kips is typically loaded on 5 axles in a 3S2 configuration. However, this configuration is not allowed to carry a 97-kip GVW. At least one additional axle will need to be added to the 3S2 configuration for this new legal weight of 97 kips. Based on this, it is determined that the truck-weight-limit change considered here is expected to affect the behavior of 3S2 vehicles (with an up limit of 80 kips) and an additional axle is needed to carry the new weight limit of 97 kips. It is also seen that a structural engi- neering background would not be adequate in making such a determination. If the user of the methodology happens to be a structural/bridge engineer without any knowledge of com- mercial vehicle regulations on weight and size, assistance will be needed from those who have such a background. The following steps are carried out to predict TWH for the alternative scenario. They use the VMT data under the Base Case as shown in Table 3.3, where each GVWk,BC interval is designated using its mid-interval value of GVW. Note that the raw VMT data in Table 3.3 needs to be normalized to obtain the TWH for the Base Case, as shown in Table 3.4. Its last col- umn shows the TWH including all the truck types, with the frequencies for all GVWk,BC intervals summed to unity. Step A. Quantitatively determine the PMGVW and TARE for the affected truck types under both the Base Case (BC) and the Alternative Scenario (AS). Based on above discussion, only the 3S2T and 3S2S trucks (both with a 3S2 configuration) are considered to have a traf- TABLE 3.1 Comparison of equivalent truck weights Weqv for Example 1 (Arkansas) (1) (2) (3) (4) Base Case Alternative Alternative Error Scenario Scenario (2)/(3)-1 (Predicted using (1986 measured) proposed method) Weqv in kN (kips) 246 (55.3) 258 (58.0) 273 (61.4) -5.5% TABLE 3.2 Comparison of equivalent truck weights Weqv for Example 2 (Idaho) (1) (2) (3) (3) Base Case Alternative Alternative Error Scenario Scenario (2)/(3)-1 (Predicted using (Average of proposed method) 1998 and 1999 measured) . Weqv in kN (kips) 302 (67.8) 319 (71.6) 313 (70.4) +1.8%

fic amount to shift away, and the CS6 trucks (with an 3S3 con- figuration) will receive an additional traffic amount. Namely, as a result of this weight-limit change, the 3S2T and 3S2S truck traffic will reduce and the CS6 truck traffic will increase. The normalized traffic amounts for these three truck types are shown in Table 3.4 as shaded. The PMGVW and TARE are determined as PMGVWBC = 80 kips, TAREBC = 30 kips, PMGVWAS = 97 kips, and TAREAS = 35 kips. Note that the user of the recommended methodology needs to provide these values for the specific Base Case and Alternative Scenario under consideration. This determination of PMGVW and TARE is also based on knowledge about trucking behavior for these truck types. The tare weight for BC may be obtained using measurement data of tare weight of the trucks. For example, WIM data may be used to extract such information. 3S2 weight data usually show a bi-modal behavior. The peak at the lower end typically represents trucks at their tare weight. The data around this peak may be used to estimate 3S2 truck tare weights. The tare weight for AS may be estimated using the tare weight for BC with an additional amount to cover the additional axle(s). For this example, the additional axle is estimated at 5 kips. The PMGVW values in this example are 33 taken as the weight up limits for the respective configura- tions (80 kips for the 3S2 configuration and 97 kips for the 3S3 configuration). This actually assumes that the weight limits can be realized with other constraints irrelevant. For example, these other restraints can be those for length and/or width. Namely, for some special types (configurations) of vehicles, the up limits of weight may not be realizable because other limits are applicable. Again these determina- tions require knowledge on the size and weight limits of trucks. Appropriate personnel may need to be consulted on these issues if the user happens to have no background in these areas. Step B. Determine the window parameters for shifting, as defined in Fig. 3.1. Note that in Fig. 3.1 c is the maximum percentage of the traffic to shift from the impacted truck types, i.e., the 3S2T and 3S2S in this example. Namely, for the traffic at a GVWk,BC equal to PMGVWBC (i.e., for GVWk,BC = PMGVWBC = 80 kips), 95% of the traffic is predicted to become a new amount of traffic at a new GVW under the Alternative Scenario. a1 and a2 indicate a range (i.e., a1 + a2) where c will be applied. In TABLE 3.3 VMT data as input for predicting TWH under alternative scenario GVWk,BC SU3 SU4 CS3 CS4 3S2T 3S2S CS6 CS7 CT4 CT5 CT6 DS5 DS6 DS7 DS8 TRP 2.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7.5 0 0 0.01 0 0 0 0 0 0.081 0 0 0 0 0 0 0 12.5 0 0 0.16 2.05 0 0 0 0 0.4185 0.015 0 0 0 0 0 0 17.5 0.8 0.048 0.91 1.37 0.0671 0.011 0.08 0 0.4861 0.177 0.033 0 3E-04 0 0 0 22.5 3.14 0.488 1.66 2.79 0.2014 0.102 0.13 0 0.7126 0.147 0.052 0.034 0.001 0.05 0 0 27.5 3.57 1.922 2.62 3 0.5369 0.116 0.31 0 0.8251 0.276 0.083 0.159 0.004 0.064 0 0 32.5 3.38 2.181 2.08 3.11 2.8861 0.122 1.41 0.012 0.8191 0.365 0.151 0.347 0.01 0.057 0 0 37.5 2.83 2.066 1.35 3.26 4.2284 0.107 1.23 0.056 0.5266 0.39 0.168 0.534 0.01 0.127 0 0 42.5 2.11 0.621 1.01 3.32 4.6983 0.093 1.18 0.164 0.3315 0.416 0.211 0.642 0.019 0.113 0.6395 0 47.5 2.63 0.859 0.71 3.26 5.7722 0.078 0.77 0.229 0.1575 0.291 0.157 0.699 0.017 0.163 0.7817 0 52.5 2.16 1.575 0.22 3.11 5.9064 0.087 0.57 0.309 0.06 0.32 0.163 0.585 0.018 0.198 0.7106 0 57.5 1.58 1.981 0.13 1.63 8.0542 0.096 0.4 0.201 0.042 0.309 0.153 0.466 0.018 0.156 1.5633 0 62.5 0.72 2.244 0.02 1.05 6.7118 0.107 0.41 0.124 0.0345 0.328 0.161 0.489 0.014 0.163 1.2791 0 67.5 0.48 2.029 0 0.32 6.1749 0.104 0.51 0.084 0.015 0.365 0.07 0.295 0.017 0.205 1.4212 0 72.5 0.05 1.504 0 0.37 4.9667 0.076 0.72 0.072 0.009 0.298 0.079 0.222 0.019 0.149 0.7817 0 77.5 0 0.788 0 0.11 4.4969 0.069 1.13 0.056 0 0.144 0.056 0.114 0.013 0.255 0.4264 0 82.5 0 0.382 0 0.05 5.5037 0.06 1.57 0.036 0 0.158 0.027 0.148 0.008 0.205 1.0659 0 87.5 0 0.239 0 0.11 7.6515 0.044 2.17 0.04 0 0.147 0.035 0.057 0.004 0.234 1.5633 0 92.5 0 0.119 0 0 5.705 0.042 2.7 0.116 0 0.151 0.01 0.017 0.002 0.156 1.137 0 97.5 0 0.048 0 0 2.6847 0.031 2.08 0.104 0 0.059 0.014 0.023 9E-04 0.191 1.7055 0 102.5 0 0 0 0 1.3424 0.007 1.2 0.12 0 0.029 0.006 0 3E-04 0.134 1.3502 0 107.5 0 0 0 0 0.6712 0.004 0.7 0.048 0 0.004 0.006 0 6E-04 0.092 0.9948 0 112.5 0 0 0 0 0 0.007 0.44 0.068 0 0 0.004 0 0 0.071 0.4974 0 117.5 0 0 0 0 0 0 0.23 0.008 0 0 0.008 0 2E-04 0.085 0.4974 0 122.5 0 0 0 0 0 0 0.09 0.008 0 0 0.002 0 0 0.057 0.6395 0 127.5 0 0 0 0 0 0 0.08 0.004 0 0 0 0 0 0.035 0.7106 0 132.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0.05 0.9238 0 137.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0.042 0.4974 0 142.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0.021 0.2132 0 147.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0.028 1.0659 0 Total 23.4 19.1 10.9 28.9 78.3 1.4 20.1 1.9 4.5 4.4 1.6 4.8 0.2 3.1 20.5 0

TABLE 3.4 Normalized VMT data from Table 3.3 and summed to TWH,BC as last column (impacted truck types 3S2T and 3S2S (shift from) and CS6 (shift to) are shaded) GVWk,BC SU3 SU4 CS3 CS4 3S2T 3S2S CS6 CS7 CT4 CT5 CT6 2.5 0 0 0 0 0 0 0 0 0 0 0 7.5 0 0 3.62E-05 0 0 0 0 0 0.000363 0 0 12.5 0 0 0.000724 0.009208 0 0 0 0 0.001877 6.61E-05 0 17.5 0.003578 0.000214 0.004088 0.006139 0.0003009 4.983E-05 0.000363 0 0.002179 0.000793 0.000148 22.5 0.0141 0.002187 0.007453 0.012513 0.0009028 0.0004584 0.0005704 0 0.003195 0.000661 0.000234 27.5 0.015994 0.008619 0.011758 0.013458 0.0024074 0.0005182 0.0014 0 0.003699 0.001239 0.000373 32.5 0.015152 0.009777 0.009334 0.01393 0.0129396 0.0005481 0.0063257 5.4E-05 0.003672 0.001635 0.000677 37.5 0.01267 0.009262 0.006042 0.014638 0.018958 0.0004784 0.0054961 0.000252 0.002361 0.001751 0.000755 42.5 0.009449 0.002782 0.004522 0.014874 0.0210645 0.0004186 0.0052887 0.000737 0.001486 0.001866 0.000946 47.5 0.011811 0.003852 0.003184 0.014638 0.0258792 0.0003488 0.003448 0.001025 0.000706 0.001305 0.000703 52.5 0.009663 0.007062 0.000977 0.01393 0.026481 0.0003887 0.0025407 0.001385 0.000269 0.001437 0.000729 57.5 0.007087 0.008882 0.000579 0.007319 0.0361105 0.0004285 0.0017888 0.000899 0.000188 0.001387 0.000686 62.5 0.003221 0.010059 0.000109 0.004722 0.0300921 0.0004784 0.0018407 0.000558 0.000155 0.00147 0.000721 67.5 0.002147 0.009096 0 0.001417 0.0276847 0.0004684 0.0022814 0.000378 6.73E-05 0.001635 0.000313 72.5 0.000215 0.006741 0 0.001653 0.0222681 0.0003388 0.0032406 0.000324 4.04E-05 0.001338 0.000356 77.5 0 0.003531 0 0.000472 0.0201617 0.0003089 0.0050813 0.000252 0 0.000644 0.000252 82.5 0 0.001712 0 0.000236 0.0246755 0.0002691 0.0070257 0.000162 0 0.00071 0.000122 87.5 0 0.00107 0 0.000472 0.034305 0.0001993 0.0097219 0.00018 0 0.000661 0.000156 92.5 0 0.000535 0 0 0.0255783 0.0001893 0.012107 0.000522 0 0.000677 4.34E-05 97.5 0 0.000214 0 0 0.0120368 0.0001395 0.009333 0.000468 0 0.000264 6.08E-05 102.5 0 0 0 0 0.0060184 2.99E-05 0.0053924 0.00054 0 0.000132 2.6E-05 107.5 0 0 0 0 0.0030092 1.993E-05 0.0031369 0.000216 0 1.65E-05 2.6E-05 112.5 0 0 0 0 0 2.99E-05 0.0019703 0.000306 0 0 1.74E-05 117.5 0 0 0 0 0 0 0.001037 3.6E-05 0 0 3.47E-05 122.5 0 0 0 0 0 0 0.0003889 3.6E-05 0 0 8.68E-06 127.5 0 0 0 0 0 0 0.000363 1.8E-05 0 0 0 132.5 0 0 0 0 0 0 0 0 0 0 0 137.5 0 0 0 0 0 0 0 0 0 0 0 142.5 0 0 0 0 0 0 0 0 0 0 0 147.5 0 0 0 0 0 0 0 0 0 0 0 Total DS5 DS6 DS7 DS8 TRP TWH,BC 0 0 0 0 0 0 0 0 0 0 0 0.000399 0 0 0 0 0 0.011874 0 1.39E-06 0 0 0 0.017853 0.000153 5.58E-06 0.000222 0 0 0.042655 0.000713 1.81E-05 0.000286 0 0 0.060482 0.001554 4.46E-05 0.000254 0 0 0.075898 0.002395 4.39E-05 0.000571 0 0 0.075673 0.002879 8.72E-05 0.000508 0.002867 0 0.069777 0.003134 7.67E-05 0.00073 0.003505 0 0.074345 0.002624 7.88E-05 0.000888 0.003186 0 0.07164 0.002089 7.88E-05 0.000698 0.007009 0 0.07523 0.002191 6.28E-05 0.00073 0.005735 0 0.062143 0.001325 7.46E-05 0.00092 0.006372 0 0.054178 0.000994 8.51E-05 0.000666 0.003505 0 0.041764 0.00051 5.65E-05 0.001142 0.001912 0 0.034323 0.000662 3.49E-05 0.00092 0.004779 0 0.041308 0.000255 1.74E-05 0.001047 0.007009 0 0.055094 7.64E-05 6.97E-06 0.000698 0.005098 0 0.045531 0.000102 4.18E-06 0.000857 0.007646 0 0.031125 0 1.39E-06 0.000603 0.006053 0 0.018796 0 2.79E-06 0.000412 0.00446 0 0.0113 0 0 0.000317 0.00223 0 0.004871 0 6.98E-07 0.000381 0.00223 0 0.003719 0 0 0.000254 0.002867 0 0.003555 0 0 0.000159 0.003186 0 0.003726 0 0 0.000222 0.004142 0 0.004364 0 0 0.00019 0.00223 0 0.002421 0 0 9.52E-05 0.000956 0 0.001051 0 0 0.000127 0.004779 0 0.004906

addition, b1 and b2 indicate another range (i.e., b1 + b2) where there is an impact. Namely, beyond this range there will not be any impact and the traffic will remain the same. The default values for these window parameters are used here: c = 95%, a1 = a2 = 10%, and b1 = b2 = 20%. Note that these default values have been tested using available data, as discussed in this report. However, the user may select other values if warranted. On the other hand, without rigorous research the user will likely use these default values. Such rigorous research will require measured truck weight data before and after the implementation of weight limit change. In addition, the data need to be gathered according to appro- priate design so that comparison can be made. For example, the sites selected need to be identical or compatible to avoid site-dependent issues; the time periods for data collection before and after the weight limit change also need to be com- patible so that seasonable changes in truck weights will not adversely affect the comparison, etc. Step C. Perform shifting, which includes the following substeps. In concept, this step starts from the TWH of the truck type under the Base Case that has been identified to shift traffic away. According to the shifting window parameters (c, a1, a2, b1, and b2), each weight interval’s traffic is examined to deter- mine what fraction of it will shift away to which weight inter- val under the Alternative Scenario. Thus, this step ends with a TWH of the truck type receiving traffic under the Alterna- tive Scenario and a residual TWH of the truck type shifting traffic away. The latter becomes the TWH of the same truck type under the Alternative Scenario. These steps are discussed in detail now. (i) Identify the intervals GVWk,BC that will have traffic to shift away, as well as their traffic amounts. (ii) Determine the intervals GVWi,AS and GVWi + 1,AS that will receive new traffic, as well as the amounts of the new traffic. Note that these traffic amounts are different from the traffic amounts shifting away because the new truck type (CS6) is allowed to carry higher payload. Tables 3.5 and 3.6 show the process of this shifting, respectively for 3S2T and 3S2S trucks. Table 3.7 shows the resulting TWH in the last col- umn as the predicted TWH for AS. The process is further detailed next. For Substep (i): For example, for interval GVWk,BC = 62.5 kips: GVWk,BC/PMGVWBC = 62.5/80 = 0.78125 which is out of the area defined by 1 − b1 = 0.8 and 1 + b2 = 1.2. Thus no shift- ing will occur, or the percentage of traffic to be shifted away is zero, as shown in column TT ′GVWk,BC in Table 3.5. For the next interval GVWk,BC = 67.5 kips: 35 GVWk,BC /PMGVWBC = 67.5/80 = 0.84375 which is within the area defined by 1 − b1 = 0.8 and 1 + b2 = 1.2. Thus a frac- tion of this interval’s traffic will shift, as determined next. This fraction is calculated according to Eq. 3.2.2.1 = c(GVWk,BC/PMGVWBC − 1 + b1)/(b1 − a1) = 0.95(.84375 − 1 + .2)(.2 − .1) = .41563, as shown in column “window-f” in Table 3.5. Thus, according to Eq. 3.2.2.1, TT ′GVWk,BC = (.41563)TTGVWk,BC = (.41563)(0.0276847) = 0.01151, as shown in column “TT ′GVWk,BC” of Table 3.5. The rest of the operating weight (GVWk,BC) intervals are treated in the same way as illustrated. The results are given in the columns “window-f” and “TT ′GVWk,BC” in Table 3.5. For Substep (ii): The following calculation is carried out only for those GVWk,BC intervals that have a traffic amount to shift away. For interval GVWk,BC = 67.5 kips: According to Eq. 3.2.2.2a, GVWAS = GVWk,BC (PMGVWAS/ PMGVWBC) = (67.5)(97)/80 = 81.844 kips (column “GVWAS” in Fig. 3.5). According to Eq. 3.2.2.2b, TTGVW,AS = TT ′GVWk,BC (GVWk,BC − TAREBC)/(GVWAS − TAREAS) = 0.01151 (67.5 − 30)/(81.844 − 35) = 0.00921 (column “TTGVW,AS” in Table 3.5). Determine the ratio of distributing TTGVW,AS = 0.00921 to two weight intervals GVWi,AS = 77.5 kips and GVWi + 1,AS = 82.5 kips (because GVWAS = 81.844 is between these two values). It is done via solving Eqs. 3.2.2.4 and 3.2.2.5 for pi and pi + 1. They are, respectively, 0.13125 and 0.86875 as shown in the column “pi and pi + 1 for GVWAS = 81.844 Eq. 3.2.2.4 and Eq. 3.2.2.5” in Table 3.5. Then perform the distribution: pi TTGVW,AS = (0.13125) (0.00921) = 0.00121 is obtained as shown in column “TTGVW,AS” in Table 3.5 for interval GVWAS = 77.5 kips. pi + 1 TTGVW,AS = (0.86875)(0.00921) = 0.00800 is obtained as shown in column “TTGVW,AS” in Table 3.5 for interval GVWAS = 82.5 kips. The rest of the impacted weight intervals are treated in the same way as illustrated. Note that one weight interval under the alternative scenario GVWj,AS may receive traffic amounts from two adjacent weight intervals of GVWk,BC. This is because each weight interval GVWk,BC shifts traffic to two weight intervals: GVWi,AS and GVWi + 1,AS. Also note that the same calculation is done for 3S2S trucks as shown in Table 3.6 in the same format.

36 TABLE 3.5 3S2T shifting calculations and results Pi,Pi+1for Pi,Pi+1for Pi,Pi+1for Pi,Pi+1for Pi,Pi+1for Pi,Pi+1for GVWAS= GVWAS= GVWAS= GVWAS= GVWAS= GVWAS= GVWk,BC TTGVWK,BC window-f TT'GVWK, GVW AS TTGVW,As81.844 87.906 93.969 100.031 106.094 112.156 TTGVWAS (column 3S2T Eq.3.2.2.1 Eq.3.2.2.2aEq.3.2.2.2bEq.3.2.2.4 Eq.3.2.2.4 Eq.3.2.2.4 Eq.3.2.2.4 Eq.3.2.2.4 Eq.3.2.2.4 of Table 3.4) Eq.3.2.2.5 Eq.3.2.2.5 Eq.3.2.2.5 Eq.3.2.2.5 Eq.3.2.2.5 Eq.3.2.2.5 2.5 0.00000 7.5 0.00000 12.5 0.00000 17.5 0.00030 22.5 0.00090 27.5 0.00241 32.5 0.01294 37.5 0.01896 -0.00230 42.5 0.02106 -0.00423 +0.000241 47.5 0.02588 -0.00383 +0.001600 52.5 0.02648 -0.00469 +0.003122 57.5 0.03611 -0.00652 +0.002455 62.5 0.03009 -0.00213 +0.002775 67.5 0.02768 0.415625 0.01151 81.844 0.00921 +0.003399 72.5 0.02227 0.95 0.02115 87.906 0.01699 +0.003908 77.5 0.02016 0.95 0.01915 93.969 0.01543 0.13125 +0.001604+ 82.5 0.02468 0.95 0.02344 100.031 0.01892 0.86875 0.00800 87.5 0.03430 0.95 0.03259 106.094 0.02636 0.91875 0.01561 92.5 0.02558 0.415625 0.01063 112.156 0.00861 0.08125 0.70625 0.01228 97.5 0.01204 0.29375 0.49375 0.01388 102.5 0.00602 0.50625 0.28125 0.01699 107.5 0.00301 0.71875 0.06875 0.01954 112.5 0.00000 0.93125 0.00802 117.5 0.00000 122.5 0.00000 127.5 0.00000 132.5 0.00000 137.5 0.00000 142.5 0.00000 147.5 0.00000 Total 0.35087 0.14218 0.09553 0.11463 GVWj,AS 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 72.5 77.5 82.5 87.5 92.5 97.5 102.5 107.5 112.5 117.5 122.5 127.5 132.5 137.5 142.5 147.5

37 TABLE 3.6 3S2S shifting calculations and results Pi,Pi+1for Pi,Pi+1for Pi,Pi+1for Pi,Pi+1for Pi,Pi+1for Pi,Pi+1for GVWAS= GVWAS= GVWAS= GVWAS= GVWAS= GVWAS= GVWk,BC TTGVWK,BC window-f TT'GVWK,BC GVW AS TTGVW,As81.844 87.906 93.969 100.031 106.094 112.156 (column 3S2S Eq.3.2.2.1 Eq.3.2.2.2aEq.3.2.2.2bEq.3.2.2.4 Eq.3.2.2.4 Eq.3.2.2.4 Eq.3.2.2.4 Eq.3.2.2.4 Eq.3.2.2.4 of Table 3.4) Eq.3.2.2.5 Eq.3.2.2.5 Eq.3.2.2.5 Eq.3.2.2.5 Eq.3.2.2.5 Eq.3.2.2.5 2.5 0.000E+00 7.5 0.000E+00 12.5 0.000E+00 17.5 4.983E-05 22.5 4.584E-04 27.5 5.182E-04 32.5 5.481E-04 37.5 4.784E-04 -3.893E-05 42.5 4.186E-04 -6.438E-05 47.5 3.488E-04 -5.870E-05 52.5 3.887E-04 -5.112E-05 57.5 4.285E-04 -3.787E-05 62.5 4.784E-04 -1.574E-05 67.5 4.684E-04 0.415625 1.947E-04 81.844 1.558E-04 72.5 3.388E-04 0.95 3.219E-04 87.906 2.586E-04 77.5 3.089E-04 0.95 2.935E-04 93.969 2.364E-04 0.13125 82.5 2.691E-04 0.95 2.556E-04 100.031 2.064E-04 0.86875 87.5 1.993E-04 0.95 1.893E-04 106.094 1.531E-04 0.91875 92.5 1.893E-04 0.415625 7.870E-05 112.156 6.375E-05 0.08125 0.70625 97.5 1.395E-04 0.29375 0.49375 102.5 2.990E-05 0.50625 0.28125 0 107.5 1.993E-05 0.71875 0.06875 112.5 2.990E-05 0.93125 117.5 0.000E+00 122.5 0.000E+00 127.5 0.000E+00 132.5 0.000E+00 137.5 0.000E+00 142.5 0.000E+00 147.5 0.000E+00 Total 0.00610901 0.001601 0.001074 TTGVWAS GVWj,AS 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 +4.091E-06 42.5 +2.708E-05 47.5 +4.751E-05 52.5 +3.759E-05 57.5 +3.427E-05 62.5 +2.951E-05 67.5 +2.289E-05 72.5 +1.187E-05 77.5 1.354E-04 82.5 2.376E-04 87.5 1.880E-04 92.5 1.713E-04 97.5 1.475E-04 102.5 1.145E-04 107.5 5.937E-05 112.5 117.5 122.5 127.5 132.5 137.5 142.5 147.5 1.289E-03

38 TABLE 3.7 Predicted TWH under alternative scenario (non-normalized) GVWj,ASSU3 SU4 CS3 CS4 3S2T 3S2S CS6 CS7 CT4 CT5 CT6 DS5 2.5 0 0 0 0 0 0 0 0 0 0 0 0 7.5 0 0 0.00807 0 0 0 0 0 0.00036 0 0 0 12.5 0 0 0.16138 2.05375 0 0 0 0 0.00188 6.6E-05 0 0 17.5 0.79797 0.04773 0.91182 1.36916 0.0003 5E-05 0.00036 0 0.00218 0.00079 0.00015 0 22.5 3.14492 0.48778 1.66225 2.79099 0.0009 0.00046 0.00057 0 0.00319 0.00066 0.00023 0.00015 27.5 3.56738 1.92241 2.62248 3.00163 0.00241 0.00052 0.0014 0 0.0037 0.00124 0.00037 0.00071 32.5 3.37962 2.18065 2.08184 3.10695 0.01294 0.00055 0.00633 5.4E-05 0.00367 0.00164 0.00068 0.00155 37.5 2.82593 2.06588 1.34755 3.26493 0.01896 0.00048 0.0055 0.00025 0.00236 0.00175 0.00076 0.00239 42.5 2.10747 0.62055 1.00865 3.31759 0.02106 0.00042 0.00529 0.00074 0.00149 0.00187 0.00095 0.00288 47.5 2.63434 0.85922 0.71009 3.26493 0.02588 0.00035 0.00345 0.00103 0.00071 0.0013 0.0007 0.00313 52.5 2.15537 1.57523 0.21787 3.10695 0.02648 0.00039 0.00254 0.00138 0.00027 0.00144 0.00073 0.00262 57.5 1.5806 1.98097 0.12911 1.63247 0.03611 0.00043 0.00179 0.0009 0.00019 0.00139 0.00069 0.00209 62.5 0.71846 2.24351 0.02421 1.0532 0.03009 0.00048 0.00184 0.00056 0.00015 0.00147 0.00072 0.00219 67.5 0.47897 2.0287 0 0.31596 0.01618 0.00027 0.00228 0.00038 6.7E-05 0.00164 0.00031 0.00132 72.5 0.0479 1.50363 0 0.36862 0.00111 1.7E-05 0.00324 0.00032 4E-05 0.00134 0.00036 0.00099 77.5 0 0.78762 0 0.10532 0.00101 1.5E-05 0.00631 0.00025 0 0.00064 0.00025 0.00051 82.5 0 0.38187 0 0.05266 0.00123 1.3E-05 0.01516 0.00016 0 0.00071 0.00012 0.00066 87.5 0 0.23867 0 0.10532 0.00172 1E-05 0.02557 0.00018 0 0.00066 0.00016 0.00025 92.5 0 0.11934 0 0 0.01495 0.00011 0.02457 0.00052 0 0.00068 4.3E-05 7.6E-05 97.5 0 0.04773 0 0 0.01204 0.00014 0.02338 0.00047 0 0.00026 6.1E-05 0.0001 102.5 0 0 0 0 0.00602 3E-05 0.02253 0.00054 0 0.00013 2.6E-05 0 107.5 0 0 0 0 0.00301 2E-05 0.02279 0.00022 0 1.7E-05 2.6E-05 0 112.5 0 0 0 0 0 3E-05 0.01005 0.00031 0 0 1.7E-05 0 117.5 0 0 0 0 0 0 0.00104 3.6E-05 0 0 3.5E-05 0 122.5 0 0 0 0 0 0 0.00039 3.6E-05 0 0 8.7E-06 0 127.5 0 0 0 0 0 0 0.00036 1.8E-05 0 0 0 0 132.5 0 0 0 0 0 0 0 0 0 0 0 0 137.5 0 0 0 0 0 0 0 0 0 0 0 0 142.5 0 0 0 0 0 0 0 0 0 0 0 0 147.5 0 0 0 0 0 0 0 0 0 0 0 0 Total Note: Traffic amount reduction = 1 - 0.976791 = 0.023209 DS6 DS7 DS8 TRP TWH,AS 0 0 0 0 0 0 0 0 0 0.0004 0 0 0 0 0.01187 1.4E-06 0 0 0 0.01785 5.6E-06 0.00022 0 0 0.04265 1.8E-05 0.00029 0 0 0.06048 4.5E-05 0.00025 0 0 0.0759 4.4E-05 0.00057 0 0 0.07567 8.7E-05 0.00051 0.00287 0 0.06978 7.7E-05 0.00073 0.0035 0 0.07435 7.9E-05 0.00089 0.00319 0 0.07164 7.9E-05 0.0007 0.00701 0 0.07523 6.3E-05 0.00073 0.00573 0 0.06214 7.5E-05 0.00092 0.00637 0 0.04248 8.5E-05 0.00067 0.0035 0 0.02029 5.6E-05 0.00114 0.00191 0 0.01611 3.5E-05 0.00092 0.00478 0 0.02575 1.7E-05 0.00105 0.00701 0 0.03817 7E-06 0.0007 0.0051 0 0.04729 4.2E-06 0.00086 0.00765 0 0.04517 1.4E-06 0.0006 0.00605 0 0.03594 2.8E-06 0.00041 0.00446 0 0.03095 0 0.00032 0.00223 0 0.01295 7E-07 0.00038 0.00223 0 0.00372 0 0.00025 0.00287 0 0.00355 0 0.00016 0.00319 0 0.00373 0 0.00022 0.00414 0 0.00436 0 0.00019 0.00223 0 0.00242 0 9.5E-05 0.00096 0 0.00105 0 0.00013 0.00478 0 0.00491 0.97679

Step D. Account for effects of tare weight changes. The default ratio of empty to loaded trips is used for this example: rE/L = 0.2. It characterizes the amounts of traffic used to deliver payload and to return to the operation base with no payload. The default value is based on data collected in previous studies. The user may change this value if data are available to support such a change. For weight interval GVWk,BC = 32.5 kips, the empty truck traffic will not change, as shown in Table 3.5 in column “TT ′GVWk,BC”. This is because the lowest impacted interval is 67.5 kips. This leads to the lowest-impacted empty traffic interval being 67.5 − 30 = 37.5 kips using the selected tare weight of 3S2T of 30 kips. Accordingly, there will be no increase of empty traffic to the CS6 trucks, as shown in col- umn “TTGVW,AS” of Table 3.5. For weight interval GVWk,BC = 37.5 kips, the empty truck traffic is determined as ∆TTTARE − GVW,BC = −rE/L TT ′VWk,BC = (0.2)(0.01151) = −0.00230, according to Eq. 3.2.2.9. This value is shown in column “TT ′GVWk,BC Eq. 3.2.2.1” of Table 3.5 with a negative sign to distinguish it from those traffic amounts for loaded trips. Similarly, the increased empty truck traffic is calculated as DTTTARE −GVW,AS = +rE/L TTGVWi,AS = +(0.2)(0.0012) = +0.000241, shown in column “TTGVW,AS” of Table 3.5. For the rest of impacted weight intervals from 42.5 to 62.5 kips, the same calculations are repeated as shown in columns “TT ′GVWk,BC Eq.3.2.2.1” and “TTGVW,AS” of Table 3.5. Note also that interval 77.5 kips has two additional amounts. The first one (0.001604) came from the empty truck traffic due to the weight interval 112.5 kips ((0.001604 = (0.2)(0.00802)). The second amount (0.0012) came from the loaded traffic increase discussed above. The calculation for the truck type 3S2S is given in Table 3.6 in the same format. Step E. Summarize the entries to the TWH for Alternative Scenario. The final resulting TWH is given in Table 3.7 in the last col- umn “TWHAS”. It was summed using the normalized traffic amounts in Table 3.4, except those for truck types 3S2T, 3S2S, and CS6 that are impacted upon. For truck types 3S2T and 3S2S, traffic shifts away to type CS6. Columns “TT ′GVWk,BC” in Tables 3.5 and 3.6 show the decreased amounts, respectively, for 3S2T and 3S2S. Columns “TTGVW,AS” in these two figures give the increased traffic amounts to truck type CS6. Table 3.7 gives the final results of these calculations. Comparison of Tables 3.4 and 3.7 shows the impact of the considered Alternative Scenario on TWH. Also note that the TWH for the Alternative Scenario in Table 3.7 has not been normalized as the sum of the frequencies is not unity. The summed traffic amount 0.9768 has a difference from unity. This difference, 0.0232, is the traffic amount reduction as a result of weight limit change, because CS6 is allowed to 39 carry a higher payload for each trip. The TWH needs to be normalized to perform other analyses that require a normal- ized TWH, such as the analysis to find the equivalent weight for steel fatigue life prediction. 3.2.5 Recommended Method for Predicting WWHs For assessing RC deck fatigue, truck wheel-weight distri- butions are needed to estimate the effects of changes in truck weight limits. Also, although outside the scope of this proj- ect, wheel weight distributions are needed to estimate pave- ment impacts. It is assumed that there is a correlation relationship between the wheel weights and the GVW. Accordingly, the concept of the recommended method is to estimate the wheel weights based on GVW. This assumption is particu- larly valid for trucks loaded to the limits, which is dominant to RC deck fatigue. When a TWH is available, possibly obtained using the method recommended above, the wheel weights can be estimated using the following empirical relationships: Wheel Weight = 0.5 Mean Axle Weight + Weight Residual = 0.5 (e + f ∗ Gross Vehicle Weight in kips) (3.2.5.1) in kips + X where e and f are regression coefficients. They form the first part at the right hand side of this equation expressing the mean wheel weight. X is a correction parameter to cover extreme wheel loads away from the mean loads. These rela- tionships have been established for all individual axles of a variety of trucks, using WIM data from California. They are shown as Data Set A-5.2.2 in Appendix A, as the default database for e and f. Note that for each configuration, coeffi- cients e sum to zero and coefficients f sum to one. This con- dition guarantees that the sum of the axle weights is equal to the gross weight. It is also recommended that agencies use their own WIM data to obtain those coefficients for typical truck types within the jurisdiction. To demonstrate the application of the equations, a con- ventional 5-axle tractor–semi-trailer carrying 77,000 pounds would have 10,760 pounds on its first axle (the steering axle). Data set A-5.2.2 gives e = 7.603 and f = 0.041. Thus, the first axle’s average weight is calculated here as 1,000 pounds ∗ (7.603 + 0.041 ∗ 77 (kips)) = 10,760 pounds = 47,860 kN. It is noted, again, that the extremely high and extremely low wheel weights are not covered in the mean or average weight obtained using that regression relationship in the first part in Eq. 3.2.5.1. This is due to the nature of regression for predict- ing the conditional mean of a function (i.e., wheel weight here) of the independent variable (i.e., GVW here). As discussed in Section 2.3.2, RC deck fatigue is greatly influenced by the highest wheel loads. Thus, X is used in Eq. 3.2.5.1 to cover

this additional amount to be added to the average wheel weight. Based on WIM data provided by the Idaho Transportation Department (ITD), this additional amount, or residual weight, is modeled here using a truncated skewed double exponential distribution (Benjamin and Cornell 1970, John- son et al. 1994). The truncated probability density function f ′X(x,λ) is expressed as follows. f ′X(x,λ) = fX(x,λ)/A where λ >0 (3.2.5.2) where X is a random variable to model the residual wheel weight. λ is its skew factor. A is the area of the skewed dou- ble exponential probability density function fX(x,λ) after truncation of the part for x > x0. x0 represents the maximum wheel weight on bridges. It is usually not the same as the legal maximum wheel load. It depends on the degree of com- pliance of truckers to wheel weight limits and effectiveness of enforcement. Using the WIM data from Idaho, x0 is found to be at 18 kips. The skewed double exponential probability density func- tion fX(x,λ) is defined as follows fX(x,λ) = 2FX(λx)fX(x) where λ >0 (3.2.5.3) and (3.2.5.4) (3.2.5.5) where µ is the mean value of random variable X, i.e., µ = E[X] (3.2.5.6) E stands for expectation. Using the WIM data from Idaho used in the application example, it is found to be zero. This is expected, because X is the residual or deviation from the regression-predicted wheel weight. b is another model para- meter related to the variance of X: 2β2 = E[X2] (3.2.5.7) The WIM data from Idaho were also used to estimate β and λ for both before and after a change in truck weight limit: β = 1.25 kips and λ = 0.1. They may be used as default data. For- tunately, they were found to be little influenced by the truck weight limit change. This perhaps is because the WIM data were from a case where the wheel weight limits did not change. F (x) x x x x X = −    − < − − −    − ≥      1 2 0 1 12 0 exp ( ) ( ) exp ( ) ( ) µ β µ µ β µ f (x) xX = − −   1 2β µ βexp 40 The above probabilistic model for the residual X is then used in generating WWHs using TWHs. The procedure is as follows: (1) For each weight interval in TWH, use Eq. 3.2.5.1 to find the mean wheel weight. (2) Then distribute the traffic of the GVW weight interval to a range of wheel weight inter- vals, with the obtained mean wheel weight at the center of the range. The traffic distribution follows the truncated skewed double exponential distribution discussed above. Note that this procedure has been implemented in the software module in Attachment 5 in the attached CD. 3.3 STEEL MEMBER FATIGUE ASSESSMENT AND FATIGUE TRUCK MODELS The analysis for this cost-impact category typically con- sists of the following steps. 1. Identify possibly vulnerable bridges. 2. Sample the possibly vulnerable bridges to reduce the number of bridges to be analyzed in details, if Level I analysis is used. 3. For the analysis of each bridge in the sample (if Level I analysis is performed), generate the TWHs under the Base Case and predict the TWHs under the Alterna- tive Scenario. Estimate remaining safe life and remain- ing mean life for both the Base Case and Alternative Scenario. Select the responding action based on the estimated remaining lives. Estimate the costs for the selected action. 4. Summarize the costs for all bridges. 5. Perform a sensitivity analysis to understand possible controlling effects of the input data. The concepts for these steps are discussed in more details next. 3.3.1 Identifying Vulnerable Bridges and Sampling Bridges to Be Analyzed Vulnerable bridges are defined here as those that have details of E and/or E′ fatigue strength category according to the AASHTO specifications (1990, 1996, 1998). (Section A-5.1.3 in Appendix A presents a set of general guidelines that can be used in this process.) Typically, the agency’s bridge inventory can be used for identification of these bridges. The NBI can be used as the default database if the agency does not have more detailed bridge inventory. Most likely, the NBI is needed when a federal-level analysis is performed. It should be noted that if fatigue-prone details other than E and E′ categories are of concern to the agency, they can be added to the analysis process to be covered. Thus the software for the recommended methodology should reserve an option for the user to include these detail types. When this is the case, the agency will need to provide all other information needed

to reach a cost estimate, such as the procedure to obtain the stress range, the repair or replacement procedure, associated unit costs, etc. When the network being analyzed is extensive including a large number of bridges, a smaller sample will be desired considering the resource constraint. This represents a Level I data requirement case because only the information on this bridge sample will be needed for detailed analysis. Such a sample should be representative for the entire population, as the estimated costs for this sample will be proportioned to the entire population. It is thus advised that sampling be done with respect to the characteristics of the bridges, because these char- acteristics influence costs, sometimes very significantly. These characteristics may include jurisdiction (state vs. local agency); functional class of the roadway; type of construction (plate girders vs. rolled beams); type of spans (simple vs. continu- ous); span length; and the year of original construction. Fortu- nately, this type of information is available in the NBI or a typ- ical state agency’s bridge inventory. Note that the identified bridges as a result of this step are pos- sibly vulnerable ones. They may or may not have the targeted E or E′ details. To confirm the presence of such targeted details, a detailed analysis needs to be performed for each possibly vul- nerable bridge (for a Level II analysis) or for each bridge in the sample selected (for a Level I analysis). This detailed analysis should proceed as specified in (AASHTO 1990) or the new AASHTO manual after its adoption, as follows. 3.3.2 Bridge Analysis for Remaining Lives For each bridge selected (resulting from the last step), the fatigue analysis should follow the AASHTO procedure (1990) to be consistent with current practice (or the new AASHTO manual after adoption). Namely, the following safe life esti- mation should be used: (3.3.2.1) where Y is the total life in years. K is a constant tabulated for each type of fatigue sensitive detail in the AASHTO specifi- cations, and f equal to 1 for safe life and 2 for mean life. C is the number of cycles for a passage of the fatigue truck. Rs is a reliability factor. Sr is stress range in ksi for a passage of the fatigue truck whose weight can be more reliably determined using WIM data according to Eq. 3.2.2.11. For the Base Case and the Alternative Scenario, this stress range should be cal- culated using respective TWHs. The Base Case TWH is based on site-specific WIM data or the default VMT data whose sample is presented in Data Set A-5.2.1 of Appendix A. The Alternative Scenario’s TWH is to be developed using the Base Cases’ TWH and the prediction method discussed in Section 3.2 above. T is the current annual daily truck volume for the outer lane. Ta is an estimated lifetime-average daily Y fKT T TC R Sa s r = ×106 3( / ) ( ) 41 truck volume in the outer lane. The AASHTO specifications (1990) provide values for these parameters or guidelines about determining them. Note that the AASHTO procedure for Ta represents an approximation, which may lead to under- or overestimates. The following formula is recommended to improve this assessment. Its derivation starts from the definition of Ta /T: (3.3.2.2) where u is the annual traffic growth rate. It may be esti- mated using information in the agency’s bridge inventory or the NBI (i.e., the latest recorded traffic volume and future traffic volume), if more specific information is not available. A is the current age of the bridge. The numerator in Eq. 3.3.2.2 represents the sum of the total traffic over the life span Y, using a constant annual growth rate u. The numerator divided by the fatigue life Y gives the life-aver- age annual traffic, except that the initial traffic volume is not included. The denominator (1 + u)A represents the cur- rent traffic at the age of A years, except the initial traffic volume. These two missing terms actually cancel each other, and thus they are not shown. According to Eq. 3.3.2.2, it appears that finding Y needs an iterative approach because the unknown Y is in both sides of the equation in Eq. 3.3.2.1. This would require some computa- tional effort. Fortunately, the summation in Eq. 3.3.2.2 can be explic- itly written as (3.3.2.3) Substituting Eq. 3.3.2.3 into Eq. 3.3.2.2 and then into Eq. 3.3.2.1 allows directly solving for Y as follows. (3.3.2.4) This formula for Y is very helpful in simplifying the calcu- lation as well as increasing its calculation speed in the soft- ware module for the recommended methodology. 3.3.3 Impact-Cost Estimation The impact costs largely depend on the action to be selected in response to the calculated remaining life changes and other factors. These factors may be the current age of the bridge, agency’s policy regarding fatigue repair failure, and so on. While the agency may decide whether more frequent Y fK TC(R S ) u(1 u) u s r A 1 = × + +   + −log log( ) 10 1 1 6 3 ( ( ) (1 1 1 1+ = + × + −  − ∑ u) u u) u i Y i 1 Y T T u Y(1 u) a i i 1 Y A= + + = ∑ ( )1

inspection (monitoring) is warranted, the expected repair and replacement costs are recommended below. As discussed above, despite the research efforts spent over the past decades on steel fatigue, there is still uncertainty in the estimation process. Therefore, steel fatigue failure is con- sidered to be a random process. Accordingly, it is recom- mended that a probability based approach be used for esti- mating the costs directly related to fatigue failure (cracking). These costs could be repair or replacement costs. The default decision in the recommended methodology will be repair, while other actions (such as monitoring) may be added. On the other hand, replacement costs for individual members depend on many factors that cannot be comprehensively cov- ered in this project. They may include, for example, whether or how many other members are to be affected or replaced, the cost-effectiveness for repair compared with other options, etc. On the other hand, repair costs are much less scattered. Some default cost data are included in Appendix A in case more specific cost data are not available. A proba- bilistic approach is recommended here to estimate the expected repair costs. The safe remaining life and the mean remaining life are needed in this approach, using Eq. 3.3.2.4. The following equation can be used to estimate the remaining life’s standard deviation σY σY/YMean Life = −β +(β2 + 2 Ln(YMean Life /YSafe Life))1/2 (3.3.3.1) where YSafe Life and YMean Life are safe and mean lives calculated using Eq. 3.3.2.4. β is the target reliability index to which the AASHTO specifications (1990) have been calibrated (Moses et al. 1987). β is equal to 2 and 3, respectively, for redundant and non-redundant components. Then the probability of fatigue failure (cracking) Pf within the considered planning period PP (in years) can be estimated as follows using a trun- cated lognormal cumulative function LOGN: Pf = (LOGN(PP + A,YMeanLife,σY) − LOGN(A,YMeanLife,σY))(1 − LOGN(A,YMeanLife,σY)) (3.3.3.2) A is the current age of the bridge. The change in this failure probability from the Base Case to the Alternative Scenario is the impact on fatigue failure risk due to the Alternative Sce- nario being investigated. Thus, the expected impact costs can be estimated as follows Expected Impact Cost = Impact Cost(Pf,AS − Pf,BC) (3.3.3.3) where subscripts AS and BC indicate respectively the Base Case and the Alternative Scenario. When the expected impact cost turns out to be negative (i.e., the failure probability under the Alternative Scenario is smaller than that under the Base Case), then the expected impact cost is taken to be zero because not impact is expected. The impact cost here depends 42 on the action selected in response to the life change. It may be for repair, replacement, monitoring, or their combinations. The default is recommended to be repair. Data Set A-5.2.4 in Appendix A provides steel fatigue repair cost estimates as the default costs data. Note that this recommended approach is consistent with the concept of the AASHTO fatigue assessment procedure, using a probability based approach. It has the following advan- tages. (1) Using the concept of expected cost equal to the cost times the probability of cost incurrence, the likelihood of fail- ure occurrence (i.e., reaching end of fatigue life) is clearly described. This reflects the nature of fatigue failure with uncer- tainty (including the RC deck fatigue to be discussed below). (2) It also avoids the difficulty in deterministicly deciding which responding action to use, in previously recommended deterministic approaches, in which different decisions could cause extremely large differences. 3.3.4 Validity of the AASHTO Fatigue Truck Model The recommended procedure above suggests the use of the current AASHTO fatigue truck, which was developed based on WIM data collected many years ago. There is a concern that truck configurations may have changed and will change as a result of truck weight limit changes. This section addresses this issue, by considering a specific scenario of truck weight limit change and providing guidelines for examining the issue for a general Alternative Scenario. 3.3.4.1 Introductory Remarks This investigation is to quantitatively evaluate the current AASHTO fatigue truck model (defined in AASHTO Guide Specifications for Fatigue Evaluation of Existing Steel Bridges 1990) under truck weight limit changes. The AASHTO fatigue truck model has a fixed configuration (with axle distances of 14 and 30 ft, and axle weights of and of GVW for 3 axles as shown in Fig. 2.1). The GVW is determined using the equivalent weight concept based on the truck weight histogram (TWH) if available: GVW = Wequivalent = (Σi = 1,2,3,… f i Wi3)1/3 (3.3.4.1) where Wi is the GVW for weight interval i which is taken at the mid-interval, and fi is the frequency for that weight inter- val. The AASHTO fatigue truck model was developed using WIM data collected in the early 1980s by Fred Moses and his colleagues (1987). Present investigation is to evaluate whether this model would be valid when truck traffic changes (mainly in configuration or the distribution of GVW among axles) under changes in truck weight limits. 4 9 1 9 4 9, ,

3.3.4.2 Alternative Scenario A specific scenario of weight limit change is selected for this investigation: legalizing GVW of 431 kN (97 kips) on 6 axles. This scenario was selected because of the following considerations. (1) Only those scenarios that legalize new weights across the board could affect the validity of current fatigue truck model. Localized legal weight-limit changes or permit-limit changes would unlikely generate such an impact because the amount of traffic to be affected would be too small. (2) The 431 kN (97 kips) legal weight has the poten- tial to be legalized. It has been, and still will be, a subject of debate at Congress. Further, this scenario is legal in Canadian provinces; this pressures border U.S. states to legalize this scenario (e.g., Michigan). Because it is not certain what axle distances will be legal- ized for the selected scenario, two 6-axle configurations are considered here. Shown in Fig. B-2.1.1 in Appendix B, these two configurations represent upper and lower bounds for the reality with respect to axle distances, especially the axle dis- tance between the tridem and the tandem. The two truck con- figurations are respectively referred to as 3S3A and 3S3B. Note that the steering axle weight is set to be constant because it does not vary significantly with GVW for this kind of configuration. When fully loaded to 431 kN (97 kips), these two configurations will have their tandem and tridem respectively weighing 151 kN (34 kips) and 227 kN (51 kips). It should be noted that, with respect to dimensions, the 3-S3A is likely more acceptable than the 3-S3B because it is shorter, requires less space, and thus is easier to be accommodated. 3.3.4.3 Approach The validity of the AASHTO fatigue truck model is deter- mined herein by understanding whether the model can ade- quately predict load effects, which relate to fatigue damage more directly than truck weights. Bending moment is used here for this purpose, because it is proportional to stress: Mequivalent = (Σ i = 1,2,3,... fi Mi3)1/3 (3.3.4.2) where M stands for moment, and the rest of the symbols have been defined in Eq. 3.3.4.1. As discussed above, Mequivalent can be used to better predict fatigue wear. In routine practice, Mequivalent is not readily available because it requires knowl- edge of trucks’ axle weights. A less direct way of estimating fatigue is the AASHTO method as described in Eq. 3.3.4.1. The difference between Mequivalent and the moment induced by the AASHTO fatigue truck model (with GVW = Wequivalent as defined in Eq. 3.3.4.1) is used here to indicate the model’s validity: Error = MAASHTO fatigue truck model /Mequivalent − 1 (3.3.4.3) 43 WIM data collected in 1996 are used here from interstate rural highways in New York, which has a significant number of steel bridges. These are the latest data from that state avail- able at FHWA. These data are used here as the load for the Base Case. The researchers then apply the recommended method for predicting TWHs presented in Section 3.2 to esti- mate the load under the Alternative Scenario (i.e., legalizing 431 kN or 97 kip GVW on 6 axles). For the midspan moment in simple spans, the results are compared using Eq. 3.3.4.3. The default window parameters are used for this investigation. All “shifting” of loads between truck configurations due to the hypothetical weight-limit change is done for the individ- ual 3-S3 trucks. Namely, each truck in the original WIM data is examined to determine whether its load will be hauled by a new 6-axle semi that could haul higher load. Only those 5-axle trucks that have a GVW close to the current weight limit are eligible to be shifted to 6-axle trucks because they are likely to be affected. Only a specified fraction of these trucks will be subject to such shifting, according to the default window parameters a1 = a2 = 10%, b1 = b2 = 20%, and c = 0.95. When a truck is confirmed to be eligible for shift- ing, deciding whether the particular truck will be shifted or not is based on random selection, which assures the specified fraction. If shifted, the truck is replaced by a new truck with 6 axles. As a result of shifting, the number of trucks is also reduced to maintain constant payload, as defined by Eq. 3.2.2.2. After all trucks have been examined and shifted if deemed necessary, they are used to find their maximum midspan moment for a simple span. A histogram of moment is then generated for calculation in Eq. 3.3.4.2, which is used to find Mequivalent for Eq. 3.3.4.3. 3.3.4.4 Results and Conclusions As discussed above, inadequate information exists on the real configurations of 6-axle semis. The 3S3-A and 3S3-B in Fig. B-2.1.1 are respectively used for this purpose to produce results as two bounds for the reality, as shown in Table 3.8 for a range of simple spans. For comparison, the Base Case data are also used in Table 3.8 to show the validity of the AASHTO fatigue truck model for current truck traffic. Sev- eral observations are made as follows for these results. 1. If most 6-axle semis have a configuration of 3S3-A, the AASHTO fatigue truck model is still valid, as shown by low errors in the column of 3S3-A in Table 3.8. The maximum error there is 3.66 percent for moment (or stress) and 11.39 percent (= 1.03663 − 1) for fatigue wear according to the cube rule. This is because the 3S3-A configuration is close to that of the AASHTO fatigue truck. Further, this validity improves with span length, because the weight distribution among axles in a truck becomes less significant to moment for longer spans.

2. If most 6-axle semis have a configuration of 3S3-B, the AASHTO fatigue truck will be less valid. For some spans, this becomes severe, as shown by larger errors in the column of 3S3-B. For example, for a span of 30 m, the error is 15.56 percent. This error will cause an overestimation of fatigue by 54.3 percent according to the cube rule. 3. The reality is understood to be between the two bounds discussed above. Depending on how close the real truck configurations are to 3S3-A and 3S3-B, the real error would rest between the two bounds (Columns of 3-S3A and 3-S3B) given in Table 3.8. 4. For very short spans, the axle, tandem, or tridem weights become governing for moment. Thus the dif- ference between different configurations becomes small, as shown in Table 3.8 for the 18-m span. This span can- not have all the axles on the bridge for the 3S3-B, and cannot have axles all contributing to moment signifi- cantly for the 3S3-A either. As a result, the tandem and tridem weights control the maximum moment. 5. For current truck traffic, the AASHTO fatigue truck still appears to be a reasonable model, as shown in the column of Base Case without shifting. The maximum error is 5.22 percent for moment, thus 16.49 percent for fatigue damage. 6. It should be noted that medium span lengths that are just long enough to have all the axles on the span and all of them making notable contributions to moment would suffer from highest approximation as shown in Table 3.8 for the 30-m (98-ft) span length. When the span length increases, this approximation becomes more acceptable. 7. In general, if the new trucks under the Alternative Scenario do not significantly differ from the current AASHTO fatigue truck in configuration (for a span length), the AASHTO fatigue truck would still be valid. Further, if the replaced traffic does not occupy a large percentage of the traffic traveling at the current weight limit (e.g., permit truck traffic or localized legal trucks), the current fatigue truck would still be valid. In other words, thereby caused approximation in fatigue assess- ment will be acceptable. 44 3.3.5 Sensitivity Analysis As alluded to or directly discussed earlier, there is a level of uncertainty involved in the AASHTO fatigue assessment procedure. It is thus critical to understand the effects of this uncertainty on the final results, the estimated expected impact costs. Due to a large number of parameters used here in the recommended methodology, a general approach to this requirement is recommended to be as follows: (1) identify those parameters or assumptions that may significantly influ- ence the final result and (2) alter the identified parameters in a realistic range and re-perform the analysis accordingly. This sensitivity analysis will help identify those parameters that have more dominant influence or higher sensitivity. These parameters may need re-examination and possibly adjustment for more reliable results. This concept is recom- mended for all four cost impact categories covered in the rec- ommended methodology. For the cost-impact category of steel fatigue, the follow- ing parameters may need to be examined for their effects on the final result in the sensitivity analysis. (1) The window parameters a1, a2, b1, b2, and c, and the parameter for exoge- nous shifting in the TWH prediction method, as defined in Section 3.2. (2) Load distribution factor used to calculate the stress range. (3) Impact factor. (4) ADTT. (5) Repair cost data. (6) Action selected by the agency user. (7) The sample bridges selected, if Level I analysis is used. 3.3.6 Secondary Bending Fatigue failure caused by secondary bending is commonly observed in the field. It results from distortion of members and partial fixity at connections that are assumed to be pinned (Moses et al. 1987). Currently there are no general quantita- tive methods for identifying and analyzing them for fatigue assessment, mainly because this type of failure is a result of local condition, which may vary significantly over the nation. Thus, it is very difficult to develop general guidelines for iden- tification and analysis to cover a large variety of situations. Section A-5.1.3 offers a general concept for addressing cost impact for this type of steel fatigue. The assumption used TABLE 3.8 Errors by using the AASHTO fatigue truck model under the alternative scenario of legalizing 431 kN (97 kips) GVW on 6 axles Error (%)(Using Eq.3.3.4.3) . Alternative Scenario Alternative Scenario Base Case Shifted to 3S3-A Shifted to 3S3-B Without Shifting Span Length in m (ft) 18 (59) - 0.88 - 0.88 - 0.96 30 (98) +3.66 +15.56 +5.22 42 (138) +2.54 +11.29 +3.57 54 (177) +2.34 +8.26 +2.71 66 (216) +1.54 +6.50 +2.17

there is that, within a jurisdiction, the variation of situation may be much smaller. This situation may make it possible to perform detailed analysis for several typical vulnerable details common within the jurisdiction. 3.4 RC DECK FATIGUE 3.4.1 The RC Deck Fatigue Model The following formula has been recommended for pre- dicting fatigue failure of RC decks: Log(P/Pu) = A + B Log(N) (3.4.1.1) where P/Pu is the ratio of the repetitively applied load P and the static ultimate strength of the concrete deck Pu. N is the number of times (cycles) load P is repetitively applied. A and B are model parameters to be determined based on reported physical testing and statistical analysis of the test results. Note that this format is very similar to that for steel fatigue discussed above, known as S-N curves: Log (S) = A + B Log(N) (3.4.1.2) where S is the stress range due to repetitively applied load. The rest of the symbols are defined the same as those in Eq. 3.4.1.1. Parameter B has been found approximately equal to −1/3 for steel fatigue, based on a large number of tests (Moses et al. 1987). Further, parameter A has been found to be dependent on the type of weld detail. The AASHTO bridge design codes (1996, 1998) classify these weld details into fatigue strength categories A through E′. Using the prin- ciple of Eq. 3.4.1.2 and the assumption of linear accumula- tion of damage (the Miner’s Law), the AASHTO specifica- tions (1990) include provisions on A for fatigue evaluation, which are being integrated into the new AASHTO evalua- tion manual under NCHRP Project 12-46 (AGLichtenstein & Associates 1999). (This AASHTO procedure has been included in the recommended methodology in Appendix A.) The latest effort of investigating RC deck fatigue was reported in Perdikaris et al. (1993) and the study was con- ducted at Case Western Reserve University. In this study for Ohio DOT, a large number of rolling wheel tests were per- formed on 1/3– and 1/6.6–scale models, as well as static tests for the ultimate capacities of these deck models. The model system, including the deck, the beams, and the wheel load, was carefully scaled according to the similitude theorem. Three reinforcement ratios were used in the testing program: the AASHTO method (0.7% in the transverse direction and 0.35% in the longitudinal direction); the Ontario method (0.3% in both directions); and isotropic 0.2% in both direc- tions. A scaled wheel load was used to apply moving (rolling) load to the deck. Two prototype beam spacings were included in the test program: 7 ft and 10 ft. The 1/3–scale models had 45 two steel beams supporting a deck, and the 1/6.6 models had 4 steel beams simulating typical U.S. highway bridges. The test program of this ODOT project represents the most comprehensive research effort to date for RC deck fatigue behavior. Cracking damage was shown resembling that in real bridges, which was also observed by other researchers independently (Matsui 1991, Kato and Goto 1984, Okada et al. 1978). The results reported in Perdikaris et al. (1993) can be sum- marized as Log(P/Pu) = −0.1737 − 0.0557 Log(N) (3.4.1.3) which is in the same format as Eq. 3.4.1.2 for steel fatigue, except that the stress S is replaced by a stress ratio P/Pu. On the other hand, the ODOT study did not cover the effects of water presence. Using full-scale models for highway bridges in Japan, Matsui (1991) found that water worsens the situation and significantly accelerates the fatigue process. The fatigue life (number of cycles) may be reduced by as much as 1,000 times as a result of water presence because water “washes” cement and sand off the cracked surfaces, enlarges the crack, and in turn increases the rate of deterioration. As discussed above, the load–life (S-N) curves are shown as straight lines in the log-log scale. By comparing these straight lines under dry and wet conditions, Matsui (1991) found that the S-N curves for dry condition are “rigidly shifted” down by an amount, almost without a change in the slope. Namely, the interception of the straight line was low- ered (i.e., parameter A in Eq. 3.4.1.1 is reduced by an amount). 3.4.2 The Recommended Fatigue Assessment Procedure Based on Eq. 3.4.1.3 the following procedure is recom- mended for assessing RC deck fatigue using a similar format to that in Article 3.2 of the AASHTO specifications (1990) for steel fatigue assessment: (3.4.2.1a) where Yd is the service life of the deck. Using the concept in Eq. 3.3.2.4, Yd can be explicitly computed as follows (3.4.2.1b) Yd will be the mean service life for the reliability factor Rd set equal to 1 and the evaluation life for Rd equal to 1.35. Ta is the life-average of daily truck volume and T is the current daily truck traffic volume for the outer lane, as used in Eq. 3.3.2.1 for steel fatigue. Cd is the average number of axles per Y K K TC (R IP P/P ) u(1 u) u d d p d d s u 17.95 A 1 = + +  + − log log( ) 1 1 Y K K (T /T)TC (R IP P/P )d d p a d d s u 17.95=

truck. P/Pu is the equivalent stress ratio caused by wheel load P defined as follows: P/Pu = [Σfi(Pi /Pu)(Pi /Pu)17.95]1/17.95 (3.4.2.2) where Pu is the ultimate shear capacity of the deck. Eq. 3.4.2.1 uses the same linear damage accumulation assumption (the Miner’s Law) as for steel fatigue. Kd is a coefficient that cov- ers the model uncertainty (with respect to the assumed Miner’s Law). For calculation convenience, the model constant A = −0.000762 in Eq. 3.4.1.3 and a constant of 365 days/year are also include in Kd. Kp addresses the difference between the time of deck failure (punch through) described by Eq. 3.4.1.3 and the time of real deck treatment. It also covers accelerated fatigue due to water presence, which is a variable over the country due to climate condition. The parameters in Eq. 3.4.2.1 may be divided into three groups: (1) load magnitude related (I, Ps, and P/Pu); (2) num- ber of stress cycles related (Ta /T, T, and Cd); and (3) model- related (Kd and Kp). These parameters are further discussed as follows. The recommended approach is based on the following concept. The useful service life of a bridge deck is a random variable that is a function of a number of other variables: load magnitudes, number of load cycles, and decision as to when it should be renewed (by overlay or replacement). Note that patching is considered to improve service to the public by providing better riding quality, but it does not increase struc- tural capacity against fatigue. Note that deciding the end of service life inherently involves uncertainty. Weyers et al. (1994) has showed, in SHRP Project C-103, that the opinions of engineers making the decision on when to overlay a bridge deck are far from uniform. When they were given the same information about the top surface condition of the same decks, their answers had significant scatter as to whether these decks have reached the end of service life or whether they need treat- 46 ment. The procedure in Eq. 3.4.2.1 has been designed to cover this factor to a certain extent. This is discussed below in more detail. 3.4.2.1 Load Magnitude Related Parameters In Eq. 3.4.2.1, the nominal impact factor I from the AASHTO specifications is used here to cover dynamic effects of truck wheels. On the other hand, the real dynamic impact is a random variable, assumed to have a mean equal to the code specified nominal value and a standard deviation equal to the mean times the coefficient of variation (COV), which is set equal to 15% (Moses et al. 1987). The parameter Ps is referred to as axle-group factor. It is to cover effective load increase due to closely spaced wheels in axle groups, such as tandems and tridems commonly used in heavy trucks. In general, this factor is deck dependent because it is a function of the deck’s relative geometry related to the following parameters: (1) deck thickness; (2) spacing of the supporting beams (i.e., the span length of the deck); and 3) span length of the supporting beams, which could deter- mine whether the deck is closer to a one-way slab or a two- way slab. Furthermore, the spacings of the wheels in a tandem or tridem are not constant. Therefore, parameter Ps describes an interactive relation between the wheel loads and the deck. The finite element analysis method was used to understand the effects of the above variables to be covered by Ps for 6 RC decks in Arizona, Alabama, and Georgia. Note that many bridges in these states have not been subjected to de-icing salt, for which the recommended method for fatigue assess- ment is applicable. The finite element modeling was vali- dated against field test data presented in Fu et al. (1997) using a bridge in New York whose deck was load tested several times over a period of 7 years. Figs. 3.4 to 3.9 show the finite element models for these bridges, with both the deck and supporting beams modeled. Two of these bridges have con- crete beams and the rest have steel beams. It is found that the Figure 3.4. Finite element analysis model for Bridge 845 in Arizona.

47 Figure 3.5. Finite element analysis model for Bridge 1596 in Arizona. Figure 3.6. Finite element analysis model for Bridge 12102420 in Georgia. Figure 3.7. Finite element analysis model for Bridge 7232 in Alabama.

shear effect increase due to closely spaced wheels varies from 2 to 9 percent. Based on this set of analysis data, the rec- ommended value for Ps is determined at 1.04 for Eq. 3.4.2.1. Ps actually can be modeled as a random variable having a mean equal to 1.04 and a COV equal to 3.5%. Pu is the nominal ultimate shear strength of the deck to be estimated as follows, according to the ACI design code (Perdikaris et al. 1993) and the AASHTO design code (1996): Pu = (2 + 4/α)(f c′)1/2 b0 dγ < 4(f c′)1/2 b0 dγ (3.4.2.3) where f c′ is the concrete compressive strength in psi. α is the ratio of the tire print’s long side to short side, set equal to 2.5 for a nominal tire print of 0.508 m by 0.203 m (20 in. by 8 in.) for dual tires. d is the deck’s effective thickness equal to the total thickness minus the bottom cover thickness. It is rec- ommended to also subtract a 0.00635–m (0.25–in.) thick layer from the nominal thickness to account for wearing observed in bridge decks. b0 is the perimeter of the critical 48 section, which is defined by the straight lines parallel to and at a distance d/2 from the edges of the tire print used. γ is a model correction parameter, which is set at 1.55 based on the test data in Perdikaris et al. (1993). It should be noted that the above parameters are nominal values of respective variables with uncertainty, as in many other cases for strength or fatigue assessment. Thus Pu can be expressed as a random variable with its bias (nominal value divided by the mean value) equal to one and a COV equal to 23%, based on the data reported in Perdikaris et al. (1993). P in Eq. 3.4.2.1 is an equivalent fatigue load that can be calculated as follows using a WWH. P = (Σfi(Pi)P i17.95)1/17.95 (3.4.2.4) where Pi is the mid-interval value of the ith interval in the WWH, and f(Pi) is the frequency for that interval. Eq. 3.4.2.4 is similar to Eq. 3.3.4.1 for steel fatigue, except that the model constant is 17.95 in the former and is 3 in the latter. Figure 3.8. Finite element analysis model for Bridge 5360 in Alabama. Figure 3.9. Finite element analysis model for Bridge 6446 in Alabama.

Here 17.95 = −1/B with B = −0.0557 taken from Eq. 3.4.1.3 based on reported physical testing results. Further note that a steering wheel usually consists of a sin- gle tire not dual tires. The wheel acts on an area that is approximately half of the dual tire print. Thus the ultimate shear capacity Pu is reduced by about 33 percent. For calcu- lation convenience, Pu can be kept as a constant with the load increased by 1/0.67. In other words, the steering wheel weights should be increased by 1/0.67 for Pu to be treated as a constant and taken out of the summation sign, as indicated in Eq. 3.4.2.4. According to previous research experience (Moses et al. 1987), the equivalent weight P can be described by a random variable with its bias equal to 1 and its COV equal to 0.15. 3.4.2.2 Load Cycle-Related Parameters These parameters include Ta/T, T, and Cd. Eq. 3.3.2.2 gives the formula to calculate Ta/T, as the ratio of the life-average truck traffic to the current truck traffic for the outer lane. As discussed there, an iterative approach should be used to reach a reliable estimation. T is the current truck traffic for the outer lane, according to the procedure given in AASHTO (1990). This includes adjustment to the total truck traffic recorded in the agency’s bridge inventory or the NBI, according to the number of traffic directions (one way or two way) and the number of lanes. Cd is the average number of axles per vehicle, which may be obtained using appropriate WIM data or the default VMT data whose sample is given in Data Set A-5.2.1 of Appendix A. Cd can be calculated using WIM data according to the following formula: Cd = Σni fi(truck typei with ni axles) (3.4.2.5) where f(a) indicates the frequency of a. When appropriate WIM data are not available, the FHWA VMT data may be used to obtain the frequencies f(a) for Eq. 3.4.2.5. In this default data set, 18 vehicle types are included besides auto- mobiles and light 4-tire trucks, which are usually excluded in bridge structure related analyses. Numbers of axles for these 18 vehicles are graphically shown in Data Set A-5.2.1 in Appendix A. They may be used to estimate Cd according to Eq. 3.4.2.5. 3.4.2.3 Model Related Parameters The model underlying the recommended procedure in Eq. 3.4.2.1, as well as that in Eq. 3.3.2.1 for steel fatigue assess- ment, is based on Miner’s law. It assumes that the fatigue life consumed by one application of a load P is inversely propor- 49 tional to the number of cycles at which a constant repetitive load P will exhaust the fatigue life. Namely for RC decks: (3.4.2.6) where N(P/Pu)17.95 = c is the S-N curve based on physical testing, which describes the same relationship as Eq. 3.4.1.1 except in a different format. This linear model is apparently a convenient approximation. Parameter Kd in Eq. 3.4.2.1 is to model the uncertainty in this prediction and is modeled as a random variable. A nominal value of Kd = 2.09 × 10−6 is recommended, based on reported test results (Perdikaris et al. 1993). Furthermore, the fatigue S-N curve for RC decks shown in Eq. 3.4.1.3 refers to ultimate failure—a cone shaped concrete cracks off (i.e., is punched through) the deck. On the other hand, most RC decks in the United States are overlaid using new concrete or replaced before ultimate failure, except for a few incidents of real failure showing deck holes. This indi- cates that there is a clear difference in the definition of end of service life between what Eq. 3.4.1.3 describes and what is recognized in practice. The practice includes an apparent safety margin for preventing serious consequence of deck failure. This difference is covered by parameter Kp in Eq. 3.4.2.1. This parameter also covers the effect of water pres- ence that accelerates deck fatigue. A nominal value of Kp = 3.16 × 10−7 is recommended, based on a calibration using 11 bridge decks in Alabama, Arizona, California, Georgia, Mis- sissippi, Nebraska, and Washington. These decks have been overlaid or have been scheduled for overlay in the near future. This influence is also modeled by a random variable, with a bias equal to 1 (i.e., unbiased or mean value equal to the nom- inal value) and a COV of 2. This COV is relatively large as observed variation in deciding deck rehabilitation and water presence. As commented on above, there is a notable scat- ter among engineers who make decisions on when a bridge deck needs overlay or replacement for the same physical deck conditions (Weyers et al. 1994). Needless to say, there are other variables beyond the physical condition that contribute to the variation in the deck overlay or replacement decision process; for example, whether other bridges on the same route would need rehabilitation in order to save mobilization costs and user costs caused by traffic disturbance. Furthermore, the reliability factor Rd in Eq. 3.4.2.1 is deter- mined using the same approach used in Moses et al. (1987) for Rs in steel fatigue assessment according to Eq. 3.3.2.1. This process takes into account the variation of the random vari- ables discussed above. The main purpose for Rd here is to pro- vide a second point on the probability distribution curve for the deck service life being estimated, beside the mean life using Rd equal to 1. This second point is referred to as evaluation life. Fatigue consumption of one application of P 1 N c P/ Pu = = 1 17 95( ) .

Rd is set equal to 1.35 being the same as Rs for steel fatigue. This Rs value corresponds to a reliability index β = 0.94, due to higher uncertainty observed than that in steel fatigue. With these two points made available, the probability of failure (i.e, the probability of reaching the end of service life) can be com- puted for any time interval. This project is interested in this probability for the pre-selected PP as discussed next. 3.4.2.4 Cost-Impact Estimation The recommended procedure defined in Eq. 3.4.2.1 offers two values of life: the mean life and the evaluation life. These two values define two points on the distribution curve for the service life of an RC deck, in the same fashion as the proce- dure for steel fatigue discussed earlier. The mean service life indicates the expected life. The true service life has equal probabilities (50%) to be higher or lower than the mean life. The evaluation life is defined to be associated with a proba- bility value of approximately 0.174 (i.e., the probability of service life smaller than the evaluation life is 0.174). Thus the safety index β = −Φ−1(0.174) = 0.94 where Φ−1 is the inverse cumulative probability function for the standard normal vari- able. This procedure for RC deck fatigue has a consistent for- mat with that for steel fatigue assessment. The expected impact cost can then be estimated in the same way as Eqs. 3.3.3.1 to 3.3.3.3: Expected Impact Cost = Impact Cost (Pf,AS − Pf,BC) (3.4.2.7) where Pf is the probability of failure or probability of reach- ing the service life end during PP years. σY is the standard deviation of the deck service life, to be calculated as follows, using Eq.3.3.3.1: σY/YMean Life = − β + (β2 + 2 Ln(YMean Life/YSafe Life ))1/2 β = 0.8 (3.4.2.8) The subscripts BC and AS in Eq. 3.4.2.7 indicate the Base Case and the Alternative Scenario, respectively. Parameter Kp in Eq. 3.4.2.1 has been calibrated using seven RC decks that have reached the end of service life (recently overlaid or have been planned to be overlaid) in Alabama, Arizona, California, and Georgia. Thus the default responding action is concrete overlay and the default impact cost above is the overlay cost. Note that deck replacement in those states that are not subjected to much snow has been much less frequent than concrete overlay. Thus data are not adequate at this point to calibrate Eq. 3.4.2.1 against replace- ment need, although it may be performed at a later time when more data become available. The mechanism of deck fatigue after concrete overlay is also not well understood at this point, particularly for those states not using much salt. As such, the following recommen- dations are made as to what action should be selected for cost estimation. Options of responding action may be (i) patching 50 and then concrete overlay, (ii) immediate concrete overlay, (iii) patching and then asphalt concrete overlay, (iv) immedi- ate asphalt concrete overlay, and (v) patching and then replace- ment. These options are discussed in more details next, offer- ing guidelines useful for corresponding cost estimation. As a principle of cost estimation stated earlier, the impact costs are those expected to incur within PP. Options (i) and (ii) correspond to the situation for which Eq. 3.4.2.1 has been calibrated. Option (i) includes patching in addition to concrete overlay, which is mainly to “buy” time but does not address the structural need of the deck. Thus it is considered to be an option of the agency, which may depend on whether funds for concrete overlay are available. Options (iii) and (iv) actually do not completely address the structural need either but they may buy more time than just patching. They are likely to be done before the deck reaches the condition needing a concrete overlay. They could be a less expensive responding action, compared with concrete overlay but for a shorter life span as a return. Assuming no further action is needed after the asphalt concrete overlay (within PP), this selection is expected to pro- vide a conservative cost estimate or under-estimate. Option (v) is certainly an option even for a deck appearing to need a concrete overlay, because the different needs for overlay and replacement sometimes are not very well defined. Other fac- tors may override their differences. For optimizing life cycle costs at the network level, replacement may be a more cost- effective option than concrete overlay. Thus this option is listed for the agency to decide according to the specific deck situation. 3.4.2.5 Generation of WWHs for RC Deck Fatigue Assessment The required WWHs can be generated using the TWHs respectively for the Base Case and the Alternative Scenario. The starting point of this process could be the truck weight data for TWH as seen in Data Set A-5.2.1 in Appendix A or WIM data. Note that the data in that table are directly taken from the FHWA VMT data that have not been normalized to satisfy the definition of histogram that the summation of all the frequencies should be 1.0. For each vehicle type, Data Set A-5.2.1 in Appendix A offers an empirical way to find the individual wheel weights if the GVW and the configuration are known: Wheel Weight = 0.5 Axle Weight = 0.5 (e + f GVW) + X (3.2.5.1) where e and f are model parameters resulting from regression analysis of wheel weights and GVW. Data Set 5.2.2 in Appen- dix A provides the default values for e and f, which was obtained using a large number measured wheel weights and GVWs. When more site-specific data are not available, this data set may be used as the default data. It is recommended that

state agencies obtain jurisdiction specific values for these pa- rameters, using available WIM data that include axle weights and distances. X is the residual from the average wheel weight predicted by the regression relationship 0.5 (e + f GVW), as modeled in Eqs. 3.2.5.2 to 3.2.5.6. This “back track” approach makes it possible to obtain WWHs based on TWHs. 3.4.2.6 An Illustration Example For illustration, an application example for Eq. 3.4.2.1 is presented here. The RC deck studied here is on Bridge No. 15420 in the state of Idaho, built in 1966. It carries 2 lanes of traffic in two directions. For a total deck thickness of 0.175 m (67/8 in.), d is taken as 0.143 m (55/8 in.): d = d′ − c − w = 0.143 m (5.625 in.) d′ = 0.175 m (67/8 in.) is the total thickness, c is the bottom cover equal to 0.0254 m (1 in.), and w accounts for wearing of the thickness, taken as 0.00635 m (0.25 in.). For concrete compressive strength f c′ = 20.68 MPa (3000 psi), the ultimate strength Pu is found to be 600 kN (134.9 kips) using a tire print of 0.2032 m × 0.508 m (8 in × 20 in), according to Eq. 3.4.2.3: Pu = (2 + 4/α)(f c′)1/2 b0dγ = (2 + 4/2.5)(3000)1/2 (2(20 + 8 + 2 × 5.625))5.625(1.55) = (3.6)(54.77)(78.5)(5.625)(1.55) (3.4.2.3) = 134.9 kips = 600 kN < 4 (f c′)1/2 b0dγ = 4(54.77)(78.5)(5.625)(1.55) The following recommended model parameters are used: Ps = 1.04 (a constant, based on calibration using decks in several states) Kd = 2.09 × 10−6 (model constant based on reported RC deck test results) Kp = 3.16 × 10−7 (model constant calibrated for US field condition for water presence and practice in service life definition) Rd = 1 for mean service life. Other parameters are calculated as follows. T = (AADT for the bridge )(Truck Percentage) × (Outer Lane Coefficient from AASHTO 1990) = (6100)(0.08)(0.6) = 292.8 trucks per day on outer lane Cd = 5.00, using WIM data and Eq. 3.4.2.5 I = 1.2 (AASHTO 1990) u = 0.0195 (annual traffic growth rate, estimated using current AADT and future AADT in the NBI) P = 54.1 kN (12.7 kips) using the WWH shown in Fig. B-1.2.1 and Eq. 3.4.2.4 for the Base Case. A = 1997 − 1966 = 31 years (current age) Thus, 51 (3.4.2.1b) It should be noted that due to uncertainty observed in reported physical test results and practice in determining end of service life, the real service life of the deck is not certain. Thus a probabilistic approach has been recommended above in Eq. 3.4.2.7 to estimate the expected impact cost as the product of the cost for the action and the probability of reach- ing the end of service life during the next PP years (which is the probability for that action to take place). 3.4.3 Sensitivity Analysis for the Recommended Procedure The recommended RC deck fatigue assessment procedure in Eq. 3.4.2.1 is based on reported test results and a calibration for U.S. practice for deck renewal by concrete overlay. These test results and agency practice have notable uncertainty. It is important to understand the effects of such uncertainty in order to guide appropriate application and interpretation of results. This section addresses this issue, by performing a sensitivity analysis. As discussed above, the terms in Eq. 3.4.2.1 with an expo- nent of 17.95 are in the same situation for their effect on the estimated life. These parameters include the reliability factor Rd, the dynamic impact factor I, the stress ratio P/Pu, and the axle-group factor Ps for closely spaced truck wheels. For the same bridge used in Section 3.4.2.6 above in the illustration example, the probability of failure (reaching the end of ser- vice life) is calculated for three cases of the dynamic impact factor value I, as follows. Probability of Deck Dynamic Impact Life Exhausted Factor I in Next 20 Years 1.25 0.279 1.20 (reference) 0.335 1.35 0.407 The reference case is that shown in the illustration example above, using the recommended value of 1.2 for I. That value is given in the AASHTO code (1990). It is seen that dynamic Y K K TC (R IP P/P ) u(1 u) u d d p d d s u 17.95 A 1 = + + + = + + − − −     log log( ) log ( . )( . ) . ( . )[ . ( . ) . ( . / )] log( . ) ( . )( . ) log( . ) . 1 1 2 09 3 16 10 292 8 5 00 1 0 1 2 1 04 54 1 600 1 0195 0 0195 1 0 0195 1 1 0195 13 17 95 31 1 = 51 1. years

impact factor plays an important role in the resulting proba- bility of failure, using Eq. 3.4.2.7 for cost estimation. Furthermore, the following results show the effects of the average number of axles per truck Cd in Eq. 3.4.2.1. This fac- tor directly affects the number of load cycles. Average Number Probability of Deck of Axle per Life Exhausted Truck Cd in Next 20 Years 4.70 0.330 5.00 (reference) 0.335 5.30 0.340 As seen, its effect on the probability of failure is much smaller than that of the dynamic impact factor I. This is because I has an exponent of 17.95. It follows that the terms with this expo- nent dominantly contribute to the uncertainty associated with the service life. This observation indicates the importance of determining the stress ratio P/Pu, the dynamic impact I, and the axle-group factor Ps in RC deck fatigue assessment. It also indicates that appropriate wheel weight limits and their enforcement are critical to RC deck service life. 3.4.4 Application of the Recommended Procedure to a Network of Bridges The Level II analysis requires that the recommended analysis of Eq. 3.4.2.1 be performed for every bridge deck in the network. However, when this becomes excessively costly, a sampling approach is recommended at Level I. The level of data requirement for this cost impact category is very similar to that for steel fatigue at the same level of data requirement and detail. It requires detailed analysis for only a small sam- ple of bridge decks considered to be representative for the entire population. The results for this sample will be used then to project to the entire population. Based on the sensitivity analysis discussed above, the dominant factors are those included in the relative load term raised to the 18th power. The exponent of 17.95 here is equiv- alent to the exponent of 3 for steel fatigue. −1/17.95 and −1/3 are the slopes of the S-N curves in the log-log scale respec- tively for RC deck and steel fatigue. Graphically, a slope of −1/17.95 means a much “flatter” straight line than one with a −1/3 slope. Physically, it indicates that the relative stress range P/Pu is much more dominant or sensitive in life pre- diction. In other words, a small change in P/Pu could cause a large change in the number of stress cycles N. For example, a 10 percent increase in P/Pu could cause fatigue accumula- tion increase by 453% (1.117.95 − 1 = 4.53), which will cause the predicted life to reduce by 82 percent (1 − 1.1−17.95). In contrast for steel fatigue, the same amount of increase in the stress range causes fatigue accumulation increase by only 33 percent (1.13 − 1 = 0.33), and the predicted life is reduced by only 25 percent (1 − 1.1−3). Thus, load is much more dom- 52 inant for RC deck fatigue accumulation. It also indicates the importance of enforcement for wheel weight limits. Accordingly, sampling the bridge deck population for Level I analysis should take into account these characteristics of RC deck fatigue. Sites with similar parameters as follows should be grouped together in the sampling process. (1) Sites sub- jected to heavy wheel loads (not necessarily GVW although wheel weights and GVW may be correlated to certain extent). (2) Bridges that have a rough road surface (perhaps with a low condition rating) causing higher dynamic impact. (3) Decks with a lower thickness and/or lower concrete strength, result- ing in a lower Pu and thus relatively high P/Pu. (4) Bridges with similar age (year built) may have similar deck design in terms of thickness and materials. Thus, a bridge (or a few of them) from the same group can well represent the group, because they likely have similar deck thickness, concrete strength, similar deterioration on the driving surface, etc. On the other hand, traffic volume has become secondary, com- pared with other factors related to the relative load. This has been discussed above in the sensitivity analysis. 3.4.5 Limitation of the Recommended Procedure and Future Research Work It should be noted that the above recommended procedure for RC deck fatigue assessment is still limited, due to the lim- ited data available and the present state of knowledge. The limits are commented here, which may be used to appropri- ately interpret the results and to determine future research. 1. Replacement after Overlay as a Responding Action The recommended procedure addresses expected cost impact for RC deck fatigue calibrated to the need for concrete overlay. It seems that overlaying an RC deck at least once before replacing has been a popular choice if not a routine practice. This makes sense when it is realized that the deck’s life span is usually shorter (some times much shorter) than the life span of the supporting beams. Based on this understanding, the first step of treatment is often overlaying instead of replacement, when the deck needs a significant renewal. While deck replacement is listed as an option for cost estimation in Section 3.4.2.4, replacement after one (or more) concrete overlay(s) has not been included as a possible option because there are no reliable data avail- able to help quantify how an overlaid concrete layer works with an old concrete deck. This approach of ignoring the future replacement is conservative in pro- ducing an under-estimate. Further, such a replacement may not very likely take place within a typical PP of 20 years for the states that use no or little salt because the life spans of concrete overlay have been estimated around 15 years in states subjected to high rebar corro- sion rates due to salt (Weyers et al. 1993, 1994), depend- ing on the type of overlay material used. Thus, the

extended end of deck life by concrete overlay is expected to be beyond the typical PP of 20 years. In other words, within this default PP, the need will unlikely occur for replacing an overlaid deck. Furthermore, an approach to this issue of estimating fatigue life for an overlaid deck is to assume that the overlaid concrete perfectly bonds to the old concrete and they form a monolithic deck. Further assuming that the renewed deck had a concrete strength equal to that of the new concrete. Then the recommended procedure in Eq. 3.4.2.1 can be applied to estimate the renewed life span staring at the end of the old life span. This would over- estimate the deck’s fatigue strength and therefore under- estimate the cost impact. 2. Patching and Asphalt Overlay as Responding Action Options Patching (using cement concrete or asphalt concrete) and then asphalt concrete overlay are considered “time buyers” without fundamentally improving the struc- tural condition of the deck. Patching buys the agency less time than the overlay. These measures are taken often because the funds needed for a longer term solu- tion are not available or other actions are not cost effec- tive based on network-level considerations. Thus there should be an option in the recommended methodology for the user to select patching or asphalt overlay if desired. On the other hand, patching needs to be fol- lowed by an overlay (using cement concrete or asphalt 53 concrete), as expected to take place within a typical PP of 20 years. After an asphalt-overlay is done, there may or may not be a need for deck replacement within the PP. This possibility is not included as an option for a conservative underestimation 3. Interaction of Rebar Corrosion Due to Salting and Load Related Fatigue The interacting deterioration between steel reinforce- ment corrosion and load-related RC fatigue is not quan- tified in the recommended procedure, because data are not available with regard to the mechanism of such inter- action. The steel rebar corrosion has been recognized as a dominant deterioration mechanism for RC decks in the areas where a large amount of salt is used in the winter for de-icing. Thus it should be a decision of the user whether steel rebar corrosion is significant for the con- cerned bridge network. Fig. 3.10 shows a map indicat- ing typical salt usage for the states, as a result of a SHRP study (Weyer et al. 1994). The states are divided into three groups, referred to as groups of minimal salt usage, moderate salt usage, and severe salt usage. Based on this map, it is recommended that states with minimal salt usage should include RC deck fatigue assessment developed herein in their application of the recom- mended methodology. Note that there may be exceptions within these states, because salt usage is not uniform even within a state. Also note that in a state belonging to the moderate or sever salt usage, there may be bridges Figure 3.10. Road salt usage in the United States (from Weyers et al., 1994).

that have been subjected to much less salt than in the rest of the state. These bridges should also be analyzed for RC deck fatigue. The user of the recommended methodology needs to make that decision according to the jurisdiction specific or site specific data. In addition, a conceptual model of interaction between the two factors is discussed below for future research. 3.4.6 Interaction Model of Fatigue Damage and Salt-Induced Rebar Corrosion Conceptually, corrosion may cause volume increase of steel rebars and thus concrete cracking. This can worsen the fatigue process and increase the rate of damage accumulation. Vice versa, fatigue damage (concrete cracking) may also worsen the damage caused by rebar corrosion. Both situations can adversely change the service life of RC decks. Accordingly, a model for the interaction is discussed here between load-induced fatigue and salt-induced corrosion in RC deck deterioration. A general format of the model is pro- posed first, with its rationality presented. In the following section, two major quantities, service lives subjected to load- induced fatigue or slat-induced corrosion only, are further elaborated including the concept of determining these quan- tities. Then the types of data needed are discussed to fully develop the interaction model. It is suggested that a separate data collection effort needs to be designed and carried out to accomplish this. 3.4.6.1 General Format of Interaction Prediction Model A literature review was conducted in this task to identify candidate models for describing the interaction focused here. As a result, no models were found in the literature directly related to the subject of interaction between load-induced fatigue and salt-induced corrosion in RC deck deterioration. Furthermore, no research work was found in the literature that investigated the interactive chemical and physical mech- anisms, except that general observation was reported that RC decks deteriorate at a higher rate when both salting and load- ing become severer or either one of them becomes severer. Note that research efforts have been reported in the litera- ture on load-induced fatigue alone, including the influence of presence of water (e.g., Perdikaris et al. 1993 and Matsui and Yonhee 1993). In addition, research was also done on salt- induced corrosion in RC decks (e.g., Weyers et al. 1993). However, the deterioration prediction model proposed by Weyers et al. was calibrated using data from decks in service condition, apparently also subjected to truck load. Further- more, these bridges’ identification was not reported nor were the truck loads and their volume, so that further tracking back becomes very difficult if not impossible. Although these 54 available data do not meet all the requirements for this task of study interaction, researchers still use this current knowl- edge here in developing the interaction model concept. On the other hand, in structural engineering practice, there have been satisfactory interaction models for, for example, interaction between moment and axial load effects to columns and between shear and moment to beams. These interaction models were developed using statistical concepts. Namely, statistical fitting was used in the model develop- ment to determine the best fitting parameters for pre-deter- mined functions using physical testing data. These models are considered in this effort of developing the interaction model. Based on this state of the art, a statistical approach is pro- posed here for modeling the focused interaction between load and corrosion, as opposed to physical and chemical descrip- tion of the microscopic deterioration process for RC decks. This approach can be described using the following equation to indicate a deck’s end of service life: (3.4.6.1) where A is the deck’s current age (in years). Yd and Yc are the predicted service lives (in years) respectively considering load fatigue and rebar corrosion due to salting only. The exponents a and b in Eq. 3.4.6.1 are model parameters. They are always positive. When the left hand side of the formula is equal to or larger than 1, the end of service life is reached. When it is less than 1, the service life has not been exhausted. Thus Eq. 3.4.6.1 indicates the surface of the deck’s failure in a space of two dimensions—one for load-induced fatigue and the other for salt-induced rebar corrosion. Figs. 3.11 to 3.13 show three examples of this kind of sur- faces. Fig. 3.11 is for a = b = 1, Fig. 3.12 for a = b = 3, and Fig. 3.13 for a = b = 0.6. They show that when a and b are 1, Eq. 3.4.6.1 is a linear function in the space of A/Yd and A/Yc. When a and b are larger than 1, the surface of Eq. 3.4.6.1 is convex away from the origin (Fig. 3.12). When a and b are smaller than 1, it is concave to the origin (Fig. 3.13). A Y A Yd a c b  +   = 1 0 0.5 1 0 0.5 1 A/Yd A/ Yc Figure 3.11. Interaction model for a = 1 and b = 1.

55 service life subjected to rebar-corrosion due to salting only. This project has developed a model to estimate Yd as follows: (3.4.2.1b) This formula has been given earlier using the same identifi- cation in Section 3.4.2. The symbols have been defined there. It is listed here for convenience. On the other hand, estimation for Yc remains to be a sub- ject of research. The latest comprehensive research work on relevant issues perhaps is (Weyers et al 1993) completed in the SHRP program. That project developed a model for esti- mating Yc. However, the data used to determine the model parameters were not differentiated according to what and how much truck load had been applied to the decks that pro- vided performance data. Thus, data from better controlled experiments are needed to complete the development of the model for Yc for the purpose here. 3.4.6.3 The Need for Data to Complete Development of the Model in Eq. 3.4.6.1 The model parameters in Eq. 3.4.6.1 (exponents a and b, and other parameters in Yd and Yc) may be calibrated using statis- tical techniques applied to data of deck deterioration subjected to load-induced fatigue and salt-induced corrosion. Ideally, both laboratory and in-service data are needed which describe deck deterioration. The former refers to those obtained in the laboratory and the latter in the service condition for decks that have reached end of service life, i.e., have experienced renewal work, such as patching, overlay, or replacement. Laboratory experiments can provide data on deck deteriora- tion in a controlled environment. Data from decks in service can include factors that the laboratory experiments cannot cover, such as temperature and humidity fluctuation. With respect to truck load and salting, three types of data are needed to fully develop the interaction model. They are deck deterioration history data (1) under load fatigue only, (2) under salt-induced rebar corrosion only, and (3) under different severity combinations of both load fatigue and salt- induced corrosion. The first and second types of environment conditions can provide data to determine the models to predict Yd and Yc in Eq. 3.4.6.1, respectively. Note that Yd and Yc are defined above as the service lives without the other deteriorating fac- tor. In other words, they provide data points for areas where A/Yd is close to 1 and A/Yc is close to zero, and vice versa A/Yd is close to zero and A/Yc is close to 1. The data under the third type of environment can be used to determine the model parameters (exponents) a and b in Eq. 3.4.6.1. They mainly describe which factor (load-induced fatigue or salt-induced corrosion) is more dominant in which Y K K TC (R IP P/P ) u(1 u) u d d p d d s u 17.95 A 1 = + +  + − log log( ) 1 1 0 0.5 1 0 0.5 1 A/Yd A/ Yc Figure 3.12. Interaction model for a = 3 and b = 3. 0 0.5 1 0 0.5 1 A/Yd A/ Yc Figure 3.13. Interaction model for a = 0.6 and b = 0.6. When one of the two failure mechanisms is considered to be irrelevant (i.e., having little influence), the respective ser- vice life can be accordingly set to infinity. Thus the model shows, mathematically, zero influence from that failure mech- anism. For example, in areas where rebar corrosion due to salting is not significant (e.g., where no salting is ever per- formed), Yc can be set to infinity. Thus A/Yc is set equal to zero. The model becomes (3.4.6.2) to indicate end of service life. On the other hand, when load fatigue is deemed to be irrelevant (e.g., in areas where an extremely large amount of salt is used for deicing). Yd can be viewed to be infinitely large so that A/Yd can be set equal to 0. The model then becomes (3.4.6.3) to mark the end of service life. 3.4.6.2 Estimation of Yd and Yc According to the above discussion, Yd in Eq. 3.4.6.1 is the deck’s service life under load fatigue only. Yc is the deck’s A Y or A Yc b c   = =1 1 A Y or A Yd a d   = =1 1

regions. They will provide data points to guide the surface’s trend in Figs. 3.11 to 3.15 between the two points (A/Yd = 1, A/Yc = 0) and (A/Yd = 0, A/Yc = 1). Fig. 3.14 shows an exam- ple of the failure surface with a = 0.2 and b = 8. It displays a concave trend when A/Yd is small and convex when A/Yd is larger and closer to 1. This would be suitable for a behav- ior of dominant influence from load-induced fatigue. In other words, this case indicates relatively less influencing salt-induced corrosion. For comparison, Fig. 3.15 shows an opposite situation with salt-induced corrosion more domi- nant and thus the exponents a and b have their values switched as a = 8 and b = 0.2. The first type of data used in this project was from Perdikaris et al. (1993) in the laboratory condition and from several state DOTs for the in-service condition. They were used to develop the model in Eq. 3.4.2.1 to estimate Yd. For the second and third types of data, the researchers have tried to use data gathered in Weyers et al. (1993) to complete a model for Yc. As commented on above, unfortunately, these data do not have the bridge deck identified and truck load recorded. The researchers also contacted several DOTs and researchers experienced in this area, the following difficulties were encountered in the order of significance. 1. Salt usage is usually not recorded. The best data retriev- able were based on very “rough” estimation. 2. WIM truck wheel weight data are not available for specific bridge decks (i.e., those that experienced renewal work). Furthermore, no laboratory data were found in the litera- ture or other sources for the second and third types of data. These data points should be in the region where A/Yd is not close to 1 in Figs. 3.11 to 3.15 and A/Yc varies from almost zero to 1. Therefore, it is recommended that laboratory experiments and field data collection be designed specifically for the pur- pose of developing the subject interaction model. It appears that gathering data from efforts with other research purposes will not meet the need here. 56 3.5 DEFICIENCY DUE TO OVERSTRESS FOR EXISTING BRIDGES 3.5.1 Level I Analysis The concept of overstress criteria recommended here is consistent with that employed by state transportation agen- cies, based on the AASHTO load rating procedure. Namely, for a specific truck, if a bridge’s rating factor is below 1.0, the bridge is considered to be overstressed for that truck. Typically, load rating requires detailed structure analysis, which would be resource consuming if every bridge needed to be analyzed in a large network. A conser- vative approach is adopted in the recommended methodol- ogy for Level I analysis, to reduce the amount of work needed for re-rating every bridge. It uses the existing rating factor as follows: RFAS = RFBC(MBC, rating vehicle /MAS, rating vehicle)/AFrating (3.5.1.1) where RFAS is the rating factor for the Alternative Scenario. RFBC is the rating factor for the Base Case (likely the existing rating factor). MBC, rating vehicle /MAS, rating vehicle is the ratio between the maximum moments due to the rating vehicle under the Base Case and due to the new rating vehicle under the Alter- native Scenario. When continuous spans are analyzed, this is the maximum ratio of those for all critical sections. This ratio should not be larger than 1, otherwise it is set at 1. This is because when this ratio is larger than 1, it means that the moment effect of the new rating vehicle is smaller than that of the current rating vehicle. Thus, the new vehicle load effect would not govern in the process of load rating. Generic spans (without specific details from the plans) may be used to find these moment ratios for the interested spans. AFrating is the ratio between the live load factors for the Base Case and the Alternative Scenario, according to Eq. 2.3.3.1: AFrating = [2WAS* + 1.41t(ADTTAS)σAS*]/[2WBC* + 1.41t(ADTTBC)σBC* ] (3.5.1.2) 0 0.5 1 0 0.5 1 A/Yd A/ Yc Figure 3.15. Interaction model for a = 8 and b = 0.2. 0 0.5 1 0 0.5 1 A/Yd A/ Yc Figure 3.14. Interaction model for a = 0.2 and b = 8.

where W* and σ* are the mean and standard deviation of the top 20 percent of the TWH, and t is a function of annual daily truck traffic (ADTT) as given in Section A-3 in Appendix A. Subscripts BS and AS respectively refer to the Base Case and the Alternative Scenario. The ADTT data for the Base Case can be taken from the agency’s bridge inventory or the NBI according to the functional class of the roadway that the bridge carries. The ADTT for the Alternative Scenario results from the prediction for future TWH using the recommended method presented in Section 2.4. Eq. 3.5.1.2 also indicates that this ratio of load factors should not be less than 1. In case the calculated value of the ratio is indeed less than 1, then it is set equal to 1. This is because the new safety factor would not be lower than then current load factor for the purpose of cost estimation here. In Eq. 3.5.1.1, MBC, rating vehicle /MAS, rating vehicle reflects the adjustment to rating due to the new truck model. As dis- cussed earlier in Chapter 2, the new truck model should rep- resent the practical maximum truck loads under the Alter- native Scenario. It is envisioned that determining this model would not be difficult at a state agency, using avail- able expertise in both areas of bridge structures and trans- portation planning. The adjustment factor AFrating is the ratio of the live load factors discussed in Section 2.3.3.2. The adjustment covers uncertainty changes in truck weight spectra, expected to result from the considered Alternative Scenario. For cost estimation, those bridges that are inadequate with RFBC < 1 under the Base Case should be excluded, because they do not contribute to the cost impact (additional costs). When a bridge is found to be inadequate or overstressed under the Alternative Scenario but adequate under the Base Case (RFBC >1 and RFAS < 1), an action needs to be selected as the basis for cost estimation. It can be, for example, post- ing, strengthening, replacing, or a combination thereof. Note that, in reality, the decision-making process requires infor- mation on a number of other factors. Such information may not be available when an application of the methodology is conducted. For example, whether this bridge is on a road that has several other bridges needing repair at the same time can be such information. Thus, this decision is to be made by the user with available information as well as engineer- ing judgment. 3.5.2 Level II Analysis This level of analysis requires more data and more analy- sis effort. It requires re-rating every bridge in the network using the rating truck model under the Alternative Scenario. Then the resulting rating factor is modified as follows to arrive at the rating factor for the Alternative Scenario: RFAS = RFBC, using AS rating vehicle /AFrating (3.5.1.3) 57 where RFBC, using AS rating vehicle is the rating factor using the new truck model under the Alternative Scenario and the live load factor under the Base Case. Comparison of Eqs. 3.5.1.3 and 3.5.1.1 indicates that the Level II analysis requires a bridge- by-bridge approach for re-rating, using the new rating vehi- cle under the Alternative Scenario. The exact critical sections will be identified and used in this process. It increases the accuracy of the result but possibly requires a larger amount of analysis work, if the network is extensive. 3.6 DEFICIENCY DUE TO OVERSTRESS FOR NEW BRIDGES As discussed in Section 2.3.4, the bridge design load is required to statistically envelope current and future truck loads over the expected life spans of the bridges to be designed. The design load needs to be updated when a significant percent- age of the trucks are to change in terms of their weight dis- tribution and total weight. This is typically expected when the considered Alternative Scenario is to legalize certain types of trucks or to permit routine overweight trucks with- out controlling the number of trips they may make. Of course the degree of impact depends on the nature of the Alternative Scenario. Eq. 2.3.4.1 indicates that this can be quantified using the mean and standard deviation of the top 20 percent of the TWH. When the design load is changed, new bridges will cost differently from those under the old design load. This con- tributes to the cost impact under this category. The recom- mended methodology uses the concept of incremental cost allocation (Saklas 1998). This approach attributes incremen- tal costs to respective groups of vehicles that trigger the increments. Accordingly, the considered Alternative Scenario is viewed responsible for the incremental costs here for a new design load. As in the cost impact category for deficient exiting bridges, a new truck model needs to be determined which is able to cover the practically possible legal or permissible vehicles. This can be the practical maximum vehicle under the Alter- native Scenario. A similar approach to that used in Cost Impact Category 3 for deficient existing bridges can be used to determine this model. It may consist of several vehicles, depending on the considered Alternative Scenario. The second step for this category is to generate the TWH for the entire network under the Base Case, and then predict the TWH for the network under the Alternative Scenario. These two TWHs will be used below to determine a live load factor ratio as part of the new design load. For a state agency, these TWHs should usually be representative for the entire state, not site specific or functional class specific as in Cost Impact Category 3, because, most likely the live load factor for design is uniform for the entire state. An exception may be that certain local bridges are considered not subject to

general heavy loads and higher truck traffic and therefore a lower design load is justifiable. 3.6.1 Level I Analysis At this level of data requirement and the related amount of analysis, new bridges constructed in recent years are used to estimate the costs that are expected if the considered Alternative Scenario is implemented. This analysis will require the following further steps: (1) Identify the new bridges constructed in the past Q years. Q needs to be deter- mined with consideration to the number of new bridges to be included. (2) Estimate the required design load for each of these bridges under the Alternative scenario. (3) Esti- mate the additional costs for each of these bridges under the new design load. Step (1) is feasible using the agency’s bridge inventory or the NBI as the default database, where the year built is recorded for each bridge in the system. Q, the number of years the analysis should track back largely depends on the number of bridges to be included. When a large number of bridges belong to this group for a relatively large network, fewer years may be used for a small Q. In contrast, if a small number of bridges were typically constructed in each year, a larger number of years would be desired to arrive at a reli- able annual cost for new bridges. Thus Q may need to be determined by iteration combined with the sensitivity analy- sis to be discussed below. Step (2) is to be accomplished using the following formula for the amount of design load change: DLCF = (MAS design vehicle /MBC design vehicle)AFdesign (3.6.1.1) MAS design vehicle /MBC design vehicle > 1 (3.6.1.2) AFdesign = (2WAS* + 6.9σAS* ) /(2WBC* + 6.9σBC* ) AFdesign > 1 (3.6.1.3) where DLCF stands for design load change factor indicating the ratio between the design loads under the Base Case and the Alternative Scenario. MAS, design vehicle /MBC, design vehicle is the ratio of the maximum moments due to the design vehicle under the Base Case and the same under the Alternative Sce- nario. Practically, it should not be lower than 1. Namely, when MAS design vehicle is smaller than MBC design vehicle, the design vehi- cle under the Base Case would be the governing load and the ratio should be taken as 1 in Eq. 3.6.1.2. This will assure that the new design load will not be lower than the current design load. AFdesign is the ratio between the live load factors under the Base Case and the Alternative Scenario. It is an adjust- ment factor for design used to cover the change in uncer- tainty associated with the considered Alternative Scenario. It 58 plays a similar role as AFrating in Eq. 3.5.1.2 for additional deficiency in existing bridges. In Level I analysis, the AASHTO HS load is assumed to be the design load for the current norm of design load, although the HL93 has been adopted in the AASHTO LRFD design specifications (1998). It is because the HS load has been the design load for many bridges in service today, which provide the data needed to project to future bridges. One of the important sets of data here is the cost increase data for design loads beyond the HS-20. The default Data Set A-5.2.7 in Appendix A refers to HS-20 as the reference for possibly higher design loads. No such data are found available refer- ring to the HL93. Thus Step (3) of this analysis is to compare the design load under the Alternative Scenario with that under the Base Case with reference to the HS load. This can be done using the DLCF resulting from Eq. 3.6.1.1, which is the mul- tiples of the new design load compared with the Base Case design load. Appendix A includes cost data for additional design load with reference to HS-20 in Data Set A-5.2.7 in Appendix A. They can be used as the default data if more specific data are not available. Note that these data may be used to extrapolate to situations where the design load is beyond the ranges given there, as appropriate. 3.6.2 Level II Analysis The recommended methodology at this level of analysis requires information about every individual bridge in the net- work. First of all, the new bridges to be constructed need to be identified using more specific information than the bridge inventory. Such information could include, but not be limited to, the agency’s capital program for the next several years, the agency’s long term plan for expenditure and candidates for new construction and replacement for the next 5 or 10 years. Next, the configurations of these future new bridges need to be identified. This information is needed to perform the analysis defined in Eq. 3.6.1.1 to 3.6.1.3. Note that this analysis can be more accurate if more details about the con- figurations can be provided by the agency. The rest of analy- sis for the Level II requirement will be identical to that for the Level I requirement. 3.6.3 Sensitivity Analysis The following parameters may need examination at the stage of sensitivity analysis. (1) The window parameters used in the TWH prediction method for the Alternative Scenario, defined in Fig. 3.1. (2) The bridge sample identified, if a Level I analysis is performed. More years of record of recent new bridges in the network may be included and averaged to produce an annual cost for this category of cost impact. (3) Possible increase of available resources due to economic

growth that result in more new bridges to be built. This may be covered at the network level by a growth factor to the total costs obtained. 3.7 TOTAL COST-IMPACT CALCULATION When all the four cost-impact categories are covered using the concept described in this chapter, individual contributions 59 from these categories need to be summarized to find the total cost. The summation process should use the principles of engineering economics. A discount rate will need to be pre- determined for this purpose, which could be one of the factors subjected to sensitivity analysis. It is recommended that all costs be converted to the same format of expression. Options of this uniform format may be present worth, annual costs for the next PP years, etc. Note that these different forms are equivalent with a discount rate consistently included.