Bridge User Cost Estimation A Synthesis of Existing Methods and Addressing the Issues of Multiple Counting, Workzones, and Traffic Capacity Limitation

Transcription

1 Bridge User Cost Estiation A Synthesis of Existing Methods and Addressing the Issues of Multiple Counting, Workzones, and Traffic Capacity Liitation Qiang Bai (Corresponding Author) Graduate Research Assistant, School of Civil Engineering Purdue University, 0 Stadiu Mall Drive West Lafayette, IN 0 Phone: () - Eail: qbai@purdue.edu Sauel Labi Assistant Professor, School of Civil Engineering Purdue University, 0 Stadiu Mall Drive West Lafayette, IN 0 Phone: () -, Fax: () - Eail: labi@purdue.edu Kuares C. Sinha Olson Distinguished Professor Purdue University School of Civil Engineering 0 Stadiu Mall Drive West Lafayette IN, 0 Phone: () -, Fax: () - Eail: sinha@purdue.edu Paul D. Thopson Consultant Hardwick Ct, Castle Rock, CO 00 Phone: (0) - Eail: pdt@pdth.co Subission Date: Noveber, 00 Nuber of Words in Text:, Nuber of Tables: Nuber of Figures: Total Equivalent Nuber of Words:, For presentation at the 0 th Annual Meeting of the Transportation Research Board and possible publication in the Transportation Research Record. TRB 0 Annual Meeting

2 0 ABSTRACT Efforts to incorporate the concerns of bridge users in project and progra evaluation are often styied by lack of a coprehensive and consistent fraework for assessing the different user cost types and coponents. There is a need to synthesize and update existing user cost estiation techniques so that the process of incorporating user costs in bridge investent evaluation can be ore consistent and strealined. Secondly, user costs during bridge workzones have rarely been considered in the literature. Thirdly, there is a need to recognize that a bridge detour ay occur for ore than one reason, thus there is a danger of ultiple counting and this could translate into overestiation of user cost. To address these issues, this paper presents a fraework for coprehensive estiation of user costs for bridge anageent, a ethodology for bridge workzone user cost estiation, and an approach to address the issue of ultiple counting. Furtherore, the paper develops a ethod to estiate the bridge user delay cost due to traffic capacity liitation. The ethodologies are deonstrated using a case study. TRB 0 Annual Meeting

3 INTRODUCTION Background Inforation and Proble Stateent Coprehensive evaluations of bridge investents need to incorporate explicitly the concerns of ajor stakeholders including bridge users (-). Bridge user cost is caused ostly by functional deficiencies of a bridge () such as liitations of load capacity or vertical clearance. These liitations cause vehicles to detour, hence increasing the cost of vehicle operations and travel tie. User costs are also incurred in the case of oveable bridge openings, where bridge users are delayed as the bridge opens up to enable ships to pass through (). The existing literature contains a fairly extensive aount of past research on the subject of bridge user cost estiation. In North Carolina and Indiana, user cost was first ipleented in the state bridge anageent systes in the 0s (,). In bridge anageent software such as PONTIS and the Indiana Bridge Manageent Syste IBMS, bridge user costs are considered in calculation of project benefits as a reduction in user cost due to bridge actions (,, ). Johnston et al. () developed user cost estiation ethods that considered accident and detour user costs, duly recognizing that accident user cost is higher where there is narrow deck width, poor alignent, or ipaired vertical clearance, and that detour cost is a function of load capacity and vertical clearance liits. Siilarly, Son and Sinha (), in their life-cycle user cost estiation ethodology for IBMS, considered detour cost due to load capacity and vertical clearance liits and also considered the travel tie cost of speed reduction due to narrow lane width. Thopson et al. (-) in developing a Pontis user cost odel for Florida DOT, incorporated bridge width, approach alignent, vertical clearance, and operating rating in user cost estiation. In presenting the different cost coponents of bridge user cost estiation, past studies focused on the costs during the period of noral operations of the bridge with little or no explicit inclusion of costs incurred during preservation workzone periods. Also, in calculating bridge user costs due to detours, ost past studies did not coprehensively account for the possibility that a vehicle ay detour for ore than one reason; as such separate calculations for each of the ultiple reasons for detouring ay yield overestiates of total user cost. Furtherore, even though soe studies considered user cost due to narrow bridges (,), their calculation structure either calculates bridge widening benefit as accident cost reduction or only presents bridge widening benefit in the for of congestion delay reduction under the situation of bridge widening without adding lanes. Thus, the estiation of bridge widening benefits due to lane addition iproveents was not addressed. Last but not least, the poor condition of the bridge wearing surface can also cause additional Vehicle Operating Cost (VOC) but is not sufficiently addressed in the literature. In recognizing the above shortcoings of the existing ethodologies for bridge user cost calculation, this paper presents an updated and coprehensive fraework for user cost calculation. This is largely based on ethods presented in the existing literature but includes refineents that consider all coponents of user cost in the bridge life-cycle horizon including noral operations periods and workzone periods. The refineents include ethodologies to reliably calculate detour user costs to avoid the proble of ultiple counting in the calculation of user benefit for all types of bridge actions, and in the calculation of user cost due to inadequate bridge traffic capacity and poor condition of TRB 0 Annual Meeting

4 0 the wearing surface. A case study is presented to deonstrate the applicability of the suggested fraework and the new refineents. Incorporating Calculated User Costs in Bridge Life-cycle Cost Analysis Life-cycle cost analysis is a widely-used technique for aalgaating the costs and benefits of bridge investents that are accrued to the bridge agency, users, and the affected counity over the long ter (,0). The accuulation of bridge user costs over tie is best appreciated in a life-cycle context. A typical life-cycle profile for a steel bridge is shown in FIGURE (). In evaluation of bridge investent alternatives, the user costs corresponding to each alternative can be first estiated in undiscounted dollars, and then brought to their present worth or equivalent unifor annual aounts for purposes of coparison (). Bridge Construction Deck Replaceent Widen & Deck Deck Replace + Raise Replaceent Bridge Replaceent th year 0 th year th year th year 0 th year FIGURE Typical bridge life-cycle profile. There are two distinct periods during the bridge life-cycle that influence the coponents of user cost that need to be considered: the noral operations period where the bridge is being used by traffic and there is no repair activity on the bridge; and the workzone period where there is a repair action on the bridge that ay ipede traffic flow. Copared to forer, the workzone period has short duration but ay have considerable user cost ipacts, often due to reduced bridge width, lower speed liit, or even coplete bridge closure. The estiation of bridge user costs should be carried out for the entire bridge life cycle, and thus should consider the various diensions of user cost incurrence: noral operations and workzone periods; road users on the highway located over the bridge or/and under the bridge; and the key coponents of user costs, naely: detour, accident, and delay costs. TRAFFIC FLOW CONTEXTS, SERVICE FEATURES, AND ASSOCIATED BRIDGE USER COSTS For purposes of user cost coputation, we herein describe two contexts of traffic flow on the road of interest relative to the bridge (FIGURE ). Road Traffic Flow over the Bridge In this case, the road is over the bridge (FIGURE (a)). The bridge passes over a water body, railway, another road, wildlife crossing, etc. Liitations in bridge width and load capacity produce road user costs. Relatively unusual is the situation where a bridge has vertical clearance liitations over the bridge: such instances include roadway on a through-trusses (where clearance is ipaired by sway bracing); roadways that, at the location of the bridge in question, also pass beneath older railroad bridges; and the even ore rare case of -level interchanges where the iddle-level bridge ay have vertical TRB 0 Annual Meeting

5 clearance liitations over the bridge. If the facility under the bridge is non-transportation related, then user cost coputation is applicable only to users of the road. However, if the service feature is a secondary road, or transporting water body, then user costs should be calculated for two classes of users: users of the road (over the bridge); and users of the facility (road, water vessels) passing beneath the bridge. Road Traffic Flow under the Bridge In this case, the road is under the bridge (FIGURE (b)). The facility over the bridge could be a pedestrian crossing, wildlife crossing, or another road. Feature over the bridge Bridge Bridge 0 0 Road FIGURE Traffic Flow Contexts TABLE presents the factors to be taken into consideration in calculating bridge user cost for the different contexts of traffic flow. It is seen that a bridge deficiency ay incur ore than one of the three cost coponents, naely, detour, accident, and delay. Besides the causes listed in Table, special events, such as incidents, also cause bridge user cost. For exaple, after an accident, the bridge ay be partially or fully closed thus causing vehicles to slow down or detour; leading to user costs of detour or delay. It is iportant to note that in certain cases of bridge replaceent or rehabilitation, there will be little or no detour (and its associated costs) because the agency provides an alternative teporary bridge structure in close proxiity to the original structure. It is also iportant to note that iediately after a bridge accident, user costs could be incurred due to the subsequent delay (due to accident clearance tie) and even to secondary accidents such as rear-end crashes. TABLE Major User Cost Coponents Traffic Flow Context Users of the road over the bridge Users of the road under the bridge Feature under the bridge (river, rail, road, etc.) (a) Road Traffic is over Bridge Periods and Causes of Cost Incurrence Road (b) Road Traffic is under Bridge Detour Cost Accident Cost Operating Periods Inadequate load capacity Yes Inadequate vertical clearance over bridge Yes Yes Inadequate horizontal clearance over bridge Yes Yes Poor alignent Yes Yes Inadequate traffic capacity Delay Cost Yes Wearing surface condition Workzone Periods Workzone-related activities Yes Yes Yes Operating Periods Inadequate vertical clearance under bridge Yes Yes Inadequate horizontal clearance under bridge Yes Yes Workzone Periods Workzone-related activities Yes Yes Yes Additional VOC Yes TRB 0 Annual Meeting

6 COMPUTATIONS FOR BRIDGE USER COST COMPONENTS Bridge users incur costs during the noral operations period or during workzone periods. During each of these periods, the specific coponents, or circustances in which the cost is incurred, include the delay and safety due to restricted bridge operating conditions or detours, as indicated in TABLE. The derivations of the expressions for the different user cost coponents share soe siilarities. In the ensuing sections of this paper, we explain the user costs due to detour, delay, accidents, and wearing surface condition. Bridge User Cost Due to Detour The cost incurred by bridge users due to detour continues to be a doinant coponent of bridge user cost (,, ). In spite of earnest efforts toward bridge iproveent, there still exists a large nuber of bridges with deficiencies that cause vehicles to detour. For instance, % bridges on Indiana s interstate syste have inventory rating under 0 tons and.% have operating rating under 0 tons. Trucks of 0 tons are allowed on the Interstate Highway Syste according to the federal coercial vehicle standards (). With regard to vertical clearance, for about,0 bridges, the feature intersected by the bridge) is another highway. Of these, % have vertical clearance under ft. There is no federal vehicle height liit, but state axius of vehicle height range fro. to. ft. As stated in earlier sections of this paper, ost detours are due to inadequacies during noral operations (such as liitations on bridge load capacity, and vertical or horizontal clearance) or operational restrictions during workzone periods. Detours cause an increase in travel distance and tie for road users, resulting in additional vehicle operating and travel tie costs. The list below presents the expressions for calculating bridge detour costs, due to the indicated deficiencies. Load capacity liit: Vehicle Operating Cost = a UVOC DL N L ; Travel Tie Cost = DL a U TTC NL SP( Vertical clearance liit (over or under): Vehicle Operating Cost = a U VOC DL NV ; Travel Tie Cost = DL a UTTC NV SP( Horizontal clearance (over or under): a Vehicle Operating Cost = U VOC DL NH ; Travel Tie Cost = DL a UTTC NH SP( Poor alignent: Vehicle Operating Cost = a UVOC DL N P ; Travel Tie Cost = DL a U TTC N P SP( Traffic flow liitations due to workzone: Vehicle Operating Cost = a UVOC DL NW ; Travel Tie Cost = DL a U TTC NW Where: DL = detour length (iles); = nuber of vehicle classes; U VOC (= unit vehicle operating cost of vehicle class i (dollars/ile); U TTC ( = unit travel tie cost of vehicle class i (dollars/hour); SP( = average speed of vehicle class i on detour (iles/hour); and SP( TRB 0 Annual Meeting

7 0 0 N a L (, N a V (, N a H (, N a P (, and N a W ( are the actual nubers of class i vehicles that detour due to load liit, vertical clearance liit, inadequate horizontal clearance liit, poor alignent, and workzone, respectively. N a L ( = N L ( ( PV L ) where N L ( = theoretical nuber of class i vehicles that ust detour due to load liit, and PV L is the percentage of load liit violators (vehicles that actually do not detour even though they exceed the load liit) Siilarly, N a V ( =N V ( (- PV V ), N a H ( =N H ( (- PV H ), ( =N P ( (-PV P ), and N a W ( =N W ( (-PV W ), where the sybols and subscripts are defined siilarly as done for load liit. In these expressions, the unit vehicle operating cost U VOC and unit travel tie cost U TTC can vary significantly depending on prevailing circustances. Derivations of U VOC and U TTC are well addressed in the literature (,, ). Rather, what is worth addressing in this paper is the deterination of the nuber of vehicles that detour, considering possible ultiple-counting issue. Subsequent sections of this paper describe in detail each reason for detouring and derive the expressions for estiating the nuber of detouring vehicles. Detour Due to Inadequate Load Capacity Vehicles with weight exceeding the posted load liit need to detour. In a deterinistic scenario, each vehicle weight is known, thus the nuber of detouring vehicles can be deterined easily. In reality, however, it is nearly ipossible to know a-priori the individual weights of the vehicles. Therefore, a probabilistic approach is appropriate. If the vehicle weight distribution is known, the probability that the bridge user detours is deterined as the shaded area under the probability density function (FIGURE ). The percentage of vehicles that detour is represented by the probability that a vehicle weight exceeds the posted load liit. Nr. or % of Vehicles N a P Distribution Function for Vehicle Weights, f(w) 0 FIGURE Deterining the proportion of vehicles detouring due to load liit. For the vehicle type i, the nuber of detouring vehicles that detour is calculated as: N L = PL Percentage( ADT () where ADT = average daily traffic on the bridge; Percentage( = percentage of vehicle type i; P L is the probability that a vehicle weight exceeds the posted load liit, and is W MAX equal to f ( w) dw,where W MAX is the axiu weight of type i vehicle PL W in Posted Load Liit, (PL) W ax Weight and f(w) is the probability density function for weights, w, of vehicle class i. TRB 0 Annual Meeting

8 Other variables have the sae eanings as presented earlier. Finally, in calculating the nuber of vehicles that detour due to inadequate load capacity, it is iportant to recognize that the load capacity itself does not reain constant over tie. The cobined and accuulated effects of truck loading, cliate, and aging culinate in deterioration, weakening, and consequently, reduction of bridge load capacity in the long ter. The load capacity of a bridge at any point in tie within its life could be deterined ostly using deterioration odels that predict load capacity at a future year. In rare and specific instances, field tests have been used to deterine load capacity. Chen et al. () presented results of research on bridge load capacity deterioration. Bridge load capacity can also be deterined using structural analysis software such as AASHTO's Virtis (), in conforance with AASHTO standards. The current standard is AASHTO s Load and Resistance Factor Rating Specifications (). In addition, it is iportant to note that the posted load liit during the workzone period ay be lower than that posted during the period of noral operations. As ay be recognized in the preceding section, estiating the nuber of detouring vehicles (Equation ()) hinges on the integrity of the vehicle weight distribution function, f(w). Fekpe and Clayton () developed a odel to predict the heavy-vehicle weight distributions. Sobanjo and Thopson () analyzed the distribution of truck weights at interstate and non-interstate highways in Florida and recognized that the truck weight distributions differ fro year-to-year and fro place-to-place. FHWA s Office of Highway Policy Inforation aintains a Vehicle Travel Inforation Syste (VTRIS) which contains data fro weigh stations in states and generates vehicle weight distributions in different highway classes. For analyzing user costs at a given bridge, a coon vehicle weight distribution generated using data fro a bridge with siilar traffic characteristics could be used for the analysis. However, the ideal situation is to collect vehicle weight data at the bridge of interest and then develop a weight distribution function specifically for that bridge. Detour Due to Inadequate Vertical Clearance Users of a highway passing on a bridge need to detour if their vehicle height exceeds the vertical clearance liit, if any, over the bridge. Typically, this is liited to only a few specific designed types or highway interchanges. Siilarly, users on a highway passing under a bridge need to detour if their vehicle height exceeds the vertical clearance liit under the bridge. Additional inforation on bridge vertical clearance inadequacies are found in the descriptions for Ites and of the Recording & Coding Guide (). The nuber of vehicles that need to detour can be calculated as: NV = PV Percentage( ADT () where P V is the percentage of vehicles whose height exceeds the bridge vertical clearance, or the probability that the height of a prospective bridge user vehicle exceeds the vertical clearance; other variables have the sae eanings as presented in Eqn (). Siilar to the case for load-related detours, not all trucks, trailers, and tractors are affected by the vertical clearance liit: only a fraction of the typically need to detour due to height liitations. The federal law does not set a height liit for trucks. However, typical state standards for truck height liit range fro. ft to. ft (). There sees to be no truck height data at the national level. However, at the state level, Sobanjo TRB 0 Annual Meeting

9 and Thopson developed truck height distribution functions for Florida s interstate and non-interstate highways (). Detour Due to Inadequacies in Horizontal Clearance For users passing over or under a bridge, detour due to horizontal clearance liitations ay be necessary ( during noral operations if the bridge is narrow and (i during workzone periods when the operating traffic lanes on the bridge are narrowed to allow repair work to be carried out on the bridge. In both cases, if the available width of the bridge is saller than the vehicle width, the vehicle detours. The nuber of detouring vehicles, which depends on the posted horizontal clearance and distribution of vehicle widths, can be calculated as follows: N H = PH Percentage( ADT () where P H = the probability that the width of a road user s vehicle exceeds the posted horizontal clearance or the percentage of vehicles whose width exceeds the posted horizontal clearance; other variables have the sae eanings as in Equation (). Detour Due to Poor Alignent In certain situations, large vehicle ight not be able to aneuver safely in crossing a bridge. This typically occurs on bridge with poor approach alignent. In such cases, the highway agency often specifies the axiu length of vehicles that can use the bridge. Vehicles whose lengths exceed this liit will need to detour. For a given bridge, the nuber of vehicles that need to detour due to poor alignent can be calculated as: N P = PP Percentage( ADT () where P P = the probability that the length of a prospective bridge user vehicle exceeds the vehicle length liit of the bridge or the percentage of vehicles whose length exceeds the vehicle length liit; other variables have the sae eanings as presented in Equation (). Detour Due to Workzone Due to repair activities associated with a bridge workzone, a bridge ay have reduced load capacity and/or vertical clearance over or under the bridge copared to the noral operation period. Also, during workzones, the highway agency often iposes liitations on the length and width of vehicles that can use the bridge. In these situations, vehicles that do not eet these standards ust detour. For calculating workzone user cost incurred by detouring vehicles due to each of these detour reasons, the procedure is siilar to that presented in previous sections for noral operations. In the case where there is a bridge closure during the workzone, all vehicles need to detour. Even though the workzone period is of relatively short duration copared to noral operations of the bridge, detour costs at such periods can be critical in investent evaluation because large nubers of detouring vehicles at a workzone typically culinate in significant overall user cost. The Issue of Multiple-Counting in Detour Coputations In practice, a vehicle ay have excessive weight and height that violate the posted weight and height liits. Such vehicles will need to detour for these two reasons siultaneously. In estiating of the benefit of bridge replaceent, Golabi et al. () considered the possible double-counting proble but did not do so for other iproveent actions. In TRB 0 Annual Meeting

10 ost other studies, the user costs of detouring vehicles were calculated separately for excess weight and for excess height, and sued to yield the total detour cost. In reality, however, one detour ay be due to both height and weight restrictions, causing double counting. Siilarly, when there are ore than two reasons for detouring, there is a possibility of ultiple counting. To illustrate our arguent about the possibility of ultiple counting and to avoid consequent overestiation of user cost, we consider the following situations: (a) Vehicles at a bridge detour for at least one of only two reasons: weight (D ) and height (D ). Traditional calculation: Fro FIGURE (a), vehicles detour due to weight and vehicles detour due to height. So overall, the traditional ethod, which does not consider the double counting issue, suggests that vehicles detour. Proposed calculation: Noticing that vehicles detour due to both reasons and should not be double counted as done in the traditional calculation, it can be seen that only vehicles detour, overall. Thus, failure to consider the issue of double counting in this exaple proble leads to a significant nuber of vehicles being counted twice, resulting in a 0% overestiation of the nr. of detouring vehicles and thus, of user cost. (b) In this case, vehicles require a detour for at least one of three reasons: weight (D ), height (D ) and width (D ). Traditional calculation: Fro FIGURE (b), vehicles detour due to weight, vehicles detour due to height, and vehicles detour due to width. So, the traditional ethod, which does not consider the triple and double counting issues in this proble, suggests that 0 vehicles detour overall. Proposed calculation: It can be noticed that (++) vehicles detour due to two reasons and should not be double counted as done in the traditional calculation. Also, vehicles detour due to three reasons siultaneously and should not be triple counted as done in the traditional calculation. The figure shows that only vehicles actually detour. Thus, failure to consider the issue of triple counting and double counting in this exaple proble leads to a significant nuber of vehicles being counted twice or thrice, resulting in % overestiation of the nuber of detouring vehicles and thus, of user cost. Where there are over three reasons for detouring, it is difficult to visualize the proble and we herein use set theory to address this. Weight (D ) 0 (a) Double counting Height (D ) Weight (D ) 0 FIGURE Multiple-counting illustration. Width (D ) Height (D ) (b) Triple counting and double counting TRB 0 Annual Meeting

11 0 0 0 We now present a generalized approach to calculate the nuber of detouring vehicles while duly accounting for the issue of ultiple counting. Based on the set theory (), the nuber of detouring vehicles due to any one or ore of n detour reasons can be derived as follows: N( D U D ULU Dn ) n = n N( D j ) N( Di I D j ) + () N( D j I Dk I Dl ) L+ ( ) N( D I D ILI Dn) j= j< k n j< k< l n where N(D j ) = the nuber of detour due to reason D j. N D U D ULU D ) of class i vehicle can be written as: N ( n ( D D ULU Dn ) P( D U D ULU Dn ) * PADT( * P( D U D ULU Dn U = ADT () where ) = the probability that a vehicle detours due to any one or ore of D to D n factors. Equation (), which is based on set theory, can be used to calculate the total percentage of vehicles that detour: P( D U D ULU Dn ) n = n P( D j ) P( Di I D j ) + () P( D j I Dk I Dl ) L+ ( ) P( D I D ILI Dn ) j= j< k n j< k< l n P(D j ) is the probability that a vehicle detours due to reason D j. P( D I D ILI Dn ) represents the probability that the vehicle detours due to all n reasons siultaneously. For exaple, if there are only two reasons for detouring (load capacity and vertical clearance), then P [( w > PL) U ( V > VC)] = P ( w > PL) + P( V > VC) P[( w > PL) I ( V > VC)] () where the PL is the posted load liit and VC is the vertical clearance liit. The rest of the variables have the sae eanings as presented for Equations () to (). Conditional probability theory is helpful for calculating P( D I D ILI Dn ). For exaple, for the case where there are two reasons for detouring, the percentage of detouring vehicles, for vehicle type j, is calculated as follows P [( w > PL) I ( V > VC)] = P [( V > VC) P[( w > PL) ( V > VC)] () or P [( w > PL) I ( V > VC)] = P [( w > PL) P[( V > VC) ( w > PL)] (0) Each variable has the sae eanings as in Equation (). Bridge User Delay Cost Due to Traffic Capacity Liitation Users of narrow bridges reduce speed at such facilities (). As such, a narrow bridge can cause additional user cost due to speed reduction (and thus, increased travel tie) during noral operations of the bridge (). Also, during periods of bridge workzone, partial lane closure at the workzone location ay reduce the bridge s traffic capacity and incur extra travel tie. Iproveents such as bridge widening ay increase the lane width or add lanes () and thus increase bridge traffic capacity. For a given traffic volue, when road capacity increases, the travel speed also increases (). Thus, the user cost due to traffic capacity liitation can be expressed as the additional travel tie cost due to speed reduction copared to a baseline or free-flow travel speed: TRB 0 Annual Meeting

12 TTCCL = ( AddedTie( k) ADT ( k)) UTTC () k = Where: U TTC ( is the unit travel tie cost of vehicle class i; ADT(k) is the traffic volue in the k th hour; AddedTie (k), the added travel tie during k th hour (hours/vehicle), is calculated as follows: Speed FreeFlow Speed Actual ( k) AddedTie ( k) = BridgeLeng th () Speed FreeFlow Speed Actual ( k) Where: Speed FreeFlow = free flow travel speed ; Speed Actual (k)= actual travel speed in the k th hour. The actual travel speed can be derived fro the Bureau of Public Roads function () as follows: SpeedFreeFlow SpeedActual( k) = b + a( V / C) () Where: a, b are paraeters representing different highway classes and different speed liits (); and V/C is the volue/capacity ratio for the k th hour. Bridge User Cost due to Accidents The user cost due to accidents is calculated as the product of the expected nuber of accidents due to bridge deficiencies or workzone and the unit onetary cost of each accident. This, which can be done for each accident type, is consistent with the North Carolina Bridge Manageent Syste procedure (, ). Thopson et al. () also developed an enhanced bridge accident odel that estiates annual accident rate as a function of nuber of bridge lanes, bridge length, road width, approaching alignent rating, deck rating, and traffic volue. Bridge User Cost due to Wearing Surface Roughness The condition of the bridge wearing surface can affect aintenance, repair, and depreciation cost of vehicles in the for of VOC. The bridge user cost due to wearing surface condition can be calculated as: U = U * BridgeLength * ADT * () Roughness VOC Where: U voc ( is the additional VOC due to wearing surface roughness for type i vehicle; ADT( = average daily traffic of type i vehicle. For calculating additional VOC due to surface roughness, Barnes and Langworthy () developed a relationship between additional VOC and International Roughness Index (IRI) and suggested that U voc (=0 when IRI<0 in/ile; U voc (=0.0 U voc (i ) when 0 in/ile IRI<0 in/ile; U voc (=0. U voc ( when 0 in/ile IRI<0 in/ile; U voc (=0. U voc ( when IRI 0 in/ile. where U voc ( is the unit VOC cost of type i vehicle. TRB 0 Annual Meeting

13 CASE STUDY The applicability of the fraework is deonstrated using a case study involving a steel bridge in the iddle level of a three-level interchange. This is a one-way Interstate bridge rap with lanes, reconstructed in. It is. long, has. ft vertical clearance over the bridge and has a load capacity of. tons. Under the bridge, there is a -lane highway with vertical clearance of. ft. The detour length for the roadway over and under the bridge is k. The 00 ADTs over and under the bridge are, and,0 respectively. To deonstrate the user cost calculation over the lifecycle, we assue that the bridge long-ter preservation strategy follows a standard lifecycle profile in IBMS (): deck replaceent in the 0 th year, th year, and th year; bridge replaceent in the 0 th year. Besides deck replaceent in the th year, bridge widening and bridge raising are also perfored (FIGURE ). The bridge widening adds one lane and the bridge raising increases the vertical clearance under the bridge fro. to.. During the rehabilitation in the 0th and the th years, the workzone is assued to narrow the operational bridge width to one lane for a -onth period. During the rehabilitation at the th year, the bridge is closed for a onth and bridge width is reduced to one lane for a - onth period. A traffic growth rate of % is assued over bridge life. As such, data on traffic volue and truck weight in Indiana were downloaded fro FHWA VTRIS to yield the following distributions: passenger car.%; bus 0.; single unit truck.%; tractor trailer.%. In VTRIS, the vehicle weight inforation only contains weights data for buses and trucks. In this case study, the load capacity of the bridge is. tons this is adequate for all autoobiles. Thus, the vehicle weight inforation in VTRIS is enough for this case study to establish the relevant vehicle weight distribution for purposes of user cost estiation. In this case study, we only consider buses, single-unit trucks, and tractor-trailers. Using the EasyFit statistical software package (0), cuulative distributions for the weights of these vehicle classes were established as follows: x.0 0. Bus weights: Burr distribution F( x) = ( + ( ) ). x. 0. Single unit truck weights: Burr distribution F ( x) = ( + ( ) ). Tractor trailer weights: Johnson SB distribution z F( x) = Φ(0.+.ln( )) z where z = x. and where Φ is the Laplace integral..00 For the vehicle height distribution, the findings of the Sobanjo and Thopson () study were used. That study provided height distribution functions for trucks on interstate highways: when the vertical clearance liit is ft, the percentage of trucks that detour is estiated using the function (.0E+) x -. ; when it is. ft, the percentage of trucks that detour is estiated by. 0.0x, where x is the vertical clearance liit in feet. The vertical clearance over/under the bridge is adequate for passenger cars and buses, thus the height distribution of trucks only is considered in the TRB 0 Annual Meeting

14 analysis. Over the bridge, since both the vertical clearance and the load capacity liit ay cause detouring of trucks, there is a need to calculate the percentage of double counted trucks. In practice, it is difficult to deterine the joint distribution of truck weights and heights. Fortunately, Sobanjo & Thopson () carried out a study that included field easureents of truck weights and heights siultaneously using Laser Range Finder and WIM stations in Florida. It is found that for trucks whose weight exceeds. ton,.% have a height exceeding. ft. Given this conditional probability, Eqn (0) can be used to calculate the percentage of double-counted trucks. The bridge load capacity deteriorates with age and the deterioration rate depends on the bridge superstructure/substructure condition (). The current superstructure and the substructure condition ratings in 00 are both. The IBMS Markov odel is used to predict the superstructure and substructure condition ratings; and the load capacity deterioration rates established in part research () are used to deterine the load capacity in future years. The unit user cost is derived fro the Son and Sinha () study. In order to be coparable, all user costs in this study are in ters of 00 US dollars. The life-cycle bridge detour user cost (due to vertical clearance liit and load capacity liit, not including the detour due to bridge closure) over the bridge are presented in FIGURE (a). In the figure, it is seen that all types of detour user costs increase fro year to year. This is ainly due to the increasing yearly traffic volues which increases the nuber of detouring vehicles. It is also seen that the detour user cost due to the vertical clearance liit exceeds that due to the load capacity liit before year 0 after which the opposite is the case. This is because the load capacity of the bridge deteriorates every year and thus causes higher percentage of vehicles to detour. However, the vertical clearance never changes, so the percentage of vehicles that have to detour due to vertical clearance does not change. As a result, ore vehicles have to detour due to the load capacity liit than due to the vertical clearance liit after Year 0. For purposes of deonstrating how the proble of ultiple counting is addressed, FIGURE (a) also presents the double-counted detour user cost that is yielded by the traditional ethod if the detours due to load capacity and vertical clearance are calculated separately. The aount of double-counted detour user cost is about.% of the detour user cost due to the load capacity liit. FIGURE (b) presents the total detour user costs (due to the vertical clearance liit, load capacity liit, and bridge closure) over the bridge. It is seen that there is a significant gap in user cost between the case with double counting and that without doubling counting. This deonstrates the iportance of addressing the double counting issue. It is also seen that while the user cost due to detour keeps increasing annually, there is a very high detour user cost in the th year: this is due to the bridge closure as work proceeds to raise and widen the bridge. This spike in user cost deonstrates the iportance of considering workzone user cost in bridge investent analysis. In fact, this case study has an additional detour due only to bridge closure in the workzone. Where the bridge also has a saller load capacity liit and lower vertical clearance during the workzone period, additional vehicles ay detour for those reasons, leading to even higher detour user costs during the workzone period. FIGURE (c) shows that before the bridge raising in the th year (0), the detour user cost due to vertical clearance increases every year (this is due to the TRB 0 Annual Meeting

15 0 increasing traffic volue); after the vertical clearance under the bridge increases to.ft, detour cost due to the vertical clearance liit under the bridge is eliinated. User Cost ($000) User Cost ($000) User Cost due to Load Capacity Liit User Cost due to Vertical Clearance Liit Double Counted User Cost Year (a)detour user costs over bridge Year (c) Detour user cost due to under bridge vertical clearance Delay Cost (SM) Year (b) Total detour user cost over bridge Year (d) User delay cost due to liited traffic capacity Year 00 (e) User delay cost under with and without lane addition action FIGURE Results of bridge user cost estiation. The delay user cost due to liited traffic capacity over the bridge life is presented in FIGURE (d). It can be observed that the cost due to delay reduces after the bridge widening action in the th year (0). It is also noted that there is a spike in delay user cost in the 0 th year (0), th year (0) and the th year (0). These are due to workzone construction in these years, further deonstrating the need to consider workzone user costs in life-cycle user cost calculations. User Cost ($000) Bridge Delay Userr Cost ($M) Without Adding One Lane User Detour Cost over the Bridge (with double counting) User Detour Cost over the Brige (without double counting) With Adding One Lane TRB 0 Annual Meeting

16 Coparing the charts in FIGURE (b) and (d), it is seen that the detour user cost is relatively sall copared with the delay cost. While the large delay cost ay be attributed to the use of the free flow speed as the base line, it is worth noting that the reduction of delay cost is uch larger than that of detour user cost in the th year. It can be concluded that the bridge widening can yield significantly greater user cost reduction copared with the bridge raising, in this case study. In presenting the evidence of the iportance of considering user delay cost due to traffic capacity liitation, FIGURE (e) presents the case where there is no lane addition in the th year. It is seen that, the delay costs for both cases are the sae before the th year. But after the th year, without adding one lane, the delay cost increase very rapidly. This is because if the traffic volue increases every year with a constant increase rate, the actual travel speed decreases every year at an increasing rate (can be derived fro Equation ()), thus the additional travel tie (Equation ()) increases at an even faster rate every year. As a result, the delay cost increases very rapidly. Coparing the two cases deonstrates the benefit of adding one ore lane in the th year. This shows the significance of considering the user delay cost caused by the traffic capacity liitation. Conclusion This paper established a fraework for coprehensive estiation of user cost for bridge investent evaluation by synthesizing existing practice and adding soe new considerations. The paper identifies the issue of ultiple counting in user cost estiation and provides a ethod to address this proble. The paper also addresses a gap in the literature by providing a ethodology for including workzone consequences in bridge user cost estiation. Furtherore, the paper develops a ethod to estiate the user delay cost due to bridge traffic capacity liitation, a critical aspect of user cost that enables the estiation of bridge widening (lane-addition) benefits. Using a case study, the applicability of the fraework is deonstrated. The results of the case study show that: the ultiple counting bias can be addressed effectively; the user cost due to workzone can be significant; and estiating of delay user cost due to traffic capacity is necessary to capture the user benefits of bridge widening. REFERENCES. Chen, C.J., D.W. Johnston. Bridge Manageent under a Level of Service Concept Providing Optiu Iproveent Action, Tie, and Budget Prediction, Final Report-. CTES, Raleigh, NC,.. Johnston, D.W., C. Chen, I. Abed-Al-Rahi. Developing User Costs for Bridge Manageent Systes. Transportation Research Circular,, pp. -.. Son, Y., K.C. Sinha. Methodology to Estiate User Costs in Indiana Bridge Manageent Syste. Transportation Research Record,, pp... Thopson, P.D, F.T. Najafi, R. Soares, and H.J. Choung. Developent of User Cost Data for Florida s BMS. Univ. of Florida, Gainesville, FL,.. Thopson, P.D., F.T. Najafi, R. Soares, and R. Kerr. Developent of Pontis User Cost Models for Florida. Transportation Research Circular, Thopson, P.D., R.H. Soares, J. Choung, F.T. Najafi, R. Kerr. User Cost Model for Bridge Manageent Systes. Transportation Research Record, 000, pp.. TRB 0 Annual Meeting

17 Saito, M., K.C. Sinha. The Developent of Optial Strategies for Maintenance, Rehabilitation and Replaceent of Highway Bridges, Final Report Vol. : Cost Analysis. Purdue University, West Lafayette, IN,.. Sobanjo, J.O., P.D. Thopson. Project Planning Models for Florida s Bridge Manageent Syste. Florida DOT, Tallahassee, FL, 00.. Golabi, K., P.D. Thopson, and W.A. Hyan. PONTIS Technical Manual. FHWA, Washington, D.C.,. 0. Thopson, P.D., J.O. Sobanjo, and R. Kerr. Florida DOT Project-Level Bridge Manageent Models, ASCE Journal of Bridge Engineering, (), 00. pp... Sinha, K.C., S. Labi, B.G. Mccullouch, Bhargava, A., Q. Bai. Updating and Enhancing the Indiana Bridge Manageent Syste (IBMS) Final Report Volue I (Technical Manual), Purdue University, IN, 00.. FHWA. FHWA Bridge Progras NBI Data. Accessed March, 00.. AASHTO. Virtis-Opis Version.0 Software for Design, Analysis and Load Rating for Concrete and Steel Bridge Superstructures. AASHTO, Washington, D.C., 00.. AASHTO. LRFD Bridge Design Specifications, Custoary U.S. Units (th Edition) with 00 and 00 U.S. Edition Interis. AASHTO, Washington, D.C., 00.. Fekpe, E.S.K., A. Calton, Prediction of Heavy-Vehicle Weight Distributions. ASCE Journal of Transportation Engineering. Vol., No.,, pp. -. FHWA. Recoding and Coding Guide for the Structure Inventory and Appraisal of the Nations Bridges. Washington, D.C.,.. Ross, S. M. Introduction to Probability Models, Elsevier, 00.. South Carolina Dept. of Transportation. Bridge Design Manual. Colubia, SC, 00.. Transportation Research Board. Highway Capacity Manual, National Research Council, Washington, D.C., MathWave Technologies. EasyFit. Accessed February 00.. Barnes, G., P. Langworthy. The Per-Mile Costs of Operating Autoobiles and Trucks. Minnesota Departent of Transportation. St. Paul, MN, 00. TRB 0 Annual Meeting